Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2014-09-01
The correlation between the void ratio of swelled montmorillonite and the vertical overburden pressure can be expressed as {e}{ m} = Kp{ s}{D{ s}-3}. The surface fractal dimension Ds of five bentonites were estimated from the swelling deformation tests according to this fractal correlation. The reliability of surface fractal dimension obtained from the swelling deformation test was confirmed by nitrogen adsorption test, with identical values of surface fractal dimension obtained from both tests. The surface fractal dimension can also be used to estimate the swelling deformation of bentonite, after calculating the swelling coefficient K from the parameters of diffuse double layer (DDL) model in the osmotic swelling phase. Comparison of the model predictions with a number of experimental results on swelling deformation of both Na dominant and Ca dominant bentonites suggests that the surface fractal model works excellent in the cases tested.
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.
NASA Astrophysics Data System (ADS)
Boness, D. A.; Canion, B.
2009-12-01
Carbonaceous soot aerosols formed in flames exhibit radiative forcing effects that are currently known only with significant uncertainty [IPCC AR4]. Better understanding of the soot aerosol range of structures, including coatings by other atmospheric constituents, and their scattering of optical radiation in relevant wavelength ranges will help constrain climate models. Numerical studies of diffusion-limited aggregation (DLA) and diffusion-limited cluster aggregation (DLCA) processes in 3D have since the 1980s indicated that the fractal dimension Df of soot aggregates is typically in the range 1.7-1.8. Multiple experimental studies, often attempting to calculate a 3D fractal dimension from electron micrograph 2D images, are in general agreement with this soot aggregate fractal dimension range. However, recent experiments find a much-wider range (with some aggregates having Df between 1.2 and 1.5) of soot aggregate fractal dimension from real combustion processes [Chakrabarty et al., Phys. Rev. Lett. 102, 235504 (2009)]. In addition, aged soot aggregates in the atmosphere may reach a fractal dimension Df substantially above 2 as they lose their filamentous nature. Several other studies have focused on the range Df from 1.7 to 1.8. We report results from undergraduate research using the T-Matrix technique to compute the optical scattering matrix elements for fractal soot aggregates over a wide range of fractal dimension (1.2 to 2.4). We generate these model aggegates using DLCA algorithms.
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.
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.
Video fire detection based on three-state Markov modal and fractal dimension calculation
NASA Astrophysics Data System (ADS)
Lei, Bo; Zhang, Zhijie; Wang, Chensheng
2012-11-01
Fire detection based on video surveillance is a very effective method for large area outdoor fire prevention, but the unpredictable place and time makes automatic fire detection a difficult problem. This paper adopts a loose color selection and frame differential to narrow down possible fire regions, where every pixel's temporal color variations are analyzed by 3-state Markov modals. One of the Markov modal is used for brightness variation examination and the other one is used for fire color likeness that is measured by color difference. In order to eliminate false detections, the fractal dimension calculation and texture match are performed. Experimental results prove the proposed method is feasible and suitable for outdoor or indoor fire detection in surveillance videos.
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.
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.
Exterior dimension of fat fractals
NASA Technical Reports Server (NTRS)
Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.
1985-01-01
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
Box-covering algorithm for fractal dimension of weighted networks
NASA Astrophysics Data System (ADS)
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-10-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.
Fractal Dimension in Epileptic EEG Signal Analysis
NASA Astrophysics Data System (ADS)
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include greater speed and the criterion to choose the maximum and minimum values for time intervals. Comparisons with the other waveform fractal dimension algorithms are also demonstrated. In order to discriminate the Healthy and the Epileptic EEGs, an improved method of Multifractal Measure such as Generalized Fractal Dimensions (GFD) is also proposed. Finally we conclude that there are significant differences between the Healthy and Epileptic Signals in the designed method than the GFD through graphical and statistical tools. The improved multifractal measure is very efficient technique to analyze the EEG Signals and to compute the state of illness of the Epileptic patients.
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
A Fractal Dimension Survey of Active Region Complexity
NASA Technical Reports Server (NTRS)
McAteer, R. T. James; Gallagher, Peter; Ireland, Jack
2005-01-01
A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .
Fractal dimension of bioconvection patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1990-01-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.
Fractal dimension of cerebral surfaces using magnetic resonance images
Majumdar, S.; Prasad, R.R.
1988-11-01
The calculation of the fractal dimension of the surface bounded by the grey matter in the normal human brain using axial, sagittal, and coronal cross-sectional magnetic resonance (MR) images is presented. The fractal dimension in this case is a measure of the convolutedness of this cerebral surface. It is proposed that the fractal dimension, a feature that may be extracted from MR images, may potentially be used for image analysis, quantitative tissue characterization, and as a feature to monitor and identify cerebral abnormalities and developmental changes.
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...
Fractal Dimensions and Martin Boundary of Graphs
Telcs, András
theorem for Markov chains with spatial symmetric Green's function. Spatial symmetry and behaviour symmet- ric Green's function with polynomial behavior at the infinity. This property is referred Fractal Dimensions and Martin Boundary of Graphs
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
DETERMINING THE FRACTAL DIMENSION OF A TIME SERIES WITH A NEURAL NET
Danon, Yaron
and require expert interaction for interpreting the calculated fractal dimension. Artificial neural nets (ANN by Barnsley [Barnsley, 1988]. Two artificial neural nets were trained with the backpropagation algorithm
Application of Fractal Dimension on Palsar Data
NASA Astrophysics Data System (ADS)
Singh, Dharmendra; Pant, Triloki
Study of land cover is the primal task of remote sensing where microwave imaging plays an important role. As an alternate of optical imaging, microwave, in particular, Synthetic Aperture Radar (SAR) imaging is very popular. With the advancement of technology, multi-polarized images are now available, e.g., ALOS-PALSAR (Phased Array type L-band SAR), which are beneficial because each of the polarization channel shows different sensitivity to various land features. Further, using the textural features, various land classes can be classified on the basis of the textural measures. One of the textural measure is fractal dimension. It is noteworthy that fractal dimension is a measure of roughness and thus various land classes can be distinguished on the basis of their roughness. The value of fractal dimension for the surfaces lies between 2.0 and 3.0 where 2.0 represents a smooth surface while 3.0 represents drastically rough surface. The study area covers subset images lying between 2956'53"N, 7750'32"E and 2950'40"N, 7757'19"E. The PALSAR images of the year 2007 and 2009 are considered for the study. In present paper a fractal based classification of PALSAR images has been performed for identification of Water, Urban and Agricultural Area. Since fractals represent the image texture, hence the present study attempts to find the fractal properties of land covers to distinguish them from one another. For the purpose a context has been defined on the basis of a moving window, which is used to estimate the local fractal dimension and then moved over the whole image. The size of the window is an important issue for estimation of textural measures which is considered to be 55 in present study. This procedure, in response, produces a textural map called fractal map. The fractal map is constituted with the help of local fractal dimension values and can be used for contextual classification. In order to study the fractal properties of PALSAR images, the three polarization images viz. HH (Horizontal-Horizontal Polarization), VV (Vertical-Vertical Polarization) and HV (Horizontal-Vertical Polarization) are considered individually. First of all each polarized image is classified in an unsupervised way and various clusters, i.e., four clusters are identified with the help of reference data as Water, Urban and Agricultural Area. For each cluster, the fractal dimension is obtained from the fractal map. Based on the study the ranges of fractal dimension for three classes are Water: 2.0-2.17, Agricultural Area: 2.24-2.72, Urban Area: 2.63-2.92 for HH polarized image; Water: 2.0-2.21, Agricultural Area: 2.20-2.64, Urban; 2.58-2.94 for VV polarized image and Water: 2.0-2.14, Agricultural Area: 2.18-2.58, Urban: 2.46-2.94 for HV polarized image. Since the class Others represents a mixture of various classes, an explicit range of D for this class can not be determined. A closer look at the ranges of fractal dimension indicates that there is an overlapping of the values for different classes, despite of which the classes can be distinguished. Also, the class Water having low value of fractal dimension can be treated as smooth and Urban Area having higher values of fractal dimension can be considered rough in structure while the class Agricultural Area shows an intermediate roughness.
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
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 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
The Correlation Fractal Dimension of Complex Networks
NASA Astrophysics Data System (ADS)
Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei
2013-05-01
The fractality of complex networks is studied by estimating the correlation dimensions of the networks. Comparing with the previous algorithms of estimating the box dimension, our algorithm achieves a significant reduction in time complexity. For four benchmark cases tested, that is, the Escherichia coli (E. Coli) metabolic network, the Homo sapiens protein interaction network (H. Sapiens PIN), the Saccharomyces cerevisiae protein interaction network (S. Cerevisiae PIN) and the World Wide Web (WWW), experiments are provided to demonstrate the validity of our algorithm.
Cosmology in One Dimension: Fractal Dimensions from Mass Oriented Partitions
NASA Astrophysics Data System (ADS)
Miller, Bruce; Rouet, Jean-Louis; Shiozawa, Yui
2015-03-01
The distribution of visible matter in the universe has its origin in the weak fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the present universe, with its galaxies, clusters and super-clusters of galaxies indicates the absence of a natural length scale. Numerical simulations of a one-dimensional system permit us to precisely follow the evolution starting with an initial perturbation in the Hubble flow. The limitation to one dimension removes the necessity to make approximations in calculating the gravitational field and 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 we apply four different methods for computing generalized fractal dimensions Dq of the distribution of particles in configuration space. We first employ the conventional methods based on partitions of equal size and then less familiar methods based on partitions of equal mass. We show that the latter are superior for computing generalized dimensions for indices q < - 1 which characterize regions of low density.
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.
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.
NASA Astrophysics Data System (ADS)
Zhang, Baoquan; Liu, Wei; Liu, Xiufeng
2006-11-01
The surface fractal dimension was calculated by using a mathematical model and mercury intrusion data for a variety of bi- and multi-disperse porous solids including silica gels, alumina pellets, and building stones. The mathematical model was obtained by modifying the well-established scaling relation published previously [ Ind. Eng. Chem. Res., 34 (1995) 1383-1386]. It was also verified by comparing with the theoretical surface fractal dimensions for regular fractal structures ( Skerpinski tetrahedron and Menger sponge) and the calculated surface fractal dimensions for silica gel and alumina particles via the linear fitting method established previously. The calculation results for various bi- and multi-disperse porous solids have demonstrated that the scale-dependent nature of the surface fractal dimension is ubiquitous. The difference in the surface fractal dimension between pore size intervals usually exists. The estimation of the surface fractal dimension on an average stand may lead to erroneous results.
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 the galaxy distribution varying by steps?
Marie-Noelle Celerier; Reuben Thieberger
2005-04-20
The structure of the large scale distribution of the galaxies have been widely studied since the publication of the first catalogs. Since large redshift samples are available, their analyses seem to show fractal correlations up to the observational limits. The value of the fractal dimension(s) calculated by different authors have become the object of a large debate, as have been the value of the expected transition from fractality to a possible large scale homogeneity. Moreover, some authors have proposed that different scaling regimes might be discerned at different lenght scales. To go further on into this issue, we have applied the correlation integral method to the wider sample currently available. We therefore obtain a fractal dimension of the galaxy distribution which seems to vary by steps whose width might be related to the organization hierarchy observed for the galaxies. This result could explain some of the previous results obtained by other authors from the analyses of less complete catalogs and maybe reconcile their apparent discrepancy. However, the method applied here needs to be further checked, since it produces odd fluctuations at each transition scale, which need to be thoroughly explained.
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.
Multiorder boundaries among discrete domains: Relative fractal dimension and others
NASA Astrophysics Data System (ADS)
Xuan, Qi; Du, Fang; Wu, Tie-Jun
2010-03-01
In nature and society, most of competitions take place on the boundaries among a group of domains where different individuals or colonies share common resources; therefore, it is widely believed that domain boundaries play important roles in the evolution of many complex systems. Here, we first give a definition for multiorder boundaries among discrete domains and then propose a general method to calculate their relative fractal dimension, i.e., the ratio of the fractal dimension of the boundaries versus that of the domains themselves. Through analyzing three types of real-world discrete domains, several interesting results are revealed. For example, the limitation on the number of domains that an individual can join in may produce longer boundaries indicating more cruel competitions among the domains. Besides, the individuals with more social links are always considered more important in social networks, and it is found that these individuals as valuable resources of social domains are always centralized on the boundaries of higher order.
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 Fractal Dimension of a Fracture Surface by Volume Covering Method
NASA Astrophysics Data System (ADS)
Zhou, H. W.; Xue, D. J.; Jiang, D. Y.
2014-01-01
An accurate calculation of the fractal dimension of a fracture surface is of prime significance for the quantitative evaluation of mechanical behavior. A laser profilometer is employed to measure a surface roughness of a large sandstone sample. Based on the definition of Minkowski dimension, a new method, referred to as the volume covering method, is proposed to estimate fractal dimension by accurately calculating the number of virtual cell needed to cover prism-like volume. It is indicated that the method can fully use laser scanning-based elevation data of the surface and make an accurate estimation of fractal dimension of a fracture surface. Furthermore, another method related to the opposite corner is adopted to estimate the same rough surface with little error of the fractal dimension. A comparison analysis indicates that the volume covering method has the advantage of high accuracy and good applicability for a real fracture surface.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2011-03-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2010-10-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
Fractal dimension of mesospheric radar backscatter at 2. 75 MHz
Hall, C.; Armstrong, R.J.; La Hoz, C. )
1991-04-01
The authors identified the fractal dimension of radar returns from the mesopause region at 2.75 MHz. The input dataset was a time series of echo amplitude at a discrete height obtained from a partial reflection radar operating at Ramfjordmoen in northern Norway. Two different algorithms both of which yield approximations to the fractal dimension have been employed and give almost identical results. The radar echo dataset in question exhibits a dimension of between 7 and 8.
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.
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...
FRACTAL DIMENSION RESULTS FOR CONTINUOUS TIME RANDOM WALKS
Meerschaert, Mark M.; Nane, Erkan; Xiao, Yimin
2013-01-01
Continuous time random walks impose random waiting times between particle jumps. This paper computes the fractal dimensions of their process limits, which represent particle traces in anomalous diffusion. PMID:23482421
On the fractal dimension of the Duffing attractor
Mariusz Tarnopolski
2014-09-12
The box counting dimension $d_C$ and the correlation dimension $d_G$ change with the number of numerically generated points forming the attractor. At a sufficiently large number of points the fractal dimension tends to a finite value. The obtained values are $d_C\\approx 1.43$ and $d_G\\approx 1.38$.
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.
Relationship between the fractal dimension and the width to length ratio of mass movements
NASA Astrophysics Data System (ADS)
Sezer, Ebru
2009-04-01
Mass movements have some typical geometrical dimensions. One of these typical geometrical dimensions is the width to length ratio. Also, the fractal dimensions of mass movements from the inventory maps of natural mass movements can be used for their geometrical description and characterization. For this reason, in the present study, development of a computer programme for digitizing and determining the fractal dimensions of mass movements, and investigation of the relationship between the fractal dimensions and the width to length (W/L) tario of the mass movements are aimed. For the purpose of the study, a computer programme namely FRACEK for determination of fractal dimensions of amorphous areas is developed by using the JAVA computer language at first. Secondly, a database including the shapes of the mass movements was compiled from the literature and digitized. Then, their width to length ratios and fractal dimensions were calculated. Finally, a series of simple statistical analyses were performed on the data obtained and the results were interpreted. To investigate the relationships between the fractal dimensions and W/L ratios of the mass movements, a series of simple regression analysis is performed. During the regression analyses, linear, power, logarithmic and exponential functions are employed. According to the results obtained, there are some correlations between the D and the W/L ratio. When considering only debris flow data, a power relationship between the D and the W/L ratio was found and its coefficient of correlation was obtained as 0.85. The lowest coefficient of correlations were obtained from the rotational failure data. The coefficients of correlations of the power and exponential funtions were same, 0.53. A similar result was obtained for the translational failure data. Their coefficient of correlations was 0.74. When all data is evaluated together, a relatively strong correlation between the D and the W/L ratio was obtained. These results revealed that to make a differantiation among the mass movements using the fractal dimension is possible.
Time evolution of the fractal dimension of a mixing front
NASA Astrophysics Data System (ADS)
Lopez Gonzalez-Nieto, P.; Grau, J.
2009-04-01
We present a description of an experimental study of an array of turbulent plumes (from one to nine plumes), investigating the time evolution of the fractal dimension of the plumes and also the spatial evolution of the fractal dimension from one plume to other. We also investigate the effects of bouyancy (different Atwood numbers), the number of plumes and the height of the bouyancy source on the fractal dimension. The plumes are formed by injecting a dense fluid from a small source (from one to nine orifices) into a stationary body of lighter brime (saline solution) contained in a tank. The source fluid was dyed with fluorescein and we use the LIF technique. The plumes were fully turbulent and we have both momentum and bouyancy regimes. The fractal dimensions of contours of concentration were measured. The fractal analysis of the turbulent convective plumes was performed with the box counting algorithm for different intensities of evolving plume images using the special software Ima_Calc. Fractal dimensions between 1.3 and 1.35 are obtained from box counting methods for free convection and neutral boundary layers. Other results have been published which use the box counting method to analyze images of jet sections -produced from LIF techniques. The regions where most of the mixing takes place are also compared with Reactive flow experiments using phenolphthalein and acid-base interfaces performed by Redondo(1994) IMA 43. Eds M. Farge, JC Hunt and C. Vassilicos.
Fractal dimensions of rampart impact craters on Mars
NASA Technical Reports Server (NTRS)
Ching, Delwyn; Taylor, G. Jeffrey; Mouginis-Mark, Peter; Bruno, Barbara C.
1993-01-01
Ejecta blanket morphologies of Martian rampart craters may yield important clues to the atmospheric densities during impact, and the nature of target materials (e.g., hard rock, fine-grained sediments, presence of volatiles). In general, the morphologies of such craters suggest emplacement by a fluidized, ground hugging flow instead of ballistic emplacement by dry ejecta. We have quantitatively characterized the shape of the margins of the ejecta blankets of 15 rampart craters using fractal geometry. Our preliminary results suggest that the craters are fractals and are self-similar over scales of approximately 0.1 km to 30 km. Fractal dimensions (a measure of the extent to which a line fills a plane) range from 1.06 to 1.31. No correlations of fractal dimension with target type, elevation, or crater size were observed, though the data base is small. The range in fractal dimension and lack of correlation may be due to a complex interplay of target properties (grain size, volatile content), atmospheric pressure, and crater size. The mere fact that the ejecta margins are fractals, however, indicates that viscosity and yield strength of the ejecta were at least as low as those of basalts, because silicic lava flows are not generally fractals.
Fractal dimensions of 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.
Change in trabecular architecture as measured by fractal dimension
NASA Astrophysics Data System (ADS)
Berry, Joel L.; Webber, Richard L.; Jerome, Chris; Pope, Thomas L., Jr.; Zimmerman, Mark; Towers, Jeffrey D.
1994-05-01
Detection of subtle structural changes in trabecular bone is important in evaluating the load- bearing capability of whole bones. Microstructural changes in trabecular bone due to remodeling or resorption lead to changes in bone strength. Recently, fractal-based analyses of radiographs have demonstrated that a fractal model can describe trabecular bone patterns independent of mass density. In this case, the descriptor of choice is a scale-invariant measure of trabecular detail known as fractal dimension. The objective of this work was to compare two measures of the distribution of trabecular bone -- fractal dimension and mean gray level -- in a decalcifying environment. The fractal-based analysis relied upon the spatial distribution of trabecular material while the mean gray level measurements depended upon the average x- radiation attenuation over a region of interest. Data were produced from four separate slices of vertebral bone which demonstrated that a change in the spatial distribution of trabecular material may be expressed in terms of a concurrently changing estimate of the fractal dimension within a region of interest. This change was not necessarily reflected in the mean gray level estimate of mass density.
Daily variation of the fractal dimension of the velocity components in the turbulent surface layer
NASA Astrophysics Data System (ADS)
Tijera, M.; Maqueda, G.; Yagüe, C.; Cano, J. L.
2012-04-01
The turbulence is a dominant property within the Planetary Boundary Layer (PBL). It is the main characteristic of the mixing in the lower atmosphere since the atmospheric turbulent fluxes are more efficient than the molecular diffusion. Turbulence can be observed in time series of meteorological variables (wind velocity for example). The sampling rate of observation in that time series has to be high in order to detect the turbulent regime. The analysis of these series presents a self-similarity structure, so the wind velocity can be considered as a fractal magnitude. This work shows a study of the fractal dimension of the wind perturbation series u'and w'components of the wind speed. Fractal dimension of velocity components can be related to others turbulent characteristics of the fluxes close to the ground. Fluctuation of longitudinal and, specially, vertical components depend on stability and, therefore, on the solar cycle. In consequence, the behaviour of fractal dimension should be in agreement with that cycle also. These series have been obtained once it has carried out the necessary transformation to get the mean wind series in short intervals, namely 5 minutes, to ensure the consistent properties of turbulence. The original records available were taken every thirty minutes by sonic anemometers (20 Hz sampling rate) during a week of a field campaign. The data analysed was recorded in the experimental campaign SABLES-98 at the Research Centre for the Lower Atmosphere (CIBA), located in Valladolid province (Spain). It has been calculated the fractal dimension (Komolgorov capacity or box- counting dimension) of the time series of fluctuations of the velocity component along of the mean wind direction and the vertical component (u' = u-U, w' = w -W), both in the physical spaces (velocity-time). It has been studied the time evolution of the fractal dimension during several days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The fractal dimension of theu' and w' components of wind velocity series have been studied, as well as the influence of different turbulent parameters depending on daily cycle: turbulent kinetic energy, friction velocity, difference of temperature between the extreme of the layer studied close of the surface (?T50-0.22m),etc. It has been observed that there is a possible correlation between the fractal dimension and some of these turbulent parameters. Finally, it has been analysed the variation of the fractal dimension versus stability obtained from the Richardson number along of the day.
Effect of Na+ on surface fractal dimension of compacted bentonite
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2015-05-01
Compacted Tsukinuno bentonite was immersed into NaCl solutions of different concentrations in oedometers, and the surface fractal dimension of bentonite-saline association was measured by nitrogen adsorption isotherms. The application of the Frenkel-Halsey-Hill equation and the Neimark thermodynamic method to nitrogen adsorption isotherms indicated that the surface roughness was greater for the bentonite-saline association. The surface fractal dimension of bentonite increased in the NaCl solution with low Na+ concentration, but decreased at high Na+ concentration. This process was accompanied by the same tendency in specific surface area and microporosity with the presence of Na+ coating in the clay particles.
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.
Local Fractal Dimension based approaches for Colonic Polyp Classification Michael Hfnerb
Uhl, Andreas
Local Fractal Dimension based approaches for Colonic Polyp Classification Michael HÃ¤fnerb , Toru This work introduces texture analysis methods that are based on computing the local fractal dimension (or to a local fractal dimension based approach. These extensions additionally extract shape and/or gradient
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 self-similarity in Asparagus plumosus
Galán, Antonio Sarmiento
Fractal dimension and self-similarity in Asparagus plumosus J. R. Castrej#19;on Pita, 1 A of an African plant that is widely cultivated as ornamental, the Asparagus plumosus. This plant presents self. These visualizations, which present several symmet- ric bifurcations [1], encouraged us to analyze the Asparagus
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 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.
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.
Kan, An-Kang; Cao, Dan; Zhang, Xue-Lai
2015-04-01
Accurately predicting the effective thermal conductivity of the fibrous materials is highly desirable but remains to be a challenging work. In this paper, the microstructure of the porous fiber materials is analyzed, approximated and modeled on basis of the statistical self-similarity of fractal theory. A fractal model is presented to accurately calculate the effective thermal conductivity of fibrous porous materials. Taking the two-phase heat transfer effect into account, the existing statistical microscopic geometrical characteristics are analyzed and the Hertzian Contact solution is introduced to calculate the thermal resistance of contact points. Using the fractal method, the impacts of various factors, including the porosity, fiber orientation, fractal diameter and dimension, rarified air pressure, bulk thermal conductivity coefficient, thickness and environment condition, on the effective thermal conductivity, are analyzed. The calculation results show that the fiber orientation angle caused the material effective thermal conductivity to be anisotropic, and normal distribution is introduced into the mathematic function. The effective thermal conductivity of fibrous material increases with the fiber fractal diameter, fractal dimension and rarefied air pressure within the materials, but decreases with the increase of vacancy porosity. PMID:26353563
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.
NASA Astrophysics Data System (ADS)
Donadio, Carlo; Magdaleno, Fernando; Mazzarella, Adriano; Mathias Kondolf, G.
2015-07-01
By applying fractal geometry analysis to the drainage network of three large watercourses in America and Europe, we have calculated for the first time their fractal dimension. The aim is to interpret the geomorphologic characteristics to better understand the morphoevolutionary processes of these fluvial morphotypes; to identify and discriminate geomorphic phenomena responsible for any difference or convergence of a fractal dimension; to classify hydrographic patterns, and finally to compare the fractal degree with some geomorphic-quantitative indexes. The analyzed catchment of Russian (California, USA), Ebro (Spain), and Volturno (Italy) rivers are situated in Mediterranean-climate regions sensu Köppen, but with different geologic context and tectonic styles. Results show fractal dimensions ranging from 1.08 to 1.50. According to the geological setting and geomorphic indexes of these basins, the lower fractal degree indicates a prevailing tectonics, active or not, while the higher degree indicates the stronger erosion processes on inherited landscapes.
Fractal Dimension of Geologically Constrained Crater Populations of Mercury
NASA Astrophysics Data System (ADS)
Mancinelli, Paolo; Pauselli, Cristina; Perugini, Diego; Lupattelli, Andrea; Federico, Costanzo
2015-07-01
Data gathered during the Mariner10 and MESSENGER missions are collated in this paper to classify craters into four geo-chronological units constrained to the geological map produced after MESSENGER's flybys. From the global catalogue, we classify craters, constraining them to the geological information derived from the map. We produce a size frequency distribution (SFD) finding that all crater classes show fractal behaviour: with the number of craters inversely proportional to their diameter, the exponent of the SFD (i.e., the fractal dimension of each class) shows a variation among classes. We discuss this observation as possibly being caused by endogenic and/or exogenic phenomena. Finally, we produce an interpretative scenario where, assuming a constant flux of impactors, the slope variation could be representative of rheological changes in the target materials.
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.
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 $\\{\\textrm{Re}\\ s=D\\}$.
Speech Emotion Recognition Based on Parametric Filter and Fractal Dimension
NASA Astrophysics Data System (ADS)
Mao, Xia; Chen, Lijiang
In this paper, we propose a new method that employs two novel features, correlation density (Cd) and fractal dimension (Fd), to recognize emotional states contained in speech. The former feature obtained by a list of parametric filters reflects the broad frequency components and the fine structure of lower frequency components, contributed by unvoiced phones and voiced phones, respectively; the latter feature indicates the non-linearity and self-similarity of a speech signal. Comparative experiments based on Hidden Markov Model and K Nearest Neighbor methods are carried out. The results show that Cd and Fd are much more closely related with emotional expression than the features commonly used.
