NSDL National Science Digital Library
Determine the fractal dimensions of several line-deformation fractals. Input the scale factor and number of similar copies, and the dimension will be calculated. Fractal Dimensions is one of the Interactivate assessment explorers.
The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter
NASA Technical Reports Server (NTRS)
Herren, Kenneth A.; Gregory, Don A.
1999-01-01
The area of information processing has grown dramatically over the last 50 years. In the areas of image processing and information storage the technology requirements have far outpaced the ability of the community to meet demands. The need for faster recognition algorithms and more efficient storage of large quantities of data has forced the user to accept less than lossless retrieval of that data for analysis. In addition to clutter that is not the object of interest in the data set, often the throughput requirements forces the user to accept "noisy" data and to tolerate the clutter inherent in that data. It has been shown that some of this clutter, both the intentional clutter (clouds, trees, etc) as well as the noise introduced on the data by processing requirements can be modeled as fractal or fractal-like. Traditional methods using Fourier deconvolution on these sources of noise in frequency space leads to loss of signal and can, in many cases, completely eliminate the target of interest. The parameters that characterize fractal-like noise (predominately the fractal dimension) have been investigated and a technique to reduce or eliminate noise from real scenes has been developed. Examples of clutter reduced images are presented.
Calculation of multi-fractal dimensions in spin chains
Atas, Y. Y.; Bogomolny, E.
2014-01-01
It was demonstrated in Atas & Bogomolny (2012 Phys. Rev. E 86, 021104) that the ground-state wave functions for a large variety of one-dimensional spin- models are multi-fractals in the natural spin-z basis. We present here the details of analytical derivations and numerical confirmations of this statement. PMID:24344342
Fractal Dimension for Fractal Structures: A Hausdorff Approach
M. A. Sánchez-Granero; Manuel Fernández-Martínez
2010-07-22
This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our definition and the classical ones and also with fractal dimensions I & II (see http://arxiv.org/submit/0080421/pdf). Therefore, we generalize them and obtain an easy method in order to calculate the fractal dimension of strict self-similar sets which are not required to verify the open set condition.
James Theiler
1990-01-01
The nature of chaos and strange attractors is reviewed, and definitions of fractal dimensions are examined. Algorithms for estimating fractal dimensions are discussed. The implementation of box-counting algorithms and of the correlation algorithm for estimating fractal dimensions is addressed.
Effect of Image Processing of a Leaf Photograph on the Calculated Fractal Dimension of Leaf Veins
Yun Kong; Shaohui Wang; Chengwei Ma; Baoming Li; Yuncong Yao
2007-01-01
Digital photography is a promised method for estimating the fractal characteristics of leaf veins. In this study, the effects\\u000a of different threshold levels and image processing methods using Adobe Photoshop software on the fractal dimension values\\u000a were examined from a digital photo of nectarine leaf. The results showed that the nectarine leaf vein has typical fractal\\u000a characteristics and its fractal
Exterior dimension of fat fractals
C. Grebogi; S. W. McDonald; E. Ott; J. A. Yorke
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
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Thanki, Sandip [Nevada State College, Department of Physical Sciences, 1125 Nevada State Drive, Henderson, NV 89002 (United States); Rhee, George; Lepp, Stephen [Physics and Astronomy Department, University of Nevada, Las Vegas, Box 4002, Las Vegas, NV 89154 (United States)], E-mail: Sandip.Thanki@nsc.nevada.edu, E-mail: grhee@physics.unlv.edu, E-mail: lepp@physics.unlv.edu
2009-09-15
In this paper we investigate the use of the fractal dimension of galaxy isophotes in galaxy classification. We have applied two different methods for determining fractal dimensions to the isophotes of elliptical and spiral galaxies derived from CCD images. We conclude that fractal dimension alone is not a reliable tool but that combined with other parameters in a neural net algorithm the fractal dimension could be of use. In particular, we have used three parameters to segregate the ellipticals and lenticulars from the spiral galaxies in our sample. These three parameters are the correlation fractal dimension D {sub corr}, the difference between the correlation fractal dimension and the capacity fractal dimension D {sub corr} - D {sub cap}, and, thirdly, the B - V color of the galaxy.
Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series
NASA Astrophysics Data System (ADS)
Sánchez-Granero, M. J.; Fernández-Martínez, M.; Trinidad-Segovia, J. E.
2012-03-01
In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of short length series, the introduced algorithms perform much better than the classical methods. Finally, an empirical application for some stock market indexes as well as some individual stocks is presented.
Fractal Dimension and Tertiary Structure of Proteins
NASA Astrophysics Data System (ADS)
Daniel, M.; Baskar, S.; Latha, M. M.
We calculate the fractal dimensions of a set of 97 proteins selected from four different structural classes and establish their relationship with the local and global folding of the tertiary structure of these proteins.
Exterior dimension of fat fractals
NASA Astrophysics Data System (ADS)
Grebogi, C.; McDonald, S. W.; Ott, E.; Yorke, J. A.
1985-07-01
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.
Exterior dimension of fat fractals
NASA Technical Reports Server (NTRS)
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.
Box-covering algorithm for fractal dimension of weighted networks
NASA Astrophysics Data System (ADS)
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-10-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.
Fractal Dimension in Epileptic EEG Signal Analysis
NASA Astrophysics Data System (ADS)
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 greater speed and the criterion to choose the maximum and minimum values for time intervals. Comparisons with the other waveform fractal dimension algorithms are also demonstrated. In order to discriminate the Healthy and the Epileptic EEGs, an improved method of Multifractal Measure such as Generalized Fractal Dimensions (GFD) is also proposed. Finally we conclude that there are significant differences between the Healthy and Epileptic Signals in the designed method than the GFD through graphical and statistical tools. The improved multifractal measure is very efficient technique to analyze the EEG Signals and to compute the state of illness of the Epileptic patients.
Texture Segmentation Using Fractal Dimension
B. B. Chaudhuri; Nirupam Sarkar
1995-01-01
This paper deals with the problem of recognizing and segmenting textures in images. For this purpose the authors employ a technique based on the fractal dimension (FD) and the multi-fractal concept. Six FD features are based on the original image, the above average\\/high gray level image, the below average\\/low gray level image, the horizontally smoothed image, the vertically smoothed image,
Fractal dimensions of the time variation of solar radio emission
Shinichi Watari
1996-01-01
The fractal dimensions of solar radio fluxes at 245, 410, 610, 1415, 2695, 2800, 4995, 8800, and 15400 MHz are calculated for the data period 1976–1990. The fractal dimension used here is an index to quantify the time variability of radio emission. The fractal dimensions were found to have values in the range of 1.2–2.0 for time scales of =
Estimation of fractal dimension and fractal curvatures from digital images
NASA Astrophysics Data System (ADS)
Spodarev, Evgeny; Straka, Peter; Winter, Steffen
2015-06-01
Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e.\\ volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.
A Fractal Dimension Survey of Active Region Complexity
NASA Technical Reports Server (NTRS)
McAteer, R. T. James; Gallagher, Peter; Ireland, Jack
2005-01-01
A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .
Fractal dimension of bioconvection patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1990-01-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.
Dimension of a fractal streamer structure
NASA Astrophysics Data System (ADS)
Lehtinen, Nikolai G.; Østgaard, Nikolai
2015-04-01
Streamer corona plays an important role in formation of leader steps in lightning. In order to understand its dynamics, the streamer front velocity is calculated in a 1D model with curvature. We concentrate on the role of photoionization mechanism in the propagation of the streamer ionization front, the other important mechanisms being electron drift and electron diffusion. The results indicate, in particular, that the effect of photoionization on the streamer velocity for both positive and negative streamers is mostly determined by the photoionization length, with a weaker dependence on the amount of photoionization, and that the velocity is decreased for positive curvature, i.e., convex fronts. These results are used in a fractal model in which the front propagation velocity is simulated as the cluster growth probability [Niemeyer et al, 1984, doi:10.1103/PhysRevLett.52.1033]. Monte Carlo simulations of the cluster growth for various ratios of background electric field E to the breakdown field Eb show that the emerging transverse size of the streamers is of the order of the photoionization length, and at the larger scale the streamer structure is a fractal similar to the one obtained in a diffusion-limited aggregation (DLA) system. In the absence of electron attachment (Eb = 0), the fractal dimension is the same (D ˜ 1.67) as in the DLA model, and is reduced, i.e., the fractal has less branching, for Eb > 0.
Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
Michel L. Lapidus; Goran Radunovi?; Darko Žubrini?
2014-11-17
The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.
MASS FRACTAL DIMENSION OF SHRINKING SOIL AGGREGATES
Technology Transfer Automated Retrieval System (TEKTRAN)
Fractal scaling for mass of dry soil aggregates has been documented in literature. This scaling results in power-law dependencies of aggregate porosity or bulk density on aggregate size. Such dependencies if measured are used to estimate mass fractal dimensions. Changes in water content are known to...
Complex fractal dimension of the bronchial tree
NASA Astrophysics Data System (ADS)
Shlesinger, Michael F.; West, Bruce J.
1991-10-01
The architecture of the mammalian lung has been shown to be correctly described using a fractal model with a complex dimension, related to a Cantor set with random errors, for four different species. Here we provide an interpretation of that model that has implications for biological evolution. We argue that fractals are more error tolerant than other structures and therefore have an evolutionary advantage.
ERIC Educational Resources Information Center
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)
Application of Fractal Dimension on Palsar Data
NASA Astrophysics Data System (ADS)
Singh, Dharmendra; Pant, Triloki
Study of land cover is the primal task of remote sensing where microwave imaging plays an important role. As an alternate of optical imaging, microwave, in particular, Synthetic Aperture Radar (SAR) imaging is very popular. With the advancement of technology, multi-polarized images are now available, e.g., ALOS-PALSAR (Phased Array type L-band SAR), which are beneficial because each of the polarization channel shows different sensitivity to various land features. Further, using the textural features, various land classes can be classified on the basis of the textural measures. One of the textural measure is fractal dimension. It is noteworthy that fractal dimension is a measure of roughness and thus various land classes can be distinguished on the basis of their roughness. The value of fractal dimension for the surfaces lies between 2.0 and 3.0 where 2.0 represents a smooth surface while 3.0 represents drastically rough surface. The study area covers subset images lying between 2956'53"N, 7750'32"E and 2950'40"N, 7757'19"E. The PALSAR images of the year 2007 and 2009 are considered for the study. In present paper a fractal based classification of PALSAR images has been performed for identification of Water, Urban and Agricultural Area. Since fractals represent the image texture, hence the present study attempts to find the fractal properties of land covers to distinguish them from one another. For the purpose a context has been defined on the basis of a moving window, which is used to estimate the local fractal dimension and then moved over the whole image. The size of the window is an important issue for estimation of textural measures which is considered to be 55 in present study. This procedure, in response, produces a textural map called fractal map. The fractal map is constituted with the help of local fractal dimension values and can be used for contextual classification. In order to study the fractal properties of PALSAR images, the three polarization images viz. HH (Horizontal-Horizontal Polarization), VV (Vertical-Vertical Polarization) and HV (Horizontal-Vertical Polarization) are considered individually. First of all each polarized image is classified in an unsupervised way and various clusters, i.e., four clusters are identified with the help of reference data as Water, Urban and Agricultural Area. For each cluster, the fractal dimension is obtained from the fractal map. Based on the study the ranges of fractal dimension for three classes are Water: 2.0-2.17, Agricultural Area: 2.24-2.72, Urban Area: 2.63-2.92 for HH polarized image; Water: 2.0-2.21, Agricultural Area: 2.20-2.64, Urban; 2.58-2.94 for VV polarized image and Water: 2.0-2.14, Agricultural Area: 2.18-2.58, Urban: 2.46-2.94 for HV polarized image. Since the class Others represents a mixture of various classes, an explicit range of D for this class can not be determined. A closer look at the ranges of fractal dimension indicates that there is an overlapping of the values for different classes, despite of which the classes can be distinguished. Also, the class Water having low value of fractal dimension can be treated as smooth and Urban Area having higher values of fractal dimension can be considered rough in structure while the class Agricultural Area shows an intermediate roughness.
Fractal Dimension of Dielectric Breakdown
L. Niemeyer; L. Pietronero; H. J. Wiesmann
1984-01-01
It is shown that the simplest nontrivial stochastic model for dielectric breakdown naturally leads to fractal structures for the discharge pattern. Planar discharges are studied in detail and the results are compared with properly designed experiments.
FRACTAL DIMENSION ESTIMATION: EMPIRICAL MODE DECOMPOSITION VERSUS WAVELETS
GonÃ§alves, Paulo
FRACTAL DIMENSION ESTIMATION: EMPIRICAL MODE DECOMPOSITION VERSUS WAVELETS Paulo GoncÂ¸alves INRIA, France. {firstname.lastname}@ens-lyon.fr ABSTRACT We address the problem of fractal dimension estimation motions. Index Terms-- fractal dimension, regularity exponents, wavelet transform, EMD 1. MOTIVATION
Fractal dimension estimators for fractional Brownian motions
N. Gache; Patrick Flandrin; Dominique Garreau
1991-01-01
Five different fractal dimension estimators are chosen which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fractional Brownian motion samples at different observation scales on the other hand), or in a mixed time-scale domain (identification of a self-similarity parameter
The Correlation Fractal Dimension of Complex Networks
NASA Astrophysics Data System (ADS)
Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei
2013-05-01
The fractality of complex networks is studied by estimating the correlation dimensions of the networks. Comparing with the previous algorithms of estimating the box dimension, our algorithm achieves a significant reduction in time complexity. For four benchmark cases tested, that is, the Escherichia coli (E. Coli) metabolic network, the Homo sapiens protein interaction network (H. Sapiens PIN), the Saccharomyces cerevisiae protein interaction network (S. Cerevisiae PIN) and the World Wide Web (WWW), experiments are provided to demonstrate the validity of our algorithm.
Estimation of fractal dimensions from transect data
Loehle, C. [Argonne National Lab., IL (United States)
1994-04-01
Fractals are a useful tool for analyzing the topology of objects such as coral reefs, forest canopies, and landscapes. Transects are often studied in these contexts, and fractal dimensions computed from them. An open question is how representative a single transect is. Transects may also be used to estimate the dimensionality of a surface. Again the question of representativeness of the transect arises. These two issues are related. This note qualifies the conditions under which transect data may be considered to be representative or may be extrapolated, based on both theoretical and empirical results.
Fractal dimension of alumina aggregates grown in two dimensions
NASA Technical Reports Server (NTRS)
Larosa, Judith L.; Cawley, James D.
1992-01-01
The concepts of fractal geometry are applied to the analysis of 0.4-micron alumina constrained to agglomerate in two dimensions. Particles were trapped at the bottom surface of a drop of a dilute suspension, and the agglomeration process was directly observed, using an inverted optical microscope. Photographs were digitized and analyzed, using three distinct approaches. The results indicate that the agglomerates are fractal, having a dimension of approximately 1.5, which agrees well with the predictions of the diffusion-limited cluster-cluster aggregation model.
Fractal dimension in dissipative chaotic scattering Jess M. Seoane,1,
Rey Juan Carlos, Universidad
Fractal dimension in dissipative chaotic scattering JesÃºs M. Seoane,1, * Miguel A. F. SanjuÃ¡n,1 on chaotic scattering is relevant to situations of physical interest. We inves- tigate how the fractal is thus the fractal dimension of the set of singularities. For nonhyperbolic scattering, it has been known
NASA Astrophysics Data System (ADS)
Mitina, Olga V.; Nourkova, Veronica V.
In the given research we offer the technique for the calculation of the density of events which people retrieve from autobiographical memory. We wanted to prove a non-uniformity nature of memories distribution in the course of time and were interested with the law of distribution of these events during life course.
Fractal dimensions of oligonucleotide compositions of DNA sequences
Korolev, S.V.; Tumanyan, V.G. [Russian Academy of Sciences, Moscow (Russian Federation). Engelhardt Inst. of Molecular Biology
1993-12-31
Fractal dimension (FD) of oligonucleotide composition is presented as an analog of genetic text complexity. FD for prokaryotic and eukaryotic sequences are estimated. Reliable differences between FD of coding and non-coding sequences in higher organisms are demonstrated. At the same time similar value of coding regions from different sources illustrate stability of such sequences against evolutionary processes. The proposed method provides a fast calculation of FD value for sequences of any length.
Fractal dimension analyses of lava surfaces and flow boundaries
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.
1993-01-01
An improved method of estimating fractal surface dimensions has been developed. The accuracy of this method is illustrated using artificially generated fractal surfaces. A slightly different from usual concept of linear dimension is developed, allowing a direct link between that and the corresponding surface dimension estimate. These methods are applied to a series of images of lava flows, representing a variety of physical and chemical conditions. These include lavas from California, Idaho, and Hawaii, as well as some extraterrestrial flows. The fractal surface dimension estimations are presented, as well as the fractal line dimensions where appropriate.
A Surface-Based Fractal Information Dimension Method for Cortical Complexity Analysis
Yuanchao Zhang; Jiefeng Jiang; Lei Lin; Feng Shi; Yuan Zhou; Chunshui Yu; Kuncheng Li; Tianzi Jiang
2008-01-01
In this paper, we proposed a new surface-based fractal information dimension (FID) method to quantify the cortical complexity.\\u000a Unlike the traditional box-counting method to measure the capacity dimension, our method is a surface-based fractal information\\u000a dimension method, which incorporates surface area into the probability calculation and thus encapsulates more information\\u000a of the original cortical surfaces. The accuracy of the algorithm
Fractal dimension in nonhyperbolic chaotic scattering
NASA Technical Reports Server (NTRS)
Lau, Yun-Tung; Finn, John M.; Ott, Edward
1991-01-01
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
Superfluid hydrodynamic in fractal dimension space
NASA Astrophysics Data System (ADS)
Tayurskii, D. A.; Lysogorskiy, Yu V.
2012-11-01
The complex behavior of such quantum fluids like liquid 4He and liquid 3He in nanoporous media is determined by influence of randomly distributed geometrical confinement as well as by significant contribution from the surface atoms. In the present paper Fractional Schrodinger equation has been used for deriving two-fluid hydrodynamical equations for describing the motion of superfluid helium in the fractal dimension space. Nonlinear equations for oscillations of pressure and temperature are obtained and coupling of pressure and temperature oscillations is observed. Moreover coupling should disappear at very low temperatures which provide an experimental test for this theory.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2010-10-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2011-03-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
Fractal dimension of transport coefficients in a deterministic dynamical system
NASA Astrophysics Data System (ADS)
Koza, Zbigniew
2004-11-01
In many low-dimensional dynamical systems, transport coefficients are very irregular and are perhaps even fractal functions of control parameters. To analyse this phenomenon, we study a dynamical system defined by a piecewise linear map and investigate the dependence of transport coefficients on the slope of the map. We present analytical arguments, supported by numerical calculations, showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of the graphs of these functions is 1 with a logarithmic correction, and we find that the exponent ? controlling this correction is bounded from above by 1 or 2, depending on some detailed properties of the system. Using numerical techniques, we show local self-similarity of the graphs. The local self-similarity scaling transformations turn out to be dependent (irregularly) on the values of the system control parameters.
Flocculation model of cohesive sediment using variable fractal dimension
Minwoo Son; Tian-Jian Hsu
2008-01-01
A new flocculation model using variable fractal dimension is proposed and validated with several experimental data and an\\u000a existing model. The proposed model consists of two processes: aggregation and breakup due to flow turbulence. For aggregation\\u000a process, the aggregate structure is considered to have the characteristic of self-similarity, the main concept of fractal\\u000a theory. Under this assumption, a variable fractal
Shower fractal dimension analysis in a highly-granular calorimeter
Ruan, M
2015-01-01
We report on an investigation of the self-similar structure of particle showers recorded at a highly-granular calorimeter. On both simulated and experimental data, a strong correlation between the number of hits and the spatial scale of the readout channels is observed, from which we define the shower fractal dimension. The measured fractal dimension turns out to be strongly dependent on particle type, which enables new approaches for particle identification. A logarithmic dependence of the particle energy on the fractal dimension is also observed.
Fractal dimension of interstellar clouds: opacity and noise effects
Nestor Sanchez; Emilio J. Alfaro; Enrique Perez
2006-10-20
There exists observational evidence that the interstellar medium has a fractal structure in a wide range of spatial scales. The measurement of the fractal dimension (Df) of interstellar clouds is a simple way to characterize this fractal structure, but several factors, both intrinsic to the clouds and to the observations, may contribute to affect the values obtained. In this work we study the effects that opacity and noise have on the determination of Df. We focus on two different fractal dimension estimators: the perimeter-area based dimension (Dper) and the mass-size dimension (Dm). We first use simulated fractal clouds to show that opacity does not affect the estimation of Dper. However, Dm tends to increase as opacity increases and this estimator fails when applied to optically thick regions. In addition, very noisy maps can seriously affect the estimation of both Dper and Dm, decreasing the final estimation of Df. We apply these methods to emission maps of Ophiuchus, Perseus and Orion molecular clouds in different molecular lines and we obtain that the fractal dimension is always in the range 2.6 2.3) average fractal dimension for the interstellar medium, as traced by different chemical species.
Using fractal dimension to assess robot operator search skill
Jeff Craighead; James A. Haley
2009-01-01
This paper discusses a new real-time fractal path analysis (RTFPA) algorithm and its use in training robot operators. Twenty-five volunteers participated in an experiment to evaluate the use of the RTFPA algorithm as a metric for robot operator search skill. The algorithm was used to estimate the fractal dimension of a path taken by a simulated, teleoperated Inuktun Extreme VGTV
On the fractal dimension of the Duffing attractor
Mariusz Tarnopolski
2014-09-12
The box counting dimension $d_C$ and the correlation dimension $d_G$ change with the number of numerically generated points forming the attractor. At a sufficiently large number of points the fractal dimension tends to a finite value. The obtained values are $d_C\\approx 1.43$ and $d_G\\approx 1.38$.
Fractal dimensions of rampart impact craters on Mars
NASA Technical Reports Server (NTRS)
Ching, Delwyn; Taylor, G. Jeffrey; Mouginis-Mark, Peter; Bruno, Barbara C.
1993-01-01
Ejecta blanket morphologies of Martian rampart craters may yield important clues to the atmospheric densities during impact, and the nature of target materials (e.g., hard rock, fine-grained sediments, presence of volatiles). In general, the morphologies of such craters suggest emplacement by a fluidized, ground hugging flow instead of ballistic emplacement by dry ejecta. We have quantitatively characterized the shape of the margins of the ejecta blankets of 15 rampart craters using fractal geometry. Our preliminary results suggest that the craters are fractals and are self-similar over scales of approximately 0.1 km to 30 km. Fractal dimensions (a measure of the extent to which a line fills a plane) range from 1.06 to 1.31. No correlations of fractal dimension with target type, elevation, or crater size were observed, though the data base is small. The range in fractal dimension and lack of correlation may be due to a complex interplay of target properties (grain size, volatile content), atmospheric pressure, and crater size. The mere fact that the ejecta margins are fractals, however, indicates that viscosity and yield strength of the ejecta were at least as low as those of basalts, because silicic lava flows are not generally fractals.
Fractal Dimensions of In Vitro Tumor Cell Proliferation
Lambrou, George I.
2015-01-01
Biological systems are characterized by their potential for dynamic adaptation. One of the challenges for systems biology approaches is their contribution towards the understanding of the dynamics of a growing cell population. Conceptualizing these dynamics in tumor models could help us understand the steps leading to the initiation of the disease and its progression. In vitro models are useful in answering this question by providing information over the spatiotemporal nature of such dynamics. In the present work, we used physical quantities such as growth rate, velocity, and acceleration for the cellular proliferation and identified the fractal structures in tumor cell proliferation dynamics. We provide evidence that the rate of cellular proliferation is of nonlinear nature and exhibits oscillatory behavior. We also calculated the fractal dimensions of our cellular system. Our results show that the temporal transitions from one state to the other also follow nonlinear dynamics. Furthermore, we calculated self-similarity in cellular proliferation, providing the basis for further investigation in this topic. Such systems biology approaches are very useful in understanding the nature of cellular proliferation and growth. From a clinical point of view, our results may be applicable not only to primary tumors but also to tumor metastases. PMID:25883653
Fractal dimensions of niobium oxide films probed by protons and lithium ions
Pehlivan, Esat; Niklasson, Gunnar A. [Department of Physics, Faculty of Arts and Sciences, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey and Angstroem Laboratory, Department of Engineering Sciences, Uppsala University, P.O. Box 534, SE-751 21 Uppsala (Sweden); Angstroem Laboratory, Department of Engineering Sciences, Uppsala University, P.O. Box 534, SE-751 21 Uppsala (Sweden)
2006-09-01
Cyclic voltammetry (CV) and atomic force microscopy (AFM) were used to determine fractal surface dimensions of sputter deposited niobium pentoxide films. Peak currents were determined by CV measurements. Power spectral densities obtained from AFM measurements of the films were used for calculating length scale dependent root mean square roughness. In order to compare the effect of Li and H ion intercalation at the fractal surfaces, LiClO{sub 4} based as well as propionic acid electrolytes were used. The CV measurements gave a fractal dimension of 2.36 when the films were intercalated by Li ions and 1.70 when the films were intercalated by protons. AFM measurements showed that the former value corresponds to the fractal surface roughness of the films, while the latter value is close to the dimensionality of the distribution of hillocks on the surface. We conclude that the protons are preferentially intercalated at such sites.
Bulk, surface and hull fractal dimension of critical Ising clusters in d=2
C. Vanderzande; A. L. Stella
1989-01-01
The authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical Ising cluster, in d=2. The results clearly support the values d=187\\/96, ds=5\\/6 which are consistent with Ising clusters being described by tricritical q=1 Potts model exponents. From this, the hull dimension dH of critical Ising clusters is found to be dH=11\\/8, consistent
Voronoi cells, fractal dimensions and fibre composites.
Summerscales, J.; Guild, F. J.; Pearce, N. R. L.; Russell, P. M.
2001-02-01
The use of fibre-reinforced polymer matrix composite materials is growing at a faster rate than the gross domestic product (GDP) in many countries. An improved understanding of their processing and mechanical behaviour would extend the potential applications of these materials. For unidirectional composites, it is predicted that localized absence of fibres is related to longitudinal compression failure. The use of woven reinforcements permits more effective manufacture than for unidirectional fibres. It has been demonstrated experimentally that compression strengths of woven composites are reduced when fibres are clustered. Summerscales predicted that clustering of fibres would increase the permeability of the reinforcement and hence expedite the processing of these materials. Commercial fabrics are available which employ this concept using flow-enhancing bound tows. The net effect of clustering fibres is to enhance processability whilst reducing the mechanical properties. The effects reported above were qualitative correlations. To improve the design tools for reinforcement fabrics we have sought to quantify the changes in the micro/meso-structure of woven reinforcement fabrics. Gross differences in the appearance of laminate sections are apparent for different weave styles. The use of automated image analysis is essential for the quantification of subtle changes in fabric architecture. This paper considers Voronoi tessellation and fractal dimensions for the quantification of the microstructures of woven fibre-reinforced composites. It reviews our studies in the last decade of the process-property-structure relationships for commercial and experimental fabric reinforcements in an attempt to resolve the processing vs. properties dilemma. A new flow-enhancement concept has been developed which has a reduced impact on laminate mechanical properties. PMID:11207917
On the fractal dimension of the solar granulation
NASA Astrophysics Data System (ADS)
Greimel, R.; Brandt, P. N.; Guenther, E.; Mattig, W.
Fractal dimension analysis may be used to determine whether the solar granulation represents homogeneous, isotropic turblence in certain ranges of scale. Several attempts have been made to investigate fractal dimension from white light granulation pictures of high spatial resolution, e.g., Roudier and Muller (1986), Darvann and Kusoffsky (1989), and Karpinsky (1990), who find a critical scale of granule sizes, at which the fractal dimension d changes abruptly. Using material from the 'Spektro-Stratoskop' and analyzing 42,742 granules, the results published earlier could be confirmed, i.e., a fractal dimension of approx. 1.3 for the small scales and d about 2 for the large scales. However, a smooth transition is found between both regimes. Moreover, a closer inspection of the methods used reveals that, in all analyses, the fractal dimension of the granulation at small scales seems to be dominated by technical problems (i.e., the limited resolution of the material, the definition of the granules, and the finite pixel size).
Planetary boundary layer detection with fractal dimension of three-wavelength lidar signals
NASA Astrophysics Data System (ADS)
Lei, Liqiao; McCormick, M. Patrick; Su, Jia
2013-05-01
Lidar backscatter signal resulting from laser light scattering from the aerosol and molecular in the atmosphere contains various information about the geometrical and physical properties of aerosol and molecular. The lidar backscatter signal can provide information about the planetary boundary layer (PBL) stratification by using aerosol as a tracer for convective and mixing processes. A PBL height and structure detecting technique based on the fractal dimension of three-wavelength backscatter signals is advanced. In this PBL height detecting technique, the three-wavelength backscatter signals are obtained by the Hampton University (HU, 37.02° N, 76.33° W) lidar. The fractal dimension was calculated using the three-wavelength lidar signals. The PBL heights obtained from fractal dimension of threewavelength lidar signals is compared with PBL heights obtained from the potential temperature profiles which are provided by NASA Langely Research Center (10 miles from HU). And results of the two methods agree well. Moreover, fractal dimension method can reduce the influence of the geometrical form factor on the PBL detecting to expand the detecting range of PBL and remove the effect of plume. Also, the fractal dimension method can show the PBL dynamics and the PBL evolution clearly.
Matrix crack detection in spatially random composite structures using fractal dimension
NASA Astrophysics Data System (ADS)
Umesh, K.; Ganguli, Ranjan
2014-03-01
Fractal dimension based damage detection method is studied for a composite structure with random material properties. A composite plate with localized matrix crack is considered. Matrix cracks are often seen as the initial damage mechanism in composites. Fractal dimension based method is applied to the static deformation curve of the structure to detect localized damage. Static deflection of a cantilevered composite plate under uniform loading is calculated using the finite element method. Composite material shows spatially varying random material properties because of complex manufacturing processes. Spatial variation of material property is represented as a two dimensional homogeneous Gaussian random field. Karhunen-Loeve (KL) expansion is used to generate a random field. The robustness of fractal dimension based damage detection methods is studied considering the composite plate with spatial variation in material properties.
Can you hear the fractal dimension of a drum?
W. Arrighetti; G. Gerosa
2005-03-31
Electromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a `just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called ``diaperiodic'', whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of ``spectral dimension'' is given and proven to depend on diaperiodic modes only for a wide class of IFSs. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the `self-similar' spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question.
Fractal dimensions of niobium oxide films probed by protons and lithium ions
Esat Pehlivan; Gunnar A. Niklasson
2006-01-01
Cyclic voltammetry (CV) and atomic force microscopy (AFM) were used to determine fractal surface dimensions of sputter deposited niobium pentoxide films. Peak currents were determined by CV measurements. Power spectral densities obtained from AFM measurements of the films were used for calculating length scale dependent root mean square roughness. In order to compare the effect of Li and H ion
The fractal dimension of ionization cascades in the glow discharge
NASA Astrophysics Data System (ADS)
Smith, Reginald D.
2005-04-01
The glow discharge's main ionization breakdown processes have been understood for about one hundred years. The glow discharge, however, still remains an area of active research in relation to pattern formation and far-from-equilibrium systems. The primary and secondary ionization processes can be mathematically modelled as general branching processes. Not only is the Townsend breakdown criterion obtained but the ionization avalanche can be characterized as a branching set with a unique Hausdorff fractal dimension. These fractal dimensions can be utilized in applications using similarity principles and Paschen's Law.
Fractal dimension analysis in a highly granular calorimeter
Ruan, M; Brient, J.C; Jeans, D; Videau, H
2015-01-01
The concept of “particle flow” has been developed to optimise the jet energy resolution by distinguishing the different jet components. A highly granular calorimeter designed for the particle flow algorithm provides an unprecedented level of detail for the reconstruction of calorimeter showers and enables new approaches to shower analysis. In this paper the measurement and use of the fractal dimension of showers is described. The fractal dimension is a characteristic number that measures the global compactness of the shower. It is highly dependent on the primary particle type and energy. Its application in identifying particles and estimating their energy is described in the context of a calorimeter designed for the International Linear Collider.
NASA Astrophysics Data System (ADS)
Wei, Wei; Cai, Jianchao; Hu, Xiangyun; Fan, Ping; Han, Qi; Lu, Jinge; Cheng, Chu-Lin; Zhou, Feng
2015-03-01
The fractal dimension of random walker (FDRW) is an important parameter for description of electrical conductivity in porous media. However, it is somewhat empirical in nature to calculate FDRW. In this paper, a simple relation between FDRW and tortuosity fractal dimension (TFD) of current streamlines is derived, and a novel method of computing TFD for different generations of two-dimensional Sierpinski carpet and three-dimensional Sierpinski sponge models is presented through the finite element method, then the FDRW can be accordingly predicted; the proposed relation clearly shows that there exists a linear relation between pore fractal dimension (PFD) and TFD, which may have great potential in analysis of transport properties in fractal porous media.
Estimating the fractal dimension and the predictability of the atmosphere
Zeng, X.; Pielke, R.A.; Eykholt, R. (Colorado State Univ., Fort Collins, CO (United States))
1992-04-15
The fractal dimension, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily surface temperature and pressure over several regions of the US and the North Atlantic Ocean with different climatic signal-to-noise ratios. Though the total number of data points is larger than those used in previous studies, it is still too small to obtain a reliable estimate of the Grassberger-Procaccia correlation dimension because of the limitations discussed by Ruelle. It can be shown that this dimension is greater than 8. It is pointed out that most, if not all, of the previous estimates of low fractal dimensions in the atmosphere are spurious. These results lead the authors to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia are sensitive to the selection of the time delay and are thus unreliable. Geographic variability of the fractal dimension is suggested, but further verification is needed. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. Using this method, it is found that the error-doubling time is about 2-3 days in Fort Collins, Colorado, about 4-5 days in Los Angeles, California, and about 5-8 days in the North Atlantic Ocean. The difference between these estimates of error-doubling time and estimates based on general circulation models (GCMs) is discussed. It is also mentioned that the computation of the Lyapunov exponents is slightly sensitive to the selection of the time delay, possibly because the fractal dimension is very high in the atmosphere. 48 refs., 10 figs., 3 tabs.
Texas at Austin. University of
Fractal dimension unscreened angles measured for radial viscous fingering Olivier Praud Harry, USA #Received November 2004; published July 2005# have examined fractal patterns formed injection experiments. fractal dimension D 0 of pattern large r / 1.70Â±0.02. Further, generalized dimensions D pattern
Original article Structure and fractal dimensions of root systems
Paris-Sud XI, Université de
Original article Structure and fractal dimensions of root systems of four co-occurring fruit tree, Bangor, Gwynedd LL57 2UW, UK (Received 1 February 1999; accepted 29 October 1999) Abstract Coarse root-auto- matically digitized. Spatial distributions of root length were determined from the digitally
Fractal dimension and turbulence in Giant HII Regions
H. E. Caicedo-Ortiz; E. Santiago-Cortés; J. López-Bonilla; H. O. Castañeda
2015-02-16
We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble's Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.
Numerical problems with evaluating the fractal dimension of real data
ADAM SZUSTALEWICZ
A new program RIVER for evaluating the fractal dimension of real data sets was written. Its performance was compared with two programs HarFA — demo version and Coastline, available in Internet. The three programs were tested on about 50 data sets. The program RIVER yielded the maximal errors less than 3 percentages for all tested data sets, while the other
Fractal dimension, wavelet shrinkage, and anomaly detection for mine hunting
Kingsbury, Nick
Fractal dimension, wavelet shrinkage, and anomaly detection for mine hunting J. D. B. Nelson and N. G. Kingsbury james@stats.ucl.ac.uk, ngk@eng.cam.ac.uk Abstract An anomaly detection approach [4]. This initial screening or detection stage is designed to accept a large number of false
Cusp-scaling behavior in fractal dimension of chaotic scattering
Adilson E. Motter; Ying-Cheng Lai
2002-06-13
A topological bifurcation in chaotic scattering is characterized by a sudden change in the topology of the infinite set of unstable periodic orbits embedded in the underlying chaotic invariant set. We uncover a scaling law for the fractal dimension of the chaotic set for such a bifurcation. Our analysis and numerical computations in both two- and three-degrees-of-freedom systems suggest a striking feature associated with these subtle bifurcations: the dimension typically exhibits a sharp, cusplike local minimum at the bifurcation.
Construction of Fractal Surfaces by Recurrent Fractal Interpolation Curves
Chol-hui Yun; Hyong-chol O.; Hui-chol Choi
2014-08-12
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 constructions of fractal surfaces.
Estimation of Fractal Dimension in Differential Diagnosis of Pigmented Skin Lesions
NASA Astrophysics Data System (ADS)
Aralica, Gorana; Miloševi?, Danko; Konjevoda, Paško; Seiwerth, Sven; Štambuk, Nikola
Medical differential diagnosis is a method of identifying the presence of a particular entity (disease) within a set of multiple possible alternatives. The significant problem in dermatology and pathology is the differential diagnosis of malignant melanoma and other pigmented skin lesions, especially of dysplastic nevi. Malignant melanoma is the most malignant skin neoplasma, with increasing incidence in various parts of the world. It is hoped that the methods of quantitative pathology, i.e. morphometry, can help objectification of the diagnostic process, since early discovery of melanoma results in 10-year survival rate of 90%. The aim of the study was to use fractal dimension calculated from the perimeter-area relation of the cell nuclei as a tool for the differential diagnosis of pigmented skin lesions. We analyzed hemalaun-eosin stained pathohistological slides of pigmented skin lesions: intradermal naevi (n = 45), dysplastic naevi (n = 47), and malignant melanoma (n = 50). It was found that fractal dimension of malignant melanoma cell nuclei differs significantly from the intradermal and dysplastic naevi (p ? 0. 001, Steel-Dwass Multiple Comparison Test). Additionaly, ROC analysis confirmed the value of fractal dimension based evaluation. It is suggested that the estimation of fractal dimension from the perimeter-area relation of the cell nuclei may be a potentially useful morphometric parameter in the medical differential diagnosis of pigmented skin lesions.
Texas at Austin. University of
Fractal dimension and unscreened angles measured for radial viscous fingering Olivier Praud fractal patterns formed by the injection of air into oil in a thin 0.127 mm layer contained between two reaches r/b=900, are far larger than in past experiments. The fractal dimension D0 of the pattern
Fractal dimension of particle showers measured in a highly granular calorimeter.
