The Three-Point Sinuosity Method for Calculating the Fractal Dimension of Machined Surface Profile
NASA Astrophysics Data System (ADS)
Zhou, Yuankai; Li, Yan; Zhu, Hua; Zuo, Xue; Yang, Jianhua
2015-04-01
The three-point sinuosity (TPS) method is proposed to calculate the fractal dimension of surface profile accurately. In this method, a new measure, TPS is defined to present the structural complexity of fractal curves, and has been proved to follow the power law. Thus, the fractal dimension can be calculated through the slope of the fitted line in the log-log plot. The Weierstrass-Mandelbrot (W-M) fractal curves, as well as the real surface profiles obtained by grinding, sand blasting and turning, are used to validate the effectiveness of the proposed method. The calculation values are compared to those obtained from root-mean-square (RMS) method, box-counting (BC) method and variation method. The results show that the TPS method has the widest scaling region, the least fit error and the highest accuracy among the methods examined, which demonstrates that the fractal characteristics of the fractal curves can be well revealed by the proposed method.
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.
a New Method for Calculating the Fractal Dimension of Surface Topography
NASA Astrophysics Data System (ADS)
Zuo, Xue; Zhu, Hua; Zhou, Yuankai; Li, Yan
2015-06-01
A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass-Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Thanki, Sandip; Rhee, George; Lepp, Stephen E-mail: grhee@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.
Classical Liquids in Fractal Dimension
Marco Heinen; Simon K. Schnyder; John F. Brady; Hartmut Löwen
2015-08-28
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.
Classical Liquids in Fractal Dimension
NASA Astrophysics Data System (ADS)
Heinen, Marco; Schnyder, Simon K.; Brady, John F.; Löwen, Hartmut
2015-08-01
We introduce fractal liquids by generalizing classical liquids of integer dimensions d =1 ,2 ,3 to a noninteger dimension dl . The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same 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 semianalytical 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.
Classical Liquids in Fractal Dimension.
Heinen, Marco; Schnyder, Simon K; Brady, John F; Löwen, Hartmut
2015-08-28
We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension d_{l}. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same 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 semianalytical 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. PMID:26371681
Dimension of fractal basin boundaries
Park, B.S.
1988-01-01
In many dynamical systems, multiple attractors coexist for certain parameter ranges. The set of initial conditions that asymptotically approach each attractor is its basin of attraction. These basins can be intertwined on arbitrary small scales. Basin boundary can be either smooth or fractal. Dynamical systems that have fractal basin boundary show final state sensitivity of the initial conditions. A measure of this sensitivity (uncertainty exponent {alpha}) is related to the dimension of the basin boundary d = D - {alpha}, where D is the dimension of the phase space and d is the dimension of the basin boundary. At metamorphosis values of the parameter, there might happen a conversion from smooth to fractal basin boundary (smooth-fractal metamorphosis) or a conversion from fractal to another fractal basin boundary characteristically different from the previous fractal one (fractal-fractal metamorphosis). The dimension changes continuously with the parameter except at the metamorphosis values where the dimension of the basin boundary jumps discontinuously. We chose the Henon map and the forced damped pendulum to investigate this. Scaling of the basin volumes near the metamorphosis values of the parameter is also being studied for the Henon map. Observations are explained analytically by using low dimensional model map.
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.
Berntson, G. M.; Stoll, P.
1997-01-01
Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales. Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.
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.
Fractal dimension of brittle fracture
Y. Y. Kagan
1991-01-01
Summary Using a worldwide catalog of earthquakes we analyze the distribution of distances between pairs of earthquake hypocenters to determine the spatial fractal dimension d of an earthquake fracture. As the time span of the catalog increases, d asymptotically reaches the value 2.1–2.2 for shallow earthquakes. Approximately the same asymptotic value of dimension is obtained for a catalog of earthquakes
Nieckarz, Zenon; Tato?, Grzegorz; Kozerska, Magdalena; Skrzat, Janusz; Sioma, Andrzej
2015-01-01
We presented a novel approach to studies of the vascular grooves located on the inner surface of the cranial vault. A three-dimensional vision system that acquired the endocranial surface topography was used for this purpose. The acquired data were used to generate images showing the branching pattern of the middle meningeal artery. Fractal dimension was used to characterize and analyze branching pattern complexity. We discussed the usefulness of the latter method and indicated difficulties and potential errors connected to the fractal dimension application. The technique introduced for recording traits of the object surface appears to be helpful in anatomical study of morphological variation of dural vascularization. It may also be applicable in paleoneurological research based on analysis of the cranial remnants. Fractal dimension should be used carefully as a method sensitive to many aspects of data acquisition and processing. PMID:25807002
Fractal dimensions of sinkholes
NASA Astrophysics Data System (ADS)
Reams, Max W.
1992-05-01
Sinkhole perimeters are probably fractals ( D=1.209-1.558) for sinkholes with areas larger than 10,000 m 2, based on area-perimeter plots of digitized data from karst surfaces developed on six geologic units in the United States. The sites in Florida, Kentucky, Indiana and Missouri were studied using maps with a scale of 1:24, 000. Size-number distributions of sinkhole perimeters and areas may also be fractal, although data for small sinkholes is needed for verification. Studies based on small-scale maps are needed to evaluate the number and roughness of small sinkhole populations.
Fractal Dimension for Data Mining Krishna Kumaraswamy
Murphy, Robert F.
Fractal Dimension for Data Mining Krishna Kumaraswamy skkumar@cs.cmu.edu Center for Automated, PA 15213 Abstract In this project, we introduce the concept of intrinsic "fractal" dimension of these problems, we show how the performance of a method is related to the fractal dimension of the data set
Fractal Dimensions of Macromolecular Structures
Todoroff, Nickolay; Kunze, Jens; Schreuder, Herman; Hessler, Gerhard; Baringhaus, Karl-Heinz; Schneider, Gisbert
2014-01-01
Quantifying the properties of macromolecules is a prerequisite for understanding their roles in biochemical processes. One of the less-explored geometric features of macromolecules is molecular surface irregularity, or ‘roughness’, which can be measured in terms of fractal dimension (D). In this study, we demonstrate that surface roughness correlates with ligand binding potential. We quantified the surface roughnesses of biological macromolecules in a large-scale survey that revealed D values between 2.0 and 2.4. The results of our study imply that surface patches involved in molecular interactions, such as ligand-binding pockets and protein-protein interfaces, exhibit greater local fluctuations in their fractal dimensions than ‘inert’ surface areas. We expect approximately 22 % of a protein’s surface outside of the crystallographically known ligand binding sites to be ligandable. These findings provide a fresh perspective on macromolecular structure and have considerable implications for drug design as well as chemical and systems biology. PMID:26213587
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 Astrophysics Data System (ADS)
Noever, David A.
1990-10-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2× 106 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˜1.7). These agree well with the two-dimensional DLA.
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...
Estimation of fractal dimension and fractal curvatures from digital Evgeny Spodarev
Spodarev, Evgueni
Estimation of fractal dimension and fractal curvatures from digital images Evgeny Spodarev Ulm the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use
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)
Trabecular Bone Mechanical Properties and Fractal Dimension
NASA Technical Reports Server (NTRS)
Hogan, Harry A.
1996-01-01
Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again and also scanned using dual-energy X-ray absorptiometry (DEXA). Cube specimens are then cut from the slabs and tested mechanically in compression. Correlations between mechanical properties and fractal dimension will then be examined to assess and quantify the predictive capability of the fractal calculations.
[Speaker gender identification based on audio fractal dimension and pitch feature].
Wang, Zhenhua; Yang, Cuirong; Wu, Wei; Fan, Yingle
2008-08-01
Automatic speaker gender identification based on voice feature is an important task in voice processing and analysis fields. In this paper non-linear parameters such as fractal dimension are applied to be one part of feature space for improving the ability of describing speaker gender feature through conventional linear parameters method. Pitch is picked using lifting scheme, and audio fractal dimension is extracted. Then based on Takens theory, the time delay method is used to reconstruct the phase space of fractal dimension sequence. And fractal dimension complexity is obtained by calculating Approximate Entropy. Three dimension feature vectors, including the pitch, the fractal dimension and the fractal dimension complexity, are applied to speaker gender identification. Experiment results show that through adding non-linear parameters, compared with the linear parameter using one dimension only such as pitch, the proposed method is more accurate and robust, and thus provides a new way for speaker gender identification. PMID:18788284
Computational cancer cells identification by fractal dimension analysis
C. Timbó; L. A. R. Da Rosa; M. Gonçalves; S. B. Duarte
2009-01-01
In the present work a software to identify cell anomaly through their fractal dimension calculation is introduced. The cell electronic microscopic image is imported to the software in gray scale and transformed to a black and white pattern in order to eliminate possible noise due to organic material close to the cell during the acquisition of the image. The number
A procedure to Estimate the Fractal Dimension of Waveforms
Carlos Sevcik
2010-03-27
A method is described for calculating the approximate fractal dimension from a set of N values y sampled from a waveform between time zero and t. The waveform was subjected to a double linear transformation that maps it into a unit square.
Fractal dimensions of lines in chaotic advection
NASA Astrophysics Data System (ADS)
Fung, J. C. H.; Vassilicos, J. C.
1991-05-01
We release a line in a flow produced by a blinking vortex1 which is known to advect fluid elements ``chaotically'' for a certain range of parameters. This is a similar problem to that of a line evolving in phase space through the action of an area-preserving map, addressed a decade ago by Berry et al.2 These authors have classified the convolutions of such a line as being either ``tendril shaped'' or ``whorl shaped.'' Recent numerical simulations of lines released in 2-D turbulence3 have shown that they only develop ``whorls,'' i.e., spiral structures. These spiral structures are produced by the eddying regions of the flow4 and are responsible for the noninteger value of the ``fractal'' dimension DK of the line, as measured by the box-counting algorithm. This ``fractal'' dimension is actually a Kolmogorov capacity. It has also been shown recently3 that the Kolmogorov capacity is a measure of local self-similarity, whereas the Hausdorff dimension DH is a measure of global self-similarity. Spirals are a good example of locally self-similar objects, for which DK>1 but DH=1. For conciseness we call a line H fractal when DH>1, and K fractal when DK>1. Most experimental and numerical evidence to date for ``fractal'' interfaces in turbulent flows is in fact evidence showing these interfaces to be K fractal. In fact, in the numerical simulations of Vassilicos,3 lines have been found not to be H fractal. Whether a line in chaotic advection becomes K fractal or H fractal is not a trivial question. If one neglects the effect of the unsteadiness of the flow, and thinks in terms of a single vortex at a fixed point in space wrapping the line around it, then it is easy to show that the spiral thus created has a DK>1 but DH=1 (and, in particular, that DK<2; the question of whether a line in chaotic advection is space filling is therefore not a trivial one either). But if one concentrates on the similarity between Aref's blinking vortex and a two-dimensional map, then one may be reminded of the Hénon attractor5 which is known to have a transversal Cantor-like structure that is H fractal. In fact, pictures of the line in the blinking vortex flow show that line to have a comparable stretched and folded structure to that of the Hénon attractor. We measure DH by measuring the length of the line with various resolutions and find that DH grows with time above 1. By zooming into the pictures of our line we can see its self-similar structure, and are therefore inclined to conclude that lines in chaotic advection do become H fractal. We also measure DK by the box-counting algorithm, and find that it also grows with time above 1, but is not equal to DH. It is a known mathematical fact that in general DK?DH, and our findings are consistent with this requirement. But we do not yet understand what this nonvanishing difference between DK and DH means for a line in chaotic advection. Furthermore, we find that both DK and DH increase as the switching of the vortex from one location to the other becomes faster. It is not clear whether these two fractal dimensions tend, asymptotically with time, toward a value strictly smaller than 2 or not. The interest of this work is to show how efficient unsteadiness (which is the central component of 2-D chaotic advection) can be for creating H fractal structures through a process of folding that it adds to stretching of the flow.6 We compare with numerical simulations of 2-D turbulence3 where the simulated, self-similar cascade of eddies fails to produce H fractal structures, and only produces K fractals.
Robert M. Peters
1999-01-01
Background: Previous studies have shown inconsistent relationships between left atrial size and various characteristics of atrial fibrillation seen on the surface electrocardiogram. The purpose of this study was to determine if the fractal dimension of atrial fibrillation derived from the ECG would be useful in predicting left atrial size. Methods: The fractal dimension (D) was calculated using resting 12-lead ECGs
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-tree wavelets and fractal dimension to adaptively suppress sand ripples and a matched filter as an initial-class support vector machine. We also implement previous work [13] that uses fractal dimension to adaptively
Estimation of fractal dimensions from transect data
Loehle, C.
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 evolution and spatial replacement dynamics of urban growth
Yanguang Chen
2011-01-01
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to interpret the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann’s equation. For the normalized data, Boltzmann’s equation is just
Fractal dimension in dissipative chaotic scattering Jess M. Seoane,1,
Lai, Ying-Cheng
Fractal dimension in dissipative chaotic scattering Jesús M. Seoane,1, * Miguel A. F. Sanjuán,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
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.
Local Earth's gravity field in view of fractal dimension
NASA Astrophysics Data System (ADS)
Mészárosová, Katarína; Minarechová, Zuzana; Janák, Juraj
2013-04-01
The poster presents the relative roughness of chosen characteristics of the Earth's gravity field in several small regions in area of Slovakia (e.g. free-air anomaly, Bouguer anomaly, gravity disturbance...) using the values of fractal dimension. In this approach, a three dimensional box counting method and the Hurst analysis method are applied to estimate the values of fractal dimensions. Then the computed fractal dimension values are used to compare all 3D models of all chosen characteristics.
Fractal dimension of nucleic acid sequences and its relation to evolutionary level
NASA Astrophysics Data System (ADS)
Luo, Liaofu; Tsai, Lu
1988-09-01
The fractal dimension (FD) in base space of nucleic acid sequences is introduced and calculated for 14 categories of nucleic acids. It is discovered that there exists a possible statistical relation between FDs and evolutionary levels of these sequences.
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.
On the Relation Between Lacunarity and Fractal Dimension
NASA Astrophysics Data System (ADS)
Borys, P.
2009-05-01
I discuss the relation between fractal dimension and lacunarity. Commenting the known results, I propose a method for estimation of the scaling constant in the power law dependency. Additionally, I provide a simple new derivation of a known experimental relation for lacunarity and fractal dimension.
Heterogeneities Analysis Using the Generalized Fractal Dimension and Continuous Wavelet Transform
NASA Astrophysics Data System (ADS)
Ouadfeul, S.; Aliouane, L.; Boudella, A.
2012-04-01
The main goal of this work is analyze heterogeneities from well-logs data using the wavelet transform modulus maxima lines (WTMM). Firstly, the continuous wavelet transform (CWT) with sliding window is calculated. The next step consists to calculate the maxima of the modulus of the CWT and estimate the spectrum of exponents. The three generalized fractal dimensions D0, D1 and D2 are then estimated. Application of the proposed idea at the synthetic and real well-logs data of a borehole located in the Algerian Sahara shows that the fractal dimensions are very sensitive to lithological variations. The generalized fractal dimensions are a very robust tool than can be used for petroleum reservoir characterization. Keywrods: reservoir, Heterogeneities, WTMM, fractal dimension.
Spectral Asymmetry and Higuchi's Fractal Dimension Measures of Depression Electroencephalogram
Bachmann, Maie; Lass, Jaanus; Suhhova, Anna; Hinrikus, Hiie
2013-01-01
This study was aimed to compare two electroencephalogram (EEG) analysis methods, spectral asymmetry index (SASI) and Higuchi's fractal dimension (HFD), for detection of depression. Linear SASI method is based on evaluation of the balance of powers in two EEG frequency bands in one channel selected higher and lower than the alpha band spectrum maximum. Nonlinear HFD method calculates fractal dimension directly in the time domain. The resting EEG signals of 17 depressive patients and 17 control subjects were used as a database for calculations. SASI values were positive for depressive and negative for control group (P < 0.05). SASI provided the true detection rate of 88% in the depressive and 82% in the control group. The calculated HFD values detected a small (3%) increase with depression (P < 0.05). HFD provided the true detection rate of 94% in the depressive group and 76% in the control group. The rate of correct indication in the both groups was 85% using SASI or HFD. Statistically significant variations were not revealed between hemispheres (P > 0.05). The results indicated that the linear EEG analysis method SASI and the nonlinear HFD method both demonstrated a good sensitivity for detection of characteristic features of depression in a single-channel EEG. PMID:24232245
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.
Archaeon and archaeal virus diversity classification via sequence entropy and fractal dimension
NASA Astrophysics Data System (ADS)
Tremberger, George, Jr.; Gallardo, Victor; Espinoza, Carola; Holden, Todd; Gadura, N.; Cheung, E.; Schneider, P.; Lieberman, D.; Cheung, T.
2010-09-01
Archaea are important potential candidates in astrobiology as their metabolism includes solar, inorganic and organic energy sources. Archaeal viruses would also be expected to be present in a sustainable archaeal exobiological community. Genetic sequence Shannon entropy and fractal dimension can be used to establish a two-dimensional measure for classification and phylogenetic study of these organisms. A sequence fractal dimension can be calculated from a numerical series consisting of the atomic numbers of each nucleotide. Archaeal 16S and 23S ribosomal RNA sequences were studied. Outliers in the 16S rRNA fractal dimension and entropy plot were found to be halophilic archaea. Positive correlation (R-square ~ 0.75, N = 18) was observed between fractal dimension and entropy across the studied species. The 16S ribosomal RNA sequence entropy correlates with the 23S ribosomal RNA sequence entropy across species with R-square 0.93, N = 18. Entropy values correspond positively with branch lengths of a published phylogeny. The studied archaeal virus sequences have high fractal dimensions of 2.02 or more. A comparison of selected extremophile sequences with archaeal sequences from the Humboldt Marine Ecosystem database (Wood-Hull Oceanography Institute, MIT) suggests the presence of continuous sequence expression as inferred from distributions of entropy and fractal dimension, consistent with the diversity expected in an exobiological archaeal community.
Fractal dimensions and the Phenomenon of Intermittency in Quantum Dynamics
Fractal dimensions and the Phenomenon of Intermittency in Quantum Dynamics Jean-Marie Barbaroux UMR Nantes C#19;edex 03, France E-mail: Jean-Marie.Barbaroux@math.univ-nantes.fr Fran#24;cois Germinet UMR
Use of fractal dimensions to quantify coral shape
NASA Astrophysics Data System (ADS)
Martin-Garin, B.; Lathuilière, B.; Verrecchia, E. P.; Geister, J.
2007-09-01
A morphometrical method to quantify and characterize coral corallites using Richardson Plots and Kaye’s notion of fractal dimensions is presented. A Jurassic coral species ( Aplosmilia spinosa) and five Recent coral species were compared using the Box-Counting Method. This method enables the characterization of their morphologies at calicular and septal levels by their fractal dimensions (structural and textural). Moreover, it is possible to determine differences between species of Montastraea and to tackle the high phenotypic plasticity of Montastraea annularis. The use of fractal dimensions versus conventional methods (e.g., measurements of linear dimensions with a calliper, landmarks, Fourier analyses) to explore a rugged boundary object is discussed. It appears that fractal methods have the potential to considerably simplify the morphometrical and statistical approaches, and be a valuable addition to methods based on Euclidian geometry.
Fractal dimension evolution and spatial replacement dynamics of urban growth
Chen, Yanguang
2011-01-01
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city syst...
Use of fractal dimensions to quantify coral shape
B. Martin-Garin; B. Lathuilière; E. P. Verrecchia; J. Geister
2007-01-01
A morphometrical method to quantify and characterize coral corallites using Richardson Plots and Kaye’s notion of fractal\\u000a dimensions is presented. A Jurassic coral species (Aplosmilia spinosa) and five Recent coral species were compared using the Box-Counting Method. This method enables the characterization of their\\u000a morphologies at calicular and septal levels by their fractal dimensions (structural and textural). Moreover, it is
The Casimir effect for parallel plates in the spacetime with a fractal extra compactified dimension
Hongbo Cheng
2011-06-23
The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like $\\delta>1/2$ to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction.
NASA Astrophysics Data System (ADS)
Gao, Wei; Zakharov, Valery P.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Artemyev, Dmitry N.; Kornilin, Dmitry V.
2015-07-01
Optical coherence tomography (OCT) is usually employed for the measurement of retinal thickness characterizing the structural changes of tissue. However, fractal dimension (FD) could also character the structural changes of tissue. Therefore, fractal dimension changes may provide further information regarding cellular layers and early damage in ocular diseases. We investigated the possibility of OCT in detecting changes in fractal dimension from layered retinal structures. OCT images were obtained from diabetic patients without retinopathy (DM, n = 38 eyes) or mild diabetic retinopathy (MDR, n = 43 eyes) and normal healthy subjects (Controls, n = 74 eyes). Fractal dimension was calculated using the differentiate box counting methodology. We evaluated the usefulness of quantifying fractal dimension of layered structures in the detection of retinal damage. Generalized estimating equations considering within-subject intereye relations were used to test for differences between the groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of fractal dimension to discriminate between the eyes of DM, MDR and healthy eyes. Significant decreases of fractal dimension were observed in all layers in the MDR eyes compared with controls except in the inner nuclear layer (INL). Significant decreases of fractal dimension were also observed in all layers in the MDR eyes compared with DM eyes. The highest area under receiver operating characteristic curve (AUROC) values estimated for fractal dimension were observed for the outer plexiform layer (OPL) and outer segment photoreceptors (OS) when comparing MDR eyes with controls. The highest AUROC value estimated for fractal dimension were also observed for the retinal nerve fiber layer (RNFL) and OS when comparing MDR eyes with DM eyes. Our results suggest that fractal dimension of the intraretinal layers may provide useful information to differentiate pathological from healthy eyes. Further research is warranted to determine how this approach may be used to improve diagnosis of early retinal neurodegeneration.
Computing fractal dimension in supertransient systems directly, fast and reliable
Romulus Breban; Helena E. Nusse
2006-08-07
Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices and they are a common phenomena in dynamical systems. Superlong chaotic transients are caused by the presence of chaotic saddles whose stable sets have fractal dimensions that are close to phase-space dimension. For many physical systems chaotic saddles have a big impact on laboratory measurements, and it is important to compute the dimension of such stable sets including fractal basin boundaries through a direct method. In this work, we present a new method to compute the dimension of stable sets of chaotic saddles directly, fast, and reliable.
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.
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
Fractal dimension analysis of cerebellum in Chiari Malformation type I.
Akar, Engin; Kara, Sad?k; Akdemir, Hidayet; K?r??, Adem
2015-09-01
Chiari Malformation type I (CM-I) is a serious neurological disorder that is characterized by hindbrain herniation. Our aim was to evaluate the usefulness of fractal analysis in CM-I patients. To examine the morphological complexity features of this disorder, fractal dimension (FD) of cerebellar regions were estimated from magnetic resonance images (MRI) of 17 patients with CM-I and 16 healthy control subjects in this study. The areas of white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF) were calculated and the corresponding FD values were computed using a 2D box-counting method in both groups. The results indicated that CM-I patients had significantly higher (p<0.05) FD values of GM, WM and CSF tissues compared to control group. According to the results of correlation analysis between FD values and the corresponding area values, FD and area values of GM tissues in the patients group were found to be correlated. The results of the present study suggest that FD values of cerebellar regions may be a discriminative feature and a useful marker for investigation of abnormalities in the cerebellum of CM-I patients. Further studies to explore the changes in cerebellar regions with the help of 3D FD analysis and volumetric calculations should be performed as a future work. PMID:26189156
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-01
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades. PMID:26305773
Fractal dimensions of niobium oxide films probed by protons and lithium ions
Pehlivan, Esat; Niklasson, Gunnar A.
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.
Sub-optimal MCV Cover Based Method for Measuring Fractal Dimension
Tolle, Charles Robert; McJunkin, Timothy R; Gorsich, D. I.
2003-01-01
A new method for calculating fractal dimension is developed in this paper. The method is based on the box dimension concept; however, it involves direct estimation of a suboptimal covering of the data set of interest. By finding a suboptimal cover, this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm. Moreover, any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation. In general, our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box. Instead, the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set. This method involves a variant of the Fuzzy-C Means clustering algorithm, as well as the use of the Minimum Cluster Volume clustering algorithm. A variety of fractal dimension estimators using this suboptimal covering method are discussed. Finally, these methods are compared to the standard box counting algorithm and wavelet-decomposition methods for calculating fractal dimension by using one-dimensional cantor dust sets and a set of standard Brownian random fractal images.
The use of fractal dimension in engineering geology
F. J. Brosch; P. Pölsler; G. Riedmüller
1992-01-01
The method of fractal geometry allows the simulation as well as description of data of many different natural states of orientation,\\u000a distribution and consistency. This affords to the engineering geologist various new possibilities of identifying rock mass\\u000a and rock. We have tried to use fractal dimension (FD) for characterising roughness profiles of shear faces as well as fracture\\u000a trace maps
Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar
Kingsbury, Nick
1 Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar J. D. B. The method exploits a fractal model of the seabed and the connection between: dual-tree wavelets and local, directional fractal dimension; interscale energy ratios, scale invariant fre- quency localised fractal
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.
Relationship between Fractal Dimension and Agreeability of Facial Imagery
NASA Astrophysics Data System (ADS)
Oyama-Higa, Mayumi; Miao, Tiejun; Ito, Tasuo
2007-11-01
Why do people feel happy and good or equivalently empathize more, with smiling face imageries than with ones of expressionless face? To understand what the essential factors are underlying imageries in relating to the feelings, we conducted an experiment by 84 subjects asked to estimate the degree of agreeability about expressionless and smiling facial images taken from 23 young persons to whom the subjects were no any pre-acquired knowledge. Images were presented one at a time to each subject who was asked to rank agreeability on a scale from 1 to 10. Fractal dimensions of facial images were obtained in order to characterize the complexity of the imageries by using of two types of fractal analysis methods, i.e., planar and cubic analysis methods, respectively. The results show a significant difference in the fractal dimension values between expressionless faces and smiling ones. Furthermore, we found a well correlation between the degree of agreeability and fractal dimensions, implying that the fractal dimension optically obtained in relation to complexity in imagery information is useful to characterize the psychological processes of cognition and awareness.
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.
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.
Fractal dimension, wavelet shrinkage, and anomaly detection for mine hunting
Nelson, James
Fractal dimension, wavelet shrinkage, and anomaly detection for mine hunting J. D. B. Nelson and N is considered for the mine hunting in sonar imagery problem. We exploit previous work that used dual attention in the mine hunting literature [2, 3, 19]. A common approach outlined in Figure 1 requires
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.
Fractal dimension and neural network based image segmentation technique
NASA Astrophysics Data System (ADS)
Lin, QiWei; Gui, Feng
2008-04-01
A new images segmentation scheme, which is based on combining technique of fractal dimension and self-organization neural network clustering, was presented in this paper. As we know features extracting is a very important step in image segmentation. So, in order to extract more effective fractal features from images, especially in the remote sensing images, a new image feature extracting and segmentation method was developed. The method extracts fractal features from a series of images that are obtained by convolving the original image with various masks to enhance its edge, line, ripple, and spot features. After that a 5-dimension feature vector are procured, in this vector each element is the fractal dimension of original image and four convolved images. And at last, we segment the image based on the strategy that combining the nearest neighbor classifier with self-organization neural network. Applying the presented algorithm to several practical remote sensing images, the experimental results show that the proposed method can improve the feature description ability and segment the images accurately.
Liver ultrasound image classification by using fractal dimension of edge
NASA Astrophysics Data System (ADS)
Moldovanu, Simona; Bibicu, Dorin; Moraru, Luminita
2012-08-01
Medical ultrasound image edge detection is an important component in increasing the number of application of segmentation, and hence it has been subject of many studies in the literature. In this study, we have classified the liver ultrasound images (US) combining Canny and Sobel edge detectors with fractal analysis in order to provide an indicator about of the US images roughness. We intend to provide a classification rule of the focal liver lesions as: cirrhotic liver, liver hemangioma and healthy liver. For edges detection the Canny and Sobel operators were used. Fractal analyses have been applied for texture analysis and classification of focal liver lesions according to fractal dimension (FD) determined by using the Box Counting method. To assess the performance and accuracy rate of the proposed method the contrast-to-noise (CNR) is analyzed.
The fractal dimension of the spectrum of quasiperiodical schrodinger operators
Laurent Marin
2012-02-20
We study the fractal dimension of the spectrum of a quasiperiodical Schrodinger operator associated to a sturmian potential. We consider potential defined with irrationnal number verifying a generic diophantine condition. We recall how shape and box dimension of the spectrum is linked to the irrational number properties. In the first place, we give general lower bound of the box dimension of the spectrum, true for all irrational numbers. In the second place, we improve this lower bound for almost all irrational numbers. We finally recall dynamical implication of the first bound.
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.
Takehara, Takuma; Ochiai, Fumio; Watanabe, Hiroshi; Suzuki, Naoto
2013-01-01
There is currently substantial literature to suggest that facial emotion recognition is impaired when other-race or inverted faces are presented. This study examined circumplex structures for recognising facial emotions under these conditions, directly measured those structures using a fractal dimension, and examined the difference between fractal dimensions. Results established that emotion ratings for the emotion prototypes used were sufficiently accurate under all conditions. Fractal analyses showed that the fractal dimensions of the circumplexes were significantly higher for recognition of facial emotions in other races than in one's own when the faces were presented upright; the fractal dimensions of the circumplexes were also higher for recognition of emotions in inverted faces than in upright faces in the own-race condition. The results suggest that a lower level of facial emotion recognition is associated with higher fractal dimension and that an increase of fractal dimension may be characterised by lack of facial familiarity. PMID:22992194
Fractal Dimension Computation From Equal Mass Partitions
Shiozawa, Yui; Rouet, Jean-Louis
2015-01-01
While the numerical methods which utilizes partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets, two mass- oriented methods are investigated by applying them to the one-dimensional generalized Cantor set. We show that both mass-oriented methods generate relatively good results for generalized dimensions for important cases where the box-counting method is known to fail. Both the strengths and limitations of the methods are also discussed.
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.
The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
David Damanik; Mark Embree; Anton Gorodetski; Serguei Tcheremchantsev
2007-05-02
We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\\lambda \\to \\infty$, $\\dim (\\sigma(H_\\lambda)) \\cdot \\log \\lambda$ converges to an explicit constant ($\\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\\"odinger dynamics generated by the Fibonacci Hamiltonian.
