ERIC Educational Resources Information Center
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
NASA Astrophysics Data System (ADS)
Beech, M.
1989-02-01
The author discusses some of the more recent research on fractal astronomy and results presented in several astronomical studies. First, the large-scale structure of the universe is considered, while in another section one drops in scale to examine some of the smallest bodies in our solar system; the comets and meteoroids. The final section presents some thoughts on what influence the fractal ideology might have on astronomy, focusing particularly on the question recently raised by Kadanoff, "Fractals: where's the physics?"
ERIC Educational Resources Information Center
Osler, Thomas J.
1999-01-01
Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…
Stadnitski, Tatjana
2012-01-01
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 α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β 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 ([Formula: see text] 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.
Stadnitski, Tatjana
2012-01-01
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 α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β 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
ERIC Educational Resources Information Center
Gray, Shirley B.
1992-01-01
This article traces the historical development of fractal geometry from early in the twentieth century and offers an explanation of the mathematics behind the recursion formulas and their representations within computer graphics. Also included are the fundamentals behind programing for fractal graphics in the C Language with appropriate…
ERIC Educational Resources Information Center
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
ERIC Educational Resources Information Center
Jurgens, Hartmut; And Others
1990-01-01
The production and application of images based on fractal geometry are described. Discussed are fractal language groups, fractal image coding, and fractal dialects. Implications for these applications of geometry to mathematics education are suggested. (CW)
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…
NASA Astrophysics Data System (ADS)
Wuorinen, Charles
2015-03-01
Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...
Magnetohydrodynamics of fractal media
Tarasov, Vasily E.
2006-05-15
The fractal distribution of charged particles is considered. An example of this distribution is the charged particles that are distributed over the fractal. The fractional integrals are used to describe fractal distribution. These integrals are considered as approximations of integrals on fractals. Typical turbulent media could be of a fractal structure and the corresponding equations should be changed to include the fractal features of the media. The magnetohydrodynamics equations for fractal media are derived from the fractional generalization of integral Maxwell equations and integral hydrodynamics (balance) equations. Possible equilibrium states for these equations are considered.
NASA Astrophysics Data System (ADS)
Orbach, R.
1986-02-01
Random structures often exhibit fractal geometry, defined in terms of the mass scaling exponent, D, the fractal dimension. The vibrational dynamics of fractal networks are expressed in terms of the exponent d double bar, the fracton dimensionality. The eigenstates on a fractal network are spatially localized for d double bar less than or equal to 2. The implications of fractal geometry are discussed for thermal transport on fractal networks. The electron-fracton interaction is developed, with a brief outline given for the time dependence of the electronic relaxation on fractal networks. It is suggested that amorphous or glassy materials may exhibit fractal properties at short length scales or, equivalently, at high energies. The calculations of physical properties can be used to test the fractal character of the vibrational excitations in these materials.
Fractal antenna and fractal resonator primer
NASA Astrophysics Data System (ADS)
Cohen, Nathan
2015-03-01
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.
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter. PMID:28117325
ERIC Educational Resources Information Center
Barton, Ray
1990-01-01
Presented is an educational game called "The Chaos Game" which produces complicated fractal images. Two basic computer programs are included. The production of fractal images by the Sierpinski gasket and the Chaos Game programs is discussed. (CW)
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
NASA Astrophysics Data System (ADS)
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
Hashemi, S M; Jagodič, U; Mozaffari, M R; Ejtehadi, M R; Muševič, I; Ravnik, M
2017-01-24
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
ERIC Educational Resources Information Center
Fraboni, Michael; Moller, Trisha
2008-01-01
Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…
Fractal interpretation of intermittency
Hwa, R.C.
1991-12-01
Implication of intermittency in high-energy collisions is first discussed. Then follows a description of the fractal interpretation of intermittency. A basic quantity with asymptotic fractal behavior is introduced. It is then shown how the factorial moments and the G moments can be expressed in terms of it. The relationship between the intermittency indices and the fractal indices is made explicit.
NASA Technical Reports Server (NTRS)
Barnsley, Michael F.; Sloan, Alan D.
1989-01-01
Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed.
Exploring Fractals in the Classroom.
ERIC Educational Resources Information Center
Naylor, Michael
1999-01-01
Describes an activity involving six investigations. Introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Contains 14 references. (ASK)
Fractals: To Know, to Do, to Simulate.
ERIC Educational Resources Information Center
Talanquer, Vicente; Irazoque, Glinda
1993-01-01
Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)
NASA Astrophysics Data System (ADS)
Knutson, Paul; Dahlberg, E. Dan
2003-10-01
In examples of fractals such as moon craters, rivers,2 cauliflower,3 and bread,4 the actual growth process of the fractal object is missed. In the simple experiment described here, one can observe and record the growth of calcium carbonate crystals — a ubiquitous material found in marble and seashells — in real time. The video frames can be digitized and analyzed to determine the fractal dimension.
Fractal Geometry of Architecture
NASA Astrophysics Data System (ADS)
Lorenz, Wolfgang E.
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.
NASA Astrophysics Data System (ADS)
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M.
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed.
NASA Astrophysics Data System (ADS)
Strichartz, Robert S.; Usher, Michael
2000-09-01
A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of [Delta]j+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.
ERIC Educational Resources Information Center
Clark, Garry
1999-01-01
Reports on a mathematical investigation of fractals and highlights the thinking involved, problem solving strategies used, generalizing skills required, the role of technology, and the role of mathematics. (ASK)
ERIC Educational Resources Information Center
Bannon, Thomas J.
1991-01-01
Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)
NASA Astrophysics Data System (ADS)
West, Bruce J.
The proper methodology for describing the dynamics of certain complex phenomena and fractal time series is the fractional calculus through the fractional Langevin equation discussed herein and applied in a biomedical context. We show that a fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process, for example, human gait and cerebral blood flow. The goal of this talk is to make clear how certain complex phenomena, such as those that are abundantly present in human physiology, can be faithfully described using dynamical models involving fractional differential stochastic equations. These models are tested against existing data sets and shown to describe time series from complex physiologic phenomena quite well.
Radlinski, A.P.; Radlinska, E.Z.; Agamalian, M.; Wignall, G.D.; Lindner, P.; Randl, O.G.
1999-04-01
The analysis of small- and ultra-small-angle neutron scattering data for sedimentary rocks shows that the pore-rock fabric interface is a surface fractal (D{sub s}=2.82) over 3 orders of magnitude of the length scale and 10 orders of magnitude in intensity. The fractal dimension and scatterer size obtained from scanning electron microscopy image processing are consistent with neutron scattering data. {copyright} {ital 1999} {ital The American Physical Society}
NASA Astrophysics Data System (ADS)
Turcotte, Donald L.
Tectonic processes build landforms that are subsequently destroyed by erosional processes. Landforms exhibit fractal statistics in a variety of ways; examples include (1) lengths of coast lines; (2) number-size statistics of lakes and islands; (3) spectral behavior of topography and bathymetry both globally and locally; and (4) branching statistics of drainage networks. Erosional processes are dominant in the development of many landforms on this planet, but similar fractal statistics are also applicable to the surface of Venus where minimal erosion has occurred. A number of dynamical systems models for landforms have been proposed, including (1) cellular automata; (2) diffusion limited aggregation; (3) self-avoiding percolation; and (4) advective-diffusion equations. The fractal statistics and validity of these models will be discussed. Earthquakes also exhibit fractal statistics. The frequency-magnitude statistics of earthquakes satisfy the fractal Gutenberg-Richter relation both globally and locally. Earthquakes are believed to be a classic example of self-organized criticality. One model for earthquakes utilizes interacting slider-blocks. These slider block models have been shown to behave chaotically and to exhibit self-organized criticality. The applicability of these models will be discussed and alternative approaches will be presented. Fragmentation has been demonstrated to produce fractal statistics in many cases. Comminution is one model for fragmentation that yields fractal statistics. It has been proposed that comminution is also responsible for much of the deformation in the earth's crust. The brittle disruption of the crust and the resulting earthquakes present an integrated problem with many fractal aspects.
Baish, J W; Jain, R K
2000-07-15
Recent studies have shown that fractal geometry, a vocabulary of irregular shapes, can be useful for describing the pathological architecture of tumors and, perhaps more surprisingly, for yielding insights into the mechanisms of tumor growth and angiogenesis that complement those obtained by modern molecular methods. This article outlines the basic methods of fractal geometry and discusses the value and limitations of applying this new tool to cancer research.
Fractal Patterns and Chaos Games
ERIC Educational Resources Information Center
Devaney, Robert L.
2004-01-01
Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.
Building Fractal Models with Manipulatives.
ERIC Educational Resources Information Center
Coes, Loring
1993-01-01
Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)
NASA Astrophysics Data System (ADS)
Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga
2016-04-01
The effectiveness of fractal toolbox to capture the scaling or fractal probability distribution, and simply fractal statistics of main hydrocarbon reservoir attributes, was highlighted by Mandelbrot (1995) and confirmed by several researchers (Zhao et al., 2015). Notwithstanding, after more than twenty years, it's still common the opinion that fractals are not useful for the petroleum engineers and especially for Geoengineering (Corbett, 2012). In spite of this negative background, we have successfully applied the fractal and multifractal techniques to our project entitled "Petroleum Reservoir as a Fractal Reactor" (2013 up to now). The distinguishable feature of Fractal Reservoir is the irregular shapes and rough pore/solid distributions (Siler, 2007), observed across a broad range of scales (from SEM to seismic). At the beginning, we have accomplished the detailed analysis of Nelson and Kibler (2003) Catalog of Porosity and Permeability, created for the core plugs of siliciclastic rocks (around ten thousand data were compared). We enriched this Catalog by more than two thousand data extracted from the last ten years publications on PoroPerm (Corbett, 2012) in carbonates deposits, as well as by our own data from one of the PEMEX, Mexico, oil fields. The strong power law scaling behavior was documented for the major part of these data from the geological deposits of contrasting genesis. Based on these results and taking into account the basic principles and models of the Physics of Fractals, introduced by Per Back and Kan Chen (1989), we have developed new software (Muukíl Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoir models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mesh' cells was designed and calibrated in the studied area. The statistically sound power law relations were established
Fractal dynamics of earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
NASA Astrophysics Data System (ADS)
Marks-Tarlow, Terry
Linear concepts of time plus the modern capacity to track history emerged out of circular conceptions characteristic of ancient and traditional cultures. A fractal concept of time lies implicitly within the analog clock, where each moment is treated as unique. With fractal geometry the best descriptor of nature, qualities of self-similarity and scale invariance easily model her endless variety and recursive patterning, both in time and across space. To better manage temporal aspects of our lives, a fractal concept of time is non-reductive, based more on the fullness of being than on fragments of doing. By using a fractal concept of time, each activity or dimension of life is multiply and vertically nested. Each nested cycle remains simultaneously present, operating according to intrinsic dynamics and time scales. By adding the vertical axis of simultaneity to the horizontal axis of length, time is already full and never needs to be filled. To attend to time's vertical dimension is to tap into the imaginary potential for infinite depth. To switch from linear to fractal time allows us to relax into each moment while keeping in mind the whole.
Fractal radar scattering from soil.
Oleschko, Klaudia; Korvin, Gabor; Figueroa, Benjamin; Vuelvas, Marco Antonio; Balankin, Alexander S; Flores, Lourdes; Carreón, Dora
2003-04-01
A general technique is developed to retrieve the fractal dimension of self-similar soils through microwave (radar) scattering. The technique is based on a mathematical model relating the fractal dimensions of the georadargram to that of the scattering structure. Clear and different fractal signatures have been observed over four geosystems (soils and sediments) compared in this work.
Backbone fractal dimension and fractal hybrid orbital of protein structure
NASA Astrophysics Data System (ADS)
Peng, Xin; Qi, Wei; Wang, Mengfan; Su, Rongxin; He, Zhimin
2013-12-01
Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. "the local fractal dimension" and "the backbone fractal dimension" (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ⩽ N ⩽ 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Thanki, Sandip; Rhee, George; Lepp, Stephen E-mail: grhee@physics.unlv.edu
2009-09-15
In this paper we investigate the use of the fractal dimension of galaxy isophotes in galaxy classification. We have applied two different methods for determining fractal dimensions to the isophotes of elliptical and spiral galaxies derived from CCD images. We conclude that fractal dimension alone is not a reliable tool but that combined with other parameters in a neural net algorithm the fractal dimension could be of use. In particular, we have used three parameters to segregate the ellipticals and lenticulars from the spiral galaxies in our sample. These three parameters are the correlation fractal dimension D {sub corr}, the difference between the correlation fractal dimension and the capacity fractal dimension D {sub corr} - D {sub cap}, and, thirdly, the B - V color of the galaxy.
Hsü, K J; Hsü, A J
1990-01-01
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
ERIC Educational Resources Information Center
Camp, Dane R.
1991-01-01
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.…
ERIC Educational Resources Information Center
Marks, Tim K.
1992-01-01
Presents a three-lesson unit that uses fractal geometry to measure the coastline of Massachusetts. Two lessons provide hands-on activities utilizing compass and grid methods to perform the measurements and the third lesson analyzes and explains the results of the activities. (MDH)
Huang, F.; Peng, R. D.; Liu, Y. H.; Chen, Z. Y.; Ye, M. F.; Wang, L.
2012-09-15
Fractal dust grains of different shapes are observed in a radially confined magnetized radio frequency plasma. The fractal dimensions of the dust structures in two-dimensional (2D) horizontal dust layers are calculated, and their evolution in the dust growth process is investigated. It is found that as the dust grains grow the fractal dimension of the dust structure decreases. In addition, the fractal dimension of the center region is larger than that of the entire region in the 2D dust layer. In the initial growth stage, the small dust particulates at a high number density in a 2D layer tend to fill space as a normal surface with fractal dimension D = 2. The mechanism of the formation of fractal dust grains is discussed.
Fractals in geology and geophysics
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.
Multilayer adsorption on fractal surfaces.
Vajda, Péter; Felinger, Attila
2014-01-10
Multilayer adsorption is often observed in liquid chromatography. The most frequently employed model for multilayer adsorption is the BET isotherm equation. In this study we introduce an interpretation of multilayer adsorption measured on liquid chromatographic stationary phases based on the fractal theory. The fractal BET isotherm model was successfully used to determine the apparent fractal dimension of the adsorbent surface. The nonlinear fitting of the fractal BET equation gives us the estimation of the adsorption equilibrium constants and the monolayer saturation capacity of the adsorbent as well. In our experiments, aniline and proline were used as test molecules on reversed phase and normal phase columns, respectively. Our results suggest an apparent fractal dimension 2.88-2.99 in the case of reversed phase adsorbents, in the contrast with a bare silica column with a fractal dimension of 2.54.
NASA Astrophysics Data System (ADS)
Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.
2004-04-01
We study the middle cerebral artery blood flow velocity (MCAfv) in humans using transcranial Doppler ultrasonography (TCD). Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of the underlying cerebral blood flow control system. We call this effect fractal rigidity.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-08-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-[3-(triethoxysilyl)propyl]-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-01-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-(3-(triethoxysilyl)propyl)-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
Church, E.L.
1988-04-15
Surface finish measurements are usually fitted to models of the finish correlation function which are parametrized in terms of root-mean-square roughnesses, sigma, and correlation lengths, l. Highly finished optical surfaces, however, are frequently better described by fractal models, which involve inverse power-law spectra and are parametrized by spectral strengths, K/sub n/, and spectral indices, n. Analyzing measurements of fractal surfaces in terms of sigma and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudoparameters and discusses the errors involved in using them for the characterization and specification of surface finish.
Church, E.L.
1988-01-01
Surface finish measurements are usually fitted to models of the finish correlation function which are parameterized in terms of root-mean-square roughness, sigma, and correlation lengths, l. Highly-finished optical surfaces, however, are frequently better described by fractal models, which involve inverse-power-law spectra and are parameterized by spectral strengths, K/sub n/, and spectral indices, n. Analyzing measurements of fractal surfaces in terms of sigma and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudo parameters and discusses the errors involved in using them for the characterization and specification of surface finish. 30 refs., 5 figs., 1 tab.
Turbulent wakes of fractal objects.
Staicu, Adrian; Mazzi, Biagio; Vassilicos, J C; van de Water, Willem
2003-06-01
Turbulence of a windtunnel flow is stirred using objects that have a fractal structure. The strong turbulent wakes resulting from three such objects which have different fractal dimensions are probed using multiprobe hot-wire anemometry in various configurations. Statistical turbulent quantities are studied within inertial and dissipative range scales in an attempt to relate changes in their self-similar behavior to the scaling of the fractal objects.
Langevin Equation on Fractal Curves
NASA Astrophysics Data System (ADS)
Satin, Seema; Gangal, A. D.
2016-07-01
We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.
Fractal multifiber microchannel plates
NASA Technical Reports Server (NTRS)
Cook, Lee M.; Feller, W. B.; Kenter, Almus T.; Chappell, Jon H.
1992-01-01
The construction and performance of microchannel plates (MCPs) made using fractal tiling mehtods are reviewed. MCPs with 40 mm active areas having near-perfect channel ordering were produced. These plates demonstrated electrical performance characteristics equivalent to conventionally constructed MCPs. These apparently are the first MCPs which have a sufficiently high degree of order to permit single channel addressability. Potential applications for these devices and the prospects for further development are discussed.
NASA Astrophysics Data System (ADS)
Martin, Demetri
2015-03-01
Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, "Be still if I fill its ebb" minus its last letter is a palindrome, and the entire poem is a palindrome...
Fractal Properties in Economics
2000-01-01
field of science, econophysics , was established recently in 1997 [2, 3]. This is a study of economic phenomena based on the methods and approaches of...physics. Among 244 many topics in econophysics there are three topics that are closely related to the study of fractals. They are price changes in open...prices, J. of Bussiness (Chicago) 36 (1963) pp.3 9 4 -4 19 . 2. J. Kertesz and 1. Kondor (Eds.), Econophysics : an emerging science (Kluwer Academic
Darwinian Evolution and Fractals
NASA Astrophysics Data System (ADS)
Carr, Paul H.
2009-05-01
Did nature's beauty emerge by chance or was it intelligently designed? Richard Dawkins asserts that evolution is blind aimless chance. Michael Behe believes, on the contrary, that the first cell was intelligently designed. The scientific evidence is that nature's creativity arises from the interplay between chance AND design (laws). Darwin's ``Origin of the Species,'' published 150 years ago in 1859, characterized evolution as the interplay between variations (symbolized by dice) and the natural selection law (design). This is evident in recent discoveries in DNA, Madelbrot's Fractal Geometry of Nature, and the success of the genetic design algorithm. Algorithms for generating fractals have the same interplay between randomness and law as evolution. Fractal statistics, which are not completely random, characterize such phenomena such as fluctuations in the stock market, the Nile River, rainfall, and tree rings. As chaos theorist Joseph Ford put it: God plays dice, but the dice are loaded. Thus Darwin, in discovering the evolutionary interplay between variations and natural selection, was throwing God's dice!
Fractals in physiology and medicine
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.; West, Bruce J.
1987-01-01
The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.
Dimension of fractal basin boundaries
Park, B.S.
1988-01-01
In many dynamical systems, multiple attractors coexist for certain parameter ranges. The set of initial conditions that asymptotically approach each attractor is its basin of attraction. These basins can be intertwined on arbitrary small scales. Basin boundary can be either smooth or fractal. Dynamical systems that have fractal basin boundary show final state sensitivity of the initial conditions. A measure of this sensitivity (uncertainty exponent {alpha}) is related to the dimension of the basin boundary d = D - {alpha}, where D is the dimension of the phase space and d is the dimension of the basin boundary. At metamorphosis values of the parameter, there might happen a conversion from smooth to fractal basin boundary (smooth-fractal metamorphosis) or a conversion from fractal to another fractal basin boundary characteristically different from the previous fractal one (fractal-fractal metamorphosis). The dimension changes continuously with the parameter except at the metamorphosis values where the dimension of the basin boundary jumps discontinuously. We chose the Henon map and the forced damped pendulum to investigate this. Scaling of the basin volumes near the metamorphosis values of the parameter is also being studied for the Henon map. Observations are explained analytically by using low dimensional model map.
Investigating Fractal Geometry Using LOGO.
ERIC Educational Resources Information Center
Thomas, David A.
1989-01-01
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)
Fractals in biology and medicine
NASA Technical Reports Server (NTRS)
Havlin, S.; Buldyrev, S. V.; Goldberger, A. L.; Mantegna, R. N.; Ossadnik, S. M.; Peng, C. K.; Simons, M.; Stanley, H. E.
1995-01-01
Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.
Fractal analysis: methodologies for biomedical researchers.
Ristanović, Dusan; Milosević, Nebojsa T
2012-01-01
Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine. Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology of fractal analysis have partly restricted its wider and comprehensible application. It is a notable fact that fractal analysis is derived from fractal geometry, but there are some unresolved issues that need to be addressed. In this respect, we discuss several related underlying principles for fractal analysis and establish the meaningful relationship between fractal analysis and fractal geometry. Since some concepts in fractal analysis are determined descriptively and/or qualitatively, this paper provides their exact mathematical definitions or explanations. Another aim of this study is to show that nowadays fractal analysis is an independent mathematical and experimental method based on Mandelbrot's fractal geometry, Euclidean traditiontal geometry and Richardson's coastline method.
ERIC Educational Resources Information Center
Simoson, Andrew J.
2009-01-01
This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)
Terahertz spectroscopy of plasmonic fractals.
Agrawal, A; Matsui, T; Zhu, W; Nahata, A; Vardeny, Z V
2009-03-20
We use terahertz time-domain spectroscopy to study the transmission properties of metallic films perforated with aperture arrays having deterministic or stochastic fractal morphologies ("plasmonic fractals"), and compare them with random aperture arrays. All of the measured plasmonic fractals show transmission resonances and antiresonances at frequencies that correspond to prominent features in their structure factors in k space. However, in sharp contrast to periodic aperture arrays, the resonant transmission enhancement decreases with increasing array size. This property is explained using a density-density correlation function, and is utilized for determining the underlying fractal dimensionality, D(<2). Furthermore, a sum rule for the transmission resonances and antiresonances in plasmonic fractals relative to the transmission of the corresponding random aperture arrays is obtained, and is shown to be universal.