Ruan, Manqi; Jeans, Daniel; Boudry, Vincent; Brient, Jean-Claude; Videau, Henri
2014-01-10
We explore the fractal nature of particle showers using Monte Carlo simulation. We define the fractal dimension of showers measured in a high granularity calorimeter designed for a future lepton collider. The shower fractal dimension reveals detailed information of the spatial configuration of the shower. It is found to be characteristic of the type of interaction and highly sensitive to the nature of the incident particle. Using the shower fractal dimension, we demonstrate a particle identification algorithm that can efficiently separate electromagnetic showers, hadronic showers, and nonshowering tracks. We also find a logarithmic dependence of the shower fractal dimension on the particle energy. PMID:24483887
Estimating fractal dimension of profiles: A comparison of methods
John C. Gallant; Ian D. Moore; Michael F. Hutchinson; Paul Gessler
1994-01-01
This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles. The semi-variogram, roughness-length, and two spectral methods are compared using synthetic 1024-point profiles generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface. The analysis concentrates on the Hurst exponent H,which is linearly
Fractal dimensions of flocs between clay particles and HAB organisms
NASA Astrophysics Data System (ADS)
Wang, Hongliang; Yu, Zhiming; Cao, Xihua; Song, Xiuxian
2011-05-01
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 ( D 2) was 1.92 and three-dimensional fractal dimension ( D 3) 2.81, while for the modified clay, D 2 was 1.84 and D 3 was 2.50. The addition of polyaluminum chloride (PACl) lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D 2 and D 3. Due to the decrease of D 3, 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 D 3 and D 2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D 3 and D 2 of flocs were the lowest.
NASA Astrophysics Data System (ADS)
Donadio, Carlo; Magdaleno, Fernando; Mazzarella, Adriano; Mathias Kondolf, G.
2015-07-01
By applying fractal geometry analysis to the drainage network of three large watercourses in America and Europe, we have calculated for the first time their fractal dimension. The aim is to interpret the geomorphologic characteristics to better understand the morphoevolutionary processes of these fluvial morphotypes; to identify and discriminate geomorphic phenomena responsible for any difference or convergence of a fractal dimension; to classify hydrographic patterns, and finally to compare the fractal degree with some geomorphic-quantitative indexes. The analyzed catchment of Russian (California, USA), Ebro (Spain), and Volturno (Italy) rivers are situated in Mediterranean-climate regions sensu Köppen, but with different geologic context and tectonic styles. Results show fractal dimensions ranging from 1.08 to 1.50. According to the geological setting and geomorphic indexes of these basins, the lower fractal degree indicates a prevailing tectonics, active or not, while the higher degree indicates the stronger erosion processes on inherited landscapes.
Fractal Dimension of Geologically Constrained Crater Populations of Mercury
NASA Astrophysics Data System (ADS)
Mancinelli, Paolo; Pauselli, Cristina; Perugini, Diego; Lupattelli, Andrea; Federico, Costanzo
2015-07-01
Data gathered during the Mariner10 and MESSENGER missions are collated in this paper to classify craters into four geo-chronological units constrained to the geological map produced after MESSENGER's flybys. From the global catalogue, we classify craters, constraining them to the geological information derived from the map. We produce a size frequency distribution (SFD) finding that all crater classes show fractal behaviour: with the number of craters inversely proportional to their diameter, the exponent of the SFD (i.e., the fractal dimension of each class) shows a variation among classes. We discuss this observation as possibly being caused by endogenic and/or exogenic phenomena. Finally, we produce an interpretative scenario where, assuming a constant flux of impactors, the slope variation could be representative of rheological changes in the target materials.
Klatt, Jan; Gerich, Carola E; Gröbe, Alexander; Opitz, Jörg; Schreiber, Jürgen; Hanken, Henning; Salomon, Georg; Heiland, Max; Kluwe, Lan; Blessmann, Marco
2014-09-01
Early detection and complete resection of oral carcinomas is of crucial importance for patient survival. This could be significantly improved by developing a non-invasive, sensitive and real-time detection technique. Time-resolved autofluorescence measurement is state-of-the-art technology originally developed for non-destructive inspection of material. In this study, we measured time-resolved autofluorescence in tumours and healthy tissues of the oral cavity ex vivo and calculated the corresponding fractal dimension which was significantly higher in tumours than in healthy tissues (1.8 vs. 1.6, P < 0.001, unpaired t-test) with non-overlapping 95% confidential intervals 1.88-1.84 and 1.57-1.69, respectively. Very high specificity (86%) could be reached at 100% sensitivity. The area under the curve was 99%, further suggesting the superior prediction potential of fractal dimension based on time-resolved autofluorescence spectra. PMID:24444757
Static and fatigue crack growth of epoxy adhesives and fractal dimensions
Kimiyoshi Naito; Toru Fujii
1998-01-01
Fractured surfaces of epoxy adhesives under mode I static and fatigue (cyclic) loading have fractal characteristics. The effects of rubber modification, adhesive thickness and cross-head speed (static only) on both static and fatigue fracture surfaces of epoxy adhesives were examined using fractals.Under static loading, the fractal dimension becomes high due to rubber modification. It is related to the static crack
Quantification of Collagen Organization Using Fractal Dimensions and Fourier Transforms
Frisch, Kayt E.; Duenwald-Kuehl, Sarah E.; Lakes, Roderic S.; Vanderby, Ray
2011-01-01
Summary The structure of the collagen fibers that composes tendon and ligament are disrupted or damaged during injury and healing. Quantification of these changes is traditionally a laborious and subjective task. In this work we apply two automated techniques, Fourier transformation (FFT) and fractal dimension analysis (FA) to quantify the organization of collagen fibrils. Using multi-photon images we show that for healing ligament FA differentiates more clearly between the different time-points during healing. Using scanning electron microcopy images of overstretched tendon we show that combining FFT and FA measures separates the damaged and undamaged groups more clearly than either method individually. PMID:21529898
Application of Fractal Dimension in Edge Detection of Log with Rotten Knot Image
Dawei Qi; Li Li; Jingwei Song; Hongbo Mu
2007-01-01
Defects connotative feature information in the X-ray image data of log with rotten knot was studied. Adopting box counting method detected log with rotten knot image edges effectively. Describe the irregular degree of image in quantity by using the value of fractal dimension to confirm the defects position. There were differences in fractal dimension between the normal regions and the
Federico Maggi
2007-01-01
While the fractal dimension of suspended flocs of cohesive sediment is known to vary with the shear rate, electrochemical properties of the sediment and environment, geometrical restructuring, and presence of organic matter, experimental data presented in this work suggest changes in fractal dimension also during floc genesis at constant sedimentological and hydraulic conditions. A power law function is proposed to
Wang Xujing [Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294 (United States); Becker, Frederick F.; Gascoyne, Peter R. C. [Department of Molecular Pathology, University of Texas M. D. Anderson Cancer Center, Houston, Texas 77030 (United States)
2010-12-15
The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski-Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM.
Wang, Xujing; Becker, Frederick F; Gascoyne, Peter R C
2010-12-01
The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski-Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM. PMID:21198103
Wang, Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.
2010-01-01
The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski–Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM. PMID:21198103
Fractal dimension of soil aggregates as an index of soil erodibility
NASA Astrophysics Data System (ADS)
Ahmadi, Abbas; Neyshabouri, Mohammad-Reza; Rouhipour, Hassan; Asadi, Hossein
2011-04-01
SummaryAggregate stability is an influential factor governing soil erodibility. The fractal dimension of soil aggregates has been related to their size distributions and stabilities. Several fractal models have been proposed for estimating fractal dimension of soil aggregates. This study was conducted to investigate how closely the soil interrill erodibility factor in WEPP model can be correlated to and predicted from soil aggregate size distribution or from their fractal dimensions. Samples from 36 soil series with contrasting properties were collected from northwest of Iran. The fractal dimensions of soil aggregates were calculated from Rieu and Sposito ( D n), Tyler and Wheatcraft ( D mT), and Young and Crawford ( D mY) models using aggregate size distribution (ASD) data. A rainfall simulator with drainable tilting flume (1 × 0.5 m) at slope of 9% was employed and total interrill erosion ( TIE), total splashed soil ( TS) and interrill erodibility factor ( K i) were calculated for 20, 37, and 47 mm h -1 rainfall intensities. Results showed that both D n and D mT estimated from aggregate wet-sieving data characterized ASD of the examined soils and significantly ( p < 0.01) correlated to TS, TIE and K i. Values of D n and D mT estimated from dry-sieving data only correlated to TS but not to TIE and K i. Using air-dried aggregates of 4.75-8 mm size range, instead of aggregates <4.75 mm, in wet-sieving was better for estimating D n as an index for the predication of TIE, TS and K i. Correction of ASD for the particle fraction greater than lower sieve mesh size in each size class decreased the correlation coefficient between TIE, TS or K i and D n or D mT. The values of D mY were not correlated to TS, TIE and K i. The correlation coefficient TIE and K i with D n and D mT derived from wet-sieving data, were higher than those with wet-aggregate stability (WAS), mean weight diameter (MWD) and geometric mean diameter (GMD), implying that D n and D mT may be better alternative variables for empirically predicting soil erodibility factor and hence interrill erosion.
Fractal Dimensions of a Weakly Clustered Distribution and the Scale of Homogeneity
J. S. Bagla; Jaswant Yadav; T. R. Seshadri
2008-08-04
Homogeneity and isotropy of the universe at sufficiently large scales is a fundamental premise on which modern cosmology is based. Fractal dimensions of matter distribution is a parameter that can be used to test the hypothesis of homogeneity. In this method, galaxies are used as tracers of the distribution of matter and samples derived from various galaxy redshift surveys have been used to determine the scale of homogeneity in the Universe. Ideally, for homogeneity, the distribution should be a mono-fractal with the fractal dimension equal to the ambient dimension. While this ideal definition is true for infinitely large point sets, this may not be realised as in practice, we have only a finite point set. The correct benchmark for realistic data sets is a homogeneous distribution of a finite number of points and this should be used in place of the mathematically defined fractal dimension for infinite number of points (D) as a requirement for approach towards homogeneity. We derive the expected fractal dimension for a homogeneous distribution of a finite number of points. We show that for sufficiently large data sets the expected fractal dimension approaches D in absence of clustering. It is also important to take the weak, but non-zero amplitude of clustering at very large scales into account. In this paper we also compute the expected fractal dimension for a finite point set that is weakly clustered. Clustering introduces departures in the Fractal dimensions from D and in most situations the departures are small if the amplitude of clustering is small. Features in the two point correlation function, like those introduced by Baryon Acoustic Oscillations (BAO) can lead to non-trivial variations in the Fractal dimensions where the amplitude of clustering and deviations from D are no longer related in a monotonic manner.
[Using fractal dimensions of hyperspectral curves to analyze the healthy status of vegetation].
Du, Hua-Qiang; Jin, Wei; Ge, Hong-Li; Fan, Wen-Yi; Xu, Xiao-Jun
2009-08-01
The reflectance spectral curves of leaves can reflect many information of vegetation growth, and its variation maybe means that the healthy status of vegetation will change. Many spectral feature parameters such as red edge position, height of green peak, depth of red band absorption, the area of red edge and some vegetation index have been used to describe this change. However, the change of vegetation healthy status is not some feature parameters, but a comprehensive variation of the whole curve. So, a comprehensive index maybe has more value to describe the change of hyperspectral curve of vegetation and indicates its healthy status. Fractal is an appropriate mathematical tool, and fractal dimension can be used to explain the comprehensive variation of a curve. Therefore, in the present study, fractal theory was used to analyze the healthy status of different vegetation. Firstly, analytical spectral devices (ASD) were used to measure the hyperspectral curves of different vegetations with different healthy status. Secondly, spectral curves were analyzed, and some parameters which can really reflect different healthy status were obtained. Finally, the fractal dimension of reflectance spectral curves inside a spectral band zone between 450 and 780nm was computed by variation method, and the relationship between fractal dimensions and spectral feature parameters was established. The research results showed that (1) the hyperspectral curves of vegetation have fractal feature, and their fractal dimensions gradually decrease with the health deterioration of leaves, (2) fractal dimension has positive correlation with the height of green peak, the depth of red band absorption and the area of red edge, (3) multivariate analysis showed that fractal dimensions have a significant linear relationship with the three spectral feature parameters just mentioned above. So, the fractal dimension of hyperspectral curve can serve as a new comprehensive parameter to analyze quantitatively the healthy status of vegetations. PMID:19839325
Mass fractal dimension of the ribosome and implication of its dynamic characteristics
NASA Astrophysics Data System (ADS)
Lee, Chang-Yong
2006-04-01
Self-similar properties of the ribosome in terms of the mass fractal dimension are investigated. We find that both the 30S subunit and the 16S rRNA have fractal dimensions of 2.58 and 2.82, respectively; while the 50S subunit as well as the 23S rRNA has the mass fractal dimension close to 3, implying a compact three-dimensional macromolecule. This finding supports the dynamic and active role of the 30S subunit in the protein synthesis, in contrast to the pass role of the 50S subunit.
NASA Astrophysics Data System (ADS)
Davy, Philippe; Sornette, Anne; Sornette, Didier
1992-02-01
A series of experiments scaled for gravity on the formation of faults in a laboratory model of the earth's lithosphere have shown that the obtained fault patterns are self-similar and can be characterized by various fractal dimensions. By analyzing a large set of experimental results, a remarkable scaling law relating the generalized fractal dimensions Dq, the fault barycenter fractal dimension b and the exponent a of the fault length distribution was discovered, namely D0 = b is independent of a whereas D sub q not less than 1 = b + 2-a, for a = 2-3 as found in our experiments.
NASA Astrophysics Data System (ADS)
Zeng, Qiang; Luo, Mingyong; Pang, Xiaoyun; Li, Le; Li, Kefei
2013-10-01
This study investigates the surface fractal dimensions (SFDs) of pore structure of cement pastes and mortars with/without ground granulated blast-furnace slag (GGBS) incorporated into binder. The samples were subject to water curing and sealed curing. The fractal dimensions of samples are determined by Zhang’s model (Ind Eng Chem Res, 34 (1995):1383-1386) on the basis of mercury intrusion porosimetry (MIP) data. The results confirm the scale-dependent property of fractal dimension of pore structures and the micro-fractal, transition and macro-fractal regions are identified for all samples. The upper pore size range for micro-fractal regions is around 30 nm, the transition regions cover 0.5-2 magnitude orders of pore size and macro fractal regions cover 1.5-3 magnitude orders. Both curing conditions and GGBS in binder have impact on the fractal properties of pore structure, and samples incorporating GGBS have substantially larger values for micro-fractal regions.
Wu, Jia-Ming; Kuo, Chung-Ming; Chen, Ching-Jiang
2013-01-01
Purpose. This study describes how to identify the coincidence of desired planning isodose curves with film experimental results by using a mathematical fractal dimension characteristic method to avoid the errors caused by visual inspection in the intensity modulation radiation therapy (IMRT). Methods and Materials. The isodose curves of the films delivered by linear accelerator according to Plato treatment planning system were acquired using Osiris software to aim directly at a single interested dose curve for fractal characteristic analysis. The results were compared with the corresponding planning desired isodose curves for fractal dimension analysis in order to determine the acceptable confidence level between the planning and the measurement. Results. The film measured isodose curves and computer planning curves were deemed identical in dose distribution if their fractal dimensions are within some criteria which suggested that the fractal dimension is a unique fingerprint of a curve in checking the planning and film measurement results. The dose measured results of the film were presumed to be the same if their fractal dimension was within 1%. Conclusions. This quantitative rather than qualitative comparison done by fractal dimension numerical analysis helps to decrease the quality assurance errors in IMRT dosimetry verification. PMID:23956976
Beyond uniformity and independence: analysis of R-trees using the concept of fractal dimension
Christos Faloutsos; Ibrahim Kamel
1994-01-01
We propose the concept of fractal dimension of a set of points, in order to quantify the deviation from the uniformity distribution. Using measurements on real data sets (road intersections of U.S. counties, star coordinates from NASA's Infrared-Ultraviolet Explorer etc.) we provide evidence that real data indeed are skewed, and, moreover, we show that they behave as mathematical fractals, with
NASA Astrophysics Data System (ADS)
Brinkhoff, L. A.; von Savigny, C.; Randall, C. E.; Burrows, J. P.
2015-05-01
The fractal perimeter dimension is a fundamental property of clouds. It describes the cloud shape and is used to improve the understanding of atmospheric processes responsible for cloud shapes. von Savigny et al. (2011) determined the fractal perimeter dimension of noctilucent clouds (or polar mesospheric clouds) for the first time based on a limited data set of cloud images observed with the CIPS (Cloud Imaging and Particle Size) instrument on board the AIM (Aeronomy of Ice in the Mesosphere) satellite. This paper builds on von Savigny et al. (2011) by first presenting a sensitivity analysis of the determination of the fractal perimeter dimension, and secondly presenting results on the seasonal and interhemispheric differences of the perimeter dimension of noctilucent clouds (NLCs). The same method as in the earlier study is applied to an extended data set of satellite images of noctilucent cloud fields taken with the CIPS experiment. The sensitivity studies reveal that cloud holes play an important role for the area-perimeter method, since excluding clouds with holes reduces the dimension value by up to 3%. The results on the fractal perimeter dimension over six NLC seasons from 2007 to 2009 demonstrate that the dimension values of the NLCs neither show significant differences between the seasons nor between the hemispheres.
Cheng, Hongbo
2011-01-01
We discuss the Casimir effect for massless scalar fields subject to the Dirichlet boundary conditions on the parallel plates at finite temperature in the presence of one fractal extra compactified dimension. We obtain the Casimir energy density with the help of the regularization of multiple zeta function with one arbitrary exponent and further the renormalized Casimir energy density involving the thermal corrections. It is found that when the temperature is sufficiently high, the sign of the Casimir energy remains negative no matter how great the scale dimension $\\delta$ is within its allowed region. We derive and calculate the Casimir force between the parallel plates affected by the fractal additional compactified dimension and surrounding temperature. The stronger thermal influence leads the force to be stronger. The nature of the Casimir force keeps attractive.
Automatic Human Face Recognition System Using Fractal Dimension and Modified Hausdorff Distance
Kwan-ho Lin; Baofeng Guo; Kin-man Lam; Wan-chi Siu
2001-01-01
In this paper, an efficient automatic human face recognition system is proposed. Fractal dimension is an efficient representation\\u000a of texture which is used to locate the eyes in a human face. We propose a modified approach to estimate the fractal dimensions\\u000a which is less sensitive to lighting conditions and provides information about the orientation of an image under consideration.\\u000a Based
Fractal Dimension of the Universal Julia Sets for the Chebyshev-Halley Family of Methods
NASA Astrophysics Data System (ADS)
Gutiérrez, J. M.; Magreñán, Á. A.; Varona, J. L.
2011-09-01
The concept of universal Julia set introduced in [5] allows us to conclude that the dynamics of a root-finding algorithm applied to any quadratic polynomial can be understood through the analysis of a particular rational map. In this study we go a step beyond in this direction. In particular, we can define the universal fractal dimension of the aforementioned algorithms as the fractal dimension of they corresponding universal Julia sets.
Crack detection in beams in noisy conditions using scale fractal dimension analysis of mode shapes
NASA Astrophysics Data System (ADS)
Bai, R. B.; Ostachowicz, W.; Cao, M. S.; Su, Z.
2014-06-01
Fractal dimension analysis of mode shapes has been actively studied in the area of structural damage detection. The most prominent features of fractal dimension analysis are high sensitivity to damage and instant determination of damage location. However, an intrinsic deficiency is its susceptibility to measurement noise, likely obscuring the features of damage. To address this deficiency, this study develops a novel damage detection method, scale fractal dimension (SFD) analysis of mode shapes, based on combining the complementary merits of a stationary wavelet transform (SWT) and Katz’s fractal dimension in damage characterization. With this method, the SWT is used to decompose a mode shape into a set of scale mode shapes at scale levels, with damage information and noise separated into distinct scale mode shapes because of their dissimilar scale characteristics; the Katz’s fractal dimension individually runs on every scale mode shape in the noise-adaptive condition provided by the SWT to canvass damage. Proof of concept for the SFD analysis is performed on cracked beams simulated by the spectral finite element method; the reliability of the method is assessed using Monte Carlo simulation to mimic the operational variability in realistic damage diagnosis. The proposed method is further experimentally validated on a cracked aluminum beam with mode shapes acquired by a scanning laser vibrometer. The results show that the SFD analysis of mode shapes provides a new strategy for damage identification in noisy conditions.
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.
2004-01-01
The accuracy of traditional multispectral maximum-likelihood image classification is limited by the skewed statistical distributions of reflectances from the complex heterogenous mixture of land cover types in urban areas. This work examines the utility of local variance, fractal dimension and Moran's I index of spatial autocorrelation in segmenting multispectral satellite imagery. Tools available in the Image Characterization and Modeling System (ICAMS) were used to analyze Landsat 7 imagery of Atlanta, Georgia. Although segmentation of panchromatic images is possible using indicators of spatial complexity, different land covers often yield similar values of these indices. Better results are obtained when a surface of local fractal dimension or spatial autocorrelation is combined as an additional layer in a supervised maximum-likelihood multispectral classification. The addition of fractal dimension measures is particularly effective at resolving land cover classes within urbanized areas, as compared to per-pixel spectral classification techniques.
Smith, R.L., E-mail: firefan@ufl.edu; Mecholsky, J.J., E-mail: jmech@ufl.edu
2011-05-15
Fractal analysis has been used as a method to study fracture surfaces of brittle materials. However, it has not been determined if the fractal characteristics of brittle materials is consistent throughout the fracture surface. Therefore, the fractal dimensional increment of the mirror, mist, and hackle regions of the fracture surface of silica glass was determined using atomic force microscopy. The fractal dimensional increment of the mirror region (0.17-0.26) was determined to be statistically greater than that for the mist (0.08-0.12) and hackle (0.08-0.13) regions. It is thought that the increase in the fractal dimensional increment is caused by a greater tortuosity in the mirror region due to, most likely, the slower crack velocity of the propagating crack in that region and that there is a point between the mirror and mist region at which the fractal dimension decreases and becomes constant. - Research Highlights: {yields} The fracture surface of silica glass does not have a constant fractal dimension. {yields} Mirror region has greater fractal dimension than mist or hackle region. {yields} Fractal dimension decreases between mirror and mist region. {yields} Greater fractal dimension could be due to slower crack velocity in mirror region.
Fractal dimension in butterflies' wings: a novel approach to understanding wing patterns?
Castrejón-Pita, A A; Sarmiento-Galán, A; Castrejón-Pita, J R; Castrejón-García, R
2005-05-01
The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and a small number of unrelated species, thus implying the consideration of only a fraction of the wing pattern diversity. The members of the first mimicry ring, composed by the species Danaus plexippus (better known as the monarch butterfly), and the two subspecies Basilarchia archippus obsoleta (or northern viceroy) and Basilarchia archippus hoffmanni (or tropical viceroy), are found to have a very similar value for the fractal dimension of their wing patterns, even though they do not look very similar at first sight. It is also found that the female of another species (Neophasia terlootii), which looks similar to the members of the previous group, does not share the same feature, while the Lycorea ilione albescens does share it. For the members of the second group of mimicry related butterflies, the Greta nero nero and the Hypoleria cassotis, it is shown that they also have very close values for the fractal dimension of their wing patterns. Finally, it is shown that other species, which apparently have very similar wing patterns, do not have the same fractal dimension. A possible, not completely tested hypothesis is then conjectured: the formation of groups by individuals whose wing patterns have an almost equal fractal dimension may be due to the fact that they do share the same developmental raw material, and that this common feature is posteriorly modified by natural selection, possibly through predation. PMID:15614549
Leila M. V. Carvalho; Maria A. F. Silva Dias
1998-01-01
Mesoscale cloud patterns are analyzed through the application of fractal box dimensions. Verification of fractal properties in satellite infrared images is carried out by computing box dimensions with two different methods and by computing the fraction of cloudy pixels for two sets of images: 174 are considered the ''control series,'' and 178 (for verification) are considered the ''test series.'' The
Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension
Michel L. Lapidus; John A. Rock; Darko Žubrini?
2013-02-01
We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings; and the abscissae of convergence of new classes of zeta functions. Specifically, we define box-counting zeta functions of infinite bounded subsets of Euclidean space and discuss results pertaining to distance and tube zeta functions. Appealing to an analysis of these zeta functions allows for the development of theories of complex dimensions for bounded sets in Euclidean space, extending techniques and results regarding (ordinary) fractal strings obtained by the first author and van Frankenhuijsen.
Approximating the Ising model on fractal lattices of dimension below two
Codello, Alessandro; Hietanen, Ari
2015-01-01
We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of zero external magnetic field, using a generalization of the combinatorial method of Feynman and Vodvickenko. Our procedure is applicable to any fractal obtained by the removal of sites of a periodic two dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way to determine $T_c$ for any fractal of dimension below two. Critical exponents are more difficult to determine since the free energy of any periodic approximation still has a logarithmic singularity at the critical point implying $\\alpha = 0$. We also compute the correlation length as a function of the temperature and extract the relative critical exponent. ...
Approximating the Ising model on fractal lattices of dimension below two
Alessandro Codello; Vincent Drach; Ari Hietanen
2015-05-25
We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of zero external magnetic field, using a generalization of the combinatorial method of Feynman and Vodvickenko. Our procedure is applicable to any fractal obtained by the removal of sites of a periodic two dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way to determine $T_c$ for any fractal of dimension below two. Critical exponents are more difficult to determine since the free energy of any periodic approximation still has a logarithmic singularity at the critical point implying $\\alpha = 0$. We also compute the correlation length as a function of the temperature and extract the relative critical exponent. We find $\
On the Fractal Dimension of Isosurfaces Marc Khoury and Rephael Wenger
Wenger, Rephael
the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly with some scalar value sv. The grid covers some rectilinear region and partitions into cubes (or, more- tial equations such as fluid flow simulations, or can be sampled from a scalar function f : R3 R
Size and Fractal Dimension of Colloid Deposits in Model Porous Media
NASA Astrophysics Data System (ADS)
Roth, E. J.; Mays, D. C.; Gilbert, B.
2014-12-01
Colloids exert significant influence on subsurface hydrology, geochemistry, and microbiology. In particular, colloid deposits reduce permeability, triggering a reduction or realignment of flow. Since many subsurface processes are transport-limited, this reduction or realignment of flow, in turn, influences numerous chemical and biological processes. This work explores a conceptual model linking permeability with colloid deposit morphology, where deposit morphology is quantified by two metrics of the colloid deposit: (1) characteristic size and (2) fractal dimension. These two metrics are measured using static light scattering (SLS) within refractive index matched (RIM) porous media, into which a suspension of 100 nm carboxylate-modified polystyrene microspheres are eluted at constant flow. Scattering data are fitted with a two-parameter model that includes deposit fractal dimension, and with a three-parameter model that also includes deposit size. For each set of scattering measurements, the appropriate model is selected using the Akaike information criterion, and model errors are estimated using the bootstrap with 100 replicates. Results indicate two key findings. First, fractal dimensions generally decrease with time as additional colloids are eluted into the column, indicating a transition from more uniform to more dendritic deposits. Second, permeability reduction is associated with colloid deposits having smaller fractal dimensions, that is, with more dendritic and space-filling deposits. Modeling efforts are currently underway to correlate permeability with the underlying hydrodynamic and geochemical variables that determine colloid deposit morphology.
Francisco J. Esteban; Nelly Padilla; Magdalena Sanz-Cortés; Juan Ruiz de Miras; Núria Bargalló; Pablo Villoslada; Eduard Gratacós
2010-01-01
In the search for a useful parameter to detect and quantify subtle brain abnormalities in infants with intrauterine growth restriction (IUGR), we hypothesised that the analysis of the structural complexity of grey matter (GM) and white matter (WM) using the fractal dimension (FD), a measurement of the topological complexity of an object, could be established as a useful tool for
StreamGP: tracking evolving GP ensembles in distributed data streams using fractal dimension
Gianluigi Folino; Clara Pizzuti; Giandomenico Spezzano
2007-01-01
The paper presents an adaptive GP boosting ensemble method forthe classification of distributed homogeneous streaming data that comes from multiple locations. The approach is able to handle concept drift via change detection by employing a change detection strategy, based on self-similarity of the ensemble behavior, and measured by its fractal dimension. It is efficient since each nodeof the network works
The strength and fractal dimension characteristics of alum–kaolin flocs
Tao Li; Zhe Zhu; Dongsheng Wang; Chonghua Yao; Hongxiao Tang
2007-01-01
Flocs generated by various shear forces exhibit different characteristics of size, strength and structure. These properties were investigated by employing a continuous optical monitoring and a microscope with CCD camera to directly monitor aggregation under six different shear intensities. The floc structure was characterized by the fractal dimension. The results showed that the flocculation index (FI) decreased from 1.16 at
Detection of explosive lung and bowel sounds by means of fractal dimension
Leontios J. Hadjileontiadis; Ioannis T. Rekanos
2003-01-01
An efficient technique for detecting explosive lung sounds (LS) (fine\\/coarse crackles and squawks) or bowel sounds (BS) in clinical auscultative recordings is presented. The technique is based on a fractal-dimension (FD) analysis of the recorded LS and BS obtained from controls and patients with pulmonary and bowel pathology, respectively. Experimental results demonstrate the efficiency of the proposed method, since it
Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar
Nelson, James
1 Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar J. D. B. Nelson and N. G. Kingsbury Abstract--Sand ripples present a difficult challenge to current mine hunting approaches. We propose a robust and adaptive method that suppresses sand ripples prior to the detection stage
StreamGP: Tracking Evolving GP Ensembles in Distributed Data Streams using Fractal Dimension
Fernandez, Thomas
StreamGP: Tracking Evolving GP Ensembles in Distributed Data Streams using Fractal Dimension that comes from multiple locations. The approach is able to handle concept drift via change detection- populations to transfer the knowledge acquired on the own local data. StreamGP models the system
Structural and Fractal Dimensions are Reliable Determinants of Grain Yield in Soybean
Technology Transfer Automated Retrieval System (TEKTRAN)
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 ...
The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms
Geoffrey B. West; James H. Brown; Brian J. Enquist
1999-01-01
The existence of fractal-like networks effectively endows life with an additional fourth spatial dimension. This is the origin of quarter-power scaling which is so pervasive in biology. Organisms have evolved hierarchical networks which terminate in invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules, which are independent of organism size. Natural selection has tended to maximize both metabolic capacity
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
The fractal dimension of the protein aggregates can be estimated by dynamic mechanical methods when the particle aggregates are imbedded in a polymer matrix. Nanocomposites were formed by mixing hydrolyzed soy protein isolate (HSPI) nanoparticle aggregates with styrene-butadiene (SB) latex, followe...
Effect of variable fractal dimension on the floc size distribution of suspended cohesive sediment
F. Maggi; F. Mietta; J. C. Winterwerp
2007-01-01
Flocculation of suspended cohesive sediment, well-known to impact the floc size distribution and vertical fluxes, and cause morphodynamic changes of marine and riverine environments, is modelled by means of a population balance equation that implements a novel description of floc geometry: the capacity dimension of fractal flocs, normally assumed constant over the population, has recently been argued to change during
M. Høgh Jensen; Per Bak; Tomas Bohr
1983-01-01
It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension D=0.87. This exponent is found to be universal for a large class of functions.
Fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-01-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, have not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension, 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with fractal dimension 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas. PMID:23629025
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-06-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas. PMID:23629025
NASA Astrophysics Data System (ADS)
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-06-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.
Are fractal dimensions of the spatial distribution of mineral deposits meaningful?
Raines, G.L.
2008-01-01
It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.
Detection of the cardiac function by fractal dimension analysis.
Yambe, T; Nanka, S; Kobayashi, S; Tanaka, A; Owada, N; Yoshizawa, M; Abe, K; Tabayashi, K; Takeda, H; Nishihira, T; Nitta, S
1999-08-01
Nonlinearity in circulation control attracts attention because nonlinearity is thought to be essential in the function of the living body. Many investigators have pointed out that the analysis of heart rate variability in particular is important in the analysis of autonomic nerve and cardiac function evaluation. Heart rate variability shows nonlinear behavior. However, until the present, many reports have been premised on linearity; linear correlation by frequency analysis has been used by many studies. However, in terms of this methodology, there is a problem applying it to the nonlinear living body. Therefore, fractal and chaos methodology has been used. The ascertainment of cardiac function has become important in allowing the clinical stage of a ventricular assist system to be successful. The purpose of this study was cardiac function evaluation by a methodology that was premised on nonlinearity. Chaos and fractal theory was used as a nonlinear dynamic theory. As a methodology of measurement, the volume of the left ventricle was used rather than an electrocardiogram, the waveform of arterial blood pressure. The volume was measured using acoustic quantification (AQ) ultrasonic echocardiography. Using these methodologies, the time series of many patients were analyzed. For example, drug administration was attempted in this study, and it was found that some drugs like ACE inhibitors showed a significant effect upon nonlinear dynamics in the cardiovascular system. The result, which attempted cardiac function evaluation by these various methodologies, is reported. PMID:10463502
L. Ya. Kobelev
2000-06-10
In the space and the time with a fractional dimensions the Lorents transformations fulfill only as a good approach and become exact only when dimensions are integer. So the principle of relativity (it is exact when dimensions are integer) may be treated also as a good approximation and may remain valid (but modified) in case of small fractional corrections to integer dimensions of time and space. In this paper presented the gravitation field theory in the fractal time and space (based on the fractal theory of time and space developed by author early). In the theory are taken into account the alteration of Lorents transformations for case including $v=c$ and are described the real gravitational fields with spin equal 2 in the fractal time defined on the Riemann or Minkowski measure carrier. In the theory introduced the new "quasi-spin", given four equations for gravitational fields (with different "quasi spins" and real and imaginary energies). For integer dimensions the theory coincide with Einstein GR or Logunov- Mestvirichvili gravitation theory.
Huang, F. [College of Science, China Agricultural University, Beijing 100083 (China); Peng, R. D. [State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing 100083 (China); Liu, Y. H. [Institute of Complexity Science, Qingdao University, Qingdao 266071 (China); Chen, Z. Y. [Department of Physics, Beijing University of Chemical Technology, Beijing 100029 (China); Ye, M. F.; Wang, L. [Institute of Physics, Chinese Academy of Science, Beijing 100190 (China)
2012-09-15
Fractal dust grains of different shapes are observed in a radially confined magnetized radio frequency plasma. The fractal dimensions of the dust structures in two-dimensional (2D) horizontal dust layers are calculated, and their evolution in the dust growth process is investigated. It is found that as the dust grains grow the fractal dimension of the dust structure decreases. In addition, the fractal dimension of the center region is larger than that of the entire region in the 2D dust layer. In the initial growth stage, the small dust particulates at a high number density in a 2D layer tend to fill space as a normal surface with fractal dimension D = 2. The mechanism of the formation of fractal dust grains is discussed.
Low fractal dimension of clusters with repulsive interactions in colloidal aggregation
NASA Astrophysics Data System (ADS)
González, Agustín E.
2014-03-01
Monte Carlo simulations of colloidal aggregation were performed showing that, by increasing the range or the height of a repulsive barrier of non-negligible width between the particles, the fractal dimension of the formed clusters decreases, at least for not very large clusters. This result offers an explanation of old experimental findings of 2D aggregates, with a low fractal dimensionality, made of silica colloids confined on the air-water interface, as well as of very recent experimental results of low fractal dimensionality of clusters of diverse colloids in 3D, when the size of the primary particles is increased. This second case was studied through an analysis of the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential between the particles.
Gheonea, Dan Ionu?; Streba, Costin Teodor; Vere, Cristin Constantin; ?erb?nescu, Mircea; Pirici, Daniel; Com?nescu, Maria; Streba, Leti?ia Adela Maria; Ciurea, Marius Eugen; Mogoant?, Stelian; Rogoveanu, Ion
2014-01-01
Background and Aims. Hepatocellular carcinoma (HCC) remains a leading cause of death by cancer worldwide. Computerized diagnosis systems relying on novel imaging markers gained significant importance in recent years. Our aim was to integrate a novel morphometric measurement—the fractal dimension (FD)—into an artificial neural network (ANN) designed to diagnose HCC. Material and Methods. The study included 21 HCC and 28 liver metastases (LM) patients scheduled for surgery. We performed hematoxylin staining for cell nuclei and CD31/34 immunostaining for vascular elements. We captured digital images and used an in-house application to segment elements of interest; FDs were calculated and fed to an ANN which classified them as malignant or benign, further identifying HCC and LM cases. Results. User intervention corrected segmentation errors and fractal dimensions were calculated. ANNs correctly classified 947/1050 HCC images (90.2%), 1021/1050 normal tissue images (97.23%), 1215/1400 LM (86.78%), and 1372/1400 normal tissues (98%). We obtained excellent interobserver agreement between human operators and the system. Conclusion. We successfully implemented FD as a morphometric marker in a decision system, an ensemble of ANNs designed to differentiate histological images of normal parenchyma from malignancy and classify HCCs and LMs. PMID:25025042
A new way of describing meiosis that uses fractal dimension to predict metaphase I
2005-01-01
Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465
A new way of describing meiosis that uses fractal dimension to predict metaphase I.
Ross, Cynthia M
2005-01-01
Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465
Reljin, Natasa; Reyes, Bersain A.; Chon, Ki H.
2015-01-01
In this paper, we propose the use of blanket fractal dimension (BFD) to estimate the tidal volume from smartphone-acquired tracheal sounds. We collected tracheal sounds with a Samsung Galaxy S4 smartphone, from five (N = 5) healthy volunteers. Each volunteer performed the experiment six times; first to obtain linear and exponential fitting models, and then to fit new data onto the existing models. Thus, the total number of recordings was 30. The estimated volumes were compared to the true values, obtained with a Respitrace system, which was considered as a reference. Since Shannon entropy (SE) is frequently used as a feature in tracheal sound analyses, we estimated the tidal volume from the same sounds by using SE as well. The evaluation of the performed estimation, using BFD and SE methods, was quantified by the normalized root-mean-squared error (NRMSE). The results show that the BFD outperformed the SE (at least twice smaller NRMSE was obtained). The smallest NRMSE error of 15.877% ± 9.246% (mean ± standard deviation) was obtained with the BFD and exponential model. In addition, it was shown that the fitting curves calculated during the first day of experiments could be successfully used for at least the five following days. PMID:25923929
NASA Astrophysics Data System (ADS)
Ohsasa, K.; Natsume, Y.; Sekiya, T.; Hatayama, T.