Fractal dimension analysis of malignant and benign endobronchial ultrasound nodes
2014-01-01
Background Endobronchial ultrasonography (EBUS) has been applied as a routine procedure for the diagnostic of hiliar and mediastinal nodes. The authors assessed the relationship between the echographic appearance of mediastinal nodes, based on endobronchial ultrasound images, and the likelihood of malignancy. Methods The images of twelve malignant and eleven benign nodes were evaluated. A previous processing method was applied to improve the quality of the images and to enhance the details. Texture and morphology parameters analyzed were: the image texture of the echographies and a fractal dimension that expressed the relationship between area and perimeter of the structures that appear in the image, and characterizes the convoluted inner structure of the hiliar and mediastinal nodes. Results Processed images showed that relationship between log perimeter and log area of hilar nodes was lineal (i.e. perimeter vs. area follow a power law). Fractal dimension was lower in the malignant nodes compared with non-malignant nodes (1.47(0.09), 1.53(0.10) mean(SD), Mann–Whitney U test p?Fractal dimension of ultrasonographic images of mediastinal nodes obtained through endobronchial ultrasound differ in malignant nodes from non-malignant. This parameter could differentiate malignat and non-malignat mediastinic and hiliar nodes. PMID:24920158
Effect of mobile phone radiation on brain using EEG analysis by Higuichi's fractal dimension method
NASA Astrophysics Data System (ADS)
Smitha, C. K.; Narayanan, N. K.
2013-01-01
venient window on the mind, revealing synaptic action that is moderately to strongly co-relate with brain state. Fractal dimension, measure of signal complexity can be used to characterize the physiological conditions of the brain. As the EEG signal is non linear, non stationary and noisy, non linear methods will be suitable for the analysis. In this paper Higuichi's fractal method is applied to find the fractal dimension. EEGs of 5 volunteers were recorded at rest and on exposure to radiofrequency (RF) emissions from mobile phones having different SAR values. Mobiles were positioned near the ears and then near the cz position. Fractal dimensions for all conditions are calculated using Higuich's FD estimation algorithm. The result shows that there are some changes in the FD while using mobile phone. The change in FD of the signal varies from person to person. The changes in FD show the variations in EEG signal while using mobile phone, which demonstrate transformation in the activities of brain due to radiation.
Surface evaluation by estimation of fractal dimension and statistical tools.
Hotar, Vlastimil; Salac, Petr
2014-01-01
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380
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.
NASA Astrophysics Data System (ADS)
Tijera, Manuel; Maqueda, Gregorio; Cano, José L.; López, Pilar; Yagüe, Carlos
2010-05-01
The wind velocity series of the atmospheric turbulent flow in the planetary boundary layer (PBL), in spite of being highly erratic, present a self-similarity structure (Frisch, 1995; Peitgen et., 2004; Falkovich et., 2006). So, the wind velocity can be seen as a fractal magnitude. We calculate the fractal dimension (Komolgorov capacity or box-counting dimension) of the wind perturbation series (u' = u- ) in the physical spaces (namely velocity-time). It has been studied the time evolution of the fractal dimension along different days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The data analysed was recorded in the experimental campaign SABLES-98 (Cuxart et al., 2000) at the Research Centre for the Lower Atmosphere (CIBA) located in Valladolid (Spain). In this work the u, v and w components of wind velocity series have been measured by sonic anemometers (20 Hz sampling rate). The fractal dimension versus the integral length scales of the mean wind series have been studied, as well as the influence of different turbulent parameters. A method for estimating these integral scales is developed using the normalized autocorrelation function and a Gaussian fit. Finally, it will be analysed the variation of the fractal dimension versus stability parameters (as Richardson number) in order to explain some of the dominant features which are likely immersed in the fractal nature of these turbulent flows. References - Cuxart J, Yagüe C, Morales G, Terradellas E, Orbe J, Calvo J, Fernández A, Soler MR, Infante C, Buenestado P, Espinalt A, Joergensen HE, Rees JM, Vilá J, Redondo JM, Cantalapiedra IR and Conangla L (2000) Stable atmospheric boundary-layer experiment in Spain (SABLES98): a report. Boundary- Layer Meteorol 96:337-370 - Falkovich G and Kattepalli R. Sreenivasan (2006) Lessons from Hidrodynamic Turbulence. Physics Today 59: 43-49 - Frisch U (1995) Turbulence the legacy of A.N. Kolmogorov Cambridge University Press 269pp - Peitgen H, Jürgens H and Saupe D (2004) Chaos and Fractals Springer-Verlag 971pp
Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
Michel L. Lapidus; Goran Radunovi?; Darko Žubrini?
2015-08-19
We study meromorphic extensions of distance and tube zeta functions, as well as of zeta functions of fractal strings, which include perturbations of the Riemann zeta function. The distance zeta function $\\zeta_A(s):=\\int_{A_\\delta} d(x,A)^{s-N}\\mathrm{d}x$, where $\\delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$, has been introduced by the first author in 2009, extending the definition of the zeta function $\\zeta_{\\mathcal L}$ associated with bounded fractal strings $\\mathcal L=(\\ell_j)_{j\\geq 1}$ to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence $D(\\zeta_A)$ coincides with $D:=\\overline\\dim_BA$, the upper box (or Minkowski) dimension of $A$. The (visible) complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of $A$ to a suitable connected neighborhood of the "critical line" $\\{\\textrm{Re}\\ s=D\\}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $t\\to0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. Furthermore, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line $\\{\\textrm{Re}\\ s=D\\}$. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct "maximally-hyperfractal" compact subsets of $\\mathbb{R}^N$, for $N\\geq 1$ arbitrary. These are compact subsets of $\\mathbb{R}^N$ such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line $\\{\\textrm{Re}\\ s=D\\}$.
Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation
Bruce N. Miller; Jean-Louis Rouet
2010-12-08
Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so, over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position-velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. The methodology employed includes: 1) The derivation of explicit solutions for the gravitational potential and field for a one-dimensional system with periodic boundary conditions (Ewald sums for one dimension); 2) The development of a procedure for obtaining scale-free initial conditions for the growing mode in phase space for an arbitrary power-law index; 3) The evaluation of power spectra, correlation functions, and generalized fractal dimensions at different stages of the system evolution. It is shown that a simple analytic representation of the power spectra captures the main features of the evolution, including the correct time dependence of the crossover from the linear to nonlinear regime and the transition from regular to fractal geometry. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.
Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects
NASA Astrophysics Data System (ADS)
Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.
2013-04-01
This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to higher number of dimensions. Easy integration with other applications by using the very simple comma separated values file format for storing multi-dimensional images. Implementation of ?2 test as a criterion for deciding whether an object is fractal or not. User friendly graphical interface. Hyper-Fractal Analysis-Test on the Sierpinski hypertetrahedron 4D gasket (Df=ln(5)/ln(2)?2.32). Running time: In a first approximation, the algorithm is linear [2]. References: [1] V. Grossu, D. Felea, C. Besliu, Al. Jipa, C.C. Bordeianu, E. Stan, T. Esanu, Computer Physics Communications, 181 (2010) 831-832. [2] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C. C. Bordeianu, D. Felea, Computer Physics Communications, 180 (2009) 1999-2001. [3] J. Ruiz de Miras, J. Navas, P. Villoslada, F.J. Esteban, Computer Methods and Programs in Biomedicine, 104 Issue 3 (2011) 452-460.
Wen-Shiung Chen; Shang-Yuan Yuan
2003-01-01
The paper presents a novel approach, mainly based on fractal dimensions, for recognizing the iris of a human eye. The proposed system consists mainly of four modules: iris image acquisition; iris image preprocessing; iris feature extraction; iris pattern recognition. Two methods for extracting the iris features based on the fractal dimension are proposed. Recognition performances of 91.18% in accepting authentics
Edinburgh Research Explorer Retinal Vascular Fractal Dimension, Childhood IQ, and Cognitive
Millar, Andrew J.
disease is associated with dementia. Differences in the topogra- phy of the retinal vascular network mayEdinburgh Research Explorer Retinal Vascular Fractal Dimension, Childhood IQ, and Cognitive Ability Vascular Fractal Dimension, Childhood IQ, and Cognitive Ability in Old Age: The Lothian Birth Cohort Study
Analysis of fractal dimensions of rat bones from film and digital images
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.
2001-01-01
OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.
Wang Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.
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.
Low Fractal Dimension Cluster-Dilute Soot Aggregates from a Premixed Flame
Rajan K. Chakrabarty; Hans Moosmüller; W. Patrick Arnott; Mark A. Garro; Guoxun Tian; Jay G. Slowik; Eben S. Cross; Jeong-Ho Han; Paul Davidovits; Timothy B. Onasch; Douglas R. Worsnop
2009-01-01
Using a novel morphology segregation technique, we observed minority populations (≈3%) of submicron-sized, cluster-dilute fractal-like aggregates, formed in the soot-formation window (fuel-to-air equivalence ratio of 2.0-3.5) of a premixed flame, to have mass fractal dimensions between 1.2 and 1.51. Our observations disagree with previous observations of a universal mass fractal dimension of ≈1.8 for fractal-like aerosol aggregates formed in the
NASA Astrophysics Data System (ADS)
Dathe, A.; Baveye, P.
2003-04-01
The structure of many natural objects exhibits a self-similar or self-affine scaling behavior. Examples range from porous media to mountain ranges, river networks, clouds in the atmosphere or the mass distribution in the universe. The fractal dimension of those objects can be measured by image analyzing techniques. Concerning the structure of porous media, there are different features which may show a fractal behavior: The mass distribution, the pore space and the pore-solid interface. In many cases the fractal dimension of one of these features has been determined and has been taken to describe the entire system. Beyond this it is interesting what the fractal dimensions are and how they are related for the different features of the same object. The measurements for three natural systems (a silty soil structure, a dendrite and the void system of a clayey soil) are presented and compared with measurements of a textbook fractal (Sirpinski carpet). The length scale of the natural objects extends from 1 mm for SEM images of an impregnated Luvisol over the moss agate to about 400 mm for the void system. Results show different dimensions for different features of the same object with a tendency to higher fractal dimensions for the mass distribution and lower fractal dimensions for the interface. The measured results are compared with a pore-solid fractal model of soil structure.
On the fractal dimension of orbits compatible with Tsallis statistics
NASA Astrophysics Data System (ADS)
Carati, A.
2008-03-01
In a previous paper [A. Carati, Physica A 348 (2005) 110-120] it was shown how, for a dynamical system, the probability distribution function of sojourn-times in phase-space, defined in terms of the dynamical orbits (up to a given observation time), induces unambiguously a statistical ensemble in phase-space. In the present paper, the p.d.f. of the sojourn-times corresponding to a Tsallis ensemble is obtained (this, by the way, requires the solution of a problem of a general character, disregarded in paper [A. Carati, Physica A 348 (2005) 110-120]). In particular some qualitative properties, such as the fractal dimension, of the dynamical orbits compatible with the Tsallis ensembles are indicated.
Scaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers
Hans-Karl Janssen; Olaf Stenull
2012-03-13
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.
Scaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers
Janssen, Hans-Karl
2012-01-01
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.
Scaling exponents for a monkey on a tree: fractal dimensions of randomly branched polymers.
Janssen, Hans-Karl; Stenull, Olaf
2012-05-01
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results. PMID:23004722
Scaling exponents for a monkey on a tree: Fractal dimensions of randomly branched polymers
NASA Astrophysics Data System (ADS)
Janssen, Hans-Karl; Stenull, Olaf
2012-05-01
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results.
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.
Urschler, Martin; Kullnig, Peter; Stollberger, Rudolf; Kovacs, Gabor; Olschewski, Andrea; Olschewski, Horst; Bálint, Zoltán
2014-01-01
Pulmonary hypertension (PH) can result in vascular pruning and increased tortuosity of the blood vessels. In this study we examined whether automatic extraction of lung vessels from contrast-enhanced thoracic computed tomography (CT) scans and calculation of tortuosity as well as 3D fractal dimension of the segmented lung vessels results in measures associated with PH. In this pilot study, 24 patients (18 with and 6 without PH) were examined with thorax CT following their diagnostic or follow-up right-sided heart catheterisation (RHC). Images of the whole thorax were acquired with a 128-slice dual-energy CT scanner. After lung identification, a vessel enhancement filter was used to estimate the lung vessel centerlines. From these, the vascular trees were generated. For each vessel segment the tortuosity was calculated using distance metric. Fractal dimension was computed using 3D box counting. Hemodynamic data from RHC was used for correlation analysis. Distance metric, the readout of vessel tortuosity, correlated with mean pulmonary arterial pressure (Spearman correlation coefficient: ??=?0.60) and other relevant parameters, like pulmonary vascular resistance (??=?0.59), arterio-venous difference in oxygen (??=?0.54), arterial (??=??0.54) and venous oxygen saturation (??=??0.68). Moreover, distance metric increased with increase of WHO functional class. In contrast, 3D fractal dimension was only significantly correlated with arterial oxygen saturation (??=?0.47). Automatic detection of the lung vascular tree can provide clinically relevant measures of blood vessel morphology. Non-invasive quantification of pulmonary vessel tortuosity may provide a tool to evaluate the severity of pulmonary hypertension. Trial Registration ClinicalTrials.gov NCT01607489 PMID:24498123
NASA Astrophysics Data System (ADS)
Ram, Avadh; Roy, P. N. S.
2005-03-01
Several destructive earthquakes have occurred in the Kachchh region of Gujarat during the past two centuries, among them Allah Bund earthquake (M7.8) in 1819, Anjar earthquake (M6) in 1956 and the recent Bhuj earthquake (M7.6) in 2001. The Anjar earthquake was on KMF (Kachchh Mainland Fault) and the recent Bhuj events were caused by a hidden fault north of KMF. The present study discusses the fractal analysis of tectonics governing seismic activity in the region. The region has been divided into five blocks and the fractal dimension of each block has been calculated using the box-counting technique. The results show significantly low value of fractal dimension of the Kachchh rift block consisting of the KMF compared to the other surrounding blocks, which also contain faults and rifts of higher fractal dimension. This indicates that the cause of earthquakes in this block may be asperities and barriers. However, the predominance of aftershocks over foreshocks signifies that barriers may be the main cause. The other results, such as the lower value of dimension of fault clustering show that the Kachchh rift block has faults which are distributed in a clustered manner. In this context, the seismicity of this block seems to be high.
Act of CVT and EVT In The Formation of Number-Theoretic Fractals
Pabitra, Pal Choudhury; Kumar, Nayak Birendra; Sarif, Hassan Sk
2009-01-01
In this paper we have defined two functions that have been used to construct different fractals having fractal dimensions between 1 and 2. More precisely, we can say that one of our defined functions produce the fractals whose fractal dimension lies in [1.58, 2) and rest function produce the fractals whose fractal dimension lies in (1, 1.58]. Also we tried to calculate the amount of increment of fractal dimension in accordance with base of the number systems. And in switching of fractals from one base to another, the increment of fractal dimension is constant, which is 1.58, its quite surprising!
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.
Hongbo Cheng
2011-09-06
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.
Diluted networks of nonlinear resistors and fractal dimensions of percolation clusters.
Janssen, H K; Stenull, O
2000-05-01
We study random networks of nonlinear resistors, which obey a generalized Ohm's law V approximately Ir. Our renormalized field theory, which thrives on an interpretation of the involved Feynman diagrams as being resistor networks themselves, is presented in detail. By considering distinct values of the nonlinearity r, we calculate several fractal dimensions characterizing percolation clusters. For the dimension associated with the red bonds we show that dred = 1/nu at least to order O(epsilon 4), with nu being the correlation length exponent, and epsilon = 6 - d, where d denotes the spatial dimension. This result agrees with a rigorous one by Coniglio. Our result for the chemical distance, dmin = 2 - epsilon/6 - [937/588 + 45/49(ln 2 - 9/10 ln 3)](epsilon/6)2 + O(epsilon 3) verifies a previous calculation by one of us. For the backbone dimension we find DB = 2 + epsilon/21 - 172 epsilon 2/9261 + 2[-74639 + 22680 zeta(3)]epsilon 3/4084101 + O(epsilon 4), where zeta(3) = 1.202057..., in agreement to second order in epsilon with a two-loop calculation by Harris and Lubensky. PMID:11031523
The b-value and fractal dimension of local seismicity around Koyna Dam (India)
NASA Astrophysics Data System (ADS)
Kumar, Arjun; Rai, S. S.; Joshi, Anand; Mittal, Himanshu; Sachdeva, Rajiv; Kumar, Rohtash; Ghangas, Vandana
2013-11-01
Earthquakes began to occur in Koyna region (India) soon after the filling of Koyna Dam in 1962. In the present study, three datasets 1964-1993, 1993-1995, and 1996-1997 are analyzed to study the b-value and fractal dimension. The b-value is calculated using the Gutenberg-Richter relationship and fractal dimension D corr. using correlation integral method. The estimated b-value and D corr. of this region before 1993 are found to be in good agreement with previously reported studies. In the subsequent years after 1995, the b-value shows an increase. The estimated b-values of this region are found within the limits of global average. Also, the pattern of spatial clustering of earthquakes show increase in clustering and migration along the three zones called North-East Zone, South-East Zone (SEZ), and Warna Seismic Zone. The earthquake events having depth ?5 km are largely confined to SEZ. After 1993, the D corr. shows decrease, implying that earthquake activity gets clustered. This seismic clustering could be helpful for earthquake forecasting.
NASA Astrophysics Data System (ADS)
Neves, L. A.; Oliveira, F. R.; Peres, F. A.; Moreira, R. D.; Moriel, A. R.; de Godoy, M. F.; Murta Junior, L. O.
2011-03-01
This paper presents a method for the quantification of cellular rejection in endomyocardial biopsies of patients submitted to heart transplant. The model is based on automatic multilevel thresholding, which employs histogram quantification techniques, histogram slope percentage analysis and the calculation of maximum entropy. The structures were quantified with the aid of the multi-scale fractal dimension and lacunarity for the identification of behavior patterns in myocardial cellular rejection in order to determine the most adequate treatment for each case.
Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.
2001-01-01
Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.
COMPARISON OF FRACTAL DIMENSION ALGORITHMS FOR THE COMPUTATION OF EEG BIOMARKERS FOR DEMENTIA
Paris-Sud XI, Université de
Alzheimer's Disease. Keywords: Fractal Dimension, factal dimension algorithms, EEG, dementia, biomarkers INTRODUCTION Improved life expectancy due to better lifestyle and nutrition has led to a significant increase for the mild symptomatic relief of dementia of the Alzheimer's type (DAT). Nonetheless, unless sufferers
NASA Astrophysics Data System (ADS)
Pu, Yang; Wang, Wubao; Alrubaiee, M.; Gayen, S. K.; Xu, Min
2012-02-01
The optical coefficients (?s, ?a, ?'s and g)of human cancerous and normal prostate tissues were investigated and compared in the spectral range of 750nm - 860 nm. The fractal dimensional parameters including fractal dimension (Df), cutoff diameter (dmax) and the most efficient diameter (dm) between the cancerous and normal prostate tissues were determined based on the extinction and diffusion reflection intensity measurements and the determination of?s, ?a, ?'s and g. The results are in good agreement with prostate cancer evolution defined by Gleason Grades. The difference of fractal dimensional parameters and optic
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.
On the relationship between fractal dimension and encounters in three-dimensional trajectories.
Uttieri, Marco; Cianelli, Daniela; Strickler, J Rudi; Zambianchi, Enrico
2007-08-01
The encounter of individuals-prey, predators and mates-living in the surrounding environment is a fundamental process in the life of an organism. Along with the sensory abilities, this process will be regulated by the movement rules adopted by the individual. In this work we discuss the encounter-enhancement effect due to different natatorial modes by calculating the number of encounters realised by differently convoluted trajectories in two homogeneous distributions of particles. Using numerically generated trajectories representative of specific swimming behaviour, we demonstrate that high values of three-dimensional fractal dimension D(3D)(>1.9) are beneficial only at high concentration, whereas at low concentration less tortuous tracks (D(3D) approximately 1.5) are almost equally efficient. In the light of our results it is possible to better understand the behavioural adaptations evolved by individuals to thrive in their environment. PMID:17467741
Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation
NASA Astrophysics Data System (ADS)
Mistakidis, E. S.; Panagouli, O. K.; Panagiotopoulos, P. D.
The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the ``fixed points'' of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C0-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.
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. Mecholsky, J.J.
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.
Model to estimate fractal dimension for ion-bombarded materials
NASA Astrophysics Data System (ADS)
Hu, A.; Hassanein, A.
2014-03-01
Comprehensive fractal Monte Carlo model ITMC-F (Hu and Hassanein, 2012 [1]) is developed based on the Monte Carlo ion bombardment simulation code, i.e., Ion Transport in Materials and Compounds (ITMC) code (Hassanein, 1985 [2]). The ITMC-F studies the impact of surface roughness on the angular dependence of sputtering yield. Instead of assuming material surfaces to be flat or composed of exact self-similar fractals in simulation, we developed a new method to describe the surface shapes. Random fractal surfaces which are generated by midpoint displacement algorithm and support vector machine algorithm are combined with ITMC. With this new fractal version of ITMC-F, we successfully simulated the angular dependence of sputtering yield for various ion-target combinations, with the input surface roughness exponent directly depicted from experimental data (Hu and Hassanein, 2012 [1]). The ITMC-F code showed good agreement with the experimental data. In advanced, we compare other experimental sputtering yield with the results from ITMC-F to estimate the surface roughness exponent for ion-bombarded material in this research.
Salinas-Nolasco, Manlio Favio; Méndez-Vivar, Juan
2010-03-16
Among several analysis techniques applied to the study of surface passivation using dicarboxylic acids, small angle X-ray scattering (SAXS) has proved to be relevant in the physicochemical interpretation of the surface association resulting between calcium carbonate and the molecular structure of malonic acid. It is possible to establish chemical affinity principles through bidimensional geometric analysis in terms of the fractal dimension obtained experimentally by SAXS. In this Article, we present results about the adsorption of malonic acid on calcite, using theoretical and mathematical principles of the fractal dimension. PMID:20163152
Empirically derived relationships between fractal dimension and power law form frequency spectra
NASA Astrophysics Data System (ADS)
Fox, Christopher G.
1989-03-01
Fractal analysis and Fourier analysis are independent techniques for quantitatively describing the variability of natural figures. Both methods have been applied to a variety of natural phenomena. Previous analytical work has formulated relationships between the fractal dimension and power law form frequency spectrum. Mandelbrot (1985) has shown that difficulties arise when the ruler method for measuring dimensionality is applied to other than self-similar figures. Since an investigator presumably does not know in advance the dimensionality of a natural profile, it is essential to quantify the nature of the discrepancy for self-affine cases. In this study, a series of experiments are conducted in which discrete random series of specified spectral forms are analyzed using the fractal ruler method. The various parameters of the fractal measurement are related to the parameters of the spectral model. In this way, empirical relationships between the techniques can be derived for discrete, finite series which simulate the results of applying the fractal method to observational data. The results of the study indicate that there are considerable discrepancies between the results predicted by theory and those derived empirically. The fundamental power law form of length versus resolution pairs does not hold over the entire region of analysis. The predicted linear relationship between fractal dimension and exponent of the frequency spectrum does not hold, and the spectral signals can be extended beyond the limits of dimension inferred by theory. Root-mean-square variability is also shown to be linearly related to the fractal intercept term. An investigation of the effect of nonstationary sampling is conducted by generating signals composed of segments of differing spectral characteristics. Fractal analyses of these signals appear identical to those conducted on stationary series. The discrepancies between theoretical prediction and empirical results described in this study reflect the difficulties of applying analytically derived techniques to measurement data. Both Fourier and fractal techniques are formulated through rigorous mathematics, assuming various conditions for the underlying signal. When these techniques are applied to discrete, finite length, nonstationary series, certain statistical transformations must be applied to the data. Methods such as windowing, prewhitening, and anti-aliasing filters have been developed over many years for use with Fourier analysis. At present, no such statistical theory exists for use with fractal analysis. It is apparent from the results of this study that such a statistical foundation is required before the fractal ruler method can be routinely applied to observational data.
A. A. B. Shirazi; L. Nasseri
2008-01-01
A novel algorithm to classify iris images automatically is announced by this paper, which apply differential of fractal dimension (DFD) by determined fractal dimension of image with box counting method, and using neural network. First iris image is separated from other part of captured eye image then normalized. After that normalized iris images partition to sixteen boxes that size of
Fractal dimension of 3-blocks in four-, five-, and six-dimensional percolation systems.
Paul, Gerald; Stanley, H Eugene
2003-02-01
Using Monte Carlo simulations, we study the distributions of the 3-block mass N3 in four-, five-, and six-dimensional percolation systems. Because the probability of creating large 3-blocks in these dimensions is very small, we use a "go with the winners" method of statistical enhancement to simulate configurations having probability as small as 10(-30). In earlier work, the fractal dimensions of 3-blocks, d(3), in 2D (two dimensional) and 3D were found to be 1.20+/-0.1 and 1.15+/-0.1, respectively, consistent with the possibility that the fractal dimension might be the same in all dimensions. We find that the fractal dimension of 3-blocks decreases rapidly in higher dimensions, and estimate d(3)=0.7+/-0.2 (4D) and 0.5+/-0.2 (5D). At the upper critical dimension of percolation, d(c)=6, our simulations are consistent with d(3)=0 with logarithmic corrections to power-law scaling. PMID:12636744
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...
Di Ieva, Antonio; Grizzi, Fabio; Ceva-Grimaldi, Giorgia; Russo, Carlo; Gaetani, Paolo; Aimar, Enrico; Levi, Daniel; Pisano, Patrizia; Tancioni, Flavio; Nicola, Giancarlo; Tschabitscher, Manfred; Dioguardi, Nicola; Baena, Riccardo Rodriguez y
2007-01-01
It is well known that angiogenesis is a complex process that accompanies neoplastic growth, but pituitary tumours are less vascularized than normal pituitary glands. Several analytical methods aimed at quantifying the vascular system in two-dimensional histological sections have been proposed, with very discordant results. In this study we investigated the non-Euclidean geometrical complexity of the two-dimensional microvasculature of normal pituitary glands and pituitary adenomas by quantifying the surface fractal dimension that measures its space-filling property. We found a statistical significant difference between the mean vascular surface fractal dimension estimated in normal versus adenomatous tissues (P = 0.01), normal versus secreting adenomatous tissues (P = 0.0003), and normal versus non-secreting adenomatous tissues (P = 0.047), whereas the difference between the secreting and non-secreting adenomatous tissues was not statistically significant. This study provides the first demonstration that fractal dimension is an objective and valid quantitator of the two-dimensional geometrical complexity of the pituitary gland microvascular network in physiological and pathological states. Further studies are needed to compare the vascular surface fractal dimension estimates in different subtypes of pituitary tumours and correlate them with clinical parameters in order to evaluate whether the distribution pattern of vascular growth is related to a particular state of the pituitary gland. PMID:17784937
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.
Quantifying degeneration of white matter in normal aging using fractal dimension
Luduan Zhang; David Dean; Jing Z. Liu; Vinod Sahgal; Xiaofeng Wang; Guang H. Yue
2007-01-01
Although degeneration of brain white matter (WM) in aging is a well-recognized problem, its quantification has mainly relied on volumetric measurements, which lack detail in describing the degenerative adaptation. In this study, WM structural complexity was evaluated in healthy old and young adults by analyzing the three-dimensional fractal dimension (FD) of WM segmented from magnetic resonance images of brain. FDs
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.
Topological pressure and fractal dimensions of cookie-cutter-like sets
Roychowdhury, Mrinal Kanti
2012-01-01
The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, we have shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.
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.
Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke
Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca
2014-01-01
The brain is a self-organizing system which displays self-similarities at different spatial and temporal scales. Thus, the complexity of its dynamics, associated to efficient processing and functional advantages, is expected to be captured by a measure of its scale-free (fractal) properties. Under the hypothesis that the fractal dimension (FD) of the electroencephalographic signal (EEG) is optimally sensitive to the neuronal dysfunction secondary to a brain lesion, we tested the FD’s ability in assessing two key processes in acute stroke: the clinical impairment and the recovery prognosis. Resting EEG was collected in 36 patients 4–10 days after a unilateral ischemic stroke in the middle cerebral artery territory and 19 healthy controls. National Health Institute Stroke Scale (NIHss) was collected at T0 and 6 months later. Highuchi FD, its inter-hemispheric asymmetry (FDasy) and spectral band powers were calculated for EEG signals. FD was smaller in patients than in controls (1.447±0.092 vs 1.525±0.105) and its reduction was paired to a worse acute clinical status. FD decrease was associated to alpha increase and beta decrease of oscillatory activity power. Larger FDasy in acute phase was paired to a worse clinical recovery at six months. FD in our patients captured the loss of complexity reflecting the global system dysfunction resulting from the structural damage. This decrease seems to reveal the intimate nature of structure-function unity, where the regional neural multi-scale self-similar activity is impaired by the anatomical lesion. This picture is coherent with neuronal activity complexity decrease paired to a reduced repertoire of functional abilities. FDasy result highlights the functional relevance of the balance between homologous brain structures’ activities in stroke recovery. PMID:24967904
Fractal dimension of EEG activity senses neuronal impairment in acute stroke.
Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca
2014-01-01
The brain is a self-organizing system which displays self-similarities at different spatial and temporal scales. Thus, the complexity of its dynamics, associated to efficient processing and functional advantages, is expected to be captured by a measure of its scale-free (fractal) properties. Under the hypothesis that the fractal dimension (FD) of the electroencephalographic signal (EEG) is optimally sensitive to the neuronal dysfunction secondary to a brain lesion, we tested the FD's ability in assessing two key processes in acute stroke: the clinical impairment and the recovery prognosis. Resting EEG was collected in 36 patients 4-10 days after a unilateral ischemic stroke in the middle cerebral artery territory and 19 healthy controls. National Health Institute Stroke Scale (NIHss) was collected at T0 and 6 months later. Highuchi FD, its inter-hemispheric asymmetry (FDasy) and spectral band powers were calculated for EEG signals. FD was smaller in patients than in controls (1.447±0.092 vs 1.525±0.105) and its reduction was paired to a worse acute clinical status. FD decrease was associated to alpha increase and beta decrease of oscillatory activity power. Larger FDasy in acute phase was paired to a worse clinical recovery at six months. FD in our patients captured the loss of complexity reflecting the global system dysfunction resulting from the structural damage. This decrease seems to reveal the intimate nature of structure-function unity, where the regional neural multi-scale self-similar activity is impaired by the anatomical lesion. This picture is coherent with neuronal activity complexity decrease paired to a reduced repertoire of functional abilities. FDasy result highlights the functional relevance of the balance between homologous brain structures' activities in stroke recovery. PMID:24967904
The Hausdorff dimension of fractal sets and fractional quantum Hall effect
Wellington da Cruz
2003-05-27
We consider Farey series of rational numbers in terms of {\\it fractal sets} labeled by the Hausdorff dimension with values defined in the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$ and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of {\\it dual topological quantum numbers}, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that {\\it the universality classes of the quantum Hall transitions are classified in terms of $h$}. The connection between Number Theory and Physics appears naturally in this context.