Fractals analysis of cardiac arrhythmias.
Saeed, Mohammed
2005-09-06
Heart rhythms are generated by complex self-regulating systems governed by the laws of chaos. Consequently, heart rhythms have fractal organization, characterized by self-similar dynamics with long-range order operating over multiple time scales. This allows for the self-organization and adaptability of heart rhythms under stress. Breakdown of this fractal organization into excessive order or uncorrelated randomness leads to a less-adaptable system, characteristic of aging and disease. With the tools of nonlinear dynamics, this fractal breakdown can be quantified with potential applications to diagnostic and prognostic clinical assessment. In this paper, I review the methodologies for fractal analysis of cardiac rhythms and the current literature on their applications in the clinical context. A brief overview of the basic mathematics of fractals is also included. Furthermore, I illustrate the usefulness of these powerful tools to clinical medicine by describing a novel noninvasive technique to monitor drug therapy in atrial fibrillation.
Feltrin, Gian Pietro; Stramare, Roberto; Miotto, Diego; Giacomini, Dario; Saccavini, Claudio
2004-06-01
Fractal analysis is a quantitative method used to evaluate complex anatomic findings in their elementary component. Its application to biologic images, particularly to cancellous bones, has been well practiced within the past few years. The aims of these applications are to assess changes in bone and the loss of spongious architecture, indicate bone fragility, and to show the increased risk for fracture in primary or secondary osteoporosis. The applications are very promising to help complete the studies that can define bone density (bone mineral density by dual energy x-ray absorptiometry or quantitative computed tomography), and also have the capacity to distinguish the patients with a high or low risk for fracture. Their extension to the clinical fields, to define a test for fracture risk, is still limited by difficult application to the medical quantitative imaging of bones, between correct application at superficial bones and unreliable application to deep bones. The future evolution and validity do not depend upon fractal methods but upon well-detailed imaging of the bones in clinical conditions.
Fractal structure of asphaltene aggregates.
Rahmani, Nazmul H G; Dabros, Tadeusz; Masliyah, Jacob H
2005-05-15
A photographic technique coupled with image analysis was used to measure the size and fractal dimension of asphaltene aggregates formed in toluene-heptane solvent mixtures. First, asphaltene aggregates were examined in a Couette device and the fractal-like aggregate structures were quantified using boundary fractal dimension. The evolution of the floc structure with time was monitored. The relative rates of shear-induced aggregation and fragmentation/restructuring determine the steady-state floc structure. The average floc structure became more compact or more organized as the floc size distribution attained steady state. Moreover, the higher the shear rate is, the more compact the floc structure is at steady state. Second, the fractal dimensions of asphaltene aggregates were also determined in a free-settling test. The experimentally determined terminal settling velocities and characteristic lengths of the aggregates were utilized to estimate the 2D and 3D fractal dimensions. The size-density fractal dimension (D(3)) of the asphaltene aggregates was estimated to be in the range from 1.06 to 1.41. This relatively low fractal dimension suggests that the asphaltene aggregates are highly porous and very tenuous. The aggregates have a structure with extremely low space-filling capacity.
Target Detection Using Fractal Geometry
NASA Technical Reports Server (NTRS)
Fuller, J. Joseph
1991-01-01
The concepts and theory of fractal geometry were applied to the problem of segmenting a 256 x 256 pixel image so that manmade objects could be extracted from natural backgrounds. The two most important measurements necessary to extract these manmade objects were fractal dimension and lacunarity. Provision was made to pass the manmade portion to a lookup table for subsequent identification. A computer program was written to construct cloud backgrounds of fractal dimensions which were allowed to vary between 2.2 and 2.8. Images of three model space targets were combined with these backgrounds to provide a data set for testing the validity of the approach. Once the data set was constructed, computer programs were written to extract estimates of the fractal dimension and lacunarity on 4 x 4 pixel subsets of the image. It was shown that for clouds of fractal dimension 2.7 or less, appropriate thresholding on fractal dimension and lacunarity yielded a 64 x 64 edge-detected image with all or most of the cloud background removed. These images were enhanced by an erosion and dilation to provide the final image passed to the lookup table. While the ultimate goal was to pass the final image to a neural network for identification, this work shows the applicability of fractal geometry to the problems of image segmentation, edge detection and separating a target of interest from a natural background.
Losa, Gabriele A
2009-01-01
The extension of the concepts of Fractal Geometry (Mandelbrot [1983]) toward the life sciences has led to significant progress in understanding complex functional properties and architectural / morphological / structural features characterising cells and tissues during ontogenesis and both normal and pathological development processes. It has even been argued that fractal geometry could provide a coherent description of the design principles underlying living organisms (Weibel [1991]). Fractals fulfil a certain number of theoretical and methodological criteria including a high level of organization, shape irregularity, functional and morphological self-similarity, scale invariance, iterative pathways and a peculiar non-integer fractal dimension [FD]. Whereas mathematical objects are deterministic invariant or self-similar over an unlimited range of scales, biological components are statistically self-similar only within a fractal domain defined by upper and lower limits, called scaling window, in which the relationship between the scale of observation and the measured size or length of the object can be established (Losa and Nonnenmacher [1996]). Selected examples will contribute to depict complex biological shapes and structures as fractal entities, and also to show why the application of the fractal principle is valuable for measuring dimensional, geometrical and functional parameters of cells, tissues and organs occurring within the vegetal and animal realms. If the criteria for a strict description of natural fractals are met, then it follows that a Fractal Geometry of Life may be envisaged and all natural objects and biological systems exhibiting self-similar patterns and scaling properties may be considered as belonging to the new subdiscipline of "fractalomics".
Fractal features of seismic noise
NASA Astrophysics Data System (ADS)
Caserta, A.; Consolini, G.; Michelis, P. De
2003-04-01
We present experimental observations and data analysis concerning the fractal features of seismic noise in the frequency range from 1 Hz to 40 Hz. In detail, we investigate the 3D average squared soil displacement and the distribution function of its fluctuations for different near-surface geological structures. We found that the seismic noise is consistent with a persistent fractal brownian motion characterized by a Hurst exponent grather than 1/2. Moreover, a clear dependence of the fractal nature of the seismic noise on the near-surface local geology has been found.
Exterior dimension of fat fractals
NASA Technical Reports Server (NTRS)
Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.
1985-01-01
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
The transience of virtual fractals.
Taylor, R P
2012-01-01
Artists have a long and fruitful tradition of exploiting electronic media to convert static images into dynamic images that evolve with time. Fractal patterns serve as an example: computers allow the observer to zoom in on virtual images and so experience the endless repetition of patterns in a matter that cannot be matched using static images. This year's featured cover artist, Susan Lowedermilk, instead plans to employ persistence of human vision to bring virtual fractals to life. This will be done by incorporating her prints of fractal patterns into zoetropes and phenakistoscopes.
Thermal collapse of snowflake fractals
NASA Astrophysics Data System (ADS)
Gallo, T.; Jurjiu, A.; Biscarini, F.; Volta, A.; Zerbetto, F.
2012-08-01
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.
Thermodynamics of Photons on Fractals
Akkermans, Eric; Dunne, Gerald V.; Teplyaev, Alexander
2010-12-03
A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV{sub s}=U/d{sub s}, where d{sub s} is the spectral dimension and V{sub s} defines the 'spectral volume'. For regular manifolds, V{sub s} coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.
Fractal analysis of Mesoamerican pyramids.
Burkle-Elizondo, Gerardo; Valdez-Cepeda, Ricardo David
2006-01-01
A myth of ancient cultural roots was integrated into Mesoamerican cult, and the reference to architecture denoted a depth religious symbolism. The pyramids form a functional part of this cosmovision that is centered on sacralization. The space architecture works was an expression of the ideological necessities into their conception of harmony. The symbolism of the temple structures seems to reflect the mathematical order of the Universe. We contemplate two models of fractal analysis. The first one includes 16 pyramids. We studied a data set that was treated as a fractal profile to estimate the Df through variography (Dv). The estimated Fractal Dimension Dv = 1.383 +/- 0.211. In the second one we studied a data set to estimate the Dv of 19 pyramids and the estimated Fractal Dimension Dv = 1.229 +/- 0.165.
Steady laminar flow of fractal fluids
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Mena, Baltasar; Susarrey, Orlando; Samayoa, Didier
2017-02-01
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.
Applications of fractals in ecology.
Sugihara, G; M May, R
1990-03-01
Fractal models describe the geometry of a wide variety of natural objects such as coastlines, island chains, coral reefs, satellite ocean-color images and patches of vegetation. Cast in the form of modified diffusion models, they can mimic natural and artificial landscapes having different types of complexity of shape. This article provides a brief introduction to fractals and reports on how they can be used by ecologists to answer a variety of basic questions, about scale, measurement and hierarchy in, ecological systems.
Fractal processes in soil water retention
Tyler, S.W.; Wheatcraft, S.W. )
1990-05-01
The authors propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that they used for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar holes (or pores) over a wide range of scales. The authors evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. They relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet soil. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Their fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides paramters which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.
The topological insulator in a fractal space
Song, Zhi-Gang; Zhang, Yan-Yang; Li, Shu-Shen
2014-06-09
We investigate the band structures and transport properties of a two-dimensional model of topological insulator, with a fractal edge or a fractal bulk. A fractal edge does not affect the robust transport even when the fractal pattern has reached the resolution of the atomic-scale, because the bulk is still well insulating against backscattering. On the other hand, a fractal bulk can support the robust transport only when the fractal resolution is much larger than a critical size. Smaller resolution of bulk fractal pattern will lead to remarkable backscattering and localization, due to strong couplings of opposite edge states on narrow sub-edges which appear almost everywhere in the fractal bulk.
Analysis of fractals with combined partition
NASA Astrophysics Data System (ADS)
Dedovich, T. G.; Tokarev, M. V.
2016-03-01
The space—time properties in the general theory of relativity, as well as the discreteness and non-Archimedean property of space in the quantum theory of gravitation, are discussed. It is emphasized that the properties of bodies in non-Archimedean spaces coincide with the properties of the field of P-adic numbers and fractals. It is suggested that parton showers, used for describing interactions between particles and nuclei at high energies, have a fractal structure. A mechanism of fractal formation with combined partition is considered. The modified SePaC method is offered for the analysis of such fractals. The BC, PaC, and SePaC methods for determining a fractal dimension and other fractal characteristics (numbers of levels and values of a base of forming a fractal) are considered. It is found that the SePaC method has advantages for the analysis of fractals with combined partition.
Fractal study and simulation of fracture roughness
Kumar, S.; Bodvarsson, G.S. )
1990-05-01
This study examines the roughness profiles of the surfaces of fractures and faults by using concepts from fractal geometry. Relationships between fractal characteristics of profiles and isotropic surfaces are analytically developed and a deterministic representation of the roughness is examined.
The fractal aggregation of asphaltenes.
Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott
2013-07-16
This paper discusses time-resolved small-angle neutron scattering results that were used to investigate asphaltene structure and stability with and without a precipitant added in both crude oil and model oil. A novel approach was used to isolate the scattering from asphaltenes that are insoluble and in the process of aggregating from those that are soluble. It was found that both soluble and insoluble asphaltenes form fractal clusters in crude oil and the fractal dimension of the insoluble asphaltene clusters is higher than that of the soluble clusters. Adding heptane also increases the size of soluble asphaltene clusters without modifying the fractal dimension. Understanding the process of insoluble asphaltenes forming fractals with higher fractal dimensions will potentially reveal the microscopic asphaltene destabilization mechanism (i.e., how a precipitant modifies asphaltene-asphaltene interactions). It was concluded that because of the polydisperse nature of asphaltenes, no well-defined asphaltene phase stability envelope exists and small amounts of asphaltenes precipitated even at dilute precipitant concentrations. Asphaltenes that are stable in a crude oil-precipitant mixture are dispersed on the nanometer length scale. An asphaltene precipitation mechanism is proposed that is consistent with the experimental findings. Additionally, it was found that the heptane-insoluble asphaltene fraction is the dominant source of small-angle scattering in crude oil and the previously unobtainable asphaltene solubility at low heptane concentrations was measured.
Synthesis, Analysis, and Processing of Fractal Signals
1991-10-01
fractal dimension of the underlying signal , when defined. Robust estimation of the fractal dimension of 1/f processes is important in a number of...modeling errors. The resulting parameter estimation algorithms, which compute both fractal dimension parameters and the accompanying signal and noise...Synthesis, Analysis, and Processing of Fractal Signals RLE Technical Report No. 566 Gregory W. Wornell October 1991 Research Laboratory of
Fractal characterization of fracture surfaces in concrete
Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.
1990-01-01
Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.
Fractal analysis of time varying data
Vo-Dinh, Tuan; Sadana, Ajit
2002-01-01
Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.
A Novel Triangular Shaped UWB Fractal Antenna Using Circular Slot
NASA Astrophysics Data System (ADS)
Shahu, Babu Lal; Pal, Srikanta; Chattoraj, Neela
2016-03-01
The article presents the design of triangular shaped fractal based antenna with circular slot for ultra wideband (UWB) application. The antenna is fed using microstrip line and has overall dimension of 24×24×1.6 mm3. The proposed antenna is covering the wide frequency bandwidth of 2.99-11.16 GHz and is achieved using simple fractal based triangular-circular geometries and asymmetrical ground plane. The antenna is designed and parametrical studies are performed using method of moment (MOM) based Full Wave Electromagnetic (EM) software Simulator Zeland IE3D. The prototype of proposed antenna is fabricated and tested to compare the simulated and measured results of various antenna parameters. The antenna has good impedance bandwidth, nearly constant gain and stable radiation pattern. Measured return loss shows fair agreement with simulated one. Also measured group delay variation obtained is less than 1.0 ns, which proves good time domain behavior of the proposed antenna.
Fractal aggregates in Titan's atmosphere
NASA Astrophysics Data System (ADS)
Cabane, M.; Rannou, P.; Chassefiere, E.; Israel, G.
1993-04-01
The cluster structure of Titan's atmosphere was modeled by using an Eulerian microphysical model with the specific formulation of microphysical laws applying to fractal particles. The growth of aggregates in the settling phase was treated by introducing the fractal dimension as a parameter of the model. The model was used to obtain a vertical distribution of size and number density of the aggregates for different production altitudes. Results confirm previous estimates of the formation altitude of photochemical aerosols. The vertical profile of the effective radius of aggregates was calculated as a function of the visible optical depth.
Hydrodynamic behavior of fractal aggregates
NASA Astrophysics Data System (ADS)
Wiltzius, Pierre
1987-02-01
Measurements of the radius of gyration RG and the hydrodynamic radius RH of colloidal silica aggregates are reported. These aggregates have fractal geometry and RH is proportional to RG for 500 Å<=RH<=7000 Å, with a ratio RH/RG=0.72+/-0.02. The results are compared with predictions for macromolecules of various shapes. The proportionality of the two radii can be understood with use of the pair correlation function of fractal objects and hydrodynamic interactions on the Oseen level. The value of the ratio remains to be explained.
Fractal universe and quantum gravity.
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.
Image Segmentation via Fractal Dimension
1987-12-01
statistical expectation K = a proportionality constant H = the Hurst exponent , in interval [0,1] (14:249) Eq (4) is a mathematical generalization of...ease, negatively correlated (24:16). The Hurst exponent is directly related to the fractal diment.ion of the process being modelled by the relation (24...24) DzE.I -H (5) where D = the fractal dimension E m the Euclidean dimension H = the Hurst exponent The effect of N1 on a typical trace can be seen
Designing fractal nanostructured biointerfaces for biomedical applications.
Zhang, Pengchao; Wang, Shutao
2014-06-06
Fractal structures in nature offer a unique "fractal contact mode" that guarantees the efficient working of an organism with an optimized style. Fractal nanostructured biointerfaces have shown great potential for the ultrasensitive detection of disease-relevant biomarkers from small biomolecules on the nanoscale to cancer cells on the microscale. This review will present the advantages of fractal nanostructures, the basic concept of designing fractal nanostructured biointerfaces, and their biomedical applications for the ultrasensitive detection of various disease-relevant biomarkers, such microRNA, cancer antigen 125, and breast cancer cells, from unpurified cell lysates and the blood of patients.
Roughness Perception of Haptically Displayed Fractal Surfaces
NASA Technical Reports Server (NTRS)
Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)
2000-01-01
Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.
Astrophysical fractals - An overview and prospects
NASA Technical Reports Server (NTRS)
Perdang, J.
1990-01-01
Different astrophysical circumstances under which fractal structures have been identified so far, or are likely to be identified in the future, are reviewed. The observed fractals can be classified into 2 main groups: (1) fractal configurations in space-time, materializing as fractals defined over the time axis at a given position in space, or over the physical configuration space at a fixed instant in time; and (2) fractals in parameter spaces. The theoretical interpretation of the origin of the spatial fractal geometry of the most conspicuous 'irregular' astronomical bodies is still wanting in the context of standard continuum models. In contrast, the less conventional discrete models (cellular automata) naturally produce such spatially fractal structures.
Emergence of fractal scaling in complex networks.
Wei, Zong-Wen; Wang, Bing-Hong
2016-09-01
Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.
Emergence of fractal scaling in complex networks
NASA Astrophysics Data System (ADS)
Wei, Zong-Wen; Wang, Bing-Hong
2016-09-01
Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.
Fractal Characterization of Hyperspectral Imagery
NASA Technical Reports Server (NTRS)
Qiu, Hon-Iie; Lam, Nina Siu-Ngan; Quattrochi, Dale A.; Gamon, John A.
1999-01-01
Two Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) hyperspectral images selected from the Los Angeles area, one representing urban and the other, rural, were used to examine their spatial complexity across their entire spectrum of the remote sensing data. Using the ICAMS (Image Characterization And Modeling System) software, we computed the fractal dimension values via the isarithm and triangular prism methods for all 224 bands in the two AVIRIS scenes. The resultant fractal dimensions reflect changes in image complexity across the spectral range of the hyperspectral images. Both the isarithm and triangular prism methods detect unusually high D values on the spectral bands that fall within the atmospheric absorption and scattering zones where signature to noise ratios are low. Fractal dimensions for the urban area resulted in higher values than for the rural landscape, and the differences between the resulting D values are more distinct in the visible bands. The triangular prism method is sensitive to a few random speckles in the images, leading to a lower dimensionality. On the contrary, the isarithm method will ignore the speckles and focus on the major variation dominating the surface, thus resulting in a higher dimension. It is seen where the fractal curves plotted for the entire bandwidth range of the hyperspectral images could be used to distinguish landscape types as well as for screening noisy bands.
ERIC Educational Resources Information Center
Marks-Tarlow, Terry
2010-01-01
In this article, the author draws on contemporary science to illuminate the relationship between early play experiences, processes of self-development, and the later emergence of the fractal self. She argues that orientation within social space is a primary function of early play and developmentally a two-step process. With other people and with…
Exploring Fractal Geometry with Children.
ERIC Educational Resources Information Center
Vacc, Nancy Nesbitt
1999-01-01
Heightens the awareness of elementary school teachers, teacher educators, and teacher-education researchers of possible applications of fractal geometry with children and, subsequently, initiates discussion about the appropriateness of including this new mathematics in the elementary curriculum. Presents activities for exploring children's…
NASA Astrophysics Data System (ADS)
Tao, Xie; William, Perrie; Shang-Zhuo, Zhao; He, Fang; Wen-Jin, Yu; Yi-Jun, He
2016-07-01
Sea surface current has a significant influence on electromagnetic (EM) backscattering signals and may constitute a dominant synthetic aperture radar (SAR) imaging mechanism. An effective EM backscattering model for a one-dimensional drifting fractal sea surface is presented in this paper. This model is used to simulate EM backscattering signals from the drifting sea surface. Numerical results show that ocean currents have a significant influence on EM backscattering signals from the sea surface. The normalized radar cross section (NRCS) discrepancies between the model for a coupled wave-current fractal sea surface and the model for an uncoupled fractal sea surface increase with the increase of incidence angle, as well as with increasing ocean currents. Ocean currents that are parallel to the direction of the wave can weaken the EM backscattering signal intensity, while the EM backscattering signal is intensified by ocean currents propagating oppositely to the wave direction. The model presented in this paper can be used to study the SAR imaging mechanism for a drifting sea surface. Project supported by the National Natural Science Foundation of China (Grant No. 41276187), the Global Change Research Program of China (Grant No. 2015CB953901), the Priority Academic Program Development of Jiangsu Higher Education Institutions, China, the Program for the Innovation Research and Entrepreneurship Team in Jiangsu Province, China, the Canadian Program on Energy Research and Development, and the Canadian World Class Tanker Safety Service Program.
Electromagnetic field of fractal distribution of charged particles
Tarasov, Vasily E.
2005-08-15
Electric and magnetic fields of fractal distribution of charged particles are considered. The fractional integrals are used to describe fractal distribution. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of integral Maxwell equation, the simple examples of the fields of homogeneous fractal distribution are considered. The electric dipole and quadrupole moments for fractal distribution are derived.
Lung cancer-a fractal viewpoint.
Lennon, Frances E; Cianci, Gianguido C; Cipriani, Nicole A; Hensing, Thomas A; Zhang, Hannah J; Chen, Chin-Tu; Murgu, Septimiu D; Vokes, Everett E; Vannier, Michael W; Salgia, Ravi
2015-11-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed.
Order-fractal transitions in abstract paintings
Calleja, E.M. de la; Cervantes, F.; Calleja, J. de la
2016-08-15
In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.
Lung cancer—a fractal viewpoint
Lennon, Frances E.; Cianci, Gianguido C.; Cipriani, Nicole A.; Hensing, Thomas A.; Zhang, Hannah J.; Chen, Chin-Tu; Murgu, Septimiu D.; Vokes, Everett E.; W. Vannier, Michael; Salgia, Ravi
2016-01-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed. PMID:26169924
Fractal patterns of fractures in granites
Velde, B.; Dubois, J.; Moore, D.; Touchard, G.
