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1

Fractal Dimension for Fractal Structures

The main goal of this paper has a double purpose. On the one hand, we propose a new definition in order to compute the fractal dimension of a subset respect to any fractal structure, which completes the theory of classical box-counting dimension. Indeed, if we select the so called natural fractal structure on each euclidean space, then we will get the box-counting dimension as a particular case. Recall that box-counting dimension could be calculated over any euclidean space, although it can be defined over any metrizable one. Nevertheless, the new definition we present can be computed on an easy way over any space admitting a fractal structure. Thus, since a space is metrizable if and only if it supports a starbase fractal structure, our model allows to classify and distinguish a much larger number of topological spaces than the classical definition. On the other hand, our aim consists also of studying some applications of effective calculation of the fractal dimension over a kind of contexts where the box-counting dimension has no sense, like the domain of words, which appears when modeling the streams of information in Kahn's parallel computation model. In this way, we show how to calculate and understand the fractal dimension value obtained for a language generated by means of a regular expression, and also we pay attention to an empirical and novel application of fractal dimension to natural languages.

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

2010-07-22

2

Exterior dimension of fat fractals

NASA Technical Reports Server (NTRS)

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.

Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.

1985-01-01

3

Box-covering algorithm for fractal dimension of weighted networks.

Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the "Sierpinski" weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed. PMID:24157896

Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran

2013-01-01

4

Box-covering algorithm for fractal dimension of weighted networks

NASA Astrophysics Data System (ADS)

Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.

Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran

2013-10-01

5

Fractal dimensions of wildfire spreading

NASA Astrophysics Data System (ADS)

The time series data of 31 wildfires in 2012 in the US were analyzed. The fractal dimensions (FD) of the wildfires during spreading were studied and their geological features were identified. A growth model based on the cellular automata method is proposed here. Numerical study was performed and is shown to give good agreement with the fractal dimensions and scaling behaviors of the corresponding empirical data.

Wang, S.-L.; Lee, H.-I.; Li, S.-P.

2014-08-01

6

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.

7

Fractal dimension of bioconvection patterns

NASA Technical Reports Server (NTRS)

Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.

Noever, David A.

1990-01-01

8

The Dimension of Projections of Fractal Percolations

NASA Astrophysics Data System (ADS)

Fractal percolation or Mandelbrot percolation is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by a typical realization of the fractal percolation on the plane, If then for all lines ? the orthogonal projection E ? of E to ? has the same Hausdorff dimension as E,

Rams, Micha?; Simon, Károly

2014-02-01

9

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

10

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

11

Stochastic Models That Separate Fractal Dimension and Hurst Effect

self-affinity implies a linear relationship be- tween fractal dimension and Hurst coefficient fractal dimensions of pial vascular networks, thin sections of sandstones, earth resistivity, tunnel systems of subterranean termites, blood pressure rates, human cell proliferation, and cryo- sectioned

Washington at Seattle, University of

12

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

13

Output functions and fractal dimensions in dynamical systems.

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition. PMID:11290037

de Moura, A P; Grebogi, C

2001-03-26

14

Twoband Infrared Data Fusion Method Based-on Fractal Dimension

Researches indicate that the gray-scale images mapped from most nature objects accord to the fractal Brown stochastic field which has a foundation of self-similarity, meaning the image is made up of copies of itself in a reduced scale. The fractal dimension can quantifiably depict the fractal character and the property of an image. Therefore, a new method based-on fractal dimension

Yuqui Sun; Jinwen Tian; Jian Liu

2005-01-01

15

FRACTAL DIMENSION OF S&P CNX NIFTY STOCK RETURNS

A fractal is a geometrical structure that is self-similar when scaled. By assessing the fractal dimension of asset returns, the retail investors with recent innovation in financial prediction and computational power, can exploit these prices. A branch of a tree is often used as an example in Fractal. The branch is similar to the whole tree and if we break

Mahalingam Gayathri; Murugesan Selvam; Kasilingam Lingaraja; Vinayagamoorthi Vasanth; Venkatraman Karpagam

2013-01-01

16

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

17

Fractal dimension analysis of complexity in Ligeti piano pieces

NASA Astrophysics Data System (ADS)

Fractal correlation dimensional analysis has been performed with whole solo piano pieces by Gyrgy Ligeti at every 50ms interval of the pieces. The resulting curves of development of complexity represented by the fractal dimension showed up a very reasonable correlation with the perceptional density of events during these pieces. The seventh piece of Ligeti's ``Musica ricercata'' was used as a test case. Here, each new part of the piece was followed by an increase of the fractal dimension because of the increase of information at the part changes. The second piece ``Galamb borong,'' number seven of the piano Etudes was used, because Ligeti wrote these Etudes after studying fractal geometry. Although the piece is not fractal in the strict mathematical sense, the overall structure of the psychoacoustic event-density as well as the detailed event development is represented by the fractal dimension plot.

Bader, Rolf

2005-04-01

18

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

19

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

20

Characterization of border structure using fractal dimension in melanomas.

There are many characteristics that differentiate normal moles (nevi) from melanomas. One of them is their boundary irregularity, which can be quantified using Fractal Dimension. In this work, fractal dimension of normal moles and melanoma was computed using the box counting method. These measurements were used to train a linear decoder in order to predict the pathology. The average performance to discriminate normal moles from melanomas reached 85% giving some insights about the power of the fractal dimension as a candidate for automatic detection and diagnosis. PMID:21096624

Carbonetto, S H; Lew, S E

2010-01-01

21

Filtered Dynamics and Fractal Dimensions for Noisy Speech Recognition

We explore methods from fractals and dynamical systems theory for robust processing and recognition of noisy speech. A speech signal is embedded in a multidimensional phase-space and is subsequently filtered exploiting aspects of its unfolded dynamics. Invariant measures (fractal dimensions) of the filtered signal are used as features in automatic speech recognition (ASR). We evaluate the new proposed features as

Vassilis Pitsikalis; Petros Maragos

2006-01-01

22

Use of fractal dimensions to quantify coral shape

NASA Astrophysics Data System (ADS)

A morphometrical method to quantify and characterize coral corallites using Richardson Plots and Kaye’s notion of fractal dimensions is presented. A Jurassic coral species ( Aplosmilia spinosa) and five Recent coral species were compared using the Box-Counting Method. This method enables the characterization of their morphologies at calicular and septal levels by their fractal dimensions (structural and textural). Moreover, it is possible to determine differences between species of Montastraea and to tackle the high phenotypic plasticity of Montastraea annularis. The use of fractal dimensions versus conventional methods (e.g., measurements of linear dimensions with a calliper, landmarks, Fourier analyses) to explore a rugged boundary object is discussed. It appears that fractal methods have the potential to considerably simplify the morphometrical and statistical approaches, and be a valuable addition to methods based on Euclidian geometry.

Martin-Garin, B.; Lathuilière, B.; Verrecchia, E. P.; Geister, J.

2007-09-01

23

Fractal Dimensions of In Vitro Tumor Cell Proliferation

Biological systems are characterized by their potential for dynamic adaptation. One of the challenges for systems biology approaches is their contribution towards the understanding of the dynamics of a growing cell population. Conceptualizing these dynamics in tumor models could help us understand the steps leading to the initiation of the disease and its progression. In vitro models are useful in answering this question by providing information over the spatiotemporal nature of such dynamics. In the present work, we used physical quantities such as growth rate, velocity, and acceleration for the cellular proliferation and identified the fractal structures in tumor cell proliferation dynamics. We provide evidence that the rate of cellular proliferation is of nonlinear nature and exhibits oscillatory behavior. We also calculated the fractal dimensions of our cellular system. Our results show that the temporal transitions from one state to the other also follow nonlinear dynamics. Furthermore, we calculated self-similarity in cellular proliferation, providing the basis for further investigation in this topic. Such systems biology approaches are very useful in understanding the nature of cellular proliferation and growth. From a clinical point of view, our results may be applicable not only to primary tumors but also to tumor metastases.

Lambrou, George I.

2015-01-01

24

Fractal dimensions of small (15-200 ?m) particles in Eastern Pacific coastal waters

NASA Astrophysics Data System (ADS)

Particles 3-300 ?m (average length) in seawater include single cells, non-viable particles of identifiable origin (such as fecal pellets), aggregated particles formed from water column debris, and aggregated mixtures of all of these materials. While macroscopic marine snow-sized aggregates (>0.5 mm in average length) have been shown to be fractal, relatively less is known about the average characteristics of smaller particles. We calculated the fractal dimensions of microscopic particles 15-200 ?m in length through simultaneous measurements of particle size distributions as a function of solid equivalent diameter (from solid volumes measured using a Coulter Counter) and average length (from image analysis of acridine-orange stained filtered particles). Particle size distributions were measured at two eastern Pacific coastal areas, one in Monterey Bay, CA, and the other in East Sound, WA. Average fractal dimensions of particles indicated that D was highest in East Sound ( D=2.59±0.17) during a phytoplankton bloom that did not appear to be aggregating, and lowest at one site in Monterey Bay ( D=1.77±0.34), where old diatom flocs and marine snow-size aggregates were observed. There was no direct relationship between D and total particle concentration, chlorophyll a, or transparent exopolymer particles (TEP) concentration, although the highest concentration of TEP was found at the site with the lowest fractal dimension. Particles with low fractal dimensions are produced through coagulation. Our subjective assessment of the importance of aggregate formation at these sites, based on diving and microscopic observations, indicated that aggregates were more abundant at sites where particles had lower fractal dimensions. Thus, we attribute the low fractal dimensions of these small particles to be the result of their formation through coagulation processes.

Li, Xiaoyan; Passow, Uta; Logan, Bruce E.

1998-01-01

25

Fractal dimension analysis in a highly granular calorimeter

The concept of “particle flow” has been developed to optimise the jet energy resolution by distinguishing the different jet components. A highly granular calorimeter designed for the particle flow algorithm provides an unprecedented level of detail for the reconstruction of calorimeter showers and enables new approaches to shower analysis. In this paper the measurement and use of the fractal dimension of showers is described. The fractal dimension is a characteristic number that measures the global compactness of the shower. It is highly dependent on the primary particle type and energy. Its application in identifying particles and estimating their energy is described in the context of a calorimeter designed for the International Linear Collider.

Ruan, M; Brient, J.C; Jeans, D; Videau, H

2015-01-01

26

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

27

NASA Astrophysics Data System (ADS)

In the field of corrosion research, mass gain/loss, electrochemical tests and comparing the surface elemental distributions, phase constitutions as well as surface morphologies before and after corrosion are extensively applied to investigate the corrosion behavior or estimate the corrosion resistance of materials that operated in various environments. Most of the above methods are problem oriented, complex and longer-period time-consuming. However from an object oriented point of view, the corroded surfaces of materials often have self-similar characterization: fractal property which can be employed to efficiently achieve damaged surface analysis. The present work describes a strategy of comparison of the surface fractal dimensions for corrosion resistance estimation: chromizing coating was synthesized on P110 steel surface to improve its performance via pack cementation. Scanning electron microscope (SEM) was used to investigate the surface morphologies of the original and corroded samples. Surface fractal dimensions of the detected samples were calculated by binary images related to SEM images of surface morphologies with box counting algorithm method. The results showed that both surface morphologies and surface fractal dimensions of P110 steel varied greatly before and after corrosion test, but the chromizing coating changed slightly. The chromizing coating indicated better corrosion resistance than P110 steel. Comparison of surface fractal dimensions of original and corroded samples can rapidly and exactly realize the estimation of corrosion resistance.

Lin, Naiming; Guo, Junwen; Xie, Faqin; Zou, Jiaojuan; Tian, Wei; Yao, Xiaofei; Zhang, Hongyan; Tang, Bin

2014-08-01

28

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

29

Original article Structure and fractal dimensions of root systems

Original article Structure and fractal dimensions of root systems of four co-occurring fruit tree, Bangor, Gwynedd LL57 2UW, UK (Received 1 February 1999; accepted 29 October 1999) Abstract Coarse root-auto- matically digitized. Spatial distributions of root length were determined from the digitally

Paris-Sud XI, Université de

30

Fractal dimension and turbulence in Giant HII Regions

We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble's Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.

H. E. Caicedo-Ortiz; E. Santiago-Cortés; J. López-Bonilla; H. O. Castañeda

2015-02-16

31

Fractal dimension and turbulence in Giant HII Regions

We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble's Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.

Caicedo-Ortiz, H E; López-Bonilla, J; Castañeda, H O

2015-01-01

32

Numerical problems with evaluating the fractal dimension of real data

A new program RIVER for evaluating the fractal dimension of real data sets was written. Its performance was compared with two programs HarFA — demo version and Coastline, available in Internet. The three programs were tested on about 50 data sets. The program RIVER yielded the maximal errors less than 3 percentages for all tested data sets, while the other

ADAM SZUSTALEWICZ

33

Turbulence in non-integer dimensions by fractal Fourier decimation

Fractal decimation reduces the effective dimensionality of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius $k$ is proportional to $k^D$ for large $k$. At the critical dimension D=4/3 there is an equilibrium Gibbs state with a $k^{-5/3}$ spectrum, as in [V. L'vov {\\it et al.}, Phys. Rev. Lett. {\\bf 89}, 064501 (2002)]. Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as $(D-4/3)^{-2/3}$.

Uriel Frisch; Anna Pomyalov; Itamar Procaccia; Samriddhi Sankar Ray

2011-08-05

34

Liver ultrasound image classification by using fractal dimension of edge

NASA Astrophysics Data System (ADS)

Medical ultrasound image edge detection is an important component in increasing the number of application of segmentation, and hence it has been subject of many studies in the literature. In this study, we have classified the liver ultrasound images (US) combining Canny and Sobel edge detectors with fractal analysis in order to provide an indicator about of the US images roughness. We intend to provide a classification rule of the focal liver lesions as: cirrhotic liver, liver hemangioma and healthy liver. For edges detection the Canny and Sobel operators were used. Fractal analyses have been applied for texture analysis and classification of focal liver lesions according to fractal dimension (FD) determined by using the Box Counting method. To assess the performance and accuracy rate of the proposed method the contrast-to-noise (CNR) is analyzed.

Moldovanu, Simona; Bibicu, Dorin; Moraru, Luminita

2012-08-01

35

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

2012-02-20

36

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

37

NASA Astrophysics Data System (ADS)

A series of escarpments along the upland/lowland contact of Deuteronilus Mensae, Mars, were analyzed for their fractal dimensions. The escarpments have fractal dimensions ranging from 1.20 to 1.25, nearly identitical to fractals found in Earth shorelines.

Cull, S. C.

2003-03-01

38

Texture descriptor combining fractal dimension and artificial crawlers

NASA Astrophysics Data System (ADS)

Texture is an important visual attribute used to describe images. There are many methods available for texture analysis. However, they do not capture the detail richness of the image surface. In this paper, we propose a new method to describe textures using the artificial crawler model. This model assumes that agents can interact with the environment and each other. Since this swarm system alone does not achieve a good discrimination, we developed a new method to increase the discriminatory power of artificial crawlers, together with the fractal dimension theory. Here, we estimated the fractal dimension by the Bouligand-Minkowski method due to its precision in quantifying structural properties of images. We validate our method on two texture datasets and the experimental results reveal that our method leads to highly discriminative textural features. The results indicate that our method can be used in different texture applications.

Gonçalves, Wesley Nunes; Machado, Bruno Brandoli; Bruno, Odemir Martinez

2014-02-01

39

Fractal dimension of faults network in the upper silesian coal basin (Poland): Preliminary studies

NASA Astrophysics Data System (ADS)

Fractal analysis of faults network, tremor foci spatial distribution as well as the Gutenberg-Richter relationship could further explain whether the biggest seismic events are connected with recent tectonic activity. Fractality of fault systems geometry, as a first step of the analysis, was tested fro a part of the USCB embodying the main structural units. The cluster analysis and the box counting methods were employed. The calculated fractal dimension of fault network was 1.98 for the whole area yet for considered structural units it was close to 1.6. The results point to similarity of studied fault pattern to river network. Faults within selected tectonic units make separate sets which have a distinct geometry and origin. The value of 1.6 is an upper limit to the fracture geometry of rocks that can be explained on the basis of Griffith energy balance concept.

Idziak, Adam; Teper, Les?aw

1996-07-01

40

NASA Astrophysics Data System (ADS)

We report a model for the fractal dimension Ds of rough surfaces based on the fractal distribution of roughness elements on surfaces and the fractal character of surface profiles. The proposed model for the fractal dimension Ds is expressed as a function of the fractal dimensions D for conic roughness diameter/height and Dp for surface profile, maximum roughness base diameter ?max, the ratio ? of conic roughness height to its base radius as well as the ratio ?min?max of the minimum to the maximal base diameter.

Li, Jian-Hua; Yu, Bo-Ming; Zou, Ming-Qing

2009-11-01

41

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

Salac, Petr

2014-01-01

42

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

43

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

44

The VFractal: a new estimator for fractal dimension of animal movement paths

Fractal measurements of animal movement paths have been used to analyze how animals view habitats at different spatial scales. One problem has been the absence of error estimates for fractal d estimators. To address this weakness, I present and test 4 new estimators for measuring fractal dimension at different spatial scales, along with estimates of their variation. The estimators are

Vilis O. Nams

1996-01-01

45

Speech Emotion Recognition Based on Parametric Filter and Fractal Dimension

NASA Astrophysics Data System (ADS)

In this paper, we propose a new method that employs two novel features, correlation density (Cd) and fractal dimension (Fd), to recognize emotional states contained in speech. The former feature obtained by a list of parametric filters reflects the broad frequency components and the fine structure of lower frequency components, contributed by unvoiced phones and voiced phones, respectively; the latter feature indicates the non-linearity and self-similarity of a speech signal. Comparative experiments based on Hidden Markov Model and K Nearest Neighbor methods are carried out. The results show that Cd and Fd are much more closely related with emotional expression than the features commonly used.

Mao, Xia; Chen, Lijiang

46

Spectra of recurrence dimension for dynamically defined subsets of Rauzy Fractals

, Venezuela. e-mail: vsirvent@usb.ve Abstract We compute the spectra of the recurrence dimension are totally disconnected. Key words: Substitutions, Rauzy fractals, Adic systems, Dimension theory, Poincar

Sirvent, VÃctor F.

47

COMPARISON OF FRACTAL DIMENSION ALGORITHMS FOR THE COMPUTATION OF EEG BIOMARKERS FOR DEMENTIA

COMPARISON OF FRACTAL DIMENSION ALGORITHMS FOR THE COMPUTATION OF EEG BIOMARKERS FOR DEMENTIA C Goh of the Fractal Dimension of the EEG appears to be a good approach for the computation of biomarkers for dementia for computing reliable biomarkers, specifically, for the assessment of dementia. To achieve this, some

Paris-Sud XI, Université de

48

Analysis of fractal dimensions of rat bones from film and digital images

NASA Technical Reports Server (NTRS)

OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.

2001-01-01

49

Fractal Dimension in Butterflies’ Wings: a novel approach to understanding wing patterns ?

The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and

A. A. Castrejón-Pita; A. Sarmiento-Galán; J. R. Castrejón-Pita; R. Castrejón-García

2005-01-01

50

Dependence of fractal dimension of DLCA clusters on size of primary particles.

It is well known that clusters generated from colloidal aggregation driven by Brownian motion are typical fractal objects with the fractal dimension in the range of 1.75-1.85 under the diffusion-limited cluster aggregation (DLCA) conditions. In this work, we review and analyze the values of the fractal dimension for DLCA clusters experimentally determined in the literature. It is found that the value of the fractal dimension decreases significantly as the primary particle radius increases. Then, we have properly designed the DLCA experiments, using different radii of the primary particles, and determined the fractal dimensions of the generated clusters. Our results have well confirmed that the fractal dimension indeed decreases as the particle radius increases. To explore the mechanism leading to such dependence, we have performed intense computations through the full T-Matrix theory, and we conclude that this is not related to the effect of the intra-cluster multiple scattering on the slope of the scattering structure factor. The large fractal dimensions of the clusters generated by very small nanoparticles could be explained by thermal restructuring due to their low bonding energies, but no clear explanation can be given for the small fractal dimensions of the clusters made of large particles. PMID:23623300

Wu, Hua; Lattuada, Marco; Morbidelli, Massimo

2013-07-01

51

NASA Astrophysics Data System (ADS)

The structure of many natural objects exhibits a self-similar or self-affine scaling behavior. Examples range from porous media to mountain ranges, river networks, clouds in the atmosphere or the mass distribution in the universe. The fractal dimension of those objects can be measured by image analyzing techniques. Concerning the structure of porous media, there are different features which may show a fractal behavior: The mass distribution, the pore space and the pore-solid interface. In many cases the fractal dimension of one of these features has been determined and has been taken to describe the entire system. Beyond this it is interesting what the fractal dimensions are and how they are related for the different features of the same object. The measurements for three natural systems (a silty soil structure, a dendrite and the void system of a clayey soil) are presented and compared with measurements of a textbook fractal (Sirpinski carpet). The length scale of the natural objects extends from 1 mm for SEM images of an impregnated Luvisol over the moss agate to about 400 mm for the void system. Results show different dimensions for different features of the same object with a tendency to higher fractal dimensions for the mass distribution and lower fractal dimensions for the interface. The measured results are compared with a pore-solid fractal model of soil structure.

Dathe, A.; Baveye, P.

2003-04-01

52

Scaling exponents for a monkey on a tree: fractal dimensions of randomly branched polymers.

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results. PMID:23004722

Janssen, Hans-Karl; Stenull, Olaf

2012-05-01

53

Scaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.

Hans-Karl Janssen; Olaf Stenull

2012-03-13

54

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; Bálint, Zoltán

2014-01-01

55

Low fractal dimension cluster-dilute soot aggregates from a premixed flame.

Using a novel morphology segregation technique, we observed minority populations ( approximately 3%) of submicron-sized, cluster-dilute fractal-like aggregates, formed in the soot-formation window (fuel-to-air equivalence ratio of 2.0-3.5) of a premixed flame, to have mass fractal dimensions between 1.2 and 1.51. Our observations disagree with previous observations of a universal mass fractal dimension of approximately 1.8 for fractal-like aerosol aggregates formed in the dilute-limit via three-dimensional diffusion-limited cluster aggregation processes. A hypothesis is presented to explain this observation. Subject to verification of this hypothesis, it may be possible to control the fractal dimension and associated properties of aggregates in the cluster-dilute limit through application of a static electric field during the aggregation process. PMID:19658949

Chakrabarty, Rajan K; Moosmüller, Hans; Arnott, W Patrick; Garro, Mark A; Tian, Guoxun; Slowik, Jay G; Cross, Eben S; Han, Jeong-Ho; Davidovits, Paul; Onasch, Timothy B; Worsnop, Douglas R

2009-06-12

56

A new fuel-cell electrocatalyst based on highly porous carbonized polyacrylonitrile (PAN) microcellular foam with platinum particles was prepared recently in this laboratory. Its surface morphology, one of the most important aspects of a practical electrocatalyst, has been examined in terms of fractal theory and methods. The fractal dimension of the platinum particles dispersed in porous carbonized PAN foam was determined by using chronometric and rotating-disk-electrode methods in oxygen-saturated solutions. A fractal dimension smaller than 2 was obtained, which was attributed to the partially active nature of the surface of this electrocatalytic material. This value of fractal dimension is also proposed to be considered as a reaction dimension. A reaction dimension smaller than 2 may indicate that not all of the platinum particle surface is accessible to the incoming oxygen molecules.

Ye, S. [Inst. de Recherche d`Hydro-Quebec, Varennes, Quebec (Canada)]|[Inst. National de la Recherche Scientifique, Varennes, Quebec (Canada). Lab. de Recherche sur les Materiaux Avances; Vijh, A.K. [Inst. de Recherche d`Hydro-Quebec, Varennes, Quebec (Canada); Dao, L.H. [Inst. National de la Recherche Scientifique, Varennes, Quebec (Canada). Lab. de Recherche sur les Materiaux Avances

1997-05-01

57

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

58

It is demonstrated that fluorescence resonance energy transfer may be used to determine the fractal dimension of aggregates of membrane-bound proteins. Theoretical and experimental results are presented for two different experimental designs: energy transfer between proteins and energy transfer from lipids to proteins. For energy transfer between proteins the lattice spacing must be known independently for a fractal dimension to be uniquely determined, and this represents a disadvantage to this experimental design. Results are presented for the calcium ATPase and a fractal dimension of 1.9 is estimated for ATPase aggregates by assuming a lattice spacing of 50 A. Energy transfer from lipids to protein provides a means of estimating the length of the "coast-line" of the aggregate. In this case the fractal dimension is uniquely determined from a log-log plot. An analysis of data for bacteriohodopsin reconstituted in phospholipid vesicles gives a fractal dimension of 1.6. The structural basis of the value for the fractal dimension is discussed for these two systems. These techniques provide a means of assessing the nature of protein-protein interactions in membranous systems. PMID:2528385

Dewey, T G; Datta, M M

1989-01-01

59

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.

60

Characterization of melanophore morphology by fractal dimension analysis.

Fractal or focal dimension (FD) analysis is a valuable tool to identify physiologic stimuli at the cellular and tissue levels that allows for quantification of cell perimeter complexity. The FD analysis was determined on fluorescence images of caffeine- or epinephrine-treated (or untreated control) killifish Fundulus heteroclitus (Linneaus) melanophores in culture. Cell perimeters were indicated by rhodamine-phalloidin labeling of cortical microfilaments using box-counting FD analysis. Caffeine-treated melanophores displayed dispersed melanosomes in cells with less serrated edges and reduced FD and complexity. Complexity in epinephrine-treated cells was significantly higher than the caffeine-treated cells or in the control. Cytoarchitectural variability of the cell perimeter is expected because cells change shape when cued with agents. Epinephrine-treated melanophores demonstrated aggregated melanosomes in cells with more serrated edges, significantly higher FD and thus complexity. Melanophores not treated with caffeine or epinephrine produced variable distributions of melanosomes and resulted in cells with variably serrated edges and intermediate FD with a larger SE of the regression and greater range of complexity. Dispersion of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centrifugal distribution of melanosomes throughout the cell and to the periphery. The loading of melanosomes onto cortical microfilaments may provide a less complex cell contour, with the even distribution of the cytoskeleton and melanosomes. Aggregation of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centripetal distribution of melanosomes. The aggregation of melanosomes may contribute to centripetal retraction of the cytoskeleton and plasma membrane. The FD analysis is, therefore, a convenient method to measure contrasting morphologic changes within stimulated cells. PMID:15016306

Kimler, Victoria A; Tracy-Bee, Mary; Ollie, Candace D; Langer, Renee M; Montante, James M; Marks, Charles R C; Carl Freeman, D; Anton Hough, R; Taylor, John D

2004-04-01

61

Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading

NASA Technical Reports Server (NTRS)

Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.

2001-01-01

62

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

63

Fractal dimension in butterflies' wings: a novel approach to understanding wing patterns?