2015-06-01
The dendrite morphology of unidirectionally solidified Al-Si alloys was evaluated by measuring the fractal dimension and dimensionless perimeter of dendrites. In an unidirectional solidification experiment, columnar crystals grew from a bottom chill and columnar to equiaxed transition (CET) occurred at the upper part of an ingot. Then, equiaxed crystals were formed at the top of the ingot. Different dendrite morphology was observed in longitudinal, transverse and oblique sections, however, the fractal dimension or dimensionless perimiter of the dendrites in the sections with same local solidification time showed same values, and continuously decreased with increase in the local solidification time through columnar, CET and equiaxed regions. It can be considered that the fractal dimension and dimensionless perimiter of dendrites are controlled by local solidification time and irrespective of dendrite morphology. This result demonstrated the potential of the fractal dimension and dimensionless perimiter as a parameter for estimating local solidification time of an ingot in which the measurement of SDAS is difficult.
NASA Astrophysics Data System (ADS)
Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun
1995-05-01
This paper describes the quantitative measurement of the amount of fibrosis in the rat liver using the fractal dimension of the shape of power spectrum. The shape of the power spectrum of the scattered echo from biotissues is strongly affected by its internal structure. The fractal dimension, which is one of the important parameters of the fractal theory, is useful to express the complexity of shape of figures such as the power spectrum. From in vitro experiments using rat liver, it was found that this method can be used to quantitatively measure the amount of fibrosis in the liver, and has the possibility for use in the diagnosis of human liver cirrhosis.
Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S
2011-04-30
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated. (optical technologies in biophysics and medicine)
NASA Astrophysics Data System (ADS)
Ul'yanov, A. S.; Lyapina, A. M.; Ulianova, O. V.; Fedorova, V. A.; Uianov, S. S.
2011-04-01
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated.
NASA Astrophysics Data System (ADS)
Alonso, C.; Benito, R. M.; Tarquis, A. M.
2012-04-01
Satellite image data have become an important source of information for monitoring vegetation and mapping land cover at several scales. Beside this, the distribution and phenology of vegetation is largely associated with climate, terrain characteristics and human activity. Various vegetation indices have been developed for qualitative and quantitative assessment of vegetation using remote spectral measurements. In particular, sensors with spectral bands in the red (RED) and near-infrared (NIR) lend themselves well to vegetation monitoring and based on them [(NIR - RED) / (NIR + RED)] Normalized Difference Vegetation Index (NDVI) has been widespread used. Given that the characteristics of spectral bands in RED and NIR vary distinctly from sensor to sensor, NDVI values based on data from different instruments will not be directly comparable. The spatial resolution also varies significantly between sensors, as well as within a given scene in the case of wide-angle and oblique sensors. As a result, NDVI values will vary according to combinations of the heterogeneity and scale of terrestrial surfaces and pixel footprint sizes. Therefore, the question arises as to the impact of differences in spectral and spatial resolutions on vegetation indices like the NDVI. The aim of this study is to establish a comparison between two different sensors in their NDVI values at different spatial resolutions. Scaling analysis and modeling techniques are increasingly understood to be the result of nonlinear dynamic mechanisms repeating scale after scale from large to small scales leading to non-classical resolution dependencies. In the remote sensing framework the main characteristic of sensors images is the high local variability in their values. This variability is a consequence of the increase in spatial and radiometric resolution that implies an increase in complexity that it is necessary to characterize. Fractal and multifractal techniques has been proven to be useful to extract such complexities from remote sensing images and will applied in this study to see the scaling behavior for each sensor in generalized fractal dimensions. The studied area is located in the provinces of Caceres and Salamanca (east of Iberia Peninsula) with an extension of 32 x 32 km2. The altitude in the area varies from 1,560 to 320 m, comprising natural vegetation in the mountain area (forest and bushes) and agricultural crops in the valleys. Scaling analysis were applied to Landsat-5 and MODIS TERRA to the normalized derived vegetation index (NDVI) on the same region with one day of difference, 13 and 12 of July 2003 respectively. From these images the area of interest was selected obtaining 1024 x 1024 pixels for Landsat image and 128 x 128 pixels for MODIS image. This implies that the resolution for MODIS is 250x250 m. and for Landsat is 30x30 m. From the reflectance data obtained from NIR and RED bands, NDVI was calculated for each image focusing this study on 0.2 to 0.5 ranges of values. Once that both NDVI fields were obtained several fractal dimensions were estimated in each one segmenting the values in 0.20-0.25, 0.25-0.30 and so on to rich 0.45-0.50. In all the scaling analysis the scale size length was expressed in meters, and not in pixels, to make the comparison between both sensors possible. Results are discussed. Acknowledgements This work has been supported by the Spanish MEC under Projects No. AGL2010-21501/AGR, MTM2009-14621 and i-MATH No. CSD2006-00032
Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis
Metze, Konradin
2013-01-01
Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed. PMID:24063399
M. N. Piacquadio Losada
2007-11-17
The Cantor set complementary to the Devil's Staircase associated with the Circle Map has a fractal dimension d approximately equal to 0.87, a value that is universal for a wide range of maps, such results being of a numerical character. In this paper we deduce a formula for such dimensional value. The Devil's Staircase associated with the Circle Map is a function that transforms horizontal unit interval I onto vertical I, and is endowed with the Farey-Brocot (F-B) structure in the vertical axis via the rational heights of stability intervals. The underlying Cantor-dust fractal set Omega in the horizontal axis --Omega contained in I, with fractal dimension d(Omega) approx. 0.87-- has a natural covering with segments that also follow the F-B hierarchy: therefore, the staircase associates vertical I (of unit dimension) with horizontal Omega in I (of dimension approx. 0.87), i.e. it selects a certain subset Omega of I, both sets F- B structured, the selected Omega with smaller dimension than that of I. Hence, the structure of the staircase mirrors the F- B hierarchy. In this paper we consider the subset Omega-F-B of I that concentrates the measure induced by the F-B partition and calculate its Hausdorff dimension, i.e. the entropic or information dimension of the F-B measure, and show that it coincides with d(Omega) approx. 0.87. Hence, this dimensional value stems from the F-B structure, and we draw conclusions and conjectures from this fact. Finally, we calculate the statistical "Euclidean" dimension (based on the ordinary Lebesgue measure) of the F-B partition, and we show that it is the same as d(Omega-F-B), which permits conjecturing on the universality of the dimensional value d approximately equal to 0.87.
Fractal dimension to describe soil macropore structure using X ray computed tomography
NASA Astrophysics Data System (ADS)
Peyton, R. Lee; Gantzer, Clark J.; Anderson, Stephen H.; Haeffner, Brian A.; Pfeifer, Peter
1994-03-01
The size, shape, and arrangement of structured voids 1-10 mm in size play an important role in the transport of water and solutes through soil. However, these characteristics are complex and difficult to quantify. Improved methods are needed to quantify the characteristics of these voids to better understand and predict the behavior of water and solutes passing through them. This study applied fractal analysis to soil bulk density data measured by X ray computed tomography (CT), a relatively new tool for nondestructively measuring macropore-scale density in soil cores. Studies were conducted using undisturbed soil cores (7.6 cm ID) from forested and cultivated sites in the A horizon of a Menfro silt loam soil containing macropores and using two groups of soil cores which were uniformly packed with Menfro aggregates from 1-2 mm in diameter for one group and <1 mm in diameter for the other group. Samples were probed using CT to produce a 512 by 512 digital matrix of CT pixel values corresponding to bulk density. Pixels above a specified CT "cutoff" value were designated as occupied. A box-counting method was used to find the fractal dimension of the perimeters between occupied and unoccupied pixels and of the areas formed by the unoccupied pixels. For length scales from 1 to 10 mm, perimeters and areas of these regions appeared to be fractal systems. Single degree of freedom orthogonal contrast tests determined from analysis of variance showed significant differences between the fractal dimension for (1) forest and cultivated cores versus uniformly packed cores, (2) two groups of uniformly packed cores made of different aggregate sizes, and (3) forest versus cultivated cores.
The fractality of lung sounds: A comparison of three waveform fractal dimension algorithms
Moussavi, Zahra M. K.
were recorded from five females, ages 2440 years, breathing at flows between 0.353 ± 0.063 and 2- surement (flow sensor with mouthpiece and nose clip). FDs of time-domain LS were calculated within three to breath sounds in past studies for flow onset detection using tracheal sounds in healthy subjects [3
Multiscale Fractal Descriptors Applied to Nanoscale Images
Florindo, João B; Pereira, Ernesto C; Bruno, Odemir M
2012-01-01
This work proposes the application of fractal descriptors to the analysis of nanoscale materials under different experimental conditions. We obtain descriptors for images from the sample applying a multiscale transform to the calculation of fractal dimension of a surface map of such image. Particularly, we have used the}Bouligand-Minkowski fractal dimension. We applied these descriptors to discriminate between two titanium oxide films prepared under different experimental conditions. Results demonstrate the discrimination power of proposed descriptors in such kind of application.
Marco Heinen; Simon K. Schnyder; John F. Brady; Hartmut Löwen
2015-05-05
We introduce fractal liquids by generalizing classical liquids of integer dimensions $d = 1, 2, 3$ to a fractal dimension $d_f$. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same non-integer dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semi-analytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.
ERIC Educational Resources Information Center
McCartney, M.; Myers, D.; Sun, Y.
2008-01-01
The divider dimensions of a range of maps of Ireland dating from 1567 to 1893 are evaluated, and it is shown that for maps produced before 1650 the fractal dimension of the map can be correlated to its date of publication. Various classroom uses and extensions are discussed. (Contains 2 figures.)
NASA Astrophysics Data System (ADS)
Ahammer, Helmut; DeVaney, Trevor T. J.
2004-03-01
The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.
Zheng, Xiuqing; Hu, Shaoxiang; Li, Ming; Zhou, Jiliu
2013-01-01
NLMs is a state-of-art image denoising method; however, it sometimes oversmoothes anatomical features in low-dose CT (LDCT) imaging. In this paper, we propose a simple way to improve the spatial adaptivity (SA) of NLMs using pointwise fractal dimension (PWFD). Unlike existing fractal image dimensions that are computed on the whole images or blocks of images, the new PWFD, named pointwise box-counting dimension (PWBCD), is computed for each image pixel. PWBCD uses a fixed size local window centered at the considered image pixel to fit the different local structures of images. Then based on PWBCD, a new method that uses PWBCD to improve SA of NLMs directly is proposed. That is, PWBCD is combined with the weight of the difference between local comparison windows for NLMs. Smoothing results for test images and real sinograms show that PWBCD-NLMs with well-chosen parameters can preserve anatomical features better while suppressing the noises efficiently. In addition, PWBCD-NLMs also has better performance both in visual quality and peak signal to noise ratio (PSNR) than NLMs in LDCT imaging. PMID:23606907
THE FRACTAL DIMENSION OF STAR-FORMING REGIONS AT DIFFERENT SPATIAL SCALES IN M33
Sanchez, Nestor; Alfaro, Emilio J. [Instituto de Astrofisica de AndalucIa, CSIC, Apdo. 3004, E-18080, Granada (Spain); Anez, Neyda [Departamento de Fisica, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo (Venezuela, Bolivarian Republic of); Odekon, Mary Crone, E-mail: nestor@iaa.e [Department of Physics, Skidmore College, Saratoga Springs, NY 12866 (United States)
2010-09-01
We study the distribution of stars, H II regions, molecular gas, and individual giant molecular clouds in M33 over a wide range of spatial scales. The clustering strength of these components is systematically estimated through the fractal dimension. We find scale-free behavior at small spatial scales and a transition to a larger correlation dimension (consistent with a nearly uniform distribution) at larger scales. The transition region lies in the range {approx}500-1000 pc. This transition defines a characteristic size that separates the regime of small-scale turbulent motion from that of large-scale galactic dynamics. At small spatial scales, bright young stars and molecular gas are distributed with nearly the same three-dimensional fractal dimension (D {sub f,3D} {approx}< 1.9), whereas fainter stars and H II regions exhibit higher values, D {sub f,3D} {approx_equal} 2.2-2.5. Our results indicate that the interstellar medium in M33 is on average more fragmented and irregular than in the Milky Way.
Fractal dimension analysis of landscape scale variability in greenhouse gas production potentials
NASA Astrophysics Data System (ADS)
da Silva Bicalho, Elton; Spokas, Kurt; La Scala, Newton, Jr.
2015-04-01
Soil greenhouse gas emission is influenced by tillage and management practices that modify soil attributes directly related to the dynamics of soil carbon in the agricultural environment. The aim of this study was to assess the soil CO2 and N2O production potentials and their spatial variability characterized by fractal dimension in different scales, in addition to their correlation with other soil attributes. The quantification of soil CO2 and N2O production was carried out from dry soil samples collected in a grid of 50 × 50 m containing 133 points arranged symmetrically on a sugarcane area under green residue management in southern Brazil. Laboratory incubations were used to analyze greenhouse gas dynamics by gas chromatography. Soil CO2 and N2O production were correlated significantly (P < 0.05) with microbial biomass, silt and clay content, pH, available phosphorus, sum of metal cations (bases), and cation exchange capacity. Similarly, these soil attributes also were correlated with microbial biomass, supporting their role in soil microbial activity and greenhouse gas production. Furthermore, variations in the fractal dimension over the scale indicate that the pattern of the spatial variability structure of soil CO2 production potential was correlated to that observed for microbial biomass, pH, available phosphorus, sum of bases, and cation exchange capacity. On the other hand, only the spatial structure of the clay content, pH and the sum of bases were correlated with the soil N2O production. Therefore, examining the fractal dimension enables the spatially visualization of altering processes across a landscape at different scales, which highlights properties that influence greenhouse gas production and emission in agricultural areas.
Power spectrum and fractal dimension of laser backscattering from the ocean.
Churnside, James H; Wilson, James J
2006-11-01
We flew an airborne lidar perpendicular to the coastline along straight-line transects that varied in length between 230 and 280 km. The sample spacing was approximately 3 m, so we sampled almost five decades of spatial scales. Except for the return from right at the surface, the power spectra of backscattered power had a power-law dependence on spatial frequency, with a slope of approximately 1.49. This corresponds to a fractal dimension of 1.76. This implies that the distribution is not as patchy as that of a purely turbulent process. PMID:17047710
NASA Technical Reports Server (NTRS)
Garneau, S.; Plaut, J. J.
2000-01-01
The surface roughness of the Vastitas Borealis Formation on Mars was analyzed with fractal statistics. Root mean square slopes and fractal dimensions were calculated for 74 topographic profiles. Results have implications for radar scattering models.
Objective Auscultation of TCM Based on Wavelet Packet Fractal Dimension and Support Vector Machine
Yan, Jian-Jun; Wang, Yi-Qin; Liu, Guo-Ping; Yan, Hai-Xia; Xia, Chun-Ming; Shen, Xiaojing
2014-01-01
This study was conducted to illustrate that auscultation features based on the fractal dimension combined with wavelet packet transform (WPT) were conducive to the identification the pattern of syndromes of Traditional Chinese Medicine (TCM). The WPT and the fractal dimension were employed to extract features of auscultation signals of 137 patients with lung Qi-deficient pattern, 49 patients with lung Yin-deficient pattern, and 43 healthy subjects. With these features, the classification model was constructed based on multiclass support vector machine (SVM). When all auscultation signals were trained by SVM to decide the patterns of TCM syndromes, the overall recognition rate of model was 79.49%; when male and female auscultation signals were trained, respectively, to decide the patterns, the overall recognition rate of model reached 86.05%. The results showed that the methods proposed in this paper were effective to analyze auscultation signals, and the performance of model can be greatly improved when the distinction of gender was considered. PMID:24883068
NASA Technical Reports Server (NTRS)
Bazell, David; Dwek, Eli
1990-01-01
Mathis and Whiffen (1989) have recently suggested that interstellar dust particles are fluffy aggregates of submicron-size particles composed of various astronomical minerals. These dust particles should exhibit optical properties that are quite different from standard dust, characterized by spherical particles of various homogeneous mineral composition. In this paper, the discrete dipole approximation (DDA) method is used to examine the effects of chemical inhomogeneities and spatial structure on the optical properties of interstellar Mathis-Whiffen-type dust particles. The spatial structure of the dust is represented by its fractal dimension, and the chemical inhomogeneities are simulated by randomly assigning the composition of the occupied sites in the structure to be either carbon or silicate. It is found that compositional inhomogeneities are the dominant parameter affecting the shape of the 9.7 and 18 micron silicate bands. Some bands-shape variations can be attributed to the fractal dimension of the dust. The results derived here can be used to explain or constrain variations in these parameters among various astronomical objects.
Fractal Dimension Analysis in Self-Assembled Poly(dA)·poly(dT) DNA Network on Mica Surface
NASA Astrophysics Data System (ADS)
Kawano, Satoyuki
Characteristics of the self-assembled poly(dA)·poly(dT) DNA network adhered on the mica substrate are experimentally investigated based on the AFM observations and the fractal dimension analysis. Artificial B-type double stranded DNA, which consists of 50 base pairs of adenine and thymine, is specially prepared for the experiment. The manufacturing process of DNA network is done in the aqueous solution of poly(dA)·poly(dT) DNA, and the systematical experimental runs are made for various concentration of the solution. It is found that the 2D fractal dimension strongly depends on the fabrication process of the DNA network.
Effect of variable fractal dimension on the floc size distribution of suspended cohesive sediment
NASA Astrophysics Data System (ADS)
Maggi, F.; Mietta, F.; Winterwerp, J. C.
2007-09-01
SummaryFlocculation of suspended cohesive sediment, well-known to impact the floc size distribution and vertical fluxes, and cause morphodynamic changes of marine and riverine environments, is modelled by means of a population balance equation that implements a novel description of floc geometry: the capacity dimension of fractal flocs, normally assumed constant over the population, has recently been argued to change during flocculation. Our experiments have shown that a power-law function of the dimensionless floc size can conveniently be used to describe these changes. This description of floc capacity dimension is used to explore in detail the extent to which the geometrical properties of flocs affect aggregation and breakup processes, and contribute to shaping their size distribution. A comparison of experimental floc size distributions from settling column test with computed distributions for two hypotheses of floc capacity dimension (i.e., constant and variable) and two hypotheses of flocculation reactions (i.e., semi-stochastic and deterministic) are shown. This suggests that the use of variable rather than constant floc capacity dimension, and the use of semi-stochastic and asymmetric reactions rather than deterministic and symmetric, result in better predictions of the floc size distribution in the environmental conditions herein analysed.
NASA Astrophysics Data System (ADS)
Rodkin, M. V.; Shatakhtsyan, A. R.
2015-05-01
The method for calculating the fractal correlation dimension is applied for analyzing the data on the locations of large and extralarge ore deposits. The approach implemented in this study differs by a few of important points from that commonly used, e.g., in the calculations of the correlation dimension for a set of the epicenters (hypocenters) of the earthquakes. Firstly, we demonstrate the possibility and advisability of obtaining different dimension estimates for different spatial scales. Such a separation turned out to be useful in distinguishing between the regularities in the location of ore deposits on the scale of an ore cluster, ore province, and entire continent. Secondly, we introduce a new notion, a mixed correlation dimension, and use it for different types of the objects (e.g., Au and Ag). The standard formula for calculating the correlation dimension is trivially generalized on this case. It is shown that the values of the correlation dimension can be lower and higher than the dimension of the hosting medium. The cases when the correlation dimension is higher than that of the hosting medium are interpreted as a "mutual repulsion" of the deposits of the two mentioned types. In contrast, the small correlation dimensions indicate that the deposits of the corresponding types tend to have spatially close locations. The calculations are conducted for the spherical Earth. The method is applied to the data on the large and extralarge world-class ore deposits from the Largest Mineral Deposits of the World (LMDs) geoinformation system (GIS). Different patterns of the studied behavior are illustrated by the model examples.
R. Orbach
1986-01-01
Random structures often exhibit fractal geometry, defined in terms of the mass scaling exponent, D, the fractal dimension. The vibrational dynamics of fractal networks are expressed in terms of the exponent d double bar, the fracton dimensionality. The eigenstates on a fractal network are spatially localized for d double bar less than or equal to 2. The implications of fractal
Andrés Duhour; Cristina Costa; Fernando Momo; Liliana Falco; Leonardo Malacalza
2009-01-01
Soil structure degradation and its relationship with soil fauna communities is a crucial issue in soil management. The aim of this work is to analyze soil fractal dimensions and the earthworm community structure along a perturbation gradient in a Typic Argiudoll soil with different combinations of annual crops and pastures. Samples were taken in four sites: a natural grassland (NAT)
Dong Liu; Tianxiao Li; Qiang Fu
2010-01-01
In order to reveal the groundwater depth series complexity characteristic of well irrigation area in Sanjiang Plain, taking Jiansanjiang branch bureau as example, the fractal dimension estimation method of hydrological series based on continuous wavelet transform was adopted to analyze the complexity of groundwater depth series in 7 production teams of Jiansanjiang branch bureau. The results show that the wavelet
Felipe Richter Reis; Marcelo Kaminski Lenzi; Graciela Inês Bolzón de Muñiz; Silvana Nisgoski; Maria Lucia Masson
2012-01-01
This work aimed at studying the drying kinetics of yacon slices during vacuum drying and evaluating the effect of drying temperature, slice thickness, and citric acid concentration in the enzymatic inhibition solution on the fractal dimension and rehydration ratio of yacon slices. The drying kinetics was tentatively fitted to various thin-layer drying models, and the best fit was obtained using
Ferer, M. [West Virginia Univ., Morgantown, WV (United States); National Energy Technology Lab. (NETL), Morgantown, WV (United States); Bromhal, Grant S. [National Energy Technology Lab. (NETL), Morgantown, WV (United States); Smith, Duane H. [West Virginia Univ., Morgantown, WV (United States); National Energy Technology Lab. (NETL), Morgantown, WV (United States)
2009-07-01
Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact/linear flow at a characteristic crossover time, ?, in three dimensions to systems with as many as a 10^{6} pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, t >> ?. Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, t >> ?, the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, ?=1.0±0.1. We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Levy flights, the height variations across the interface appear to be random with occasional large height variations.
NASA Astrophysics Data System (ADS)
Braga, F. L.; Mattos, O. A.; Amorin, V. S.; Souza, A. B.
2015-07-01
Clusters formation models have been extensively studied in literature, and one of the main task of this research area is the analysis of the particle aggregation processes. Some work support that the main characteristics of this processes are strictly correlated to the cluster morphology, for example in DLA. It is expected that in the DLA clusters formation with particles containing different sizes the modification of the aggregation processes can be responsible for changes in the DLA morphology. The present article is going to analyze the formation of DLA clusters of particles with different sizes and show that the aggregates obtained by this approach generate an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affect the fractal dimension of the clusters.
Determination of the fractal dimension for the epitaxial n-GaAs surface in the local limit
Torkhov, N. A., E-mail: trkf@mail.ru; Bozhkova, V. G. [Scientific-Research Institute of Semiconductor Devices (Russian Federation); Ivonin, I. V.; Novikov, V. A. [Tomsk State University (Russian Federation)
2009-01-15
Atomic-force microscopy studies of epitaxial n-GaAs surfaces prepared to deposit barrier contacts showed that major relief for such surfaces is characterized by a roughness within 3-15 nm, although 'surges' up to 30-70 nm are observed. Using three independent methods for determining the spatial dimension of the surface, based on the fractal analysis for the surface (triangulation method), its section contours in the horizontal plane, and the vertical section (surface profile), it was shown that the active surface for epitaxial n-GaAs obeys all main features of behavior for fractal Brownian surfaces and, in the local approximation, can be characterized by the fractal dimension D{sub f} slightly differing for various measuring scales. The most accurate triangulation method showed that the fractal dimensions for the studied surface of epitaxial n-GaAs for measurement scales from 0.692 to 0.0186 {mu}m are in the range D{sub f} = 2.490-2.664. The real surface area S{sub real} for n-GaAs epitaxial layers was estimated using a graphical method in the approximation {delta} {sup {yields}} 0 {delta} is the measurement scale parameter). It was shown that the real surface area for epitaxial n-GaAs can significantly (ten times and more) exceed the area of the visible contact window.
NASA Astrophysics Data System (ADS)
Karemore, Gopal; Nielsen, Mads
2009-02-01
Structural texture measures are used to address the aspect of breast cancer risk assessment in screening mammograms. The current study investigates whether texture properties characterized by local Fractal Dimension (FD) and Lacunarity contribute to asses breast cancer risk. FD represents the complexity while the Lacunarity characterize the gappiness of a fractal. Our cross-sectional case-control study includes mammograms of 50 patients diagnosed with breast cancer in the subsequent 2-4 years and 50 matched controls. The longitudinal double blind placebo controlled HRT study includes 39 placebo and 36 HRT treated volunteers for two years. ROIs with same dimension (250*150 pixels) were created behind the nipple region on these radiographs. Box counting method was used to calculate the fractal dimension (FD) and the Lacunarity. Paired t-test and Pearson correlation coefficient were calculated. It was found that there were no differences between cancer and control group for FD (P=0.8) and Lacunarity (P=0.8) in crosssectional study whereas earlier published heterogeneity examination of radiographs (BC-HER) breast cancer risk score separated groups (p=0.002). In the longitudinal study, FD decreased significantly (P<0.05) in the HRT treated population while Lacunarity remained insignificant (P=0.2). FD is negatively correlated to Lacunarity (-0.74, P<0.001), BIRADS (-0.34, P<0.001) and Percentage Density (-0.41, P<0.001). FD is invariant to the mammographic texture change from control to cancer population but marginally varying in HRT treated population. This study yields no evidence that lacunarity or FD are suitable surrogate markers of mammographic heterogeneity as they neither pick up breast cancer risk, nor show good sensitivity to HRT.
Jiménez, J; López, A M; Cruz, J; Esteban, F J; Navas, J; Villoslada, P; Ruiz de Miras, J
2014-10-01
This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimer's disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud. PMID:24909817
ERIC Educational Resources Information Center
House, Garvey; Zelhart, Paul F.
The complexity (fractal dimension value) of responses to the Rey-Osterrieth Complex Figure Test (ROCFT) between 10 undergraduate students with learning disabilities and a comparison group of 10 students without learning disabilities were compared. The fractal value of responses was assessed under three conditions (copy, immediate, and delay) by…
Generalized Fibonacci Description of Fractal aggregates
NASA Astrophysics Data System (ADS)
Sorensen, Chris; Heinson, William; Chakrabarti, Amit
2009-10-01
We present a theory for calculating the fractal dimension of Diffusion Limited Cluster Aggregates (DLCA) based on cluster shape preservation. The shape is described by a d-dimensional Golden Mean, which is the ratio of consecutive d-dimensional Fibonacci numbers. For d =2 the canonical Fibonacci series is found with the Golden Mean value known since antiquity, phi = 1.618 to yield a fractal dimension of 1.44, in agreement with simulations and experiment. Generalizations to other dimensions are equally successful. Recent computer simulations also yield accurate values for the fractal aggregate prefactor, thus completing the theory.
Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension
NASA Astrophysics Data System (ADS)
Backes, André Ricardo; Cavaleri Gerhardinger, Leandro; do Espírito Santo Batista Neto, João; Martinez Bruno, Odemir
2015-02-01
Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered.
Entanglement in permutation symmetric states, fractal dimensions, and geometric quantum mechanics
NASA Astrophysics Data System (ADS)
Castro-Alvaredo, Olalla A.; Doyon, Benjamin
2013-02-01
We study the von Neumann and Rényi bipartite entanglement entropies in the thermodynamic limit of many-body quantum states with spin-s sites that possess full symmetry under exchange of sites. It turns out that there is essentially a one-to-one correspondence between such thermodynamic states and probability measures on CP2s. Let a measure be supported on a set of possibly fractal real dimension d with respect to the Study-Fubini metric of CP2s. Let m be the number of sites in a subsystem of the bipartition. We give evidence that in the limit m ? ?, the entanglement entropy diverges like (d/2)logm. Further, if the measure is supported on a submanifold of CP2s and can be described by a density f with respect to the metric induced by the Study-Fubini metric, we give evidence that the correction term is simply related to the entropy associated with f: the geometric entropy of geometric quantum mechanics. This extends results obtained by the authors in a recent letter where the spin-\\frac{1}{2} case was considered. Here we provide more examples as well as detailed accounts of the ideas and computations leading to these general results. For special choices of the state in the spin-s situation, we recover the scaling behaviour previously observed by Popkov et al, showing that their result is but a special case of a more general scaling law.
Building Fractal Models with Manipulatives.
ERIC Educational Resources Information Center
Coes, Loring
1993-01-01
Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)
Fractal Geometric Characterization of Functionally Graded Materials
Ostoja-Starzewski, Martin
Fractal Geometric Characterization of Functionally Graded Materials A. Saharan1 ; M. Ostoja graded materials (FGM) is studied from the standpoint of fractal geometry. First, upon introducing fractals, and an interfacial fractal dimension is estimated for varying degrees of fineness. Avariation
2010-01-01
Background Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony. Methods Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization. Results A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total. Conclusions The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process. PMID:20846449
Electromagnetic time-domain calculations in two and three dimensions
Chan, K.C.D.
1988-01-01
For some time, computer codes have been available for time-domain calculations of the beam-induced electromagnetic fields in axially symmetric structures (two dimensions). Recently, these codes have been extended to three-dimensional geometries. Time-domain calculations are complementary to frequency-domain calculations in accelerator designs and represent a better approach in some areas. Some of these areas will be reviewed in this paper and an introduction to the computer codes will be given.
NASA Astrophysics Data System (ADS)
Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.
2012-04-01
In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe in Japan, Merapi in Indonesia. Preliminary results reveal that the fractal dimension of natural areas, being related only to the roughness of the observed surface, is very stable as the radar illumination geometry, the resolution and the wavelength change, thus holding a very unique property in SAR data inversion. Such a behavior is not verified in case of non-natural objects. As a matter of fact, when the fractal estimation is performed in the presence of either man-made objects or SAR image features depending on geometrical distortions due to the SAR system acquisition (i.e. layover, shadowing), fractal dimension (D) values outside the range of fractality of natural surfaces (2 < D < 3) are retrieved. These non-fractal characteristics show to be heavily dependent on sensor acquisition parameters (e.g. view angle, resolution). In this work, the behaviour of the maps generated starting from the C- and X- band SAR data, relevant to all the considered volcanoes, is analyzed: the distribution of the obtained fractal dimension values is investigated on different zones of the maps. In particular, it is verified that the fore-slope and back-slope areas of the image share a very similar fractal dimension distribution that is placed around the mean value of D=2.3. We conclude that, in this context, the fractal dimension could be considered as a signature of the identification of the volcano growth as a natural process. The COSMO-SkyMed data used in this study have been processed at IREA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0.
Comparison of different fractal dimension measuring algorithms for RE-TM M-O films
NASA Technical Reports Server (NTRS)
Bernacki, Bruce E.; Mansuripur, M.
1991-01-01
Noise in magneto-optical recording devices is discussed. In general, it appears that either the divider technique or amplitude spectrum technique may be used interchangeably to measure the fractal dimension (D) in the domain wall structure of ideal images. However, some caveats must be observed for best results. The divider technique is attractive for its simplicity and relatively modest computation requirements. However, it is sensitive to noise, in that noise pixels that touch the domain boundary are interpreted as being part of the boundary, skewing the measurement. Also, it is not useful in measuring nucleation-dominated films or domains that have significant amounts of structure within the interior of the domain wall. The amplitude spectrum method is more complex, and less intuitive than the divider method, and somewhat more expensive to implement computationally. However, since the camera noise tends to be white, the noise can be avoided in the measurement of D by avoiding that portion of the curve that is flat (due to the white noise) when the least squares line is fit to the plot. Also, many image processing software packages include a Fast Fourier Transformation (FFT) facility, while the user will most likely have to write his own edge extraction routine for the divider method. The amplitude spectrum method is a true two dimensional technique that probes the interior of the domain wall, and in fact, can measure arbitrary clusters of domains. It can also be used to measure grey-level images, further reducing processing steps needed to threshold the image.
Fractal Geometry and Spatial Phenomena A Bibliography
California at Santa Barbara, University of
Fractal Geometry and Spatial Phenomena A Bibliography January 1991 Mark MacLennan, A. Stewart. MEASUREMENT ISSUES........................................................... 8 II.1 ESTIMATION OF FRACTAL DIMENSION - GENERAL ISSUES .......... 8 II.2 ESTIMATION OF FRACTAL DIMENSION FOR CURVES/PROFILES ... 9 II.3
Lin, Fan; Zhu, Pengli; Huang, Feng; Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Yu, Peng; Lin, Jing; Chen, Falin
2015-05-01
The objective of this study was to evaluate the association of the central retinal arteriolar equivalent (CRAE) and the retinal vascular fractal dimension, two quantitative parameters that reflect microcirculation, with aortic stiffness. In this cross-sectional study, we identified the cardiovascular risk factors in 2169 subjects using a health questionnaire, physical examinations and laboratory examinations. We evaluated the aortic stiffness using noninvasive brachial-ankle pulse wave velocity (baPWV) and assessed the microcirculatory alterations with CRAE and retinal vascular fractal dimension, which were measured using fundus photography and semiautomatic quantitative software, respectively. The increase in baPWV (Q1-Q4) correlated with an increased likelihood of the central retinal artery narrowing and a reduction in the retinal vascular fractal dimension. Further adjustment of the cardiovascular risk factors diminished the association between baPWV and CRAE, but increased the association between baPWV and retinal vascular fractal dimension. Elevated baPWV correlates with reduced CRAE and retinal vascular fractal dimension. Such a finding supports macrocirculation- and microcirculation-associated hypotheses. PMID:25716651
NASA Astrophysics Data System (ADS)
Roth, E. J.; Mays, D. C.
2013-12-01
Clogging is an important limitation to essentially any technology or environmental process involving flow in porous media. Examples include (1) groundwater remediation, (2) managed or natural aquifer recharge, (3) hydrocarbon reservoir damage, (4) head loss in water treatment filters, (5) fouling in porous media reactors, and (6) nutrient flow for plants or bacteria. Clogging, that is, a detrimental reduction in permeability, is a common theme in each of these examples. Clogging results from a number of mechanisms, including deposition of colloidal particles (such as clay minerals), which is the focus of this research. Colloid deposits reduce porosity, which is recognized to play an important role in clogging, as expressed in the Kozeny-Carman equation. However, recent research has demonstrated that colloid deposit morphology is also a crucial variable in the clogging process. Accordingly, this presentation reports an ongoing series of laboratory experiments whose goal is to quantify deposit morphology as a fractal dimension, using an innovative technique based on static light scattering (SLS) in refractive index matched (RIM) porous media. For experiments conducted at constant flow, with constant influent suspension concentration, and initially clean porous media, results indicate that clogging is associated with colloid deposits having smaller fractal dimensions, that is, more dendritic and space-filling deposits. This result is consistent with previous research that quantified colloid deposit morphology using an empirical parameter. Clogging by colloid deposits also provides insight into the more complex clogging mechanisms of bioclogging, mineralization, and biomineralization. Although this line of work was originally motivated by problems of clogging in groundwater remediation, the methods used and the insight gained by correlating clogging with fractal dimension are expected to have relevance to other areas where flow in porous media overlaps with colloid science: Hydrogeology, petrology, water treatment, and chemical engineering.
Preliminary calculation of cylinder dimensions for aircraft engines
NASA Technical Reports Server (NTRS)
Schwager, Otto
1921-01-01
It is extremely important in building aircraft engines to determine the requisite cylinder dimensions as accurately as possible, in order that the weight required for a given power shall not be excessive. This report presents a calculation method that depends on the air requirement of the fuel.
Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions
Gospodinov, Dragomir [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria); Geophysical Institute of Bulgarian Academy of Sciences, Akad. G. Bonchev Str., bl.3, Sofia (Bulgaria); Marekova, Elisaveta; Marinov, Alexander [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria)
2010-01-21
A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/r{sup D}propor tor{sup -D} where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis.The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are: - Simple way to quantify scale-invariant distributions of complex objects or phenomena by a small number of parameters. - It is becoming evident that the applicability of fractal distributions to geological problems could have a more fundamental basis. Chaotic behaviour could underlay the geotectonic processes and the applicable statistics could often be fractal.The application of fractal distribution analysis has, however, some specific aspects. It is usually difficult to present an adequate interpretation of the obtained values of fractal coefficients for earthquake epicenter or hypocenter distributions. That is why in this paper we aimed at other goals - to verify how a fractal coefficient depends on different spatial distributions. We simulated earthquake spatial data by generating randomly points first in a 3D space - cube, then in a parallelepiped, diminishing one of its sides. We then continued this procedure in 2D and 1D space. For each simulated data set we calculated the points' fractal coefficient (correlation fractal dimension of epicentres) and then checked for correlation between the coefficients values and the type of spatial distribution.In that way one can obtain a set of standard fractal coefficients' values for varying spatial distributions. These then can be used when real earthquake data is analyzed by comparing the real data coefficients values to the standard fractal coefficients. Such an approach can help in interpreting the fractal analysis results through different types of spatial distributions.
A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes
NASA Technical Reports Server (NTRS)
Hsui, Albert T.; Rust, Kelly A.; Klein, George D.
1993-01-01
Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.
NSDL National Science Digital Library
2007-12-12
Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.
NSDL National Science Digital Library
Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.
NSDL National Science Digital Library
2010-01-01
This lesson is designed to develop students' understanding of fractals and fractal dimension. This lesson provides links to discussions and activities related to fractals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.
Stanley, H. Eugene
1477 (http://iopscience.iop.org/0305-4470/21/6/024) Download details: IP Address: 129 dimension of the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b). The members of the family are labelled and characterised by an integer b, 2 s b
Benoit B. Mandelbrot
1975-01-01
The degree of irregularity in oceanic coastlines and in vertical sections of the Earth, the distribution of the numbers of islands according to area, and the commonality of global shape between continents and islands, all suggest that the Earth's surface is statistically self-similar. The preferred parameter, one which increases with the degree of irregularity, is the fractal dimension, D, of
NASA Astrophysics Data System (ADS)
Yao, Yanbin; Liu, Dameng; Tang, Dazhen; Tang, Shuheng; Huang, Wenhui; Liu, Zhihua; Che, Yao
2009-06-01
To better understand the characteristics of seepage-pores (pore radius larger than 100 nanometers) and their influence on the permeability of coals, we have conducted fractal analyses for 34 fresh coal samples (mean maximum vitrinite reflectance Ro,max from 0.43% to 4.21%) from North, Northwest and Northeast China. Mercury porosimetry data indicate that the coals are fractal, with pore radius ranging from 0.1 to 50 ?m. Calculated fractal dimensions of these coals range from 2.61 to 2.98, higher than those from other kinds of rocks such as sandstone, shale, and carbonate. The data suggest that the coals have more complicated and inhomogeneous pore structures than other rocks. The fractal dimension has a negative correlation with the petrologic permeability of coals, particularly for higher rank coals (with 1.47-4.21% Ro,max), from which a strong negative linear correlation ( R2=0.85) between fractal dimension and permeability is observed. A 'U-shaped' trend between fractal dimensions and coal ranks is observed, with the minimum fractal dimensions occurring at 1.1-1.3% Ro,max. The sub-bituminous, high volatile bituminous, semi-anthracite, and anthracite have higher fractal dimensions. The effects of coal rank upon fractal dimensions are mainly due to the variety of micropore contents and aromaticity of coals with coalification.