1991-01-01
Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: 1. (1)|The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). 2. (2)|Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. 3. (3)|Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis. ?? 1991.
Fractal structures and fractal functions as disease indicators
Escos, J.M; Alados, C.L.; Emlen, J.M.
1995-01-01
Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.
Study on Huber fractal image compression.
Jeng, Jyh-Horng; Tseng, Chun-Chieh; Hsieh, Jer-Guang
2009-05-01
In this paper, a new similarity measure for fractal image compression (FIC) is introduced. In the proposed Huber fractal image compression (HFIC), the linear Huber regression technique from robust statistics is embedded into the encoding procedure of the fractal image compression. When the original image is corrupted by noises, we argue that the fractal image compression scheme should be insensitive to those noises presented in the corrupted image. This leads to a new concept of robust fractal image compression. The proposed HFIC is one of our attempts toward the design of robust fractal image compression. The main disadvantage of HFIC is the high computational cost. To overcome this drawback, particle swarm optimization (PSO) technique is utilized to reduce the searching time. Simulation results show that the proposed HFIC is robust against outliers in the image. Also, the PSO method can effectively reduce the encoding time while retaining the quality of the retrieved image.
Fractal applications to complex crustal problems
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
Complex scale-invariant problems obey fractal statistics. The basic definition of a fractal distribution is that the number of objects with a characteristic linear dimension greater than r satisfies the relation N = about r exp -D where D is the fractal dimension. Fragmentation often satisfies this relation. The distribution of earthquakes satisfies this relation. The classic relationship between the length of a rocky coast line and the step length can be derived from this relation. Power law relations for spectra can also be related to fractal dimensions. Topography and gravity are examples. Spectral techniques can be used to obtain maps of fractal dimension and roughness amplitude. These provide a quantitative measure of texture analysis. It is argued that the distribution of stress and strength in a complex crustal region, such as the Alps, is fractal. Based on this assumption, the observed frequency-magnitude relation for the seismicity in the region can be derived.
Fractal Weyl law for Linux Kernel architecture
NASA Astrophysics Data System (ADS)
Ermann, L.; Chepelianskii, A. D.; Shepelyansky, D. L.
2011-01-01
We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates.
Fractal model of anomalous diffusion.
Gmachowski, Lech
2015-12-01
An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion.
Chaos, Fractals and Their Applications
NASA Astrophysics Data System (ADS)
Thompson, J. Michael T.
2016-12-01
This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.
Fractal properties of financial markets
NASA Astrophysics Data System (ADS)
Budinski-Petković, Lj.; Lončarević, I.; Jakšić, Z. M.; Vrhovac, S. B.
2014-09-01
We present an analysis of the USA stock market using a simple fractal function. Financial bubbles preceding the 1987, 2000 and 2007 crashes are investigated using the Besicovitch-Ursell fractal function. Fits show a good agreement with the S&P 500 data when a complete financial growth is considered, starting at the threshold of the abrupt growth and ending at the peak. Moving the final time of the fitting interval towards earlier dates causes growing discrepancy between two curves. On the basis of a detailed analysis of the financial index behavior we propose a method for identifying the stage of the current financial growth and estimating the time in which the index value is going to reach the maximum.
Fractal globule as a molecular machine
NASA Astrophysics Data System (ADS)
Avetisov, V. A.; Ivanov, V. A.; Meshkov, D. A.; Nechaev, S. K.
2013-10-01
A fractal (crumpled) polymer globule, which is an unusual equilibrium state of a condensed unknotted macromolecule that is experimentally found in the DNA folding in human chromosomes, has been formed through the hierarchical collapse of a polymer chain. The relaxation dynamics of the elastic network constructed through the contact matrix of the fractal globule has been studied. It has been found that the fractal globule in its dynamic properties is similar to a molecular machine.
Fractal scattering of microwaves from soils.
Oleschko, K; Korvin, G; Balankin, A S; Khachaturov, R V; Flores, L; Figueroa, B; Urrutia, J; Brambila, F
2002-10-28
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.
Comparison of two fractal interpolation methods
NASA Astrophysics Data System (ADS)
Fu, Yang; Zheng, Zeyu; Xiao, Rui; Shi, Haibo
2017-03-01
As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass-Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass-Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass-Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass-Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass-Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass-Mandelbrot fractal function method has
Lorimer, N.D.; Haight, R.G.; Leary, R.A.
1994-07-20
Fractal geometry is a tool for describing and analyzing irregularity. Because most of what we measure in the forest is discontinuous, jagged, and fragmented, fractal geometry has potential for improving the precision of measurement and description. The study reviews the literature on fractal geometry and its applications to forest measurements.
Fractal properties of quantum spacetime.
Benedetti, Dario
2009-03-20
We show that, in general, a spacetime having a quantum group symmetry has also a scale-dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what is observed in some approaches to quantum gravity. In particular, we analyze the cases of a quantum sphere and of kappa-Minkowski spacetime, the latter being relevant in the context of quantum gravity.
Fractal Metrology for biogeosystems analysis
NASA Astrophysics Data System (ADS)
Torres-Argüelles, V.; Oleschko, K.; Tarquis, A. M.; Korvin, G.; Gaona, C.; Parrot, J.-F.; Ventura-Ramos, E.
2010-11-01
The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes of this complex system. In the present research, we studied the aggregation process as self-organizing and operating near a critical point. The structural pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate" Clay) and compared in terms of roughness of the gray-intensity distribution quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them measured in terms of standard deviation. Some of the applied methods are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc.) while the others have been recently developed by our Group. The combination of these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM). We show the usefulness of FM for complex systems analysis through a case study of the soil's physical and chemical degradation applying the selected toolbox to describe and compare the structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
Fractal metrology for biogeosystems analysis
NASA Astrophysics Data System (ADS)
Torres-Argüelles, V.; Oleschko, K.; Tarquis, A. M.; Korvin, G.; Gaona, C.; Parrot, J.-F.; Ventura-Ramos, E.
2010-06-01
The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes. In the present research, this pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate'' Clay) and compared in terms of roughness of the gray-intensity distribution (the measurand) quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them and to the measurement function which best fits to the experimental results. Some of the applied techniques are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc.) while the others have been recently developed by our Group. The combination of all these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM). We show the usefulness of FM through a case study of soil physical and chemical degradation applying the selected toolbox to describe and compare the main structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
Fractal analysis of complex microstructure in castings
Lu, S.Z.; Lipp, D.C.; Hellawell, A.
1995-12-31
Complex microstructures in castings are usually characterized descriptively which often raises ambiguity and makes it difficult to relate the microstructure to the growth kinetics or mechanical properties in processing modeling. Combining the principle of fractal geometry and computer image processing techniques, it is feasible to characterize the complex microstructures numerically by the parameters of fractal dimension, D, and shape factor, a, without ambiguity. Procedures of fractal measurement and analysis are described, and a test case of its application to cast irons is provided. The results show that the irregular cast structures may all be characterized numerically by fractal analysis.
Generalized Mandelbrot rule for fractal sections
NASA Astrophysics Data System (ADS)
Meisel, L. V.
1992-01-01
Mandelbrot's rule for sections is generalized to apply to the Hentschel and Procaccia fractal dimension at arbitrary q and on arbitrary sections. It is shown that for almost all (n-m)-dimensional sections, Dn(q)=Dn-m(q)+m, where the Dr(q) are box-counting, Hentschel, and Procaccia generalized fractal dimensions of r-dimensional sections of homogeneous fractal point sets in Rn and Dn-m(q)>0. The rule applies for finite ``thickness'' sections as well as ``true'' sections and can be interpreted for inhomogenous fractal sets.
Stability limits for bioconvective fractals - Microgravity prospects
NASA Technical Reports Server (NTRS)
Noever, David A.
1992-01-01
Fractal objects are delicate aggregates which show self-similar behavior and vanishing density for increasing length scales. In practice real fractals in nature however possess only a limited region of verifiable self-similarity. As natural fractal objects increase in size, they become easier to disrupt mechanically. Herein the effects of thermal vibrations and gravity are investigated as deforming forces on fractal aggregation. Example calculations are carried out on a biological fractal formed from the surface aggregation of various cells such as alga and bacteria. For typical cell parameters, the predicted diameter of this so-called 'bioconvective' fractal agrees well with the observed limits of about 5 cm. On earth, this size represents an experimental maximum for finding bioconvective fractal objects. To extend this size range of fractals available for statistical study, a reduced gravity environment offers one way to achieve larger fractals. For these enhanced sizes, the present scaling predicts that microgravity can yield up to a 35-fold improvement in extending statistical resolution.
Estimation of Surface Soil Moisture Using Fractal
NASA Astrophysics Data System (ADS)
Chen, Yen Chang; He, Chun Hsuan
2016-04-01
This study establishes the relationship between surface soil moisture and fractal dimension. The surface soil moisture is one of important factors in the hydrological cycle of surface evaporation. It could be used in many fields, such as reservoir management, early drought warning systems, irrigation scheduling and management, and crop yield estimations. Soil surface cracks due to dryness can be used to describe drought conditions. Soil cracking phenomenon and moisture have a certain relationship, thus this study makes used the fractal theory to interpret the soil moisture represented by soil cracks. The fractal dimension of surface soil cracking is a measure of the surface soil moisture. Therefore fractal dimensions can also be used to indicate how dry of the surface soil is. This study used the sediment in the Shimen Reservoir to establish the fractal dimension and soil moisture relation. The soil cracking is created under the control of temperature and thickness of surface soil layers. The results show the increase in fractal dimensions is accompanied by a decreases in surface soil moisture. However the fractal dimensions will approach a constant even the soil moisture continually decreases. The sigmoid function is used to fit the relation of fractal dimensions and surface soil moistures. The proposed method can be successfully applied to estimate surface soil moisture. Only a photo taken from the field is needed and is sufficient to provide the fractal dimension. Consequently, the surface soil moisture can be estimated quickly and accurately.
Fractal signatures in the aperiodic Fibonacci grating.
Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam
2014-05-01
The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.
Computer simulation of microwave propagation in heterogeneous and fractal media
NASA Astrophysics Data System (ADS)
Korvin, Gabor; Khachaturov, Ruben V.; Oleschko, Klaudia; Ronquillo, Gerardo; Correa López, María de jesús; García, Juan-josé
2017-03-01
Maxwell's equations (MEs) are the starting point for all calculations involving surface or borehole electromagnetic (EM) methods in Petroleum Industry. In well-log analysis numerical modeling of resistivity and induction tool responses has became an indispensable step of interpretation. We developed a new method to numerically simulate electromagnetic wave propagation through heterogeneous and fractal slabs taking into account multiple scattering in the direction of normal incidence. In simulation, the gray-scale image of the porous medium is explored by monochromatic waves. The gray-tone of each pixel can be related to the dielectric permittivity of the medium at that point by two different equations (linear dependence, and fractal or power law dependence). The wave equation is solved in second order difference approximation, using a modified sweep technique. Examples will be shown for simulated EM waves in carbonate rocks imaged at different scales by electron microscopy and optical photography. The method has wide ranging applications in remote sensing, borehole scanning and Ground Penetrating Radar (GPR) exploration.
Fractal Music: The Mathematics Behind "Techno" Music
ERIC Educational Resources Information Center
Padula, Janice
2005-01-01
This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on…
Undergraduate Experiment with Fractal Diffraction Gratings
ERIC Educational Resources Information Center
Monsoriu, Juan A.; Furlan, Walter D.; Pons, Amparo; Barreiro, Juan C.; Gimenez, Marcos H.
2011-01-01
We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics…
The Fractal Nature of Relevance: A Hypothesis.
ERIC Educational Resources Information Center
Ottaviani, J. S.
1994-01-01
Discusses precision and recall in information science and proposes a new model based on fractal geometry for clusters of relevant documents. Search strategies for retrieving a group of relevant documents are reviewed; fractal sets and chaotic processes are described; and the new model is explained. (Contains 43 references.) (LRW)
Fractal Geometry in Elementary School Mathematics.
ERIC Educational Resources Information Center
Vacc, Nancy Nesbitt
1992-01-01
Reports a case study to evaluate whether basic concepts of fractal geometry are teachable to elementary school children and to determine the effectiveness of having an elementary school student present a lesson to inservice and preservice teachers. Concludes that simple concepts of fractal geometry appear appropriate for the elementary school…
Fractal nanoparticle plasmonics: the Cayley tree.
Gottheim, Samuel; Zhang, Hui; Govorov, Alexander O; Halas, Naomi J
2015-03-24
There has been strong, ongoing interest over the past decade in developing strategies to design and engineer materials with tailored optical properties. Fractal-like nanoparticles and films have long been known to possess a remarkably broad-band optical response and are potential nanoscale components for realizing spectrum-spanning optical effects. Here we examine the role of self-similarity in a fractal geometry for the design of plasmon line shapes. By computing and fabricating simple Cayley tree nanostructures of increasing fractal order N, we are able to identify the principle behind how the multimodal plasmon spectrum of this system develops as the fractal order is increased. With increasing N, the fractal structure acquires an increasing number of modes with certain degeneracies: these modes correspond to plasmon oscillations on the different length scales inside a fractal. As a result, fractals with large N exhibit broad, multipeaked spectra from plasmons with large degeneracy numbers. The Cayley tree serves as an example of a more general, fractal-based route for the design of structures and media with highly complex optical line shapes.
A Classroom Demonstration of Electrodeposited Fractal Patterns.
ERIC Educational Resources Information Center
Garcia, Edelfredo; Liu, C. H.
1995-01-01
Presents an inexpensive laboratory experiment that combines the recommended techniques for teaching fractal geometry in the classroom with the standard procedures for studying electrochemical deposition of ramified patterns in the regime of low solution concentration and low applied constant driving force. Introduces students to fractal growth…
Fractal Trigonometric Polynomials for Restricted Range Approximation
NASA Astrophysics Data System (ADS)
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
Segmentation of histological structures for fractal analysis
NASA Astrophysics Data System (ADS)
Dixon, Vanessa; Kouznetsov, Alexei; Tambasco, Mauro
2009-02-01
Pathologists examine histology sections to make diagnostic and prognostic assessments regarding cancer based on deviations in cellular and/or glandular structures. However, these assessments are subjective and exhibit some degree of observer variability. Recent studies have shown that fractal dimension (a quantitative measure of structural complexity) has proven useful for characterizing structural deviations and exhibits great potential for automated cancer diagnosis and prognosis. Computing fractal dimension relies on accurate image segmentation to capture the architectural complexity of the histology specimen. For this purpose, previous studies have used techniques such as intensity histogram analysis and edge detection algorithms. However, care must be taken when segmenting pathologically relevant structures since improper edge detection can result in an inaccurate estimation of fractal dimension. In this study, we established a reliable method for segmenting edges from grayscale images. We used a Koch snowflake, an object of known fractal dimension, to investigate the accuracy of various edge detection algorithms and selected the most appropriate algorithm to extract the outline structures. Next, we created validation objects ranging in fractal dimension from 1.3 to 1.9 imitating the size, structural complexity, and spatial pixel intensity distribution of stained histology section images. We applied increasing intensity thresholds to the validation objects to extract the outline structures and observe the effects on the corresponding segmentation and fractal dimension. The intensity threshold yielding the maximum fractal dimension provided the most accurate fractal dimension and segmentation, indicating that this quantitative method could be used in an automated classification system for histology specimens.
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.
2015-04-01
This paper is devoted to the mechanics of fractally heterogeneous media. A model of fractal continuum with a fractional number of spatial degrees of freedom and a fractal metric is suggested. The Jacobian matrix of the fractal continuum deformation is defined and the kinematics of deformations is elucidated. The symmetry of the Cauchy stress tensor for continua with the fractal metric is established. A homogenization framework accounting for the connectivity, topological, and metric properties of fractal domains in heterogeneous materials is developed. The mapping of mechanical problems for fractal media into the corresponding problems for the fractal continuum is discussed. Stress and strain distributions in elastic fractal bars are analyzed. An approach to fractal bar optimization is proposed. Some features of acoustic wave propagation and localization in fractal media are briefly highlighted.
Applications of Variance Fractal Dimension: a Survey
NASA Astrophysics Data System (ADS)
Phinyomark, Angkoon; Phukpattaranont, Pornchai; Limsakul, Chusak
2012-04-01
Chaotic dynamical systems are pervasive in nature and can be shown to be deterministic through fractal analysis. There are numerous methods that can be used to estimate the fractal dimension. Among the usual fractal estimation methods, variance fractal dimension (VFD) is one of the most significant fractal analysis methods that can be implemented for real-time systems. The basic concept and theory of VFD are presented. Recent research and the development of several applications based on VFD are reviewed and explained in detail, such as biomedical signal processing and pattern recognition, speech communication, geophysical signal analysis, power systems and communication systems. The important parameters that need to be considered in computing the VFD are discussed, including the window size and the window increment of the feature, and the step size of the VFD. Directions for future research of VFD are also briefly outlined.
Designing Phoxonic Metamaterials with Fractal Geometry
NASA Astrophysics Data System (ADS)
Ni, Sisi; Koh, Cheong Yang; Kooi, Steve; Thomas, Edwin
2012-02-01
Recently, the concepts of fractal geometry have been introduced into electromagnetic and plasmonic metamaterials. With their self-similarity, structures based on fractal geometry should exhibit multi-band character with high Q factors due to the scaling law. However, there exist few studies of phononic metamaterials based on fractal geometry. We use COMSOL to investigate the wave propagation in two dimensional systems possessing fractal geometries. The simulations of these systems, guided by our recently developed general design framework, help to understand the role of design in determining the phononic properties of the structures. Proposed structures are being fabricated via standard lithographic or 3D printing techniques. The wave behavior of the structures can be characterized using Brillouin Light Scattering, Scanning Acoustic Microscope and Near-field Scanning Optical Microscopy. Due to their sparse spatial distribution, fractal phononic structures show potential fir ``smart skin'', where multifunctional components can be fabricated on the same platform.
Fractal structures in nonlinear plasma physics.
Viana, R L; da Silva, E C; Kroetz, T; Caldas, I L; Roberto, M; Sanjuán, M A F
2011-01-28
Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.
Random sequential adsorption on fractals.
Ciesla, Michal; Barbasz, Jakub
2012-07-28
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions.
Fragmentation of Fractal Random Structures
NASA Astrophysics Data System (ADS)
Elçi, Eren Metin; Weigel, Martin; Fytas, Nikolaos G.
2015-03-01
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.
Fractal lattice of gelatin nanoglobules
NASA Astrophysics Data System (ADS)
Novikov, D. V.; Krasovskii, A. N.
2012-11-01
The globular structure of polymer coatings on a glass, which were obtained from micellar solutions of gelatin in the isooctane-water-sodium (bis-2-ethylhexyl) sulfosuccinate system, has been studied using electron microscopy. It has been shown that an increase in the average globule size is accompanied by the formation of a fractal lattice of nanoglobules and a periodic physical network of macromolecules in the coating. The stability of such system of the "liquid-in-a-solid" type is limited by the destruction of globules and the formation of a homogeneous network structure of the coating.
Fractal dimension of bioconvection patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1990-01-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.
Fractal Dimension of Bioconvection Patterns
NASA Astrophysics Data System (ADS)
Noever, David A.
1990-10-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2× 106 organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching and a fractal dimension (d˜1.7). These agree well with the two-dimensional DLA.
Pre-Service Teachers' Concept Images on Fractal Dimension
ERIC Educational Resources Information Center
Karakus, Fatih
2016-01-01
The analysis of pre-service teachers' concept images can provide information about their mental schema of fractal dimension. There is limited research on students' understanding of fractal and fractal dimension. Therefore, this study aimed to investigate the pre-service teachers' understandings of fractal dimension based on concept image. The…
Fractal gait patterns are retained after entrainment to a fractal stimulus.
Rhea, Christopher K; Kiefer, Adam W; Wittstein, Matthew W; Leonard, Kelsey B; MacPherson, Ryan P; Wright, W Geoffrey; Haran, F Jay
2014-01-01
Previous work has shown that fractal patterns in gait can be altered by entraining to a fractal stimulus. However, little is understood about how long those patterns are retained or which factors may influence stronger entrainment or retention. In experiment one, participants walked on a treadmill for 45 continuous minutes, which was separated into three phases. The first 15 minutes (pre-synchronization phase) consisted of walking without a fractal stimulus, the second 15 minutes consisted of walking while entraining to a fractal visual stimulus (synchronization phase), and the last 15 minutes (post-synchronization phase) consisted of walking without the stimulus to determine if the patterns adopted from the stimulus were retained. Fractal gait patterns were strengthened during the synchronization phase and were retained in the post-synchronization phase. In experiment two, similar methods were used to compare a continuous fractal stimulus to a discrete fractal stimulus to determine which stimulus type led to more persistent fractal gait patterns in the synchronization and post-synchronization (i.e., retention) phases. Both stimulus types led to equally persistent patterns in the synchronization phase, but only the discrete fractal stimulus led to retention of the patterns. The results add to the growing body of literature showing that fractal gait patterns can be manipulated in a predictable manner. Further, our results add to the literature by showing that the newly adopted gait patterns are retained for up to 15 minutes after entrainment and showed that a discrete visual stimulus is a better method to influence retention.
An Approach to Study Elastic Vibrations of Fractal Cylinders
NASA Astrophysics Data System (ADS)
Steinberg, Lev; Zepeda, Mario
2016-11-01
This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.
Snow metamorphism: A fractal approach.
Carbone, Anna; Chiaia, Bernardino M; Frigo, Barbara; Türk, Christian
2010-09-01
Snow is a porous disordered medium consisting of air and three water phases: ice, vapor, and liquid. The ice phase consists of an assemblage of grains, ice matrix, initially arranged over a random load bearing skeleton. The quantitative relationship between density and morphological characteristics of different snow microstructures is still an open issue. In this work, a three-dimensional fractal description of density corresponding to different snow microstructure is put forward. First, snow density is simulated in terms of a generalized Menger sponge model. Then, a fully three-dimensional compact stochastic fractal model is adopted. The latter approach yields a quantitative map of the randomness of the snow texture, which is described as a three-dimensional fractional Brownian field with the Hurst exponent H varying as continuous parameters. The Hurst exponent is found to be strongly dependent on snow morphology and density. The approach might be applied to all those cases where the morphological evolution of snow cover or ice sheets should be conveniently described at a quantitative level.
