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1

Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation

NASA Astrophysics Data System (ADS)

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.

Xiang, G. S.; Xu, Y. F.; Jiang, H.

2014-09-01

2

Fractal Dimension for Fractal Structures: A Hausdorff Approach

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.

M. A. Sánchez-Granero; Manuel Fernández-Martínez

2010-07-22

3

The nature of chaos and strange attractors is reviewed, and definitions of fractal dimensions are examined. Algorithms for estimating fractal dimensions are discussed. The implementation of box-counting algorithms and of the correlation algorithm for estimating fractal dimensions is addressed.

James Theiler

1990-01-01

4

FRACTAL DIMENSION OF GALAXY ISOPHOTES

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.

Thanki, Sandip [Nevada State College, Department of Physical Sciences, 1125 Nevada State Drive, Henderson, NV 89002 (United States); Rhee, George; Lepp, Stephen [Physics and Astronomy Department, University of Nevada, Las Vegas, Box 4002, Las Vegas, NV 89154 (United States)], E-mail: Sandip.Thanki@nsc.nevada.edu, E-mail: grhee@physics.unlv.edu, E-mail: lepp@physics.unlv.edu

2009-09-15

5

Application of general fractal dimension to rolling bearing diagnosis

The multi-fractal theory is applied for proposing the calculation formula of general fractal dimension according to the characteristic of rolling bearing vibration signal. The general dimension of measured non-stationary time-domain signal is calculated and analysed, the box dimension, the information dimension and the correlation dimension are obtained. Combined general dimension sequence value and mathematics method, the fractal diagnosis classification theory

Meng Li; Xiujuan Liu; Huadong Yu; Ping Zhao

2009-01-01

6

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales. Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.

Berntson, G. M.; Stoll, P.

1997-01-01

7

Fractal Dimension in Epileptic EEG Signal Analysis

NASA Astrophysics Data System (ADS)

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.

Uthayakumar, R.

8

Texture Segmentation Using Fractal Dimension

This paper deals with the problem of recognizing and segmenting textures in images. For this purpose the authors employ a technique based on the fractal dimension (FD) and the multi-fractal concept. Six FD features are based on the original image, the above average\\/high gray level image, the below average\\/low gray level image, the horizontally smoothed image, the vertically smoothed image,

B. B. Chaudhuri; Nirupam Sarkar

1995-01-01

9

Fractal dimension and frequency response of fractal shaped antennas

We have used the fractal nature of the geometry in obtaining approximate design equations for dipole antennas with Hilbert curve geometry. We have also reported that in the case of Koch curves, the fractal dimension can be varied by changing the indentation angle, and the resonant frequencies of the resultant antenna follow a close relation with the fractal dimension. It

K. J. Vinoy; J. K. Abraham; V. K. Varadan

2003-01-01

10

Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums

The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.

Michel L. Lapidus; Goran Radunovi?; Darko Žubrini?

2014-07-30

11

Fractal dimensions of individual flocs and floc populations in streams

NASA Astrophysics Data System (ADS)

The fractal dimension of an individual floc is a measure of the complexity of its external shape. Fractal dimensions can also be used to characterize floc populations, in which case the fractal dimension indicates how the shape of the smaller flocs relates to that of the larger flocs. The objective of this study is to compare the fractal dimensions of floc populations with those of individual flocs, and to evaluate how well both indicate contributions of sediment sources and reflect the nature and extent of flocculation in streams.Suspended solids were collected prior to and during snowmelt at upstream and downstream sites in two southern Ontario streams with contrasting riparian zones. An image analysis system was used to determine area, longest axis and perimeter of flocs. The area-perimeter relationship was used to calculate the fractal dimension, D, that characterizes the floc population. For each sample, the fractal dimension, Di , of the 28 to 30 largest individual flocs was determined from the perimeter-step-length relationship. Prior to snowmelt, the mean value of Di ranged from 1·19 (Cedar Creek, downstream) to 1·22 (Strawberry Creek, upstream and downstream). A comparison of the means using t-tests indicates that most samples on this day had comparable mean values of Di . During snowmelt, there was no significant change in the mean value of Di at the Cedar Creek sites. In contrast, for Strawberry Creek the mean value of Di at both sites increased significantly, from 1·22 prior to snowmelt to 1·34 during snowmelt. This increase reflects the contribution of sediment-laden overland flow to the sediment load. At three of the sampling sites, the increase in fractal dimensions was accompanied by a decreases in effective particle size, which can be explained by an increase in bed shear stress. A comparison of fractal dimensions of individual flocs in a sample with the fractal dimensions of the floc populations indicates that both fractal dimensions provide similar information about the temporal changes in sediment source contributions, about the contrasting effectiveness of the riparian buffer zones in the two basins, and about the hydraulic conditions in the streams. Nevertheless, determining the individual fractal dimensions of a set of large flocs in a sample is very time consuming. Using fractal dimensions of floc populations is therefore the preferred method to characterize suspended matter.

de Boer, Dirk H.; Stone, Mike; Lévesque, Lucie M. J.

2000-03-01

12

Classification of surface EMG signal with fractal dimension.

Surface EMG (electromyography) signal is a complex nonlinear signal with low signal to noise ratio (SNR). This paper is aimed at identifying different patterns of surface EMG signals according to fractal dimension. Two patterns of surface EMG signals are respectively acquired from the right forearm flexor of 30 healthy volunteers during right forearm supination (FS) or forearm pronation (FP). After the high frequency noise is filtered from surface EMG signal by a low-pass filter, fractal dimension is calculated from the filtered surface EMG signal. The results showed that the fractal dimensions of filtered FS surface EMG signals and those of filtered FP surface EMG signals distribute in two different regions, so the fractal dimensions can represent different patterns of surface EMG signals. PMID:16052721

Hu, Xiao; Wang, Zhi-zhong; Ren, Xiao-mei

2005-08-01

13

Classification of surface EMG signal with fractal dimension*

Surface EMG (electromyography) signal is a complex nonlinear signal with low signal to noise ratio (SNR). This paper is aimed at identifying different patterns of surface EMG signals according to fractal dimension. Two patterns of surface EMG signals are respectively acquired from the right forearm flexor of 30 healthy volunteers during right forearm supination (FS) or forearm pronation (FP). After the high frequency noise is filtered from surface EMG signal by a low-pass filter, fractal dimension is calculated from the filtered surface EMG signal. The results showed that the fractal dimensions of filtered FS surface EMG signals and those of filtered FP surface EMG signals distribute in two different regions, so the fractal dimensions can represent different patterns of surface EMG signals. PMID:16052721

Hu, Xiao; Wang, Zhi-zhong; Ren, Xiao-mei

2005-01-01

14

The Fractal Dimension of Projected Clouds

The interstellar medium seems to have an underlying fractal structure which can be characterized through its fractal dimension. However, interstellar clouds are observed as projected two-dimensional images, and the projection of a tri-dimensional fractal distorts its measured properties. Here we use simulated fractal clouds to study the relationship between the tri-dimensional fractal dimension (D_f) of modeled clouds and the dimension resulting from their projected images. We analyze different fractal dimension estimators: the correlation and mass dimensions of the clouds, and the perimeter-based dimension of their boundaries (D_per). We find the functional forms relating D_f with the projected fractal dimensions, as well as the dependence on the image resolution, which allow to estimatethe "real" D_f value of a cloud from its projection. The application of these results to Orion A indicates in a self-consistent way that 2.5 < D_f < 2.7 for this molecular cloud, a value higher than the result D_per+1 = 2.3 some times assumed in literature for interstellar clouds.

Nestor Sanchez; Emilio J. Alfaro; Enrique Perez

2005-01-26

15

Applications of Variance Fractal Dimension: a Survey

NASA Astrophysics Data System (ADS)

Chaotic dynamical systems are pervasive in nature and can be shown to be deterministic through fractal analysis. There are numerous methods that can be used to estimate the fractal dimension. Among the usual fractal estimation methods, variance fractal dimension (VFD) is one of the most significant fractal analysis methods that can be implemented for real-time systems. The basic concept and theory of VFD are presented. Recent research and the development of several applications based on VFD are reviewed and explained in detail, such as biomedical signal processing and pattern recognition, speech communication, geophysical signal analysis, power systems and communication systems. The important parameters that need to be considered in computing the VFD are discussed, including the window size and the window increment of the feature, and the step size of the VFD. Directions for future research of VFD are also briefly outlined.

Phinyomark, Angkoon; Phukpattaranont, Pornchai; Limsakul, Chusak

2012-04-01

16

Trabecular Bone Mechanical Properties and Fractal Dimension

NASA Technical Reports Server (NTRS)

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.

Hogan, Harry A.

1996-01-01

17

Fractal Dimension of Dielectric Breakdown

It is shown that the simplest nontrivial stochastic model for dielectric breakdown naturally leads to fractal structures for the discharge pattern. Planar discharges are studied in detail and the results are compared with properly designed experiments.

L. Niemeyer; L. Pietronero; H. J. Wiesmann

1984-01-01

18

Estimating of fractal and correlation dimension from 2-D and 3-D images

NASA Astrophysics Data System (ADS)

If the recursive algorithm or the dynamical system that generate the fractal shape is known, then the fractal dimension can be calculated. Many natural shapes are fractals. However, one can usually not estimate exactly the corresponding dimension from images, because the images are formed of finite amount of values at discrete points. In here, one algorithm to estimate fractal dimension and two algorithms to estimate the correlation dimension, which is a lower bound of fractal dimension, from a 2D or 3D image are introduced. The calculated dimension could be used in order to solve many practical problems. For example, one could use the dimension image on icebreakers in order to determine the type of ice and in order to find cracks in the ice using satellite images. Also one can use dimension image in image compression and in pattern recognition.

Vepsalainen, Ari M.; Ma, Jun

1989-11-01

19

Fractal Dimension in Eeg Signals during Muscle Fatigue

NASA Astrophysics Data System (ADS)

Fractal dimension (FD) has been successfully used to characterize signals in the format of time series. In this study, we calculated FD of EEG signals recorded during human muscle fatigue as a measure of changes in the EEG signal complexity along fatigue. Subjects performed 200 intermittent handgrip contractions at 100contraction level. Each contraction lasted 2 s, followed by a 5-s rest. EEG data were recorded from the scalp along with handgrip force and muscle EMG signals. The FD computation was based on measurements of the length (Lk) of the signal at 6 different temporal resolutions (k = 1, 2, ¡, 6). FD was determined from the relationship between Lk and k using the least square fit. The results showed that: (1) EEG fractal dimension associated with the motor performance was significantly higher than that during the rest period; (2) changes in the fractal dimension along the process of fatigue showed a significant correlation with the decline in force and EMG signals.

Huang, Haibin; Yao, Bin; Yue, Guang; Brown, Robert; Jing, Liu

2003-10-01

20

A procedure to Estimate the Fractal Dimension of Waveforms

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.

Carlos Sevcik

2010-03-27

21

Fractal dimension of microbead assemblies used for protein detection.

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 ?m, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles. PMID:25195559

Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos; Kopelman, Raoul

2014-11-10

22

FRACTAL DIMENSION ESTIMATION: EMPIRICAL MODE DECOMPOSITION VERSUS WAVELETS

FRACTAL DIMENSION ESTIMATION: EMPIRICAL MODE DECOMPOSITION VERSUS WAVELETS Paulo GoncÂ¸alves INRIA, France. {firstname.lastname}@ens-lyon.fr ABSTRACT We address the problem of fractal dimension estimation motions. Index Terms-- fractal dimension, regularity exponents, wavelet transform, EMD 1. MOTIVATION

GonÃ§alves, Paulo

23

Fractal Dimensions and Entropies of Meragi Songs

NASA Astrophysics Data System (ADS)

Melodies can be treated as time series

Aydemir, Adnan; Gündüz, Güngör

24

Fractal dimension estimators for fractional Brownian motions

Five different fractal dimension estimators are chosen which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fractional Brownian motion samples at different observation scales on the other hand), or in a mixed time-scale domain (identification of a self-similarity parameter

N. Gache; Patrick Flandrin; Dominique Garreau

1991-01-01

25

The Correlation Fractal Dimension of Complex Networks

NASA Astrophysics Data System (ADS)

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.

Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei

2013-05-01

26

Fractal dimension of alumina aggregates grown in two dimensions

NASA Technical Reports Server (NTRS)

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.

Larosa, Judith L.; Cawley, James D.

1992-01-01

27

Estimation of fractal dimensions from transect data

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.

Loehle, C. [Argonne National Lab., IL (United States)

1994-04-01

28

Multi-Bands Image Analysis Using Local Fractal Dimension

An important application of fractals is image texture analysis. The main aspect of fractal geometry used in this application is the concept of fractal dimensions to characterize the texture-scaling behavior. A new idea is presented: the use of fractals for texture identification in multiband image analysis. This is not a simple extension of the usual characterization of multifractal from its

Aura Conci; Éldman De Oliveira Nunes

2001-01-01

29

Fractal dimension in dissipative chaotic scattering Jess M. Seoane,1,

Fractal dimension in dissipative chaotic scattering JesÃºs M. Seoane,1, * Miguel A. F. SanjuÃ¡n,1 on chaotic scattering is relevant to situations of physical interest. We inves- tigate how the fractal is thus the fractal dimension of the set of singularities. For nonhyperbolic scattering, it has been known

Rey Juan Carlos, Universidad

30

Fractal dimensions of the galaxy distribution varying by steps?

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.

Marie-Noelle Celerier; Reuben Thieberger

2005-04-20

31

Fractal dimension analyses of lava surfaces and flow boundaries

NASA Technical Reports Server (NTRS)

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.

Cleghorn, Timothy F.

1993-01-01

32

Fractal dimension in nonhyperbolic chaotic scattering

NASA Technical Reports Server (NTRS)

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.

Lau, Yun-Tung; Finn, John M.; Ott, Edward

1991-01-01

33

NSDL National Science Digital Library

This set of lessons guides students in the making of the Sierpinski Triangle, Jurassic Park fractal, and Koch Snowflake, both with paper and pencil, and with Java applets. Accompanying articles, written for children, explain the properties of fractals (self-similarity, fractional dimension, formation by iteration) and answer the questions "Why study fractals? What's so hot about fractals, anyway?"

Lanius, Cynthia

2000-01-01

34

An efficient approach to compute fractal dimension in texture image

Fractal dimension is a feature used to characterize roughness and self-similarity in a picture. This feature is used in texture segmentation and classification, shape analysis and other problems. An efficient differential box-counting approach to fractal dimension estimation is proposed and compared with four other methods

B. B. Chaudhuri; N. Sarkar

1992-01-01

35

Spatial-temporal variability of coastline in Bohai Rim based on fractal dimension

NASA Astrophysics Data System (ADS)

This paper extracted the spatial distribution of the continental coastline of Bohai Rim utilizing Remote Sensing and GIS spatial analysis techniques, and calculated the fractal dimension of the coastline by boxcounting method, with a time from 1990 to 2010. Moreover, we analyzed the characteristics of spatialtemporal variability of the coastline's length and fractal dimension, the relationship between the large scales length change and fractal dimension change. During the research period, the coastline length of the study area increased progressively and the most significant change in coastline length was found in Tianjin Municipality. Especially after 2000, the coastline length entered a period of rapid growth. In addition, the fractal dimension of the overall coastline of the study area was between the fractal dimensions of the regional coastlines and was close to the maximum fractal dimensions of these regional coastlines. The fractal dimension of the coastline in Bohai Rim was increasing during the research period, large scale project such as ports construction, reduced tortuous degree of the coastline.

Xu, Ning; Gao, Zhiqiang; Ning, Jicai; Liu, Xiangyang

2014-10-01

36

Heterogeneities Analysis Using the Generalized Fractal Dimension and Continuous Wavelet Transform

NASA Astrophysics Data System (ADS)

The main goal of this work is analyze heterogeneities from well-logs data using the wavelet transform modulus maxima lines (WTMM). Firstly, the continuous wavelet transform (CWT) with sliding window is calculated. The next step consists to calculate the maxima of the modulus of the CWT and estimate the spectrum of exponents. The three generalized fractal dimensions D0, D1 and D2 are then estimated. Application of the proposed idea at the synthetic and real well-logs data of a borehole located in the Algerian Sahara shows that the fractal dimensions are very sensitive to lithological variations. The generalized fractal dimensions are a very robust tool than can be used for petroleum reservoir characterization. Keywrods: reservoir, Heterogeneities, WTMM, fractal dimension.

Ouadfeul, S.; Aliouane, L.; Boudella, A.

2012-04-01

37

NASA Astrophysics Data System (ADS)

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.

Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.

2011-03-01

38

NASA Astrophysics Data System (ADS)

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.

Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.

2010-10-01

39

Fractal dimension of interstellar clouds: opacity and noise effects

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.

Nestor Sanchez; Emilio J. Alfaro; Enrique Perez

2006-10-20

40

Archaeon and archaeal virus diversity classification via sequence entropy and fractal dimension

NASA Astrophysics Data System (ADS)

Archaea are important potential candidates in astrobiology as their metabolism includes solar, inorganic and organic energy sources. Archaeal viruses would also be expected to be present in a sustainable archaeal exobiological community. Genetic sequence Shannon entropy and fractal dimension can be used to establish a two-dimensional measure for classification and phylogenetic study of these organisms. A sequence fractal dimension can be calculated from a numerical series consisting of the atomic numbers of each nucleotide. Archaeal 16S and 23S ribosomal RNA sequences were studied. Outliers in the 16S rRNA fractal dimension and entropy plot were found to be halophilic archaea. Positive correlation (R-square ~ 0.75, N = 18) was observed between fractal dimension and entropy across the studied species. The 16S ribosomal RNA sequence entropy correlates with the 23S ribosomal RNA sequence entropy across species with R-square 0.93, N = 18. Entropy values correspond positively with branch lengths of a published phylogeny. The studied archaeal virus sequences have high fractal dimensions of 2.02 or more. A comparison of selected extremophile sequences with archaeal sequences from the Humboldt Marine Ecosystem database (Wood-Hull Oceanography Institute, MIT) suggests the presence of continuous sequence expression as inferred from distributions of entropy and fractal dimension, consistent with the diversity expected in an exobiological archaeal community.

Tremberger, George, Jr.; Gallardo, Victor; Espinoza, Carola; Holden, Todd; Gadura, N.; Cheung, E.; Schneider, P.; Lieberman, D.; Cheung, T.

2010-09-01

41

Fractal dimension of cluster boundaries in porous polycrystalline HTSC materials

NASA Astrophysics Data System (ADS)

The fractal dimension of the boundaries of clusters formed by pores and granules in polycrystalline materials is shown to be determined by the sample density and crystallite sizes. The dependence of the fractal dimension on the density has a maximum. It is shown that the maximum diamagnetic response can be obtained in a porous high-temperature superconductor with a porosity of 50-60% and small crystallite sizes.

Bykov, A. A.; Terent'ev, K. Yu.; Gokhfeld, D. M.; Petrov, M. I.

2012-10-01

42

Fractal-Dimension Crossovers in Turbulent Passive Scalar Signals

NASA Astrophysics Data System (ADS)

The fractal dimension $\\delta_g^{(1)}$ of turbulent passive scalar signals is calculated from the fluid dynamical equation. $\\delta_g^{(1)}$ depends on the scale. For small Prandtl (or Schmidt) number $Pr<10^{-2}$ one gets two ranges, $\\delta_g^{(1)}=1$ for small scale r and $\\delta_g^{(1)}$=5/3 for large r, both as expected. But for large $Pr> 1$ one gets a third, intermediate range in which the signal is extremely wrinkled and has $\\delta_g^{(1)}=2$. In that range the passive scalar structure function $D_\\theta(r)$ has a plateau. We calculate the $Pr$-dependence of the crossovers. Comparison with a numerical reduced wave vector set calculation gives good agreement with our predictions.

Grossmann, S.; Lohse, D.

1994-08-01

43

The Casimir effect for parallel plates in the spacetime with a fractal extra compactified dimension

The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like $\\delta>1/2$ to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction.

Hongbo Cheng

2011-06-23

44

Sub-optimal MCV Cover Based Method for Measuring Fractal Dimension

A new method for calculating fractal dimension is developed in this paper. The method is based on the box dimension concept; however, it involves direct estimation of a suboptimal covering of the data set of interest. By finding a suboptimal cover, this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm. Moreover, any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation. In general, our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box. Instead, the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set. This method involves a variant of the Fuzzy-C Means clustering algorithm, as well as the use of the Minimum Cluster Volume clustering algorithm. A variety of fractal dimension estimators using this suboptimal covering method are discussed. Finally, these methods are compared to the standard box counting algorithm and wavelet-decomposition methods for calculating fractal dimension by using one-dimensional cantor dust sets and a set of standard Brownian random fractal images.

Tolle, Charles Robert; McJunkin, Timothy R; Gorsich, D. I.

2003-01-01

45

Fractal dimensions of niobium oxide films probed by protons and lithium ions

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.

Pehlivan, Esat; Niklasson, Gunnar A. [Department of Physics, Faculty of Arts and Sciences, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey and Angstroem Laboratory, Department of Engineering Sciences, Uppsala University, P.O. Box 534, SE-751 21 Uppsala (Sweden); Angstroem Laboratory, Department of Engineering Sciences, Uppsala University, P.O. Box 534, SE-751 21 Uppsala (Sweden)

2006-09-01

46

Planetary boundary layer detection with fractal dimension of three-wavelength lidar signals

NASA Astrophysics Data System (ADS)

Lidar backscatter signal resulting from laser light scattering from the aerosol and molecular in the atmosphere contains various information about the geometrical and physical properties of aerosol and molecular. The lidar backscatter signal can provide information about the planetary boundary layer (PBL) stratification by using aerosol as a tracer for convective and mixing processes. A PBL height and structure detecting technique based on the fractal dimension of three-wavelength backscatter signals is advanced. In this PBL height detecting technique, the three-wavelength backscatter signals are obtained by the Hampton University (HU, 37.02° N, 76.33° W) lidar. The fractal dimension was calculated using the three-wavelength lidar signals. The PBL heights obtained from fractal dimension of threewavelength lidar signals is compared with PBL heights obtained from the potential temperature profiles which are provided by NASA Langely Research Center (10 miles from HU). And results of the two methods agree well. Moreover, fractal dimension method can reduce the influence of the geometrical form factor on the PBL detecting to expand the detecting range of PBL and remove the effect of plume. Also, the fractal dimension method can show the PBL dynamics and the PBL evolution clearly.

Lei, Liqiao; McCormick, M. Patrick; Su, Jia

2013-05-01

47

Effect of solidification conditions on fractal dimension of dendrites

NASA Astrophysics Data System (ADS)

Dendrites are complex, three-dimensional structures that have conventionally been characterized by measuring the secondary dendrite arm spacing or the primary spacing in a dendritic network, but these global measures do not adequately describe the branched appearance of secondary and tertiary arms. This work focuses on the integral measurement of fractal dimension, a measure of complexity relatively unexplored in dendrites, in addition to specific surface area. Measurements were made on aluminum dendrites in directionally solidified Al-Si alloys of varying composition and solidification velocity, with and without induced convection currents. Contrary to expectations, average fractal dimension was found to be relatively insensitive to changes in solidification velocity and fluid flow within the ranges observed, compared to the variation in fractal dimension measured within any individual data set. Specific surface area was found to increase linearly with solidification velocity.

Genau, Amber L.; Freedman, Alex C.; Ratke, Lorenz

2013-01-01

48

Glass Transition Temperature and Fractal Dimension of Protein Free Energy Landscapes

The free-energy landscape of two peptides is evaluated at various temperatures and an estimate for its fractal dimension at these temperatures calculated. We show that monitoring this quantity as a function of temperature allows to determine the glass transition temperature.

Nelson A. Alves; Ulrich H. E. Hansmann

2000-01-13

49

Comparison of Two Numerical Methods for Computing Fractal Dimensions

NASA Astrophysics Data System (ADS)

From cosmology to economics, the examples of fractals can be found virtually everywhere. However, since few fractals permit the analytical evaluation of generalized fractal dimensions or R'enyi dimensions, the search for effective numerical methods is inevitable. In this project two promising numerical methods for obtaining generalized fractal dimensions, based on the distribution of distances within a set, are examined. They can be applied, in principle, to any set even if no closed-form expression is available. The biggest advantage of these methods is their ability to generate a spectrum of generalized dimensions almost simultaneously. It should be noted that this feature is essential to the analysis of multifractals. As a test of their effectiveness, here the methods were applied to the generalized Cantor set and the multiplicative binomial process. The generalized dimensions of both sets can be readily derived analytically, thus enabling the accuracy of the numerical methods to be verified. Here we will present a comparison of the analytical results and the predictions of the methods. We will show that, while they are effective, care must be taken in their interpretation.

Shiozawa, Yui; Miller, Bruce; Rouet, Jean-Louis

2012-10-01

50

Estimating the fractal dimension and the predictability of the atmosphere

The fractal dimension, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily surface temperature and pressure over several regions of the US and the North Atlantic Ocean with different climatic signal-to-noise ratios. Though the total number of data points is larger than those used in previous studies, it is still too small to obtain a reliable estimate of the Grassberger-Procaccia correlation dimension because of the limitations discussed by Ruelle. It can be shown that this dimension is greater than 8. It is pointed out that most, if not all, of the previous estimates of low fractal dimensions in the atmosphere are spurious. These results lead the authors to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia are sensitive to the selection of the time delay and are thus unreliable. Geographic variability of the fractal dimension is suggested, but further verification is needed. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. Using this method, it is found that the error-doubling time is about 2-3 days in Fort Collins, Colorado, about 4-5 days in Los Angeles, California, and about 5-8 days in the North Atlantic Ocean. The difference between these estimates of error-doubling time and estimates based on general circulation models (GCMs) is discussed. It is also mentioned that the computation of the Lyapunov exponents is slightly sensitive to the selection of the time delay, possibly because the fractal dimension is very high in the atmosphere. 48 refs., 10 figs., 3 tabs.

Zeng, X.; Pielke, R.A.; Eykholt, R. (Colorado State Univ., Fort Collins, CO (United States))

1992-04-15

51

Fractal dimension unscreened angles measured for radial viscous fingering Olivier Praud Harry, USA #Received November 2004; published July 2005# have examined fractal patterns formed injection experiments. fractal dimension D 0 of pattern large r / 1.70Â±0.02. Further, generalized dimensions D pattern

Texas at Austin. University of

52

INCIDENT DETECTION BY FRACTAL DIMENSION ANALYSIS OF LOOP DETECTOR DATA

This paper describes a research project which aimed to demonstrate the feasibility of using Fractal Dimension analysis of speed, occupancy and flow data for automatic incident detection (AID). Non-recurrent congestion resulting from accidents, breakdowns and other incidents accounts for about 60% of the delays on freeways (Dia and Rose, 1997). Therefore, the sooner an appropriate incident response is implemented, the

Kim Thomas; Hussein Dia

2000-01-01

53

Stochastic Branching Models of Fault Surfaces and Estimated Fractal Dimensions

Stochastic Branching Models of Fault Surfaces and Estimated Fractal Dimensions ERIC LIBICKI 1 of representative points such as hypocenter distributions. Key words: Fault structures, stochastic branching) or mathematical models such as stochastic branching and probability distributions (e.g., HARRIS, 1963; KARLIN

Ben-Zion, Yehuda

54

Fractal dimension, wavelet shrinkage, and anomaly detection for mine hunting

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

Kingsbury, Nick

55

Fractal dimension and neural network based image segmentation technique

NASA Astrophysics Data System (ADS)

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.

Lin, QiWei; Gui, Feng

2008-04-01

56

The fractal dimension of the spectrum of quasiperiodical schrodinger operators

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.

Laurent Marin

2010-06-21

57

Estimation of Fractal Dimension in Differential Diagnosis of Pigmented Skin Lesions

NASA Astrophysics Data System (ADS)

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.

Aralica, Gorana; Miloševi?, Danko; Konjevoda, Paško; Seiwerth, Sven; Štambuk, Nikola

58

Fractal dimension computation from equal mass partitions.

Numerical methods which utilize partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets. However, it is known that mass-oriented methods generate relatively good results for computing generalized dimensions for important cases where the box-counting method is known to fail. Here, we revisit two mass-oriented methods and discuss their strengths and limitations. PMID:25273186

Shiozawa, Yui; Miller, Bruce N; Rouet, Jean-Louis

2014-09-01

59

Method for Measuring Effective Density and Fractal Dimension of Aerosol Agglomerates

A method to find particle effective density and the fractal dimension, based on simultaneous size distribution measurements with SMPS and ELPI, is introduced. A fitting procedure is used to find the particle density as a function of particle size and the fractal dimension. The method was tested by simulation and by experimental measurements of particles with varying morphology. For fractal

Annele Virtanen; Jyrki Ristimäki; Jorma Keskinen

2004-01-01

60

Fractal dimension and unscreened angles measured for radial viscous fingering Olivier Praud fractal patterns formed by the injection of air into oil in a thin 0.127 mm layer contained between two reaches r/b=900, are far larger than in past experiments. The fractal dimension D0 of the pattern

Texas at Austin. University of

61

Fractal Dimension of Trajectory as Invariant of Genetic Algorithms

Convergence properties of genetic algorithms are investigated. For them some measures are introduced. A classification procedure\\u000a is proposed for genetic algorithms based on a conjecture: the entropy and the fractal dimension of trajectories produced by\\u000a them are quantities that characterize the classes of the algorithms. The role of these quantities as invariants of the algorithm\\u000a classes is presented. The present

Stefan Kotowski; Witold Kosinski; Zbigniew Michalewicz; Jakub Nowicki; Bartosz Przepiórkiewicz

2008-01-01

62

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\\lambda \\to \\infty$, $\\dim (\\sigma(H_\\lambda)) \\cdot \\log \\lambda$ converges to an explicit constant ($\\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\\"odinger dynamics generated by the Fibonacci Hamiltonian.

David Damanik; Mark Embree; Anton Gorodetski; Serguei Tcheremchantsev

2007-05-02

63

Fractal dimension analysis of malignant and benign endobronchial ultrasound nodes

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

2014-01-01

64

Infrared image quality assessment based on fractal dimension method

NASA Astrophysics Data System (ADS)

The operation and observation experience of users is affected by the quality of infrared images which are collected by infrared imager. And image quality is a significant indicator for the performance of image processing algorithm and the optimization of system parameters as well. An image quality reduced reference assessment model is put forward to evaluate the degree of infrared image quality reduction. The detail characteristic of infrared image texture is extracted by the fractal dimension analysis method proposed in this paper as the representation of image quality. The method computes the fractal dimension of every pixel one by one with a multi-scale window over the entire image to get the information of corresponding image block. A quality information image is mapped from the fractal dimension of all pixels to describe the infrared image quality. The parameters of the quality information image combined with the peak SNR of original infrared image are adopted as the metric of infrared image quality. The method can be embedded into image processing system to optimize image processing algorithms and parameters settings, and provide reference for fault diagnosis.

Zhang, Zhijie; Zhang, Jufeng; Yue, Song; Wang, Chensheng

2012-12-01

65

NASA Astrophysics Data System (ADS)

The use of fractal statistics for characterizing and synthesizing scenes and signals has in recent time been demonstrated as feasible. Traditionally global fractal dimensions based on morphological coverings were used to quantify the texture of sampled data sets. This texture could be used to describe the second order statistics in 2D scenes or in the jaggedness and fine structure of time series. With the realization of the benefits of fractal analysis has come a need for faster and more efficient computational algorithms. ROSETA is an algorithm which yields substantial computational performance improvements by calculating entropy based statistics instead of morphological geometric statistics. ROSETA may be used as a robust general purpose analytical tool and several examples of its implementation are described.