The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and a small number of unrelated species, thus implying the consideration of only a fraction of the wing pattern diversity. The members of the first mimicry ring, composed by the species Danaus plexippus (better known as the monarch butterfly), and the two subspecies Basilarchia archippus obsoleta (or northern viceroy) and Basilarchia archippus hoffmanni (or tropical viceroy), are found to have a very similar value for the fractal dimension of their wing patterns, even though they do not look very similar at first sight. It is also found that the female of another species (Neophasia terlootii), which looks similar to the members of the previous group, does not share the same feature, while the Lycorea ilione albescens does share it. For the members of the second group of mimicry related butterflies, the Greta nero nero and the Hypoleria cassotis, it is shown that they also have very close values for the fractal dimension of their wing patterns. Finally, it is shown that other species, which apparently have very similar wing patterns, do not have the same fractal dimension. A possible, not completely tested hypothesis is then conjectured: the formation of groups by individuals whose wing patterns have an almost equal fractal dimension may be due to the fact that they do share the same developmental raw material, and that this common feature is posteriorly modified by natural selection, possibly through predation. PMID:15614549

Castrejón-Pita, A A; Sarmiento-Galán, A; Castrejón-Pita, J R; Castrejón-García, R

2005-05-01

64

Fractal dimensions of aggregates formed in different fluid mechanical environments

The fractal properties of aggregates formed under two different fluid mechanical environments, a paddle mixer and a rolling cylinder, were measured using three different techniques: a non-steady state method requiring both volume and length size distributions, a steady state size distribution method, and an aggregate property scaling method. Based on cumulative size distributions and the non-steady state method, aggregates produced

Bruce E. Logan; John R. Kilps

1995-01-01

65

Rationale and Objectives To investigate whether using fractal dimension as an objective index (quantitative measure) to assess and control the “visual” or “texture” similarity of reference image regions selected by a CBIR (content-based image retrieval) scheme will (or will not) affect the performance of the scheme in classification between image regions depicting suspicious breast masses. Materials and Methods An image dataset depicting 1500 verified mass regions and 1500 false-positive mass regions was used. We computed 14 morphological and intensity distribution based features and a fractal dimension. A CBIR scheme using a k-nearest neighbor classifier was applied and two experiments were conducted. In the first experiment, we evaluated our CBIR scheme using all 15 features. In the second experiment, we used the fractal dimension as a prescreening feature to guide the CBIR scheme to search for the most similar reference images that have similar measure in the fractal dimension. Results The CBIR scheme achieved classification performance with area under ROC curve (AZ) of 0.857 with 95% confidence interval (CI) of [0.844, 0.870] using 14 features and 0.866 with 95% CI of [0.853, 0.879] after adding fractal dimension. The p-value of two classification results was 0. 005. After using fractal dimension as a prescreening feature, the CBIR scheme achieved AZ = 0.851 with 95% CI of [0.837, 0.864] without significant difference as comparing with the previous result using the original 14 features (p = 0.120). The difference of fractal dimension values between the selected similar reference images was reduced by 56.7% indicating the improvement of image texture similarity. In addition, more than half of references were early discarded without similarity comparison indicating the improvement of searching efficiency. Conclusions This study demonstrated the feasibility of applying the fractal dimension as an objective (quantitative) and efficient search index to assess and maintain texture similarity of reference mass regions selected by the CBIR schemes without reducing the scheme performance in classifying between suspicious breast masses. PMID:19524455

Park, Sang Cheol; Wang, Xiao-Hui; Zheng, Bin

2009-01-01

66

Fractal dimension of 3-blocks in four-, five-, and six-dimensional percolation systems.

Using Monte Carlo simulations, we study the distributions of the 3-block mass N3 in four-, five-, and six-dimensional percolation systems. Because the probability of creating large 3-blocks in these dimensions is very small, we use a "go with the winners" method of statistical enhancement to simulate configurations having probability as small as 10(-30). In earlier work, the fractal dimensions of 3-blocks, d(3), in 2D (two dimensional) and 3D were found to be 1.20+/-0.1 and 1.15+/-0.1, respectively, consistent with the possibility that the fractal dimension might be the same in all dimensions. We find that the fractal dimension of 3-blocks decreases rapidly in higher dimensions, and estimate d(3)=0.7+/-0.2 (4D) and 0.5+/-0.2 (5D). At the upper critical dimension of percolation, d(c)=6, our simulations are consistent with d(3)=0 with logarithmic corrections to power-law scaling. PMID:12636744

Paul, Gerald; Stanley, H Eugene

2003-02-01

67

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 approaches. We propose a robust and adaptive method that suppresses sand ripples prior to the detection stage

Nelson, James

68

Analysis of the hemispheric asymmetry using fractal dimension of a skeletonized cerebral surface

We investigated hemispheric asymmetry using the fractal dimension (FD) of the skeletonized cerebral surface. Sixty-two T1-weighted magnetic resonance imaging volumes from normal Korean adults were used. The skeletonization of binary volume data, which corresponded to the union of the gray matter and cerebrospinal flow classified by fuzzy clustering, was performed slice by slice in the sagittal direction, and then skeletonized

Jong-Min Lee; Uicheul Yoon; Jae-Jin Kim; In Young Kim; Dong Soo Lee; Jun Soo Kwon; S. I. Kim

2004-01-01

69

Generalized fractal dimensions of laser Doppler flowmetry signals recorded from glabrous and nonglabrous skin Benjamin Buarda Groupe ESAIP, 18 rue du 8 mai 1945, BP 80022, 49180 Saint Barthélémy d Purpose: The technique of laser Doppler flowmetry LDF is commonly used to have a peripheral view

Chapeau-Blondeau, François

70

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions Received 15 December 2007; published 29 February 2008 The relationship between capillary pressure been substantiated by assuming that capillary pressure is directly related to the pore radius. When

Deinert, Mark

71

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

72

Are fractal dimensions of the spatial distribution of mineral deposits meaningful?

It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.

Raines, G.L.

2008-01-01

73

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

74

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

75

Improving accuracy and precision in estimating fractal dimension of animal movement paths.

It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step. PMID:16823606

Nams, Vilis O

2006-01-01

76

Fractal dust grains of different shapes are observed in a radially confined magnetized radio frequency plasma. The fractal dimensions of the dust structures in two-dimensional (2D) horizontal dust layers are calculated, and their evolution in the dust growth process is investigated. It is found that as the dust grains grow the fractal dimension of the dust structure decreases. In addition, the fractal dimension of the center region is larger than that of the entire region in the 2D dust layer. In the initial growth stage, the small dust particulates at a high number density in a 2D layer tend to fill space as a normal surface with fractal dimension D = 2. The mechanism of the formation of fractal dust grains is discussed.

Huang, F. [College of Science, China Agricultural University, Beijing 100083 (China); Peng, R. D. [State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing 100083 (China); Liu, Y. H. [Institute of Complexity Science, Qingdao University, Qingdao 266071 (China); Chen, Z. Y. [Department of Physics, Beijing University of Chemical Technology, Beijing 100029 (China); Ye, M. F.; Wang, L. [Institute of Physics, Chinese Academy of Science, Beijing 100190 (China)

2012-09-15

77

Fractal Dimension of Mie Scattering Spectra for the Appraisal of Infected HeLa Cells in Cultures

With the aim of evaluating viral infection effects on cellular structures three fractal methods (Petrosian, Hurst exponent and Height-height correlation) were performed as an expression of quantitative measure of cell nuclei size distribution. Cell nuclei size distribution was determined measuring Mie scattering spectra registered using an original LSS system. Fractal dimension for interest area of Mie scattering spectra was computed

Radu Dobrescu; Loretta Ichim

2007-01-01

78

Fractional Brownian surfaces have been widely discussed as an appropriate model for the statistical behavior of topographic surfaces. The fractals model proposes that topographic surfaces are statistically self-similar, and that a single parameter, the fractal dimension, applies at all scales. This paper presents the results of empirical examinations of 17 topographic samples. Only one of these samples shows the statistical

David M. Mark; Peter B. Aronson

1984-01-01

79

Zone Specific Fractal Dimension of Retinal Images as Predictor of Stroke Incidence

Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, ? = 0.05) compared with SFD (H = 0.51, P = 0.475, ? = 0.05) and BC (H = 0.41, P = 0.520, ? = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed. PMID:25485298

Kumar, Dinesh Kant; Hao, Hao; Unnikrishnan, Premith; Kawasaki, Ryo; Mitchell, Paul

2014-01-01

80

Fractal dimension of debris-avalanche deposits in the Hawaiian submarine landslide deposits

NASA Astrophysics Data System (ADS)

17 landslide deposits on the flanks of the southern Hawaiian Ridge have been classified into two major types: SLUMPS, which moved slowly as a coherent mass, and DEBRIS AVALANCHES, which moved quickly.The debris-avalanche deposits are predominant on submarine flanks of volcanic ocean islands elsewhere in the world. Such huge landslides are considered to produce giant tsunamis and megaturbidites covering large areas of abyssal plains. Based on the small scale topographic elements, we reinvestigated the distribution areas and emplacement styles of the debris-avalanche deposits, which differ from those previously proposed from GLORIA images without benefit of detailed bathymetric data or direct seafloor observations. There are several types of small scale topographic elements in the debris-avalanche deposits previously proposed: source amphitheater, toppled blocks, marginal levee, slide-emplaced blocks, chute, mud wave, hummocky terrain. They are very similar to those appeared in subaerial volcanic debris-avalanche fields. However, no correlation between the collapse height and runout distance are observed in the submarine debris-avalanche deposits. The hummocky terrains can be classified into two types: FLAT-TYPE, which is distributed in the nearly flat abyssal plain, less than 0.5 degree, and SLOPE-TYPE, which located on the lower part of the submarine flanks, greater than 1 degree. The size of hummocks in a slope-type hummocky terrain have an unimodal distribution pattern with a broad peak in the number of hummocks versus height category diagram. On the contrary, the size of hummocks in flat-type hummocky terrains have a power law distribution pattern in the same diagram. The fractal dimensions calculated from these diagrams are 1.19 (Nuuanu landslide), 2.32 (Ka Lae landslide) and 2.96 (Alika 2 debris-avalanche), respectively. They are expected to reflect the processes and degree of fragmentation. Therefore, among the debris_]avalanche deposits proposed previously around Hawaiian ridge, only three debris-avalanche deposits, Nuuanu, Alika 2, Ka Lae, could be a huge landslide deposits accompanied with huge tsunamis. Because fractal dimension indicates degree of the fragmentation, Alika 2 debris flow could be the most powerful turbulent flow among others. The bending trend in the power law distribution pattern of the Nuuanu landslide imply that the hummocks were produced by two different fragmentation: turbulent flows at the toe of the debris-avalanche and translational disruption at proximal part. The hummocks without a power law distribution in the terrain have been produced by a overlapping of small scale debris flow deposits rather than a huge landslide failure. Unimodal size distribution with a broad peak may be interpreted as a gravel-rich submarine fans rather than a huge landslide deposit.

Yokose, H.; Yamato, S.

2005-12-01

81

Transition of fractal dimension in a latticed dynamical system

We study a recursion relation that manifests two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigenbaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns are reproduced in this new route. In the oscillatory instability regime the fracal dimension saturates at D/sub F/ approx. = 2.6 with imbedding dimensions while in the turbulent regime D/sub F/ saturates at 6.0. 19 refs., 3 figs.

Duong-van, M.

1986-03-01

82

Can one hear the dimension of a fractal?

NASA Astrophysics Data System (ADS)

We consider the spectrum of the Laplacian in a bounded open domain of ? n with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function. The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones.

Brossard, Jean; Carmona, René

1986-03-01

83

A new way of describing meiosis that uses fractal dimension to predict metaphase I

Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465

2005-01-01

84

Fractal dimension of intersection sets in the dielectric breakdown model

Clusters grown with the dielectric breakdown model (DBM) in the cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translational invariant in the main growth direction. The box-counting dimension of one-dimensional intersection sets of the clusters is studied for six different values of the growth

C. Evertsz

1989-01-01

85

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

86

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

87

NASA Astrophysics Data System (ADS)

This paper describes the quantitative measurement of the amount of fibrosis in the rat liver using the fractal dimension of the shape of power spectrum. The shape of the power spectrum of the scattered echo from biotissues is strongly affected by its internal structure. The fractal dimension, which is one of the important parameters of the fractal theory, is useful to express the complexity of shape of figures such as the power spectrum. From in vitro experiments using rat liver, it was found that this method can be used to quantitatively measure the amount of fibrosis in the liver, and has the possibility for use in the diagnosis of human liver cirrhosis.

Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun

1995-05-01

88

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

89

Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis

Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed. PMID:24063399

Metze, Konradin

2013-01-01

90

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

91

ERIC Educational Resources Information Center

The divider dimensions of a range of maps of Ireland dating from 1567 to 1893 are evaluated, and it is shown that for maps produced before 1650 the fractal dimension of the map can be correlated to its date of publication. Various classroom uses and extensions are discussed. (Contains 2 figures.)

McCartney, M.; Myers, D.; Sun, Y.

2008-01-01

92

Fractal Feature Analysis Of Beef Marblingpatterns

NASA Astrophysics Data System (ADS)

The purpose of this study is to investigate fractal behavior of beef marbling patterns and to explore relationships between fractal dimensions and marbling scores. Authors firstly extracted marbling images from beef rib-eye crosssection images using computer image processing technologies and then implemented the fractal analysis on these marbling images based on the pixel covering method. Finally box-counting fractal dimension (BFD) and informational fractal dimension (IFD) of one hundred and thirty-five beef marbling images were calculated and plotted against the beef marbling scores. The results showed that all beef marbling images exhibit fractal behavior over the limited range of scales accessible to analysis. Furthermore, their BFD and IFD are closely related to the score of beef marbling, suggesting that fractal analyses can provide us a potential tool to calibrate the score of beef marbling.

Chen, Kunjie; Qin, Chunfang

93

NASA Astrophysics Data System (ADS)

The box-counting algorithm is the most commonly used method for evaluating the fractal dimension D of natural images. However, its application may easily lead to erroneous results. In a previous paper (Gonzato et al. 1998) we demonstrated that a crucial bias is introduced by insufficient sampling and/or by uncritical application of the regression technique. This bias turns out to be common in many practical applications. Here it is shown that an equally important additional bias is introduced by the orientation, placement and length of the digitized object relative to that of the initial box. Some additional problems are introduced by objects containing unconnected parts, since the discontinuities may or may not be indicative of a fractal pattern. Last, but certainly not least in magnitude, the thickness of the digitized profile, which is implicitly controlled by the scanner resolution versus the image line thickness, plays a fundamental role. All of these factors combined introduce systematic errors in determining D, the magnitudes of which are found to exceed 50 per cent in some cases, crucially affecting classification. To study these errors and minimize them, a program that accounts for image digitization, zooming and automatic box counting has been developed and tested on images of known dimension. The code automatically extracts the unconnected parts from a digitized shape given as input, zooms each part as optimally as possible, and performs the box-counting algorithm on a virtual screen. The size of the screen can be set to meet the sampling requirement needed to produce stable and reliable results. However, this code does not provide image vectorization, which must be performed prior to running this program. A number of image vectorizing codes are available that successfully reduce the thickness of the image parts to one pixel. Image vectorization applied prior to the application of our code reduces the sampling bias for objects with known fractal dimension to around 10-20 per cent. Since this bias is always positive, this effect can be readily compensated by a multiplying factor, and estimates of the fractal dimension accurate to about 10 per cent are effectively possible.

Gonzato, Guido; Mulargia, Francesco; Ciccotti, Matteo

2000-07-01

94

Purpose Cerebral microvascular disease is associated with dementia. Differences in the topography of the retinal vascular network may be a marker for cerebrovascular disease. The association between cerebral microvascular state and non-pathological cognitive ageing is less clear, particularly because studies are rarely able to adjust for pre-morbid cognitive ability level. We measured retinal vascular fractal dimension (Df) as a potential marker of cerebral microvascular disease. We examined the extent to which it contributes to differences in non-pathological cognitive ability in old age, after adjusting for childhood mental ability. Methods Participants from the Lothian Birth Cohort 1936 Study (LBC1936) had cognitive ability assessments and retinal photographs taken of both eyes aged around 73 years (n = 648). IQ scores were available from childhood. Retinal vascular Df was calculated with monofractal and multifractal analysis, performed on custom-written software. Multiple regression models were applied to determine associations between retinal vascular Df and general cognitive ability (g), processing speed, and memory. Results Only three out of 24 comparisons (two eyes × four Df parameters × three cognitive measures) were found to be significant. This is little more than would be expected by chance. No single association was verified by an equivalent association in the contralateral eye. Conclusions The results show little evidence that fractal measures of retinal vascular differences are associated with non-pathological cognitive ageing. PMID:25816017

Taylor, Adele M.; MacGillivray, Thomas J.; Henderson, Ross D.; Ilzina, Lasma; Dhillon, Baljean; Starr, John M.; Deary, Ian J.

2015-01-01

95

NASA Technical Reports Server (NTRS)

The surface roughness of the Vastitas Borealis Formation on Mars was analyzed with fractal statistics. Root mean square slopes and fractal dimensions were calculated for 74 topographic profiles. Results have implications for radar scattering models.

Garneau, S.; Plaut, J. J.

2000-01-01

96

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

97

Fractal approach to measuring roughness of geomembranes

Fractal analysis was used to evaluate the roughness of four commercially available geomembranes used in waste-containment systems. Fractal analysis uses the concept of the fractal dimension D as a way to calculate the roughness of simple and complex profiles. In the present study, it was determined that the fractal dimension D of the geomembrane profiles increased as the roughness of their profiles increased. For example, the smoothest of the four geomembrane profiles had a fractal dimension D equal to 1.001; the roughest of the geomembrane profiles had a fractal dimension D equal to 1.1345. The difference in the fractal dimension D for the four geomembrane profiles was found to be relatively small. Thus, when the fractal dimension D is used to evaluate the roughness of geomembranes, at least four-digit precision after the decimal point is advisable. The sensitivity of the fractal dimension to inputs such as the direction of measurement and cut as well as the size of the segment length used to cover the geomembrane profiles were also analyzed.

Vallejo, L.E.; Zhou, Y. [Univ. of Pittsburgh, PA (United States). Dept. of Civil Engineering] [Univ. of Pittsburgh, PA (United States). Dept. of Civil Engineering

1995-05-01

98

Having a classifier of cell types in a breast cancer microscopic image (BCMI), obtained with immunohistochemical staining, is required as part of a computer-aided system that counts the cancer cells in such BCMI. Such quantitation by cell counting is very useful in supporting decisions and planning of the medical treatment of breast cancer. This study proposes and evaluates features based on texture analysis by fractal dimension (FD), for the classification of histological structures in a BCMI into either cancer cells or non-cancer cells. The cancer cells include positive cells (PC) and negative cells (NC), while the normal cells comprise stromal cells (SC) and lymphocyte cells (LC). The FD feature values were calculated with the box-counting method from binarized images, obtained by automatic thresholding with Otsu's method of the grayscale images for various color channels. A total of 12 color channels from four color spaces (RGB, CIE-L*a*b*, HSV, and YCbCr) were investigated, and the FD feature values from them were used with decision tree classifiers. The BCMI data consisted of 1,400, 1,200, and 800 images with pixel resolutions 128?×?128, 192?×?192, and 256?×?256, respectively. The best cross-validated classification accuracy was 93.87%, for distinguishing between cancer and non-cancer cells, obtained using the Cr color channel with window size 256. The results indicate that the proposed algorithm, based on fractal dimension features extracted from a color channel, performs well in the automatic classification of the histology in a BCMI. This might support accurate automatic cell counting in a computer-assisted system for breast cancer diagnosis. SCANNING 37:145-151, 2015. © 2015 Wiley Periodicals, Inc. PMID:25689353

Jitaree, Sirinapa; Phinyomark, Angkoon; Boonyaphiphat, Pleumjit; Phukpattaranont, Pornchai

2015-03-01

99

Small-angle scattering from fat fractals

NASA Astrophysics Data System (ADS)

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.

Anitas, Eugen M.

2014-06-01

100

Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones. PMID:25603473

Puškaš, Nela; Zaletel, Ivan; Stefanovi?, Bratislav D; Ristanovi?, Dušan

2015-03-01

101

NASA Astrophysics Data System (ADS)

Structural texture measures are used to address the aspect of breast cancer risk assessment in screening mammograms. The current study investigates whether texture properties characterized by local Fractal Dimension (FD) and Lacunarity contribute to asses breast cancer risk. FD represents the complexity while the Lacunarity characterize the gappiness of a fractal. Our cross-sectional case-control study includes mammograms of 50 patients diagnosed with breast cancer in the subsequent 2-4 years and 50 matched controls. The longitudinal double blind placebo controlled HRT study includes 39 placebo and 36 HRT treated volunteers for two years. ROIs with same dimension (250*150 pixels) were created behind the nipple region on these radiographs. Box counting method was used to calculate the fractal dimension (FD) and the Lacunarity. Paired t-test and Pearson correlation coefficient were calculated. It was found that there were no differences between cancer and control group for FD (P=0.8) and Lacunarity (P=0.8) in crosssectional study whereas earlier published heterogeneity examination of radiographs (BC-HER) breast cancer risk score separated groups (p=0.002). In the longitudinal study, FD decreased significantly (P<0.05) in the HRT treated population while Lacunarity remained insignificant (P=0.2). FD is negatively correlated to Lacunarity (-0.74, P<0.001), BIRADS (-0.34, P<0.001) and Percentage Density (-0.41, P<0.001). FD is invariant to the mammographic texture change from control to cancer population but marginally varying in HRT treated population. This study yields no evidence that lacunarity or FD are suitable surrogate markers of mammographic heterogeneity as they neither pick up breast cancer risk, nor show good sensitivity to HRT.

Karemore, Gopal; Nielsen, Mads

2009-02-01

102

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

103

NASA Astrophysics Data System (ADS)

The volcanic unit of Montaña Reventada is an example of magma mixing in Tenerife (Canary Islands, Spain). The eruptive process has been detonated by a basanite intruding into a phonolite magma chamber. This eruption started with a basanite followed by a phonolite. Montaña Reventada phonolite is characterized by the presence of mafic enclaves. These enclaves represent about the 2% of the outcrop and have been classified like basanites, phono-tephrite and tephri-phonolite. The enclaves have different morphologies, from rounded to complex fingers-like structures, and usually exhibit cuspate terminations. This study aims to provide a new perspective on the 1100 AD Montaña Reventada eruption quantifying the textural heterogeneities related to the enclaves generated by the mixing process. The textural study was carried out using a fractal geometry approach, and its results were used to calculate some parameters related to magma chamber dynamics. Photographs of 67 samples were taken normal to the surface of the enclaves with the aim of delineating the contact between the enclaves and the host rocks. The resulted pictures were processed with the NIH (National Institutes of Health) image analysis software to generate binary images in which enclaves and host rock were replaced by black and white pixels, respectively. The fractal dimension (Dbox) has been computed by using the box-counting method in order to quantify the complexity of the enclaves morphology. Viscosity ratio (?R) between the phonolite and the enclaves has been calculated as follows: log(?R) = 0.013e3.34Dbox PIC The viscosity of the enclaves has been calculated according to the ?Rvalue with the higher frequency and to the calculated viscosity of the phonolite between 900° and 1200° . We hypothesized that this value corresponds to the amount of mafic magma present in the system, while the other values represent different degrees of mingling and chemical diffusion. Viscosity of the basanite can be computed like: ?enclave = (%phonolite *?phonolite)+ (%basanite *?basanite) PIC ?enclaves--(%phonolite *?phonolite) ?basanite = %basanite PIC The minimum percentages which satisfy the relation are 69.5% of basanite and 30.5% of phonolite. Although the amount of mafic magma reaches the 69.5%, the presence of enclaves in the phonolite is just the ?1% and the amount of basanite erupted before could correspond to the 15% of the phonolite (estimated from stratigraphic sections). Probably a magma body of basanite was still stored in the magma chamber. The volume of basanite still stored during this time may have evolved to a more explosive magma and hence increases the volcanic risk in the area.

Albert, Helena; Perugini, Diego; Martí, Joan

2014-05-01

104

Structural investigations of fat fractals using small-angle scattering

NASA Astrophysics Data System (ADS)

Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.

Anitas, Eugen M.

2015-01-01

105

NASA Astrophysics Data System (ADS)

As part of an ongoing undergraduate research project of light scattering calculations involving fractal carbonaceous soot aggregates relevant to current anthropogenic and natural sources in Earth's atmosphere, we have read with interest a recent paper [E.T. Wolf and O.B Toon,Science 328, 1266 (2010)] claiming that the Faint Young Sun paradox discussed four decades ago by Carl Sagan and others can be resolved without invoking heavy CO2 concentrations as a greenhouse gas warming the early Earth enough to sustain liquid water and hence allow the origin of life. Wolf and Toon report that a Titan-like Archean Earth haze, with a fractal haze aggregate nature due to nitrogen-methane photochemistry at high altitudes, should block enough UV light to protect the warming greenhouse gas NH3 while allowing enough visible light to reach the surface of the Earth. To test this hypothesis, we have employed a rigorous T-Matrix arbitrary-particle light scattering technique, to avoid the simplifications inherent in Mie-sphere scattering, on haze fractal aggregates at UV and visible wavelenths of incident light. We generate these model aggregates using diffusion-limited cluster aggregation (DLCA) algorithms, which much more closely fit actual haze fractal aggregates than do diffusion-limited aggregation (DLA) algorithms.

Boness, D. A.; Terrell-Martinez, B.

2010-12-01

106

The diffusion-limited binding kinetics of analyte in solution to receptor immobilized on a biosensor surface is analysed within a fractal framework. Both a single- as well as a dual-fractal analysis are utilized. Antigen-antibody and analyte-receptor systems are analysed. For the antigen-antibody and analyte-receptor systems where a single- or a dual-fractal analysis was used, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction. The binding rate coefficient expressions obtained as a function of the fractal dimension indicate the high sensitivity of the binding rate coefficient with respect to the fractal dimension. For example, for a single-fractal analysis and for the binding of (a) 1 microM BSA in solution to the anti-BSA-protein fused to a biosensor surface, and for (b) the binding of m-xylene-saturated STE buffer solution to the microorganism immobilized to the fiber-optic end and covered with a polycarbonate membrane, the orders of dependence of the binding rate coefficient on the fractal dimension were 5.535 and 3.314, respectively. This emphasizes the importance of the degree of heterogeneity on the biosensor surface and its impact on the binding rate coefficient, k. This high sensitivity is also indicated for a dual-fractal analysis, at least for the binding rate coefficient, k2. For example, during regeneration runs and for the binding of polymerase chain-reaction amplified DNA in solution to DNA capture protein immobilized on a fiber-optic biosensor, the order of dependence of k2 on Df2 was 3.399. The fractional order of dependence of the binding rate coefficient(s) on the fractal dimension(s) further reinforces the fractal nature of the system. The binding rate coefficient expressions developed as a function of the fractal dimension for both single-fractal analysis and dual-fractal analysis systems are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface. Also, the importance of the nature of the surface on biosensor performance is emphasized in a quantitative sense. PMID:9842708

Sadana, A

1998-11-01

107

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

108

Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.

Mossotti, Victor G.; Eldeeb, A. Raouf

2000-01-01

109

Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension.

Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered. PMID:25586375

Backes, André Ricardo; Gerhardinger, Leandro Cavaleri; Batista Neto, João do Espírito Santo; Bruno, Odemir Martinez

2015-02-01

110

Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension

NASA Astrophysics Data System (ADS)

Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered.

Backes, André Ricardo; Cavaleri Gerhardinger, Leandro; do Espírito Santo Batista Neto, João; Martinez Bruno, Odemir

2015-02-01

111

The aim of this study was to determine the pattern of bone remodeling after maxillary sinus lifting in humans by means of fractal dimension (FD) and histomorphometric analysis. Therefore, the correlation between FD and the histomorphometric findings was evaluated. Sixteen patients with posterior edentulous maxilla were enrolled in this study. Maxillary sinus lifting was performed using autogenous bone grafted from the mandibular retromolar area. Three direct digital panoramic radiographs were obtained: before surgery (Group 1), immediately postoperatively (Group 2) and after 6 months of healing (Group 3) for FD analysis. Biopsies were taken after 6 months, processed and submitted to histological and histomorphometric analysis. Data were analyzed by Shapiro-Wilk test and ANOVA test followed by a Tukey test (a = 0.05). The bone volume fraction of newly trabecular bone (TB) and medullary area (MA) was measured as 62.75% ± 17.16% and 37.25 ± 17.16%, respectively. Significant difference in FD analysis was measured between Group 1 and Group 3. No significant difference was found in the correlation between FD and histomorphometric analysis for TB and MA (p = 0.84). In conclusion, all performed analyses were effective in assessing the bone-remodeling pattern in the maxillary sinus, offering complementary information about healing and predictable outcomes. There were no correlations between FD and histomorphometric analysis. PMID:25672378

de Molon, Rafael Scaf; de Paula, Wagner Nunes; Spin-Neto, Rubens; Verzola, Mario Henrique Arruda; Tosoni, Guilherme Monteiro; Lia, Raphael Carlos Comelli; Scaf, Gulnara; Marcantonio, Elcio

2015-01-01

112

Assessing severity of obstructive sleep apnea by fractal dimension sequence analysis of sleep EEG

NASA Astrophysics Data System (ADS)

Different sleep stages are associated with distinct dynamical patterns in EEG signals. In this article, we explored the relationship between the sleep architecture and fractal dimension (FD) of sleep EEG. In particular, we applied the FD analysis to the sleep EEG of patients with obstructive sleep apnea-hypopnea syndrome (OSAHS), which is characterized by recurrent oxyhemoglobin desaturation and arousals from sleep, a disease which received increasing public attention due to its significant potential impact on health. We showed that the variation of FD reflects the macrostructure of sleep. Furthermore, the fast fluctuation of FD, as measured by the zero-crossing rate of detrended FD (zDFD), is a useful indicator of sleep disturbance, and therefore, correlates with apnea-hypopnea index (AHI), and hourly number of blood oxygen saturation (SpO 2) decreases greater than 4%, as obstructive apnea/hypopnea disturbs sleep architecture. For practical purpose, a modified index combining zDFD of EEG and body mass index (BMI) may be useful for evaluating the severity of OSAHS symptoms.