NASA Astrophysics Data System (ADS)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to be studied, and use the commands on the package to (for instance) calculate the fractal dimension of a certain boundary, without knowing or worrying about a single line of C programming. So the package combines the flexibility and friendly aspect of Maple with the fast and robust numerical integration of the compiled (for example C) basin. The package is old, but the problems it was designed to dealt with are still there. Since Maple evolved, the package stopped working, and we felt compelled to produce this version, fully compatible with the latest version of Maple, to make it again available to the Maple user. Summary of revisions Deprecated Maple Packages and Commands: Paraphrasing the Maple in-built help files, "Some Maple commands and packages are deprecated. A command (or package) is deprecated when its functionality has been replaced by an improved implementation. The newer command is said to supersede the older one, and use of the newer command is strongly recommended". So, we have examined our code to see if some of these occurrences could be dangerous for it. For example, the "readlib" command is unnecessary, and we have removed its occurrences from our code. We have checked and changed all the necessary commands in order for us to be safe in respect to danger from this source. Another change we had to make was related to the tools we have implemented in order to use the interface for performing the numerical integration in C, externally, via the use of the Maple command "ssystem". In the past, we had used, for the external C integration, the DJGPP system. But now we present the package with (free) Borland distribution. The compilation and compiling commands are now slightly changed. For example, to compile only, we had used "gcc-c"; now, we use "bcc32-c", etc. All this installation (Borland) is explained on a "README" file we are submitting here to help the potential user. Restrictions Besides the inherent restrictions of numerical integration methods, this version of the package only deals w
Huang, F.; Peng, R. D.; Liu, Y. H.; Chen, Z. Y.; Ye, M. F.; Wang, L.
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.
Improving accuracy and precision in estimating fractal dimension of animal movement paths.
Nams, Vilis O
2006-01-01
It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step. PMID:16823606
Landmine detection using IR image segmentation by means of fractal dimension analysis
NASA Astrophysics Data System (ADS)
Abbate, Horacio A.; Gambini, Juliana; Delrieux, Claudio; Castro, Eduardo H.
2009-05-01
This work is concerned with buried landmines detection by long wave infrared images obtained during the heating or cooling of the soil and a segmentation process of the images. The segmentation process is performed by means of a local fractal dimension analysis (LFD) as a feature descriptor. We use two different LFD estimators, box-counting dimension (BC), and differential box counting dimension (DBC). These features are computed in a per pixel basis, and the set of features is clusterized by means of the K-means method. This segmentation technique produces outstanding results, with low computational cost.
Stanley, H. Eugene
1995-01-01
ELSEVIER Journal of Neuroscience Methods 56 (1995) 133-144 Determination of fractal dimension Elsevier Science B.V. All rights reserved SFDI O165-0270(94)00115-4 #12;134 F. Caserta et al. /Journal of Neuroscience Methods 56 (199s) 133-l 44 mathematically related. Third, we extend the method of fractal analysis
Fractal Dimension and Vessel Complexity in Patients with Cerebral Arteriovenous Malformations
Reishofer, Gernot; Koschutnig, Karl; Enzinger, Christian; Ebner, Franz; Ahammer, Helmut
2012-01-01
The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs. PMID:22815946
Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; Dou, Jie; Wang, Cheng
2015-08-12
The morphological properties of kaolin flocs were investigated in a Couette-flow experiment at the steady state under seven shear flow conditions (shear rates of 5.36, 9.17, 14, 24, 31, 41 and 53 s-1). These properties include a one-dimensional (1-D) fractal dimension (D1), a two-dimensional (2-D) fractal dimension (D2), a perimeter-based fractal dimension (Dpf) and an aspect ratio (AR). They were calculated based on the projected area (A), equivalent size, perimeter (P) and length (L) of the major axis of the floc determined through sample observation and an image analysis system. The parameter D2, which characterizes the relationship between the projectedmore »area and the length of the major axis using a power function, A ? LD2, increased from 1.73 ± 0.03, 1.72 ± 0.03, and 1.75 ± 0.04 in the low shear rate group (G = 5.36, 9.17, and 14 s-1) to 1.92 ± 0.03, 1.82 ± 0.02, 1.85 ± 0.02, and 1.81 ± 0.02 in the high shear rate group (24, 31, 41 and 53 s-1), respectively. The parameter D1 characterizes the relationship between the perimeter and length of the major axis by the function P ? LD1 and decreased from 1.52 ± 0.02, 1.48 ± 0.02, 1.55 ± 0.02, and 1.63 ± 0.02 in the low shear group (5.36, 9.17, 14 and 24 s-1) to 1.45 ± 0.02, 1.39 ± 0.02, and 1.39 ± 0.02 in the high shear group (31, 41 and 53 s-1), respectively. The results indicate that with increasing shear rates, the flocs become less elongated and that their boundary lines become tighter and more regular, caused by more breakages and possible restructurings of the flocs. The parameter Dpf, which is related to the perimeter and the projected area through the function , decreased as the shear rate increased almost linearly. The parameter AR, which is the ratio of the length of the major axis and equivalent diameter, decreased from 1.56, 1.59, 1.53 and 1.51 in the low shear rate group to 1.43, 1.47 and 1.48 in the high shear rate group. These changes in Dpf and AR show that the flocs become less convoluted and more symmetrical and that their boundaries become smoother and more regular in the high shear rate group than in the low shear rate group due to breakage and possible restructuring processes. To assess the effects of electrolyte and sediment concentration, 0.1 mol/L calcium chloride (CaCl2) and initial sediment concentration from 7.87 × 10-5 to 1.57 × 10-5 were used in this preliminary study. The addition of electrolyte and increasing sediment concentration could produce more symmetrical flocs with less convoluted and simpler boundaries. In addition, some new information on the temporal variation of the median size of the flocs during the flocculation process is presented.« less
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
Transition of fractal dimension in a latticed dynamical system
Duong-van, M.
1986-03-01
We study a recursion relation that manifests two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigenbaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns are reproduced in this new route. In the oscillatory instability regime the fracal dimension saturates at D/sub F/ approx. = 2.6 with imbedding dimensions while in the turbulent regime D/sub F/ saturates at 6.0. 19 refs., 3 figs.
Earthquake frequency-magnitude distribution and fractal dimension in mainland Southeast Asia
NASA Astrophysics Data System (ADS)
Pailoplee, Santi; Choowong, Montri
2014-12-01
The 2004 Sumatra and 2011 Tohoku earthquakes highlighted the need for a more accurate understanding of earthquake characteristics in both regions. In this study, both the a and b values of the frequency-magnitude distribution (FMD) and the fractal dimension ( D C ) were investigated simultaneously from 13 seismic source zones recognized in mainland Southeast Asia (MLSEA). By using the completeness earthquake dataset, the calculated values of b and D C were found to imply variations in seismotectonic stress. The relationships of D C -b and D C -( a/ b) were investigated to categorize the level of earthquake hazards of individual seismic source zones, where the calibration curves illustrate a negative correlation between the D C and b values ( D c = 2.80 - 1.22 b) and a positive correlation between the D C and a/ b ratios ( D c = 0.27( a/ b) - 0.01) with similar regression coefficients ( R 2 = 0.65 to 0.68) for both regressions. According to the obtained relationships, the Hsenwi-Nanting and Red River fault zones revealed low-stress accumulations. Conversely, the Sumatra-Andaman interplate and intraslab, the Andaman Basin, and the Sumatra fault zone were defined as high-tectonic stress regions that may pose risks of generating large earthquakes in the future.
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
Can one hear the dimension of a fractal?
Jean Brossard; René Carmona
1986-01-01
We consider the spectrum of the Laplacian in a bounded open domain of Rn with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the
Pankaj M. Bhattacharya; J. R. Kayal; Saurabh Baruah; S. S. Arefiev
2010-01-01
We have imaged earthquake source zones beneath the northeast India region by seismic tomography, fractal dimension and b value mapping. 3D P-wave velocity (Vp) structure is imaged by the Local Earthquake Tomography (LET) method. High precision P-wave (3,494) and S-wave (3,064) travel times of 980 selected earthquakes, m d >= 2.5, are used. The events were recorded by 77 temporary\\/permanent
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…
Characterization of the irregularity of a terrain using fractal dimension of lakes' boundaries
Karle, Nakul N
2014-01-01
Even though many objects and phenomena of importance in geophysics have been shown to have fractal character, there are still many of them which show self-similar character and yet to be studied. The objective of the present work is to demonstrate that the fractal dimension of the boundary of a natural water body can be used to shed light on irregularity as well as other properties of a region. Owing to easy availability of satellite images and image processing softwares this turns out to be a handy tool. In this study, we have analyzed several lakes in India mostly around the Western Ghats region. We find that the fractal dimension of their boundaries for the length scales between around 40 meters to 2 kilometers, in general, has broad variation from 1.2 to 1.6. But when they are grouped into three categories, viz., lakes along the ridge of Western Ghats, lakes in the planes and lakes in the mountain region, we find the first two groups to have a narrower distribution of dimensions.
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.
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.
Limitation of the Least Square Method in the Evaluation of Dimension of Fractal Brownian Motions
Qiao, Bingqiang; Zeng, Houdun; Li, Xiang; Dai, Benzhong
2015-01-01
With the standard deviation for the logarithm of the re-scaled range $\\langle |F(t+\\tau)-F(t)|\\rangle$ of simulated fractal Brownian motions $F(t)$ given in a previous paper \\cite{q14}, the method of least squares is adopted to determine the slope, $S$, and intercept, $I$, of the log$(\\langle |F(t+\\tau)-F(t)|\\rangle)$ vs $\\rm{log}(\\tau)$ plot to investigate the limitation of this procedure. It is found that the reduced $\\chi^2$ of the fitting decreases with the increase of the Hurst index, $H$ (the expectation value of $S$), which may be attributed to the correlation among the re-scaled ranges. Similarly, it is found that the errors of the fitting parameters $S$ and $I$ are usually smaller than their corresponding standard deviations. These results show the limitation of using the simple least square method to determine the dimension of a fractal time series. Nevertheless, they may be used to reinterpret the fitting results of the least square method to determine the dimension of fractal Brownian motions more...
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.)
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.
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.; Anez, Neyda; Odekon, Mary Crone
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.
Act of CVT in the Formation of Music Fractals
Choudhury, Pabitra Pal; Sahoo, Sudhakar; Nayak, Birendra Kumar
2009-01-01
In this paper we have defined one function that have been used to construct different fractals having fractal dimensions between 1.58 and 2. Also, we tried to calculate the amount of increment of fractal dimension in accordance with the base of the number systems. And in switching of fractals from one base to another, the increment of fractal dimension is constant, which is 1.58, its quite surprising. Further, interestingly enough, these very fractals could be a frame of lyrics for the musicians, as we know the fractal dimension of music is around 1.65 and varies between a high of 1.68 and a low of 1.60. further, at the end we conjecture that the switching form one music fractal to another is nothing but enhancing a constant amount of musical notes in various orientations.
Pankaj M. Bhattacharya; J. R. Kayal; Saurabh Baruah; S. S. Arefiev
2010-01-01
We have imaged earthquake source zones beneath the northeast India region by seismic tomography, fractal dimension and b value mapping. 3D P-wave velocity (Vp) structure is imaged by the Local Earthquake Tomography (LET) method. High precision\\u000a P-wave (3,494) and S-wave (3,064) travel times of 980 selected earthquakes, m\\u000a d ? 2.5, are used. The events were recorded by 77 temporary\\/permanent seismic stations
Pankaj M. Bhattacharya; J. R. Kayal; Saurabh Baruah; S. S. Arefiev
\\u000a We have imaged earthquake source zones beneath the northeast India region by seismic tomography, fractal dimension and b value mapping. 3D P-wave velocity (Vp) structure is imaged by the Local Earthquake Tomography (LET) method. High precision P-wave (3,494) and S-wave (3,064) travel\\u000a times of 980 selected earthquakes, m\\u000a d ? 2.5, are used. The events were recorded by 77 temporary\\/permanent
Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy
NASA Astrophysics Data System (ADS)
Kristoufek, Ladislav; Vosvrda, Miloslav
2014-07-01
We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [L. Kristoufek, M. Vosvrda, Physica A 392, 184 (2013)]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock market indices across the world, we find that the most efficient markets are situated in the Eurozone (the Netherlands, France and Germany) and the least efficient ones in the Latin America (Venezuela and Chile).
Grizzi, Fabio; Russo, Carlo; Colombo, Piergiuseppe; Franceschini, Barbara; Frezza, Eldo E; Cobos, Everardo; Chiriva-Internati, Maurizio
2005-01-01
Background Modeling the complex development and growth of tumor angiogenesis using mathematics and biological data is a burgeoning area of cancer research. Architectural complexity is the main feature of every anatomical system, including organs, tissues, cells and sub-cellular entities. The vascular system is a complex network whose geometrical characteristics cannot be properly defined using the principles of Euclidean geometry, which is only capable of interpreting regular and smooth objects that are almost impossible to find in Nature. However, fractal geometry is a more powerful means of quantifying the spatial complexity of real objects. Methods This paper introduces the surface fractal dimension (Ds) as a numerical index of the two-dimensional (2-D) geometrical complexity of tumor vascular networks, and their behavior during computer-simulated changes in vessel density and distribution. Results We show that Ds significantly depends on the number of vessels and their pattern of distribution. This demonstrates that the quantitative evaluation of the 2-D geometrical complexity of tumor vascular systems can be useful not only to measure its complex architecture, but also to model its development and growth. Conclusions Studying the fractal properties of neovascularity induces reflections upon the real significance of the complex form of branched anatomical structures, in an attempt to define more appropriate methods of describing them quantitatively. This knowledge can be used to predict the aggressiveness of malignant tumors and design compounds that can halt the process of angiogenesis and influence tumor growth. PMID:15701176
Taylor, Adele M.; MacGillivray, Thomas J.; Henderson, Ross D.; Ilzina, Lasma; Dhillon, Baljean; Starr, John M.; Deary, Ian J.
2015-01-01
Purpose Cerebral microvascular disease is associated with dementia. Differences in the topography of the retinal vascular network may be a marker for cerebrovascular disease. The association between cerebral microvascular state and non-pathological cognitive ageing is less clear, particularly because studies are rarely able to adjust for pre-morbid cognitive ability level. We measured retinal vascular fractal dimension (Df) as a potential marker of cerebral microvascular disease. We examined the extent to which it contributes to differences in non-pathological cognitive ability in old age, after adjusting for childhood mental ability. Methods Participants from the Lothian Birth Cohort 1936 Study (LBC1936) had cognitive ability assessments and retinal photographs taken of both eyes aged around 73 years (n = 648). IQ scores were available from childhood. Retinal vascular Df was calculated with monofractal and multifractal analysis, performed on custom-written software. Multiple regression models were applied to determine associations between retinal vascular Df and general cognitive ability (g), processing speed, and memory. Results Only three out of 24 comparisons (two eyes × four Df parameters × three cognitive measures) were found to be significant. This is little more than would be expected by chance. No single association was verified by an equivalent association in the contralateral eye. Conclusions The results show little evidence that fractal measures of retinal vascular differences are associated with non-pathological cognitive ageing. PMID:25816017
NASA Astrophysics Data System (ADS)
Guo, Long; Cai, XU
2009-08-01
It is shown that many real complex networks share distinctive features, such as the small-world effect and the heterogeneous property of connectivity of vertices, which are different from random networks and regular lattices. Although these features capture the important characteristics of complex networks, their applicability depends on the style of networks. To unravel the universal characteristics many complex networks have in common, we study the fractal dimensions of complex networks using the method introduced by Shanker. We find that the average 'density' (?(r)) of complex networks follows a better power-law function as a function of distance r with the exponent df, which is defined as the fractal dimension, in some real complex networks. Furthermore, we study the relation between df and the shortcuts Nadd in small-world networks and the size N in regular lattices. Our present work provides a new perspective to understand the dependence of the fractal dimension df on the complex network structure.
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.
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.
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
Pore size distribution in porous glass: fractal dimension obtained by calorimetry
NASA Astrophysics Data System (ADS)
Neffati, R.; Rault, J.
2001-05-01
By differential Scanning Calorimetry (DSC), at low heating rate and using a technique of fractionation, we have measured the equilibrium DSC signal (heat flow) J q 0 of two families of porous glass saturated with water. The shape of the DSC peak obtained by these techniques is dependent on the sizes distribution of the pores. For porous glass with large pore size distribution, obtained by sol-gel technology, we show that in the domain of ice melting, the heat flow Jq is related to the melting temperature depression of the solvent, ? T m , by the scaling law: J q 0˜? T m - (1 + D). We suggest that the exponent D is of the order of the fractal dimension of the backbone of the pore network and we discuss the influence of the variation of the melting enthalpy with the temperature on the value of this exponent. Similar D values were obtained from small angle neutron scattering and electronic energy transfer measurements on similar porous glass. The proposed scaling law is explained if one assumes that the pore size distribution is self similar. In porous glass obtained from mesomorphic copolymers, the pore size distribution is very sharp and therefore this law is not observed. One concludes that DSC, at low heating rate ( q? 2°C/min) is the most rapid and less expensive method for determining the pore distribution and the fractal exponent of a porous material.
Huang, Wen Lai; Cui, Shi Hua; Liang, Kai Ming; Gu, Shou Ren; Yuan, Zhang Fu
2002-02-01
Silica xerogels were prepared by thermal drying wet gels in an electric oven (70 degrees C) after certain duration of ambient drying, and the relevant effect is investigated on the mesopore structures and surface fractal dimensions of the resultant xerogels. The silica gels were derived from a hydrochloric acid-catalyzed TEOS (tetraethylorthaosilicate) system, and both magnetic stirring and ultrasonic vibration were adopted during sol preparation. The percentage mesoporosity and surface fractal dimensions are evaluated using image analysis methods, based on FE-SEM (field emission gun-scanning electron microscopy) images. The results show that the mesoporosity of the resultant xerogels decreases with the duration of ambient drying for samples prepared using magnetic stirring and low-intensity ultrasonic vibration, while samples subjected to high-intensity ultrasound show a somewhat reverse trend. Samples prepared with magnetic stirring have almost constant surface fractal dimensions (nearly 3), irrespective of the ambient drying before thermal drying. The surface fractal dimensions of samples prepared using ultrasound increase with the duration of ambient drying. PMID:16290393
Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Chen, Falin
2014-01-01
Background This study aimed to investigate the correlation between quantitative retinal vascular parameters such as central retinal arteriolar equivalent (CRAE) and retinal vascular fractal dimension (D(f)), and cardiovascular risk factors in the Chinese Han population residing in the in islands of southeast China. Methodology/Principle Findings In this cross-sectional study, fundus photographs were collected and semi-automated analysis software was used to analyze retinal vessel diameters and fractal dimensions. Cardiovascular risk factors such as relevant medical history, blood pressure (BP), lipids, and blood glucose data were collected. Subjects had a mean age of 51.9±12.0 years and included 812 (37.4%) males and 1,357 (62.6%) females. Of the subjects, 726 (33.5%) were overweight, 226 (10.4%) were obese, 272 (12.5%) had diabetes, 738 (34.0%) had hypertension, and 1,156 (53.3%) had metabolic syndrome. After controlling for the effects of potential confounders, multivariate analyses found that age (??=?0.06, P?=?0.008), sex (??=?1.33, P?=?0.015), mean arterial blood pressure (??=??0.12, P<0.001), high-sensitivity C-reactive protein (??=??0.22, P?=?0.008), and CRVE (??=?0.23, P<0.001) were significantly associated with CRAE. Age (??=??0.0012, P<0.001), BP classification (prehypertension: ??=??0.0075, P?=?0.014; hypertension: ??=??0.0131, P?=?0.002), and hypertension history (??=??0.0007, P?=?0.009) were significantly associated with D(f). Conclusions/Significance D(f) exhibits a stronger association with BP than CRAE. Thus, D(f) may become a useful indicator of cardiovascular risk. PMID:25188273
Puškaš, Nela; Zaletel, Ivan; Stefanovi?, Bratislav D; Ristanovi?, Dušan
2015-03-01
Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones. PMID:25603473
Determination of the fractal dimension for the epitaxial n-GaAs surface in the local limit
Torkhov, N. A. Bozhkova, V. G.; Ivonin, I. V.; Novikov, V. A.
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.
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
NASA Astrophysics Data System (ADS)
Albert, Helena; Perugini, Diego; Martí, Joan
2014-05-01
The volcanic unit of Montaña Reventada is an example of magma mixing in Tenerife (Canary Islands, Spain). The eruptive process has been detonated by a basanite intruding into a phonolite magma chamber. This eruption started with a basanite followed by a phonolite. Montaña Reventada phonolite is characterized by the presence of mafic enclaves. These enclaves represent about the 2% of the outcrop and have been classified like basanites, phono-tephrite and tephri-phonolite. The enclaves have different morphologies, from rounded to complex fingers-like structures, and usually exhibit cuspate terminations. This study aims to provide a new perspective on the 1100 AD Montaña Reventada eruption quantifying the textural heterogeneities related to the enclaves generated by the mixing process. The textural study was carried out using a fractal geometry approach, and its results were used to calculate some parameters related to magma chamber dynamics. Photographs of 67 samples were taken normal to the surface of the enclaves with the aim of delineating the contact between the enclaves and the host rocks. The resulted pictures were processed with the NIH (National Institutes of Health) image analysis software to generate binary images in which enclaves and host rock were replaced by black and white pixels, respectively. The fractal dimension (Dbox) has been computed by using the box-counting method in order to quantify the complexity of the enclaves morphology. Viscosity ratio (?R) between the phonolite and the enclaves has been calculated as follows: log(?R) = 0.013e3.34Dbox PIC The viscosity of the enclaves has been calculated according to the ?Rvalue with the higher frequency and to the calculated viscosity of the phonolite between 900° and 1200° . We hypothesized that this value corresponds to the amount of mafic magma present in the system, while the other values represent different degrees of mingling and chemical diffusion. Viscosity of the basanite can be computed like: ?enclave = (%phonolite *?phonolite)+ (%basanite *?basanite) PIC ?enclaves--(%phonolite *?phonolite) ?basanite = %basanite PIC The minimum percentages which satisfy the relation are 69.5% of basanite and 30.5% of phonolite. Although the amount of mafic magma reaches the 69.5%, the presence of enclaves in the phonolite is just the ?1% and the amount of basanite erupted before could correspond to the 15% of the phonolite (estimated from stratigraphic sections). Probably a magma body of basanite was still stored in the magma chamber. The volume of basanite still stored during this time may have evolved to a more explosive magma and hence increases the volcanic risk in the area.
Structural investigations of fat fractals using small-angle scattering
NASA Astrophysics Data System (ADS)
Anitas, Eugen M.
2015-01-01
Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.
Manera, M; Dezfuli, B S; Borreca, C; Giari, L
2014-11-01
Fractal analysis is a reliable method for describing, summarizing object complexity and heterogeneity and has been widely used in biology and medicine to deal with scale, size and shape management problems. The aim of present survey was to use fractal analysis as a complexity measure to characterize mast cells (MCs) degranulation in a rainbow trout ex vivo model (isolated organ bath). Compound 48/80, a condensation product of N-methyl-p-methoxyphenethylamine with formaldehyde, was adopted as MCs degranulation agent in trout intestinal strips. Fractal dimension (D), as a measure of complexity, 'roughness' and lacunarity (?), as a measure of rotational and translational invariance, heterogeneity, in other words, of the texture, were compared in MCs images taken from intestinal strips before and after compound 48/80 addition to evaluate if and how they were affected by degranulation. Such measures were also adopted to evaluate their discrimination efficacy between compound 48/80 degranulated group and not degranulated group and the results were compared with previously reported data obtained with conventional texture analysis (image histogram, run-length matrix, co-occurrence matrix, autoregressive model, wavelet transform) on the same experimental material. Outlines, skeletons and original greyscale images were fractal analysed to evaluate possible significant differences in the measures values according to the analysed feature. In particular, and considering outline and skeleton as analysed features, fractal dimensions from compound 48/80 treated intestinal strips were significantly higher than the corresponding untreated ones (paired t and Wilcoxon test, p < 0.05), whereas corresponding lacunarity values were significantly lower (paired Wilcoxon test, p < 0.05) but only for outline as analysed feature. Outlines roughness increase is consistent with an increased granular mediators interface, favourable for their biological action; while lacunarity (image heterogeneity) reduction is consistent with the biological informative content decrease, due to granule content depletion. In spite of the significant differences in fractal dimension and lacunarity values registered according to the analysed feature (greyscale obtained values were, on average, lower than those obtained from outlines and skeletons; General Linear Model, p < 0.01), the discrimination power between not degranulated and degranulated MCs was, on average, the same and fully comparable with previously performed texture analysis on the same experimental material (outline and skeleton misclassification error, 20% [two false negative cases]; greyscale misclassification error, 30% [two false negative cases and one false positive case]). Fractal analysis proved to be a reliable and objective method for the characterization of MCs degranulation. PMID:25087582
Fractal Dimension and Size Scaling of Domains in Thin Films of Multiferroic BiFeO3
G. Catalan; H. Béa; S. Fusil; M. Bibes; P. Paruch; A. Barthélémy; J. F. Scott
2008-01-01
Domains in ferroelectric films are usually smooth, stripelike, very thin compared with magnetic ones, and satisfy the Landau-Lifshitz-Kittel scaling law (width proportional to square root of film thickness). However, the ferroelectric domains in very thin films of multiferroic BiFeO3 have irregular domain walls characterized by a roughness exponent 0.5 0.6 and in-plane fractal Hausdorff dimension H||=1.4±0.1, and the domain size
Unbiased estimation of multi-fractal dimensions of finite data sets
A. J. Roberts; A. Cronin
1996-02-01
We present a novel method for determining multi-fractal properties from experimental data. It is based on maximising the likelihood that the given finite data set comes from a particular set of parameters in a multi-parameter family of well known multi-fractals. By comparing characteristic correlations obtained from the original data with those that occur in artificially generated multi-fractals with the {\\em same} number of data points, we expect that predicted multi-fractal properties are unbiased by the finiteness of the experimental data.
2013-01-01
Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522
Mossotti, Victor G.; Eldeeb, A. Raouf
2000-01-01
Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.
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)
Assessing severity of obstructive sleep apnea by fractal dimension sequence analysis of sleep EEG
NASA Astrophysics Data System (ADS)
Zhang, J.; Yang, X. C.; Luo, L.; Shao, J.; Zhang, C.; Ma, J.; Wang, G. F.; Liu, Y.; Peng, C.-K.; Fang, J.
2009-10-01
Different sleep stages are associated with distinct dynamical patterns in EEG signals. In this article, we explored the relationship between the sleep architecture and fractal dimension (FD) of sleep EEG. In particular, we applied the FD analysis to the sleep EEG of patients with obstructive sleep apnea-hypopnea syndrome (OSAHS), which is characterized by recurrent oxyhemoglobin desaturation and arousals from sleep, a disease which received increasing public attention due to its significant potential impact on health. We showed that the variation of FD reflects the macrostructure of sleep. Furthermore, the fast fluctuation of FD, as measured by the zero-crossing rate of detrended FD (zDFD), is a useful indicator of sleep disturbance, and therefore, correlates with apnea-hypopnea index (AHI), and hourly number of blood oxygen saturation (SpO 2) decreases greater than 4%, as obstructive apnea/hypopnea disturbs sleep architecture. For practical purpose, a modified index combining zDFD of EEG and body mass index (BMI) may be useful for evaluating the severity of OSAHS symptoms.
Gene Entropy-Fractal Dimension Informatics with Application to Mouse-Human Translational Medicine
Holden, T.; Cheung, E.; Dehipawala, S.; Ye, J.; Tremberger, G.; Lieberman, D.; Cheung, T.
2013-01-01
DNA informatics represented by Shannon entropy and fractal dimension have been used to form 2D maps of related genes in various mammals. The distance between points on these maps for corresponding mRNA sequences in different species is used to study evolution. By quantifying the similarity of genes between species, this distance might be indicated when studies on one species (mouse) would tend to be valid in the other (human). The hypothesis that a small distance from mouse to human could facilitate mouse to human translational medicine success is supported by the studied ESR-1, LMNA, Myc, and RNF4 sequences. ID1 and PLCZ1 have larger separation. The collinearity of displacement vectors is further analyzed with a regression model, and the ID1 result suggests a mouse-chimp-human translational medicine approach. Further inference was found in the tumor suppression gene, p53, with a new hypothesis of including the bovine PKM2 pathways for targeting the glycolysis preference in many types of cancerous cells, consistent with quantum metabolism models. The distance between mRNA and protein coding CDS is proposed as a measure of the pressure associated with noncoding processes. The Y-chromosome DYS14 in fetal micro chimerism that could offer protection from Alzheimer's disease is given as an example. PMID:23586047
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
Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
Joe P. Chen; Stanislav Molchanov; Alexander Teplyaev
2015-07-28
We establish an asymptotic formulas for the eigenvalue counting function of the Schr\\"odinger operator $-\\Delta +V$ for some unbounded potentials $V$ on several types of unbounded fractal spaces. We give sufficient conditions for Bohr's formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.
Spectral dimension and Bohr's formula for Schrödinger operators on unbounded fractal spaces
NASA Astrophysics Data System (ADS)
Chen, Joe P.; Molchanov, Stanislav; Teplyaev, Alexander
2015-09-01
We establish an asymptotic formula for the eigenvalue counting function of the Schrödinger operator -{{? }}+V for some unbounded potentials V on several types of unbounded fractal spaces. We give sufficient conditions for Bohr’s formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.
Zhang, Lihui; Duan, Feng; Huang, Yaji; Chyang, Chiensong
2015-12-01
The changes in pore structure characteristics of sewage sludge particles under effect of calcium magnesium acetate (CMA) during combustion were investigated, the samples were characterized by N2 isothermal absorption method, and the data were used to analyze the fractal properties of the obtained samples. Results show that reaction time and the mole ratio of calcium to sulfur (Ca/S ratio) have notable impact on the pore structure and morphology of solid sample. The Brunauer-Emmett-Teller (BET) specific surface area (SBET) of sample increases with Ca/S ratio, while significant decreases with reaction time. The fractal dimension D has the similar trend with that of SBET, indicating that the surface roughness of sludge increases under the effect of CMA adding, resulting in improved the sludge combustion and the desulfurization process. PMID:26342334
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.
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.
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
Biometric feature extraction using local fractal auto-correlation
NASA Astrophysics Data System (ADS)
Chen, Xi; Zhang, Jia-Shu
2014-09-01
Image texture feature extraction is a classical means for biometric recognition. To extract effective texture feature for matching, we utilize local fractal auto-correlation to construct an effective image texture descriptor. Three main steps are involved in the proposed scheme: (i) using two-dimensional Gabor filter to extract the texture features of biometric images; (ii) calculating the local fractal dimension of Gabor feature under different orientations and scales using fractal auto-correlation algorithm; and (iii) linking the local fractal dimension of Gabor feature under different orientations and scales into a big vector for matching. Experiments and analyses show our proposed scheme is an efficient biometric feature extraction approach.