Jaenisch, Holger M.; Barton, Philip E.; Carruth, R. T.

1993-09-01

66

Effect of mobile phone radiation on brain using EEG analysis by Higuichi's fractal dimension method

NASA Astrophysics Data System (ADS)

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.

Smitha, C. K.; Narayanan, N. K.

2013-01-01

67

Multiresolution estimation of fractal dimension from images affected by signal-dependent noise

NASA Astrophysics Data System (ADS)

A well-suited approach to calculate the fractal dimension of digital images stems from the power spectrum of a fractional Brownian motion: the ratio between powers at different scales is related to the persistence parameter H and, thus, to the fractal dimension D equals 3 - H. The signal- dependent nature of the speckle noise, however, prevents from a correct estimation of fractal dimension from Synthetic Aperture Radar (SAR) images. Here, we propose and assess a novel method to obtain D based on the multiscale decomposition provided by the normalized Laplacian pyramid, which is a bandpass representation obtained by dividing the layers of an LP by its expanded baseband and is designed to force the noise to become signal-independent. Extensive experiments on synthetic fractal textures, both noise-free and noisy, corroborate the underlying assumptions and show the performances, in terms of both accuracy and confidence of estimation, of pyramid methods compared with the well- established method based on the wavelet transform. Preliminary results on true SAR images from ERS-1 look promising as well.

Aiazzi, Bruno; Alparone, Luciano; Baronti, Stefano; Garzelli, Andrea

1999-10-01

68

Surface evaluation by estimation of fractal dimension and statistical tools.

Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380

Hotar, Vlastimil; Salac, Petr

2014-01-01

69

Mapping soil fractal dimension in agricultural fields with GPR

NASA Astrophysics Data System (ADS)

We documented that the mapping of the fractal dimension of the backscattered Ground Penetrating Radar traces (Fractal Dimension Mapping, FDM) accomplished over heterogeneous agricultural fields gives statistically sound combined information about the spatial distribution of Andosol' dielectric permittivity, volumetric and gravimetric water content, bulk density, and mechanical resistance under seven different management systems. The roughness of the recorded traces was measured in terms of a single number H, the Hurst exponent, which integrates the competitive effects of volumetric water content, pore topology and mechanical resistance in space and time. We showed the suitability to combine the GPR traces fractal analysis with routine geostatistics (kriging) in order to map the spatial variation of soil properties by nondestructive techniques and to quantify precisely the differences under contrasting tillage systems. Three experimental plots with zero tillage and 33, 66 and 100% of crop residues imprinted the highest roughness to GPR wiggle traces (mean HR/S=0.15), significantly different to Andosol under conventional tillage (HR/S=0.47).

Oleschko, K.; Korvin, G.; Muñoz, A.; Velazquez, J.; Miranda, M. E.; Carreon, D.; Flores, L.; Martínez, M.; Velásquez-Valle, M.; Brambila, F.; Parrot, J.-F.; Ronquillo, G.

2008-09-01

70

Fractal Dimension of Geologically Constrained Crater Populations of Mercury

NASA Astrophysics Data System (ADS)

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.

Mancinelli, Paolo; Pauselli, Cristina; Perugini, Diego; Lupattelli, Andrea; Federico, Costanzo

2014-08-01

71

Purpose To assess bony trabecular changes potentially caused by loading stress around dental implants using fractal dimension analysis. Methods Fractal dimensions were measured in 48 subjects by comparing radiographs taken immediately after prosthesis delivery with those taken 1 year after functional loading. Regions of interest were isolated, and fractal analysis was performed using the box-counting method with Image J 1.42 software. Wilcoxon signed-rank test was used to analyze the difference in fractal dimension before and after implant loading. Results The mean fractal dimension before loading (1.4213±0.0525) increased significantly to 1.4329±0.0479 at 12 months after loading (P<0.05). Conclusions Fractal dimension analysis might be helpful in detecting changes in peri-implant alveolar trabecular bone patterns in clinical situations. PMID:24236242

Mu, Teh-Jing; Lee, Dong-Won; Park, Kwang-Ho

2013-01-01

72

Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation

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.

Bruce N. Miller; Jean-Louis Rouet

2010-04-01

73

Fractal Dimensions for Continuous Time Random Walk Limits

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.

Mark M. Meerschaert; Erkan Nane; Yimin Xiao

2011-02-02

74

On the Fractal Dimension of Isosurfaces Marc Khoury and Rephael Wenger

On the Fractal Dimension of Isosurfaces Marc Khoury and Rephael Wenger 2 2.2 2.4 2.6 2.8 3 0 50 100 150 200 250 1 10 100 1000 10000 0 50 100 150 200 250 Fractal dimension Topological Noise (Number of Components) Isovalue 60 Isovalue 68 Isovalue 72 Fig. 1: Visible male data set (www.stereofx.org): Fractal box

Wenger, Rephael

75

Novel method for the global characterization of time series, based on the calculation of fractal dimension of a two-dimensional recurrence plots is proposed. The method is used for the characterization of differences between regular and chaotic systems and for the analysis of human electrocardiogram.

P. Babinec; M. Ku?era; M. Babincová

2005-01-01

76

NASA Astrophysics Data System (ADS)

The fluctuation of the dynamic scattered light of particles was characterized with fractal dimensions, and the influence of temperature on the fractal dimension was discussed. In the experiments, the fractal dimensions of dynamic scattered light intensity signal of particles with the diameter of 60nm, 90nm, 200nm, 300nm and 450nm, were obtained under the temperatures of 18°C, 20°C, 22°C, 24°C, 26°C, 28°C, 30°C. The experimental results shown the monotony relationship between fractal dimension of scattered light intensity signal of particles and the temperature in the particle system, which indicate that the fractal dimension of scattered light signal correlate well not only with the particle size but also the temperature of the suspension. Under the condition of constant temperature in the cuvette, the smaller the particles, the larger their fractal dimensions. For the same particle system, the higher the temperature, the larger the fractal dimensions of dynamic scattered light. By using two-dimensional interpolation surface chart of fractal dimensions, the polydisperse particle system were measured.

Shen, Jin; Tan, Boxue; Ding, Qiang; Yang, Shulian

2008-03-01

77

Density, fractal angle, and fractal dimension in linear Zn electrodeposition morphology

NASA Astrophysics Data System (ADS)

In this paper, we explore some aspects of the formation of tree-like structures (dendrites) with branching and ramifications in Zn electrodeposition systems. A special cell is designed wherein dendrites are grown between two parallel electrodes. All measurements are carried out at fixed voltage (9.5 V). As the concentration is varied at that fixed potential, the fractal box-count dimension is exactly anti-correlated with the pattern density. The latter is measured by the fraction covered by the electrodeposits, of a total area delimited by the cathode and the farthest dendritic tip. At relatively high concentration, the obtuse angle of lateral branching and the box dimension show a good correlation. The pattern density decreases with interelectrode distance, thus displaying a perfect correlation with the dependence of the current on the latter parameter. The time evolution of the pattern density exhibits an interesting oscillation caused by the preferential growth of individualized trees.

Saab, Rana; Sultan, Rabih

2005-11-01

78

The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski-Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM.

Wang Xujing [Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294 (United States); Becker, Frederick F.; Gascoyne, Peter R. C. [Department of Molecular Pathology, University of Texas M. D. Anderson Cancer Center, Houston, Texas 77030 (United States)

2010-12-15

79

Fractal Dimension of Particle Showers Measured in a Highly Granular Calorimeter

We explore the fractal nature of particle showers using Monte-Carlo simulation. We define the fractal dimension of showers measured in a high granularity calorimeter designed for a future lepton collider. The shower fractal dimension reveals detailed information of the spatial configuration of the shower. %the information hidden in the details of shower spatial configuration, It is found to be characteristic of the type of interaction and highly sensitive to the nature of the incident particle. Using the shower fractal dimension, we demonstrate a particle identification algorithm that can efficiently separate electromagnetic showers, hadronic showers and non-showering tracks. We also find a logarithmic dependence of the shower fractal dimension on the particle energy.

Manqi Ruan; Daniel Jeans; Vincent Boudry; Jean-Claude Brient; Henri Videau

2013-12-30

80

Fractal Dimensions of a Weakly Clustered Distribution and the Scale of Homogeneity

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.

J. S. Bagla; Jaswant Yadav; T. R. Seshadri

2007-12-18

81

[Using fractal dimensions of hyperspectral curves to analyze the healthy status of vegetation].

The reflectance spectral curves of leaves can reflect many information of vegetation growth, and its variation maybe means that the healthy status of vegetation will change. Many spectral feature parameters such as red edge position, height of green peak, depth of red band absorption, the area of red edge and some vegetation index have been used to describe this change. However, the change of vegetation healthy status is not some feature parameters, but a comprehensive variation of the whole curve. So, a comprehensive index maybe has more value to describe the change of hyperspectral curve of vegetation and indicates its healthy status. Fractal is an appropriate mathematical tool, and fractal dimension can be used to explain the comprehensive variation of a curve. Therefore, in the present study, fractal theory was used to analyze the healthy status of different vegetation. Firstly, analytical spectral devices (ASD) were used to measure the hyperspectral curves of different vegetations with different healthy status. Secondly, spectral curves were analyzed, and some parameters which can really reflect different healthy status were obtained. Finally, the fractal dimension of reflectance spectral curves inside a spectral band zone between 450 and 780nm was computed by variation method, and the relationship between fractal dimensions and spectral feature parameters was established. The research results showed that (1) the hyperspectral curves of vegetation have fractal feature, and their fractal dimensions gradually decrease with the health deterioration of leaves, (2) fractal dimension has positive correlation with the height of green peak, the depth of red band absorption and the area of red edge, (3) multivariate analysis showed that fractal dimensions have a significant linear relationship with the three spectral feature parameters just mentioned above. So, the fractal dimension of hyperspectral curve can serve as a new comprehensive parameter to analyze quantitatively the healthy status of vegetations. PMID:19839325

Du, Hua-Qiang; Jin, Wei; Ge, Hong-Li; Fan, Wen-Yi; Xu, Xiao-Jun

2009-08-01

82

Pulmonary hypertension (PH) can result in vascular pruning and increased tortuosity of the blood vessels. In this study we examined whether automatic extraction of lung vessels from contrast-enhanced thoracic computed tomography (CT) scans and calculation of tortuosity as well as 3D fractal dimension of the segmented lung vessels results in measures associated with PH. In this pilot study, 24 patients (18 with and 6 without PH) were examined with thorax CT following their diagnostic or follow-up right-sided heart catheterisation (RHC). Images of the whole thorax were acquired with a 128-slice dual-energy CT scanner. After lung identification, a vessel enhancement filter was used to estimate the lung vessel centerlines. From these, the vascular trees were generated. For each vessel segment the tortuosity was calculated using distance metric. Fractal dimension was computed using 3D box counting. Hemodynamic data from RHC was used for correlation analysis. Distance metric, the readout of vessel tortuosity, correlated with mean pulmonary arterial pressure (Spearman correlation coefficient: ??=?0.60) and other relevant parameters, like pulmonary vascular resistance (??=?0.59), arterio-venous difference in oxygen (??=?0.54), arterial (??=??0.54) and venous oxygen saturation (??=??0.68). Moreover, distance metric increased with increase of WHO functional class. In contrast, 3D fractal dimension was only significantly correlated with arterial oxygen saturation (??=?0.47). Automatic detection of the lung vascular tree can provide clinically relevant measures of blood vessel morphology. Non-invasive quantification of pulmonary vessel tortuosity may provide a tool to evaluate the severity of pulmonary hypertension. Trial Registration ClinicalTrials.gov NCT01607489 PMID:24498123

Urschler, Martin; Kullnig, Peter; Stollberger, Rudolf; Kovacs, Gabor; Olschewski, Andrea; Olschewski, Horst; Balint, Zoltan

2014-01-01

83

Purpose. This study describes how to identify the coincidence of desired planning isodose curves with film experimental results by using a mathematical fractal dimension characteristic method to avoid the errors caused by visual inspection in the intensity modulation radiation therapy (IMRT). Methods and Materials. The isodose curves of the films delivered by linear accelerator according to Plato treatment planning system were acquired using Osiris software to aim directly at a single interested dose curve for fractal characteristic analysis. The results were compared with the corresponding planning desired isodose curves for fractal dimension analysis in order to determine the acceptable confidence level between the planning and the measurement. Results. The film measured isodose curves and computer planning curves were deemed identical in dose distribution if their fractal dimensions are within some criteria which suggested that the fractal dimension is a unique fingerprint of a curve in checking the planning and film measurement results. The dose measured results of the film were presumed to be the same if their fractal dimension was within 1%. Conclusions. This quantitative rather than qualitative comparison done by fractal dimension numerical analysis helps to decrease the quality assurance errors in IMRT dosimetry verification. PMID:23956976

Wu, Jia-Ming; Kuo, Chung-Ming; Chen, Ching-Jiang

2013-01-01

84

Fractal dimension of line width roughness and its effects on transistor performance

NASA Astrophysics Data System (ADS)

The effects of Line Width Roughness (LWR) on transistor performance are one of the hottest issues in semiconductor industry. However, in most related studies, LWR is considered as the fluctuations of gate lengths and not of resist lines. In this paper, we examine the direct effects of one of the spatial resist LWR parameters, the fractal dimension, on transistor off current deviations for various correlation lengths and gate widths. The aim is to exploit the fractality of LWR in order to link the gap between the LWR of long resist lines and the gate length roughness that affects transistor performance. The used methodology is based on the simulation of both resist lines and transistor operation. The results of the two step methodology are presented for both narrow and wide gates. For the first, it is found that for all correlation lengths, higher fractal dimension (smaller roughness exponent) of the resist line leads to off state currents closer to the nominal value. For wide gates, an interesting differentiation is found at the dependence of the standard deviation from the fractal dimension as correlation length decreases. For sufficiently low correlation length, the behavior is reversed and the low fractal dimension are more beneficial that the higher ones. An explanation of that reverse is provided by means of the dependence of the CD variation on gate width for various fractal dimensions. Finally, the implications of these findings on the dependencies of the yield of transistors on fractal dimension and correlation length are also discussed.

Constantoudis, V.; Gogolides, E.

2008-03-01

85

Image fusion algorithm based on fractal dimension and contrast in multi-wavelet transform domain

In order to make full use of texture features of image when fusing images, and taking into account the inherent advantages of fractal theory in this respect, a novel image fusion algorithm, which used fractal dimension and directional contrast, based on multi-wavelet transform was proposed in this paper. This fusion algorithm decomposed original images needed to be fused by multi-wavelet

Zhihui Wang; Yong Tie; Shuhua Li; Dong Li

2011-01-01

86

Beyond uniformity and independence: analysis of R-trees using the concept of fractal dimension

We propose the concept of fractal dimension of a set of points, in order to quantify the deviation from the uniformity distribution. Using measurements on real data sets (road intersections of U.S. counties, star coordinates from NASA's Infrared-Ultraviolet Explorer etc.) we provide evidence that real data indeed are skewed, and, moreover, we show that they behave as mathematical fractals, with

Christos Faloutsos; Ibrahim Kamel

1994-01-01

87

Optical coherence tomography (OCT) images of left-descending coronary tissues harvested from three porcine specimens were acquired with a home-build swept-source OCT setup. Despite the fact that OCT is capable of acquiring high resolution circumferential images of vessels, many distinct histological features of a vessel have comparable optical properties leading to poor contrast in OCT images. Two classification methods were tested in this report for the purpose of enhancing contrast between soft-tissue components of porcine coronary vessels. One method involved analyzing the attenuation of the OCT signal as a function of light penetration into the tissue. We demonstrated that by analyzing the signal attenuation in this manner we were able to differentiate two media sub-layers with different orientations of the smooth muscle cells. The other classification method used in our study was fractal analysis. Fractal analysis was implemented in a box-counting (fractal dimension) image-processing code and was used as a tool to differentiate and quantify variations in tissue texture at various locations in the OCT images. The calculated average fractal dimensions had different values in distinct regions of interest (ROI) within the imaged coronary samples. When compared to the results obtained by using the attenuation of the OCT signal, the method of fractal analysis demonstrated better classification potential for distinguishing amongst the tissue ROI. PMID:20360632

Flueraru, C; Popescu, D P; Mao, Y; Chang, S; Sowa, M G

2010-04-21

88

NASA Astrophysics Data System (ADS)

Optical coherence tomography (OCT) images of left-descending coronary tissues harvested from three porcine specimens were acquired with a home-build swept-source OCT setup. Despite the fact that OCT is capable of acquiring high resolution circumferential images of vessels, many distinct histological features of a vessel have comparable optical properties leading to poor contrast in OCT images. Two classification methods were tested in this report for the purpose of enhancing contrast between soft-tissue components of porcine coronary vessels. One method involved analyzing the attenuation of the OCT signal as a function of light penetration into the tissue. We demonstrated that by analyzing the signal attenuation in this manner we were able to differentiate two media sub-layers with different orientations of the smooth muscle cells. The other classification method used in our study was fractal analysis. Fractal analysis was implemented in a box-counting (fractal dimension) image-processing code and was used as a tool to differentiate and quantify variations in tissue texture at various locations in the OCT images. The calculated average fractal dimensions had different values in distinct regions of interest (ROI) within the imaged coronary samples. When compared to the results obtained by using the attenuation of the OCT signal, the method of fractal analysis demonstrated better classification potential for distinguishing amongst the tissue ROI.

Flueraru, C.; Popescu, D. P.; Mao, Y.; Chang, S.; Sowa, M. G.

2010-04-01

89

NASA Astrophysics Data System (ADS)

We present a method for generating fractal surfaces of dimension between two and three. By using the method, five fractal surfaces with dimension 2.262, 2.402, 2.524, 2.631, and 2.771 are created. For each of these surfaces, the reaction of carbon monoxide and oxygen is simulated by using a Monte Carlo method based on the ZGB model [Phys. Rev. Lett. 24 (1986) 2553]. The results show that the catalytic CO oxidation proceeds more efficiently on a surface with higher fractal dimension. It is also found that as the fractal dimension of the surface becomes higher, the first-order kinetic phase transition point (y 2) is shifted to a higher partial pressure of CO. This implies that poisoning of the catalyst surface due to CO segregation sets in at a higher CO partial pressure for surfaces with more complexity.

Park, Hwangseo; Kim, Hojing; Lee, Sangyoub

1997-05-01

90

We discuss the Casimir effect for massless scalar fields subject to the Dirichlet boundary conditions on the parallel plates at finite temperature in the presence of one fractal extra compactified dimension. We obtain the Casimir energy density with the help of the regularization of multiple zeta function with one arbitrary exponent and further the renormalized Casimir energy density involving the thermal corrections. It is found that when the temperature is sufficiently high, the sign of the Casimir energy remains negative no matter how great the scale dimension $\\delta$ is within its allowed region. We derive and calculate the Casimir force between the parallel plates affected by the fractal additional compactified dimension and surrounding temperature. The stronger thermal influence leads the force to be stronger. The nature of the Casimir force keeps attractive.

Hongbo Cheng

2011-09-06

91

Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.

An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants. PMID:25234864

Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter

2014-11-20

92

Processing of gray scale images in order to determine the corresponding fractal dimension is very important due to widespread use of imaging technologies and application of fractal analysis in many areas of science, technology, and medicine. To this end, many methods for estimation of fractal dimension from gray scale images have been developed and routinely used. Unfortunately different methods (dimension estimators) often yield significantly different results in a manner that makes interpretation difficult. Here, we report results of comparative assessment of performance of several most frequently used algorithms/methods for estimation of fractal dimension. To that purpose, we have used scanning electron microscope images of aluminum oxide surfaces with different fractal dimensions. The performance of algorithms/methods was evaluated using the statistical Z-score approach. The differences between performances of six various methods are discussed and further compared with results obtained by electrochemical impedance spectroscopy on the same samples. The analysis of results shows that the performance of investigated algorithms varies considerably and that systematically erroneous fractal dimensions could be estimated using certain methods. The differential cube counting, triangulation, and box counting algorithms showed satisfactory performance in the whole investigated range of fractal dimensions. Difference statistic is proved to be less reliable generating 4% of unsatisfactory results. The performances of the Power spectrum, Partitioning and EIS were unsatisfactory in 29%, 38%, and 75% of estimations, respectively. The results of this study should be useful and provide guidelines to researchers using/attempting fractal analysis of images obtained by scanning microscopy or atomic force microscopy. PMID:23483485

Risovi?, Dubravko; Pavlovi?, Zivko

2013-01-01

93

The b-value and fractal dimension of local seismicity around Koyna Dam (India)

NASA Astrophysics Data System (ADS)

Earthquakes began to occur in Koyna region (India) soon after the filling of Koyna Dam in 1962. In the present study, three datasets 1964-1993, 1993-1995, and 1996-1997 are analyzed to study the b-value and fractal dimension. The b-value is calculated using the Gutenberg-Richter relationship and fractal dimension D corr. using correlation integral method. The estimated b-value and D corr. of this region before 1993 are found to be in good agreement with previously reported studies. In the subsequent years after 1995, the b-value shows an increase. The estimated b-values of this region are found within the limits of global average. Also, the pattern of spatial clustering of earthquakes show increase in clustering and migration along the three zones called North-East Zone, South-East Zone (SEZ), and Warna Seismic Zone. The earthquake events having depth ?5 km are largely confined to SEZ. After 1993, the D corr. shows decrease, implying that earthquake activity gets clustered. This seismic clustering could be helpful for earthquake forecasting.

Kumar, Arjun; Rai, S. S.; Joshi, Anand; Mittal, Himanshu; Sachdeva, Rajiv; Kumar, Rohtash; Ghangas, Vandana

2013-11-01

94

Fractal dimensions of silica gels generated using reactive molecular dynamics simulations

We have used molecular dynamics simulations based on a three-body potential with charge transfer to generate nanoporous silica aerogels. Care was taken to reproduce the sol-gel condensation reaction that forms the gel backbone as realistically as possible and to thereby produce credible gel structures. The self-similarity of aerogel structures was investigated by evaluating their fractal dimension from geometric correlations. For comparison, we have also generated porous silica glasses by rupturing dense silica and computed their fractal dimension. The fractal dimension of the porous silica structures was found to be process dependent. Finally, we have determined that the effect of supercritical drying on the fractal nature of condensed silica gels is not appreciable.

Bhattacharya, Sudin; Kieffer, John [Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2158 (United States); Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 (United States)

2005-03-01

95

NASA Astrophysics Data System (ADS)

This paper presents a method for the quantification of cellular rejection in endomyocardial biopsies of patients submitted to heart transplant. The model is based on automatic multilevel thresholding, which employs histogram quantification techniques, histogram slope percentage analysis and the calculation of maximum entropy. The structures were quantified with the aid of the multi-scale fractal dimension and lacunarity for the identification of behavior patterns in myocardial cellular rejection in order to determine the most adequate treatment for each case.

Neves, L. A.; Oliveira, F. R.; Peres, F. A.; Moreira, R. D.; Moriel, A. R.; de Godoy, M. F.; Murta Junior, L. O.

2011-03-01

96

Fractal Dimension Characterization of in-vivo Laser Doppler Flowmetry signals

NASA Astrophysics Data System (ADS)

Laser Doppler Blood Flow meter uses tissue backscattered light to non-invasively assess the blood flow rate. qualitatively. As there is large spatial variability and the temporal heterogeneity in tissue microvasculature, the measured blood flow rate is expressed in relative units. A non-linear approach in order to understand the dynamics of the microcirculation led to the fractal characterization of the blood flow signals. The study presented in the paper aims to analyze the fractal behavior of Laser Doppler Flow (LDF) signals and to quantitatively estimate the fractal dimension of waveforms using Box-Counting method. The measured Fractal dimension is an estimate of temporal variability of tissue perfusion. The rate at which fractal dimension varies as a function of location between individuals, exhibits a weak correlation with time. Further studies with a larger number of subjects are necessary to test the generality of the findings and if changes in dimension are reproducible in given individuals. In conclusion, the fractal dimension determined by Box-counting method may be useful for characterizing LDF time series signals. Future experiments evaluating whether the technique can be used to quantify microvascular dysfunction, as commonly occurring in conditions such as Diabetes, Raynaud's phenomenon, Erythromelalgia and Achenbach syndrome needs to be evaluated.

Srinivasan, Gayathri; Sujatha, N.

97

The fractal dimension of a test signal: implications for system identification procedures.

The experimental identification of a non-linear biologic transducer is often approached via consideration of its response to a stochastic test ensemble, such as Gaussian white noise (Marmarelis and Marmarelis 1978). In this approach, the input-output relationship a deterministic transducer is described by an orthogonal series of functionals. Laboratory implementation of such procedures requires the use of a particular test signal drawn from the idealized stochastic ensemble; the statistics of the particular test signal necessarily deviate from the statistics of the ensemble. The notion of a fractal dimension (specifically the capacity dimension) is a means to characterize a complex time series. It characterizes one aspect of the difference between a specific example of a test signal and the test ensemble from which it is drawn: the fractal dimension of ideal Gaussian white noise is infinite, while the fractal dimension of a particular test signal is finite. This paper shows that the fractal dimension of a test signal is a key descriptor of its departure from ideality: the fractal dimension of the test signal bounds the number of terms that can reliably be identified in the orthogonal functional series of an unknown transducer. PMID:3435730

Victor, J D

1987-01-01

98

Crack detection in beams in noisy conditions using scale fractal dimension analysis of mode shapes

NASA Astrophysics Data System (ADS)

Fractal dimension analysis of mode shapes has been actively studied in the area of structural damage detection. The most prominent features of fractal dimension analysis are high sensitivity to damage and instant determination of damage location. However, an intrinsic deficiency is its susceptibility to measurement noise, likely obscuring the features of damage. To address this deficiency, this study develops a novel damage detection method, scale fractal dimension (SFD) analysis of mode shapes, based on combining the complementary merits of a stationary wavelet transform (SWT) and Katz’s fractal dimension in damage characterization. With this method, the SWT is used to decompose a mode shape into a set of scale mode shapes at scale levels, with damage information and noise separated into distinct scale mode shapes because of their dissimilar scale characteristics; the Katz’s fractal dimension individually runs on every scale mode shape in the noise-adaptive condition provided by the SWT to canvass damage. Proof of concept for the SFD analysis is performed on cracked beams simulated by the spectral finite element method; the reliability of the method is assessed using Monte Carlo simulation to mimic the operational variability in realistic damage diagnosis. The proposed method is further experimentally validated on a cracked aluminum beam with mode shapes acquired by a scanning laser vibrometer. The results show that the SFD analysis of mode shapes provides a new strategy for damage identification in noisy conditions.

Bai, R. B.; Ostachowicz, W.; Cao, M. S.; Su, Z.

2014-06-01

99

Electroconvulsive therapy (ECT), in which electrical current is used to induce seizures, is an effective treatment in psychiatry. Different methods of analyzing the electroencephalogram (EEG) changes during ECT have been studied for predicting clinical outcome. Analysis of the fractal dimension (FD) is one such method. Mid-seizure and post-seizure FD has been shown to correlate with antidepressant effect. In this study, we examined whether the highest fractal dimension achieved during each ECT session changed over the course of 6 ECTs. The sample for this study came from a randomized controlled trial, comparing the efficacy of bifrontal and bitemporal electrode placements in schizophrenia. EEG was recorded using bilateral frontal pole leads during all ECT sessions. In 40 of the 114 randomized patients, we could obtain artifact-free EEGs for the first 6 ECT sessions. FD was calculated using standardized algorithms. For each session, the average of 5 highest FDs was calculated. The change in this value over a course of 6 ECTs was analyzed using repeated-measures analysis of variance. The average highest FD remained virtually unchanged across the 6 ECT sessions. Means (standard deviations) average maximum FDs over the 6 sessions were 1.57 (0.075), 1.57 (0.064), 1.56 (0.064), 1.57 (0.062), 1.55 (0.07), and 1.56 (0.067); occasion effect, F = 0.5, P = .75. Group effect (F = 0.01, P = .92) and group × occasion interaction effect (F = 1.88, P = .1) were not significant, suggesting no influence of electrode placement on maximum FD. Seizure duration, however, showed significant decline over the course of ECT. Maximum FD of ECT-induced EEG seizure remains fairly constant over frontal poles across the first 6 ECT sessions, which is true irrespective of ECT electrode placements. PMID:23760035

Rakesh, Gopalkumar; Abhishekh, Hulegar A; Thirthalli, Jagadisha; Phutane, Vivek H; Muralidharan, Kesavan; Candade, Vittal S; Gangadhar, Bangalore N

2014-04-01

100

Multi-fractal analysis of weighted networks

In many real complex networks, the fractal and self-similarity properties have been found. The fractal dimension is a useful method to describe fractal property of complex networks. Fractal analysis is inadequate if only taking one fractal dimension to study complex networks. In this case, multifractal analysis of complex networks are concerned. However, multifractal dimension of weighted networks are less involved. In this paper, multifractal dimension of weighted networks is proposed based on box-covering algorithm for fractal dimension of weighted networks (BCANw). The proposed method is applied to calculate the fractal dimensions of some real networks. Our numerical results indicate that the proposed method is efficient for analysis fractal property of weighted networks.

Daijun Wei; Xiaowu Chen; Cai Gao; Haixin Zhang; Bo Wei; Yong Deng

2014-02-28

101

NASA Technical Reports Server (NTRS)

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.

Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.

2004-01-01

102

Use of the fractal dimension for the analysis of electroencephalographic time series

. ?Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their\\u000a fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by\\u000a means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals\\u000a are needed and consequently only long-term events can be analysed;

A. Accardo; M. Affinito; M. Carrozzi; F. Bouquet

1997-01-01

103

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.

Smith, R.L., E-mail: firefan@ufl.edu; Mecholsky, J.J., E-mail: jmech@ufl.edu

2011-05-15

104

Temporal evolution of characteristic length and fractal dimension for a non-Euclidean system

NASA Astrophysics Data System (ADS)

In order to understand some yet incomprehensible experimental observations on temporal evolution of characteristic length and fractal dimension of a dynamical system, a model simulation has been attempted. It has been shown that the present simulation corroborates well with the trend of the experimentally observed temporal evolutions of fractal dimension and characteristic length during light water hydration of calcium silicates ( C3S and C2S ) and ordinary Portland cement. The dynamical scaling of the scattering functions was found to be valid during hydration, particularly in intermediate and in late stages of hydration process.

Sen, D.; Mazumder, S.; Bahadur, J.