Zhang, J.; Yang, X. C.; Luo, L.; Shao, J.; Zhang, C.; Ma, J.; Wang, G. F.; Liu, Y.; Peng, C.-K.; Fang, J.

2009-10-01

113

Background Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony. Methods Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization. Results A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total. Conclusions The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process. PMID:20846449

2010-01-01

114

Comparison of different fractal dimension measuring algorithms for RE-TM M-O films

NASA Technical Reports Server (NTRS)

Noise in magneto-optical recording devices is discussed. In general, it appears that either the divider technique or amplitude spectrum technique may be used interchangeably to measure the fractal dimension (D) in the domain wall structure of ideal images. However, some caveats must be observed for best results. The divider technique is attractive for its simplicity and relatively modest computation requirements. However, it is sensitive to noise, in that noise pixels that touch the domain boundary are interpreted as being part of the boundary, skewing the measurement. Also, it is not useful in measuring nucleation-dominated films or domains that have significant amounts of structure within the interior of the domain wall. The amplitude spectrum method is more complex, and less intuitive than the divider method, and somewhat more expensive to implement computationally. However, since the camera noise tends to be white, the noise can be avoided in the measurement of D by avoiding that portion of the curve that is flat (due to the white noise) when the least squares line is fit to the plot. Also, many image processing software packages include a Fast Fourier Transformation (FFT) facility, while the user will most likely have to write his own edge extraction routine for the divider method. The amplitude spectrum method is a true two dimensional technique that probes the interior of the domain wall, and in fact, can measure arbitrary clusters of domains. It can also be used to measure grey-level images, further reducing processing steps needed to threshold the image.

Bernacki, Bruce E.; Mansuripur, M.

1991-01-01

115

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

116

Vol. 7,No. 6/June 1990/J. Opt. Soc. Am. A 1055 Estimating fractal dimension

electrochemical deposi- tion,1 ,2 viscous fingering,3 and dielectric breakdown4 as well as porous rocks5 diffusion-limited aggregates,8 and iterated func- tion systems9 provide mathematical models of fractals

Theiler, James

117

The goal of this study was to determine material properties for the anterior cortex and subcortical regions of human patellae and relate those properties to mineral density and fractal dimension of the bone. Ten human patellae were obtained from eight fresh frozen human cadavers and subjected to anteriorly-directed spherical indentation-relaxation experiments using two different sized indenters to two different indentation depths. Response data were fit to a three-mode viscoelastic model obtained through elastic-viscoelastic correspondence of the Hertzian contact relation for spherical indentation. A location-specific effective bone density measurement that more heavily weighted bone material close to the indentation site (by von Mises stress distribution) was determined from micro-computed tomography (38µm resolution) data captured for each specimen. The same imagery data were used to compute location specific fractal dimension estimates for each indentation site. Individual and averaged patella material models verified the hypothesis that when the larger indenter and greater indentation depth is used to engage the surface and deeper (trabecular) bone, the bone exhibits a more compliant response than when only the surface (cortical) bone was engaged (instantaneous elastic modulus was 325MPa vs. 207MPa, p<0.05). Effective bone mineral density was shown to be a significant predictor of the elastic modulus for both small and large indentation types (p<0.05) despite relatively low correlations. Exponential regressions of fractal dimension on elastic modulus showed significant relationships with high correlation for both the small (R(2)=0.93) and large (R(2)=0.97) indentations. PMID:23972564

Kerrigan, Jason R; Sanchez-Molina, David; Neggers, Jan; Arregui-Dalmases, Carlos; Velazquez-Ameijide, Juan; Crandall, Jeff R

2014-05-01

118

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

119

Fractal analysis of motor imagery recognition in the BCI research

NASA Astrophysics Data System (ADS)

A fractal approach is employed for the brain motor imagery recognition and applied to brain computer interface (BCI). The fractal dimension is used as feature extraction and SVM (Support Vector Machine) as feature classifier for on-line BCI applications. The modified Inverse Random Midpoint Displacement (mIRMD) is adopted to calculate the fractal dimensions of EEG signals. The fractal dimensions can effectively reflect the complexity of EEG signals, and are related to the motor imagery tasks. Further, the SVM is employed as the classifier to combine with fractal dimension for motor-imagery recognition and use mutual information to show the difference between two classes. The results are compared with those in the BCI 2003 competition and it shows that our method has better classification accuracy and mutual information (MI).

Chang, Chia-Tzu; Huang, Han-Pang; Huang, Tzu-Hao

2011-12-01

120

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (Df) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the Df developed for the antigen-antibody binding systems, indicate the high sensitivity of the k on the Df when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5 beta in solution to gp120 peptide immobilized on a BIAcore biosensor, the order of dependence on the Df was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k1 and k2, on the fractal dimensions, Df1 and Df2, were 7.6335 and -11.55, respectively. The fractional order of dependence of the k(s) on the Df(s) further reinforces the fractal nature of the system. The k(s) expressions developed as a function of the Df(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance. PMID:9779572

Sadana, A

1998-01-01

121

NSDL National Science Digital Library

Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.

2007-12-12

122

Fractal-Based Intrinsic Dimension Estimation and Its Application in Dimensionality Reduction

Dimensionality reduction is an important step in knowledge discovery in databases. Intrinsic dimension indicates the number of variables necessary to describe a data set. Two methods, box-counting dimension and correlation dimension, are commonly used for intrinsic dimension estimation. However, the robustness of these two methods has not been rigorously studied. This paper demonstrates that correlation dimension is more robust with

Dengyao Mo; Samuel H. Huang

2012-01-01

123

NASA Astrophysics Data System (ADS)

This article extends a signal-based approach formerly proposed by the authors, which utilizes the fractal dimension of time frequency feature (FDTFF) of displacements, for earthquake damage detection of moment resist frame (MRF), and validates the approach with shaking table tests. The time frequency feature (TFF) of the relative displacement at measured story is defined as the real part of the coefficients of the analytical wavelet transform. The fractal dimension (FD) is to quantify the TFF within the fundamental frequency band using box counting method. It is verified that the FDTFFs at all stories of the linear MRF are identical with the help of static condensation method and modal superposition principle, while the FDTFFs at the stories with localized nonlinearities due to damage will be different from those at the stories without nonlinearities using the reverse-path methodology. By comparing the FDTFFs of displacements at measured stories in a structure, the damage-induced nonlinearity of the structure under strong ground motion can be detected and localized. Finally shaking table experiments on a 1:8 scale sixteen-story three-bay steel MRF with added frictional dampers, which generate local nonlinearities, are conducted to validate the approach.

Tao, Dongwang; Mao, Chenxi; Zhang, Dongyu; Li, Hui

2014-12-01

124

Efficient market theory is always the cornerstone to establish and study modern financial theory. But its explanatory power for financial markets is weakening with the appearing of financial crisis. Rapid development of nonlinear science such as wavelet analysis and fractal theory provides new theoretical tools for groping financial market. Taking intraday closing price of soybean futures on Dalian Commodity Exchange

Yu Zhao

2009-01-01

125

NASA Astrophysics Data System (ADS)

The Basin and Range fault blocks, which were formed by an extensional event around 17 Ma, have continuously been deforming by younger, diachronous system of cross normal faults in southwest Montana and southeastern Idaho since 16.6 Ma. Reactivation of these two mid-Tertiary-Quaternary systems of normal faults, and two older, approximately N-S and E-W sets of regional normal faults, has evolved into a seismically active block faulted terrain. For both fault systems, high fractal dimensions occur in areas characterized by a large number of fault traces, high fault trace linear density, and maximum fault trace azimuthal variation. The major axis of the anisotropy ellipse of the fractal dimensions for each set of the two normal fault systems is sub-perpendicular to the linear directional mean of the faults, and gives an estimate for the direction of extension. Indentations on the point distribution on the anisotropy ellipse of fractal dimensions indicate heterogeneities due to the presence of several fault sets and/or variation in their trend. Domains in which there is only one set of faults produce smooth, well-defined fractal anisotropy ellipses with no indentations. The axial ratio of the anisotropy ellipse provides a measure for the range of variation in the trend of the faults. The trace length, linear density, and fractal dimension of the cross normal faults, decrease, in a direction across and away from the Snake River Plain (SRP), suggesting a diminishing effect of faulting probably due to the attenuation of the Yellowstone hotspot-related thermal doming with distance from centers of eruption. The spatio-temporal distribution of the trajectories of the minor axes of the anisotropy ellipses of fractal dimensions and the linear directional mean of the cross faults define a set of asymmetric, sub-parabolic spatio-temporal pattern about the axis of the SRP, with their apices located on diachronous centers of eruption.

Davarpanah, Armita; Babaie, Hassan A.

2013-11-01

126

NSDL National Science Digital Library

This lesson is designed to develop students' understanding of fractals and fractal dimension. This lesson provides links to discussions and activities related to fractals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

2011-05-23

127

There are persistent difficulties in monitoring nonpoint source pollution and in the related field of hydrology. The problems stem from variations in spatial distribution which are poorly understood and difficult to model with established methods. Two recent developments may offer a solution, if they are combined with care. The first development is the increasing capability of computer mapping, called geographic information systems (GIS). These systems can store, retrieve, and manipulate data with an explicit spatial structure. The second development is the field of fractal mathematics. Fractal mathematics includes geometric sets which have simple descriptions, despite complex appearances. One family of such fractal sets are the Brownian surfaces, which capture many of the qualities of natural land surfaces in a simple statistical model. Up until now, the Brownian models have been constrained by the assumption that the same statistical relationship holds over the entire surface. This is called the constraint of stationarity. The need to study how the landscape differs by location leads to relaxing the constraint of stationarity. This, in turn, causes some profound changes in the model. A special computer program applies the new model to a set of three-dimensional digital maps of natural terrain (DEMs). The model performs well, and highlights differences in landforms. This suggests several new approaches to spatial variation.

Wagenseil, R.

1991-01-01

128

NASA Astrophysics Data System (ADS)

The properties of turbulence in galaxies are a fundamental part of our understanding of the interstellar medium (ISM) as a complex and dynamic system. Turbulence changes the proportions of gas in the warm and cold phases and affects the regulation of star formation. Supersonic turbulence is known to have many sources, ranging from supernovae and stellar winds to the interaction between rotational shear and galactic magnetic fields. However, the details of these mechanisms and their interactions are not well understood. Several studies have linked turbulence with self-similarity, a property of the mathematical objects known as fractals. This study compares the fractal dimension of contours of three components of the ISM in nearby galaxies. These components sample the atomic, molecular and gas phases by way of the 21cm line of atomic Hydrogen (HI), the CO J=2?J1 transition and mid-IR 70?m dust emission. The THINGS (HI), SINGS (IR) and HERACLES (CO) surveys share a common galaxy sample from which five galaxies are selected for analysis. We present the results of this study.

Bowman, Lorraine; Ott, J.; Westpfahl, D.

2014-01-01

129

Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel Received 25 July 2000 Diffusion limited of coordinates in d-dimensions, and follows the creation of a cluster by releasing random walkers from infinity for the spontaneous creation of fractal objects in na- ture, and also as a paradigm for a family of related ``harmonic

Levermann, Anders

130

A generalized volume dimension of complex networks

NASA Astrophysics Data System (ADS)

The fractal and self-similarity properties are investigated in many real complex networks. The volume dimension method is an effective tool to measure the fractal property of complex networks. In this paper, a new volume dimension measure is proposed based on the node degree of complex networks. We apply the proposed method to calculate the fractal dimension of some real networks and Newman–Watts (NW) small-world. The results show that the proposed method is effective when dealing with the fractal dimension problem of complex networks. In addition, we find that the fractal dimension is mainly influenced by the probability of ‘adding edges’ and the average length of the small-world network.

Wei, Daijun; Wei, Bo; Zhang, Haixin; Gao, Cai; Deng, Yong

2014-10-01

131

Fractal dimension of sparkles in automotive metallic coatings by multispectral imaging measurements.

Sparkle in surface coatings is a property of mirror-like pigment particles that consists of remarkable bright spots over a darker surround under unidirectional illumination. We developed a novel nondestructive method to characterize sparkles based on the multispectral imaging technique, and we focused on automotive metallic coatings containing aluminum flake pigments. Multispectral imaging was done in the visible spectrum at different illumination angles around the test sample. Reflectance spectra at different spatial positions were mapped to color coordinates and visualized in different color spaces. Spectral analysis shows that sparkles exhibit higher reflectance spectra and narrower bandwidths. Colorimetric analysis indicates that sparkles present higher lightness values and are far apart from the bulk of color coordinates spanned by the surround. A box-counting procedure was applied to examine the fractal organization of color coordinates in the CIE 1976 L*a*b* color space. A characteristic noninteger exponent was found at each illumination position. The exponent was independent of the illuminant spectra. Together, these results demonstrate that sparkles are extreme deviations relative to the surround and that their spectral properties can be described as fractal patterns within the color space. Multispectral reflectance imaging provides a powerful, noninvasive method for spectral identification and classification of sparkles from metal flake pigments on the micron scale. PMID:24945784

Medina, José M; Díaz, José A; Vignolo, Carlos

2014-07-23

132

Cake porosity analysis using 1D-3D fractal dimensions in coagulation-microfiltration of NOM.

Fouling during coagulation-ceramic microfiltration of natural organic matter was investigated. Two process configurations (inline coagulation (IC) and tank coagulation (TC)) and two process conditions (types of coagulants-aluminum-based PAX and iron-based PIX-and G-values) were studied. The rate of irreversible fouling corresponding to the increase of initial transmembrane pressure after backwash of IC-PAX was lowest followed by TC-PAX and TC-PIX, while the performance of IC-PIX was found worst. The 1D and 2D fractal analysis revealed that flocs from IC were morphologically different from those of TC, leading to different filtration characteristics. The 3D fractal analysis revealed two groups of morphologically similar flocs: one led to successful filtration experiments, whereas the other led to unsuccessful ones. Cake porosity was found dependent on the floc morphology. Thus, such an approach was found complementary with fouling analysis by means of a membrane fouling model and minimization of fouling phenomenon was achieved by combining the two approaches. PMID:25768221

Raspati, G S; Leiknes, T O

2015-03-01

133

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

134

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

135

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

136

Theory of Laplacian fractals: Diffusion limited aggregation and dielectric breakdown model

We describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the Laplace equation and a stochastic field. This new theory provides a systematic method for the calculation of the fractal dimension D and of the multifractal spectrum of the growth probability (f;(alpha)). A detailed application to the dielectric breakdown model and

L. Pietronero; A. Erzan; C. Evertsz

1988-01-01

137

NASA Astrophysics Data System (ADS)

Fractal geometry can describe the complex texture of many natural objects, the description being summarized by a key parameter: fractal dimension. In previous work, the fractal dimension of several ceramic fracture surfaces was empirically related to fracture toughness by rm K_{Ic} ~ (D-1)^ {1/2} where D is the dimension of a contour line of the fracture surface. This equation states that an increase in fractal dimension is accompanied by an increase in fracture toughness. A fractal modification of Griffith's energy balance approach is derived and results in a theoretical relationship between toughness and fractal dimension of the fracture surface:rm K_ {Ic} = sqrt{2Egamma_{s }} sqrt{(D-1)} sqrt {{bf c}^{D-2}} sqrt{a_sp{0}{2-D }}where gamma_ {rm s} is the thermodynamic surface energy, c is the flaw size, a_0 is a characteristic length parameter and D is the dimension of the fracture surface. This equation differs from the empirical K_{rm Ic} -D relationship, but does verify a toughness-fractal dimension relationship. This thesis investigates the use of fractal dimension as a descriptive parameter in brittle fracture. Several key issues are addressed including the relationship between contour and surface dimensions, a derivation of the fractal Griffith equation, an experimental investigation of dimension as a function of flaw-to-grain size ratio, and the derivation of Richardson's equation. Contour dimensions are shown to be one less than surface dimensions while no such relation holds for profile dimensions. Furthermore, the contour dimensions will remain the same at various altitudes through the fracture surface. A change in angle of the cutting plane does alter the measured dimension, but does not do so until a critical angle is reached. The critical angle will depend upon the dimension of the surface; lower dimensional surfaces having smaller critical angles than high dimensional surfaces. The fractal dimension of fracture surfaces of ZnS are found to follow the same flaw-to-grain-size behavior as the corresponding fracture toughness. Finally, the fractal Griffith equation explains the single-to-polycrystalline fracture transitions that are noted in the experimental results.

Mackin, Thomas James

1990-01-01

138

Introduction Solutions that cause elective cardiac arrest are constantly evolving, but the ideal compound has not yet been found. The authors compare a new cardioplegic solution with histidine-tryptophan-glutamate (Group 2) and other one with histidine-tryptophan-cetoglutarate (Group 1) in a model of isolated rat heart. Objective To quantify the fractal dimension and Shannon entropy in rat myocytes subjected to cardioplegia solution using histidine-tryptophan with glutamate in an experimental model, considering the caspase markers, IL-8 and KI-67. Methods Twenty male Wistar rats were anesthetized and heparinized. The chest was opened, the heart was withdrawn and 40 ml/kg of cardioplegia (with histidine-tryptophan-cetoglutarate or histidine-tryptophan-glutamate solution) was infused. The hearts were kept for 2 hours at 4ºC in the same solution, and thereafter placed in the Langendorff apparatus for 30 min with Ringer-Locke solution. Analyzes were performed for immunohistochemical caspase, IL-8 and KI-67. Results The fractal dimension and Shannon entropy were not different between groups histidine-tryptophan-glutamate and histidine-tryptophan-acetoglutarate. Conclusion The amount of information measured by Shannon entropy and the distribution thereof (given by fractal dimension) of the slices treated with histidine-tryptophan-cetoglutarate and histidine-tryptophan-glutamate were not different, showing that the histidine-tryptophan-glutamate solution is as good as histidine-tryptophan-acetoglutarate to preserve myocytes in isolated rat heart. PMID:25140464

de Oliveira, Marcos Aurélio Barboza; Brandi, Antônio Carlos; dos Santos, Carlos Alberto; Botelho, Paulo Henrique Husseni; Cortez, José Luís Lasso; de Godoy, Moacir Fernandes; Braile, Domingo Marcolino

2014-01-01

139

Purpose Over the past decade, linear and non-linear surface electromyography descriptors for central and peripheral components of fatigue have been developed. In the current study, we tested fractal dimension (FD) and conduction velocity (CV) as myoelectric descriptors of central and peripheral fatigue, respectively. To this aim, we analyzed FD and CV slopes during sustained fatiguing contractions of the quadriceps femoris in healthy humans. Methods A total of 29 recreationally active women (mean age±standard deviation: 24±4 years) and two female elite athletes (one power athlete, age 24 and one endurance athlete, age 30 years) performed two knee extensions: (1) at 20% maximal voluntary contraction (MVC) for 30 s, and (2) at 60% MVC held until exhaustion. Surface EMG signals were detected from the vastus lateralis and vastus medialis using bidimensional arrays. Results Central and peripheral fatigue were described as decreases in FD and CV, respectively. A positive correlation between FD and CV (R=0.51, p<0.01) was found during the sustained 60% MVC, probably as a result of simultaneous motor unit synchronization and a decrease in muscle fiber CV during the fatiguing task. Conclusions Central and peripheral fatigue can be described as changes in FD and CV, at least in young, healthy women. The significant correlation between FD and CV observed at 60% MVC suggests that a mutual interaction between central and peripheral fatigue can arise during submaximal isometric contractions. PMID:25880369

Beretta-Piccoli, Matteo; D’Antona, Giuseppe; Barbero, Marco; Fisher, Beth; Dieli-Conwright, Christina M.; Clijsen, Ron; Cescon, Corrado

2015-01-01

140

The increasing number of applications of fractal theory in the environmental sciences reflects the recognized Importance of spatial and temporal scale to the study of ecological systems and processes. In this paper, we summarize the various algorithms that have been developed for estimating the fractal dimenSIon of such natural phenomena as landscapes, soils, plant root systems, paths of foraging animals,

N. C. Kenkel; D. J. Walker

1993-01-01

141

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

142

Discrimination of walking patterns using wavelet-based fractal analysis.

In this paper, we attempted to classify the acceleration signals for walking along a corridor and on stairs by using the wavelet-based fractal analysis method. In addition, the wavelet-based fractal analysis method was used to evaluate the gait of elderly subjects and patients with Parkinson's disease. The triaxial acceleration signals were measured close to the center of gravity of the body while the subject walked along a corridor and up and down stairs continuously. Signal measurements were recorded from 10 healthy young subjects and 11 elderly subjects. For comparison, two patients with Parkinson's disease participated in the level walking. The acceleration signal in each direction was decomposed to seven detailed signals at different wavelet scales by using the discrete wavelet transform. The variances of detailed signals at scales 7 to 1 were calculated. The fractal dimension of the acceleration signal was then estimated from the slope of the variance progression. The fractal dimensions were significantly different among the three types of walking for individual subjects (p < 0.01) and showed a high reproducibility. Our results suggest that the fractal dimensions are effective for classifying the walking types. Moreover, the fractal dimensions were significantly higher for the elderly subjects than for the young subjects (p < 0.01). For the patients with Parkinson's disease, the fractal dimensions tended to be higher than those of healthy subjects. These results suggest that the acceleration signals change into a more complex pattern with aging and with Parkinson's disease, and the fractal dimension can be used to evaluate the gait of elderly subjects and patients with Parkinson's disease. PMID:12503784

Sekine, Masaki; Tamura, Toshiyo; Akay, Metin; Fujimoto, Toshiro; Togawa, Tatsuo; Fukui, Yasuhiro

2002-09-01

143

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

144

Fractal dynamics in chaotic quantum transport

NASA Astrophysics Data System (ADS)

Despite several experiments on chaotic quantum transport, corresponding ab initio quantum simulations have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. Applying a large set of magnetic fields yields a complete picture of the magnetoconductance that indicates fractal scaling on intermediate time scales. Two methods that originate from different fields of physics are used to analyze the scaling exponent and the fractal dimension. They lead to consistent results that, in turn, qualitatively agree with the previous experimental data.

Rasanen, Esa; Kotimaki, Ville; Hennig, Holger; Heller, Eric

2013-03-01

145

Fractal dynamics in chaotic quantum transport

NASA Astrophysics Data System (ADS)

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis—a widely used method in time-series analysis—and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.

Kotimäki, V.; Räsänen, E.; Hennig, H.; Heller, E. J.

2013-08-01

146

Fractal dynamics in chaotic quantum transport.

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data. PMID:24032907

Kotimäki, V; Räsänen, E; Hennig, H; Heller, E J

2013-08-01

147

Fractal Dynamics in Chaotic Quantum Transport

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis -- a widely used method in time series analysis -- and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that, in turn, qualitatively agree with the previous experimental data.

Ville Kotimaki; Esa Rasanen; Holger Hennig; Eric J. Heller

2013-07-28

148

Cluster-Cluster Aggregation Calculations of Fractal Haze Particles: Titan and the Early Earth

NASA Astrophysics Data System (ADS)

The atmosphere of the Archean Earth (3.8 to 2.5 billion years ago) is thought to have been dominated by a thick hydrocarbon haze similar to that of Titan's current atmosphere. To understand radiative transport in the atmospheres of the early Earth and of Titan, it is necessary to compute light scattering in UV, visible, and IR wavelength ranges for realistic fractal aggregate hydrocarbon aerosol particles. We report preliminary work on MATLAB, True BASIC, and Fortran programs to simulate the growth of fractal aggregate aerosols through diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) physical processes. The results of these computations are being used with a T-Matrix light scattering program to test recently published, widely-reported conclusions about the early Earth and the faint young Sun paradox [E. T. Wolf and O. B. Toon, Science 328, 1266 (2010)]. This modeling is also relevant to understanding atmospheric carbonaceous soot aerosol anthropogenic and natural effects on climate change of Earth today.

Terrell-Martinez, Bernice; Boness, David

2010-10-01

149

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

150

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

151

Fractal analysis of time varying data

Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.

Vo-Dinh, Tuan (Knoxville, TN); Sadana, Ajit (Oxford, MS)

2002-01-01

152

Fractal characterization of fracture surfaces in concrete

Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.

Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.

1990-01-01

153

Rheological and fractal hydrodynamics of aerobic granules.

The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U?l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area. PMID:25836036

Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini

2015-06-01

154

The nature of fractals and the use of fractals instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described. The mathematical development of fractals is reviewed, and examples of natural fractals are cited. Relationships among power laws, noise, and fractal time signals are examined

William Deering; Bruce J. West

1992-01-01

155

Self-avoiding walk on fractal complex networks: Exactly solvable cases

NASA Astrophysics Data System (ADS)

We study the self-avoiding walk on complex fractal networks called the (u ,v ) -flower by mapping it to the N -vector model in a generating function formalism. First, we analytically calculate the critical exponent ? and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ? is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u ,u ) -flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension.

Hotta, Yoshihito

2014-11-01

156

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.

157

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

158

Porosity is an important factor to consider in a large variety of materials. Porosity can be visualized in bone or 3D synthetic biomaterials by microcomputed tomography (microCT). Blocks of porous poly(2-hydroxyethyl methacrylate) were prepared with polystyrene beads of different diameter (500, 850, 1160 and 1560 ?m) and analysed by microCT. On each 2D binarized microCT section, pixels of the pores which belong to the same image column received the same pseudo-colour according to a look up table. The same colour was applied on the same column of a frontal plane image which was constructed line by line from all images of the microCT stack. The fractal dimension Df of the frontal plane image was measured as well as the descriptors of the 3D models (porosity, 3D fractal dimension D3D , thickness, density and separation of material walls. Porosity, thickness Df and D3D increased with the size of the porogen beads. A linear correlation was observed between Df and D3D . This method provides quantitative and qualitative analysis of porosity on a single frontal plane image of a porous object. PMID:25556606

Chappard, Daniel; Stancu, Izabela-Cristina

2015-04-01

159

MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.

Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert

1998-01-01

160

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

161

Roughness Perception of Haptically Displayed Fractal Surfaces

NASA Technical Reports Server (NTRS)

Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.

Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)

2000-01-01

162

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

163

Fractal analysis of narwhal space use patterns.

Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice. PMID:16351924

Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R

2004-01-01

164

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

165

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

2012-03-02

166

FractalNet: A Neural Network Approach to Fractal Geometry

This paper presents a multiply connected neural network designed to estimate the fractal dimension (Df) using the Box-counting method (BCM). Fractal analysis is a powerful shape recognition tool and has been applied to many pattern recognition problems. Additionally, the Box-Counting Method is one of the most popular methods for estimating Df. However, traditional methods used to estimate Df are sequential

Ronald Marsh

167

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

168

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

169

Target Detection Using Fractal Geometry

NASA Technical Reports Server (NTRS)

The concepts and theory of fractal geometry were applied to the problem of segmenting a 256 x 256 pixel image so that manmade objects could be extracted from natural backgrounds. The two most important measurements necessary to extract these manmade objects were fractal dimension and lacunarity. Provision was made to pass the manmade portion to a lookup table for subsequent identification. A computer program was written to construct cloud backgrounds of fractal dimensions which were allowed to vary between 2.2 and 2.8. Images of three model space targets were combined with these backgrounds to provide a data set for testing the validity of the approach. Once the data set was constructed, computer programs were written to extract estimates of the fractal dimension and lacunarity on 4 x 4 pixel subsets of the image. It was shown that for clouds of fractal dimension 2.7 or less, appropriate thresholding on fractal dimension and lacunarity yielded a 64 x 64 edge-detected image with all or most of the cloud background removed. These images were enhanced by an erosion and dilation to provide the final image passed to the lookup table. While the ultimate goal was to pass the final image to a neural network for identification, this work shows the applicability of fractal geometry to the problems of image segmentation, edge detection and separating a target of interest from a natural background.

Fuller, J. Joseph

1991-01-01

170

The degree of irregularity in oceanic coastlines and in vertical sections of the Earth, the distribution of the numbers of islands according to area, and the commonality of global shape between continents and islands, all suggest that the Earth's surface is statistically self-similar. The preferred parameter, one which increases with the degree of irregularity, is the fractal dimension, D, of the coastline; it is a fraction between 1 (limit of a smooth curve) and 2 (limit of a plane-filling curve). A rough Poisson-Brown stochastic model gives a good first approximation account of the relief, by assuming it to be created by superposing very many, very small cliffs, placed along straight faults and statistically independent. However, the relative area predicted for the largest islands is too small, and the irregularity predicted for the relief is excessive for most applications; so is indeed the value of the dimension, which is D = 1.5. Several higher approximation self-similar models are described. Any can be matched to the empirically observed D, and can link all the observations together, but the required self-similarity cannot yet be fully explained. Images PMID:16578734

Mandelbrot, Benoit B.