Wei, Mao-Hong; Lin, Hui-Long
2014-03-01
The alpine meadow in the source region of the Yangtze and Yellow River is suffering serious deterioration. Though great efforts have been put into, the restoration for the degraded grassland is far from being effective, mainly due to poor understanding of the degradation mechanism of alpine meadow in this region. In order to clarify the formation mechanism of degradation grassland and provide the new ideas for restoration, degradation sequences of the alpine meadow in the source region of the Yangtze and Yellow River were taken as target systems to analyze the soil particle size distribution, the fractal dimension of the soil particle size, and the relationship between soil erosion modulus and fractal dimension. The results showed that, with increasing grassland degradation, the percentage contents of clay increased while the percentage contents of silt sand and very fine sand showed a decreasing trend. The fractal dimension presented a positive correlation with clay among the degradation sequences while negative correlations were found with very fine sand and silt sand. The curvilinear regression of fractal dimension and erosion modulus fitted a quadratic function. Judged by the function, fractal dimension 2.81 was the threshold value of soil erosion. The threshold value has an indicative meaning on predicting the breakout of grazing-induced erosion and on restoration of the degraded grassland. Taking fractal dimension of 2.81 as the restoration indicator, adoption of corresponding measures to make fractal dimension less than 2.81, would an effective way to restore the degradation grassland. PMID:24984483
NASA Astrophysics Data System (ADS)
Chadee, X. T.
2007-05-01
The fractal dimension, Lyapunov-exponent spectrum, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily wind speeds over the Caribbean region. It can be shown that this dimension is greater than 8. However, the number of data points may be too small to obtain a reliable estimate of the Grassberger-Procaccia (1983a) correlation dimension because of the limitations discussed by Ruelle (1990). These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia (1983b) are sensitive to the selection of the time delay and are thus unreliable. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. (1991). Using this method, it is found that the error-doubling time is 2-3 days for time series over the Caribbean region. This is comparable to the predictability time found by Waelbrock (1995) for a single station in Mexico. The predictability time over land is slightly less than that over ocean which tends to have higher climatic signal-to-noise ratio. This analysis impacts on the selection of prediction tools (deterministic chaotic linear and non-linear maps or linear stochastic modeling) for wind speeds in the short term for wind energy farm resource planning and management. We conclude that short term wind predictions in the Caribbean region, for a few days ahead, may be best done with a stochastic model instead of a deterministic chaotic model. References Grassberger, P., and I. Procaccia. 1983a. Measuring the strangeness of attractors. Physica D 9: 189-208. Grassberger, P., and I. Procaccia. 1983b. Estimating the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28: 2591-2593. Ruelle, D. 1990. Deterministic chaos: the science and the fiction. Proc. Royal Soc. Lond. A 427: 241-248. Waelbrock, H. 1995. Deterministic chaos in tropical atmospheric dynamics. J. Atmos. Sci. 52: 2404-2415. Zeng, X., R. Eykholt, and R. A. Pielke. 1991. Estimating the Lyapunov-exponent spectrum from short time series of low precision. Phys. Rev. Lett. 66: 3229-3232.
Fractal radar scattering from soil.
Oleschko, Klaudia; Korvin, Gabor; Figueroa, Benjamin; Vuelvas, Marco Antonio; Balankin, Alexander S; Flores, Lourdes; Carreón, Dora
2003-04-01
A general technique is developed to retrieve the fractal dimension of self-similar soils through microwave (radar) scattering. The technique is based on a mathematical model relating the fractal dimensions of the georadargram to that of the scattering structure. Clear and different fractal signatures have been observed over four geosystems (soils and sediments) compared in this work. PMID:12786363
Fractal radar scattering from soil
Klaudia Oleschko; Gabor Korvin; Benjamin Figueroa; Marco Antonio Vuelvas; Alexander S. Balankin; Lourdes Flores; Dora Carreón
2003-01-01
A general technique is developed to retrieve the fractal dimension of self-similar soils through microwave (radar) scattering. The technique is based on a mathematical model relating the fractal dimensions of the georadargram to that of the scattering structure. Clear and different fractal signatures have been observed over four geosystems (soils and sediments) compared in this work.
NASA Astrophysics Data System (ADS)
Mostafa, Mostafa E.
2008-03-01
In this paper, we present the finite cube elements method (FCEM); a novel numerical tool for calculating the gravity anomaly g and structural index SI of solid models with defined boundaries and variable density distributions, tilted or in normal position (e.g. blocks, faulted blocks, cylinders, spheres, hemispheres, triaxial ellipsoids). Extending the calculation to fractal objects, such as Menger sponges of different orders and bodies defined by polyhedrons, demonstrates the robustness of FCEM. In addition, approximating the cube element by a sphere of equal volume makes the calculation of gravitation and related derivatives much simpler. In gravity modelling of a sphere, cubes with edges of 100 m and 200 m achieve a good compromise between running time and overall error. Displaying the distribution of SI of the studied models on contour maps and profiles will have a strong impact on the forward and inverse modelling of potential field data, especially for Euler deconvolution. For Menger sponges, plots of gravity elements g and its derivatives show similar patterns independent of fractal order. Moreover, both the pattern and magnitude of SI are independent of fractal order, allowing the use of SI as a new invariant measure for fractal objects. However, SI pattern and magnitude strongly depend on the depth to the buried bodies as do other elements In this study, we also present a new type of plot; the structural index against distance variation diagrams from which we extract the three critical SI (CSI) values, one per axis. The inversion of gravity anomaly data at CSI values gives the optimal mean location of the buried body.
The Fractal Distribution of HII Regions in Disk Galaxies
Nestor Sanchez; Emilio J. Alfaro
2008-04-29
It is known that the gas has a fractal structure in a wide range of spatial scales with a fractal dimension that seems to be a constant around Df = 2.7. It is expected that stars forming from this fractal medium exhibit similar fractal patterns. Here we address this issue by quantifying the degree to which star-forming events are clumped. We develop, test, and apply a precise and accurate technique to calculate the correlation dimension Dc of the distribution of HII regions in a sample of disk galaxies. We find that the determination of Dc is limited by the number of HII regions, since if there are fractal dimension among galaxies, contrary to a universal picture sometimes claimed in literature. The fractal dimension exhibits a weak but significant correlation with the absolute magnitude and, to a lesser extent, with the galactic radius. The faintest galaxies tend to distribute their HII regions in more clustered (less uniform) patterns. The fractal dimension for the brightest HII regions within the same galaxy seems to be smaller than for the faintest ones suggesting some kind of evolutionary efffect, but the obtained correlation remains unchanged if only the brightest regions are taken into account.
Fractals in complexity and geometry
Xiaoyang Gu
2009-01-01
Fractal dimensions have been used by mathematicians and physicists to study properties of dynamic systems and geometry for around a century and have been used by computer scientists to study complexity classes for two decades. But in computer science, the usefulness of fractal dimensions was very limited before Lutz effectivized classical Hausdorff dimension to effective dimensions in 2000. With effective
Fractal dimension of the topological charge density distribution in SU(2) lattice gluodynamics
P. V. Buividovich; T. Kalaydzhyan; M. I. Polikarpov
2012-10-21
We study the effect of cooling on the spatial distribution of the topological charge density in quenched SU(2) lattice gauge theory with overlap fermions. We show that as the gauge field configurations are cooled, the Hausdorff dimension of regions where the topological charge is localized gradually changes from d = 2..3 towards the total space dimension. Therefore, the cooling procedure destroys some of the essential properties of the topological charge distribution.
A study on the computerized fractal analysis of architectural distortion in screening mammograms
Georgia D Tourassi; David M Delong; Carey E Floyd Jr
2006-01-01
Architectural distortion (AD) is a sign of malignancy often missed during mammographic interpretation. The purpose of this study was to explore the application of fractal analysis to the investigation of AD in screening mammograms. The study was performed using mammograms from the Digital Database for Screening Mammography (DDSM). The fractal dimension (FD) of mammographic regions of interest (ROIs) was calculated
de Oliveira, Marcos Aurélio Barboza; Brandi, Antônio Carlos; dos Santos, Carlos Alberto; Botelho, Paulo Henrique Husseni; Cortez, José Luís Lasso; de Godoy, Moacir Fernandes; Braile, Domingo Marcolino
2014-01-01
Introduction Solutions that cause elective cardiac arrest are constantly evolving, but the ideal compound has not yet been found. The authors compare a new cardioplegic solution with histidine-tryptophan-glutamate (Group 2) and other one with histidine-tryptophan-cetoglutarate (Group 1) in a model of isolated rat heart. Objective To quantify the fractal dimension and Shannon entropy in rat myocytes subjected to cardioplegia solution using histidine-tryptophan with glutamate in an experimental model, considering the caspase markers, IL-8 and KI-67. Methods Twenty male Wistar rats were anesthetized and heparinized. The chest was opened, the heart was withdrawn and 40 ml/kg of cardioplegia (with histidine-tryptophan-cetoglutarate or histidine-tryptophan-glutamate solution) was infused. The hearts were kept for 2 hours at 4ºC in the same solution, and thereafter placed in the Langendorff apparatus for 30 min with Ringer-Locke solution. Analyzes were performed for immunohistochemical caspase, IL-8 and KI-67. Results The fractal dimension and Shannon entropy were not different between groups histidine-tryptophan-glutamate and histidine-tryptophan-acetoglutarate. Conclusion The amount of information measured by Shannon entropy and the distribution thereof (given by fractal dimension) of the slices treated with histidine-tryptophan-cetoglutarate and histidine-tryptophan-glutamate were not different, showing that the histidine-tryptophan-glutamate solution is as good as histidine-tryptophan-acetoglutarate to preserve myocytes in isolated rat heart. PMID:25140464
NASA Astrophysics Data System (ADS)
Foresti, M. L.; Valle, G.; Bonetto, R.; Ferreira, M. L.; Briand, L. E.
2010-01-01
Lipase B from Candida Antarctica (also known as Candida antarctica lipase B or CALB) was immobilized onto titanium dioxide (TiO 2) in a buffer-free, bidistilled aqueous medium. The adsorption isotherm was determined by UV-vis analysis of supernatant solution at 280 nm, revealing that in 7 h 98% of the theoretical lipase monolayer on the TiO 2 (with 45.7 m 2/g BET area) was achieved. Samples withdrawn from the supernatant immobilization medium were analyzed by Fourier-transform infrared spectroscopy in the 1700-1600 cm -1 range (where the Amide I Proteins band appears) in order to obtain quantitative information on the evolution of the secondary-structure elements of the protein. The analysis performed revealed that lipase conformation suffers only minor changes during its adsorption onto TiO 2. However, water associated to the lipase may interact of several ways with the surface of the hydrated oxide. Characterization of the immobilized biocatalyst (CALB/TiO 2) implied SEM, fractal dimension analysis and FTIR techniques. A proposal of lipase-hydrated oxide interaction is presented.
Fractal Pore Structure Model and Multilayer Fractal Adsorption in Shale
NASA Astrophysics Data System (ADS)
Zhang, Liehui; Li, Jianchao; Tang, Hongming; Guo, Jingjing
2014-09-01
The complex structure and surface property of porous media have significant impact on its accumulation and adsorption capacity. Based on the fractal theory, this paper presents a fractal pore structure model for shales. The effect of different pore structures on fractal dimension is discussed, and the influence of fractal dimension and pore size distribution on porosity is also analyzed. It is shown that the fractal dimension D decreases with the increase of structure parameter q/m for a certain pore diameter ratio, and porosity has positive relationship with fractal dimension. This paper also presents a multilayer fractal adsorption model which takes into account the roughness of adsorption surface by using fractal theory. With the introduction of pseudo-saturated vapor pressure in the supercritical temperature condition, the proposed adsorption model can be applied into a wider range of temperature. Based on the low-pressure nitrogen adsorption and methane isothermal adsorption experiments, the effect of fractal dimension on the adsorption behavior of shales is discussed. Fractal dimension has significant impact on the surface adsorption property and adsorption layer number n. The monolayer saturated adsorption volume Vm increases with the increase of D, while parameter C has the opposite variation trend. Finally, the optimal combination of fractal parameters for describing pore structure of shale samples is selected.
DIFFUSION IN FRACTAL (TURBULENT) ENVIRONMENTS
H. Isliker; L. Vlahos
We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e. of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite
Random walk through fractal environments
H. Isliker; L. Vlahos
2003-01-01
We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite
Fractal characteristics of ozonometric network
NASA Technical Reports Server (NTRS)
Gruzdev, Alexander N.
1994-01-01
The fractal (correlation) dimensions are calculated which characterize the distribution of stations in the ground-based total ozone measuring network and the distribution of nodes in a latitude-longitude grid. The dimension of the ground-based ozonometric network equals 1.67 +/- 0.1 with an appropriate scaling in the 60 to 400 km range. For the latitude-longitude grid two scaling regimes are revealed. One regime, with the dimension somewhat greater than one, is peculiar to smaller scales and limited from a larger scale by the latitudinal resolution of the grid. Another scaling regime, with the dimension equal 1.84, ranges up to 15,000 km scale. The fact that the dimension of a measuring network is less than two possesses problems in observing sparse phenomena. This has to have important consequences for ozone statistics.
2014-01-01
Background The evaluation of intestinal trophism, mainly the mucosal layer, is an important issue in various conditions associated with injury, atrophy, recovery, and healing of the gut. The aim of the present study was to evaluate the kinetics of the proliferation and apoptosis of enterocytes by immunohistochemistry and to assess the complexity of intestinal mucosa by fractal dimension (FD) analysis in Solea solea fed different experimental diets. Results Histomorphological evaluation of all intestinal segments did not show signs of degeneration or inflammation. Cell proliferation index and FD were significantly reduced with a diet high in mussel meal (MM; p?=?0.0034 and p?=?0.01063, respectively), while apoptotic index did not show any significant difference for the same comparison (p?=?0.3859). Linear regression analysis between apoptotic index (independent variable) and FD (dependent variable) showed a statistically significant inverse relationship (p?=?0.002528). Linear regression analysis between cell proliferation index (independent variable) and FD (dependent variable) did not show any significant correlation (p?=?0.131582). Conclusions The results demonstrated that diets containing increasing levels of mussel meal in substitution of fishmeal did not incite a hyperplastic response of the intestinal mucosa. The mussel meal, which is derived from molluscs, could mimic the characteristics of the sole’s natural prey, being readily digestible, even without increasing the absorptive surface of intestinal mucosa. Interestingly, from this study emerged that FD could be used as a numeric indicator complementary to in situ quantification methods to measure intestinal trophism, in conjunction with functional parameters. PMID:24997003
Electromagnetism on Anisotropic Fractals
Martin Ostoja-Starzewski
2011-06-08
We derive basic equations of electromagnetic fields in fractal media which are specified by three indepedent fractal dimensions {\\alpha}_{i} in the respective directions x_{i} (i=1,2,3) of the Cartesian space in which the fractal is embedded. To grasp the generally anisotropic structure of a fractal, we employ the product measure, so that the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving the {\\alpha}_{i}'s. First, a formulation based on product measures is shown to satisfy the four basic identities of vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Amp\\`ere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.
Prusinkiewicz, Przemyslaw
Chapter 8 Fractal properties of plants What is a fractal? In his 1982 book, Mandelbrot defines it as a set with Fractals vs. finite curvesHausdorff-Besicovitch dimension DH strictly exceeding the topological dimension DT [95, page 15]. In this sense, none of the figures presented in this book are fractals
NASA Technical Reports Server (NTRS)
Wiscombe, W.
1999-01-01
The purpose of this paper is discuss the concept of fractal dimension; multifractal statistics as an extension of this; the use of simple multifractal statistics (power spectrum, structure function) to characterize cloud liquid water data; and to understand the use of multifractal cloud liquid water models based on real data as input to Monte Carlo radiation models of shortwave radiation transfer in 3D clouds, and the consequences of this in two areas: the design of aircraft field programs to measure cloud absorptance; and the explanation of the famous "Landsat scale break" in measured radiance.
Fractal analysis: A new remote sensing tool for lava flows
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Many important quantitative parameters have been developed that relate to the rheology and eruption and emplacement mechanics of lavas. This research centers on developing additional, unique parameters, namely the fractal properties of lava flows, to add to this matrix of properties. There are several methods of calculating the fractal dimension of a lava flow margin. We use the 'structured walk' or 'divider' method. In this method, we measure the length of a given lava flow margin by walking rods of different lengths along the margin. Since smaller rod lengths transverse more smaller-scaled features in the flow margin, the apparent length of the flow outline will increase as the length of the measuring rod decreases. By plotting the apparent length of the flow outline as a function of the length of the measuring rod on a log-log plot, fractal behavior can be determined. A linear trend on a log-log plot indicates that the data are fractal. The fractal dimension can then be calculated from the slope of the linear least squares fit line to the data. We use this 'structured walk' method to calculate the fractal dimension of many lava flows using a wide range of rod lengths, from 1/8 to 16 meters, in field studies of the Hawaiian islands. We also use this method to calculate fractal dimensions from aerial photographs of lava flows, using lengths ranging from 20 meters to over 2 kilometers. Finally, we applied this method to orbital images of extraterrestrial lava flows on Venus, Mars, and the Moon, using rod lengths up to 60 kilometers.
Fractal analysis of time varying data
Vo-Dinh, Tuan (Knoxville, TN); Sadana, Ajit (Oxford, MS)
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 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.
William Deering; Bruce J. West
1992-01-01
The nature of fractals and the use of fractals instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described. The mathematical development of fractals is reviewed, and examples of natural fractals are cited. Relationships among power laws, noise, and fractal time signals are examined
Testing Fractal Methods on Observed and Simulated Solar Magnetograms
NASA Technical Reports Server (NTRS)
Adams, M.; Falconer, D. A.; Lee, J. K.; Jones, C.
2003-01-01
The term "magnetic complexity" has not been sufficiently quantified. To accomplish this, we must understand the relationship between the observed magnetic field of solar active regions and fractal dimension measurements. Using data from the Marshall Space Flight Center's vector magnetograph ranging from December 1991 to July 2001, we compare the results of several methods of calculating a fractal dimension, e.g., Hurst coefficient, the Higuchi method, power spectrum, and 2-D Wavelet Packet Analysis. In addition, we apply these methods to synthetic data, beginning with representations of very simple dipole regions, ending with regions that are magnetically complex.
Routes to fractality and entropy in Liesegang systems
NASA Astrophysics Data System (ADS)
Kalash, Leen; Sultan, Rabih
2014-06-01
Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M. [National Simulation Resource, Department of Bioengineering, University of Washington, Seattle, Washington 98195 (United States)
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Roughness Perception of Haptically Displayed Fractal Surfaces
NASA Technical Reports Server (NTRS)
Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)
2000-01-01
Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.
Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background
Chiang, Lung-Yih
Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background Naoki KOBAYASHI and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the fractal dimension De of the entire pattern of isothermal lines, the fractal dimension Dc of a single
Anomalous thermal conduction in one dimension: a quantum calculation.
Santhosh, G; Kumar, Deepak
2007-08-01
In this paper, we study the thermal conductivity of an anharmonically coupled chain of atoms. Numerical studies using classical dynamics have shown that the conductivity of a chain with nearest neighbor couplings diverges with chain length L as L(alpha); earlier studies found alpha approximately = 0.4 under a range of conditions, but a recent study on longer chains claims alpha = 1/3. Analytically, this problem has been studied by calculating the relaxation rate gamma(q) of the normal modes of vibration as a function of its wave vector q. Two theoretical studies of classical chains, one using the mode-coupling formulation and the other the Boltzmann equation method, led to gamma(q) proportional to q(5/3), which is consistent with alpha = 0.4. Here we study the problem for a quantum anharmonic chain with quartic anisotropy. We develop a low-temperature expansion for gamma(q) and find that, in the regime Dirac's constant omega(q) < k(B)T, gamma(q) is proportional to q(5/3)T2, where omega(q) is the frequency of the mode. In our analysis, the relaxation arises due to umklapp scattering processes. We further evaluate the thermal conductivity of the chain using the Kubo formula, which enables us to take into account the transport relaxation time through vertex corrections for the current-current correlator. This calculation also yields alpha = 0.4. PMID:17930004
Won-Jin Yi; Min-Suk Heo; Sam-Sun Lee; Soon-Chul Choi; Kyung-Hoe Huh
2007-01-01
The mechanical quality of trabecular bone depends on both its stiffness and its strength characteristics, which can be predicted\\u000a indirectly by the combination of bone volume fraction and architectural anisotropy. To analyze the directional anisotropy\\u000a of the trabecular bone, we applied the fractal geometry technique to plain radiographs. The anisotropy of the bone was quantified\\u000a from an ellipse, based on
Donor-acceptor one step energy transfer via exchange coupling on a fractal lattice
NASA Astrophysics Data System (ADS)
Yang, C. L.; El-Sayed, M. A.
1986-07-01
Temporal behavior of donor intensity, I sub D (t), resulting from one step donor-acceptor electronic excitation energy transfer process via exchange mechanism is calculated on fractal lattices, with discrete dilation symmetry, of Euclidean dimension, d = 2, and fractal dimension, D, ranging from 1.99 to 1.0. I sub D (t) is fitted to the approximate equation of Klafter-Blumen (K-B) which is useful in fitting experimental results to determine fractal dimension from the slope of the expected straight line obtained by plotting in(-in I sub d (t)), vs. in 1 sub D t). The result for fractal lattice with D = 1.99 indicates that the approximation is appropriate for the time range used in our calculations. The results for different fractal lattices also show that the K-B equation indeed gives a straight line for structure when D/d is not much smaller than unity. As this ratio decreases, deviation from the expected straight line results and an oscillatory behavior is observed. From the latter structural information (i.e., fractal dimensionality, and geometrical parameters characterizing the fractal lattices) as well as the molecular interaction parameter gamma, the distance dependence of the exchange interaction can be determined.
NASA Astrophysics Data System (ADS)
Kopelman, Raoul
1988-09-01
Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal ``memories.'' The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of ``fractal-like kinetics'' are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds. Diffusion-controlled reactions with geometrical constraints, as found in heterogeneous kinetics, may be described by reactions on fractal domains. The hallmarks of ``fractal-like'' reactions are anomalous reaction orders and time-dependent reaction rate ``constants.'' These anomalies stem from the nonrandomness of the reactant distributions in low dimensions. For homo-bimolecular reactions (A + A --> Pr) the distribution is partially ordered, for example, quasi-periodic. However, for hetero-bimolecular reactions (A + B --> Pr) the reactants segregate. Theory, simulations, and experiments are interrelated through the formalism of fractal reaction kinetics (42).
Juergens, H.; Peitgen, H.O.; Saupe, D. (Univ. of Bremen (West Germany))
1990-08-01
The pathological structures conjured up by 19th-century mathematicians have, in recent years, taken the form of fractals, mathematical figures that have fractional dimension rather than the integral dimensions of familiar geometric figures (such as one-dimensional lines or two-dimensional planes). Fractals are much more than a mathematical curiosity. They offer an extremely compact method for describing objects and formations. Many structures have an underlying geometric regularity, known as scale invariance or self-similarity. If one examines these objects at different size scales, one repeatedly encounters the same fundamental elements. The repetitive pattern defines the fractional, or fractal, dimension of the structure. Fractal geometry seems to describe natural shapes and forms more gracefully and succinctly than does Euclidean geometry. Scale invariance has a noteworthy parallel in contemporary chaos theory, which reveals that many phenomena, even though they follow strict deterministic rules, are in principle unpredictable. Chaotic events, such as turbulence in the atmosphere or the beating of a human heart, show similar patterns of variation on different time scales, much as scale-invariant objects show similar structural patterns on different spatial scales. The correspondence between fractals and chaos is no accident. Rather it is a symptom of a deep-rooted relation: fractal geometry is the geometry of chaos.
Rangaraj M. Rangayyan; Shormistha Prajna; Fábio J. Ayres; J. E. Leo Desautels
2008-01-01
Objective Mammography is a widely used screening tool for the early detection of breast cancer. One of the commonly missed signs of\\u000a breast cancer is architectural distortion. The purpose of this study is to explore the application of fractal analysis and\\u000a texture measures for the detection of architectural distortion in screening mammograms taken prior to the detection of breast\\u000a cancer.\\u000a \\u000a \\u000a \\u000a Materials
NASA Technical Reports Server (NTRS)
Maasch, Kirk A.
1989-01-01
The Grassberger-Procaccia method of calculating dimension from a time series is applied to 14 late Pleistocene delta O-18 records. A step-by-step sequence leading from data to the Grassberger-Procaccia dimension is outlined, and the problems encountered when dealing with observed (as opposed to theoretical) data are discussed; for the climatic proxy data these problems include situations where the time series is not very long, is noisy and/or smoothed, and is not sampled at a constant time interval. The delta O-18 records to be used are described, and the results are presented and compared with previously published dimension calculations. New dimension interpretations are assessed, and an example using a synthetic time series that illustrates the possible error due to inconsistencies in the time scale is analyzed.
NASA Astrophysics Data System (ADS)
Yang, C. L.; Evesque, P.; El-Sayed, M. A.
1985-07-01
A simulation calculation is carried out for the time dependence of the donor intensity as one-step acceptor trapping process on a simple cubic lattice of interconnected rods of dimensions comparable to those found for the pores of Vycor (Corning Glass 7930). The donor and acceptor are allowed to occupy random sites on the surface of the pore. The results are fitted using the equations of Blumen and Klafter for energy transfer on a fractal structure. An apparent fractal dimension, (d'), is determined which is shown to result from an excluded volume effect and is not due to a real fractal structure. (D') is found to depend slightly on time, the acceptor concentration, and the length and/or width of the pores.
Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface
NASA Astrophysics Data System (ADS)
Xie, Tao; He, Chao; William, Perrie; Kuang, Hai-Lan; Zou, Guang-Hui; Chen, Wei
2010-02-01
In recent years, linear fractal sea surface models have been developed for the sea surface in order to establish an electromagnetic backscattering model. Unfortunately, the sea surface is always nonlinear, particularly at high sea states. We present a nonlinear fractal sea surface model and derive an electromagnetic backscattering model. Using this model, we numerically calculate the normalized radar cross section (NRCS) of a nonlinear sea surface. Comparing the averaged NRCS between linear and nonlinear fractal models, we show that the NRCS of a linear fractal sea surface underestimates the NRCS of the real sea surface, especially for sea states with high fractal dimensions, and for dominant ocean surface gravity waves that are either very short or extremely long.
Texture Analysis In Cytology Using Fractals
NASA Astrophysics Data System (ADS)
Basu, Santanu; Barba, Joseph; Chan, K. S.
1990-01-01
We present some preliminary results of a study aimed to assess the actual effectiveness of fractal theory in the area of medical image analysis for texture description. The specific goal of this research is to utilize "fractal dimension" to discriminate between normal and cancerous human cells. In particular, we have considered four types of cells namely, breast, bronchial, ovarian and uterine. A method based on fractal Brownian motion theory is employed to compute the "fractal dimension" of cells. Experiments with real images reveal that the range of scales over which the cells exhibit fractal property can be used as the discriminatory feature to identify cancerous cells.
Box-counting dimension without boxes: Computing D0 from average expansion rates Ernest Barreto,1
Barreto, Ernest
Box-counting dimension without boxes: Computing D0 from average expansion rates Paul So,1 Ernest for calculating the box-counting capacity dimension of a chaotic attractor in terms of its average expansion rates is the box-counting dimen- sion or capacity dimension D0 15 . Given a fractal set in a d
Fractal Function Estimation via Wavelet Shrinkage Yazhen Wang
Wang, Yazhen
Fractal Function Estimation via Wavelet Shrinkage Yazhen Wang University of Missouri studies objects are often very rough. Mathematically these rough objects are modeled by fractal functions, and fractal dimension is usually used to measure their roughness. The present paper investigates fractal
Fractal algebras of discretization sequences Steffen Roch (TU Darmstadt)
Potts, Daniel
Abstract Fractal algebras of discretization sequences Steffen Roch (TU Darmstadt) First a warning: Fractality, in the sense of these lectures, has nothing to do with fractal geometries or broken dimensions or other involved things. Rather, the notion fractal algebra had been chosen in order to emphasize
Target Detection Using Fractal Geometry
NASA Technical Reports Server (NTRS)
Fuller, J. Joseph
1991-01-01
The concepts and theory of fractal geometry were applied to the problem of segmenting a 256 x 256 pixel image so that manmade objects could be extracted from natural backgrounds. The two most important measurements necessary to extract these manmade objects were fractal dimension and lacunarity. Provision was made to pass the manmade portion to a lookup table for subsequent identification. A computer program was written to construct cloud backgrounds of fractal dimensions which were allowed to vary between 2.2 and 2.8. Images of three model space targets were combined with these backgrounds to provide a data set for testing the validity of the approach. Once the data set was constructed, computer programs were written to extract estimates of the fractal dimension and lacunarity on 4 x 4 pixel subsets of the image. It was shown that for clouds of fractal dimension 2.7 or less, appropriate thresholding on fractal dimension and lacunarity yielded a 64 x 64 edge-detected image with all or most of the cloud background removed. These images were enhanced by an erosion and dilation to provide the final image passed to the lookup table. While the ultimate goal was to pass the final image to a neural network for identification, this work shows the applicability of fractal geometry to the problems of image segmentation, edge detection and separating a target of interest from a natural background.
Application of the fractal theory on the study of filter cake constructure
Xu, X.; Xu, J.; Deng, C.; Qian, L. [Northeastern Univ., Shenyang (China); Yan, K.
1995-12-31
Cake filtration is a complex process and the cake constructure is very difficult to describe in theory. Cake constructure parameters, such as the cake porosity, pore size shape and even its distribution, are main factors influencing the filtration results but have not been thoroughly understood yet. In this paper the fractal theory, an effective mathematical method in describing the self-similar phenomenon is used to investigate the filter cake constructure, and the scanning electron microscope and automatic image analyzer are used to measure the cake constructure. Cakes which formed in different conditions are examined and the fractal dimension of the cake are calculated. The study shows that the constructure of the filter cake can be approximated by Sierpinski fractal geometry and that the fractal dimension of filter cake, related to the particle characteristics, slurry concentration and filtration pressure is a good parameter to describe the pore size distribution and the cake penetrability.
Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E; Bertoldo, Alessandra
2015-02-21
Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders. PMID:25633275
NASA Astrophysics Data System (ADS)
Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E.; Bertoldo, Alessandra
2015-02-01
Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.
Fractals in Quantum Information Process
NASA Astrophysics Data System (ADS)
Bi, Feng; Li, Chuan-Feng
2013-01-01
In the recent work of Kiss et al. [Phys. Rev. Lett. 107 (2011) 100501], the evolvement of two-qubit quantum states in a measurement-based purification process is studied. As they pointed out, the purification results manifest sensitivity to the applied initial states. The convergence regions to different stable circles are depicted on a complex plane. Because of the result patterns' likeness to typical fractals, we make further study on the interesting patterns' connection to fractals. Finally, through a numerical method we conclude that the boundaries of different islands of the patterns are fractals, which possess a non-integral fractal dimension. Also, we show that the fractal dimension would vary with the change of the portion of the noise added to the initial states.
Kopelman, R
1988-09-23
Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds. PMID:17820893
NASA Astrophysics Data System (ADS)
Xu, Ye; Lee, Michael C.; Boroczky, Lilla; Cann, Aaron D.; Borczuk, Alain C.; Kawut, Steven M.; Powell, Charles A.
2009-02-01
Features calculated from different dimensions of images capture quantitative information of the lung nodules through one or multiple image slices. Previously published computer-aided diagnosis (CADx) systems have used either twodimensional (2D) or three-dimensional (3D) features, though there has been little systematic analysis of the relevance of the different dimensions and of the impact of combining different dimensions. The aim of this study is to determine the importance of combining features calculated in different dimensions. We have performed CADx experiments on 125 pulmonary nodules imaged using multi-detector row CT (MDCT). The CADx system computed 192 2D, 2.5D, and 3D image features of the lesions. Leave-one-out experiments were performed using five different combinations of features from different dimensions: 2D, 3D, 2.5D, 2D+3D, and 2D+3D+2.5D. The experiments were performed ten times for each group. Accuracy, sensitivity and specificity were used to evaluate the performance. Wilcoxon signed-rank tests were applied to compare the classification results from these five different combinations of features. Our results showed that 3D image features generate the best result compared with other combinations of features. This suggests one approach to potentially reducing the dimensionality of the CADx data space and the computational complexity of the system while maintaining diagnostic accuracy.
NSDL National Science Digital Library
National Council of Teachers of Mathematics
2009-01-01
Using this tool, students build these classic fractals: the Koch snowflake, a fractal tree, a reduced square, and the Sierpinksi triangle. As these shapes grow and change using an iterative process, students can observe patterns in the images created and in the table of values as the fractals progress through several stages.
NASA Astrophysics Data System (ADS)
Kayal, J. R.; Das, Vishal; Ghosh, Uma
2012-12-01
We examined seismic characteristics, b value and fractal dimension of the aftershock sequence of the January 26, 2001 Bhuj earthquake (Mw 7.7) that occurred in the Kutch failed rift basin, western margin of the Stable Continental Region (SCR) of India. A total of about 2,000 events (M ? 2.0) were recorded within two and a half months, immediately after the main shock. Some 795 events were precisely relocated by simultaneous inversion. These relocated events are used for mapping the frequency-magnitude relation ( b value) and fractal correlation dimension (Dc) to understand the seismic characteristics of the aftershocks and the source zone of the main shock. The surface maps of the b value and Dc reveal two distinct tectonic arms or zones of the V-shaped aftershock area, western zone and eastern zone. The b value is relatively higher (~1.6) in the western zone compared to a lower value (~1.4) in the eastern zone. The Dc map also shows a higher value (1.2-1.35) in the western zone compared to a lower Dc (0.80-1.15) in the eastern zone; this implies a positive correlation between Dc and b value. Two cross sections, E-W and N-S, are examined. The E-W sections show similar characteristics, higher b value and higher Dc in the western zone and lower in the eastern zone with depth. The N-S sections across the fault zones, however, show unique features; it imaged both the b and Dc characteristics convincingly to identify two known faults, the Kutch Mainland fault and the South Wagad fault (SWF), one stepping over the other with a seismogenic source zone at depth (20-35 km). The source zone at depth is imaged with a relatively lower b and higher Dc at the `fault end' of the SWF showing a negative correlation. These observations, corroborated with the seismic tomography as well as with the proposed geological/tectonic model, shed a new light to our understanding on seismogenesis of the largest SCR earthquake in India in the recent years.
Clustering in stock market based on fractal theory
Zeng Xiu; Peng Hong; Zeng Zhen
2009-01-01
The K-line, which reflects the trend of stock, is fractal graphics with a stable fractal dimension; we use the stable fractal dimension as an important parameter in the research of stock cluster analysis. We have made an empirical research on the A-stock market of Shanghai. And the result shows that the same kind of stocks, which is clustered by the
Fractal feature analysis and classification in medical imaging
CHI-CHANG CHEN; JOHN S. DAPONTE; MARTIN D. FOX
1989-01-01
Following B.B. Mandelbrot's fractal theory (1982), it was found that the fractal dimension could be obtained in medical images by the concept of fractional Brownian motion. An estimation concept for determination of the fractal dimension based upon the concept of fractional Brownian motion is discussed. Two applications are found: (1) classification; (2) edge enhancement and detection. For the purpose of
Characterization and Measurement of Random Fractals
Richard F. Voss
1986-01-01
Mandelbrot's fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A
Relativistic Fractal Cosmologies
Marcelo B. Ribeiro
2009-10-26
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 fractal model we find that all Friedmann models look inhomogeneous along the backward null cone, with a departure from the observable homogeneous region at relatively close ranges. It is also shown that with these same observational relations the Einstein-de Sitter model can have an interpretation where it has zero global density, a result consistent with the "zero global density postulate" advanced by Wertz for hierarchical cosmologies and conjectured by Pietronero for fractal cosmological models. The article ends with a brief discussion on the possible link between this model and nonlinear and chaotic dynamics.
Fractal Musicand Fractal Music Lab
NSDL National Science Digital Library
2007-12-12
This first website offers a collection of fractal music using images created by G.W.F. Albrecht. The technology and mathematics which this presentation draws on is described on the second website. The second website, developed by David Strohbeen, offers some basic information about fractals and fractal music. He has also posted some samples of his music and invites visitors to download software for creating fractal music and to submit their own compositions.
Milovanovic, Petar; Djuric, Marija; Rakocevic, Zlatko
2012-01-01
There is an increasing interest in bone nano-structure, the ultimate goal being to reveal the basis of age-related bone fragility. In this study, power spectral density (PSD) data and fractal dimensions of the mineralized bone matrix were extracted from atomic force microscope topography images of the femoral neck trabeculae. The aim was to evaluate age-dependent differences in the mineralized matrix of human bone and to consider whether these advanced nano-descriptors might be linked to decreased bone remodeling observed by some authors and age-related decline in bone mechanical competence. The investigated bone specimens belonged to a group of young adult women (n = 5, age: 20–40 years) and a group of elderly women (n = 5, age: 70–95 years) without bone diseases. PSD graphs showed the roughness density distribution in relation to spatial frequency. In all cases, there was a fairly linear decrease in magnitude of the power spectra with increasing spatial frequencies. The PSD slope was steeper in elderly individuals (?2.374 vs. ?2.066), suggesting the dominance of larger surface morphological features. Fractal dimension of the mineralized bone matrix showed a significant negative trend with advanced age, declining from 2.467 in young individuals to 2.313 in the elderly (r = 0.65, P = 0.04). Higher fractal dimension in young women reflects domination of smaller mineral grains, which is compatible with the more freshly remodeled structure. In contrast, the surface patterns in elderly individuals were indicative of older tissue age. Lower roughness and reduced structural complexity (decreased fractal dimension) of the interfibrillar bone matrix in the elderly suggest a decline in bone toughness, which explains why aged bone is more brittle and prone to fractures. PMID:22946475
Haitao, Sun; Ning, Li; Lijun, Guo; Fei, Gao
2011-01-01
Objective The aim of this study was to use fractal dimension (FD) analysis on multidetector CT (MDCT) images for quantifying the morphological changes of the pulmonary artery tree in patients with pulmonary hypertension (PH). Materials and Methods Fourteen patients with PH and 17 patients without PH as controls were studied. All of the patients underwent contrast-enhanced helical CT and transthoracic echocardiography. The pulmonary artery trees were generated using post-processing software, and the FD and projected image area of the pulmonary artery trees were determined with ImageJ software in a personal computer. The FD, the projected image area and the pulmonary artery pressure (PAP) were statistically evaluated in the two groups. Results The FD, the projected image area and the PAP of the patients with PH were higher than those values of the patients without PH (p < 0.05, t-test). There was a high correlation of FD with the PAP (r = 0.82, p < 0.05, partial correlation analysis). There was a moderate correlation of FD with the projected image area (r = 0.49, p < 0.05, partial correlation analysis). There was a correlation of the PAP with the projected image area (r = 0.65, p < 0.05, Pearson correlation analysis). Conclusion The FD of the pulmonary arteries in the PH patients was significantly higher than that of the controls. There is a high correlation of FD with the PAP. PMID:21603288
Frankel, A.