2009-04-01

105

A robust marker to describe mass, hydrophobicity and polarizability distribution holds the key to deciphering structural and folding constraints within proteins. Since each of these distributions is inhomogeneous in nature, the construct should be sensitive in describing the patterns therein. We show, for the first time, that the hydrophobicity and polarizability distributions in protein interior follow fractal scaling. It is

Anirban Banerji; Indira Ghosh; Sotirios Koutsopoulos

2009-01-01

106

parameters. We can in- terpret horizontal, but not vertical, correlation lengths and fractal dimensions from and fractal dimension interpretation from seismic data using variograms and power spectra Ken Mela and John N survey and analyzed it for the statistical parameters of correlation length and fractal dimension

107

Detection of explosive lung and bowel sounds by means of fractal dimension

An efficient technique for detecting explosive lung sounds (LS) (fine\\/coarse crackles and squawks) or bowel sounds (BS) in clinical auscultative recordings is presented. The technique is based on a fractal-dimension (FD) analysis of the recorded LS and BS obtained from controls and patients with pulmonary and bowel pathology, respectively. Experimental results demonstrate the efficiency of the proposed method, since it

Leontios J. Hadjileontiadis; Ioannis T. Rekanos

2003-01-01

108

Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar

1 Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar J. D. B. Nelson and N. G. Kingsbury Abstract--Sand ripples present a difficult challenge to current mine hunting mine hunting. Manual inspection of such data can be a time consuming task that requires significant

Nelson, James

109

The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms

The existence of fractal-like networks effectively endows life with an additional fourth spatial dimension. This is the origin of quarter-power scaling which is so pervasive in biology. Organisms have evolved hierarchical networks which terminate in invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules, which are independent of organism size. Natural selection has tended to maximize both metabolic capacity

Geoffrey B. West; James H. Brown; Brian J. Enquist

1999-01-01

110

NASA Astrophysics Data System (ADS)

In this paper, we use the so-called the Wavelet Transform Modulus Maxima lines (WTMM) technique for estimation of the capacity, the information and the correlation fractal dimensions of the Intermagnet Observatories time series. Analysis of Hermanus, Baker-Lake, Kakioka, Albibag and Wingst observatories data shows that the correlation and the information dimensions can be used a supplementary indexes for geomagnetic disturbances identification.

Ouadfeul, S.-A.; Aliouane, L.; Tourtchine, V.

2013-09-01

111

Fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model

The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, have not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension, 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with fractal dimension 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas. PMID:23629025

Jo, Junghyo; Hornblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul

2013-01-01

112

NASA Astrophysics Data System (ADS)

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.

Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul

2013-06-01

113

Fractal dimension of EEG activity senses neuronal impairment in acute stroke.

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

Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca

2014-01-01

114

NASA Astrophysics Data System (ADS)

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

Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.

2012-09-01

115

Physical Consequences of Complex Dimensions of Fractals Eric Akkermans1

use and extend these results to study the resulting (log-periodic) oscillations in the heat kernel than, greater than, or equal to the critical dimension 2. Log-periodic oscillations have a long history

Teplyaev, Alexander

116

Overlapping-box-covering method for the fractal dimension of complex networks.

The fractality and self-similarity of complex networks have been widely investigated by evaluating the fractal dimension, the crux of which is how to locate the optimal solution or how to tile the network with the fewest boxes. The results yielded by the box-covering method with separated boxes possess great randomness or large errors. In this paper, we adopt the overlapping box to tile the entire network, called the overlapping-box-covering method. In such a case, for verifying its validity, we propose an overlapping-box-covering algorithm; we first apply it to three deterministic networks, then to four real-world fractal networks. It produces optimums or more accurate fractal dimension for the former; the quantities of boxes finally obtained for the latter are fewer and more deterministic, with the redundant box reaching up to 33.3%. The experimental results show that the overlapping-box-covering method is available and that the overlapping box outperforms the previous case, rendering the errors smaller. Moreover, we conclude that the overlapping box is an important determinant to acquire the fewest boxes for complex networks. PMID:24827295

Sun, Yuanyuan; Zhao, Yujie

2014-04-01

117

Overlapping-box-covering method for the fractal dimension of complex networks

NASA Astrophysics Data System (ADS)

The fractality and self-similarity of complex networks have been widely investigated by evaluating the fractal dimension, the crux of which is how to locate the optimal solution or how to tile the network with the fewest boxes. The results yielded by the box-covering method with separated boxes possess great randomness or large errors. In this paper, we adopt the overlapping box to tile the entire network, called the overlapping-box-covering method. In such a case, for verifying its validity, we propose an overlapping-box-covering algorithm; we first apply it to three deterministic networks, then to four real-world fractal networks. It produces optimums or more accurate fractal dimension for the former; the quantities of boxes finally obtained for the latter are fewer and more deterministic, with the redundant box reaching up to 33.3%. The experimental results show that the overlapping-box-covering method is available and that the overlapping box outperforms the previous case, rendering the errors smaller. Moreover, we conclude that the overlapping box is an important determinant to acquire the fewest boxes for complex networks.

Sun, Yuanyuan; Zhao, Yujie

2014-04-01

118

Fractal Dimension and Vessel Complexity in Patients with Cerebral Arteriovenous Malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs. PMID:22815946

Reishofer, Gernot; Koschutnig, Karl; Enzinger, Christian; Ebner, Franz; Ahammer, Helmut

2012-01-01

119

Linear correlation between fractal dimension of EEG signal and handgrip force.

Fractal dimension (FD) has been proved useful in quantifying the complexity of dynamical signals in biology and medicine. In this study, we measured FDs of human electroencephalographic (EEG) signals at different levels of handgrip forces. EEG signals were recorded from five major motor-related cortical areas in eight normal healthy subjects. FDs were calculated using three different methods. The three physiological periods of handgrip (command preparation, movement and holding periods) were analyzed and compared. The results showed that FDs of the EEG signals during the movement and holding periods increased linearly with handgrip force, whereas FD during the preparation period had no correlation with force. The results also demonstrated that one method (Katz's) gave greater changes in FD, and thus, had more power in capturing the dynamic changes in the signal. The linear increase of FD, together with results from other EEG and neuroimaging studies, suggest that under normal conditions the brain recruits motor neurons at a linear progress when increasing the force. PMID:16028075

Liu, J Z; Yang, Q; Yao, B; Brown, R W; Yue, G H

2005-08-01

120

Oscillations in the evaluation of fractal dimension of RR intervals time series.

Previously, we have reported the presence of oscillations in the graphs we have used to evaluate the Higuchi's fractal dimension in RR intervals time series of congestive heart failure (CHF) patients in the sleep phase but these oscillations hardly appear in all the six hours of the awake phase. In this paper we report the same analysis for heart rate time series for different groups of healthy subjects; we are looking for the presence of this kind of oscillations in other situations. We analyzed all the time series in the Exaggerated Heart Rate Oscillations database of Physionet during two meditation techniques: volunteers with spontaneous breathing, subjects in meditation, volunteers in a metronomic breathing group and elite athletes. We have found oscillations in the graphs of the Higuchi's fractal dimension in the heart rate time series of subjects in meditation and metronomic breathing and this fact coincides with previous reported results. PMID:21095797

Muñoz Diosdado, A; Gálvez Coyt, G; Pérez Uribe, B M

2010-01-01

121

Typical clinical evaluation of the tumour microvascular environment is performed by dynamic contrast enhanced MRI (dceMRI). However, this approach has been defined by numerous mathematical models each with their own physiologic assumptions which often leads to inconclusive results in the assessment of microstructure. Alternatively, Blood Oxygen Level Dependent (BOLD) Magnetic Resonance Imaging (MRI) is already known to be sensitive to the microvascular environment through fluctuation in the oxyhemoglobin to deoxyhemoglobin ratio. Consequently, quantification of the BOLD signal's temporal complexity using a fractal dimension index allows maps to be generated that are physiologically distinctive in nature and allow potential insight into tumour microvasculature with no a priori assumptions as in dceMRI. Here, using rectal carcinoma as an example, we present this novel approach to tumour microvascular evaluation using fractal dimension parametric mapping. PMID:18294894

Wardlaw, Graeme; Wong, Raimond; Noseworthy, Michael D

2008-06-01

122

NASA Astrophysics Data System (ADS)

For the system with colored multiplicative noise the nonlinearity of the synergetic potential like ?^{2+m} model in Langevin equation was shown to be capable of providing the expanse of the stochastic system phase space. The concrete system of the population dynamics with the noise correlation time ?_cto? is examined. The fractal dimension of that kind of a system is defined as D=m, in contrast to the system with a white noise were D=0.

Kharchenko, D. O.

123

Calculation of a static potential created by plane fractal cluster

NASA Astrophysics Data System (ADS)

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: rn = r0?n and en = e0bn, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5,6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster ( r0 ? r ? rN-1 ), where r0, rN-1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously ( b > 0) and heterogeneously ( b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.

Nigmatullin, Raoul R.; Alekhin, Alexander P.

2011-12-01

124

Surface fractal dimensions and textural properties of mesoporous alkaline-earth hydroxyapatites

NASA Astrophysics Data System (ADS)

This work examines the surface fractal dimensions (Df) and textural properties of three different alkaline-earth hydroxyapatites. Calcium, strontium and barium hydroxyapatite compounds were successfully synthesized via chemical precipitation method and characterized using X-ray diffraction, scanning electron microscopy, energy dispersive X-ray spectrometry, Fourier transform infrared spectroscopy, and N2-physisorption measurements. Surface fractal dimensions were determined using single N2-adsorption/desorption isotherms method to quantify the irregular surface of as-prepared compounds. The obtained materials were also characterized through their surface hydroxyl group content, determined by the mass titration method. It was found that the Df values for the three materials covered the range of 0.77 ± 0.04-2.33 ± 0.11; these results indicated that the materials tend to have smooth surfaces, except the irregular surface of barium hydroxyapatite. Moreover, regarding the synthesized calcium hydroxyapatite exhibited better textural properties compared with the synthesized strontium and barium hydroxyapatites for adsorbent purposes. However, barium hydroxyapatite shows irregular surface, indicating a high population of active sites across the surface, in comparison with the others studied hydroxyapatites. Finally, the results showed a linear correlation between the surface hydroxyl group content at the external surface of materials and their surface fractal dimensions.

Vilchis-Granados, J.; Granados-Correa, F.; Barrera-Díaz, C. E.

2013-08-01

125

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)

Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S

2011-04-30

126

NASA Astrophysics Data System (ADS)

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.

Ul'yanov, A. S.; Lyapina, A. M.; Ulianova, O. V.; Fedorova, V. A.; Uianov, S. S.

2011-04-01

127

A Brief Historical Introduction to Fractals and Fractal Geometry

ERIC Educational Resources Information Center

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…

Debnath, Lokenath

2006-01-01

128

The Cantor set complementary to the Devil's Staircase associated with the Circle Map has a fractal dimension d approximately equal to 0.87, a value that is universal for a wide range of maps, such results being of a numerical character. In this paper we deduce a formula for such dimensional value. The Devil's Staircase associated with the Circle Map is a function that transforms horizontal unit interval I onto vertical I, and is endowed with the Farey-Brocot (F-B) structure in the vertical axis via the rational heights of stability intervals. The underlying Cantor-dust fractal set Omega in the horizontal axis --Omega contained in I, with fractal dimension d(Omega) approx. 0.87-- has a natural covering with segments that also follow the F-B hierarchy: therefore, the staircase associates vertical I (of unit dimension) with horizontal Omega in I (of dimension approx. 0.87), i.e. it selects a certain subset Omega of I, both sets F- B structured, the selected Omega with smaller dimension than that of I. Hence, the structure of the staircase mirrors the F- B hierarchy. In this paper we consider the subset Omega-F-B of I that concentrates the measure induced by the F-B partition and calculate its Hausdorff dimension, i.e. the entropic or information dimension of the F-B measure, and show that it coincides with d(Omega) approx. 0.87. Hence, this dimensional value stems from the F-B structure, and we draw conclusions and conjectures from this fact. Finally, we calculate the statistical "Euclidean" dimension (based on the ordinary Lebesgue measure) of the F-B partition, and we show that it is the same as d(Omega-F-B), which permits conjecturing on the universality of the dimensional value d approximately equal to 0.87.

M. N. Piacquadio Losada

2007-11-17

129

NASA Astrophysics Data System (ADS)

Fractal analysis of the total magnetic field (TMF) time series from 1997 to 2003 at Popocatépetl Volcano is performed and compared with the TMF-series of the Teoloyucan Magnetic Observatory, 100 km away. Using Higuchi's fractal dimension method (D). The D changes over time for both series were computed. It was observed, when the time windows used to compute D increase in length, both series show nearly the same behavior. Some criteria of comparison were employed to discriminate the local effects inherent to volcano-magnetism. The simultaneous maximum in D (1.8) of the TMF series at Popocatépetl Volcano and the recovered volcanic activity indicates a scaling relation of the TMF at Popocatépetl Volcano and demonstrates a link between the magnetic field and volcanic activity.

Flores-Marquez, E. L.; Galvez-Coyt, G.; Cifuentes-Nava, G.

2012-12-01

130

NASA Astrophysics Data System (ADS)

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.

Ahammer, Helmut; DeVaney, Trevor T. J.

2004-03-01

131

The fractal nature of the DNA arrangement has been postulated to be a common feature of all cell nuclei. We investigated the prognostic importance of the fractal dimension (FD) of chromatin in blasts of patients with acute precursor B lymphoblastic leukemia (B-ALL). In 28 patients, gray scale transformed pseudo-3D images of 100 nuclei (May-Grünwald-Giemsa stained bone marrow smears) were analyzed. FD was determined by the Minkowski-Bouligand method extended to three dimensions. Goodness-of-fit of FD was estimated by the R2 values in the log-log plots. Whereas FD presented no prognostic relevance, patients with higher R2 values showed a prolonged survival. White blood cell count (WBC), age and mean fluorescence intensity of CD45 (MFICD45) were all unfavorable prognostic factors in univariate analyses. In a multivariate Cox-regression, R2, WBC, and MFICD45, entered the final model, which showed to be stable in a bootstrap resampling study. Blasts with lower R2 values, equivalent to accentuated "coarseness" of the chromatin pattern, which may reflect profound changes of the DNA methylation, indicated a poor prognosis. In conclusion the goodness-of-fit of the Minkowski-Bouligand dimension of chromatin can be regarded as a new and biologically relevant prognostic factor for patients with B-ALL. PMID:16675881

Adam, Randall L; Silva, Rosana C; Pereira, Fernanda G; Leite, Neucimar J; Lorand-Metze, Irene; Metze, Konradin

2006-01-01

132

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

Zheng, Xiuqing; Hu, Shaoxiang; Li, Ming; Zhou, Jiliu

2013-01-01

133

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

Zheng, Xiuqing; Liao, Zhiwu; Hu, Shaoxiang; Li, Ming; Zhou, Jiliu

2013-01-01

134

Objective Auscultation of TCM Based on Wavelet Packet Fractal Dimension and Support Vector Machine

This study was conducted to illustrate that auscultation features based on the fractal dimension combined with wavelet packet transform (WPT) were conducive to the identification the pattern of syndromes of Traditional Chinese Medicine (TCM). The WPT and the fractal dimension were employed to extract features of auscultation signals of 137 patients with lung Qi-deficient pattern, 49 patients with lung Yin-deficient pattern, and 43 healthy subjects. With these features, the classification model was constructed based on multiclass support vector machine (SVM). When all auscultation signals were trained by SVM to decide the patterns of TCM syndromes, the overall recognition rate of model was 79.49%; when male and female auscultation signals were trained, respectively, to decide the patterns, the overall recognition rate of model reached 86.05%. The results showed that the methods proposed in this paper were effective to analyze auscultation signals, and the performance of model can be greatly improved when the distinction of gender was considered. PMID:24883068

Yan, Jian-Jun; Wang, Yi-Qin; Liu, Guo-Ping; Yan, Hai-Xia; Xia, Chun-Ming; Shen, Xiaojing

2014-01-01

135

Objective Auscultation of TCM Based on Wavelet Packet Fractal Dimension and Support Vector Machine.

This study was conducted to illustrate that auscultation features based on the fractal dimension combined with wavelet packet transform (WPT) were conducive to the identification the pattern of syndromes of Traditional Chinese Medicine (TCM). The WPT and the fractal dimension were employed to extract features of auscultation signals of 137 patients with lung Qi-deficient pattern, 49 patients with lung Yin-deficient pattern, and 43 healthy subjects. With these features, the classification model was constructed based on multiclass support vector machine (SVM). When all auscultation signals were trained by SVM to decide the patterns of TCM syndromes, the overall recognition rate of model was 79.49%; when male and female auscultation signals were trained, respectively, to decide the patterns, the overall recognition rate of model reached 86.05%. The results showed that the methods proposed in this paper were effective to analyze auscultation signals, and the performance of model can be greatly improved when the distinction of gender was considered. PMID:24883068

Yan, Jian-Jun; Guo, Rui; Wang, Yi-Qin; Liu, Guo-Ping; Yan, Hai-Xia; Xia, Chun-Ming; Shen, Xiaojing

2014-01-01

136

Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact /linear flow at a characteristic crossover time, tau , in three dimensions to systems with as many as a 10(6) pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, ttau. Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, ttau, the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, beta=1.0+/-0.1. We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Lévy flights, the height variations across the interface appear to be random with occasional large height variations. PMID:19658710

Ferer, M; Bromhal, Grant S; Smith, Duane H

2009-07-01

137

NASA Astrophysics Data System (ADS)

The microwave radiation effects on EEG-signal have been studied by comparison with photo-stimulaton. The study of photos-stimulation effects at 16 Hz frequency and microwave radiation stimulation effects at 450 MHz modulated with 7 Hz frequency show fractal dimension increase.

Lipping, T.; Olejarczyk, E.; Parts, M.

2004-07-01

138

Random structures often exhibit fractal geometry, defined in terms of the mass scaling exponent, D, the fractal dimension. The vibrational dynamics of fractal networks are expressed in terms of the exponent d double bar, the fracton dimensionality. The eigenstates on a fractal network are spatially localized for d double bar less than or equal to 2. The implications of fractal

R. Orbach

1986-01-01

139

The aim of this study was to explore new techniques in analysing postural control using nonlinear time-series analysis and to relate these results with the clinical knowledge on the postural system in Down syndrome (DS) subjects. In order to achieve the goal, we analysed the time domain and the frequency domain behaviour, the fractal dimension and the entropy of the centre of pressure signal in both directions during quiet standing in 35 participants with DS, comparing the results with a control population. DS patients evidenced a lack in postural control in anterior-posterior direction due to the impairment both in the high organisation and synergies and in the impairments due to ligament laxity and hypotonia. Maintaining posture is a task achieved by the integration of visual, vestibular and somatosensory receptors and the dynamical nature of this signal gives fundamental data about the lack of postural control in specific pathological condition. PMID:22657255

Rigoldi, C; Galli, M; Mainardi, L; Albertini, G

2014-04-01

140

An improved vulnerability index of complex networks based on fractal dimension

With an increasing emphasis on network security, much more attention has been attracted to the vulnerability of complex networks. The multi-scale evaluation of vulnerability is widely used since it makes use of combined powers of the links' betweenness and has an effective evaluation to vulnerability. However, how to determine the coefficient in existing multi-scale evaluation model to measure the vulnerability of different networks is still an open issue. In this paper, an improved model based on the fractal dimension of complex networks is proposed to obtain a more reasonable evaluation of vulnerability with more physical significance. Not only the structure and basic physical properties of networks is characterized, but also the covering ability of networks, which is related to the vulnerability of the network, is taken into consideration in our proposed method. The numerical examples and real applications are used to illustrate the efficiency of our proposed method.

Gou, Li; Sadiq, Rehan; Mahadevan, Sankaran; Deng, Yong

2014-01-01

141

Journal of Coastal Research 22 5 1300Â1304 West Palm Beach, Florida September 2006 Fractal Analysis(5), 1300Â1304. West Palm Beach (Florida), ISSN 0749-0208. Average fractal dimensions (D) are calculated

Perfect, Ed

142

Determination of the fractal dimension for the epitaxial n-GaAs surface in the local limit

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.

Torkhov, N. A., E-mail: trkf@mail.ru; Bozhkova, V. G. [Scientific-Research Institute of Semiconductor Devices (Russian Federation); Ivonin, I. V.; Novikov, V. A. [Tomsk State University (Russian Federation)

2009-01-15

143

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

Jiménez, J; López, A M; Cruz, J; Esteban, F J; Navas, J; Villoslada, P; Ruiz de Miras, J

2014-10-01

144

The Fractal Geometrical Properties of Nuclei

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.

W. H. Ma; J. S. Wang; Q. Wang; S. Mukherjee; L. Yang; Y. Y. Yang; M. R. Huang; Y. J. Zhou

2014-06-06

145

The heterogeneity of regional pulmonary blood flow (RPBF) can be assessed by fractal analysis. The fractal dimension (FD) is a scale-independent measure of spatial heterogeneity of blood flow. The relative dispersion (RD) is often used to obtain the heterogeneity of RPBF but it is influenced by the resolution of measurement. The Blood Flow Analysis (BFA) System was developed in Delphi

D. A. Bottino; M. Kleen; G. Kemming; F. Meisner; O. Habler; K. Messmer

2001-01-01

146

ERIC Educational Resources Information Center

The complexity (fractal dimension value) of responses to the Rey-Osterrieth Complex Figure Test (ROCFT) between 10 undergraduate students with learning disabilities and a comparison group of 10 students without learning disabilities were compared. The fractal value of responses was assessed under three conditions (copy, immediate, and delay) by…

House, Garvey; Zelhart, Paul F.

147

Modelling Applicability of Fractal Analysis to Efficiency of Soil Exploration by Roots

• Background and Aims Fractal analysis allows calculation of fractal dimension, fractal abundance and lacunarity. Fractal analysis of plant roots has revealed correlations of fractal dimension with age, topology or genotypic variation, while fractal abundance has been associated with root length. Lacunarity is associated with heterogeneity of distribution, and has yet to be utilized in analysis of roots. In this study, fractal analysis was applied to the study of root architecture and acquisition of diffusion?limited nutrients. The hypothesis that soil depletion and root competition are more closely correlated with a combination of fractal parameters than by any one alone was tested. • Model The geometric simulation model SimRoot was used to dynamically model roots of various architectures growing for up to 16 d in three soil types with contrasting nutrient mobility. Fractal parameters were calculated for whole roots, projections of roots and vertical slices of roots taken at 0, 2·5 and 5 cm from the root origin. Nutrient depletion volumes, competition volumes, and relative competition were regressed against fractal parameters and root length. • Key Results Root length was correlated with depletion volume, competition volume and relative competition at all times. In analysis of three?dimensional, projected roots and 0 cm slices, log(fractal abundance) was highly correlated with log(depletion volume) when times were pooled. Other than this, multiple regression yielded better correlations than regression with single fractal parameters. Correlations decreased with age of roots and distance of vertical slices from the root origin. Field data were also examined to see if fractal dimension, fractal abundance and lacunarity can be used to distinguish common bean genotypes in field situations. There were significant differences in fractal dimension and fractal abundance, but not in lacunarity. • Conclusions These results suggest that applying fractal analysis to research of soil exploration by root systems should include fractal abundance, and possibly lacunarity, along with fractal dimension. PMID:15145791

WALK, THOMAS C.; VAN ERP, ERIK; LYNCH, JONATHAN P.

2004-01-01

148

Fractal dimension and size scaling of domains in thin films of multiferroic BiFeO3.

Domains in ferroelectric films are usually smooth, stripelike, very thin compared with magnetic ones, and satisfy the Landau-Lifshitz-Kittel scaling law (width proportional to square root of film thickness). However, the ferroelectric domains in very thin films of multiferroic BiFeO3 have irregular domain walls characterized by a roughness exponent 0.5-0.6 and in-plane fractal Hausdorff dimension H||=1.4+/-0.1, and the domain size scales with an exponent 0.59+/-0.08 rather than 1/2. The domains are significantly larger than those of other ferroelectrics of the same thickness, and closer in size to those of magnetic materials, which is consistent with a strong magnetoelectric coupling at the walls. A general model is proposed for ferroelectrics, ferroelastics or ferromagnetic domains which relates the fractal dimension of the walls to domain size scaling. PMID:18232925

Catalan, G; Béa, H; Fusil, S; Bibes, M; Paruch, P; Barthélémy, A; Scott, J F

2008-01-18

149

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

Manera, M; Dezfuli, B S; Borreca, C; Giari, L

2014-11-01

150

Fractal Dimension and Size Scaling of Domains in Thin Films of Multiferroic BiFeO3

Domains in ferroelectric films are usually smooth, stripelike, very thin compared with magnetic ones, and satisfy the Landau-Lifshitz-Kittel scaling law (width proportional to square root of film thickness). However, the ferroelectric domains in very thin films of multiferroic BiFeO3 have irregular domain walls characterized by a roughness exponent 0.5 0.6 and in-plane fractal Hausdorff dimension H||=1.4±0.1, and the domain size

G. Catalan; H. Béa; S. Fusil; M. Bibes; P. Paruch; A. Barthélémy; J. F. Scott

2008-01-01

151

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

2013-01-01

152

Gene Entropy-Fractal Dimension Informatics with Application to Mouse-Human Translational Medicine

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

Holden, T.; Cheung, E.; Dehipawala, S.; Ye, J.; Tremberger, G.; Lieberman, D.; Cheung, T.

2013-01-01

153

Gene entropy-fractal dimension informatics with application to mouse-human translational medicine.

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

Holden, T; Cheung, E; Dehipawala, S; Ye, J; Tremberger, G; Lieberman, D; Cheung, T

2013-01-01

154

Ictal EEG fractal dimension in ECT predicts outcome at 2 weeks in schizophrenia.

Studies of electroconvulsive therapy (ECT) have found an association between ictal electroencephalographic (EEG) measures and clinical outcome in depression. Such studies are lacking in schizophrenia. Consenting schizophrenia patients receiving ECT were assessed using the Brief Psychiatric Rating Scale (BPRS) before and 2 weeks after the start of ECT. The patients' seizure was monitored using EEG. In 26 patients, completely artifact-free EEG derived from the left frontal-pole (FP1) channel and electrocardiography (ECG) were available. The fractal dimension (FD) was computed to assess 4-s EEG epochs, and the maximal value from the earliest ECT session (2nd, 3rd or 4th) was used for analysis. There was a significant inverse correlation between the maximum FD and the total score following 6th ECT. An inverse Inverse correlation was also observed between the maximum FD and the total number of ECTs administered as well as the maximum heart rate (HR) and BPRS scores following 6th ECT. In patients with schizophrenia greater intensity of seizures (higher FD) during initial sessions of ECT is associated with better response at the end of 2 weeks. PMID:23261182

Abhishekh, Hulegar A; Thirthalli, Jagadisha; Manjegowda, Anusha; Phutane, Vivek H; Muralidharan, Kesavan; Gangadhar, Bangalore N

2013-09-30

155

Urban Malignancy: Similarity in the Fractal Dimensions of Urban Morphology and Malignant Neoplasms

All contemporary landscapes on the planet feature the aggressive growth of cities and other urbanizations. In 1990, we suggested that the similarity between urban forms and malignant lesions could be studied with the use of fractal geometry. Two separate disciplines have emerged since then: the study of urban morphology using various fractal analyses, and \\

Warren M. Hern

2008-01-01

156

NASA Astrophysics Data System (ADS)

Two silver ion conducting solid electrolytes, Ag2HgI4 and Ag2CdI4, representing a wide class of AgI-based halogenide superionics have been the subjects of study by means of electrical impedance spectroscopy, SEM, porosity measurements and fractal dimension analysis. Even though both materials have been obtained by the same method under strictly identical conditions they were found to exhibit certain differences at the microstructural level. Thus, by the direct measurements of porosity and density it was found that the grain boundaries are better developed in silver mercuric iodide. On the assumption that pore geometry in the materials under study displays fractal character it was shown that the fractal dimension of the pore contours is larger in the case of Ag2HgI4. These results are in agreement with electrical studies which indicated that the grain boundary capacitance in Ag2CdI4 is two orders of magnitude smaller than that of the silver mercuric iodide.

Bellucci, S.; Bolesta, I.; Karbovnyk, I.; Hrytskiv, R.; Fafilek, G.; Popov, A. I.

2008-11-01

157

Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection

NASA Astrophysics Data System (ADS)

Fractal dimension (FD) is a natural measure of the irregularity of a curve. In this study the performances of three waveform FD estimation algorithms (i.e. Katz's, Higuchi's and the k-nearest neighbour (k-NN) algorithm) were compared in terms of their ability to detect the onset of epileptic seizures in scalp electroencephalogram (EEG). The selection of parameters involved in FD estimation, evaluation of the accuracy of the different algorithms and assessment of their robustness in the presence of noise were performed based on synthetic signals of known FD. When applied to scalp EEG data, Katz's and Higuchi's algorithms were found to be incapable of producing consistent changes of a single type (either a drop or an increase) during seizures. On the other hand, the k-NN algorithm produced a drop, starting close to the seizure onset, in most seizures of all patients. The k-NN algorithm outperformed both Katz's and Higuchi's algorithms in terms of robustness in the presence of noise and seizure onset detection ability. The seizure detection methodology, based on the k-NN algorithm, yielded in the training data set a sensitivity of 100% with 10.10 s mean detection delay and a false positive rate of 0.27 h-1, while the corresponding values in the testing data set were 100%, 8.82 s and 0.42 h-1, respectively. The above detection results compare favourably to those of other seizure onset detection methodologies applied to scalp EEG in the literature. The methodology described, based on the k-NN algorithm, appears to be promising for the detection of the onset of epileptic seizures based on scalp EEG.

Polychronaki, G. E.; Ktonas, P. Y.; Gatzonis, S.; Siatouni, A.; Asvestas, P. A.; Tsekou, H.; Sakas, D.; Nikita, K. S.

2010-08-01

158

NASA Astrophysics Data System (ADS)

In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe in Japan, Merapi in Indonesia. Preliminary results reveal that the fractal dimension of natural areas, being related only to the roughness of the observed surface, is very stable as the radar illumination geometry, the resolution and the wavelength change, thus holding a very unique property in SAR data inversion. Such a behavior is not verified in case of non-natural objects. As a matter of fact, when the fractal estimation is performed in the presence of either man-made objects or SAR image features depending on geometrical distortions due to the SAR system acquisition (i.e. layover, shadowing), fractal dimension (D) values outside the range of fractality of natural surfaces (2 < D < 3) are retrieved. These non-fractal characteristics show to be heavily dependent on sensor acquisition parameters (e.g. view angle, resolution). In this work, the behaviour of the maps generated starting from the C- and X- band SAR data, relevant to all the considered volcanoes, is analyzed: the distribution of the obtained fractal dimension values is investigated on different zones of the maps. In particular, it is verified that the fore-slope and back-slope areas of the image share a very similar fractal dimension distribution that is placed around the mean value of D=2.3. We conclude that, in this context, the fractal dimension could be considered as a signature of the identification of the volcano growth as a natural process. The COSMO-SkyMed data used in this study have been processed at IREA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0.

Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.

2012-04-01

159

Fractal Geometry and Spatial Phenomena A Bibliography

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

California at Santa Barbara, University of

160

Biometric feature extraction using local fractal auto-correlation

NASA Astrophysics Data System (ADS)

Image texture feature extraction is a classical means for biometric recognition. To extract effective texture feature for matching, we utilize local fractal auto-correlation to construct an effective image texture descriptor. Three main steps are involved in the proposed scheme: (i) using two-dimensional Gabor filter to extract the texture features of biometric images; (ii) calculating the local fractal dimension of Gabor feature under different orientations and scales using fractal auto-correlation algorithm; and (iii) linking the local fractal dimension of Gabor feature under different orientations and scales into a big vector for matching. Experiments and analyses show our proposed scheme is an efficient biometric feature extraction approach.

Chen, Xi; Zhang, Jia-Shu

2014-09-01

161

NASA Astrophysics Data System (ADS)

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.

Roth, E. J.; Mays, D. C.

2013-12-01

162

Fractal measurements of sandstones, shales, and carbonates

NASA Astrophysics Data System (ADS)

Measurements were made of the fractal properties of sandstones, shales, and carbonates using a statistical analysis of structural features on fracture surfaces. Fractal behavior is associated with power law behavior for the number of features as a function of the feature size on the pore-rock interface. Only one sedimentary rock, a novaculite, was found not to have a fractal structure. The fractal dimensions range from 2.27 to 2.89, and the long-length limits to the fractal regime range from 2 ?m to over 50 ?m. In all cases, the fractal behavior extends to less than 0.2 ?m which is the measurement resolution. The porosity associated with the fractal pore-rock interface can be calculated from the fractal parameters. Some of the samples have additional porosity not associated with power law behavior. Photographs and other evidence are used to show that the fractal structures are the result of diagenesis. Fractal diagenetic structures include euhedral quartz overgrowths, druse quartz, calcite, dolomite, clays, and chert.