1975-01-01

171

Application of the fractal theory on the study of filter cake constructure

Cake filtration is a complex process and the cake constructure is very difficult to describe in theory. Cake constructure parameters, such as the cake porosity, pore size shape and even its distribution, are main factors influencing the filtration results but have not been thoroughly understood yet. In this paper the fractal theory, an effective mathematical method in describing the self-similar phenomenon is used to investigate the filter cake constructure, and the scanning electron microscope and automatic image analyzer are used to measure the cake constructure. Cakes which formed in different conditions are examined and the fractal dimension of the cake are calculated. The study shows that the constructure of the filter cake can be approximated by Sierpinski fractal geometry and that the fractal dimension of filter cake, related to the particle characteristics, slurry concentration and filtration pressure is a good parameter to describe the pore size distribution and the cake penetrability.

Xu, X.; Xu, J.; Deng, C.; Qian, L. [Northeastern Univ., Shenyang (China); Yan, K.

1995-12-31

172

NSDL National Science Digital Library

Using this tool, students build these classic fractals: the Koch snowflake, a fractal tree, a reduced square, and the Sierpinksi triangle. As these shapes grow and change using an iterative process, students can observe patterns in the images created and in the table of values as the fractals progress through several stages.

National Council of Teachers of Mathematics

2009-01-01

173

NASA Astrophysics Data System (ADS)

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E.; Bertoldo, Alessandra

2015-02-01

174

Two-Dimensional Fractal Characteristics of the Martian Surface

NASA Astrophysics Data System (ADS)

We present global maps of two-dimensional fractal statistics for Mars topography calculated by applying the two-dimensional Fourier spectral approach to MOLA altimetry measurements over spatial scales extending from approximately 450 meters to 15 kilometers. Three global maps were generated: 1) surface (two-dimensional) fractal dimension, 2) roughness amplitude at a scale of one kilometer, and 3) linear model fit error in the log-log relation of mean power spectral density to radial wavenumber. The linear model fit error is a convenient way to judge the appropriateness of the fractal model. Examination of the fractal dimension and model error maps reveals that a majority of the surface is well modeled by fractal geometry. This is evidenced by minimal systematic spatial variation in fractal dimension and low model fit errors, with the northern plains exhibiting slightly higher overall error than the cratered highlands. There are also several spatially coherent regions in the fractal dimension map that have enhanced values. These regions include Amazonis Planitia and southeast Elysium Planitia. On the other hand, Isidis Planitia and portions of the Olympus Mons aureole exhibit high model fit errors which imply a lower applicability of fractal geometry to these terrains. The one kilometer roughness amplitude map exhibits a tremendous amount of spatial detail and clearly delineates differing roughness terrains. The portions of Amazonis Planitia and southeast Elysium Planitia with enhanced fractal dimension have roughness amplitudes significantly below the global mean, while the Valles Marineris system, the circum-Argyre region, and the chaotic and heavily eroded terrains located along the crustal dichotomy boundary exhibit elevated roughness values. The Tharsis region is particularly rich in detail, displaying a wide range of spatially contiguous roughness provinces that are traceable to known surface units. Comparison of the roughness amplitude map to the MOLA pulse width-derived roughness data (75 meter baseline) reveals a strong correlation with a few notable exceptions. The circum-polar debris mantle located 30 to 45 degrees bilaterally from the equator and a small yet distinct terrain located northwest of Olympus Mons are both evident in the 75 m pulse width data but are not expressed in the longer wavelength roughness amplitude map. This implies that the surface processes responsible for producing these terrains are dominant only at shorter length scales.

Seelos, F. P.; Deal, K. S.; Arvidson, R. E.; Neumann, G. A.

2003-12-01

175

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

176

Fractal Musicand Fractal Music Lab

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.

2007-12-12

177

ERIC Educational Resources Information Center

After introducing the two-dimensional Koch curve, which is generated by simple recursions on an equilateral triangle, the process is extended to three dimensions with simple recursions on a regular tetrahedron. Included, for both fractal sequences, are iterative formulae, illustrations of the first several iterations, and a sample PASCAL program.…

Camp, Dane R.

1991-01-01

178

Self-organized stiffness in regular fractal polymer structures

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals:

Marco Werner; Jens-Uwe Sommer

2011-01-01

179

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

180

Fractal topology of hand-crumpled paper.

We study the statistical topology of folding configurations of hand folded paper balls. Specifically, we are studying the distribution of two sides of the sheet along the ball surface and the distribution of sheet fragments when the ball is cut in half. We found that patterns obtained by mapping of ball surface into unfolded flat sheet exhibit the fractal properties characterized by two fractal dimensions which are independent on the sheet size and the ball diameter. The mosaic patterns obtained by sheet reconstruction from fragments of two parts (painted in two different colors) of the ball cut in half also possess a fractal scale invariance characterized by the box fractal dimension DBF=1.68 ± 0.04 , which is independent on the sheet size. Furthermore, we noted that DBF, at least numerically, coincide with the universal fractal dimension of the intersection of hand folded paper ball with a plane. Some other fractal properties of folding configurations are recognized. PMID:20866397

Balankin, Alexander S; Ochoa, Didier Samayoa; Miguel, Israel Andrés; Ortiz, Julián Patiño; Cruz, Miguel Angel Martínez

2010-06-01

181

in the backbone, arises in connection with its equivalence to the thermal scaling power, (3)de, the fractal-Stauffer conjecture, (6)dw, the fractal dimension of a random walk, (7)do, the fractal dimension of growth sitesPublishingCorporation #12;844 Stanley powers, tl) If there are nine extrinsic fractal dimensions, then there are poten

Stanley, H. Eugene

182

Fractal generation of textures and backgrounds

BACKGROUND. 2. 1 Terminology . 2. 1. 1 Self-similarity. . 2. 1. 2 Fractal Dimension. 2. 1. 3 Fractional Brownian Motion. . 2. 1. 4 Self-affinity . . 2. 2 Random Fractal Techniques 2. 3 Fractal Textures 2. 4 Fractal Landscapes . . 2. 5 Fractal Botany.... 2. 6 Conclusion. III COMPLEX DYNAMICAL SYSTEMS . . 3. 1 Julia Sets. . 3. 2 Mandelbrot Set . 3. 3 Interface IV TEXTURES . 4. 1 General Textures. 4. 2 Planetary Surface Textures. 4. 3 Color. V SKYSCAPES . Page 1V V11 10 12 13 16 17 20...

Reuter, Kevin Duane

1999-01-01

183

NSDL National Science Digital Library

Paul Bourke of the Astrophysics and Supercomputing department at Swinburne University of Technology is the author of this massive resource on fractals and chaos. He gives examples of many different kinds and classes of fractals, including the Mandelbrot set and various attractors; and brief explanations accompany each one. A substantial introduction to fractals covers the underlying principles and connection to chaos theory. Many stunning, high resolution fractal image galleries show elaborate patterns and colors. Examples of C and PovRay code used to create the remarkable images are provided. Bourke's homepage has many other sections of tutorials, papers, and notes on a diverse range of subjects.

2003-01-01

184

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

185

Fractal dynamics of earthquakes

Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).

Bak, P.; Chen, K. [Brookhaven National Lab., Upton, NY (United States). Dept. of Physics

1995-05-01

186

Introduction to Fractals: Geometric Fractals

NSDL National Science Digital Library

This lesson is designed to continue developing students' knowledge of fractals by introducing them to some popular examples of fractals, Sierpinski's carpet and Sierpinski's triangle. This lesson provides links to discussions and activities related to fractals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

2011-05-24

187

NASA Astrophysics Data System (ADS)

We investigated time evolution of shear moduli in the physical gelation process of 1,3:2,4-bis-O-(p-methylbenzylidene)-D-sorbitol in polystyrene melt. At the gel point, storage and loss shear moduli, G' and G?, were described by the power law of frequency ?, G'˜G?˜?n, with the critical exponent n being nearly equal to 2/3, in agreement with the value predicted by the percolation theory. We also investigated the structure factor over two decades in length scale at gel point by using ultra-small-angle X-ray scattering, and small-angle X-ray scattering. We found the power-law behavior in low-q region, indicating that the gel network forms the self-similar structure with mass-fractal dimension. Comparison between the exponent of mass-fractal dimension from structure factor and that from viscoelasticity indicates that hydrodynamic interactions are completely screened out and the excluded volume effects are dominant in the gel. The gel strength was found to increase with the decrease in the lower limit length scale of fractality.

Takenaka, Mikihito; Kobayashi, Toshiaki; Saijo, Kenji; Tanaka, Hirokazu; Iwase, Naoki; Hashimoto, Takeji; Takahashi, Masaoki

2004-08-01

188

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

189

ERIC Educational Resources Information Center

Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…

Osler, Thomas J.

1999-01-01

190

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

191

Talbot image of two-dimensional fractal grating

NASA Astrophysics Data System (ADS)

Talbot effect of two-dimensional fractal grating built by square aperture arrays is studied theoretically and experimentally in this paper. The amplitude fractal gratings are produced by use of the spatial light modulator, and the diffraction intensity distributions of fractal gratings with different fractal level in Fresnel diffraction field are measured with the help of the two-dimensional CCD. Talbot images of fractal gratings with 1-level and 2-level fractal are obtained in practical experiment. The analytic expression of Fresnel diffraction intensity of the fractal gratings is derived through decomposing fractal gratings into the sum of many periodic gratings. Theoretic results predict the self-image of fractal grating reappears at some certain distance. The numerical calculations also show the Talbot image and the fractional Talbot image of fractal grating. These results may extend the application of fractal grating in the optical processing of information and laser measurement.

Teng, Shuyun; Wang, Junhong; Li, Furui; Zhang, Wei

2014-03-01

192

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

NASA Astrophysics Data System (ADS)

This paper performs a fractal analysis of the galaxy distribution and presents evidence that it can be described as a fractal system within the redshift range of the FORS Deep Field (FDF) galaxy survey data. The fractal dimension D was derived by means of the galaxy number densities calculated by Iribarrem et al. (2012) using the FDF luminosity function parameters and absolute magnitudes obtained by Gabasch et al. (2004, 2006) in the spatially homogeneous standard cosmological model with ?m0 = 0.3, ??0 = 0.7 and H0 = 70 kms-1Mpc-1. Under the supposition that the galaxy distribution forms a fractal system, the ratio between the differential and integral number densities ? and ?? obtained from the red and blue FDF galaxies provides a direct method to estimate D and implies that ? and ?? vary as power-laws with the cosmological distances, feature which provides a second method for calculating D. The luminosity distance dL, galaxy area distance dG and redshift distance dz were plotted against their respective number densities to calculate D by linear fitting. It was found that the FDF galaxy distribution is better characterized by two single fractal dimensions at successive distance ranges, that is, two scaling ranges in the fractal dimension. Two straight lines were fitted to the data, whose slopes change at z ? 1.3 or z ? 1.9 depending on the chosen cosmological distance. The average fractal dimension calculated using ?? changes from < D > = 1 .4-0.6+0.7 to < D > = 0 .5-0.4+1.2 for all galaxies. Besides, D evolves with z, decreasing as the redshift increases. Small values of D at high z mean that in the past galaxies and galaxy clusters were distributed much more sparsely and the large-scale structure of the universe was then possibly dominated by voids.

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

2015-01-01

193

Texture Analysis And Characterization Using Probability Fractal Descriptors

A gray-level image texture descriptors based on fractal dimension estimation is proposed in this work. The proposed method estimates the fractal dimension using probability (Voss) method. The descriptors are computed applying a multiscale transform to the fractal dimension curves of the texture image. The proposed texture descriptor method is evaluated in a classification task of well known benchmark texture datasets. The results show the great performance of the proposed method as a tool for texture images analysis and characterization.

Florindo, J B

2012-01-01

194

Fractal Geometric Characterization of Functionally Graded Materials

graded materials (FGM) is studied from the standpoint of fractal geometry. First, upon introducing the fineness as the number of grains of either phase across the FGM, the two-phase FGM is char- acterized using in local fractal dimension is considered across or along the FGM domain, and it is characterized by Fourier

Ostoja-Starzewski, Martin

195

Large-dimension configuration-interaction calculations of positron binding to the group-II atoms

The configuration-interaction (CI) method is applied to the calculation of the structures of a number of positron binding systems, including e{sup +}Be, e{sup +}Mg, e{sup +}Ca, and e{sup +}Sr. These calculations were carried out in orbital spaces containing about 200 electron and 200 positron orbitals up to l=12. Despite the very large dimensions, the binding energy and annihilation rate converge slowly with l, and the final values do contain an appreciable correction obtained by extrapolating the calculation to the l{yields}{infinity} limit. The binding energies were 0.00317 hartree for e{sup +}Be, 0.0170 hartree for e{sup +}Mg, 0.0189 hartree for e{sup +}Ca, and 0.0131 hartree for e{sup +}Sr.

Bromley, M. W. J.; Mitroy, J. [Department of Physics, San Diego State University, San Diego, California 92182 (United States); Faculty of Technology, Charles Darwin University, Darwin NT 0909 (Australia)

2006-03-15

196

Fractal geometry of aggregate snowflakes revealed by triple-wavelength radar measurements

NASA Astrophysics Data System (ADS)

Radar reflectivity measurements from three different wavelengths are used to retrieve information about the shape of aggregate snowflakes in deep stratiform ice clouds. Dual-wavelength ratios are calculated for different shape models and compared to observations at 3, 35, and 94 GHz. It is demonstrated that many scattering models, including spherical and spheroidal models, do not adequately describe the aggregate snowflakes that are observed. The observations are consistent with fractal aggregate geometries generated by a physically based aggregation model. It is demonstrated that the fractal dimension of large aggregates can be inferred directly from the radar data. Fractal dimensions close to 2 are retrieved, consistent with previous theoretical models and in situ observations.

Stein, T. H. M.; Westbrook, C. D.; Nicol, J. C.

2015-01-01

197

Self-avoiding walk on fractal complex networks: Exactly solvable cases.

We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent ? and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ? is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension. PMID:25493847

Hotta, Yoshihito

2014-11-01

198

NSDL National Science Digital Library

This webpage is dedicated to all things fractals, and is organized and updated by Professor Paul Bourke. Visitors will find all kinds of useful tools for discovering and learning about fractals including: Fractal forms found by using Google Earth, Natural Fractals in Grand Canyon National Park, Introduction to fractals, a gallery of fractals, and much, much more.

Bourke, Paul

199

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they\\u000a are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify\\u000a a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure\\u000a with predefined statistical and

Hongbing Sun; Manfred Koch

1998-01-01

200

Fractal texture analysis of bread crumb digital images

A fractal texture analysis technique was applied to bread crumb digital images. Fractal dimensions obtained from several methods\\u000a (fractional Brownian motion, frequency domain, relative differential box-counting, morphological fractal, mass fractal and\\u000a random walks methods) were investigated in order to determine their capability to accurately describe the surface roughness\\u000a of bread crumb images or the visual appearance of bread crumb in

Ursula Gonzales-Barron; Francis Butler

2008-01-01

201

Calculating nasoseptal flap dimensions: a cadaveric study using cone beam computed tomography.

We hypothesize that three-dimensional imaging using cone beam computed tomography (CBCT) is suitable for calculating nasoseptal flap (NSF) dimensions. To evaluate our hypothesis, we compared CBCT NSF dimensions with anatomical dissections. The NSF reach and vascularity were studied. In an anatomical study (n = 10), CBCT NSF length and surface were calculated and compared with anatomical dissections. The NSF position was evaluated by placing the NSF from the anterior sphenoid sinus wall and from the sella along the skull base towards the frontal sinus. To visualize the NSF vascularity in CBCT, the external carotic arteries were perfused with colored Iomeron. Correlations between CBCT NSFs and anatomical dissections were strongly positive (r > 0.70). The CBCT NSF surface was 19.8 cm(2) [16.6-22.3] and the left and right CBCT NSF lengths were 78.3 mm [73.2-89.5] and 77.7 mm [72.2-88.4] respectively. Covering of the anterior skull base was possible by positioning the NSF anterior to the sphenoid sinus. If the NSF was positioned from the sella along the skull base towards the frontal sinus, the NSF reached partially into the anterior ethmoidal sinuses. CBCT is a valuable technique for calculating NSF dimensions. CBCT to demonstrate septum vascularity in cadavers proved to be less suitable. The NSF reach for covering the anterior skull base depends on positioning. This study encourages preoperative planning of a customized NSF, in an attempt to spare septal mucosa. In the concept of minimal invasive surgery, accompanied by providing customized care, this can benefit the patients' postoperative complaints. PMID:25359192

Ten Dam, Ellen; Korsten-Meijer, Astrid G W; Schepers, Rutger H; van der Meer, Wicher J; Gerrits, Peter O; van der Laan, Bernard F A M; Feijen, Robert A

2014-10-31

202

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

203

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

204

Three-dimensional metrology and fractal analysis of dendritic nanostructures

NASA Astrophysics Data System (ADS)

Three-dimensional quantification and fractal dimension analysis are performed on a nanoscale dendrite, grown by precipitation as a crystal from a glass matrix. In order to process the entire three-dimensional (3D) volume and surface properties, a reconstruction from multiple projections has been achieved using electron tomography. Digital evaluation of the 3D volume and surface of the reconstructed dendritic nanoparticle allows quantifying its surface-volume ratio, convexity, solidity, and fractal dimension. The structure is found fractal across much of the nanoscale, with a fractal dimension estimated to 2.4.

Saghi, Zineb; Xu, Xiaojing; Möbus, Günter

2008-11-01

205

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

206

Fractals related to Pascal's triangle

A precise definition of a fractalFpr1 derived from Pascal's triangle modulopr (p prime) is given. The number of nonzero terms in the firstps lines of Pascal's triangle modulopr is computed. From this result the Hausdorff dimension and Hausdorff measure ofFpr1 are deduced. The nonself-similarty ofFpr1,r=2, is also discussed.

I. Jiménez Calvo; J. Muñoz Masqué

1996-01-01

207

Fractal Theory for Permeability Prediction, Venezuelan and USA Wells

NASA Astrophysics Data System (ADS)

Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.

Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana

2014-05-01

208

Determination of fish gender using fractal analysis of ultrasound images.

The gender of cod Gadus morhua can be determined by considering the complexity in their gonadal ultrasonographic appearance. The fractal dimension (D(B)) can be used to describe this feature in images. B-mode gonadal ultrasound images in 32 cod, where gender was known, were collected. Fractal analysis was performed on these images and D(B) was determined using the box counting method. A receiver-operating curve (ROC) was drawn for D(B) as a test for male fish. Using a range of D(B) values, the maximum accuracy for this test was calculated and compared with the accuracy for identifying male fish by subjective analysis alone. The mean (and standard deviation) of the fractal dimension D(B) for male fish was 1.554 (0.073) while for female fish it was 1.468 (0.061); the difference was statistically significant (P = 0.001). The area under the ROC curve was 0.84 indicating the value of fractal analysis in gender determination in cod. Maximum accuracy (0.84) for D(B) as a test for male fish was obtained using the threshold value D(B) = 1.5058 compared with an accuracy of 0.78 for subjective image evaluation. The use of two thresholds, D(B) < 1.4475 (females) and D(B) > 1.5054 (males) gives an 80% certainty in the classification result. Fractal analysis is useful for gender determination in cod. This or a similar form of analysis may have wide application in veterinary imaging as a tool for quantification of complexity in images. PMID:19788038

McEvoy, Fintan J; Tomkiewicz, Jonna; Støttrup, Josianne G; Overton, Julie L; McEvoy, Conni; Svalastoga, Eiliv

2009-01-01

209

NSDL National Science Digital Library

This website from the Department of Physics and Astronomy at Johns Hopkins University introduces chaos and describes how it appears in animal populations and weather models. The site also describes fractals and explains the butterfly effect. Images provide representations of chaotic behavior.

Bradley, Larry

210

The possibility of fractal benzenoids built upon a (deterministically) regular self-similar structure are proposed. Of various possibilities considered a simpler class of “trigonal” structures is identified. Possible directed synthetic schemes are described. Further, Kekulé structure counts are made as well as conjugated-circuits estimates of resonance energies. Results, for the general trigonal class as well as for particular cases are given.

D. J. Klein; M. J. Cravey; G. E. Hite

1991-01-01

211

Calculus on Fractal Curves in R^n

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.

Abhay Parvate; Seema Satin; A. D. Gangal

2010-04-06

212

Perturbative calculations in space-time having extra dimensions: The 6D single axial box anomaly

NASA Astrophysics Data System (ADS)

A detailed investigation about the 6D single axial box anomalous amplitude is presented. The superficial degree of divergence involved, in the one-loop perturbative calculations, is quadratic and the corresponding theory is nonrenormalizable. In spite of this, we show that the phenomenon of anomaly can be clearly characterized in a completely analogous way as that of 4D single axial triangle anomaly. The required calculations are made within the context of a novel calculational strategy where the amplitudes are not modified in intermediary steps. Divergent integrals are, in fact, not really solved. Adequate representations for the internal propagators are adopted according to the degree of divergence involved, so that when the last Feynman rule is taken (integration over the loop momentum) all the dependence on the internal (arbitrary) momenta are placed only in finite integrals. In the divergent structures emerging, no physical parameter is present and such objects are not really integrated. Only very general properties are assumed for such quantities which are universal (all space-time dimensions). The consistency of the perturbative calculations fixes some relations among the divergent integrals so that all the potentially ambiguous terms can be automatically removed. In spite of the absence of ambiguities, the emerging results allow us to give a clear and transparent description of the anomaly. The present investigation confirms the point of view stated by the same prescription for the well-known 2D axial-vector (AV) two-point and 4D single (AVV) and triple (AAA) axial-vector anomalies: the anomalous amplitudes need not be assumed as ambiguous quantities to allow an adequate description of the anomalies. We show also that a surprising, but natural, connection between the coupling of fermions with a pseudoscalar tensor field is found. In addition, we show that the crucial mathematical aspects of the problem are deeply related to a recently arisen controversy involving the evaluation of the Higgs Boson decay and the question of unicity in the dimensional regularization.

Fonseca, M. V. S.; Dallabona, G.; Battistel, O. A.

2014-11-01

213

Elasticity of fractal materials using the continuum model with non-integer dimensional space

NASA Astrophysics Data System (ADS)

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Tarasov, Vasily E.

2015-01-01

214

Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Tarasov, Vasily E

2015-01-01

215

Reduced Dimension Rovibrational Variational Calculations of the S_1 State of C_2H_2

NASA Astrophysics Data System (ADS)

The bending and torsional degrees of freedom in S_1 acetylene, C_2H_2, are subject to severe vibrational resonances and rovibrational interactions, which result in the low-energy vibrational polyad structure of these modes. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local equilibrium geometries. Experimental spectra in the region near the cis-trans transition state exhibit these complicated new patterns. In order to rationalize our near-barrier observations and predict the detailed effects of cis-trans isomerization on the rovibrational energy structure, we have performed reduced dimension rovibrational variational calculations of the S_1 state. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended-CNPI group theoretical treatment of a multivalued internal coordinate system that is appropriate for bending and torsional large amplitude motions. We will discuss these results and the insights they offer on understanding both large-scale features and spectroscopic details, such as tunneling staggerings, of barrier-proximal rovibrational levels of the S_1 state. We will also discuss spectral features by which barriers can be located and characterized in general polyatomic systems.

Changala, P. B.; Baraban, J. H.; Field, R. W.; Stanton, J. F.; Merer, A. J.

2013-06-01

216

Analytical 2D model to invert hydraulic pumping tests in fractured rocks with fractal behavior

NASA Astrophysics Data System (ADS)

A solution to interference pumping tests in fractal fractured media of Euclidean dimension two has been developed. It is proposed in dimensioned variables with a pre-conditioning of its most sensitive parameter and a Gauss-Newton inversion. The method allows for a rapid identification of hydrodynamic parameters by fitting experimental data. The fractal dimension and the scale exponent of the hydraulic diffusion are also determined without any other calculation or reference to a theoretical medium. Thus, the results provide a reliable appraisal of how the hydrodynamic parameters evolve with the size of the system. This feature has important applications in hydrology and petroleum engineering especially when up-scaling approaches are needed.

Delay, Frederick; Porel, Gilles; Bernard, Stephane

2004-08-01

217

Statistical fractal analysis of 25 young star clusters

A large sample of young stellar groups is analysed aiming to investigate their clustering properties and dynamical evolution. A comparison of the Q statistical parameter, measured for the clusters, with the fractal dimension estimated for the projected clouds shows that 52% of the sample has substructures and tends to follow the theoretically expected relation between clusters and clouds, according to calculations for artificial distribution of points. The fractal statistics was also compared to structural parameters revealing that clusters having radial density profile show a trend of parameter s increasing with mean surface stellar density. The core radius of the sample, as a function of age, follows a distribution similar to that observed in stellar groups of Milky Way and other galaxies. They also have dynamical age, indicated by their crossing time that is similar to unbound associations. The statistical analysis allowed us to separate the sample into two groups showing different clustering characteristi...

Gregorio-Hetem, J; Santos-Silva, T; Fernandes, B

2015-01-01

218

NASA Astrophysics Data System (ADS)

Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...

Wuorinen, Charles

2015-03-01

219

NSDL National Science Digital Library

This applet lets students explore connections among geometry, measurement and number patterns. Students select one of four preset fractal processes, including the Koch snowflake and the Sierpinski triangle, and observe the stages of complexity. The number of parts and an area or linear measure are shown in a table. The tool supports the lesson How Many Triangles Can You Construct? in the unit Building with Triangles (cataloged separately).

2011-01-01

220

The model of the universe is considered in which background of the universe is not defined by the matter but is a priori specified as a homogenous and isotropic flat space. The scale factor of the universe follows the linear law. The scale of mass changes proportional to the scale factor. This leads to that the universe has the fractal structure with a power index of 2.

D. L. Khokhlov

1999-01-15

221

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

222

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

223

Microtopographic Inspection and Fractal Analysis of Skin Neoplasia

NASA Astrophysics Data System (ADS)

Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh) corresponding to some neoplasia is higher (1.334+/-0.072) than those for healthy skin (1.091+/-0.082). A significant difference between the fractal dimensions of neoplasia and healhty skin (>0.001) was registered. The FD of microtopography maps (FDm) can also distinguish between healthy and malignant tissue in general (2.277+/-0.070 to 2.309+/-0.040), but not discriminate the different types of skin neoplasias. The combination of the rugometric evaluation and fractal geometry characterization provides valuable information about the malignity of skin lesions and type of lesion.

Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón

2008-04-01

224

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

225

Fractal analysis of DNA sequence data

DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the [open quote]sandbox method[close quote]. Analysis of 164 human DNA sequences compared to three types of control sequences (random, base-content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than to invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.

Berthelsen, C.L.

1993-01-01

226

Fractal morphometry of cell complexity.

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage. PMID:12449683

Losa, Gabriele A

2002-01-01

227

Characterization of branch complexity by fractal analyses

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

228

Causal Dynamical Triangulations in Four Dimensions

NASA Astrophysics Data System (ADS)

Recent results obtained within a non-perturbative approach to quantum gravity based on the method of four-dimensional Causal Dynamical Triangulations are described. The phase diagram of the model consists of three phases. In the physically most interesting phase, the time-translational symmetry is spontaneously broken. Calculations of expectation values required introducing procedures taking into account the inhomogeneity of configurations. It was shown that the dynamically emerged four-dimensional background geometry corresponds to a Euclidean de Sitter space and reveals no fractality at large distances. Measurements of the covariance matrix of scale factor fluctuations allowed to reconstruct the effective action, which remained in agreement with the discrete minisuperspace action. Values of the Hausdorff dimension and spectral dimension of three-dimensional spatial slices suggest their fractal nature, which was confirmed by a direct analysis of triangulation structure. The Monte Carlo algorithm used to obtain presented results is described.

Görlich, Andrzej

2011-11-01

229

In this work, the effect of the electromagnetic radiation generated by mobile phone, on the heart rate variability (HRV) has been investigated using correlation dimension calculation which is a nonlinear analysis method. The 17 volunteer subjects participated to our work and the experiment is designed as three periods and each period have 7 minutes. The electrocardiogram (ECG) signals were recorded

Derya Yilmaz; Metin Yildiz

2009-01-01

230

Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.