Vee-Liem Saw; Lock Yue Chew
2015-02-14
We formulate the helicaliser, which replaces a given smooth curve by another curve that winds around it. In our analysis, we relate this formulation to the geometrical properties of the self-similar circular fractal (the discrete version of the curved helical fractal). Iterative applications of the helicaliser to a given curve yields a set of helicalisations, with the infinitely helicalised object being a fractal. We derive the Hausdorff dimension for the infinitely helicalised straight line and circle, showing that it takes the form of the self-similar dimension for a self-similar fractal, with lower bound of 1. Upper bounds to the Hausdorff dimension as functions of $\\omega$ have been determined for the linear helical fractal, curved helical fractal and circular fractal, based on the no-self-intersection constraint. For large number of windings $\\omega\\rightarrow\\infty$, the upper bounds all have the limit of 2. This would suggest that carrying out a topological analysis on the structure of chromosomes by modelling it as a two-dimensional surface may be beneficial towards further understanding on the dynamics of DNA packaging.
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.
NASA Astrophysics Data System (ADS)
Saji, Ryoya; Konno, Hidetoshi
2000-02-01
We have studied local irregularity of brain waves using “local fractal dimensions (LFDs)” for two groups of elderly people, one healthy and the other affected by senile dementia. It is determined that (a) the probability distribution of the LFDs for both groups is subject to the universal law of the beta distribution; (b) the stochastic processes of LFDs of the two groups show a marked difference. We have demonstrated the applicability of the present statistical method based on the LFD for estimating the degree of progression of dementia.
NASA Astrophysics Data System (ADS)
Wu, Yongfeng; Batuski, D. J.; Khalil, A.
2007-12-01
The fractal dimension of the spatial distribution of galaxies can be characterized by various statistical and topological methods, such as the box counting and the two-point correlation function. Here we develop a new way to get fractal information, that is the Metric Space Technique (MST). It allows multiple measures to be simultaneously applied for quantitative analysis of any type of structure distribution. All such distributions are considered to be elements of multi-parameter space, and the analysis is based on considering a sample's output functions, which characterize the distributions in multi-parameter space. We use a dozen slices of a volume of space containing many newly measured galaxies from Sloan Digital Sky Survey Data Release 5. We compare results with that of mock samples of galaxies from N-body simulation with current best estimates of cosmological parameters and nested-pairs simulations, and random catalogs. By systematically studying those slices including hundreds of thousands of galaxies, we demonstrated that in the local universe there exists a fractal structure from MST. We also apply the method to 2MASS and WMAP surveys and get interesting results.
Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions
Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander
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.
Schweizer, K.S.; Curro, J.G. )
1991-03-01
Recently developed methods for obtaining exact and approximate analytical solutions of the reference interaction site model-mean spherical approximation (RISM-MSA) integral equations for liquid mixtures composed of long, flexible polymers are applied to study the critical temperature {ital T}{sub {ital c}} for phase separation of symmetric isotopic binary blends as a function of degree of polymerization {ital N}, spatial dimension {ital D}, and fractal dimension {ital d}{sub {ital f}} of the individual macromolecules. For ideal random walk coils, the theory predicts a nonclassical behavior given by {ital T}{sub {ital c}}{proportional to}{ital N}{sup ({ital D}{minus}2)/2} in two and three dimensions, and the classical Flory--Huggins mean field {ital T}{sub {ital c}}{proportional to}{ital N} law is recovered in four and higher dimensions. For arbitrary interpenetrating polymeric fractals, the theory predicts {ital T}{sub {ital c}}{proportional to}{ital N}{sup ({ital D}{minus}{ital d}{sub {ital f}})/{ital d}{sub {ital f}}} for spatial dimensions below 2{ital d}{sub {ital f}} and Flory--Huggins behavior for {ital D}{gt}2{ital d}{sub {ital f}}. These novel scaling laws for isotopic mixtures are a consequence of a consistent treatment of chain connectivity on all length scales, intermolecular excluded volume, and a short range unfavorable interaction between hydrogenated and deuterated polymers. A general, closure-independent physical argument based on a renormalization of the bare chi parameter by relatively long range correlated fluctuations in the blend is proposed which reproduces all the qualitative predictions of the RISM-MSA integral equation theory. Analogies with nonclassical critical fluctuation effects are established. Application of the analytical approach to purely athermal blends is also presented.
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.
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.
NASA Astrophysics Data System (ADS)
Cámara, Joaquín; Gómez-Miguel, Vicente; Martín, Miguel Ángel
2015-07-01
Geologists know that drainage networks can exhibit different drainage patterns depending on the hydrogeological properties of the underlying materials. Geographic Information System (GIS) technologies and the increasing availability and resolution of digital elevation data have greatly facilitated the delineation, quantification, and study of drainage networks. This study investigates the possibility of inferring geological information of the underlying material from fractal and linear parameters describing drainage networks automatically extracted from 5-m-resolution LiDAR digital terrain model (DTM) data. According to the lithological information (scale 1:25,000), the study area is comprised of 30 homogeneous bedrock lithologies, the lithological map units (LMUs). These are mostly igneous and metamorphic rocks, but also include some sedimentary rocks. A statistical classification model of the LMUs by rock type has been proposed based on both the fractal dimension and drainage density of the overlying drainage networks. The classification model has been built using 16 LMUs, and it has correctly classified 13 of the 14 LMUs used for its validation. Results for the study area show that LMUs, with areas ranging from 177.83 ± 0.01 to 3.16 ± 0.01 km2, can be successfully classified by rock type using the fractal dimension and the drainage density of the drainage networks derived from medium resolution LiDAR DTM data with different flow support areas. These results imply that the information included in a 5-m-resolution LiDAR DTM and the appropriate techniques employed to manage it are the only inputs required to identify the underlying geological materials.
NASA Astrophysics Data System (ADS)
Tao, Dongwang; Mao, Chenxi; Zhang, Dongyu; Li, Hui
2014-12-01
This article extends a signal-based approach formerly proposed by the authors, which utilizes the fractal dimension of time frequency feature (FDTFF) of displacements, for earthquake damage detection of moment resist frame (MRF), and validates the approach with shaking table tests. The time frequency feature (TFF) of the relative displacement at measured story is defined as the real part of the coefficients of the analytical wavelet transform. The fractal dimension (FD) is to quantify the TFF within the fundamental frequency band using box counting method. It is verified that the FDTFFs at all stories of the linear MRF are identical with the help of static condensation method and modal superposition principle, while the FDTFFs at the stories with localized nonlinearities due to damage will be different from those at the stories without nonlinearities using the reverse-path methodology. By comparing the FDTFFs of displacements at measured stories in a structure, the damage-induced nonlinearity of the structure under strong ground motion can be detected and localized. Finally shaking table experiments on a 1:8 scale sixteen-story three-bay steel MRF with added frictional dampers, which generate local nonlinearities, are conducted to validate the approach.
Zouein, Fouad A; Kurdi, Mazen; Booz, George W; Fuseler, John W
2014-08-01
Hearts of mice with reduction of function mutation in STAT3 (SA/SA) develop fibrotic collagen foci and reduced systolic function with hypertension. This model was used to determine if fractal dimension and image analysis can provide a quantitative description of myocardial fibrosis using routinely prepared trichome-stained material. Collagen was characterized by relative density [integrated optical density/area (IOD/A)] and fractal dimension (D), an index of complexity. IOD/A of collagen in wild type mice increased with hypertension while D decreased, suggesting tighter collagen packing that could eventually stiffen the myocardium as in diastolic heart failure. Reduced STAT3 function caused modest collagen fibrosis with increased IOD/A and D, indicating more tightly packed, but more disorganized collagen than normotensive and hypertensive controls. Hypertension in SA/SA mice resulted in large regions where myocytes were lost and replaced by fibrotic collagen characterized by decreased density and increased disorder. This indicates that collagen associated with reparative fibrosis in SA/SA hearts experiencing hypertension was highly disorganized and more space filling. Loss of myocytes and their replacement by disordered collagen fibers may further weaken the myocardium leading to systolic heart failure. Our findings highlight the utility of image analysis in revealing importance of a cellular protein for normal and reparative extracellular matrix deposition. PMID:25410603
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.
Cosmology in one dimension: Symmetry role in dynamics, mass oriented approaches to fractal analysis
Miller, Bruce N; Shiozawa, Yui
2015-01-01
The distribution of visible matter in the universe, such as galaxies and galaxy clusters, has its origin in the week fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the universe, with its galaxies, clusters and super-clusters of galaxies indicates the absence of a natural length scale. In the Newtonian formulation, numerical simulations of a one-dimensional system permit us to precisely follow the evolution of an ensemble of particles starting with an initial perturbation in the Hubble flow. The limitation of the investigation to one dimension removes the necessity to make approximations in calculating the gravitational field and, on the whole, the system dynamics. It is then possible to accurately follow the trajectories of particles for a long time. The simulations show the emergence of a self-similar hierarchical structure in both the phase space and the configuration space and invites the implementation of a multifractal analysis. Here, after showing th...
Shimada, Hirohiko
2015-01-01
The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the $O(N)$ models from $N=1$ (Ising model) to $N=0$ (polymer). The unitarity bound below $N=1$ of the scaling dimension for the the $O(N)$-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as $N$ tends to zero, an emerging asymmetric minimum in the current central charge $C_J$ can be used to locate the CFT. It is pointed out that certain level degeneracies at the $O(N)$ CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the $\\mathcal{N}=1$ supersymmetry, which could be useful for numerically locating the tricritical point in the phase diagram.
Wagenseil, R.
1991-01-01
There are persistent difficulties in monitoring nonpoint source pollution and in the related field of hydrology. The problems stem from variations in spatial distribution which are poorly understood and difficult to model with established methods. Two recent developments may offer a solution, if they are combined with care. The first development is the increasing capability of computer mapping, called geographic information systems (GIS). These systems can store, retrieve, and manipulate data with an explicit spatial structure. The second development is the field of fractal mathematics. Fractal mathematics includes geometric sets which have simple descriptions, despite complex appearances. One family of such fractal sets are the Brownian surfaces, which capture many of the qualities of natural land surfaces in a simple statistical model. Up until now, the Brownian models have been constrained by the assumption that the same statistical relationship holds over the entire surface. This is called the constraint of stationarity. The need to study how the landscape differs by location leads to relaxing the constraint of stationarity. This, in turn, causes some profound changes in the model. A special computer program applies the new model to a set of three-dimensional digital maps of natural terrain (DEMs). The model performs well, and highlights differences in landforms. This suggests several new approaches to spatial variation.
NASA Astrophysics Data System (ADS)
Bowman, Lorraine; Ott, J.; Westpfahl, D.
2014-01-01
The properties of turbulence in galaxies are a fundamental part of our understanding of the interstellar medium (ISM) as a complex and dynamic system. Turbulence changes the proportions of gas in the warm and cold phases and affects the regulation of star formation. Supersonic turbulence is known to have many sources, ranging from supernovae and stellar winds to the interaction between rotational shear and galactic magnetic fields. However, the details of these mechanisms and their interactions are not well understood. Several studies have linked turbulence with self-similarity, a property of the mathematical objects known as fractals. This study compares the fractal dimension of contours of three components of the ISM in nearby galaxies. These components sample the atomic, molecular and gas phases by way of the 21cm line of atomic Hydrogen (HI), the CO J=2?J1 transition and mid-IR 70?m dust emission. The THINGS (HI), SINGS (IR) and HERACLES (CO) surveys share a common galaxy sample from which five galaxies are selected for analysis. We present the results of this study.
Mincione, Gabriella; Di Nicola, Marta; Di Marcantonio, Maria Carmela; Muraro, Raffaella; Piattelli, Adriano; Rubini, Corrado; Penitente, Enrico; Piccirilli, Marcello; Aprile, Giuseppe; Perrotti, Vittoria; Artese, Luciano
2015-10-01
Fractal dimension (FD) in tissue specimens from patients with oral squamous cell carcinoma (OSCC) was evaluated. FD values in different stages of OSCC, and the correlations with clinicopathological variables and patient survival were investigated. Histological sections from OSCC and control non-neoplastic mucosa specimens were stained with hematoxylin-eosin for pathological analysis and with Feulgen for nuclear evaluation. FD in OSCC groups vs. controls revealed statistically significant differences (P < 0.001). In addition, a progressive increase of FD from stage I and II lesions and stage III and IV lesions was observed, with statistically significant differences (P = 0.003). Moreover, different degrees of tumor differentiation showed a significant difference in the average nuclear FD values (P = 0.001). A relationship between FD and patients' survival was also detected with lower FD values associated to longer survival time and higher FD values with shorter survival time (P = 0.034). These data showed that FD significantly increased during OSCC progression. Thus, FD could represent a novel prognostic tool for OSCC, as FD values significantly correlated with patient survival. Fractal geometry could give insights into tumor morphology and could become an useful tool for analyzing irregular tumor growth patterns. PMID:25367085
Non-extensive block entropy statistics of Cantor fractal sets
A. Provata
2007-01-01
By using non-extensive block entropy statistics, we demonstrate analytically that the static structures of deterministic Cantor sets with fractal dimension df are characterised by a non-extensive q-exponent q=1\\/(df-d), for dfd (where d is the embedding dimensions of the fractal set). To calculate the Sq entropy we use the block entropy method based on non-overlapping windows and standard exact enumeration on
Fractal analysis of high-resolution CT images as a tool for quantification of lung diseases
Uppaluri, R.; Mitsa, T.; Galvin, J.R.
1995-12-31
Fractal geometry is increasingly being used to model complex naturally occurring phenomena. There are two types of fractals in nature-geometric fractals and stochastic fractals. The pulmonary branching structure is a geometric fractal and the intensity of its grey scale image is a stochastic fractal. In this paper, the authors attempt to quantify the texture of CT lung images using properties of both types of fractals. A simple algorithm for detecting of abnormality in human lungs, based on 2-D and 3-D fractal dimensions, is presented. This method involves calculating the local fractal dimensions, based on intensities, in the 2-D slice to air edge enhancement. Following this, grey level thresholding is performed and a global fractal dimension, based on structure, for the entire data is estimated in 2-D and 3-D. High Resolution CT images of normal and abnormal lungs were analyzed. Preliminary results showed that classification of normal and abnormal images could be obtained based on the differences between their global fractal dimensions.
K. Murase
2004-01-01
Changes in form of hypocenter distribution preceding the 2003 Tokachi-oki Earthquake (MJ = 8.0) were investigated by using the analysis of temporal variation in spatial fractal dimension D. In this study, it was found that the D value began to decrease in 1998, and had been very small for about one year before the main shock occurrence. Such a D
Hierarchical fractal structure of perfect single-layer grapheme
NASA Astrophysics Data System (ADS)
Zhang, T.; Ding, K.
2013-12-01
The atomic lattice structure of perfect singlelayer graphene that can actually be regarded as a kind of hierarchical fractal structure from the perspective of fractal geometry was studied for the first time. Three novel and special discoveries on hierarchical fractal structure and sets were unveiled upon examination of the regular crystal lattices of the single-layer graphene. The interior fractaltype structure was discovered to be the fifth space-filling curve from physical realm. Two efficient methods for calculating the fractal dimension of this fresh member was also provided. The outer boundary curve had a fractal dimension equal to one, and a multi-fractal structure from a naturally existing material was found for the first time. A series of strict self-similar hexagons comprised a rotating fractal set. These hexagons slewed at a constant counterclockwise angle ? of 19.1° when observed from one level to the next higher level. From the perspective of fractal geometry, these pioneering discoveries added three new members to the existing regular fractal structures and sets. A fundamental example of a multi-fractal structure was also presented.
Cake porosity analysis using 1D-3D fractal dimensions in coagulation-microfiltration of NOM.
Raspati, G S; Leiknes, T O
2015-01-01
Fouling during coagulation-ceramic microfiltration of natural organic matter was investigated. Two process configurations (inline coagulation (IC) and tank coagulation (TC)) and two process conditions (types of coagulants-aluminum-based PAX and iron-based PIX-and G-values) were studied. The rate of irreversible fouling corresponding to the increase of initial transmembrane pressure after backwash of IC-PAX was lowest followed by TC-PAX and TC-PIX, while the performance of IC-PIX was found worst. The 1D and 2D fractal analysis revealed that flocs from IC were morphologically different from those of TC, leading to different filtration characteristics. The 3D fractal analysis revealed two groups of morphologically similar flocs: one led to successful filtration experiments, whereas the other led to unsuccessful ones. Cake porosity was found dependent on the floc morphology. Thus, such an approach was found complementary with fouling analysis by means of a membrane fouling model and minimization of fouling phenomenon was achieved by combining the two approaches. PMID:25768221
Theoretical concepts fpr fractal growth
NASA Astrophysics Data System (ADS)
Pietronero, L.
1989-09-01
After the introduction of fractal geometry by Benoit Mandelbrot the key problem is to understand why nature gives rise to fractal structures. This implies the formulation of models of fractal growth based on physical phenomena and the subsequent understanding of their mathematical structure in the same sense as the renormalization group has allowed to understand sing-type models. The models of diffusion-limited aggregation and the more general dielectric breakdown model, based on iterative processes governed by the Laplace equation and a stochastic field, have a clear physical meaning and they spontaneously evolve into random fractal structures of great complexity. From a theoretical point of view however it is not possible to describe them within usual concepts. Recently we have introduced a new theoretical framework for this class of problems. This clarifies the origin of fractal structures in these models and provides a systematic method for the calculation of the fractal dimension and the multifractal properties. Here we summarize the basic ideas of this new approach and report about recent developments.
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and
Washington at Seattle, University of
, in extensive finite sample simulation studies, and in a data example on arctic sea-ice profiles. For time dimension; Gaussian process; Hausdorff dimension; Mado- gram; Power variation; Robustness; Sea-ice thickness = lim 0 log N( ) log(1/ ) , (1) where N( ) denotes the smallest number of cubes of width in Rd which can
Analysis of bone x-rays using morphological fractals
Samarabandu, J.; Acharya, R. ); Hausmann, E.; Allen, K. . Dept. of Oral Biology)
1993-09-01
The authors have applied mathematical morphology for fractal analysis of bone x-ray images. The digitized gray level image is treated as a three-dimensional surface whose fractal dimension is calculated by performing a series of dilations on this surface and plotting the area of the resulting set of surfaces against the size of the structuring element. This approach gives the added advantage of encoding structural information via the use of a structuring element. The algorithm has been applied to several bone radiographs and the results demonstrate that the fractal dimension using mathematical morphology gives us a robust texture measure of trabecular bone structures.
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.
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.
Fault diagnosis of diesel engine based on adaptive wavelet packets and EEMD-fractal dimension
NASA Astrophysics Data System (ADS)
Wang, Xia; Liu, Changwen; Bi, Fengrong; Bi, Xiaoyang; Shao, Kang
2013-12-01
In this paper a novel method for de-noising nonstationary vibration signal and diagnosing diesel engine faults is presented. The method is based on the adaptive wavelet threshold (AWT) de-noising, ensemble empirical mode decomposition (EEMD) and correlation dimension (CD). A new adaptive wavelet packet (WP) thresholding function for vibration signal de-noising is used in this paper. To alleviate the mode mixing problem occurring in EMD, ensemble empirical mode decomposition (EEMD) is presented. With EEMD, the components with truly physical meaning can be extracted from the signal. Utilizing the advantage of EEMD, this paper proposes a new AWT-EEMD-based method for fault diagnosis of diesel engine. A study of correlation dimension in engine condition monitoring is reported also. Some important influencing factors relating directly to the computational precision of correlation dimension are discussed. Industrial engine normal and fault vibration signals measured from different operating conditions are analyzed using the above method.
Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension
Benjamin Steinhurst
2008-01-01
In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincar\\\\'e inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger's work these upper gradients can be used to
Jurczyszyn, Kamil; Osiecka, Beata J.; Zió?kowski, Piotr
2012-01-01
Fractal dimension analysis (FDA) is modern mathematical method widely used to describing of complex and chaotic shapes when classic methods fail. The main aim of this study was evaluating the influence of photodynamic therapy (PDT) with cystein proteases inhibitors (CPI) on the number and morphology of blood vessels inside tumor and on increase of effectiveness of combined therapy in contrast to PDT and CPI used separately. Animals were divided into four groups: control, treated using only PDT, treated using only CPI and treated using combined therapy, PDT and CPI. Results showed that time of animal survival and depth of necrosis inside tumor were significantly higher in CPI+PDT group in contrast to other groups. The higher value of fractal dimension (FD) was observed in control group, while the lowest value was found in the group which was treated by cystein protease inhibitors. The differences between FD were observed in CPI group and PDT+CPI group in comparison to control group. Our results revealed that fractal dimension analysis is a very useful tool in estimating differences between irregular shapes like blood vessels in PDT treated tumors. Thus, the implementation of FDA algorithms could be useful method in evaluating the efficacy of PDT. PMID:22991578
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)
Tomczak, P.
1996-01-01
The Suzuki-Takano quantum decimation technique is applied to the antiferromagnetic, nearest-neighbor, frustrated Heisenberg spin-1/2 system attached to a lattice with dimension d=ln3/ln2. Some thermodynamical functions are calculated. The temperature dependence of the specific heat is very similar to that obtained for the Heisenberg spin system on a kagomé lattice.
Beretta-Piccoli, Matteo; D’Antona, Giuseppe; Barbero, Marco; Fisher, Beth; Dieli-Conwright, Christina M.; Clijsen, Ron; Cescon, Corrado
2015-01-01
Purpose Over the past decade, linear and non-linear surface electromyography descriptors for central and peripheral components of fatigue have been developed. In the current study, we tested fractal dimension (FD) and conduction velocity (CV) as myoelectric descriptors of central and peripheral fatigue, respectively. To this aim, we analyzed FD and CV slopes during sustained fatiguing contractions of the quadriceps femoris in healthy humans. Methods A total of 29 recreationally active women (mean age±standard deviation: 24±4 years) and two female elite athletes (one power athlete, age 24 and one endurance athlete, age 30 years) performed two knee extensions: (1) at 20% maximal voluntary contraction (MVC) for 30 s, and (2) at 60% MVC held until exhaustion. Surface EMG signals were detected from the vastus lateralis and vastus medialis using bidimensional arrays. Results Central and peripheral fatigue were described as decreases in FD and CV, respectively. A positive correlation between FD and CV (R=0.51, p<0.01) was found during the sustained 60% MVC, probably as a result of simultaneous motor unit synchronization and a decrease in muscle fiber CV during the fatiguing task. Conclusions Central and peripheral fatigue can be described as changes in FD and CV, at least in young, healthy women. The significant correlation between FD and CV observed at 60% MVC suggests that a mutual interaction between central and peripheral fatigue can arise during submaximal isometric contractions. PMID:25880369
Targets detection in smoke-screen image sequences using fractal and rough set theory
NASA Astrophysics Data System (ADS)
Yan, Xiaoke
2015-08-01
In this paper, a new algorithm for the detection of moving targets in smoke-screen image sequences is presented, which can combine three properties of pixel: grey, fractal dimensions and correlation between pixels by Rough Set. The first step is to locate and extract regions that may contain objects in an image by locally grey threshold technique. Secondly, the fractal dimensions of pixels are calculated, Smoke-Screen is done at different fractal dimensions. Finally, according to temporal and spatial correlations between different frames, the singular points can be filtered. The experimental results show that the algorithm can effectively increase detection probability and has robustness.
Extended fractal analysis method and its application for linear rivers
NASA Astrophysics Data System (ADS)
Wang, Liqin; Long, Yi; Cui, Shilin
2008-10-01
Extended fractal analysis method can analyze the fractal character (i.e. self-similarity) objectively, especially the difference and change of the shape and the structure in different observation scale intervals. As one of the common fractal objects, river on the map can be surveyed its length and quantified the complexity of its shape and structure as well as its partial details with Extended Fractal Dimension Analysis method (abbreviated as EFDA). Compared to the traditional method, EFDA has unparalleled advantages. Considering the extended fractal character with scaling variance, and based on its simulating function adopting the Inverse Logistic Model, the paper gained the extended fractal function for quantifying the length of the river depending on the different observing scales. Furthermore, based on the mathematical derivation of its simulating function and fractal analysis, the paper obtained the relevant parameter for establishing Meta Fractal Dimension (abbreviated as MFD) Model to quantify the local complexity of the river on the map. Several experiments based on the China's seven major rivers done indicate that this method is easy to operate and has a relatively high calculation precision and a logical result of spatial analysis.
On the Fractal Distribution of HII Regions in Disk Galaxies
Nestor Sanchez; Emilio J. Alfaro
2008-10-02
In this work we quantify the degree to which star-forming events are clumped. We apply a precise and accurate technique to calculate the correlation dimension Dc of the distribution of HII regions in a sample of disk galaxies. Our reliable results are distributed in the range 1.5fractal dimension among galaxies, contrary to a universal picture sometimes claimed in literature. The faintest galaxies tend to distribute their HII regions in more clustered (less uniform) patterns. Moreover, 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 effect.
Rhythmic precipitate patterns and fractal structure
NASA Astrophysics Data System (ADS)
Sultan, Rabih F.
2011-02-01
Liesegang patterns of parallel precipitate bands are obtained when solutions containing co-precipitate ions interdiffuse in a 1D gel matrix. The sparingly soluble salt formed, displays a beautiful stratification of discs of precipitate perpendicular to the 1D tube axis. The Liesegang structures are analyzed from the viewpoint of their fractal nature. Geometric Liesegang patterns are constructed in conformity with the well-known empirical laws such as the time, band spacing and band width laws. The dependence of the band spacing on the initial concentrations of diffusing (outer) and immobile (inner) electrolytes ( A 0 and B 0, respectively) is taken to follow the Matalon-Packter law. Both mathematical fractal dimensions and box-count dimensions are calculated. The fractal dimension is found to increase with increasing A 0 and decreasing B 0. We also analyze mosaic patterns with random distribution of crystallites, grown under different conditions than the classical Liesegang gel method, and report on their fractal properties. Finally, complex Liesegang patterns wherein the bands are grouped in multiplets are studied, and it is shown that the fractal nature increases with the multiplicity.
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.
Zhang, Luduan; Butler, Andrew J.; Sun, Chang-Kai; Sahgal, Vinod; Wittenberg, George F.; Yue, Guang H.
2008-01-01
Little is known about the association between brain white matter (WM) structure and motor function in humans. This study investigated complexity of brain WM interior shape as determined by magnetic resonance imaging (MRI) and its relationship with upper-extremity (UE) motor function in patients post stroke. We hypothesized that (1) the WM complexity would decrease following stroke, and (2) higher WM complexity in non-affected cortical areas would be related to greater UE motor function. Thirty-eight stroke patients (16 with left-hemisphere lesions) underwent MRI anatomical brain scans. Fractal dimension (FD), a quantitative shape metric, was applied onto skeletonized brain WM images to evaluate WM internal structural complexity. Wolf Motor Function Test (WMFT) and Fugl-Meyer Motor Assessment (FM) scores were measured to assess motor function of the affected limb. The WM complexity was lower in the stroke-affected hemisphere. The FD was associated with better motor function in two subgroups: with left-subcortical lesions, FD values of the lesion-free areas of the left hemisphere were associated with better FM scores; with right-cortical lesions, FD values of lesion-free regions were robustly associated with better WMFT scores. These findings suggest that greater residual WM complexity is associated with less impaired UE motor function, which is more robust in patients with right-hemisphere lesions. No correlations were found between lesion volume and WMFT or FM scores. This study addressed WM complexity in stroke patients and its relationship with UE motor function. Measurement of brain WM reorganization may be a sensitive correlate of UE function in people recovering from stroke. PMID:18590710
Gravitation theory in a fractal space-time
Agop, M.; Gottlieb, I.
2006-05-15
Assimilating the physical space-time with a fractal, a general theory is built. For a fractal dimension D=2, the virtual geodesics of this space-time implies a generalized Schroedinger type equation. Subsequently, a geometric formulation of the gravitation theory on a fractal space-time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equation are calculated. It results, by means of a dilation operator, the equivalence of this model with quantum Einstein gravity.
Fractional hydrodynamic equations for fractal media
Tarasov, Vasily E.
for the fractal media and derive the fractional generalization of the equations of balance of mass density media. We use the procedure of replacement of the fractal medium with fractal mass dimension by someFractional hydrodynamic equations for fractal media Vasily E. Tarasov * Skobeltsyn Institute
NASA Astrophysics Data System (ADS)
Rodríguez Pascua, M. A.; De Vicente, G.; Calvo, J. P.; Pérez-López, R.
2003-05-01
A paleoseismic data set derived from the relationship between the thickness of seismites, 'mixed layers' in lacustrine Miocene deposits and the magnitude of the earthquakes is presented. The relationship between both parameters was calibrated by the threshold of fluidification limits in the interval of magnitude 5 and 5.5. The mixed layers (deformational sediment structures due to seismic activity) were observed in varved sediments from three Neogene lacustrine basins near Hell?´n (Albacete, Spain), El Cenajo, Elche de la Sierra and H?´jar, and are interpreted as liquefaction features due to seismic phenomena. These paleoseismic structures were dated (relative values) by measurements of cyclic annual sedimentation in the varved sediments. From these observations, we are able to establish a recurrence interval of 130 years with events for magnitude bigger than or equal to four. Both paleoseismicity and instrumental seismicity data sets obey the Gutenberg-Richter law and the 'b' value is close to 0.86. The fractal dimension (dimension of capacity) of spatial distribution of potentially active faults (faults oriented according to the stress tensor regime in the area) was measured by the box-counting technique ( D0=1.73). According to the Aki empirical relation ( D0=2 b) for the instrumental seismicity and paleoseismic data sets in the area, the fractal dimension is close to 1.72. The similar value of the fractal dimension obtained by both techniques shows homogeneous seismic dynamics during the studied time interval. Moreover, the better established 'b' value of the paleoseismic data sets (0.86) compared with the 'b' value for the incomplete historic seismicity (<0.5) in the area increases the seismic series beyond the historic seismic record.
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.
Map of fluid flow in fractal porous medium into fractal continuum flow.
Balankin, Alexander S; Elizarraraz, Benjamin Espinoza
2012-05-01
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided. PMID:23004869
Map of fluid flow in fractal porous medium into fractal continuum flow
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Elizarraraz, Benjamin Espinoza
2012-05-01
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow ds is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.
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.
NASA Astrophysics Data System (ADS)
Davarpanah, A.; Babaie, H. A.