1991-01-01
The high-frequency falloff ??-y of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R-D, D being the fractal dimension. In the model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R??, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (?? = 0) produces a main shock with an ??-2 falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to "islands' of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 - ??. Thus D = 2 corresponds to constant stress drop scaling (?? = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. -from Author
Edge detection and image segmentation of space scenes using fractal analyses
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.; Fuller, J. J.
1992-01-01
A method was developed for segmenting images of space scenes into manmade and natural components, using fractal dimensions and lacunarities. Calculations of these parameters are presented. Results are presented for a variety of aerospace images, showing that it is possible to perform edge detections of manmade objects against natural background such as those seen in an aerospace environment.
R. Sarathi; R. K. Sahu; T. Tanaka
2008-01-01
In the present study, the influence of water ageing on the surface characteristics of epoxy nanocomposites was analyzed through atomic force microscopy (AFM) studies. The hydrophobic properties of the epoxy nanocomposite material were analyzed through contact angle and diffusion coefficient measurements. Fractal dimension were calculated by adopting a multi resolution signal decomposition (MRSD) to the 1D-AFM signal through power spectral
NASA Astrophysics Data System (ADS)
West, Bruce J.
The natural variability in physiological structure is herein related to the geometric concept of a fractal. The average dimensions of the branches in the tracheobronchial tree, long thought to be exponential, are shown to be an inverse power law of the generation number modulated by a harmonic variation. A similar functional form is found for the power spectrum of the QRS-complex of the healthy human heart. These results follow from the assumption that the bronchial tree and the cardiac conduction system are fractal forms. The fractal concept provides a mechanism for the morphogenesis of complex structures which are more stable than those generated by classical scaling (i.e., they are more error tolerant).
NSDL National Science Digital Library
2003-01-01
Paul Bourke of the Astrophysics and Supercomputing department at Swinburne University of Technology is the author of this massive resource on fractals and chaos. He gives examples of many different kinds and classes of fractals, including the Mandelbrot set and various attractors; and brief explanations accompany each one. A substantial introduction to fractals covers the underlying principles and connection to chaos theory. Many stunning, high resolution fractal image galleries show elaborate patterns and colors. Examples of C and PovRay code used to create the remarkable images are provided. Bourke's homepage has many other sections of tutorials, papers, and notes on a diverse range of subjects.
Thermodynamics of Photons on Fractals
Akkermans, Eric [Department of Physics, Technion Israel Institute of Technology, 32000 Haifa (Israel); Dunne, Gerald V. [Department of Physics, University of Connecticut, Storrs, Connecticut 06269 (United States); Teplyaev, Alexander [Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269 (United States)
2010-12-03
A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV{sub s}=U/d{sub s}, where d{sub s} is the spectral dimension and V{sub s} defines the 'spectral volume'. For regular manifolds, V{sub s} coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.
Introduction to Fractals: Geometric Fractals
NSDL National Science Digital Library
2010-01-01
This lesson is designed to continue developing students' knowledge of fractals by introducing them to some popular examples of fractals, Sierpinski's carpet and Sierpinski's triangle. This lesson provides links to discussions and activities related to fractals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.
Fractal dynamics of bioconvective patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1991-01-01
Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.
ERIC Educational Resources Information Center
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.…
NASA Astrophysics Data System (ADS)
Chen, C.-C.; Lee, Ya-Ting; Hasumi, Tomohiro
2009-04-01
We calculated the Hurst exponent H and the power-law scaling exponent B for data of avalanche sizes in a new modification of sandpile models, i.e. the long-range connective sandpile (LRCS) models. The LRCS model is introduced by considering the random distant connection between two separated (instead of neighboring) cells. We explore the relationships between those two exponents H and B, and find the strong dependence upon the system size for such relationships. As the system size L of sandpile model decreases, the LRCS model can demonstrate a transition from the negative to positive correlations between the H- and B-values. While the negative and null correlations are associated with the fractional Gaussian noise and generalized Cauchy processes, respectively, the regime with the positive correlation between the Hurst and power-law scaling exponents may suggest an unknown, interesting class of the stationary Gaussian processes.
Fractal Coagulation Bruce E. Logan
Fractal Coagulation Kinetics Bruce E. Logan Department of Civil & Environmental Engineering can basis of calculations be improved? #12;Coagulation No collisions Unsuccessful collisions Successful collisions & coagulation #12;Coagulation Theory ·Coagulation theory is quite old, dating back
Geoffrey Dougherty; Geoffrey M. Henebry
2001-01-01
Fractal analysis is a method of characterizing complex shapes such as the trabecular structure of bone. Numerous algorithms for estimating fractal dimension have been described, but the Fourier power spectrum method is particularly applicable to self-affine fractals, and facilitates corrections for the effects of noise and blurring in an image. We found that it provided accurate estimates of fractal dimension
Diffusion of magnetic flux elements on a fractal geometry
J. K. Lawrence
1991-01-01
Recent observations have indicated that magnetic field elements are distributed on the Sun in fractal patterns with dimension D < 2. We suggest that the transport of magnetic field elements across the solar surface should be treated as diffusion on a fractal geometry. We review a semi-analytical, theoretical treatment of fractal diffusion. Comparison with observations of small-scale motions of solar
The synthesis and rendering of eroded fractal terrains
F. Kenton Musgrave; Craig E. Kolb; Robert S. Mace
1989-01-01
In standard fractal terrain models based on fractional Brownian motion the statistical character of the surface is, by design, the same everywhere. A new approach to the synthesis of fractal terrain height fields is presented which, in contrast to previous techniques, features locally independent control of the frequencies composing the surface, and thus local control of fractal dimension and other
Breaks in Fractal Scaling of Real and Synthetic Earthquake Catalogues
NASA Astrophysics Data System (ADS)
Khademi, M. H.; McCloskey, J.
2004-12-01
Earthquake generation within the crust is the result of a series of complicated spacio-temporal interactions between different tectonic blocks and units. The end product of the process is a function of both long term deterministic-chaotic processes in a regional scale and short-term Self-Organized Critical (SOC) processes of a local nature [e.g. McCloskey and Bean,1994; ]. In the past three to four decades many models of seismicity have been developed [e.g. Burridge and Knopoff, 1967; Huang and Turcotte, 1990; McCloskey, 1993; Ben-Zion, 1996] trying to model the observed patterns of earthquake generation and seismicity. Some of these studies have shown that it is possible to reproduce the main features of the real earthquake populations. In this study the fractal dimension of SCSN, JMA and ISC seismicity catalogues have been studied. the aim was to see whether all the different sizes of earthquakes within a catalogue (i.e. a single spacio-temporal window) belong to the same population and whether any breaks in fractal scaling exists within the catalogue concerned. Furthermore, a selection of synthetic earthquake models were analyzed with the same approach to determine whether they are able to reproduce the same results as empirical ones. Subsequent analysis of the data have revealed several distinct breaks in fractal scaling of earthquakes of different magnitudes. In other words, it emerged that small and large earthquakes in each catalogue are obeying different fractal dimensions hence belonging to different earthquake populations. It is possible to associate one of the breaks, observed in the SCSN catalogue to the average thickness of the seismogenic crust of California ( ˜ 15 km as calculated by Nazareth and Hauksson, 2004). With the same technique used for the empirical catalogues, three different synthetic catalogues [McCloskey, 1993; Ben-Zion, 1996; Khademi and McCloskey, this study ] were analyzed. Results have shown that all the models are able to predict the fractal dimension of the empirical catalogues to some degree and further, two latter models are able to simulate the breaks in the fractal scaling. The fractal dimensions from models F, U, M and A of Ben-Zion [1996] are generally in good agreement with observed dimensions of empirical catalogues, though the observed breaks in the empirical catalogues cannot be seen in these synthetic models. However, a gradual decrease in the fractal dimension with increasing treshold magnitude can be observed. The McCloskey [1993] Chaotic-SOC hierarchical model, with reasonable accuracy, predicts both the fractal dimensions and dimension breaks, observed in the empirical catalogues. The model's success in the prediction of behaviour of empirical data is particularly due to the combination of low-dimensional chaotic behaviour of bigger blocks (i.e. larger events) and high-dimensional SOC behaviour of smaller blocks (i.e. small events resulting from activity of smaller portions of the main fault or adjacent minor discontinuities). And finally the Khademi-McCloskey (KMC) model is able to reproduce both the dimension and one of the breaks of scaling. But, the model is unable to produce more than one break in scaling (i.e. to distinguish more than two earthquake populations within the same dataset). It is concluded that hierarchical earthquake models (though with some modifications) can be used to extend the temporally limited empirical catalogues to much longer time spans and to overcome the temporal limitations of the existing empirical catalogues.
Optical wave diffraction on fractal objects
G. Chabassier; B. Angeli; F. Heliodore; A. LeMehaute
1992-01-01
In the frame of a generalized functions analysis, it is shown that the light scattered by fractal objects has fractal properties. The difference between mass diffraction and surface diffraction is presented and illustrated through clear and easy experiments based on the fundamental difference between the concepts of dimension and codimension.
NSDL National Science Digital Library
Frame, Michael
This is one of the best online resources about fractals, and is "meant to support a first course in fractal geometry for students without especially strong mathematical preparation." The site is incredibly deep, providing everything from the most basic definitions and non-technical discussions to involved mathematical formulations. Interactive Java applets, downloadable software for the PC and Macintosh, and laboratory activities are also presented. A particularly interesting section of the site explores about 100 places in nature and society where fractals are found.
Fractal Universe and Quantum Gravity
Calcagni, Gianluca [Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Muehlenberg 1, D-14476 Golm (Germany)
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.
Fractal universe and quantum gravity.
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed. PMID:20867360
Fractal structure of lunar topography: An interpretation of topographic characteristics
NASA Astrophysics Data System (ADS)
Cao, Wei; Cai, Zhanchuan; Tang, Zesheng
2015-06-01
Over the years, fractal geometry has been applied extensively in many fields of geoscience. Based on the global gridded data generated from the Lunar Reconnaissance Orbiter, we carry out our fractal measure to interpret lunar fractures by using qualitative (similar ratio) and quantitative (fractal dimension) approaches of fractal geometry. We find that most of the lunar surface exhibits fractal behavior over the given scales ranging from 1 to 256 m. Lunar maria have higher fractal dimensions than other geological units, while those of volcanic areas and highlands are lower than their surroundings. Simple and flat surfaces have low values of similar ratios and these areas indicate low surface roughness and young ages. Older-aged areas, such as the Hertzsprung basin, have low fractal dimensions and high similar ratios by their complicated topography.
Fractal characterization of fracture networks: An improved box-counting technique
Perfect, Ed
Fractal characterization of fracture networks: An improved box-counting technique Ankur Roy,1 fracture networks as fractals and estimating their fractal dimensions (D). If this analysis yields a power and r is the box size, then the network is considered to be fractal. However, researchers are divided
Fractal generation of surface area of porous media
Hongbing Sun; Manfred Koch
1998-01-01
Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they\\u000a are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify\\u000a a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure\\u000a with predefined statistical and
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. PMID:24784044
NSDL National Science Digital Library
Bourke, Paul
This webpage is dedicated to all things fractals, and is organized and updated by Professor Paul Bourke. Visitors will find all kinds of useful tools for discovering and learning about fractals including: Fractal forms found by using Google Earth, Natural Fractals in Grand Canyon National Park, Introduction to fractals, a gallery of fractals, and much, much more.
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. PMID:23808932
Fractals and cosmological large-scale structure.
Luo, X; Schramm, D N
1992-04-24
Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a "limited" fractal with dimension D approximately 1.2. This is not a "pure" fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process. PMID:17787947
Fractals and cosmological large-scale structure
NASA Technical Reports Server (NTRS)
Luo, Xiaochun; Schramm, David N.
1992-01-01
Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a 'limited' fractal with dimension D of about 1.2. This is not a 'pure' fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process.
Fractal Theory for Permeability Prediction, Venezuelan and USA Wells
NASA Astrophysics Data System (ADS)
Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana
2014-05-01
Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.
Fractal characterization of brain lesions in CT images
Jauhari, Rajnish K.; Trivedi, Rashmi; Munshi, Prabhat; Sahni, Kamal [Indian Institute of Technology, Kanpur (India); J.K. Cancer Institute, G.S.V.M. Medical College, Kanpur (India)
2005-12-15
Fractal Dimension (FD) is a parameter used widely for classification, analysis, and pattern recognition of images. In this work we explore the quantification of CT (computed tomography) lesions of the brain by using fractal theory. Five brain lesions, which are portions of CT images of diseased brains, are used for the study. These lesions exhibit self-similarity over a chosen range of scales, and are broadly characterized by their fractal dimensions.
FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES
Chapeau-Blondeau, François
FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES Case.chapeau-blondeau}@univ-angers.fr Keywords: Color image, Color histogram, Fractal, Self-similarity, Capacity dimension, Correlation dimension of algorithms which can characterize fractal organizations in the support and population of their three
FRACTAL COMPLEXITY OF THE HUMAN CORTEX IS INCREASED IN WILLIAMS SYNDROME
Thompson, Paul
FRACTAL COMPLEXITY OF THE HUMAN CORTEX IS INCREASED IN WILLIAMS SYNDROME 1 Paul M. Thompson, 1 algorithm to measure the fractal dimension, or complexity, of the human cerebral cortex. Cortical complexity, the proposed fractal dimension takes into account the full 3D cortical surface geometry, and is independent
Application of fractal geometry to damage development and brittle fracture in materials
T ANDERSON
1989-01-01
The concepts of fractal geometry are applicable to a wide range of problems in materials science. One obvious application is the characterization of irregular surfaces (e.g., fracture surfaces) by means of a fractal dimension. Several papers on this subject have recently appeared. A somewhat less obvious use of the fractal dimension involves characterization of the fragmented (nonuniform) nature of microstructural
Fractal Characterization of Hyperspectral Imagery
NASA Technical Reports Server (NTRS)
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.
The Possible Role of Fractal Geometry in Tribology
Frederick F. Ling
1989-01-01
Fractal geometry, in which infinite numbers of fractional dimensions are permitted in contradistinction to the three integer dimensions in Euclidean geometry, has been applied to the study of surface roughness. A tentative conclusion is that fractal geometry offers yet another vehicle for the physical chemist to meet the mechanical engineer on solving problems in the boundary lubrication regime of tribology.
Morphological Modeling Using Fractal Geometries
NASA Astrophysics Data System (ADS)
Nelson, Thomas R.
1988-06-01
The application of fractal concepts to the analysis of non-linear dynamics and morphology has expanded our insight into many diverse natural phenomena. Fractal mathematics provides new methods of analysis also applicable to biophysical phenomena including the structure and function of systems comprising the human body. The brain, heart and the tracheo-bronchial tree possess characteristics common to fractal objects including: (a) a large degree of heterogeneity, (b) self-similar structures over many size scales, and (c) no well defined (characteristic) scale of measure. The fractal dimension, DF is a measure of the structural complexity. This paper presents an overview of some of the general concepts underlying fractals and their relationship to non-linear dynamics and morphology. Areas of investigation that benefit from the application of these concepts to biological phenomena and modeling are discussed and an algorithm for modeling lung development based on fractal concepts is presented. Structures that are in good agreement with actual morphological data may be generated using simple recursive algorithms and constraints.
Petite, Samuel
Fractal-e-s Barbara Schapira Enseignante-chercheuse au L.A.M.F.A., UniversitÂ´e de Picardie Jules Verne http ://www.mathinfo.u-picardie.fr/schapira/ #12;Historique #12;Historique Â· Premiers fractals mathÂ´ematiques : Julia et Fatou dÂ´ebut 20`eme. #12;Historique Â· Premiers fractals math
Calculus on Fractal Curves in R^n
Abhay Parvate; Seema Satin; A. D. Gangal
2010-04-06
A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
Fractal PatternsFractal Patterns in Chaotic Mixingin Chaotic Mixing
Anlage, Steven
Fractal PatternsFractal Patterns in Chaotic Mixingin Chaotic Mixing Amir Ali Ahmadi, UniversityTREND 2005 #12;What is a Fractal? Romanesco broccoli Fractal an object which has variation://upload.wikimedia.org/wikipedia/en/thumb/8/8a/800px-Fractal_Broccoli.jpg #12;Fractal Example http://colos1.fri.uni-lj.si/~sis/GRAFIKA/FRACTALS/FRACTAL
Walter, M.Todd
Influence of image resolution and thresholding on the apparent mass fractal characteristics choices regarding image resolution, the definition adopted for the "fractal" dimension statistical significance, with R 0.999. Of the various parameters subject to choice, image resolution seems
2008-01-01
Self-similar patterns are frequently observed in Nature. Their reproduction is possible on a length scale 102–105 nm with lithographic methods, but seems impossible on the nanometer length scale. It is shown that this goal may be achieved via a multiplicative variant of the multi-spacer patterning technology, in this way permitting the controlled preparation of fractal surfaces.
NASA Astrophysics Data System (ADS)
Cerofolini, G. F.; Narducci, D.; Amato, P.; Romano, E.
2008-10-01
Self-similar patterns are frequently observed in Nature. Their reproduction is possible on a length scale 102 105 nm with lithographic methods, but seems impossible on the nanometer length scale. It is shown that this goal may be achieved via a multiplicative variant of the multi-spacer patterning technology, in this way permitting the controlled preparation of fractal surfaces.
NSDL National Science Digital Library
Math Forum
2000-01-01
Fractal images made for the most part using a software application called Flarium24. Galleries contain about 15 images each and should be viewed in hi-color or truecolor settings. Tilable images that can be downloaded?for wallpaper are also available.
Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space
Tarasov, Vasily E
2015-01-01
Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.
Fractal Propagators in QED and QCD and Implications for the Problem of Confinement
S. Gulzari; Y. N. Srivastava; J. Swain; A. Widom
2006-12-09
We show that QED radiative corrections change the propagator of a charged Dirac particle so that it acquires a fractional anomalous exponent connected with the fine structure constant. The result is a nonlocal object which represents a particle with a roughened trajectory whose fractal dimension can be calculated. This represents a significant shift from the traditional Wigner notions of asymptotic states with sharp well-defined masses. Non-abelian long-range fields are more difficult to handle, but we are able to calculate the effects due to Newtonian gravitational corrections. We suggest a new approach to confinement in QCD based on a particle trajectory acquiring a fractal dimension which goes to zero in the infrared as a consequence of self-interaction, representing a particle which, in the infrared limit, cannot propagate.
Sridhar Poosapadi Arjunan; Dinesh Kant Kumar
2008-01-01
This research paper reports the use of fractal features based technique in physiological signals like surface electromyogram (sEMG), electroencephalogram (EEG) which has gained increasing attention in biosignal processing for medical and healthcare applications. This research reports the use of fractal dimension, a fractal complexity measure in physiological signals and also reports identification of a new feature of sEMG, maximum fractal
NSDL National Science Digital Library
Spanky Fractal Database: fractal images, programs, documents, papers, code examples, and other fractal related material. Submitted by contributors or hunted down from various nooks and crannies on the net. Enjoy and discover.
Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique
NASA Technical Reports Server (NTRS)
Qiu, Hong-Lie; Lam, Nina Siu-Ngan; Quattrochi, Dale
1997-01-01
With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.
A fractal-like resistive network
NASA Astrophysics Data System (ADS)
Saggese, A.; De Luca, R.
2014-11-01
The equivalent resistance of a fractal-like network is calculated by means of approaches similar to those employed in defining the equivalent resistance of an infinite ladder. Starting from an elementary triangular circuit, a fractal-like network, named after Saggese, is developed. The equivalent resistance of finite approximations of this network is measured, and the didactical implications of the model are highlighted.
Kober, Martin; Bleicher, Marcus [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Koch, Benjamin [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Frankfurt Institute for Advanced Studies (FIAS), Max-von-Laue-Str. 1, 60438 Frankfurt (Germany)
2007-12-15
We calculate the inclusive cross section of double Z-boson production within large extra dimensions at the Large Hadron Collider (LHC). Using perturbatively quantized gravity in the ADD (Arkani-Hamed, Dvali, Dimopoulos) model we perform a first order calculation of the graviton mediated contribution to the pp{yields}ZZ+x cross section. At low energies (e.g. Tevatron) this additional contribution is very small, making it virtually unobservable, for a fundamental mass scale above 2500 GeV. At LHC energies, however, the calculation indicates that the ZZ-production rate within the ADD model should differ significantly from the standard model if the new fundamental mass scale would be below 15000 GeV. A comparison with the observed production rate at the LHC might therefore provide direct hints on the number and structure of the extra dimensions.
NASA Astrophysics Data System (ADS)
Wuorinen, Charles
2015-03-01
Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...
Microtopographic Inspection and Fractal Analysis of Skin Neoplasia
NASA Astrophysics Data System (ADS)
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) corresponding to some neoplasia is higher (1.334+/-0.072) than those for healthy skin (1.091+/-0.082). A significant difference between the fractal dimensions of neoplasia and healhty skin (>0.001) was registered. The FD of microtopography maps (FDm) can also distinguish between healthy and malignant tissue in general (2.277+/-0.070 to 2.309+/-0.040), but not discriminate the different types of skin neoplasias. The combination of the rugometric evaluation and fractal geometry characterization provides valuable information about the malignity of skin lesions and type of lesion.
Anomalous relaxation in fractal structures
Fujiwara, S.; Yonezawa, F. (Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223 (Japan))
1995-03-01
For the purpose of studying some interesting properties of anomalous relaxation in fractal structures, we carry out Monte Carlo simulations of random walks on two-dimensional fractal structures (Sierpinski carpets with different cutouts and site-percolation clusters in a square lattice at the critical concentration). We find that the relaxation is of the Cole-Cole type [J. Chem. Phys. 9, 341 (1941)], which is one of the empirical laws of anomalous relaxation. Scaling properties are found in the relaxation function as well as in the particle density. We also find that, in strucures with almost the same fractal dimension, relaxation in structures with dead ends is slower than that in structures without them. This paper ascertains that the essential aspects of the anomalous relaxation due to many-body effects can be explained in the framework of the one-body model.
Fractal Substructure of a Nanopowder
Thomas Schwager; Dietrich E. Wolf; Thorsten Poeschel
2008-02-25
The structural evolution of a nano-powder by repeated dispersion and settling can lead to characteristic fractal substructures. This is shown by numerical simulations of a two-dimensional model agglomerate of adhesive rigid particles. The agglomerate is cut into fragments of a characteristic size l, which then are settling under gravity. Repeating this procedure converges to a loosely packed structure, the properties of which are investigated: a) The final packing density is independent of the initialization, b) the short-range correlation function is independent of the fragment size, c) the structure is fractal up to the fragmentation scale l with a fractal dimension close to 1.7, and d) the relaxation time increases linearly with l.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
NSDL National Science Digital Library
2011-01-01
This applet lets students explore connections among geometry, measurement and number patterns. Students select one of four preset fractal processes, including the Koch snowflake and the Sierpinski triangle, and observe the stages of complexity. The number of parts and an area or linear measure are shown in a table. The tool supports the lesson How Many Triangles Can You Construct? in the unit Building with Triangles (cataloged separately).
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 $G''$.
Pedro Patricio; Catarina R. Leal; Jorge Duarte; Cristina Januario
2015-06-05
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 $G''$.
3 FROM FRACTAL OBJECTS TO FRACTAL SPACES 49 Excerpt from
Nottale, Laurent
3 FROM FRACTAL OBJECTS TO FRACTAL SPACES 49 Excerpt from FRACTAL SPACE-TIME AND MICROPHYSICS.3-3.6 Chapter 3 FROM FRACTAL OBJECTS TO FRACTAL SPACES 3.3. Fractal Curves in a Plane. Let us now come to our first attempts to define fractals in an intrinsic way and to deal with infinities and with their non
Hull/EPIC3 linked Eulerian/Lagrangian calculation in three dimension
NASA Astrophysics Data System (ADS)
Matuska, D. A.; Osborn, J. J.
1981-09-01
This report documents a demonstration calculation performed to evaluate the three dimensional Eulerian/Lagrangian linked hydrocode developed under this contract. The calculation consists of a staballoy rod impacting an armor plate at obliquity 65 degrees and velocity 1 km/sec. Calculational results compared favorably with experimental data. The linked calculation was completed with the use of about nine total CDC 7600 computer hours as compared to the estimated 25 hours if the calculation had been run with only the Eulerian code.
Characterization of branch complexity by fractal analyses
Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.
1999-01-01
The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.
Stability limits for bioconvective fractals - Microgravity prospects
NASA Technical Reports Server (NTRS)
Noever, David A.
1992-01-01
Fractal objects are delicate aggregates which show self-similar behavior and vanishing density for increasing length scales. In practice real fractals in nature however possess only a limited region of verifiable self-similarity. As natural fractal objects increase in size, they become easier to disrupt mechanically. Herein the effects of thermal vibrations and gravity are investigated as deforming forces on fractal aggregation. Example calculations are carried out on a biological fractal formed from the surface aggregation of various cells such as alga and bacteria. For typical cell parameters, the predicted diameter of this so-called 'bioconvective' fractal agrees well with the observed limits of about 5 cm. On earth, this size represents an experimental maximum for finding bioconvective fractal objects. To extend this size range of fractals available for statistical study, a reduced gravity environment offers one way to achieve larger fractals. For these enhanced sizes, the present scaling predicts that microgravity can yield up to a 35-fold improvement in extending statistical resolution.
Two-loop calculation of the anomalous dimensions of four-fermion operators with static heavy quarks
V. Giménez
1993-01-01
We compute in the heavy-quark effective theory the two-loop anomalous dimension of the effective four-fermion operator ODeltaB = 2 whose matrix element determines the B-parameter of the B-meson. Dimensional reduction is the regularization method chosen in this calculation, which is performed using the techniques already developed by us to treat the two-loop renormalization in standard dimensional regularization of the heavy-quark
FRACTAL DESCRIPTION OF ROUGH SURFACES FOR HAPTIC DISPLAY
Michael Allan Costa; Oussama Khatib; John Kenneth Salisbury
Abstract This thesis develops a structure suitable to study the roughness perception of natural rough surfaces rendered on a,haptic display system using fractals. A background,on traditional methods,for describing surface roughness is given. Fractals are used to characterize one- dimensional surface profiles using two parameters, the amplitude coefficient and the frac- tal dimension. Synthesized fractal profiles are compared,to the profiles of
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
M. N. Chernodub; Stephane Ouvry
2015-04-09
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
Chernodub, M N
2015-01-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.
NASA Astrophysics Data System (ADS)
Subramaniam, Raji; Sullivan, R.; Schneider, P. S.; Flamholz, A.; Cheung, E.; Tremberger, G., Jr.; Wong, P. K.; Lieberman, D. H.; Cheung, T. D.; Garcia, F.; Bewry, N.; Yee, A.
2006-10-01
Images of packaged raw chicken purchased in neighborhood supermarkets were captured via a digital camera in laboratory and home settings. Each image contained the surface reflectivity information of the chicken tissue. The camera's red, green and blue light signals fluctuated and each spectral signal exhibited a random series across the surface. The Higuchi method, where the length of each increment in time (or spatial) lag is plotted against the lag, was used to explore the fractal property of the random series. (Higuchi, T., "Approach to an irregular time series on the basis of fractal theory", Physica D, vol 31, 277-283, 1988). The fractal calculation algorithm was calibrated with the Weierstrass function. The standard deviation and fractal dimension were shown to correlate with the time duration that a package was left at room temperature within a 24-hour period. Comparison to packaged beef results suggested that the time dependence could be due microbial spoilage. The fractal dimension results in this study were consistent with those obtained from yeast cell, mammalian cell and bacterial cell studies. This analysis method can be used to detect the re-refrigeration of a "left-out" package of chicken. The extension to public health issues such as consumer shopping is also discussed.
FRACTAL ANTENNAS Philip Felber
FRACTAL ANTENNAS by Philip Felber A literature study as a project for ECE 576 Illinois Institute of Technology December 12, 2000 (Revised: January 16, 2001) #12;2 Felber: "Fractal Antennas" Abstract 3 Introduction 3 Chronology 3 Background 4 Fractals 5 Antennas 6 Fractal Antennas 7 Applications 9 Classic
Fuzzy fractals, chaos, and noise
Zardecki, A.
1997-05-01
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.
Magnetohydrodynamics of fractal media
Tarasov, Vasily E. [Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992 (Russian Federation)
2006-05-15
The fractal distribution of charged particles is considered. An example of this distribution is the charged particles that are distributed over the fractal. The fractional integrals are used to describe fractal distribution. These integrals are considered as approximations of integrals on fractals. Typical turbulent media could be of a fractal structure and the corresponding equations should be changed to include the fractal features of the media. The magnetohydrodynamics equations for fractal media are derived from the fractional generalization of integral Maxwell equations and integral hydrodynamics (balance) equations. Possible equilibrium states for these equations are considered.
Fractal Interrelationships in Field and Seismic Data
Wilson, T.H.; Dominic, Jovita; Halverson, Joel
1997-10-01
Size scaling interrelationships are evaluated in this study using a fractal model. Fractal models of several geologic variables are examined and include fracture patterns, reflection travel times, structural relief, drainage, topographic relief and active fault patterns. The fractal properties of structural relief inferred from seismic data and structural cross sections provide a quantitative means to characterize and compare complex structural patterns. Studies were conducted using seismic data from the Granny Creek oil field in the Appalachian Plateau. Previous studies of the field reveal that subtle detached structures present on the limb of a larger structure are associated with enhanced production from the field. Vertical increases of fractal dimension across the zone of detachment provide a measure of the extent to which detachment has occurred. The increases of fractal dimension are greatest in the more productive areas of the field. A result with equally important ramifications is that fracture systems do not appear to be intrinsically fractal as is often suggested in the literature. While examples of nearly identical patterns can be found at different scales supporting the idea of self-similarity, these examples are often taken from different areas and from different lithologies. Examination of fracture systems at different scales in the Valley and Ridge Province suggest that their distribution become increasingly sparse with scale reduction, and therefore are dissimilar or non-fractal. Box counting data in all cases failed to yield a fractal regime. The results obtained from this analysis bring into question the general applicability of reservoir simulations employing fractal models of fracture distribution. The same conclusions were obtained from the analysis of 1D fracture patterns such as those that might appear in a horizontal well.
Fractal analysis of scatter imaging signatures to distinguish breast pathologies
NASA Astrophysics Data System (ADS)
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.
Fractal properties of quantum spacetime
Dario Benedetti
2009-03-25
We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of $\\k$-Minkowski, the latter being relevant in the context of quantum gravity.
NASA Astrophysics Data System (ADS)
Chang, Kuo-En; Lin, Tang-Huang; Lien, Wei-Hung
2015-04-01
Anthropogenic pollutants or smoke from biomass burning contribute significantly to global particle aggregation emissions, yet their aggregate formation and resulting ensemble optical properties are poorly understood and parameterized in climate models. Particle aggregation refers to formation of clusters in a colloidal suspension. In clustering algorithms, many parameters, such as fractal dimension, number of monomers, radius of monomer, and refractive index real part and image part, will alter the geometries and characteristics of the fractal aggregation and change ensemble optical properties further. The cluster-cluster aggregation algorithm (CCA) is used to specify the geometries of soot and haze particles. In addition, the Generalized Multi-particle Mie (GMM) method is utilized to compute the Mie solution from a single particle to the multi particle case. This computer code for the calculation of the scattering by an aggregate of spheres in a fixed orientation and the experimental data have been made publicly available. This study for the model inputs of optical determination of the monomer radius, the number of monomers per cluster, and the fractal dimension is presented. The main aim in this study is to analyze and contrast several parameters of cluster aggregation aforementioned which demonstrate significant differences of optical properties using the GMM method finally. Keywords: optical properties, fractal aggregation, GMM, CCA
Fractal Characterization of Fracture Networks: An Improved Box-counting Technique
Ankur Roy; Edmund Perfect; William M. Dunne; Larry D. McKay
2007-01-01
Box counting is widely used for characterizing fracture networks as fractals and estimating their fractal dimensions (D). If this analysis yields a power law distribution given by N $\\\\propto$ r?D, where N is the number of boxes containing one or more fractures and r is the box size, then the network is considered to be fractal. However, researchers are divided
Fractal characterization of fracture networks: An improved box-counting technique
Ankur Roy; Edmund Perfect; William M. Dunne; Larry D. McKay
2007-01-01
Box counting is widely used for characterizing fracture networks as fractals and estimating their fractal dimensions (D). If this analysis yields a power law distribution given by N$\\\\propto$r?D, where N is the number of boxes containing one or more fractures and r is the box size, then the network is considered to be fractal. However, researchers are divided in their
Fractal characterisation of high-pressure and hydrogen-enriched CH4air turbulent premixed flames
Gülder, Ömer L.
Fractal characterisation of high-pressure and hydrogen-enriched CH4air turbulent premixed flames measurements were performed to obtain the flame front images, which were further analyzed for fractal of the flame front curvature as a function of the pressure. Fractal dimension showed a strong dependence
Fractal structure of the Horsehead nebula (B 33)
S. Datta
2003-01-01
Analysis of the CCD image of the Horsehead nebula (B 33), taken in the H alpha (6561 Å) using the 2.34 m Vainu Bappu Telescope (VBT) at Kavalur, India, is performed to test its fractal structure. Ten sample readings of the box dimension of this image were taken using a fractal analysis software, giving an average value of 1.6965725. The
Fractal geometry of spinglass models J. F. Fontanari
Stadler, Peter F.
Fractal geometry of spinÂglass models J. F. Fontanari Instituto de Fâ??ï¿½sica de Sâ?ao Carlos through saddle s, and D is the fractal dimension of the phase space. PACS 75.10.Nr (principal), 87.23.Kg
Fractal simulation of the resistivity and capacitance of arsenic selenide
Balkhanov, V. K., E-mail: ballar@yandex.ru; Bashkuev, Yu. B. [Russian Academy of Sciences, Division of Physical Problems, Buryat Scientific Center, Siberian Branch (Russian Federation)
2010-03-15
The temperature dependences of the ac resistivity R and ac capacitance C of arsenic selenide were measured more than four decades ago [V. I. Kruglov and L. P. Strakhov, in Problems of Solid State Electronics, Vol. 2 (Leningrad Univ., Leningrad, 1968)]. According to these measurements, the frequency dependences are R {proportional_to} {omega}{sup -0.80{+-}0.01} and {Delta}C {proportional_to} {omega}{sup -0.120{+-}0.006} ({omega} is the circular frequency and {Delta}C is measured from the temperature-independent value C{sub 0}). According to fractal-geometry methods, R {proportional_to} {omega}{sup 1-3/h} and {Delta}C {proportional_to} {omega}{sup -2+3/h}, where h is the walk dimension of the electric current in arsenic selenide. Comparison of the experimental and theoretical results indicates that the walk dimensions calculated from the frequency dependences of resistivity and capacitance are h{sub R} = 1.67 {+-} 0.02 and h{sub C} = 1.60 {+-} 0.08, which are in agreement with each other within the measurement errors. The fractal dimension of the distribution of conducting sections is D = 1/h = 0.6. Since D < 1, the conducting sections are spatially separated and form a Cantor set.
Fractal analysis of yeast cell optical speckle
NASA Astrophysics Data System (ADS)
Flamholz, A.; Schneider, P. S.; Subramaniam, R.; Wong, P. K.; Lieberman, D. H.; Cheung, T. D.; Burgos, J.; Leon, K.; Romero, J.
2006-02-01
Steady state laser light propagation in diffuse media such as biological cells generally provide bulk parameter information, such as the mean free path and absorption, via the transmission profile. The accompanying optical speckle can be analyzed as a random spatial data series and its fractal dimension can be used to further classify biological media that show similar mean free path and absorption properties, such as those obtained from a single population. A population of yeast cells can be separated into different portions by centrifuge, and microscope analysis can be used to provide the population statistics. Fractal analysis of the speckle suggests that lower fractal dimension is associated with higher cell packing density. The spatial intensity correlation revealed that the higher cell packing gives rise to higher refractive index. A calibration sample system that behaves similar as the yeast samples in fractal dimension, spatial intensity correlation and diffusion was selected. Porous silicate slabs with different refractive index values controlled by water content were used for system calibration. The porous glass as well as the yeast random spatial data series fractal dimension was found to depend on the imaging resolution. The fractal method was also applied to fission yeast single cell fluorescent data as well as aging yeast optical data; and consistency was demonstrated. It is concluded that fractal analysis can be a high sensitivity tool for relative comparison of cell structure but that additional diffusion measurements are necessary for determining the optimal image resolution. Practical application to dental plaque bio-film and cam-pill endoscope images was also demonstrated.
Shimizu, Wataru; Nakamura, Satoshi; Sato, Takaaki; Murakami, Yasushi
2012-08-21
Amorphous titanium dioxide (TiO(2)) thin films exhibiting high refractive indices (n ? 2.1) and high transparency were fabricated by spin-coating titanium oxide liquid precursors having a weakly branched polymeric structure. The precursor solution was prepared from titanium tetra-n-butoxide (TTBO) via the catalytic sol-gel process with hydrazine monohydrochloride used as a salt catalyst, which serves as a conjugate acid-base pair catalyst. Our unique catalytic sol-gel technique accelerated the overall polycondensation reaction of partially hydrolyzed alkoxides, which facilitated the formation of liner polymer-like titanium oxide aggregates having a low fractal dimension of ca. (5)/(3), known as a characteristic of the so-called "expanded polymer chain". Such linear polymeric features are essential to the production of highly dense amorphous TiO(2) thin films; mutual interpenetration of the linear polymeric aggregates avoided the creation of void space that is often generated by the densification of high-fractal-dimension (particle-like) aggregates produced in a conventional sol-gel process. The mesh size of the titanium oxide polymers can be tuned either by water concentration or the reaction time, and the smaller mesh size in the liquid precursor led to a higher n value of the solid thin film, thanks to its higher local electron density. The reaction that required no addition of organic ligand to stabilize titanium alkoxides was advantageous to overcoming issues from organic residues such as coloration. The dense amorphous film structure suppressed light scattering loss owing to its extremely smooth surface and the absence of inhomogeneous grains or particles. Furthermore, the fabrication can be accomplished at a low heating temperature of <80 °C. Indeed, we successfully obtained a transparent film with a high refractive index of n = 2.064 (at ? = 633 nm) on a low-heat-resistance plastic, poly(methyl methacrylate), at 60 °C. The result offers an efficient route to high-refractive-index amorphous TiO(2) films as well as base materials for a wider range of applications. PMID:22817104
Fractal characterization of neural correlates of consciousness
NASA Astrophysics Data System (ADS)
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.