Fractal Dimensions for Continuous Time Random Walk Limits
Meerschaert, Mark M; Xiao, Yimin
2011-01-01
In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW scaling limits are time-changed processes that model anomalous diffusion. The outer process describes particle jumps, and the non-Markovian inner process (or time change) accounts for waiting times between jumps. This paper studies fractal properties of the sample functions of a time-changed process, and establishes some general results on the Hausdorff and packing dimensions of its range and graph. Then those results are applied to CTRW scaling limits.
Structure and fractal dimension of protein-detergent complexes
NASA Astrophysics Data System (ADS)
Chen, Sow-Hsin; Teixeira, José
1986-11-01
Small-angle neutron-scattering experiments were made on bovine serum albumin (BSA)-lithium dodecyl sulfate (LDS) complexes in buffer solutions. As increasing amounts of LDS are added, the scattering data indicate that BSA molecules are successively transformed into random coil conformations with LDS forming globular micelles randomly decorating the polypeptide backbones. A cross-section formula is developed which successfully fits small-angle neutron-scattering spectra over the entire Q range. The fractal dimension, the micellar size, and the extent of the denatured protein are simultaneously extracted.
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.
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.
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
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.
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
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.
NASA Astrophysics Data System (ADS)
Zeng, Qiang; Luo, Mingyong; Pang, Xiaoyun; Li, Le; Li, Kefei
2013-10-01
This study investigates the surface fractal dimensions (SFDs) of pore structure of cement pastes and mortars with/without ground granulated blast-furnace slag (GGBS) incorporated into binder. The samples were subject to water curing and sealed curing. The fractal dimensions of samples are determined by Zhang’s model (Ind Eng Chem Res, 34 (1995):1383-1386) on the basis of mercury intrusion porosimetry (MIP) data. The results confirm the scale-dependent property of fractal dimension of pore structures and the micro-fractal, transition and macro-fractal regions are identified for all samples. The upper pore size range for micro-fractal regions is around 30 nm, the transition regions cover 0.5-2 magnitude orders of pore size and macro fractal regions cover 1.5-3 magnitude orders. Both curing conditions and GGBS in binder have impact on the fractal properties of pore structure, and samples incorporating GGBS have substantially larger values for micro-fractal regions.
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.
Seizure detection method based on fractal dimension and gradient boosting.
Zhang, Yanli; Zhou, Weidong; Yuan, Shasha; Yuan, Qi
2015-02-01
Automatic seizure detection technology is necessary and crucial for the long-term electroencephalography (EEG) monitoring of patients with epilepsy. This article presents a patient-specific method for the detection of epileptic seizures. The fractal dimensions of preprocessed multichannel EEG were firstly estimated using a k-nearest neighbor algorithm. Then, the feature vector constructed for each epoch was fed into a trained gradient boosting classifier. After a series of postprocessing, including smoothing, threshold processing, collar operation, and union of seizure detections in a short time interval, a binary decision was made to determine whether the epoch belonged to seizure status or not. Both the epoch-based and event-based assessments were used for the performance evaluation of this method on the EEG data of 21 patients from the Freiburg dataset. An average epoch-based sensitivity of 91.01% and a specificity of 95.77% were achieved. For the event-based assessment, this method obtained an average sensitivity of 94.05%, with a false detection rate of 0.27/h. PMID:25549952
Local fractal dimension based approaches for colonic polyp classification.
Häfner, Michael; Tamaki, Toru; Tanaka, Shinji; Uhl, Andreas; Wimmer, Georg; Yoshida, Shigeto
2015-12-01
This work introduces texture analysis methods that are based on computing the local fractal dimension (LFD; or also called the local density function) and applies them for colonic polyp classification. The methods are tested on 8 HD-endoscopic image databases, where each database is acquired using different imaging modalities (Pentax's i-Scan technology combined with or without staining the mucosa) and on a zoom-endoscopic image database using narrow band imaging. In this paper, we present three novel extensions to a LFD based approach. These extensions additionally extract shape and/or gradient information of the image to enhance the discriminativity of the original approach. To compare the results of the LFD based approaches with the results of other approaches, five state of the art approaches for colonic polyp classification are applied to the employed databases. Experiments show that LFD based approaches are well suited for colonic polyp classification, especially the three proposed extensions. The three proposed extensions are the best performing methods or at least among the best performing methods for each of the employed databases. The methods are additionally tested by means of a public texture image database, the UIUCtex database. With this database, the viewpoint invariance of the methods is assessed, an important features for the employed endoscopic image databases. Results imply that most of the LFD based methods are more viewpoint invariant than the other methods. However, the shape, size and orientation adapted LFD approaches (which are especially designed to enhance the viewpoint invariance) are in general not more viewpoint invariant than the other LFD based approaches. PMID:26385078
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.
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.
Development of methods of the Fractal Dimension estimation for the ecological data analysis
Jura, Jakub; Mironovová, Martina
2015-01-01
This paper deals with an estimating of the Fractal Dimension of a hydrometeorology variables like an Air temperature or humidity at a different sites in a landscape (and will be further evaluated from the land use point of view). Three algorithms and methods of an estimation of the Fractal Dimension of a hydrometeorology time series were developed. The first results indicate that developed methods are usable for the analysis of a hydrometeorology variables and for a testing of the relation with autoregulation functions of ecosystem
Reconstructing the fractal dimension of granular aggregates from light intensity spectra.
Tang, Fiona H M; Maggi, Federico
2015-11-25
There has been growing interest in using the fractal dimension to study the hierarchical structures of soft materials after realising that fractality is an important property of natural and engineered materials. This work presents a method to quantify the internal architecture and the space-filling capacity of granular fractal aggregates by reconstructing the three-dimensional capacity dimension from their two-dimensional optical projections. Use is made of the light intensity of the two-dimensional aggregate images to describe the aggregate surface asperities (quantified by the perimeter-based fractal dimension) and the internal architecture (quantified by the capacity dimension) within a mathematical framework. This method was tested on control aggregates of diffusion-limited (DLA), cluster-cluster (CCA) and self-correlated (SCA) types, stereolithographically-fabricated aggregates, and experimentally-acquired natural sedimentary aggregates. Statistics of the reconstructed capacity dimension featured correlation coefficients R ? 98%, residuals NRMSE ? 10% and percent errors PE ? 4% as compared to controls, and improved earlier approaches by up to 50%. PMID:26414181
Approximating the Ising model on fractal lattices of dimension less than two
NASA Astrophysics Data System (ADS)
Codello, Alessandro; Drach, Vincent; Hietanen, Ari
2015-11-01
We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of zero external magnetic field, using a generalization of the combinatorial method of Feynman and Vodvickenko. Our procedure is applicable to any fractal obtained by the removal of sites of a periodic two dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way to determine $T_c$ for any fractal of dimension below two. Critical exponents are more difficult to determine since the free energy of any periodic approximation still has a logarithmic singularity at the critical point implying $\\alpha = 0$. We also compute the correlation length as a function of the temperature and extract the relative critical exponent. We find $\
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...
Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions
Deinert, Mark
Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions been substantiated by assuming that capillary pressure is directly related to the pore radius. When capillary pressure and pore volume are directly propor- tional to the pore radius. If a system's pore space
Fractal Dimension of Down Fibre Assemblies Jing Gao1,2
Pan, Ning
-resolution desk-top micro-CT system made in SkyScan Ltd. of Belgium to observe the microstructures of down fiber of polyester film, as shown in figure 2. Then scan the vessels cross sections in this micro-CT system. So fiber assembly. This work presents a unified treatment using the tool of "local fractal dimensions
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.
Structural and Fractal Dimensions are Reliable Determinants of Grain Yield in Soybean
Technology Transfer Automated Retrieval System (TEKTRAN)
Reliable models are needed to describe plants with complex geometric structures, quantify the impact of management strategies on the plant’s geometric distribution in space and time, and predict yield as a function of fractal dimension. We measured growth and development variables on single soybean ...
Roth, Eric J; Gilbert, Benjamin; Mays, David C
2015-10-20
Experiments reveal a wide discrepancy between the permeability of porous media containing colloid deposits and the available predictive equations. Evidence suggests that this discrepancy results, in part, from the predictive equations failing to account for colloid deposit morphology. This article reports a series of experiments using static light scattering (SLS) to characterize colloid deposit morphology within refractive index matched (RIM) porous media during flow through a column. Real time measurements of permeability, specific deposit, deposit fractal dimension, and deposit radius of gyration, at different vertical positions, were conducted with initially clean porous media at various ionic strengths and fluid velocities. Decreased permeability (i.e., increased clogging) corresponded with higher specific deposit, lower fractal dimension, and smaller radius of gyration. During deposition, fractal dimension, radius of gyration, and permeability decreased with increasing specific deposit. During flushing with colloid-free fluid, these trends reversed, with increased fractal dimension, radius of gyration, and permeability. These observations suggest a deposition scenario in which large and uniform aggregates become deposits, which reduce porosity, lead to higher fluid shear forces, which then decompose the deposits, filling the pore space with small and dendritic fragments of aggregate. PMID:26412205
Cusp-scaling behavior in fractal dimension of chaotic scattering Adilson E. Motter1
Lai, Ying-Cheng
Cusp-scaling behavior in fractal dimension of chaotic scattering Adilson E. Motter1 and Ying in chaotic scattering is characterized by a sudden change in the topology of the infinite set of unstable of important physical processes, such as chaotic scattering 1,2 . A scattering process is chaotic
Fractal dimensions: A new paradigm to assess spatial memory and learning using Morris water maze.
Singh, Surjeet; Kaur, Harpreet; Sandhir, Rajat
2016-02-15
Morris water maze has been widely used for analysis of cognitive functions and relies on the time taken by animal to find the platform i.e. escape latency as a parameter to quantify spatial memory and learning. However, escape latency is confounded by swimming speed which is not necessarily a cognitive factor. Rather, path length may be a more appropriate and reliable parameter to assess spatial learning. This paper presents fractal dimension as a new paradigm to assess spatial memory and learning in animals. Male wistar rats were administrated with pentylenetetrazole and scopolamine to induce chronic epilepsy and dementia respectively. Fractal dimension of the random path followed by the animals on Morris water maze was analyzed and statistically compared among different experimental groups; the results suggest that fractal dimension is more reliable and accurate parameter to assess cognitive deficits compared to escape latency. Thus, the present study suggests that fractal dimensions could be used as an independent parameter to assess spatial memory and learning in animals using Morris water maze. PMID:26592165
NASA Astrophysics Data System (ADS)
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-06-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.
Are fractal dimensions of the spatial distribution of mineral deposits meaningful?
Raines, G.L.
2008-01-01
It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.
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
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
Di Ieva, Antonio; Grizzi, Fabio; Tschabitscher, Manfred; Colombo, Piergiuseppe; Casali, Massimiliano; Simonelli, Matteo; Widhalm, Georg; Muzzio, Pier Carlo; Matula, Christian; Chiti, Arturo; Rodriguez y Baena, Riccardo
2010-09-01
Neuroradiological and metabolic imaging is a fundamental diagnostic procedure in the assessment of patients with primary and metastatic brain tumors. The correlation between objective parameters capable of quantifying the neoplastic angioarchitecture and imaging data may improve our understanding of the underlying physiopathology and make it possible to evaluate treatment efficacy in brain tumors. Only a few studies have so far correlated the quantitative parameters measuring the neovascularity of brain tumors with the metabolic profiles measured by means of amino acid uptake in positron emission tomography (PET) scans. Fractal geometry offers new mathematical tools for the description and quantification of complex anatomical systems, including microvascularity. In this study, we evaluated the microvascular network complexity of six cases of human glioblastoma multiforme quantifying the surface fractal dimension on CD34 immunostained specimens. The microvascular fractal dimension was estimated by applying the box-counting algorithm. As the fractal dimension depends on the density, size and shape of the vessels, and their distribution pattern, we defined it as an index of the whole complexity of microvascular architecture and compared it with the uptake of (11)C-methionine (MET) assessed by PET. The different fractal dimension values observed showed that the same histological category of brain tumor had different microvascular network architectures. Fractal dimension ranged between 1.19 and 1.77 (mean: 1.415+/-0.225), and the uptake of (11)C-methionine ranged between 1.30 and 5.30. A statistically significant direct correlation between the microvascular fractal dimension and the uptake of (11)C-methionine (p=0.02) was found. Our preliminary findings indicate that that vascularity (estimated on the histologic specimens by means of the fractal dimension) and (11)C-methionine uptake (assessed by PET) closely correlate in glioblastoma multiforme and that microvascular fractal dimension can be a useful parameter to objectively describe and quantify the geometrical complexity of the microangioarchitecture in glioblastoma multiforme. PMID:20394759
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.
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
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
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
NASA Astrophysics Data System (ADS)
Ohsasa, K.; Natsume, Y.; Sekiya, T.; Hatayama, T.
2015-06-01
The dendrite morphology of unidirectionally solidified Al-Si alloys was evaluated by measuring the fractal dimension and dimensionless perimeter of dendrites. In an unidirectional solidification experiment, columnar crystals grew from a bottom chill and columnar to equiaxed transition (CET) occurred at the upper part of an ingot. Then, equiaxed crystals were formed at the top of the ingot. Different dendrite morphology was observed in longitudinal, transverse and oblique sections, however, the fractal dimension or dimensionless perimiter of the dendrites in the sections with same local solidification time showed same values, and continuously decreased with increase in the local solidification time through columnar, CET and equiaxed regions. It can be considered that the fractal dimension and dimensionless perimiter of dendrites are controlled by local solidification time and irrespective of dendrite morphology. This result demonstrated the potential of the fractal dimension and dimensionless perimiter as a parameter for estimating local solidification time of an ingot in which the measurement of SDAS is difficult.
Criscuoli, S; Ermolli, I; Centrone, M; Criscuoli, Serena; Rast, Mark; Ermolli, Ilaria; Centrone, Mauro
2006-01-01
Several studies have investigated the fractal and multifractal nature of magnetic features in the solar photosphere and its variation with the solar magnetic activity cycle. Here we extend those studies by examining the fractal geometry of bright magnetic features at higher atmospheric levels, specifically in the solar chromosphere. We analyze structures identified in CaIIK images obtained with the Precision Solar Photometric Telescopes (PSPTs) at Osservatorio Astronomico di Roma (OAR) and Mauna Loa Solar Observatory (MLSO). Fractal dimension estimates depend on the estimator employed, the quality of the images, and the structure identification techniques used. We examine both real and simulated data and employ two different perimeter-area estimators in order to understand the sensitivity of the deduced fractal properties to pixelization and image quality. The fractal dimension of bright 'magnetic' features in CaIIK images ranges between values of 1.2 and 1.7 for small and large structures respectively. This ...
Characterization of the Irregularity of a Terrain Using Fractal Dimension of Lakes' Boundaries
NASA Astrophysics Data System (ADS)
Karle, Nakul N.; Kolwankar, Kiran M.
2015-12-01
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 m to 2 km, 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.
NASA Astrophysics Data System (ADS)
Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun
1995-05-01
This paper describes the quantitative measurement of the amount of fibrosis in the rat liver using the fractal dimension of the shape of power spectrum. The shape of the power spectrum of the scattered echo from biotissues is strongly affected by its internal structure. The fractal dimension, which is one of the important parameters of the fractal theory, is useful to express the complexity of shape of figures such as the power spectrum. From in vitro experiments using rat liver, it was found that this method can be used to quantitatively measure the amount of fibrosis in the liver, and has the possibility for use in the diagnosis of human liver cirrhosis.
Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S
2011-04-30
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated. (optical technologies in biophysics and medicine)
NASA Astrophysics Data System (ADS)
Ul'yanov, A. S.; Lyapina, A. M.; Ulianova, O. V.; Fedorova, V. A.; Uianov, S. S.
2011-04-01
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated.
NASA Astrophysics Data System (ADS)
Alonso, C.; Benito, R. M.; Tarquis, A. M.
2012-04-01
Satellite image data have become an important source of information for monitoring vegetation and mapping land cover at several scales. Beside this, the distribution and phenology of vegetation is largely associated with climate, terrain characteristics and human activity. Various vegetation indices have been developed for qualitative and quantitative assessment of vegetation using remote spectral measurements. In particular, sensors with spectral bands in the red (RED) and near-infrared (NIR) lend themselves well to vegetation monitoring and based on them [(NIR - RED) / (NIR + RED)] Normalized Difference Vegetation Index (NDVI) has been widespread used. Given that the characteristics of spectral bands in RED and NIR vary distinctly from sensor to sensor, NDVI values based on data from different instruments will not be directly comparable. The spatial resolution also varies significantly between sensors, as well as within a given scene in the case of wide-angle and oblique sensors. As a result, NDVI values will vary according to combinations of the heterogeneity and scale of terrestrial surfaces and pixel footprint sizes. Therefore, the question arises as to the impact of differences in spectral and spatial resolutions on vegetation indices like the NDVI. The aim of this study is to establish a comparison between two different sensors in their NDVI values at different spatial resolutions. Scaling analysis and modeling techniques are increasingly understood to be the result of nonlinear dynamic mechanisms repeating scale after scale from large to small scales leading to non-classical resolution dependencies. In the remote sensing framework the main characteristic of sensors images is the high local variability in their values. This variability is a consequence of the increase in spatial and radiometric resolution that implies an increase in complexity that it is necessary to characterize. Fractal and multifractal techniques has been proven to be useful to extract such complexities from remote sensing images and will applied in this study to see the scaling behavior for each sensor in generalized fractal dimensions. The studied area is located in the provinces of Caceres and Salamanca (east of Iberia Peninsula) with an extension of 32 x 32 km2. The altitude in the area varies from 1,560 to 320 m, comprising natural vegetation in the mountain area (forest and bushes) and agricultural crops in the valleys. Scaling analysis were applied to Landsat-5 and MODIS TERRA to the normalized derived vegetation index (NDVI) on the same region with one day of difference, 13 and 12 of July 2003 respectively. From these images the area of interest was selected obtaining 1024 x 1024 pixels for Landsat image and 128 x 128 pixels for MODIS image. This implies that the resolution for MODIS is 250x250 m. and for Landsat is 30x30 m. From the reflectance data obtained from NIR and RED bands, NDVI was calculated for each image focusing this study on 0.2 to 0.5 ranges of values. Once that both NDVI fields were obtained several fractal dimensions were estimated in each one segmenting the values in 0.20-0.25, 0.25-0.30 and so on to rich 0.45-0.50. In all the scaling analysis the scale size length was expressed in meters, and not in pixels, to make the comparison between both sensors possible. Results are discussed. Acknowledgements This work has been supported by the Spanish MEC under Projects No. AGL2010-21501/AGR, MTM2009-14621 and i-MATH No. CSD2006-00032
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…
Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis
Metze, Konradin
2013-01-01
Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed. PMID:24063399
Modified box dimension and average weighted receiving time on the weighted fractal networks
Dai, Meifeng; Sun, Yanqiu; Shao, Shuxiang; Xi, Lifeng; Su, Weiyi
2015-01-01
In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is. PMID:26666355
Nuclear Fractal Dimensions as a Tool for Prognostication of Oral Squamous Cell Carcinoma
Yinti, Shanmukha Raviteja; Boaz, Karen; Lewis, Amitha J; Ashokkumar, Pandya Jay; Kapila, Supriya Nikita
2015-01-01
Background Carcinogenesis follows complex molecular alterations, which are triggered by subtle chromatin architectural changes that are imperceptible to the human eye. As the treatment decisions in Oral Squamous Cell Carcinoma (OSCC) are hindered by the imprecise clinical stage determination and inter-observer variability in histological grading, focus in recent years has shifted to discovering identifiers related to neoplastic cell morphology studied through computer-aided image analysis. One such approach is the assessment of fractal geometry, a technique first described by Mandelbrot, which aids in precise assessment of architecture of natural objects. Assessment and quantification of degree of complexity of these fractal objects (self-similarities in structural complexity at different magnifying scales) is described as fractal dimension (FD). Aim To evaluate the nuclear fractal dimension (NFD) in OSCC using computer-aided image analysis. Materials and Methods Histological sections of 14 selected cases of Oral Squamous Cell Carcinoma (OSCC) and 6 samples of normal buccal mucosa (as control) were stained with Haematoxylin-Eosin and Feulgen stain for histopathological examination and evaluation of nuclear complexity respectively. Fifteen HPF at Invasive Tumour Front (ITF) and Tumour Proper (TP) of Feulgen-stained sections were selected and photographed in test and control samples. At ITF, TP and normal buccal mucosa 200 nuclei each were selected and analyzed using Image J software to quantify FD. The test and control groups were compared statistically using Independent sample t-test and One-way ANOVA. Results Nuclear FD increased progressively towards worst tumour staging as compared to normal buccal mucosa. Conclusion Nuclear FD can be considered for quantification of nuclear architectural changes as a prognostic indicator in OSCC. PMID:26674013
ERIC Educational Resources Information Center
McCartney, M.; Myers, D.; Sun, Y.
2008-01-01
The divider dimensions of a range of maps of Ireland dating from 1567 to 1893 are evaluated, and it is shown that for maps produced before 1650 the fractal dimension of the map can be correlated to its date of publication. Various classroom uses and extensions are discussed. (Contains 2 figures.)
NASA Astrophysics Data System (ADS)
Ahammer, Helmut; DeVaney, Trevor T. J.
2004-03-01
The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.
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.
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
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.
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).
Serena Criscuoli; Mark Rast; Ilaria Ermolli; Mauro Centrone
2006-09-27
Several studies have investigated the fractal and multifractal nature of magnetic features in the solar photosphere and its variation with the solar magnetic activity cycle. Here we extend those studies by examining the fractal geometry of bright magnetic features at higher atmospheric levels, specifically in the solar chromosphere. We analyze structures identified in CaIIK images obtained with the Precision Solar Photometric Telescopes (PSPTs) at Osservatorio Astronomico di Roma (OAR) and Mauna Loa Solar Observatory (MLSO). Fractal dimension estimates depend on the estimator employed, the quality of the images, and the structure identification techniques used. We examine both real and simulated data and employ two different perimeter-area estimators in order to understand the sensitivity of the deduced fractal properties to pixelization and image quality. The fractal dimension of bright 'magnetic' features in CaIIK images ranges between values of 1.2 and 1.7 for small and large structures respectively. This size dependency largely reflects the importance of image pixelization in the measurement of small objects. The fractal dimension of chromospheric features does not show any clear systematic variation with time over the period examined, the descending phase of solar cycle 23. These conclusions, and the analysis of both real and synthetic images on which they are based, are important in the interpretation of previously reported results.
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 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.
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.
Chol-Hui Yun; Hui-Chol Choi; Hyong-Chol O
2013-07-10
We consider a construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal interpolation surface (RFIS) is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given data set on rectangular grids, we construct general recurrent iterated function systems with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.
Pan, Ning
Golden Mean and Fractal Dimension of Goose Down Jing Gao a* , Ning Pan b , Weidong Yu a a Textile College2001jing@mail.dhu.edu.cn Abstract Goose down is widely used in textile thermal products. The fractal dimension of goose down is studied in this paper both theoretically and experimentally, revealing its value
Small-angle scattering from fat fractals
NASA Astrophysics Data System (ADS)
Anitas, Eugen M.
2014-06-01
A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.
NASA Astrophysics Data System (ADS)
Braga, F. L.; Mattos, O. A.; Amorin, V. S.; Souza, A. B.
2015-07-01
Clusters formation models have been extensively studied in literature, and one of the main task of this research area is the analysis of the particle aggregation processes. Some work support that the main characteristics of this processes are strictly correlated to the cluster morphology, for example in DLA. It is expected that in the DLA clusters formation with particles containing different sizes the modification of the aggregation processes can be responsible for changes in the DLA morphology. The present article is going to analyze the formation of DLA clusters of particles with different sizes and show that the aggregates obtained by this approach generate an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affect the fractal dimension of the clusters.
2013-01-01
Background Prostate cancer is a serious public health problem that affects quality of life and has a significant mortality rate. The aim of the present study was to quantify the fractal dimension and Shannon’s entropy in the histological diagnosis of prostate cancer. Methods Thirty-four patients with prostate cancer aged 50 to 75 years having been submitted to radical prostatectomy participated in the study. Histological slides of normal (N), hyperplastic (H) and tumor (T) areas of the prostate were digitally photographed with three different magnifications (40x, 100x and 400x) and analyzed. The fractal dimension (FD), Shannon’s entropy (SE) and number of cell nuclei (NCN) in these areas were compared. Results FD analysis demonstrated the following significant differences between groups: T vs. N and H vs. N groups (p?
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.
NASA Astrophysics Data System (ADS)
Karemore, Gopal; Nielsen, Mads
2009-02-01
Structural texture measures are used to address the aspect of breast cancer risk assessment in screening mammograms. The current study investigates whether texture properties characterized by local Fractal Dimension (FD) and Lacunarity contribute to asses breast cancer risk. FD represents the complexity while the Lacunarity characterize the gappiness of a fractal. Our cross-sectional case-control study includes mammograms of 50 patients diagnosed with breast cancer in the subsequent 2-4 years and 50 matched controls. The longitudinal double blind placebo controlled HRT study includes 39 placebo and 36 HRT treated volunteers for two years. ROIs with same dimension (250*150 pixels) were created behind the nipple region on these radiographs. Box counting method was used to calculate the fractal dimension (FD) and the Lacunarity. Paired t-test and Pearson correlation coefficient were calculated. It was found that there were no differences between cancer and control group for FD (P=0.8) and Lacunarity (P=0.8) in crosssectional study whereas earlier published heterogeneity examination of radiographs (BC-HER) breast cancer risk score separated groups (p=0.002). In the longitudinal study, FD decreased significantly (P<0.05) in the HRT treated population while Lacunarity remained insignificant (P=0.2). FD is negatively correlated to Lacunarity (-0.74, P<0.001), BIRADS (-0.34, P<0.001) and Percentage Density (-0.41, P<0.001). FD is invariant to the mammographic texture change from control to cancer population but marginally varying in HRT treated population. This study yields no evidence that lacunarity or FD are suitable surrogate markers of mammographic heterogeneity as they neither pick up breast cancer risk, nor show good sensitivity to HRT.
Jiménez, J; López, A M; Cruz, J; Esteban, F J; Navas, J; Villoslada, P; Ruiz de Miras, J
2014-10-01
This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimer's disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud. PMID:24909817
The Fractal Geometrical Properties of Nuclei
W. H. Ma; J. S. Wang; Q. Wang; S. Mukherjee; L. Yang; Y. Y. Yang; M. R. Huang; Y. J. Zhou
2014-06-06
We present a new idea to understand the structure of nuclei, which is comparing to the liquid drop model. After discussing the probability that the nuclear system may be a fractal object with the characteristic of self-similarity, the nuclear irregular structure properties and the self-similarity characteristic are considered to be an intrinsic aspects of nuclear structure properties. For the description of nuclear geometric properties, nuclear fractal dimension is an irreplaceable variable similar to the nuclear radius. In order to determine these two variables, a new nuclear potential energy formula which is related to the fractal dimension is put forward and the phenomenological semi-empirical Bethe-Weizsacker binding energy formula is modified using the fractal geometric theory. And one important equation set with two equations is obtained, which is related to the conception that the fractal dimension should be a dynamical parameter in the process of nuclear synthesis. The fractal dimensions of the light nuclei are calculated and their physical meanings are discussed. We have compared the nuclear fractal mean density radii with the radii calculated by the liquid drop model for the light stable and unstable nuclei using rational nuclear fractal structure types. In the present model of fractal nuclear structure there is an obvious feature comparing to the liquid drop model, since the present model can reflect the geometric informations of the nuclear structure, especially for the nuclei with clusters, such as the {\\alpha}-cluster nuclei and halo nuclei.