2012-12-01
The interaction of the thermally induced stress field of the Yellowstone hotspot (YHS) with existing Basin and Range (BR) fault blocks, over the past 17 m.y., has produced a new, spatially and temporally variable system of normal faults around the Snake River Plain (SRP) in Idaho and Wyoming-Montana area. Data about the trace of these new cross faults (CF) and older BR normal faults were acquired from a combination of satellite imageries, DEM, and USGS geological maps and databases at scales of 1:24,000, 1:100,000, 1:250,000, 1:1000, 000, and 1:2,500, 000, and classified based on their azimuth in ArcGIS 10. The box-counting fractal dimension (Db) of the BR fault traces, determined applying the Benoit software, and the anisotropy intensity (ellipticity) of the fractal dimensions, measured with the modified Cantor dust method applying the AMOCADO software, were measured in two large spatial domains (I and II). The Db and anisotropy of the cross faults were studied in five temporal domains (T1-T5) classified based on the geologic age of successive eruptive centers (12 Ma to recent) of the YHS along the eastern SRP. The fractal anisotropy of the CF system in each temporal domain was also spatially determined in the southern part (domain S1), central part (domain S2), and northern part (domain S3) of the SRP. Line (fault trace) density maps for the BR and CF polylines reveal a higher linear density (trace length per unit area) for the BR traces in the spatial domain I, and a higher linear density of the CF traces around the present Yellowstone National Park (S1T5) where most of the seismically active faults are located. Our spatio-temporal analysis reveals that the fractal dimension of the BR system in domain I (Db=1.423) is greater than that in domain II (Db=1.307). It also shows that the anisotropy of the fractal dimension in domain I is less eccentric (axial ratio: 1.242) than that in domain II (1.355), probably reflecting the greater variation in the trend of the BR system in domain I. The CF system in the S1T5 domain has the highest fractal dimension (Db=1.37) and the lowest anisotropy eccentricity (1.23) among the five temporal domains. These values positively correlate with the observed maxima on the fault trace density maps. The major axis of the anisotropy ellipses is consistently perpendicular to the average trend of the normal fault system in each domain, and therefore approximates the orientation of extension for normal faulting in each domain. This fact gives a NE-SW and NW-SE extension direction for the BR system in domains I and II, respectively. The observed NE-SW orientation of the major axes of the anisotropy ellipses in the youngest T4 and T5 temporal domains, oriented perpendicular to the mean trend of the normal faults in the these domains, suggests extension along the NE-SW direction for cross faulting in these areas. The spatial trajectories (form lines) of the minor axes of the anisotropy ellipses, and the mean trend of fault traces in the T4 and T5 temporal domains, define a large parabolic pattern about the axis of the eastern SRP, with its apex at the Yellowstone plateau.
Statistical analysis of the fractal nature of turbulent premixed flames
Dae Hoon Lee; Sejin Kwon
2003-01-01
Since the introduction of fractal geometry, it has been widely accepted that naturally arising geometries governed by a nonlinear process exhibit fractal aspects. Numerous measurements on interfaces subject to turbulent motion demonstrated that those surfaces indeed display a fractal nature. Turbulent premixed flame surface is an example of such an interface. Past fractal analyses of turbulent flames emphasized fractal dimensions
Fractal Statistics, Fractal Index and Fractons
NASA Astrophysics Data System (ADS)
da Cruz, W.
2001-02-01
The concept of fractal index is introduced in connection with the idea of universal class h of particles or quasiparticles termed fractons which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge c[?] and the particle-hole duality ? <-> {1 / ? }, for integer-value ? of the statistical parameter. The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. We also establish a contact between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.
Dimension of chaotic attractors
Farmer, J.D.; Ott, E.; Yorke, J.A.
1982-09-01
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on probabilistic properties (that is, they depend on the frequency with which a typical trajectory visits different regions of the attractor). Both our example and the previous work that we review support the conclusion that all of the probabilistic dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
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.
Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert
1998-01-01
MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.
Routes to fractality and entropy in Liesegang systems
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 PbF{sub 2} and PbI{sub 2} 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 PbF{sub 2} and PbI{sub 2} 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.
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.
Fractional FokkerPlanck equation for fractal media Vasily E. Tarasova
Tarasov, Vasily E.
that defines the medium mass. We prove that the fractional integrals can be used to describe the media media with noninteger mass dimensions. The fractional integrals can be used not only to calculate the mass dimensions of fractal media. Fractional integra- tion can be used to describe the dynamical
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
NASA Astrophysics Data System (ADS)
Jiang, Xiao-Dian; Yu, Zeng-Hui
1998-03-01
This is a quantitative method for studying the fractal character of magnetic anomaly fields and delineating tectonic elements using fractal theory. For an area with different geological and geophysical features, the definition of Hausdorff dimension is used to calculate the fractal dimension of the area’s magnetic field, then delineate geological tectonic elements by comparing this value. Use of 1978 1986 geophysical survey data in this method applied to the South China Sea yieled three first grade tectonic elements and nine second grade ones, in six of which subsecond grade elements were found.
Naik, Ganesh R; Arjunan, Sridhar; Kumar, Dinesh
2011-06-01
The surface electromyography (sEMG) signal separation and decphompositions has always been an interesting research topic in the field of rehabilitation and medical research. Subtle myoelectric control is an advanced technique concerned with the detection, processing, classification, and application of myoelectric signals to control human-assisting robots or rehabilitation devices. This paper reviews recent research and development in independent component analysis and Fractal dimensional analysis for sEMG pattern recognition, and presents state-of-the-art achievements in terms of their type, structure, and potential application. Directions for future research are also briefly outlined. PMID:21416388
Flocculation control study based on fractal theory*
Chang, Ying; Liu, Qian-jun; Zhang, Jin-song
2005-01-01
A study on flocculation control based on fractal theory was carried out. Optimization test of chemical coagulant dosage confirmed that the fractal dimension could reflect the flocculation degree and settling characteristics of aggregates and the good correlation with the turbidity of settled effluent. So that the fractal dimension can be used as the major parameter for flocculation system control and achieve self-acting adjustment of chemical coagulant dosage. The fractal dimension flocculation control system was used for further study carried out on the effects of various flocculation parameters, among which are the dependency relationship among aggregates fractal dimension, chemical coagulant dosage, and turbidity of settled effluent under the conditions of variable water quality and quantity. And basic experimental data were obtained for establishing the chemical coagulant dosage control model mainly based on aggregates fractal dimension. PMID:16187420
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.
Monte Carlo Simulation Study of Lattice Gas Diffusion in a Box Fractal
NASA Astrophysics Data System (ADS)
Knowlton, Daniel P.; Johnson, James L.; Wentworth, Christopher D.
2008-10-01
In this investigation we study a simple model of diffusion of a concentrated lattice gas in a box fractal structure. The model involves a fixed concentration of particles that undergo random hopping to nearest-neighbor sites with equal probability. The particles do not interact except that double-occupancy of a lattice site is not allowed. The particles move in a lattice of box fractal structure, which has a fractal dimension of 1.465. The mean-square displacement of a tracer particle as a function of time is calculated from the simulation. The simulation suggests anomalous diffusion occurs in this lattice structure.
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
Mandelbrot, Benoit B.
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 the coastline; it is a fraction between 1 (limit of a smooth curve) and 2 (limit of a plane-filling curve). A rough Poisson-Brown stochastic model gives a good first approximation account of the relief, by assuming it to be created by superposing very many, very small cliffs, placed along straight faults and statistically independent. However, the relative area predicted for the largest islands is too small, and the irregularity predicted for the relief is excessive for most applications; so is indeed the value of the dimension, which is D = 1.5. Several higher approximation self-similar models are described. Any can be matched to the empirically observed D, and can link all the observations together, but the required self-similarity cannot yet be fully explained. Images PMID:16578734
Bies, Alexander; Taylor, Richard; Sereno, Margaret
2015-09-01
Edges are significant, ubiquitous features of natural scenes. Basic properties of visual stimuli such as edges should be controlled for in experiments and reported in the literature. Currently, no commonly reported image statistics describe natural scenes' edges. An edge's fractal dimension (Df) could serve as a statistic that quantifies edge roughness in an image across scales. Researchers have often relied on hand tracing to isolate edges in natural scenes for box-counting, a Df measurement technique. For a typical experiment's stimulus set, this would be unfeasibly time consuming. To expedite the process, we developed an algorithm to isolate selected edges of a natural scene for fractal analysis. Our algorithm consists of a three-step manual component (select specific color channels and average their intensity maps, apply an intensity-based threshold, and choose a set of binary objects to retain) followed by a two-step automated component (draw the edges and perform a box-count). We implemented our algorithm in Matlab and applied it to 89 images of clouds. We found that clouds as viewed from the ground have mean Df=1.34 (SD=0.11). We also computed the slope (?) of the radially averaged power spectrum for each image to test for a relationship between Df and ?. We found no significant correlation between Df and ? (r(89)=0.145, p=0.175). This implies that an image's textures may be independent from the Df of the textures' borders. This distinction is important because ? can be computed with full automation. While computing Df for natural image's objects' edges has been time-intensive, our algorithm allows for quick determination of this critical scene statistic. Df could be used characterize the roughness of edges in visually presented natural scene stimuli. Studying how multi-scale contours affect visual processing would complement the literature on the visual processing of texture. Meeting abstract presented at VSS 2015. PMID:26326457
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.
Fractal applications to complex crustal problems
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
Complex scale-invariant problems obey fractal statistics. The basic definition of a fractal distribution is that the number of objects with a characteristic linear dimension greater than r satisfies the relation N = about r exp -D where D is the fractal dimension. Fragmentation often satisfies this relation. The distribution of earthquakes satisfies this relation. The classic relationship between the length of a rocky coast line and the step length can be derived from this relation. Power law relations for spectra can also be related to fractal dimensions. Topography and gravity are examples. Spectral techniques can be used to obtain maps of fractal dimension and roughness amplitude. These provide a quantitative measure of texture analysis. It is argued that the distribution of stress and strength in a complex crustal region, such as the Alps, is fractal. Based on this assumption, the observed frequency-magnitude relation for the seismicity in the region can be derived.
Quantitative Characterization Of Basaltic Tephra Using The Fractal Spectrum Technique
NASA Astrophysics Data System (ADS)
Maria, A.
2006-12-01
Geologists have studied volcanic eruptions on Hawaii more closely than anywhere else. Even so, processes of magma fragmentation during Hawaiian style eruptions (e.g. lava fountaining) are not well understood. Furthermore, the products of these eruptions have not been fully characterized. Analysis of tephra shape is particularly useful for understanding the nature of eruptions, as particle morphology reflects numerous volcanic parameters (e.g. magma viscosity, volatile content, interaction with water, transport processes). The technique applied in this study, based on fractal geometry, uses data produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Previous applications of this technique have proven helpful for characterizing basaltic to rhyolitic products of specific fragmentation processes, and modes of volcanic transport/deposition. In this study, all the samples are basaltic, eliminating the variable of composition, and include material from two Hawaiian lava-fountaining events (Mauna Ulu, 1969; Kilauea Iki, 1959), as well as material from Masaya, Nicaragua (San Judas Formation) that is thought to have been unusually explosive, for comparison. One of our goals is to identify characteristic particle shapes formed during the lava-fountain events. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions, and also allows use of multivariate statistical techniques. Cluster analysis provides a visual display of the similarities of particles in a sample, and facilitates identification of the types of shapes that are most characteristic of a given deposit. Use of principal components analysis to summarize the data as accurately as possible using a few components, facilitates comparison between samples.
Electromagnetism on Anisotropic Fractals
Ostoja-Starzewski, Martin
2011-01-01
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 con...
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.
Electromagnetism on anisotropic fractal media
NASA Astrophysics Data System (ADS)
Ostoja-Starzewski, Martin
2013-04-01
Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the 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 Ampère 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, so as 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 in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.
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
Performance bounds for fractal coding
Bernd Hiirtgen; Rwth Aachen
1995-01-01
Reports on investigations concerning the performance of fractal transforms. Emerging from the structural constraints of fractal coding schemes, lower bounds for the reconstruction error are given without regarding quantization noise. This implies finding an at least locally optimal transformation matrix. A full search approach is by definition optimal but also intractable for practical implementations. In order to simplify the calculation
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.
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 earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
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.
Thermodynamics of Photons on Fractals
Akkermans, Eric; Dunne, Gerald V.; Teplyaev, Alexander
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.
Alexandrov, S E; Kretusheva, I V; Mishin, M V; Yasenovets, G M
2011-09-01
Fractal structures were formed on silicon substrates from SiO2 nanoparticles homogeneously synthesized in low temperature atmospheric pressure plasma from tetraethoxysilane (TEOS). RF discharge (power absorbed was about 10 W) sustained between two parallel mesh electrodes was used to generate plasma. The average size of nanoparticles was in the range of 8-20 nm and was determined by process parameters. The obtained products were analyzed by SEM (scanning electron microscopy) and XPS (X-ray photoelectron spectroscopy). Values of fractal dimension parameter of bidimensionals agglomerates formed on the substrate surface from nanoparticles were calculated with the use of Gwyddion and others. It was found that values of this parameter of the deposited structures varied in the range of 1.48-2 and were determined by combination of the process parameters. An empirical model explaining mechanism of the fractal structures formation and variation of the fractal dimension parameter with the process parameters was proposed. PMID:22097514
Calculation of the anomalous dimensions of the octet baryon currents: two-loop approximation
Pivovarov, A. A.; Surguladze, L. R.
1988-12-01
In this paper we calculate the renormalization constants and anomalous dimensions of the operators of baryon currents with octet quantum numbers in the two/minus/loop approximation. By taking into account the corrections to the coefficient functions, it becomes possible to complete the sum/minus/rule analysis for the proton in the next/minus/to/minus/leading approximation in the strong coupling constant ..cap alpha../sub s/. (AIP)
NASA Astrophysics Data System (ADS)
Popescu, Dan P.; Flueraru, Costel; Mao, Youxin; Chang, Shoude; Sowa, Michael G.
2010-02-01
Two methods for analyzing OCT images of arterial tissues are tested. These methods are applied toward two types of samples: segments of arteries collected from atherosclerosis-prone Watanabe heritable hyper-lipidemic rabbits and pieces of porcine left descending coronary arteries without atherosclerosis. The first method is based on finding the attenuation coefficients for the OCT signal that propagates through various regions of the tissue. The second method involves calculating the fractal dimensions of the OCT signal textures in the regions of interest identified within the acquired images. A box-counting algorithm is used for calculating the fractal dimensions. Both parameters, the attenuation coefficient as well as the fractal dimension correlate very well with the anatomical features of both types of samples.
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.…
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.
The Zeta Function Approach to Casimir Energy Calculations in Higher Dimensions
Richard Obousy
2011-08-30
The vacuum fluctuations give rise to a number of phenomena; however, the the Casimir Effect is arguably the most salient manifestation of the quantum vacuum. In its most basic form it is realized through the interaction of a pair of neutral parallel conducting plates. The presence of the plates modifies the quantum vacuum, and this modifcation causes the plates to be pulled toward each other. The Casimir Effect has also been explored in the context of higher dimensional theories. The non-trivial boundary conditions imposed by compactified periodic higher dimensions is know to alter the vacuum in a quantifiable way, and is a possible solution to the issue of modulus stabilization, namely - the stabilization of higher dimensions. Typical in Casimir energy calculations are renormalization techniques which are used to tame the infinite sums and integrals that arise. These calculations are usally fairly involved, and explicit pedagogical material is sparse. The purpose of this paper is to introduce Dimensional Regularization techniques specific to Casimir energy calculations with an additional compactified spatial dimension.
NASA Astrophysics Data System (ADS)
Bicalho, E. S.; Teixeira, D. B.; Panosso, A. R.; Perillo, L. I.; Iamaguti, J. L.; Pereira, G. T.; La Scala, N., Jr.
2012-04-01
Soil CO2 emission (FCO2) is influenced by chemical, physical and biological factors that affect the production of CO2 in the soil and its transport to the atmosphere, varying in time and space depending on environmental conditions, including the management of agricultural area. The aim of this study was to investigate the structure of spatial variability of FCO2 and soil properties by using fractal dimension (DF), derived from isotropic variograms at different scales, and construction of fractograms. The experimental area consisted of a regular grid of 60 × 60 m on sugarcane area under green management, containing 141 points spaced at minimum distances ranging from 0.5 to 10 m. Soil CO2 emission, soil temperature and soil moisture were evaluated over a period of 7 days, and soil chemical and physical properties were determined by sampling at a depth of 0.0 to 0.1 m. FCO2 showed an overall average of 1.51 µmol m-2 s-1, correlated significantly (p < 0.05) with soil physical factors such as soil bulk density, air-filled pore space, macroporosity and microporosity. Significant DF values were obtained in the characterization of FCO2 in medium and large scales (from 20 m). Variations in DF with the scale, which is the fractogram, indicate that the structure of FCO2 variability is similar to that observed for the soil temperature and total pore volume, and reverse for the other soil properties, except for macroporosity, sand content, soil organic matter, carbon stock, C/N ratio and CEC, which fractograms were not significantly correlated to the FCO2 fractogram. Thus, the structure of spatial variability for most soil properties, characterized by fractogram, presents a significant relationship with the structure of spatial variability of FCO2, generally with similar or dissimilar behavior, indicating the possibility of using the fractogram as tool to better observe the behavior of the spatial dependence of the variables along the scale.
Hearing the Hausdorff dimension
Dutkay, Dorin Ervin; Sun, Qiyu; Weber, Eric
2009-01-01
We study Fourier frames of exponentials on fractal measures. We prove that, for affine iterated function system measures, the Beurling dimension of a Fourier frame must coincide with the Hausdorff dimension of the fractal. We present necessary and/or sufficient conditions for a set of frequencies to form a Bessel sequence or a frame of exponential functions.
On the properties of fractal cloud complexes
Nestor Sanchez; Emilio J. Alfaro; Enrique Perez
2005-12-09
We study the physical properties derived from interstellar cloud complexes having a fractal structure. We first generate fractal clouds with a given fractal dimension and associate each clump with a maximum in the resulting density field. Then, we discuss the effect that different criteria for clump selection has on the derived global properties. We calculate the masses, sizes and average densities of the clumps as a function of the fractal dimension (D_f) and the fraction of the total mass in the form of clumps (epsilon). In general, clump mass does not fulfill a simple power law with size of the type M_cl ~ (R_cl)**(gamma), instead the power changes, from gamma ~ 3 at small sizes to gammasizes. The number of clumps per logarithmic mass interval can be fitted to a power law N_cl ~ (M_cl)**(-alpha_M) in the range of relatively large masses, and the corresponding size distribution is N_cl ~ (R_cl)**(-alpha_R) at large sizes. When all the mass is forming clumps (epsilon=1) we obtain that as D_f increases from 2 to 3 alpha_M increases from ~0.3 to ~0.6 and alpha_R increases from ~1.0 to ~2.1. Comparison with observations suggests that D_f ~ 2.6 is roughly consistent with the average properties of the ISM. On the other hand, as the fraction of mass in clumps decreases (epsilon<1) alpha_M increases and alpha_R decreases. When only ~10% of the complex mass is in the form of dense clumps we obtain alpha_M ~ 1.2 for D_f=2.6 (not very different from the Salpeter value 1.35), suggesting this a likely link between the stellar initial mass function and the internal structure of molecular cloud complexes.
Fractal analysis of the galaxy distribution in the redshift range 0.45 ? z ? 5.0
NASA Astrophysics Data System (ADS)
Conde-Saavedra, G.; Iribarrem, A.; Ribeiro, Marcelo B.
2015-01-01
This paper performs a fractal analysis of the galaxy distribution and presents evidence that it can be described as a fractal system within the redshift range of the FORS Deep Field (FDF) galaxy survey data. The fractal dimension D was derived by means of the galaxy number densities calculated by Iribarrem et al. (2012) using the FDF luminosity function parameters and absolute magnitudes obtained by Gabasch et al. (2004, 2006) in the spatially homogeneous standard cosmological model with ?m0 = 0.3, ??0 = 0.7 and H0 = 70 kms-1Mpc-1. Under the supposition that the galaxy distribution forms a fractal system, the ratio between the differential and integral number densities ? and ?? obtained from the red and blue FDF galaxies provides a direct method to estimate D and implies that ? and ?? vary as power-laws with the cosmological distances, feature which provides a second method for calculating D. The luminosity distance dL, galaxy area distance dG and redshift distance dz were plotted against their respective number densities to calculate D by linear fitting. It was found that the FDF galaxy distribution is better characterized by two single fractal dimensions at successive distance ranges, that is, two scaling ranges in the fractal dimension. Two straight lines were fitted to the data, whose slopes change at z ? 1.3 or z ? 1.9 depending on the chosen cosmological distance. The average fractal dimension calculated using ?? changes from < D > = 1 .4-0.6+0.7 to < D > = 0 .5-0.4+1.2 for all galaxies. Besides, D evolves with z, decreasing as the redshift increases. Small values of D at high z mean that in the past galaxies and galaxy clusters were distributed much more sparsely and the large-scale structure of the universe was then possibly dominated by voids.
Fractal Distribution of Experimentally Generated Pyroclasts
NASA Astrophysics Data System (ADS)
Kueppers, U.; Perugini, D.; Dingwell, D. B.
2005-12-01
Despite recent advances by means of experiments and high-resolution surveys, volcanic eruptions remain highly unpredictable in terms of the type of activity and the duration an imminent eruption will probably exhibit. This uncertainty hinders hazard assessment tremendously. In an effort to counter this problem, a comparison of natural deposits and pyroclasts from laboratory experiments has been undertaken in order to enable estimation of the physical conditions during volcanic eruptions. Three sample sets of Unzen volcano, Japan, have been investigated in order to evaluate the influence of open porosity in combination with applied gas overpressure on the fragmentation behaviour and on the pyroclast generation (fragmentation efficiency). All experiments have been performed at 850 °C and at initial pressure values above the respective fragmentation threshold. The set-up allowed for accurate simulation of explosive volcanic fragmentation whilst investigating the resulting pyroclast generation. The generated pyroclasts have been analysed for their grain-size distribution and the fractality of that distribution. The grain-size distribution was analysed by dry sieving for particles bigger than 250 ?m and laser refraction of the suspended particles smaller than 250 ?m. Laser refraction was found to be applicable to the size analysis of pyroclasts from natural samples. The grain-size analysis exhibits a clear dependence of applied pressure and open porosity on the resulting pyroclasts: i.e. the fragmentation efficiency was found to have increased with increasing potential energy for fragmentation (gas fraction × applied pressure). The fractal fragmentation theory was applied to the achieved grain-size distribution. The fractal dimension of fragmentation (Df) was calculated for all experiments for samples with different open porosity. Results show a general linear increase of Df, i.e. intensity of fragmentation, as the pressure increases. An additional important point is the variation of intercept of linear fitting of data. In particular, the intercept increases with the open porosity of the samples indicating that the intensity of the fragmentation process increases with the open porosity of the samples. These results indicate that fractal fragmentation theory may allow for quantifying fragmentation processes during explosive volcanic eruptions, a feature that is difficult to study by using classical statistical methods. The results may help in evaluating volcanic risk by estimating the explosivity (e.g. pressure in the conduit and possibly other parameters) from the value of fractal dimension of grain-size distribution of natural deposits. This may give the opportunity to draw iso-Df or iso-explosivity contour maps based on fractal statistics.
Large-dimension configuration-interaction calculations of positron binding to the group-II atoms
Bromley, M. W. J.; Mitroy, J. [Department of Physics, San Diego State University, San Diego, California 92182 (United States); Faculty of Technology, Charles Darwin University, Darwin NT 0909 (Australia)
2006-03-15
The configuration-interaction (CI) method is applied to the calculation of the structures of a number of positron binding systems, including e{sup +}Be, e{sup +}Mg, e{sup +}Ca, and e{sup +}Sr. These calculations were carried out in orbital spaces containing about 200 electron and 200 positron orbitals up to l=12. Despite the very large dimensions, the binding energy and annihilation rate converge slowly with l, and the final values do contain an appreciable correction obtained by extrapolating the calculation to the l{yields}{infinity} limit. The binding energies were 0.00317 hartree for e{sup +}Be, 0.0170 hartree for e{sup +}Mg, 0.0189 hartree for e{sup +}Ca, and 0.0131 hartree for e{sup +}Sr.
Kmetyk, L.N.; Yarrington, P.
1989-05-01
Calculations were performed with the CTH and HULL finite difference wavecodes to evaluate computational capabilities for predicting depth and diameter of target cavities produced in high velocity penetration events. The calculations simulated selected tests in a set of armor penetration experiments conducted by the US Army Ballistic Research Laboratory and reported earlier in the literature. The tests and simulations involved penetration of semi-infinite targets by long rod projectiles over a range of impact velocities from 1.3 to 4.5 km/sec. Comparisons are made between the calculated and measured dimensions of the target cavities, and the sensitivity of the predicted results to target property variations is investigated. 9 refs., 18 figs., 3 tabs.
The scattering of electromagnetic waves in fractal media
Zhen-song Wang; Bao-wei Lu
1994-01-01
The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave,
A Fractal Galaxy Distribution in a Homogeneous Universe?
Ruth Durrer; Francesco Sylos Labini
1998-10-06
In this letter we present an idea which reconciles a homogeneous and isotropic Friedmann universe with a fractal distribution of galaxies. We use two observational facts: The flat rotation curves of galaxies and the (still debated) fractal distribution of galaxies with fractal dimension D=2. Our idea can also be interpreted as a redefinition of the notion of bias.
Fractal Weyl Laws for Chaotic Open Systems S. Sridhar,1
Sridhar, Srinivas
Fractal Weyl Laws for Chaotic Open Systems W.T. Lu,1 S. Sridhar,1 and Maciej Zworski2 1 Department a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension. A notable example is the conjecture by Berry [2] for the density of states of closed systems with fractal
ERIC Educational Resources Information Center
Jurgens, Hartmut; And Others
1990-01-01
The production and application of images based on fractal geometry are described. Discussed are fractal language groups, fractal image coding, and fractal dialects. Implications for these applications of geometry to mathematics education are suggested. (CW)
Calculating nasoseptal flap dimensions: a cadaveric study using cone beam computed tomography.
Ten Dam, Ellen; Korsten-Meijer, Astrid G W; Schepers, Rutger H; van der Meer, Wicher J; Gerrits, Peter O; van der Laan, Bernard F A M; Feijen, Robert A
2015-09-01
We hypothesize that three-dimensional imaging using cone beam computed tomography (CBCT) is suitable for calculating nasoseptal flap (NSF) dimensions. To evaluate our hypothesis, we compared CBCT NSF dimensions with anatomical dissections. The NSF reach and vascularity were studied. In an anatomical study (n = 10), CBCT NSF length and surface were calculated and compared with anatomical dissections. The NSF position was evaluated by placing the NSF from the anterior sphenoid sinus wall and from the sella along the skull base towards the frontal sinus. To visualize the NSF vascularity in CBCT, the external carotic arteries were perfused with colored Iomeron. Correlations between CBCT NSFs and anatomical dissections were strongly positive (r > 0.70). The CBCT NSF surface was 19.8 cm(2) [16.6-22.3] and the left and right CBCT NSF lengths were 78.3 mm [73.2-89.5] and 77.7 mm [72.2-88.4] respectively. Covering of the anterior skull base was possible by positioning the NSF anterior to the sphenoid sinus. If the NSF was positioned from the sella along the skull base towards the frontal sinus, the NSF reached partially into the anterior ethmoidal sinuses. CBCT is a valuable technique for calculating NSF dimensions. CBCT to demonstrate septum vascularity in cadavers proved to be less suitable. The NSF reach for covering the anterior skull base depends on positioning. This study encourages preoperative planning of a customized NSF, in an attempt to spare septal mucosa. In the concept of minimal invasive surgery, accompanied by providing customized care, this can benefit the patients' postoperative complaints. PMID:25359192
H. Samavati; A. Hajimiri; A. R. Shahani; G. N. Nasserbakht; T. H. Lee
1998-01-01
A linear capacitor structure using fractal geometries is described. This capacitor exploits both lateral and vertical electric fields to increase the capacitance per unit area. Compared to standard parallel-plate capacitors, the parasitic bottom-plate capacitance is reduced. Unlike conventional metal-to-metal capacitors, the capacitance density increases with technology scaling. A classic fractal structure is implemented with 0.6-?m metal spacing, and a factor
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.
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
Fractal Boundaries of Complex Networks , Sergey V. Buldyrev2,1
Cohen, Reuven
Fractal Boundaries of Complex Networks Jia Shao1 , Sergey V. Buldyrev2,1 , Reuven Cohen3 , MaksimÂ¨os-RÂ´enyi and scale-free model networks, as well as for several real networks, the boundary has fractal properties are fractals with a fractal dimension df 2. We present analytical and numerical evidence supporting
Optics on a fractal surface and the photometry of the regoliths
NASA Astrophysics Data System (ADS)
Drossart, P.
1993-05-01
The light scattered by a rough surface is calculated in a model where the surface is simulated by a mathematical fractal of dimension (D(H) between 2 and 3) and fractal density in the projected area towards the observer rho(H) (rho(H) between 0 and 1). The reflectance on such a surface is calculated in the special case of a 'hemispherical' fractal, in both the geometric optics approximation and a more general diffraction regime. By using a two-parameter phase function (single scattering albedo omega-sub-0 and asymmetry parameter g-sub-0), and including multiple scattering, this four-parameter model is found to reproduce within a good accuracy the phase function of several classes of atmosphereless bodies in the solar system, in good agreement with previous photometric models. The main effect of the diffraction is to reduce the width of the opposition surge by roughly a factor of 2. Another prediction of the model is that the single-scattering contribution due to the fractal part of the surface can be reduced, for nonzero phase angle, to an arbitrarily small amount, for high enough fractal dimension and density. This effect could give a new interpretation of the strong opposition effect observed on some objects, and also of the very low brightness of many solar system bodies.
Fractal Geometry in the High School Classroom.
ERIC Educational Resources Information Center
Camp, Dane R.
1995-01-01
Discusses classroom activities that involve applications of fractal geometry. Includes an activity sheet that explores Pascal's triangle, Sierpinsky's gasket, and modular arithmetic in two and three dimensions. (Author/MKR)
NASA Astrophysics Data System (ADS)
Burdzy, Krzysztof; Ho?yst, Robert; Pruski, ?ukasz
2013-05-01
We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n×n with periodic boundary conditions. A fraction p?20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (nlnn)2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by M? where ?=-1.18 for p=0.1, decreasing to ?=-1.28 for p=0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a=1.56 for p=0.20 to a=1.42 for p=0.001 (for n=4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.
NASA Technical Reports Server (NTRS)
Huang, J.; Turcotte, D. L.
1989-01-01
The concept of fractal mapping is introduced and applied to digitized topography of Arizona. It is shown that the fractal statistics satisfy the topography of the state to a good approximation. The fractal dimensions and roughness amplitudes from subregions are used to construct maps of these quantities. It is found that the fractal dimension of actual two-dimensional topography is not affected by the adding unity to the fractal dimension of one-dimensional topographic tracks. In addition, consideration is given to the production of fractal maps from synthetically derived topography.
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.
Fractons and Fractal Statistics
Wellington da Cruz
2000-07-31
Fractons are anyons classified into equivalence classes and they obey a specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension $h$. We consider this approach in the context of the Fractional Quantum Hall Effect (FQHE) and the concept of duality between such classes, defined by $\\tilde{h}=3-h$ shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes $h$ and the modular group for the quantum phase transitions of the FQHE is also obtained. A $\\beta-$function is defined for a complex conductivity which embodies the classes $h$. The thermodynamics is also considered for a gas of fractons $(h,\
Measurement Based Quantum Computation on Fractal Lattices
Damian Markham; Janet Anders; Michal Hajdušek; Vlatko Vedral
2010-06-08
In this article we extend on work which establishes an analology between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater than one which do all act as good resources for one-way quantum computation, and sets of fractal lattices with dimension greater than one all of which do not. The difference is put down to other topological factors such as ramification and connectivity. This work adds confidence to the analogy and highlights new features to what we require for universal resources for one-way quantum computation.