Krohn, Christine E.

1988-04-01

163

Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions

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.

Gospodinov, Dragomir [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria); Geophysical Institute of Bulgarian Academy of Sciences, Akad. G. Bonchev Str., bl.3, Sofia (Bulgaria); Marekova, Elisaveta; Marinov, Alexander [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria)

2010-01-21

164

A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes

NASA Technical Reports Server (NTRS)

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.

Hsui, Albert T.; Rust, Kelly A.; Klein, George D.

1993-01-01

165

Discrete scale invariance, complex fractal dimensions, and log-periodic fluctuations in seismicity

We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described

H. Saleur; C. G. Sammis; D. Sornette

1996-01-01

166

the grey level to regenerate the image. By choosing the background as complete dark and the goose down = 1, black crp pgrey level = 0, white A goose down image thus enhanced is seen in Figure 4 Golden Mean and Fractal Dimension of Goose Down Jing Gao a* , Ning Pan b , Weidong Yu a a Textile College

Pan, Ning

167

Image reconstruction in multislice spiral/helical computed tomography (MSCT) consists of a package of data on the arbitrary direction of the Z-axis that can be collected by active detector arrays. Thus the recombined data vary with each spiral pitch. In certain cases of spiral pitch, data compression can occur, and the spiral artifacts that are characteristic of MSCT would change. In our study, we evaluated image complications by fractal dimensions, because the geometrical patterns from a conic phantom are closely related to data transfer in the direction of the Z-axis in spiral pitches. We hoped to establish useful spiral pitches and slice collimation for clinical use in a 4-row MSCT scanner. By employing a conic phantom of 120 mm in diameter and a cone angle of 100 degrees, we measured the fractal dimension of the conic phantom image by making a binary to outline from 2.0 to 8.0 of various slice collimations. Moreover, in order to evaluate the correlation between fractal dimensions and image artifacts, we confirmed the influence of spiral pitch and reconstruction slice thickness for clinical use. We found that, when the reconstruction slice thickness was the same, the cross section of the conic phantom that was from thin-slice collimations was more similar to an actual circle than that of wide-slice collimations. The former deserved a low value and showed slight changes, and, therefore, its fractal dimensions were fixed. As a phenomenon worthy of attention, when we employed wide-slice collimations (4x5.0 mm) during peculiar low spiral pitches of 2.5 to 3.0 fractal dimensions remained low and similar to an actual circle. By these analyses of the influence of data transfer in the direction of the Z-axis, we found that spiral pitch influenced the rate of slice collimation used for data acquisition closely to the reconstruction slice thickness. Based on these findings, when slice collimations and reconstruction slice thickness should be made equal, we estimated that spiral pitches of low image artifacts in 4row MSCT ranged from 2.5 to 3.0 using fractal dimensions. We consider that a new adaptation of fractal dimension analysis is possible when it is used as an index in determining protocols. PMID:12743520

Hara, Takanori; Kato, Hideki; Tsuzaka, Masatoshi

2003-04-01

168

Fractal analysis of high-resolution CT images as a tool for quantification of lung diseases

Fractal geometry is increasingly being used to model complex naturally occurring phenomena. There are two types of fractals in nature-geometric fractals and stochastic fractals. The pulmonary branching structure is a geometric fractal and the intensity of its grey scale image is a stochastic fractal. In this paper, the authors attempt to quantify the texture of CT lung images using properties of both types of fractals. A simple algorithm for detecting of abnormality in human lungs, based on 2-D and 3-D fractal dimensions, is presented. This method involves calculating the local fractal dimensions, based on intensities, in the 2-D slice to air edge enhancement. Following this, grey level thresholding is performed and a global fractal dimension, based on structure, for the entire data is estimated in 2-D and 3-D. High Resolution CT images of normal and abnormal lungs were analyzed. Preliminary results showed that classification of normal and abnormal images could be obtained based on the differences between their global fractal dimensions.

Uppaluri, R.; Mitsa, T.; Galvin, J.R. [Univ. of Iowa, Iowa City, IA (United States)

1995-12-31

169

NASA Astrophysics Data System (ADS)

Visual media processing is becoming increasingly important because of the wide variety of image and video based applications. Recently, several architectures have been reported in the literature to implement image and video processing algorithms. They range from programmable DSP processors to application specific integrated circuits (ASICs). DSPs have to be software programed to execute individual operations in image and video processing. However they are not suitable for real-time execution of highly compute intensive applications such as fractal block processing (FBP). On the other hand, dedicated architectures and ASICs are designed to implement specific functions. Since they are optimized for a specific task, they cannot be used in a wide variety of applications. In this paper, we propose a parallel and pipelined architecture called fractal engine to implement the operations in FBP. Fractal engine is simple, modular, scaleable and is optimized to execute both low level and mid level operations. We note that implementation of the basic operations by fractal engine enables efficient execution of a majority of visual computing tasks. These include spatial filtering, contrast enhancement, frequency domain operations, histogram calculation, geometric transforms, indexing, vector quantization, fractal block coding, motion estimation, etc. The individual modules of fractal engine have been implemented in VHDL (VHSIC hardware description language). We have chosen to demonstrate the real-time execution capability of fractal engine by mapping a fractal block coding (FBC) algorithm onto the proposed architecture.

Fatemi, Omid; Panchanathan, Sethuraman

1997-01-01

170

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

Berry, Hugues

2002-01-01

171

The scalp distribution of the fractal dimension of the EEG and its variation with mental tasks

Summary The insights gained by the concept of deterministic chaos for the EEG is that this seemingly disordered process may be governed by relatively few simple laws which could be determined. One of the quantitative measures of a complex dynamical system is that of its dimension. The term ‘dimension’ refers to the ability of a space to contain a set

W. Lutzenberger; T. Elbert; N. Birbaumer; W. J. Ray; H. Schupp

1992-01-01

172

Fractal dimension of the topological charge density distribution in SU(2) lattice gluodynamics

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.

P. V. Buividovich; T. Kalaydzhyan; M. I. Polikarpov

2011-11-29

173

The Fractal Distribution of HII Regions in Disk Galaxies

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.

Nestor Sanchez; Emilio J. Alfaro

2008-04-29

174

Electrocardiogram signal classification based on fractal features

Atrial fibrillation ECG signals have been classified with fractal features only. The fractal features -fractal dimension, mass dimension and lacunarities were estimated by a new box counting algorithm; called the true box counting method. The classification result and stepwise discriminant analysis for these fractal features were determined. It was seen that lacunarities based on higher mass moments were more important

A. N. Esgiar; P. K. Chakravorty

2004-01-01

175

Fault diagnosis of diesel engine based on adaptive wavelet packets and EEMD-fractal dimension

NASA Astrophysics Data System (ADS)

In this paper a novel method for de-noising nonstationary vibration signal and diagnosing diesel engine faults is presented. The method is based on the adaptive wavelet threshold (AWT) de-noising, ensemble empirical mode decomposition (EEMD) and correlation dimension (CD). A new adaptive wavelet packet (WP) thresholding function for vibration signal de-noising is used in this paper. To alleviate the mode mixing problem occurring in EMD, ensemble empirical mode decomposition (EEMD) is presented. With EEMD, the components with truly physical meaning can be extracted from the signal. Utilizing the advantage of EEMD, this paper proposes a new AWT-EEMD-based method for fault diagnosis of diesel engine. A study of correlation dimension in engine condition monitoring is reported also. Some important influencing factors relating directly to the computational precision of correlation dimension are discussed. Industrial engine normal and fault vibration signals measured from different operating conditions are analyzed using the above method.

Wang, Xia; Liu, Changwen; Bi, Fengrong; Bi, Xiaoyang; Shao, Kang

2013-12-01

176

P-adic coverage method in fractal analysis of showers

NASA Astrophysics Data System (ADS)

Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/ k parts of the space studied.

Dedovich, T. G.; Tokarev, M. V.

2011-11-01

177

Fractal structure of the Kashubian hydrographic system

NASA Astrophysics Data System (ADS)

SummaryThe research included seven drainage basins comprising of 30 sub-basins of the Kashubian hydrographic system (ksh), part of the Pomeranian Lake District in northern Poland. In the study maps of 1:50 000 scale were used. The river network is developing according to the Horton's laws. The network as identified with the method of hydrographic interpretation, while its ordering was based on the methods of Horton-Strahler (Horton, 1945) and of Drwal (Drwal, 1982). The fractal dimension was calculated by two methods: one of them is based on the bifurcation ratio and the stream length ratio and is called geomorphic fractal dimension, and second method to estimate the fractal dimension of river networks is functional box counting (is called raster fractal dimension). Ordering of a network by the results of the Drwal method means reducing the bifurcation ratio (Rb) and the average length ratio (Rl) values in comparison with the analogous values obtained with the Horton-Strahler method. In the network analysis more reliable results were obtained by using Drwal's method, as the values of the geomorphic fractal dimension obtained with the Horton-Strahler method do not always fall within the range specified for the fractal dimension, e.g. between 1 and 2. However, in the box-counting method (raster dimension) it is important to adopt as short measurement lines as possible or as many measurement line sections as possible, and not only two. The ksh river network is still in the organization stage. Its most mature stage is represented by the network of the catchments A2 (D = 1.89), C3 (D = 1.97) and D (D = 1.97) on the northern slope and the network G (D = 1.79) on the southern slope.

Fac-Beneda, Joanna

2013-04-01

178

NASA Astrophysics Data System (ADS)

Lipase B from Candida Antarctica (also known as Candida antarctica lipase B or CALB) was immobilized onto titanium dioxide (TiO 2) in a buffer-free, bidistilled aqueous medium. The adsorption isotherm was determined by UV-vis analysis of supernatant solution at 280 nm, revealing that in 7 h 98% of the theoretical lipase monolayer on the TiO 2 (with 45.7 m 2/g BET area) was achieved. Samples withdrawn from the supernatant immobilization medium were analyzed by Fourier-transform infrared spectroscopy in the 1700-1600 cm -1 range (where the Amide I Proteins band appears) in order to obtain quantitative information on the evolution of the secondary-structure elements of the protein. The analysis performed revealed that lipase conformation suffers only minor changes during its adsorption onto TiO 2. However, water associated to the lipase may interact of several ways with the surface of the hydrated oxide. Characterization of the immobilized biocatalyst (CALB/TiO 2) implied SEM, fractal dimension analysis and FTIR techniques. A proposal of lipase-hydrated oxide interaction is presented.

Foresti, M. L.; Valle, G.; Bonetto, R.; Ferreira, M. L.; Briand, L. E.

2010-01-01

179

Fractal Pore Structure Model and Multilayer Fractal Adsorption in Shale

NASA Astrophysics Data System (ADS)

The complex structure and surface property of porous media have significant impact on its accumulation and adsorption capacity. Based on the fractal theory, this paper presents a fractal pore structure model for shales. The effect of different pore structures on fractal dimension is discussed, and the influence of fractal dimension and pore size distribution on porosity is also analyzed. It is shown that the fractal dimension D decreases with the increase of structure parameter q/m for a certain pore diameter ratio, and porosity has positive relationship with fractal dimension. This paper also presents a multilayer fractal adsorption model which takes into account the roughness of adsorption surface by using fractal theory. With the introduction of pseudo-saturated vapor pressure in the supercritical temperature condition, the proposed adsorption model can be applied into a wider range of temperature. Based on the low-pressure nitrogen adsorption and methane isothermal adsorption experiments, the effect of fractal dimension on the adsorption behavior of shales is discussed. Fractal dimension has significant impact on the surface adsorption property and adsorption layer number n. The monolayer saturated adsorption volume Vm increases with the increase of D, while parameter C has the opposite variation trend. Finally, the optimal combination of fractal parameters for describing pore structure of shale samples is selected.

Zhang, Liehui; Li, Jianchao; Tang, Hongming; Guo, Jingjing

2014-09-01

180

NASA Astrophysics Data System (ADS)

Following an earlier development of a Fokker-Planck equation (FP) for modeling fractal-scale-dependent transport of solutes in one-dimensional subsurface flow of heterogeneous porous media, this technical note extends the FP to three dimensions, and presents a two-dimensional (2-D) FP by reducing the 3-D FP with the aid of the Dupuit approximation. The 2-D FP is derived by including two fractal dispersivities in the convective-dispersive equation leading to a generalized Feller-Fokker-Planck equation (GFFP) featuring both the generalized Feller equation (GF) and FP. Similarity solutions of the 2-D GFP with two linear-scale-dependent dispersivities are presented which can be used as a kernel in the convolution integral to yield an output on a real timescale, and the input function can be derived by a procedure known as the inverse problem with the aid of a Laplace transform.

Su, Ninghu

1997-05-01

181

Rationale A widely applicable model of emphysema that allows efficient and sensitive quantification of injury is needed to compare potential therapies. Objectives To establish such a model, we studied the relationship between elastase dose and the severity of emphysema in female C57BL/6J mice. We compared alveolar fractal box dimension (DB), a new measure which is an assessment of the complexity of the tissue, with mean linear intercept (Lm), which is commonly used to estimate airspace size, for sensitivity and efficiency of measurement. Methods Emphysema was induced in female C57BL/6J mice by administering increasing intratracheal doses of porcine pancreatic elastase (PPE). Changes in morphology and static lung compliance (CL) were examined 21 days later. Correlation of DB with Lm was determined in histological sections of lungs exposed to PPE. The inverse relationship between DB and Lm was supported by examining similar morphological sections from another experiment where the development of emphysema was studied 1 to 3 weeks after instillation of human neutrophil elastase (HNE). Results Lm increased with PPE dose in a sigmoidal curve. CL increased after 80 or 120 U/kg body weight (P < 0.05), but not after 40 U/kg, compared with the control. DB progressively declined from 1.66 ± 0.002 (standard error of the mean) in controls, to 1.47 ± 0.006 after 120 U PPE/kg (P < 0.0001). After PPE or HNE instillation, DB was inversely related to Lm (R = ?0.95, P < 0.0001 and R = ?0.84, P = 0.01, respectively), with a more negative slope of the relationship using HNE (P < 0.0001). Conclusion Intratracheal instillation of increasing doses of PPE yields a scale of progression from mild to severe emphysema. DB correlates inversely with Lm after instillation of either PPE or HNE and yields a rapid, sensitive measure of emphysema after elastase instillation. PMID:22500123

Andersen, Mary P; Parham, A Read; Waldrep, J Clifford; McKenzie, Wayland N; Dhand, Rajiv

2012-01-01

182

Background The evaluation of intestinal trophism, mainly the mucosal layer, is an important issue in various conditions associated with injury, atrophy, recovery, and healing of the gut. The aim of the present study was to evaluate the kinetics of the proliferation and apoptosis of enterocytes by immunohistochemistry and to assess the complexity of intestinal mucosa by fractal dimension (FD) analysis in Solea solea fed different experimental diets. Results Histomorphological evaluation of all intestinal segments did not show signs of degeneration or inflammation. Cell proliferation index and FD were significantly reduced with a diet high in mussel meal (MM; p?=?0.0034 and p?=?0.01063, respectively), while apoptotic index did not show any significant difference for the same comparison (p?=?0.3859). Linear regression analysis between apoptotic index (independent variable) and FD (dependent variable) showed a statistically significant inverse relationship (p?=?0.002528). Linear regression analysis between cell proliferation index (independent variable) and FD (dependent variable) did not show any significant correlation (p?=?0.131582). Conclusions The results demonstrated that diets containing increasing levels of mussel meal in substitution of fishmeal did not incite a hyperplastic response of the intestinal mucosa. The mussel meal, which is derived from molluscs, could mimic the characteristics of the sole’s natural prey, being readily digestible, even without increasing the absorptive surface of intestinal mucosa. Interestingly, from this study emerged that FD could be used as a numeric indicator complementary to in situ quantification methods to measure intestinal trophism, in conjunction with functional parameters. PMID:24997003

2014-01-01

183

On the Fractal Distribution of HII Regions in Disk Galaxies

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.

Nestor Sanchez; Emilio J. Alfaro

2008-10-02

184

Fractal characteristics of an asphaltene deposited heterogeneous surface

NASA Astrophysics Data System (ADS)

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.

Amin, J. Sayyad; Ayatollahi, Sh.; Alamdari, A.

185

Gravitation theory in a fractal space-time

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.

Agop, M.; Gottlieb, I. ['Gh. Asachi' Department of Physics, Technical University, Blvd. Mangeron, 700029, Iasi (Romania); Faculty of Physics, Department of Theoretical Physics, 'Al.I.Cuza' University, Blvd. Carol No.1, 700506, Iasi (Romania)

2006-05-15

186

Fractal analysis of RF signals scattered by small-scale ionospheric irregularities

NASA Astrophysics Data System (ADS)

The fractal dimension as a characteristic of an experimental data series is considered. The correlation integral method is used for dimension calculation. It is shown that by the fractal dimension one can identify a variety of ionospheric processes even when the conventional spectral analysis fails. It is found that the realizations corresponding to volume scattering by natural and artificial irregularities have finite dimension, which is significantly different. A technique for preparing experimental data to be processed by the correlation integral method is developed. The influence of the data sampling rate and signal-to-noise ratio on the dimension is analyzed.

Bulgakov, S. A.; Ponomarenko, P. V.; Yampolsky, Yu. M.

1995-06-01

187

Small-angle scattering from generalized self-similar Vicsek fractals

NASA Astrophysics Data System (ADS)

An analytical approach for calculating the small-angle X-ray or neutron scattering (SAXS/SANS) from generalized self-similar Vicsek fractals (GSSVF) is presented; each fractal consists of spherical subunits. The system considered is a mass-fractal, generated iteratively from a regular 3D Vicsek fractal structure. Its fractal dimension is controllable and increases with increasing the value of the scaling factor. Small-angle scattering (SAS) intensity is determined from a set of non-interacting, randomly oriented and uniformly distributed GSSVF fractals. It is shown that in the fractal region, the curve I(q)qD is approximately log-periodic with the period equal to the scaling factor of fractal; here D and I(q) are the fractal dimension and the SAS intensity, respectively. In particular, the positions of deepest minima and highest maxima are log-periodic, and their number coincides with the number of fractal iterations. The log-periodicity of the scattering curves is a consequence of the self-similarity of GSSVF.

Cherny, Alexander Yu; Anitas, Eugen M.; Osipov, Vladimir A.; Kuklin, Alexander I.

2012-03-01

188

NASA Astrophysics Data System (ADS)

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.

Davarpanah, A.; Babaie, H. A.

2012-12-01

189

Map of fluid flow in fractal porous medium into fractal continuum flow.

This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided. PMID:23004869

Balankin, Alexander S; Elizarraraz, Benjamin Espinoza

2012-05-01

190

Map of fluid flow in fractal porous medium into fractal continuum flow

NASA Astrophysics Data System (ADS)

This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow ds is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.

Balankin, Alexander S.; Elizarraraz, Benjamin Espinoza

2012-05-01

191

Electromagnetism on Anisotropic Fractals

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.

Martin Ostoja-Starzewski

2011-06-08

192

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

Prusinkiewicz, Przemyslaw

193

NASA Astrophysics Data System (ADS)

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).

Maria, Anton; Carey, Steven

2007-03-01

194

Fractal analysis: A new remote sensing tool for lava flows

NASA Technical Reports Server (NTRS)

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.

Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.

1992-01-01

195

[Application of fractal theory to spectral signal recognition].

The fractal theory is a discipline that studies a kind of irregular and chaotic object with similarity of its part and whole. The fractal dimension is a basic index mark of irregularity and self-similarity in the fractal theory. The present paper takes the spectral signal according with Lambert-beer' law as the object, introduces the basic theory of fractal geometry in brief, puts forward the method of fractal dimension as the feature of spectral signal recognition, makes use of reconstructing phase space to gain the fractal dimension of spectral signal, compares different values of the fractal dimension to recognize different spectral signal, and gives an example for explanation. PMID:16836159

Xiong, Yu-hong; Wen, Zhi-yu; Zhang, Liu-qiang; Wen, Zhong-quan; Liang, Yu-qian

2006-04-01

196

Fractal and multifractal properties of a family of fractal networks

NASA Astrophysics Data System (ADS)

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the ?(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension

Li, Bao-Gen; Yu, Zu-Guo; Zhou, Yu

2014-02-01

197

Fractal antenna engineering: the theory and design of fractal antenna arrays

A fractal is a recursively generated object having a fractional dimension. Many objects, including antennas, can be designed using the recursive nature of a fractal. In this article, we provide a comprehensive overview of recent developments in the field of fractal antenna engineering, with particular emphasis placed on the theory and design of fractal arrays. We introduce some important properties

Douglas H. Werner; R. L. Haupt; P. L. Werner

1999-01-01

198

The Use of Fractals for the Study of the Psychology of Perception:

NASA Astrophysics Data System (ADS)

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.

Mitina, Olga V.; Abraham, Frederick David

199

We investigate from the fractal viewpoint the way in which the dark matter is grouped at z = 0 in the Millennium dark matter cosmological simulation. The determination of the cross to homogeneity in the Millennium Simulation data is described from the behaviour of the fractal dimension and the lacunarity. We use the sliding window technique to calculate the fractal mass-radius dimension, the pre-factor F and the lacunarity of this fractal relation. Besides, we determinate the multi-fractal dimension and the lacunarity spectrum, including their dependence with radial distance. This calculations show a radial distance dependency of all the fractal quantities, with heterogeneity clustering of dark matter haloes up to depths of 100 Mpc/h. The dark matter haloes clustering in the Millennium Simulation shows a radial distance dependency, with two regions clearly defined. The lacunarity spectrum for values of the structure parameter q >= 1 shows regions with relative maxima, revealing the formation of clusters and voids in the dark matter haloes distribution. With the use of the multi-fractal dimension and the lacunarity spectrum, the transition to homogeneity at depths between 100 Mpc/h and 120 Mpc/h for the Millennium Simulation dark matter haloes is detected.

César A. Chacón-Cardona; Rigoberto A. Casas-Miranda

2012-09-12

200

Fractal patterns in the human retina and their physiological correlates

NASA Astrophysics Data System (ADS)

The biological mechanism for the formation of retinal vessel patterns in the developing human eye is unknown even though it is a question of importance. The current hypothesis is based on the existence of a variable oxygen gradient across the developing photoreceptors which stimulates the release of angiogenic factors which diffuse in the plane of the retina and result in the growth of retinal vessels. This implies that the rate-limiting step in the formation of the vessel pattern is a diffusion process. In order to analyze this hypothesis the fractal dimension of the retinal blood vessel patterns was determined. Several methods were used to calculate the fractal dimension. Red-free fundus images of normal humans were traced and two methods of analysis were used: the massradius relation and the scaling relation of the two-point densitydensity correlation. In addition the vessel patterns were digitized using a digital image processing system and the number of pixels corresponding to the retinal vessels were determined within circles of various diameters. The slope of the log of the size of the circle versus the log of the number of voxels yielded the fractal dimension. All three methods applied to retinal patterns derived from 15 different normal humans (age 14-65) showed that the human retinal blood vessels have a self-similar structure with a fractal dimension of about 1 . 7. This is the same fractal dimension found for

Masters, Barry R.

1991-04-01

201

been to the study of fractal objects [11. An increasing used extensively to analyze small experimental the extremely large numbers of iterates needed for the conver- gence of the algorithms. We study quantitatively-dimensional diffusion-limited aggregates [5,6] and strange dynamical systems as well as on delay differential attractors

Texas at Austin. University of

202

Fractal Geometry of Architecture

NASA Astrophysics Data System (ADS)

In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.

Lorenz, Wolfgang E.

203

Fractal structures and processes

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.}

Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M. [National Simulation Resource, Department of Bioengineering, University of Washington, Seattle, Washington 98195 (United States)

1996-06-01

204

Fractal analysis of remotely sensed images: A review of methods and applications

Mandelbrot's fractal geometry has sparked considerable interest in the remote sensing community since the publication of his highly influential book in 1977. Fractal models have been used in several image processing and pattern recognition applications such as texture analysis and classification. Applications of fractal geometry in remote sensing rely heavily on estimation of the fractal dimension. The fractal dimension (D)

W. Sun; G. Xu; P. Gong; S. Liang

2006-01-01

205

Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background

Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background Naoki KOBAYASHI and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the fractal dimension De of the entire pattern of isothermal lines, the fractal dimension Dc of a single

Chiang, Lung-Yih

206

NASA Astrophysics Data System (ADS)

We initially prepared films of poly(vinylidene fluoride-co-hexafluoropropylene)/poly(ethyl methacrylate)-ammonium trifluorome-thanesulfonate dispersed with various wt.% of chromium oxide to study their properties and potential application in electrochemical devices. However, a few months later the nanocomposite polymer electrolyte membranes were found to become a sort of medium for fractal growth. This discovery led to a simulation of the fractals observed in these polymer electrolyte films using a diffusion-limited aggregation model that is based on Brownian motion theory (random walk). The fractal dimensions, D, of the fractal patterns obtained from experimental and simulation work were calculated using the box-counting method. The fractal patterns and dimensions of the simulated fractal patterns were comparable with those obtained from the original fractals observed in the polymer electrolyte films.

Amir, S.; Hashim Ali, S. A.; Mohamed, N. S.

2011-10-01

207

NASA Astrophysics Data System (ADS)

The differentiation between normal and abnormal photoreceptor rearrangement before and after treatments may improve understanding on the sequence of events involved in the visual field defects. In this study, we evaluated a fractal analysis approach to quantify photoreceptor rearrangement and vision restoration. We analyzed Optical Coherence Tomography (OCT) data from an individual with outer retinopathy before and after treatment. The outer nuclear layer (ONL) was delineated from the rest of the retinal structure by using a custom-built segmentation algorithm. We then determined the fractal box dimension of the ONL's outline using the box counting method. Thickness and reflectance of the ONL were also calculated. Our results showed that the ONL's fractal dimension, thickness and relative reflectivity decreased after treatment. These early results showed that ONL's fractal dimension could be used as an index of photoreceptor rearrangement, which might lead to a more effective approach to therapy and improved diagnosis.

Cabrera Debuc, Delia; Tchitnga, Robert

2009-03-01

208

Anomalous thermal conduction in one dimension: a quantum calculation.

In this paper, we study the thermal conductivity of an anharmonically coupled chain of atoms. Numerical studies using classical dynamics have shown that the conductivity of a chain with nearest neighbor couplings diverges with chain length L as L(alpha); earlier studies found alpha approximately = 0.4 under a range of conditions, but a recent study on longer chains claims alpha = 1/3. Analytically, this problem has been studied by calculating the relaxation rate gamma(q) of the normal modes of vibration as a function of its wave vector q. Two theoretical studies of classical chains, one using the mode-coupling formulation and the other the Boltzmann equation method, led to gamma(q) proportional to q(5/3), which is consistent with alpha = 0.4. Here we study the problem for a quantum anharmonic chain with quartic anisotropy. We develop a low-temperature expansion for gamma(q) and find that, in the regime Dirac's constant omega(q) < k(B)T, gamma(q) is proportional to q(5/3)T2, where omega(q) is the frequency of the mode. In our analysis, the relaxation arises due to umklapp scattering processes. We further evaluate the thermal conductivity of the chain using the Kubo formula, which enables us to take into account the transport relaxation time through vertex corrections for the current-current correlator. This calculation also yields alpha = 0.4. PMID:17930004

Santhosh, G; Kumar, Deepak

2007-08-01

209

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. We focus on the images sent by the Cassini spacecraft mission: slide #42 "Mapping Clumps in Saturn's Rings" and slide #54 "Scattered Sunshine". Using the box-counting method, we determine the fractal dimension of rings seen here (and in several other images from the same source) to be consistently about 1.6~1.7. This supports many conjectures put forth over several decades that Saturn's rings are indeed fractal.

Li, Jun

2012-01-01

210

Higuchi Dimension of Digital Images

There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied. PMID:21931854

Ahammer, Helmut

2011-01-01

211

The pathological structures conjured up by 19th-century mathematicians have, in recent years, taken the form of fractals, mathematical figures that have fractional dimension rather than the integral dimensions of familiar geometric figures (such as one-dimensional lines or two-dimensional planes). Fractals are much more than a mathematical curiosity. They offer an extremely compact method for describing objects and formations. Many structures have an underlying geometric regularity, known as scale invariance or self-similarity. If one examines these objects at different size scales, one repeatedly encounters the same fundamental elements. The repetitive pattern defines the fractional, or fractal, dimension of the structure. Fractal geometry seems to describe natural shapes and forms more gracefully and succinctly than does Euclidean geometry. Scale invariance has a noteworthy parallel in contemporary chaos theory, which reveals that many phenomena, even though they follow strict deterministic rules, are in principle unpredictable. Chaotic events, such as turbulence in the atmosphere or the beating of a human heart, show similar patterns of variation on different time scales, much as scale-invariant objects show similar structural patterns on different spatial scales. The correspondence between fractals and chaos is no accident. Rather it is a symptom of a deep-rooted relation: fractal geometry is the geometry of chaos.

Juergens, H.; Peitgen, H.O.; Saupe, D. (Univ. of Bremen (West Germany))

1990-08-01

212

The surface electromyography (sEMG) signal separation and decphompositions has always been an interesting research topic in the field of rehabilitation and medical research. Subtle myoelectric control is an advanced technique concerned with the detection, processing, classification, and application of myoelectric signals to control human-assisting robots or rehabilitation devices. This paper reviews recent research and development in independent component analysis and Fractal dimensional analysis for sEMG pattern recognition, and presents state-of-the-art achievements in terms of their type, structure, and potential application. Directions for future research are also briefly outlined. PMID:21416388

Naik, Ganesh R; Arjunan, Sridhar; Kumar, Dinesh

2011-06-01

213

Objective Mammography is a widely used screening tool for the early detection of breast cancer. One of the commonly missed signs of\\u000a breast cancer is architectural distortion. The purpose of this study is to explore the application of fractal analysis and\\u000a texture measures for the detection of architectural distortion in screening mammograms taken prior to the detection of breast\\u000a cancer.\\u000a \\u000a \\u000a \\u000a Materials

Rangaraj M. Rangayyan; Shormistha Prajna; Fábio J. Ayres; J. E. Leo Desautels

2008-01-01

214

Metamaterial model of fractal time

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.

Igor I. Smolyaninov

2011-10-11

215

Fractal Geometric Characterization of Functionally Graded Materials

Fractal Geometric Characterization of Functionally Graded Materials A. Saharan1 ; M. Ostoja. Author keywords: Fractal; Fractal dimension; Functionally graded materials (FGM); Scale dependence, since the 1980s, there has been a major research effort on the so-called functionally graded materials

Ostoja-Starzewski, Martin

216

Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface

NASA Astrophysics Data System (ADS)

In recent years, linear fractal sea surface models have been developed for the sea surface in order to establish an electromagnetic backscattering model. Unfortunately, the sea surface is always nonlinear, particularly at high sea states. We present a nonlinear fractal sea surface model and derive an electromagnetic backscattering model. Using this model, we numerically calculate the normalized radar cross section (NRCS) of a nonlinear sea surface. Comparing the averaged NRCS between linear and nonlinear fractal models, we show that the NRCS of a linear fractal sea surface underestimates the NRCS of the real sea surface, especially for sea states with high fractal dimensions, and for dominant ocean surface gravity waves that are either very short or extremely long.