Fractal analysis of bone structure with applications to osteoporosis and microgravity effects
Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E. [State Univ. of New York, Buffalo, NY (United States). Biomedical Imaging Group; LeBlanc, A.; Lin, C. [Baylor Coll. of Medicine, Houston, TX (United States); Shackelford, L. [National Aeronautics and Space Administration, Houston, TX (United States). Johnson Space Center
1995-12-31
The authors characterize the trabecular structure with the aid of fractal dimension. The authors use Alternating Sequential filters to generate a nonlinear pyramid for fractal dimension computations. The authors do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of the scheme is the rudimentary definition of self similarity. This allows them the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, the authors have shown that the ASF methods outperform other existing methods for fractal dimension estimation. They have shown that the fractal dimension remains the same when computed with both the X-Ray images and the MRI images of the patella. They have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, the authors have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, they have shown that the subject`s prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.
Fractal Strings and Multifractal Zeta Functions
Michel L. Lapidus; Jacques Levy Vehel; John A. Rock
2009-02-09
For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.
Multiscale Fractal Characterization of Three-Dimensional Gene Expression Data
Edson Tadeu Monteiro Manoel; Luciano Da Fontoura Costa; Johannes Streicher; Gerd B. Müller
2002-01-01
Abstract. This article reports on the ,application of the recently ,introduced concept of multiscale ,fractal dimension,(MFD) as a ,resource for quantifying three-dimensional ,gene expression patterns in embryonic development. While ,traditional fractal dimensions ,provide interesting possibilities for quantifying pattern complexity, as defined by the intensity in which the pattern interacts with its surrounding space, those approaches,fail to take into account the
NASA Astrophysics Data System (ADS)
García, Alejandro; Aldana, Milagrosa; Cabrera, Ana
2013-04-01
In this work, we have applied a Wavelet Based Fractal Analysis (WBFA) to well logs and seismic data at the Teapot Dome Field, Natrona Country, Wyoming-USA, trying to characterize a reservoir using fractal parameters, as intercept (b), slope (m) and fractal dimension (D), and to correlate them with the sedimentation processes and/or the lithological characteristics of the area. The WBFA was first applied to the available logs (Gamma Ray, Spontaneous Potential, Density, Neutron Porosity and Deep Resistivity) from 20 wells located at sectors 27, 28, 33 and 34 of the 3D seismic of the Teapot Dome field. Also the WBFA was applied to the calculated curve of water saturation (Sw). At a second step, the method was used to analyze a set of seismic traces close to the studied wells, extracted from the 3D seismic data. Maps of the fractal parameters were obtained. A spectral analysis of the seismic data was also performed in order to identify seismic facies and to establish a possible correlation with the fractal results. The WBFA results obtained for the wells logs indicate a correlation between fractal parameters and the lithological content in the studied interval (i.e. top-base of the Frontier Formation). Particularly, for the Gamma Ray logs the fractal dimension D can be correlated with the sand-shale content: values of D lower than 0.9 are observed for those wells with more sand content (sandy wells); values of D between 0.9 and 1.1 correspond to wells where the sand packs present numerous inter-bedded shale layers (sandy-shale wells); finally, wells with more shale content (shaly wells) have D values greater than 1.1. The analysis of the seismic traces allowed the discrimination of shaly from sandy zones. The D map generated for the seismic traces indicates that this value can be associated with the shale content in the area. The iso-frequency maps obtained from the seismic spectral analysis show trends associated to the lithology of the field. These trends are similar to those observed in the maps of the fractal parameters, indicating that both analyses respond to lithological and/or sedimentation features in the area.
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
ERIC Educational Resources Information Center
Barton, Ray
1990-01-01
Presented is an educational game called "The Chaos Game" which produces complicated fractal images. Two basic computer programs are included. The production of fractal images by the Sierpinski gasket and the Chaos Game programs is discussed. (CW)
Fractal analysis of the galaxy distribution in the redshift range 0.45 < z < 5.0
G. Conde-Saavedra; A. Iribarrem; Marcelo B. Ribeiro
2014-09-18
Evidence is presented that the galaxy distribution can be described as a fractal system in the redshift range of the FDF galaxy survey. The fractal dimension $D$ was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with $\\Omega_{m_0}=0.3$, $\\Omega_{\\Lambda_0}=0.7$ and $H_0=70 \\; \\mbox{km} \\; {\\mbox{s}}^{-1} \\; {\\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number densities $\\gamma$ and $\\gamma^\\ast$ obtained from the red and blue FDF galaxies provides a direct method to estimate $D$, implying that $\\gamma$ and $\\gamma^\\ast$ vary as power-laws with the cosmological distances. The luminosity distance $d_{\\scriptscriptstyle L}$, galaxy area distance $d_{\\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against their respective number densities to calculate $D$ by linear fitting. It was found that the FDF galaxy distribution is characterized by two single fractal dimensions at successive distance ranges. Two straight lines were fitted to the data, whose slopes change at $z \\approx 1.3$ or $z \\approx 1.9$ depending on the chosen cosmological distance. The average fractal dimension calculated using $\\gamma^\\ast$ changes from $\\langle D \\rangle=1.4^{\\scriptscriptstyle +0.7}_{\\scriptscriptstyle -0.6}$ to $\\langle D \\rangle=0.5^{\\scriptscriptstyle +1.2}_{\\scriptscriptstyle -0.4}$ for all galaxies, and $D$ decreases as $z$ increases. Small values of $D$ at high $z$ mean that in the past galaxies were distributed much more sparsely and the large-scale galaxy structure was then possibly dominated by voids. Results of Iribarrem et al. (2014, arXiv:1401.6572) indicating similar fractal features with $\\langle D \\rangle =0.6 \\pm 0.1$ in the far-infrared sources of the Herschel/PACS evolutionary probe (PEP) at $1.5 \\lesssim z \\lesssim 3.2$ are also mentioned.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.
1998-01-01
Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data. A software package known as the Image Characterization and Modeling System (ICAMS) was used to explore how fractal dimension is related to surface texture and pattern. The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was measured to investigate the relationship between texture and resolution for different land covers.
B. Sapoval; Th. Gobron; A. Margolina
1991-01-01
Fractal boundary conditions drastically alter wave excitations. The low-frequency vibrations of a membrane bounded by a rigid fractal contour are observed and localized modes are found. The first lower eigenmodes are computed using an analogy between the wave and the diffusion equations. The fractal frontier induces a strong confinement of the wave analogous to superlocalization. The wave forms exhibit singular
ERIC Educational Resources Information Center
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.…
Hidden Markov Models and Gaussian Mixture Models for Bearing Fault Detection Using Fractals
Tshilidzi Marwala; Unathi Mahola; Fulufhelo Vincent Nelwamondo
2006-01-01
Bearing vibration signals features are extracted using time domain fractal based feature extraction technique. This technique uses multi-scale fractal dimension (MFD) estimated using box-counting dimension. The extracted features are then used to classify faults using Gaussian mixture models (GMM) and hidden Markov models (HMM). The results obtained show that the proposed feature extraction technique does extract fault specific information. Furthermore,
Measurement of normal contact stiffness of fractal rough surfaces
Chongpu Zhai; Sébastien Bevand; Yixiang Gan; Dorian Hanaor; Gwénaëlle Proust; Bruno Guelorget; Delphine Retraint
2014-09-03
We investigate the effects of roughness and fractality on the normal contact stiffness of rough surfaces. Samples of isotropically roughened aluminium surfaces are considered. The roughness and fractal dimension were altered through blasting using different sized particles. Subsequently, surface mechanical attrition treatment (SMAT) was applied to the surfaces in order to modify the surface at the microscale. The surface topology was characterised by interferometry based profilometry. The normal contact stiffness was measured through nanoindentation with a flat tip utilising the partial unloading method. We focus on establishing the relationships between surface stiffness and roughness, combined with the effects of fractal dimension. The experimental results, for a wide range of surfaces, showed that the measured contact stiffness depended very closely on surfaces' root mean squared (RMS) slope and their fractal dimension, with correlation coefficients of around 90\\%, whilst a relatively weak correlation coefficient of 57\\% was found between the contact stiffness and RMS roughness.
Fat fractal percolation and k-fractal percolation Erik Bromana
Meester, Ronald
Fat fractal percolation and k-fractal percolation Erik Bromana Tim van de Brugb Federico Camiab fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided . This is analogous to the result of Falconer and Grimmett in [8] that the critical value for Mandelbrot fractal
Retinal vascular fractals in Behçet’s Disease: A screening method?
Norouzpour, Amir; Mehdizadeh, Alireza
2015-01-01
Objective The branching pattern of retinal vessels may be affected in Behçet’s Disease (BD). Fractal analysis can be used as a new method to quantify the changes of the vascular branching pattern. In this study, we examined, for the first time, the relationship between retinal fractal dimension (Df) and retinal vascular changes seen in patients with BD. Methods We conducted a retrospective study of 10 new cases of BD with clinically ocular involvement. Color fundus images taken from both eyes of the participants have been analyzed, and Df of the whole retinal vasculature was quantified using a novel computer-based program. The resultant Df was compared with that of healthy individuals. Results The mean Df, calculated from 20 fundus images of cases with BD, was 1.59 ± 0.064. It was lower than that of healthy participants (1.65 ± 0.060) significantly (P = 0.013). Conclusion Retinal fractal analysis of cases with BD has been performed for the first time, and the results showed that early retinal vascular changes seen in new cases of BD are associated with lower retinal Df. Retinal fractal analysis in BD can be practically utilized as a potential tool for screening of retinal involvement, evaluating the prognosis and the response to treatment. PMID:26155081
McConathy, R.K.
1983-03-01
The study describes the gradients of stomatal size and density in the crown of a mature forest-grown tulip-poplar (Liriodendron tulipifera L.) in eastern Tennessee. These data are used to predict leaf resistance to vapor diffusion in relation to stomatal width and boundary layer resistance. Stomatal density on individual leaves did not vary, but density increased with increasing crown height. Stomatal size decreased with increasing height of leaves within the crown. Stomatal size and density variations interacted to result in a constant number of stomata per leaf at all crown heights. Stomatal diffusive resistance values calculated from stomatal measurements and varying environmental parameters indicated that stomatal resistance controlled transpiration water losses only at small apertures (<0.6 ..mu..m). Boundary layer resistance was controlling at large stomatal apertures (>0.6 ..mu..m) and at low wind speeds (approx.100 cm/s). Under normal forest conditions tulip-poplar stomatal resistance exercised more control over transpiration than did boundary layer resistance.
Analytical estimation of the correlation dimension of integer lattices
Lucas Lacasa; Jesús Gómez-Gardeñes
2014-07-07
Recently [L. Lacasa and J. G\\'omez-Garde\\~nes, Phys. Rev. Lett. {\\bf 110}, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called {\\em correlation dimension}, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order the correlation dimension $\\beta$ of integer lattices $\\mathbb{Z}^d$ coincides with the Haussdorf dimension of their coarsely-equivalent Euclidean spaces, $\\beta=d$.
Fractal structure of the Horsehead nebula (B 33)
NASA Astrophysics Data System (ADS)
Datta, S.
2003-04-01
Analysis of the CCD image of the Horsehead nebula (B 33), taken in the H alpha (6561 Å) using the 2.34 m Vainu Bappu Telescope (VBT) at Kavalur, India, is performed to test its fractal structure. Ten sample readings of the box dimension of this image were taken using a fractal analysis software, giving an average value of 1.6965725. The sample dimensions were found to be different from the topological dimension of one. Importantly, the box dimension of B 33 was not found to be significantly different from that of the Julia set (box dimension 1.679594) with c = -0.745429 + 0.113008i. This provides compelling evidence to show that the structure of the Horsehead nebula is not only fractal, but also that its geometry can be described by the Julia function f(z) = z2 + c, where both z and c are complex numbers.
Svozil, Karl
Fractal Analysis: An Objective Method for Identifying Atypical Nuclei in Dysplastic Lesions, Vienna, Austria Received October 15, 1998 Objectives. Fractal geometry is a tool used to characterize.g., the human renal artery tree), but also to derive parameters such as the fractal dimension in order
Beaucage, Gregory
Fractal Analysis of Flame-Synthesized Nanostructured Silica and Titania Powders Using Small-Angle X these powders display mass-fractal morphologies, which are composed of ramified aggregates of nanoscale primary particles. Primary particle size, aggregate size, fractal dimension, and specific surface area are obtained
Jing Li; Wensheng Deng
2010-01-01
Taking Jiangxia District in Wuhan City as the research region by using RS and GIS comprehensively. According to the country secondary category and classification standard interpretation. Reveal the characteristics of the fractal structures of the distribution of land use types. Explore the significance of the fractal on land use types. The fractal dimensions (D) and stability indexes (S) of first
Fractal analysis of the structural complexity of the connective tissue in human carotid bodies
Guidolin, Diego; Porzionato, Andrea; Tortorella, Cinzia; Macchi, Veronica; De Caro, Raffaele
2014-01-01
The carotid body (CB) may undergo different structural changes during perinatal development, aging, or in response to environmental stimuli. In the previous literature, morphometric approaches to evaluate these changes have considered quantitative first order parameters, such as volumes or densities, while changes in spatial disposition and/or complexity of structural components have not yet been considered. In the present study, different strategies for addressing morphological complexity of CB, apart from the overall amount of each tissue component, were evaluated and compared. In particular, we considered the spatial distribution of connective tissue in the carotid bodies of young control subjects, young opiate-related deaths and aged subjects, through analysis of dispersion (Morisita's index), gray level co-occurrence matrix (entropy, angular second moment, variance, correlation), and fractal analysis (fractal dimension, lacunarity). Opiate-related deaths and aged subjects showed a comparable increase in connective tissue with respect to young controls. However, the Morisita's index (p < 0.05), angular second moment (p < 0.05), fractal dimension (p < 0.01), and lacunarity (p < 0.01) permitted to identify significant differences in the disposition of the connective tissue between these two series. A receiver operating characteristic (ROC) curve was also calculated to evaluate the efficiency of each parameter. The fractal dimension and lacunarity, with areas under the ROC curve of 0.9651 (excellent accuracy) and 0.8835 (good accuracy), respectively, showed the highest discriminatory power. They evidenced higher level of structural complexity in the carotid bodies of opiate-related deaths than old controls, due to more complex branching of intralobular connective tissue. Further analyses will have to consider the suitability of these approaches to address other morphological features of the CB, such as different cell populations, vascularization, and innervation. PMID:25414672
Fractal boundaries in magnetotail particle dynamics
NASA Technical Reports Server (NTRS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-01-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Fractal Structure of Molecular Clouds
Srabani Datta
2001-05-02
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 the production of these clouds seem to be turbulence followed by Brownian motion till high densities are reached, at which point structure formation is dictated by gravity. The author presents image analysis of four varied examples, namely those of the Horsehead nebula, Eagle nebula, Rosette nebula and Paley I nebula to prove her hypothesis. The images of these nebulae are analyzed for their box dimension using fractal analysis software and comparisons are made with the given Julia set.
Triangular constellations in fractal measures
NASA Astrophysics Data System (ADS)
Wilkinson, Michael; Grant, John
2014-09-01
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 \\varepsilon{:}\\ {\\cal N}\\sim \\varepsilon^D . It is desirable to characterise the 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 ? and approximately uniform below a critical flow compressibility \\beta_{\\text{c}} , which we estimate. For \\beta>\\beta_{\\text{c}} the distribution appears to be described by two power laws: P(z)\\sim z^{\\alpha_1} when 1\\gg z\\gg z_{\\text{c}}(\\varepsilon) , and P(z)\\sim z^{\\alpha_2} when z\\ll z_{\\text{c}}(\\varepsilon) .
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 $z\\ll z_{\\rm c}(\\epsilon)$.
A fractal model for crustal deformation
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1986-01-01
It is hypothesized that crustal deformation occurs on a scale-invariant matrix of faults. For simplicity, a two-dimensional pattern of hexagons on which strike-slip faulting occurs is considered. The behavior of the system is controlled by a single parameter, the fractal dimension. Deformation occurs on all scales of faults. The fractal dimension determines the fraction of the total displacement that occurs on the first-order or primary faults. The value of the fractal dimension can be obtained from the frequency-magnitude relation for earthquakes. The results are applied to the San Andreas fault system in central California. Earthquake studies give D = 1.90. The main strand of the San Andreas fault is associated with the primary faults of the fractal system. It is predicted that the relative velocity across the main strand is 2.93 cm/yr. The remainder of the relative velocity of 5.5 cm/yr between the Pacific and North American plates occurs on higher-order faults. The predicted value is in reasonably good agreement with the value 3.39 + or - 0.29 cm/yr obtained from geological studies.
Fractals: To Know, to Do, to Simulate.
ERIC Educational Resources Information Center
Talanquer, Vicente; Irazoque, Glinda
1993-01-01
Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)
Exploring Fractals in the Classroom.
ERIC Educational Resources Information Center
Naylor, Michael
1999-01-01
Describes an activity involving six investigations. Introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Contains 14 references. (ASK)
Paul B. Slater
2007-03-26
Wu and Sprung (Phys. Rev. E 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded -- and similarly van Zyl and Hutchinson (Phys. Rev. E 67, 066211 (2003)) -- that the potential possesses a fractal structure of dimension d=3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting d=3/2, we estimate the frequency parameter (gamma), plus an overall scaling parameter (sigma) we introduce. We search for that pair of parameters (gamma,sigma) which minimizes the least-squares fit S_{n}(gamma,sigma) of the lowest n eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the lowest n Riemann zeros for n =25. For the additional cases we study, n=50 and 75, we simply set sigma=1. The fits obtained are compared to those gotten by using just the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement -- 5.7261 vs. 6.39207 (n=25), 11.2672 vs. 11.7002 (n=50) and 16.3119 vs. 16.6809 (n=75) -- is found in our (non-optimized, computationally-bound) search procedures. The improvements are relatively strong in the vicinities of gamma=3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher-order corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential.
Optical diffraction of fractal figures: random Sierpinski carpets
NASA Astrophysics Data System (ADS)
Berger, Denise; Chamaly, Stéphane; Perreau, Michel; Mercier, Daniel; Monceau, Pascal; Levy, Jean-Claude Serge
1991-10-01
The optical diffraction patterns of random Sierpinski carpets of different fractal dimensions at different levels of iteration are shown and analyzed. The sensitivity of such an analysis to long range correlations, is demonstrated theoretically by means of the transfer matrix formalism of fractals, T.M.F. The relation between the subdimensions defined in T.M.F. and diffraction patterns is outlined. Finally an analysis of experimental diffraction patterns is proposed in order to measure these new theoretical subdimensions. On présente ici les clichés de diffraction optique de tapis de Sierpinski aléatoires de différentes dimensions fractales, pris à des niveaux d'itération différents. Au moyen du formalisme de la matrice de transfert dans les fractals, on montre la sensibilité de cette analyse expérimentale aux corrélations à moyenne et longue portée. Ainsi la relation entre les sous-dimensions fractales du F.M.T. et les rapports d'intensité entre les clichés de diffraction de figures fractales à des niveaux d'itération différents est soulignée. Enfin on esquisse le principe d'une analyse expérimentale de ces nouvelles dimensions théoriques.
Two-loop calculation of the anomalous dimensions of four-fermion operators with static heavy quarks
NASA Astrophysics Data System (ADS)
Giménez, V.
1993-07-01
We compute in the heavy-quark effective theory the two-loop anomalous dimension of the effective four-fermion operator overlineO?B = 2 whose matrix element determines the B-parameter of the B-meson. Dimensional reduction is the regularization method chosen in this calculation, which is performed using the techniques already developed by us to treat the two-loop renormalization in standard dimensional regularization of the heavy-quark axial current. For N = 3, we find ? overlineO?B = 2(2) = - {1}/{36}[202+26?(2)-16n F] , with ?(2) = {1}/{6}? 2. This anomalous dimension turns out to be equal to the one in standard dimensional regularization. We employ ? overlineO?B = 2(2) to derive the complete scheme-independent order ?s correction to therelation between O?B = 2 in QCD and its counterpart overlineO?B = 2 in the effective theory. For the heavy-quark axial current, we recalculate the corresponding analogous order- ?s relation in dimensional reduction and the 't Hooft-Veltman dimensional regularization scheme. We show explicitly that this is renormalization-scheme independent, which completes the study carried out by us in our previous work. Moreover, these relations are used to obtain the renormalization constants for the lattice values of the B-meson decay constant, ƒ B, and B-parameter, BB, whose product enters the theoretical prediction of the B0- overlineB0 mixing. As an extra result, the theoretic subtleties of dimensional reduction applied to the heavy-quark effective theory are exposed and resolved.
Generation and display of geometric fractals in 3-D
Alan Norton
1982-01-01
We present some straightforward algorithms for the generation and display in 3-D of fractal shapes. These techniques are very general and particularly adapted to shapes which are much more costly to generate than to display, such as those fractal surfaces defined by iteration of algebraic transformations. In order to deal with the large space and time requirements of calculating these
Scaling laws for slippage on superhydrophobic fractal surfaces
Cottin-Bizonne, C; Bocquet, L
2012-01-01
We study the slippage on hierarchical fractal superhydrophobic surfaces, and find an unexpected rich behavior for hydrodynamic friction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension, and is found to be always smaller on fractal surfaces as compared to surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractal surface: depending on the underlying geometry of the roughness, strongly superhydrophobic surfaces may in some cases be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractal surfaces, in order to estimate the Kapitsa surface resistance on patterned surfaces, as well as to the question of trapping of diff...
Edges of Saturn's rings are fractal.
Li, Jun; Ostoja-Starzewski, Martin
2015-01-01
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63?~?1.78. This clarifies in what sense Saturn's rings are fractal. PMID:25883885
Fractal generalized zone plates
Omel Mendoza-Yero; Mercedes Fernández-Alonso; Gladys Mínguez-Vega; Jesús Lancis; Vicent Climent; Juan A. Monsoriu
2009-01-01
The construction of fractal generalized zone plates (FraGZPs) from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us to increase the number of foci of a conventional fractal zone plate (FraZP), keeping the self-similarity property within the axial irradiance. The focusing properties of these fractal diffractive optical elements for points not only along but
Helene Porchon
2012-01-25
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topological space.
NSDL National Science Digital Library
Gries, Daniel
While some may know fractals primarily from their use in abstract painting and African art, fractals are important elements within the world of mathematics. For those who seek to learn more about the construction of fractals and their uses, this very nice Flash-enabled feature from Daniel Gries at Merrimack College will definitely come in handy. This particular Flash applet draws fractals by means of a recursive algorithm, using a simple "generator" that users draw in the space that it is provided. Before using the application, users may wish read the overview offered online, and also take the time to read the instructions thoroughly.
Optical diffraction of fractal figures: random Sierpinski carpets
Denise Berger; Stéphane Chamaly; Michel Perreau; Daniel Mercier; Pascal Monceau; Jean-Claude Serge Levy
1991-01-01
The optical diffraction patterns of random Sierpinski carpets of different fractal dimensions at different levels of iteration are shown and analyzed. The sensitivity of such an analysis to long range correlations, is demonstrated theoretically by means of the transfer matrix formalism of fractals, T.M.F. The relation between the subdimensions defined in T.M.F. and diffraction patterns is outlined. Finally an analysis
Design of modified Sierpinski fractal antenna for multiband application
Zhang Hu; Guobin Wan; Changjie Sun; Huiling Zhao
2009-01-01
A modified Sierpinski fractal broadband antenna for multiband application is investigated, simulated, and measured in this paper. The perturbed fractal patch and the modified groundplane are employed to obtain the wider bandwidth at the resonance frequencies. The implemented antenna, with nearly omnidirectional radiation pattern, has been designed with a total dimension of 50.8??69??1.6 mm3. According to the measured results, it
Fractal geometry of some Martian lava flow margins: Alba Patera
NASA Technical Reports Server (NTRS)
Kauhanen, K.
1993-01-01
Fractal dimension for a few lava flow margins on the gently sloping flanks of Alba Patera were measured using the structured walk method. Fractal behavior was observed at scales ranging from 20 to 100 pixels. The upper limit of the linear part of log(margin length) vs. log(scale) profile correlated well to the margin length. The lower limit depended on resolution and flow properties.
Iterated Function Systems and the Global Construction of Fractals
M. F. Barnsley; S. Demko
1985-01-01
Iterated function systems (i.f.ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i.f.ss and occur as the supports of probability measures associated with functional equations. The existence of certain `p-balanced' measures for i.f.ss is established, and these measures are uniquely characterized for hyperbolic i.f.ss. The Hausdorff-Besicovitch dimension for some
Fourier and fractal analysis of cytoskeletal morphology altered by xenobiotics
Giovanni F. Crosta; Chiara Urani; Laura Fumarola
2003-01-01
The cytoskeletal microtubules (MTs) of rat hepatocytes treated by Benomyl (a fungicide) were imaged by means of immunofluorescent staining and optical microscopy. Images of untreated, or control (C), and of treated (T) cells were processed both by fractal and Fourier analysis. The C-MTs had contour fractal dimensions higher (>= 1.4) than those of T-MTs (<=1.3). Fourier analysis included computation of
Fractal images induce fractal pupil dilations and constrictions
Taylor, Richard
1 Fractal images induce fractal pupil dilations and constrictions P. Moon, J. Muday, S. Raynor, J. Schirillo Wake Forest University C. Boydston, M. S. Fairbanks, R.P. Taylor University of Oregon Fractals revealed fractal patterns in many natural and physiological processes. This article investigates pupillary
Fractal Weyl law for quantum fractal eigenstates D. L. Shepelyansky
Shepelyansky, Dima
Fractal Weyl law for quantum fractal eigenstates D. L. Shepelyansky Laboratoire de Physique of such states is described by the fractal Weyl law, and their Husimi distributions closely follow the strange, and the concept of the fractal Weyl law has been introduced to describe the dependence of the number of resonant
NASA Technical Reports Server (NTRS)
Hudson, Richard K.; Anderson, Steven W.; McColley, Shawn; Fink, Jonathan H.
2004-01-01
Fractals are objects that are generally self similar at all scales. Coastlines, mountains, river systems, planetary orbits and some mathematical objects are all examples of fractals. Bruno et al. used the structured walk model of Richardson to establish that lava flows are fractals and that lava flow morphology could be determined by looking at the fractal dimension of flow margins. They determined that Hawaiian a.a flows have fractal dimensions that range from 1.05 to 1.09 and that the pahoehoe lava flows have a fractal dimension from 1.13 to 1.23. We have analyzed a number of natural and simulated lava flow margins and find that the fractal dimension varies according to the number and length of rod lengths used in the structured walk method. The potential variation we find in our analyses is sufficiently large so that unambiguous determination of lava flow morphology is problematic for some flows. We suggest that the structured walk method can provide meaningful fractal dimensions if rod lengths employed in the analysis provide a best-fit residual of greater than 0.98, as opposed to the 0.95 cutoff used in previous studies. We also find that the use of more than 4 rod lengths per analysis also reduces ambiguity in the results.
Fractal Segmentation and Clustering Analysis for Seismic Time Slices
NASA Astrophysics Data System (ADS)
Ronquillo, G.; Oleschko, K.; Korvin, G.; Arizabalo, R. D.
2002-05-01
Fractal analysis has become part of the standard approach for quantifying texture on gray-tone or colored images. In this research we introduce a multi-stage fractal procedure to segment, classify and measure the clustering patterns on seismic time slices from a 3-D seismic survey. Five fractal classifiers (c1)-(c5) were designed to yield standardized, unbiased and precise measures of the clustering of seismic signals. The classifiers were tested on seismic time slices from the AKAL field, Cantarell Oil Complex, Mexico. The generalized lacunarity (c1), fractal signature (c2), heterogeneity (c3), rugosity of boundaries (c4) and continuity resp. tortuosity (c5) of the clusters are shown to be efficient measures of the time-space variability of seismic signals. The Local Fractal Analysis (LFA) of time slices has proved to be a powerful edge detection filter to detect and enhance linear features, like faults or buried meandering rivers. The local fractal dimensions of the time slices were also compared with the self-affinity dimensions of the corresponding parts of porosity-logs. It is speculated that the spectral dimension of the negative-amplitude parts of the time-slice yields a measure of connectivity between the formation's high-porosity zones, and correlates with overall permeability.
Carroll O. Bennett; W. Curtis Conner
1995-01-01
A brief outline of the application of fractals to porous solids is given. The variation of porosity and surface area as functions of particle size and pore size are discussed. For F-20 Alcoa alumina most of the surface is contained in pores of radius near 20 Å. However, fractal behavior is observed if probe molecules are used that are too
Landscape roughness analysis of Mt. Etna volcanic complex detected via fractal geometry
NASA Astrophysics Data System (ADS)
De Luca, Claudio; Bonfante, Antonello; Di Martino, Gerardo; Iodice, Antonio; Manzo, Mariarosaria; Pepe, Antonio; Pepe, Susi; Riccio, Daniele; Sansosti, Eugenio; Tizzani, Pietro; Zinno, Ivana
2013-04-01
During the last years several aspects relevant to volcanic activity have been analyzed in fractal context. These studies have been aimed at identifying the power laws that govern the magma fragmentation processes and/or the classification of different geological processes. In this work we exploit the algorithm proposed by Di Martino et al. (2012) that allows retrieving the fractal dimension of a natural surface starting from its corresponding Synthetic Aperture Radar (SAR) image. Such an algorithm is based on an analytical model that links the stochastic characterization of a single SAR amplitude image to the fractal dimension of the observed surface, modeled via a fractional Brownian motion (fBm) process. The considered SAR image processing provides - as an output product - the pixel by pixel map of the fractal dimension of the scene observed by the sensor. Previous works demonstrated that the fractal dimension of lava flows is strictly connected to the natural surface roughness. Moreover, Pepe et al. (2012) showed the possibility of characterizing the single volcanic structures by means of the fractal dimension values retrieved from the corresponding SAR images. In the present work we consider a data-set of Cosmo-SkyMed high resolution images acquired over the Mt. Etna volcanic complex (South Italy), spanning the 2009 - 2011 time period. Starting from the SAR amplitude images of the considered data-set, we generated the corresponding fractal dimension maps that were subsequently co-registered each other, thus retrieving the fractal dimension time-series of the Mt. Etna volcano. Then, by averaging the so-computed fractal dimension maps with respect to time we generated a map of the mean fractal dimension of the investigated area. This procedure allows significantly improving the quality of the final fractal dimension map, as the average operation reduces the noise (due to the speckle effect on SAR images) present on each fractal map. Besides, the so-obtained mean fractal dimension map was quantized in order to identify and aggregate areas that are homogeneous from a fractal viewpoint, i.e., areas having a similar surface roughness. The comparison with the geological map of Mt. Etna volcanic complex reveals that the level of roughness is strictly connected with the lava flow age. More specifically, the roughness of lava flows, which is measured by the fractal dimension, decreases with the increasing of time. In other words, our analysis demonstrates that the volcanic depositional processes of the lava flows produce surfaces with high fractal dimension (roughness), whereas the exogenous phenomena, as the erosion processes, significantly reduce the fractal dimension associated with eruptive mechanisms. The COSMO-SkyMed and ALOS data used in this study have been processed at REA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0. G. Di Martino, D. Riccio, I. Zinno, "SAR Imaging of Fractal Surfaces," IEEE Trans. Geosci. Remote Sens., vol.50, no.2, pp. 630-644, Feb. 2012. S. Pepe, G. Di Martino, A. Iodice, M. Manzo, A. Pepe, D. Riccio, G. Ruello, E. Sansosti, P. Tizzani, I. Zinno; "Analysis of the fractal dimension of volcano geomorphology through Synthetic Aperture Radar (SAR) amplitude images acquired in C and X band", Geophysical Research Abstracts Vol. 14, EGU2012-6573-1, 2012, EGU General Assembly 2012.
Spatial Pattern of Biological Soil Crust with Fractal Geometry
NASA Astrophysics Data System (ADS)
Ospina, Abelardo; Florentino, Adriana; Tarquis, Ana M.
2015-04-01
Soil surface characteristics are subjected to changes driven by several interactions between water, air, biotic and abiotic components. One of the examples of such interactions is provided through biological soil crusts (BSC) in arid and semi-arid environments. BSC are communities composed of cyanobacteria, fungi, mosses, lichens, algae and liverworts covering the soil surface and play an important role in ecosystem functioning. The characteristics and formation of these BSC influence the soil hydrological balance, control the mass of eroded sediment, increase stability of soil surface, and influence plant productivity through the modification of nitrogen and carbon cycle. This study focus on characterize the spatial arrangements of the BSC based on image analysis and fractal concepts. To this end, RGB images of different types of biological soil crust where taken, each image corresponding to an area of 3.6 cm2 with a resolution of 1024x1024 pixels. For each image and channel, mass dimension and entropy were calculated. Preliminary results indicate that fractal methods are useful to describe changes associated to different types of BSC. Further research is necessary to apply these methodologies to several situations.
Statistical fractal analysis of 25 young star clusters
NASA Astrophysics Data System (ADS)
Gregorio-Hetem, J.; Hetem, A.; Santos-Silva, T.; Fernandes, B.
2015-04-01
A large sample of young stellar groups is analysed to investigate their clustering properties and dynamical evolution. A comparison of the Q statistical parameter, measured for the clusters, with the fractal dimension estimated for the projected clouds, shows that 52 per cent of the sample has substructures and tends to follow the theoretically expected relation between clusters and clouds, according to calculations for the artificial distribution of points. The fractal statistics was also compared to structural parameters, revealing that clusters having a radial density profile show a trend of parameter overline{s} increasing with mean surface stellar density. The core radius of the sample, as a function of age, follows a similar distribution to that observed in stellar groups of the Milky Way and other galaxies. They also have dynamical age, indicated by their crossing time, which is similar to unbound associations. The statistical analysis allowed us to separate the sample into two groups showing different clustering characteristics. However, they have the same dynamical evolution, since the whole sample has been revealed as expanding objects, for which the substructures seem to have not been erased. These results are in agreement with simulations that adopt low surface densities and models under supervirial conditions.
Fractal nature of multiple shear bands in severely deformed metallic glass
Sun, B. A.; Wang, W. H. [Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China)
2011-05-16
We present an analysis of fractal geometry of extensive and complex shear band patterns in a severely deformed metallic glass. We show that the shear band patterns have fractal characteristics, and the fractal dimensions are determined by the stress noise induced by the interaction between shear bands. A theoretical model of the spatial evolution of multiple shear bands is proposed in which the collective shear bands slide is considered as a stochastic process far from thermodynamic equilibrium.
T. Hawa; M. R. Zachariah
2007-01-01
A simple modification to the Frenkel sintering law is developed for nanoparticle fractal aggregates, based on molecular dynamics (MD) simulations. The fractal aggregates investigated consist of up to 110 primary particles, with primary particles of 2.5nm in diameter, and the fractal dimension of 1 (wire), 1.9 (complex), and 3 (compact). In addition simple prototype L- and T-shape aggregate were considered.
Crystallization of space: Space-time fractals from fractal arithmetics
Diederik Aerts; Marek Czachor; Maciej Kuna
2015-06-22
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.
Crystallization of space: Space-time fractals from fractal arithmetics
Aerts, Diederik; Kuna, Maciej
2015-01-01
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.
The internet differential equation and fractal networks
NASA Astrophysics Data System (ADS)
Baker, R. G. V.
2013-02-01
The Internet is an example of a general physical problem dealing with motion near the speed of light relative to different time frames of reference. The second order differential equation (DE) takes the form of 'time diffusion' near the speed of light or alternatively, considered as a complex variable with real time and imaginary longitudinal components. Congestion waves are generated by peak global traffic from different time zones following the Earth's revolution defined by spherical harmonics and a day/night bias. The DE is essentially divided into space and time operators constrained by the speed of light c, band capacity w and a fractal dimension Z (Hausdorff dimension). This paper explores the relationship between the dynamics and the network including the addition of fractal derivatives to the DE for regional networks for 0 < Z < 1.
Michel L. Lapidus; Goran Radunovi?; Darko Žubrini?
2014-11-24
We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.
On the Weierstrass-Mandelbrot Fractal Function
M. V. Berry; Z. V. Lewis
1980-01-01
The function W(t) equiv sum^?n=-? [(1 - eigamma^nt)eiphi_n]\\/gamma(2-D)n (1 < D 1, phi_n = arbitrary phases), is continuous but non-differentiable and possesses no scale. The graph of ReW or ImW has Hausdorff-Besicovitch (fractal) dimension D. Choosing phi_n = mu n gives a deterministic W the scaling properties of which can be studied analytically in terms of a representation obtained by
Fractal surface synthesis based on two dimensional discrete Fourier transform
NASA Astrophysics Data System (ADS)
Zhou, Chao; Gao, Chenghui; Huang, Jianmeng
2013-11-01
The discrete Fourier transform(DFT) is used for fractional Brownian motion(FBM) surface synthesis in tribology(i.e., contact, sliding, and sealing, etc). However, the relationship between fractal parameters(fractal dimension and scale factor) and traditional parameters, the influence of fractal parameters on surface appearance, have not been deeply discussed yet. These lead to some kind of difficulty to ensure the synthesized surfaces with ideal fractal characteristic, required traditional parameters and geometric appearance. A quantitative relationship between fractal parameters and the root mean square deviation of surface ( Sq) is derived based on the energy conservation property between the space and frequency domain of DFT. Under the stability assumption, the power spectrum of a FBM surface is composed of concentric circles strictly, a series of FBM surfaces with prescribed Sq could be synthesized with given fractal dimension, scale factor, and sampling numbers, but the ten-point height( Sz), the skewness( Ssk) and the kurtosis( Sku) are still in random, where the probability distributions of Sz and Ssk are approximately normal distribution. Furthermore, by iterative searching, a surface with desired Abbott-Firestone curve could be obtained among those surfaces. An intuitive explanation for the influence of fractal dimension and scale factor on surface appearance is obtained by discussing the effects on the ratio of energy between high and low frequency components. Based on the relationship between Sq and surface energy, a filtering method of surface with controllable Sq is proposed. The proposed research ensures the synthesized surfaces possess ideal FBM properties with prescribed Sq, offers a method for selecting desired Abbott-Firestone curve of synthesized fractal surfaces, and makes it possible to control the Sq of surfaces after filtering.