Chernodub, M N

2015-01-01

231

Efficient Desalination with Fractal Absorbers

NASA Astrophysics Data System (ADS)

A class of Ramified graphs (RG) is introduced as Iterated Function Systems (IFS) to optimally design networks for efficient reverse osmosis desalination in deep seawater. Different forms of the IFS are presented, along with a corresponding contractivity factor sc, in order to identify the attractors of the systems and their fractal dimension. Using the analogy to electrostatics, the diffusion equation is solved for the desalination systems under three different boundary conditions, i) all nodes having the same pressure difference across the absorbers, ii) all nodes producing permeate at identical rates, and iii) each node having the same salt node strength. Optimal branching angles and branch length ratios are found by phase-space and discrete simulated annealing search techniques for each boundary condition, which either maximize production of permeate or minimize expenditure of energy for different fixed numbers of absorbers. Dependence of desalination recovery ratios on the geometry and fractal dimension of the RG is also explored.

Singleton, Martin; Heiss, Gregor; Hubler, Alfred

2008-03-01

232

Fuzzy fractals, chaos, and noise

To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.

Zardecki, A.

1997-05-01

233

We calculate the inclusive cross section of double Z-boson production within large extra dimensions at the Large Hadron Collider (LHC). Using perturbatively quantized gravity in the ADD (Arkani-Hamed, Dvali, Dimopoulos) model we perform a first order calculation of the graviton mediated contribution to the pp{yields}ZZ+x cross section. At low energies (e.g. Tevatron) this additional contribution is very small, making it virtually unobservable, for a fundamental mass scale above 2500 GeV. At LHC energies, however, the calculation indicates that the ZZ-production rate within the ADD model should differ significantly from the standard model if the new fundamental mass scale would be below 15000 GeV. A comparison with the observed production rate at the LHC might therefore provide direct hints on the number and structure of the extra dimensions.

Kober, Martin; Bleicher, Marcus [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Koch, Benjamin [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Frankfurt Institute for Advanced Studies (FIAS), Max-von-Laue-Str. 1, 60438 Frankfurt (Germany)

2007-12-15

234

Fractal analysis of scatter imaging signatures to distinguish breast pathologies

NASA Astrophysics Data System (ADS)

Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.

Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.

2013-02-01

235

NASA Astrophysics Data System (ADS)

Images of packaged raw chicken purchased in neighborhood supermarkets were captured via a digital camera in laboratory and home settings. Each image contained the surface reflectivity information of the chicken tissue. The camera's red, green and blue light signals fluctuated and each spectral signal exhibited a random series across the surface. The Higuchi method, where the length of each increment in time (or spatial) lag is plotted against the lag, was used to explore the fractal property of the random series. (Higuchi, T., "Approach to an irregular time series on the basis of fractal theory", Physica D, vol 31, 277-283, 1988). The fractal calculation algorithm was calibrated with the Weierstrass function. The standard deviation and fractal dimension were shown to correlate with the time duration that a package was left at room temperature within a 24-hour period. Comparison to packaged beef results suggested that the time dependence could be due microbial spoilage. The fractal dimension results in this study were consistent with those obtained from yeast cell, mammalian cell and bacterial cell studies. This analysis method can be used to detect the re-refrigeration of a "left-out" package of chicken. The extension to public health issues such as consumer shopping is also discussed.

Subramaniam, Raji; Sullivan, R.; Schneider, P. S.; Flamholz, A.; Cheung, E.; Tremberger, G., Jr.; Wong, P. K.; Lieberman, D. H.; Cheung, T. D.; Garcia, F.; Bewry, N.; Yee, A.

2006-10-01

236

Tortuosity of unsaturated porous fractal materials

NASA Astrophysics Data System (ADS)

The tortuosity of a capillary-condensed film of inviscid fluid adsorbed onto fractal substrates as a function of the filling fraction of the fluid has been calculated numerically. This acts as a way of probing the multiscale structure of the objects. It is found that the variation of tortuosity ? with filling fraction ? is found to follow a power law of the form ?˜?-? for both deterministic and stochastic fractals. These numerically calculated exponents are compared to exponents obtained from a phenomenological scaling and good agreement is found, particularly for the stochastic fractals.

Coleman, S. W.; Vassilicos, J. C.

2008-07-01

237

Statistical analysis of fractal-based brain tumor detection algorithms

Fractals are geometric objects that have a noninteger fractal dimension (FD). The FD has been exploited for various biomedical recognition applications such as breast tumor and lung tumor detection. Our previous work shows that the FD is useful in the detection of brain tumors when a reference nontumor image is available. In this work, we extend our previous work by

Justin M. Zook; Khan M. Iftekharuddin

2005-01-01

238

Fractal image analysis - Application to the topography of Oregon and synthetic images.

NASA Technical Reports Server (NTRS)

Digitized topography for the state of Oregon has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with the results for Oregon. This approach can be easily applied to any digitzed image that obeys fractal statistics.

Huang, Jie; Turcotte, Donald L.

1990-01-01

239

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

240

Fractal characterization of neural correlates of consciousness

NASA Astrophysics Data System (ADS)

In this work we present a novel experimental paradigm, based on binocular rivalry, to address the study of internally and externally generated conscious percepts. Assuming the nonlinear nature of the EEG signals, we propose the use of fractal dimension to characterize the complexity of the EEG associated with each percept. Data analysis showed significant differences in complexity between the internally and externally generated percepts. Moreover, EEG complexity of auditory and visual percepts was unequal. These results support fractal dimension analyses as a new tool to characterize conscious perception.

Ibañez-Molina, A. J.; Iglesias-Parro, S.

2013-01-01

241

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

242

Light Scattering From Fractal Titania Aggregates

NASA Astrophysics Data System (ADS)

We studied the fractal morphology of titania aggregates by light scattering. Titanium dioxide particles were generated by the thermal decomposition of titanium tetra-isopropoxide(TTIP) in a glass furnace at various temperatures in the range of 100 - 500^o C. We scattered vertically polarized He-Ne laser (? = 6328Ålight from a laminar aerosol stream of particles and measured the optical structure factor. This structure factor shows Rayleigh, Guinier, fractal and Porod regimes. The radius of gyration Rg was determined from the Guinier analysis. The data were then fit to the Fisher-Burford form to determine the fractal dimension of about 2.0. This fit also delineated the crossover from the fractal to Porod regime, which can be used to determine the monomer particle size of about 0.1 ?m. These optical measurements will be compared to electron microscope analysis of aggregates collected from the aerosol. This work was supported by NSF grant CTS-9908153.

Pande, Rajiv; Sorensen, Christopher M.

1996-03-01

243

Quantitative Assessment of Early Diabetic Retinopathy Using Fractal Analysis

OBJECTIVE—Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes. RESEARCH DESIGN AND METHODS—We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12–20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol. RESULTS—In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023–1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02–7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21–1.56]). This association remained after additional adjustment for retinal vascular caliber. CONCLUSIONS—Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage. PMID:18835945

Cheung, Ning; Donaghue, Kim C.; Liew, Gerald; Rogers, Sophie L.; Wang, Jie Jin; Lim, Shueh-Wen; Jenkins, Alicia J.; Hsu, Wynne; Li Lee, Mong; Wong, Tien Y.

2009-01-01

244

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

245

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

246

Fractal antenna and fractal resonator primer

NASA Astrophysics Data System (ADS)

Self-similarity and fractals have opened new and important avenues for antenna and electronic solutions over the last 25 years. This primer provides an introduction to the benefits provided by fractal geometry in antennas, resonators, and related structures. Such benefits include, among many, wider bandwidths, smaller sizes, part-less electronic components, and better performance. Fractals also provide a new generation of optimized design tools, first used successfully in antennas but applicable in a general fashion.

Cohen, Nathan

2015-03-01

247

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.

248

To obtain more accurate correlation dimension estimations for chaotic time series, a novel scaling region identification method\\u000a is developed. First, points that obviously do not belong to the scaling region associated with the whole double logarithm\\u000a correlation integral curve are removed using the K-means algorithm. Second, a point-slope-error algorithm is developed to\\u000a recognize a possible scaling region. Third, the K-means

CuiCui Ji; Hua Zhu; Wei Jiang

2011-01-01

249

ERIC Educational Resources Information Center

Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…

Fraboni, Michael; Moller, Trisha

2008-01-01

250

NASA Astrophysics Data System (ADS)

In this work, we have applied a Wavelet Based Fractal Analysis (WBFA) to well logs and seismic data at the Teapot Dome Field, Natrona Country, Wyoming-USA, trying to characterize a reservoir using fractal parameters, as intercept (b), slope (m) and fractal dimension (D), and to correlate them with the sedimentation processes and/or the lithological characteristics of the area. The WBFA was first applied to the available logs (Gamma Ray, Spontaneous Potential, Density, Neutron Porosity and Deep Resistivity) from 20 wells located at sectors 27, 28, 33 and 34 of the 3D seismic of the Teapot Dome field. Also the WBFA was applied to the calculated curve of water saturation (Sw). At a second step, the method was used to analyze a set of seismic traces close to the studied wells, extracted from the 3D seismic data. Maps of the fractal parameters were obtained. A spectral analysis of the seismic data was also performed in order to identify seismic facies and to establish a possible correlation with the fractal results. The WBFA results obtained for the wells logs indicate a correlation between fractal parameters and the lithological content in the studied interval (i.e. top-base of the Frontier Formation). Particularly, for the Gamma Ray logs the fractal dimension D can be correlated with the sand-shale content: values of D lower than 0.9 are observed for those wells with more sand content (sandy wells); values of D between 0.9 and 1.1 correspond to wells where the sand packs present numerous inter-bedded shale layers (sandy-shale wells); finally, wells with more shale content (shaly wells) have D values greater than 1.1. The analysis of the seismic traces allowed the discrimination of shaly from sandy zones. The D map generated for the seismic traces indicates that this value can be associated with the shale content in the area. The iso-frequency maps obtained from the seismic spectral analysis show trends associated to the lithology of the field. These trends are similar to those observed in the maps of the fractal parameters, indicating that both analyses respond to lithological and/or sedimentation features in the area.

García, Alejandro; Aldana, Milagrosa; Cabrera, Ana

2013-04-01

251

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data. A software package known as the Image Characterization and Modeling System (ICAMS) was used to explore how fractal dimension is related to surface texture and pattern. The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was measured to investigate the relationship between texture and resolution for different land covers.

Emerson, Charles W.

1998-01-01

252

Fractals in art and nature: why do we like them?

NASA Astrophysics Data System (ADS)

Fractals have experienced considerable success in quantifying the visual complexity exhibited by many natural patterns, and continue to capture the imagination of scientists and artists alike. Fractal patterns have also been noted for their aesthetic appeal, a suggestion further reinforced by the discovery that the poured patterns of the American abstract painter Jackson Pollock are also fractal, together with the findings that many forms of art resemble natural scenes in showing scale-invariant, fractal-like properties. While some have suggested that fractal-like patterns are inherently pleasing because they resemble natural patterns and scenes, the relation between the visual characteristics of fractals and their aesthetic appeal remains unclear. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Whereas our previous preference studies focused on fractal images consisting of black shapes on white backgrounds, here we extend our investigations to include grayscale images in which the intensity variations exhibit scale invariance. This scale-invariance is generated using a 1/f frequency distribution and can be tuned by varying the slope of the rotationally averaged Fourier amplitude spectrum. Thresholding the intensity of these images generates black and white fractals with equivalent scaling properties to the original grayscale images, allowing a direct comparison of preferences for grayscale and black and white fractals. We found no significant differences in preferences between the two groups of fractals. For both set of images, the visual preference peaked for images with the amplitude spectrum slopes from 1.25 to 1.5, thus confirming and extending the previously observed relationship between fractal characteristics of images and visual preference.

Spehar, Branka; Taylor, Richard P.

2013-03-01

253

Riemann zeros, prime numbers, and fractal potentials

Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension.

Brandon P. van Zyl; David A. Hutchinson

2003-01-01

254

NASA Astrophysics Data System (ADS)

An algorithm to estimate the average local intrinsic dimension (

Hediger, T.; Passamante, A.; Farrell, Mary Eileen

1990-05-01

255

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

256

Fractal analysis of seismogenic ULF emissions

NASA Astrophysics Data System (ADS)

Fractal analysis has been performed on ultra-low-frequency (ULF) geomagnetic field data observed at the Izu peninsula. The first attempt of using fractal analysis for ULF geomagnetic field data during the large Guam and Biak earthquakes has been based on FFT. In order to quantitatively estimate the FFT-based fractal analysis, we first study a few fractal analyses, including the Burlaga-Klein method and the Higuchi method, as compared with the former FFT-based method. The accuracies of these three methods have been evaluated by applying them to the test data-sets of fractional Brownian motion with white noise. We conclude that the Higuchi method is superior to the FFT-based method and the Burlaga-Klein method. The S/ N effect was also discussed. Then, we have applied the Higuchi method to the ULF geomagnetic field data during a big earthquake swarm (during June-August 2000) in the Izu peninsula. It is found that the fractal dimension exhibits a significant increase just before earthquakes with a magnitude Mj>6.0 associated with the Izu islands swarm with any of these three fractal analyses. This experimental finding will lend further convincing support to the presence of precursory ULF emissions.

Gotoh, Kaoru; Hayakawa, Masashi; Smirnova, Natalia A.; Hattori, Katsumi

257

NASA Astrophysics Data System (ADS)

Spatial and temporal variability of soil moisture content has been frequently evaluated using statistical and geostatistical methods for several issues. For example, the statistical study of the temporal persistence or temporal stability in spatial patterns of soil moisture content has found interest to improve soil water monitoring strategies and to correct the average soil water content for missing data. Fractal analysis and graph theory are additional tools that can provide information and further insight to assess and to model indirect or hidden interactions in soil moisture content. In fractal analysis the fractal dimension (D) is an indicator of the pattern and extent of spatial and/or temporal variability. Large D values indicate the importance of short-range variation, while small D values reflect the importance of long-range variation when spatial and temporal data sets are analyzed. Moreover, for spatial and temporal variability, D can range from 1 to 2 for a profile and from 2 to 3 for a two dimensional network. Moreover, as the fractal dimension value increases the degree of roughness also increases. Graph theory tools take into account network structure by modelling pair wise relations between objects, which allow considering explicitly spatial-temporal connectivity of a given data set. The objective of this study was to use fractal analysis and graph theory to characterize the pattern of spatial and temporal variability of soil moisture content. The experimental field was located at Ottawa, Canada. Volumetric water content was monitored using Time Domain Reflectometry (TDR) during 34 dates at 164 locations per date. The depth of the TDR probes was 20 cm. The first and last measurements were 21 month apart and no data were taken in winter when the soil was covered by snow. The fractal dimension, D, was estimated from the slope of the regression line of log semivariogram versus distance for each of studied data sets. Using graph theory various parameters were calculated from the data measured in the 164 experimental vertices including edges, disconnected pair's number, average degree and clustering, etc.; calculations were performed for 21 groups of sets measured during three successive dates. Fractal dimension, D, ranged from 2.589 to 2.910, so that the smallest and the largest values indicate domination of long- and short-range variation respectively. Interestingly there was no correlation between fractal dimension, D, and coefficient of variation. Highest D values were recorded in spring and summer time. Parameters derived from graphs also allowed discrimination of the structure corresponding to successive data sets measured in three successive dates. For example, clustering varied from 0.406 to 0.836, given a correlation coefficient of 0.995. Different degrees of connectivity corresponded to different seasons. Parameters derived from fractal analysis and graph theory were useful to characterize the pattern and extent of spatial and temporal variability of soil moisture content. Acknowledgement: This work was partly supported by Spanish Ministry of Education (Project PHB2009-0094-PC.)

Vieira, Sidney R.; Vidal Vázquez, Eva; Miranda, José G. V.; Paz Ferreiro, Jorge; Topp, George C.

2010-05-01

258

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

2014-09-03

259

Fragmentation of Fractal Random Structures

NASA Astrophysics Data System (ADS)

We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.

Elçi, Eren Metin; Weigel, Martin; Fytas, Nikolaos G.

2015-03-01

260

Fractal Structure of Molecular Clouds

Compelling evidence exists to show that the structure of molecular clouds is fractal in nature. In this paper, the author reiterates this view and, in addition, asserts that not only is cloud geometry fractal, but that they also have a common characteristic - they are similar in shape to the Horsehead nebula in Orion. This shape can be described by the Julia function f(x)= z^2 + c,where both z and c are complex quantities and c = -0.745429 + 0.113008i. The dynamical processes responsible for the production of these clouds seem to be turbulence followed by Brownian motion till high densities are reached, at which point structure formation is dictated by gravity. The author presents image analysis of four varied examples, namely those of the Horsehead nebula, Eagle nebula, Rosette nebula and Paley I nebula to prove her hypothesis. The images of these nebulae are analyzed for their box dimension using fractal analysis software and comparisons are made with the given Julia set.

Srabani Datta

2001-05-02

261

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

262

NASA Astrophysics Data System (ADS)

The bending and torsional degrees of freedom in S1 acetylene, C2H2, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S1 state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.

Changala, P. Bryan

2014-01-01

263

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

264

How Fractal are Coastlines Really? Observation and Theory

NASA Astrophysics Data System (ADS)

Rocky coastlines have been held up as a prime example of fractal geometry since Mandelbrot introduced the concept. However, we will present a map of the fractal dimensions measured for the contiguous United States coastline which shows that many open-ocean sand--and even rocky--coastlines have fractal dimensions close to one; i.e. they tend to not be very fractal. The fractal nature of rocky coastlines likely represents an inherited fluvial or glacial signature that tends to be erased by coastal processes. Recent theoretical and numerical-modeling developments indicate that wave-driven coastal processes on sandy shores tend to produce one-dimensional coastlines. Gradients in alongshore sediment flux tend to smooth a shoreline, as long as the local wave climate is dominated by 'low-angle' waves (waves that approach the coastline in deep water from angles, relative to the coastline orientation, that are lower than the sediment-flux- maximizing angle). Even when a regional wave climate is dominated by high-angle waves--which produce an instability in plan-view shoreline shape--on the large scale, coastlines self organize in a way that produces locally low-angle-dominated wave climates almost everywhere. These processes explain why wave-dominated sandy coastlines, such as the Carolina and Texas coasts, exhibit fractal dimensions barely above one; wave- driven alongshore transport is an anti-fractal landsculpting agent over a range of scales greater than 0.2 km. In contrast, fluvial landsculpting produces famously fractal topography. When rapid sea-level rise causes the approximately horizontal plane of sea level to intersect a fractal fluvial topography, a fractal coastline results. Where wave energy is low, relative to rock erodibility, the fluvial fractal signature can persist. However, on the rocky West Coast of the US, fractal dimensions are relatively low (1.1 - 1.2), suggesting modification by wave-driven processes; that the production and rearrangement of sediment into ever-expanding pocket beaches has been reducing the fractality of this high-wave-energy, relatively easily eroded coastline. Glacially carved coastlines, such as that of Maine (and some parts of western Britain and Norway), exhibit high fractal dimensions (approximately 1.5), where erodibility is low enough the self-similarity of the intersection of sea-level with a glacially sculpted topography remains. Although wave-driven coastal processes tend to generate low-fractal-dimension shorelines, on sandy coastlines dominated by tidal currents, coastal processes also etch a fractal dendritic network of channels into the coastline. Tidally dominated coastlines, such as those in the Georgia Bight (Southeastern US), sport highly fractal shapes as a result (fractal dimensions approximately 1.5).

Murray, A.; Barton, C. C.

2007-12-01

265

Fractal Coagulation Bruce E. Logan

orange staining shows large holes in non-spherical biological aggregates... #12;...and acridine orange staining reveals interesting shapes of fractal objects! #12;Â· Mathematically define "fractal" and "fractal

266

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. PMID:25414672

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

2014-01-01

267

Stabilogram diffusion analysis algorithm to estimate the Hurst exponent of high-dimensional fractals

NASA Astrophysics Data System (ADS)

We suggest an algorithm for the estimation of the Hurst exponent that is based on the results of the well-known stabilogram diffusion analysis method of Hurst exponent estimation for one-dimensional fractals. Our algorithm can be applied to Hurst exponent estimation for fractals with two or more dimensions. To assess the efficiency of this algorithm, we compare its calculation results to those of the well-known Hurst exponent estimation detrending moving average analysis algorithm. In this paper, the computation of the Hurst exponent has been performed for two-dimensional domains of various sizes, which were generated by the Cholesky-Levinson factorization algorithm. The surrogate surfaces have Hurst exponents of H = 0.1, 0.5, and 0.9. It has been established that the detrending moving average analysis algorithm is more sensitive to high-frequency components, while the stabilogram diffusion analysis tends to be sensitive to low-frequency components.

Gorshkov, Oleg

2012-04-01

268

Size distribution effect on the power law regime of the structure factor of fractal aggregates.

We consider the large qR(g), where q is the magnitude of the scattering wave vector and R(g) is the aggregate radius of gyration, part of the structure factor of fractal aggregates, and quantify the coefficient C of the power law, S(q) approximately C(qR(g))(-D), where D is the fractal dimension, for various structure factors proposed in the literature. With the aid of earlier work, we conclude the most accurate structure factors have C=1.0. We then calculate the effects of polydispersity on this coefficient, and show the effects are significant, enough so to allow a measurement of the distribution width. These concepts are accurately supported with scattering data from a diffusion limited aerosol and a reaction limited colloid. PMID:11970655

Sorensen, C M; Wang, G M

1999-12-01

269

Size distribution effect on the power law regime of the structure factor of fractal aggregates

NASA Astrophysics Data System (ADS)

We consider the large qRg, where q is the magnitude of the scattering wave vector and Rg is the aggregate radius of gyration, part of the structure factor of fractal aggregates, and quantify the coefficient C of the power law, S(q)~C(qRg)-D, where D is the fractal dimension, for various structure factors proposed in the literature. With the aid of earlier work, we conclude the most accurate structure factors have C=1.0. We then calculate the effects of polydispersity on this coefficient, and show the effects are significant, enough so to allow a measurement of the distribution width. These concepts are accurately supported with scattering data from a diffusion limited aerosol and a reaction limited colloid.

Sorensen, C. M.; Wang, G. M.

1999-12-01

270

Calculation of grey level co-occurrence matrix-based seismic attributes in three dimensions

NASA Astrophysics Data System (ADS)

Seismic interpretation can be supported by seismic attribute analysis. Common seismic attributes use mathematical relationships based on the geometry and the physical properties of the subsurface to reveal features of interest. But they are mostly not capable of describing the spatial arrangement of depositional facies or reservoir properties. Textural attributes such as the grey level co-occurrence matrix (GLCM) and its derived attributes are able to describe the spatial dependencies of seismic facies. The GLCM - primary used for 2D data - is a measure of how often different combinations of pixel brightness values occur in an image. We present in this paper a workflow for full three-dimensional calculation of GLCM-based seismic attributes that also consider the structural dip of the seismic data. In our GLCM workflow we consider all 13 possible space directions to determine GLCM-based attributes. The developed workflow is applied onto various seismic datasets and the results of GLCM calculation are compared to common seismic attributes such as coherence.

Eichkitz, Christoph Georg; Amtmann, Johannes; Schreilechner, Marcellus Gregor

2013-10-01

271

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

272

Analytical estimation of the correlation dimension of integer lattices

Recently [L. Lacasa and J. G\\'omez-Garde\\~nes, Phys. Rev. Lett. {\\bf 110}, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called {\\em correlation dimension}, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order the correlation dimension $\\beta$ of integer lattices $\\mathbb{Z}^d$ coincides with the Haussdorf dimension of their coarsely-equivalent Euclidean spaces, $\\beta=d$.

Lucas Lacasa; Jesús Gómez-Gardeñes

2014-07-07

273

Analytical estimation of the correlation dimension of integer lattices.

Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 110, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called correlation dimension, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution, we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order, the correlation dimension ? of integer lattices ?(d) coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, ??=?d. PMID:25554021

Lacasa, Lucas; Gómez-Gardeñes, Jesús

2014-12-01

274

Fractal and Multifractal Analysis of Human Gait

NASA Astrophysics Data System (ADS)

We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.

Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.

2003-09-01

275

Edges of Saturn's rings are fractal.

The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63?~?1.78. This clarifies in what sense Saturn's rings are fractal. PMID:25883885

Li, Jun; Ostoja-Starzewski, Martin

2015-01-01

276

NSDL National Science Digital Library

While some may know fractals primarily from their use in abstract painting and African art, fractals are important elements within the world of mathematics. For those who seek to learn more about the construction of fractals and their uses, this very nice Flash-enabled feature from Daniel Gries at Merrimack College will definitely come in handy. This particular Flash applet draws fractals by means of a recursive algorithm, using a simple "generator" that users draw in the space that it is provided. Before using the application, users may wish read the overview offered online, and also take the time to read the instructions thoroughly.

Gries, Daniel

277

Fractal Structures Driven by Self-gravity: Molecular Clouds and the Universe

NASA Astrophysics Data System (ADS)

In the interstellar medium, as well as in the Universe, large density fluctuations are observed, that obey power-law density distributions and correlation functions. These structures are hierarchical, chaotic, turbulent, but are also self-organizing. The apparent disorder is not random noise, but can be described by a fractal, with a deterministic fractal dimension. We present a new theory of the self-gravity thermodynamics, that could explain the existence of these fractal structures, and predict their fractal dimension. The media obeying scaling laws can be considered critical, as in second order phase transitions for instance.

Combes, Francoise

1998-09-01

278

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

279

Control of multiscale systems with constraints. 2. Fractal nuclear isomers and clusters

We consider the influence of the Fermi statistics of nucleons on the binding energy of a new type of nuclear structures such as fractal nuclear clusters (fractal isomers of nuclei). It is shown that the fractal nuclear isomers possess a wide spectrum of binding energies that exceed, in many cases, the values known at the present time. The transition of the nuclear matter in the form of ordinary nuclei (drops of the nuclear fluid) in the state with the fractal structure or in the form of bubble nuclei opens new sources of energy and has huge perspectives. This transition is based on a new state of matter - collective coherently correlated state. It manifests itself, first of all, in the property of nonlocality of nuclear multiparticle processes. We develop a phenomenological theory of the binding energy of nuclear fractal structures and modify the Bethe - Weizs\\"acker formula for nuclear clusters with the mass number A, charge Z, and fractal dimension D_f. The consideration of fractal nuclear isomers allows one to interpret the experimental results on a new level of the comprehension of processes of the nuclear dynamics. The possibility to determine the fractal dimension of nuclear systems with the help of the method of nuclear dipole resonance for fractal isomers is discussed. The basic relations for fractal electroneutral structures such as the electron-nucleus plasma of fractal isomers are presented.

S. Adamenko; V. Bolotov; V. Novikov

2013-07-17

280

Control of spin-wave excitations in deterministic fractals

NASA Astrophysics Data System (ADS)

We study spin-wave spectra of mesoscopic ferromagnetic Sierpinski carpets by means of broadband-ferromagnetic resonance measurements and micromagnetic simulations. Sierpinski carpets are self-similar fractals with noninteger Hausdorff dimension that are constructed via a deterministic iteration process. The number of quantized spin-wave modes in the spectra increases with the iteration level of the carpets and the frequency splitting resembles bandpass characteristics known from fractal antennas. We find that the splitting is sensitive to the fractal dimension as well as to the relative alignment of the magnetic field and the sides of the fractals. Micromagnetic simulations provide the localization of individual spin-wave modes determined by the confinement and the inhomogeneity of the internal field.

Swoboda, Christian; Martens, Michael; Meier, Guido

2015-02-01

281

Evaluation of Two Fractal Methods for Magnetogram Image Analysis

NASA Technical Reports Server (NTRS)

Fractal and multifractal techniques have been applied to various types of solar data to study the fractal properties of sunspots as well as the distribution of photospheric magnetic fields and the role of random motions on the solar surface in this distribution. Other research includes the investigation of changes in the fractal dimension as an indicator for solar flares. Here we evaluate the efficacy of two methods for determining the fractal dimension of an image data set: the Differential Box Counting scheme and a new method, the Jaenisch scheme. To determine the sensitivity of the techniques to changes in image complexity, various types of constructed images are analyzed. In addition, we apply this method to solar magnetogram data from Marshall Space Flight Centers vector magnetograph.

Stark, B.; Adams, M.; Hathaway, D. H.; Hagyard, M. J.