Xie, Tao; He, Chao; William, Perrie; Kuang, Hai-Lan; Zou, Guang-Hui; Chen, Wei

2010-02-01

217

[Lithology feature extraction of CASI hyperspectral data based on fractal signal algorithm].

Hyperspectral data is characterized by combination of image and spectrum and large data volume dimension reduction is the main research direction. Band selection and feature extraction is the primary method used for this objective. In the present article, the authors tested methods applied for the lithology feature extraction from hyperspectral data. Based on the self-similarity of hyperspectral data, the authors explored the application of fractal algorithm to lithology feature extraction from CASI hyperspectral data. The "carpet method" was corrected and then applied to calculate the fractal value of every pixel in the hyperspectral data. The results show that fractal information highlights the exposed bedrock lithology better than the original hyperspectral data The fractal signal and characterized scale are influenced by the spectral curve shape, the initial scale selection and iteration step. At present, research on the fractal signal of spectral curve is rare, implying the necessity of further quantitative analysis and investigation of its physical implications. PMID:25095444

Tang, Chao; Chen, Jian-Ping; Cui, Jing; Wen, Bo-Tao

2014-05-01

218

Fractal Function Estimation via Wavelet Shrinkage Yazhen Wang

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

Wang, Yazhen

219

Fractal algebras of discretization sequences Steffen Roch (TU Darmstadt)

Abstract Fractal algebras of discretization sequences Steffen Roch (TU Darmstadt) First a warning: Fractality, in the sense of these lectures, has nothing to do with fractal geometries or broken dimensions or other involved things. Rather, the notion fractal algebra had been chosen in order to emphasize

Potts, Daniel

220

The Fractal Geometrical Properties of Nuclei

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 ...

Ma, W H; Wang, Q; Mukherjee, S; Yang, L; Yang, Y Y; Huang, M R; Zhou, Y J

2014-01-01

221

Fractal applications to complex crustal problems

NASA Technical Reports Server (NTRS)

Complex scale-invariant problems obey fractal statistics. The basic definition of a fractal distribution is that the number of objects with a characteristic linear dimension greater than r satisfies the relation N = about r exp -D where D is the fractal dimension. Fragmentation often satisfies this relation. The distribution of earthquakes satisfies this relation. The classic relationship between the length of a rocky coast line and the step length can be derived from this relation. Power law relations for spectra can also be related to fractal dimensions. Topography and gravity are examples. Spectral techniques can be used to obtain maps of fractal dimension and roughness amplitude. These provide a quantitative measure of texture analysis. It is argued that the distribution of stress and strength in a complex crustal region, such as the Alps, is fractal. Based on this assumption, the observed frequency-magnitude relation for the seismicity in the region can be derived.

Turcotte, Donald L.

1989-01-01

222

Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds. PMID:17820893

Kopelman, R

1988-09-23

223

We report protocols and techniques to image and mechanically manipulate individual fibrin fibers, which are key structural components of blood clots. Using atomic force microscopy-based lateral force manipulations we determined the rupture force, FR, of fibrin fibers as a function of their diameter, D, in ambient conditions. As expected, the rupture force increases with increasing diameter; however, somewhat unexpectedly, it increases as FR ? D1.30±0.06. Moreover, using a combined atomic force microscopy-fluorescence microscopy instrument, we determined the light intensity, I, of single fibers, that were formed with fluorescently labeled fibrinogen, as a function of their diameter, D. Similar to the force data, we found that the light intensity, and thus the number of molecules per cross section, increases as I ? D1.25±0.11. Based on these findings we propose that fibrin fibers are fractals for which the number of molecules per cross section increases as about D1.3. This implies that the molecule density varies as ?(D) ? D?0.7, i.e., thinner fibers are denser than thicker fibers. Such a model would be consistent with the observation that fibrin fibers consist of 70–80% water and only 20–30% protein, which also suggests that fibrin fibers are very porous. PMID:15465869

Guthold, M.; Liu, W.; Stephens, B.; Lord, S. T.; Hantgan, R. R.; Erie, D. A.; Taylor, R. M.; Superfine, R.

2004-01-01

224

Fractal Physiology And Nuclear Medicine Scans

NASA Astrophysics Data System (ADS)

Measurement of the power spectra of liver scans reveals that the radiocolloid distribution in the human liver behaves as a fractal object. Analysis of the power spectra suggests that the fractal dimension of the functional units of the liver changes with disease state, and that power spectral slope may be a useful classifier for the presence of disease. Models are proposed that relate the power spectral slope to the fractal dimension of the liver parenchyma.

Cargill, E. B.; Barrett, H. H.; Fiete, R. D.; Ker, M.; Patton, D. D.; Seeley, G. W.

1988-06-01

225

Fractal scattering of microwaves from soils.

Using a combination of laboratory experiments and computer simulation we show that microwaves reflected from and transmitted through soil have a fractal dimension correlated to that of the soil's hierarchic permittivity network. The mathematical model relating the ground-penetrating radar record to the mass fractal dimension of soil structure is also developed. The fractal signature of the scattered microwaves correlates well with some physical and mechanical properties of soils. PMID:12398644

Oleschko, K; Korvin, G; Balankin, A S; Khachaturov, R V; Flores, L; Figueroa, B; Urrutia, J; Brambila, F

2002-10-28

226

Quantitative Characterization Of Basaltic Tephra Using The Fractal Spectrum Technique

NASA Astrophysics Data System (ADS)

Geologists have studied volcanic eruptions on Hawaii more closely than anywhere else. Even so, processes of magma fragmentation during Hawaiian style eruptions (e.g. lava fountaining) are not well understood. Furthermore, the products of these eruptions have not been fully characterized. Analysis of tephra shape is particularly useful for understanding the nature of eruptions, as particle morphology reflects numerous volcanic parameters (e.g. magma viscosity, volatile content, interaction with water, transport processes). The technique applied in this study, based on fractal geometry, uses data produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Previous applications of this technique have proven helpful for characterizing basaltic to rhyolitic products of specific fragmentation processes, and modes of volcanic transport/deposition. In this study, all the samples are basaltic, eliminating the variable of composition, and include material from two Hawaiian lava-fountaining events (Mauna Ulu, 1969; Kilauea Iki, 1959), as well as material from Masaya, Nicaragua (San Judas Formation) that is thought to have been unusually explosive, for comparison. One of our goals is to identify characteristic particle shapes formed during the lava-fountain events. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions, and also allows use of multivariate statistical techniques. Cluster analysis provides a visual display of the similarities of particles in a sample, and facilitates identification of the types of shapes that are most characteristic of a given deposit. Use of principal components analysis to summarize the data as accurately as possible using a few components, facilitates comparison between samples.

Maria, A.

2006-12-01

227

Fractal feature analysis and classification in medical imaging

Following B.B. Mandelbrot's fractal theory (1982), it was found that the fractal dimension could be obtained in medical images by the concept of fractional Brownian motion. An estimation concept for determination of the fractal dimension based upon the concept of fractional Brownian motion is discussed. Two applications are found: (1) classification; (2) edge enhancement and detection. For the purpose of

CHI-CHANG CHEN; JOHN S. DAPONTE; MARTIN D. FOX

1989-01-01

228

Calculation of the dimensions of dosage forms with release controlled by diffusion for in vivo use.

Using numerical models and data obtained from in vitro experiments, the dimensions of diffusion controlled release dosage forms to achieve desired in vivo levels are predicted. Monolithic polymer-drug devices are considered, the release of the drug being controlled by transient diffusion with constant diffusivity. The dimensions of the devices are calculated for various shapes (e.g. spheres, parallelepipeds, cylinders), so that 85% of the drug is released within 6 or 24 h, respectively. Caffeine, diltiazem HCl, and theophylline are studied in ethylcellulose (EC), plasticized with dibutyl sebacate (DBS) or acetyltributyl citrate (ATBC), respectively. The dosage forms are to be administered orally once a day. The resulting drug levels in the plasma are calculated using a numerical model that takes into account: the kinetics of drug release and the pharmacokinetic data of these dosage forms and drugs. Plasma levels resulting from immediate release dosage forms are also calculated, serving as reference. PMID:11154899

Ainaoui, A; Siepmann, J; Bodmeier, R; Vergnaud, J M

2001-01-01

229

Relativistic Fractal Cosmologies

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.

Marcelo B. Ribeiro

2009-10-26

230

Aim of this study was to investigate the application of normalized mean of the empirical Higuchi fractal dimension (FD) distributions, as a new approach to analyze the spontaneous bioelectrical activity of garden snail (Helix pomatia) Br neuron. The effect of ouabain on modulation of Br neuron bursting activity before and after the exposure to 10 mT static magnetic field (SMF) was observed by analyzing the following parameters: action potential (AP), interspike interval (ISI) and interbursting interval (IBI) components. Normalized mean of the empirical FD distributions were formed for the following experimental conditions: Control 1, Ouabain 1, Control 2, SMF 2, ASMF 2, Control 3, SMF 3 and Ouabain ASMF 3. Our main results have shown that ouabain without SMF induced increase in participation of AP and a decrease in participation of IBI components compared to the first control condition. However, in the presence of 10 mT SMF, ouabain-induced changes of measured parameters of Br neuron activity were less pronounced compared to the third control condition. We have shown that normalized mean of the empirical FD distributions is a useful method for detecting the changes in AP, ISI, and IBI components of complex bursting activity in altered physiological states. PMID:24968407

Kesi?, Srdjan; Nikoli?, Ljiljana; Savi?, Aleksandar G; Petkovi?, Branka; Spasi?, Sladjana Z

2014-01-01

231

Fractal characteristics of dendrite in aluminum alloys

NASA Astrophysics Data System (ADS)

The fractal dimensions of dendrites in Al-Si and Al-Cu binary alloys were measured under furnace cooling and casting experiments. The fractal dimension of the Al-Si alloy increased from 1.275 to 1.495 along with increase in Si content. The fractal dimension of the Al-Cu alloy increased from 1.139 to 1.486 along with increase in Cu content. The fractal dimension of the binary alloys also increased with increase in cooling rate during solidification. Phase-field simulations for the evolution of the dendrites in the binary aluminium alloys were carried out and a same tendency as the experimental results was obtained. The permeability of an Al-5mass%Si alloy was estimated from the measured fractal dimension of an experimentally observed dendrite structure. The estimated permeability agreed well with reported permeability of an Al-Si alloy.

Ohsasa, K.; Katsumi, T.; Sugawara, R.; Natsume, Y.

2012-07-01

232

Fractal characteristics and microstructure evolution of magnetron sputtering Cu thin films

NASA Astrophysics Data System (ADS)

How to describe surface morphology characteristic and microstructure evolution are the hottest researches of current thin film researches. But in traditional characterization of surface morphology, the roughness parameters are scale related. And the microstructure evolution of thin film during post-treatment is usually not considered in detail. To give a better understanding of the roughness of thin films topography, fractal method is carried out. In addition, microstructure evolution of thin films is analyzed based on the crystallography and energy theory. Cu thin films are deposited on Si (100) substrates by magnetron sputtering, and then annealed at different temperatures. Surface topography is characterized by atomic force microscope (AFM). Triangular prism surface area (TPSA) algorithm is used to calculate the fractal dimension of the AFM images. Apparent scale effect exists between the surface morphology roughness and film thickness. Relationship between the fractal dimension and roughness is analyzed by linear regression method and linear relationship exists between fractal dimension and surface roughness root mean square (RMS). Fractal dimension can be characterized as a scale independence parameter to represent the complex degree and roughness level of surface. With the increase of annealing temperature, surface roughness and fractal dimension decrease. But when the annealing temperature exceeds the recrystallization temperature, due to the agglomeration and coalescence of Cu grain, surface roughness and fractal dimension increase. Scale effect and changing regularity of grain growth and shape evolution for different film thickness under different annealing temperatures are analyzed. Based on minimum total free energy, regularity of grain growth and changing is proposed. The proposed research has some theory significance and applicative value of Cu interconnect process and development of MEMS.

Du, Shiwen; Li, Yongtang

2013-01-01

233

Electromagnetism on anisotropic fractal media

NASA Astrophysics Data System (ADS)

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

Ostoja-Starzewski, Martin

2013-04-01

234

. Vertical increases of fractal dimension across the zone of detachment provide a measure of the extent to which detachment has occurred. The increases of fractal dimension are greatest in the more productive relief exhibits fractal behavior. Along strike variations in the fractal dimension of structural cross

Wilson, Thomas H.

235

The Zeta Function Approach to Casimir Energy Calculations in Higher Dimensions

The vacuum fluctuations give rise to a number of phenomena; however, the the Casimir Effect is arguably the most salient manifestation of the quantum vacuum. In its most basic form it is realized through the interaction of a pair of neutral parallel conducting plates. The presence of the plates modifies the quantum vacuum, and this modifcation causes the plates to be pulled toward each other. The Casimir Effect has also been explored in the context of higher dimensional theories. The non-trivial boundary conditions imposed by compactified periodic higher dimensions is know to alter the vacuum in a quantifiable way, and is a possible solution to the issue of modulus stabilization, namely - the stabilization of higher dimensions. Typical in Casimir energy calculations are renormalization techniques which are used to tame the infinite sums and integrals that arise. These calculations are usally fairly involved, and explicit pedagogical material is sparse. The purpose of this paper is to introduce Dimensional Reg...

Obousy, Richard

2011-01-01

236

Fractal Analysis of Optical Coherence Tomography of Normal and Malignant Breast Tissue

NASA Astrophysics Data System (ADS)

Optical coherence tomography (OCT) provides real-time imaging of tissue several mean free photon paths into tissue by heterodyne detection of backscattered light. OCT can potentially be used to rapidly assess tumor margins during breast cancer resection, however, currently it is difficult to differentiate between normal and malignant tissues with OCT. Because cancer is characterized morphologically by increasing disorder, we investigated the fractal dimension of OCT images of normal and cancerous breast tissue. 3D OCT images of 44 specimens were collected, then tissues were histologically processed to independently determine distinct regions of adipose, stroma and cancer. The fractal dimension of each tissue type was then calculated with a one-dimensional box-counting algorithm applied to the OCT axial scans. We found that the fractal dimensions of stromal tissues were significantly higher than those of cancer (P<10-6), while those of adipose tissue were significantly lower than those of cancer (P<10-4).

Sullivan, Amanda C.; Hunt, John P.; Oldenburg, Amy L.

2011-03-01

237

The Zeta Function Approach to Casimir Energy Calculations in Higher Dimensions

The vacuum fluctuations give rise to a number of phenomena; however, the the Casimir Effect is arguably the most salient manifestation of the quantum vacuum. In its most basic form it is realized through the interaction of a pair of neutral parallel conducting plates. The presence of the plates modifies the quantum vacuum, and this modifcation causes the plates to be pulled toward each other. The Casimir Effect has also been explored in the context of higher dimensional theories. The non-trivial boundary conditions imposed by compactified periodic higher dimensions is know to alter the vacuum in a quantifiable way, and is a possible solution to the issue of modulus stabilization, namely - the stabilization of higher dimensions. Typical in Casimir energy calculations are renormalization techniques which are used to tame the infinite sums and integrals that arise. These calculations are usally fairly involved, and explicit pedagogical material is sparse. The purpose of this paper is to introduce Dimensional Regularization techniques specific to Casimir energy calculations with an additional compactified spatial dimension.

Richard Obousy

2011-08-30

238

Performance bounds for fractal coding

Reports on investigations concerning the performance of fractal transforms. Emerging from the structural constraints of fractal coding schemes, lower bounds for the reconstruction error are given without regarding quantization noise. This implies finding an at least locally optimal transformation matrix. A full search approach is by definition optimal but also intractable for practical implementations. In order to simplify the calculation

Bernd Hiirtgen; Rwth Aachen

1995-01-01

239

Fractal structures were formed on silicon substrates from SiO2 nanoparticles homogeneously synthesized in low temperature atmospheric pressure plasma from tetraethoxysilane (TEOS). RF discharge (power absorbed was about 10 W) sustained between two parallel mesh electrodes was used to generate plasma. The average size of nanoparticles was in the range of 8-20 nm and was determined by process parameters. The obtained products were analyzed by SEM (scanning electron microscopy) and XPS (X-ray photoelectron spectroscopy). Values of fractal dimension parameter of bidimensionals agglomerates formed on the substrate surface from nanoparticles were calculated with the use of Gwyddion and others. It was found that values of this parameter of the deposited structures varied in the range of 1.48-2 and were determined by combination of the process parameters. An empirical model explaining mechanism of the fractal structures formation and variation of the fractal dimension parameter with the process parameters was proposed. PMID:22097514

Alexandrov, S E; Kretusheva, I V; Mishin, M V; Yasenovets, G M

2011-09-01

240

NASA Astrophysics Data System (ADS)

Two methods for analyzing OCT images of arterial tissues are tested. These methods are applied toward two types of samples: segments of arteries collected from atherosclerosis-prone Watanabe heritable hyper-lipidemic rabbits and pieces of porcine left descending coronary arteries without atherosclerosis. The first method is based on finding the attenuation coefficients for the OCT signal that propagates through various regions of the tissue. The second method involves calculating the fractal dimensions of the OCT signal textures in the regions of interest identified within the acquired images. A box-counting algorithm is used for calculating the fractal dimensions. Both parameters, the attenuation coefficient as well as the fractal dimension correlate very well with the anatomical features of both types of samples.

Popescu, Dan P.; Flueraru, Costel; Mao, Youxin; Chang, Shoude; Sowa, Michael G.

2010-02-01

241

Instrumented postural control analysis plays an important role in evaluating the effects of injury on dynamic stability during balance tasks, and is often conveyed with measures based on the displacement of the center-of-pressure (COP) assessed with a force platform. However, the desired outcome of the task is frequently characterized by a loss of dynamic stability, secondary to injury. Typically, these failed trials are discarded during research investigations, with the potential loss of informative data pertaining to task success. The novelty of the present study is that COP characteristics of failed trials in injured participants are compared to successful trial data in another injured group, and a control group of participants, using the fractal dimension (FD) method. Three groups of participants attempted a task of eyes closed single limb stance (SLS): twenty-nine participants with acute ankle sprain successfully completed the task on their non-injured limb (successful injury group); twenty eight participants with acute ankle sprain failed their attempt on their injured limb (failed injury group); sixteen participants with no current injury successfully completed the task on their non-dominant limb (successful non-injured group). Between trial analyses of these groups revealed significant differences in COP trajectory FD (successful injury group: 1.58±0.06; failed injury group: 1.54±0.07; successful non-injured group: 1.64±0.06) with a large effect size (0.27). These findings demonstrate that successful eyes-closed SLS is characterized by a larger FD of the COP path when compared to failed trials, and that injury causes a decrease in COP path FD. PMID:24746034

Doherty, Cailbhe; Bleakley, Chris; Hertel, Jay; Caulfield, Brian; Ryan, John; Delahunt, Eamonn

2014-05-01

242

Thermodynamics of Photons on Fractals

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.

Akkermans, Eric [Department of Physics, Technion Israel Institute of Technology, 32000 Haifa (Israel); Dunne, Gerald V. [Department of Physics, University of Connecticut, Storrs, Connecticut 06269 (United States); Teplyaev, Alexander [Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269 (United States)

2010-12-03

243

NASA Technical Reports Server (NTRS)

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.

Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.

1992-01-01

244

Fractal Electronic Circuits Assembled From Nanoclusters

NASA Astrophysics Data System (ADS)

Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.

Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.

2009-07-01

245

Calculation of the dimensions of drug-polymer devices based on diffusion parameters.

The release kinetics of a polymeric-controlled release device are determined by its geometry and dimensions. A method to calculate the required size and shape of diffusion-controlled dosage forms to achieve a particular release profile is presented. The diffusion parameters are determined for various drugs (theophylline, diltiazem hydrochloride and caffeine) with thin ethyl cellulose (EC) films, containing different plasticizers [dibutyl sebacate (DBS) and acetyl tributyl citrate (ATBC)]. Computer simulations are then used to predict the drug release kinetics from various dosage forms (e.g. microparticles and cylinders). The practical benefit of these simulations is to optimize the geometry and dimensions of a controlled release device without the need of experimental studies. To verify the theoretical predictions, the release kinetics of theophylline from EC/ATBC microparticles of different size have also been determined experimentally. Good agreement is found between theory and experiment, proving the validity of the presented method. PMID:9649350

Siepmann, J; Ainaoui, A; Vergnaud, J M; Bodmeier, R

1998-07-01

246

When investigating fractal phenomena, the following questions are fundamental for the applied researcher: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent ?, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent ? of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators (d^ML, power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series. PMID:22586408

Stadnitski, Tatjana

2012-01-01

247

Generalized dimensions applied to speaker identification

NASA Astrophysics Data System (ADS)

This paper describes an application of fractal dimensions to speech processing and speaker identification. There are several dimensions that can be used to characterize speech signals such as box dimension, correlation dimension, etc. We are mainly concerned with the generalized dimensions of speech signals as they provide more information than individual dimensions. Generalized dimensions of arbitrary orders are used in speaker identification in this work. Based on the experimental data, the artificial phase space is generated and smooth behavior of correlation integral is obtained in a straightforward and accurate analysis. Using the dimension D(2) derived from the correlation integral, the generalized dimension D(q) of an arbitrary order q is calculated. Moreover, experiments applying the generalized dimension in speaker identification have been carried out. A speaker recognition dedicated Chinese language speech corpus with PKU-SRSC, recorded by Peking University, was used in the experiments. The results are compared to a baseline speaker identification that uses MFCC features. Experimental results have indicated the usefulness of fractal dimensions in characterizing speaker's identity.

Hou, Limin; Wang, Shuozhong

2004-08-01

248

The synthesis and rendering of eroded fractal terrains

In standard fractal terrain models based on fractional Brownian motion the statistical character of the surface is, by design, the same everywhere. A new approach to the synthesis of fractal terrain height fields is presented which, in contrast to previous techniques, features locally independent control of the frequencies composing the surface, and thus local control of fractal dimension and other

F. Kenton Musgrave; Craig E. Kolb; Robert S. Mace

1989-01-01

249

Fractal Universe and Quantum Gravity

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.

Calcagni, Gianluca [Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Muehlenberg 1, D-14476 Golm (Germany)

2010-06-25

250

Evolution of a fractal system with conserved order parameter under thermal annealing

Mesoscopic structural evolution under thermal annealing of yttrium aluminium garnet fractal aggregates has been investigated by small-angle neutron scattering. Fractal dimension remains invariant with sintering temperature but the extent of the fractal realm is narrowed down significantly. A Monte Carlo simulation, based on Ostwald-ripening type of relaxation of fractal aggregates for a mass conserved system, has been attempted in order

J. Bahadur; S. Mazumder; D. Sen; S. Ramanathan

2010-01-01

251

DOI 10.1007/s10659-011-9333-6 Waves in Fractal Media

equations for fractal media depend explicitly on fractal dimensions and reduce to conven- tional formsJ Elast DOI 10.1007/s10659-011-9333-6 Waves in Fractal Media Paul N. Demmie ï¿½ Martin Ostoja-Starzewski Received: 11 December 2010 ï¿½ Springer Science+Business Media B.V. 2011 Abstract The term fractal was coined

Ostoja-Starzewski, Martin

252

Fractal characterization of fracture networks: An improved box-counting technique

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

Perfect, Ed

253

Fractal signatures in the aperiodic Fibonacci grating.

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

Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam

2014-05-01

254

Quantum statistical mechanics on infinitely ramified fractals

NASA Astrophysics Data System (ADS)

I present the thermodynamics of identical particles confined in infinitely ramified, exactly self-similar fractals, such as the Sierpinski carpet (in 2D) and the Menger sponge (in 3D). Recent results from analysis on fractals have established that the heat kernel associated with the Laplacian on such fractals satisfy, in the short-time regime, a scaling relation with exponent dS/2 (where dS is the spectral dimension) modulated by log-periodic oscillations. I explain how such a scaling affects the partition function, and the resultant thermodynamics associated with blackbody radiation [1], Casimir effect, and electrons in the fractal box.

Chen, Joe P.

2011-03-01

255

Optics on a fractal surface and the photometry of the regoliths

NASA Astrophysics Data System (ADS)

The light scattered by a rough surface is calculated in a model where the surface is simulated by a mathematical fractal of dimension (D(H) between 2 and 3) and fractal density in the projected area towards the observer rho(H) (rho(H) between 0 and 1). The reflectance on such a surface is calculated in the special case of a 'hemispherical' fractal, in both the geometric optics approximation and a more general diffraction regime. By using a two-parameter phase function (single scattering albedo omega-sub-0 and asymmetry parameter g-sub-0), and including multiple scattering, this four-parameter model is found to reproduce within a good accuracy the phase function of several classes of atmosphereless bodies in the solar system, in good agreement with previous photometric models. The main effect of the diffraction is to reduce the width of the opposition surge by roughly a factor of 2. Another prediction of the model is that the single-scattering contribution due to the fractal part of the surface can be reduced, for nonzero phase angle, to an arbitrarily small amount, for high enough fractal dimension and density. This effect could give a new interpretation of the strong opposition effect observed on some objects, and also of the very low brightness of many solar system bodies.

Drossart, P.

1993-05-01

256

The fractal aggregation of asphaltenes.

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

Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott

2013-07-16

257

Applications of fractal analysis to physiology

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism. PMID:1885430

Glenny, Robb W.; Robertson, H. Thomas; Yamashiro, Stanley; Bassingthwaighte, James B.

2010-01-01

258

NASA Astrophysics Data System (ADS)

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.

Burdzy, Krzysztof; Ho?yst, Robert; Pruski, ?ukasz

2013-05-01

259

Fractons and Fractal Statistics

NASA Astrophysics Data System (ADS)

Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by ~ {h}=3-h shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A ?-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,?) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

da Cruz, Wellington

260

Fractal characterization of brain lesions in CT images

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.

Jauhari, Rajnish K.; Trivedi, Rashmi; Munshi, Prabhat; Sahni, Kamal [Indian Institute of Technology, Kanpur (India); J.K. Cancer Institute, G.S.V.M. Medical College, Kanpur (India)

2005-12-15

261

FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES

FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES Case.chapeau-blondeau}@univ-angers.fr Keywords: Color image, Color histogram, Fractal, Self-similarity, Capacity dimension, Correlation dimension of algorithms which can characterize fractal organizations in the support and population of their three

Chapeau-Blondeau, FranÃ§ois

262

Fractal analysis of surface EMG signals from the biceps

Nonlinear analysis techniques are necessary to understand the complexity of the EMG. The purpose of the present study was to determine the fractal dimension of surface EMO obtained from the biceps brachii of normal subjects during isokinetic flexion-extension of the arm. The measurements were obtained with different loading conditions on the arm and for various rates of flexion-extension. Fractal dimensions

Vineet Gupta; Srikanth Suryanarayanan; Narender P. Reddy

1997-01-01

263

Emission of terahertz radiations from fractal antennas

NASA Astrophysics Data System (ADS)

We investigate the emission of terahertz radiation from a photoconductive fractal antenna fabricated on a semi-insulating gallium arsenide substrate. Owing to the self-similarity of fractal structures, our fractal antenna shows a multiband emission of terahertz radiation. The emission intensity at peak frequency is about twice that from a bow-tie antenna. We also investigate the mechanism of the multiband emission by using the finite-difference time-domain calculation.

Miyamaru, F.; Saito, Y.; Takeda, M. W.; Liu, L.; Hou, B.; Wen, W.; Sheng, Ping

2009-11-01

264

Fractal characterization and frequency properties of near-fault ground motions

NASA Astrophysics Data System (ADS)

This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T m. Meanwhile, the box dimension of the velocity histories D vel is negatively correlated with the characteristic period T c and improved characteristic period T gi.

Yang, Dixiong; Zhang, Changgeng

2013-12-01

265

93FRACTAL SOLIDS, PRODUCT MEASURES AND FRACTIONAL WAVE EQUATIONSLi - Ostoja-Starzewski ABSTRACT. This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length

Ostoja-Starzewski, Martin

266

Deposition of fractal-like aerosol aggregates in a model of human nasal cavity.

Toxicity of diesel exhaust is related to the inhalation of nano-sized fractal-like aerosol aggregates. Their complex behavior (in comparison to spherical particles) should be taken into account in deposition modeling. The deposition of aerosol fractal-like aggregates in the model of a human nose was studied numerically for the flow rate corresponding to breathing conditions. The simplified geometry of the human nasal replica was implemented in the computational fluid dynamics (CFD) code (FLUENT) used for calculation of the three-dimensional airflow structure. A Brownian dynamic (BD) algorithm was applied for determination of the aggregates deposition in the nasal cavity during inhalation. These calculations were carried out for several populations of aggregates. The values of parameters used in the BD simulations for characterization of fractal-like aggregates, that is, fractal dimension (Df) and the radius of gyration (Rg), were in the range of 1.7-2.1 and 0.24-0.36 microm, respectively. These are the representative values for soot aggregates emitted from diesel engines. The results of computation show approximately 20% penetration of submicrometer aggregates through the nose and a weak dependence of deposition efficiency on Df and Rg values. The proposed methodology may lead to a more realistic description of deposition of nonspherical aerosol particles in the respiratory system. A more sophisticated approach for description of fractal-like aggregates dynamics is suggested for future studies. PMID:16774861

Moskal, Arkadiusz; Makowski, Lukasz; Sosnowski, Tomasz R; Grado?, Leon

2006-09-01

267

A fractal description of grain boundaries in a sintered powder metallurgical sample

In this paper, the authors have presented the concept of using the fractal dimension as a descriptor for the tortuosity of a grain boundary. The implication of fractal concepts in reality is not fully understood. But in their view, estimation of the fractal dimension of the grain boundaries might be beneficial to quantify the surface area related parameters. This would be of importance in materials where the surface properties play a dominant role such as in catalysis. In the synthesis of powder particles, an idea about the fractal dimension might also help in optimizing process parameters to produce powder particles having large values of fractal dimensions. This in turn could ensure better sinterability and surface activity of powders. The grain boundary traces of sintered YBa[sub 2]Cu[sub 3]O[sub 7] high [Tc] superconductor showed fractal structure and the fractal dimension was found to be dependent on grain orientation.

Ramakrishnan, K.N.; Venkadesan, S.; Murthy, K.P.N. (Indira Gandhi Centre for Atomic Research, Tamilnadu (India). Metallurgy and Materials Group)

1995-03-01

268

Fractal Propagators in QED and QCD and Implications for the Problem of Confinement

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.

S. Gulzari; Y. N. Srivastava; J. Swain; A. Widom

2006-12-09

269

Fractal Boundaries of Complex Networks

We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We find that for both Erd\\"{o}s-R\\'{e}nyi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes {\\it B} follows a power-law probability density function which scales as $B^{-2}$. The clusters formed by the boundary nodes are fractals with a fractal dimension $d_{f} \\approx 2$. We present analytical and numerical evidence supporting these results for a broad class of networks. Our findings imply potential applications for epidemic spreading.

Jia Shao; Sergey V. Buldyrev; Reuven Cohen; Maksim Kitsak; Shlomo Havlin; H. Eugene Stanley

2008-04-11

270

Fractal Substructure of a Nanopowder

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.

Thomas Schwager; Dietrich E. Wolf; Thorsten Poeschel

2008-02-25

271

A fractal-like resistive network

NASA Astrophysics Data System (ADS)

The equivalent resistance of a fractal-like network is calculated by means of approaches similar to those employed in defining the equivalent resistance of an infinite ladder. Starting from an elementary triangular circuit, a fractal-like network, named after Saggese, is developed. The equivalent resistance of finite approximations of this network is measured, and the didactical implications of the model are highlighted.