Fractal characterization of brain lesions in CT images
Jauhari, Rajnish K.; Trivedi, Rashmi; Munshi, Prabhat; Sahni, Kamal
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 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.
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.
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
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
Spectral and Fractal Analysis of Biosignals and Coloured Noise
S. Spasic; Despot Stephan Boulevard
2007-01-01
The amplitude power spectra and fractal dimensions of surrogate data designed as coloured noise were compared with spectra and FDs of biosignals in order to improve the tools for brain activity analysis. It is interesting if biosignals yielded spectral indices and fractal dimension values different from almost all of those of surrogate data, could we have more confidence that original
Perturbative calculations in space-time having extra dimensions: The 6D single axial box anomaly
NASA Astrophysics Data System (ADS)
Fonseca, M. V. S.; Dallabona, G.; Battistel, O. A.
2014-11-01
A detailed investigation about the 6D single axial box anomalous amplitude is presented. The superficial degree of divergence involved, in the one-loop perturbative calculations, is quadratic and the corresponding theory is nonrenormalizable. In spite of this, we show that the phenomenon of anomaly can be clearly characterized in a completely analogous way as that of 4D single axial triangle anomaly. The required calculations are made within the context of a novel calculational strategy where the amplitudes are not modified in intermediary steps. Divergent integrals are, in fact, not really solved. Adequate representations for the internal propagators are adopted according to the degree of divergence involved, so that when the last Feynman rule is taken (integration over the loop momentum) all the dependence on the internal (arbitrary) momenta are placed only in finite integrals. In the divergent structures emerging, no physical parameter is present and such objects are not really integrated. Only very general properties are assumed for such quantities which are universal (all space-time dimensions). The consistency of the perturbative calculations fixes some relations among the divergent integrals so that all the potentially ambiguous terms can be automatically removed. In spite of the absence of ambiguities, the emerging results allow us to give a clear and transparent description of the anomaly. The present investigation confirms the point of view stated by the same prescription for the well-known 2D axial-vector (AV) two-point and 4D single (AVV) and triple (AAA) axial-vector anomalies: the anomalous amplitudes need not be assumed as ambiguous quantities to allow an adequate description of the anomalies. We show also that a surprising, but natural, connection between the coupling of fermions with a pseudoscalar tensor field is found. In addition, we show that the crucial mathematical aspects of the problem are deeply related to a recently arisen controversy involving the evaluation of the Higgs Boson decay and the question of unicity in the dimensional regularization.
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
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 characterization and frequency properties of near-fault ground motions
NASA Astrophysics Data System (ADS)
Yang, Dixiong; Zhang, Changgeng
2013-12-01
This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T m. Meanwhile, the box dimension of the velocity histories D vel is negatively correlated with the characteristic period T c and improved characteristic period T gi.
Elasticity of fractal materials using the continuum model with non-integer dimensional space
NASA Astrophysics Data System (ADS)
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 the 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.
Jeffrey L. Anderson; Labros A. Karagounis; Kenneth M. Stein; Fidela L. Moreno; Robert Ledingham; Alfred Hallstrom
1997-01-01
Objectives. Our objective was to test fractal dimension (D), a measure of clustering of ventricular premature complexes (VPCs), on entry Holter recording as a predictor of future arrhythmic death and other-cause mortality in postinfarction patients in the Cardiac Arrhythmic Suppression Trial (CAST).Background. Nonlinear dynamic methods of signal processing are being applied in medicine to provide new insights into apparently “chaotic”
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...
Space time fractal properties of the forest-fire series in central Italy
NASA Astrophysics Data System (ADS)
Telesca, Luciano; Amatucci, Giuseppe; Lasaponara, Rosa; Lovallo, Michele; Rodrigues, Maria Joao
2007-10-01
The space-time fractality of the forest-fire sequence (1997-2003) occurred in the Tuscany Region (central Italy), one of the most vulnerable to wildfires in Italy, has been approached by using spatial and temporal fractal tools. The fractal exponent ?, estimated by the Fano factor method, characterises the time-clustering behaviour of the set of fires, while the correlation dimension Dc, calculated by means of the correlation integral method, gives information on the space-clustering behaviour of the sequence of fires. We found that (i) the investigated fire set is globally characterized by space-time clustering behaviour; (ii) ? and Dc decreases and increases, respectively with the increase of the threshold size of burned area; (iii) the time variation of ? shows a tendency towards Poissonian processes in correspondence of the largest events.
D. L. Khokhlov
1999-01-15
The model of the universe is considered in which background of the universe is not defined by the matter but is a priori specified as a homogenous and isotropic flat space. The scale factor of the universe follows the linear law. The scale of mass changes proportional to the scale factor. This leads to that the universe has the fractal structure with a power index of 2.
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.
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.
Fractal analysis applied to characteristic segments of the San Andreas Fault
NASA Astrophysics Data System (ADS)
Aviles, C. A.; Scholz, C. H.; Boatwright, John
1987-01-01
Fractal theory is applied in a quantitative analysis of the San Andreas fault (SAF) geometry. The method, which directly measures the increase in total fault length with a decrease in ruler size, gives the fractal dimension D and scaling properties for the chosen length band 0.5-1000 km. A physical interpretation of D is that it measures the irregularity of the fault trace in the selected band. The fault is subdivided into six segments of distinctive seismic behavior. A "main" fault trace which shows either maximum coseismic slip or creeping was selected for analysis, with three alternative branches examined for the SAF system south of San Bernardino. Branches of the fault trace are not considered in this work. Fractal dimensions calculated for the different segments range from 1.0008 to 1.0191, and are different from 1.0 to the 95% confidence interval. These small changes reflect the overall smoothness of the main fault trace. Significant variations in D among segments indicate heterogeneities in the fault smoothness along strike. D also changes significantly from the short-length band to the long-length band where the demarcation point ranges from 1 to 2 km. The short-length band has larger D values. A slight correlation is obtained between the fractal dimension of the main trace and the extent of subparallel faulting. This indicates some correspondence between the main fault trace irregularity and the complexity of subsidiary fault traces in plan view.
Nucleation of squat cracks in rail, calculation of crack initiation angles in three dimensions
NASA Astrophysics Data System (ADS)
Naeimi, Meysam; Li, Zili; Dollevoet, Rolf
2015-07-01
A numerical model of wheel-track system is developed for nucleation of squat-type fatigue cracks in rail material. The model is used for estimating the angles of squat cracks in three dimensions. Contact mechanics and multi-axial fatigue analysis are combined to study the crack initiation mechanism in rails. Nonlinear material properties, actual wheel-rail geometries and realistic loading conditions are considered in the modelling process. Using a 3D explicit finite element analysis the transient rolling contact behaviour of wheel on rail is simulated. Employing the critical plane concept, the material points with the largest possibility of crack initiation are determined; based on which, the 3D orientations/angles of the possible squat cracks are estimated. Numerical estimations are compared with sample results of experimental observations on a rail specimen with squat from the site. The findings suggest a proper agreement between results of modelling and experiment. It is observed that squat cracks initiate at an in-plane angle around 13°-22° relative to the rail surface. The initiation angle seen on surface plane is calculated around 29°-48°, while the crack tend to initiate in angles around 25°-31° in the rail cross-section.
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.
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.
Random sequential adsorption on fractals
NASA Astrophysics Data System (ADS)
Ciesla, Michal; Barbasz, Jakub
2012-07-01
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions.
Random sequential adsorption on fractals.
Ciesla, Michal; Barbasz, Jakub
2012-07-28
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions. PMID:22852643
Computerized analysis of mammographic parenchymal patterns using fractal analysis
NASA Astrophysics Data System (ADS)
Li, Hui; Giger, Maryellen L.; Huo, Zhimin; Olopade, Olufunmilayo I.; Chinander, Michael R.; Lan, Li; Bonta, Ioana R.
2003-05-01
Mammographic parenchymal patterns have been shown to be associated with breast cancer risk. Fractal-based texture analyses, including box-counting methods and Minkowski dimension, were performed within parenchymal regions of normal mammograms of BRCA1/BRCA2 gene mutation carriers and within those of women at low risk for developing breast cancer. Receiver Operating Characteristic (ROC) analysis was used to assess the performance of the computerized radiographic markers in the task of distinguishing between high and low-risk subjects. A multifractal phenomenon was observed with the fractal analyses. The high frequency component of fractal dimension from the conventional box-counting technique yielded an Az value of 0.84 in differentiating between two groups, while using the LDA to estimate the fractal dimension yielded an Az value of 0.91 for the high frequency component. An Az value of 0.82 was obtained with fractal dimensions extracted using the Minkowski algorithm.
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.
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
2015-08-03
We model the cytoskeleton as a fractal network by identifying each segment 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, which may support stress, without relaxing. 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, $\\cal L\\sim\\tau^{\\alpha}$. 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 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
Chaos vs linear instability in the Vlasov equation: A fractal analysis characterization
Atalmi, A.; Baldo, M.; Burgio, G.F.; Rapisarda, A. [Centro Siciliano di Fisica Nucleare e Struttura della Materia, c.so Italia 57, I-95129 Catania (Italy)] [Centro Siciliano di Fisica Nucleare e Struttura della Materia, c.so Italia 57, I-95129 Catania (Italy); [Dipartimento di Fisica Universita di Catania, c.so Italia 57, I-95129 Catania (Italy); [I.N.F.N. Sezione di Catania, c.so Italia 57, I-95129 Catania (Italy)
1996-05-01
In this paper we discuss the most recent results concerning the Vlasov dynamics inside the spinodal region. The chaotic behavior which follows an initial regular evolution is characterized through the calculation of the fractal dimension of the distribution of the final modes excited. The ambiguous role of the largest Lyapunov exponent for unstable systems is also critically reviewed. This investigation seems to confirm the crucial role played by deterministic chaos in nuclear multifragmentation. {copyright} {ital 1996 The American Physical Society.}
Fractal sets of dual topological quantum numbers
Wellington da Cruz
2004-06-18
The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval $1 < h < 2$ and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of the quantum paths of particles termed fractons which obey a fractal distribution function associated with a fractal von Neumann entropy. This way, the quantum Hall transitions satisfy some properties related to the Farey sequences of rational numbers and so our theoretical description of the FQHE establishes a connection between physics, fractal geometry and number theory. The FQHE as a convenient physical system for a possible prove of the Riemann hypothesis is suggested.
Analysis Of Lung Scans Using Fractals
NASA Astrophysics Data System (ADS)
Cargill, Ellen B.; Donohoe, Kevin; Kolodny, Gerald M.; Parker, J. Anthony; Zimmerman, Robert E.
1989-05-01
Measurement of the average power spectra of perfusion lung scans shows that the distribution of macroaggregated albumin in the walls of the lung lobules behaves as a fractal object. Measures related to the power spectral slope for all views in a study, mean slope and slope standard deviation, have a high correlation with the visual interpretation of the radiologist. Application of appropriate fractal models reveals that the surface dimension of normal lung parenchyma range between 2.7 and 3.0.
Fractal Analysis of DNA Sequence Data
NASA Astrophysics Data System (ADS)
Berthelsen, Cheryl Lynn
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the "sandbox method." Analysis of 164 human DNA sequences compared to three types of control sequences (random, base -content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than do invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Fractal nature of humic materials
Rice, J.A.; Lin, J.S.
1992-03-01
Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this ``disorder within disorder.`` Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.
Fractal nature of humic materials
Rice, J.A. . Dept. of Chemistry); Lin, J.S. )
1992-01-01
Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this disorder within disorder.'' Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.
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.
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
Magnetohydrodynamics of fractal media
Tarasov, Vasily E.
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 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 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)
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 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 study of magnetic domain patterns
NASA Astrophysics Data System (ADS)
Han, Bao-Shan; Li, Dan; Zheng, De-Juan; Zhou, Yan
2002-07-01
Fractal geometry is introduced into the analysis of ``two-phase'' magnetic domain patterns. The line-measuring dimension Dline is selected to quantitatively describe the ``line structure'' patterns of the multi-branched domains (MBD's) formed in garnet bubble films, and a meaningful Dline can be related to the numbers of vertical Bloch lines in their walls, i.e., to the hardness of the MBD's. For quantitatively describing the ``plane-filling'' domain patterns of magnetic materials with uniaxial anisotropy, such as corrugation and spike, even ``flower,'' domains, the box-counting dimension Dbox is selected. For describing the series of domains of Co and Dy-NdFeB single crystals due to branching process, Dline and Dbox are used in section. Our results show that two phase domain patterns possess fractal natures, and can be described by fractal dimensions.
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 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
Synthetic aperture radar imagery scene segmentation using fractal processing
Clayton V. Stewart; Baback Moghaddam; Kenneth J. Hintz
1992-01-01
This paper demonstrates the application of fractal random process models and their related scaling parameters as features in the analysis and segmentation of clutter in high-resolution polarimetric synthetic aperture radar (SAR) imagery. Specifically, the fractal dimension of natural clutter sources, such as grass and trees, is computed and used as a texture feature for a Bayesian classifier. The SAR shadows
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. Bashkuev, Yu. B.
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 image analysis - Application to the topography of Oregon and synthetic images.
NASA Technical Reports Server (NTRS)
Huang, Jie; Turcotte, Donald L.
1990-01-01
Digitized topography for the state of Oregon has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with the results for Oregon. This approach can be easily applied to any digitzed image that obeys fractal statistics.
Hexagonal and Pentagonal Fractal Multiband Antennas
NASA Technical Reports Server (NTRS)
Tang, Philip W.; Wahid, Parveen
2005-01-01
Multiband dipole antennas based on hexagonal and pentagonal fractals have been analyzed by computational simulations and functionally demonstrated in experiments on prototypes. These antennas are capable of multiband or wide-band operation because they are subdivided into progressively smaller substructures that resonate at progressively higher frequencies by virtue of their smaller dimensions. The novelty of the present antennas lies in their specific hexagonal and pentagonal fractal configurations and the resonant frequencies associated with them. These antennas are potentially applicable to a variety of multiband and wide-band commercial wireless-communication products operating at different frequencies, including personal digital assistants, cellular telephones, pagers, satellite radios, Global Positioning System receivers, and products that combine two or more of the aforementioned functions. Perhaps the best-known prior multiband antenna based on fractal geometry is the Sierpinski triangle antenna (also known as the Sierpinski gasket), shown in the top part of the figure. In this antenna, the scale length at each iteration of the fractal is half the scale length of the preceding iteration, yielding successive resonant frequencies related by a ratio of about 2. The middle and bottom parts of the figure depict the first three iterations of the hexagonal and pentagonal fractals along with typical dipole-antenna configuration based on the second iteration. Successive resonant frequencies of the hexagonal fractal antenna have been found to be related by a ratio of about 3, and those of the pentagonal fractal antenna by a ratio of about 2.59.
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 plate reconstructions with spreading asymmetry
NASA Astrophysics Data System (ADS)
Pilger, Rex H.
2012-06-01
Information theory and fractal analysis are the basis of a novel fitting criterion for simultaneous plate tectonic reconstructions of magnetic isochrons and fracture zone crossings of a range of ages, rather than a single isochron age. Accretionary boundaries are modeled as two-dimensional fractal structures including both contemporary spreading boundaries and reconstructed magnetic isochron and fracture zone crossings. Each model incorporates reconstruction parameters which describe the full accretionary history, including asymmetry. The reconstruction parameters are derived by spline interpolation in time of trial rotation pseudovectors, including variable asymmetric spreading between ridge segments. Iterative algorithms, without partial derivative constraints, converge on a nominally optimal model by minimizing the sum of two-dimensional fractal bins, over the range of bin-spacings, and produce thereby progressively refined fractal spectra. The new method can incorporate all isochron identifications from the selected plates and age range in the iterative calculation set. The solution set also provides continuous instantaneous rotation parameters, including asymmetries. An example data set illustrates the methodology and model results. The rationale for an optimal fractal criterion is rooted in recent developments in information theory: fractal structures maximize Shannon information entropy distributed over a range of scales. The fractal measure is the sum of bins occupied by reconstructed data points for each bin spacing. The fitting criterion utilized in this work is, in turn, the grand sum of the fractal measures over all calculated bin spacings. The optimal fractal measure for the grand sum has minimal integrated "fractality" relative to non-optimal sets while maximizing entropy for the optimal parameters for each bin spacing.
Analysis of transient flow and starting pressure gradient of power-law fluid in fractal porous media
NASA Astrophysics Data System (ADS)
Tan, Xiao-Hua; Li, Xiao-Ping; Zhang, Lie-Hui; Liu, Jian-Yi; Cai, Jianchao
2015-09-01
A transient flow model for power-law fluid in fractal porous media is derived by combining transient flow theory with the fractal properties of tortuous capillaries. Pressure changes of transient flow for power-law fluid in fractal porous media are related to pore fractal dimension, tortuosity fractal dimension and the power-law index. Additionally, the starting pressure gradient model of power-law fluid in fractal porous media is established. Good agreement between the predictions of the present model and that of the traditional empirical model is obtained, the sensitive parameters that influence the starting pressure gradient are specified and their effects on the starting pressure gradient are discussed.
Ferretti, R.; Zhang, J.; Buffle, J. [Univ. of Geneva (Switzerland)] [Univ. of Geneva (Switzerland)
1998-12-15
The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (M{sub w} = 1.36 {times} 10{sup 6}, M{sub w}/M{sub n} = 1.53; M{sub w} = 3.69 {times} 10{sup 4}, M{sub w}/M{sub n} = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, D{sub f} = 1.84 {+-} 0.02 and 1.73 {+-} 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension D{sub f} {approx} 1.8. Moreover, for all three regimes, the D{sub f} values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure.
Light Scattering From Fractal Titania Aggregates
NASA Astrophysics Data System (ADS)
Pande, Rajiv; Sorensen, Christopher M.
1996-03-01
We studied the fractal morphology of titania aggregates by light scattering. Titanium dioxide particles were generated by the thermal decomposition of titanium tetra-isopropoxide(TTIP) in a glass furnace at various temperatures in the range of 100 - 500^o C. We scattered vertically polarized He-Ne laser (? = 6328Ålight from a laminar aerosol stream of particles and measured the optical structure factor. This structure factor shows Rayleigh, Guinier, fractal and Porod regimes. The radius of gyration Rg was determined from the Guinier analysis. The data were then fit to the Fisher-Burford form to determine the fractal dimension of about 2.0. This fit also delineated the crossover from the fractal to Porod regime, which can be used to determine the monomer particle size of about 0.1 ?m. These optical measurements will be compared to electron microscope analysis of aggregates collected from the aerosol. This work was supported by NSF grant CTS-9908153.
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.
Fractal behavior of a wrinkled annular diffusion flame--
Gutmark, E.; Hanson-Parr, D.M.; Parr, T.P.; Schadow, K.C. . Research Dept.)
1990-01-01
Fractal geometry is used to describe quantitatively the random character of a turbulent wrinkled diffusion flame. The article describes how basic ideas of mathematics of fractals can be applied to study the changes in the irregularity of the flame's surface at various axial positions. In the present work, the planar laser-induced-fluorescence (PLIF) imaging technique has been used to map the hydroxyl concentration in an annular diffusion flame with propane fuel. The short pulse duration (18 ns) of the laser sheet enabled instantaneous pictures of the flame structure to be obtained with high spatial resolution of 0.5 {times} 0.5 {times} 0.5 mm. The isopleths of concentration were highly wrinkled and random in character, both in space and time. They were measured in flames that were forced at different instability frequencies, in order to stabilize the roll-up of the vortices in the flame shear layer and to simulate vortex merging. Measurements were done from the flameholder to the flame's end at many axial positions, in order to study the evolution of the flame structure. The fractal dimension was calculated for all these conditions, using two methods: the Kolmogorov-Hansdorff definition and the perimeter area relation.
Fractal study of polystyrene latex and silica particle aggregates
NASA Astrophysics Data System (ADS)
Zhou, Zukang; Chu, Benjamin
1991-09-01
Aggregation kinetics and fractal structures of salt-induced fractal aggregates of polystyrene colloid and cationic-surfactant (TPB)-induced fractal silica aggregates have been studied. For polystyrene colloid in the quasi-fast regime of aggregation, the fractal dimensions ( df) largely deviate from the diffusion limited cluster aggregation (DLCA) value of 1.75. For TPB-induced silica clusters, all the df values obtained under different regimes of aggregation have essentially the same value 2.10 ± 0.05. The question of restructuring is raised.
Fractal analysis of bone structure with applications to osteoporosis and microgravity effects
Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E.; LeBlanc, A.; Lin, C.; Shackelford, L.
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.
Theoretical study of statistical fractal model with applications to mineral resource prediction
NASA Astrophysics Data System (ADS)
Wei, Shen; Pengda, Zhao
2002-04-01
The statistical estimation of fractal dimensions is an important topic of investigation. Current solutions emphsize visual straight-line fitting, but nonlinear statistical modeling has the potential of making valuable contributions in this field. In this paper, we present the concepts of generalized fractal models and generalized fractal dimension and conclude that many geological models are special cases of the generalized models. We show that the power-function distribution possesses the fractal property of scaling invariance under upper truncation, which may help in lead statistical fractal modeling. A new method is developed on the basis of nonlinear regression to estimate fractal parameters. This method has advantages with respect to the traditional method based on linear regression for estimating the fractal dimension. Finally, the new method is illustrated by means of application to a real data set.
Array Patterns of Fractal Linear Array Antennas Based on Cantor Set
NASA Astrophysics Data System (ADS)
Deepika Rani, N.; Sri Devi, P. V.
2012-03-01
A fractal is a recursively generated object having a fractional dimension. Antennas can be designed using the recursive nature of a fractal. In this paper general expression for array factor of fractal linear array based on cantor set was compared with conventional linear array. The similarity of the radiation patterns and their fractal features are examined for various iterations with the simulated results using MATLAB.
Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems.
Eberspächer, Alexander; Main, Jörg; Wunner, Günter
2010-10-01
The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations. PMID:21230359
NASA Astrophysics Data System (ADS)
Prajna, Shormistha; Rangayyan, Rangaraj M.; Ayres, Fábio J.; Desautels, J. E. Leo
2008-03-01
Mammography is a widely used screening tool for the early detection of breast cancer. One of the commonly missed signs of breast cancer is architectural distortion. The purpose of this study is to explore the application of fractal analysis and texture measures for the detection of architectural distortion in screening mammograms taken prior to the detection of breast cancer. A method based on Gabor filters and phase portrait analysis was used to detect initial candidates of sites of architectural distortion. A total of 386 regions of interest (ROIs) were automatically obtained from 14 "prior mammograms", including 21 ROIs related to architectural distortion. The fractal dimension of the ROIs was calculated using the circular average power spectrum technique. The average fractal dimension of the normal (false-positive) ROIs was higher than that of the ROIs with architectural distortion. For the "prior mammograms", the best receiver operating characteristics (ROC) performance achieved was 0.74 with the fractal dimension and 0.70 with fourteen texture features, in terms of the area under the ROC curve.
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)
NSDL National Science Digital Library
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.
Fractal structure of the interplanetary magnetic field
NASA Technical Reports Server (NTRS)
Burlaga, L. F.; Klein, L. W.
1985-01-01
Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.
Wellington da Cruz
2002-10-01
We consider the {\\it fractal von Neumann entropy} associated with the {\\it fractal distribution function} and we obtain for some {\\it universal classes h of fractons} their entropies. We obtain also for each of these classes a {\\it fractal-deformed Heisenberg algebra}. This one takes into account the braid group structure of these objects which live in two-dimensional multiply connected space.
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.…
A novel fractal monocular and stereo video codec based on MCP and DCP
Shiping Zhu; Zaikuo Wang; Kamel Belloulata
2010-01-01
In the paper, a novel fractal monocular video codec is proposed which includes: using new macroblock partition scheme instead of classical quadtree partition scheme; reducing the block searching strategy and range, thus increasing the calculation speed greatly; using homo-I-frame like in H.264; reducing repeating calculation. The fractal monocular video codec uses the MCP structure. And the fractal stereo video codec
Fractal statistics, fractal index and fractons
Wellington da Cruz
2000-10-10
The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge $c[\
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.
Self-affine fractal model for a metal-electrolyte interface
Kaplan, T.; Gray, L.J.; Liu, S.H.
1987-04-01
Fractal models of rough interfaces have provided a possible explanation for the observed ac response of an interface between a metal and an electrolyte with blocking contacts which displays the constant-phase-angle (CPA) behavior. For this system, two different relations between the exponent eta of the frequency dependence of the CPA element and the fractal dimension d/sub s/ of the interface have been proposed. By solving for the properties of a self-affine fractal model of an interface and using Mandelbrot's box dimension for the fractal dimension it is demonstrated that there is no universal relation in which eta is simply a function of d/sub s/.
Fractals in art and nature: why do we like them?
NASA Astrophysics Data System (ADS)
Spehar, Branka; Taylor, Richard P.
2013-03-01
Fractals have experienced considerable success in quantifying the visual complexity exhibited by many natural patterns, and continue to capture the imagination of scientists and artists alike. Fractal patterns have also been noted for their aesthetic appeal, a suggestion further reinforced by the discovery that the poured patterns of the American abstract painter Jackson Pollock are also fractal, together with the findings that many forms of art resemble natural scenes in showing scale-invariant, fractal-like properties. While some have suggested that fractal-like patterns are inherently pleasing because they resemble natural patterns and scenes, the relation between the visual characteristics of fractals and their aesthetic appeal remains unclear. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Whereas our previous preference studies focused on fractal images consisting of black shapes on white backgrounds, here we extend our investigations to include grayscale images in which the intensity variations exhibit scale invariance. This scale-invariance is generated using a 1/f frequency distribution and can be tuned by varying the slope of the rotationally averaged Fourier amplitude spectrum. Thresholding the intensity of these images generates black and white fractals with equivalent scaling properties to the original grayscale images, allowing a direct comparison of preferences for grayscale and black and white fractals. We found no significant differences in preferences between the two groups of fractals. For both set of images, the visual preference peaked for images with the amplitude spectrum slopes from 1.25 to 1.5, thus confirming and extending the previously observed relationship between fractal characteristics of images and visual preference.
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
NASA Astrophysics Data System (ADS)
Vieira, Sidney R.; Vidal Vázquez, Eva; Miranda, José G. V.; Paz Ferreiro, Jorge; Topp, George C.
2010-05-01
Spatial and temporal variability of soil moisture content has been frequently evaluated using statistical and geostatistical methods for several issues. For example, the statistical study of the temporal persistence or temporal stability in spatial patterns of soil moisture content has found interest to improve soil water monitoring strategies and to correct the average soil water content for missing data. Fractal analysis and graph theory are additional tools that can provide information and further insight to assess and to model indirect or hidden interactions in soil moisture content. In fractal analysis the fractal dimension (D) is an indicator of the pattern and extent of spatial and/or temporal variability. Large D values indicate the importance of short-range variation, while small D values reflect the importance of long-range variation when spatial and temporal data sets are analyzed. Moreover, for spatial and temporal variability, D can range from 1 to 2 for a profile and from 2 to 3 for a two dimensional network. Moreover, as the fractal dimension value increases the degree of roughness also increases. Graph theory tools take into account network structure by modelling pair wise relations between objects, which allow considering explicitly spatial-temporal connectivity of a given data set. The objective of this study was to use fractal analysis and graph theory to characterize the pattern of spatial and temporal variability of soil moisture content. The experimental field was located at Ottawa, Canada. Volumetric water content was monitored using Time Domain Reflectometry (TDR) during 34 dates at 164 locations per date. The depth of the TDR probes was 20 cm. The first and last measurements were 21 month apart and no data were taken in winter when the soil was covered by snow. The fractal dimension, D, was estimated from the slope of the regression line of log semivariogram versus distance for each of studied data sets. Using graph theory various parameters were calculated from the data measured in the 164 experimental vertices including edges, disconnected pair's number, average degree and clustering, etc.; calculations were performed for 21 groups of sets measured during three successive dates. Fractal dimension, D, ranged from 2.589 to 2.910, so that the smallest and the largest values indicate domination of long- and short-range variation respectively. Interestingly there was no correlation between fractal dimension, D, and coefficient of variation. Highest D values were recorded in spring and summer time. Parameters derived from graphs also allowed discrimination of the structure corresponding to successive data sets measured in three successive dates. For example, clustering varied from 0.406 to 0.836, given a correlation coefficient of 0.995. Different degrees of connectivity corresponded to different seasons. Parameters derived from fractal analysis and graph theory were useful to characterize the pattern and extent of spatial and temporal variability of soil moisture content. Acknowledgement: This work was partly supported by Spanish Ministry of Education (Project PHB2009-0094-PC.)
The fractal energy measurement and the singularity energy spectrum analysis
NASA Astrophysics Data System (ADS)
Xiong, Gang; Zhang, Shuning; Yang, Xiaoniu
2012-12-01
The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.
Fractal Coagulation Bruce E. Logan
Fractal Coagulation Kinetics Bruce E. Logan Department of Civil & Environmental Engineering paradigm shift is needed to explain the formation of marine snow? #12;Birth of Fractal Geometry ·In 1982, Benoit Mandelbrot publishes "Fractal Geometry" and fractal mathematics is born. ·Fractal scaling
Boyd, O.S.
2006-01-01
We have created a second-order finite-difference solution to the anisotropic elastic wave equation in three dimensions and implemented the solution as an efficient Matlab script. This program allows the user to generate synthetic seismograms for three-dimensional anisotropic earth structure. The code was written for teleseismic wave propagation in the 1-0.1 Hz frequency range but is of general utility and can be used at all scales of space and time. This program was created to help distinguish among various types of lithospheric structure given the uneven distribution of sources and receivers commonly utilized in passive source seismology. Several successful implementations have resulted in a better appreciation for subduction zone structure, the fate of a transform fault with depth, lithospheric delamination, and the effects of wavefield focusing and defocusing on attenuation. Companion scripts are provided which help the user prepare input to the finite-difference solution. Boundary conditions including specification of the initial wavefield, absorption and two types of reflection are available. ?? 2005 Elsevier Ltd. All rights reserved.
Spin Transport in Multiply Connected Fractal Conductors
NASA Astrophysics Data System (ADS)
Lee, Bo-Ray; Chang, Ching-Ray; Klik, Ivo
2014-12-01
We consider spin and charge transport in a Sierpinski planar carpet; the interest here is its unique geometry. We analyze the fractal conductor as a combination of multiply connected quantum wires, and we observe the evolution of the transmission envelope in different fractal generations. For a fractal conductor dominated by resonant modes the transmission is characterized by strong fluctuations and conduction gaps. We show that charge and spin transport have different responses both to the presence of defects and to applied bias. At a high bias, or in a high-order fractal generation, spin accumulation is separated from charge accumulation because the larger drift velocity needs a longer polarization length, and the sample may turn into an insulator by the action of the defects. Our results are calculated numerically using the Keldysh Green function within the tight-binding framework.