Antenna Miniaturization Using Koch Snowflake Fractal Geometry
NASA Astrophysics Data System (ADS)
Minal, Dhama, Nitin
2010-11-01
The Wireless Industry is witnessing an volatile emergence today in present era. Also requires the performance over several frequency bands or are reconfigurable as the demands on the system changes. This Paper Presents Rectangular, Koch Fractal Patch Antennas on Single and Multilayer Substrate With and Without Air-Gap using Advanced Design System Simulator (ADS). Fractal Antenna provides Miniaturization over conventional microstrip Antennas. The Antennas Have Been Designed on FR4 substrate with ∈ = 4.2, h = 1.53 and the initial Dimension of the simple Rectangular Patch is 36.08 * 29.6 mm. The experimental Resonant Frequencies of the Fractal Patch with 1st, 2nd & 3rd are observed 2.22, 2.14 & 2.02 GHz Respectively in comparison to Rectangular Patch with 2.43 GHz. The reduced Impedance bandwidth of the Fractal Patch has been improved by designing the patch over multilayer substrate with varying Air-gap between two Substrate. As we increase the air- gap between the two substrate layer further enhancement in impedance bandwidth of Fractal antenna has been Obtained. The Radiation pattern of Koch Fractal antenna is as similar to rectangular patch antenna but with better H-plane Cross Polarization for fractal patch. The all simulated Results are in close Agreement with experimental Results.
Hexagonal and Pentagonal Fractal Multiband Antennas
NASA Technical Reports Server (NTRS)
Tang, Philip W.; Wahid, Parveen
2005-01-01
Multiband dipole antennas based on hexagonal and pentagonal fractals have been analyzed by computational simulations and functionally demonstrated in experiments on prototypes. These antennas are capable of multiband or wide-band operation because they are subdivided into progressively smaller substructures that resonate at progressively higher frequencies by virtue of their smaller dimensions. The novelty of the present antennas lies in their specific hexagonal and pentagonal fractal configurations and the resonant frequencies associated with them. These antennas are potentially applicable to a variety of multiband and wide-band commercial wireless-communication products operating at different frequencies, including personal digital assistants, cellular telephones, pagers, satellite radios, Global Positioning System receivers, and products that combine two or more of the aforementioned functions. Perhaps the best-known prior multiband antenna based on fractal geometry is the Sierpinski triangle antenna (also known as the Sierpinski gasket), shown in the top part of the figure. In this antenna, the scale length at each iteration of the fractal is half the scale length of the preceding iteration, yielding successive resonant frequencies related by a ratio of about 2. The middle and bottom parts of the figure depict the first three iterations of the hexagonal and pentagonal fractals along with typical dipole-antenna configuration based on the second iteration. Successive resonant frequencies of the hexagonal fractal antenna have been found to be related by a ratio of about 3, and those of the pentagonal fractal antenna by a ratio of about 2.59.
Kinetic properties of fractal stellar media
NASA Astrophysics Data System (ADS)
Chumak, O. V.; Rastorguev, A. S.
2017-01-01
Kinetic processes in fractal stellar media are analysed in terms of the approach developed in our earlier paper involving a generalization of the nearest neighbour and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time-scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case, kinetic parameters depend on spatial scalelength and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties of the spatial stellar density distribution. Also derived are its limit forms that can be used to describe small departures of fractal gravitating medium from equilibrium.
Molecular dynamics simulation of fractal aggregate diffusion
NASA Astrophysics Data System (ADS)
Pranami, Gaurav; Lamm, Monica H.; Vigil, R. Dennis
2010-11-01
The diffusion of fractal aggregates constructed with the method by Thouy and Jullien [J. Phys. A 27, 2953 (1994)10.1088/0305-4470/27/9/012] comprised of Np spherical primary particles was studied as a function of the aggregate mass and fractal dimension using molecular dynamics simulations. It is shown that finite-size effects have a strong impact on the apparent value of the diffusion coefficient (D) , but these can be corrected by carrying out simulations using different simulation box sizes. Specifically, the diffusion coefficient is inversely proportional to the length of a cubic simulation box, and the constant of proportionality appears to be independent of the aggregate mass and fractal dimension. Using this result, it is possible to compute infinite dilution diffusion coefficients (Do) for aggregates of arbitrary size and fractal dimension, and it was found that Do∝Np-1/df , as is often assumed by investigators simulating Brownian aggregation of fractal aggregates. The ratio of hydrodynamic radius to radius of gyration is computed and shown to be independent of mass for aggregates of fixed fractal dimension, thus enabling an estimate of the diffusion coefficient for a fractal aggregate based on its radius of gyration.
On the ubiquitous presence of fractals and fractal concepts in pharmaceutical sciences: a review.
Pippa, Natassa; Dokoumetzidis, Aristides; Demetzos, Costas; Macheras, Panos
2013-11-18
Fractals have been very successful in quantifying nature's geometrical complexity, and have captured the imagination of scientific community. The development of fractal dimension and its applications have produced significant results across a wide variety of biomedical applications. This review deals with the application of fractals in pharmaceutical sciences and attempts to account the most important developments in the fields of pharmaceutical technology, especially of advanced Drug Delivery nano Systems and of biopharmaceutics and pharmacokinetics. Additionally, fractal kinetics, which has been applied to enzyme kinetics, drug metabolism and absorption, pharmacokinetics and pharmacodynamics are presented. This review also considers the potential benefits of using fractal analysis along with considerations of nonlinearity, scaling, and chaos as calibration tools to obtain information and more realistic description on different parts of pharmaceutical sciences. As a conclusion, the purpose of the present work is to highlight the presence of fractal geometry in almost all fields of pharmaceutical research.
Fractal and multifractal analysis: a review.
Lopes, R; Betrouni, N
2009-08-01
Over the last years, fractal and multifractal geometries were applied extensively in many medical signal (1D, 2D or 3D) analysis applications like pattern recognition, texture analysis and segmentation. Application of this geometry relies heavily on the estimation of the fractal features. Various methods were proposed to estimate the fractal dimension or multifractal spectral of a signal. This article presents an overview of these algorithms, the way they work, their benefits and their limits. The aim of this review is to explain and to categorize the various algorithms into groups and their application in the field of medical signal analysis.
Characterization of strange attractors as inhomogeneous fractals
NASA Astrophysics Data System (ADS)
Paladin, G.; Vulpiani, A.
1984-09-01
The geometry of strange attractors of chaotic dynamical systems is investigated analytically within the framework of fractal theory. A set of easily computable exponents which generalize the fractal dimensionality and characterize the inhomogeneity of the fractals of strange attractors is derived, and sample computations are shown. It is pointed out that the fragmentation process described is similar to models of intermittency in fully developed turbulence. The exponents for the sample problems are computed in the same amount of CPU time as the computation of nu by the method of Grassberger and Procaccia (1983) but provide more information; less time is required than for the nu(n) computation of Hentschel and Procaccia (1983).
Fractal-based image edge detection
NASA Astrophysics Data System (ADS)
Luo, Huiguo; Zhu, Yaoting; Zhu, Guang-Xi; Wan, Faguang; Zhang, Ping
1993-08-01
Image edge is an important feature of image. Usually, we use Laplacian or Sober operator to get an image edge. In this paper, we use fractal method to get the edge. After introducing Fractal Brownian Random (FBR) field, we give the definition of Discrete Fractal Brownian Increase Random (DFBIR) field and discuss its properties, then we apply the DFBIR field to detect the edge of an image. According to the parameters H and D of DFBIR, we give a measure M equals (alpha) H + (beta) D. From the M value of each pixel, we can detect the edge of image.
Application of Fractal Dimension on Palsar Data
NASA Astrophysics Data System (ADS)
Singh, Dharmendra; Pant, Triloki
Study of land cover is the primal task of remote sensing where microwave imaging plays an important role. As an alternate of optical imaging, microwave, in particular, Synthetic Aperture Radar (SAR) imaging is very popular. With the advancement of technology, multi-polarized images are now available, e.g., ALOS-PALSAR (Phased Array type L-band SAR), which are beneficial because each of the polarization channel shows different sensitivity to various land features. Further, using the textural features, various land classes can be classified on the basis of the textural measures. One of the textural measure is fractal dimension. It is noteworthy that fractal dimension is a measure of roughness and thus various land classes can be distinguished on the basis of their roughness. The value of fractal dimension for the surfaces lies between 2.0 and 3.0 where 2.0 represents a smooth surface while 3.0 represents drastically rough surface. The study area covers subset images lying between 2956'53"N, 7750'32"E and 2950'40"N, 7757'19"E. The PALSAR images of the year 2007 and 2009 are considered for the study. In present paper a fractal based classification of PALSAR images has been performed for identification of Water, Urban and Agricultural Area. Since fractals represent the image texture, hence the present study attempts to find the fractal properties of land covers to distinguish them from one another. For the purpose a context has been defined on the basis of a moving window, which is used to estimate the local fractal dimension and then moved over the whole image. The size of the window is an important issue for estimation of textural measures which is considered to be 55 in present study. This procedure, in response, produces a textural map called fractal map. The fractal map is constituted with the help of local fractal dimension values and can be used for contextual classification. In order to study the fractal properties of PALSAR images, the three polarization images
Recurrence Quantification of Fractal Structures
Webber, Charles L.
2012-01-01
By definition, fractal structures possess recurrent patterns. At different levels repeating patterns can be visualized at higher magnifications. The purpose of this chapter is threefold. First, general characteristics of dynamical systems are addressed from a theoretical mathematical perspective. Second, qualitative and quantitative recurrence analyses are reviewed in brief, but the reader is directed to other sources for explicit details. Third, example mathematical systems that generate strange attractors are explicitly defined, giving the reader the ability to reproduce the rich dynamics of continuous chaotic flows or discrete chaotic iterations. The challenge is then posited for the reader to study for themselves the recurrent structuring of these different dynamics. With a firm appreciation of the power of recurrence analysis, the reader will be prepared to turn their sights on real-world systems (physiological, psychological, mechanical, etc.). PMID:23060808
Fractal cartography of urban areas
Encarnação, Sara; Gaudiano, Marcos; Santos, Francisco C.; Tenedório, José A.; Pacheco, Jorge M.
2012-01-01
In a world in which the pace of cities is increasing, prompt access to relevant information is crucial to the understanding and regulation of land use and its evolution in time. In spite of this, characterization and regulation of urban areas remains a complex process, requiring expert human intervention, analysis and judgment. Here we carry out a spatio-temporal fractal analysis of a metropolitan area, based on which we develop a model which generates a cartographic representation and classification of built-up areas, identifying (and even predicting) those areas requiring the most proximate planning and regulation. Furthermore, we show how different types of urban areas identified by the model co-evolve with the city, requiring policy regulation to be flexible and adaptive, acting just in time. The algorithmic implementation of the model is applicable to any built-up area and simple enough to pave the way for the automatic classification of urban areas worldwide. PMID:22829981
Fractal cartography of urban areas.
Encarnação, Sara; Gaudiano, Marcos; Santos, Francisco C; Tenedório, José A; Pacheco, Jorge M
2012-01-01
In a world in which the pace of cities is increasing, prompt access to relevant information is crucial to the understanding and regulation of land use and its evolution in time. In spite of this, characterization and regulation of urban areas remains a complex process, requiring expert human intervention, analysis and judgment. Here we carry out a spatio-temporal fractal analysis of a metropolitan area, based on which we develop a model which generates a cartographic representation and classification of built-up areas, identifying (and even predicting) those areas requiring the most proximate planning and regulation. Furthermore, we show how different types of urban areas identified by the model co-evolve with the city, requiring policy regulation to be flexible and adaptive, acting just in time. The algorithmic implementation of the model is applicable to any built-up area and simple enough to pave the way for the automatic classification of urban areas worldwide.
Fractal characteristics of ozonometric network
NASA Technical Reports Server (NTRS)
Gruzdev, Alexander N.
1994-01-01
The fractal (correlation) dimensions are calculated which characterize the distribution of stations in the ground-based total ozone measuring network and the distribution of nodes in a latitude-longitude grid. The dimension of the ground-based ozonometric network equals 1.67 +/- 0.1 with an appropriate scaling in the 60 to 400 km range. For the latitude-longitude grid two scaling regimes are revealed. One regime, with the dimension somewhat greater than one, is peculiar to smaller scales and limited from a larger scale by the latitudinal resolution of the grid. Another scaling regime, with the dimension equal 1.84, ranges up to 15,000 km scale. The fact that the dimension of a measuring network is less than two possesses problems in observing sparse phenomena. This has to have important consequences for ozone statistics.
Nanoflow over a fractal surface
NASA Astrophysics Data System (ADS)
Papanikolaou, Michail; Frank, Michael; Drikakis, Dimitris
2016-08-01
This paper investigates the effects of surface roughness on nanoflows using molecular dynamics simulations. A fractal model is employed to model wall roughness, and simulations are performed for liquid argon confined by two solid walls. It is shown that the surface roughness reduces the velocity in the proximity of the walls with the reduction being accentuated when increasing the roughness depth and wettability of the solid wall. It also makes the flow three-dimensional and anisotropic. In flows over idealized smooth surfaces, the liquid forms parallel, well-spaced layers, with a significant gap between the first layer and the solid wall. Rough walls distort the orderly distribution of fluid layers resulting in an incoherent formation of irregularly shaped fluid structures around and within the wall cavities.
Fractal Signals & Space-Time Cartoons
NASA Astrophysics Data System (ADS)
Oetama, -Hc, Jakob, , Dr; Maksoed, Wh-
2016-03-01
In ``Theory of Scale Relativity'', 1991- L. Nottale states whereas ``scale relativity is a geometrical & fractal space-time theory''. It took in comparisons to ``a unified, wavelet based framework for efficiently synthetizing, analyzing ∖7 processing several broad classes of fractal signals''-Gregory W. Wornell:``Signal Processing with Fractals'', 1995. Furthers, in Fig 1.1. a simple waveform from statistically scale-invariant random process [ibid.,h 3 ]. Accompanying RLE Technical Report 566 ``Synthesis, Analysis & Processing of Fractal Signals'' as well as from Wornell, Oct 1991 herewith intended to deducts =a Δt + (1 - β Δ t) ...in Petersen, et.al: ``Scale invariant properties of public debt growth'',2010 h. 38006p2 to [1/{1- (2 α (λ) /3 π) ln (λ/r)}depicts in Laurent Nottale,1991, h 24. Acknowledgment devotes to theLates HE. Mr. BrigadierGeneral-TNI[rtd].Prof. Ir. HANDOJO.
Fractal Geometry in the High School Classroom.
ERIC Educational Resources Information Center
Camp, Dane R.
1995-01-01
Discusses classroom activities that involve applications of fractal geometry. Includes an activity sheet that explores Pascal's triangle, Sierpinsky's gasket, and modular arithmetic in two and three dimensions. (Author/MKR)
Fractal geometry is heritable in trees.
Bailey, Joseph K; Bangert, Randy K; Schweitzer, Jennifer A; Trotter, R Talbot; Shuster, Stephen M; Whitham, Thomas G
2004-09-01
Understanding the genetic basis to landscape vegetation structure is an important step that will allow us to examine ecological and evolutionary processes at multiple spatial scales. Here for the first time we show that the fractal architecture of a dominant plant on the landscape exhibits high broad-sense heritability and thus has a genetic basis. The fractal architecture of trees is known to influence ecological communities associated with them. In a unidirectional cottonwood-hybridizing complex (Populus angustifolia x P. fremontii) pure and hybrid cottonwoods differed significantly in their fractal architecture, with phenotypic variance among backcross hybrids exceeding that of F1 hybrids and of pure narrowleaf cottonwoods by two-fold. This result provides a crucial link between genes and fractal scaling theory, and places the study of landscape ecology within an evolutionary framework.
Fractal analysis of fracture in glass ceramics
Mecholsky, J.J. Jr.
1995-12-01
The application of fractal geometry offers the potential to establish the scaling relationships between critical energies at all levels during fracture. The fracture energy, {gamma}, is directly related to the fractal dimensional increment, D*, and the elastic modulus, E: {gamma} = 0.5 E D* a{sub 0}, where the characteristic atomic parameter, a{sub 0}, can be interpreted as the fractal generator (on the atomic scale) or the process zone size. D* can be shown to be related to the crack/fracture-mirror size ratio. The purpose of this paper is to show the relationship between fractal geometry, fractography and the fracture process in the analysis of the fracture of glass ceramics. Examples of the fracture analysis of glass ceramics used as materials for dental restorations, radomes and electron generators will be presented.
Fractals in petroleum geology and earth processes
Barton, C.C.; La Pointe, P.R.
1995-12-31
The editors of this book chose a diverse spectrum of papers written by pioneers in the field of fractals and their application to the exploration and production of hydrocarbons. The geology of the Earth`s crust is complex, chaotic, and unpredictable. Fractal geometry can quantify the spatial heterogeneity of the different geologic patterns and ultimately help improve the results of both production and exploration. To this goal the book has accomplished such an objective with diverse, well-chosen contributions from a variety of experts in the field. The book starts with a chapter introducing the basics, with a short historical foot-note by Benoit Mandelbrot, who is considered the {open_quotes}father of fractals.{close_quotes} Mandelbrot emphasized that geologic processes not only exhibit fractal properties but also are strongly connected to the economic system. This paved the way for the next three chapters that deal with the size and spatial distribution of hydrocarbon reserves and their importance in economic evaluations. The following four chapters deal with the fractal processes as related to sedimentologic, stratigraphic, and geomorphologic systems. Chapter five is an interesting one that deals with stratigraphic models and how their fractal processes can be tied with the inter-well correlation and reconstruct depositional environments. The next three chapters are concerned with porous and fractured rocks and how they affect the flow of fluids. The last two chapters (chapters 13 and 14) are of particular interest. Chapter 13 deals with the vertical vs. horizontal well-log variability and application to fractal reservoir modeling. Chapter 14 illustrates how fractal geometry brings mathematical order to geological and geophysical disorder. This is evident when dealing with geophysical modeling and inversion.
Fractal analysis of DNA sequence data
Berthelsen, C.L.
1993-01-01
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.
Fractal dimension and architecture of trabecular bone.
Fazzalari, N L; Parkinson, I H
1996-01-01
The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P < 0.0001), BS/TV (r = 0.74, P < 0.0001), Tb.Th (r = 0.50, P < 0.02), Tb.Sp (r = -0.81, P < 0.0001), and Tb.N (r = 0.76, P < 0.0001). This method for calculating fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
Pulse regime in formation of fractal fibers
NASA Astrophysics Data System (ADS)
Smirnov, B. M.
2016-11-01
The pulse regime of vaporization of a bulk metal located in a buffer gas is analyzed as a method of generation of metal atoms under the action of a plasma torch or a laser beam. Subsequently these atoms are transformed into solid nanoclusters, fractal aggregates and then into fractal fibers if the growth process proceeds in an external electric field. We are guided by metals in which transitions between s and d-electrons of their atoms are possible, since these metals are used as catalysts and filters in interaction with gas flows. The resistance of metal fractal structures to a gas flow is evaluated that allows one to find optimal parameters of a fractal structure for gas flow propagation through it. The thermal regime of interaction between a plasma pulse or a laser beam and a metal surface is analyzed. It is shown that the basic energy from an external source is consumed on a bulk metal heating, and the efficiency of atom evaporation from the metal surface, that is the ratio of energy fluxes for vaporization and heating, is 10-3-10-4 for transient metals under consideration. A typical energy flux ( 106 W/cm2), a typical surface temperature ( 3000 K), and a typical pulse duration ( 1 μs) provide a sufficient amount of evaporated atoms to generate fractal fibers such that each molecule of a gas flow collides with the skeleton of fractal fibers many times.
Fractal Image Informatics: from SEM to DEM
NASA Astrophysics Data System (ADS)
Oleschko, K.; Parrot, J.-F.; Korvin, G.; Esteves, M.; Vauclin, M.; Torres-Argüelles, V.; Salado, C. Gaona; Cherkasov, S.
2008-05-01
In this paper, we introduce a new branch of Fractal Geometry: Fractal Image Informatics, devoted to the systematic and standardized fractal analysis of images of natural systems. The methods of this discipline are based on the properties of multiscale images of selfaffine fractal surfaces. As proved in the paper, the image inherits the scaling and lacunarity of the surface and of its reflectance distribution [Korvin, 2005]. We claim that the fractal analysis of these images must be done without any smoothing, thresholding or binarization. Two new tools of Fractal Image Informatics, firmagram analysis (FA) and generalized lacunarity (GL), are presented and discussed in details. These techniques are applicable to any kind of image or to any observed positive-valued physical field, and can be used to correlate between images. It will be shown, by a modified Grassberger-Hentschel-Procaccia approach [Phys. Lett. 97A, 227 (1983); Physica 8D, 435 (1983)] that GL obeys the same scaling law as the Allain-Cloitre lacunarity [Phys. Rev. A 44, 3552 (1991)] but is free of the problems associated with gliding boxes. Several applications are shown from Soil Physics, Surface Science, and other fields.
Fractal analysis of deformation-induced dislocation patterns
Zaiser, M. ); Bay, K. . Inst. fuer Theoretische und Angewandte Physik); Haehner, P. . Joint Research Centre TU Braunschweig . Inst. fuer Metallphysik und Nukleare Festkoerperphysik)
1999-06-22
The paper reports extensive analyses of the fractal geometry of cellular dislocation structures observed in Cu deformed in multiple-slip orientation. Several methods presented for the determination of fractal dimensions are shown to give consistent results. Criteria are formulated which allow the distinguishing of fractal from non-fractal patterns, and implications of fractal dislocation patterning for quantitative metallography are discussed in detail. For an interpretation of the findings a theoretical model is outlined according to which dislocation cell formation is associated to a noise-induced structural transition far from equilibrium. This allows relating the observed fractal dimensions to the stochastic properties of deformation by collective dislocation glide.