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.
NASA Astrophysics Data System (ADS)
Boness, D. A.; Terrell-Martinez, B.
2010-12-01
As part of an ongoing undergraduate research project of light scattering calculations involving fractal carbonaceous soot aggregates relevant to current anthropogenic and natural sources in Earth's atmosphere, we have read with interest a recent paper [E.T. Wolf and O.B Toon,Science 328, 1266 (2010)] claiming that the Faint Young Sun paradox discussed four decades ago by Carl Sagan and others can be resolved without invoking heavy CO2 concentrations as a greenhouse gas warming the early Earth enough to sustain liquid water and hence allow the origin of life. Wolf and Toon report that a Titan-like Archean Earth haze, with a fractal haze aggregate nature due to nitrogen-methane photochemistry at high altitudes, should block enough UV light to protect the warming greenhouse gas NH3 while allowing enough visible light to reach the surface of the Earth. To test this hypothesis, we have employed a rigorous T-Matrix arbitrary-particle light scattering technique, to avoid the simplifications inherent in Mie-sphere scattering, on haze fractal aggregates at UV and visible wavelenths of incident light. We generate these model aggregates using diffusion-limited cluster aggregation (DLCA) algorithms, which much more closely fit actual haze fractal aggregates than do diffusion-limited aggregation (DLA) algorithms.
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.
Hinrikus, Hiie; Bachmann, Maie; Karai, Deniss; Klonowski, W?odzimierz; Lass, Jaanus; Stepien, Pavel; Stepien, Robert; Tuulik, Viiu
2011-05-01
This study addresses application of Higuchi's fractal dimension (FD) as a measure to evaluate the effect of external periodic stressor on electrical oscillations in the brain. Modulated microwave radiation was applied as a weak periodic stressor with strongly inhomogeneous distribution inside the brain. Experiments were performed on a group of 14 volunteers. Ten cycles (1 min on, 1 min off) of 450-MHz microwave radiation modulated at 40 Hz were applied. Higuchi's FD was calculated in eight symmetric electroencephalographic (EEG) channels located in frontal, temporal, parietal, and occipital areas. FD values averaged over a group detected a small (1-2%) but statistically significant increase with exposure in all EEG channels. FD increased for 12, decreased for one, and was constant for one subject. FD showed the most remarkable effect in temporal and parietal regions of the left hemisphere where the microwave field was maximal. Changes of FD in these regions of the right hemisphere were much higher than expected in accordance with the field distribution. Correlation of FD between different EEG channels was high and retained its value in exposed conditions. Spreading of disturbance between different brain areas is supposed to be crucial for the effect of exposure on the electrical oscillations in the brain. PMID:21465274
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 geometrical properties of nuclei
NASA Astrophysics Data System (ADS)
Ma, Wei-Hu; Wang, Jian-Song; Wang, Qi; Mukherjee, S.; Yang, Lei; Yang, Yan-Yun; Huang, Mei-Rong
2015-10-01
We present a new idea to understand the structure of nuclei and compare it to the liquid drop model. After discussing the probability that the nuclear system may be a fractal object with the characteristic of self-similarity, the irregular nuclear structure properties and the self-similarity characteristic are considered to be an intrinsic aspect of the nuclear structure properties. For the description of nuclear geometric properties, the nuclear fractal dimension is an irreplaceable variable similar to the nuclear radius. In order to determine these two variables, a new nuclear potential energy formula which is related to the fractal dimension is put forward and the phenomenological semiempirical Bethe-Weizsäcker binding energy formula is modified using the fractal geometric theory. One important equation set with two equations is obtained, which is related to the concept that the fractal dimension should be a dynamic parameter in the process of nuclear synthesis. The fractal dimensions of the light nuclei are calculated and their physical meanings are discussed. We compare the nuclear fractal mean density radii with the radii calculated by the liquid drop model for the light stable and unstable nuclei using rational nuclear fractal structure types. In the present model of fractal nuclear structure there is an obvious additional feature compared to the liquid drop model, since the present model can reflect the geometric information of the nuclear structure, especially for nuclei with clusters, such as the ?-cluster nuclei and halo nuclei. Supported by National Basic Research Program of China (973 Program) (2014CB845405, 2013CB8344x), National Natural Science Foundation of China (U1432247, 11205209, 11205221)
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
Gómez, Carlos; Mediavilla, Angela; Hornero, Roberto; Abásolo, Daniel; Fernández, Alberto
2009-04-01
Alzheimer's disease (AD) is an irreversible brain disorder of unknown aetiology that gradually destroys brain cells and represents the most prevalent form of dementia in western countries. The main aim of this study was to analyse the magnetoencephalogram (MEG) background activity from 20 AD patients and 21 elderly control subjects using Higuchi's fractal dimension (HFD). This non-linear measure can be used to estimate the dimensional complexity of biomedical time series. Before the analysis with HFD, the stationarity and the non-linear structure of the signals were proved. Our results showed that MEG signals from AD patients had lower HFD values than control subjects' recordings. We found significant differences between both groups at 71 of the 148 MEG channels (p<0.01; Student's t-test with Bonferroni's correction). Additionally, five brain regions (anterior, central, left lateral, posterior and right lateral) were analysed by means of receiver operating characteristic curves, using a leave-one-out cross-validation procedure. The highest accuracy (87.8%) was achieved when the mean HFD over all channels was analysed. To sum up, our results suggest that spontaneous MEG rhythms are less complex in AD patients than in healthy control subjects, hence indicating an abnormal type of dynamics in AD. PMID:18676171
Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps.
Cao, K F; Peng, S L
1999-09-01
The dual-star composition rule of doubly superstable (DSS) sequences presents a complete renormalizable algebraic structure for studying Feigenbaum's metric universality and self-similar classification of DSS sequences in symbolic dynamics of bimodal maps of the interval. Here an important feature is that the complete combinations of up- and down-star products create all the generalized Feigenbaum's routes of transitions to chaos. These routes can be classified into two types: one consists of countably infinitely many regular routes which preserve Feigenbaum's metric universality; another consists of uncountably infinitely many universal nonscaling routes described by the irregularly mixed dual-star products, which break Feigenbaum's asymptotically convergent metric universality although they are structurally universal. The combinatorial complexity of dual-star products may increase the grammatical complexity of languages of symbolic dynamics. Moreover, it is found that there exists a global regularity between the fractal dimensions d and the scaling factors [alpha(C),alpha(D)] for Feigenbaum-type attractors: d(Z)log(/Z/)/alpha(C)(Z)alpha(D)(Z)/=beta((2)), where beta((2)) is independent of the concrete DSS sequences Z. PMID:11970079
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
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.
Chen, Yanguang
2015-01-01
An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels. By analogy with Horton-Strahler's laws of river composition, three exponential laws have been put forward. These exponential laws can be reconstructed and transformed into three linear scaling laws, which can be named composition laws of blood vessels network. From these linear scaling laws it follows a set of power laws, including the three-parameter Zipf's law on the rank-size distribution of blood vessel length and the allometric scaling law on the length-diameter relationship of blood vessels in different orders. The models are applied to the observed data on human beings and animals early given by other researchers, and an interesting finding is that human bodies more conform to natural r...
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
Meng, Zhiyong; Hashmi, Sara M; Elimelech, Menachem
2013-02-15
The time-evolutions of nanoparticle hydrodynamic radius and aggregate fractal dimension during the aggregation of fullerene (C(60)) nanoparticles (FNPs) were measured via simultaneous multiangle static and dynamic light scattering. The FNP aggregation behavior was determined as a function of monovalent (NaCl) and divalent (CaCl(2)) electrolyte concentration, and the impact of addition of dissolved natural organic matter (humic acid) to the solution was also investigated. In the absence of humic acid, the fractal dimension decreased over time with monovalent and divalent salts, suggesting that aggregates become slightly more open and less compact as they grow. Although the aggregates become slightly more open, the magnitude of the fractal dimension suggests intermediate aggregation between the diffusion- and reaction-limited regimes. We observed different aggregation behavior with monovalent and divalent salts upon the addition of humic acid to the solution. For NaCl-induced aggregation, the introduction of humic acid significantly suppressed the aggregation rate of FNPs at NaCl concentrations lower than 150mM. In this case, the aggregation was intermediate or reaction-limited even at NaCl concentrations as high as 500mM, giving rise to aggregates with a fractal dimension of 2.0. For CaCl(2)-induced aggregation, the introduction of humic acid enhanced the aggregation of FNPs at CaCl(2) concentrations greater than about 5mM due to calcium complexation and bridging effects. Humic acid also had an impact on the FNP aggregate structure in the presence of CaCl(2), resulting in a fractal dimension of 1.6 for the diffusion-limited aggregation regime. Our results with CaCl(2) indicate that in the presence of humic acid, FNP aggregates have a more open and loose structure than in the absence of humic acid. The aggregation results presented in this paper have important implications for the transport, chemical reactivity, and toxicity of engineered nanoparticles in aquatic environments. PMID:23211871
Lung cancer-a fractal viewpoint.
Lennon, Frances E; Cianci, Gianguido C; Cipriani, Nicole A; Hensing, Thomas A; Zhang, Hannah J; Chen, Chin-Tu; Murgu, Septimiu D; Vokes, Everett E; Vannier, Michael W; Salgia, Ravi
2015-11-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed. PMID:26169924
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.
Comprehensive Fractal Description of Porosity of Coal of Different Ranks
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10?nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100?nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03?nm and 361.14?nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10?nm and 100?nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407
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
Földes-Papp, Z; Peng, W G; Seliger, H; Kleinschmidt, A K
1995-06-21
Oligonucleotides are becoming more and more important in molecular biomedicine; for example, they are used as defined primers in polymerase chain reaction and as antisense oligonucleotides in gene therapy. In this paper, we model the dynamics of polymer-supported oligonucleotide synthesis to an inverse power law of driven multi-cycle synthesis on fixed starting sites. The mathematical model is employed by presenting the accompanying view of error sequences dynamics. This model is a practical one, and is applicable beyond oligonucleotide synthesis to dynamics of biological diversity. Computer simulations show that the polymer support synthesis of oligonucleotides and single-stranded DNA sequences in iterated cyclic format can be assumed as scale-invariant. This synthesis is quantitatively described by nonlinear equations. From these the fractal dimension Da (N,d) is derived as the growth term (N = number of target nucleotides, d = coupling probability function). Da(N,d) is directly measurable from oligonucleotide yields via high-performance liquid chromatography or capillary electrophoresis, and quantitative gel electrophoresis. Different oligonucleotide syntheses, including those with large-scale products can be directly compared with regard to error sequences dynamics. In addition, for short sequences the fractal dimension Da (N,d) is characteristic for the efficiency with which a polymer support of a given load allows oligonucleotide chain growth. We analyze the results of separations of crude oligonucleotide product from the synthesis of a 30 mer. Preliminary analysis of a 238 mer single-stranded DNA sequence is consistent with a simulated estimate of crude synthesis product, although the target sequence itself is not detectable. We characterize the oligonucleotide support syntheses by simulated and experimentally determined values of the fractal dimension Da (N,d0) within limitations (d0 = constant (average) coupling probability). PMID:7666672
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.
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.
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.
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.
ACCELERATION OF A PROCEDURE TO GENERATE FRACTAL CURVES OF A GIVEN DIMENSION THROUGH THE
Alfonseca, Manuel
described the use of grammatical evolution to automatically generate L Systems (LS) representing fractal; c) application of a grammar-evolution based genetic algorithm to get a grammar representing, by means of a representation scheme: vector graphics or turtle graphics. In a previous work [Alfonseca
NASA Astrophysics Data System (ADS)
Lemasters, R. D.; Larsen, M.
2014-12-01
There is empirical evidence that large raindrop arrival times are not perfectly random. As a way to characterize this behavior, it has been suggested that raindrop arrival times may be well described by fractal statistics. Two-Dimensional Video Disdrometer data with millisecond resolution was used to construct an ensemble of time series, each of which was associated with a small range of drop diameters for an individual storm. The monofractal dimension was determined for each of these time series. It is apparent that the fractal dimension has a clear dependence on drop size. This drop size dependence, as well as storm to storm variabilitiy, is explored.
NASA Astrophysics Data System (ADS)
Davarpanah, Armita; Babaie, Hassan A.
2013-11-01
The Basin and Range fault blocks, which were formed by an extensional event around 17 Ma, have continuously been deforming by younger, diachronous system of cross normal faults in southwest Montana and southeastern Idaho since 16.6 Ma. Reactivation of these two mid-Tertiary-Quaternary systems of normal faults, and two older, approximately N-S and E-W sets of regional normal faults, has evolved into a seismically active block faulted terrain. For both fault systems, high fractal dimensions occur in areas characterized by a large number of fault traces, high fault trace linear density, and maximum fault trace azimuthal variation. The major axis of the anisotropy ellipse of the fractal dimensions for each set of the two normal fault systems is sub-perpendicular to the linear directional mean of the faults, and gives an estimate for the direction of extension. Indentations on the point distribution on the anisotropy ellipse of fractal dimensions indicate heterogeneities due to the presence of several fault sets and/or variation in their trend. Domains in which there is only one set of faults produce smooth, well-defined fractal anisotropy ellipses with no indentations. The axial ratio of the anisotropy ellipse provides a measure for the range of variation in the trend of the faults. The trace length, linear density, and fractal dimension of the cross normal faults, decrease, in a direction across and away from the Snake River Plain (SRP), suggesting a diminishing effect of faulting probably due to the attenuation of the Yellowstone hotspot-related thermal doming with distance from centers of eruption. The spatio-temporal distribution of the trajectories of the minor axes of the anisotropy ellipses of fractal dimensions and the linear directional mean of the cross faults define a set of asymmetric, sub-parabolic spatio-temporal pattern about the axis of the SRP, with their apices located on diachronous centers of eruption.
NASA Astrophysics Data System (ADS)
Cymberknop, L.; Legnani, W.; Pessana, F.; Bia, D.; Zócalo, Y.; Armentano, R. L.
2011-12-01
The advent of vascular diseases, such as hypertension and atherosclerosis, is associated to significant alterations in the physical properties of arterial vessels. Evaluation of arterial biomechanical behaviour is related to the assessment of three representative indices: arterial compliance, arterial distensibility and arterial stiffness index. Elasticity is the most important mechanical property of the arterial wall, whose natures is strictly non-linear. Intervention of elastin and collagen fibres, passive constituent elements of the arterial wall, is related to the applied wall stress level. Concerning this, appropriate tools are required to analyse the temporal dynamics of the signals involved, in order to characterize the whole phenomenon. Fractal geometry can be mentioned as one of those techniques. The aim of this study consisted on arterial pressure and diameter signals processing, by means of nonlinear techniques based on fractal geometry. Time series morphology was related to different arterial stiffness states, generated by means of blood flow variations, during experiences performed in vitro.
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.
Sutherland, Scott
Math 331, Fall 2002: Problems 21-24 21. (expires 11/22) [No Maple] Compute the box counting dimension of the fractal in the figure below: 22. (expires 11/22) Suppose that a turtle is moving and after 100 seconds. 23. (expires 11/22) Consider the recursively defined sequence Sn = S2 n-1 - 4Sn-1 + 6
Fractal fragmentation and small-angle scattering
NASA Astrophysics Data System (ADS)
Anitas, E. M.
2015-09-01
The small-angle scattering form factor of a three-dimensional idealized fragmentation model based on the concept of renormalization is calculated. The system consists of randomly oriented microscopic fractal objects whose positions are uncorrelated. It is shown that in the fractal region, the monodisperse form factor is characterized by a succession of maxima and minima superimposed on a simple power-law decay, and whose scattering exponent coincide with the fractal dimension ofthe scatterer. The results could be used to obtain additional structural information about systems obtained through fragmentation processes at microscale.
Fractal Spectrum Technique for Quantitative Analysis of Volcanic Particle Shapes
NASA Astrophysics Data System (ADS)
Maria, A. H.; Carey, S. N.
2001-12-01
The shapes of volcanic particles reflect numerous eruptive parameters (e.g. magma viscosity, volatile content, degree of interaction with water) and are useful for understanding fragmentation and transport processes associated with volcanic eruptions. However, quantitative analysis of volcanic particle shapes has proven difficult due to their morphological complexity and variability. Shape analysis based on fractal geometry has been successfully applied to a wide variety of particles and appears to be well suited for describing complex features. The technique developed and applied to volcanic particles in this study uses fractal 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. Quantitative comparisons are carried out using multivariate statistical techniques such as cluster and principal components analysis. Compared with previous fractal methods that express shape in terms of only one or two fractal dimensions, use of multiple fractal dimensions results in more effective discrimination between samples. In addition, the technique eliminates the subjectivity associated with selecting linear segments on Richardson plots for fractal dimension calculation, and allows direct comparison of particles as long as instantaneous dimensions used as input to multivariate analyses are selected at the same scales for each particle. Applications to samples from well documented eruptions (e.g. Mt. St. Helens, Tambora, Surtsey) indicate that the fractal spectrum technique provides a useful means of characterizing volcanic particles and can be helpful for identifying the products of specific fragmentation processes (volatile exsolution, phreatomagmatic, quench granulation) and modes of volcanic deposition (tephra fall, pyroclastic flow, blast/surge).
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.
Fractal dimension of sparkles in automotive metallic coatings by multispectral imaging measurements.
Medina, José M; Díaz, José A; Vignolo, Carlos
2014-07-23
Sparkle in surface coatings is a property of mirror-like pigment particles that consists of remarkable bright spots over a darker surround under unidirectional illumination. We developed a novel nondestructive method to characterize sparkles based on the multispectral imaging technique, and we focused on automotive metallic coatings containing aluminum flake pigments. Multispectral imaging was done in the visible spectrum at different illumination angles around the test sample. Reflectance spectra at different spatial positions were mapped to color coordinates and visualized in different color spaces. Spectral analysis shows that sparkles exhibit higher reflectance spectra and narrower bandwidths. Colorimetric analysis indicates that sparkles present higher lightness values and are far apart from the bulk of color coordinates spanned by the surround. A box-counting procedure was applied to examine the fractal organization of color coordinates in the CIE 1976 L*a*b* color space. A characteristic noninteger exponent was found at each illumination position. The exponent was independent of the illuminant spectra. Together, these results demonstrate that sparkles are extreme deviations relative to the surround and that their spectral properties can be described as fractal patterns within the color space. Multispectral reflectance imaging provides a powerful, noninvasive method for spectral identification and classification of sparkles from metal flake pigments on the micron scale. PMID:24945784
Berry, Hugues
2002-01-01
Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes. PMID:12324410
Critical behavior of loops and biconnected clusters on fractals of dimension d < 2
Dibyendu Das; Supravat Dey; Jesper Lykke Jacobsen; Deepak Dhar
2008-09-09
We solve the O(n) model, defined in terms of self- and mutually avoiding loops coexisting with voids, on a 3-simplex fractal lattice, using an exact real space renormalization group technique. As the density of voids is decreased, the model shows a critical point, and for even lower densities of voids, there is a dense phase showing power-law correlations, with critical exponents that depend on n, but are independent of density. At n=-2 on the dilute branch, a trivalent vertex defect acts as a marginal perturbation. We define a model of biconnected clusters which allows for a finite density of such vertices. As n is varied, we get a line of critical points of this generalized model, emanating from the point of marginality in the original loop model. We also study another perturbation of adding local bending rigidity to the loop model, and find that it does not affect the universality class.
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.
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 based sand ripple suppression for mine hunting with sidescan sonar
Kingsbury, Nick
dimension, and a novel wavelet shrinkage approach. Tests on a reasonably large, real synthetic aperture. Williams and Coiras [4] recently proposed a filter bank with 6 directions and 6 scale levels, which, they ventured a hardware, as opposed to our proposed software, ripple suppression approach. Unfortunately, owing
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.
Clinical relevance of the fractal dimension of F0 perturbations computed by the box-counting method.
Boek, W; Wieneke, G H; Dejonckere, P H
1997-12-01
The box-counting method for determining the fractal dimension (Dfj) was applied to the fundamental frequency (F0) perturbations in normal and pathological voices in order to assess its clinical value. The upper limit of these Dfj values was similar for both groups, but the distribution for pathological voices extended to lower values than for the normal voices. However, these lower values were most probably the result of one or a few outlying frequency points due to incorrect determination of the vocal period. The Dfj of normal voices were within the range of values found for randomly varying F0 values. It was concluded, that the vocal perturbations in pathological voices are also more or less randomly distributed. So, the Dfj, at that least determined with the box-counting method, do not contain clinically relevant information in addition to the traditional measures for the extent of the vocal period perturbations. An exception is special perturbation types like diplophonia. The result of the computation is very sensitive for voice breaks and vibrato and depends on the number of periods. PMID:9422278
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
Davies, Nia Anne; Harrison, Nicholas Kim; Morris, Roger H Keith; Noble, Simon; Lawrence, Matthew James; D'Silva, Lindsay Antonio; Broome, Laura; Brown, Martin Rowan; Hawkins, Karl M; Williams, Phylip Rhodri; Davidson, Simon; Evans, Phillip Adrian
2015-11-25
Venous thromboembolism (VTE) is common in cancer patients, and is the second commonest cause of death associated with the disease. Patients with chronic inflammation, such as cancer, have been shown to have pathological clot structures with modulated mechanical properties. Fractal dimension (df) is a new technique which has been shown to act as a marker of the microstructure and mechanical properties of blood clots, and can be performed more readily than current methods such as scanning electron microscopy (SEM). We measured df in 87 consecutive patients with newly diagnosed lung cancer prior to treatment and 47 matched-controls. Mean group values were compared for all patients with lung cancer vs controls and for limited disease vs extensive disease. Results were compared with conventional markers of coagulation, fibrinolysis and SEM images. Significantly higher values of df were observed in lung cancer patients compared with controls and patients with extensive disease had higher values than those with limited disease (p
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.
Teerikorpi, P; Theureau, G; Baryshev, Yu V; Paturel, G; Bottinelli, L; Gouguenheim, L; Baryshev, Yu.
1998-01-01
We have studied using the KLUN sample of 5171 spiral galaxies having Tully-Fisher distance moduli, the average radial space distribution of galaxies out to a distance of about 200 Mpc (for H_0=50 km/s/Mpc). One motivation came from the current debate on the fractal dimension D and maximum scale of fractality. To study this question, we used a new method based on photometric TF distances, independent of redshift, to construct the number density distribution. Our main results are: (1) While scattered below about 20 Mpc, at larger distances the radial distribution starts to follow, in terms of distance modulus mu_TF, the law log N = 0.46 mu + const., using diameter TF relation, and log N = 0.40 mu + const. for magnitudes. These are the predictions based on fractal dimensions 2.3 and 2.0, respectively. These radial density gradients are valid up to the limits of KLUN, or about 200 Mpc. (2) We have tried to understand the derived radial density behaviour as a result of some bias in KLUN or our analysis, however, w...
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.
Ali, Zulfiqar; Elamvazuthi, Irraivan; Alsulaiman, Mansour; Muhammad, Ghulam
2016-01-01
Voice disorders are associated with irregular vibrations of vocal folds. Based on the source filter theory of speech production, these irregular vibrations can be detected in a non-invasive way by analyzing the speech signal. In this paper we present a multiband approach for the detection of voice disorders given that the voice source generally interacts with the vocal tract in a non-linear way. In normal phonation, and assuming sustained phonation of a vowel, the lower frequencies of speech are heavily source dependent due to the low frequency glottal formant, while the higher frequencies are less dependent on the source signal. During abnormal phonation, this is still a valid, but turbulent noise of source, because of the irregular vibration, affects also higher frequencies. Motivated by such a model, we suggest a multiband approach based on a three-level discrete wavelet transformation (DWT) and in each band the fractal dimension (FD) of the estimated power spectrum is estimated. The experiments suggest that frequency band 1-1562 Hz, lower frequencies after level 3, exhibits a significant difference in the spectrum of a normal and pathological subject. With this band, a detection rate of 91.28 % is obtained with one feature, and the obtained result is higher than all other frequency bands. Moreover, an accuracy of 92.45 % and an area under receiver operating characteristic curve (AUC) of 95.06 % is acquired when the FD of all levels is fused. Likewise, when the FD of all levels is combined with 22 Multi-Dimensional Voice Program (MDVP) parameters, an improvement of 2.26 % in accuracy and 1.45 % in AUC is observed. PMID:26531753
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.
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.
Fractal characteristics of an asphaltene deposited heterogeneous surface
NASA Astrophysics Data System (ADS)
Amin, J. Sayyad; Ayatollahi, Sh.; Alamdari, A.
2009-10-01
Several methods have been employed in recent years to investigate homogeneous surface topography based on image analysis, such as AFM (atomic force microscopy) and SEM (scanning electron microscopy). Fractal analysis of the images provides fractal dimension of the surface which is used as one of the most common surface indices. Surface topography has generally been considered to be mono-fractal. On the other hand, precipitation of organic materials on a rough surface and its irregular growth result in morphology alteration and converts a homogeneous surface to a heterogeneous one. In this case a mono-fractal description of the surface does not completely describe the nature of the altered surface. This work aims to investigate the topography alteration of a glass surface as a result of asphaltene precipitation and its growth at various pressures using a bi-fractal approach. The experimental results of the deposited surfaces were clearly indicating two regions of micro- and macro-asperities namely, surface types I and II, respectively. The fractal plots were indicative of bi-fractal behavior and for each surface type one fractal dimension was calculated. The topography information of the surfaces was obtained by two image analyses, AFM and SEM imaging techniques. Results of the bi-fractal analysis demonstrated that topography alteration in surface type II (macro-asperities) is more evident than that in surface type I (micro-asperities). Compared to surface type II, a better correlation was observed between the fractal dimensions inferred from the AFM images ( DA) and those of the SEM images ( DS) in surface type I.
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.
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.
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.
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.
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 Astrophysics Data System (ADS)
Maria, Anton; Carey, Steven
2007-03-01
The morphology of volcanic particles can yield insight into magma fragmentation, transport processes, and style of eruption. However, the complexity and variability of volcanic particle shapes make quantitative characterization difficult. The technique applied in this study is based on fractal geometry, which has been successfully used to characterize a wide variety of particles and shapes. Here, fractal data is 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. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions. Quantitative comparisons are carried out using multivariate statistical techniques such as cluster and principal components analysis. Applications to samples from well-documented eruptions (e.g. Mt. St. Helens 1980, Tambora 1815, Surtsey 1963-64) indicate that the fractal spectrum technique provides a useful means of characterizing volcanic particles and can be helpful for identifying the products of specific fragmentation processes (volatile exsolution, phreatomagmatic, quench granulation) and modes of volcanic transport/deposition (tephra fall, pyroclastic flow, blast/surge).