1997-01-01

282

FORTRAN programs for calculating nonlinear seismic ground response in two dimensions

The programs described here were designed for calculating the nonlinear seismic response of a two-dimensional configuration of soil underlain by a semi-infinite elastic medium representing bedrock. There are two programs. One is for plane strain motions, that is, motions in the plane perpendicular to the long axis of the structure, and the other is for antiplane strain motions, that is motions parallel to the axis. The seismic input is provided by specifying what the motion of the rock-soil boundary would be if the soil were absent and the boundary were a free surface. This may be done by supplying a magnetic tape containing the values of particle velocity for every boundary point at every instant of time. Alternatively, a punch card deck may be supplied giving acceleration values at every instant of time. In the plane strain program it is assumed that the acceleration values apply simultaneously to every point on the boundary; in the antiplane strain program it is assumed that the acceleration values characterize a plane shear wave propagating upward in the underlying elastic medium at a specified angle with the vertical. The nonlinear hysteretic behavior of the soil is represented by a three-dimensional rheological model. A boundary condition is used which takes account of finite rigidity in the elastic substratum. The computations are performed by an explicit finite-difference scheme that proceeds step by step in space and time. Computations are done in terms of stress departures from an unspecified initial state. Source listings are provided here along with instructions for preparing the input. A more detailed discussion of the method is presented elsewhere.

Joyner, W.B.

1978-01-01

283

Psychological and physiological benefits of viewing nature have been extensively studied for some time. More recently it has been suggested that some of these positive effects can be explained by nature's fractal properties. Virtually all studies on human responses to fractals have used stimuli that represent the specific form of fractal geometry found in nature, i.e. statistical fractals, as opposed to fractal patterns which repeat exactly at different scales. This raises the question of whether human responses like preference and relaxation are being driven by fractal geometry in general or by the specific form of fractal geometry found in nature. In this study we consider both types of fractals (statistical and exact) and morph one type into the other. Based on the Koch curve, nine visual stimuli were produced in which curves of three different fractal dimensions evolve gradually from an exact to a statistical fractal. The patterns were shown for one minute each to thirty-five subjects while qEEG was continuously recorded. The results showed that the responses to statistical and exact fractals differ, and that the natural form of the fractal is important for inducing alpha responses, an indicator of a wakefully relaxed state and internalized attention. PMID:25575556

Hagerhall, C M; Laike, T; Kuller, M; Marcheschi, E; Boydston, C; Taylor, R P

2015-01-01

284

Lattice animals, fractality and criticality in hadronic and partonic systems

NASA Astrophysics Data System (ADS)

The cluster description of near coexistence phases (e.g. Fisher theory) requires an evaluation of cluster surface entropy. This surface degeneracy can be estimated with lattice models where clusters appear. The maximum probability lies near the maximum cluster surface. At low temperatures, clusters are forced to be nearly spherical by the surface energy and the associated Boltzmann factor. At higher temperatures and near criticality, the fractal dimension of clusters changes so that clusters become fractal. In the MIT bag model, where there is no surface energy, bags are always fractal.

Moretto, L. G.; Elliot, J. B.; Lake, P. T.; Phair, L.

2011-01-01

285

Signatures of fractal clustering of aerosols advected under gravity.

Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example, at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension D(0) of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology. PMID:17677314

Vilela, Rafael D; Tél, Tamás; de Moura, Alessandro P S; Grebogi, Celso

2007-06-01

286

Fractal pharmacokinetics of the drug mibefradil in the liver.

We explore the ramifications of the fractal geometry of the key organ for drug elimination, the liver, on pharmacokinetic data analysis. A formalism is developed for the use of a combination of well-stirred Euclidean and fractal compartments in the body. Perturbation analysis is carried out to obtain analytical solutions for the drug concentration time evolution. These results are then fitted to experimental data collected from clinically instrumented dogs [see, A. Skerjanec et al., J. Pharm. Sci. 85, 189 (1995)] using the drug mibefradil. The thus obtained spectral fractal dimension has a range of values that is consistent with the value found in independently performed ultrasound experiments on the liver. PMID:12241211

Fuite, J; Marsh, R; Tuszy?ski, J

2002-08-01

287

Fractal images induce fractal pupil dilations and constrictions.

Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed. PMID:24978815

Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P

2014-09-01

288

Observation of two different fractal structures in nanoparticle, protein and surfactant complexes

Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.

Mehan, Sumit, E-mail: sumit.mehan@gmail.com; Kumar, Sugam, E-mail: sumit.mehan@gmail.com; Aswal, V. K., E-mail: sumit.mehan@gmail.com [Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai-400085 (India)

2014-04-24

289

On fractal analysis of cardiac interbeat time series

NASA Astrophysics Data System (ADS)

In recent years the complexity of a cardiac beat-to-beat time series has been taken as an auxiliary tool to identify the health status of human hearts. Several methods has been employed to characterize the time series complexity. In this work we calculate the fractal dimension of interbeat time series arising from three groups: 10 young healthy persons, 8 elderly healthy persons and 10 patients with congestive heart failures. Our numerical results reflect evident differences in the dynamic behavior corresponding to each group. We discuss these results within the context of the neuroautonomic control of heart rate dynamics. We also propose a numerical simulation which reproduce aging effects of heart rate behavior.

Guzmán-Vargas, L.; Calleja-Quevedo, E.; Angulo-Brown, F.

2003-09-01

290

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

291

Fractal dimension-based EEG biofeedback system

Biofeedback plays an increasingly important role in mainstream computer applications, including hands-free human-machine interaction. The most obvious use of this technology is to help disabled people interact with their environment. The main challenge in brain computer interfaces is to identify the particular EEG signal components that can be successfully used as control commands. In this study, we show that the

A. Bashashati; R. K. Ward; G. E. Birch; M. R. Hashemi; M. A. Khalilzadeh

2003-01-01

292

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

293

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

294

Statistical fractal analysis of 25 young star clusters

NASA Astrophysics Data System (ADS)

A large sample of young stellar groups is analysed to investigate their clustering properties and dynamical evolution. A comparison of the Q statistical parameter, measured for the clusters, with the fractal dimension estimated for the projected clouds, shows that 52 per cent of the sample has substructures and tends to follow the theoretically expected relation between clusters and clouds, according to calculations for the artificial distribution of points. The fractal statistics was also compared to structural parameters, revealing that clusters having a radial density profile show a trend of parameter overline{s} increasing with mean surface stellar density. The core radius of the sample, as a function of age, follows a similar distribution to that observed in stellar groups of the Milky Way and other galaxies. They also have dynamical age, indicated by their crossing time, which is similar to unbound associations. The statistical analysis allowed us to separate the sample into two groups showing different clustering characteristics. However, they have the same dynamical evolution, since the whole sample has been revealed as expanding objects, for which the substructures seem to have not been erased. These results are in agreement with simulations that adopt low surface densities and models under supervirial conditions.

Gregorio-Hetem, J.; Hetem, A.; Santos-Silva, T.; Fernandes, B.

2015-04-01

295

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

296

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

297

Riemann zeros, prime numbers, and fractal potentials

Using two distinct inversion techniques, the local one-dimensional potentials\\u000afor the Riemann zeros and prime number sequence are reconstructed. We establish\\u000athat both inversion techniques, when applied to the same set of levels, lead to\\u000athe same fractal potential. This provides numerical evidence that the potential\\u000aobtained by inversion of a set of energy levels is unique in one-dimension. We

Brandon P. van Zyl; David A. W. Hutchinson

2003-01-01

298

Fractal characterization of an earthquake sequence

The generalized fractal dimensions for the distribution of interoccurrence time and the relation of cumulative frequency versus interoccurrence time of earthquakes are studied. The data set used is a complete one consisting of M ? 6 earthquakes occurring in the north-south seismic belt of mainland China during the 1900–1990 period. The log-log plot of Cq(t) versus t, where Cq(t) is

Wang Jeen-Hwa; Lee Chung-Wein

1995-01-01

299

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

300

Tissue as a self-organizing system with fractal dynamics

NASA Astrophysics Data System (ADS)

Cell is a supramolecular dynamic network. Screening of tissue-specific cDNA library and results of Relative RT-PCR indicate that the relationship between genotype, (i.e., dynamic network of genes and their protein regulatory elements) and phenotype is non-bijective, and mendelian inheritance is a special case only. This implies non-linearity, complexity, and quasi-determinism, (i.e., co-existence of deterministic and non-deterministic events) of dynamic cellular network; prerequisite conditions for the existence of fractal structure. Indeed, the box counting method reveals that morphological patterns of the higher order, such as gland-like structures or populations of differentiating cancer cells possess fractal dimension and self-similarity. Since fractal space is not filled out randomly, a variety of morphological patterns of functional states arises. The expansion coefficient characterizes evolution of fractal dynamics. The coefficient indicates what kind of interactions occurs between cells, and how far from the limiting integer dimension of the Euclidean space the expanding population of cells is. We conclude that cellular phenomena occur in the fractal space; aggregation of cells is a supracollective phenomenon (expansion coefficient > 0), and differentiation is a collective one (expansion coefficient < 0). Fractal dimension or self-similarity are lost during tumor progression. The existence of fractal structure in a complex tissue system denotes that dynamic cellular phenomena generate an attractor with the appropriate organization of space-time. And vice versa, this attractor sets up physical limits for cellular phenomena during their interactions with various fields. This relationship can help to understand the emergence of extraterrestial forms of life. Although those forms can be composed of non-carbon molecules, fractal structure appears to be the common feature of all interactive biosystems.

Waliszewski, P.; Konarski, J.

2001-01-01

301

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

302

Self-organized stiffness in regular fractal polymer structures

NASA Astrophysics Data System (ADS)

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.

Werner, Marco; Sommer, Jens-Uwe

2011-05-01

303

NSDL National Science Digital Library

Cynthia Lanius, a former mathematics teacher who currently serves as Technology Integration Specialist for Sinton Independent School District in Sinton, Texas, has posted numerous lessons online. This website features a Fractals Unit for elementary and middle school students (although adults are also welcome to enjoy the lesson). The lesson includes a discussion on why one might study fractals and then provides step-by-step explanations on how to make fractals using Java, along with some challenging mathematics questions to consider. Samples of student work are also posted. A section for teachers provides an overview of the unit objectives along with links to other resources and materials to use in the classroom.

304

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

305

Electrical response of fractal and porous interfaces

The electrical response of porous electrodes is calculated in several particular cases, which permit one to approach the response of a realistic model for a porous interface. The case of nonblocking surfaces and the case of the diffusion impedance of a fractal electrode are also considered. The use of Bode diagrams is shown to provide a very simple means for

B. Sapoval; J.-N. Chazalviel; J. Peyrière

1988-01-01

306

Modeling of fractal patterns in matrix acidizing and their impact on well performance

This paper describes a model where wormholes, the primary feature of carbonate acidizing, are considered as fractals. The influences of acid volume, injection rate, fractal dimension, porosity, and the ratio of undamaged to damaged permeabilities on well performance are studied. Exact expressions of post-treatment skin effects are developed for vertical and horizontal wells.

Frick, T.P.; Kuermayr, M.; Economides, M.J.

1994-02-01

307

DOES FRACTAL CHARACTERISTICS OF HEMOGLOBIN CHANGE FROM ONE ORGANISM TO ANOTHER?

Within this study we reveal the fractal aspects of hemoglobins belonging to eight different species and those of four of its mutants respectively. The mean values of the surface fractal dimensions for the hemoglobins belonging to the eight species are DS = 2.233 ) 0.141 for the A chain and DS = 2.228 ) 0.107 for the B chain, those

D. CRACIUN; A. ISVORAN; N. M. AVRAM

2009-01-01

308

ERIC Educational Resources Information Center

Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)

Bannon, Thomas J.

1991-01-01

309

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.

2011-01-20

310

Spectral dimension of a quantum universe

In this paper, we calculate in a transparent way the spectral dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a spectral dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the spectral dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.

Modesto, Leonardo; Nicolini, Piero [Perimeter Institute for Theoretical Physics, 31 Caroline Street, Waterloo, ON N2L 2Y5 (Canada); Frankfurt Institute for Advanced Studies (FIAS), Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main (Germany)

2010-05-15

311

Study of Fractal Features of Geomagnetic Activity Through an MHD Shell Model

NASA Astrophysics Data System (ADS)

Studies on complexity have been of great interest in plasma physics, because they provide new insights and reveal possible universalities on issues such as geomagnetic activity, turbulence in laboratory plasmas, physics of the solar wind, etc. [1, 2]. In particular, various studies have discussed the relationship between the fractal dimension, as a measure of complexity, and physical processes in magnetized plasmas such as the Sun's surface, the solar wind and the Earth's magnetosphere, including the possibility of forecasting geomagnetic activity [3, 4, 5]. Shell models are low dimensional dynamical models describing the main statistical properties of magnetohydrodynamic (MHD) turbulence [6]. These models allow us to describe extreme parameter conditions hence reaching very high Reynolds (Re) numbers. In this work a MHD shell model is used to describe the dissipative events which are taking place in the Earth's magnetosphere and causing geomagnetic storms. The box-counting fractal dimension (D) [7] is calculated for the time series of the magnetic energy dissipation rate obtained in this MHD shell model. We analyze the correlation between D and the energy dissipation rate in order to make a comparison with the same analysis made on the geomagnetic data. We show that, depending on the values of the viscosity and the diffusivity, the fractal dimension and the occurrence of bursts exhibit correlations similar as those observed in geomagnetic and solar data, [8] suggesting that the latter parameters could play a fundamental role in these processes. References [1] R. O. Dendy, S. C. Chapman, and M. Paczuski, Plasma Phys. Controlled Fusion 49, A95 (2007). [2] T. Chang and C. C. Wu, Phys. Rev. E 77, 045401 (2008). [3] R. T. J. McAteer, P. T. Gallagher, and J. Ireland, Astrophys. J. 631, 628 (2005). [4] V. M. Uritsky, A. J. Klimas, and D. Vassiliadis, Adv. Space Res. 37, 539 (2006). [5] S. C. Chapman, B. Hnat, and K. Kiyani, Nonlinear Proc. Geophys. 15, 445 (2008). [6] G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and A. Vulpiani, Phys. Rev. Lett. 83, 4662 (1999). [7] P. S. Addison, Fractals and Chaos, an Illustrated Course, vol. 1 (Institute of Physics Publishing, Bristol and Philadelphia, 1997), second ed. [8] M. Domínguez, V. Muñoz, and J. A. Valdivia, Temporal evolution of fractality in the Earth's magnetosphere and the solar photosphere, in preparation.

Dominguez, M.; Nigro, G.; Munoz, V.; Carbone, V.

2013-12-01

312

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 universe by suitably choosing the fractal parameter $\\beta$.

Ahmad Sheykhi; Zeinab Teimoori; Bin Wang

2013-01-12

313

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

314

Quantum critical behavior of the quantum Ising model on fractal lattices

NASA Astrophysics Data System (ADS)

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpi?ski carpet, Sierpi?ski gasket, and Sierpi?ski tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpi?ski tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

Yi, Hangmo

2015-01-01

315

Antigen-antibody binding kinetics for biosensor applications. A dual-fractal analysis.

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The fit obtained by a dual-fractal analysis is compared with that obtained from a single-fractal analysis. In some cases, the dual-fractal analysis provides an improved fit when compared with a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (San Rafael, CA). These examples are presented. It is of interest to note that the state of disorder (or the fractal dimension) and the binding rate coefficient both increase (or decrease, a single example is presented for this case) as the reaction progresses on the biosensor surface. For example, for the binding of monoclonal antibody MAb 49 in solution to surface-immobilized antigen, a 90.4% increase in the fractal dimension (Df1 to Df2) from 1.327 to 2.527 leads to an increase in the binding rate coefficient (k1 to k2) by a factor of 9.4 from 11.74 to 110.3. The different examples analyzed and presented together provide a means by which the antigen-antibody reactions may be better controlled by noting the magnitude of the changes in the fractal dimension and in the binding rate coefficient as the reaction progresses on the biosensor surface. PMID:9170257

Sadana, A; Suturia, M

1997-01-01

316

Silica wet gels were prepared from hydrolysis of tetraethoxysilane (TEOS) with additions of sodium dodecyl sulfate (SDS). The surfactant was removed after gelation. Wet gels exhibited mass-fractal structure with mass-fractal dimension D (typically around 2.25) in a length scale extending from a characteristic size ? (typically about 10 nm) of the mass-fractal domains to a characteristic size a0 (typically between 0.3 and 0.4 nm) of the primary particles building up the fractal domains. ? increased while D and a0 diminished slightly as the SDS quantity increased. Aerogels with typical specific surface of 1000 m(2)/g and density of 0.20 g/cm(3) were obtained by supercritical drying of the wet gels after washing with ethanol and n-hexane. The pore volume and the mean pore size increased with the increase of the SDS quantity. The aerogels presented most of the mass-fractal characteristics of the original wet gels at large length scales and exhibited at a higher resolution level at about 0.7 nm a crossover to a mass-surface fractal structure, with apparent mass-fractal dimension Dm ? 2.4 and surface-fractal dimension Ds ? 2.6, as inferred from small-angle X-ray scattering (SAXS) and nitrogen adsorption data. PMID:25513729

Perissinotto, Amanda P; Awano, Carlos M; Donatti, Dario A; de Vicente, Fabio S; Vollet, Dimas R

2015-01-13

317

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

318

SAR image post-processing for the estimation of fractal parameters

NASA Astrophysics Data System (ADS)

In this paper a fractal based processing for the analysis of SAR images of natural surfaces is presented. Its definition is based on a complete direct imaging model developed by the authors. The application of this innovative algorithm to SAR images makes possible to obtain complete maps of the two key parameters of a fractal scene: the fractal dimension and the increment standard deviation. The fractal parameters extraction is based on the estimation of the power spectral density of the SAR amplitude image. From a theoretic point of view, the attention is focused on the retrieving procedure of the increment standard deviation, here presented for the first time. In the last section of the paper, the application of the introduced processing to high resolution SAR images is presented, with the relevant maps of the fractal dimension and of the increment standard deviation.

Di Martino, Gerardo; Riccio, Daniele; Ruello, Giuseppe; Zinno, Ivana

2011-11-01

319

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

320

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330-345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in {{mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts are proved.

Zhang, Xicheng

2012-04-01

321

Three-dimensional fractal homeomorphisms

NASA Astrophysics Data System (ADS)

We provide a simple introduction to fractal transformations and fractal homeomorphisms. We introduce tri-affine iterated function systems on R3, and illustrate associated fractal homeomorphisms applied to three dimensional graphical data sets. We also comment on the algorithms used.

Barnsley, Michael F.; Harding, Brendan

2015-03-01

322

Variable Topology on Fractal Manifold

In this paper, we study the topology associated to the fractal manifold model. It turns out that this topology is actually a family of topologies that gives to the fractal manifold a structure of variable topological space. Additionally, we prove that using the fractal manifold as model for the universe dynamic, the universe expansion is intimately correlated to the variation of the topology.

Helene Porchon

2012-11-15

323

Fractal and grey level co-occurrence matrix (GLCM) analysis represent two mathematical computer-assisted algorithms that are today thought to be able to accurately detect and quantify changes in tissue architecture during various physiological and pathological processes. However, despite their numerous applications in histology and pathology, their sensitivity, specificity and validity regarding evaluation of brain tissue remain unclear. In this article we present the results indicating that certain parameters of fractal and GLCM analysis have high discriminatory ability in distinguishing two morphologically similar regions of rat hippocampus: stratum lacunosum-moleculare and stratum radiatum. Fractal and GLCM algorithms were performed on a total of 240 thionine-stained hippocampus micrographs of 12 male Wistar albino rats. 120 digital micrographs represented stratum lacunosum-moleculare, and another 120 stratum radiatum. For each image, 7 parameters were calculated: fractal dimension, lacunarity, GLCM angular second moment, GLCM contrast, inverse difference moment, GLCM correlation, and GLCM variance. GLCM variance (VAR) resulted in the largest area under the Receiver operating characteristic (ROC) curve of 0.96, demonstrating an outstanding discriminatory power in analysis of stratum lacunosum-moleculare (average VAR equaled 478.1±179.8) and stratum radiatum (average VAR of 145.9±59.2, p<0.0001). For the criterion VAR?227.5, sensitivity and specificity were 90% and 86.7%, respectively. GLCM correlation as a parameter also produced large area under the ROC curve of 0.95. Our results are in accordance with the findings of our previous study regarding brain white mass fractal and textural analysis. GLCM algorithm as an image analysis method has potentially high applicability in structural analysis of brain tissue cytoarcitecture. PMID:25665716

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

2015-04-01

324

An Analytical Solution to an Archard-Type Fractal Rough Surface Contact Model

Due to the existence of multiple scales of features on surfaces, during the past two decades models have been developed that employ a fractal description of the rough surface profile. Most of these models use a truncation calculation to predict the contact area. The load corresponding to the truncated area of the fractal surface is then calculated using elastic and

Robert L. Jackson

2010-01-01

325

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

326

ERIC Educational Resources Information Center

Five activities are presented in this student workbook on using the electronic calculator. Following the directions for using the machine, problems are given on multiplying and dividing, finding percentages, calculating the area of assorted polygons, changing fractions to decimals, and finding squares and square roots. (JH)

Parma City School District, OH.

327

The Relationship Between Mass Factal Dimensions of Solid Matrix and Pore Space in Porous Media

NASA Astrophysics Data System (ADS)

Measuring fractal dimensions by image analyzing techniques has become a common practice to describe structural properties of porous media. Depending on the object of interest, different features of the structure can be measured: solid matrix, pores, and the interface between them. In many cases the fractal dimension of one of these features has been determined and taken to describe the entire system. The question arises whether these dimensions are independent from each other or whether they can be related to an underlying property of the structure or image, respectively. For a variety of porous media we measured the fractal dimension of the matrix, the pore space, and the interface between them simultaneously using the box counting method. The images were obtained from soil thin sections, a void system in a clayey soil, and a moss agate, which is a dendrite structure within a calcite matrix. Measured fractal dimensions were compared with fractal dimensions estimated by the pore-solid fractal (PSF) model, which derives the fractal properties of the matrix and the pore space completely as a function of the porosity, the size of the initiator and the fractal dimension of the interface. Measured results agree well with values obtained from the PSF model. A clear relationship between the mass fractal dimensions of the two phases (solid matrix and pore space) of a porous media was observed. For all images the smallest fractal dimensions were found for the interface between matrix and pores. Values for the fractal dimension of the two phases were between those for the interface and the Euclidian space with the phase with the lower mass fraction always having the smaller dimension. Model results also predict a dependency of the dimension of the phases on the spatial resolution of the analyzed image.

Dathe, A.; Thullner, M.

2004-05-01

328

Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model

A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent renormalization group prediction of an upper critical Î·{sub c}=4 , at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.

M. B. Hastings

2001-01-01

329

Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model

A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent renormalization group prediction of an upper critical etac = 4, at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.

M. B. Hastings

2001-01-01

330

Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model

A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent RG prediction of an upper critical $\\eta_c=4$, at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.

M. B. Hastings

2001-06-01

331

Fractal characterization of hematite aggregates by X-ray microscopy

NASA Astrophysics Data System (ADS)

X-ray microscopy supplies the actual morphology of hematite aggregates in an aqueous dispersion medium. The fractal dimension of hematite aggregates has been determined below and above the critical coagulation concentration. The box-counting method has been used as a morphometric tool. The values obtained are not in accordance with the values produced by numerical approaches.

Thieme, J.; Niemeyer, J.

1996-12-01

332

Statistical errors in the fractal analysis of flame boundaries

A high speed tomographic technique is used to evaluate the effect of spatial resolution, and requirements for statistical convergence on the fractal analysis of a turbulent, premixed, stoichiometric methane/air flame at high Damkoehler number. The gas velocity at the nozzle exit is 5 m/s, the turbulence intensity is 7%, the integral length scale 3 mm and hence the turbulence Reynolds number is 70. The light source is a copper vapor laser which produces 20ns, 5 mJ pulses at a 4KHz repetition rate. Cylindrical lenses transform the 38mm circular laser beam to a sheet 50 mm high and 0.6 mm thick. A high speed Fastax camera is used to record the tomographic images formed by the scattering of light from oil droplets seeded in the reactant flow. The films are digitized and the flame front extracted from the images by a thresholding technique. Digitization noise, which appears in the fractal plots at approximately twice the pixel resolution, can obscure the inner cutoff. Simple smoothing can remove this problem if the spatial resolution is sufficient. At insufficient resolution smoothing produces plausible resolutes are produced which in fact erroneous. If the inner cutoff is ambiguous the range over which the fractal dimension is determined will be unclear. The wide distribution of fractal dimensions obtained from the individual images indicates the necessity of ensemble averaging the fractal plots if reliable statistical results are to be obtained. 8 refs., 6 figs.

Shepherd, I.G.; Cheng, R.K.

1990-10-01

333

A cake collapse model was developed by taking the combined effects of fractal dimension, relaxation ratio, coordination number, and aggregate diameter into consideration. The cake porosity including intraaggregate and interaggregate porosities was modeled successively by three typical coordination numbers (n = 6, 8, and 12). Accordingly, an inversion method made it possible to deduce the coordination number using the measured cake porosities, and the reverse-calculated value with minimum error and the corresponding relaxation ratios were applied as the parameters for the model. As a result, the profiles of intraaggregate and interaggregate porosities and cake porosity were respectively predicted in contrast to the integrated variation of the relaxation ratio and the fractal dimension. Furthermore, a comparison between the model predictions of the cake pressure drop gradients with and without aggregate compression was conducted to validate the presence of cake collapse. The results show that the predictions based on the proposed collapse model are in agreement with the experiments, and the coordination number is one of the key factors that must be incorporated into the cake collapse models. PMID:21488606

Zhang, Wei; Li, Cai-Ting; Wei, Xian-Xun; Gao, Hong-Liang; Wen, Qing-Bo; Fan, Xiao-Peng; Shu, Xin; Zeng, Guang-Ming; Wei, Wei; Zhai, Yun-Bo; He, Yi-De; Li, Shan-Hong

2011-05-15

334

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

335

A method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the detrending moving average algorithm [A. Carbone, Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three-dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc. PMID:20365674

Türk, Christian; Carbone, Anna; Chiaia, Bernardino M

2010-02-01

336

Post-Critically Finite fractals are self-similar structures with the boundary ... form on a vector space F of real valued functions on X, and E(u) = 0 if and ..... zero seminorm elements and completing. ... Magnetic Schrödinger operators. Classically.

Daniel J. Kelleher

2014-08-31

337

NASA Astrophysics Data System (ADS)

The back-propagation neural (BPN) network was proposed to model the relationship between the parameters of the dieless drawing process and the microstructures of the QSi3-1 silicon bronze alloy. Combined with image processing techniques, grain sizes and grain-boundary morphologies were respectively determined by the quantitative metallographic method and the fractal theory. The outcomes obtained show that the deformed microstructures exhibit typical fractal features, and the boundaries can be characterized quantitatively by fractal dimensions. With the temperature of 600-800°C and the drawing speed of 0.67-1.00 mm·s-1, either a lower temperature or a higher speed will cause a smaller grain size together with an elevated fractal dimension. The developed model can be capable for forecasting the microstructure evolution with a minimum error. The average relative errors between the predicted results and the experimental values of grain size and fractal dimension are 3.9% and 0.9%, respectively.

Wang, Zhen; Liu, Xue-Feng; He, Yong; Xie, Jian-Xin

2010-12-01

338

Background Fractal geometry has been the basis for the development of a diagnosis of preneoplastic and neoplastic cells that clears up the undetermination of the atypical squamous cells of undetermined significance (ASCUS). Methods Pictures of 40 cervix cytology samples diagnosed with conventional parameters were taken. A blind study was developed in which the clinic diagnosis of 10 normal cells, 10 ASCUS, 10 L-SIL and 10 H-SIL was masked. Cellular nucleus and cytoplasm were evaluated in the generalized Box-Counting space, calculating the fractal dimension and number of spaces occupied by the frontier of each object. Further, number of pixels occupied by surface of each object was calculated. Later, the mathematical features of the measures were studied to establish differences or equalities useful for diagnostic application. Finally, the sensibility, specificity, negative likelihood ratio and diagnostic concordance with Kappa coefficient were calculated. Results Simultaneous measures of the nuclear surface and the subtraction between the boundaries of cytoplasm and nucleus, lead to differentiate normality, L-SIL and H-SIL. Normality shows values less than or equal to 735 in nucleus surface and values greater or equal to 161 in cytoplasm-nucleus subtraction. L-SIL cells exhibit a nucleus surface with values greater than or equal to 972 and a subtraction between nucleus-cytoplasm higher to 130. L-SIL cells show cytoplasm-nucleus values less than 120. The rank between 120–130 in cytoplasm-nucleus subtraction corresponds to evolution between L-SIL and H-SIL. Sensibility and specificity values were 100%, the negative likelihood ratio was zero and Kappa coefficient was equal to 1. Conclusions A new diagnostic methodology of clinic applicability was developed based on fractal and euclidean geometry, which is useful for evaluation of cervix cytology. PMID:24742118

2014-01-01

339

Fractal Analysis of AFM Images of the Surface of Bowman's Membrane of the Human Cornea.