Saggese, A.; De Luca, R.

2014-11-01

272

Random sequential adsorption on fractals

NASA Astrophysics Data System (ADS)

Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions.

Ciesla, Michal; Barbasz, Jakub

2012-07-01

273

Random walks of oriented particles on fractals

NASA Astrophysics Data System (ADS)

Random walks of point particles on fractals exhibit subdiffusive behavior, where the anomalous diffusion exponent is smaller than one, and the corresponding random walk dimension is larger than two. This is due to the limited space available in fractal structures. Here, we endow the particles with an orientation and analyze their dynamics on fractal structures. In particular, we focus on the dynamical consequences of the interactions between the local surrounding fractal structure and the particle orientation, which are modeled using an appropriate move class. These interactions can lead to particles becoming temporarily or permanently stuck in parts of the structure. A surprising finding is that the random walk dimension is not affected by the orientation while the diffusion constant shows a variety of interesting and surprising features.

Haber, René; Prehl, Janett; Hoffmann, Karl Heinz; Herrmann, Heiko

2014-04-01

274

3 FROM FRACTAL OBJECTS TO FRACTAL SPACES 49 Excerpt from

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

Nottale, Laurent

275

NASA Astrophysics Data System (ADS)

Pore structure is one of important factors affecting the properties of porous media, but it is difficult to describe the complexity of pore structure exactly. Fractal theory is an effective and available method for quantifying the complex and irregular pore structure. In this paper, the fractal dimension calculated by box-counting method based on fractal theory was applied to characterize the pore structure of artificial cores. The microstructure or pore distribution in the porous material was obtained using the nuclear magnetic resonance imaging (MRI). Three classical fractals and one sand packed bed model were selected as the experimental material to investigate the influence of box sizes, threshold value, and the image resolution when performing fractal analysis. To avoid the influence of box sizes, a sequence of divisors of the image was proposed and compared with other two algorithms (geometric sequence and arithmetic sequence) with its performance of partitioning the image completely and bringing the least fitted error. Threshold value selected manually and automatically showed that it plays an important role during the image binary processing and the minimum-error method can be used to obtain an appropriate or reasonable one. Images obtained under different pixel matrices in MRI were used to analyze the influence of image resolution. Higher image resolution can detect more quantity of pore structure and increase its irregularity. With benefits of those influence factors, fractal analysis on four kinds of artificial cores showed the fractal dimension can be used to distinguish the different kinds of artificial cores and the relationship between fractal dimension and porosity or permeability can be expressed by the model of D = a - bln(x + c).

Wang, Heming; Liu, Yu; Song, Yongchen; Zhao, Yuechao; Zhao, Jiafei; Wang, Dayong

2012-11-01

276

[Chaos and fractals and their applications in electrocardial signal research].

Chaos and fractals are ubiquitous phenomena of nature. A system with fractal structure usually behaves chaos. As a complicated nonlinear dynamics system, heart has fractals structure and behaves as chaos. The deeper inherent mechanism of heart can be opened out when the chaos and fractals theory is utilized in the research of the electrical activity of heart. Generally a time series of a system was used for describing the status of the strange attractor of the system. The indices include Poincare plot, fractals dimension, Lyapunov exponent, entropy, scaling exponent, Hurst index and so on. In this article, the basic concepts and the methods of chaos and fractals were introduced firstly. Then the applications of chaos and fractals theories in the study of electrocardial signal were expounded with example of how they are used for ventricular fibrillation. PMID:19634696

Jiao, Qing; Guo, Yongxin; Zhang, Zhengguo

2009-06-01

277

A Fractal Approach to Assess the Risks of Nitroamine Explosives

To the best of our knowledge, this work represents the first thermal conductivity theory for fractal energetic particle groups to combine fractal and hot-spot theories. We considered the influence of the fractal dimensions of particles on their thermal conductivity and even on the sensitivity of the explosive. Based on this theory, two types of nitroamine explosives (hexahydro-1,3,5-trinitro-1,3,5-triazine [RDX] and hexanitrohexaazaisowurtzitane

Xiaolan Song; Fengsheng Li; Yi Wang; Chongwei An; Jingyu Wang; Jinglin Zhang

2012-01-01

278

The relationship between fractal properties of solid matrix and pore space in porous media

Measuring fractal dimensions has become a common practice for describing structural properties of porous media. Depending on the object of interest, different features of the structure can be measured: solid matrix, pores, and the interface between them. However, when measuring the fractal dimension of all these features, the question arises whether these dimensions are independent from each other or whether

Annette Dathea

279

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

280

Magnetohydrodynamics of fractal media

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.

Tarasov, Vasily E. [Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992 (Russian Federation)

2006-05-15

281

Dynamic critical phenomena in fractals

NASA Astrophysics Data System (ADS)

Dynamic critical phenomena are investigated, via the spin-flip kinetic Ising model, on two finitely ramified fractals: the Sierpinski gasket (SG) and the Koch curve. We show, using the Kawasaki inequality, that the dynamic critical exponent of the SG satisfies z>=df, the lower bound forming the conventional value. We also formulate a lower bound for the characteristic decay time. For the Koch curve we show exactly that z=2df=dw, where dw is the random-walk dimension.

Luscombe, James H.; Desai, Rashmi C.

1985-07-01

282

Fractal properties of quantum spacetime

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.

Dario Benedetti

2008-11-10

283

Fractality of eroded coastlines of correlated landscapes.

Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines. PMID:21867252

Morais, P A; Oliveira, E A; Araújo, N A M; Herrmann, H J; Andrade, J S

2011-07-01

284

The sensitivity of optical coherence tomography images to sample morphology is tested by two methods. The first method estimates the attenuation of the OCT signal from various regions of the probed tissue. The second method uses a box-counting algorithm to calculate the fractal dimensions in the regions of interest identified in the images. Although both the attenuation coefficient as well as the fractal dimension correlate very well with the anatomical features of the probed samples; the attenuation method provides a better sensitivity. Two types of samples are used in this study: segments of arteries collected from atherosclerosis–prone Watanabe rabbits (WHHL-MI) and healthy segments of porcine coronary arteries. PMID:21258464

Popescu, Dan P.; Flueraru, Costel; Mao, Youxin; Chang, Shoude; Sowa, Michael G.

2010-01-01

285

The sensitivity of optical coherence tomography images to sample morphology is tested by two methods. The first method estimates the attenuation of the OCT signal from various regions of the probed tissue. The second method uses a box-counting algorithm to calculate the fractal dimensions in the regions of interest identified in the images. Although both the attenuation coefficient as well as the fractal dimension correlate very well with the anatomical features of the probed samples; the attenuation method provides a better sensitivity. Two types of samples are used in this study: segments of arteries collected from atherosclerosis-prone Watanabe rabbits (WHHL-MI) and healthy segments of porcine coronary arteries. PMID:21258464

Popescu, Dan P; Flueraru, Costel; Mao, Youxin; Chang, Shoude; Sowa, Michael G

2010-01-01

286

Fault Diagnosis of Bearing Based on Fractal Method

Fractal geometry is a new method to apply in analyzing fault signals. After researching the characteristic of rolling bearings, a new quantificational definition about fault signals of rolling bearings is proposed. Based on fractal theory and the conception of box dimension it can describe both non-stationary and non-linear signals of vibration signals generated by rolling bearings. Experiment results shows that

Lu Shuang; Liu Jing

2006-01-01

287

Dimensionality reduction of hyperspectral data using spectral fractal feature

A new approach for dimensionality reduction of hyperspectral data has been proposed in this article. The method is based on extraction of fractal-based features from the hyperspectral data. The features have been generated using spectral fractal dimension of the spectral response curves (SRCs) after smoothing, interpolating and segmenting the curves. The new features so generated have then been used to

Kriti Mukherjee; Jayanta Kumar Ghosh; Ramesh Chand Mittal

2011-01-01

288

Fractal analysis of the retinal vascular network in fundus images

Complexity of the retinal vascular network is quantified through the measurement of fractal dimension. A computerized approach enhances and segments the retinal vasculature in digital fundus images with an accuracy of 94% in comparison to the gold standard of manual tracing. Fractal analysis was performed on skeletonized versions of the network in 40 images from a study of stroke. Mean

T. J. MacGillivray; N. Patton; F. N. Doubal; C. Graham; J. M. Wardlaw

2007-01-01

289

Neuromorphological Phenotyping in Transgenic Mice: A Multiscale Fractal Analysis

, 04109 Leipzig, Germany In: Mathematical Modeling of Biological Systems, Volume II. A. Deutsch, R. Bravo the multiscale fractal dimension (MFD) of reconstructed neuronal cells. Changes in the complexity of neuronal morNeuromorphological Phenotyping in Transgenic Mice: A Multiscale Fractal Analysis Andreas

Schierwagen, Andreas

290

Math Article Review Symmetries of Fractal Tilings Crista Moreno

seem strange and exotic, fractals are inherent in nature. In the formation of clouds, mountain ranges as stated by Mandelbrot [3]. Definition 1. A fractal is a set for which the Hausdorff Besicovitch dimension displays a refinement, from the union of blue colored sets to the union of green colored sets

Morrow, James A.

291

Fractal geometry in the San Andreas fault system

It has been noted that the spatial distribution of earthquakes and the mode of strain release in the San Andreas fault system is related to the complexity of fault geometry. Because of their rough appearance over many length scales, faults can be regarded as fractal surfaces. Direct estimates of fractal dimension D of portions of the San Andreas fault system

Paul G. Okubo; Keiiti Aki

1987-01-01

292

Fractal Approach to Large-Scale Galaxy Distribution

We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. Angular correlation function was used as a general instrument for the structure analysis. It was realized later that a normalization condition for the reduced correlation function estimator results in distorted values for both R_{hom} and fractal dimension D. Moreover, according to a theorem on projections of fractals, galaxy angular catalogues can not be used for detecting a structure with the fractal dimension D>2. For this 3-d maps are required, and indeed modern extensive redshift-based 3-d maps have revealed the ``hidden'' fractal dimension of about 2, and have confirmed superclustering at scales even up to 500 Mpc (e.g. the Sloan Great Wall). On scales, where the fractal analysis is possible in completely embedded spheres, a power--law density field has been found. The fractal dimension D =2.2 +- 0.2 was directly obtained from 3-d maps and R_{hom} has expanded from 10 Mpc to scales approaching 100 Mpc. In concordance with the 3-d map results, modern all sky galaxy counts in the interval 10^m - 15^m give a 0.44m-law which corresponds to D=2.2 within a radius of 100h^{-1}_{100} Mpc. We emphasize that the fractal mass--radius law of galaxy clustering has become a key phenomenon in observational cosmology.

Yurij Baryshev; Pekka Teerikorpi

2005-05-10

293

Fractal simulation of the resistivity and capacitance of arsenic selenide

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.

Balkhanov, V. K., E-mail: ballar@yandex.ru; Bashkuev, Yu. B. [Russian Academy of Sciences, Division of Physical Problems, Buryat Scientific Center, Siberian Branch (Russian Federation)

2010-03-15

294

Subcooled pool boiling heat transfer in fractal nanofluids: A novel analytical model

NASA Astrophysics Data System (ADS)

A novel analytical model to determine the heat flux of subcooled pool boiling in fractal nanofluids is developed. The model considers the fractal character of nanofluids in terms of the fractal dimension of nanoparticles and the fractal dimension of active cavities on the heated surfaces; it also takes into account the effect of the Brownian motion of nanoparticles, which has no empirical constant but has parameters with physical meanings. The proposed model is expressed as a function of the subcooling of fluids and the wall superheat. The fractal analytical model is verified by a reasonable agreement with the experimental data and the results obtained from existing models.

Xiao, Bo-Qi; Yang, Yi; Xu, Xiao-Fu

2014-02-01

295

The fractal characteristic of vibration signals in different milling tool wear periods

NASA Astrophysics Data System (ADS)

There are a wide variety of condition monitoring techniques currently used for the recognition and diagnosis of machinery faults. Tool wear often results in chaotics on milling process. Little research has been carried out about the occurrence and detection of chaotic behavior in time series signal of tool vibration. In the paper the vibration acceleration signal based on the operating stages of tool wear is established for the analysis of the correlation dimension of the operating stages of tool wear. Correlation dimension is calculated to recognize the tool wear operating conditions. Finally ,some experimental results from the fractal characteristic show that there are distinct differences in the correlation dimension in different tool wear conditions and close the correlation dimension in same tool wear conditions. The correlation dimension not only can be used as important scientific basis for monitoring tool wear, but also complement of other characteristic picking up method.

Xu, Chuangwen; Cheng, Hualing; Liu, Limei

2008-10-01

296

Characterization of synthetic seismicity distributions with spatially fractal properties

NASA Astrophysics Data System (ADS)

Assessment of seismic hazard in a country is the first important step for the definition of a seismic building code. The goal of probabilistic seismic hazard assessment (PSHA) is to quantify the rate of exceeding specific ground-motion levels at a site, given all possible earthquakes. A critical step in PSHA is the accurate definition and characterization of relevant seismic sources. This is particularly challenging in low-seismicity regions, because observation periods are relatively short, seismicity is often diffuse, and active faults are difficult to identify. For these reasons, large source zones are commonly used with spatially uniformly distributed seismicity inside. Observed seismicity, however, is generally not uniformly distributed, but reflects seismotectonic forces and tectonic structure. Rather, observed seismicity even in subregions defined as seismic sources is clustered in space: seismicity tends to aggregate on or close to major fault structures. Thus the hypothesis of uniform distribution of events inside a source zone does not relate well to observed seismicity and could overestimate or underestimate the value of ground-acceleration on the PSHA. Seismicity is a classical example of a complex phenomenon that can be quantified using fractal concepts. In particular, fault networks and epicenter distributions are know to have fractal properties. The fractal dimension is an extension of the Euclidean dimension and measure the degree of clustering of earthquakes. In this study, we move towards a more realistic characterization of spatial distribution of seismicity within each source zone. We quantify differences between different spatial characterization of seismicity and validate a more realistic method for the generation of synthetic seismicity on a source zone as input for PSHA, extending the concepts described in Beauval et al. (BSSA, 2006). We calculate differences in terms of hazard curves and hazard maps for synthetic catalogs characterized by a uniform (D = 2.0) or a clustered (fractal, D < 2.0) distribution of events on a hypothetical square source zone. We find that the assumption of D = 2.0 overestimates the resulting hazard in some parts of the source zones. This overestimation is larger for low probability levels; it can typically reach 10 percent. We then apply these results to the current seismic zonation model for Switzerland, which consist of about 25 zones of equal seismic potential. We measure the fractal dimension of instrumental seismicity for the past 25 years as D = 1.5 and build a seismic hazard model using this value. Preliminary results show that the impact for such a case is negligible, with the possible exception of very low probability levels.

Spada, M.; Wiemer, S.; Kissling, E. H.

2009-12-01

297

Antenna Miniaturization Using Koch Snowflake Fractal Geometry

NASA Astrophysics Data System (ADS)

The Wireless Industry is witnessing an volatile emergence today in present era. Also requires the performance over several frequency bands or are reconfigurable as the demands on the system changes. This Paper Presents Rectangular, Koch Fractal Patch Antennas on Single and Multilayer Substrate With and Without Air-Gap using Advanced Design System Simulator (ADS). Fractal Antenna provides Miniaturization over conventional microstrip Antennas. The Antennas Have Been Designed on FR4 substrate with ? = 4.2, h = 1.53 and the initial Dimension of the simple Rectangular Patch is 36.08 * 29.6 mm. The experimental Resonant Frequencies of the Fractal Patch with 1st, 2nd & 3rd are observed 2.22, 2.14 & 2.02 GHz Respectively in comparison to Rectangular Patch with 2.43 GHz. The reduced Impedance bandwidth of the Fractal Patch has been improved by designing the patch over multilayer substrate with varying Air-gap between two Substrate. As we increase the air- gap between the two substrate layer further enhancement in impedance bandwidth of Fractal antenna has been Obtained. The Radiation pattern of Koch Fractal antenna is as similar to rectangular patch antenna but with better H-plane Cross Polarization for fractal patch. The all simulated Results are in close Agreement with experimental Results.

Minal, Dhama, Nitin

2010-11-01

298

Hexagonal and Pentagonal Fractal Multiband Antennas

NASA Technical Reports Server (NTRS)

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.

Tang, Philip W.; Wahid, Parveen

2005-01-01

299

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

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.

Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E. [State Univ. of New York, Buffalo, NY (United States). Biomedical Imaging Group; LeBlanc, A.; Lin, C. [Baylor Coll. of Medicine, Houston, TX (United States); Shackelford, L. [National Aeronautics and Space Administration, Houston, TX (United States). Johnson Space Center

1995-12-31

300

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

NASA Astrophysics Data System (ADS)

We characterize the trabecular structure with the aid of fractal dimension. We use alternating sequential filters (ASF) to generate a nonlinear pyramid for fractal dimension computations. We do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of our scheme is the rudimentary definition of self-similarity. This allows us the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, we have shown that the ASF methods outperform other existing methods for fractal dimension estimation. We have shown that the fractal dimension remains the same when computed with both the x-ray images and the MRI images of the patella. We have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, we have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, we have shown that the subject's prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.

Acharya, Raj S.; LeBlanc, Adrian; Shackelford, Linda; Swarnakar, Vivek; Krishnamurthy, Ram; Hausman, E.; Lin, Chin-Shoou

1995-05-01

301

Array Patterns of Fractal Linear Array Antennas Based on Cantor Set

NASA Astrophysics Data System (ADS)

A fractal is a recursively generated object having a fractional dimension. Antennas can be designed using the recursive nature of a fractal. In this paper general expression for array factor of fractal linear array based on cantor set was compared with conventional linear array. The similarity of the radiation patterns and their fractal features are examined for various iterations with the simulated results using MATLAB.

Deepika Rani, N.; Sri Devi, P. V.

2012-03-01

302

Fractal behavior of a wrinkled annular diffusion flame--

Fractal geometry is used to describe quantitatively the random character of a turbulent wrinkled diffusion flame. The article describes how basic ideas of mathematics of fractals can be applied to study the changes in the irregularity of the flame's surface at various axial positions. In the present work, the planar laser-induced-fluorescence (PLIF) imaging technique has been used to map the hydroxyl concentration in an annular diffusion flame with propane fuel. The short pulse duration (18 ns) of the laser sheet enabled instantaneous pictures of the flame structure to be obtained with high spatial resolution of 0.5 {times} 0.5 {times} 0.5 mm. The isopleths of concentration were highly wrinkled and random in character, both in space and time. They were measured in flames that were forced at different instability frequencies, in order to stabilize the roll-up of the vortices in the flame shear layer and to simulate vortex merging. Measurements were done from the flameholder to the flame's end at many axial positions, in order to study the evolution of the flame structure. The fractal dimension was calculated for all these conditions, using two methods: the Kolmogorov-Hansdorff definition and the perimeter area relation.

Gutmark, E.; Hanson-Parr, D.M.; Parr, T.P.; Schadow, K.C. (Naval Weapons Center, China Lake, CA (USA). Research Dept.)

1990-01-01

303

FAST TRACK COMMUNICATION: Weyl law for fat fractals

NASA Astrophysics Data System (ADS)

It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.

Spina, María E.; García-Mata, Ignacio; Saraceno, Marcos

2010-10-01

304

Fractal Strings and Multifractal Zeta Functions

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.

Michel L. Lapidus; Jacques Levy Vehel; John A. Rock

2006-10-06

305

1 / 20 Lois d'Ã©chelle et transitions fractal Â non fractal en gÃ©ographie Scaling laws and fractal Â non fractal transitions in geography Maxime Forriez*, Philippe Martin* et Laurent Nottale concept de transition fractal Â non fractal fut introduit initialement dans un cadre de physique

Paris-Sud XI, UniversitÃ© de

306

Search on a fractal lattice using a quantum random walk

NASA Astrophysics Data System (ADS)

The spatial search problem on regular lattice structures in integer number of dimensions d?2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

Patel, Apoorva; Raghunathan, K. S.

2012-07-01

307

Fractal Analysis of Cervical Intraepithelial Neoplasia

Introduction Cervical intraepithelial neoplasias (CIN) represent precursor lesions of cervical cancer. These neoplastic lesions are traditionally subdivided into three categories CIN 1, CIN 2, and CIN 3, using microscopical criteria. The relation between grades of cervical intraepithelial neoplasia (CIN) and its fractal dimension was investigated to establish a basis for an objective diagnosis using the method proposed. Methods Classical evaluation of the tissue samples was performed by an experienced gynecologic pathologist. Tissue samples were scanned and saved as digital images using Aperio scanner and software. After image segmentation the box counting method as well as multifractal methods were applied to determine the relation between fractal dimension and grades of CIN. A total of 46 images were used to compare the pathologist's neoplasia grades with the predicted groups obtained by fractal methods. Results Significant or highly significant differences between all grades of CIN could be found. The confusion matrix, comparing between pathologist's grading and predicted group by fractal methods showed a match of 87.1%. Multifractal spectra were able to differentiate between normal epithelium and low grade as well as high grade neoplasia. Conclusion Fractal dimension can be considered to be an objective parameter to grade cervical intraepithelial neoplasia. PMID:25302712

Fabrizii, Markus; Moinfar, Farid; Jelinek, Herbert F.; Karperien, Audrey; Ahammer, Helmut

2014-01-01

308

Deterministic fractals: extracting additional information from small-angle scattering data.

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed. PMID:22060471

Cherny, A Yu; Anitas, E M; Osipov, V A; Kuklin, A I

2011-09-01

309

Deterministic fractals: Extracting additional information from small-angle scattering data

NASA Astrophysics Data System (ADS)

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)qD is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)qD in the momentum space is related to the log-periodicity of the quantity g(r)r3-D in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.

Cherny, A. Yu.; Anitas, E. M.; Osipov, V. A.; Kuklin, A. I.

2011-09-01

310

Improved techniques to calculate two-loop anomalous dimensions in QCD

NASA Astrophysics Data System (ADS)

In this thesis, the calculation of the full flavour non- singlet Altarelli-Parisi splitting functions as well as the Nc2-part of the gluon-gluon splitting functions in next-to-leading order is presented. The calculation has been performed by employing the method of Curci, Furmanski and Petronzio (CFP), which is based on the light-cone gauge. In previous calculations relying on the CFP method, the spurious poles of the gluon propagator in light-cone gauge always had been regularized by using the 'principal value' (PV) prescription. As the PV prescription is formally unsatisfactory in several respects, it entails the application of some 'phenomenological rules', whose working principles are not really understood, to obtain the correct result. The calculation presented here has been done by applying the Mandelstam-Leibbrandt (ML) prescription, which has a solid field-theoretical foundation, to regulate the gauge induced poles. As a consequence, the phenomenological rules needed in the PV case became obsolete. On the other hand, the use of the ML prescription increased the complexity of the calculation mainly due to the fact that unitarity requires the inclusion of so-called 'axial ghost' degrees of freedom. The calculation can be organized by studying gauge invariant subparts defined by a certain colour structure. The part proportional to CF2, being of Abelian nature, constitutes an opportunity to study the effects of the ML prescription in isolation from other complications. The non-Abelian part proportional to CFNc turned out to be much more involved, revealing new features concerning the cancellation mechanism of the spurious poles. For colour structure CFTf, the calculation of the two-loop quark selfenergy has been included, thus being able to extract the full endpoint contribution at x = 1. In this way it was possible to check the consistency with the sum rules expressing fermion number conservation, which constitutes a new test not only of the ML prescription, but also of the CFP method itself. In order to investigate the viability of the ML prescription in all possible one-loop structures of QCD, the Nc2-part of the gluon-gluon splitting function, which contains the highly nontrivial one-loop three-gluon vertex, also has been calculated. Using the methods developed for the CFNc-part, the usefulness and reliability of the ML prescription in this context again could be confirmed. In summary, having established the CFP method with ML prescription as a method without conceptual loopholes, this work might serve as a powerful tool to extend the calculation to three loops.

Heinrich, Gudrun Marlen

311

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept\\u000a of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim\\u000a of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially,\\u000a there

Anirban Banerji; Indira Ghosh

2011-01-01

312

Fractal structure of the interplanetary magnetic field

NASA Technical Reports Server (NTRS)

Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.

Burlaga, L. F.; Klein, L. W.

1985-01-01

313

Rapid calculation method for Frenkel-type two-exciton states in one to three dimensions.

Biexciton and two-exciton dissociated states of Frenkel-type excitons are well described by a tight-binding model with a nearest-neighbor approximation. Such two-exciton states in a finite-size lattice are usually calculated by numerical diagonalization of the Hamiltonian, which requires an increasing amount of computational time and memory as the lattice size increases. I develop here a rapid, memory-saving method to calculate the energies and wave functions of two-exciton states by employing a bisection method. In addition, an attractive interaction between two excitons in the tight-binding model can be obtained directly so that the biexciton energy agrees with the observed energy, without the need for the trial-and-error procedure implemented in the numerical diagonalization method. PMID:25053304

Ajiki, Hiroshi

2014-07-21

314

Propagation of Low-Energy Cosmic Rays in Molecular Clouds: Calculations in Two Dimensions

We calculate cosmic ray transport with a collisional Boltzmann Transport Equation, including E-M forces. We apply the Crank-Nicholson Method to solve this equation. At each time step, the spatial distribution of cosmic rays is applied to the ZEUS 2D magnetohydrodynamics model, which is then utilized to calculate the resulting E-M field. Finally, the field is applied to the initial equation. This sequence is repeated over many time steps until a steady state is reached. We consider results from t = 0 until steady state for an isotropic low energy cosmic ray flux, and also for an enhanced cosmic ray flux impinging on one side of a molecular cloud. This cosmic ray flux is used to determine an ionization rate of interstellar hydrogen by cosmic rays, zeta. Astrochemical implications are briefly mentioned.

Rimmer, Paul

2011-01-01

315

Rapid calculation method for Frenkel-type two-exciton states in one to three dimensions

NASA Astrophysics Data System (ADS)

Biexciton and two-exciton dissociated states of Frenkel-type excitons are well described by a tight-binding model with a nearest-neighbor approximation. Such two-exciton states in a finite-size lattice are usually calculated by numerical diagonalization of the Hamiltonian, which requires an increasing amount of computational time and memory as the lattice size increases. I develop here a rapid, memory-saving method to calculate the energies and wave functions of two-exciton states by employing a bisection method. In addition, an attractive interaction between two excitons in the tight-binding model can be obtained directly so that the biexciton energy agrees with the observed energy, without the need for the trial-and-error procedure implemented in the numerical diagonalization method.

Ajiki, Hiroshi

2014-07-01

316

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.

Boyd, O.S.

2006-01-01

317

NSDL National Science Digital Library

This first website offers a collection of fractal music using images created by G.W.F. Albrecht. The technology and mathematics which this presentation draws on is described on the second website. The second website, developed by David Strohbeen, offers some basic information about fractals and fractal music. He has also posted some samples of his music and invites visitors to download software for creating fractal music and to submit their own compositions.

318

Results on the numerical analysis of the Sierpinski and Koch fractal antennas are presented. It is shown that self-similarity\\u000a of fractal structures affects electromagnetic properties of antenna structures created on the basis of these fractals. It\\u000a is demonstrated that the Sierpinski and Koch fractal antennas are multiband structures; therefore, these antennas can be used\\u000a for the development of radar and

S. V. Krupenin

2006-01-01

319

Fractal analysis of the galaxy distribution in the redshift range 0.45 < z < 5.0

Evidence is presented that the galaxy distribution can be described as a fractal system in the redshift range of the FDF galaxy survey. The fractal dimension $D$ was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with $\\Omega_{m_0}=0.3$, $\\Omega_{\\Lambda_0}=0.7$ and $H_0=70 \\; \\mbox{km} \\; {\\mbox{s}}^{-1} \\; {\\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number densities $\\gamma$ and $\\gamma^\\ast$ obtained from the red and blue FDF galaxies provides a direct method to estimate $D$, implying that $\\gamma$ and $\\gamma^\\ast$ vary as power-laws with the cosmological distances. The luminosity distance $d_{\\scriptscriptstyle L}$, galaxy area distance $d_{\\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against their respective number densities to calculate $D$ by linear fitting. It was found that the FDF galaxy distribution is characterized by two single fractal dimensions at successive distance ranges. Two straight lines were fitted to the data, whose slopes change at $z \\approx 1.3$ or $z \\approx 1.9$ depending on the chosen cosmological distance. The average fractal dimension calculated using $\\gamma^\\ast$ changes from $\\langle D \\rangle=1.4^{\\scriptscriptstyle +0.7}_{\\scriptscriptstyle -0.6}$ to $\\langle D \\rangle=0.5^{\\scriptscriptstyle +1.2}_{\\scriptscriptstyle -0.4}$ for all galaxies, and $D$ decreases as $z$ increases. Small values of $D$ at high $z$ mean that in the past galaxies were distributed much more sparsely and the large-scale galaxy structure was then possibly dominated by voids. Results of Iribarrem et al. (2014, arXiv:1401.6572) indicating similar fractal features with $\\langle D \\rangle =0.6 \\pm 0.1$ in the far-infrared sources of the Herschel/PACS evolutionary probe (PEP) at $1.5 \\lesssim z \\lesssim 3.2$ are also mentioned.

G. Conde-Saavedra; A. Iribarrem; Marcelo B. Ribeiro

2014-09-18

320

Hexagonal fractal multiband antenna

The design of a new fractal multiband antenna, based on the hexagon, is presented. Three iterations of the hexagonal fractal multiband antenna, arranged in the dipole configuration, are examined. Experimental results are compared with those obtained using the method of moments and the fractal antenna is found to possess predictable multiband characteristics.

P. W. Tang; P. F. Wahid

2004-01-01

321

Load-flow fractals draw clues to erratic behavior

Fractal images have been discovered in many areas of science and engineering in the past two decades, so it is not surprising that they have also appeared in power system literature. This article provides a brief introduction to fractals and presents some fractal patterns produced by Newton-Raphson load-flow calculations for a small power system. If one imagines performing load-flow calculations from a dense grid of initial conditions, the region of initial conditions that converge to a particular equilibrium point using the Newton-Raphson method is seen to have a fractal boundary.

Thorp, J.S.; Naqavi, S.A. [Cornell Univ., Ithaca, NY (United States)] [Cornell Univ., Ithaca, NY (United States)

1997-01-01

322

Calculation of the transfer matrix T in six dimensions for an rf-deflector element

One possible element for funneling two beams together is a deflector with a constant or time-varying electric-field strength. With such an element, arbitrary beams can be brought together and maintained on the axis, if the appropriate combination of deflector parameters is chosen. A parallel beam can be handled only with a time-varying voltage of the deflector. The six-dimensional transfer matrices are calculated for constant or time-varying fields; all the results are correct in first-order approximation.

Bongardt, K.

1981-01-01

323

Continuous Medium Model for Fractal Media

We consider the description of the fractal media that uses the fractional integrals. We derive the fractional generalizations of the equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. The fractional equation of continuity is considered.