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 characterization of fracture surfaces in concrete
Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.
1990-01-01
Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.
Fractal analysis of time varying data
Vo-Dinh, Tuan (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.
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U?l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area. PMID:25836036
Chappard, Daniel; Stancu, Izabela-Cristina
2015-04-01
Porosity is an important factor to consider in a large variety of materials. Porosity can be visualized in bone or 3D synthetic biomaterials by microcomputed tomography (microCT). Blocks of porous poly(2-hydroxyethyl methacrylate) were prepared with polystyrene beads of different diameter (500, 850, 1160 and 1560 ?m) and analysed by microCT. On each 2D binarized microCT section, pixels of the pores which belong to the same image column received the same pseudo-colour according to a look up table. The same colour was applied on the same column of a frontal plane image which was constructed line by line from all images of the microCT stack. The fractal dimension Df of the frontal plane image was measured as well as the descriptors of the 3D models (porosity, 3D fractal dimension D3D, thickness, density and separation of material walls. Porosity, thickness Df and D3D increased with the size of the porogen beads. A linear correlation was observed between Df and D3D. This method provides quantitative and qualitative analysis of porosity on a single frontal plane image of a porous object. PMID:25556606
The Use of Fractals for the Study of the Psychology of Perception:
NASA Astrophysics Data System (ADS)
Mitina, Olga V.; Abraham, Frederick David
The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber-Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
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.
Testing Fractal Methods on Observed and Simulated Solar Magnetograms
NASA Technical Reports Server (NTRS)
Adams, M.; Falconer, D. A.; Lee, J. K.; Jones, C.
2003-01-01
The term "magnetic complexity" has not been sufficiently quantified. To accomplish this, we must understand the relationship between the observed magnetic field of solar active regions and fractal dimension measurements. Using data from the Marshall Space Flight Center's vector magnetograph ranging from December 1991 to July 2001, we compare the results of several methods of calculating a fractal dimension, e.g., Hurst coefficient, the Higuchi method, power spectrum, and 2-D Wavelet Packet Analysis. In addition, we apply these methods to synthetic data, beginning with representations of very simple dipole regions, ending with regions that are magnetically complex.
Routes to fractality and entropy in Liesegang systems.
Kalash, Leen; Sultan, Rabih
2014-06-01
Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures. PMID:24985435
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.
Fractal model of anomalous diffusion.
Gmachowski, Lech
2015-12-01
An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion. PMID:26129728
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M.
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Roughness Perception of Haptically Displayed Fractal Surfaces
NASA Technical Reports Server (NTRS)
Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)
2000-01-01
Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.
Fractal processes in soil water retention
Tyler, S.W.; Wheatcraft, S.W. )
1990-05-01
The authors propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that they used for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar holes (or pores) over a wide range of scales. The authors evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. They relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet soil. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Their fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides paramters which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.
Fractal analysis of circulating platelets in type 2 diabetic patients.
Bianciardi, G; Tanganelli, I
2015-10-28
This paper investigates the use of computerized fractal analysis for objective characterization by means of transmission electron microscopy of the complexity of circulating platelets collected from healthy individuals and from type 2 diabetic patients, a pathologic condition in which platelet hyperreactivity has been described. Platelet boundaries were extracted by means of automatically image analysis. Local fractal dimension by box counting (measure of geometric complexity) was automatically calculated. The results showed that the platelet boundary observed by electron microscopy is fractal and that the shape of the circulating platelets is significantly more complex in the diabetic patients in comparison to healthy subjects (p?< ?0.01), with 100% correct classification. In vitro activated platelets from healthy subjects show an analogous increase of geometric complexity. Computerized fractal analysis of platelet shape by transmission electron microscopy can provide accurate, quantitative, data to study platelet activation in diabetes mellitus. PMID:25335814
Metamaterial model of fractal time
Igor I. Smolyaninov
2012-03-02
While numerous examples of fractal spaces may be found in various fields of science, the flow of time is typically assumed to be one-dimensional and smooth. Here we present a metamaterial-based physical system, which can be described by effective three-dimensional (2+1) Minkowski spacetime. The peculiar feature of this system is that its time-like variable has fractal character. The fractal dimension of the time-like variable appears to be D=2.
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
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.
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.
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.
Frankel, A.
1991-01-01
The high-frequency falloff ??-y of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R-D, D being the fractal dimension. In the model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R??, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (?? = 0) produces a main shock with an ??-2 falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to "islands' of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 - ??. Thus D = 2 corresponds to constant stress drop scaling (?? = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. -from Author
Edge detection and image segmentation of space scenes using fractal analyses
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.; Fuller, J. J.
1992-01-01
A method was developed for segmenting images of space scenes into manmade and natural components, using fractal dimensions and lacunarities. Calculations of these parameters are presented. Results are presented for a variety of aerospace images, showing that it is possible to perform edge detections of manmade objects against natural background such as those seen in an aerospace environment.
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.
Absolute free energy calculations by thermodynamic integration in four spatial dimensions
NASA Astrophysics Data System (ADS)
Rodinger, Tomas; Howell, P. Lynne; Pomès, Régis
2005-07-01
An optimized technique for calculating the excess chemical potential of small molecules in dense liquids and the binding affinity of molecular ligands to biomolecules is reported. In this method, a molecular species is coupled to the system of interest via a nonphysical fourth spatial dimension w through which insertion or extraction can be carried out [R. Pomès, E. Eisenmesser, C. B. Post et al., J. Chem. Phys. 111, 3387 (1999)]. Molecular simulations are used to compute the potential of mean force (PMF) acting on the solute molecule in the fourth dimension. The excess chemical potential of that molecule is obtained as the difference in the PMF between fully coupled and fully decoupled systems. The simplicity, efficiency, and generality of the method are demonstrated for the calculation of the hydration free energies of water and methanol as well as sodium, cesium, and chloride ions. A significant advantage over other methods is that the 4D-PMF approach provides a single effective and general route for decoupling all nonbonded interactions (i.e., both Lennard-Jones and Coulombic) at once for both neutral and charged solutes. Direct calculation of the mean force from thermodynamic integration is shown to be more computationally efficient than calculating the PMF from umbrella sampling. Statistical error analysis suggests a simple strategy for optimizing sampling. The detailed analysis of systematic errors arising from the truncation of Coulombic interactions in a solvent droplet of finite size leads to straightforward corrections to ionic hydration free energies.
Xiao, Yimin
2011-01-01
Let X= {X_t, t \\ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {\\mu_e \\in [1, \\infty), e \\in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\\ge 3. Let R = {x \\in Z^d: X_t = x for some t \\ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \\subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily large t>0 is given in terms of dim_H(A). Similar results for Bouchoud's trap model in Z^d (d \\ge 3) are also proven.
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.
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.
Large-dimension configuration-interaction calculations of positron binding to the group-II atoms
Bromley, M. W. J.; Mitroy, J.
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.
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.
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.
Algebraic calculation of the resolvent of a generalized quantum oscillator in a space of dimension D
NASA Astrophysics Data System (ADS)
Karpov, K. S.; Pismak, Yu. M.
2015-10-01
We consider the formalism based on using the sl(2) algebra instead of the conventional Heisenberg algebra for isotropic models of quantum mechanics. The operators of the squared momentum p 2 and squared coordinates q 2 and also the dilation operator H = i( pq + qp) are used as its generators. This allows calculating with the space dimension D as an arbitrary, not necessarily integer parameter. We obtain integral representations for the resolvent and its trace for a generalized harmonic oscillator with the Hamiltonian H( a, b, c) = ap 2+ bq 2+ cH and any D and study their analytic properties for different model parameter values.
Fractal Methods in Radioactivity Measurements.
NASA Astrophysics Data System (ADS)
Semkow, Thomas M.
1996-10-01
This work is in the category of applications of physics. We describe a concept of Minkowski volume (sausage) and show how an expectation value of a physical process can be calculated on a fractal object. These concepts are applied to the radioactivity in nature. The Minkowski volume is used to study the distribution of ^238U and ^232Th radioactive series as well as ^40K in environmental particles such as soils and coal fly ash.(T.M. Semkow, Environ. Intern. (1996) to be published.) The surface and internal radioactivity concentrations are obtained from the fits to experimental data, in addition to the fractal dimensions of the surfaces and thicknesses of the surface layers. We also study ^222Rn emanation from solid materials and show that the radon emanating power is proportional to V(R), where V is the Minkowski volume and R is the ?-recoil range from the decay of ^226Ra, if Ra is distributed uniformly in the solid. (T.M. Semkow, Phys. Rev. Lett. 66 (1991) 3012.) The dependence of emanating power on surface roughness and Ra distribution is also discussed.
Scaling of light scattered from fractal aggregates at resonance
NASA Astrophysics Data System (ADS)
Ortiz, Guillermo P.; Mochán, W. Luis
2003-05-01
Due to the scale invariance of fractal aggregates, light scattered from them often decays as a power of the scattering wave vector. The exponent in this power law has been usually interpreted as the geometrical fractal dimension. However, the validity of this interpretation is questionable for frequencies close to the resonances of the system, for which multiple scattering becomes important. In this work we calculate the dipole moments optically induced in fractal aggregates and the corresponding self-consistent field, as well as the electromagnetic normal modes. To this end, we develop a multiresolution hierarchical representation of the aggregate that allows the study of large systems taking fully into account the long range of the interactions. We analyze the scaling properties of the dynamically induced dipolar distribution. We find that under resonant conditions, scaling with the geometric fractal dimension is only observed for systems much larger than a length scale that is related to the linewidth of each individual resonance. The relevance to this result for the interpretation of light scattering experiments is discussed.
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
Fractal geometry of aggregate snowflakes revealed by triple-wavelength radar measurements
NASA Astrophysics Data System (ADS)
Stein, T. H. M.; Westbrook, C. D.; Nicol, J. C.
2015-01-01
Radar reflectivity measurements from three different wavelengths are used to retrieve information about the shape of aggregate snowflakes in deep stratiform ice clouds. Dual-wavelength ratios are calculated for different shape models and compared to observations at 3, 35, and 94 GHz. It is demonstrated that many scattering models, including spherical and spheroidal models, do not adequately describe the aggregate snowflakes that are observed. The observations are consistent with fractal aggregate geometries generated by a physically based aggregation model. It is demonstrated that the fractal dimension of large aggregates can be inferred directly from the radar data. Fractal dimensions close to 2 are retrieved, consistent with previous theoretical models and in situ observations.
Computational Complexity of Fractal Sets Kamo Hiroyasu
Kawamura, Kiko
Computational Complexity of Fractal Sets Kamo Hiroyasu Faculty of Science, Nara Women's University of computational complexity is indepen- dent of the classi#12;cation by means of fractal dimension. In this pa- per provides us examples of sets whose computational complexity are polynomial time computable, and which have
Random Walk and Broad Distributions on Fractal Curves
Seema Satin; A. D. Gangal
2011-03-27
In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves.
Application of fractal geometry to dissolution kinetic study of a sweetener excipient.
Tromelin, A; Hautbout, G; Pourcelot, Y
2001-08-14
In the context of relationship study between dissolution kinetic and particle morphology using the fractal geometry tool, we use a commercially available quality of saccharin powder. The characterization of molecular feature and image analysis study allows us to conclude to the statistic self-similarity of particles of four sieved particles size fractions, permitting the fractal approach. Calculation of reactive fractal dimension is performed using two forms of mass transfer equation: -dQ/dt=kQ(D(R)/3)DeltaC and -dQ/dt=k'R(D(R)-3)DeltaC, with DeltaC=(C(f)/[lnC(s)/(C(s)-C(f))]). Based on comparison of the surface fractal dimension D(S) on the two values of reactive fractal dimension D(R), a dissolution mechanism can be drawn: the dissolution starts at the whole surface of particles and is further governed by digging into holes that involve inner mass of particles. S.E.M. observations confirm this hypothesis. The confrontation between the D(R) values provided by the two ways of determination is essential for a good prediction of the mechanism. PMID:11472822
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 universe and quantum gravity.
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed. PMID:20867360
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)
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
Fractal characterization of fracture networks: An improved box-counting technique
Kah, Linda
Fractal characterization of fracture networks: An improved box-counting technique Ankur Roy,1 fracture networks as fractals and estimating their fractal dimensions (D). If this analysis yields a power and r is the box size, then the network is considered to be fractal. However, researchers are divided
The fractal aggregation of asphaltenes.
Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott
2013-07-16
This paper discusses time-resolved small-angle neutron scattering results that were used to investigate asphaltene structure and stability with and without a precipitant added in both crude oil and model oil. A novel approach was used to isolate the scattering from asphaltenes that are insoluble and in the process of aggregating from those that are soluble. It was found that both soluble and insoluble asphaltenes form fractal clusters in crude oil and the fractal dimension of the insoluble asphaltene clusters is higher than that of the soluble clusters. Adding heptane also increases the size of soluble asphaltene clusters without modifying the fractal dimension. Understanding the process of insoluble asphaltenes forming fractals with higher fractal dimensions will potentially reveal the microscopic asphaltene destabilization mechanism (i.e., how a precipitant modifies asphaltene-asphaltene interactions). It was concluded that because of the polydisperse nature of asphaltenes, no well-defined asphaltene phase stability envelope exists and small amounts of asphaltenes precipitated even at dilute precipitant concentrations. Asphaltenes that are stable in a crude oil-precipitant mixture are dispersed on the nanometer length scale. An asphaltene precipitation mechanism is proposed that is consistent with the experimental findings. Additionally, it was found that the heptane-insoluble asphaltene fraction is the dominant source of small-angle scattering in crude oil and the previously unobtainable asphaltene solubility at low heptane concentrations was measured. PMID:23808932
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. PMID:23767500
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.
Khalili, N. R.; Pan, M.; Sandi, G.; Chemistry; Illinois Inst. of Tech.
2000-01-01
The total surface area, micropore volume, and fractal dimensions of five different carbons (Sorbonorite 4, GAC 1240, and three amorphous carbons) were evaluated from analysis of gas (N{sub 2}) and liquid (phenanthrene) adsorption isotherm data. The modified BET and fractal Frenkel-Halsey-Hill (FHH) models were used to estimate surface fractal dimensions. Micropore volumes were estimated from Dubinin-Radushkevich (DR) plots and were compared to those calculated from standard N{sub 2} adsorption isotherm data using de Boer's t-method. The estimated surface fractal dimensions using the modified BET and FHH models (D{sub s}=3+3h, and P/P{sub 0} from 0.0 to 0.4) were (2.7, 2.6, 2.1, 2.4, and 2.1) and (2.5, 2.6, 1.9, 2.4, and 1.9), respectively. The FHH fractal analysis suggested that van der Waals forces are the dominant interaction forces between nitrogen and carbon surfaces. Depending on the method of analysis, the fractal dimensions of the carbons with suggested micropore structure, Sorbonorite 4 and GAC 1240, were 2.5-2.9 and 2.6-2.9, respectively. Analysis of the adsorption-desorption data suggested that amorphous carbons with fractal dimensions of 2.1 (from the modified BET model) have smooth surfaces, with respect to their micropore structure. Further analysis of the adsorption data showed that the slopes of the linear segment of the plots of adsorption potential versus relative amount adsorbed are dependent on the pore size range and surface structure (fractal dimension) of the carbons.
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.
Effect of disorder on a fractal model for the ac response of a rough interface
NASA Astrophysics Data System (ADS)
Kaplan, Theodore; Gray, L. J.
1985-12-01
The effect of disorder on the fractal model of Liu for a rough interface between two materials of very different conductivities is examined. It is found that the average admittance has the experimentally observed property of a constant-phase-angle (CPA) element. The exponent ? of the frequency dependence of the CPA element depends on the specific form of the probability distribution of the scaling. Furthermore, the fractal dimension d¯s of the random surface area of the interface is calculated, and ?=3-d¯s.
Fun with Fractals Dr. Bori Mazzag
Mazzag, Borbala "Bori"
sequences are colored #12;Creating a fractal Recursion with pictures Generating the Koch snowflake #12;Creating a fractal Recursion with pictures Generating the Koch snowflake Generating Sierpinski's gasket #12;Calculating the perimenter of the the Koch snowflake #12;Calculating the perimenter
Ferromagnetism in fractal-based complexes
NASA Astrophysics Data System (ADS)
Ugajin, Ryuichi
2002-11-01
Ferromagnetism in fractal-based complexes, which are generated using the dielectric-breakdown model with appropriate controls of their fractal dimension, is investigated using the standard Monte Carlo simulations. The difference in the fractal dimensions of a nerve-cell-like complex creates a heterotic phase in which the spin-ordered Gibbs state of a somatic nucleus and the spin-disordered Gibbs state of dendritic portions are orchestrated. On the other hand, a nebulalike complex in which many sites are grown on a dendritic substrate behaves as a single ferromagnetic system and is characterized by a particular Curie temperature.
Molecular dynamics simulation of fractal aggregate diffusion
NASA Astrophysics Data System (ADS)
Pranami, Gaurav; Lamm, Monica H.; Vigil, R. Dennis
2010-11-01
The diffusion of fractal aggregates constructed with the method by Thouy and Jullien [J. Phys. A 27, 2953 (1994)10.1088/0305-4470/27/9/012] comprised of Np spherical primary particles was studied as a function of the aggregate mass and fractal dimension using molecular dynamics simulations. It is shown that finite-size effects have a strong impact on the apparent value of the diffusion coefficient (D) , but these can be corrected by carrying out simulations using different simulation box sizes. Specifically, the diffusion coefficient is inversely proportional to the length of a cubic simulation box, and the constant of proportionality appears to be independent of the aggregate mass and fractal dimension. Using this result, it is possible to compute infinite dilution diffusion coefficients (Do) for aggregates of arbitrary size and fractal dimension, and it was found that Do?Np-1/df , as is often assumed by investigators simulating Brownian aggregation of fractal aggregates. The ratio of hydrodynamic radius to radius of gyration is computed and shown to be independent of mass for aggregates of fixed fractal dimension, thus enabling an estimate of the diffusion coefficient for a fractal aggregate based on its radius of gyration.
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 Theory for Permeability Prediction, Venezuelan and USA Wells
NASA Astrophysics Data System (ADS)
Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana
2014-05-01
Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.
Fractal Characterization of 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 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
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
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.
Calculus on Fractal Curves in R^n
Abhay Parvate; Seema Satin; A. D. Gangal
2010-04-06
A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
Fractal PatternsFractal Patterns in Chaotic Mixingin Chaotic Mixing
Anlage, Steven
Fractal PatternsFractal Patterns in Chaotic Mixingin Chaotic Mixing Amir Ali Ahmadi, UniversityTREND 2005 #12;What is a Fractal? Romanesco broccoli Fractal an object which has variation://upload.wikimedia.org/wikipedia/en/thumb/8/8a/800px-Fractal_Broccoli.jpg #12;Fractal Example http://colos1.fri.uni-lj.si/~sis/GRAFIKA/FRACTALS/FRACTAL
Compact Polymers on Fractal Lattices
NASA Astrophysics Data System (ADS)
Elezovi?-Hadži?, Sun?ica; Mar?eti?, Dušanka; Maleti?, Slobodan
2007-04-01
We study compact polymers, modelled by Hamiltonian walks (HWs), i.e. self-avoiding walks that visit every site of the lattice, on various fractal lattices: Sierpinski gasket (SG), Given-Mandelbrot family of fractals, modified SG fractals, and n-simplex fractals. Self-similarity of these lattices enables establishing exact recursion relations for the numbers of HWs conveniently divided into several classes. Via analytical and numerical analysis of these relations, we find the asymptotic behaviour of the number of HWs and calculate connectivity constants, as well as critical exponents corresponding to the overall number of open and closed HWs. The nonuniversality of the HW critical exponents, obtained for some homogeneous lattices is confirmed by our results, whereas the scaling relations for the number of HWs, obtained here, are in general different from the relations expected for homogeneous lattices.
Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space
Tarasov, Vasily E
2015-01-01
Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.
Statistical fractal analysis of 25 young star clusters
Gregorio-Hetem, J; Santos-Silva, T; Fernandes, B
2015-01-01
A large sample of young stellar groups is analysed aiming to investigate their clustering properties and dynamical evolution. A comparison of the Q statistical parameter, measured for the clusters, with the fractal dimension estimated for the projected clouds shows that 52% of the sample has substructures and tends to follow the theoretically expected relation between clusters and clouds, according to calculations for artificial distribution of points. The fractal statistics was also compared to structural parameters revealing that clusters having radial density profile show a trend of parameter s increasing with mean surface stellar density. The core radius of the sample, as a function of age, follows a distribution similar to that observed in stellar groups of Milky Way and other galaxies. They also have dynamical age, indicated by their crossing time that is similar to unbound associations. The statistical analysis allowed us to separate the sample into two groups showing different clustering characteristi...
NASA Technical Reports Server (NTRS)
Pandey, Apoorva; Chakrabarty, Rajan K.; Liu, Li; Mishchenko, Michael I.
2015-01-01
Soot aggregates (SAs)-fractal clusters of small, spherical carbonaceous monomers-modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension Df = 1.8) and aged (Df = 2.6) SAs at 550 nm wavelength using the numericallyexact superposition T-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships.
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 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.
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...
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.
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.
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.
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
Study of morphology of reactive dissolution interface using fractal geometry.
Tromelin, A; Gnanou, J C; Andrès, C; Pourcelot, Y; Chaillot, B
1996-09-01
The determination of reactive fractal dimension was carried out using two forms of the Noyes-Whitney equation, -dQ/dt = K(Q/Q0)DR/3 and -dQ/dt = K'RDR-3 using the Richardson plot on the basis of previous data obtained by dissolution of an orthoboric acid powder. The correlation of the results provided by the two ways of calculation allows proposal of the hypothesis that dissolution begins on a specific population of reactive sites and probably promotes the formation of microporous volumes or cracks. PMID:8877880
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.
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
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
NASA Astrophysics Data System (ADS)
Chernodub, Maxim N.; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum 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 web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics.
Chernodub, Maxim N; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum 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 web in contrast to the Hofstadter butterfly for unbiased motion. PMID:26565163
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.
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.
Bender, C M; Carl M Bender; Stefan Boettcher
1994-01-01
This paper extends an earlier high-temperature lattice calculation of the renormalized Green's functions of a D-dimensional Euclidean scalar quantum field theory in the Ising limit. The previous calculation included all graphs through sixth order. Here, we present the results of an eleventh-order calculation. The extrapolation to the continuum limit in the previous calculation was rather clumsy and did not appear to converge when D>2. Here, we present an improved extrapolation which gives uniformly good results for all real values of the dimension between D=0 and D=4. We find that the four-point Green's function has the value 0.620 \\pm 0.007 when D=2 and 0.98 \\pm 0.01 when D=3 and that the six-point Green's function has the value 0.96 \\pm 0.03 when D=2 and 1.2 \\pm 0.2 when D=3.
Carl M. Bender; Stefan Boettcher
1994-05-06
This paper extends an earlier high-temperature lattice calculation of the renormalized Green's functions of a $D$-dimensional Euclidean scalar quantum field theory in the Ising limit. The previous calculation included all graphs through sixth order. Here, we present the results of an eleventh-order calculation. The extrapolation to the continuum limit in the previous calculation was rather clumsy and did not appear to converge when $D>2$. Here, we present an improved extrapolation which gives uniformly good results for all real values of the dimension between $D=0$ and $D=4$. We find that the four-point Green's function has the value $0.620 \\pm 0.007$ when $D=2$ and $0.98 \\pm 0.01$ when $D=3$ and that the six-point Green's function has the value $0.96 \\pm 0.03$ when $D=2$ and $1.2 \\pm 0.2$ when $D=3$.
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 properties of quantum spacetime.
Benedetti, Dario
2009-03-20
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 is observed in some approaches to quantum gravity. In particular, we analyze the cases of a quantum sphere and of kappa-Minkowski spacetime, the latter being relevant in the context of quantum gravity. PMID:19392189
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.
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.
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
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 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.
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.
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.
Fractal structure of a white cauliflower
Sang-Hoon Kim
2004-09-30
The fractal structure of a white cauliflower is investigated by box-counting method of its cross-section. The capacity dimension of the cross-section is $1.88 \\pm 0.02$ independent of directions. From the result, we predict that the capacity dimension of the cauliflower is about 2.8. The vertical cross-section of the cauliflower is modeled into a self-similar set of a rectangular tree. We discuss the condition of the fractal object in the tree, and show that the vertical cross-section has an angle of 67 degrees in our model.
Fractal Measure and Microscopic Modeling of Osseointegration.
Santos, Leonardo Cavalcanti Bezerra; Carvalho, Alessandra Albuquerque; Leão, Jair Carneiro; Neto, Paulo Jose; Stosic, Tatijana; Stosic, Borko
2015-01-01
In this study, the process of osseointegration on titanium implant surfaces with different physicochemical treatments subjected to a simulated corporal fluid submersion was evaluated using the concept of fractal dimension. It was found that different treatments led to rather different calcium phosphate crystal growth patterns, with fractal dimension ranging from 1.68 to 1.93. The observed crystal patterns may be explained by a general deposition, diffusion, and aggregation growth mechanism, where diffusing particle sticking probability plays a fundamental role. PMID:26509989
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 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.
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.
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)
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.
Fractal Analysis of Stress Sensitivity of Permeability in Porous Media
NASA Astrophysics Data System (ADS)
Tan, Xiao-Hua; Li, Xiao-Ping; Liu, Jian-Yi; Zhang, Lie-Hui; Cai, Jianchao
2015-12-01
A permeability model for porous media considering the stress sensitivity is derived based on mechanics of materials and the fractal characteristics of solid cluster size distribution. The permeability of porous media considering the stress sensitivity is related to solid cluster fractal dimension, solid cluster fractal tortuosity dimension, solid cluster minimum diameter and solid cluster maximum diameter, Young's modulus, Poisson's ratio, as well as power index. Every parameter has clear physical meaning without the use of empirical constants. The model predictions of permeability show good agreement with those obtained by the available experimental expression. The proposed model may be conducible to a better understanding of the mechanism for flow in elastic porous media.
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.
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.…
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
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.
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.
Wenseleers, Wim
traveling plasmon waves, aggregates of metal particles pos- sessing a fractal geometry (noninteger dimension) near-field fluorescence imaging. Photodecomposition of pyridine and other aromatic molecules
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.
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
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
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
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.