The utility of fractal analysis in clinical neuroscience.
John, Ann M; Elfanagely, Omar; Ayala, Carlos A; Cohen, Michael; Prestigiacomo, Charles J
2015-01-01
Physicians and scientists can use fractal analysis as a tool to objectively quantify complex patterns found in neuroscience and neurology. Fractal analysis has the potential to allow physicians to make predictions about clinical outcomes, categorize pathological states, and eventually generate diagnoses. In this review, we categorize and analyze the applications of fractal theory in neuroscience found in the literature. We discuss how fractals are applied and what evidence exists for fractal analysis in neurodegeneration, neoplasm, neurodevelopment, neurophysiology, epilepsy, neuropharmacology, and cell morphology. The goal of this review is to introduce the medical community to the utility of applying fractal theory in clinical neuroscience.
Fractal Dimension in Epileptic EEG Signal Analysis
NASA Astrophysics Data System (ADS)
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include
Fractal Gait Patterns Are Retained after Entrainment to a Fractal Stimulus
Rhea, Christopher K.; Kiefer, Adam W.; Wittstein, Matthew W.; Leonard, Kelsey B.; MacPherson, Ryan P.; Wright, W. Geoffrey; Haran, F. Jay
2014-01-01
Previous work has shown that fractal patterns in gait can be altered by entraining to a fractal stimulus. However, little is understood about how long those patterns are retained or which factors may influence stronger entrainment or retention. In experiment one, participants walked on a treadmill for 45 continuous minutes, which was separated into three phases. The first 15 minutes (pre-synchronization phase) consisted of walking without a fractal stimulus, the second 15 minutes consisted of walking while entraining to a fractal visual stimulus (synchronization phase), and the last 15 minutes (post-synchronization phase) consisted of walking without the stimulus to determine if the patterns adopted from the stimulus were retained. Fractal gait patterns were strengthened during the synchronization phase and were retained in the post-synchronization phase. In experiment two, similar methods were used to compare a continuous fractal stimulus to a discrete fractal stimulus to determine which stimulus type led to more persistent fractal gait patterns in the synchronization and post-synchronization (i.e., retention) phases. Both stimulus types led to equally persistent patterns in the synchronization phase, but only the discrete fractal stimulus led to retention of the patterns. The results add to the growing body of literature showing that fractal gait patterns can be manipulated in a predictable manner. Further, our results add to the literature by showing that the newly adopted gait patterns are retained for up to 15 minutes after entrainment and showed that a discrete visual stimulus is a better method to influence retention. PMID:25221981
Fractal and multifractal analyses of bipartite networks.
Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua
2017-03-31
Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.
MRI Image Processing Based on Fractal Analysis
Marusina, Mariya Y; Mochalina, Alexandra P; Frolova, Ekaterina P; Satikov, Valentin I; Barchuk, Anton A; Kuznetcov, Vladimir I; Gaidukov, Vadim S; Tarakanov, Segrey A
2017-01-01
Background: Cancer is one of the most common causes of human mortality, with about 14 million new cases and 8.2 million deaths reported in in 2012. Early diagnosis of cancer through screening allows interventions to reduce mortality. Fractal analysis of medical images may be useful for this purpose. Materials and Methods: In this study, we examined magnetic resonance (MR) images of healthy livers and livers containing metastases from colorectal cancer. The fractal dimension and the Hurst exponent were chosen as diagnostic features for tomographic imaging using Image J software package for image processings FracLac for applied for fractal analysis with a 120x150 pixel area. Calculations of the fractal dimensions of pathological and healthy tissue samples were performed using the box-counting method. Results: In pathological cases (foci formation), the Hurst exponent was less than 0.5 (the region of unstable statistical characteristics). For healthy tissue, the Hurst index is greater than 0.5 (the zone of stable characteristics). Conclusions: The study indicated the possibility of employing fractal rapid analysis for the detection of focal lesions of the liver. The Hurst exponent can be used as an important diagnostic characteristic for analysis of medical images.
Characterization of branch complexity by fractal analyses
Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.
1999-01-01
The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U∝l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area.
Fractal modeling of natural fracture networks
Ferer, M.; Dean, B.; Mick, C.
1995-06-01
West Virginia University will implement procedures for a fractal analysis of fractures in reservoirs. This procedure will be applied to fracture networks in outcrops and to fractures intersecting horizontal boreholes. The parameters resulting from this analysis will be used to generate synthetic fracture networks with the same fractal characteristics as the real networks. Recovery from naturally fractured, tight-gas reservoirs is controlled by the fracture network. Reliable characterization of the actual fracture network in the reservoir is severely limited. The location and orientation of fractures intersecting the borehole can be determined, but the length of these fractures cannot be unambiguously determined. Because of the lack of detailed information about the actual fracture network, modeling methods must represent the porosity and permeability associated with the fracture network, as accurately as possible with very little a priori information. In the sections following, the authors will (1) present fractal analysis of the MWX site, using the box-counting procedure; (2) review evidence testing the fractal nature of fracture distributions and discuss the advantages of using the fractal analysis over a stochastic analysis; and (3) present an efficient algorithm for producing a self-similar fracture networks which mimic the real MWX outcrop fracture network.
Fractal and multifractal analyses of bipartite networks
Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua
2017-01-01
Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions. PMID:28361962
Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique
NASA Technical Reports Server (NTRS)
Qiu, Hong-Lie; Lam, Nina Siu-Ngan; Quattrochi, Dale
1997-01-01
With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.
Construction of fractal nanostructures based on Kepler-Shubnikov nets
Ivanov, V. V. Talanov, V. M.
2013-05-15
A system of information codes for deterministic fractal lattices and sets of multifractal curves is proposed. An iterative modular design was used to obtain a series of deterministic fractal lattices with generators in the form of fragments of 2D structures and a series of multifractal curves (based on some Kepler-Shubnikov nets) having Cantor set properties. The main characteristics of fractal structures and their lacunar spectra are determined. A hierarchical principle is formulated for modules of regular fractal structures.
Fractals and the irreducibility of consciousness in plants and animals.
Gardiner, John
2013-08-01
In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness.
Investigation into How 8th Grade Students Define Fractals
ERIC Educational Resources Information Center
Karakus, Fatih
2015-01-01
The analysis of 8th grade students' concept definitions and concept images can provide information about their mental schema of fractals. There is limited research on students' understanding and definitions of fractals. Therefore, this study aimed to investigate the elementary students' definitions of fractals based on concept image and concept…
Fresnel diffraction of fractal grating and self-imaging effect.
Wang, Junhong; Zhang, Wei; Cui, Yuwei; Teng, Shuyun
2014-04-01
Based on the self-similarity property of fractal, two types of fractal gratings are produced according to the production and addition operations of multiple periodic gratings. Fresnel diffractions of fractal grating are analyzed theoretically, and the general mathematic expressions of the diffraction intensity distributions of fractal grating are deduced. The gray-scale patterns of the 2D diffraction distributions of fractal grating are provided through numerical calculations. The diffraction patterns take on the periodicity along the longitude and transverse directions. The 1D diffraction distribution at some certain distances shows the same structure as the fractal grating. This indicates that the self-image of fractal grating is really formed in the Fresnel diffraction region. The experimental measurement of the diffraction intensity distribution of fractal grating with different fractal dimensions and different fractal levels is performed, and the self-images of fractal grating are obtained successfully in experiments. The conclusions of this paper are helpful for the development of the application of fractal grating.
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-07
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades.
Fractal dynamics in chaotic quantum transport.
Kotimäki, V; Räsänen, E; Hennig, H; Heller, E J
2013-08-01
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.
Retinal fractals and acute lacunar stroke.
Cheung, Ning; Liew, Gerald; Lindley, Richard I; Liu, Erica Y; Wang, Jie Jin; Hand, Peter; Baker, Michelle; Mitchell, Paul; Wong, Tien Y
2010-07-01
This study aimed to determine whether retinal fractal dimension, a quantitative measure of microvascular branching complexity and density, is associated with lacunar stroke. A total of 392 patients presenting with acute ischemic stroke had retinal fractal dimension measured from digital photographs, and lacunar infarct ascertained from brain imaging. After adjusting for age, gender, and vascular risk factors, higher retinal fractal dimension (highest vs lowest quartile and per standard deviation increase) was independently and positively associated with lacunar stroke (odds ratio [OR], 4.27; 95% confidence interval [CI], 1.49-12.17 and OR, 1.85; 95% CI, 1.20-2.84, respectively). Increased retinal microvascular complexity and density is associated with lacunar stroke.
Fractal characterization of wear-erosion surfaces
Rawers, James C.; Tylczak, Joseph H.
1999-12-01
Wear erosion is a complex phenomenon resulting in highly distorted and deformed surface morphologies. Most wear surface features have been described only qualitatively. In this study wear surfaces features were quantified using fractal analysis. The ability to assign numerical values to wear-erosion surfaces makes possible mathematical expressions that will enable wear mechanisms to be predicted and understood. Surface characterization came from wear-erosion experiments that included varying the erosive materials, the impact velocity, and the impact angle. Seven fractal analytical techniques were applied to micrograph images of wear-erosion surfaces. Fourier analysis was the most promising. Fractal values obtained were consistent with visual observations and provided a unique wear-erosion parameter unrelated to wear rate.
Fractal characterization of wear-erosion surfaces
Rawers, J.; Tylczak, J.
1999-12-01
Wear erosion is a complex phenomenon resulting in highly distorted and deformed surface morphologies. Most wear surface features have been described only qualitatively. In this study wear surfaces features were quantified using fractal analysis. The ability to assign numerical values to wear-erosion surfaces makes possible mathematical expressions that will enable wear mechanisms to be predicted and understood. Surface characterization came from wear-erosion experiments that included varying the erosive materials, the impact velocity, and the impact angle. Seven fractal analytical techniques were applied to micrograph images of wear-erosion surfaces. Fourier analysis was the most promising. Fractal values obtained were consistent with visual observations and provided a unique wear-erosion parameter unrelated to wear rate. In this study stainless steel was evaluated as a function of wear erosion conditions.
Scaling and fractal behaviour underlying meiotic recombination.
Waxman, D; Stoletzki, N
2010-01-01
In this paper we investigate some of the mathematical properties of meiotic recombination. Working within the framework of a genetic model with n loci, where alpha alleles are possible at each locus, we find that the proportion of all possible diploid parental genotypes that can produce a particular haploid gamete is exp[-n log(alpha(2)/[2alpha-1])]. We show that this proportion connects recombination with a fractal geometry of dimension log(2alpha-1)/log(alpha). The fractal dimension of a geometric object manifests itself when it is measured at increasingly smaller length scales. Decreasing the length scale of a geometric object is found to be directly analogous, in a genetics problem, to specifying a multilocus haplotype at a larger number of loci, and it is here that the fractal dimension reveals itself.
Fractal structure of the interplanetary magnetic field
NASA Technical Reports Server (NTRS)
Burlaga, L. F.; Klein, L. W.
1985-01-01
Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.
Chaos, fractals, and our concept of disease.
Varela, Manuel; Ruiz-Esteban, Raul; Mestre de Juan, Maria Jose
2010-01-01
The classic anatomo-clinic paradigm based on clinical syndromes is fraught with problems. Nevertheless, for multiple reasons, clinicians are reluctant to embrace a more pathophysiological approach, even though this is the prevalent paradigm under "which basic sciences work. In recent decades, nonlinear dynamics ("chaos theory") and fractal geometry have provided powerful new tools to analyze physiological systems. However, these tools are embedded in the pathophysiological perspective and are not easily translated to our classic syndromes. This article comments on the problems raised by the conventional anatomo-clinic paradigm and reviews three areas in which the influence of nonlinear dynamics and fractal geometry can be especially prominent: disease as a loss of complexity, the idea of homeostasis, and fractals in pathology.
Finite transformers for construction of fractal curves
Lisovik, L.P.
1995-01-01
In this paper we continue the study of infinite R{sup n}-transformers that can be used to define real functions and three-dimensional curves. An R{sup n}-transformer A generates an output n-tuple A(x) = (Y{sub 1},...,Y{sub n}), consisting of output binary representations. We have previously shown that finite R{sup n}-transformers with n = 1, 2 can be used to define a continuous, nowhere differentiable function and a Peano curve. Curves of this kind are objects of fractal geometry. Here we show that some other fractal curves, which are analogs of the Koch curve and the Sierpinski napkin, can be defined by finite R{sup 2}-transformers. R{sup n}-transformers (and also finite R{sup n}-transformers) thus provide a convenient tool for definition of fractal curves.
Higuchi fractal properties of onset epilepsy electroencephalogram.
Khoa, Truong Quang Dang; Ha, Vo Quang; Toi, Vo Van
2012-01-01
Epilepsy is a medical term which indicates a common neurological disorder characterized by seizures, because of abnormal neuronal activity. This leads to unconsciousness or even a convulsion. The possible etiologies should be evaluated and treated. Therefore, it is necessary to concentrate not only on finding out efficient treatment methods, but also on developing algorithm to support diagnosis. Currently, there are a number of algorithms, especially nonlinear algorithms. However, those algorithms have some difficulties one of which is the impact of noise on the results. In this paper, in addition to the use of fractal dimension as a principal tool to diagnose epilepsy, the combination between ICA algorithm and averaging filter at the preprocessing step leads to some positive results. The combination which improved the fractal algorithm become robust with noise on EEG signals. As a result, we can see clearly fractal properties in preictal and ictal period so as to epileptic diagnosis.
Edges of Saturn's rings are fractal.
Li, Jun; Ostoja-Starzewski, Martin
2015-01-01
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn's rings are fractal.
Fractal and Multifractal Analysis of Human Gait
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.
2003-09-01
We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.
``the Human BRAIN & Fractal quantum mechanics''
NASA Astrophysics Data System (ADS)
Rosary-Oyong, Se, Glory
In mtDNA ever retrieved from Iman Tuassoly, et.al:Multifractal analysis of chaos game representation images of mtDNA''.Enhances the price & valuetales of HE. Prof. Dr-Ing. B.J. HABIBIE's N-219, in J. Bacteriology, Nov 1973 sought:'' 219 exist as separate plasmidDNA species in E.coli & Salmonella panama'' related to ``the brain 2 distinct molecular forms of the (Na,K)-ATPase..'' & ``neuron maintains different concentration of ions(charged atoms'' thorough Rabi & Heisenber Hamiltonian. Further, after ``fractal space time are geometric analogue of relativistic quantum mechanics''[Ord], sought L.Marek Crnjac: ``Chaotic fractals at the root of relativistic quantum physics''& from famous Nottale: ``Scale relativity & fractal space-time:''Application to Quantum Physics , Cosmology & Chaotic systems'',1995. Acknowledgements to HE. Mr. H. TUK SETYOHADI, Jl. Sriwijaya Raya 3, South-Jakarta, INDONESIA.
Trabecular Pattern Analysis Using Fractal Dimension
NASA Astrophysics Data System (ADS)
Ishida, Takayuki; Yamashita, Kazuya; Takigawa, Atsushi; Kariya, Komyo; Itoh, Hiroshi
1993-04-01
Feature extraction from a digitized image is advantageous for the detection of signs of disease. In this work, we attempted to evaluate bone trabecular pattern changes in osteoporosis using the fractal dimension and the root mean square (RMS) values. The relationship between the fractal dimension and the 1st moment of the power spectrum is explored, and we investigated the relationship between the results of this analysis and the bone mineral density (BMD) value which was measured using dual-energy X-ray absorptiometry (DEXA). As a result, we were able to extract useful information, using the fractal dimension and the RMS value of the radiographs (lateral view of the lumbar vertebrae), for the diagnosis of osteoporosis. Abnormal clinical cases were separated from normal cases based on the evaluation values. Negligible correlation between the BMD value and these indexes was observed.
Dynamic structure factor of vibrating fractals.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-02-10
Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.
Fractal design concepts for stretchable electronics.
Fan, Jonathan A; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J; Huang, Yonggang; Rogers, John A
2014-01-01
Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.
Fractal design concepts for stretchable electronics
NASA Astrophysics Data System (ADS)
Fan, Jonathan A.; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J.; Huang, Yonggang; Rogers, John A.
2014-02-01
Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.
Compressive optical remote sensing via fractal classification
NASA Astrophysics Data System (ADS)
Sun, Quan-sen; Liu, Ji-xin
2015-11-01
High resolution and large field of view are two major development trends in optical remote sensing imaging. But these trends will cause the difficult problem of mass data processing and remote sensor design under the limitation of conventional sampling method. Therefore, we will propose a novel optical remote sensing imaging method based on compressed sensing theory and fractal feature extraction in this study. We could utilize the result of fractal classification to realize the selectable partitioned image recovery with undersampling measurement. The two experiments illustrate the availability and feasibility of this new method.
The Fractal Simulation Of Biological Shapes
NASA Astrophysics Data System (ADS)
Pickover, Clifford A.
1989-04-01
This paper provides a light introduction to simple graphics techniques for visualizing a large class of biological shapes generated from recursive algorithms. In order to capture some of the structural richness inherent in organisms, the algorithms produce not only extreme variability but also a high level of organization. The material primarily comes from previous published works of the author. For a general background on fractal methods in mathematics and science, see Mandelbrot's famous book. For research on the fractal characterization of other biological structures, such as the lung's bronchial tree and the surfaces of protein molecules.
Fractal boundaries in magnetotail particle dynamics
NASA Technical Reports Server (NTRS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-01-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Estimation of fractal dimensions from transect data
Loehle, C.
1994-04-01
Fractals are a useful tool for analyzing the topology of objects such as coral reefs, forest canopies, and landscapes. Transects are often studied in these contexts, and fractal dimensions computed from them. An open question is how representative a single transect is. Transects may also be used to estimate the dimensionality of a surface. Again the question of representativeness of the transect arises. These two issues are related. This note qualifies the conditions under which transect data may be considered to be representative or may be extrapolated, based on both theoretical and empirical results.
Fractal characterization of neural correlates of consciousness
NASA Astrophysics Data System (ADS)
Ibañez-Molina, A. J.; Iglesias-Parro, S.
2013-01-01
In this work we present a novel experimental paradigm, based on binocular rivalry, to address the study of internally and externally generated conscious percepts. Assuming the nonlinear nature of the EEG signals, we propose the use of fractal dimension to characterize the complexity of the EEG associated with each percept. Data analysis showed significant differences in complexity between the internally and externally generated percepts. Moreover, EEG complexity of auditory and visual percepts was unequal. These results support fractal dimension analyses as a new tool to characterize conscious perception.
Unifying iteration rule for fractal objects
NASA Astrophysics Data System (ADS)
Kittel, A.; Parisi, J.; Peinke, J.; Baier, G.; Klein, M.; Rössler, O. E.
1997-03-01
We introduce an iteration rule for real numbers capable to generate attractors with dragon-, snowflake-, sponge-, or Swiss-flag-like cross sections. The idea behind it is the mapping of a torus into two (or more) shrunken and twisted tori located inside the previous one. Three distinct parameters define the symmetry, the dimension, and the connectedness or disconnectedness of the fractal object. For some selected triples of parameter values, a couple of well known fractal geometries (e.g. the Cantor set, the Sierpinski gasket, or the Swiss flag) can be gained as special cases.
Preparation and characterization of fractal elastomer surfaces.
Nonomura, Yoshimune; Seino, Eri; Abe, Saya; Mayama, Hiroyuki
2013-01-01
The elastomer materials with hierarchical structure and suitable wettability are useful as biological surface model. In the present study, urethane resin and silicone resin elastomers with hierarchical rough surfaces were prepared and referred to as "fractal elastomers". We found a hierarchy of small projections that existed over larger ones on these surfaces. These elastomers were synthesized by transferring a fractal surface structure of alkylketene dimer. The rough structure enhanced the hydrophobicity and weakened friction resistance of the elastomer surfaces. These materials can be useful for artificial skin with biomimetic surface properties.
On the Classification of Fractal Squares
NASA Astrophysics Data System (ADS)
Luo, Jun Jason; Liu, Jing-Cheng
2016-01-01
In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc. 155 (2013) 73-86], Lau, Luo and Rao completely classified the topological structure of so called fractal square F defined by F = (F + 𝒟)/n, where 𝒟 ⊊ {0, 1,…,n - 1}2,n ≥ 2. In this paper, we further provide simple criteria for the F to be totally disconnected, then we discuss the Lipschitz classification of F in the case n = 3, which is an attempt to consider non-totally disconnected sets.
Tomomitsu, Tatsushi; Mimura, Hiroaki; Murase, Kenya; Tamada, Tsutomu; Sone, Teruki; Fukunaga, Masao
2005-06-20
Many analyses of bone microarchitecture using three-dimensional images of micro CT (microCT) have been reported recently. However, as extirpated bone is the subject of measurement on microCT, various kinds of information are not available clinically. Our aim is to evaluate usefulness of fractal dimension as an index of bone strength different from bone mineral density in in-vivo, to which microCT could not be applied. In this fundamental study, the relation between pixel size and the slice thickness of images was examined when fractal analysis was applied to clinical images. We examined 40 lumbar spine specimens extirpated from 16 male cadavers (30-88 years; mean age, 60.8 years). Three-dimensional images of the trabeculae of 150 slices were obtained by a microCT system under the following conditions: matrix size, 512 x 512; slice thickness, 23.2 em; and pixel size, 18.6 em. Based on images of 150 slices, images of four different matrix sizes and nine different slice thicknesses were made using public domain software (NIH Image). The threshold value for image binarization, and the relation between pixel size and the slice thickness of an image used for two-dimensional and three-dimensional fractal analyses were studied. In addition, the box counting method was used for fractal analysis. One hundred forty-five in box counting was most suitable as the threshold value for image binarization on the 256 gray levels. The correlation coefficients between two-dimensional fractal dimensions of processed images and three-dimensional fractal dimensions of original images were more than 0.9 for pixel sizes < or =148.8 microm at a slice thickness of 1 mm, and < or =74.4 microm at one of 2 mm. In terms of the relation between the three-dimensional fractal dimension of processed images and three-dimensional fractal dimension of original images, when pixel size was less than 74.4 microm, a correlation coefficient of more than 0.9 was obtained even for the maximal slice thickness
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-19
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
[Chaos and fractals and their applications in electrocardial signal research].