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
Fragmentation of fractal random structures.
Elçi, Eren Metin; Weigel, Martin; Fytas, Nikolaos G
2015-03-20
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects. PMID:25839290
Fragmentation of Fractal Random Structures
NASA Astrophysics Data System (ADS)
Elçi, Eren Metin; Weigel, Martin; Fytas, Nikolaos G.
2015-03-01
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.
Triangular Constellations in Fractal Measures
Wilkinson, Michael
2014-01-01
The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\\cal N}$ in a ball to its radius $\\epsilon$: ${\\cal N}\\sim \\epsilon^D$. It is desirable to characterise the {\\em shapes} of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio $z$ of its area to the radius of gyration squared. We show that the probability density of $z$ has a phase transition: $P(z)$ is independent of $\\epsilon$ and approximately uniform below a critical flow compressibility $\\beta_{\\rm c}$, but for $\\beta>\\beta_{\\rm c}$ it is described by two power laws: $P(z)\\sim z^{\\alpha_1}$ when $1\\gg z\\gg z_{\\rm c}(\\epsilon)$, and $P(z)\\sim z^{\\alpha_2}$ when $z\\ll z_{\\rm c}(\\epsilon)$.
Fractal 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.
Calculation of grey level co-occurrence matrix-based seismic attributes in three dimensions
NASA Astrophysics Data System (ADS)
Eichkitz, Christoph Georg; Amtmann, Johannes; Schreilechner, Marcellus Gregor
2013-10-01
Seismic interpretation can be supported by seismic attribute analysis. Common seismic attributes use mathematical relationships based on the geometry and the physical properties of the subsurface to reveal features of interest. But they are mostly not capable of describing the spatial arrangement of depositional facies or reservoir properties. Textural attributes such as the grey level co-occurrence matrix (GLCM) and its derived attributes are able to describe the spatial dependencies of seismic facies. The GLCM - primary used for 2D data - is a measure of how often different combinations of pixel brightness values occur in an image. We present in this paper a workflow for full three-dimensional calculation of GLCM-based seismic attributes that also consider the structural dip of the seismic data. In our GLCM workflow we consider all 13 possible space directions to determine GLCM-based attributes. The developed workflow is applied onto various seismic datasets and the results of GLCM calculation are compared to common seismic attributes such as coherence.
Atomic scale fractal dimensionality in proteins
NASA Astrophysics Data System (ADS)
Medini, Duccio; Widom, Allan
2003-02-01
The soft condensed matter of biological organisms exhibits atomic motions whose properties depend strongly on temperature and hydration conditions. Due to the superposition of rapidly fluctuating alternative motions at both very low temperatures (quantum effects) and very high temperatures (classical Brownian motion regime), the dimension of an atomic "path" is in reality different from unity. In the intermediate temperature regime and under environmental conditions which sustain active biological functions, the fractal dimension of the sets upon which atoms reside is an open question. Measured values of the fractal dimension of the sets on which the hydrogen atoms reside within the azurin protein macromolecule are reported. The distribution of proton positions was measured employing thermal neutron elastic scattering from azurin protein targets. As the temperature was raised from low to intermediate values, a previously known and biologically relevant dynamical transition was verified for the azurin protein only under hydrated conditions. The measured fractal exponent of the geometrical sets on which protons reside in the biologically relevant temperature regime is given by D=(0.65±0.1). The relationship between fractal dimensionality and biological function is qualitatively discussed.
How Fractal are Coastlines Really? Observation and Theory
NASA Astrophysics Data System (ADS)
Murray, A.; Barton, C. C.
2007-12-01
Rocky coastlines have been held up as a prime example of fractal geometry since Mandelbrot introduced the concept. However, we will present a map of the fractal dimensions measured for the contiguous United States coastline which shows that many open-ocean sand--and even rocky--coastlines have fractal dimensions close to one; i.e. they tend to not be very fractal. The fractal nature of rocky coastlines likely represents an inherited fluvial or glacial signature that tends to be erased by coastal processes. Recent theoretical and numerical-modeling developments indicate that wave-driven coastal processes on sandy shores tend to produce one-dimensional coastlines. Gradients in alongshore sediment flux tend to smooth a shoreline, as long as the local wave climate is dominated by 'low-angle' waves (waves that approach the coastline in deep water from angles, relative to the coastline orientation, that are lower than the sediment-flux- maximizing angle). Even when a regional wave climate is dominated by high-angle waves--which produce an instability in plan-view shoreline shape--on the large scale, coastlines self organize in a way that produces locally low-angle-dominated wave climates almost everywhere. These processes explain why wave-dominated sandy coastlines, such as the Carolina and Texas coasts, exhibit fractal dimensions barely above one; wave- driven alongshore transport is an anti-fractal landsculpting agent over a range of scales greater than 0.2 km. In contrast, fluvial landsculpting produces famously fractal topography. When rapid sea-level rise causes the approximately horizontal plane of sea level to intersect a fractal fluvial topography, a fractal coastline results. Where wave energy is low, relative to rock erodibility, the fluvial fractal signature can persist. However, on the rocky West Coast of the US, fractal dimensions are relatively low (1.1 - 1.2), suggesting modification by wave-driven processes; that the production and rearrangement of sediment into ever-expanding pocket beaches has been reducing the fractality of this high-wave-energy, relatively easily eroded coastline. Glacially carved coastlines, such as that of Maine (and some parts of western Britain and Norway), exhibit high fractal dimensions (approximately 1.5), where erodibility is low enough the self-similarity of the intersection of sea-level with a glacially sculpted topography remains. Although wave-driven coastal processes tend to generate low-fractal-dimension shorelines, on sandy coastlines dominated by tidal currents, coastal processes also etch a fractal dendritic network of channels into the coastline. Tidally dominated coastlines, such as those in the Georgia Bight (Southeastern US), sport highly fractal shapes as a result (fractal dimensions approximately 1.5).
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)
Fractal structure and cluster statistics of zinc-metal trees de- posited on a line electrode
NASA Astrophysics Data System (ADS)
Matsushita, M.; Hayakawa, Y.; Sawada, Y.
1985-12-01
Zinc-metal ``trees'' are grown two dimensionally from a line electrode by electrodeposition. The deposits, which consist of trees of all sizes, bear close resemblance to the patterns of two-dimensional (d=2) diffusion-limited deposition on a one-dimensional (db=1) substrate computer simulated first by Meakin. The fractal nature of the deposits is confirmed with the fractal dimension df(2,1)=0.70+/-0.06. The number of deposits (trees) of size s is also found to scale as ns~s-? with ?apeq21.54. These values are in excellent agreement with the theoretical values calculated for the case of the diffusion-limited deposition with d=2 and db=1 and with the large-scale computer-simulation results.
NASA Astrophysics Data System (ADS)
Qing-Feng, Hou; Xian-Cai, Lu; Xian-Dong, Liu; Bai-Xing, Hu; Zhi-Jun, Lu; Jian, Shen
2005-02-01
The fractal analysis is carried out to study the influence of adsorption of polyoxyethylene sorbitan monooleate (Tween 80) on the surface properties of graphite. The surface fractal dimension ( d), BET surface area ( S) and pore size distribution (PSD) are calculated from low temperature nitrogen adsorption isotherms. The decline in the d of graphite surface is found as the adsorption amount of Tween 80 increases, which suggests that the adsorbed Tween 80 smoothes the graphite surface. Additionally, the observation of atomic force microscopy (AFM) proves that the original slit pores in pure graphite are blocked up and the step defect sites are screened by Tween 80, which may result in the reduction of graphite roughness. The PSD pattern of graphite changes after the adsorption due to the pore blocking effect. S of the graphite decreases as the adsorption amount of Tween 80 increases, which is attributed to both pore blocking effect and surface screening effect.
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.
NASA Technical Reports Server (NTRS)
Barnsley, Michael F.; Sloan, Alan D.
1989-01-01
Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed.
Black carbon fractal morphology and short-wave radiative impact: a modelling study
NASA Astrophysics Data System (ADS)
Kahnert, M.; Devasthale, A.
2011-11-01
We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of different error sources. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material. Finally, we found that the spectral variation of the absorption cross section of black carbon significantly deviates from a simple 1/? scaling law. We therefore discourage the use of single-wavelength absorption measurements in conjunction with a 1/? scaling relation in broadband radiative forcing simulations of black carbon.
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.
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.
Full dimension Rb2He ground triplet potential energy surface and quantum scattering calculations.
Guillon, Grégoire; Viel, Alexandra; Launay, Jean-Michel
2012-05-01
We have developed a three-dimensional potential energy surface for the lowest triplet state of the Rb(2)He complex. A global analytic fit is provided as in the supplementary material [see supplementary material at http://dx.doi.org/10.1063/1.4709433 for the corresponding Fortran code]. This surface is used to perform quantum scattering calculations of (4)He and (3)He colliding with (87)Rb(2) in the partial wave J = 0 at low and ultralow energies. For the heavier helium isotope, the computed vibrational relaxation probabilities show a broad and strong shape resonance for a collisional energy of 0.15 K and a narrow Feshbach resonance at about 17 K for all initial Rb(2) vibrational states studied. The broad resonance corresponds to an efficient relaxation mechanism that does not occur when (3)He is the colliding partner. The Feshbach resonance observed at higher collisional energy is robust with respect to the isotopic substitution. However, its effect on the vibrational relaxation mechanism is faint for both isotopes. PMID:22583230
Resistance of Feynman diagrams and the percolation backbone dimension.
Janssen, H K; Stenull, O; Oerding, K
1999-06-01
We present an alternative view of Feynman diagrams for the field theory of random resistor networks, in which the diagrams are interpreted as being resistor networks themselves. This simplifies the field theory considerably as we demonstrate by calculating the fractal dimension D(B) of the percolation backbone to three loop order. Using renormalization group methods we obtain D(B)=2+epsilon/21-172epsilon(2)/9261+2epsilon(3)[-74 639+22 680zeta(3)]/4 084 101, where epsilon=6-d with d being the spatial dimension and zeta(3)=1.202 057... . PMID:11969728
Analytical estimation of the correlation dimension of integer lattices
NASA Astrophysics Data System (ADS)
Lacasa, Lucas; Gómez-Gardeñes, Jesús
2014-12-01
Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 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 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 ? of integer lattices ? d coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, ? = d.
Analytical estimation of the correlation dimension of integer lattices
Lacasa, Lucas; Gómez-Gardeñes, Jesús
2014-12-01
Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 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 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 ? of integer lattices ?{sup d} coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, ??=?d.
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...
Fractal and Multifractal Analysis of Human Gait
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.
2003-09-01
We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.
FORTRAN programs for calculating nonlinear seismic ground response in two dimensions
Joyner, W.B.
1978-01-01
The programs described here were designed for calculating the nonlinear seismic response of a two-dimensional configuration of soil underlain by a semi-infinite elastic medium representing bedrock. There are two programs. One is for plane strain motions, that is, motions in the plane perpendicular to the long axis of the structure, and the other is for antiplane strain motions, that is motions parallel to the axis. The seismic input is provided by specifying what the motion of the rock-soil boundary would be if the soil were absent and the boundary were a free surface. This may be done by supplying a magnetic tape containing the values of particle velocity for every boundary point at every instant of time. Alternatively, a punch card deck may be supplied giving acceleration values at every instant of time. In the plane strain program it is assumed that the acceleration values apply simultaneously to every point on the boundary; in the antiplane strain program it is assumed that the acceleration values characterize a plane shear wave propagating upward in the underlying elastic medium at a specified angle with the vertical. The nonlinear hysteretic behavior of the soil is represented by a three-dimensional rheological model. A boundary condition is used which takes account of finite rigidity in the elastic substratum. The computations are performed by an explicit finite-difference scheme that proceeds step by step in space and time. Computations are done in terms of stress departures from an unspecified initial state. Source listings are provided here along with instructions for preparing the input. A more detailed discussion of the method is presented elsewhere.
Zhou, Jack
"Fractal Growth Modeling of Electrochemical Deposition in Solid Freeform Fabrication," J. G. Zhou, August, 1999. FRACTAL GROWTH MODELING OF ELECTROCHEMICAL DEPOSITION IN SOLID FREEFORM FABRICATION Jack G deposition among metal particles during ECLD-SFF is a fractal growth process. The fractal dimension
Frequency-dependent viscous flow in channels with fractal rough surfaces
NASA Astrophysics Data System (ADS)
Cortis, Andrea; Berryman, James 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 ? (but has a finite cutoff), and the universal scaling of the dynamic permeability, k(? ), needs modification. We performed accurate numerical computations of k(? ) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k(? ) 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.
Fractal Aspects of Miscible Displacement in Rough Fractures: AN Experimental Approach
NASA Astrophysics Data System (ADS)
Korfanta, M.; Babadagli, T.; Develi, K.
2015-02-01
Experiments were performed to study the effect of fracture surface roughness on fluid distribution during miscible displacement. The transparent replicas of single fractures obtained from seven different rocks were prepared and the surface roughness of each sample was described by fractal dimensions using the variogram, power spectral, and triangular prism (TP) techniques. Then, the effect of flow rate and viscosity on the geometry of the displacement front during miscible radial injection was investigated experimentally. The fractal dimensions of the fronts were obtained using box counting fractal analysis at different time lapses. The fractal values of invasion front varied from lithology to lithology, due to different surface roughnesses controlled by the lithology of the rocks. Although fluctuations of fractal values were observed during the growth of the front, fractal dimensions typically yielded an increasing trend. Fractal dimension became more stable with increasing flow rate and developed modestly with increasing viscosity. Finally, relationships between the fractal dimensions of displacement fronts and fracture surfaces were quantitatively analyzed and correlated in order to improve the prediction of fluid distribution within a single fracture during miscible displacement. Overall, correlations were observed between the surface characteristics and front fractal dimension values with some exceptions. In summary, to determine the probable distribution of miscible fluid and development of the front, all parameters except power spectral density (PSD) fractal dimension can be applied in the case of high viscosity ratios. In the case of low injection rates, TP could be applicable. No fractal behavior was present at extreme injection and low viscosity ratios, thus no correlation can be determined for the miscible displacement.
Fourier transforms and fractals in the food and agricultural industry
NASA Astrophysics Data System (ADS)
Zwiggelaar, Reyer; Bull, Christine R.
1994-11-01
Links between the fractal Hausdorff-dimension, the Fourier transform of 2D scenes and image segmentation by texture are discussed. It is shown that the fractal Hausdorff-dimension can be derived by integration of the intensity of the spatial frequency domain (i.e. the Fourier plane) over a set of different band-limited spatial filters. The difference between a computational and optical approach to determine the Hausdorff-dimension are shown, with advantages of both methods discussed. Possible future directions of research/improvements are mentioned. Natural and simulated scenes are considered which apply to a wide range of situations in the agricultural and food industry.
Facilitated diffusion of proteins through crumpled fractal DNA globules
NASA Astrophysics Data System (ADS)
Smrek, Jan; Grosberg, Alexander Y.
2015-07-01
We explore how the specific fractal globule conformation, found for the chromatin fiber of higher eukaryotes and topologically constrained dense polymers, affects the facilitated diffusion of proteins in this environment. Using scaling arguments and supporting Monte Carlo simulations, we relate DNA looping probability distribution, fractal dimension, and protein nonspecific affinity for the DNA to the effective diffusion parameters of the proteins. We explicitly consider correlations between subsequent readsorption events of the proteins, and we find that facilitated diffusion is faster for the crumpled globule conformation with high intersegmental surface dimension than in the case of dense fractal conformations with smooth surfaces. As a byproduct, we obtain an expression for the macroscopic conductivity of a hypothetic material consisting of conducting fractal nanowires immersed in a weakly conducting medium.
Evaluation of Two Fractal Methods for Magnetogram Image Analysis
NASA Technical Reports Server (NTRS)
Stark, B.; Adams, M.; Hathaway, D. H.; Hagyard, M. J.
1997-01-01
Fractal and multifractal techniques have been applied to various types of solar data to study the fractal properties of sunspots as well as the distribution of photospheric magnetic fields and the role of random motions on the solar surface in this distribution. Other research includes the investigation of changes in the fractal dimension as an indicator for solar flares. Here we evaluate the efficacy of two methods for determining the fractal dimension of an image data set: the Differential Box Counting scheme and a new method, the Jaenisch scheme. To determine the sensitivity of the techniques to changes in image complexity, various types of constructed images are analyzed. In addition, we apply this method to solar magnetogram data from Marshall Space Flight Centers vector magnetograph.
Scaling of voids and fractality in the galaxy distribution
Jose Gaite; Susanna C. Manrubia
2002-05-14
We study here, from first principles, what properties of voids are to be expected in a fractal point distribution and how the void distribution is related to its morphology. We show this relation in various examples and apply our results to the distribution of galaxies. If the distribution of galaxies forms a fractal set, then this property results in a number of scaling laws to be fulfilled by voids. Consider a fractal set of dimension $D$ and its set of voids. If voids are ordered according to decreasing sizes (largest void has rank R=1, second largest R=2 and so on), then a relation between size $\\Lambda$ and rank of the form $\\Lambda (R) \\propto R^{-z}$ must hold, with $z = d/D$, and where $d$ is the euclidean dimension of the space where the fractal is embedded. The physical restriction $D 1$ in a fractal set. The average size $\\bar \\Lambda$ of voids depends on the upper ($\\Lambda_u$) and the lower ($\\Lambda_l$) cut-off as ${\\bar \\Lambda} \\propto \\Lambda_u^{1-D/d} \\Lambda_l^{D/d}$. Current analysis of void sizes in the galaxy distribution do not show evidence of a fractal distribution, but are insufficient to rule it out. We identify possible shortcomings of current void searching algorithms, such as changes of shape in voids at different scales or merging of voids, and propose modifications useful to test fractality in the galaxy distribution.
Fractal Generation on GPU Most fractal generation software uses
Lu, Enyue "Annie"
Fractal Generation on GPU Abstract Most fractal generation software uses shortcuts and display of fractals would be much more easily done on a graphics card. My work is to start applying the shortcuts and functionality of free fractal software to code that runs on the GPU using the CUDA programming
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
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 diffraction elements with variable transmittance and phase shift
NASA Astrophysics Data System (ADS)
Muzychenko, Ya. B.; Zinchik, A. A.; Stafeev, S. C.; Tomilin, M. G.
2011-09-01
The new type of diffraction fractal elements is presented and optical fields properties, obtained from these elements are discussed. Fractal diffraction elements based on well-known fractals, possess exact or statistical selfsimilarity, but have managed amplitude transmittance and phase shift, which are correlated with fractal spatial characteristics. The fractal dimension is not enough for these objects description, and the correlation coefficient between phase/amplitude and spatial characteristic is needed. For this reason the fractal objects were called multifractal structures (MFS). It is shown that the MFS diffraction spectrum possess prevailing power of high frequencies in comparison with spectra of regular two-dimensional or fractal structures with binary transmittance and phase shift. This property could be applied for spatial filtering and transparent objects phase heterogeneities detection. Modeling results for different MFS types are presented and it is shown that MFS application allows detecting the value of initial object distortion with high accuracy. The description of fractal zone plates (FraZP) with variable transmittance and/or phase shift is also presented. The results of Fresnel diffraction modeling from FraZPs with MFS show that the correlation coefficient value has influence on the focal point position.
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.
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed. PMID:24978815
On fractal analysis of cardiac interbeat time series
NASA Astrophysics Data System (ADS)
Guzmán-Vargas, L.; Calleja-Quevedo, E.; Angulo-Brown, F.
2003-09-01
In recent years the complexity of a cardiac beat-to-beat time series has been taken as an auxiliary tool to identify the health status of human hearts. Several methods has been employed to characterize the time series complexity. In this work we calculate the fractal dimension of interbeat time series arising from three groups: 10 young healthy persons, 8 elderly healthy persons and 10 patients with congestive heart failures. Our numerical results reflect evident differences in the dynamic behavior corresponding to each group. We discuss these results within the context of the neuroautonomic control of heart rate dynamics. We also propose a numerical simulation which reproduce aging effects of heart rate behavior.
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
Mehan, Sumit, E-mail: sumit.mehan@gmail.com; Kumar, Sugam, E-mail: sumit.mehan@gmail.com; Aswal, V. K., E-mail: sumit.mehan@gmail.com [Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai-400085 (India)
2014-04-24
Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.
A Fractal Analysis Approach for the Evaluation of Hybridization Kinetics in Biosensors.
Sadana, Ajit; Ramakrishnan, Anand
2001-02-01
The diffusion-limited hybridization kinetics of analyte in solution to a receptor immobilized on a biosensor or immunosensor surface is analyzed within a fractal framework. The data may be analyzed by a single- or a dual-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (Sigmaplot, Scientific Graphing Software, User's Manual, Jandel Scientific, CA, 1993). It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes, in general, in the same direction for both the single-fractal and the dual-fractal analysis examples presented. The binding rate coefficient expression developed as a function of the analyte concentration in solution and the fractal dimension is of particular value since it provides a means to better control biosensor or immunosensor performance. Copyright 2001 Academic Press. PMID:11161484
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.
Heterogeneity description using fractal concepts
NASA Astrophysics Data System (ADS)
Yortsos, Yanis C.
A brief overview of basic aspects of fractal geometry is presented and arguments for its relevance to porous media are given. Three different classes of application are considered: characterization and properties of fractal pore surfaces, gradient transport over fractal objects (e.g. fracture networks), and the representation of property heterogeneity (e.g. permeability) by fractal statistics (fractional Brownian motion).
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.
Electromagnetic fields in fractal continua
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Mena, Baltasar; Patiño, Julián; Morales, Daniel
2013-04-01
Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum ?D3?E3 with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space F? accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed.
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.
Soliton fractals in the Korteweg-de Vries equation.
Zamora-Sillero, Elias; Shapovalov, A V
2007-10-01
We have studied the process of creation of solitons and generation of fractal structures in the Korteweg-de Vries (KdV) equation when the relation between the nonlinearity and dispersion is abruptly changed. We observed that when this relation is changed nonadiabatically the solitary waves present in the system lose their stability and split up into ones that are stable for the set of parameters. When this process is successively repeated the trajectories of the solitary waves create a fractal treelike structure where each branch bifurcates into others. This structure is formed until the iteration where two solitary waves overlap just before the breakup. By means of a method based on the inverse scattering transformation, we have obtained analytical results that predict and control the number, amplitude, and velocity of the solitary waves that arise in the system after every change in the relation between the dispersion and the nonlinearity. This complete analytical information allows us to define a recursive L system which coincides with the treelike structure, governed by KdV, until the stage when the solitons start to overlap and is used to calculate the Hausdorff dimension and the multifractal properties of the set formed by the segments defined by each of the two "brothers" solitons before every breakup. PMID:17995132
Flocs vs granules: Differentiation by fractal dimension
M. Bellouti; M. M. Alves; J. M. Novais; M. Mota
1997-01-01
High rate anaerobic wastewater treatment systems usually give rise to biomass structured in different types of aggregates, depending on prevalent environmental conditions. Although highly dependent on wastewater characteristics, granules are generally formed and found in UASB reactors, whereas flocs are mainly found in fixed bed reactors. Different structures usually have different shapes and surface roughness. The aim of this work
Super Water and Oil-Repellent Surfaces Resulting from Fractal Structure
S. Shibuichi; T. Yamamoto; T. Onda; K. Tsujii
1998-01-01
Super water- and oil-repellent surfaces have been made utilizing the fractal structure of the solid surfaces. Super water-repellent surfaces showing a contact angle of 160° for water droplets have been made of anodically oxidized aluminum surfaces treated with hydrophobic surface coupling agent (1H,1H,2H,2H-perfluorooctyltrichlorosilane). The fractal dimension of this surface was evaluated to be 2.19 by box counting fractal analysis. The
Fractal nature of multiple shear bands in severely deformed metallic glass
Sun, B. A.; Wang, W. H.
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.
Fractal Index, Central Charge and Fractons
NASA Astrophysics Data System (ADS)
da Cruz, Wellington; de Oliveira, Rosevaldo
We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge c[?] and the particle-hole duality ?<-->1/?, for integer-value ? of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as v ~(c[? ])/(? +1). The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.
Thermodynamics of Fractal Universe
Ahmad Sheykhi; Zeinab Teimoori; Bin Wang
2013-01-12
We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $ \\delta Q=T_h d{S_h}$, where $ \\delta Q $ and $ T_{h} $ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon. We find that the entropy $S_h$ consists two terms, the first one which obeys the usual area law and the second part which is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \\leq q thermodynamics can be secured in a fractal universe by suitably choosing the fractal parameter $\\beta$.
NASA Astrophysics Data System (ADS)
McAteer, R. T. J.
2013-06-01
When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve into an even stronger link in the near future.
Fractal analysis of the spatial distribution of earthquakes along the Hellenic Subduction Zone
NASA Astrophysics Data System (ADS)
Papadakis, Giorgos; Vallianatos, Filippos; Sammonds, Peter
2014-05-01
The Hellenic Subduction Zone (HSZ) is the most seismically active region in Europe. Many destructive earthquakes have taken place along the HSZ in the past. The evolution of such active regions is expressed through seismicity and is characterized by complex phenomenology. The understanding of the tectonic evolution process and the physical state of subducting regimes is crucial in earthquake prediction. In recent years, there is a growing interest concerning an approach to seismicity based on the science of complex systems (Papadakis et al., 2013; Vallianatos et al., 2012). In this study we calculate the fractal dimension of the spatial distribution of earthquakes along the HSZ and we aim to understand the significance of the obtained values to the tectonic and geodynamic evolution of this area. We use the external seismic sources provided by Papaioannou and Papazachos (2000) to create a dataset regarding the subduction zone. According to the aforementioned authors, we define five seismic zones. Then, we structure an earthquake dataset which is based on the updated and extended earthquake catalogue for Greece and the adjacent areas by Makropoulos et al. (2012), covering the period 1976-2009. The fractal dimension of the spatial distribution of earthquakes is calculated for each seismic zone and for the HSZ as a unified system using the box-counting method (Turcotte, 1997; Robertson et al., 1995; Caneva and Smirnov, 2004). Moreover, the variation of the fractal dimension is demonstrated in different time windows. These spatiotemporal variations could be used as an additional index to inform us about the physical state of each seismic zone. As a precursor in earthquake forecasting, the use of the fractal dimension appears to be a very interesting future work. Acknowledgements Giorgos Papadakis wish to acknowledge the Greek State Scholarships Foundation (IKY). References Caneva, A., Smirnov, V., 2004. Using the fractal dimension of earthquake distributions and the slope of the recurrence curve to forecast earthquakes in Colombia. Earth Sci. Res. J., 8, 3-9. Makropoulos, K., Kaviris, G., Kouskouna, V., 2012. An updated and extended earthquake catalogue for Greece and adjacent areas since 1900. Nat. Hazards Earth Syst. Sci., 12, 1425-1430. Papadakis, G., Vallianatos, F., Sammonds, P., 2013. Evidence of non extensive statistical physics behavior of the Hellenic Subduction Zone seismicity. Tectonophysics, 608, 1037-1048. Papaioannou, C.A., Papazachos, B.C., 2000. Time-independent and time-dependent seismic hazard in Greece based on seismogenic sources. Bull. Seismol. Soc. Am., 90, 22-33. Robertson, M.C., Sammis, C.G., Sahimi, M., Martin, A.J., 1995. Fractal analysis of three-dimensional spatial distributions of earthquakes with a percolation interpretation. J. Geophys. Res., 100, 609-620. Turcotte, D.L., 1997. Fractals and chaos in geology and geophysics. Second Edition, Cambridge University Press. Vallianatos, F., Michas, G., Papadakis, G., Sammonds, P., 2012. A non-extensive statistical physics view to the spatiotemporal properties of the June 1995, Aigion earthquake (M6.2) aftershock sequence (West Corinth rift, Greece). Acta Geophys., 60, 758-768.
Fractal aggregates in tennis ball systems
NASA Astrophysics Data System (ADS)
Sabin, J.; Bandín, M.; Prieto, G.; Sarmiento, F.
2009-09-01
We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the images of the cluster of balls, following Forrest and Witten's pioneering studies on the aggregation of smoke particles, to estimate their fractal dimension.
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.
POROSITIES AND DIMENSIONS OF MEASURES SATISFYING THE DOUBLING JEANPIERRE ECKMANN
POROSITIES AND DIMENSIONS OF MEASURES SATISFYING THE DOUBLING CONDITION JEANÂPIERRE ECKMANN doubling condition. Our interest in the question of voids in fractals (porosity) was raised by questions the fractal dimension can be estimated. This paper provides a new such tool, namely the porosity
Tissue as a self-organizing system with fractal dynamics.
Waliszewski, P; Konarski, J
2001-01-01
Cell is a supramolecular dynamic network. Screening of tissue-specific cDNA library and results of Relative RT-PCR indicate that the relationship between genotype, (i.e., dynamic network of genes and their protein regulatory elements) and phenotype is non-bijective, and mendelian inheritance is a special case only. This implies non-linearity, complexity, and quasi-determinism, (i.e., co-existence of deterministic and non-deterministic events) of dynamic cellular network; prerequisite conditions for the existence of fractal structure. Indeed, the box counting method reveals that morphological patterns of the higher order, such as gland-like structures or populations of differentiating cancer cells possess fractal dimension and self-similarity. Since fractal space is not filled out randomly, a variety of morphological patterns of functional states arises. The expansion coefficient characterizes evolution of fractal dynamics. The coefficient indicates what kind of interactions occurs between cells, and how far from the limiting integer dimension of the Euclidean space the expanding population of cells is. We conclude that cellular phenomena occur in the fractal space; aggregation of cells is a supracollective phenomenon (expansion coefficient > 0), and differentiation is a collective one (expansion coefficient < 0). Fractal dimension or self-similarity are lost during tumor progression. The existence of fractal structure in a complex tissue system denotes that dynamic cellular phenomena generate an attractor with the appropriate organization of space-time. And vice versa, this attractor sets up physical limits for cellular phenomena during their interactions with various fields. This relationship can help to understand the emergence of extraterrestial forms of life. Although those forms can be composed of non-carbon molecules, fractal structure appears to be the common feature of all interactive biosystems. PMID:11799986
The thermodynamics of fractals revisited with wavelets
NASA Astrophysics Data System (ADS)
Arneodo, A.; Bacry, E.; Muzy, J. F.
1995-02-01
The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f( ?) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.