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 analysis of the structural complexity of the connective tissue in human carotid bodies
Guidolin, Diego; Porzionato, Andrea; Tortorella, Cinzia; Macchi, Veronica; De Caro, Raffaele
2014-01-01
The carotid body (CB) may undergo different structural changes during perinatal development, aging, or in response to environmental stimuli. In the previous literature, morphometric approaches to evaluate these changes have considered quantitative first order parameters, such as volumes or densities, while changes in spatial disposition and/or complexity of structural components have not yet been considered. In the present study, different strategies for addressing morphological complexity of CB, apart from the overall amount of each tissue component, were evaluated and compared. In particular, we considered the spatial distribution of connective tissue in the carotid bodies of young control subjects, young opiate-related deaths and aged subjects, through analysis of dispersion (Morisita's index), gray level co-occurrence matrix (entropy, angular second moment, variance, correlation), and fractal analysis (fractal dimension, lacunarity). Opiate-related deaths and aged subjects showed a comparable increase in connective tissue with respect to young controls. However, the Morisita's index (p < 0.05), angular second moment (p < 0.05), fractal dimension (p < 0.01), and lacunarity (p < 0.01) permitted to identify significant differences in the disposition of the connective tissue between these two series. A receiver operating characteristic (ROC) curve was also calculated to evaluate the efficiency of each parameter. The fractal dimension and lacunarity, with areas under the ROC curve of 0.9651 (excellent accuracy) and 0.8835 (good accuracy), respectively, showed the highest discriminatory power. They evidenced higher level of structural complexity in the carotid bodies of opiate-related deaths than old controls, due to more complex branching of intralobular connective tissue. Further analyses will have to consider the suitability of these approaches to address other morphological features of the CB, such as different cell populations, vascularization, and innervation. PMID:25414672
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)
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.
A fractal approach to probabilistic seismic hazard assessment
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1989-01-01
The definition of a fractal distribution is that the number of objects (events) N with a characteristic size greater than r satisfies the relation N proportional to r exp - D is the fractal dimension. The applicability of a fractal relation implies that the underlying physical process is scale-invariant over the range of applicability of the relation. The empirical frequency-magnitude relation for earthquakes defining a b-value is a fractal relation with D = 2b. Accepting the fractal distribution, the level of regional seismicity can be related to the rate of regional strain and the magnitude of the largest characteristic earthquake. High levels of seismic activity indicate either a large regional strain or a low-magnitude maximum characteristic earthquake (or both). If the regional seismicity has a weak time dependence, the approach can be used to make probabilistic seismic hazard assessments.
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...
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.
Edges of Saturn's rings are fractal.
Li, Jun; Ostoja-Starzewski, Martin
2015-01-01
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63?~?1.78. This clarifies in what sense Saturn's rings are fractal. PMID:25883885
Fractal 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.
Control of multiscale systems with constraints. 2. Fractal nuclear isomers and clusters
S. Adamenko; V. Bolotov; V. Novikov
2013-07-17
We consider the influence of the Fermi statistics of nucleons on the binding energy of a new type of nuclear structures such as fractal nuclear clusters (fractal isomers of nuclei). It is shown that the fractal nuclear isomers possess a wide spectrum of binding energies that exceed, in many cases, the values known at the present time. The transition of the nuclear matter in the form of ordinary nuclei (drops of the nuclear fluid) in the state with the fractal structure or in the form of bubble nuclei opens new sources of energy and has huge perspectives. This transition is based on a new state of matter - collective coherently correlated state. It manifests itself, first of all, in the property of nonlocality of nuclear multiparticle processes. We develop a phenomenological theory of the binding energy of nuclear fractal structures and modify the Bethe - Weizs\\"acker formula for nuclear clusters with the mass number A, charge Z, and fractal dimension D_f. The consideration of fractal nuclear isomers allows one to interpret the experimental results on a new level of the comprehension of processes of the nuclear dynamics. The possibility to determine the fractal dimension of nuclear systems with the help of the method of nuclear dipole resonance for fractal isomers is discussed. The basic relations for fractal electroneutral structures such as the electron-nucleus plasma of fractal isomers are presented.
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.
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.
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.
Linear chains and chain-like fractals from electrostatic heteroaggregation.
Kim, Anthony Y; Hauch, Kip D; Berg, John C; Martin, James E; Anderson, Robert A
2003-04-01
The internal structure of materials prepared by aggregation of oppositely charged polystyrene spheres (electrostatic heteroaggregation) is investigated by static light scattering, optical microscopy, and Brownian dynamics simulation. Light scattering indicates ultralow mass fractal dimensions, as low as 1.2. Such low fractal dimensions, approaching the theoretical limit of a linear object, imply a chaining mechanism. Optical micrographs reveal linear chains with the particle charge alternating down the chains. Brownian dynamics simulation gives additional support for a chaining mechanism. For the polystyrene system (120-nm primary particle diameters), the fractal dimension is found to increase from 1.2 to 1.7 as the background electrolyte is increased. In terms of electrostatic screening, the results match those reported recently for larger polystyrene spheres. The low fractal dimensions appear to represent a crossover from linear chains to a structure of diffusion-limited aggregates; however, experiments under density-neutral conditions imply that sedimentation plays an important role in the formation of ultralow fractal dimensions. The practical implication is that microcomposites with a locally uniform distribution of starting materials and almost any degree of branching can be prepared from oppositely charged particles. PMID:12742045
Turbulence on a Fractal Fourier set
Alessandra Sabina Lanotte; Roberto Benzi; Luca Biferale; Shiva Kumar Malapaka; Federico Toschi
2015-12-04
A novel investigation of the nature of intermittency in incompressible, homogeneous and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension, D, from the original three dimensional case to a strongly decimated system with D=2.5, where only about $3\\%$ of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, 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 intermittency of the vorticity field is observed to be quasi-singular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at $D \\sim 2.98$. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.
Correlated fractal percolation and the Palis conjecture
Michel Dekking; Henk Don
2009-10-30
Let F1 and F2 be independent copies of correlated fractal percolation, with Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will contain an interval? The well known Palis conjecture states that `generically' this should be true. Recent work by Kuijvenhoven and the first author (arXiv:0811.0525) on random Cantor sets can not answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of (arXiv:0811.0525) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.
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 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
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
Mehan, Sumit Kumar, Sugam Aswal, V. K.
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.
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.
NASA Technical Reports Server (NTRS)
Hudson, Richard K.; Anderson, Steven W.; McColley, Shawn; Fink, Jonathan H.
2004-01-01
Fractals are objects that are generally self similar at all scales. Coastlines, mountains, river systems, planetary orbits and some mathematical objects are all examples of fractals. Bruno et al. used the structured walk model of Richardson to establish that lava flows are fractals and that lava flow morphology could be determined by looking at the fractal dimension of flow margins. They determined that Hawaiian a.a flows have fractal dimensions that range from 1.05 to 1.09 and that the pahoehoe lava flows have a fractal dimension from 1.13 to 1.23. We have analyzed a number of natural and simulated lava flow margins and find that the fractal dimension varies according to the number and length of rod lengths used in the structured walk method. The potential variation we find in our analyses is sufficiently large so that unambiguous determination of lava flow morphology is problematic for some flows. We suggest that the structured walk method can provide meaningful fractal dimensions if rod lengths employed in the analysis provide a best-fit residual of greater than 0.98, as opposed to the 0.95 cutoff used in previous studies. We also find that the use of more than 4 rod lengths per analysis also reduces ambiguity in the results.
Fractal 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
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.
Crystallization of space: Space-time fractals from fractal arithmetic
Diederik Aerts; Marek Czachor; Maciej Kuna
2015-12-09
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.
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
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.
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 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.
Spectral dimension of a quantum universe
Modesto, Leonardo; Nicolini, Piero
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.
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
Study of Fractal Features of Geomagnetic Activity Through an MHD Shell Model
NASA Astrophysics Data System (ADS)
Dominguez, M.; Nigro, G.; Munoz, V.; Carbone, V.
2013-12-01
Studies on complexity have been of great interest in plasma physics, because they provide new insights and reveal possible universalities on issues such as geomagnetic activity, turbulence in laboratory plasmas, physics of the solar wind, etc. [1, 2]. In particular, various studies have discussed the relationship between the fractal dimension, as a measure of complexity, and physical processes in magnetized plasmas such as the Sun's surface, the solar wind and the Earth's magnetosphere, including the possibility of forecasting geomagnetic activity [3, 4, 5]. Shell models are low dimensional dynamical models describing the main statistical properties of magnetohydrodynamic (MHD) turbulence [6]. These models allow us to describe extreme parameter conditions hence reaching very high Reynolds (Re) numbers. In this work a MHD shell model is used to describe the dissipative events which are taking place in the Earth's magnetosphere and causing geomagnetic storms. The box-counting fractal dimension (D) [7] is calculated for the time series of the magnetic energy dissipation rate obtained in this MHD shell model. We analyze the correlation between D and the energy dissipation rate in order to make a comparison with the same analysis made on the geomagnetic data. We show that, depending on the values of the viscosity and the diffusivity, the fractal dimension and the occurrence of bursts exhibit correlations similar as those observed in geomagnetic and solar data, [8] suggesting that the latter parameters could play a fundamental role in these processes. References [1] R. O. Dendy, S. C. Chapman, and M. Paczuski, Plasma Phys. Controlled Fusion 49, A95 (2007). [2] T. Chang and C. C. Wu, Phys. Rev. E 77, 045401 (2008). [3] R. T. J. McAteer, P. T. Gallagher, and J. Ireland, Astrophys. J. 631, 628 (2005). [4] V. M. Uritsky, A. J. Klimas, and D. Vassiliadis, Adv. Space Res. 37, 539 (2006). [5] S. C. Chapman, B. Hnat, and K. Kiyani, Nonlinear Proc. Geophys. 15, 445 (2008). [6] G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and A. Vulpiani, Phys. Rev. Lett. 83, 4662 (1999). [7] P. S. Addison, Fractals and Chaos, an Illustrated Course, vol. 1 (Institute of Physics Publishing, Bristol and Philadelphia, 1997), second ed. [8] M. Domínguez, V. Muñoz, and J. A. Valdivia, Temporal evolution of fractality in the Earth's magnetosphere and the solar photosphere, in preparation.
A fractal transition in the two dimensional shear layer
NASA Technical Reports Server (NTRS)
Jimenez, Javier; Martel, Carlos
1990-01-01
The dependence of product generation with the Peclet and Reynolds number in a numerically simulated, reacting, two dimensional, temporally growing mixing layer is used to compute the fractal dimension of passive scalar interfaces. A transition from a low dimension of 4/3 to a higher one of 5/3 is identified and shown to be associated to the kinematic distortion on the flow field during the first pairing interaction. It is suggested that the structures responsible for this transition are non-deterministic, non-random, inhomogeneous fractals. Only the large scales are involved. No further transition is found for Reynolds numbers up to 20,000.
Quantification of the fractal nature of mycelial aggregation in Aspergillus niger submerged cultures
Papagianni, Maria
2006-01-01
Background Fractal geometry estimates have proven useful in studying the growth strategies of fungi in response to different environments on soil or on agar substrates, but their use in mycelia grown submerged is still rare. In the present study, the effects of certain important fermentation parameters, such as the spore inoculum level, phosphate and manganese concentrations in the medium, on mycelial morphology of the citric acid producer Aspergillus niger were determined by fractal geometry. The value of employing fractal geometry to describe mycelial structures was examined in comparison with information from other descriptors including classic morphological parameters derived from image analysis. Results Fractal analysis of distinct morphological forms produced by fermentation conditions that influence fungal morphology and acid production, showed that the two fractal dimensions DBS (box surface dimension) and DBM (box mass dimension) are very sensitive indexes, capable of describing morphological differences. The two box-counting methods applied (one applied to the whole mass of the mycelial particles and the other applied to their surface only) enabled evaluation of fractal dimensions for mycelial particles in this analysis in the region of DBS = 1.20–1.70 and DBM = 1.20–2.70. The global structure of sufficiently branched mycelia was described by a single fractal dimension D, which did not exceed 1.30. Such simple structures are true mass fractals (DBS = DBM = D) and they could be young mycelia or dispersed forms of growth produced by very dense spore inocula (108–109 spores/ml) or by addition of manganese in the medium. Mycelial clumps and pellets were effectively discriminated by fractal analysis. Fractal dimension values were plotted together with classic morphological parameters derived from image analysis for comparisons. Their sensitivity to treatment was analogous to the sensitivity of classic morphological parameters suggesting that they could be equally used as morphological descriptors. Conclusion Starting from a spore, the mycelium develops as a mass fractal and, depending on culture conditions, it either turns to a surface fractal or remains a mass fractal. Since fractal dimensions give a measure of the degree of complexity and the mass filling properties of an object, it may be possible that a large number of morphological parameters which contribute to the overall complexity of the particles, could be replaced by these indexes effectively. PMID:16472407
Fractal contact spot and its application in the contact model of isotropic surfaces
NASA Astrophysics Data System (ADS)
Zhou, An'an; Chen, Tianning; Wang, Xiaopeng; Xi, Yanhui
2015-10-01
All engineering surfaces are rough and composed of asperities with different scales. In conventional fractal contact models, asperities are used to build contact models between two rough surfaces. However, it is found that the deformation of asperities in the contact process is rather complex because only a small part of the asperities deform elastically or plastically. Moreover, the asperities used in conventional fractal contact models are not constructed exactly from contact mechanism and, of course, they are not fractal or self-affine. So, these fractal contact models seem like statistical models using fractal parameters rather than pure-fractal contact models. In the present paper, instead of asperities, fractal contact spots are utilized to describe the contact process of isotropic rough surfaces. The differences between fractal contact spots and asperities are analyzed to prove that fractal contact spots are more suitable to reveal the nature of contact process. Numerical method is used to simulate the contact process and study the characteristics of fractal contact spots, and the load-separation relationship of fractal contact spots is obtained using a fractal method. Then, a pure-fractal contact model is proposed and extended to calculate the contact stiffness of rough surfaces. To verify the contact model proposed, an experiment of stiffness identification of isotropic metal surfaces is carried out. The identified stiffness is in good agreement with that calculated by the contact model proposed, which indicates that the pure-fractal contact model applying the fractal contact spots is more suitable to reveal the contact mechanism of isotropic surfaces.
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.
Fractal analysis of replication site images of the human cell nucleus.
Bhattacharya, S; Malyavantham, K S; Acharya, R; Berezney, R
2004-01-01
Epi-fluorescent microscopic images of the mammalian cell nucleus taken during the early, mid and late S (synthetic) phase of the cell cycle suggest that the mass of replicating DNA that belong to the cell nucleus can be characterized as a space filling fractal curve. We reason from a biological standpoint and our understanding of naturally occurring fractals that our microscopic images reveal portions of the spatially complex DNA molecule and present methods for computing the fractal dimensions of the images. Results presented here suggest that our methodology based on fractal properties can distinguish replication of DNA occurring in early versus mid or late S-phase. PMID:17271966
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 universe by suitably choosing the fractal parameter $\\beta$.
Contact Kinetics in Fractal Macromolecules
Dolgushev, Maxim; Blumen, Alexander; Bénichou, Olivier; Voituriez, Raphaël
2015-01-01
We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the Mean First Contact Time (MFCT) for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the non-equilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the MFCT between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.
Contact Kinetics in Fractal Macromolecules
NASA Astrophysics Data System (ADS)
Dolgushev, Maxim; Guérin, Thomas; Blumen, Alexander; Bénichou, Olivier; Voituriez, Raphaël
2015-11-01
We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the mean first contact time for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the nonequilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the mean first contact time between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.
Contact Kinetics in Fractal Macromolecules.
Dolgushev, Maxim; Guérin, Thomas; Blumen, Alexander; Bénichou, Olivier; Voituriez, Raphaël
2015-11-13
We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the mean first contact time for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the nonequilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the mean first contact time between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations. PMID:26613478
Hydrodynamics of fractal continuum flow.
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. PMID:22463270
Fractal Themes at Every Level Kenneth G. Monks
Monks, Kenneth
Fractal Themes at Every Level Kenneth G. Monks University of Scranton August 19, 1998 OK I admit it. I love fractals. Fractal programs, fractal tee-shirts, fractal notebooks, fractal screen savers... What other
Fractal Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks. PMID:21423355
Fractal 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.
NASA Astrophysics Data System (ADS)
Bender, Carl M.; Boettcher, Stefan
1995-02-01
This paper extends an earlier high-temperature lattice calculation of the renormalized Green's function of a D-dimensional Euclidean scalar quantum field theory in the Ising limit. The previous calculation included all graphs through sixth order. Here, we present the results of an eleventh-order calculation. The extrapolation to the continuum limit in the previous calculation was rather clumsy and did not appear to converge when D>2. Here, we present an improved extrapolation which gives uniformly good results for all real values of the dimension between D=0 and D=4. We find that the four-point Green's function has the value 0.620+/-0.007 when D=2 and 0.98+/-0.01 when D=3 and that the six-point Green's function has the value 0.96+/-0.03 when D=2 and 1.2+/-0.2 when D=3.
Fractal and multifractal analysis of pore-scale images of soil
NASA Astrophysics Data System (ADS)
Bird, Nigel; Díaz, M. Cruz; Saa, Antonio; Tarquis, Ana M.
2006-05-01
We examine critically the fractal and multifractal analysis of two-dimensional images of soil sections. We demonstrate that, dependent on the porosity displayed in the image, both a fractal dimension and a multifractal spectrum can be extracted from such images irrespective of whether these images exhibit fractal structures and multifractal scaling of local density and porosity. We suggest ways to transform the data arising from the analysis in order to differentiate better between fractal and non-fractal images. We examine three soil images and conclude that there is no compelling evidence of scaling properties associated with mass fractal and multifractal structures. Our results point to a need for alternative methods for characterizing soil pore structures and to extend our modelling of complex and multiscale porous media to cases where scaling symmetries are relaxed.
Wilson, T.H.; Dominic, J.; Halverson, J.
1996-12-31
The primary goal of this study is to evaluate the possibility that the fractal characteristics of reservoir fracture systems might be inferred from the fractal characteristics of the reservoir reflector. Results discussed in the summary below provide support for such a view. The matter will, however, remain unresolved until fracture data acquired from core or FMS logs can be compared to reflection seismic data from the core areas. A series of cross sections along the Middle Mountain syncline and Elkhorn Mountain anticline were evaluated. Near-surface deformation in the Middle Mountain and Elkhorn mountain area of the Valley and Ridge province is significant. In this area the fractal dimension of topography is linearly related to the fractal dimension of underlying structure. Comparison of the fractal variability of Valley and Ridge structures with those observed in seismic data from the Plateau indicate that the increased fractal dimension of reflection events implies greater relative abundance of higher order or smaller wavelength structures. Results from the seismic evaluation suggest that fractal analysis might provide a useful exploration tool in cases where one is interested in locating subtle detached structures or identifying fractured reservoirs. Results from the Valley and a Ridge area suggest that, in active tectonic areas, fractal analysis may provide a means to assess the relative frequency of earthquake activity over time periods that extend beyond the historical record.
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.
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.
The fractal structure of the ventral scales in legless reptiles
Abdel-Aal, Hisham A
2015-01-01
Surface constructs in snakes reflect desirable design traits for technical surface engineering. Their micro-textural patterns, however, do not lend themselves easily to unified analysis due to species-specific variations in surface geometry and topology. Fractal description is useful in this context since it accentuates the correspondence between patterns especially when responding to tribological phenomena. In this work we examine the surface construction of 14 snake species, representing five families, and evaluate the fractal dimension for each of the skins (both the dorsal and ventral sides) using three different computational algorithms. Our results indicate first that all of the examined species share a common fractal dimension (with a very small variation between species in the order 4-5%). This finding implies that despite the different micro-geometry of texture among species, the skin as a unit responds in a similar manner to many interfacial influences.
Fractal 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
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.
Micromeasure distributions and applications for conformally generated fractals
NASA Astrophysics Data System (ADS)
Fraser, Jonathan M.; Pollicott, Mark
2015-11-01
We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.
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.
Fractal frontiers of bursts and cracks in a fiber bundle model of creep rupture
Zsuzsa Danku; Ferenc Kun; Hans J. Herrmann
2015-12-03
We investigate the geometrical structure of breaking bursts generated during the creep rupture of heterogeneous materials. Based on a fiber bundle model with localized load sharing we show that bursts are compact geometrical objects, however, their external frontier has a fractal structure which reflects their growth dynamics. The perimeter fractal dimension of bursts proved to have the universal value 1.25 independent of the external load and of the amount of disorder in the system. We conjecture that according to their geometrical features breaking bursts fall in the universality class of loop-erased self-avoiding random walks with perimeter fractal dimension 5/4 similar to the avalanches of Abelian sand pile models. The fractal dimension of the growing crack front along which bursts occur proved to increase from 1 to 1.25 as bursts gradually cover the entire front.
Fractal Magneto-conductance Fluctuations in Mesoscopic Semiconductor Billiards
Adam P. Micolich
2002-09-22
Negatively biased surface-gates allow electrostatic depletion of selected regions of a 2DEG, forming confined regions of specific geometry called billiards, in which ballistic transport occurs. At millikelvin temperatures, the electron phase coherence length is sufficient that quantum interference effects produce reproducible magneto-conductance fluctuations (MCF) that act as a 'magneto-fingerprint' of the scattering dynamics in the billiard. It has been predicted that billiard MCF are fractal. Fractal MCF in mesoscopic semiconductor billiards are investigated experimentally. The MCF of a Sinai billiard displayed exact self-similarity (ESS). A correlation function analysis is used to quantify the presence of ESS. A model for the Sinai billiard MCF based on a Weierstrass function is presented. Using a bridging interconnect, a continuous transition between the Sinai and an empty square geometry is achieved. The removal of the circle induces a transition from ESS to statistical self-similarity (SSS), suggesting that ESS is due to the presence of an obstacle at the center of the billiard. The physical dependencies of SSS are investigated and show variation in the fractal dimension, rather than the fractal scaling range. SSS obeys a unified picture where the fractal dimension depends only on the ratio between the average spacing and broadening of billiard energy levels, irrespective of other billiard parameters. The semiclassical origin of SSS is demonstrated and the suppression of SSS is observed in both the quantum and classical limits. The influence of soft-wall potential profile on fractal MCF is investigated using double-2DEG billiards. Detailed reviews of semiconductor billiard fabrication, low-temperature electrical measurements and fractal analysis are also presented.
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
Fractal analysis of vascular networks: insights from morphogenesis.
Lorthois, Sylvie; Cassot, Francis
2010-02-21
Considering their extremely complicated and hierarchical structure, a long standing question in vascular physio-pathology is how to characterize blood vessels patterns, including which parameters to use. Another question is how to define a pertinent taxonomy, with applications to normal development and to diagnosis and/or staging of diseases. To address these issues, fractal analysis has been applied by previous investigators to a large variety of healthy or pathologic vascular networks whose fractal dimensions have been sought. A review of the results obtained on healthy vascular networks first shows that no consensus has emerged about whether normal networks must be considered as fractals or not. Based on a review of previous theoretical work on vascular morphogenesis, we argue that these divergences are the signature of a two-step morphogenesis process, where vascular networks form via progressive penetration of arterial and venous quasi-fractal arborescences into a pre-existing homogeneous capillary mesh. Adopting this perspective, we study the multi-scale behavior of generic patterns (model structures constructed as the superposition of homogeneous meshes and quasi-fractal trees) and of healthy intracortical networks in order to determine the artifactual and true components of their multi-scale behavior. We demonstrate that, at least in the brain, healthy vascular structures are a superposition of two components: at low scale, a mesh-like capillary component which becomes homogeneous and space-filling over a cut-off length of order of its characteristic length; at larger scale, quasi-fractal branched (tree-like) structures. Such complex structures are consistent with all previous studies on the multi-scale behavior of vascular structures at different scales, resolving the apparent contradiction about their fractal nature. Consequences regarding the way fractal analysis of vascular networks should be conducted to provide meaningful results are presented. Finally, consequences for vascular morphogenesis or hemodynamics are discussed, as well as implications in case of pathological conditions, such as cancer. PMID:19913557
La Russa, Daniel J; Rogers, D W O
2008-12-01
In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at 60Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at 60Co energies within a accuracy of a few percent or less. PMID:19175120
On the Estimation of Pointwise Dimension
Shohei Hidaka; Neeraj Kashyap
2014-01-16
Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. The first is the insensitivity of correlation dimension itself to differences in dimensionality over data, which we term "dimension blindness". The second comes from the reliance of the method on the inference of limiting behavior from finite data. We propose pointwise dimension as an object for estimation in response to the dimension blindness of correlation dimension. Pointwise dimension is a local quantity, and the distribution of pointwise dimensions over the data contains the information to which correlation dimension is blind. We use a "limit-free" description of pointwise dimension to develop a new estimator. We conclude by discussing potential applications of our estimator as well as some challenges it raises.
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).
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.
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.
Changala, Peter Bryan
Reduced dimension variational calculations have been performed for the rovibrational level structure of the S[subscript 1] state of acetylene. The state exhibits an unusually complicated level structure, for various reasons. ...
Changala, P. Bryan
2014-01-14
The bending and torsional degrees of freedom in S{sub 1} acetylene, C{sub 2}H{sub 2}, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S{sub 1} state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.
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.
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
Collapse of loaded fractal trees
NASA Astrophysics Data System (ADS)
Turcotte, D. L.; Smalley, R. F.; Solla, Sara A.
1985-02-01
Mandelbrot1 has argued that a wide range of natural objects and phenomena are fractals; examples of fractal trees include actual trees, plants such as a cauliflower, river systems and the cardiovascular system. Here we apply the renormalization group approach2 to the collapse of fractal trees, which may be applicable to a variety of problems including cardiac arrest, failure of bronchial systems, failure of electrical distribution systems and the instability resulting in earthquakes.
Temperature induced smoothing of initially fractal grain boundaries
Streitenberger, P.; Foerster, D.; Kolbe, G.; Veit, P.
1996-01-01
Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fractal geometry has attracted great attention. It has been reported that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. In the present paper, for the first time, measurements of the change in the roughness of initially fractal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially fractal dimension.
A fractal analysis of pathogen detection by biosensors
NASA Astrophysics Data System (ADS)
Doke, Atul M.; Sadana, Ajit
2006-05-01
A fractal analysis is presented for the detection of pathogens such as Franscisela tularensis, and Yersinia pestis (the bacterium that causes plague) using a CANARY (cellular analysis and notification of antigens risks and yields) biosensor (Rider et al., 2003). In general, the binding and dissociation rate coefficients may be adequately described by either a single- or a dual-fractal analysis. An attempt is made to relate the binding rate coefficient to the degree of heterogeneity (fractal dimension value) present on the biosensor surface. Binding and dissociation rate coefficient values obtained are presented. The kinetics aspects along with the affinity values presented are of interest, and should along with the rate coefficients presented for the binding and the dissociation phase be of significant interest in help designing better biosensors for an application area that is bound to gain increasing importance in the future.