Jiao, Qing; Guo, Yongxin; Zhang, Zhengguo
2009-06-01
Chaos and fractals are ubiquitous phenomena of nature. A system with fractal structure usually behaves chaos. As a complicated nonlinear dynamics system, heart has fractals structure and behaves as chaos. The deeper inherent mechanism of heart can be opened out when the chaos and fractals theory is utilized in the research of the electrical activity of heart. Generally a time series of a system was used for describing the status of the strange attractor of the system. The indices include Poincare plot, fractals dimension, Lyapunov exponent, entropy, scaling exponent, Hurst index and so on. In this article, the basic concepts and the methods of chaos and fractals were introduced firstly. Then the applications of chaos and fractals theories in the study of electrocardial signal were expounded with example of how they are used for ventricular fibrillation.
Plant microtubule cytoskeleton complexity: microtubule arrays as fractals.
Gardiner, John; Overall, Robyn; Marc, Jan
2012-01-01
Biological systems are by nature complex and this complexity has been shown to be important in maintaining homeostasis. The plant microtubule cytoskeleton is a highly complex system, with contributing factors through interactions with microtubule-associated proteins (MAPs), expression of multiple tubulin isoforms, and post-translational modification of tubulin and MAPs. Some of this complexity is specific to microtubules, such as a redundancy in factors that regulate microtubule depolymerization. Plant microtubules form partial helical fractals that play a key role in development. It is suggested that, under certain cellular conditions, other categories of microtubule fractals may form including isotropic fractals, triangular fractals, and branched fractals. Helical fractal proteins including coiled-coil and armadillo/beta-catenin repeat proteins and the actin cytoskeleton are important here too. Either alone, or in combination, these fractals may drive much of plant development.
Fractality à la carte: a general particle aggregation model
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
Fractality à la carte: a general particle aggregation model
NASA Astrophysics Data System (ADS)
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
Fractal superconductivity near localization threshold
Feigel'man, M.V.; Ioffe, L.B.; Kravtsov, V.E.; Cuevas, E.
2010-07-15
We develop a semi-quantitative theory of electron pairing and resulting superconductivity in bulk 'poor conductors' in which Fermi energy E{sub F} is located in the region of localized states not so far from the Anderson mobility edge E{sub c}. We assume attractive interaction between electrons near the Fermi surface. We review the existing theories and experimental data and argue that a large class of disordered films is described by this model. Our theoretical analysis is based on analytical treatment of pairing correlations, described in the basis of the exact single-particle eigenstates of the 3D Anderson model, which we combine with numerical data on eigenfunction correlations. Fractal nature of critical wavefunction's correlations is shown to be crucial for the physics of these systems. We identify three distinct phases: 'critical' superconductive state formed at E{sub F} = E{sub c}, superconducting state with a strong pseudo-gap, realized due to pairing of weakly localized electrons and insulating state realized at E{sub F} still deeper inside a localized band. The 'critical' superconducting phase is characterized by the enhancement of the transition temperature with respect to BCS result, by the inhomogeneous spatial distribution of superconductive order parameter and local density of states. The major new feature of the pseudo-gapped state is the presence of two independent energy scales: superconducting gap {Delta}, that is due to many-body correlations and a new 'pseudo-gap' energy scale {Delta}{sub P} which characterizes typical binding energy of localized electron pairs and leads to the insulating behavior of the resistivity as a function of temperature above superconductive T{sub c}. Two gap nature of the pseudo-gapped superconductor is shown to lead to specific features seen in scanning tunneling spectroscopy and point-contact Andreev spectroscopy. We predict that pseudo-gapped superconducting state demonstrates anomalous behavior of the optical
Fractal nature of hydrocarbon deposits. 2. Spatial distribution
Barton, C.C.; Schutter, T.A; Herring, P.R.; Thomas, W.J. ); Scholz, C.H. )
1991-03-01
Hydrocarbons are unevenly distributed within reservoirs and are found in patches whose size distribution is a fractal over a wide range of scales. The spatial distribution of the patches is also fractal and this can be used to constrain the design of drilling strategies also defined by a fractal dimension. Fractal distributions are scale independent and are characterized by a power-law scaling exponent termed the fractal dimension. The authors have performed fractal analyses on the spatial distribution of producing and showing wells combined and of dry wells in 1,600-mi{sup 2} portions of the Denver and Powder River basins that were nearly completely drilled on quarter-mile square-grid spacings. They have limited their analyses to wells drilled to single stratigraphic intervals so that the map pattern revealed by drilling is representative of the spatial patchiness of hydrocarbons at depth. The fractal dimensions for the spatial patchiness of hydrocarbons in the two basins are 1.5 and 1.4, respectively. The fractal dimension for the pattern of all wells drilled is 1.8 for both basins, which suggests a drilling strategy with a fractal dimension significantly higher than the dimensions 1.5 and 1.4 sufficient to efficiently and economically explore these reservoirs. In fact, the fractal analysis reveals that the drilling strategy used in these basins approaches a fractal dimension of 2.0, which is equivalent to random drilling with no geologic input. Knowledge of the fractal dimension of a reservoir prior to drilling would provide a basis for selecting and a criterion for halting a drilling strategy for exploration whose fractal dimension closely matches that of the spatial fractal dimension of the reservoir, such a strategy should prove more efficient and economical than current practice.
A Fractal Perspective on Scale in Geography
NASA Astrophysics Data System (ADS)
Jiang, Bin; Brandt, S.
2016-06-01
Scale is a fundamental concept that has attracted persistent attention in geography literature over the past several decades. However, it creates enormous confusion and frustration, particularly in the context of geographic information science, because of scale-related issues such as image resolution, and the modifiable areal unit problem (MAUP). This paper argues that the confusion and frustration mainly arise from Euclidean geometric thinking, with which locations, directions, and sizes are considered absolute, and it is time to reverse this conventional thinking. Hence, we review fractal geometry, together with its underlying way of thinking, and compare it to Euclidean geometry. Under the paradigm of Euclidean geometry, everything is measurable, no matter how big or small. However, geographic features, due to their fractal nature, are essentially unmeasurable or their sizes depend on scale. For example, the length of a coastline, the area of a lake, and the slope of a topographic surface are all scale-dependent. Seen from the perspective of fractal geometry, many scale issues, such as the MAUP, are inevitable. They appear unsolvable, but can be dealt with. To effectively deal with scale-related issues, we introduce topological and scaling analyses based on street-related concepts such as natural streets, street blocks, and natural cities. We further contend that spatial heterogeneity, or the fractal nature of geographic features, is the first and foremost effect of two spatial properties, because it is general and universal across all scales. Keywords: Scaling, spatial heterogeneity, conundrum of length, MAUP, topological analysis
Fractal analysis of the Navassa Island seascape
Zawada, David G.
2011-01-01
This release provides the numerical results of the fractal analyses discussed in Zawada and others (2010) for the Navassa Island reefscape. The project represents the continuation of a U.S. Geological Survey (USGS) research effort begun in 2006 (Zawada and others, 2006) to understand the patterns and scalability of roughness and topographic complexity from individual corals to complete reefscapes.
Ghost DBI-essence in fractal geometry
NASA Astrophysics Data System (ADS)
Acikgoz, I.; Binbay, F.; Salti, M.; Aydogdu, O.
2016-05-01
Focusing on a fractal geometric ghost dark energy, we reconstruct the Dirac-Born-Infeld (DBI)-essence-type scalar field and find exact solutions of the potential and warped brane tension. We also discuss statefinders for the selected dark energy description to make it distinguishable among others.
Generating Fractals through Self-Replication.
ERIC Educational Resources Information Center
Reinstein, David; And Others
1997-01-01
Describes a classroom activity designed to give students hands-on experience using technology and geometric visualization, as well as to explore fractal geometry in a cooperative classroom environment. Natural phenomena is the context of these activities. Enriches understanding of Euclidean geometry and infinite sequences. Lists materials,…
Measuring fractal dimension of metro systems
NASA Astrophysics Data System (ADS)
Deng, S.; Li, W.; Gu, J.; Zhu, Y.; Zhao, L.; Han, J.
2015-04-01
We discuss cluster growing method and box-covering method as well as their connection to fractal geometry. Our measurements show that for small network systems, box-covering method gives a better scaling relation. We then measure both unweighted and weighted metro networks with optimal box-covering method.
Pond fractals in a tidal flat.
Cael, B B; Lambert, Bennett; Bisson, Kelsey
2015-11-01
Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.
Fractals Illustrate the Mathematical Way of Thinking.
ERIC Educational Resources Information Center
Nievergelt, Yves
1991-01-01
Presented are exercises that demonstrate the application of standard concepts in the design of algorithms for plotting certain fractals. The exercises can be used in any course that explains the concepts of bounded or unbounded planar sets and may serve as an application in a course on complex analysis. (KR)
Flames in fractal grid generated turbulence
NASA Astrophysics Data System (ADS)
Goh, K. H. H.; Geipel, P.; Hampp, F.; Lindstedt, R. P.
2013-12-01
Twin premixed turbulent opposed jet flames were stabilized for lean mixtures of air with methane and propane in fractal grid generated turbulence. A density segregation method was applied alongside particle image velocimetry to obtain velocity and scalar statistics. It is shown that the current fractal grids increase the turbulence levels by around a factor of 2. Proper orthogonal decomposition (POD) was applied to show that the fractal grids produce slightly larger turbulent structures that decay at a slower rate as compared to conventional perforated plates. Conditional POD (CPOD) was also implemented using the density segregation technique and the results show that CPOD is essential to segregate the relative structures and turbulent kinetic energy distributions in each stream. The Kolmogorov length scales were also estimated providing values ∼0.1 and ∼0.5 mm in the reactants and products, respectively. Resolved profiles of flame surface density indicate that a thin flame assumption leading to bimodal statistics is not perfectly valid under the current conditions and it is expected that the data obtained will be of significant value to the development of computational methods that can provide information on the conditional structure of turbulence. It is concluded that the increase in the turbulent Reynolds number is without any negative impact on other parameters and that fractal grids provide a route towards removing the classical problem of a relatively low ratio of turbulent to bulk strain associated with the opposed jet configuration.
Paradigms of Complexity: Fractals and Structures in the Sciences
NASA Astrophysics Data System (ADS)
Novak, Miroslav M.
The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic
Aesthetic Responses to Exact Fractals Driven by Physical Complexity.
Bies, Alexander J; Blanc-Goldhammer, Daryn R; Boydston, Cooper R; Taylor, Richard P; Sereno, Margaret E
2016-01-01
Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a
Aesthetic Responses to Exact Fractals Driven by Physical Complexity
Bies, Alexander J.; Blanc-Goldhammer, Daryn R.; Boydston, Cooper R.; Taylor, Richard P.; Sereno, Margaret E.
2016-01-01
Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a
Link between truncated fractals and coupled oscillators in biological systems.
Paar, V; Pavin, N; Rosandić, M
2001-09-07
This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.
Quantitative evaluation of midpalatal suture maturation via fractal analysis
Kwak, Kyoung Ho; Kim, Yong-Il; Kim, Yong-Deok
2016-01-01
Objective The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. Methods The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman's correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. Results The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235. Conclusions There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation. PMID:27668195
Solar Flare Geometries. I. The Area Fractal Dimension
NASA Astrophysics Data System (ADS)
Aschwanden, Markus J.; Aschwanden, Pascal D.
2008-02-01
In this study we investigate for the first time the fractal dimension of solar flares and find that the flare area observed in EUV wavelengths exhibits a fractal scaling. We measure the area fractal dimension D2, also called the Hausdorff dimension, with a box-counting method, which describes the fractal area as A(L) ~ LD2. We apply the fractal analysis to a statistical sample of 20 GOES X- and M-class flares, including the Bastille Day 2000 July 14 flare, one of the largest flares ever recorded. We find that the fractal area (normalized by the time-integrated flare area Af) varies from near zero at the beginning of the flare to a maximum of A(t)/Af = 0.65 +/- 0.12 after the peak time of the flare, which corresponds to an area fractal dimension in the range of 1.0lesssim D2(t) lesssim 1.89 +/- 0.05. We find that the total EUV flux Ftot(t) is linearly correlated with the fractal area A(t) . From the area fractal dimension D2, the volume fractal dimension D3 can be inferred (subject of Paper II), which is crucial to inferring a realistic volume filling factor, which affects the derived electron densities, thermal energies, and cooling times of solar and stellar flares.
Fractals in art and nature: why do we like them?
NASA Astrophysics Data System (ADS)
Spehar, Branka; Taylor, Richard P.
2013-03-01
Fractals have experienced considerable success in quantifying the visual complexity exhibited by many natural patterns, and continue to capture the imagination of scientists and artists alike. Fractal patterns have also been noted for their aesthetic appeal, a suggestion further reinforced by the discovery that the poured patterns of the American abstract painter Jackson Pollock are also fractal, together with the findings that many forms of art resemble natural scenes in showing scale-invariant, fractal-like properties. While some have suggested that fractal-like patterns are inherently pleasing because they resemble natural patterns and scenes, the relation between the visual characteristics of fractals and their aesthetic appeal remains unclear. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Whereas our previous preference studies focused on fractal images consisting of black shapes on white backgrounds, here we extend our investigations to include grayscale images in which the intensity variations exhibit scale invariance. This scale-invariance is generated using a 1/f frequency distribution and can be tuned by varying the slope of the rotationally averaged Fourier amplitude spectrum. Thresholding the intensity of these images generates black and white fractals with equivalent scaling properties to the original grayscale images, allowing a direct comparison of preferences for grayscale and black and white fractals. We found no significant differences in preferences between the two groups of fractals. For both set of images, the visual preference peaked for images with the amplitude spectrum slopes from 1.25 to 1.5, thus confirming and extending the previously observed relationship between fractal characteristics of images and visual preference.
Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics.
Rhea, Christopher K; Kiefer, Adam W; D'Andrea, Susan E; Warren, William H; Aaron, Roy K
2014-08-01
Fractal patterns characterize healthy biological systems and are considered to reflect the ability of the system to adapt to varying environmental conditions. Previous research has shown that fractal patterns in gait are altered following natural aging or disease, and this has potential negative consequences for gait adaptability that can lead to increased risk of injury. However, the flexibility of a healthy neurological system to exhibit different fractal patterns in gait has yet to be explored, and this is a necessary step toward understanding human locomotor control. Fifteen participants walked for 15min on a treadmill, either in the absence of a visual stimulus or while they attempted to couple the timing of their gait with a visual metronome that exhibited a persistent fractal pattern (contained long-range correlations) or a random pattern (contained no long-range correlations). The stride-to-stride intervals of the participants were recorded via analog foot pressure switches and submitted to detrended fluctuation analysis (DFA) to determine if the fractal patterns during the visual metronome conditions differed from the baseline (no metronome) condition. DFA α in the baseline condition was 0.77±0.09. The fractal patterns in the stride-to-stride intervals were significantly altered when walking to the fractal metronome (DFA α=0.87±0.06) and to the random metronome (DFA α=0.61±0.10) (both p<.05 when compared to the baseline condition), indicating that a global change in gait dynamics was observed. A variety of strategies were identified at the local level with a cross-correlation analysis, indicating that local behavior did not account for the consistent global changes. Collectively, the results show that a gait dynamics can be shifted in a prescribed manner using a visual stimulus and the shift appears to be a global phenomenon.
Imai, K; Ikeda, M; Enchi, Y; Niimi, T
2009-12-01
The purposes of our studies are to examine whether or not fractal-feature distance deduced from virtual volume method can simulate observer performance indices and to investigate the physical meaning of pseudo fractal dimension and complexity. Contrast-detail (C-D) phantom radiographs were obtained at various mAs values (0.5 - 4.0 mAs) and 140 kVp with a computed radiography system, and the reference image was acquired at 13 mAs. For all C-D images, fractal analysis was conducted using the virtual volume method that was devised with a fractional Brownian motion model. The fractal-feature distances between the considered and reference images were calculated using pseudo fractal dimension and complexity. Further, we have performed the C-D analysis in which ten radiologists participated, and compared the fractal-feature distances with the image quality figures (IQF). To clarify the physical meaning of the pseudo fractal dimension and complexity, contrast-to-noise ratio (CNR) and standard deviation (SD) of images noise were calculated for each mAs and compared with the pseudo fractal dimension and complexity, respectively. A strong linear correlation was found between the fractal-feature distance and IQF. The pseudo fractal dimensions became large as CNR increased. Further, a linear correlation was found between the exponential complexity and image noise SD.
ZnS:Cr Nanostructures Building Fractals and Their Properties
Gogoi, D. P.; Das, U.; Mohanta, D.; Ahmed, G. A.; Choudhury, A.
2010-10-04
Cr doped ZnS nanostructures have been fabricated through colloidal solution route by using Polyvinyl alcohol (-C{sub 2}H{sub 4}O){sub n} and Polyvinyl pyrrolidone k30 (C{sub 6}H{sub 9}NO){sub x} as dielectric hosts. Growth of fractal structures have been observed through Transmission Electron Microscopy. Higher magnification TEM study reveals that these fractals actually a organize structure of ZnS:Cr nanostructures. The structural study of these nanostructures in the fractals is done by X-Ray Diffraction, UV-Visible spectroscopy, Photoluminescence spectroscopy AFM and MFM. These investigations allow us to form a comprehensive explanation of fractal as well as nanostructure growth. We have done dimensional study of these fractals and the reason behind the formation of these fractals.
Evaluation of 3D Printer Accuracy in Producing Fractal Structure.
Kikegawa, Kana; Takamatsu, Kyuuichirou; Kawakami, Masaru; Furukawa, Hidemitsu; Mayama, Hiroyuki; Nonomura, Yoshimune
2017-01-01
Hierarchical structures, also known as fractal structures, exhibit advantageous material properties, such as water- and oil-repellency as well as other useful optical characteristics, owing to its self-similarity. Various methods have been developed for producing hierarchical geometrical structures. Recently, fractal structures have been manufactured using a 3D printing technique that involves computer-aided design data. In this study, we confirmed the accuracy of geometrical structures when Koch curve-like fractal structures with zero to three generations were printed using a 3D printer. The fractal dimension was analyzed using a box-counting method. This analysis indicated that the fractal dimension of the third generation hierarchical structure was approximately the same as that of the ideal Koch curve. These findings demonstrate that the design and production of fractal structures can be controlled using a 3D printer. Although the interior angle deviated from the ideal value, the side length could be precisely controlled.
[Fractal analysis in the diagnosis of breast tumors].
Crişan, D A; Lesaru, M; Dobrescu, R; Vasilescu, C
2007-01-01
Last years studies made by researchers from over the world show that fractal geometry is a viable alternative for image analysis. Fractal features of natural forms give to fractal analysis new valences in various fields, medical imaging being a very important one. This paper intend to prove that fractal dimension, as a way to characterize the complexity of a form, can be used for diagnosis of mammographic lesions classified BI-RADS 4, further investigations being not necessary. The experiments made on 30 cases classified BI-RADS 4 confirmed that 89% of benign lesions have an average fractal dimension under the threshold 1.4, meanwhile malign lesions are characterized, in a similar percentage, by an average fractal dimension over that threshold.
Fractal dimension analysis of complexity in Ligeti piano pieces
NASA Astrophysics Data System (ADS)
Bader, Rolf
2005-04-01
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.
Automatic detection of microcalcifications with multi-fractal spectrum.
Ding, Yong; Dai, Hang; Zhang, Hang
2014-01-01
For improving the detection of micro-calcifications (MCs), this paper proposes an automatic detection of MC system making use of multi-fractal spectrum in digitized mammograms. The approach of automatic detection system is based on the principle that normal tissues possess certain fractal properties which change along with the presence of MCs. In this system, multi-fractal spectrum is applied to reveal such fractal properties. By quantifying the deviations of multi-fractal spectrums between normal tissues and MCs, the system can identify MCs altering the fractal properties and finally locate the position of MCs. The performance of the proposed system is compared with the leading automatic detection systems in a mammographic image database. Experimental results demonstrate that the proposed system is statistically superior to most of the compared systems and delivers a superior performance.
Investigations of human EEG response to viewing fractal patterns.
Hagerhall, Caroline M; Laike, Thorbjörn; Taylor, Richard P; Küller, Marianne; Küller, Rikard; Martin, Theodore P
2008-01-01
Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.
Fractal image compression: A resolution independent representation for imagery
NASA Technical Reports Server (NTRS)
Sloan, Alan D.
1993-01-01
A deterministic fractal is an image which has low information content and no inherent scale. Because of their low information content, deterministic fractals can be described with small data sets. They can be displayed at high resolution since they are not bound by an inherent scale. A remarkable consequence follows. Fractal images can be encoded at very high compression ratios. This fern, for example is encoded in less than 50 bytes and yet can be displayed at resolutions with increasing levels of detail appearing. The Fractal Transform was discovered in 1988 by Michael F. Barnsley. It is the basis for a new image compression scheme which was initially developed by myself and Michael Barnsley at Iterated Systems. The Fractal Transform effectively solves the problem of finding a fractal which approximates a digital 'real world image'.
A tutorial introduction to adaptive fractal analysis.
Riley, Michael A; Bonnette, Scott; Kuznetsov, Nikita; Wallot, Sebastian; Gao, Jianbo
2012-01-01
The authors present a tutorial description of adaptive fractal analysis (AFA). AFA utilizes an adaptive detrending algorithm to extract globally smooth trend signals from the data and then analyzes the scaling of the residuals to the fit as a function of the time scale at which the fit is computed. The authors present applications to synthetic mathematical signals to verify the accuracy of AFA and demonstrate the basic steps of the analysis. The authors then present results from applying AFA to time series from a cognitive psychology experiment on repeated estimation of durations of time to illustrate some of the complexities of real-world data. AFA shows promise in dealing with many types of signals, but like any fractal analysis method there are special challenges and considerations to take into account, such as determining the presence of linear scaling regions.