Vasily E. Tarasov

2005-06-06

324

Agglomeration due to Brownian motion of fractal-structured combustion aerosols

A dynamic Monte-Carlo type lattice model has been developed to simulate the agglomeration of non-spherical chain-line aggregate combustion aerosols due to Brownian motion. Simulations are carried out in the free molecular and continuum regimes, for both initial monodisperse and initial log-normally distributed aerosols, with and without source mechanisms. Preservation of the chain-like structure of the aggregate is accomplished throughout the simulation by describing the agglomerate as fractal, that is, scale-invariant, self-similar with a noninteger dimensionality. Simulation results indicate that cluster growth is more rapid in the free molecular regime than in the continuum. Aerosols and log-normal distributions retain their log-normal characteristics even after long coagulation times. The effect of the clusters' fractal dimension on the cluster growth rate is determined; the rate of agglomeration increases when the structure of the agglomerate is more fragmented (lower fractal dimension). An analytical solution to the coagulation equation is obtained for log-normal aerosols by calculating moments of the distribution and solving sets of moment equations to determine the size distribution parameters. Condition numbers are employed to determine which moments should be calculated to most accurately determine these parameters. Excellent agreement is obtained between the simulations and the solution to the moment equations. Experimental measurements of soot particle velocity in a premixed methane/air flame are made using laser Doppler velocimetry.

Kaplan, C.H.

1987-01-01

325

Hidden Markov Models and Gaussian Mixture Models for Bearing Fault Detection Using Fractals

Bearing vibration signals features are extracted using time domain fractal based feature extraction technique. This technique uses multi-scale fractal dimension (MFD) estimated using box-counting dimension. The extracted features are then used to classify faults using Gaussian mixture models (GMM) and hidden Markov models (HMM). The results obtained show that the proposed feature extraction technique does extract fault specific information. Furthermore,

Tshilidzi Marwala; Unathi Mahola; Fulufhelo Vincent Nelwamondo

2006-01-01

326

Fractal Representation of Turbulent Dispersing Plumes.

NASA Astrophysics Data System (ADS)

Fractal analysis techniques have been applied to the concentration fields from large-eddy simulations of plume dispersion in a turbulent boundary layer. Fractal dimensions between 1.3 and 1.35 are obtained from area-perimeter and box-counting analyses for neutral and convective conditions. These values are close to previous estimates from atmospheric data. Methods for generating fractal fields with given statistical moments are examined and the simplest of these, the recursive refinement technique, is shown to be inadequate. The problem is shown to be the interpolation step of the procedure, which intrinsically reduces the variance with each refinement. Accurate statistical representation is obtained by replacing the interpolation step of the refinement technique with a sum of random pulses of appropriate width and random location. The pulse technique can easily he adapted to generate either clipped-normal or lognormal one-point probability distributions. Results from the fractal generation technique using simulated mean statistics are compared with realizations of instantaneous plume cross sections from the large-eddy simulations. The simulated probability distributions lie between the clipped normal and the lognormal, so the fractal fields cannot match the realizations precisely. Larger-scale features of the plumes are generally well represented by the fractal method, however.

Sykes, R. I.; Gabruk, R. S.

1994-06-01

327

Charging of Fractal Dust Agglomerates in a Plasma Environment

The charge on micron-sized dust grains plays a crucial role in the structure and evolution of forming aggregates within the dust population during the coagulation process. The manner in which the charge is arranged on developing irregular structures can affect the fractal dimension of aggregates formed during collisions, which in turn influences the coagulation rate and size evolution of the dust cloud. Preliminary models for the charge evolution on fractal aggregates immersed in a plasma environment calculated using a modification to the orbital-motion-limited (OML) theory are presented in this paper. The model calculates currents to each point on the aggregate surface using a line-of-sight (LOS) approximation: only those electron or ion trajectories which are not blocked by another grain within the aggregate contribute to the charging current. Both the total charge and the dipole moment are calculated for the dust aggregate. While most coagulation theories assume that it is difficult for like-charged grains to coagulate, the OML_LOS approximation indicates that the electric potentials of aggregate structures are often reduced enough to allow significant coagulation.

L. S. Matthews; T. W. Hyde

2007-07-25

328

Measurement of normal contact stiffness of fractal rough surfaces

We investigate the effects of roughness and fractality on the normal contact stiffness of rough surfaces. Samples of isotropically roughened aluminium surfaces are considered. The roughness and fractal dimension were altered through blasting using different sized particles. Subsequently, surface mechanical attrition treatment (SMAT) was applied to the surfaces in order to modify the surface at the microscale. The surface topology was characterised by interferometry based profilometry. The normal contact stiffness was measured through nanoindentation with a flat tip utilising the partial unloading method. We focus on establishing the relationships between surface stiffness and roughness, combined with the effects of fractal dimension. The experimental results, for a wide range of surfaces, showed that the measured contact stiffness depended very closely on surfaces' root mean squared (RMS) slope and their fractal dimension, with correlation coefficients of around 90\\%, whilst a relatively weak correlation coefficient of 57\\% was found between the contact stiffness and RMS roughness.

Chongpu Zhai; Sébastien Bevand; Yixiang Gan; Dorian Hanaor; Gwénaëlle Proust; Bruno Guelorget; Delphine Retraint

2014-08-26

329

Fat fractal percolation and k-fractal percolation Erik Bromana

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

Meester, Ronald

330

The fractal energy measurement and the singularity energy spectrum analysis

NASA Astrophysics Data System (ADS)

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.

Xiong, Gang; Zhang, Shuning; Yang, Xiaoniu

2012-12-01

331

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

Beaucage, Gregory

332

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 Â·In 1982, Benoit Mandelbrot publishes "Fractal Geometry" and fractal mathematics is born. Â·Fractal scaling

333

Dynamics of swollen fractal networks

The dynamics of swollen fractal networks (Rouse model) has been studied through computer simulations. The fluctuation-relaxation theorem was used instead of the usual Langevin approach to Brownian dynamics. We measured the equivalent of the mean square displacement $\\langle \\vec r^{\\,2} \\rangle$ and the coefficient of self-diffusion $D$ of two-and three-dimensional Sierpinski networks and of the two-dimensional percolation network. The results showed an anomalous diffusion, i. e., a power law for $D$, decreasing with time with an exponent proportional to the spectral dimension of the network.

Teixeira, Alvaro V N C

2014-01-01

334

Fractal analysis of the structural complexity of the connective tissue in human carotid bodies

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.

Guidolin, Diego; Porzionato, Andrea; Tortorella, Cinzia; Macchi, Veronica; De Caro, Raffaele

2014-01-01

335

Triangular constellations in fractal measures

NASA Astrophysics Data System (ADS)

The local structure of a fractal set is described by its dimension D, which is the exponent of a power-law relating the mass {\\cal N} in a ball to its radius \\varepsilon{:}\\ {\\cal N}\\sim \\varepsilon^D . It is desirable to characterise the shapes of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio z of its area to the radius of gyration squared. We show that the probability density of z has a phase transition: P(z) is independent of ? and approximately uniform below a critical flow compressibility \\beta_{\\text{c}} , which we estimate. For \\beta>\\beta_{\\text{c}} the distribution appears to be described by two power laws: P(z)\\sim z^{\\alpha_1} when 1\\gg z\\gg z_{\\text{c}}(\\varepsilon) , and P(z)\\sim z^{\\alpha_2} when z\\ll z_{\\text{c}}(\\varepsilon) .

Wilkinson, Michael; Grant, John

2014-09-01

336

Triangular Constellations in Fractal Measures

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 $z\\ll z_{\\rm c}(\\epsilon)$.

Wilkinson, Michael

2014-01-01

337

Black carbon fractal morphology and short-wave radiative impact: a modelling study

NASA Astrophysics Data System (ADS)

We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of differences in the optical cross sections and asymmetry parameters. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material.

Kahnert, M.; Devasthale, A.

2011-08-01

338

Black carbon fractal morphology and short-wave radiative impact: a modelling study

NASA Astrophysics Data System (ADS)

We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of different error sources. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material. Finally, we found that the spectral variation of the absorption cross section of black carbon significantly deviates from a simple 1/? scaling law. We therefore discourage the use of single-wavelength absorption measurements in conjunction with a 1/? scaling relation in broadband radiative forcing simulations of black carbon.

Kahnert, M.; Devasthale, A.

2011-11-01

339

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.

Paul B. Slater

2006-06-01

340

Fractal structures in casting films from chlorophyll

NASA Astrophysics Data System (ADS)

Chlorophyll (Chl) molecules are important because they can act as natural light-harvesting devices during the photosynthesis. In addition, they have potential for application as component of solar cell. In this work, we have prepared casting films from chlorophyll (Chl) and demonstrated the occurrence of fractal structures when the films were submitted to different concentrations. By using optical microscopy and the box-count method, we have found that the fractal dimension is Df = 1.55. This value is close to predicted by the diffusion-limited aggregation (DLA) model. This suggests that the major mechanism - which determines the growth of the fractal structures from Chl molecules - is the molecular diffusion. Since the efficiencies of solar cells depend on the morphology of their interfaces, these finds can be useful to improve this kind of device.

Pedro, G. C.; Gorza, F. D. S.; de Souza, N. C.; Silva, J. R.

2014-04-01

341

Generation and display of geometric fractals in 3-D

We present some straightforward algorithms for the generation and display in 3-D of fractal shapes. These techniques are very general and particularly adapted to shapes which are much more costly to generate than to display, such as those fractal surfaces defined by iteration of algebraic transformations. In order to deal with the large space and time requirements of calculating these

Alan Norton

1982-01-01

342

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.

Helene Porchon

2012-01-25

343

The Fractal Ratio as a Metric of Nanostructure Development in Hydrating Cement Paste

It is necessary to have appropriate metrics to quantify the development of the nanostructure in Portland cement paste. The\\u000a fractal ratio, calculated from Small Angle Neutron Scattering (SANS) data, serves as such a metric. It expresses the proportion\\u000a of the volume-fractal surface area of calcium-silicate-hydrate gel (C-S-H) to the surface-fractal surface area. The volume\\u000a fractal develops in the scale range

R. A. Livingston; W. Bumrongjaroen; A. J. Allen

344

A MultiFractal Spectrum Analysis of Turbulence Data and the DNA of Worms

This paper discusses the physical meanings of various parameters in multi-fractal spectrum and offers the thermo-mechanical formula in calculating the multi-fractal spectrum. Besides it works out the sets of multi-fractal Cantor, the multi-fractal spectrum of turbulence data of Rayleigh-Benard convection and DNA series by means of wavelet transform maximum modulus (WTMM). In the end, it arrives at the conclusions that

Q. Fu; Z. F. Chen; Y. H. Zhou; L. Wang

2011-01-01

345

Fractal aspects and convergence of Newton`s method

Newton`s Method is a widely established iterative algorithm for solving non-linear systems. Its appeal lies in its great simplicity, easy generalization to multiple dimensions and a quadratic local convergence rate. Despite these features, little is known about its global behavior. In this paper, we will explain a seemingly random global convergence pattern using fractal concepts and show that the behavior of the residual is entirely explicable. We will also establish quantitative results for the convergence rates. Knowing the mechanism of fractal generation, we present a stabilization to the orthodox Newton method that remedies the fractal behavior and improves convergence.

Drexler, M. [Oxford Univ. Computing Lab. (United Kingdom)

1996-12-31

346

Robust fractal characterization of 1D and 2D signals

NASA Astrophysics Data System (ADS)

Fractal characterization of signals is well suited in analysis of some time series data and in classification of natural shapes and textures. A maximum likelihood estimator is used to measure the parameter H which is directly related to the fractal dimension. The robustness of the estimator and the performance of the method are demonstrated on datasets generated using a variety of techniques. Finally the characterization is used in segmentation of composite images of natural textures.

Avadhanam, Niranjan; Mitra, Sunanda

1993-10-01

347

Iterated Function Systems and the Global Construction of Fractals

Iterated function systems (i.f.ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i.f.ss and occur as the supports of probability measures associated with functional equations. The existence of certain `p-balanced' measures for i.f.ss is established, and these measures are uniquely characterized for hyperbolic i.f.ss. The Hausdorff-Besicovitch dimension for some

M. F. Barnsley; S. Demko

1985-01-01

348

A Modified Sierpinski Fractal Antenna for Multiband Application

A broadband planar Sierpinski fractal antenna for multiband application is proposed, designed, and tested. The perturbed Sierpinski fractal patch and slotted ground plane are employed to achieve broadband characteristics. The implemented antenna including the ground plane has a total dimension of 100 times 53.7 times 0.8 mm3. The measured 10-dB return loss bandwidths are 808-1008 MHz (22%) and 1581-2760 MHz

Kuem C. Hwang

2007-01-01

349

Fractal images induce fractal pupil dilations and constrictions

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

Taylor, Richard

350

Fractal Weyl law for quantum fractal eigenstates D. L. Shepelyansky

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

Shepelyansky, Dima

351

Fractal Segmentation and Clustering Analysis for Seismic Time Slices

NASA Astrophysics Data System (ADS)

Fractal analysis has become part of the standard approach for quantifying texture on gray-tone or colored images. In this research we introduce a multi-stage fractal procedure to segment, classify and measure the clustering patterns on seismic time slices from a 3-D seismic survey. Five fractal classifiers (c1)-(c5) were designed to yield standardized, unbiased and precise measures of the clustering of seismic signals. The classifiers were tested on seismic time slices from the AKAL field, Cantarell Oil Complex, Mexico. The generalized lacunarity (c1), fractal signature (c2), heterogeneity (c3), rugosity of boundaries (c4) and continuity resp. tortuosity (c5) of the clusters are shown to be efficient measures of the time-space variability of seismic signals. The Local Fractal Analysis (LFA) of time slices has proved to be a powerful edge detection filter to detect and enhance linear features, like faults or buried meandering rivers. The local fractal dimensions of the time slices were also compared with the self-affinity dimensions of the corresponding parts of porosity-logs. It is speculated that the spectral dimension of the negative-amplitude parts of the time-slice yields a measure of connectivity between the formation's high-porosity zones, and correlates with overall permeability.

Ronquillo, G.; Oleschko, K.; Korvin, G.; Arizabalo, R. D.

2002-05-01

352

Fractal Analysis of Prime Indian STOCK Market Indices

NASA Astrophysics Data System (ADS)

The purpose of the present work is to study the fractal behaviour of prime Indian stock exchanges, namely Bombay Stock Exchange Sensitivity Index (BSE Sensex) and National Stock Exchange (NSE). To analyze the monofractality of these indices we have used Higuchi method and Katz method separately. By applying Mutifractal Detrended Fluctuation Analysis (MFDFA) technique we have calculated the generalized Hurst exponents, multifractal scaling exponents and generalized multifractal dimensions for the present indices. We have deduced Hölder exponents as well as singularity spectra for BSE and NSE. It has been observed that both the stock exchanges are possessing self-similarity at different small ranges separately and inhomogeneously. By comparing the multifractal behaviour of the BSE and NSE indices, we have found that the second one exhibits a richer multifractal feature than the first one.

Samadder, Swetadri; Ghosh, Koushik; Basu, Tapasendra

2013-03-01

353

A Multi-Fractal Spectrum Analysis of Turbulence Data and the DNA of Worms

NASA Astrophysics Data System (ADS)

This paper discusses the physical meanings of various parameters in multi-fractal spectrum and offers the thermo-mechanical formula in calculating the multi-fractal spectrum. Besides it works out the sets of multi-fractal Cantor, the multi-fractal spectrum of turbulence data of Rayleigh-Benard convection and DNA series by means of wavelet transform maximum modulus (WTMM). In the end, it arrives at the conclusions that the means of WTMM is plausible on the application of multi-fractal research, multi-fractal spectrum offers function ?˜f(?), which describes all the sub-collections' characters. So it gives specific information of a system. Turbulence is similar to DNA's multi-fractal character. The latter is more uneven and complex. Multi-fractal spectrum analysis can reveal the uneven overall distribution information of the series (or sets). But it has a limited description of the local position information of signal singularity and concrete local structure.

Fu, Q.; Chen, Z. F.; Zhou, Y. H.; Wang, L.

2011-09-01

354

Highly Miniaturized Fractal Antennas

In this paper, the use of fractal elements in miniaturized antenna applications is discussed. Hilbert grounded wires are studied and a simple design routine is outlined. Using commercial software tools, the design of groundless fractals is shown to be simple and fast. We present grounded and groundless Hilbert and Meander antennas for 250 MHz, 400 MHz and 2 GHz applications.

George S. A. Shaker; Safieddin Safavi-Naeini

2007-01-01

355

Landscape roughness analysis of Mt. Etna volcanic complex detected via fractal geometry

NASA Astrophysics Data System (ADS)

During the last years several aspects relevant to volcanic activity have been analyzed in fractal context. These studies have been aimed at identifying the power laws that govern the magma fragmentation processes and/or the classification of different geological processes. In this work we exploit the algorithm proposed by Di Martino et al. (2012) that allows retrieving the fractal dimension of a natural surface starting from its corresponding Synthetic Aperture Radar (SAR) image. Such an algorithm is based on an analytical model that links the stochastic characterization of a single SAR amplitude image to the fractal dimension of the observed surface, modeled via a fractional Brownian motion (fBm) process. The considered SAR image processing provides - as an output product - the pixel by pixel map of the fractal dimension of the scene observed by the sensor. Previous works demonstrated that the fractal dimension of lava flows is strictly connected to the natural surface roughness. Moreover, Pepe et al. (2012) showed the possibility of characterizing the single volcanic structures by means of the fractal dimension values retrieved from the corresponding SAR images. In the present work we consider a data-set of Cosmo-SkyMed high resolution images acquired over the Mt. Etna volcanic complex (South Italy), spanning the 2009 - 2011 time period. Starting from the SAR amplitude images of the considered data-set, we generated the corresponding fractal dimension maps that were subsequently co-registered each other, thus retrieving the fractal dimension time-series of the Mt. Etna volcano. Then, by averaging the so-computed fractal dimension maps with respect to time we generated a map of the mean fractal dimension of the investigated area. This procedure allows significantly improving the quality of the final fractal dimension map, as the average operation reduces the noise (due to the speckle effect on SAR images) present on each fractal map. Besides, the so-obtained mean fractal dimension map was quantized in order to identify and aggregate areas that are homogeneous from a fractal viewpoint, i.e., areas having a similar surface roughness. The comparison with the geological map of Mt. Etna volcanic complex reveals that the level of roughness is strictly connected with the lava flow age. More specifically, the roughness of lava flows, which is measured by the fractal dimension, decreases with the increasing of time. In other words, our analysis demonstrates that the volcanic depositional processes of the lava flows produce surfaces with high fractal dimension (roughness), whereas the exogenous phenomena, as the erosion processes, significantly reduce the fractal dimension associated with eruptive mechanisms. The COSMO-SkyMed and ALOS data used in this study have been processed at REA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0. G. Di Martino, D. Riccio, I. Zinno, "SAR Imaging of Fractal Surfaces," IEEE Trans. Geosci. Remote Sens., vol.50, no.2, pp. 630-644, Feb. 2012. S. Pepe, G. Di Martino, A. Iodice, M. Manzo, A. Pepe, D. Riccio, G. Ruello, E. Sansosti, P. Tizzani, I. Zinno; "Analysis of the fractal dimension of volcano geomorphology through Synthetic Aperture Radar (SAR) amplitude images acquired in C and X band", Geophysical Research Abstracts Vol. 14, EGU2012-6573-1, 2012, EGU General Assembly 2012.

De Luca, Claudio; Bonfante, Antonello; Di Martino, Gerardo; Iodice, Antonio; Manzo, Mariarosaria; Pepe, Antonio; Pepe, Susi; Riccio, Daniele; Sansosti, Eugenio; Tizzani, Pietro; Zinno, Ivana

2013-04-01

356

Analysis of root fractal characteristics in remote areas of the Taklimakan desert, China

Fractal geometry is a potential new approach to analyze the root architecture, which may offer improved ways to quantify and\\u000a summarize root system complexity as well as yield ecological and physiological insights into the functional relevance of specific\\u000a architectural patterns. Fractal analysis is a sensitive measure of root branching intensity and fractal dimension expresses\\u000a the “space filling” properties of a

Xiao-lin Yang; Xi-ming Zhang; Yi-ling Li; Shao-cai Li; Hai-long Sun

2008-01-01

357

Fractals, Vol. 11, No. 4 (2003) 345352 c World Scientific Publishing Company

is in fact a map-specified Moran fractal. As a fractal set, the various dimensions of the Moran fractal have is constructed in a good way so that it can be well encoded. Let = {0, 1, . . . , r} where r is a positive inte(||)(x). In this case, F is actually the self-similar set deter- mined by {hj : j }, which satisfies the open set

Li, Wenxia

358

Fractal nature of multiple shear bands in severely deformed metallic glass

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.

Sun, B. A.; Wang, W. H. [Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China)

2011-05-16

359

Electromagnetic fields in fractal continua

NASA Astrophysics Data System (ADS)

Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum ?D3?E3 with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space F? accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed.

Balankin, Alexander S.; Mena, Baltasar; Patiño, Julián; Morales, Daniel

2013-04-01

360

Fractal analysis of the spatial distribution of earthquakes along the Hellenic Subduction Zone

NASA Astrophysics Data System (ADS)

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.

Papadakis, Giorgos; Vallianatos, Filippos; Sammonds, Peter

2014-05-01

361

Fractal aggregates in tennis ball systems

NASA Astrophysics Data System (ADS)

We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the images of the cluster of balls, following Forrest and Witten's pioneering studies on the aggregation of smoke particles, to estimate their fractal dimension.

Sabin, J.; Bandín, M.; Prieto, G.; Sarmiento, F.

2009-09-01

362

On the Weierstrass-Mandelbrot Fractal Function

The function W(t) equiv sum^?n=-? [(1 - eigamma^nt)eiphi_n]\\/gamma(2-D)n (1 < D 1, phi_n = arbitrary phases), is continuous but non-differentiable and possesses no scale. The graph of ReW or ImW has Hausdorff-Besicovitch (fractal) dimension D. Choosing phi_n = mu n gives a deterministic W the scaling properties of which can be studied analytically in terms of a representation obtained by

M. V. Berry; Z. V. Lewis

1980-01-01

363

Counterexamples in Fractal Roughness Analysis and Their Physical Properties

NASA Astrophysics Data System (ADS)

We disprove the widely held notion that a surface with nontrivial roughness exponents fluctuates at all scales ("structure within structure") and has nontrivial fractal dimension. Strong counterexamples are Cantor staircases, which have nontrivial roughness exponents, do not fluctuate at all, and have trivial fractal dimension. Weak counterexamples fluctuate intermittently and have nontrivial fractal dimension. Characteristic of all counterexamples is: (i) they consist of terraces of all sizes and exhibit scaling over the entire range of terrace sizes. (ii) they have roughness exponents Hq that vary strongly with order q; (iii) they are self affine, but not all affinities are invertible. The strong variation of Hq drives a strongly varying surface response to different external interactions (different interactions are governed by different orders q) and abrupt changes similar to a phase transition, with Hq playing the role of temperature. A summary of this extraordinary functional tunability and its applications is given.

Pfeifer, Peter; Gheorghiu, Stefan

364

Selective modulation of cell response on engineered fractal silicon substrates

A plethora of work has been dedicated to the analysis of cell behavior on substrates with ordered topographical features. However, the natural cell microenvironment is characterized by biomechanical cues organized over multiple scales. Here, randomly rough, self-affinefractal surfaces are generated out of silicon,where roughness Ra and fractal dimension Df are independently controlled. The proliferation rates, the formation of adhesion structures, and the morphology of 3T3 murine fibroblasts are monitored over six different substrates. The proliferation rate is maximized on surfaces with moderate roughness (Ra ~ 40?nm) and large fractal dimension (Df ~ 2.4); whereas adhesion structures are wider and more stable on substrates with higher roughness (Ra ~ 50?nm) and lower fractal dimension (Df ~ 2.2). Higher proliferation occurson substrates exhibiting densely packed and sharp peaks, whereas more regular ridges favor adhesion. These results suggest that randomly roughtopographies can selectively modulate cell behavior. PMID:23492898

Gentile, Francesco; Medda, Rebecca; Cheng, Ling; Battista, Edmondo; Scopelliti, Pasquale E.; Milani, Paolo; Cavalcanti-Adam, Elisabetta A.; Decuzzi, Paolo

2013-01-01

365

The mass of the dominant particle in a fractal universe

An empirically validated, phenomenological model relating the parameters of an astronomical body to the stochastic fluctuations of its granular components is generalized in terms of fractal scaling laws. The mass of the particle constituting the preponderance of the mass of a typical galaxy is determined from the generalized model as a function of the fractal dimension. For a fractal dimension between 1 and 3 the mass of the dominant particle in galaxies is, roughly, between the Planck mass and 1eV. If the dimension is near 2 then the fractal model is identical to the original stochastic model, and the mass of the dominant particle must be of order near the nucleon mass. Two additional expressions for the mass of the dominant particle in the universe are obtained from basic quantum considerations and from the existence of a cosmological constant. It follows that the fractal dimension 2 is favored and that the mass of the dominant particle is proportional to sixth root of the cosmological constant and of order near the nucleon mass.

Scott Funkhouser; Nicola Pugno

2008-04-10

366

NASA Astrophysics Data System (ADS)

When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve into an even stronger link in the near future.

McAteer, R. T. J.

2013-06-01

367

Thermodynamics of Fractal Universe

We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $ \\delta Q=T_h d{S_h}$, where $ \\delta Q $ and $ T_{h} $ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon. We find that the entropy $S_h$ consists two terms, the first one which obeys the usual area law and the second part which is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \\leq q thermodynamics can be secured in a fractal universe by suitably choosing the fractal parameter $\\beta$.

Ahmad Sheykhi; Zeinab Teimoori; Bin Wang

2012-10-29

368

Retinal Vascular Fractals and Cognitive Impairment

Background Retinal microvascular network changes have been found in patients with age-related brain diseases such as stroke and dementia including Alzheimer's disease. We examine whether retinal microvascular network changes are also present in preclinical stages of dementia. Methods This is a cross-sectional study of 300 Chinese participants (age: ?60 years) from the ongoing Epidemiology of Dementia in Singapore study who underwent detailed clinical examinations including retinal photography, brain imaging and neuropsychological testing. Retinal vascular parameters were assessed from optic disc-centered photographs using a semiautomated program. A comprehensive neuropsychological battery was administered, and cognitive function was summarized as composite and domain-specific Z-scores. Cognitive impairment no dementia (CIND) and dementia were diagnosed according to standard diagnostic criteria. Results Among 268 eligible nondemented participants, 78 subjects were categorized as CIND-mild and 69 as CIND-moderate. In multivariable adjusted models, reduced retinal arteriolar and venular fractal dimensions were associated with an increased risk of CIND-mild and CIND-moderate. Reduced fractal dimensions were associated with poorer cognitive performance globally and in the specific domains of verbal memory, visuoconstruction and visuomotor speed. Conclusion A sparser retinal microvascular network, represented by reduced arteriolar and venular fractal dimensions, was associated with cognitive impairment, suggesting that early microvascular damage may be present in preclinical stages of dementia. PMID:25298774

Ong, Yi-Ting; Hilal, Saima; Cheung, Carol Yim-lui; Xu, Xin; Chen, Christopher; Venketasubramanian, Narayanaswamy; Wong, Tien Yin; Ikram, Mohammad Kamran

2014-01-01

369

An experimental study to determine fractal parameters for lean premixed flames

In this study the fractal parameters of a lean, premixed methane-air flame were determined over a range of turbulence conditions. The focus of the present work was to improve the experimental technique so as to resolve the inner cutoff scale, the outer cutoff scale, and the fractal dimension. By adjusting the flow velocity through a set of three interchangeable grids

A. K. Das; R. L. Evans

1997-01-01

370

Fractal behaviour of carbon black and smectite dispersions by small angle neutron scattering

From dielectric spectroscopy on mechanically sheared dispersions of carbon black in mineral oil it was known that the distribution of relaxation times was invariant irrespective of the applied shear rate. From this observation fractal behaviour was suspected and a small angle neutron scattering experiment was set up for confirmation. Indeed, by this experiment a fractal dimension has been found of

J. A. Helsen; J. Teixeira

1986-01-01

371

The paper is aimed at the fractal analysis of the real (i.e., observed experimentally) creeping discharge patterns propagating over pressboard immersed in mineral and vegetable rape-seed oils, under lightning impulse voltages, using a pointplane electrode arrangement. By using the box counting method, we show that the discharge patterns present a fractal dimension D which depends on the thickness of pressboard

A. Beroual; Viet-Hung Dang

2011-01-01

372

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the ?-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to '?/? toroid' (all-? class) and 'PLP-dependent transferase-like' domains (?/? class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the ?/? class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-? class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, ?/?-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties. PMID:21614471

Banerji, Anirban; Ghosh, Indira

2011-08-01

373

Fractal Spacetime Structure in Asymptotically Safe Gravity

Four-dimensional Quantum Einstein Gravity (QEG) is likely to be an asymptotically safe theory which is applicable at arbitrarily small distance scales. On sub-Planckian distances it predicts that spacetime is a fractal with an effective dimensionality of 2. The original argument leading to this result was based upon the anomalous dimension of Newton's constant. In the present paper we demonstrate that also the spectral dimension equals 2 microscopically, while it is equal to 4 on macroscopic scales. This result is an exact consequence of asymptotic safety and does not rely on any truncation. Contact is made with recent Monte Carlo simulations.

O. Lauscher; M. Reuter

2005-08-26

374

Clumpy and fractal shocks, and the generation of a velocity dispersion in molecular clouds

We present an alternative explanation for the nature of turbulence in molecular clouds. Often associated with classical models of turbulence, we instead interpret the observed gas dynamics as random motions, induced when clumpy gas is subject to a shock. From simulations of shocks, we show that a supersonic velocity dispersion occurs in the shocked gas provided the initial distribution of gas is sufficiently non-uniform. We investigate the velocity size-scale relation $\\sigma \\propto r^{\\alpha}$ for simulations of clumpy and fractal gas, and show that clumpy shocks can produce realistic velocity size-scale relations with mean $\\alpha \\thicksim 0.5$. For a fractal distribution, with a fractal dimension of 2.2 similar to what is observed in the ISM, we find $\\sigma \\propto r^{0.4}$. The form of the velocity size-scale relation can be understood as due to mass loading, i.e. the post-shock velocity of the gas is determined by the amount of mass encountered as the gas enters the shock. We support this hypothesis with analytical calculations of the velocity dispersion relation for different initial distributions. A prediction of this model is that the line-of sight velocity dispersion should depend on the angle at which the shocked gas is viewed.

Clare Dobbs; Ian Bonnell

2006-10-24

375

In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obtained, that the embedding of compact influences on characteristics of wave in final state. Thus, the embedding of Cantor set in Euclidean space is additional property of a fractal which can be important both for applications of fractal electrodynamics and for physics of strong interactions.

V. A. Okorokov; E. V. Sandrakova

2012-06-03

376

An Analysis of Mine Water Inrush Based on Fractal and Non-Darcy Seepage Theory

NASA Astrophysics Data System (ADS)

Mining rock mechanics is a new cross subject of mechanics and mining engineering, the seepage theory is one of the important research directions. This paper combines Wu-fractal/Ergun high-speed flow theory and dynamic system instability, reveals the influence factors of mine water inrush. Research shows that: Mine water inrush related to rock porosity, particle size, shape, fractal dimension, ratio of pore and throat, and other factors. Compared the critical Reynolds number which are got from Wu-fractal model and Ergun equation, Wu-fractal model can reveal more influence factors of mine water inrush than Ergun equation.

Wu, Jinsui; Cai, Jianchao; Zhao, Dongyun; Chen, Xuexi

2014-09-01

377

Spatial log-periodic oscillations of first-passage observables in fractals.