Magnetic Reconnection Rate in Space Plasmas: A Fractal Approach
Materassi, Massimo; Consolini, Giuseppe
2007-10-26
Magnetic reconnection is generally discussed via a fluid description. Here, we evaluate the reconnection rate assuming a fractal topology of the reconnection region. The central idea is that the fluid hypothesis may be violated at the scales where reconnection takes place. The reconnection rate, expressed as the Alfven Mach number of the plasma moving toward the diffusion region, is shown to depend on the fractal dimension and on the sizes of the reconnection or diffusion region. This mechanism is more efficient than prediction of the Sweet-Parker model and even Petschek's model for finite magnetic Reynolds number. A good agreement also with rates given by Hall MHD models is found. A discussion of the fractal assumption on the diffusion region in terms of current microstructures is proposed. The comparison with in-situ satellite observations suggests the reconnection region to be a filamentary domain.
Kim, Tae Hyung
2009-05-15
and natural fractures were investigated in this study using an X-Ray CT Scanner. Fractal dimension, D, and amplitude parameter, A, of fracture aperture approaches a constant value with increased sampling area, similar to the behavior of fracture roughness...
NASA Astrophysics Data System (ADS)
Muto, Jun; Nakatani, Tsurugi; Nishikawa, Osamu; Nagahama, Hiroyuki
2015-05-01
The size distributions of particle in pulverized rocks from the San Andreas fault and the Arima-Takatsuki Tectonic Line were measured. The rocks are characterized by the development of opening mode fractures with an apparent lack of shear. Fragments in the rocks in both fault zones show a fractal size distribution down to the micron scale. Fractal dimensions, dependent on mineral type, decrease from 2.92 to 1.97 with increasing distance normal to the fault core. The fractal dimensions of the rocks are higher than those of both natural and experimentally created fault gouges measured in previous studies. Moreover, the dimensions are higher than the theoretically estimated upper fractal limit under confined comminution. Dimensions close to 3.0 have been reported in impact loading experiments. The observed characteristics indicate that pulverization is likely to have occurred by a dynamic stress pulse with instantaneous volumetric expansion, possibly during seismic rupture propagation similar to impact loading.
NASA Astrophysics Data System (ADS)
Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.
2004-04-01
We study the middle cerebral artery blood flow velocity (MCAfv) in humans using transcranial Doppler ultrasonography (TCD). Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of the underlying cerebral blood flow control system. We call this effect fractal rigidity.
The Fractal Behavior of Crystal Distribution of la Gloria Pluton, Chile
NASA Astrophysics Data System (ADS)
Gutiérrez, F. J.; Payacán, I. J.; Pasten, D.; Aravena, A.; Gelman, S. E.; Bachmann, O.; Parada, M. A.
2013-12-01
We utilize fractal analysis to study the spatial distributions of crystals in a 10 Ma granitic intrusion (La Gloria pluton) located in the central Chilean Andes. Previous work determined the crystal size distribution (CSD) and anisotropy of magnetic susceptibility (AMS) tensors throughout this pluton. Using orthogonal thin sections oriented along the AMS tensor axes, we have applied fractal analysis in three magmatic crystal families: plagioclase, ferromagnesian minerals (biotite and amphibole), and Fe-Ti oxides (magnetite with minor ilmenite). We find that plagioclase and ferromagnesian minerals have a Semi-logarithmic CSD (S-CSD), given by: log(n/n0)= -L/C (1) where n [mm-4], n0 [mm-4], L [mm] and C [mm] are crystal density, intercept (nucleation density; L=0), size of crystals (three axes) and characteristic length, respectively. In contrast, Fe-Ti oxides have a Fractal CSD (F-CSD, power law size distribution), given by: log(n)= - Dn log(L) + n1 (2) where Dn and n1 [log(mm-4)] are a non-dimensional proportionality constant and the logarithm of the initial crystallization density (n1 = log(n(L=1 mm))), respectively. Finally, we calculate the fractal dimension (D0) by applying the box-counting method on each crystal thin section image, using: log(N) = -D0 log(?) (3) where N and ? are the number of boxes occupied by minerals and the length of the square box, respectively. Results indicate that D0 values (eq. 3) are well defined for all minerals, and are higher for plagioclase than for ferromagnesian minerals and lowest for Fe-Ti oxides. D0 values are correlated with n0 and -1/C for S-CSD (eq. 1), and with n1 values for F-CSD (eq. 2). These correlations between fractal dimensions with CSD parameters suggest crystal growth follows a fractal behaviour in magmatic systems. Fractal behaviour of CSD means that the spatial distribution of crystals follows an all-scale pattern as part of a self-organized magmatic system. We interpret S-CSD of plagioclase and ferromagnesian minerals as consequence of early to intermediate crystal growth, whereas F-CSD of magnetite is also a consequence of late magmatic equilibration by increasing of fine magnetite crystals (e.g. reaction of hornblende to magnetite plus actinolite, biotite and titanite). Acknowledgments. This research has been developed by the FONDECYT N°11100241 and PBCT-PDA07 projects granted by CONICYT (Chilean National Commission for Science and Technology). I.P. is supported by CONICYT magister grant N°22130729. F.G. and I.P. thank to FONDAP N°15090013 for supporting during the conference. D.P. acknowledges FONDECYT grant N° 3120237.
Fractals in physiology and medicine
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.; West, Bruce J.
1987-01-01
The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.
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.
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.
On the Fractal Structure of the Universe
P V Grujic; V D Pankovic
2009-07-13
Despite the observational evidence that the Universe appears hierarchically structured up to a distance of at least 30 Mpc/h (and possibly up to 100 Mpc/h), the fractal paradigm has not yet been recognized by the majority of cosmologists today. In this work we provide a brief overview of the recent observational and theoretical advances relevant to the question of the global cosmic structure and present some simple calculations which indicate how the hierarchical structure may pass over to the homogeneous Universe at very large scale. We show that the fractal structure may be derived from the moderately nonuniform matter distribution. We address a number of epistemological questions relevant to a general outlook of the Cosmos at large too.
Intrinsic half-metallicity in fractal carbon nitride honeycomb lattices.
Wang, Aizhu; Zhao, Mingwen
2015-09-14
Fractals are natural phenomena that exhibit a repeating pattern "exactly the same at every scale or nearly the same at different scales". Defect-free molecular fractals were assembled successfully in a recent work [Shang et al., Nature Chem., 2015, 7, 389-393]. Here, we adopted the feature of a repeating pattern in searching two-dimensional (2D) materials with intrinsic half-metallicity and high stability that are desirable for spintronics applications. Using first-principles calculations, we demonstrate that the electronic properties of fractal frameworks of carbon nitrides have stable ferromagnetism accompanied by half-metallicity, which are highly dependent on the fractal structure. The ferromagnetism increases gradually with the increase of fractal order. The Curie temperature of these metal-free systems estimated from Monte Carlo simulations is considerably higher than room temperature. The stable ferromagnetism, intrinsic half-metallicity, and fractal characteristics of spin distribution in the carbon nitride frameworks open an avenue for the design of metal-free magnetic materials with exotic properties. PMID:26105981
Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis
??lu, ?tefan; C?lug?ru, Dan Mihai; Lupa?cu, Carmen Alina
2015-01-01
AIM To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method. METHODS This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software. RESULTS It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions). CONCLUSION The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from healthy individuals. PMID:26309878
Frequency-dependent viscous flow in channels with fractal rough surfaces
Cortis, A.; Berryman, J.G.
2010-05-01
The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to {infinity} (but has a finite cutoff), and the universal scaling of the dynamic permeability, k({omega}), needs modification. We performed accurate numerical computations of k({omega}) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k({omega}) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.
Fractal geometry in an expanding, one-dimensional, Newtonian universe.
Miller, Bruce N; Rouet, Jean-Louis; Le Guirriec, Emmanuel
2007-09-01
Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the mu (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure. PMID:17930359
Scaling laws for slippage on superhydrophobic fractal surfaces
C. Cottin-Bizonne; C. Barentin; L. Bocquet
2012-01-24
We study the slippage on hierarchical fractal superhydrophobic surfaces, and find an unexpected rich behavior for hydrodynamic friction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension, and is found to be always smaller on fractal surfaces as compared to surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractal surface: depending on the underlying geometry of the roughness, strongly superhydrophobic surfaces may in some cases be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractal surfaces, in order to estimate the Kapitsa surface resistance on patterned surfaces, as well as to the question of trapping of diffusing particles by patchy hierarchical surfaces, in the context of chemoreception.
Fractal approach to the description of the auroral region
Chernyshov, A. A. Mogilevsky, M. M.; Kozelov, B. V.
2013-07-15
The plasma of the auroral region, where energetic particles precipitate from the magnetosphere into the ionosphere, is highly inhomogeneous and nonstationary. In this case, traditional methods of classical plasma physics turn out to be inapplicable. In order to correctly describe the dynamic regimes, transition processes, fluctuations, and self-similar scalings in this region, nonlinear dynamics methods based of the concepts of fractal geometry and percolation theory can be used. In this work, the fractal geometry and percolation theory are used to describe the spatial structure of the ionospheric conductivity. The topological properties, fractal dimensions, and connective indices characterizing the structure of the Pedersen and Hall conductivities on the nightside auroral zone are investigated theoretically. The restrictions imposed on the fractal estimates by the condition of ionospheric current percolation are analyzed. It is shown that the fluctuation scalings of the electric fields and auroral glow observed in the auroral zone fit well the restrictions imposed by the critical condition on the percolation of the Pedersen current. Thus, it is demonstrated that the fractal approach is a promising and convenient method for studying the properties of the ionosphere.
Fractal Heteroaggregation of Oppositely Charged Colloids.
Kim; Berg
2000-09-15
Floc structures resulting from selective heteroaggregation of positively and negatively charged colloids are investigated as a function of number ratio and shear conditions at pH 6. Negatively charged silica and positively charged alumina-coated silica undergo rapid aggregation due to attractive electrostatic interactions. At either extreme in number ratio, growth is terminated at an early stage, presumably because the aggregates acquire the same sign of charge, eliminating the driving force for further aggregation. For intermediate number ratios, extensive growth occurs, since the distribution of positive and negative charges is more uniform. Structure evolution of large heteroaggregates is assessed by static light scattering. Shear strongly influences the packing geometry and the tendency for the aggregates to undergo restructuring. At high shear (N(Re)>2000), heteroaggregates show relatively dense packing and do not restructure. Fractal dimension D(f) decreases from 2.64 to 2.26 as the number of positive particles is increased. At low shear (N(Re)<200), packing of the particles is more open and restructuring occurs. The lowest observed fractal dimension is 1.79. In the absence of applied shear, heteroaggregates with D(f)=1.79 rearrange to more compact structures with D(f)=1.88. Copyright 2000 Academic Press. PMID:10985842
Multispectral image fusion based on fractal features
NASA Astrophysics Data System (ADS)
Tian, Jie; Chen, Jie; Zhang, Chunhua
2004-01-01
Imagery sensors have been one indispensable part of the detection and recognition systems. They are widely used to the field of surveillance, navigation, control and guide, et. However, different imagery sensors depend on diverse imaging mechanisms, and work within diverse range of spectrum. They also perform diverse functions and have diverse circumstance requires. So it is unpractical to accomplish the task of detection or recognition with a single imagery sensor under the conditions of different circumstances, different backgrounds and different targets. Fortunately, the multi-sensor image fusion technique emerged as important route to solve this problem. So image fusion has been one of the main technical routines used to detect and recognize objects from images. While, loss of information is unavoidable during fusion process, so it is always a very important content of image fusion how to preserve the useful information to the utmost. That is to say, it should be taken into account before designing the fusion schemes how to avoid the loss of useful information or how to preserve the features helpful to the detection. In consideration of these issues and the fact that most detection problems are actually to distinguish man-made objects from natural background, a fractal-based multi-spectral fusion algorithm has been proposed in this paper aiming at the recognition of battlefield targets in the complicated backgrounds. According to this algorithm, source images are firstly orthogonally decomposed according to wavelet transform theories, and then fractal-based detection is held to each decomposed image. At this step, natural background and man-made targets are distinguished by use of fractal models that can well imitate natural objects. Special fusion operators are employed during the fusion of area that contains man-made targets so that useful information could be preserved and features of targets could be extruded. The final fused image is reconstructed from the composition of source pyramid images. So this fusion scheme is a multi-resolution analysis. The wavelet decomposition of image can be actually considered as special pyramid decomposition. According to wavelet decomposition theories, the approximation of image (formula available in paper) at resolution 2j+1 equal to its orthogonal projection in space , that is, where Ajf is the low-frequency approximation of image f(x, y) at resolution 2j and , , represent the vertical, horizontal and diagonal wavelet coefficients respectively at resolution 2j. These coefficients describe the high-frequency information of image at direction of vertical, horizontal and diagonal respectively. Ajf, , and are independent and can be considered as images. In this paper J is set to be 1, so the source image is decomposed to produce the son-images Af, D1f, D2f and D3f. To solve the problem of detecting artifacts, the concepts of vertical fractal dimension FD1, horizontal fractal dimension FD2 and diagonal fractal dimension FD3 are proposed in this paper. The vertical fractal dimension FD1 corresponds to the vertical wavelet coefficients image after the wavelet decomposition of source image, the horizontal fractal dimension FD2 corresponds to the horizontal wavelet coefficients and the diagonal fractal dimension FD3 the diagonal one. These definitions enrich the illustration of source images. Therefore they are helpful to classify the targets. Then the detection of artifacts in the decomposed images is a problem of pattern recognition in 4-D space. The combination of FD0, FD1, FD2 and FD3 make a vector of (FD0, FD1, FD2, FD3), which can be considered as a united feature vector of the studied image. All the parts of the images are classified in the 4-D pattern space created by the vector of (FD0, FD1, FD2, FD3) so that the area that contains man-made objects could be detected. This detection can be considered as a coarse recognition, and then the significant areas in each son-images are signed so that they can be dealt with special rules. There has been various fusion rules developed wit
Breki, Christina-Marina; Hassel, Jessica; Theoharis, Theoharis; Sachpekidis, Christos; Pan, Leyun; Provata, Astero
2016-01-01
PET/CT with F-18-Fluorodeoxyglucose (FDG) images of patients suffering from metastatic melanoma have been analysed using fractal and multifractal analysis to assess the impact of monoclonal antibody ipilimumab treatment with respect to therapy outcome. Our analysis shows that the fractal dimensions which describe the tracer dispersion in the body decrease consistently with the deterioration of the patient therapeutic outcome condition. In 20 out-of 24 cases the fractal analysis results match those of the medical records, while 7 cases are considered as special cases because the patients have non-tumour related medical conditions or side effects which affect the results. The decrease in the fractal dimensions with the deterioration of the patient conditions (in terms of disease progression) are attributed to the hierarchical localisation of the tracer which accumulates in the affected lesions and does not spread homogeneously throughout the body. Fractality emerges as a result of the migration patterns which t...
Dynamic structure factor of vibrating fractals: Proteins as a case study
NASA Astrophysics Data System (ADS)
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-01-01
We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kRg?1, where Rg is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u?i(t)-u?j(0)]2>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u?i(t)-u?i(0)]2>˜t?, where ? is the anomalous diffusion exponent that depends on the spectral dimension ds and fractal dimension df. As a result, for wave numbers obeying k2?1, S(k,t) effectively decays as a stretched exponential S(k,t)?S(k)e-(?kt)? with ???, where the relaxation rate is ?k˜(kBT/m?o2)1/?k2/?, T is the temperature, and m?o2 the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)˜k-df, with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent ? to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ?. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics.
Anisotropic fractal media by vector calculus in non-integer dimensional space
Tarasov, Vasily E.
2014-08-15
A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
Vasily E. Tarasov
2015-03-09
A review of different approaches to describe anisotropic fractal media is proposed. In this paper differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
Fractal dynamics of heartbeat time series of young persons with metabolic syndrome
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.
2012-10-01
Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.
Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems
Martin J. Körber; Matthias Michler; Arnd Bäcker; Roland Ketzmerick
2013-09-16
In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.
Fractal mechanism for characterizing singularity of mode shape for damage detection
Cao, M. S.; Ostachowicz, W.; Faculty of Automotive and Construction Machinery, Warsaw University of Technology, Narbutta 84, 02-524 Warsaw ; Bai, R. B.; Radzie?ski, M.
2013-11-25
Damage is an ordinary physical phenomenon jeopardizing structural safety; damage detection is an ongoing interdisciplinary issue. Waveform fractal theory has provided a promising resource for detecting damage in plates while presenting a concomitant problem: susceptibility to false features of damage. This study proposes a fractal dimension method based on affine transformation to address this problem. Physical experiments using laser measurement demonstrate that this method can substantially eliminate false features of damage and accurately identify complex cracks in plates, providing a fundamental mechanism that brings the merits of waveform fractal theory into full play in structural damage detection applications.
Applications of Fractal Analytical Techniques in the Estimation of Operational Scale
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Quattrochi, Dale A.
2000-01-01
The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.
Fractal Wallpaper Patterns Douglas Dunham
Dunham, Doug
Fractal Wallpaper Patterns Douglas Dunham Department of Computer Science University of Minnesota and triangles that tile the plane, which yields wallpaper patterns that are locally fractal in nature "wallpaper" patterns. Figure 1 shows a randomly created circle pattern with symmetry group p6mm. In order
NASA Astrophysics Data System (ADS)
Brothers, Harlan J.
2015-03-01
Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.
Image steganography in fractal compression
NASA Astrophysics Data System (ADS)
Chen, Mei-Ching; Agaian, Sos S.; Chen, C. L. Philip; Rodriguez, Benjamin M.
2009-05-01
This paper proposes a steganographic scheme utilizing within and/or after fractal encoding procedures on images for data security. Fractal generation exploits the concepts of iterated function systems (IFS) consisting of a collection of contractive transformations. Fractal images make use of partitioned iterated function systems (PIFS) to determine self similarity within images along with approximating the original uncompressed image. The transformed coefficients are stored in a fractal code table in order to decode the image as an alternative to storing or transmitting image pixel values directly. The proposed steganographic algorithm conceals secret information in the contrast/scaling and brightness/shifting coefficients in the code table, resulting in a stego fractal code table. Using fractal transform as a means of steganography provides a new embedding domain other than the existing steganography tools. The advantages of using fractal compression for securing information within images are: no current fractal detection methods exist, the hidden information is disseminated throughout the image in the spatial domain, the capacity of the image can be increased, and the decoding of the stego table results in a visually undistorted image.
ERIC Educational Resources Information Center
Simoson, Andrew J.
2009-01-01
This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)
Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis
Mantica, Giorgio
2015-01-01
Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical role of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non--analytic Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.
Fractal images induce fractal pupil dilations and constrictions , J. Muday a
Raynor, Sarah G.
Fractal images induce fractal pupil dilations and constrictions P. Moon a , J. Muday a , S. Raynor June 2014 Keywords: Pupillary oscillations Fractals Autonomic nervous system Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has re- vealed fractal
Simulations of fractal electronic circuits
NASA Astrophysics Data System (ADS)
Montgomery, R. D.; Fairbanks, M. S.; Brown, S. A.; Taylor, R. P.
2010-03-01
Many natural structures make use of fractal geometry's inherent properties, which can include very high surface area to volume ratios, connectivity, and dispersion. Recent research and technological applications have begun to leverage these same properties in artificial structures including antenna and capacitor designs [1]. Here we present DC electrical simulations as a first step toward circuits in which the components themselves have fractal character. Our results show that such `fractal circuits' produce complicated differential resistance curves (in response to a simple electrostatic gating scheme) that is unique to the underlying fractal geometry. Finally, we will discuss potential applications of these devices as well as candidate systems for fractal circuit fabrication. [4pt] [1] Cohen, N. L. Communications Quarterly Summer, 9 (1995).; Samavati, H., Hajimiri, A., Shahani, A. R., et al. IEEE J Sol St Circ 33 2035 - 2041 (1998).
A study on the computerized fractal analysis of architectural distortion in screening mammograms
NASA Astrophysics Data System (ADS)
Tourassi, Georgia D.; Delong, David M.; Floyd, Carey E., Jr.
2006-03-01
Architectural distortion (AD) is a sign of malignancy often missed during mammographic interpretation. The purpose of this study was to explore the application of fractal analysis to the investigation of AD in screening mammograms. The study was performed using mammograms from the Digital Database for Screening Mammography (DDSM). The fractal dimension (FD) of mammographic regions of interest (ROIs) was calculated using the circular average power spectrum technique. Initially, the variability of the FD estimates depending on ROI location, mammographic view and breast side was studied on normal mammograms. Then, the estimated FD was evaluated using receiver operating characteristics (ROC) analysis to determine if it can discriminate ROIs depicting AD from those depicting normal breast parenchyma. The effect of several factors such as ROI size, image subsampling and breast density was studied in detail. Overall, the average FD of the normal ROIs was statistically significantly higher than that of the ROIs with AD. This result was consistent across all factors studied. For the studied set of implementation parameters, the best ROC performance achieved was 0.89 ± 0.02. The generalizability of these conclusions across different digitizers was also demonstrated.
NASA Astrophysics Data System (ADS)
Zhang, Changjiang; Chen, Yuan; Lu, Juan
2014-11-01
An efficient algorithm for typhoon center location is proposed using fractal feature and gradient of infrared satellite cloud image. The centers are generally located in this region for a typhoon except the latter disappearing typhoon. The characteristics of dense cloud region are smoother texture and higher gray values than those of marginal clouds. So the window analysis method is used to select an appropriate cloud region. The window whose difference value between the sum of the gray-gradient co-occurrence matrix and fractal dimension is the biggest is chosen as the dense cloud region. The temperature gradient of the region, which is near typhoon center except typhoon eye, is small. Thus the gradient information is strengthened and is calculated by canny operator. Then we use a window to traverse the dense cloud region. If there is a closed curve, the region of curve is considered as the typhoon center region. Otherwise, the region in which there is the most texture intersection and the biggest density is considered as the typhoon center region. Finally, the geometric center of the center region is determined as the typhoon center location. The effectiveness is test by Chinese FY-2C stationary satellite cloud image. And the result is compared with the typhoon center location in the "tropical cyclone yearbook" which was compiled by Shanghai typhoon institute of China meteorological administration. Experimental results show that the high location accuracy can be obtained.
Rheology of the cytoskeleton as a fractal network
NASA Astrophysics Data System (ADS)
Patrício, P.; Leal, C. R.; Duarte, J.; Januário, C.
2015-10-01
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 one, two, or three dimensions, we are able to express the viscoelasticity of the network as an effective generalized Kelvin-Voigt model with a power law spectrum of retardation times L ˜?? . We relate the parameter ? with the fractal dimension of the gel. In some regimes (0
Multiresolution processing for fractal analysis of airborne remotely sensed data
NASA Technical Reports Server (NTRS)
Jaggi, S.; Quattrochi, D.; Lam, N.
1992-01-01
Images acquired by NASA's Calibrated Airborne Multispectral Scanner are used to compute the fractal dimension as a function of spatial resolution. Three methods are used to determine the fractal dimension: Shelberg's (1982, 1983) line-divider method, the variogram method, and the triangular prism method. A description of these methods and the result of applying these methods to a remotely-sensed image is also presented. The scanner data was acquired over western Puerto Rico in January, 1990 over land and water. The aim is to study impacts of man-induced changes on land that affect sedimentation into the near-shore environment. The data were obtained over the same area at three different pixel sizes: 10 m, 20 m, and 30 m.
Fractal characterization of rain-gauge networks and precipitations: an application in Central Italy
Capecchi, Valerio; Melani, Samantha; Morabito, Marco; Politi, Paolo
2011-01-01
The measuring stations of a geophysical network are often spatially distributed in an inhomogeneous manner. The areal inhomogeneity can be well characterized by the fractal dimension D_H of the network, which is usually smaller than the euclidean dimension of the surface, this latter equal to 2. The resulting dimensional deficit, (2-D_H), is a measure of precipitating events which cannot be detected by the network. The aim of the present study is to estimate the fractal dimension of a rain-gauge network in Tuscany (Central Italy) and to relate its dimension to the dimensions of daily rainfall events detected by a mixed satellite/radar methodology. We find that D_H = 1.85, while typical summer precipitations are characterized by a dimension much greater than the dimensional deficit 0.15.
Fractal analysis of the hierarchic structure of fossil coal surface
Alekseev, A.D.; Vasilenko, T.A.; Kirillov, A.K.
2008-05-15
The fractal analysis is described as method of studying images of surface of fossil coal, one of the natural sorbent, with the aim of determining its structural surface heterogeneity. The deformation effect as a reduction in the dimensions of heterogeneity boundaries is considered. It is shown that the theory of nonequilibrium dynamic systems permits to assess a formation level of heterogeneities involved into a sorbent composition by means of the Hurst factor.
Method of incorporating prior information on the structure of random fractal surfaces
NASA Astrophysics Data System (ADS)
Blackledge, J. M.
1990-08-01
Random fractal surfaces (Mandeibrot surfaces) are finding more and more applications in computer graphics, image analysis and the simulation of naturally occurring topologies. A random fractal as a fractional geometry whose statistical properties are scale invarient. In other words, the object looks similar (statistically) at all magnifications. The generation of a random fractal surface involves the user having to input two essential parameters: (i) the Fractal Dimension (a decimal number D where 2 < D < 3) which controls the surface roughness and (ii) the seed of a random number generator which determines the structure of the surface. By changing the seed, the user can generate different surfaces and by increasing the fractal dimension the surface roughness can be increased. In practice, algorithms of this type do not allow the user to construct a random fractal with specific topological features. Hence, in respect of the surface obtained, the user is ultimately at the mercy of a random number generator. In this paper, we address the problem of how to incorporate a priori information into a Mandelbrot surface in such a way that the end product is still fractal. A solution is provided to this problem which provides the user with control over the general topology of the surface. We demonstrate its application for incorporating low resolution data obtained from geographical/geological survey maps on the topology of a given area. Also, we show how the method can be used to generate synthetic terrain databases for the validation of certain surveying algorithms. The technique employs the Fourier Synthesis Method for generating Mandelbrot surfaces and is based on transmitting a predetermined proportion of the complex Fourier coefficients used to describe a given topology. In addition to its use as a complex terrain modeller, it is also shown how the same technique can be used for data compression of general topologies. The idea here is to describe a surface in terms of a few essential coordinate parameters (a prior information), a given seed and a specific fractal dimension.
Collective and fractal properties of pion jets in the four-velocity space at intermediate energies
V. A. Okorokov; A. K. Ponosov; F. M. Sergeev
2010-02-02
Experimental results are presented for study of collective and fractal properties of soft pion jets in the space of relative four-dimensional velocities. Significant decreasing is obtained for mean square of second particle distances from jet axis for pion-proton interactions at initial energies $\\sim 3$ GeV in comparison with hadron-nuclear collisions at close energies. The decreasing results in power dependence of distance variable on collision energy for range $\\sim 2 - 4$ GeV. The observation allows us to estimate the low boundary of manifestation of color degree of freedom in pion jet production. Cluster dimension values were deduced for pion jets in various reactions. Fractional values of this dimension indicate on the manifestation of fractal-like properties by pion jets. Changing of mean kinetic energy of jet particles and fractal dimension with initial energy increasing is consistent with suggestion for presence of color degrees of freedom in pion jet production at intermediate energies.