A tutorial introduction to adaptive fractal analysis
Riley, Michael A.; Bonnette, Scott; Kuznetsov, Nikita; Wallot, Sebastian; Gao, Jianbo
2012-01-01
The authors present a tutorial description of adaptive fractal analysis (AFA). AFA utilizes an adaptive detrending algorithm to extract globally smooth trend signals from the data and then analyzes the scaling of the residuals to the fit as a function of the time scale at which the fit is computed. The authors present applications to synthetic mathematical signals to verify the accuracy of AFA and demonstrate the basic steps of the analysis. The authors then present results from applying AFA to time series from a cognitive psychology experiment on repeated estimation of durations of time to illustrate some of the complexities of real-world data. AFA shows promise in dealing with many types of signals, but like any fractal analysis method there are special challenges and considerations to take into account, such as determining the presence of linear scaling regions. PMID:23060804
Static friction between rigid fractal surfaces.
Alonso-Marroquin, Fernando; Huang, Pengyu; Hanaor, Dorian A H; Flores-Johnson, E A; Proust, Gwénaëlle; Gan, Yixiang; Shen, Luming
2015-09-01
Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.
Region-based fractal video coding
NASA Astrophysics Data System (ADS)
Zhu, Shiping; Belloulata, Kamel
2008-10-01
A novel video sequence compression scheme is proposed in order to realize the efficient and economical transmission of video sequence, and also the region-based functionality of MPEG-4. The CPM and NCIM fractal coding scheme is applied on each region independently by a prior image segmentation map (alpha plane) which is exactly the same as defined in MPEG-4. The first n frames of video sequence are encoded as a "set" using the Circular Prediction Mapping (CPM) and encode the remaining frames using the Non Contractive Interframe Mapping (NCIM). The CPM and NCIM accomplish the motion estimation and compensation, which can exploit the high temporal correlations between the adjacent frames of video sequence. The experimental results with the monocular video sequences provide promising performances at low bit rate coding, such as the application in video conference. We believe the proposed fractal video codec will be a powerful and efficient technique for the region-based video sequence coding.
Static friction between rigid fractal surfaces
NASA Astrophysics Data System (ADS)
Alonso-Marroquin, Fernando; Huang, Pengyu; Hanaor, Dorian A. H.; Flores-Johnson, E. A.; Proust, Gwénaëlle; Gan, Yixiang; Shen, Luming
2015-09-01
Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.
Fractal Tempo Fluctuation and Pulse Prediction
Rankin, Summer K.; Large, Edward W.; Fink, Philip W.
2010-01-01
WE INVESTIGATED PEOPLES’ ABILITY TO ADAPT TO THE fluctuating tempi of music performance. In Experiment 1, four pieces from different musical styles were chosen, and performances were recorded from a skilled pianist who was instructed to play with natural expression. Spectral and rescaled range analyses on interbeat interval time-series revealed long-range (1/f type) serial correlations and fractal scaling in each piece. Stimuli for Experiment 2 included two of the performances from Experiment 1, with mechanical versions serving as controls. Participants tapped the beat at ¼- and ⅛-note metrical levels, successfully adapting to large tempo fluctuations in both performances. Participants predicted the structured tempo fluctuations, with superior performance at the ¼-note level. Thus, listeners may exploit long-range correlations and fractal scaling to predict tempo changes in music. PMID:25190901
Fractal characteristics for binary noise radar waveform
NASA Astrophysics Data System (ADS)
Li, Bing C.
2016-05-01
Noise radars have many advantages over conventional radars and receive great attentions recently. The performance of a noise radar is determined by its waveforms. Investigating characteristics of noise radar waveforms has significant value for evaluating noise radar performance. In this paper, we use binomial distribution theory to analyze general characteristics of binary phase coded (BPC) noise waveforms. Focusing on aperiodic autocorrelation function, we demonstrate that the probability distributions of sidelobes for a BPC noise waveform depend on the distances of these sidelobes to the mainlobe. The closer a sidelobe to the mainlobe, the higher the probability for this sidelobe to be a maximum sidelobe. We also develop Monte Carlo framework to explore the characteristics that are difficult to investigate analytically. Through Monte Carlo experiments, we reveal the Fractal relationship between the code length and the maximum sidelobe value for BPC waveforms, and propose using fractal dimension to measure noise waveform performance.
Fractal characterization of bpn weights evolution.
Gunasekaran, S; Venkatesh, B; Sagar, B S D
2004-04-01
Training methodology of the Back Propagation Network (BPN) is well documented. One aspect of BPN that requires investigation is whether or not the BPN would get trained for a given training data set and architecture. In this paper the behavior of the BPN is analyzed during its training phase considering convergent and divergent training data sets. Evolution of the weights during the training phase was monitored for the purpose of analysis. The evolution of weights was plotted as return map and was characterized by means of fractal dimension. This fractal dimensional analysis of the weight evolution trajectories is used to provide a new insight to understand the behavior of BPN and dynamics in the evolution of weights.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
NASA Astrophysics Data System (ADS)
Chernodub, Maxim N.; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
The fractal structure of the mitochondrial genomes
NASA Astrophysics Data System (ADS)
Oiwa, Nestor N.; Glazier, James A.
2002-08-01
The mitochondrial DNA genome has a definite multifractal structure. We show that loops, hairpins and inverted palindromes are responsible for this self-similarity. We can thus establish a definite relation between the function of subsequences and their fractal dimension. Intriguingly, protein coding DNAs also exhibit palindromic structures, although they do not appear in the sequence of amino acids. These structures may reflect the stabilization and transcriptional control of DNA or the control of posttranscriptional editing of mRNA.
The albedo of fractal stratocumulus clouds
NASA Technical Reports Server (NTRS)
Cahalan, Robert F.; Ridgway, William; Wiscombe, Warren J.; Bell, Thomas L.; Snider, Jack B.
1994-01-01
An increase in the planetary albedo of the earth-atmosphere system by only 10% can decrease the equilibrium surface temperature to that of the last ice age. Nevertheless, albedo biases of 10% or greater would be introduced into large regions of current climate models if clouds were given their observed liquid water amounts, because of the treatment of clouds as plane parallel. The focus on marine stratocumulus clouds is due to their important role in cloud radiative forcing and also that, of the wide variety of earth's cloud types, they are most nearly plane parallel, so that they have the least albedo bias. The fractal model employed here reproduces both the probability distribution and the wavenumber spectrum of the stratocumulus liquid water path, as observed during the First ISCCP Regional Experiment (FIRE). A single new fractal parameter 0 less than or equal to f less than or equal to 1, is introduced and determined empirically by the variance of the logarithm of the vertically integrated liquid water. The reduced reflectivity of fractal stratocumulus clouds is approximately given by the plane-parallel reflectivity evaluated at a reduced 'effective optical thickness,' which when f = 0.5 is tau(sub eff) approximately equal to 10. Study of the diurnal cycle of stratocumulus liquid water during FIRE leads to a key unexpected result: the plane-parallel albedo bias is largest when the cloud fraction reaches 100%, that is, when any bias associated with the cloud fraction vanishes. This is primarily due to the variability increase with cloud fraction. Thus, the within-cloud fractal structure of stratocumulus has a more significant impact on estimates of its mesoscale-average albedo than does the cloud fraction.
The Correlation Fractal Dimension of Complex Networks
NASA Astrophysics Data System (ADS)
Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei
2013-05-01
The fractality of complex networks is studied by estimating the correlation dimensions of the networks. Comparing with the previous algorithms of estimating the box dimension, our algorithm achieves a significant reduction in time complexity. For four benchmark cases tested, that is, the Escherichia coli (E. Coli) metabolic network, the Homo sapiens protein interaction network (H. Sapiens PIN), the Saccharomyces cerevisiae protein interaction network (S. Cerevisiae PIN) and the World Wide Web (WWW), experiments are provided to demonstrate the validity of our algorithm.
Lagrangian and Hamiltonian Mechanics on Fractals Subset of Real-Line
NASA Astrophysics Data System (ADS)
Golmankhaneh, Alireza Khalili; Golmankhaneh, Ali Khalili; Baleanu, Dumitru
2013-11-01
A discontinuous media can be described by fractal dimensions. Fractal objects has special geometric properties, which are discrete and discontinuous structure. A fractal-time diffusion equation is a model for subdiffusive. In this work, we have generalized the Hamiltonian and Lagrangian dynamics on fractal using the fractional local derivative, so one can use as a new mathematical model for the motion in the fractal media. More, Poisson bracket on fractal subset of real line is suggested.
Dynamic contact interactions of fractal surfaces
NASA Astrophysics Data System (ADS)
Jana, Tamonash; Mitra, Anirban; Sahoo, Prasanta
2017-01-01
Roughness parameters and material properties have significant influence on the static and dynamic properties of a rough surface. In the present paper, fractal surface is generated using the modified two-variable Weierstrass-Mandelbrot function in MATLAB and the same is imported to ANSYS to construct the finite element model of the rough surface. The force-deflection relationship between the deformable rough fractal surface and a contacting rigid flat is studied by finite element analysis. For the dynamic analysis, the contacting system is represented by a single degree of freedom spring mass-damper-system. The static force-normal displacement relationship obtained from FE analysis is used to determine the dynamic characteristics of the rough surface for free, as well as for forced damped vibration using numerical methods. The influence of fractal surface parameters and the material properties on the dynamics of the rough surface is also analyzed. The system exhibits softening property for linear elastic surface and the softening nature increases with rougher topography. The softening nature of the system increases with increase in tangent modulus value. Above a certain value of yield strength the nature of the frequency response curve is observed to change its nature from softening to hardening.
Simulation of geological surfaces using fractals
Yfantis, E.A.; Flatman, G.T.; Englund, E.J.
1988-08-01
Methods suggests in the past for simulated ore concentration or pollution concentration over an area of interest, subject to the condition that the simulated surface is passing through specifying points, are based on the assumption of normality. A new method is introduced here which is a generalization of the subdivision method used in fractals. This method is based on the construction of a fractal plane-to-line function f(x, y, R, e, u), where (x, y) is in (a, b) x (c, d), R is the autocorrelation function, e is the resolution limit, and u is a random real function on (-l, l). The simulation using fractals escapes from any distribution assumptions of the data. The given network of points is connected to form quadrilaterals; each one of the quadrilaterals is split based on ways which are extensions of the well-known subdivision method. The quadrilaterals continue to split and grow until resolution obtained in both x and y directions is smaller than a prespecified resolution. If the x coordinate of the ith quadrilateral is in (a/sub i/, b/sub i/) and the y coordinate is in (c/sub i/, d/sub i/), the growth of this quadrilateral is a function of (b/sub i/ - a/sub i/) and (d/sub i/ - c/sub i/); the quadrilateral could grow toward the positive or negative z axis with equal probability forming four new quadrilaterals having a common vertex.
Fractal analysis of Xylella fastidiosa biofilm formation
NASA Astrophysics Data System (ADS)
Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.
2009-07-01
We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.
The role of the circadian system in fractal neurophysiological control
Pittman-Polletta, Benjamin R.; Scheer, Frank A.J.L.; Butler, Matthew P.; Shea, Steven A.; Hu, Kun
2013-01-01
Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system’s role in fractal regulation. PMID:23573942
Fractal analysis of scatter imaging signatures to distinguish breast pathologies
NASA Astrophysics Data System (ADS)
Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.
2013-02-01
Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.
Application study of fractal theory in mechanical transmission
NASA Astrophysics Data System (ADS)
Zhao, Han; Wu, Qilin
2016-09-01
Mechanical transmissions are applied widely in various electrical and mechanical products, but some qualities of some high-end products can't meet people's demand, and need to be improved with some new methods or theories. The fractal theory is a new mathematic tool, which provides a new approach for the further study in the area of the mechanical transmission, and helps to solve some problems. The basic contents of the fractal theory are introduced firstly, especially the two important concepts, the self-similar fractal and the fractal dimension. Then, the deferent application of the fractal theory in this area are given to display how to further the study and improve some important characteristics of the mechanical transmission, such as contact surfaces, manufacturing precise, friction and wear, stiffness, strength, dynamics, fault diagnosis, etc. Finally, the problems of the fractal theory and its application are discussed, and some weaknesses, such as the calculation capacity of the fractal theory is not strong, are pointed out. Some new solutions are suggested, such as combining the fractal theory with the fuzzy theory, the chaos theory and so on. The new application fields of the fractal theory in the area of the mechanical transmission are proposed.
The role of the circadian system in fractal neurophysiological control.
Pittman-Polletta, Benjamin R; Scheer, Frank A J L; Butler, Matthew P; Shea, Steven A; Hu, Kun
2013-11-01
Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system's role in fractal regulation.
GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION
NASA Astrophysics Data System (ADS)
Ramírez, José L.; Rubiano, Gustavo N.; Zlobec, Borut Jurčič
2015-10-01
In this paper, we introduce the p-circle inversion which generalizes the classical inversion with respect to a circle (p = 2) and the taxicab inversion (p = 1). We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the p-circle inversion.
SANS spectra of the fractal supernucleosomal chromatin structure models
NASA Astrophysics Data System (ADS)
Ilatovskiy, Andrey V.; Lebedev, Dmitry V.; Filatov, Michael V.; Petukhov, Michael G.; Isaev-Ivanov, Vladimir V.
2012-03-01
The eukaryotic genome consists of chromatin—a nucleoprotein complex with hierarchical architecture based on nucleosomes, the organization of higher-order chromatin structures still remains unknown. Available experimental data, including SANS spectra we had obtained for whole nuclei, suggested fractal nature of chromatin. Previously we had built random-walk supernucleosomal models (up to 106 nucleosomes) to interpret our SANS spectra. Here we report a new method to build fractal supernucleosomal structure of a given fractal dimension or two different dimensions. Agreement between calculated and experimental SANS spectra was significantly improved, especially for model with two fractal dimensions—3 and 2.
Reinforcement of rubber by fractal aggregates
NASA Astrophysics Data System (ADS)
Witten, T. A.; Rubinstein, M.; Colby, R. H.
1993-03-01
Rubber is commonly reinforced with colloidal aggregates of carbon or silica, whose structure has the scale invariance of a fractal object. Reinforced rubbers support large stresses, which often grow faster than linearly with the strain. We argue that under strong elongation the stress arises through lateral compression of the aggregates, driven by the large bulk modulus of the rubber. We derive a power-law relationship between stress and elongation λ when λgg 1. The predicted power p depends on the fractal dimension D and a second structural scaling exponent C. For diffusion-controlled aggregates this power p should lie beween 0.9 and 1.1 ; for reaction-controlled aggregates p should lie between 1.8 and 2.4. For uniaxial compression the analogous powers lie near 4. Practical rubbers filled with fractal aggregates should approach the conditions of validity for these scaling laws. On renforce souvent le caoutchouc avec des agrégats de carbone ou de silice dont la structure a l'invariance par dilatation d'un objet fractal. Les caoutchoucs ainsi renforcés supportent de grandes contraintes qui croissent souvent plus vite que l'élongation. Nous prétendons que, sous élongation forte, cette contrainte apparaît à cause d'une compression latérale des agrégats induite par le module volumique important du caoutchouc. Nous établissons une loi de puissance reliant la contrainte et l'élongation λ quand λgg 1. Cet exposant p dépend de la dimension fractale D et d'un deuxième exposant structural C. Pour des agrégats dont la cinétique de formation est limitée par diffusion, p vaut entre 0,9 et 1,1. Si la cinétique est limitée par le soudage local des particules, p vaut entre 1,8 et 2,4. Sous compression uniaxiale, les puissances homologues valent environ 4. Des caoutchoucs pratiques chargés de tels agrégats devraient approcher des conditions où ces lois d'échelle sont valables.
Influence of buoyancy on drainage of a fractal porous medium
NASA Astrophysics Data System (ADS)
Huinink, H. P.; Michels, M. A.
2002-10-01
The influence of stabilizing hydrostatic pressure gradients on the drainage of a fractal porous medium is studied. The invasion process is treated with invasion percolation (IP) in a gradient. Fractality is mimicked by randomly closing bonds of a network. Two length scales govern the problem: the characteristic length of the pore structure ξs and a length scale ξg above which buoyancy determines the structure of the cluster. When ξs<ξg the local structure of the invading cluster is governed by the interplay of capillarity and the fractal properties of the pore space. Only parts of the backbone of the pore structure can be invaded. Therefore, the obtained fractal dimension for small systems L<ξs is much lower (1.40) than the one for ordinary IP (1.82). On larger length scales, ξs
The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter
NASA Technical Reports Server (NTRS)
Herren, Kenneth A.; Gregory, Don A.
1999-01-01
The area of information processing has grown dramatically over the last 50 years. In the areas of image processing and information storage the technology requirements have far outpaced the ability of the community to meet demands. The need for faster recognition algorithms and more efficient storage of large quantities of data has forced the user to accept less than lossless retrieval of that data for analysis. In addition to clutter that is not the object of interest in the data set, often the throughput requirements forces the user to accept "noisy" data and to tolerate the clutter inherent in that data. It has been shown that some of this clutter, both the intentional clutter (clouds, trees, etc) as well as the noise introduced on the data by processing requirements can be modeled as fractal or fractal-like. Traditional methods using Fourier deconvolution on these sources of noise in frequency space leads to loss of signal and can, in many cases, completely eliminate the target of interest. The parameters that characterize fractal-like noise (predominately the fractal dimension) have been investigated and a technique to reduce or eliminate noise from real scenes has been developed. Examples of clutter reduced images are presented.
Fractal behavior of traffic volume on urban expressway through adaptive fractal analysis
NASA Astrophysics Data System (ADS)
He, Hong-di; Wang, Jun-li; Wei, Hai-rui; Ye, Cheng; Ding, Yi
2016-02-01
In this paper, we investigate the fractal behavior of traffic volume in urban expressway based on a newly developed adaptive fractal analysis (AFA), which has a number of advantages over traditional method of detrended fluctuation analysis (DFA). Before fractal analysis, autocorrelation function was first adopted on traffic volume data and the long-range correlation behavior was found to be existed in both on-ramp and off-ramp situations. Then AFA as well as DFA was applied to further examine the fractal behavior. The results showed that the multifractality and the long-range anti-persistent behavior existed on both on-ramp and off-ramp. Additionally, multifractal analysis on weekdays and weekends are performed respectively and the results show that the degree of multifractality on weekdays is higher than that on weekends, implying that long-range correlation behaviors were more obvious on weekdays. Finally, the source of multifractality is examined with randomly shuffled and the surrogated series. Long-range correlation behaviors are identified in both on-ramp and off-ramp situations and fat-tail distributions were found to make little in the contributions of multifractality.
Fractal and Multifractal Models Applied to Porous Media - Editorial
Technology Transfer Automated Retrieval System (TEKTRAN)
Given the current high level of interest in the use of fractal geometry to characterize natural porous media, a special issue of the Vadose Zone Journal was organized in order to expose established fractal analysis techniques and cutting-edge new developments to a wider Earth science audience. The ...
The fractal nature of vacuum arc cathode spots
Anders, Andre
2005-05-27
Cathode spot phenomena show many features of fractals, for example self-similar patterns in the emitted light and arc erosion traces. Although there have been hints on the fractal nature of cathode spots in the literature, the fractal approach to spot interpretation is underutilized. In this work, a brief review of spot properties is given, touching the differences between spot type 1 (on cathodes surfaces with dielectric layers) and spot type 2 (on metallic, clean surfaces) as well as the known spot fragment or cell structure. The basic properties of self-similarity, power laws, random colored noise, and fractals are introduced. Several points of evidence for the fractal nature of spots are provided. Specifically power laws are identified as signature of fractal properties, such as spectral power of noisy arc parameters (ion current, arc voltage, etc) obtained by fast Fourier transform. It is shown that fractal properties can be observed down to the cutoff by measurement resolution or occurrence of elementary steps in physical processes. Random walk models of cathode spot motion are well established: they go asymptotically to Brownian motion for infinitesimal step width. The power spectrum of the arc voltage noise falls as 1/f {sup 2}, where f is frequency, supporting a fractal spot model associated with Brownian motion.
An application of geostatistics and fractal geometry for reservoir characterization
Aasum, Y.; Kelkar, M.G. ); Gupta, S.P. )
1991-03-01
This paper presents an application of geostatistics and fractal geometry concepts for 2D characterization of rock properties (k and {phi}) in a dolomitic, layered-cake reservoir. The results indicate that lack of closely spaced data yield effectively random distributions of properties. Further, incorporation of geology reduces uncertainties in fractal interpolation of wellbore properties.
a Fractal Network Model for Fractured Porous Media
NASA Astrophysics Data System (ADS)
Xu, Peng; Li, Cuihong; Qiu, Shuxia; Sasmito, Agus Pulung
2016-04-01
The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.
A comparison of the fractal and JPEG algorithms
NASA Technical Reports Server (NTRS)
Cheung, K.-M.; Shahshahani, M.
1991-01-01
A proprietary fractal image compression algorithm and the Joint Photographic Experts Group (JPEG) industry standard algorithm for image compression are compared. In every case, the JPEG algorithm was superior to the fractal method at a given compression ratio according to a root mean square criterion and a peak signal to noise criterion.
Fractal Modeling and Scaling in Natural Systems - Editorial
Technology Transfer Automated Retrieval System (TEKTRAN)
The special issue of Ecological complexity journal on Fractal Modeling and Scaling in Natural Systems contains representative examples of the status and evolution of data-driven research into fractals and scaling in complex natural systems. The editorial discusses contributions to understanding rela...
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
A Fractal Dimension Survey of Active Region Complexity
NASA Technical Reports Server (NTRS)
McAteer, R. T. James; Gallagher, Peter; Ireland, Jack
2005-01-01
A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
Perceptual and Physiological Responses to Jackson Pollock's Fractals.