The mass of the dominant particle in a fractal universe
Scott Funkhouser; Nicola Pugno
2008-11-06
An empirically validated, phenomenological model relating the parameters of an astronomical body to the stochastic fluctuations of its granular components is generalized in terms of fractal scaling laws. The mass of the particle constituting the preponderance of the mass of a typical galaxy is determined from the generalized model as a function of the fractal dimension. For a fractal dimension between 1 and 3 the mass of the dominant particle in galaxies is, roughly, between the Planck mass and 1eV. If the dimension is near 2 then the fractal model is identical to the original stochastic model, and the mass of the dominant particle must be of order near the nucleon mass. Two additional expressions for the mass of the dominant particle in the universe are obtained from basic quantum considerations and from the existence of a cosmological constant. It follows that the fractal dimension 2 is favored and that the mass of the dominant particle is proportional to sixth root of the cosmological constant and of order near the nucleon mass.
Retinal Vascular Fractals and Cognitive Impairment
Ong, Yi-Ting; Hilal, Saima; Cheung, Carol Yim-lui; Xu, Xin; Chen, Christopher; Venketasubramanian, Narayanaswamy; Wong, Tien Yin; Ikram, Mohammad Kamran
2014-01-01
Background Retinal microvascular network changes have been found in patients with age-related brain diseases such as stroke and dementia including Alzheimer's disease. We examine whether retinal microvascular network changes are also present in preclinical stages of dementia. Methods This is a cross-sectional study of 300 Chinese participants (age: ?60 years) from the ongoing Epidemiology of Dementia in Singapore study who underwent detailed clinical examinations including retinal photography, brain imaging and neuropsychological testing. Retinal vascular parameters were assessed from optic disc-centered photographs using a semiautomated program. A comprehensive neuropsychological battery was administered, and cognitive function was summarized as composite and domain-specific Z-scores. Cognitive impairment no dementia (CIND) and dementia were diagnosed according to standard diagnostic criteria. Results Among 268 eligible nondemented participants, 78 subjects were categorized as CIND-mild and 69 as CIND-moderate. In multivariable adjusted models, reduced retinal arteriolar and venular fractal dimensions were associated with an increased risk of CIND-mild and CIND-moderate. Reduced fractal dimensions were associated with poorer cognitive performance globally and in the specific domains of verbal memory, visuoconstruction and visuomotor speed. Conclusion A sparser retinal microvascular network, represented by reduced arteriolar and venular fractal dimensions, was associated with cognitive impairment, suggesting that early microvascular damage may be present in preclinical stages of dementia. PMID:25298774
Reinforcement of rubber by fractal aggregates
NASA Astrophysics Data System (ADS)
Witten, T. A.; Rubinstein, M.; Colby, R. H.
1993-03-01
Rubber is commonly reinforced with colloidal aggregates of carbon or silica, whose structure has the scale invariance of a fractal object. Reinforced rubbers support large stresses, which often grow faster than linearly with the strain. We argue that under strong elongation the stress arises through lateral compression of the aggregates, driven by the large bulk modulus of the rubber. We derive a power-law relationship between stress and elongation ? when ?gg 1. The predicted power p depends on the fractal dimension D and a second structural scaling exponent C. For diffusion-controlled aggregates this power p should lie beween 0.9 and 1.1 ; for reaction-controlled aggregates p should lie between 1.8 and 2.4. For uniaxial compression the analogous powers lie near 4. Practical rubbers filled with fractal aggregates should approach the conditions of validity for these scaling laws. On renforce souvent le caoutchouc avec des agrégats de carbone ou de silice dont la structure a l'invariance par dilatation d'un objet fractal. Les caoutchoucs ainsi renforcés supportent de grandes contraintes qui croissent souvent plus vite que l'élongation. Nous prétendons que, sous élongation forte, cette contrainte apparaît à cause d'une compression latérale des agrégats induite par le module volumique important du caoutchouc. Nous établissons une loi de puissance reliant la contrainte et l'élongation ? quand ?gg 1. Cet exposant p dépend de la dimension fractale D et d'un deuxième exposant structural C. Pour des agrégats dont la cinétique de formation est limitée par diffusion, p vaut entre 0,9 et 1,1. Si la cinétique est limitée par le soudage local des particules, p vaut entre 1,8 et 2,4. Sous compression uniaxiale, les puissances homologues valent environ 4. Des caoutchoucs pratiques chargés de tels agrégats devraient approcher des conditions où ces lois d'échelle sont valables.
Utilisation de nouveaux paramtres base de fractale pour la discrimination des fontes arabes
Paris-Sud XI, Université de
Utilisation de nouveaux paramètres à base de fractale pour la discrimination des fontes arabes S,benabdelhafid}@univ-lehavre.fr Résumé : La méthode présentée dans cet article permet l'identification des fontes arabes dans des images-clés : AOFR (Reconnaissance Optique de Fontes Arabes), OCR, analyse de texture, dimension fractale, k
Modeling of fractal patterns in matrix acidizing and their impact on well performance
Frick, T.P.; Kuermayr, M.; Economides, M.J.
1994-02-01
This paper describes a model where wormholes, the primary feature of carbonate acidizing, are considered as fractals. The influences of acid volume, injection rate, fractal dimension, porosity, and the ratio of undamaged to damaged permeabilities on well performance are studied. Exact expressions of post-treatment skin effects are developed for vertical and horizontal wells.
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. PMID:26091207
NSDL National Science Digital Library
2007-12-12
Cynthia Lanius, a former mathematics teacher who currently serves as Technology Integration Specialist for Sinton Independent School District in Sinton, Texas, has posted numerous lessons online. This website features a Fractals Unit for elementary and middle school students (although adults are also welcome to enjoy the lesson). The lesson includes a discussion on why one might study fractals and then provides step-by-step explanations on how to make fractals using Java, along with some challenging mathematics questions to consider. Samples of student work are also posted. A section for teachers provides an overview of the unit objectives along with links to other resources and materials to use in the classroom.
Criticality of two- and three-spin Ising model in an external field on a fractal family
NASA Astrophysics Data System (ADS)
Redinz, José Arnaldo; de Magalhães, Aglaé Cristina Navarro
1997-02-01
We study the Ising model with pair and alternate triplet interactions subjected to an external magnetic field on a family of infinitely ramified fractal lattices with a triangular topology. The three-dimensional phase diagram and correlation length critical exponents are calculated within an exact real-space renormalization group framework. The zero-field results for the ferromagnetic model show that, although the pure triplet case and the pure nearest-neighbor pair interaction model are in different universality classes, there is no crossover phenomenon since the system becomes paramagnetic in the mixed case. In the pure nearest-neighbor antiferromagnetic model, the appearance of an unusual Berker and Kadanoff's-phase type (with a power-law decay of correlations) when the fractal dimension is sufficiently high is destroyed by the application of a magnetic field or a triplet interaction field.
Study of Fractal Features of Geomagnetic Activity Through an MHD Shell Model
NASA Astrophysics Data System (ADS)
Dominguez, M.; Nigro, G.; Munoz, V.; Carbone, V.
2013-12-01
Studies on complexity have been of great interest in plasma physics, because they provide new insights and reveal possible universalities on issues such as geomagnetic activity, turbulence in laboratory plasmas, physics of the solar wind, etc. [1, 2]. In particular, various studies have discussed the relationship between the fractal dimension, as a measure of complexity, and physical processes in magnetized plasmas such as the Sun's surface, the solar wind and the Earth's magnetosphere, including the possibility of forecasting geomagnetic activity [3, 4, 5]. Shell models are low dimensional dynamical models describing the main statistical properties of magnetohydrodynamic (MHD) turbulence [6]. These models allow us to describe extreme parameter conditions hence reaching very high Reynolds (Re) numbers. In this work a MHD shell model is used to describe the dissipative events which are taking place in the Earth's magnetosphere and causing geomagnetic storms. The box-counting fractal dimension (D) [7] is calculated for the time series of the magnetic energy dissipation rate obtained in this MHD shell model. We analyze the correlation between D and the energy dissipation rate in order to make a comparison with the same analysis made on the geomagnetic data. We show that, depending on the values of the viscosity and the diffusivity, the fractal dimension and the occurrence of bursts exhibit correlations similar as those observed in geomagnetic and solar data, [8] suggesting that the latter parameters could play a fundamental role in these processes. References [1] R. O. Dendy, S. C. Chapman, and M. Paczuski, Plasma Phys. Controlled Fusion 49, A95 (2007). [2] T. Chang and C. C. Wu, Phys. Rev. E 77, 045401 (2008). [3] R. T. J. McAteer, P. T. Gallagher, and J. Ireland, Astrophys. J. 631, 628 (2005). [4] V. M. Uritsky, A. J. Klimas, and D. Vassiliadis, Adv. Space Res. 37, 539 (2006). [5] S. C. Chapman, B. Hnat, and K. Kiyani, Nonlinear Proc. Geophys. 15, 445 (2008). [6] G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and A. Vulpiani, Phys. Rev. Lett. 83, 4662 (1999). [7] P. S. Addison, Fractals and Chaos, an Illustrated Course, vol. 1 (Institute of Physics Publishing, Bristol and Philadelphia, 1997), second ed. [8] M. Domínguez, V. Muñoz, and J. A. Valdivia, Temporal evolution of fractality in the Earth's magnetosphere and the solar photosphere, in preparation.
A fractal transition in the two dimensional shear layer
NASA Technical Reports Server (NTRS)
Jimenez, Javier; Martel, Carlos
1990-01-01
The dependence of product generation with the Peclet and Reynolds number in a numerically simulated, reacting, two dimensional, temporally growing mixing layer is used to compute the fractal dimension of passive scalar interfaces. A transition from a low dimension of 4/3 to a higher one of 5/3 is identified and shown to be associated to the kinematic distortion on the flow field during the first pairing interaction. It is suggested that the structures responsible for this transition are non-deterministic, non-random, inhomogeneous fractals. Only the large scales are involved. No further transition is found for Reynolds numbers up to 20,000.
Fractal Spacetime Structure in Asymptotically Safe Gravity
O. Lauscher; M. Reuter
2005-08-26
Four-dimensional Quantum Einstein Gravity (QEG) is likely to be an asymptotically safe theory which is applicable at arbitrarily small distance scales. On sub-Planckian distances it predicts that spacetime is a fractal with an effective dimensionality of 2. The original argument leading to this result was based upon the anomalous dimension of Newton's constant. In the present paper we demonstrate that also the spectral dimension equals 2 microscopically, while it is equal to 4 on macroscopic scales. This result is an exact consequence of asymptotic safety and does not rely on any truncation. Contact is made with recent Monte Carlo simulations.
Fractal Spacetime Structure in Asymptotically Safe Gravity
Lauscher, O
2005-01-01
Four-dimensional Quantum Einstein Gravity (QEG) is likely to be an asymptotically safe theory which is applicable at arbitrarily small distance scales. On sub-Planckian distances it predicts that spacetime is a fractal with an effective dimensionality of 2. The original argument leading to this result was based upon the anomalous dimension of Newton's constant. In the present paper we demonstrate that also the spectral dimension equals 2 microscopically, while it is equal to 4 on macroscopic scales. This result is an exact consequence of asymptotic safety and does not rely on any truncation. Contact is made with recent Monte Carlo simulations.
Size effects of exchange cation on the pore structure and surface fractality of montmorillonite
Lee, J.F.; Lee, C.K.; Juang, L.C.
1999-09-01
Ca-montmorillonites were exchanged with both metal cations (manganese and copper) and organic cations (tetramethylammonium (TMA) and hexadecyltrimmethylammonium (HDTMA)) to study the cation size effects on the pore structure and surface roughness of montmorillonite based on the classical and fractal analysis of their nitrogen adsorption isotherms. The surface fractal dimension D was calculated from their nitrogen isotherms with the aid of the recently proposed Neimark equation. The decrease of BET surface area of montmorillonite induced by the larger size of exchange cation was interpreted with both the coverture of some surface roughness (surface screening effect) and the inhibition of nitrogen molecule into some pores (pore blocking effect). The pore blocking effect was examined with the changes of mean pore size. Meanwhile, the D value was used to examine whether or not the surface screening effect existed. It was concluded that the combination of classical and fractal analyses of nitrogen isotherms may facilitate understanding of the evolution of pore and surface structures of clay exchanged with different cations.
NASA Astrophysics Data System (ADS)
Mattingly, Margarita Claudia Krieghoff
The space-time development of hadron-nucleus interactions is examined using bubble chamber and downstream particle identifier data from the hybrid spectrometer of Fermilab experiment E597. 5583 events representing 12 interactions are studied with conventional and fractal techniques. Comparisons are made to simulated events from the Lund Monte Carlo FRITIOF 1.6. Multiplicities are studied conventionally. Negative binomial descriptions of produced particle multiplicities are interpreted in terms of clusters and cascading and in terms of partial stimulated emission; forward-backward correlations, in terms of short- and long-range correlations and multiple scattering. Multiplicities are consistent with a multiple collision view of multiparticle production mechanisms and are investigated in terms of the number of collisions nu. Rapidity density fluctuations are studied fractally. The possibility of new dynamics is considered on the basis of event-by-event studies of spike phenomena, intermittency, and fractal dimensions. Results from these exploratory studies are consistent with predictions made for quark-gluon plasma transitions. 131 spike events are analyzed; intermittency is investigated with normalized factorial moments and cumulants; and fractal dimensions and correlations dimensions are calculated. Seagull effects and production region sizes from Bose-Einstein pion interferometry are also considered.
Quantum critical behavior of the quantum Ising model on fractal lattices.
Yi, Hangmo
2015-01-01
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpi?ski carpet, Sierpi?ski gasket, and Sierpi?ski tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpi?ski tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry. PMID:25679581
Quantum critical behavior of the quantum Ising model on fractal lattices
NASA Astrophysics Data System (ADS)
Yi, Hangmo
2015-01-01
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpi?ski carpet, Sierpi?ski gasket, and Sierpi?ski tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpi?ski tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
ERIC Educational Resources Information Center
Clark, Garry
1999-01-01
Reports on a mathematical investigation of fractals and highlights the thinking involved, problem solving strategies used, generalizing skills required, the role of technology, and the role of mathematics. (ASK)
NSDL National Science Digital Library
National Council of Teachers of Mathematics (NCTM)
2002-01-01
This online activity challenges students to explore the relationship between the number of triangles and the sum of the triangle perimeters in each of the first three iterations of the Sierpinski triangle fractal. The activity is one of 80 mathematical challenges featured on the Figure This! web site. In this activity, students are encouraged to use two problem-solving strategies: investigate a simpler problem and make a chart. For other sections of the activity, students find the general rule for determining the amount of paint needed to cover the increasing number of triangles in iterations of the Sierpinski triangle and investigate similar area and perimeter questions with square fractals. The activity includes information about self-similarity, a key characteristic of fractals, and about how fractals can model natural phenomena. Copyright 2005 Eisenhower National Clearinghouse
Quantifying left ventricular trabeculae function – application of image-based fractal analysis
Moore, Brandon; Prasad Dasi, Lakshmi
2013-01-01
The ventricular-blood interface is geometrically complex due to the presence of ventricular trabeculae carneae (VTC). We introduce a new image-based framework to quantify VTC function using high-resolution computed tomography (CT) imaging and offer new insights into the active role of VTCs during ejection. High-resolution Cine CT scans of a patient with normal cardiac function were acquired at a resolution of 0.77 mm per pixel at 10 phases of the cardiac cycle. The images were segmented and the VTC surface was obtained by triangulating the segmented data. Fractal dimension of the VTC surface was calculated for each cardiac phase as a function of scale size using the box-counting algorithm. The fractal dimension, D corresponding to VTCs ranged between 2.05 and 2.2 and varied as a function of time during the cardiac cycle. Fractal dimension is highest at diastole and lowest at peak systole with the change being significantly different (P < 0.003). This variation of D when plotted against stroke volume (i.e., D-V loop) revealed an active VTC role due to hysteresis in the loop. Physically the hysteresis in the D-V loop indicates a new mechanical function of VTCs as structures that provide mechanical leverage during early systolic ejection through contraction. VTC relaxation is noted to occur during late diastole at larger ventricular volume. D-V loop of VTCs quantifies VTC function. A new dynamic physical role of VTCs is suggested by way of mechanical leverage, as opposed to the traditionally accepted passive role. PMID:24303149
Extended Fractal Fits to Riemann Zeros
Paul B. Slater
2007-05-21
We extend to the first 300 Riemann zeros, the form of analysis reported by us in arXiv:math-ph/0606005, in which the largest study had involved the first 75 zeros. Again, we model the nonsmooth fluctuating part of the Wu-Sprung potential, which reproduces the Riemann zeros, by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting the fractal dimension equal to 3/2. we estimate the frequency parameter (g), plus an overall scaling parameter (s) introduced. We search for that pair of parameters (g,s) which minimizes the least-squares fit of the lowest 300 eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the first 300 Riemann zeros. We randomly sample values within the rectangle 0 fractal supplementation. Some limited improvement is again found. There are two (primary and secondary) quite distinct subdomains, in which the values giving improvements in fit are concentrated.
NSDL National Science Digital Library
2011-01-20
This lesson is designed to introduce students to the idea of finding patterns in the generation of several different types of fractals. This lesson provides links to discussions and activities related to patterns and fractals as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.
Fractal Themes at Every Level Kenneth G. Monks
Monks, Kenneth
Fractal Themes at Every Level Kenneth G. Monks University of Scranton August 19, 1998 OK I admit it. I love fractals. Fractal programs, fractal tee-shirts, fractal notebooks, fractal screen savers... What other
Fractal physiology and the fractional calculus: a perspective.
West, Bruce J
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks. PMID:21423355
Fractal Relativity, Generalized Noether Theorem and New Research of Space-Time
Yi-Fang Chang
2007-07-02
First, let the fractal dimension D=n(integer)+d(decimal), so the fractal dimensional matrix was represented by a usual matrix adds a special decimal row (column). We researched that mathematics, for example, the fractal dimensional linear algebra, and physics may be developed to fractal and the complex dimension extended from fractal. From this the fractal relativity is discussed, which connects with self-similarity Universe and the extensive quantum theory. The space dimension has been extended from real number to superreal and complex number. Combining the quaternion, etc., the high dimensional time is introduced. Such the vector and irreversibility of time are derived. Then the fractal dimensional time is obtained, and space and time possess completely symmetry. It may be constructed preliminarily that the higher dimensional, fractal, complex and supercomplex space-time theory covers all. We propose a generalized Noether theorem, and irreversibility of time should correspond to non-conservation of a certain quantity. Resumed reversibility of time and possible decrease of entropy are discussed. Finally, we obtain the quantitative relations between energy-mass and space-time, which is consistent with the space-time uncertainty relation in string theory.
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.
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...
Surface structures of equilibrium restricted curvature model on two fractal substrates
NASA Astrophysics Data System (ADS)
Song, Li-Jian; Tang, Gang; Zhang, Yong-Wei; Han, Kui; Xun, Zhi-Peng; Xia, Hui; Hao, Da-Peng; Li, Yan
2014-01-01
With the aim to probe the effects of the microscopic details of fractal substrates on the scaling of discrete growth models, the surface structures of the equilibrium restricted curvature (ERC) model on Sierpinski arrowhead and crab substrates are analyzed by means of Monte Carlo simulations. These two fractal substrates have the same fractal dimension df, but possess different dynamic exponents of random walk zrw. The results show that the surface structure of the ERC model on fractal substrates are related to not only the fractal dimension df, but also to the microscopic structures of the substrates expressed by the dynamic exponent of random walk zrw. The ERC model growing on the two substrates follows the well-known Family—Vicsek scaling law and satisfies the scaling relations 2? + df asymp z asymp 2zrw. In addition, the values of the scaling exponents are in good agreement with the analytical prediction of the fractional Mullins—Herring equation.
Fractal signatures in analogs of interplanetary dust particles
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Banerjee, Varsha; Puri, Sanjay
2014-10-01
Interplanetary dust particles (IDPs) are an important constituent of the earths stratosphere, interstellar and interplanetary medium, cometary comae and tails, etc. Their physical and optical characteristics are significantly influenced by the morphology of silicate aggregates which form the core in IDPs. In this paper we reinterpret scattering data from laboratory analogs of cosmic silicate aggregates created by Volten et al. (2007) [1] to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension dm?1.75. The same fractal dimension also characterizes clusters obtained from diffusion limited aggregation (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically transported silicate particles.
Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Zhang, Xicheng
2012-04-01
In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330-345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in {{mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts are proved.
The fractal structure of the ventral scales in legless reptiles
Abdel-Aal, Hisham A
2015-01-01
Surface constructs in snakes reflect desirable design traits for technical surface engineering. Their micro-textural patterns, however, do not lend themselves easily to unified analysis due to species-specific variations in surface geometry and topology. Fractal description is useful in this context since it accentuates the correspondence between patterns especially when responding to tribological phenomena. In this work we examine the surface construction of 14 snake species, representing five families, and evaluate the fractal dimension for each of the skins (both the dorsal and ventral sides) using three different computational algorithms. Our results indicate first that all of the examined species share a common fractal dimension (with a very small variation between species in the order 4-5%). This finding implies that despite the different micro-geometry of texture among species, the skin as a unit responds in a similar manner to many interfacial influences.
Fractal Patterns and Chaos Games
ERIC Educational Resources Information Center
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.
Fractal oscillations of self-adjoint and damped linear differential equations of second-order
Mervan Paši?; Satoshi Tanaka
2011-01-01
For a prescribed real number s?[1,2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y=y(x), y?C2((0,T]) of the linear differential equation (p(x)y?)?+q(x)y=0 on (0,T], is bounded and fractal oscillatory near x=0 with the fractal dimension equal to s. This means that y oscillates near x=0 and the fractal (box-counting) dimension of the graph
Santacruz-Vázquez, Claudia; Santacruz-Vázquez, Verónica
2015-02-01
Changes in the concentration profiles of ?-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 ?-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 have an important effect on the impregnation phenomenon, generating irregular concentration profiles of ?-carotene, which are numerically described by the fractal dimension FDPROFILES and are related to the diffusion process during impregnation in dried edible tissue. PMID:25694678
Fractal-wavelet image denoising
Mohsen Ghazel; Edward R. Vrscay; George H. Freeman
2002-01-01
In this paper, we propose a simple yet effective fractal-wavelet scheme for edge-preserving smoothing of noisy images. Over the past decade, there has been significant interest in fractal coding for the purpose of image compression. Fractal-wavelet transforms were introduced in an effort to reduce the blockiness and com- putational complexity that are inherent in fractal image compres- sion. Applications of
Monte Carlo Sampling in Fractal Landscapes
NASA Astrophysics Data System (ADS)
Leitão, Jorge C.; Lopes, J. M. Viana Parente; Altmann, Eduardo G.
2013-05-01
We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.
NASA Technical Reports Server (NTRS)
Lam, Nina Siu-Ngan; Qiu, Hong-Lie; Quattrochi, Dale A.; Emerson, Charles W.; Arnold, James E. (Technical Monitor)
2001-01-01
The rapid increase in digital data volumes from new and existing sensors necessitates the need for efficient analytical tools for extracting information. We developed an integrated software package called ICAMS (Image Characterization and Modeling System) to provide specialized spatial analytical functions for interpreting remote sensing data. This paper evaluates the three fractal dimension measurement methods: isarithm, variogram, and triangular prism, along with the spatial autocorrelation measurement methods Moran's I and Geary's C, that have been implemented in ICAMS. A modified triangular prism method was proposed and implemented. Results from analyzing 25 simulated surfaces having known fractal dimensions show that both the isarithm and triangular prism methods can accurately measure a range of fractal surfaces. The triangular prism method is most accurate at estimating the fractal dimension of higher spatial complexity, but it is sensitive to contrast stretching. The variogram method is a comparatively poor estimator for all of the surfaces, particularly those with higher fractal dimensions. Similar to the fractal techniques, the spatial autocorrelation techniques are found to be useful to measure complex images but not images with low dimensionality. These fractal measurement methods can be applied directly to unclassified images and could serve as a tool for change detection and data mining.
Fractal Superconductivity near Localization Threshold
Fominov, Yakov
Fractal Superconductivity near Localization Threshold Mikhail Feigel'man Landau Institute, Moscow-electron states are extended but fractal and populate small fraction of the whole volume How BCS theory should be modified to account for eigenstates fractality ? #12;Mean-Field Eq. for Tc #12;#12;3D Anderson model: = 0
Towards a new potential field theory of fractal objects
Mostafa E. Mostafa
2010-01-01
The Potential Field Anomaly (PFA) data of the self similar Fractal Objects (FOs) include gravity and magnetic fields and potentials along with the related derivatives. These elements are calculated on\\u000a grids due to buried FOs at different fractal orders. The objects have variable physical property distributions; while in magnetic, the orientation\\u000a and magnitude of polarization or earth magnetic field is
Comparison between spectral and fractal EEG analyses of sleeping newborns
A. P. Accardo; M. Affinito; M. Carrozzi; S. Cisint; F. Bouquet
1998-01-01
Spectral parameters (power spectrum in the delta, theta, alpha, beta1 and beta2 bands) and the fractal dimension were estimated for each two seconds frame of the EEG sleep time series at the awake condition or during one of the four EEG sleep states: active sleep (two stages: mixed and low voltage irregular) and quiet sleep (two stages: quiet sleep high
Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model
Hastings, M. B.
2001-10-22
A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent renormalization group prediction of an upper critical {eta}{sub c}=4 , at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.
Fractal Generalized Zone Plates
Mendoza-Yero, Omel; Minguez-Vega, Gladys; Lancis, Jesus; Climent, Vicent; Monsoriu, Juan A
2008-01-01
The construction of fractal generalized zone plates (FraGZPs) from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us to increase the number of foci of a conventional fractal zone plate (FraZP), keeping the self-similarity property within the axial irradiance. The focusing properties of these fractal diffractive optical elements for points not only along but also in the close vicinity of the optical axis are investigated. In both cases analytical expressions for the irradiance are derived. Numerical simulations of the energetic efficiency of FraGZPs under plane wave illumination are carried out. In addition, some effects on the axial irradiance caused by the variation in area of their transparent rings are shown.
QSPR prediction of chromatographic retention times of pesticides: partition and fractal indices.
Torrens, Francisco; Castellano, Gloria
2014-06-01
The high-performance liquid-chromatographic retentions of red-wine pesticide residues are modeled by structure-property relationships. The effect of different types of features is analyzed: geometric, lipophilic, etc. The properties are fractal dimensions, partition coefficient, etc., in linear and nonlinear correlation models. Biological plastic evolution is an evolutionary perspective conjugating the effect of acquired characters and relations that emerge among the principles of evolutionary indeterminacy, morphological determination and natural selection. It is applied to design the co-ordination index that is used to characterize pesticide retentions. The parameters used to calculate the co-ordination index are the molar formation enthalpy, molecular weight and surface area. The morphological and co-ordination indices barely improve the correlations. The fractal dimension averaged for non?buried atoms, partition coefficient, etc. distinguishes the pesticide molecular structures. The structural and constituent classification is based on nonplanarity, and the number of cycles, and O, S, N and Cl atoms. Different behavior depends on the number of cycles. PMID:24762177
Fractal-like exciton kinetics in porous glasses, organic membranes, and filter papers
NASA Astrophysics Data System (ADS)
Kopelman, Raoul; Parus, Stephen; Prasad, Jagdish
1986-04-01
We have measured the exciton (triplet) recombination (fusion, annihilation) characteristics of naphthalene-doped microporous materials. This technique yields the dynamic (spectral, fracton) dimension of the embedded naphthalene structure or the effective random-walk dimension of the porous network. Temperature studies separate the energetic and geometric features of the pore space. The geometric dynamic (spectral) dimensions are mostly between 1 and 2, i.e., fractal-like, and are consistent with previous results on the static (fractal or Euclidean) dimensions of the porous Vycor glass samples.
Fractal analysis of vascular networks: insights from morphogenesis.
Lorthois, Sylvie; Cassot, Francis
2010-02-21
Considering their extremely complicated and hierarchical structure, a long standing question in vascular physio-pathology is how to characterize blood vessels patterns, including which parameters to use. Another question is how to define a pertinent taxonomy, with applications to normal development and to diagnosis and/or staging of diseases. To address these issues, fractal analysis has been applied by previous investigators to a large variety of healthy or pathologic vascular networks whose fractal dimensions have been sought. A review of the results obtained on healthy vascular networks first shows that no consensus has emerged about whether normal networks must be considered as fractals or not. Based on a review of previous theoretical work on vascular morphogenesis, we argue that these divergences are the signature of a two-step morphogenesis process, where vascular networks form via progressive penetration of arterial and venous quasi-fractal arborescences into a pre-existing homogeneous capillary mesh. Adopting this perspective, we study the multi-scale behavior of generic patterns (model structures constructed as the superposition of homogeneous meshes and quasi-fractal trees) and of healthy intracortical networks in order to determine the artifactual and true components of their multi-scale behavior. We demonstrate that, at least in the brain, healthy vascular structures are a superposition of two components: at low scale, a mesh-like capillary component which becomes homogeneous and space-filling over a cut-off length of order of its characteristic length; at larger scale, quasi-fractal branched (tree-like) structures. Such complex structures are consistent with all previous studies on the multi-scale behavior of vascular structures at different scales, resolving the apparent contradiction about their fractal nature. Consequences regarding the way fractal analysis of vascular networks should be conducted to provide meaningful results are presented. Finally, consequences for vascular morphogenesis or hemodynamics are discussed, as well as implications in case of pathological conditions, such as cancer. PMID:19913557
La Russa, Daniel J.; Rogers, D. W. O. [Carleton Laboratory for Radiotherapy Physics, Ottawa Carleton Institute of Physics, Carleton University Campus, Ottawa, Ontario K1S 5B6 (Canada)
2008-12-15
In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at {sup 60}Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at {sup 60}Co energies within a accuracy of a few percent or less.
La Russa, Daniel J; Rogers, D W O
2008-12-01
In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at 60Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at 60Co energies within a accuracy of a few percent or less. PMID:19175120
Fractal Based Image Steganography
Paul Davern; Michael Scott
1996-01-01
This paper describes a new and novel steganographic method for inserting secret information into image files. The method uses fractal image compression techniques in the production of these steganographic image files. The method allows a user to specify a visual key when hiding the secret information. The visual key must then be used when retrieving the hidden data. The paper
Sheqi Zhang; Bingzhi Li; Yunpeng Liu; Linsen Zhang; Ziqing Wang; Mingyu Han
2011-01-01
The structural diagrams of apple trees are the comprehensive reflection of the effects of their training and pruning as well as their physiological and ecological characteristics and yield. However, there have been few research reports on the characteristics of the structural diagrams of apple trees. The study investigated the fractal dimension numbers and fractal characteristics of the two-dimensional images of
Ht-Index for Quantifying the Fractal or Scaling Structure of Geographic Features
Bin Jiang; Junjun Yin
2013-06-22
Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is false, is mainly attributed to the strict definition of fractal dimension as a measure or index for characterizing the complexity of fractals. In this paper, we propose an alternative, the ht-index, to quantify the fractal or scaling structure of geographic features. A geographic feature has ht-index h if the pattern of far more small things than large ones recurs (h-1) times at different scales. The higher the ht-index, the more complex the geographic feature. We conduct three case studies to illustrate how the computed ht-indices capture the complexity of different geographic features. We further discuss how the ht-index is complementary to fractal dimension, and elaborate on a dynamic view behind the ht-index that enables better understanding of geographic forms and processes. Keywords: Scaling of geographic space, fractal dimension, Richardson plot, nested rank-size plots, and head/tail breaks
Fractal analysis of AFM images of the surface of Bowman's membrane of the human cornea.
??lu, ?tefan; Stach, Sebastian; Sueiras, Vivian; Ziebarth, Noël Marysa
2015-04-01
The objective of this study is to further investigate the ultrastructural details of the surface of Bowman's membrane of the human cornea, using atomic force microscopy (AFM) images. One representative image acquired of Bowman's membrane of a human cornea was investigated. The three-dimensional (3-D) surface of the sample was imaged using AFM in contact mode, while the sample was completely submerged in optisol solution. Height and deflection images were acquired at multiple scan lengths using the MFP-3D AFM system software (Asylum Research, Santa Barbara, CA), based in IGOR Pro (WaveMetrics, Lake Oswego, OR). A novel approach, based on computational algorithms for fractal analysis of surfaces applied for AFM data, was utilized to analyze the surface structure. The surfaces revealed a fractal structure at the nanometer scale. The fractal dimension, D, provided quantitative values that characterize the scale properties of surface geometry. Detailed characterization of the surface topography was obtained using statistical parameters, in accordance with ISO 25178-2: 2012. Results obtained by fractal analysis confirm the relationship between the value of the fractal dimension and the statistical surface roughness parameters. The surface structure of Bowman's membrane of the human cornea is complex. The analyzed AFM images confirm a fractal nature of the surface, which is not taken into account by classical surface statistical parameters. Surface fractal dimension could be useful in ophthalmology to quantify corneal architectural changes associated with different disease states to further our understanding of disease evolution. PMID:25266935
Lapelosa, Mauro; Abrams, Cameron F.
2013-01-01
Rare events between states in complex systems are fundamental in many scientific fields and can be studied by building reaction pathways. A theoretical framework to analyze reaction pathways is provided by transition-path theory (TPT). The central object in TPT is the committor function, which is found by solution of the backward-Kolmogorov equation on a given potential. Once determined, the committor can be used to calculate reactive fluxes and rates, among other important quantities. We demonstrate here that the committor can be calculated using the method of finite elements on non-uniform meshes. We show that this approach makes it feasible to perform TPT calculations on 3D potentials because it requires many fewer degrees of freedom than a regular-mesh finite-difference approach. In various illustrative 2D and 3D problems, we calculate the committor function and reaction rates at different temperatures, and we discuss effects of temperatures and simple entropic barriers on the structure of the committor and the reaction rate constants. PMID:24014889
Zhonghe Ye; Wei Zhang; Qinghai Huang; Chuanming Chen
2006-01-01
A new and simple method is used to derive the formulae for the tooth profile of the small rotor in a Cycloid rotor pump. Undercutting is avoided by calculating the maximum radius of curvature of the small rotor tooth profile on the convex section. It is found that undercutting can be avoided so long as the maximum value is not
The Sound of Fractal Strings and the Riemann Hypothesis
Michel L. Lapidus
2015-05-07
We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments, including the theory of (fractal) complex dimensions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his collaborators).
Changala, P. Bryan
Reduced dimension variational calculations have been performed for the rovibrational level structure of the S[subscript 1] state of acetylene. The state exhibits an unusually complicated level structure, for various reasons. ...
Fractional diffusion on a fractal grid comb.
Sandev, Trifce; Iomin, Alexander; Kantz, Holger
2015-03-01
A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N=1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1/2. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure. PMID:25871055
Fractal chemical kinetics: Simulations and experiments
NASA Astrophysics Data System (ADS)
Anacker, L. W.; Kopelman, R.
1984-12-01
We relate the effective order X of the elementary binary reaction, A+A? products, on a fractal, to the spectral dimension. For ds<2: X=1+2/ds. We regain the calsssical value (X=2) for the cubic lattice but get X=3 for the linear lattice. Supercomputer simulations (transient and steady state) and exciton fusion experiments are in quantitative agreement with the above. For percolating clusters X=2.5 and for the Sierpinski gasket X=2.45. Relevance to heterogeneous kinetics is discussed.
Fractional diffusion on a fractal grid comb
NASA Astrophysics Data System (ADS)
Sandev, Trifce; Iomin, Alexander; Kantz, Holger
2015-03-01
A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N =1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1 /2 . We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.
Reflectivity of planar metallic fractal patterns
NASA Astrophysics Data System (ADS)
Zhou, Lei; Wen, Weijia; Chan, C. T.; Sheng, Ping
2003-02-01
We studied the reflective properties of a small dielectric plate covered with a fractal-like metallic pattern generated by a particular type of space-filling curves. We found, both experimentally and theoretically, that the plate can reflect electromagnetic waves in a multitude of frequencies, generated from a near-field monopole antenna. Some of the reflected waves have wavelengths much larger than the lateral dimension of the plate. In comparison, a metal plate of the same size failed to reflect when its lateral size was smaller than half of the corresponding wavelength.
Error Assessment in Modeling with Fractal Brownian Motions
NASA Astrophysics Data System (ADS)
Qiao, Bingqiang; Liu, Siming
2013-12-01
To model a given time series F(t) with fractal Brownian motions (fBms), it is necessary to have appropriate error assessment for related quantities. Usually the fractal dimension D is derived from the Hurst exponent H via the relation D = 2-H, and the Hurst exponent can be evaluated by analyzing the dependence of the rescaled range <|F(t + ?) - F(t)|> on the time span ?. For fBms, the error of the rescaled range not only depends on data sampling but also varies with H due to the presence of long term memory. This error for a given time series then can not be assessed without knowing the fractal dimension. We carry out extensive numerical simulations to explore the error of rescaled range of fBms and find that for 0 < H < 0.5, |F(t + ?) - F(t)| can be treated as independent for time spans without overlap; for 0.5 < H < 1, the long term memory makes |F(t + ?) - F(t)| correlated and an approximate method is given to evaluate the error of <|F(t + ?) - F(t)|>. The error and fractal dimension can then be determined self-consistently in the modeling of a time series with fBms.
Manic, Vesna; Manic, Goran; Nikezic, Dragoslav; Krstic, Dragana
2012-12-01
The absorbed gamma dose rate in indoor air due to natural radionuclides in concrete as a building material was determined in this work. The dose rate conversion factors for (238)U, (232)Th and (40)K, for standard rooms as well as rooms with different sets of dimensions, were evaluated by the point kernel technique, using Harima (geometric progression) build-up factors. The values of the conversion factors, in units (nGy h(-1) (Bq kg(-1))(-1)) calculated for the standard room are: 0.76, 0.91 and 0.070, respectively for (238)U, (232)Th and (40)K. The fitting formula was obtained for dose rate conversion factors, enabling them to be conveniently calculated for a room with arbitrary dimensions. For concrete block samples collected in the area of Niš, Serbia, the measurement of the radionuclide activity concentrations was also carried out. The evaluated absorbed dose rate conversion factors were then applied in the assessment of corresponding indoor gamma dose rates, finding that all the concrete samples fulfilled the usage requirement. PMID:22522178
NASA Astrophysics Data System (ADS)
Huang, Jun; Zhang, Jianhui; Wang, Shouyin; Liu, Weidong
2014-05-01
Microchannel heat sink with high heat transfer coefficients has been extensively investigated due to its wide application prospective in electronic cooling. However, this cooling system requires a separate pump to drive the fluid transfer, which is uneasy to minimize and reduces their reliability and applicability of the whole system. In order to avoid these problems, valveless piezoelectric pump with fractal-like Y-shape branching tubes is proposed. Fractal-like Y-shape branching tube used in microchannel heat sinks is exploited as no-moving-part valve of the valveless piezoelectric pump. In order to obtain flow characteristics of the pump, the relationship between tube structure and flow rate of the pump is studied. Specifically, the flow resistances of fractal-like Y-shape branching tubes and flow rate of the pump are analyzed by using fractal theory. Then, finite element software is employed to simulate the flow field of the tube, and the relationships between pressure drop and flow rate along merging and dividing flows are obtained. Finally, valveless piezoelectric pumps with fractal-like Y-shape branching tubes with different fractal dimensions of diameter distribution are fabricated, and flow rate experiment is conducted. The experimental results show that the flow rate of the pump increases with the rise of fractal dimension of the tube diameter. When fractal dimension is 3, the maximum flow rate of the valveless pump is 29.16 mL/min under 100 V peak to peak (13 Hz) power supply, which reveals the relationship between flow rate and fractal dimensions of tube diameter distribution. This paper investigates the flow characteristics of valveless piezoelectric pump with fractal-like Y-shape branching tubes, which provides certain references for valveless piezoelectric pump with fractal-like Y-shape branching tubes in application on electronic chip cooling.