Fractal interfaces and product generation in the two-dimensional mixing layer
NASA Technical Reports Server (NTRS)
Jimenez, Javier; Martel, Carlos
1991-01-01
The dependence of product generation on Peclet and Reynolds numbers in a numerically simulated, reacting, two-dimensional, temporally growing mixing layer is related theoretically to the fractal dimension of the passive scalar interfaces. This reaction is verified using product generation measurements and fractal dimensions derived from the box counting technique. A transition from a low initial dimension to a higher one of approximately 5/3 is identified and shown to be associated to the kinematic distortion of the flow field during the first pairing interaction. It is suggested that the structures reponsible for this transition are nondeterministic, nonrandom, inhomogeneous fractals. In the range of Schmidt numbers investigated (0.25-4), only the large scales are involved. No further transitions, either in the spectra of the vorticity field or in the mixing behavior, are found for Reynolds numbers up to 90,000.
Asphalt flocculation and deposition: 2: Formation and growth of fractal aggregates
Rassamdana, H.; Sahimi, M.
1996-12-01
Extensive small-angle X-ray and neutron-scattering data, as well as results of precipitation measurements, are analyzed to delineate the structure of the asphalt and asphaltene aggregates that are formed when a solvent is injected into a system containing crude oil. The two types of data strongly suggest that both small and large aggregates have a fractal structure, with well-defined fractal dimensions. If the system is aged long enough at low enough temperature, large asphalt particles will have the structure of diffusion-limited cluster-cluster aggregates with a fractal dimension D{sub f} {approx_equal} 1.8, while the small ones are similar to diffusion-limited particle aggregates with a fractal dimension d{sub f} {approx_equal} 2.5. High temperatures increase the rotational motion of the particles, disturb the structure and mechanical stability of the aggregates, and decrease their fractal dimension. Aging and concentration effects of the asphalts in the solution, and the type of the solvent on the structure of the aggregates are also investigated. Implications of these results for the structure, mechanical stability, and molecular-weight distribution of asphalts and asphaltenes are detailed. A new molecular-weight distribution for asphalt aggregates predicts the experimental data excellently.
Fractal analysis of urban environment: land use and sewer system
NASA Astrophysics Data System (ADS)
Gires, A.; Ochoa Rodriguez, S.; Van Assel, J.; Bruni, G.; Murla Tulys, D.; Wang, L.; Pina, R.; Richard, J.; Ichiba, A.; Willems, P.; Tchiguirinskaia, I.; ten Veldhuis, M. C.; Schertzer, D. J. M.
2014-12-01
Land use distribution are usually obtained by automatic processing of satellite and airborne pictures. The complexity of the obtained patterns which are furthermore scale dependent is enhanced in urban environment. This scale dependency is even more visible in a rasterized representation where only a unique class is affected to each pixel. A parameter commonly analysed in urban hydrology is the coefficient of imperviousness, which reflects the proportion of rainfall that will be immediately active in the catchment response. This coefficient is strongly scale dependent with a rasterized representation. This complex behaviour is well grasped with the help of the scale invariant notion of fractal dimension which enables to quantify the space occupied by a geometrical set (here the impervious areas) not only at a single scale but across all scales. This fractal dimension is also compared to the ones computed on the representation of the catchments with the help of operational semi-distributed models. Fractal dimensions of the corresponding sewer systems are also computed and compared with values found in the literature for natural river networks. This methodology is tested on 7 pilot sites of the European NWE Interreg IV RainGain project located in France, Belgium, Netherlands, United-Kingdom and Portugal. Results are compared between all the case study which exhibit different physical features (slope, level of urbanisation, population density...).
On the fractal properties microaccelerations
A. V. Sedelnikov
2012-04-19
In this paper the fractal property of the internal environment of space laboratory microaccelerations that occur. Changing the size of the space lab leads to the fact that the dependence of microaccelerations from time to time has the property similar to the self-affinity of fractal functions. With the help of microaccelerations, based on the model of the real part of the fractal Weierstrass-Mandelbrot function is proposed to form the inertial-mass characteristics of laboratory space with a given level of microaccelerations.
Niu, Xiang; Gao, Peng; Wang, Bing; Liu, Yu
2015-01-01
Based on fractal theory, the fractal characteristics of soil particle size distribution (PSD) and soil water retention curve (WRC) under the five vegetation types were studied in the mountainous land of Northern China. Results showed that: (1) the fractal parameters of soil PSD and soil WRC varied greatly under each different vegetation type, with Quercus acutissima Carr. and Robina pseudoacacia Linn. mixed plantation (QRM) > Pinus thunbergii Parl. and Pistacia chinensis Bunge mixed plantation (PPM) > Pinus thunbergii Parl. (PTP) > Juglans rigia Linn. (JRL) > abandoned grassland (ABG); (2) the soil fractal dimensions of woodlands (QRM, PPM, PTP and JRL) were significantly higher than that in ABG, and mixed forests (QRM and PPM) were higher than that in pure forests (PTP and JRL); (3) the fractal dimension of soil was positively correlated with the silt and clay content but negatively correlated with the sand content; and (4) the fractal dimension of soil PSD was positively correlated with the soil WRC. These indicated that the fractal parameters of soil PSD and soil WRC could act as quantitative indices to reflect the physical properties of the soil, and could be used to describe the influences of the Return Farmland to Forests Projects on soil structure. PMID:26633458
Niu, Xiang; Gao, Peng; Wang, Bing; Liu, Yu
2015-01-01
Based on fractal theory, the fractal characteristics of soil particle size distribution (PSD) and soil water retention curve (WRC) under the five vegetation types were studied in the mountainous land of Northern China. Results showed that: (1) the fractal parameters of soil PSD and soil WRC varied greatly under each different vegetation type, with Quercus acutissima Carr. and Robina pseudoacacia Linn. mixed plantation (QRM) > Pinus thunbergii Parl. and Pistacia chinensis Bunge mixed plantation (PPM) > Pinus thunbergii Parl. (PTP) > Juglans rigia Linn. (JRL) > abandoned grassland (ABG); (2) the soil fractal dimensions of woodlands (QRM, PPM, PTP and JRL) were significantly higher than that in ABG, and mixed forests (QRM and PPM) were higher than that in pure forests (PTP and JRL); (3) the fractal dimension of soil was positively correlated with the silt and clay content but negatively correlated with the sand content; and (4) the fractal dimension of soil PSD was positively correlated with the soil WRC. These indicated that the fractal parameters of soil PSD and soil WRC could act as quantitative indices to reflect the physical properties of the soil, and could be used to describe the influences of the Return Farmland to Forests Projects on soil structure. PMID:26633458
NASA Astrophysics Data System (ADS)
Risovi?, Dubravko; Frka, Sanja; Kozarac, Zlatica
2011-01-01
The aim of this study was to investigate the connection between the lipid/amphiphile monolayer structure at the interface and its macroscopic/rheological properties, in particular, to establish the link between the fractality of the monolayer structure and its compressibility modulus. To that purpose we have used fractal analysis of images obtained by Brewster angle microscopy to infer the fractal dimension of the monolayer structure and relate its change to the corresponding changes in compressibility derived from a simultaneously measured ?-A isotherm. The results of the study confirmed the starting assumption based on theoretical considerations that the fractal dimension of an amphiphilic monolayer and its compressibility should be correlated. We have shown that there exists a strong correlation between the fractal dimension and the corresponding compressibility modulus of different amphiphilic materials. Thus, confirming the link between the short ordered structure on the molecular level and the macroscopic property—compressibility of the monolayer. The established correlation between the fractal dynamics and compressibility modulus of the monolayer enabled identification of onset of percolation—a second-order phase transition that is otherwise not easy and unambiguously detectable. We have found that the signature of percolation in a monolayer, regardless of its composition, is the occurrence of a sharp increase (a jump) of compressibility modulus (at macroscopic level) at the characteristic value of the corresponding fractal dimension D = 1.89. This is the result of the abrupt establishment of a connected structure on the molecular level, consequently involving a change in the elastic properties of the monolayer on a macroscopic scale. The results of this investigation provide means for unambiguous identification of the onset of percolation in the Langmuir layer and should facilitate a more efficient application of the percolation theory in further study of processes and structures at the interface during the monolayer compression.
Donatti, D.A.; Vollet, D.R.; Ibanez Ruiz, A.; Mesquita, A.; Silva, T.F.P.
2005-01-01
A sample series of silica sonogels was prepared using different water-tetraethoxysilane molar ratio (r{sub w}) in the gelation step of the process in order to obtain aerogels with different bulk densities after the supercritical drying. The samples were analyzed by means of small-angle x-ray-scattering (SAXS) and nitrogen-adsorption techniques. Wet sonogels exhibit mass fractal structure with fractal dimension D increasing from {approx}2.1 to {approx}2.4 and mass-fractal correlation length {xi} diminishing from {approx}13 nm to {approx}2 nm, as r{sub w} is changed in the nominal range from 66 to 6. The process of obtaining aerogels from sonogels and heat treatment at 500 deg. C, in general, increases the mass-fractal dimension D, diminishes the characteristic length {xi} of the fractal structure, and shortens the fractal range at the micropore side for the formation of a secondary structured particle, apparently evolved from the original wet structure at a high resolution level. The overall mass-fractal dimension D of aerogels was evaluated as {approx}2.4 and {approx}2.5, as determined from SAXS and from pore-size distribution by nitrogen adsorption, respectively. The fine structure of the 'secondary particle' developed in the obtaining of aerogels could be described as a surface-mass fractal, with the correlated surface and mass-fractal dimensions decreasing from {approx}2.4 to {approx}2.0 and from {approx}2.7 to {approx}2.5, respectively, as the aerogel bulk density increases from 0.25 (r{sub w}=66) up to 0.91 g/cm{sup 3} (r{sub w}=6)
Controlling Molecular Growth between Fractals and Crystals on Surfaces.
Zhang, Xue; Li, Na; Gu, Gao-Chen; Wang, Hao; Nieckarz, Damian; Szabelski, Pawe?; He, Yang; Wang, Yu; Xie, Chao; Shen, Zi-Yong; Lü, Jing-Tao; Tang, Hao; Peng, Lian-Mao; Hou, Shi-Min; Wu, Kai; Wang, Yong-Feng
2015-12-22
Recent studies demonstrate that simple functional molecules, which usually form two-dimensional (2D) crystal structures when adsorbed on solid substrates, are also able to self-assemble into ordered openwork fractal aggregates. To direct and control the growth of such fractal supramolecules, it is necessary to explore the conditions under which both fractal and crystalline patterns develop and coexist. In this contribution, we study the coexistence of Sierpi?ski triangle (ST) fractals and 2D molecular crystals that were formed by 4,4?-dihydroxy-1,1':3',1?-terphenyl molecules on Au(111) in ultrahigh vacuum. Growth competition between the STs and 2D crystals was realized by tuning substrate and molecular surface coverage and changing the functional groups of the molecular building block. Density functional theory calculations and Monte Carlo simulations are used to characterize the process. Both experimental and theoretical results demonstrate the possibility of steering the surface self-assembly to generate fractal and nonfractal structures made up of the same molecular building block. PMID:26502984
Fractal aircraft trajectories and nonclassical turbulent exponents.
Lovejoy, S; Schertzer, D; Tuck, A F
2004-09-01
The dimension (D) of aircraft trajectories is fundamental in interpreting airborne data. To estimate D, we studied data from 18 trajectories of stratospheric aircraft flights 1600 km long taken during a "Mach cruise" (near constant Mach number) autopilot flight mode of the ER-2 research aircraft. Mach cruise implies correlated temperature and wind fluctuations so that DeltaZ approximately Deltax (H(z) ) where Z is the (fluctuating) vertical and x the horizontal coordinate of the aircraft. Over the range approximately 3-300 km , we found H(z) approximately 0.58+/-0.02 close to the theoretical 5/9=0.56 and implying D=1+ H(z) =14/9 , i.e., the trajectories are fractal. For distances <3 km aircraft inertia smooths the trajectories, for distances >300 km , D=1 again because of a rise of 1 m/km due to fuel consumption. In the fractal regime, the horizontal velocity and temperature exponents are close to the nonclassical value 1/2 (rather than 1/3 ). We discuss implications for aircraft measurements as well as for the structure of the atmosphere. PMID:15524632
Effect of 475 C embrittlement on fractal behavior and tensile properties of a duplex stainless steel
Hilders, O.A.; Ramos, M.; Pena, N.D.; Saenz, L.
1999-02-01
The fractal dimension variations of several tension surfaces of a duplex stainless steel broken at room temperature has been studied after several aging treatments performed at 475 C for 1, 2, 6.5, 12, 24, 40, and 120 h. A dimple type of fracture mode was observed for small aging times and transgranular as well as dimple rupture for 24, 40, and 120 h of aging. The higher the time of aging is, the smaller the fractal dimension and the true fracture strain. An expected reduction of the strength with the time of aging was observed.
Chamousis, RL; Chang, LL; Watterson, WJ; Montgomery, RD; Taylor, RP; Moule, AJ; Shaheen, SE; Ilan, B; van de Lagemaat, J; Osterloh, FE
2014-08-21
Living organisms use fractal structures to optimize material and energy transport across regions of differing size scales. Here we test the effect of fractal silver electrodes on light distribution and charge collection in organic semiconducting polymer films made of P3HT and PCBM. The semiconducting polymers were deposited onto electrochemically grown fractal silver structures (5000 nm x 500 nm; fractal dimension of 1.71) with PEDOT:PSS as hole-selective interlayer. The fractal silver electrodes appear black due to increased horizontal light scattering, which is shown to improve light absorption in the polymer. According to surface photovoltage spectroscopy, fractal silver electrodes outperform the flat electrodes when the BHJ film thickness is large (>400 nm, 0.4 V photovoltage). Photocurrents of up to 200 microamperes cm(-2) are generated from the bulk heterojunction (BHJ) photoelectrodes under 435 nm LED (10-20 mW cm(-2)) illumination in acetonitrile solution containing 0.005 M ferrocenium hexafluorophosphate as the electron acceptor. The low IPCE values (0.3-0.7%) are due to slow electron transfer to ferrocenium ion and due to shunting along the large metal-polymer interface. Overall, this work provides an initial assessment of the potential of fractal electrodes for organic photovoltaic cells.
Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder
NASA Astrophysics Data System (ADS)
Ahmadlou, Mehran; Adeli, Hojjat; Adeli, Amir
2012-10-01
Recently, the visibility graph (VG) algorithm was proposed for mapping a time series to a graph to study complexity and fractality of the time series through investigation of the complexity of its graph. The visibility graph algorithm converts a fractal time series to a scale-free graph. VG has been used for the investigation of fractality in the dynamic behavior of both artificial and natural complex systems. However, robustness and performance of the power of scale-freeness of VG (PSVG) as an effective method for measuring fractality has not been investigated. Since noise is unavoidable in real life time series, the robustness of a fractality measure is of paramount importance. To improve the accuracy and robustness of PSVG to noise for measurement of fractality of time series in biological time-series, an improved PSVG is presented in this paper. The proposed method is evaluated using two examples: a synthetic benchmark time series and a complicated real life Electroencephalograms (EEG)-based diagnostic problem, that is distinguishing autistic children from non-autistic children. It is shown that the proposed improved PSVG is less sensitive to noise and therefore more robust compared with PSVG. Further, it is shown that using improved PSVG in the wavelet-chaos neural network model of Adeli and c-workers in place of the Katz fractality dimension results in a more accurate diagnosis of autism, a complicated neurological and psychiatric disorder.
Surface topography characterization of automotive cylinder liner surfaces using fractal methods
NASA Astrophysics Data System (ADS)
Lawrence K, Deepak; Ramamoorthy, B.
2013-09-01
This paper explores the use of fractal approaches for the possible characterization of automotive cylinder bore surface topography by employing methods such as differential box counting method, power spectral method and structure function method. Three stage plateau honing experiments were conducted to manufacture sixteen cylinder liner surfaces with different surface topographies, for the study. The three fractal methods are applied on the image data obtained using a computer vision system and 3-D profile data obtained using vertical scanning white light interferometer from the cylinder liner surfaces. The computed fractal parameters (fractal dimension and topothesy) are compared and correlated with the measured 3-D Abbott-Firestone curve parameters (Sk, Spk, Svk, Sr1 and Sr2) that are currently used for the surface topography characterization cylinder liner surfaces. The analyses of the results indicated that the fractal dimension (D) computed using the vision data as well as 3-D profile data by employing three different fractal methods consistantly showed a negative correlation with the functional surface topographical parameters that represents roughness at peak (Spk),core (Sk) and valley (Svk) regions and positive correlation with the upper bearing area (Sr1) and lower bearing area (Sr2) of the automotive of cylinder bore surface.
Field Fractal Cosmological Model As an Example of Practical Cosmology Approach
Yu. V. Baryshev
2015-03-11
The idea of the global gravitational effect as the source of cosmological redshift was considered by de Sitter (1916, 1917), Eddington (1923), Tolman (1929) and Bondi (1947). Also Hubble (1929) called the discovered distance-redshift relation as "De Sitter effect". For homogeneous matter distribution cosmological gravitational redshift is proportional to square of distance: z_grav ~ r^2. However for a fractal matter distribution having the fractal dimension D=2 the global gravitational redshift is the linear function of distance: z_grav ~ r, which gives possibility for interpretation of the Hubble law without the space expansion. Here the field gravity fractal cosmological model (FGF) is presented, which based on two initial principles. The first assumption is that the Feynman's field gravity approach describes the gravitational interaction, which delivers a natural basis for the conceptual unity of all fundamental physical interactions within the framework of the relativistic and quantum fields in Minkowski space. The second hypothesis is that the spatial distribution of gravitating matter is a fractal at all scales up to the Hubble radius. The fractal dimension of matter distribution is assumed to be D = 2, which implies that the global gravitational redshift is the explanation of the observed linear Hubble law. In the frame of the FGF all three phenomena - the cosmic background radiation, the fractal large scale structure, and the Hubble law, - could be the consequence of a unique large scale structure evolution process of the initially homogeneous ordinary matter without nonbaryonic matter and dark energy.
Fractal character of the distribution of surface potential irregularities in epitaxial n-GaAs (100)
Torkhov, N. A. Bozhkov, V. G.
2009-05-15
The fractal geometric properties of the relief of the surface potential of a heavily doped n{sup +}-GaAs (100) wafer are studied by Kelvin's method of atomic force microscopy. The average fractal dimensionalities determined by the triangulation method (D{sub f}), the method of horizontal cross sections (D{sub c}), and the method of similarity (D{sub s}) are rather close to each other, which is indicative of a unified nature of the fractal relief of the surface potential. In general, the fractal dimensionalities determined in the study suggest that the relative arrangement of local irregularities of the potential profile of the heavily doped n{sup +}-GaAs (100) wafer subjected to chemical and dynamic polishing is similar to the pattern corresponding to the fractal curve known as Serpinsky's napkin. It is found that the fractal irregularities of the potential vary much more gradually than it happens in the twodimensional case: the variations are proportional to linear dimensions to the power 2/D{sub c} (1 < D{sub c} < 2) rather than to the square of linear dimensions of the regions under study.
"Fractal Expression" in Chinese Calligraphy
Yuelin Li
2008-12-19
We show that from historical record and mathematic analysis, "fractal expressions" may have been a conscious pursuit at least one thousand years ago as an element of beauty in ancient Chinese calligraphy.
Anomalous diffusion in fractal globules.
Tamm, M V; Nazarov, L I; Gavrilov, A A; Chertovich, A V
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ?X(2)(t)??t(?F) with ?F close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing. PMID:25978267
NASA Astrophysics Data System (ADS)
Eliazar, Iddo; Klafter, Joseph
2008-09-01
The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results - extending them from affine to general nonlinear scaling schemes.
Song, Zilin; Zhang, Chao; Liu, Guobin; Qu, Dong; Xue, Sha
2015-01-01
The application of fractal geometry to describe soil structure is an increasingly useful tool for better understanding the performance of soil systems. Only a few studies, however, have focused on the structure of rhizospheric zones, where energy flow and nutrient recycling most frequently occur. We used fractal dimensions to investigate the characteristics of particle-size distribution (PSD) in the rhizospheres and bulk soils of six croplands abandoned for 1, 5, 10, 15, 20, and 30 years on the Loess Plateau of China and evaluated the changes over successional time. The PSDs of the rhizospheres and the fractal dimensions between rhizosphere soil and bulk soils during the natural succession differed significantly due to the influence of plant roots. The rhizospheres had higher sand (0.05–1.00 mm) contents, lower silt (<0.002 mm) contents, and lower fractal dimensions than the bulk soils during the early and intermediate successional stages (1–15 years). The fractal dimensions of the rhizosphere soil and bulk soil ranged from 2.102 to 2.441 and from 2.214 to 2.459, respectively, during the 30-year restoration. Rhizospheric clay and silt contents and fractal dimension tended to be higher and sand content tended to be lower as abandonment age increased, but the bulk soils had the opposite trend. Linear regression analysis indicated that the fractal dimensions of both the rhizospheres and bulk soils were significantly linearly correlated with clay, sand, organic-carbon, and total-nitrogen contents, with R2 ranging from 0.526 to 0.752 (P<0.001). In conclusion, PSD differed significantly between the rhizosphere soil and bulk soil. The fractal dimension was a sensitive and useful index for quantifying changes in the properties of the different soil zones. This study will greatly aid the application of the fractal method for describing soil structure and nutrient status and the understanding of the performance of rhizospheric zones during ecological restoration. PMID:26368339
Song, Zilin; Zhang, Chao; Liu, Guobin; Qu, Dong; Xue, Sha
2015-01-01
The application of fractal geometry to describe soil structure is an increasingly useful tool for better understanding the performance of soil systems. Only a few studies, however, have focused on the structure of rhizospheric zones, where energy flow and nutrient recycling most frequently occur. We used fractal dimensions to investigate the characteristics of particle-size distribution (PSD) in the rhizospheres and bulk soils of six croplands abandoned for 1, 5, 10, 15, 20, and 30 years on the Loess Plateau of China and evaluated the changes over successional time. The PSDs of the rhizospheres and the fractal dimensions between rhizosphere soil and bulk soils during the natural succession differed significantly due to the influence of plant roots. The rhizospheres had higher sand (0.05-1.00 mm) contents, lower silt (<0.002 mm) contents, and lower fractal dimensions than the bulk soils during the early and intermediate successional stages (1-15 years). The fractal dimensions of the rhizosphere soil and bulk soil ranged from 2.102 to 2.441 and from 2.214 to 2.459, respectively, during the 30-year restoration. Rhizospheric clay and silt contents and fractal dimension tended to be higher and sand content tended to be lower as abandonment age increased, but the bulk soils had the opposite trend. Linear regression analysis indicated that the fractal dimensions of both the rhizospheres and bulk soils were significantly linearly correlated with clay, sand, organic-carbon, and total-nitrogen contents, with R2 ranging from 0.526 to 0.752 (P<0.001). In conclusion, PSD differed significantly between the rhizosphere soil and bulk soil. The fractal dimension was a sensitive and useful index for quantifying changes in the properties of the different soil zones. This study will greatly aid the application of the fractal method for describing soil structure and nutrient status and the understanding of the performance of rhizospheric zones during ecological restoration. PMID:26368339
On the scattering properties of fractal tholin particle
NASA Astrophysics Data System (ADS)
Skorov, Y. V.; Keller, H. U.; Tomasko, M.; Doose, L.; DISR Team
2005-08-01
As is well-known, the measurements carried out by the pioneer and Voyager spacecrafts showed simultaneous high degree polarization in light scattered from Titan near 90o phase angle and essential forward scattering. The former observation could only be matched by spherical particles having radii less about 0.1 ? m. The latter observation implies presence of particles with an effective radius of about 0.35? m. This inconsistency was successfully remedied in a series of theoretical models of the Titan haze published since the early nineties. These models included the use of fractal aggregate haze particles consisting of small monomers of the required (prescribed) radius. Up to the Huygens mission there had not been enough observational data for a more developed haze model. The new DISR observations give a measurement of the monomer radius of 0.05 microns, in good agreement with previous estimates. The number of monomeres to fit the observed radiation field, however, is considerably higher (ca. 250) than microphysics models produce. The analysis uses essentially only the monomere radius and the number of monomeres. The scattering parameters of fractal particles depend on a multitude of additional physical characteristics such as the fractal dimension, fractal prefactor, different refractive indices (depending on the type of tholin assumed). Results of a systematic investigation of the resulting scattering properties based on the T-matrix method will be presented.
Skin Depth of Electromagnetic Wave through Fractal Crustal Rocks
NASA Astrophysics Data System (ADS)
Takahara, Kazutaka; Muto, Jun; Nagahama, Hiroyuki
Skin depth of electromagnetic (EM) wave depends on frequency of EM wave ? and electrical properties of rocks and minerals. Previous studies have theoretically assumed that the skin depth L?(?) can be expressed as a function of frequency ? by L?(?) ? ? -? and ? = 1 at high frequency or ? = 1/2 at low frequency. Based on fractal theory of rocks, we point out that the frequency exponent ? reflects internal fractal structures (i.e., occupancy, distribution and connectivity) of dielectric/conductive matrices of rocks such as pores, cracks, grain boundaries, inclusions and various fluids. Laboratory measurements of dielectric constant and conductivity of granite and previous studies on various rocks as a function of frequency show that ? is an exponent ranging from 1/4 to 1. By extrapolation of the skin depth by laboratory measurements at a given frequency into at other frequencies, the skin depth with variation in ? becomes longer or shorter than that by previous studies. Moreover, at a given frequency, the skin depth decreases with increasing a fractal dimension of fracture systems (decreasing ?). Thus, the skin depth of EM wave through the crust for detecting seismo-EM radiations and through rock salt domes for detecting ultra-high energy neutrinos depends on fractal structures of dielectric/conductive matrices in heterogeneous crust.
NASA Technical Reports Server (NTRS)
Jaggi, S.; Quattrochi, Dale A.; Lam, Nina S.-N.
1993-01-01
Fractal geometry is increasingly becoming a useful tool for modeling natural phenomena. As an alternative to Euclidean concepts, fractals allow for a more accurate representation of the nature of complexity in natural boundaries and surfaces. The purpose of this paper is to introduce and implement three algorithms in C code for deriving fractal measurement from remotely sensed data. These three methods are: the line-divider method, the variogram method, and the triangular prism method. Remote-sensing data acquired by NASA's Calibrated Airborne Multispectral Scanner (CAMS) are used to compute the fractal dimension using each of the three methods. These data were obtained as a 30 m pixel spatial resolution over a portion of western Puerto Rico in January 1990. A description of the three methods, their implementation in PC-compatible environment, and some results of applying these algorithms to remotely sensed image data are presented.
Entanglement and area law with a fractal boundary in a topologically ordered phase
Hamma, Alioscia; Lidar, Daniel A.; Severini, Simone
2010-01-15
Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy S associated to a bipartition scales with the boundary p between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions (A,B) of the system and this yields an upper bound for the entire topological phase. When the A-B boundary is regular we have S/p=1 for large p. When the boundary is a fractal of the Hausdorff dimension D, we show that the entanglement between the two parts scales as S/p=gamma<=1/D, and gamma depends on the fractal considered.