The objective of this study is to further investigate the ultrastructural details of the surface of Bowman's membrane of the human cornea, using atomic force microscopy (AFM) images. One representative image acquired of Bowman's membrane of a human cornea was investigated. The three-dimensional (3-D) surface of the sample was imaged using AFM in contact mode, while the sample was completely submerged in optisol solution. Height and deflection images were acquired at multiple scan lengths using the MFP-3D AFM system software (Asylum Research, Santa Barbara, CA), based in IGOR Pro (WaveMetrics, Lake Oswego, OR). A novel approach, based on computational algorithms for fractal analysis of surfaces applied for AFM data, was utilized to analyze the surface structure. The surfaces revealed a fractal structure at the nanometer scale. The fractal dimension, D, provided quantitative values that characterize the scale properties of surface geometry. Detailed characterization of the surface topography was obtained using statistical parameters, in accordance with ISO 25178-2: 2012. Results obtained by fractal analysis confirm the relationship between the value of the fractal dimension and the statistical surface roughness parameters. The surface structure of Bowman's membrane of the human cornea is complex. The analyzed AFM images confirm a fractal nature of the surface, which is not taken into account by classical surface statistical parameters. Surface fractal dimension could be useful in ophthalmology to quantify corneal architectural changes associated with different disease states to further our understanding of disease evolution. PMID:25266935

??lu, ?tefan; Stach, Sebastian; Sueiras, Vivian; Ziebarth, Noël Marysa

2015-04-01

340

Technology Transfer Automated Retrieval System (TEKTRAN)

In order to explore the effect of changes in plant communities and land use on soil properties, as a result of anthropogenic disturbances, we apply the theory of fractals and soil physics as a means to better quantify changes in particle-size distribution (PSD) and soil porosity. Fractal dimension a...

341

Fractional diffusion on a fractal grid comb

NASA Astrophysics Data System (ADS)

A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N =1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1 /2 . We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.

Sandev, Trifce; Iomin, Alexander; Kantz, Holger

2015-03-01

342

Fractal analysis of Xylella fastidiosa biofilm formation

NASA Astrophysics Data System (ADS)

We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.

Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.

2009-07-01

343

Superexponential droplet fractalization as a hierarchical formation of dissipative compactons.

We study the dynamics of a thin film over a substrate heated from below in a framework of a strongly nonlinear one-dimensional Cahn-Hilliard equation. The evolution leads to a fractalization into smaller and smaller scales. We demonstrate that a primitive element in the appearing hierarchical structure is a dissipative compacton. Both direct simulations and the analysis of a self-similar solution show that the compactons appear at superexponentially decreasing scales, which means vanishing dimension of the fractal. PMID:20866766

Shklyaev, Sergey; Straube, Arthur V; Pikovsky, Arkady

2010-08-01

344

Fractal patterns of insect movement in microlandscape mosaics

How individuals move, whether in short-term searching behavior or long-term dispersal influences the probability that individuals will experience physiological stress or encounter appropriate habitat, potential mates, prey, or predators. Because of variety and complexity, it is often difficult to make sense of movements. Because the fractal dimension of a movement pathway is scale independent, however, it may provide a useful measure for comparing dissimilar taxa. The authors use fractal measures to compare the movement pathways of individual beetles occupying semiarid shortgrass steppe in north-central Colorado. 20 refs., 1 fig., 1 tab.

Wiens, J.A. [Colorado State Univ., Fort Collins, CO (United States); Crist, T.O. [Colorado State Univ., Fort Collins, CO (United States)]|[Miami Univ., Oxford, OH (United States); With, K.A. [Colorado State Univ., Fort Collins, CO (United States)]|[Oak Ridge National Lab., TN (United States); Milne, B.T. [Univ. of New Mexico, Albuquerque, NM (United States)

1995-03-01

345

Fractal Mining - Self Similarity-based Clustering and its Applications

NASA Astrophysics Data System (ADS)

Self-similarity is the property of being invariant with respect to the scale used to look at the data set. Self-similarity can be measured using the fractal dimension. Fractal dimension is an important charactaristics for many complex systems and can serve as a powerful representation technique. In this chapter, we present a new clustering algorithm, based on self-similarity properties of the data sets, and also its applications to other fields in Data Mining, such as projected clustering and trend analysis. Clustering is a widely used knowledge discovery technique. The new algorithm which we call Fractal Clustering (FC) places points incrementally in the cluster for which the change in the fractal dimension after adding the point is the least. This is a very natural way of clustering points, since points in the same clusterhave a great degree of self-similarity among them (and much less self-similarity with respect to points in other clusters). FC requires one scan of the data, is suspendable at will, providing the best answer possible at that point, and is incremental. We show via experiments that FC effectively deals with large data sets, high-dimensionality and noise and is capable of recognizing clusters of arbitrary shape.

Barbara, Daniel; Chen, Ping

346

Fractal nature of the sea ice draft profile

NASA Astrophysics Data System (ADS)

The fractal dimension is examined as a descriptor of ice roughness for more than 3000 km of under-ice draft submarine sonar data. The data can be considered to constitute a fractal set within a limited range of scales, as defined by the Hurst parameter H. It was found that 0.55 < H < 0.78 for scales of 3-15 m and 0.15 < H < 0.45 for scales of 15-75 m, beyond which H is near unity. From this it is seen quantitatively that sea ice on the large scale is smooth. The fractal dimension, D=2-H, at the smaller scales is similar to that measured by other investigators for individual ice features such as keels. The fractal dimension did not show any changing spatial pattern across ice regions, indicating that the scaling relationship is similar even when first-order measures such as the mean and variance of ice draft change. Therefore, D does not appear to be useful for partitioning the transect into homogeneous ice areas in the draft data examined.

Key, J.; McLaren, A. S.

347

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The fit obtained by a dual-fractal analysis is compared with that obtained from a single-fractal analysis. In some cases, the dual-fractal analysis provides an improved fit when compared with a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (46). It is of interest to note that the state of disorder (or the fractal dimension) and the binding rate coefficient both increase as the reaction progresses on the biosensor surface. For example, for the binding of HIV-1 p24 in solution to monoclonal antibody (MAb) 18 covalently attached to a biosensor surface (49), an increase in the fractal dimension by 59% from a value of Df1 equal to 1.91 to Df2 equal to 2.95 leads to an increase in the binding rate coefficient by a factor of 15 from k1 equal to 21.1 to k2 equal to 339. Also, the binding of MAb 6301 and 6303 in solution to insulin growth factor binding protein-1 (IGFBP-1) covalently attached to the sensor surface is adequately described by a single-fractal analysis (48). The binding of MAb 6302 to IGFBP-1, however, requires dual fractals. This indicates a difference in the binding mechanisms of these MAbs. The different examples analyzed and presented together provide a means by which the antigen-antibody reactions may be better controlled by noting the magnitude of the changes in the fractal dimension and in the binding rate coefficient as the reaction progresses on the biosensor surface. Also, the magnitude of the changes in the binding rate coefficients (k1 and k2) and in the fractal dimensions (Df1 and Df2) as different parameters are changed for the different biosensor applications are of particular value. PMID:9245322

Milum; Sadana

1997-03-01

348

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

349

The influence of Hausdorff dimension on plasmonic antennas with Pascal's triangle geometry

We introduce fractal geometry to the common bowtie antenna and investigate the influence of a key fractal parameter, Hausdorff dimension, on the broadband spectral response of the antenna. Length scaling trends are presented for antennas having various Hausdorff dimensions. We show that antennas with Pascal's triangle geometry accommodate resonances that are red-shifted when compared to a standard bowtie antenna having

S. Sederberg; A. Y. Elezzabi

2011-01-01

350

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

351

Effects of multiple scattering on radiative properties of soot fractal aggregates

NASA Astrophysics Data System (ADS)

The in situ optical characterization of smokes composed of soot particles relies on light extinction, angular static light scattering (SLS), or laser induced incandescence (LII). These measurements are usually interpreted by using the Rayleigh-Debye-Gans theory for Fractal Aggregates (RDG-FA). RDG-FA is simple to use but it completely neglects the impact of multiple scattering (MS) within soot aggregates. In this paper, based on a scaling approach that takes into account MS effects, an extended form of the RDG-FA theory is proposed in order to take into account these effects. The parameters of this extended theory and their dependency on the number of primary sphere inside the aggregate (1

Yon, Jérôme; Liu, Fengshan; Bescond, Alexandre; Caumont-Prim, Chloé; Rozé, Claude; Ouf, François-Xavier; Coppalle, Alexis

2014-01-01

352

Single- and dual-fractal analysis of hybridization binding kinetics: biosensor applications.

The diffusion-limited hybridization kinetics of analyte in solution to a receptor immobilized on a biosensor or immunosensor surface is analyzed within a fractal framework. The data may be analyzed by a single- or a dual-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes in the same direction for both the single-fractal and the dual-fractal analysis examples presented. For example, for a single-fractal analysis and for the hybridization of 10 nM 16CFl (oligonucleotide) to 16B immobilized via sulfosuccinimidyl-6-(biotinamido)hexanoate and streptavidin using chemical and thermal regeneration (Abel, A. P.; Weller, M. G.; Duveneck, G. L.; Ehrat, M. Widmer, H. M. Anal. Chem. 1996, 68, 2905-2912), an increase in the fractal dimension, Df from 1.211 (chemical regeneration) to 1.394 (thermal regeneration), leads to an increase in the binding rate coefficient, k, from 86.53 (chemical regeneration) to 100.0 (thermal regeneration). An increase in the degree of heterogeneity on the biosensor surface leads to an increase in the binding rate coefficient. When a dual-fractal analysis was utilized, an increase in the fractal dimension value from Df1 to Df2 leads to an increase in the binding rate coefficient value from k1 to k2. The fractional order of dependence of the binding rate coefficient, k1, on (a) the analyte (rRNA) concentration in solution and (b) on the fractal dimension, Df1, for the hybridization kinetics to detect Listeria species (Fliss, R.; St-Laurent, M.; Emond, E.; Simard, R. E.; Lemieux, R.; Ettriki, A.; Pandian, S. Appl. Microbiol. Biotechnol. 1995, 43, 717-724.) further reinforces the fractal nature of the system. The binding rate coefficient(s) expressions developed as a function of the analyte concentration in solution and the fractal dimension are of particular value since they provide a means to better control of biosensor or immunosensor performance. PMID:9758669

Sadana, A; Vo-Dinh, T

1998-01-01

353

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

2003-01-01

354

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

355

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

356

Fractal properties of propagating fronts in a strongly stirred fluid

NASA Astrophysics Data System (ADS)

The fractal properties of propagating aqueous autocatalytic chemical reaction fronts are measured in a capillary-wave (CW) flow at values of the ratio of the RMS intensity of the fluid velocity fluctuation (u') to the laminar propagation rate of the front (SL) up to 220. The images of the fronts are found to exhibit fractal behavior with a fractal dimension (d) of 1.31±0.06, which is very similar to some measurements in gaseous flame fronts, as well as isoscalar contours of passive dyes in CW and other randomly stirred flows. These results suggest that u'/SL, thermal expansion, variations of viscosity and diffusivity across the flame front, and the turbulence spectrum do not significantly affect d in randomly stirred flows.

Haslam, B. D.; Ronney, P. D.

1995-08-01

357

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

358

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

359

Investigating Fractal Geometry Using LOGO.

ERIC Educational Resources Information Center

Discusses dimensionality in Euclidean geometry. Presents methods to produce fractals using LOGO. Uses the idea of self-similarity. Included are program listings and suggested extension activities. (MVL)

Thomas, David A.

1989-01-01

360

NASA Astrophysics Data System (ADS)

Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, "Be still if I fill its ebb" minus its last letter is a palindrome, and the entire poem is a palindrome...

Martin, Demetri

2015-03-01

361

Exploring the fractal terrain.

January/February CG&A artist Paul Griffitts does all his work with a free fractal rendering program called Mandelbulb 3D (MB3D). Especially with Labyrinth, the cover image, he implements various parameters for specific effects. Griffitts says he likes the effects that can be created through various coloring options and the application of different maps-height maps, color maps, light maps. PMID:25666595

Singh, Gary

2015-01-01

362

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The data is adequately described by a single- or a dual-fractal analysis. Initially, the data was modelled by a single-fractal analysis. If an inadequate fit was obtained then a dual-fractal analysis was utilized. The regression analysis provided by Sigmaplot, 1993 (Scientific Graphing Software: User's Manual. Jandel Scientific, San Rafael, CA) was utilized to determine if a single-fractal analysis is sufficient, or a dual-fractal analysis is required. In general, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction (except for a single example) for the antigen-antibody systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the antigen-antibody binding systems indicate a high sensitivity of the binding rate coefficient on the fractal dimension when both a single -as well as a dual-fractal analysis is used. For example, for a single-fractal analysis and for the binding of human endothelin-1 (ET-1) antibody in solution to ET-1(15-21) x BSA (bovine serum albumin) immobilised on a surface plasmon resonance surface, the order of dependence of the binding rate coefficient, k on the fractal dimension, Df is 7.0945. Similarly, for a dual-fractal analysis and for the binding of parasite L. donovani diluted pooled sera in solution to fluorescein isothiocyanate-labeled anti-human immunoglobulin IgG immobilized on an optical fibre, the order of dependence of k1 and k2 on Df1 and Df2 were 6.8018 and -4.393, respectively. Binding rate coefficient expressions are also developed as a function of the analyte (antigen or antibody) concentration in solution. The binding rate coefficient expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface, and emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance. PMID:11459097

Sadana, A

1999-06-30

363

The diffusion-limited binding kinetics of analyte in solution to either a receptor immobilized on a surface or to a receptorless surface is analyzed within a fractal framework for a surface plasmon resonance biosensor. The data is adequately described by a single- or a dual-fractal analysis. Initially, the data was modeled by a single-fractal analysis. If an inadequate fit was obtained then a dual-fractal analysis was utilized. The regression analysis provided by Sigmaplot (32) was used to determine if a single fractal analysis is sufficient or if a dual-fractal analysis is required. In general, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction (except for a single example) for the analyte-receptor systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the analyte-receptor binding systems indicate, in general, the high sensitivity of the binding rate coefficient on the fractal dimension when both a single- and a dual-fractal analysis is used. For example, for a single-fractal analysis and for the binding of human endothelin-1 (ET-1) antibody in solution to ET-115-21.BSA immobilized on a surface plasmon resonance (SPR) surface (33), the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, is 6.4405. Similarly, for a dual-fractal analysis and for the binding of 10(-6) to 10(-4) M bSA in solution to a receptorless surface (direct binding to SPR surface) (41) the order of dependence of k1 and k2 on Df1 and Df2 were -2.356 and 6.241, respectively. Binding rate coefficient expressions are also developed as a function of the analyte concentration in solution. The binding rate coefficient expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control SPR biosensor performance by linking it to the degree of heterogeneity that exists on the SPR biosensor surface. Copyright 1999 Academic Press. PMID:10222088

Ramakrishnan; Sadana

1999-05-15

364

In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at 60Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at 60Co energies within a accuracy of a few percent or less. PMID:19175120

La Russa, Daniel J; Rogers, D W O

2008-12-01

365

In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at {sup 60}Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at {sup 60}Co energies within a accuracy of a few percent or less.

La Russa, Daniel J.; Rogers, D. W. O. [Carleton Laboratory for Radiotherapy Physics, Ottawa Carleton Institute of Physics, Carleton University Campus, Ottawa, Ontario K1S 5B6 (Canada)

2008-12-15

366

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

367

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

368

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

369

Frequency-dependent viscous flow in channels with fractal rough surfaces

The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to {infinity} (but has a finite cutoff), and the universal scaling of the dynamic permeability, k({omega}), needs modification. We performed accurate numerical computations of k({omega}) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k({omega}) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.

Cortis, A.; Berryman, J.G.

2010-05-01

370

Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis

The purpose of this project is to apply a modified fractal analysis technique to high-resolution T1 weighted magnetic resonance images in order to quantify the alterations in the shape of the cerebral cortex that occur in patients with Alzheimer’s disease. Images were selected from the Alzheimer’s Disease Neuroimaging Initiative database (Control N=15, Mild-Moderate AD N=15). The images were segmented using a semi-automated analysis program. Four coronal and three axial profiles of the cerebral cortical ribbon were created. The fractal dimensions (Df) of the cortical ribbons were then computed using a box-counting algorithm. The mean Df of the cortical ribbons from AD patients were lower than age-matched controls on six of seven profiles. The fractal measure has regional variability which reflects local differences in brain structure. Fractal dimension is complementary to volumetric measures and may assist in identifying disease state or disease progression. PMID:20740072

George, Anuh T.; Jeon, Tina; Hynan, Linda S.; Youn, Teddy S.; Kennedy, David N.; Dickerson, Bradford

2010-01-01

371

Fractal analysis: methodologies for biomedical researchers.

Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine. Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology of fractal analysis have partly restricted its wider and comprehensible application. It is a notable fact that fractal analysis is derived from fractal geometry, but there are some unresolved issues that need to be addressed. In this respect, we discuss several related underlying principles for fractal analysis and establish the meaningful relationship between fractal analysis and fractal geometry. Since some concepts in fractal analysis are determined descriptively and/or qualitatively, this paper provides their exact mathematical definitions or explanations. Another aim of this study is to show that nowadays fractal analysis is an independent mathematical and experimental method based on Mandelbrot's fractal geometry, Euclidean traditiontal geometry and Richardson's coastline method. PMID:23757956

Ristanovi?, Dusan; Milosevi?, Nebojsa T

2012-01-01

372

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-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.

Tarasov, Vasily E.

2014-08-01

373

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

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.

Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru [Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991 (Russian Federation)

2014-08-15

374

Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space

A review of different approaches to describe anisotropic fractal media is proposed. In this paper differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.

Vasily E. Tarasov

2015-03-09

375

Fractal dynamics of heartbeat time series of young persons with metabolic syndrome

NASA Astrophysics Data System (ADS)

Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.

Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.

2012-10-01

376

Fractal mechanism for characterizing singularity of mode shape for damage detection

Damage is an ordinary physical phenomenon jeopardizing structural safety; damage detection is an ongoing interdisciplinary issue. Waveform fractal theory has provided a promising resource for detecting damage in plates while presenting a concomitant problem: susceptibility to false features of damage. This study proposes a fractal dimension method based on affine transformation to address this problem. Physical experiments using laser measurement demonstrate that this method can substantially eliminate false features of damage and accurately identify complex cracks in plates, providing a fundamental mechanism that brings the merits of waveform fractal theory into full play in structural damage detection applications.

Cao, M. S. [Department of Engineering Mechanics, Hohai University, Nanjing 210098 (China)] [Department of Engineering Mechanics, Hohai University, Nanjing 210098 (China); Ostachowicz, W. [Institute of Fluid-Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk (Poland) [Institute of Fluid-Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk (Poland); Faculty of Automotive and Construction Machinery, Warsaw University of Technology, Narbutta 84, 02-524 Warsaw (Poland); Bai, R. B., E-mail: bairunbo@gmail.com [Department of Engineering Mechanics, Shandong Agricultural University, Taian 271000 (China); Radzie?ski, M. [Institute of Fluid-Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk (Poland)] [Institute of Fluid-Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk (Poland)

2013-11-25

377

Applications of Fractal Analytical Techniques in the Estimation of Operational Scale

NASA Technical Reports Server (NTRS)

The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.

Emerson, Charles W.; Quattrochi, Dale A.

2000-01-01

378

Some features of the fractal structure of the developed small-scale ionospheric turbulence

We present the results of the first studies of the fractal structure of the developed small-scale ionospheric turbulence (SSIT)\\u000a during special experiments on radio-raying of the midlatitude ionosphere by signals from orbital satellites in 2005–2006.\\u000a It is established that under conditions of developed turbulence, typical values of the fractal dimension of the space occupied\\u000a by natural SSIT inhomogeneities are, as

V. A. Alimov; F. I. Vybornov; A. V. Rakhlin

2008-01-01

379

Applicability of Fractals to the Analysis of the Projection of Small Flocs

The validity of the concept of fractal structure for the analysis of the projection of small flocs was examined using the data obtained from a table-tennis-ball simulation and a coagulation experiment with polystyrene latex particles. Two methods defining a fractal dimension, the box-counting technique and the enumeration of primary particles in an enclosing circle as a function of the radius,

Y. Adachi; M. Kobayashi; S. Ooi

1998-01-01

380

NASA Astrophysics Data System (ADS)

Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.

Brothers, Harlan J.

2015-03-01

381

Fractal Analysis of Trabecular Bone

NSDL National Science Digital Library

Fractals are unusual geometric structures that can be used to analyze many biologic structures not amenable to conventional analysis. The purpose of this exhibit is to teach some of the fundamentals of fractal analysis, and to show how they can be applied to analysis of trabecular bone.

Gillespy, Thurman

382

Fractal growth of liquid crystals as a hysteresis phenomenon

NASA Astrophysics Data System (ADS)

Fractal percolation growth of liquid crystal phases within a supercooled isotropic liquid medium has been observed in recent years. Notable examples include the B2 phase of `banana' mesogens [1] and the smectic C phase of a calamitic hydrogen-bonding liquid crystal [2]. Here we present a dynamical model that describes such fractal growth as well as the spherical growth conventionally observed for nematics and cholesterics. The essential idea is that the supercooled medium does not fully respond to the temperature quench immediately (hysteresis). Its fraction of space available for the phase transition only relaxes from 0 to 1 at some finite rate. Depending on the coupling between the relaxation and growth rates, the liquid crystal phase either grows as a percolation cluster of fractal dimension D 1.89 or approaches a spherical shape of Euclidean dimension D -> 2. The crossover behaviour from relatively slow to fast relaxation is thoroughly investigated. Possible causes of the hysteresis for fractal growth will be discussed. [1] I. Dierking, Liq. Cryst. Today 12(1), (2003), 1 [2] I. Dierking, Chan H. K., Culfaz F., McQuire S., Phys. Rev. E 70, (2004), 051701

Chan, Ho-Kei; Dierking, Ingo

2006-03-01

383

Drip paintings and fractal analysis.

It has been claimed that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has led us to a result in fractal analysis of more general scientific significance: we show that the statistics of the "covering staircase" (closely related to the box-counting staircase) provide a way to characterize geometry and distinguish fractals from Euclidean objects. Finally we present a discussion of the composite of two fractals, a problem that was first investigated by Muzy. We show that the composite is not generally scale invariant and that it exhibits complex multifractal scaling in the small distance asymptotic limit. PMID:19518305

Jones-Smith, Katherine; Mathur, Harsh; Krauss, Lawrence M

2009-04-01

384

An introduction to the Apollonian fractal

Abstract This paper provides an introduction to the Apollonian fractal, also known by some as the curvilinear Sierpinski gasket. This fractal is not particularly well known, perhaps because it is not as straightforward to construct as many other fractals such as the related Sierpinski gasket or the Menger sponge. The brief history and a general description of the fractal will

Paul D. Bourke

2006-01-01

385

Fractal properties of medium and seismoelectric phenomena

NASA Astrophysics Data System (ADS)

Electrokinetic phenomena in a water-porous medium with a fractal structure above percolation threshold are theoretically investigated. Fracture zone with space-variable porosity is considered as a model of an earthquake hypocenter zone in which the electrokinetic current results from fluid filtration in a fractal pore network. A critical exponent of the streaming potential coefficient is found to depend on both the transport critical exponent and correlation length critical exponent. In this model, logarithmic dependence of electric field amplitude E on the earthquake magnitude M is derived which is compatible with the one observed by the VAN group. Without fractal properties, this form of dependence contradicts the empirical data. The electromagnetic field far from the hypocenter is calculated, which leads to the prediction of weak magnetic field variations. To explain the observed amplitude of VAN's Seismic Electric Signals (SES), the electric source must be at a distance of about 10 km from the registration point if the medium is homogeneous. Therefore, some conductive channel(s) are needed to explain the long distance selective SES transmission.

Surkov, V. V.; Uyeda, S.; Tanaka, H.; Hayakawa, M.

2002-07-01

386

Scorpion venom complexity fractal analysis. Its relevance for comparing venoms.

We analyzed the venom elution pattern of 15 scorpions species. Data were scanned at 1 Hz and stored digitally. Approximate fractal dimension (D) [Sevcik (1998)] was calculated for minutes 0-60 of the elutions. D was calculated for either the whole time range, or calculated using a window of 500 points, which was displaced by one time increment recursively, and stored [(t(i),D(i)) sets]. We avoid the term complexity as much as possible since defining complexity is difficult; instead we propose the term contortedness and represent it by the variable Q=D-1. To compare venom contortednesses of different species, a phase plot with their (t(i),Q(i)) sets was constructed and determination coefficient (d(s)) were calculated squaring the Spearman rank correlation coefficient. (t(i),Q(i)) sets of several elutions of the same species were averaged and compared with other species finding that some were amazingly similar (Tityus clathratus vs Tityus caripitensis, d(s) = 0.813). Tityus discrepans was similar to 6 of 8 species of the same genus (d(s) ranging from 0.23 to 0.49), and also similar to Centruroides gracilis and Chactas laevipes (d(s) 0.54 and 0.49, respectively). Serendipitously,T. discrepans was chosen many years ago to produce anti-Tityus antivenom in Venezuela; perhaps the clinical success in neutralizing the venom of the other known Venezuelan Tityus, stems from the mimetism of this venom with the remaining species' venom. PMID:20833185

D'Suze, Gina; Sevcik, Carlos

2010-12-01

387

Purpose This study was performed to evaluate possible variations in maxillary and mandibular bone texture of normal population using the fractal analysis, particles count, and area fraction in intraoral radiographs. Materials and Methods Periapical radiographs of patients who had full mouth intraoral radiographs were collected. Regions of interest (100×100 pixels) were located between the teeth of the maxillary anterior, premolar, and molar area, as well as the mandibular anterior, premolar, and molar areas. The fractal dimension (FD) was calculated by using the box counting method. The particle count (PC) and area fraction (AF) analyses were also performed. Results There was no significant difference in the FD values among the different groups of age, gender, upper, and lower jaws. The mean FD value was 1.49±0.01. The mean PC ranged from 44 to 54, and the mean AF ranged from 10.92 to 11.85. The values of FD, PC, and AF were significantly correlated with each other except for the upper molar area. Conclusion According to the results, patients with normal trabecular pattern showed a FD of approximately 1.5. Based on these results, further investigation would be recommended if the FD value of patient significantly differenct from this number, since the alteration of this value indicates microstructural modification of trabecular pattern of the jaws. Additionally, with periapical radiographs, simple and cost-effective, PC and AF could be used to assess the deviation from the normal. PMID:22474642

Amer, Maha Eshak; Brooks, Sharon L; Benavides, Erika

2012-01-01

388

Fractal analysis for reduced reference image quality assessment.

In this paper, multifractal analysis is adapted to reduced-reference image quality assessment (RR-IQA). A novel RR-QA approach is proposed, which measures the difference of spatial arrangement between the reference image and the distorted image in terms of spatial regularity measured by fractal dimension. An image is first expressed in Log-Gabor domain. Then, fractal dimensions are computed on each Log-Gabor subband and concatenated as a feature vector. Finally, the extracted features are pooled as the quality score of the distorted image using l1 distance. Compared with existing approaches, the proposed method measures image quality from the perspective of the spatial distribution of image patterns. The proposed method was evaluated on seven public benchmark data sets. Experimental results have demonstrated the excellent performance of the proposed method in comparison with state-of-the-art approaches. PMID:25794391

Xu, Yong; Liu, Delei; Quan, Yuhui; Le Callet, Patrick

2015-07-01

389

NASA Astrophysics Data System (ADS)

Nature diamond cutters are important tools to manufacture high precision optics glasses, and it is a normal method to make such cutter that soldering diamond grain with titanium coating on tool base. However, surface characteristics of titanium coating determine how much force that diamond grain joined with tool base. This paper introduces the research of surface characteristics of titanium coating on diamond grain based on AFM which uses its contacting mode to get measuring data of surface topography. Firstly, the measuring data are analyzed using 2D power spectrum algorithm to obtain spectrum energy distribution about spatial frequency. Fractal dimension is then calculated from the radius spectrum, and surface characteristics of titanium coating are evaluated using stationary wavelet transform where feature separation thresholds takes as an important role based on the fractal dimension. Coating experiments show that such method can reveal quality information of titanium coating on diamond grain comprehensively and thoroughly, thus it can offer good experimental reference to optimizing titanium coating parameters.