The effects of fractality on hydrogen permeability across meso-porous membrane
NASA Astrophysics Data System (ADS)
Helwani, Z.; Wiheeb, A. D.; Shamsudin, I. K.; Kim, J.; Othman, M. R.
2015-06-01
A fractal theory employing a box-counting method was used to describe hydrogen gas diffusion into membrane pores in the meso-porosity regime. The diffusion of the gas into the membrane pore network confirmed the existence of fractal structure in the system. Two fractal identities to represent irregularity and roughness of pore surface and tortuosity of the membrane were obtained and analyzed. Their influences on hydrogen permeability were also evaluated. The fractal permeability model that reflects different hydrogen diffusion mechanisms was calculated and compared with that of the state of the art Kozeny-Carman equation.
Multi-scale dynamic rupture simulation on fractal patch model
NASA Astrophysics Data System (ADS)
Ide, S.; Aochi, H.
2004-12-01
We carried out multi-scale full-dynamic rupture simulations, using our new calculation scheme (Aochi and Ide, GRL, 2004) and a fractal patch model as an approximation of realistic heterogeneity. A basic assumption of this model is that a local slip weakening distance (or fracture energy) at a point is proportional to the size of the minimum asperity which includes that point. Since typical topography of fault surface obeys self-affine fractal statistics, we assumed that the asperity distribution is also represented by a power law. For simplicity we prepared seven different sizes of circular patches as discretized representation of asperities. When the patch radius increases by two, the number of patches decreases by four, where the fractal dimension is 2. The whole model space is a fault plane of 4096x4096 square grids, on which the circular patches are distributed randomly. This space is represented by four 64x64 subspaces on different scales and each subspace is connected to the subspaces on the larger and/or smaller scales by renormalization. The assumed values of initial, yield, and residual stresses are homogeneous across the fault plane. We begin each dynamic rupture simulation with breaking one of the patches of the minimum level. In most cases, the rupture stops immediately after the initiation. Sometimes, the rupture coalesces with adjacent patches, propagates into a patch of next level. Frequency-size distribution of these events is approximated by a power law, which is explained by the probability of interaction between asperities. The probability of triggering of dense patch distribution is high and resultant slope of the power law is less steep. Whole rupture process is spontaneous based on exact elasto-dynamics and slip-weakening law except for the nucleation in the minimum level. Thus we observed very heterogeneous process during the rupture: Rupture directivity, rupture front shape, slip distribution, and moment release functions. Some moment rate functions increase irregularly, which resemble to so-called initial phases observed in real seismic waves. We cannot distinguish small and large events from the initial rise of moment rate functions.
Fractal topology of gene promoter networks at phase transitions
Aldrich, Preston R.; Horsley, Robert K.; Ahmed, Yousuf A.; Williamson, Joseph J.; Turcic, Stefan M.
2010-01-01
Much is known regarding the structure and logic of genetic regulatory networks. Less understood is the contextual organization of promoter signals used during transcription initiation, the most pivotal stage during gene expression. Here we show that promoter networks organize spontaneously at a dimension between the 1-dimension of the DNA and 3-dimension of the cell. Network methods were used to visualize the global structure of E. coli sigma (?) recognition footprints using published promoter sequences (RegulonDB). Footprints were rendered as networks with weighted edges representing bp-sharing between promoters (nodes). Serial thresholding revealed phase transitions at positions predicted by percolation theory, and nuclei denoting short steps through promoter space with geometrically constrained linkages. The network nuclei are fractals, a power-law organization not yet described for promoters. Genome-wide promoter abundance also scaled as a power-law. We propose a general model for the development of a fractal nucleus in a transcriptional grammar. PMID:20703327
Fractal characteristic of laser zone remelted Al 2O 3/YAG eutectic in situ composite
NASA Astrophysics Data System (ADS)
Zhang, Jun; Su, Haijun; Tang, Bo; Liu, Lin; Fu, Hengzhi
2008-01-01
Directionally solidified alumina-based eutectic in situ composite has complex, irregular and self-similar microstructure composed of two or more phases to form three dimensionally interpenetrating network as "Chinese Script" type structure. In this paper, the fractal analysis is applied to Al 2O 3/Y 3Al 5O 12 (YAG) eutectic in situ composite prepared by laser zone remelting in order to quantitatively characterize the microstructure morphology and mechanical property. The fractal dimension of the eutectic microstructure is determined by the box counting technique according to the scanning electron microscope micrograph. The relationships between the fractal dimension and the laser processing parameter, eutectic spacing, and mechanical properties are studied. The results indicate that the eutectic morphology of Al 2O 3/YAG in situ composite has typical fractal characteristic. The fractal dimension changes from 1.61 to 1.82 when the laser-scanning rate increases from 10 to 400 ?m/s. With the increase of the fractal dimension, the eutectic spacing decreases while the hardness and the fracture toughness all increase.
Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel
Shanhua, Xu; Songbo, Ren; Youde, Wang
2015-01-01
To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel. PMID:26121468
Kim, Tae Hyung
2009-05-15
and natural fractures were investigated in this study using an X-Ray CT Scanner. Fractal dimension, D, and amplitude parameter, A, of fracture aperture approaches a constant value with increased sampling area, similar to the behavior of fracture roughness...
Kim, Tae Hyung
2009-05-15
and natural fractures were investigated in this study using an X-Ray CT Scanner. Fractal dimension, D, and amplitude parameter, A, of fracture aperture approaches a constant value with increased sampling area, similar to the behavior of fracture roughness...
NASA Astrophysics Data System (ADS)
Muto, Jun; Nakatani, Tsurugi; Nishikawa, Osamu; Nagahama, Hiroyuki
2015-05-01
The size distributions of particle in pulverized rocks from the San Andreas fault and the Arima-Takatsuki Tectonic Line were measured. The rocks are characterized by the development of opening mode fractures with an apparent lack of shear. Fragments in the rocks in both fault zones show a fractal size distribution down to the micron scale. Fractal dimensions, dependent on mineral type, decrease from 2.92 to 1.97 with increasing distance normal to the fault core. The fractal dimensions of the rocks are higher than those of both natural and experimentally created fault gouges measured in previous studies. Moreover, the dimensions are higher than the theoretically estimated upper fractal limit under confined comminution. Dimensions close to 3.0 have been reported in impact loading experiments. The observed characteristics indicate that pulverization is likely to have occurred by a dynamic stress pulse with instantaneous volumetric expansion, possibly during seismic rupture propagation similar to impact loading.
Temperature induced smoothing of initially fractal grain boundaries
Streitenberger, P.; Foerster, D.; Kolbe, G.; Veit, P. [Otto-von-Guericke-Univ. Magdeburg (Germany). Inst. fuer Experimentelle Physik] [Otto-von-Guericke-Univ. Magdeburg (Germany). Inst. fuer Experimentelle Physik
1996-01-01
Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fractal geometry has attracted great attention. It has been reported that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. In the present paper, for the first time, measurements of the change in the roughness of initially fractal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially fractal dimension.
A fractal analysis of pathogen detection by biosensors
NASA Astrophysics Data System (ADS)
Doke, Atul M.; Sadana, Ajit
2006-05-01
A fractal analysis is presented for the detection of pathogens such as Franscisela tularensis, and Yersinia pestis (the bacterium that causes plague) using a CANARY (cellular analysis and notification of antigens risks and yields) biosensor (Rider et al., 2003). In general, the binding and dissociation rate coefficients may be adequately described by either a single- or a dual-fractal analysis. An attempt is made to relate the binding rate coefficient to the degree of heterogeneity (fractal dimension value) present on the biosensor surface. Binding and dissociation rate coefficient values obtained are presented. The kinetics aspects along with the affinity values presented are of interest, and should along with the rate coefficients presented for the binding and the dissociation phase be of significant interest in help designing better biosensors for an application area that is bound to gain increasing importance in the future.
Magnetic Reconnection Rate in Space Plasmas: A Fractal Approach
Materassi, Massimo [Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, V. Madonna del Piano 10, I-50019 Sesto Fiorentino (Italy); Consolini, Giuseppe [Istituto di Fisica dello Spazio Interplanetario, Istituto Nazionale di Astrofisica, V. Fosso del Cavaliere 100, I-00133 Rome (Italy)
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 Alfven 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.
Power-law hereditariness of hierarchical fractal bones.
Deseri, Luca; Di Paola, Mario; Zingales, Massimiliano; Pollaci, Pietro
2013-12-01
In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ? ? ?1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power law. PMID:23836622
Collapse of loaded fractal trees
NASA Astrophysics Data System (ADS)
Turcotte, D. L.; Smalley, R. F.; Solla, Sara A.
1985-02-01
Mandelbrot1 has argued that a wide range of natural objects and phenomena are fractals; examples of fractal trees include actual trees, plants such as a cauliflower, river systems and the cardiovascular system. Here we apply the renormalization group approach2 to the collapse of fractal trees, which may be applicable to a variety of problems including cardiac arrest, failure of bronchial systems, failure of electrical distribution systems and the instability resulting in earthquakes.
THE MULTIFRACTAL BOX DIMENSIONS OF TYPICAL MEASURES FREDERIC BAYART
Paris-Sud XI, Université de
THE MULTIFRACTAL BOX DIMENSIONS OF TYPICAL MEASURES FR´ED´ERIC BAYART Abstract. We compute the typical (in the sense of Baire's category theorem) multi- fractal box dimensions of measures on a compact subset of Rd . Our results are new even in the context of box dimensions of measures. 1. Introduction 1
Reaction kinetics on fractals: Random-walker simulations and excition experiments
NASA Astrophysics Data System (ADS)
Kopelman, R.; Klymko, P. W.; Newhouse, J. S.; Anacker, L. W.
1984-03-01
Both computer simulations and laboratory experiments on binary reactions of random walkers on fractal spaces bear out a recent conjecture: The time development of the reaction is dominated by the intrinsic fractal (fracton, spectral) dimension. For the Sierpinski gasket the effective spectral dimension for reactions is d's=1.38 (actual spectral dimension ds=1.365). For the percolating cluster (60%, square lattice) d's=1.34 (ds=1.333). From the exciton percolation laboratory experiments d's=1.5, based on triplet-triplet annihilation in naphthalene isotopic mixed crystals at 2 K.
Fractals in physiology and medicine
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.; West, Bruce J.
1987-01-01
The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.
Fractal multifiber microchannel plates
NASA Technical Reports Server (NTRS)
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.
NASA Astrophysics Data System (ADS)
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...
Frequency-dependent viscous flow in channels with fractal rough surfaces
Cortis, A.; Berryman, J.G.
2010-05-01
The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to {infinity} (but has a finite cutoff), and the universal scaling of the dynamic permeability, k({omega}), needs modification. We performed accurate numerical computations of k({omega}) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k({omega}) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.
Fractals and quantum mechanics
NASA Astrophysics Data System (ADS)
Laskin, Nick
2000-12-01
A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Lévy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrödinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Lévy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics.
Fractals and quantum mechanics.
Laskin, Nick
2000-12-01
A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics. PMID:12779428
Dynamic structure factor of vibrating fractals: Proteins as a case study
NASA Astrophysics Data System (ADS)
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-01-01
We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kRg?1, where Rg is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u?i(t)-u?j(0)]2>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u?i(t)-u?i(0)]2>˜t?, where ? is the anomalous diffusion exponent that depends on the spectral dimension ds and fractal dimension df. As a result, for wave numbers obeying k2?1, S(k,t) effectively decays as a stretched exponential S(k,t)?S(k)e-(?kt)? with ???, where the relaxation rate is ?k˜(kBT/m?o2)1/?k2/?, T is the temperature, and m?o2 the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)˜k-df, with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent ? to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ?. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics.
Dynamic structure factor of vibrating fractals: proteins as a case study.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-01-01
We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)?1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u(?>)(i)(t)-u(?>)(j)(0)](2)>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u(?>)(i)(t)-u(?>)(i)(0)](2)> ~t(?), where ? is the anomalous diffusion exponent that depends on the spectral dimension d(s) and fractal dimension d(f). As a result, for wave numbers obeying k(2))(2)>?1, S(k,t) effectively decays as a stretched exponential S(k,t)?S(k)e(-(?(k)t)(?)) with ???, where the relaxation rate is ?(k)~(k(B)T/m?(o)(2))(1/?)k(2/?), T is the temperature, and m?(o)(2) the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)~k(-d(f)), with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent ? to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ?. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics. PMID:22400590
Scaling laws for slippage on superhydrophobic fractal surfaces
C. Cottin-Bizonne; C. Barentin; L. Bocquet
2012-01-24
We study the slippage on hierarchical fractal superhydrophobic surfaces, and find an unexpected rich behavior for hydrodynamic friction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension, and is found to be always smaller on fractal surfaces as compared to surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractal surface: depending on the underlying geometry of the roughness, strongly superhydrophobic surfaces may in some cases be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractal surfaces, in order to estimate the Kapitsa surface resistance on patterned surfaces, as well as to the question of trapping of diffusing particles by patchy hierarchical surfaces, in the context of chemoreception.
Anisotropic fractal media by vector calculus in non-integer dimensional space
NASA Astrophysics Data System (ADS)
Tarasov, Vasily E.
2014-08-01
A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
Applications of Fractal Analytical Techniques in the Estimation of Operational Scale
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Quattrochi, Dale A.
2000-01-01
The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.
Fractal dynamics of heartbeat time series of young persons with metabolic syndrome
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.
2012-10-01
Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.
Fractal analysis: methodologies for biomedical researchers.
Ristanovi?, Dusan; Milosevi?, 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. PMID:23757956
Buried mine detection using fractal geometry analysis to the LWIR successive line scan data image
NASA Astrophysics Data System (ADS)
Araki, Kan
2012-06-01
We have engaged in research on buried mine/IED detection by remote sensing method using LWIR camera. A IR image of a ground, containing buried objects can be assumed as a superimposed pattern including thermal scattering which may depend on the ground surface roughness, vegetation canopy, and effect of the sun light, and radiation due to various heat interaction caused by differences in specific heat, size, and buried depth of the objects and local temperature of their surrounding environment. In this cumbersome environment, we introduce fractal geometry for analyzing from an IR image. Clutter patterns due to these complex elements have oftentimes low ordered fractal dimension of Hausdorff Dimension. On the other hand, the target patterns have its tendency of obtaining higher ordered fractal dimension in terms of Information Dimension. Random Shuffle Surrogate method or Fourier Transform Surrogate method is used to evaluate fractional statistics by applying shuffle of time sequence data or phase of spectrum. Fractal interpolation to each line scan was also applied to improve the signal processing performance in order to evade zero division and enhance information of data. Some results of target extraction by using relationship between low and high ordered fractal dimension are to be presented.
Fractal Structure of Molecular Clouds
Srabani Datta
2002-01-01
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
Fractal Analysis of Trabecular Bone
NSDL National Science Digital Library
Gillespy, Thurman
Fractals are unusual geometric structures that can be used to analyze many biologic structures not amenable to conventional analysis. The purpose of this exhibit is to teach some of the fundamentals of fractal analysis, and to show how they can be applied to analysis of trabecular bone.
ERIC Educational Resources Information Center
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.)
Fractal analysis of cytoskeleton rearrangement in cardiac muscle during head-down tilt.
Thomason, D B; Anderson, O; Menon, V
1996-10-01
Head-down tilt by tail suspension of the rat produces a volume, but not pressure, load on the heart. One response of the heart is cytoskeleton rearrangement, a phenomenon commonly referred to as disruption. In these experiments, we used fractal analysis as a means to measure complexity of the microtubule structures at 8 and 18 h after imposition of head-down tilt. Microtubules in whole tissue cardiac myocytes were stained with fluorescein colchicine and were visualized by confocal microscopy. The fractal dimensions (D) of the structures were calculated by the dilation method, which involves successively dilating the outline perimeter of the microtubule structures and measuring the area enclosed. The head-down tilt resulted in a progressive decrease in D (decreased complexity) when measured at small dilations of the perimeter, but the maximum D (maximum complexity) of the microtubule structures did not change with treatment. Analysis of the fold change in complexity as a function of the dilation indicates an almost twofold decrease in microtubule complexity at small kernel dilations. This decrease in complexity is associated with a more Gaussian distribution of microtubule diameters, indicating a less structured microtubule cytoskeleton. We interpret these data as a microtubule rearrangement, rather than erosion, because total tubulin flourescence was not different between groups. This conclusion is supported by F-actin fluorescence data indicating a dispersed structure without loss of actin. PMID:8904563
Drip paintings and fractal analysis
NASA Astrophysics Data System (ADS)
Jones-Smith, Katherine; Mathur, Harsh; Krauss, Lawrence M.
2009-04-01
It has been claimed that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has led us to a result in fractal analysis of more general scientific significance: we show that the statistics of the “covering staircase” (closely related to the box-counting staircase) provide a way to characterize geometry and distinguish fractals from Euclidean objects. Finally we present a discussion of the composite of two fractals, a problem that was first investigated by Muzy. We show that the composite is not generally scale invariant and that it exhibits complex multifractal scaling in the small distance asymptotic limit.
Drip paintings and fractal analysis.
Jones-Smith, Katherine; Mathur, Harsh; Krauss, Lawrence M
2009-04-01
It has been claimed that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has led us to a result in fractal analysis of more general scientific significance: we show that the statistics of the "covering staircase" (closely related to the box-counting staircase) provide a way to characterize geometry and distinguish fractals from Euclidean objects. Finally we present a discussion of the composite of two fractals, a problem that was first investigated by Muzy. We show that the composite is not generally scale invariant and that it exhibits complex multifractal scaling in the small distance asymptotic limit. PMID:19518305
NASA Astrophysics Data System (ADS)
Spada, M.; Wiemer, S.; Kissling, E.
2009-04-01
The calculation of seismic hazard for a country is the first important step for the definition of a seismic building code. The goal of probabilistic seismic hazard assessment (PSHA) is to quantify the rate of exceeding various ground-motion levels at a site, given all possible earthquakes. A critical step in PSHA is the accurate definition and characterization of relevant seismic sources. This is particularly challenging in low-seismicity regions, because observation periods are relatively short, seismicity is often diffuse, and active faults are difficult to identify. For these reasons, large source zones are commonly used with spatially uniformly distributed seismicity inside. Observed seismicity, however, is generally not uniformly distributed, but reflects seismotectonic forces and tectonic structure. Rather, observed seismicity even in subregions defined as seismic sources is clustered in space: seismicity tends to aggregate on or close to major fault structures. Thus the hypothesis of uniform distribution of events inside a source zone does not relate well to observed seismicity and could overestimate or underestimate the value of ground-acceleration on the PSHA. Seismicity is a classical example of a complex phenomenon that can be quantified using fractal concepts. In particular, fault networks and epicenter distributions are know to have fractal properties. The fractal dimension is an extension of the Euclidean dimension and measure the degree of clustering of earthquakes. In this study, we move towards a more realistic characterization of spatio-temporal distribution of seismicity within each source zone. As first step, we quantified differences between different spatial characterization of seismicity and validate a more realistic method for the generation of synthetic seismicity on a source zone as input for PSHA, extending the concepts described in Beauval et al. (BSSA, 2006). We calculate differences in terms of hazard curves (annual probability of exceedance as a function of ground-motion) for synthetic catalogs characterized by a uniform or a clustered distribution of events on a hypothetical square source zone. Then we apply the method in the case of a typical low seismicity region such as Switzerland. We computed the fractal dimension of the observed seismicity from the ECOS catalog (D ˜ 1.5) using a box-counting method. Next we generated sets of synthetic catalogs characterized by a given fractal dimension for each source zone of the current PSHA area source model for Switzerland. Finally, from these synthetic catalogs, we compute hazard curves for eight sites in Switzerland, assuming the same activity rate but with the traditional uniform (D = 2.0) and various fractal distributions. We find that the assumption of D = 2.0 indeed overestimates the resulting hazard even for realistic zonation models. This overestimation is larger for low probability levels; it can typically reach 10 percent.
Enhanced Fractal-Wavelet Image Denoising
Jian Lu; Yuru Zou; Zhongxing Ye
2008-01-01
This paper presents an enhanced fractal-wavelet image denoising (EFWID) algorithm by adopting quadratic function for fractal scale prediction in wavelet domain. It consists in an extension of an enhanced fractal image denoising (EFID) algorithm proposed by the authors for denoising the images degraded by additive white Gaussian noise (AWGN). In terms of the quality of the fractal-wavelet representation of the
Joint fractal-wavelet image denoising interpolation
M. Ghazel; E. R. Vrscay; G. H. Freeman; R. K. Ward; R. Abugharbieh
2005-01-01
In this paper, we propose a simple and effective fractal-wavelet simultaneous image denoising and interpolation scheme. The denoising is performed during the encoding process while the interpolation is performed during the decoding process. The fractal-based image denoising involves predicting the fractal code of the original noiseless image from the statistics of the noisy observation. This fractal code can then be
Fractal Theory on Drying: A Review
Peng Xu; Arun S. Mujumdar; Boming Yu
2008-01-01
Fractal geometry has been widely used in various dried materials and drying processes. This review summarizes the related studies and identifies the opportunities for future investigation. The application of fractal concept on drying can be categorized into describing microscopic and macroscopic structure of material in drying with fractal geometry and theoretical models with fractal theory for drying mechanism. And also,
Fractal analysis for reduced reference image quality assessment.
Xu, Yong; Liu, Delei; Quan, Yuhui; Le Callet, Patrick
2015-07-01
In this paper, multifractal analysis is adapted to reduced-reference image quality assessment (RR-IQA). A novel RR-QA approach is proposed, which measures the difference of spatial arrangement between the reference image and the distorted image in terms of spatial regularity measured by fractal dimension. An image is first expressed in Log-Gabor domain. Then, fractal dimensions are computed on each Log-Gabor subband and concatenated as a feature vector. Finally, the extracted features are pooled as the quality score of the distorted image using l1 distance. Compared with existing approaches, the proposed method measures image quality from the perspective of the spatial distribution of image patterns. The proposed method was evaluated on seven public benchmark data sets. Experimental results have demonstrated the excellent performance of the proposed method in comparison with state-of-the-art approaches. PMID:25794391
Multiresolution processing for fractal analysis of airborne remotely sensed data
NASA Technical Reports Server (NTRS)
Jaggi, S.; Quattrochi, D.; Lam, N.
1992-01-01
Images acquired by NASA's Calibrated Airborne Multispectral Scanner are used to compute the fractal dimension as a function of spatial resolution. Three methods are used to determine the fractal dimension: Shelberg's (1982, 1983) line-divider method, the variogram method, and the triangular prism method. A description of these methods and the result of applying these methods to a remotely-sensed image is also presented. The scanner data was acquired over western Puerto Rico in January, 1990 over land and water. The aim is to study impacts of man-induced changes on land that affect sedimentation into the near-shore environment. The data were obtained over the same area at three different pixel sizes: 10 m, 20 m, and 30 m.
Fractal functions and interpolation
Michael F. Barnsley
1986-01-01
Let a data set {(xi,yi) ?I×R;i=0,1,?,N} be given, whereI=[x0,xN]?R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf:I?R, which interpolate the data according tof(xi)=yi fori e {0,1,?,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some
CALCULUS ON FRACTAL SUBSETS OF REAL LINE — I: FORMULATION
ABHAY PARVATE; A. D. GANGAL
2009-01-01
A new calculus based on fractal subsets of the real line is formulated. In\\u000athis calculus, an integral of order $\\\\alpha, 0 < \\\\alpha \\\\leq 1$, called\\u000a$F^\\\\alpha$-integral, is defined, which is suitable to integrate functions with\\u000afractal support $F$ of dimension $\\\\alpha$. Further, a derivative of order\\u000a$\\\\alpha, 0 < \\\\alpha \\\\leq 1$, called $F^\\\\alpha$-derivative, is defined, which\\u000aenables
Fractal analysis of the hierarchic structure of fossil coal surface
Alekseev, A.D.; Vasilenko, T.A.; Kirillov, A.K. [National Academy of Sciences, Donetsk (Ukraine)
2008-05-15
The fractal analysis is described as method of studying images of surface of fossil coal, one of the natural sorbent, with the aim of determining its structural surface heterogeneity. The deformation effect as a reduction in the dimensions of heterogeneity boundaries is considered. It is shown that the theory of nonequilibrium dynamic systems permits to assess a formation level of heterogeneities involved into a sorbent composition by means of the Hurst factor.
On fractal properties of small-scale ionospheric irregularities
V. A. Alimov; F. I. Vybornov; A. V. Rakhlin
2007-01-01
We consider the problem of relating the local structure of small-scale ionospheric turbulence to the measured frequency-spectrum\\u000a indices and fractal dimensions of amplitude records of the signals received on the Earth during remote sensing of the ionosphere\\u000a onboard the satellites. It is shown that knowledge of these parameters permits one to determine the true values of the local-spectrum\\u000a indices of
Fractal interfaces and product generation in the two-dimensional mixing layer
NASA Technical Reports Server (NTRS)
Jimenez, Javier; Martel, Carlos
1991-01-01
The dependence of product generation on Peclet and Reynolds numbers in a numerically simulated, reacting, two-dimensional, temporally growing mixing layer is related theoretically to the fractal dimension of the passive scalar interfaces. This reaction is verified using product generation measurements and fractal dimensions derived from the box counting technique. A transition from a low initial dimension to a higher one of approximately 5/3 is identified and shown to be associated to the kinematic distortion of the flow field during the first pairing interaction. It is suggested that the structures reponsible for this transition are nondeterministic, nonrandom, inhomogeneous fractals. In the range of Schmidt numbers investigated (0.25-4), only the large scales are involved. No further transitions, either in the spectra of the vorticity field or in the mixing behavior, are found for Reynolds numbers up to 90,000.
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 Dcentral place models are converted into fractal central place models. If we further integrate the chance factors into the i...
Fractal-wavelet image denoising revisited
Mohsen Ghazel; George H. Freeman; Edward R. Vrscay
2006-01-01
The essence of fractal image denoising is to predict the fractal code of a noiseless image from its noisy observation. From the predicted fractal code, one can generate an estimate of the original image. We show how well fractal-wavelet denoising predicts parent wavelet subtress of the noiseless image. The per- formance of various fractal-wavelet denoising schemes (e.g., fixed partitioning, quadtree
Thermoreversibly aggregated microgels: fractal structure and aggregation mechanism
NASA Astrophysics Data System (ADS)
Gu, Zhenyu; Cao, Rong; Armitage, Bruce; Patterson, Gary
2003-03-01
Multifunctional molecules were designed to produce thermoreversible microgels with specific structures. Both static and dynamic light scattering were employed to determine the fractal dimension (Df) of the microgels. Avidin binds four biotin moieties. Biotin was attached covalently to engineered peptide nucleic acid (PNA) oligomers. Three designed DNA oligomers self-assembled to produce a three-way junction(TWJ) with single stranded ends that were complementary to the PNA sequence. The DNA-PNA helices thermoreversibly melt at specific temperatures in the range 20-50 oC. The fractal dimension was obtained from the angular dependence of the scattered intensity. When the microgels were formed by cooling from a temperature above the melting point of the PNA-DNA helices, reversible structures with Df 1.85 were formed, which is consistent with a cluster-cluster aggregation mechanism. When the microgels were formed by slowly adding avidin to a solution of biotinylated-PNA bound to the TWJ at room temperature, the observed fractal dimension was around 2.60, which is consistent with a point-cluster aggregation mechanism.
NASA Astrophysics Data System (ADS)
Xia, Dehong; Guo, Shanshan; Ren, Ling
2010-02-01
In this paper, it is proved that the internal porous structure of alumina-silicate refractory fiber has fractal characteristics, which is reconstructed by the computer and the reconstructed structure further proved to have fractal characteristics. Based on the reconstructed structure, the network-thermal-resistance model is established to calculate the thermal conductivity of the fiber. It is shown that the calculated results agree well with the previous experimental ones, proving the correctness of the method.
Fractal dust grains around R Coronae Borealis stars
Wright, E.L. (California Univ., Los Angeles (USA))
1989-11-01
Discrete dipole approximation calculations of the optical properties of random fractal aggregates of graphite spheroids show a UV absorption feature that is too wide and centered at too long a wavelength to fit the observed interstellar 2200-A feature, but which is a good match to the 2400-A feature seen in the hydrogen-deficient R CrB stars reported by Hecht et al. (1984). Graphite fractal grains also match the UV bump and large long-wavelenvth extinction seen in laboratory studies of carbon smoke published by Bussoletti et al. (1987), which are usually attributed to amorphous carbon. 16 refs.
On the fractal properties microaccelerations
A. V. Sedelnikov
2012-04-19
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.
Donatti, D.A.; Vollet, D.R.; Ibanez Ruiz, A.; Mesquita, A.; Silva, T.F.P. [Unesp-Universidade Estadual Paulista, IGCE, Departamento de Fisica, P.O. Box 178 CEP, 13500-970 Rio Claro, Sao Paulo (Brazil)
2005-01-01
A sample series of silica sonogels was prepared using different water-tetraethoxysilane molar ratio (r{sub w}) in the gelation step of the process in order to obtain aerogels with different bulk densities after the supercritical drying. The samples were analyzed by means of small-angle x-ray-scattering (SAXS) and nitrogen-adsorption techniques. Wet sonogels exhibit mass fractal structure with fractal dimension D increasing from {approx}2.1 to {approx}2.4 and mass-fractal correlation length {xi} diminishing from {approx}13 nm to {approx}2 nm, as r{sub w} is changed in the nominal range from 66 to 6. The process of obtaining aerogels from sonogels and heat treatment at 500 deg. C, in general, increases the mass-fractal dimension D, diminishes the characteristic length {xi} of the fractal structure, and shortens the fractal range at the micropore side for the formation of a secondary structured particle, apparently evolved from the original wet structure at a high resolution level. The overall mass-fractal dimension D of aerogels was evaluated as {approx}2.4 and {approx}2.5, as determined from SAXS and from pore-size distribution by nitrogen adsorption, respectively. The fine structure of the 'secondary particle' developed in the obtaining of aerogels could be described as a surface-mass fractal, with the correlated surface and mass-fractal dimensions decreasing from {approx}2.4 to {approx}2.0 and from {approx}2.7 to {approx}2.5, respectively, as the aerogel bulk density increases from 0.25 (r{sub w}=66) up to 0.91 g/cm{sup 3} (r{sub w}=6)
Fractal morphology of the flickering of the cataclysmic variable star KR Aurigae
NASA Astrophysics Data System (ADS)
Georgiev, Ts.; Antov, A.; Bachev, R.; Boeva, S.; Latev, G.; Spasov, B.; Stoyanov, K.; Tsvetkova, S.
We apply fractal analysis on 29 light curves of KR Aur in the system UBVRI, 20 in high brightness light state and 9 in low state. The method has been tested on normal and uniform random processes, on light and colour curves of SN 2007gh, on the Sun spot numbers and on the flickering of the cataclysmic variable V425 Cas. The brightness of KR Aur is 13-18 mag, the monitoring time is 1-10 hours and the single exposure time is 0.5-5 min. Two fractal dimensions, based on the local amplitudes (MAX-MIN), 1.2
NASA Astrophysics Data System (ADS)
Risovi?, Dubravko; Frka, Sanja; Kozarac, Zlatica
2011-01-01
The aim of this study was to investigate the connection between the lipid/amphiphile monolayer structure at the interface and its macroscopic/rheological properties, in particular, to establish the link between the fractality of the monolayer structure and its compressibility modulus. To that purpose we have used fractal analysis of images obtained by Brewster angle microscopy to infer the fractal dimension of the monolayer structure and relate its change to the corresponding changes in compressibility derived from a simultaneously measured ?-A isotherm. The results of the study confirmed the starting assumption based on theoretical considerations that the fractal dimension of an amphiphilic monolayer and its compressibility should be correlated. We have shown that there exists a strong correlation between the fractal dimension and the corresponding compressibility modulus of different amphiphilic materials. Thus, confirming the link between the short ordered structure on the molecular level and the macroscopic property—compressibility of the monolayer. The established correlation between the fractal dynamics and compressibility modulus of the monolayer enabled identification of onset of percolation—a second-order phase transition that is otherwise not easy and unambiguously detectable. We have found that the signature of percolation in a monolayer, regardless of its composition, is the occurrence of a sharp increase (a jump) of compressibility modulus (at macroscopic level) at the characteristic value of the corresponding fractal dimension D = 1.89. This is the result of the abrupt establishment of a connected structure on the molecular level, consequently involving a change in the elastic properties of the monolayer on a macroscopic scale. The results of this investigation provide means for unambiguous identification of the onset of percolation in the Langmuir layer and should facilitate a more efficient application of the percolation theory in further study of processes and structures at the interface during the monolayer compression.
Fractal properties of active region and flare
NASA Astrophysics Data System (ADS)
Golovko, A. A.; Salakhutdinova, I. I.; Khlystova, A. I.
2009-12-01
A fractal analysis of narrowband images of the chromosphere and transition layer has been performed in order to study regimes of turbulence and their variations during time-varying processes in active regions. The NOAA 10039 and 10050 activity complexes on July 31, 2002, were observed at Baikal astrophysical observatory of ISZF SO RAN in the H-? line using a chromospheric telescope equipped with a Halle birefringent filter (BF) with a passband of 0.5 Å. Images of the same activity complexes in the spectral band centered at the FeXI 171 Å line, obtained at TRACE space observatory, have been processed using the same technique. The method of structure functions has been used to compute the time series of the scaling parameters. The power spectra of two-dimensional images have been used to compute the time variations in the fractal dimension of the considered activity complex. It has been indicated that the parameters of a multifractal structure (intermittent turbulence) demonstrate jump-like and quasiperiodic time variations correlating with flares. These variations were detected in the H-? and FeXI 171 Å lines of the transition zone, using the ground-based and onboard measurements, which demonstrates that they are of the solar origin.
Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder
NASA Astrophysics Data System (ADS)
Ahmadlou, Mehran; Adeli, Hojjat; Adeli, Amir
2012-10-01
Recently, the visibility graph (VG) algorithm was proposed for mapping a time series to a graph to study complexity and fractality of the time series through investigation of the complexity of its graph. The visibility graph algorithm converts a fractal time series to a scale-free graph. VG has been used for the investigation of fractality in the dynamic behavior of both artificial and natural complex systems. However, robustness and performance of the power of scale-freeness of VG (PSVG) as an effective method for measuring fractality has not been investigated. Since noise is unavoidable in real life time series, the robustness of a fractality measure is of paramount importance. To improve the accuracy and robustness of PSVG to noise for measurement of fractality of time series in biological time-series, an improved PSVG is presented in this paper. The proposed method is evaluated using two examples: a synthetic benchmark time series and a complicated real life Electroencephalograms (EEG)-based diagnostic problem, that is distinguishing autistic children from non-autistic children. It is shown that the proposed improved PSVG is less sensitive to noise and therefore more robust compared with PSVG. Further, it is shown that using improved PSVG in the wavelet-chaos neural network model of Adeli and c-workers in place of the Katz fractality dimension results in a more accurate diagnosis of autism, a complicated neurological and psychiatric disorder.
Field Fractal Cosmological Model As an Example of Practical Cosmology Approach
Yu. V. Baryshev
2015-03-11
The idea of the global gravitational effect as the source of cosmological redshift was considered by de Sitter (1916, 1917), Eddington (1923), Tolman (1929) and Bondi (1947). Also Hubble (1929) called the discovered distance-redshift relation as "De Sitter effect". For homogeneous matter distribution cosmological gravitational redshift is proportional to square of distance: z_grav ~ r^2. However for a fractal matter distribution having the fractal dimension D=2 the global gravitational redshift is the linear function of distance: z_grav ~ r, which gives possibility for interpretation of the Hubble law without the space expansion. Here the field gravity fractal cosmological model (FGF) is presented, which based on two initial principles. The first assumption is that the Feynman's field gravity approach describes the gravitational interaction, which delivers a natural basis for the conceptual unity of all fundamental physical interactions within the framework of the relativistic and quantum fields in Minkowski space. The second hypothesis is that the spatial distribution of gravitating matter is a fractal at all scales up to the Hubble radius. The fractal dimension of matter distribution is assumed to be D = 2, which implies that the global gravitational redshift is the explanation of the observed linear Hubble law. In the frame of the FGF all three phenomena - the cosmic background radiation, the fractal large scale structure, and the Hubble law, - could be the consequence of a unique large scale structure evolution process of the initially homogeneous ordinary matter without nonbaryonic matter and dark energy.
Fractal character of the distribution of surface potential irregularities in epitaxial n-GaAs (100)
Torkhov, N. A., E-mail: trkf@mail.ru; Bozhkov, V. G. [Scientific Research Institute of Semiconductor Devices (Russian Federation)
2009-05-15
The fractal geometric properties of the relief of the surface potential of a heavily doped n{sup +}-GaAs (100) wafer are studied by Kelvin's method of atomic force microscopy. The average fractal dimensionalities determined by the triangulation method (D{sub f}), the method of horizontal cross sections (D{sub c}), and the method of similarity (D{sub s}) are rather close to each other, which is indicative of a unified nature of the fractal relief of the surface potential. In general, the fractal dimensionalities determined in the study suggest that the relative arrangement of local irregularities of the potential profile of the heavily doped n{sup +}-GaAs (100) wafer subjected to chemical and dynamic polishing is similar to the pattern corresponding to the fractal curve known as Serpinsky's napkin. It is found that the fractal irregularities of the potential vary much more gradually than it happens in the twodimensional case: the variations are proportional to linear dimensions to the power 2/D{sub c} (1 < D{sub c} < 2) rather than to the square of linear dimensions of the regions under study.