Fractal Holography: a geometric re-interpretation of cosmological large scale structure
J. R. Mureika
2007-05-17
The fractal dimension of large-scale galaxy clustering has been demonstrated to be roughly $D_F \\sim 2$ from a wide range of redshift surveys. If correct, this statistic is of interest for two main reasons: fractal scaling is an implicit representation of information content, and also the value itself is a geometric signature of area. It is proposed that the fractal distribution of galaxies may thus be interpreted as a signature of holography (``fractal holography''), providing more support for current theories of holographic cosmologies. Implications for entropy bounds are addressed. In particular, because of spatial scale invariance in the matter distribution, it is shown that violations of the spherical entropy bound can be removed. This holographic condition instead becomes a rigid constraint on the nature of the matter density and distribution in the Universe. Inclusion of a dark matter distribution is also discussed, based on theoretical considerations of possible universal CDM density profiles.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Quattrochi, Dale A.; Luvall, Jeffrey C.
1997-01-01
Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely scale-independent. Self-similarity is a property of curves or surfaces where each part is indistinguishable from the whole. The fractal dimension D of remote sensing data yields quantitative insight on the spatial complexity and information content contained within these data. Analyses 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(l0 to 80 meters). The forested scene behaves as one would expect-larger pixel sizes decrease the complexity of the image as individual clumps of trees are averaged into larger blocks. The increased complexity of the agricultural image with increasing pixel size results from the loss of homogeneous groups of pixels in the large fields to mixed pixels composed of varying combinations of NDVI values that correspond to roads and vegetation. The same process occur's in the urban image to some extent, but the lack of large, homogeneous areas in the high resolution NDVI image means the initially high D value is maintained as pixel size increases. The slope of the fractal dimension-resolution relationship provides indications of how image classification or feature identification will be affected by changes in sensor resolution.
Monte Carlo Sampling in Fractal Landscapes
Jorge C. Leitão; João M. Viana Parente Lopes; Eduardo G. Altmann
2013-05-30
We propose a flat-histogram Monte Carlo method to efficiently sample fractal landscapes such as escape time functions of open chaotic systems. This is achieved by using a random-walk step which depends on the height of the landscape via the largest Lyapunov exponent of the associated chaotic system. By generalizing the Wang-Landau algorithm, we obtain a method which simultaneously constructs the density of states (escape time distribution) and the correct step-length distribution. As a result, averages are obtained in polynomial computational time, a dramatic improvement over the exponential scaling of traditional uniform sampling. Our results are not limited by the dimensionality of the phase space and are confirmed numerically for dimensions as large as 30.
Fractal structure enables temporal prediction in music.
Rankin, Summer K; Fink, Philip W; Large, Edward W
2014-10-01
1/f serial correlations and statistical self-similarity (fractal structure) have been measured in various dimensions of musical compositions. Musical performances also display 1/f properties in expressive tempo fluctuations, and listeners predict tempo changes when synchronizing. Here the authors show that the 1/f structure is sufficient for listeners to predict the onset times of upcoming musical events. These results reveal what information listeners use to anticipate events in complex, non-isochronous acoustic rhythms, and this will entail innovative models of temporal synchronization. This finding could improve therapies for Parkinson's and related disorders and inform deeper understanding of how endogenous neural rhythms anticipate events in complex, temporally structured communication signals. PMID:25324107
Fractal analysis for classification of breast carcinoma in optical coherence tomography
NASA Astrophysics Data System (ADS)
Sullivan, Amanda C.; Hunt, John P.; Oldenburg, Amy L.
2011-06-01
The accurate and rapid assessment of tumor margins during breast cancer resection using optical coherence tomography (OCT) has the potential to reduce patient risk. However, it is difficult to subjectively distinguish cancer from normal fibroglandular stromal tissues in OCT images, and an objective measure is needed. In this initial study, we investigate the potential of a one-dimensional fractal box-counting method for cancer classification in OCT. We computed the fractal dimension, a measure of the self-similarity of an object, along the depth axis of 44 ultrahigh-resolution OCT images of human breast tissues obtained from 4 cancer patients. Correlative histology was employed to identify distinct regions of adipose, stroma, and cancer in the OCT images. We report that the fractal dimension of stroma is significantly higher than that of cancer (P < 10-5, t-test). Furthermore, by adjusting the cutoff values of fractal dimension between cancer, stroma, and adipose tissues, sensitivities and specificities of either 82.4% and 88.9%, or 88.2% and 81.5%, are obtained, respectively, for cancer classification. The use of fractal analysis with OCT could potentially provide automated identification of tumor margins during breast-sparing surgery.
Fractal Structure of Loop Quantum Gravity
Leonardo Modesto
2008-12-11
In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at hight energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity.
Generation of fractals from incursive automata, digital diffusion and wave equation systems.
Dubois, D M
1997-01-01
This paper describes modelling tools for formal systems design in the fields of information and physical systems. The concept and method of incursion and hyperincursion are first applied to the fractal machine, an hyperincursive cellular automata with sequential computations with exclusive or where time plays a central role. Simulations show the generation of fractal patterns. The computation is incursive, for inclusive recursion, in the sense that an automaton is computed at future time t + 1 as a function of its neighbouring automata at the present and/or past time steps but also at future time t + 1. The hyperincursion is an incursion when several values can be generated for each time step. External incursive inputs cannot be transformed to recursion. This is really a practical example of the final cause of Aristotle. Internal incursive inputs defined at the future time can be transformed to recursive inputs by self-reference defining then a self-referential system. A particular case of self-reference with the fractal machine shows a non deterministic hyperincursive field. The concepts of incursion and hyperincursion can be related to the theory of hypersets where a set includes itself. Secondly, the incursion is applied to generate fractals with different scaling symmetries. This is used to generate the same fractal at different scales like the box counting method for computing a fractal dimension. The simulation of fractals with an initial condition given by pictures is shown to be a process similar to a hologram. Interference of the pictures with some symmetry gives rise to complex patterns. This method is also used to generate fractal interlacing. Thirdly, it is shown that fractals can also be generated from digital diffusion and wave equations, that is to say from the modulo N of their finite difference equations with integer coefficients. PMID:9231908
A Fractal Model for the Capacitance of Lunar Dust and Lunar Dust Aggregates
NASA Technical Reports Server (NTRS)
Collier, Michael R.; Stubbs, Timothy J.; Keller, John W.; Farrell, William M.; Marshall, John; Richard, Denis Thomas
2011-01-01
Lunar dust grains and dust aggregates exhibit clumping, with an uneven mass distribution, as well as features that span many spatial scales. It has been observed that these aggregates display an almost fractal repetition of geometry with scale. Furthermore, lunar dust grains typically have sharp protrusions and jagged features that result from the lack of aeolian weathering (as opposed to space weathering) on the Moon. A perfectly spherical geometry, frequently used as a model for lunar dust grains, has none of these characteristics (although a sphere may be a reasonable proxy for the very smallest grains and some glasses). We present a fractal model for a lunar dust grain or aggregate of grains that reproduces (1) the irregular clumpy nature of lunar dust, (2) the presence of sharp points, and (3) dust features that span multiple scale lengths. We calculate the capacitance of the fractal lunar dust analytically assuming fixed dust mass (i.e. volume) for an arbitrary number of fractal levels and compare the capacitance to that of a non-fractal object with the same volume, surface area, and characteristic width. The fractal capacitance is larger than that of the equivalent non-fractal object suggesting that for a given potential, electrostatic forces on lunar dust grains and aggregates are greater than one might infer from assuming dust grains are sphericaL Consequently, electrostatic transport of lunar dust grains, for example lofting, appears more plausible than might be inferred by calculations based on less realistic assumptions about dust shape and associated capacitance.
Jones, A P
2015-01-01
Context. A porous and/or fractal description can generally be applied where particles have undergone coagulation into aggregates. Aims. To characterise finite-sized, porous and fractal particles and to understand the possible limitations of these descriptions. Methods. We use simple structure, lattice and network considerations to determine the structural properties of irregular particles. Results. We find that, for finite-sized aggregates, the terms porosity and fractal dimension may be of limited usefulness and show with some critical and limiting assumptions, that highly-porous aggregates (porosity > 80%) may not be constructable. We also investigate their effective cross-sections using a simple cubic model. Conclusions. In place of the terms porosity and fractal dimension, for finite-sized aggregates, we propose the readily-determinable quantities of inflation, I (a measure of the solid filling factor and size), and dimensionality, D (a measure of the shape). These terms can be applied to characterise any...
NASA Astrophysics Data System (ADS)
Wang, Shifang; Wu, Tao; Qi, Hongyan; Zheng, Qiusha; Zheng, Qian
2015-11-01
The fractal theory and technology has been applied to determine the flow rate, the average flow velocity, and the effective permeability for the power-law fluid in porous media composed of a number of tortuous capillaries/pores with arbitrary shapes, incorporating the tortuosity characteristic of flow paths. The fractal permeability and average flow velocity expressions are found to be a function of geometrical shape factors of capillaries, material constants, the fractal dimensions, microstructural parameters. The effects of the porosity, the tortuosity fractal dimension, material constants, and geometrical shape factors on the effective permeability are also analyzed in detail. To verify the validity of the present model, our proposed model is compared with the available macroscopic model and experimental data and there is good agreement between them.
Evaluation of Dewatering Performance and Fractal Characteristics of Alum Sludge
Sun, Yongjun; Fan, Wei; Zheng, Huaili; Zhang, Yuxin; Li, Fengting; Chen, Wei
2015-01-01
The dewatering performance and fractal characteristics of alum sludge from a drinking-water treatment plant were investigated in this study. Variations in residual turbidity of supernatant, dry solid content (DS), specific resistance to filtration (SRF), floc size, fractal dimension, and zeta potential were analyzed. Sludge dewatering efficiency was evaluated by measuring both DS and SRF. Results showed that the optimum sludge dewatering efficiency was achieved at 16 mg?L-1 flocculant dosage and pH 7. Under these conditions, the maximum DS was 54.6%, and the minimum SRF was 0.61 × 1010 m?kg-1. Floc-size measurements demonstrated that high flocculant dosage significantly improved floc size. Correlation analysis further revealed a strong correlation between fractal dimension and floc size after flocculation. A strong correlation also existed between floc size and zeta potential, and flocculants with a higher cationic degree had a larger correlation coefficient between floc size and zeta potential. In the flocculation process, the main flocculation mechanisms involved adsorption bridging under an acidic condition, and a combination between charge neutralization and adsorption-bridging interaction under neutral and alkaline conditions. PMID:26121132
Fractal Nature of Regional Myocardial Blood Flow Heterogeneity
Bassingthwaighte, James B.; King, Richard B.; Roger, Stephen A.
2010-01-01
Spatial variation in regional flows within the heart, skeletal muscle, and in other organs, and temporal variations in local arteriolar velocities and flows is measurable even with low resolution techniques. A problem in the assessment of the importance of such variations has been that the observed variance increases with increasing spatial or temporal resolution in the measurements. This resolution-dependent variance is now shown to be described by the fractal dimension, D. For example, the relative dispersion (RD=SD/mean) of the spatial distribution of flows for a given spatial resolution, is given by: RD(m)=RD(mref)·(mmref)1?Dg where m is the mass of the pieces of tissue in grams, and the reference level of dispersion, RD(mref), is taken arbitrarily to be the RD found using pieces of mass mref, which is chosen to be 1 g. Thus, the variation in regional flow within an organ can be described with two parameters, RD(mref) and the slope of the logarithmic relationship defined by the spatial fractal dimension Ds. In the heart, this relation has been found to hold over a wide range of piece sizes, the fractal Ds being about 1.2 and the correlation coefficient 0.99. A Ds of 1.2 suggests moderately strong correlation between local flows; a Ds=1.0 indicates uniform flow and a Ds= 1.5 indicates complete randomness. PMID:2766485
A physically based connection between fractional calculus and fractal geometry
Butera, Salvatore; Di Paola, Mario
2014-11-15
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
Evaluation of Dewatering Performance and Fractal Characteristics of Alum Sludge.
Sun, Yongjun; Fan, Wei; Zheng, Huaili; Zhang, Yuxin; Li, Fengting; Chen, Wei
2015-01-01
The dewatering performance and fractal characteristics of alum sludge from a drinking-water treatment plant were investigated in this study. Variations in residual turbidity of supernatant, dry solid content (DS), specific resistance to filtration (SRF), floc size, fractal dimension, and zeta potential were analyzed. Sludge dewatering efficiency was evaluated by measuring both DS and SRF. Results showed that the optimum sludge dewatering efficiency was achieved at 16 mg?L(-1) flocculant dosage and pH 7. Under these conditions, the maximum DS was 54.6%, and the minimum SRF was 0.61 × 10(10) m?kg(-1). Floc-size measurements demonstrated that high flocculant dosage significantly improved floc size. Correlation analysis further revealed a strong correlation between fractal dimension and floc size after flocculation. A strong correlation also existed between floc size and zeta potential, and flocculants with a higher cationic degree had a larger correlation coefficient between floc size and zeta potential. In the flocculation process, the main flocculation mechanisms involved adsorption bridging under an acidic condition, and a combination between charge neutralization and adsorption-bridging interaction under neutral and alkaline conditions. PMID:26121132
A physically based connection between fractional calculus and fractal geometry
NASA Astrophysics Data System (ADS)
Butera, Salvatore; Di Paola, Mario
2014-11-01
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
Fractal analysis of radiologists' visual scanning pattern in screening mammography
NASA Astrophysics Data System (ADS)
Alamudun, Folami T.; Yoon, Hong-Jun; Hudson, Kathy; Morin-Ducote, Garnetta; Tourassi, Georgia
2015-03-01
Several researchers have investigated radiologists' visual scanning patterns with respect to features such as total time examining a case, time to initially hit true lesions, number of hits, etc. The purpose of this study was to examine the complexity of the radiologists' visual scanning pattern when viewing 4-view mammographic cases, as they typically do in clinical practice. Gaze data were collected from 10 readers (3 breast imaging experts and 7 radiology residents) while reviewing 100 screening mammograms (24 normal, 26 benign, 50 malignant). The radiologists' scanpaths across the 4 mammographic views were mapped to a single 2-D image plane. Then, fractal analysis was applied on the composite 4- view scanpaths. For each case, the complexity of each radiologist's scanpath was measured using fractal dimension estimated with the box counting method. The association between the fractal dimension of the radiologists' visual scanpath, case pathology, case density, and radiologist experience was evaluated using fixed effects ANOVA. ANOVA showed that the complexity of the radiologists' visual search pattern in screening mammography is dependent on case specific attributes (breast parenchyma density and case pathology) as well as on reader attributes, namely experience level. Visual scanning patterns are significantly different for benign and malignant cases than for normal cases. There is also substantial inter-observer variability which cannot be explained only by experience level.
Cael, B. B.
Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal ...
The topological insulator in a fractal space
Song, Zhi-Gang; Zhang, Yan-Yang; Li, Shu-Shen
2014-06-09
We investigate the band structures and transport properties of a two-dimensional model of topological insulator, with a fractal edge or a fractal bulk. A fractal edge does not affect the robust transport even when the fractal pattern has reached the resolution of the atomic-scale, because the bulk is still well insulating against backscattering. On the other hand, a fractal bulk can support the robust transport only when the fractal resolution is much larger than a critical size. Smaller resolution of bulk fractal pattern will lead to remarkable backscattering and localization, due to strong couplings of opposite edge states on narrow sub-edges which appear almost everywhere in the fractal bulk.
Fractal distributions of stress and strength and variations of b-value
NASA Technical Reports Server (NTRS)
Huang, J.; Turcotte, D. L.
1988-01-01
A two-dimensional planar fault zone on which the difference between stress and strength follows a fractal distribution is simulated to study the variation of the frequency-magnitude b-value under different distributions of heterogeneities and ambient stress levels. It is suggested that earthquakes occur in regions where this difference exceeds a specified value and that the size of these regions is a measure of the magnitude of the associated earthquake. A systematic variation in b-value is observed. It is found that the b-value has a positive correlation with the fractal dimension of the distribution and is inversely related to the ambient stress level. Observational data are compared with the simulation, showing that the observed b-value variation before and during earthquake sequences is a result of changes in both the ambient stress level and in the fractal dimension of the stress-strength distribution.
Automatic Method to Classify Images Based on Multiscale Fractal Descriptors and Paraconsistent Logic
NASA Astrophysics Data System (ADS)
Pavarino, E.; Neves, L. A.; Nascimento, M. Z.; Godoy, M. F.; Arruda, P. F.; Neto, D. S.
2015-01-01
In this study is presented an automatic method to classify images from fractal descriptors as decision rules, such as multiscale fractal dimension and lacunarity. The proposed methodology was divided in three steps: quantification of the regions of interest with fractal dimension and lacunarity, techniques under a multiscale approach; definition of reference patterns, which are the limits of each studied group; and, classification of each group, considering the combination of the reference patterns with signals maximization (an approach commonly considered in paraconsistent logic). The proposed method was used to classify histological prostatic images, aiming the diagnostic of prostate cancer. The accuracy levels were important, overcoming those obtained with Support Vector Machine (SVM) and Best- first Decicion Tree (BFTree) classifiers. The proposed approach allows recognize and classify patterns, offering the advantage of giving comprehensive results to the specialists.
A CSRR-Fed SIW Cavity-Backed Fractal Patch Antenna for Wireless Energy Harvesting and Communication
Cao, Hailin; Jiang, Fen; Liu, Jiujiu; Cai, Wenbin; Tang, Mingchun; Tan, Xiaoheng; Yang, Shizhong
2015-01-01
This paper presents a novel compact dual-band and dual-polarized complementary split-ring resonator (CSRR)-fed substrate-integrated waveguide (SIW) cavity-backed fractal patch antenna for wireless energy harvesting and communication. The proposed antenna is composed of a Giuseppe Peano fractal radiation patch with a backed SIW cavity. To enhance the bandwidth and minimize the dimensions, the CSRR structure is designed to feed the Giuseppe Peano fractal patch orthogonally. A prototype of the proposed antenna is simulated, fabricated and measured. The proposed antenna exhibits good directionality and high cross-polarization level with especially compact size. PMID:26343663
A CSRR-Fed SIW Cavity-Backed Fractal Patch Antenna for Wireless Energy Harvesting and Communication.
Cao, Hailin; Jiang, Fen; Liu, Jiujiu; Cai, Wenbin; Tang, Mingchun; Tan, Xiaoheng; Yang, Shizhong
2015-01-01
This paper presents a novel compact dual-band and dual-polarized complementary split-ring resonator (CSRR)-fed substrate-integrated waveguide (SIW) cavity-backed fractal patch antenna for wireless energy harvesting and communication. The proposed antenna is composed of a Giuseppe Peano fractal radiation patch with a backed SIW cavity. To enhance the bandwidth and minimize the dimensions, the CSRR structure is designed to feed the Giuseppe Peano fractal patch orthogonally. A prototype of the proposed antenna is simulated, fabricated and measured. The proposed antenna exhibits good directionality and high cross-polarization level with especially compact size. PMID:26343663
Fractal kinetics in drug release from finite fractal matrices
NASA Astrophysics Data System (ADS)
Kosmidis, Kosmas; Argyrakis, Panos; Macheras, Panos
2003-09-01
We have re-examined the random release of particles from fractal polymer matrices using Monte Carlo simulations, a problem originally studied by Bunde et al. [J. Chem. Phys. 83, 5909 (1985)]. A certain population of particles diffuses on a fractal structure, and as particles reach the boundaries of the structure they are removed from the system. We find that the number of particles that escape from the matrix as a function of time can be approximated by a Weibull (stretched exponential) function, similar to the case of release from Euclidean matrices. The earlier result that fractal release rates are described by power laws is correct only at the initial stage of the release, but it has to be modified if one is to describe in one picture the entire process for a finite system. These results pertain to the release of drugs, chemicals, agrochemicals, etc., from delivery systems.
Fractal Structures and Correlations in Hadronic Multiparticle Distributions
NASA Astrophysics Data System (ADS)
Carruthers, P.
Multihadron rapidity distributions exhibit highly irregular (perhaps fractal) event structure. Application of methods used in nonlinear dynamics to hadronic data is made uncertain by the relatively small number of particles in a given event. We analyze three standard methods to assess their applicability: the Hausdorff (box-counting) method, the correlation dimension, and the information dimension. The Hausdorff method can work for large multiplicities if the fractal set (strange attractor) has simple self-similar behavior. It is noted that addition of points from different events will doubtless erase sponge-like structure characteristic of the attractor. This defect is even greater for the information dimension, whose proper definition requires probabilities whose evaluation involves event averaging. For a fixed attractor, this is of no consequence; however, individual collisions differ by a "noise" effect. The correlation dimension, which has the merit of rapid computational convergence, depends only on relative rapidities |yi - yj| and therefore should not depend on overall rapidity shifts between events. Possible statistical independence of different subsets of a given partition of n particles is analyzed using factorial cumulant moments and information entropy. Additivity of constituent cumulants and entropies is characteristic of statistical independence. Conditional entropies are introduced and used to generalize conventional definitions of information entropy and its (Renyi) generalization.
Branched Polymers on the Given-Mandelbrot family of fractals
Deepak Dhar
2005-04-10
We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $ log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $ \\lambda^n exp(b n ^{\\psi})$, where $\\lambda$ and $b$ are some constants, and $ 0 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\\bf B 35} (1987) 4988].
International trade network: fractal properties and globalization puzzle
Karpiarz, Mariusz; Fronczak, Agata
2014-01-01
Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue, that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade.
International trade network: fractal properties and globalization puzzle.
Karpiarz, Mariusz; Fronczak, Piotr; Fronczak, Agata
2014-12-12
Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, the box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade. PMID:25541810
International Trade Network: Fractal Properties and Globalization Puzzle
NASA Astrophysics Data System (ADS)
Karpiarz, Mariusz; Fronczak, Piotr; Fronczak, Agata
2014-12-01
Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, the box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade.
A fractal version of the pinwheel tiling
Natalie Priebe Frank; Michael F. Whittaker
2011-01-25
We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually locally derivable from the pinwheel tiling space. Interesting rotational properties, symmetries, and relative tile frequency are discussed for the tiling space associated with the fractal pinwheel tiling.
Fractal structures in nonlinear dynamics Jacobo Aguirre*
Rey Juan Carlos, Universidad
Fractal structures in nonlinear dynamics Jacobo Aguirre* Departamento de Física, Universidad Rey March 2009 In addition to the striking beauty inherent in their complex nature, fractals have become, fractals have been detected in nature and in most fields of science, with even a certain influence
Fractal-Based Spatial Analysis of Radiotelemetry
Kenkel, Norm
CHAPTER 6 Fractal-Based Spatial Analysis of Radiotelemetry Data CHRISTIAN A. HAGEN, NORM C. KENKEL of Spatial Contagion? What Factors Might Contribute to the Degree of Contagion Observed? Fractal Analysis of Spatial Pattern Box Counting Generalized Entropy Dilution Effect and Monte Carlo Test Modeling Fractal
FRACTAL APPROXIMATION AND COMPRESSION USING PROJECTED IFS
GuÃ©rin, Eric
FRACTAL APPROXIMATION AND COMPRESSION USING PROJECTED IFS Ã?ric GuÃ©rin, Ã?ric Tosan and Atilla, or images) with fractal models is an important center of interest for research. The general inverse problem.The most known of them is the fractal image compression method introduced by Jacquin. Generally speaking
FRACTAL TILING MICHAEL BARNSLEY AND ANDREW VINCE
Vince, Andrew
FRACTAL TILING MICHAEL BARNSLEY AND ANDREW VINCE Abstract. A simple, yet unifying method be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transformation on the entire space upon which the IFS acts. 1
Random Fractal Measures via the Contraction Method
RÃ¼schendorf, Ludger
Random Fractal Measures via the Contraction Method John E. Hutchinson Australian National mapping method to prove various existence and uniqueness properties of (selfÂsimilar) random fractal in order to establish a.s. exponential convergence to the unique random fractal measure. The arguments used
Foucher, F.; Mounaim-Rousselle, C.
2005-11-01
The burning rate in the vicinity of a piston is estimated from a fractal analysis. The fractal parameters are determined from laser sheet tomography flame images for methane-air mixtures with three equivalence ratios (1, 0.9, 0.8) in a transparent spark-ignition engine. Two imaging configurations were used: five horizontal planes placed at different distances from the piston (0, 1, 2, 3, and 5 mm) and a vertical one passed through the center of the combustion chamber. The methodology proposed by Foucher et al. [F. Foucher, S. Burnel, C. Mounaim-Rousselle, Proc. Combust. Inst. 29 (2002) 751-757] allows the effect of cyclic variations to be avoided. The fractal formulation is modified to take into account the flame-piston distance and flame quenching. Far from the piston, evolution of the fractal dimension versus q{sup '}/S{sub L}{sup 0} is found to be in good agreement with literature results. Near the piston, the fractal dimension evolves significantly when the distance is about twice the integral length scale and tends toward 2, the fractal dimension of a laminar flame front. The quenching ratio parameter Q{sub R} is introduced to consider the quenching of the flame by the piston. Finally, the burning rate is determined as a function of the distance between the wall and the mean flame contour and compared to a flame density approach, and similar results are found.
Fractal Geometry of Iso-Surfaces of a Passive Scalar in a Turbulent Boundary Layer
.2.5 The fractal dimension in theory and practice . . . . . . . . . . 26 iv #12;2.2.6 Dimensional classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Image calibration . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Choice-surfaces . . . . . . . . . . . . . 47 3.2.4 Noise considerations . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.5 Box
The morphology of built-up landscapes in Wallonia (Belgium): a classification using fractal indices
Biernacki, Christophe
15 the history of urbanisation. Urban sprawl seems to affect most communes, even the remotest residential practices have affected the built landscape. In other words, how much urban sprawl has modified, as well as for modelling and planning urban realities.20 Keywords: fractal dimension, built-up geometry
Yu, Junbao; Lv, Xiaofei; Bin, Ma; Wu, Huifeng; Du, Siyao; Zhou, Mo; Yang, Yanming; Han, Guangxuan
2015-01-01
The characteristic of particle size distribution (PSD) in the newly formed wetlands in coast has seldom been studied. We applied fractal-scaling theory in assessing soil particle size distribution (PSD) features of newly formed wetlands in the Yellow River Delta (YRD), China. The singular fractal dimensions (D) values ranged from 1.82 to 1.90, the capacity dimension (D0) values ranged from 0.84 to 0.93, and the entropy dimension (D1) values ranged from 0.66 to 0.84. Constrained corresponding analysis revealed that 43.5% of the variance in soil PSD can be explained by environmental factors, including 14.7% by seasonal variation, 8.6% by soil depth, and 8.0% by vegetation type. The fractal dimensions D and D1 were sensitive with fine particles with size ranging less than 126 ?m, and D0 was sensitive with coarse particles with size ranging between 126 ?m to 2000 ?m. Fractal analysis makes full use of soil PSD information, and offers a useful approach to quantify and assess the soil physical attributes in the newly formed wetland. PMID:26014107