Du, Wenhao; Yang, Wenmao; Sun, Tao; Wang, Baorui

2010-10-01

390

On fractal properties of small-scale ionospheric irregularities

We consider the problem of relating the local structure of small-scale ionospheric turbulence to the measured frequency-spectrum\\u000a indices and fractal dimensions of amplitude records of the signals received on the Earth during remote sensing of the ionosphere\\u000a onboard the satellites. It is shown that knowledge of these parameters permits one to determine the true values of the local-spectrum\\u000a indices of

V. A. Alimov; F. I. Vybornov; A. V. Rakhlin

2007-01-01

391

NANOFLARE STATISTICS FROM FIRST PRINCIPLES: FRACTAL GEOMETRY AND TEMPERATURE SYNTHESIS

parameters, the power index a of the length distribution, NÃ°lÃ? / lÃ?a, and the fractal Haussdorff dimension D between length scales l and flare areas, AÃ°lÃ? / lD. For values that are consistent with the data, iEÃ? / EÃ? with a power-law coefficient of Â¼ 1:54 Ã? 0:11. As an observational test, we perform statistics

Parnell, Clare E.

392

Fractal structures derivable from the generalisations of the Pascal triangle

Generalisations, of order K>or=2, of the Pascal triangle are used to construct generalised Pascal-Sierpinski gaskets of orders (K, L>or=2). It is shown that all such gaskets are self-affine fractals, but when K=2 and L is prime then the gasket is rigorously self-similar and possesses a similarity dimension. The evolutionary morphology of the gaskets of orders (K, L prime) bears a

A. Lakhtakia; R. Messier; V. K. Varadan; V. V. Varadan

1987-01-01

393

Thermal collapse of snowflake fractals

NASA Astrophysics Data System (ADS)

Snowflakes are thermodynamically unstable structures that would ultimately become ice balls. To investigate their dynamics, we mapped atomistic molecular dynamics simulations of small ice crystals - built as filled von Koch fractals - onto a discrete-time random walk model. Then the walkers explored the thermal evolution of high fractal generations. The in silico experiments showed that the evolution is not entirely random. The flakes step down one fractal generation before forfeiting their architecture. The effect may be used to trace the thermal history of snow.

Gallo, T.; Jurjiu, A.; Biscarini, F.; Volta, A.; Zerbetto, F.

2012-08-01

394

Anomalous diffusion in fractal globules

Fractal globule state is widely believed to be the best known model to describe the chromatin packing in the eucaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics (DPD) computer simulation for the thermal motion of monomers in the fractal globule state. We show this motion to be subdiffusive described by $\\langle X^2 (t)\\rangle \\sim t^{\\alpha_F}$ with $\\alpha_F$ close to 0.4. We also suggest a novel way to construct a fractal globule state in computer simulation, and provide simulation evidence supporting the conjecture that different initial entanglement-free states of a polymer chain converge as they are thermally annealed.

Mikhail V. Tamm; Leonid I. Nazarov; Alexey A. Gavrilov; Alexander V. Chertovich

2014-04-09

395

The Fibonacci fractal is a new fractal type

NASA Astrophysics Data System (ADS)

We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator.

Yudin, V. V.; Startzev, E. S.

2012-10-01

396

Self-organization and fractal dynamics in turbulence

NASA Astrophysics Data System (ADS)

Results of analysis of the field of helicity, obtained in three different turbulent laboratory flows (grid-flow, boundary layer and jet) and a simple helical fracton model has been used in order to provide a quantitative explanation of anomalous turbulent diffusion in the troposphere and in the ocean. It is shown that Kolmogorov turbulence is critical in respect to the localization effects of subregions with large helicity (helical fractons) and it breaks up into helical fractons under the condition Df?2, where Df=2 d/ dw is the so called fracton dimension ( D is the fractal dimension of the turbulent fractal and Dw is the dimension of random walks on this fractal). For strictly Kolmogorov turbulence D1=2. We study the internal structure of helical fractons and demonstrate that they are characterized by D f= {4}/{3}. Finally, we look at the influence of helical fractons on diffusion of a passive scalar in turbulence. It is shown that their influence is manifested in the scaling law for the turbulent diffusivity in the form K? l{8}/{7} in both three-dimensional and quasi-two-dimensional situations. This (anomalous) law is in a very good agreement with a large number of experimental data of different authors in the troposphere and in the upper ocean.

Bershadskii, A.; Kit, E.; Tsinober, A.

1993-11-01

397

Thermoreversibly aggregated microgels: fractal structure and aggregation mechanism

NASA Astrophysics Data System (ADS)

Multifunctional molecules were designed to produce thermoreversible microgels with specific structures. Both static and dynamic light scattering were employed to determine the fractal dimension (Df) of the microgels. Avidin binds four biotin moieties. Biotin was attached covalently to engineered peptide nucleic acid (PNA) oligomers. Three designed DNA oligomers self-assembled to produce a three-way junction(TWJ) with single stranded ends that were complementary to the PNA sequence. The DNA-PNA helices thermoreversibly melt at specific temperatures in the range 20-50 oC. The fractal dimension was obtained from the angular dependence of the scattered intensity. When the microgels were formed by cooling from a temperature above the melting point of the PNA-DNA helices, reversible structures with Df 1.85 were formed, which is consistent with a cluster-cluster aggregation mechanism. When the microgels were formed by slowly adding avidin to a solution of biotinylated-PNA bound to the TWJ at room temperature, the observed fractal dimension was around 2.60, which is consistent with a point-cluster aggregation mechanism.

Gu, Zhenyu; Cao, Rong; Armitage, Bruce; Patterson, Gary

2003-03-01

398

NASA Astrophysics Data System (ADS)

The aim of this study was to investigate the connection between the lipid/amphiphile monolayer structure at the interface and its macroscopic/rheological properties, in particular, to establish the link between the fractality of the monolayer structure and its compressibility modulus. To that purpose we have used fractal analysis of images obtained by Brewster angle microscopy to infer the fractal dimension of the monolayer structure and relate its change to the corresponding changes in compressibility derived from a simultaneously measured ?-A isotherm. The results of the study confirmed the starting assumption based on theoretical considerations that the fractal dimension of an amphiphilic monolayer and its compressibility should be correlated. We have shown that there exists a strong correlation between the fractal dimension and the corresponding compressibility modulus of different amphiphilic materials. Thus, confirming the link between the short ordered structure on the molecular level and the macroscopic property—compressibility of the monolayer. The established correlation between the fractal dynamics and compressibility modulus of the monolayer enabled identification of onset of percolation—a second-order phase transition that is otherwise not easy and unambiguously detectable. We have found that the signature of percolation in a monolayer, regardless of its composition, is the occurrence of a sharp increase (a jump) of compressibility modulus (at macroscopic level) at the characteristic value of the corresponding fractal dimension D = 1.89. This is the result of the abrupt establishment of a connected structure on the molecular level, consequently involving a change in the elastic properties of the monolayer on a macroscopic scale. The results of this investigation provide means for unambiguous identification of the onset of percolation in the Langmuir layer and should facilitate a more efficient application of the percolation theory in further study of processes and structures at the interface during the monolayer compression.

Risovi?, Dubravko; Frka, Sanja; Kozarac, Zlatica

2011-01-01

399

Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder

NASA Astrophysics Data System (ADS)

Recently, the visibility graph (VG) algorithm was proposed for mapping a time series to a graph to study complexity and fractality of the time series through investigation of the complexity of its graph. The visibility graph algorithm converts a fractal time series to a scale-free graph. VG has been used for the investigation of fractality in the dynamic behavior of both artificial and natural complex systems. However, robustness and performance of the power of scale-freeness of VG (PSVG) as an effective method for measuring fractality has not been investigated. Since noise is unavoidable in real life time series, the robustness of a fractality measure is of paramount importance. To improve the accuracy and robustness of PSVG to noise for measurement of fractality of time series in biological time-series, an improved PSVG is presented in this paper. The proposed method is evaluated using two examples: a synthetic benchmark time series and a complicated real life Electroencephalograms (EEG)-based diagnostic problem, that is distinguishing autistic children from non-autistic children. It is shown that the proposed improved PSVG is less sensitive to noise and therefore more robust compared with PSVG. Further, it is shown that using improved PSVG in the wavelet-chaos neural network model of Adeli and c-workers in place of the Katz fractality dimension results in a more accurate diagnosis of autism, a complicated neurological and psychiatric disorder.

Ahmadlou, Mehran; Adeli, Hojjat; Adeli, Amir

2012-10-01

400

Control of multiscale systems with constraints. 2. Fractal nuclear isomers and clusters

We consider the influence of the Fermi statistics of nucleons on the binding energy of a new type of nuclear structures such as fractal nuclear clusters (fractal isomers of nuclei). It is shown that the fractal nuclear isomers possess a wide spectrum of binding energies that exceed, in many cases, the values known at the present time. The transition of the nuclear matter in the form of ordinary nuclei (drops of the nuclear fluid) in the state with the fractal structure or in the form of bubble nuclei opens new sources of energy and has huge perspectives. This transition is based on a new state of matter - collective coherently correlated state. It manifests itself, first of all, in the property of nonlocality of nuclear multiparticle processes. We develop a phenomenological theory of the binding energy of nuclear fractal structures and modify the Bethe - Weizs\\"acker formula for nuclear clusters with the mass number A, charge Z, and fractal dimension D_f. The consideration of fractal nuclear isomers allows on...

Adamenko, S; Novikov, V

2013-01-01

401

Fractal Analysis of The ULF Geomagnetic Data In Relation To The Nearby Earthquakes

NASA Astrophysics Data System (ADS)

In our recent papers we applied fractal methods to extract the earthquake precursory signatures from scaling characteristics of the ULF geomagnetic data, obtained in seis- mic active region of Guam Island during the large earthquake of August 8, 1993. We found specific dynamics of their fractal characteristics (spectral exponents and frac- tal dimensions) before the earthquake: appearance of the flicker- noise signatures and increase of the time series fractal dimension. We confirmed the revealed effects by three methods of analysis: spectral method - FFT procedure, Burlaga-Klein approach and Higuchi method. Here we expand our consideration over ULF geomagnetic data obtained in a seismic active region of Izu Peninsula, Japan during the field experi- ment of 2000. At that period, a swarm of the large nearby earthquakes with magnitude M>6.0 took place in June, July and August. We apply the same methodic of fractal analysis. We found the common features and specific peculiarities in the behavior of fractal characteristics of the ULF time series before the Izu and Guam earthquakes. The results obtained in both regions are discussed. We reason also upon the advantage of the multifractal methods with respect to the mono- fractal analysis for study of the eathquake precursory signatures. The work was supported by Russian Programme "Intergeophysica" and by Grant IN- TAS 99-1102.

Smirnova, N.; Hayakawa, M.; Gotoh, K.

402

Field Fractal Cosmological Model As an Example of Practical Cosmology Approach

The idea of the global gravitational effect as the source of cosmological redshift was considered by de Sitter (1916, 1917), Eddington (1923), Tolman (1929) and Bondi (1947). Also Hubble (1929) called the discovered distance-redshift relation as "De Sitter effect". For homogeneous matter distribution cosmological gravitational redshift is proportional to square of distance: z_grav ~ r^2. However for a fractal matter distribution having the fractal dimension D=2 the global gravitational redshift is the linear function of distance: z_grav ~ r, which gives possibility for interpretation of the Hubble law without the space expansion. Here the field gravity fractal cosmological model (FGF) is presented, which based on two initial principles. The first assumption is that the Feynman's field gravity approach describes the gravitational interaction, which delivers a natural basis for the conceptual unity of all fundamental physical interactions within the framework of the relativistic and quantum fields in Minkowski space. The second hypothesis is that the spatial distribution of gravitating matter is a fractal at all scales up to the Hubble radius. The fractal dimension of matter distribution is assumed to be D = 2, which implies that the global gravitational redshift is the explanation of the observed linear Hubble law. In the frame of the FGF all three phenomena - the cosmic background radiation, the fractal large scale structure, and the Hubble law, - could be the consequence of a unique large scale structure evolution process of the initially homogeneous ordinary matter without nonbaryonic matter and dark energy.

Yu. V. Baryshev

2015-03-11

403

The Twinkling Fractal Theory of the Glass Transition: Applications to Soft Matter

NASA Astrophysics Data System (ADS)

The Twinkling Fractal Theory (TFT) of the glass transition has recently been demonstrated experimentally [J.F. Stanzione et al., J. Non Cryst. Sol., (2011, 357,311]. The hard to-soft matter transition is characterized by the presence of solid fractal clusters with liquid-like pools that are dynamically interchanging via their anharmonic intermolecular potentials with Boltzmann energy populations with a characteristic temperature dependent vibrational density of states g(?) ˜ ?^df . The twinkling fractal frequencies ? cover a range of 10^12 Hz to 10-10Hz and the fractal solid clusters of size R have a lifetime ? ˜ R^Df/df, where the fractal dimension Df 2.4 and the fracton dimension df = 4/3. Here we explore its application to a number of soft matter issues. These include (a) Confinement effects on Tg reduction in thin films of thickness h, where by virtue of large cluster exclusion, ?Tg ˜ 1/h^Df/df; (b) Tg gradients near bulk surfaces, where the smaller clusters on the surface have a faster relaxation time; (c) Effect of twinkling surfaces on cell growth, where at T Tg + 20 C, there exists a twinkling fractal range that leads to bell-shaped enhancement of cell growth and chemical up-regulation via the twinkling surfaces ``communicating `` with the cells through their vibrations; and (d) adhesion above and below Tg where topological fluctuations associated with g(?) promotes the development of nano-nails at the interface.

Wool, Richard

2012-02-01

404

Experimental investigation of the contact mechanics of rough fractal surfaces.

The nonstationary character of roughness is a widely recognized property of surface morphology and suggests modeling several solid surfaces by fractal geometry. In the field of contact mechanics, this demands novel investigations attempting to clarify the role of multiscale roughness during physical contact. Here we review the results we recently obtained in the characterization of the contact mechanics of fractal surfaces by depth-sensing indentation. One class of experiments was conducted on organic thin films, load-displacement curves being acquired by atomic force microscopy using custom-designed tips. Another class of experiments focused on well-defined crystalline and mechanically polished ceramic substrates probed by a traditional nanoindenter. We observed the first-loading cycle to be considerably affected by surface roughness. Plastic failure was found to dominate incipient contact while contact stiffness increased on decreasing fractal dimension and roughness. Our findings suggest fractal parameters to drive contact mechanics whenever the penetration depth is kept below the interface width. PMID:15382640

Buzio, R; Malyska, K; Rymuza, Z; Boragno, C; Biscarini, F; De Mongeot, F Buatier; Valbusa, U

2004-03-01

405

The aim of the study was to evaluate the effect of two lossy image compression methods on fractal dimension (FD) calculation. Ten periapical images of the posterior teeth with no restorations or previous root canal therapy were obtained using storage phosphor plates and were saved in TIF format. Then, all images were compressed with lossy JPEG and JPEG2000 compression methods at five compression levels, i.e., 90, 70, 50, 30, and 10. Compressed file sizes from all images and compression ratios were calculated. On each image, two regions of interest (ROIs) containing healthy trabecular bone in the posterior periapical area were selected. The FD of each ROI on the original and compressed images was calculated using differential box counting method. Both image compression and analysis were performed by a public domain software. Altogether, the FD of 220 ROIs was calculated. FDs were compared using ANOVA and Dunnett tests. The FD decreased gradually with compression level. A statistically significant decrease of the FD values was found for JPEG 10, JPEG2000 10, and JPEG2000 30 compression levels (p?calculation. PMID:21465294

Baksi, B Güniz; Fidler, Aleš

2011-12-01

406

NASA Astrophysics Data System (ADS)

Reduced dimension variational calculations have been performed for the rovibrational level structure of the S1 state of acetylene. The state exhibits an unusually complicated level structure, for various reasons. First, the potential energy surface has two accessible conformers, trans and cis. The cis conformer lies about 2700 cm-1 above the trans, and the barrier to cis-trans isomerization lies about 5000 cm-1 above the trans minimum. The trans vibrations ?4 (torsion) and ?6 (asym. bend) interact very strongly by Darling-Dennison and Coriolis resonances, such that their combination levels and overtones form polyads with unexpected structures. Both conformers exhibit very large x36 cross-anharmonicity since the pathway to isomerization is a combination of ?6 and ?3 (sym. bend). Near the isomerization barrier, the vibrational levels show an even-odd K-staggering of their rotational levels as a result of quantum mechanical tunneling through the barrier. The present calculations address all of these complications, and reproduce the observed K-structures of the bending and C-C stretching levels with good qualitative accuracy. It is expected that they will assist with the assignment of the irregular patterns near the isomerization barrier.

Changala, P. Bryan; Baraban, Joshua H.; Stanton, John F.; Merer, Anthony J.; Field, Robert W.

2014-01-01

407

Cosmological distances and fractal statistics of galaxy distribution

NASA Astrophysics Data System (ADS)

This paper studies the effect of the distance choice in radial (non-average) statistical tools} used for fractal characterization of galaxy distribution. After reviewing the basics of measuring distances of cosmological sources, various distance definitions are used to calculate the differential density ? and the integral differential density ?* of the dust distribution in the Einstein-de Sitter cosmology. The main results are as follows: (1) the choice of distance plays a crucial role in determining the scale where relativistic corrections must be taken into account, as both ? and ?* are strongly affected by such a choice; (2) inappropriate distance choices may lead to failure to find evidence of a galaxy fractal structure when one calculates those quantities, even if such a structure does occur in the galaxy distribution; (3) the comoving distance and the distance given by Mattig's formula are unsuitable to probe for a possible fractal pattern as they render ? and ?* constant for all redshifts; (4) a possible galaxy fractal system at scales larger than 100 Mpc (z ? 0.03) may only be found if those statistics are calculated with the luminosity or redshift distances, as they are the ones where ? and ?* decrease at higher redshifts; (5) Célérier & Thieberger's (\\cite{ct01}) critique of Ribeiro's (\\cite{r95}) earlier study are rendered impaired as their objections were based on misconceptions regarding relativistic distance definitions.

Ribeiro, M. B.

2005-01-01

408

The Complexity of Sequences Generated by the Arc-Fractal System

We study properties of the symbolic sequences extracted from the fractals generated by the arc-fractal system introduced earlier by Huynh and Chew. The sequences consist of only a few symbols yet possess several nontrivial properties. First using an operator approach, we show that the sequences are not periodic, even though they are constructed from very simple rules. Second by employing the ?-machine approach developed by Crutchfield and Young, we measure the complexity and randomness of the sequences and show that they are indeed complex, i.e. neither periodic nor random, with the value of complexity measure being significant as compared to the known example of logistic map at the edge of chaos. The complexity and randomness of the sequences are then discussed in relation with the properties of associated fractal objects, such as their fractal dimension, symmetry and orientations of the arcs. PMID:25700034

Huynh, Hoai Nguyen; Pradana, Andri; Chew, Lock Yue

2015-01-01

409

NASA Technical Reports Server (NTRS)

Fractal geometry is increasingly becoming a useful tool for modeling natural phenomena. As an alternative to Euclidean concepts, fractals allow for a more accurate representation of the nature of complexity in natural boundaries and surfaces. The purpose of this paper is to introduce and implement three algorithms in C code for deriving fractal measurement from remotely sensed data. These three methods are: the line-divider method, the variogram method, and the triangular prism method. Remote-sensing data acquired by NASA's Calibrated Airborne Multispectral Scanner (CAMS) are used to compute the fractal dimension using each of the three methods. These data were obtained as a 30 m pixel spatial resolution over a portion of western Puerto Rico in January 1990. A description of the three methods, their implementation in PC-compatible environment, and some results of applying these algorithms to remotely sensed image data are presented.

Jaggi, S.; Quattrochi, Dale A.; Lam, Nina S.-N.

1993-01-01

410

Fractal analysis for classification of breast carcinoma in optical coherence tomography

NASA Astrophysics Data System (ADS)

The accurate and rapid assessment of tumor margins during breast cancer resection using optical coherence tomography (OCT) has the potential to reduce patient risk. However, it is difficult to subjectively distinguish cancer from normal fibroglandular stromal tissues in OCT images, and an objective measure is needed. In this initial study, we investigate the potential of a one-dimensional fractal box-counting method for cancer classification in OCT. We computed the fractal dimension, a measure of the self-similarity of an object, along the depth axis of 44 ultrahigh-resolution OCT images of human breast tissues obtained from 4 cancer patients. Correlative histology was employed to identify distinct regions of adipose, stroma, and cancer in the OCT images. We report that the fractal dimension of stroma is significantly higher than that of cancer (P < 10-5, t-test). Furthermore, by adjusting the cutoff values of fractal dimension between cancer, stroma, and adipose tissues, sensitivities and specificities of either 82.4% and 88.9%, or 88.2% and 81.5%, are obtained, respectively, for cancer classification. The use of fractal analysis with OCT could potentially provide automated identification of tumor margins during breast-sparing surgery.

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

2011-06-01

411

A Fractal Model for the Capacitance of Lunar Dust and Lunar Dust Aggregates

NASA Technical Reports Server (NTRS)

Lunar dust grains and dust aggregates exhibit clumping, with an uneven mass distribution, as well as features that span many spatial scales. It has been observed that these aggregates display an almost fractal repetition of geometry with scale. Furthermore, lunar dust grains typically have sharp protrusions and jagged features that result from the lack of aeolian weathering (as opposed to space weathering) on the Moon. A perfectly spherical geometry, frequently used as a model for lunar dust grains, has none of these characteristics (although a sphere may be a reasonable proxy for the very smallest grains and some glasses). We present a fractal model for a lunar dust grain or aggregate of grains that reproduces (1) the irregular clumpy nature of lunar dust, (2) the presence of sharp points, and (3) dust features that span multiple scale lengths. We calculate the capacitance of the fractal lunar dust analytically assuming fixed dust mass (i.e. volume) for an arbitrary number of fractal levels and compare the capacitance to that of a non-fractal object with the same volume, surface area, and characteristic width. The fractal capacitance is larger than that of the equivalent non-fractal object suggesting that for a given potential, electrostatic forces on lunar dust grains and aggregates are greater than one might infer from assuming dust grains are sphericaL Consequently, electrostatic transport of lunar dust grains, for example lofting, appears more plausible than might be inferred by calculations based on less realistic assumptions about dust shape and associated capacitance.

Collier, Michael R.; Stubbs, Timothy J.; Keller, John W.; Farrell, William M.; Marshall, John; Richard, Denis Thomas

2011-01-01

412

The topological insulator in a fractal space

We investigate the band structures and transport properties of a two-dimensional model of topological insulator, with a fractal edge or a fractal bulk. A fractal edge does not affect the robust transport even when the fractal pattern has reached the resolution of the atomic-scale, because the bulk is still well insulating against backscattering. On the other hand, a fractal bulk can support the robust transport only when the fractal resolution is much larger than a critical size. Smaller resolution of bulk fractal pattern will lead to remarkable backscattering and localization, due to strong couplings of opposite edge states on narrow sub-edges which appear almost everywhere in the fractal bulk.

Song, Zhi-Gang; Zhang, Yan-Yang; Li, Shu-Shen [SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083 (China)

2014-06-09

413

Scale-free networks embedded in fractal space

NASA Astrophysics Data System (ADS)

The impact of an inhomogeneous arrangement of nodes in space on a network organization cannot be neglected in most real-world scale-free networks. Here we propose a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to ?=2 regardless of the fractal dimension. We show that this phenomenon is related to the transition from a noncompact to compact phase of the network and that this transition accompanies a drastic change of the network efficiency. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.

Yakubo, K.; Korošak, D.

2011-06-01

414

Scale-free networks embedded in fractal space.

The impact of an inhomogeneous arrangement of nodes in space on a network organization cannot be neglected in most real-world scale-free networks. Here we propose a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to ? = 2 regardless of the fractal dimension. We show that this phenomenon is related to the transition from a noncompact to compact phase of the network and that this transition accompanies a drastic change of the network efficiency. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture. PMID:21797445

Yakubo, K; Korošak, D

2011-06-01

415

A physically based connection between fractional calculus and fractal geometry

NASA Astrophysics Data System (ADS)

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.

Butera, Salvatore; Di Paola, Mario

2014-11-01

416

Dimensions, maximal growth sites, and optimization in the dielectric breakdown model

We study the growth of fractal clusters in the dielectric breakdown model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent alphamin ) as a function of the growth exponent eta of the DBM model. We do not find evidence for a phase transition from

Joachim Mathiesen; Mogens H. Jensen; Jan Øystein Haavig Bakke

2008-01-01

417

Automatic Method to Classify Images Based on Multiscale Fractal Descriptors and Paraconsistent Logic

NASA Astrophysics Data System (ADS)

In this study is presented an automatic method to classify images from fractal descriptors as decision rules, such as multiscale fractal dimension and lacunarity. The proposed methodology was divided in three steps: quantification of the regions of interest with fractal dimension and lacunarity, techniques under a multiscale approach; definition of reference patterns, which are the limits of each studied group; and, classification of each group, considering the combination of the reference patterns with signals maximization (an approach commonly considered in paraconsistent logic). The proposed method was used to classify histological prostatic images, aiming the diagnostic of prostate cancer. The accuracy levels were important, overcoming those obtained with Support Vector Machine (SVM) and Best- first Decicion Tree (BFTree) classifiers. The proposed approach allows recognize and classify patterns, offering the advantage of giving comprehensive results to the specialists.

Pavarino, E.; Neves, L. A.; Nascimento, M. Z.; Godoy, M. F.; Arruda, P. F.; Neto, D. S.

2015-01-01

418

NASA Astrophysics Data System (ADS)

The concept of z-scaling previously developed for analysis of inclusive reactions in proton-proton collisions is applied for description of processes with polarized protons at the planned Spin Physics Detector NICA in Dubna. Hypotheses of self-similarity and fractality of the proton spin structure are discussed. The possibilities to extract information on spin-dependent fractal dimensions of hadrons and fragmentation process from asymmetries and coefficients of polarization transfer are justified. The double longitudinal spin asymmetry A LL of ?0 meson production and the coefficient of the polarization transfer D LL of ? hyperon production in proton-proton collisions measured at RHIC are analyzed in the framework of z-scaling. The spin-dependent fractal dimensions of proton and fragmentation process with polarized ? hyperon are estimated. A study of the spin-dependent constituent energy loss as a function of transverse momentum of the inclusive hadron and collision energy is suggested.

Tokarev, M. V.; Zborovsk, I.; Aparin, A.

2015-01-01

419

Applicability of Fractals to the Analysis of the Projection of Small Flocs.

The validity of the concept of fractal structure for the analysis of the projection of small flocs was examined using the data obtained from a table-tennis-ball simulation and a coagulation experiment with polystyrene latex particles. Two methods defining a fractal dimension, the box-counting technique and the enumeration of primary particles in an enclosing circle as a function of the radius, were tested for the result of numerical simulations. Comparison of the two results revealed their qualitative tendencies and the limits of their applicability. The projection of flocs was also examined using the relation between the maximum distance in the projected floc, Dm, and the number of primary particles composing the floc, i. The decrease of fractal dimension, D, in the lower limit of i was demonstrated both numerically and experimentally. Copyright 1998 Academic Press. PMID:9820784

Adachi; Kobayashi; Ooi

1998-12-01

420

Boundary Corrections in Fractal Analysis of Galaxy Surveys

The analysis of redshift surveys with fractal tools requires one to apply some form of statistical correction for galaxies lying near the geometric boundary of the sample. In this paper we compare three different methods of performing such a correction upon estimates of the correlation integral in order to assess the extent to which estimates may be biased by boundary terms. We apply the corrections illustrative examples, including a simple fractal set (L\\'{e}vy flight), a random $\\beta$-model, and a subset of the CfA2 Southern Cap survey. This study shows that the new ``angular'' correction method we present is more generally applicable than the other methods used to date: the conventional ``capacity'' correction imposes a bias towards homogeneity, and the ``deflation'' method discards large-scale information, and consequently reduces the statistical usefulness of data sets. The ``angular'' correction method is effective at recovering true fractal dimensions, although the extent to which boundary corrections are important depends on the form of fractal distribution assumed as well as the details of the survey geometry. We also show that the CfA2 Southern sample does not show any real evidence of a transition to homogeneity. We then revisit the IRAS PSCz survey and ``mock'' PSCz catalogues made using N-body simulations of two different cosmologies. The results we obtain from the PSCz survey are not significantly affected by the form of boundary correction used, confirming that the transition from fractal to homogeneous behaviour reported by Pan & Coles (2000) is real.

Jun Pan; Peter Coles