Selective modulation of cell response on engineered fractal silicon substrates
Gentile, Francesco; Medda, Rebecca; Cheng, Ling; Battista, Edmondo; Scopelliti, Pasquale E.; Milani, Paolo; Cavalcanti-Adam, Elisabetta A.; Decuzzi, Paolo
2013-01-01
A plethora of work has been dedicated to the analysis of cell behavior on substrates with ordered topographical features. However, the natural cell microenvironment is characterized by biomechanical cues organized over multiple scales. Here, randomly rough, self-affinefractal surfaces are generated out of silicon,where roughness Ra and fractal dimension Df are independently controlled. The proliferation rates, the formation of adhesion structures, and the morphology of 3T3 murine fibroblasts are monitored over six different substrates. The proliferation rate is maximized on surfaces with moderate roughness (Ra ~ 40?nm) and large fractal dimension (Df ~ 2.4); whereas adhesion structures are wider and more stable on substrates with higher roughness (Ra ~ 50?nm) and lower fractal dimension (Df ~ 2.2). Higher proliferation occurson substrates exhibiting densely packed and sharp peaks, whereas more regular ridges favor adhesion. These results suggest that randomly roughtopographies can selectively modulate cell behavior. PMID:23492898
Bi-Phase Box Counting: AN Improved Method for Fractal Analysis of Binary Images
NASA Astrophysics Data System (ADS)
Perfect, E.; Donnelly, B.
2015-02-01
Many natural systems are irregular and/or fragmented, and have been interpreted to be fractal. An important parameter needed for modeling such systems is the fractal dimension, D. This parameter is often estimated from binary images using the box-counting method. However, it is not always apparent which fractal model is the most appropriate. This has led some researchers to report different D values for different phases of an analyzed image, which is mathematically untenable. This paper introduces a new method for discriminating between mass fractal, pore fractal, and Euclidean scaling in images that display apparent two-phase fractal behavior when analyzed using the traditional method. The new method, coined "bi-phase box counting", involves box-counting the selected phase and its complement, fitting both datasets conjointly to fractal and/or Euclidean scaling relations, and examining the errors from the resulting regression analyses. Use of the proposed technique was demonstrated on binary images of deterministic and stochastic fractals with known D values. Traditional box counting was unable to differentiate between the fractal and Euclidean phases in these images. In contrast, bi-phase box counting unmistakably identified the fractal phase and correctly estimated its D value. The new method was also applied to three binary images of soil thin sections. The results indicated that two of the soils were pore-fractals, while the other was a mass fractal. This outcome contrasted with the traditional box counting method which suggested that all three soils were mass fractals. Reclassification has important implications for modeling soil structure since different fractal models have different scaling relations. Overall, bi-phase box counting represents an improvement over the traditional method. It can identify the fractal phase and it provides statistical justification for this choice.
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
A simple model for fractalization of fault gouge
NASA Astrophysics Data System (ADS)
Hatano, Takahiro
2010-05-01
A constitutive law of fault gouge is essential in describing the stability/instability of the motion of a fault. Extensive experiments and simulations have been conducted to reveal that the frictional properties of granular matter, such as the fluctuation of frictional force, are significantly affected by the particle-size distribution (PSD). For example, see Marone and Scholz 1989, Morgan and Boettcher 1999, Mair et al. 2002, and Abe and Mair 2005. Because the PSD depends on the nature of comminution in a fault, comminution is one of the key processes that dominate the frictional properties of fault gouge. However, we still cannot answer the following fundamental questions: Why is gouge fractal? What sets the fractal dimension? Here we show a simple (perhaps the simplest) model that can give answers to the above questions. Our model is somewhat similar to a classical model by Sammis et al. (1989), but quite different in modeling successive fragmentation processes. Our model involves the time evolution equation of PSD so that we can discuss any transient states of PSD during the comminution process. One of the important results is that the PSD for a steady state is always fractal irrespective of the fracture criterion of each particle. This makes a quite contrast to the model of Sammis et al, which requires a certain condition to the fracture criterion in order to reproduce the fractal PSD; otherwise the lognormal PSD is obtained in their model. Another important prediction of our model is the fractal dimension. It is found that the fractal dimension depends on the fracture criterion. To reproduce the universal value 2.6, the fracture probability of a single particle should be proportional to d-0.4, where d denotes a dimension of a particle. We can explain this fracture probability by taking two ingredients into account: the strength of single particle and the statistics of force chains. Furthermore, by means of discrete element simulation, we investigate the rate dependence of friction coefficient of granular matter that has fractal PSD. It is found that the (time-averaged) friction coefficient of fractal systems is lower than that of a non-fractal system, but is insensitive to the fractal dimension.
An effective algorithm for quick fractal analysis of movement biosignals in ambulatory monitoring
A. Ripoli; A. Belardinelli; R. Bedini
1998-01-01
The problem of numerically classifying patterns, of crucial importance in the biomedical field, is here faced by means of their fractal dimension. A new simple algorithm was developed to characterise biomedical monodimensional signals avoiding computing expensive methods, generally required by the classical approach of the fractal theory. The algorithm produces a number related to the geometric behaviour of the pattern
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
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
O. Lauscher; M. Reuter
2005-11-25
The asymptotic safety scenario of Quantum Einstein Gravity, the quantum field theory of the spacetime metric, is reviewed and it is argued that the theory is likely to be nonperturbatively renormalizable. It is also shown that asymptotic safety implies that spacetime is a fractal in general, with a fractal dimension of 2 on sub-Planckian length scales.
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.
ERIC Educational Resources Information Center
Bannon, Thomas J.
1991-01-01
Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)
Hurd, A.J.
1989-01-01
The realization that structures in Nature often can be described by Mandelbrot's ''fractals'' has led to a revolution in many areas of physics. The interaction of waves with fractal systems has, understandably, become intensely studied since scattering is the method of choice to probe delicate fractal structures such as chainlike particle aggregates. Not all of these waves are electromagnetic: neutron scattering, for example, is an important complementary tool to structural studies by x-ray and light scattering. Since the phenomenology of small-angle neutron scattering (SANS), as it is applied to fractal systems, is identical to that of small-angle x-ray scattering (SAXS), it falls within the scope of this Working Paper. 9 refs.
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)
Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.
Captur, Gabriella; Karperien, Audrey L; Li, Chunming; Zemrak, Filip; Tobon-Gomez, Catalina; Gao, Xuexin; Bluemke, David A; Elliott, Perry M; Petersen, Steffen E; Moon, James C
2015-01-01
Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools. PMID:26346700
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.
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.
NASA Astrophysics Data System (ADS)
Huang, C.; Yang, C. Z.
1999-03-01
A polymer-matrix nanocomposite containing copper particles has been prepared by in situ chemical reduction within a Cu2+-poly(itaconic acid-co-acrylic acid) complex solid film. The copper particle size in the order of 10 nm is controlled by the initial content of the metal ions in the complex. Their fractal pattern and the value of the fractal dimension indicate that there exists a cluster-cluster aggregation process in the present system. Optical absorption spectra of copper-polymer nanocomposites show distinct plasma absorption bands and quantum size effect in the samples. The calculated blueshift of the resonance peak based on a quantum-sphere model gives remarkable agreement with the experimental data as the size of copper particles embedded in the polymer becomes smaller.
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.
Spectral dimension of a quantum universe
Modesto, Leonardo; Nicolini, Piero [Perimeter Institute for Theoretical Physics, 31 Caroline Street, Waterloo, ON N2L 2Y5 (Canada); Frankfurt Institute for Advanced Studies (FIAS), Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main (Germany)
2010-05-15
In this paper, we calculate in a transparent way the spectral dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a spectral dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the spectral dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.
Hydrodynamics of fractal continuum flow
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Elizarraraz, Benjamin Espinoza
2012-02-01
A model of fractal continuum flow employing local fractional differential operators is suggested. The generalizations of the Green-Gauss divergence and Reynolds transport theorems for a fractal continuum are suggested. The fundamental conservation laws and hydrodynamic equations for an anisotropic fractal continuum flow are derived. Some physical implications of the long-range correlations in the fractal continuum flow are briefly discussed. It is noteworthy to point out that the fractal (quasi)metric defined in this paper implies that the flow of an isotropic fractal continuum obeying the Mandelbrot rule of thumb for intersection is governed by conventional hydrodynamic equations.
NASA Astrophysics Data System (ADS)
Zelenyuk, A.; Yang, J.; Imre, D.
2007-12-01
Single particle mass spectrometers (SPMS) have been developed and deployed to characterize, in real-time the size and chemical compositions of individual ambient particles. SPMS have undeniably proven to be extremely valuable, improving our understanding of the properties and evolution of atmospheric aerosols. At the same time these instruments have also persistently been subject to criticism on a number of fronts. Recently we have completed the construction of our second generation single particle mass spectrometer - SPLAT II, in which we have made significant improvements that directly address the shortcomings that are often associated with SPMS. We will present the results of instrument characterization experiments demonstrating that considerable improvements in instrument performance have been accomplished. SPLAT II offers significantly improved detection sensitivity of small particles, such that 30% of all 100 nm particles that enter the instrument are detected and characterized. This represents an improvement of nearly 2 orders of magnitude over the previous instrument. The efficiency of detection of particles larger than 150 nm is 100%. SPLAT II can measure and record the vacuum aerodynamic diameters of up to 300 particles per second and record 100 individual particle mass spectra per second - an increase of a factor of 10 in temporal resolution over the previous instrument. SPLAT II uses an IR laser to evaporate the semivolatile fraction of the particle and a time delayed UV laser to generate ions. The IR and UV lasers are configured to assure that the individual particle mass spectra represent the complete composition of each particle. The semivolatile fraction is ionized in the gas phase, while the nonvolatile fraction is concurrently ablated. This approach yields extremely reproducible, high quality mass spectral signatures while providing complete particle characterization. The two step - IR/UV - approach is particularly suitable for the characterization of organic molecules in fine and ultrafine particles. SPLAT II uses a data acquisition system with a 14 bit dynamic range- an increase of a factor of 128 in dynamic range over the previous instrument. This allows the instrument to record mass spectra of internally mixed particles that contain a range of substances with very different ionization probabilities. SPLAT II offers ultra-sensitive detection capabilities, making it possible to combine it with a DMA or a tandem DMA. The SPLAT II/DMA can simultaneously measure, in-situ individual particle size, composition, density, effective density, dynamic shape factor, fractal dimension, and hygroscopicity at a rate that yields statistically robust data. The SPLAT II/DMA system can also provide quantitative individual particle composition. To take a full advantage of the vast amounts of detailed data produced by SPLAT II we have developed two software packages: SpectraMiner and ClusterSculptor. SpectraMiner is a statistical analysis and data mining and visualization program that offers the user a wide range of visually driven controls to explore the data. ClusterSculptor is a software package that is coupled to SpectraMiner and is designed to overcome the limitation of statistics by offering the user the ability to insert his/hers scientific knowledge into the data organization phase.
An Analysis of Mine Water Inrush Based on Fractal and Non-Darcy Seepage Theory
NASA Astrophysics Data System (ADS)
Wu, Jinsui; Cai, Jianchao; Zhao, Dongyun; Chen, Xuexi
2014-09-01
Mining rock mechanics is a new cross subject of mechanics and mining engineering, the seepage theory is one of the important research directions. This paper combines Wu-fractal/Ergun high-speed flow theory and dynamic system instability, reveals the influence factors of mine water inrush. Research shows that: Mine water inrush related to rock porosity, particle size, shape, fractal dimension, ratio of pore and throat, and other factors. Compared the critical Reynolds number which are got from Wu-fractal model and Ergun equation, Wu-fractal model can reveal more influence factors of mine water inrush than Ergun equation.
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.
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
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 AC circuits and propagating waves on fractals
Akkermans, Eric; Dunne, Gerald; Rogers, Luke G; Teplyaev, Alexander
2015-01-01
We extend Feynman's analysis of the infinite ladder AC circuit to fractal AC circuits. We show that the characteristic impedances can have positive real part even though all the individual impedances inside the circuit are purely imaginary. This provides a physical setting for analyzing wave propagation of signals on fractals, by analogy with the Telegrapher's Equation, and generalizes the real resistance metric on a fractal, which provides a measure of distance on a fractal, to complex impedances.
Fractal AC circuits and propagating waves on fractals
Eric Akkermans; Joe P. Chen; Gerald Dunne; Luke G. Rogers; Alexander Teplyaev
2015-07-21
We extend Feynman's analysis of the infinite ladder AC circuit to fractal AC circuits. We show that the characteristic impedances can have positive real part even though all the individual impedances inside the circuit are purely imaginary. This provides a physical setting for analyzing wave propagation of signals on fractals, by analogy with the Telegrapher's Equation, and generalizes the real resistance metric on a fractal, which provides a measure of distance on a fractal, to complex impedances.
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.
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...
Turbulence on a Fractal Fourier set
Alessandra Sabina Lanotte; Roberto Benzi; Luca Biferale; Shiva Kumar Malapaka; Federico Toschi
2015-05-29
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-triad nonlinear energy transfer mechanism.
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.
An electrical conductivity model for fractal porous media
NASA Astrophysics Data System (ADS)
Wei, Wei; Cai, Jianchao; Hu, Xiangyun; Han, Qi
2015-06-01
Archie's equation is an empirical electrical conductivity-porosity model that has been used to predict the formation factor of porous rock for more than 70 years. However, the physical interpretation of its parameters, e.g., the cementation exponent m, remains questionable. In this study, a theoretical electrical conductivity equation is derived based on the fractal characteristics of porous media. The proposed model is expressed in terms of the tortuosity fractal dimension (DT), the pore fractal dimension (Df), the electrical conductivity of the pore liquid, and the porosity. The empirical parameter m is then determined from physically based parameters, such as DT and Df. Furthermore, a distinct interrelationship between DT and Df is obtained. We find a reasonably good match between the predicted formation factor by our model and experimental data.
Variable Topology on Fractal Manifold
Helene Porchon
2012-11-15
In this paper, we study the topology associated to the fractal manifold model. It turns out that this topology is actually a family of topologies that gives to the fractal manifold a structure of variable topological space. Additionally, we prove that using the fractal manifold as model for the universe dynamic, the universe expansion is intimately correlated to the variation of the topology.
Fractal Wallpaper Patterns Douglas Dunham
Dunham, Doug
Fractal Wallpaper Patterns Douglas Dunham Department of Computer Science University of Minnesota smaller shapes, producing a fractal pattern. In this paper we extend that algorithm to fill rectangles and triangles that tile the plane, which yields wallpaper patterns that are locally fractal in nature
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
Three-dimensional fractal homeomorphisms
NASA Astrophysics Data System (ADS)
Barnsley, Michael F.; Harding, Brendan
2015-03-01
We provide a simple introduction to fractal transformations and fractal homeomorphisms. We introduce tri-affine iterated function systems on R3, and illustrate associated fractal homeomorphisms applied to three dimensional graphical data sets. We also comment on the algorithms used.
Fractal scaling and fluid flow in fracture networks in rock
Barton, C.C.
1996-12-31
Recovery of oil and gas resources and injection of toxic waste materials requires quantitative models to describe and predict the movement of fluids in rock. Existing models based on pore-space flow are inappropriate for study of the more rapid process of fluid flow through fracture networks. This type of flow is not a simple function of the fracture characteristics at any particular scale, but rather the integration of fracture contributions at all scales. The mathematical constructs of fractal geometry are well suited to quantify and model relationships within complex systems that are statistically self-similar over a wide range of scales. Analyses show that fracture traces mapped on two-dimensional slices through three-dimensional nature fracture networks in rock follow a fractal scaling law over six orders of magnitude. Detailed measurements of 17 two-dimensional samples of fracture networks (at diverse scales in rocks of dissimilar age, lithology, and tectonic setting) show similar fractal dimensions in the range 1.3-1.7. The range in fractal dimension implies that a single physical process of rock fracturing operates over a wide range of scales, from microscopic cracks to large, regional fault systems. The knowledge that rock-fracture networks are fractal allows the use of data from a one-dimensional drill-hole sample to predict the two- and three-dimensional scaling of the fracture system. The spacing of fractures in drill holes is a fractal Cantor distribution, and the range of fractal dimension is 0.4-0.6, which is an integer dimension less than that of fracture-trace patterns exposed on two-dimensional, planar sections. A reconstruction of the fracture history at the point of initial connectivity across the network (percolation) has a fractal dimension of 1.35 as compared to a dimension of 1.9 for the percolation cluster in a two-dimensional model. Paleo flow was mapped based on the deposition of aqueous minerals on the fracture surface.
Fractal scaling and fluid flow in fracture networks in rock
Barton, C.C. )
1996-01-01
Recovery of oil and gas resources and injection of toxic waste materials requires quantitative models to describe and predict the movement of fluids in rock. Existing models based on pore-space flow are inappropriate for study of the more rapid process of fluid flow through fracture networks. This type of flow is not a simple function of the fracture characteristics at any particular scale, but rather the integration of fracture contributions at all scales. The mathematical constructs of fractal geometry are well suited to quantify and model relationships within complex systems that are statistically self-similar over a wide range of scales. Analyses show that fracture traces mapped on two-dimensional slices through three-dimensional nature fracture networks in rock follow a fractal scaling law over six orders of magnitude. Detailed measurements of 17 two-dimensional samples of fracture networks (at diverse scales in rocks of dissimilar age, lithology, and tectonic setting) show similar fractal dimensions in the range 1.3-1.7. The range in fractal dimension implies that a single physical process of rock fracturing operates over a wide range of scales, from microscopic cracks to large, regional fault systems. The knowledge that rock-fracture networks are fractal allows the use of data from a one-dimensional drill-hole sample to predict the two- and three-dimensional scaling of the fracture system. The spacing of fractures in drill holes is a fractal Cantor distribution, and the range of fractal dimension is 0.4-0.6, which is an integer dimension less than that of fracture-trace patterns exposed on two-dimensional, planar sections. A reconstruction of the fracture history at the point of initial connectivity across the network (percolation) has a fractal dimension of 1.35 as compared to a dimension of 1.9 for the percolation cluster in a two-dimensional model. Paleo flow was mapped based on the deposition of aqueous minerals on the fracture surface.
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 Traffic: Measurements, Modelling and Performance Ronald G. Addie
Zukerman, Moshe
Fractal Traffic: Measurements, Modelling and Performance Evaluation Ronald G. Addie University. We are not con- cerned with proving or exploiting this self-similarity property as such, but only, there is a need for effective tools for dimensioning and connection admis- sion control. In fact, problems related
Fractal simulation of three-dimensional pore architectures
R. C. Faucette; L. E. Borgman
1991-01-01
A Fortran program has been developed that uses the Hausdorf-Besicovitch fractal number to model sedimentary pore systems in three dimensions. A high-resolution grid o points is established in a parallelepiped, and solid particles are randomly inserted into the grid until a predetermined porosity value is reached. The particle shapes used are spheres, ellipsoids, and randomly generated shapes with a fixed
BADLY APPROXIMABLE VECTORS ON FRACTALS DMITRY KLEINBOCK AND BARAK WEISS
Weiss, Barak
BADLY APPROXIMABLE VECTORS ON FRACTALS DMITRY KLEINBOCK AND BARAK WEISS Revised version, July 2004 of badly approximable vec- tors has the same Hausdorff dimension as C. The sets are described in terms. 1. Introduction We say that x Rn is badly approximable if there is c > 0 such that for any p Zn
Collision Frequencies of Fractal Bacterial Aggregates with Small
aggregates of 3-300 µm diameter with an average fractal dimension of D ) 2.52. To determine the rate coagulation model but 5 orders of magnitude higher than predicted using the curvilinear model. Similar. Settling and sheared aggregates of particles can also scavenge dissolved and colloidal material (3, 4
The Relationship Between Mass Factal Dimensions of Solid Matrix and Pore Space in Porous Media
NASA Astrophysics Data System (ADS)
Dathe, A.; Thullner, M.
2004-05-01
Measuring fractal dimensions by image analyzing techniques has become a common practice to describe structural properties of porous media. Depending on the object of interest, different features of the structure can be measured: solid matrix, pores, and the interface between them. In many cases the fractal dimension of one of these features has been determined and taken to describe the entire system. The question arises whether these dimensions are independent from each other or whether they can be related to an underlying property of the structure or image, respectively. For a variety of porous media we measured the fractal dimension of the matrix, the pore space, and the interface between them simultaneously using the box counting method. The images were obtained from soil thin sections, a void system in a clayey soil, and a moss agate, which is a dendrite structure within a calcite matrix. Measured fractal dimensions were compared with fractal dimensions estimated by the pore-solid fractal (PSF) model, which derives the fractal properties of the matrix and the pore space completely as a function of the porosity, the size of the initiator and the fractal dimension of the interface. Measured results agree well with values obtained from the PSF model. A clear relationship between the mass fractal dimensions of the two phases (solid matrix and pore space) of a porous media was observed. For all images the smallest fractal dimensions were found for the interface between matrix and pores. Values for the fractal dimension of the two phases were between those for the interface and the Euclidian space with the phase with the lower mass fraction always having the smaller dimension. Model results also predict a dependency of the dimension of the phases on the spatial resolution of the analyzed image.
Two-dimensional fractal geometry, critical phenomena and conformal invariance
Bertrand Duplantier
1989-01-01
The universal properties of critical geometrical systems in two dimensions (2D) like the O(n) and Potts models, are described in the framework of Coulomb gas methods and conformal invariance. The conformal spectrum of geometrical critical systems obtained is made of a discrete infinite series of scaling dimensions. Specific applications involve the fractal properties of self-avoiding walks, percolation clusters, and also
Twinkling Fractal Analysis of PolyVinyl Acetate (PVAc)
NASA Astrophysics Data System (ADS)
Zhang, Yutao; Wool, Richard P.
2014-03-01
In amorphous polystyrene melts we have shown by Atomic Force Microscopy (Height and Phase) that dynamic rigid fractal clusters form in equilibrium with the fractal liquid and their relaxation behavior determines the kinetic nature of Tg [J. Non Cryst Solids 357(2): 311-319 2011]. The fractal clusters of size R ~ 1-100 nm have relaxation times ? ~ R1.8 (solid-to-liquid) where the exponent is related to the Fractal dimension Df and Fracton dimension df via Df/df = 1.8. Israeloff et al (2006) showed nanoscale spatio-temporal thermo fluctuations in PVAc using a non-contact Dielectric Force Microscopy technique; PVAC shows similar dynamic clustering using both phase and height tapping AFM modes. The dynamic clusters are clearly evident in the range 1-700 nm. The cluster relaxation behavior was explored in both height and phase modes and found to be different. The fractal clusters have a TFT vibrational density of states G(w) ~ wdf-1 with eigenvalues (frequencies) and eigenvectors (displacements) and these are expected to manifest differently in these AFM studies on PVAc thin films. We examine the cluster relaxation functions C(t) ~ t- 4 / 3 predicted by the TFT and look for the presence of highly mobile layers near surfaces and holes in nanothin films. These results are in accord with computer simulations of anharmonically interacting particles and the recent observation of ``Dancing Molecules'' in strained ceramic glass (Huang et al, Science Oct 2013), as predicted by the TFT.
2014-01-01
Background Fractal geometry has been the basis for the development of a diagnosis of preneoplastic and neoplastic cells that clears up the undetermination of the atypical squamous cells of undetermined significance (ASCUS). Methods Pictures of 40 cervix cytology samples diagnosed with conventional parameters were taken. A blind study was developed in which the clinic diagnosis of 10 normal cells, 10 ASCUS, 10 L-SIL and 10 H-SIL was masked. Cellular nucleus and cytoplasm were evaluated in the generalized Box-Counting space, calculating the fractal dimension and number of spaces occupied by the frontier of each object. Further, number of pixels occupied by surface of each object was calculated. Later, the mathematical features of the measures were studied to establish differences or equalities useful for diagnostic application. Finally, the sensibility, specificity, negative likelihood ratio and diagnostic concordance with Kappa coefficient were calculated. Results Simultaneous measures of the nuclear surface and the subtraction between the boundaries of cytoplasm and nucleus, lead to differentiate normality, L-SIL and H-SIL. Normality shows values less than or equal to 735 in nucleus surface and values greater or equal to 161 in cytoplasm-nucleus subtraction. L-SIL cells exhibit a nucleus surface with values greater than or equal to 972 and a subtraction between nucleus-cytoplasm higher to 130. L-SIL cells show cytoplasm-nucleus values less than 120. The rank between 120–130 in cytoplasm-nucleus subtraction corresponds to evolution between L-SIL and H-SIL. Sensibility and specificity values were 100%, the negative likelihood ratio was zero and Kappa coefficient was equal to 1. Conclusions A new diagnostic methodology of clinic applicability was developed based on fractal and euclidean geometry, which is useful for evaluation of cervix cytology. PMID:24742118
Gravitational Field of Fractal Distribution of Particles
Vasily E. Tarasov
2006-04-24
In this paper we consider the gravitational field of fractal distribution of particles. To describe fractal distribution, we use the fractional integrals. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of the Gauss's law, we consider the simple examples of the fields of homogeneous fractal distribution. The examples of gravitational moments for fractal distribution are considered.
Lamplighter random walks on fractals Takashi Kumagai
3, c4 > 0 such that c1 ndf /dw exp c2 d(x, y)dw n 1/(dw 1) ! h2n(x, y) c3 ndf /dw exp c4 d(x, y)dw n 1/(dw 1) ! (1.3) holds for all d(x, y) 2n (note that h2n(x, y) = 0 when d(x, y) > 2n), where d(Â·, Â·) is the graph distance, df is the volume growth exponent of the fractal graph and dw is called a walk dimension
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.
Entropy computing via integration over fractal measures.
S?omczynski, Wojciech; Kwapien, Jaros?aw; Zyczkowski, Karol
2000-03-01
We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics. PMID:12779373
Fractal analysis of Xylella fastidiosa biofilm formation
NASA Astrophysics Data System (ADS)
Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.
2009-07-01
We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.
Technology Transfer Automated Retrieval System (TEKTRAN)
In order to explore the effect of changes in plant communities and land use on soil properties, as a result of anthropogenic disturbances, we apply the theory of fractals and soil physics as a means to better quantify changes in particle-size distribution (PSD) and soil porosity. Fractal dimension a...
Fractal patterns of insect movement in microlandscape mosaics
Wiens, J.A.; Crist, T.O. |; With, K.A. |; Milne, B.T.
1995-03-01
How individuals move, whether in short-term searching behavior or long-term dispersal influences the probability that individuals will experience physiological stress or encounter appropriate habitat, potential mates, prey, or predators. Because of variety and complexity, it is often difficult to make sense of movements. Because the fractal dimension of a movement pathway is scale independent, however, it may provide a useful measure for comparing dissimilar taxa. The authors use fractal measures to compare the movement pathways of individual beetles occupying semiarid shortgrass steppe in north-central Colorado. 20 refs., 1 fig., 1 tab.
Person identification using fractal analysis of retina images
NASA Astrophysics Data System (ADS)
Ungureanu, Constantin; Corniencu, Felicia
2004-10-01
Biometric is automated method of recognizing a person based on physiological or behavior characteristics. Among the features measured are retina scan, voice, and fingerprint. A retina-based biometric involves the analysis of the blood vessels situated at the back of the eye. In this paper we present a method, which uses the fractal analysis to characterize the retina images. The Fractal Dimension (FD) of retina vessels was measured for a number of 20 images and have been obtained different values of FD for each image. This algorithm provides a good accuracy is cheap and easy to implement.
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).
NASA Astrophysics Data System (ADS)
Sharma, P.; Byrne, S.
2011-12-01
Lacustrine features have been observed in both the north and south polar regions of Titan, by multiple instruments onboard the NASA Cassini orbiter, including the Radio Detection and Ranging (RADAR) instrument, the Visual and Infrared Mapping Spectrometer (VIMS) and the Imaging Science Subsystem (ISS). Although the RADAR and Infrared data have provided a number of lines of evidence for these features being potential lakes, their formation mechanism is currently not well understood. Using Cassini RADAR data, we have performed a statistical analysis of the shorelines of Titan's north polar lakes and found them to be closely approximated by fractal shapes, a property also demonstrated by terrestrial lake shorelines. We calculated the fractal dimensions of the shorelines via two methods: the divider/ruler method and the box-counting method, at length scales of (1-10) km and found them to average 1.27 and 1.32, respectively. The inferred power-spectral exponent of Titan's topography (?) was found to be ? 2, which is lower than the values obtained from the global topography of the Earth or Venus. In order to interpret the fractal dimensions of Titan's shorelines in terms of the surficial processes at work, we repeated the same fractal analysis with terrestrial analogues using C-band radar backscatter data from the Shuttle Radar Topography Mission (SRTM). We calculated and compared statistical parameters including fractal dimension, shoreline development index and a linearity index and found different lake generation mechanisms on the Earth produce shorelines with overlapping ranges of these statistical parameters. On the basis of our statistical analyses, we concluded there is no one mechanism or set of mechanisms that can be deduced to be responsible for forming the depressions enclosing the lakes on Titan. Also, irrespective of the surface process responsible for their initial formation, these Titanian lake shorelines, like their terrestrial counterparts, could have been subsequently modified by many processes which have been observed to be active on Titan. These processes include fluvial and aeolian action, tectonic activity, impact cratering, cryovolcanism and mantling (fallout of solid material from the atmosphere which blankets the surface). Many of these surface processes create lakes with relatively smooth shorelines that are initially not fractal. Over time, however, fluvial modification can introduce small-scale roughness that leads to more rugged shorelines as channels erode and deposit to create embayments along the shoreline. Landscape evolution modeling has proven to be very useful for testing alternative hypotheses for surface change and for determining the linkages between form and process on both Earth and other solar system bodies. We intend to simulate several processes in our landscape evolution modeling for Titan. The results of this modeling, in conjunction with the statistical analysis of Titan's shorelines and terrestrial analogs, will be used to constrain the spatial distribution of surface process types and study the evolution of lake shorelines on Titan.
Fractals in geology and geophysics
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.
Research on the band gaps of the two-dimensional Sierpinski fractal phononic crystals
NASA Astrophysics Data System (ADS)
Gao, Nansha; Wu, Jiu Hui; Jing, Li
2015-08-01
In this paper, we study the band gaps (BGs) of the two-dimensional (2D) Sierpinski fractal phononic crystals (SFPGs) embedded in the homogenous matrix. The BGs structure, transmission spectra and displacement fields of eigenmodes of the proposed structures are calculated by using finite element method (FEM). Due to the simultaneous mechanisms of the Bragg scattering, the structure can exhibit low-frequency BGs, which can be effectively shifted by changing the inclusion rotation angle. The initial stress values can compress the BGs is proposed for the first time. Through the calculation, it is shown that, in the 2D solid-solid SFPG, the multi-frequency BGs exist. The whole BGs would incline to the low-frequency range with the increase of the fractal dimension. The SFPGs with different shape inclusions, can modulate the number, width and location of BGs. The study in this paper is relevant to the design of tuning BGs and isolators in the low-frequency range.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-08-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-[3-(triethoxysilyl)propyl]-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-01-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-(3-(triethoxysilyl)propyl)-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
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
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...
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.
Fractals in biology and medicine
NASA Technical Reports Server (NTRS)
Havlin, S.; Buldyrev, S. V.; Goldberger, A. L.; Mantegna, R. N.; Ossadnik, S. M.; Peng, C. K.; Simons, M.; Stanley, H. E.
1995-01-01
Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.