For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain an expression for the source-target distance dependence of the MFPT that exhibits both the leading power-law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log-periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior. PMID:23367911

Akkermans, Eric; Benichou, Olivier; Dunne, Gerald V; Teplyaev, Alexander; Voituriez, Raphael

2012-12-01

378

Spatial log-periodic oscillations of first-passage observables in fractals

NASA Astrophysics Data System (ADS)

For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain an expression for the source-target distance dependence of the MFPT that exhibits both the leading power-law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log-periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior.

Akkermans, Eric; Benichou, Olivier; Dunne, Gerald V.; Teplyaev, Alexander; Voituriez, Raphael

2012-12-01

379

Extended Fractal Fits to Riemann Zeros

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.

Paul B. Slater

2007-05-21

380

Fractal Geometry of Architecture

\\u000a In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately,\\u000a scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects\\u000a of the twentieth century called for an overall idea that is mirrored in every single detail, but also

Wolfgang E. Lorenz

381

NSDL National Science Digital Library

This lesson is designed to introduce students to the idea of finding patterns in the generation of several different types of fractals. This lesson provides links to discussions and activities related to patterns and fractals as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

2010-01-01

382

Fractal analysis of the galaxy distribution in the redshift range 0.45 < z < 5.0

Evidence is presented that the galaxy distribution can be described as a fractal system in the redshift range of the FDF galaxy survey. The fractal dimension $D$ was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with $\\Omega_{m_0}=0.3$, $\\Omega_{\\Lambda_0}=0.7$ and $H_0=70 \\; \\mbox{km} \\; {\\mbox{s}}^{-1} \\; {\\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number densities $\\gamma$ and $\\gamma^\\ast$ obtained from the red and blue FDF galaxies provides a direct method to estimate $D$, implying that $\\gamma$ and $\\gamma^\\ast$ vary as power-laws with the cosmological distances. The luminosity distance $d_{\\scriptscriptstyle L}$, galaxy area distance $d_{\\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against their respective number densities to calculate $D$ by linear fitting. It was found that the FDF galaxy distribution is characterized by two single fractal dimensions at successive distance ranges. Two straight l...

Conde-Saavedra, G; Ribeiro, Marcelo B

2014-01-01

383

NASA Astrophysics Data System (ADS)

Ordered and disordered CoSi 2 surface structures observed in scanning tunneling microscopy (STM) were investigated by means of wavelet, Fourier and fractal analysis. The relationship between fractal dimensions and degree of surface cobalt disilicide ordering was established and the correlation between Fourier and fractal analysis was shown. Two methods were used for determining fractal dimensions for ordered and disordered structures, and the results, obtained by these methods, were compared. Wavelet analysis of spatially ordered structures and an attempt to determine the form of these structures, depending on wavelet parameters, were performed.

Gerasimenko, N. N.; Pavluchenko, M. N.; Levchenko, V. D.; Djamanbalin, K. K.; Troitskiy, V. Yu.; Skobochkin, A. M.

2003-05-01

384

Statistical mechanics of a fractal lattice: Renormalization-group analysis of the Sierpinski gasket

NASA Astrophysics Data System (ADS)

Exact renormalization-group recursion relations are constructed for several thermodynamic properties of the Sierpinski gasket (SG). A model with one, two, and three spin interactions is studied. The interactions are restricted to be explicitly part of the fractal geometry. The specific heat, paramagnetic susceptibility, and near-neighbor correlation function are calculated exactly both near and away from criticality which occurs at zero temperature (infinite near-neighbor coupling). With one exception, our results agree with the asymptotic critical properties of the SG as discussed by Gefen et al. As a finitely ramified structure, the critical exponents of the SG are essentially one dimensional. The nonuniversal properties of the SG, however, are markedly different from those in one dimension. The scale-invariant, fractal structure of the SG leads to strong critical fluctuations and, therefore, to a rather wide critical region. We find a logarithmic correction to the leading singularity in the correlation length which is necessary to account for the approach to the critical region. The specific heat of the SG is qualitatively different from that in one dimension. A continuous line of fixed points at zero temperature is also found. This is due to a marginal operator associated with a competition between the two (odd-spin) symmetry-breaking fields. In the absence of a two-spin interaction, the SG becomes frustrated when the two odd-spin interactions compete directly. The phase diagram in this limit is also calculated.

Luscombe, James H.; Desai, Rashmi C.

1985-08-01

385

Fractal Themes at Every Level Kenneth G. Monks

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

Monks, Kenneth

386

Fractal Relativity, Generalized Noether Theorem and New Research of Space-Time

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.

Yi-Fang Chang

2007-07-02

387

Texture and fractal methods for analyzing the characteristics of unsteady gas flows in pipelines

Turbulent structures in two-dimensional model gas flows are analyzed by estimating the fractal dimension and the texture characteristics\\u000a of the gas flows. The fractal dimension is estimated using the self-similarity indices of the power spectra for the following\\u000a characteristics of the gas flows: the Coriolis and Boussinesq coefficients, average pressure, hydraulic resistance coefficient,\\u000a velocity loss in a pipeline element, and

O. B. Butusov; V. P. Meshalkin

2006-01-01

388

Fractal analysis of InGaN self-assemble quantum dots grown by MOCVD

Recently, it has been shown that nitride nanostructures can be self-assembled using growth interruption during the metal-organic chemical vapor deposition (MOCVD) growth. In this work, strain-induced InGaN-GaN self-assembled quantum dot (SAQD) samples were grown by MOCVD. A fractal dimension processing has been applied to characterize the surface roughness of uncapped InGaN-GaN SAQDs. The fractal dimension D can be used to

K. T. Lam; L. W. Ji

2007-01-01

389

Fractal Weyl law for quantum fractal eigenstates

The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.

D. L. Shepelyansky

2007-09-14

390

Prediction of heat transfer of nanofluid on critical heat flux based on fractal geometry

NASA Astrophysics Data System (ADS)

Analytical expressions for nucleate pool boiling heat transfer of nanofluid in the critical heat flux (CHF) region are derived taking into account the effect of nanoparticles moving in liquid based on the fractal geometry theory. The proposed fractal model for the CHF of nanofluid is explicitly related to the average diameter of the nanoparticles, the volumetric nanoparticle concentration, the thermal conductivity of nanoparticles, the fractal dimension of nanoparticles, the fractal dimension of active cavities on the heated surfaces, the temperature, and the properties of the fluid. It is found that the CHF of nanofluid decreases with the increase of the average diameter of nanoparticles. Each parameter of the proposed formulas on CHF has a clear physical meaning. The model predictions are compared with the existing experimental data, and a good agreement between the model predictions and experimental data is found. The validity of the present model is thus verified. The proposed fractal model can reveal the mechanism of heat transfer in nanofluid.

Xiao, Bo-Qi

2013-01-01

391

Prediction of Effective Thermal Conductivity of Porous Media with Fractal-Monte Carlo Simulations

NASA Astrophysics Data System (ADS)

On the basis of the fractal scaling laws of pore distribution in natural porous media, a probability model is developed for thermal conductivity in porous media by combining fractal theory and Monte Carlo technique. The current numerical model, which was validated by comparison with the existing experimental data, shows that the thermal conductivity of porous media is a function of the thermal conductivities of volume fraction, pore area fractal dimension, tortuosity fractal dimension and random number. The effect of microstructure parameters on the effective thermal conductivity of porous media is studied. The proposed fractal Monte Carlo simulation technique has advantages compared with conventional numerical methods and may have the potential in analyzing other transport properties of porous media.

Xu, Yousheng; Zheng, Youqu; Kou, Jianlong

2014-09-01

392

This paper extends the velocity dependent Krook (VDK) model, developed at NRL over the last 4 years, to two dimensions and presents a variety of calculations. One dimensional spherical calculations presented here investigate shock ignition. Comparing VDK calculations to a flux limit calculation shows that the laser profile has to be retuned and some gain is sacrificed due to preheat of the fuel. However, preheat is by no means a show stopper for laser fusion. The recent foil acceleration experiments at the University of Rochester Laboratory for Laser Energetics are modeled with two-dimensional simulations. The radial loss is very important to consider in modeling the foil acceleration. Once this is done, the VDK model gives the best agreement with the experiment.

Manheimer, Wallace; Colombant, Denis; Schmitt, Andrew J. [Laser Plasma Branch, Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375 (United States)

2012-05-15

393

Fractal analysis of the retinal vascular network in fundus images.

Complexity of the retinal vascular network is quantified through the measurement of fractal dimension. A computerized approach enhances and segments the retinal vasculature in digital fundus images with an accuracy of 94% in comparison to the gold standard of manual tracing. Fractal analysis was performed on skeletonized versions of the network in 40 images from a study of stroke. Mean fractal dimension was found to be 1.398 (with standard deviation 0.024) from 20 images of the hypertensives sub-group and 1.408 (with standard deviation 0.025) from 18 images of the non-hypertensives subgroup. No evidence of a significant difference in the results was found for this sample size. However, statistical analysis showed that to detect a significant difference at the level seen in the data would require a larger sample size of 88 per group. PMID:18003503

Macgillivray, T J; Patton, N; Doubal, F N; Graham, C; Wardlaw, J M

2007-01-01

394

Fractal signatures in analogs of interplanetary dust particles

NASA Astrophysics Data System (ADS)

Interplanetary dust particles (IDPs) are an important constituent of the earths stratosphere, interstellar and interplanetary medium, cometary comae and tails, etc. Their physical and optical characteristics are significantly influenced by the morphology of silicate aggregates which form the core in IDPs. In this paper we reinterpret scattering data from laboratory analogs of cosmic silicate aggregates created by Volten et al. (2007) [1] to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension dm?1.75. The same fractal dimension also characterizes clusters obtained from diffusion limited aggregation (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically transported silicate particles.

Katyal, Nisha; Banerjee, Varsha; Puri, Sanjay

2014-10-01

395

On the Question of the Magnetic Susceptibility of Fractal Ferromagnetic Wires

NASA Astrophysics Data System (ADS)

With the help of a well-grounded measure, we have constructed an algorithm for calculating the longitudinal component of the magnetic susceptibility tensor of a fractal ferromagnetic wire. The calculations are based on the semiclassical kinetic equation for the magnon distribution function. The temperature and frequency dependences of the longitudinal magnetic susceptibility of a fractal wire is found.

Gladkov, S. O.; Bogdanova, S. B.

2014-08-01

396

NASA Technical Reports Server (NTRS)

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.

Lam, Nina Siu-Ngan; Qiu, Hong-Lie; Quattrochi, Dale A.; Emerson, Charles W.; Arnold, James E. (Technical Monitor)

2001-01-01

397

Monte Carlo Sampling in Fractal Landscapes

NASA Astrophysics Data System (ADS)

We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.

Leitão, Jorge C.; Lopes, J. M. Viana Parente; Altmann, Eduardo G.

2013-05-01

398

Based on the first law of thermodynamics and the law of hydrokinetics by this paper, the mathematic model of the pressure characteristic in different nozzle throat dimensions is established for the 252 kV SF6 circuit breaker, and the pressure characteristic curve of the arc-extinguishing chamber can be obtained through computation. In addition, the result of the pressure characteristic in different

Lin Xin; Liu Ke-bin

2008-01-01

399

Fractal Superconductivity near Localization Threshold

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

Fominov, Yakov

400

Generalized Sierpinski fractal multiband antenna

A new set of fractal multiband antennas called mod-p Sierpinski gaskets is presented. Mod-p Sierpinski fractal antennas derive from the Pascal triangle and present a log-periodic behavior, which is a consequence of their self-similarity properties. Mod-p Sierpinski fractal antennas constitute a generalization of the classical Sierpinski antenna

Jordi Romeu; Jordi Soler

2001-01-01

401

Fractal analysis for classification of breast carcinoma in optical coherence

University of North Carolina at Chapel Hill, Department of Physics and Astronomy, Biomedical Research Imaging Center fibroglandular stromal tissues in OCT images, and an objective measure is needed. In this initial study, we. We computed the fractal dimension, a measure of the self-similarity of an object, along the depth

Oldenburg, Amy

402

Modeling the scattering properties of mineral aerosols using concave fractal polyhedra.

The single-scattering properties of concave fractal polyhedra are investigated, with particle size parameters ranging from the Rayleigh to geometric-optics regimes. Two fractal shape parameters, irregularity and aspect ratio, are used to iteratively construct "generations" of irregular fractal particles. The pseudospectral time-domain (PSTD) method and the improved geometric-optics method (IGOM) are combined to compute the single-scattering properties of fractal particles over the range of size parameters. The effects of fractal generation, irregularity, and aspect ratio on the single-scattering properties of fractals are investigated. The extinction efficiency, absorption efficiency, and asymmetry factor, calculated by the PSTD method for fractal particles, with small-to-moderate size parameters, smoothly bridges the gap between those size parameters and size parameters for which solutions given by the IGOM may be used. Somewhat surprisingly, excellent agreement between values of the phase function of randomly oriented fractal particles calculated by the two numerical methods is found, not only for large particles, but in fact extends as far down in equivalent-projected-area size parameters as 25. The agreement in the case of other nonzero phase matrix elements is not as good at so small a size. Furthermore, the numerical results of ensemble-averaged phase matrix elements of a single fractal realization are compared with dust particle measurements, and good agreement is found by using the fractal particle model to represent data from a study of feldspar aerosols. PMID:23385901

Liu, Chao; Panetta, R Lee; Yang, Ping; Macke, Andreas; Baran, Anthony J

2013-02-01

403

Fractal characterisation of sea-scattered signals and detection of sea-surface targets

NASA Astrophysics Data System (ADS)

Fractal theory is applied to the analysis of real radar signals which are scattered from rough sea surfaces. The databases formed by sampling the radar signals include the two general cases, i.e. both forward-scattered and backscattered signals. The signals for the two cases were recorded using two entirely different radar systems and at two entirely different geographic locations. The box counting method is used to estimate the fractal dimension of the scattered signals. To corroborate this result, a computation of the fractal dimension is based on the index alpha in the power spectrum relation. The estimates derived from both methods are consistent. It is observed that the forward-scattered and back-scattered radar signals have very similar fractal dimensions. Finally, it is shown that there is a detectable variation in the fractal dimension when a target is present. Based on this variation, it is therefore possible to detect the presence of a target by observing the fractal dimension of the radar returns.

Lo, T.; Leung, H.; Litva, J.; Haykin, S.

1993-08-01

404

Asymptotic Infrared Fractal Structure of the Propagator for a Charged Fermion

It is well known that the long-range nature of the Coulomb interaction makes the definition of asymptotic ``in'' and ``out'' states of charged particles problematic in quantum field theory. In particular, the notion of a simple particle pole in the vacuum charged particle propagator is untenable and should be replaced by a more complicated branch cut structure describing an electron interacting with a possibly infinite number of soft photons. Previous work suggests a Dirac propagator raised to a fractional power dependent upon the fine structure constant, however the exponent has not been calculated in a unique gauge invariant manner. It has even been suggested that the fractal ``anomalous dimension'' can be removed by a gauge transformation. Here, a gauge invariant non-perturbative calculation will be discussed yielding an unambiguous fractional exponent. The closely analogous case of soft graviton exponents is also briefly explored.

Gulzari, S; Widom, A

2006-01-01

405

Asymptotic Infrared Fractal Structure of the Propagator for a Charged Fermion

It is well known that the long-range nature of the Coulomb interaction makes the definition of asymptotic ``in'' and ``out'' states of charged particles problematic in quantum field theory. In particular, the notion of a simple particle pole in the vacuum charged particle propagator is untenable and should be replaced by a more complicated branch cut structure describing an electron interacting with a possibly infinite number of soft photons. Previous work suggests a Dirac propagator raised to a fractional power dependent upon the fine structure constant, however the exponent has not been calculated in a unique gauge invariant manner. It has even been suggested that the fractal ``anomalous dimension'' can be removed by a gauge transformation. Here, a gauge invariant non-perturbative calculation will be discussed yielding an unambiguous fractional exponent. The closely analogous case of soft graviton exponents is also briefly explored.

S. Gulzari; J. Swain; A. Widom

2006-05-31

406

A novel multiband antenna: fractal antenna

Fractal-shaped antennas have already been proved to have some unique characteristics that are linked to the geometrical properties of the fractals. The purpose of this article is to introduce the concept of the fractal, review the progress in fractal antenna study and implementation, compare different types of fractal antenna elements and arrays and discuss the challenge and future of this

Tian Tiehong; Zhou Zheng

2003-01-01

407

Fractal Parameters of Pore Space from CT Images of Soils Under Contrasting Management Practices

NASA Astrophysics Data System (ADS)

Soil structure plays an important role in flow and transport phenomena, and a quantitative characterization of the spatial heterogeneity of the pore space geometry is beneficial for prediction of soil physical properties. Morphological features such as pore-size distribution, pore space volume or pore-solid surface can be altered by different soil management practices. Irregularity of these features and their changes can be described using fractal geometry. In this study, we focus primarily on the characterization of soil pore space as a 3D geometrical shape by fractal analysis and on the ability of fractal dimensions to differentiate between two a priori different soil structures. We analyze X-ray computed tomography (CT) images of soils samples from two nearby areas with contrasting management practices. Within these two different soil systems, samples were collected from three depths. Fractal dimensions of the pore-size distributions were different depending on soil use and averaged values also differed at each depth. Fractal dimensions of the volume and surface of the pore space were lower in the tilled soil than in the natural soil but their standard deviations were higher in the former as compared to the latter. Also, it was observed that soil use was a factor that had a statistically significant effect on fractal parameters. Fractal parameters provide useful complementary information about changes in soil structure due to changes in soil management.

Muñoz, F. J.; San José Martínez, F.; Caniego, F. J.

2014-09-01

408

Fractal Based Image Steganography

This paper describes a new and novel steganographic method for inserting secret information into image files. The method uses fractal image compression techniques in the production of these steganographic image files. The method allows a user to specify a visual key when hiding the secret information. The visual key must then be used when retrieving the hidden data. The paper

Paul Davern; Michael Scott

1996-01-01

409

Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot. PMID:11607061

Hsü, K J; Hsü, A J

1990-01-01

410

Fractal polyzirconosiloxane cluster coatings.

National Technical Information Service (NTIS)

Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-(3-(triethoxysilyl)propyl)-4,5-dihydroimidazole, Zr(OC(sub 3)...

T. Sugama

1992-01-01

411

Color constancy using fractals

NASA Astrophysics Data System (ADS)

We combine fractal decompression and the Retinex algorithm do devise a new color constancy method. We show that by using this approach, we can achieve color constancy and image compression simultaneously. Experimental results are included that show that this approach is quite promising.

Rising, Hawley K., III; Baqai, Farhan A.

2005-03-01

412

NASA Astrophysics Data System (ADS)

Rare events between states in complex systems are fundamental in many scientific fields and can be studied by building reaction pathways. A theoretical framework to analyze reaction pathways is provided by transition-path theory (TPT). The central object in TPT is the committor function, which is found by solution of the backward-Kolmogorov equation on a given potential. Once determined, the committor can be used to calculate reactive fluxes and rates, among other important quantities. We demonstrate here that the committor can be calculated using the method of finite elements on non-uniform meshes. We show that this approach makes it feasible to perform TPT calculations on 3D potentials because it requires many fewer degrees of freedom than a regular-mesh finite-difference approach. In various illustrative 2D and 3D problems, we calculate the committor function and reaction rates at different temperatures, and we discuss effects of temperatures and simple entropic barriers on the structure of the committor and the reaction rate constants.

Lapelosa, Mauro; Abrams, Cameron F.

2013-10-01

413

Rare events between states in complex systems are fundamental in many scientific fields and can be studied by building reaction pathways. A theoretical framework to analyze reaction pathways is provided by transition-path theory (TPT). The central object in TPT is the committor function, which is found by solution of the backward-Kolmogorov equation on a given potential. Once determined, the committor can be used to calculate reactive fluxes and rates, among other important quantities. We demonstrate here that the committor can be calculated using the method of finite elements on non-uniform meshes. We show that this approach makes it feasible to perform TPT calculations on 3D potentials because it requires many fewer degrees of freedom than a regular-mesh finite-difference approach. In various illustrative 2D and 3D problems, we calculate the committor function and reaction rates at different temperatures, and we discuss effects of temperatures and simple entropic barriers on the structure of the committor and the reaction rate constants. PMID:24014889

Lapelosa, Mauro; Abrams, Cameron F.

2013-01-01

414

Error Assessment in Modeling with Fractal Brownian Motions

NASA Astrophysics Data System (ADS)

To model a given time series F(t) with fractal Brownian motions (fBms), it is necessary to have appropriate error assessment for related quantities. Usually the fractal dimension D is derived from the Hurst exponent H via the relation D = 2-H, and the Hurst exponent can be evaluated by analyzing the dependence of the rescaled range <|F(t + ?) - F(t)|> on the time span ?. For fBms, the error of the rescaled range not only depends on data sampling but also varies with H due to the presence of long term memory. This error for a given time series then can not be assessed without knowing the fractal dimension. We carry out extensive numerical simulations to explore the error of rescaled range of fBms and find that for 0 < H < 0.5, |F(t + ?) - F(t)| can be treated as independent for time spans without overlap; for 0.5 < H < 1, the long term memory makes |F(t + ?) - F(t)| correlated and an approximate method is given to evaluate the error of <|F(t + ?) - F(t)|>. The error and fractal dimension can then be determined self-consistently in the modeling of a time series with fBms.

Qiao, Bingqiang; Liu, Siming

2013-12-01

415

NASA Astrophysics Data System (ADS)

Microchannel heat sink with high heat transfer coefficients has been extensively investigated due to its wide application prospective in electronic cooling. However, this cooling system requires a separate pump to drive the fluid transfer, which is uneasy to minimize and reduces their reliability and applicability of the whole system. In order to avoid these problems, valveless piezoelectric pump with fractal-like Y-shape branching tubes is proposed. Fractal-like Y-shape branching tube used in microchannel heat sinks is exploited as no-moving-part valve of the valveless piezoelectric pump. In order to obtain flow characteristics of the pump, the relationship between tube structure and flow rate of the pump is studied. Specifically, the flow resistances of fractal-like Y-shape branching tubes and flow rate of the pump are analyzed by using fractal theory. Then, finite element software is employed to simulate the flow field of the tube, and the relationships between pressure drop and flow rate along merging and dividing flows are obtained. Finally, valveless piezoelectric pumps with fractal-like Y-shape branching tubes with different fractal dimensions of diameter distribution are fabricated, and flow rate experiment is conducted. The experimental results show that the flow rate of the pump increases with the rise of fractal dimension of the tube diameter. When fractal dimension is 3, the maximum flow rate of the valveless pump is 29.16 mL/min under 100 V peak to peak (13 Hz) power supply, which reveals the relationship between flow rate and fractal dimensions of tube diameter distribution. This paper investigates the flow characteristics of valveless piezoelectric pump with fractal-like Y-shape branching tubes, which provides certain references for valveless piezoelectric pump with fractal-like Y-shape branching tubes in application on electronic chip cooling.

Huang, Jun; Zhang, Jianhui; Wang, Shouyin; Liu, Weidong

2014-05-01

416

This aim of this study was to assess the discriminatory value of fractal and grey level co-occurrence matrix (GLCM) analysis methods in standard microscopy analysis of two histologically similar brain white mass regions that have different nerve fiber orientation. A total of 160 digital micrographs of thionine-stained rat brain white mass were acquired using a Pro-MicroScan DEM-200 instrument. Eighty micrographs from the anterior corpus callosum and eighty from the anterior cingulum areas of the brain were analyzed. The micrographs were evaluated using the National Institutes of Health ImageJ software and its plugins. For each micrograph, seven parameters were calculated: angular second moment, inverse difference moment, GLCM contrast, GLCM correlation, GLCM variance, fractal dimension, and lacunarity. Using the Receiver operating characteristic analysis, the highest discriminatory value was determined for inverse difference moment (IDM) (area under the receiver operating characteristic (ROC) curve equaled 0.925, and for the criterion IDM?0.610 the sensitivity and specificity were 82.5 and 87.5%, respectively). Most of the other parameters also showed good sensitivity and specificity. The results indicate that GLCM and fractal analysis methods, when applied together in brain histology analysis, are highly capable of discriminating white mass structures that have different axonal orientation. PMID:24967845

Pantic, Igor; Dacic, Sanja; Brkic, Predrag; Lavrnja, Irena; Pantic, Senka; Jovanovic, Tomislav; Pekovic, Sanja

2014-10-01

417

Comparing the fractality of European urban neighbourhoods: do national contexts matter?

NASA Astrophysics Data System (ADS)

The objective of this paper is to show that morphological similarities between built-up urban surfaces are greater across borders than within cities in Europe: living, architectural and planning trends are international. The spatial arrangement of built-up areas is analysed here by means of fractal indices using a set of 97 town sections selected from 18 European urban agglomerations. The fractal dimension is estimated by correlation techniques. Results confirm that morphological similarities are higher across countries/cities than within. Moreover, two types of fractal laws are considered: one uses the basic fractal scaling law; the other introduces a prefactor a that is often called a "form factor" in the fractal literature. Differences in the results obtained by both laws are explained empirically as well as theoretically, and suggestions are made for further measurements.

Thomas, Isabelle; Frankhauser, Pierre; Badariotti, Dominique

2012-04-01

418

Fractals in geology and geophysics

NASA Technical Reports Server (NTRS)

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.

Turcotte, Donald L.

1989-01-01

419

A fractal analysis of pathogen detection by biosensors

NASA Astrophysics Data System (ADS)

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.

Doke, Atul M.; Sadana, Ajit

2006-05-01

420

Magnetic Reconnection Rate in Space Plasmas: A Fractal Approach

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.

Materassi, Massimo [Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, V. Madonna del Piano 10, I-50019 Sesto Fiorentino (Italy); Consolini, Giuseppe [Istituto di Fisica dello Spazio Interplanetario, Istituto Nazionale di Astrofisica, V. Fosso del Cavaliere 100, I-00133 Rome (Italy)

2007-10-26

421

Analysis of fractal-shaped antennas using the multiperiodic traveling wave Vee model

A novel analytical model to predict the radiation performance of the Sierpinski fractal multiband antenna is presented. The theoretical development allows one to accurately predict the antenna response to flare angle variations and also to calculate the radiation patterns of other fractal-shaped antennas

Carles Puente; Jordi Soler

2001-01-01

422

Fractal Clusters and Stable Distribution

Size distribution of fractal clusters is shown to follow a one-sided stable distribution. Its characteristic exponent alpha is determined by the fractal relation between the cluster's surface (S) and volume (N) under the scale change: S-->aS, N-->a1\\/alphaN. This result means that an asymptotic power law size distribution p(N){\\\\propto}N-alpha-1 is deduced from the above geometrical fractal relation.

Hideki Takayasu

1988-01-01

423

Langevin Equation on Fractal Curves

We analyse a random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, hence plays an important role in this analysis. A Langevin equation with a particular noise model is thus proposed and solved using techniques of the newly developed $F^\\alpha$-Calculus .

Seema Satin; A. D. Gangal

2014-04-28

424

Super wideband fractal antenna design

Fractals have very unique properties, therefore in recent years, antenna designers use fractal geometry in multi-band and broad-band antennas designing. In this paper, I have achieved to a 40 GHz super wideband antenna with applying a fractal geometry to a wire square loop antenna and choosing appropriate size and location for feeding. Modelling and simulation is performed via SuperNEC electromagnetic

Abolfazl Azari

2009-01-01

425

Fractals in biology and medicine

NASA Technical Reports Server (NTRS)

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.

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

426

Fractals in physiology and medicine

NASA Technical Reports Server (NTRS)

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.

Goldberger, Ary L.; West, Bruce J.

1987-01-01

427

NASA Astrophysics Data System (ADS)

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.

Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.

2004-04-01

428

Fractal structure of a three-dimensional brownian motion on an attractive plane.

Consider a brownian particle in three dimensions which is attracted by a plane with a strength proportional to some dimensionless parameter ?. We investigate the fractal spatial structure of the visited lattice sites in a cubic lattice by the particle around and on the attractive plane. We compute the fractal dimensions of the set of visited sites both in three dimensions and on the attractive plane, as a function of the strength of attraction ?. We also investigate the scaling properties of the size distribution of the clusters of nearest-neighbor visited sites on the attractive plane and compute the corresponding scaling exponent ? as a function of ?. The fractal dimension of the curves surrounding the clusters is also computed for different values of ?, which, in the limit ???, tends to that of the outer perimeter of planar brownian motion, i.e., the self-avoiding random walk (SAW). We find that all measured exponents depend significantly on the strength of attraction. PMID:21928955

Saberi, Abbas Ali

2011-08-01

429

On the Fractal Structure of the Universe

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.

P V Grujic; V D Pankovic

2009-07-13

430

Loop-erased random walk on a percolation cluster: Crossover from Euclidean to fractal geometry

NASA Astrophysics Data System (ADS)

We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ?pc, in two and three dimensions. We find that the fractal dimensions of LERWp are close to normal LERW in a Euclidean lattice, for all p >pc. However, our results reveal that LERW on critical incipient percolation clusters is fractal with df=1.217±0.002 for d =2 and 1.43±0.02 for d =3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior of the LERWp crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to pc. For finite systems, two crossover exponents and a scaling relation can be derived. This work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.

Daryaei, E.; Rouhani, S.

2014-06-01

431

A Fractal Nature for Polymerized Laminin

Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system. PMID:25296244

Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana

2014-01-01

432

A fractal nature for polymerized laminin.

Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system. PMID:25296244

Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana

2014-01-01

433

Fractal approach to the description of the auroral region

NASA Astrophysics Data System (ADS)

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.

Chernyshov, A. A.; Mogilevsky, M. M.; Kozelov, B. V.

2013-07-01

434

Fractal properties and denoising of lidar signals from cirrus clouds

NASA Astrophysics Data System (ADS)

Airborne lidar signals of cirrus clouds are analyzed to determine the cloud structure. Climate modeling and numerical weather prediction benefit from accurate modeling of cirrus clouds. Airborne lidar measurements of the European Lidar in Space Technology Experiment (ELITE) campaign were analyzed by combining shots to obtain the backscatter at constant altitude. The signal at high altitude was analyzed for horizontal structure of cirrus clouds. The power spectrum and the structure function show straight lines on a double logarithmic plot. This behavior is characteristic for a Brownian fractal. Wavelet analysis using the Haar wavelet confirms the fractal aspects. It is shown that the horizontal structure of cirrus can be described by a fractal with a dimension of 1.8 over length scales that vary 4 orders of magnitude. We use the fractal properties in a new denoising method. Denoising is required for future lidar measurements from space that have a low signal to noise ratio. Our wavelet denoising is based on the Haar wavelet and uses the statistical fractal properties of cirrus clouds in a method based on the maximum a posteriori (MAP) probability. This denoising based on wavelets is tested on airborne lidar signals from ELITE using added Gaussian noise. Superior results with respect to averaging are obtained.

van den Heuvel, J. C.; Driesenaar, M. L.; Lerou, R. J. L.

2000-02-01

435

Fractal approach to the description of the auroral region

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.

Chernyshov, A. A., E-mail: achernyshov@iki.rssi.ru; Mogilevsky, M. M. [Russian Academy of Sciences, Space Research Institute (Russian Federation)] [Russian Academy of Sciences, Space Research Institute (Russian Federation); Kozelov, B. V. [Russian Academy of Sciences, Polar Geophysical Institute, Kola Science Center (Russian Federation)] [Russian Academy of Sciences, Polar Geophysical Institute, Kola Science Center (Russian Federation)

2013-07-15

436

Anisotropic fractal media by vector calculus in non-integer dimensional space

NASA Astrophysics Data System (ADS)

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