Taylor, Richard P; Spehar, Branka; Van Donkelaar, Paul; Hagerhall, Caroline M
2011-01-01
Fractals have been very successful in quantifying the visual complexity exhibited by many natural patterns, and have captured the imagination of scientists and artists alike. Our research has shown that the poured patterns of the American abstract painter Jackson Pollock are also fractal. This discovery raises an intriguing possibility - are the visual characteristics of fractals responsible for the long-term appeal of Pollock's work? To address this question, we have conducted 10 years of scientific investigation of human response to fractals and here we present, for the first time, a review of this research that examines the inter-relationship between the various results. The investigations include eye tracking, visual preference, skin conductance, and EEG measurement techniques. We discuss the artistic implications of the positive perceptual and physiological responses to fractal patterns.
Computerized analysis of mammographic parenchymal patterns using fractal analysis
NASA Astrophysics Data System (ADS)
Li, Hui; Giger, Maryellen L.; Huo, Zhimin; Olopade, Olufunmilayo I.; Chinander, Michael R.; Lan, Li; Bonta, Ioana R.
2003-05-01
Mammographic parenchymal patterns have been shown to be associated with breast cancer risk. Fractal-based texture analyses, including box-counting methods and Minkowski dimension, were performed within parenchymal regions of normal mammograms of BRCA1/BRCA2 gene mutation carriers and within those of women at low risk for developing breast cancer. Receiver Operating Characteristic (ROC) analysis was used to assess the performance of the computerized radiographic markers in the task of distinguishing between high and low-risk subjects. A multifractal phenomenon was observed with the fractal analyses. The high frequency component of fractal dimension from the conventional box-counting technique yielded an Az value of 0.84 in differentiating between two groups, while using the LDA to estimate the fractal dimension yielded an Az value of 0.91 for the high frequency component. An Az value of 0.82 was obtained with fractal dimensions extracted using the Minkowski algorithm.
A Tutorial Review on Fractal Spacetime and Fractional Calculus
NASA Astrophysics Data System (ADS)
He, Ji-Huan
2014-11-01
This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.
Fractal analysis of motor imagery recognition in the BCI research
NASA Astrophysics Data System (ADS)
Chang, Chia-Tzu; Huang, Han-Pang; Huang, Tzu-Hao
2011-12-01
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).
Comparison of ictal and interictal EEG signals using fractal features.
Wang, Yu; Zhou, Weidong; Yuan, Qi; Li, Xueli; Meng, Qingfang; Zhao, Xiuhe; Wang, Jiwen
2013-12-01
The feature analysis of epileptic EEG is very significant in diagnosis of epilepsy. This paper introduces two nonlinear features derived from fractal geometry for epileptic EEG analysis. The features of blanket dimension and fractal intercept are extracted to characterize behavior of EEG activities, and then their discriminatory power for ictal and interictal EEGs are compared by means of statistical methods. It is found that there is significant difference of the blanket dimension and fractal intercept between interictal and ictal EEGs, and the difference of the fractal intercept feature between interictal and ictal EEGs is more noticeable than the blanket dimension feature. Furthermore, these two fractal features at multi-scales are combined with support vector machine (SVM) to achieve accuracies of 97.58% for ictal and interictal EEG classification and 97.13% for normal, ictal and interictal EEG classification.
Perceptual and Physiological Responses to Jackson Pollock's Fractals
Taylor, Richard P.; Spehar, Branka; Van Donkelaar, Paul; Hagerhall, Caroline M.
2011-01-01
Fractals have been very successful in quantifying the visual complexity exhibited by many natural patterns, and have captured the imagination of scientists and artists alike. Our research has shown that the poured patterns of the American abstract painter Jackson Pollock are also fractal. This discovery raises an intriguing possibility – are the visual characteristics of fractals responsible for the long-term appeal of Pollock's work? To address this question, we have conducted 10 years of scientific investigation of human response to fractals and here we present, for the first time, a review of this research that examines the inter-relationship between the various results. The investigations include eye tracking, visual preference, skin conductance, and EEG measurement techniques. We discuss the artistic implications of the positive perceptual and physiological responses to fractal patterns. PMID:21734876
Laser light scattering as a probe of fractal colloid aggregates
NASA Technical Reports Server (NTRS)
Weitz, David A.; Lin, M. Y.
1989-01-01
The extensive use of laser light scattering is reviewed, both static and dynamic, in the study of colloid aggregation. Static light scattering enables the study of the fractal structure of the aggregates, while dynamic light scattering enables the study of aggregation kinetics. In addition, both techniques can be combined to demonstrate the universality of the aggregation process. Colloidal aggregates are now well understood and therefore represent an excellent experimental system to use in the study of the physical properties of fractal objects. However, the ultimate size of fractal aggregates is fundamentally limited by gravitational acceleration which will destroy the fractal structure as the size of the aggregates increases. This represents a great opportunity for spaceborne experimentation, where the reduced g will enable the growth of fractal structures of sufficient size for many interesting studies of their physical properties.
Determining Effective Thermal Conductivity of Fabrics by Using Fractal Method
NASA Astrophysics Data System (ADS)
Zhu, Fanglong; Li, Kejing
2010-03-01
In this article, a fractal effective thermal conductivity model for woven fabrics with multiple layers is developed. Structural models of yarn and plain woven fabric are derived based on the fractal characteristics of macro-pores (gap or channel) between the yarns and micro-pores inside the yarns. The fractal effective thermal conductivity model can be expressed as a function of the pore structure (fractal dimension) and architectural parameters of the woven fabric. Good agreement is found between the fractal model and the thermal conductivity measurements in the general porosity ranges. It is expected that the model will be helpful in the evaluation of thermal comfort for woven fabric in the whole range of porosity.
Fractal branching pattern of the monopodial canine airway.
Wang, Ping M; Kraman, Steve S
2004-06-01
Unlike the human lung, monopodial canine airway branching follows an irregular dichotomized pattern with fractal features. We studied three canine airway molds and found a self-similarity feature from macro- to microscopic scales, which formed a fractal set up to seven scales in the airways. At each fractal scale, lateral branches evenly lined up along an approximately straight main trunk to form three to four two-dimensional structures, and each lateral branch was the monopodial main trunk of the next fractal scale. We defined this pattern as the fractal main lateral-branching pattern, which exhibited similar structures from macro- to microscopic scales, including lobes, sublobes, sub-sublobes, etc. We speculate that it, rather than a mother-daughter pattern, could better describe the actual asymmetrical architecture of the monopodial canine airway.
A Fractal Nature for Polymerized Laminin
Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana
2014-01-01
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
Pyramidal fractal dimension for high resolution images.
Mayrhofer-Reinhartshuber, Michael; Ahammer, Helmut
2016-07-01
Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥1024×1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.
Fractals and dynamics in art and design.
Guastello, Stephen J
2015-01-01
Many styles of visual art that build on fractal imagery and chaotic dynamics in the creative process have been examined in NDPLS in recent years. This article presents a gallery of artwork turned into design that appeared in the promotional products of the Society for Chaos Theory in Psychology & Life Sciences. The gallery showcases a variety of new imaging styles, including photography, that reflect a deepening perspective on nonlinear dynamics and art. The contributing artworks in design formats combine to render the verve that transcends the boundaries between the artistic and scientific communities.
On the Conditional Matching of Fractal Networks
NASA Astrophysics Data System (ADS)
Wang, Yanchun; Sun, Weigang; Zhang, Jingyuan; Qin, Sen
2016-12-01
In this paper, we propose a new matching (called a conditional matching), where the condition refers to the matching of the new constructed network which includes all the nodes in the original network. We then enumerate the conditional matchings of the new network and prove that the number of conditional matchings is just the product of degree sequences of the original network. We choose two families of fractal networks to show our obtained results, including the pseudofractal network and Cayley tree. Finally, we calculate the entropy of the conditional matchings on the considered networks and see that the entropy of Cayley tree is smaller than that of the pseudofractal network.
Trabecular Bone Mechanical Properties and Fractal Dimension
NASA Technical Reports Server (NTRS)
Hogan, Harry A.
1996-01-01
Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again
NASA Astrophysics Data System (ADS)
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-05-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
Fractal Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
Drawing conformal diagrams for a fractal landscape
Winitzki, Sergei
2005-06-15
Generic models of cosmological inflation and the recently proposed scenarios of a recycling universe and the string theory landscape predict spacetimes whose global geometry is a stochastic, self-similar fractal. To visualize the complicated causal structure of such a universe, one usually draws a conformal (Carter-Penrose) diagram. I develop a new method for drawing conformal diagrams, applicable to arbitrary 1+1-dimensional spacetimes. This method is based on a qualitative analysis of intersecting lightrays and thus avoids the need for explicit transformations of the spacetime metric. To demonstrate the power and simplicity of this method, I present derivations of diagrams for spacetimes of varying complication. I then apply the lightray method to three different models of an eternally inflating universe (scalar-field inflation, recycling universe, and string theory landscape) involving the nucleation of nested asymptotically flat, de Sitter and/or anti-de Sitter bubbles. I show that the resulting diagrams contain a characteristic fractal arrangement of lines.
Fractal aircraft trajectories and nonclassical turbulent exponents.
Lovejoy, S; Schertzer, D; Tuck, A F
2004-09-01
The dimension (D) of aircraft trajectories is fundamental in interpreting airborne data. To estimate D, we studied data from 18 trajectories of stratospheric aircraft flights 1600 km long taken during a "Mach cruise" (near constant Mach number) autopilot flight mode of the ER-2 research aircraft. Mach cruise implies correlated temperature and wind fluctuations so that DeltaZ approximately Deltax (H(z) ) where Z is the (fluctuating) vertical and x the horizontal coordinate of the aircraft. Over the range approximately 3-300 km , we found H(z) approximately 0.58+/-0.02 close to the theoretical 5/9=0.56 and implying D=1+ H(z) =14/9 , i.e., the trajectories are fractal. For distances <3 km aircraft inertia smooths the trajectories, for distances >300 km , D=1 again because of a rise of 1 m/km due to fuel consumption. In the fractal regime, the horizontal velocity and temperature exponents are close to the nonclassical value 1/2 (rather than 1/3 ). We discuss implications for aircraft measurements as well as for the structure of the atmosphere.
Fractal avalanche ruptures in biological membranes
NASA Astrophysics Data System (ADS)
Gözen, Irep; Dommersnes, Paul; Czolkos, Ilja; Jesorka, Aldo; Lobovkina, Tatsiana; Orwar, Owe
2010-11-01
Bilayer membranes envelope cells as well as organelles, and constitute the most ubiquitous biological material found in all branches of the phylogenetic tree. Cell membrane rupture is an important biological process, and substantial rupture rates are found in skeletal and cardiac muscle cells under a mechanical load. Rupture can also be induced by processes such as cell death, and active cell membrane repair mechanisms are essential to preserve cell integrity. Pore formation in cell membranes is also at the heart of many biomedical applications such as in drug, gene and short interfering RNA delivery. Membrane rupture dynamics has been studied in bilayer vesicles under tensile stress, which consistently produce circular pores. We observed very different rupture mechanics in bilayer membranes spreading on solid supports: in one instance fingering instabilities were seen resulting in floral-like pores and in another, the rupture proceeded in a series of rapid avalanches causing fractal membrane fragmentation. The intermittent character of rupture evolution and the broad distribution in avalanche sizes is consistent with crackling-noise dynamics. Such noisy dynamics appear in fracture of solid disordered materials, in dislocation avalanches in plastic deformations and domain wall magnetization avalanches. We also observed similar fractal rupture mechanics in spreading cell membranes.
Fractal Profit Landscape of the Stock Market
Grönlund, Andreas; Yi, Il Gu; Kim, Beom Jun
2012-01-01
We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q Stocks are sold and bought if the log return is bigger than p and less than –q, respectively. Repetition of this simple strategy for a long time gives the profit defined in the underlying two-dimensional parameter space of p and q. It is revealed that the local maxima in the profit landscape are spread in the form of a fractal structure. The fractal structure implies that successful strategies are not localized to any region of the profit landscape and are neither spaced evenly throughout the profit landscape, which makes the optimization notoriously hard and hypersensitive for partial or limited information. The concrete implication of this property is demonstrated by showing that optimization of one stock for future values or other stocks renders worse profit than a strategy that ignores fluctuations, i.e., a long-term buy-and-hold strategy. PMID:22558079
Fractal profit landscape of the stock market.
Grönlund, Andreas; Yi, Il Gu; Kim, Beom Jun
2012-01-01
We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repetition of this simple strategy for a long time gives the profit defined in the underlying two-dimensional parameter space of p and q. It is revealed that the local maxima in the profit landscape are spread in the form of a fractal structure. The fractal structure implies that successful strategies are not localized to any region of the profit landscape and are neither spaced evenly throughout the profit landscape, which makes the optimization notoriously hard and hypersensitive for partial or limited information. The concrete implication of this property is demonstrated by showing that optimization of one stock for future values or other stocks renders worse profit than a strategy that ignores fluctuations, i.e., a long-term buy-and-hold strategy.
ABC of multi-fractal spacetimes and fractional sea turtles
NASA Astrophysics Data System (ADS)
Calcagni, Gianluca
2016-04-01
We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with q-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.
Fractal dimensions of rampart impact craters on Mars
NASA Technical Reports Server (NTRS)
Ching, Delwyn; Taylor, G. Jeffrey; Mouginis-Mark, Peter; Bruno, Barbara C.
1993-01-01
Ejecta blanket morphologies of Martian rampart craters may yield important clues to the atmospheric densities during impact, and the nature of target materials (e.g., hard rock, fine-grained sediments, presence of volatiles). In general, the morphologies of such craters suggest emplacement by a fluidized, ground hugging flow instead of ballistic emplacement by dry ejecta. We have quantitatively characterized the shape of the margins of the ejecta blankets of 15 rampart craters using fractal geometry. Our preliminary results suggest that the craters are fractals and are self-similar over scales of approximately 0.1 km to 30 km. Fractal dimensions (a measure of the extent to which a line fills a plane) range from 1.06 to 1.31. No correlations of fractal dimension with target type, elevation, or crater size were observed, though the data base is small. The range in fractal dimension and lack of correlation may be due to a complex interplay of target properties (grain size, volatile content), atmospheric pressure, and crater size. The mere fact that the ejecta margins are fractals, however, indicates that viscosity and yield strength of the ejecta were at least as low as those of basalts, because silicic lava flows are not generally fractals.
Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.
Captur, Gabriella; Karperien, Audrey L; Li, Chunming; Zemrak, Filip; Tobon-Gomez, Catalina; Gao, Xuexin; Bluemke, David A; Elliott, Perry M; Petersen, Steffen E; Moon, James C
2015-09-07
Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools.
Comprehensive fractal description of porosity of coal of different ranks.
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure.
Measuring the Fractal Dimensions of Empirical Cartographic Curves,
1982-01-01
ahhf, by Week smnber) Fractal dimension, Chord length, Line length, Linear regression U.ASTRACF (CO1 o M 6 #d itw 4000"s If Rea.5 I.R OF Wleek amber...The fractal dimension of a curve Is a measure of Its geometric complexity and - can be any men-integer value between 1 and 2 depend ing upon the...curve’s level pair of dividers along a curve, used to calculate the fractal diummlons ofS3 cuvs. It also discusses the *hole of chord length and the wabor
Spatial behavior analysis at the global level using fractal geometry.
Sambrook, Roger C
2008-01-01
Previous work has suggested that an estimate of fractal dimension can provide a useful metric for quantifying settlement patterns. This study uses fractal methods to investigate settlement patterns at a global scale showing that the scaling behavior of the pattern of the world's largest cities corresponds to that typically observed for coastlines and rivers. This serves to validate the use of fractal dimension as a scale-independent measure of settlement patterns which can be correlated with other physical features. Such a measure may be a useful validation criterion for models of human settlement and spatial behavior.
Emergence of fractals in aggregation with stochastic self-replication.
Hassan, Md Kamrul; Hassan, Md Zahedul; Islam, Nabila
2013-10-01
We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse. Furthermore, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given, and the relationship between this fractal dimension and a conserved quantity is pointed out.
Fractal atomic-level percolation in metallic glasses.
Chen, David Z; Shi, Crystal Y; An, Qi; Zeng, Qiaoshi; Mao, Wendy L; Goddard, William A; Greer, Julia R
2015-09-18
Metallic glasses are metallic alloys that exhibit exotic material properties. They may have fractal structures at the atomic level, but a physical mechanism for their organization without ordering has not been identified. We demonstrated a crossover between fractal short-range (<2 atomic diameters) and homogeneous long-range structures using in situ x-ray diffraction, tomography, and molecular dynamics simulations. A specific class of fractal, the percolation cluster, explains the structural details for several metallic-glass compositions. We postulate that atoms percolate in the liquid phase and that the percolating cluster becomes rigid at the glass transition temperature.
Fractal Image Filters for Specialized Image Recognition Tasks
2010-02-11
Mendoza , Cambridge University Press, Cambridge, 1995. [20] J . Kigami, Harmonic Calculus on p.c.f. Self-similar Sets, Trans. Amer. Math. Soc. 335 (1993) 721...Transformations between fractals, Progress in Probability, 61 (2009) 227-250. [4] M. F. Barnsley, J . Hutchinson, Ö. Sten�o, V-variable fractals: fractals with...convex body K R2 such that F (K) K. The associated Hilbert metric dH is de�ned on K by dH(x; y) = ln jR (x; y; a; b) j for all x; y 2 K with x 6
Fractal-geometry simulation of a lightning discharge
NASA Astrophysics Data System (ADS)
Balkhanov, V. K.; Bashkuev, Yu. B.
2012-12-01
It is suggested that a wideband lightning discharge be approximated by a damped periodic oscillation. With such an approach, the oscillation frequency and relaxation time are introduced and it is found that lightning radiates over a distance of several tens of kilometers. This length is much greater than the lightning bolt's apparent length (several kilometers). The difference between the lengths is explained using fractal geometry. In terms of fractal geometry, the lightning discharge is so tortuous that an actually very long lightning bolt is accommodated by a short straight line. An attempt is made to determine the fractal dimension of tortuous and intermittent lightning bolts.
Fractal aspects and convergence of Newton`s method
Drexler, M.
1996-12-31
Newton`s Method is a widely established iterative algorithm for solving non-linear systems. Its appeal lies in its great simplicity, easy generalization to multiple dimensions and a quadratic local convergence rate. Despite these features, little is known about its global behavior. In this paper, we will explain a seemingly random global convergence pattern using fractal concepts and show that the behavior of the residual is entirely explicable. We will also establish quantitative results for the convergence rates. Knowing the mechanism of fractal generation, we present a stabilization to the orthodox Newton method that remedies the fractal behavior and improves convergence.
Gravitation theory in a fractal space-time
Agop, M.; Gottlieb, I.
2006-05-15
Assimilating the physical space-time with a fractal, a general theory is built. For a fractal dimension D=2, the virtual geodesics of this space-time implies a generalized Schroedinger type equation. Subsequently, a geometric formulation of the gravitation theory on a fractal space-time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equation are calculated. It results, by means of a dilation operator, the equivalence of this model with quantum Einstein gravity.
Fractal dimension of cerebral surfaces using magnetic resonance images
Majumdar, S.; Prasad, R.R.
1988-11-01
The calculation of the fractal dimension of the surface bounded by the grey matter in the normal human brain using axial, sagittal, and coronal cross-sectional magnetic resonance (MR) images is presented. The fractal dimension in this case is a measure of the convolutedness of this cerebral surface. It is proposed that the fractal dimension, a feature that may be extracted from MR images, may potentially be used for image analysis, quantitative tissue characterization, and as a feature to monitor and identify cerebral abnormalities and developmental changes.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.
1998-01-01
Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and
Fractal pharmacokinetics of the drug mibefradil in the liver
NASA Astrophysics Data System (ADS)
Fuite, J.; Marsh, R.; Tuszyński, J.
2002-08-01
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.
Fractal Globules: A New Approach to Artificial Molecular Machines
Avetisov, Vladik A.; Ivanov, Viktor A.; Meshkov, Dmitry A.; Nechaev, Sergei K.
2014-01-01
The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path. PMID:25418305
Process for applying control variables having fractal structures
Bullock, J.S. IV; Lawson, R.L.
1996-01-23
A process and apparatus are disclosed for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform. 3 figs.
Process for applying control variables having fractal structures
Bullock, IV, Jonathan S.; Lawson, Roger L.
1996-01-01
A process and apparatus for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform.
Fractal patterns applied to implant surface: definitions and perspectives.
Dohan Ehrenfest, David M
2011-10-01
Fractal patterns are frequently found in nature, but they are difficult to reproduce in artificial objects such as implantable materials. In this article, a definition of the concept of fractals for osseointegrated surfaces is suggested, based on the search for quasi-self-similarity on at least 3 scales of investigation: microscale, nanoscale, and atomic/crystal scale. Following this definition, the fractal dimension of some surfaces may be defined (illustrated here with the Intra-Lock Ossean surface). However the biological effects of this architecture are still unknown and should be examined carefully in the future.
NASA Astrophysics Data System (ADS)
Besselink, R.; Stawski, T. M.; Van Driessche, A. E. S.; Benning, L. G.
2016-12-01
Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ṡ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.
A fractal model of the Universe
NASA Astrophysics Data System (ADS)
Gottlieb, Ioan
The book represents a revisioned, extended, completed and translated version of the book "Superposed Universes. A scientific novel and a SF story" (1995). The book contains a hypothesis by the author concerning the complexity of the Nature. An introduction to the theories of numbers, manyfolds and topology is given. The possible connection with the theory of evolution of the Universe is discussed. The book contains also in the last chapter a SF story based on the hypothesis presented. A connection with fractals theory is given. A part of his earlier studies (1955-1956) were subsequently published without citation by Ali Kyrala (Phys. Rev. vol.117, No.5, march 1, 1960). The book contains as an important appendix the early papers (some of which are published in the coauthoprship with his scientific advisors): 1) T.T. Vescan, A. Weiszmann and I.Gottlieb, Contributii la studiul problemelor geometrice ale teoriei relativitatii restranse. Academia R.P.R. Baza Timisoara. Lucrarile consfatuirii de geometrie diferentiala din 9-12 iunie 1955. In this paper the authors show a new method of the calculation of the metrics. 2) Jean Gottlieb, L'hyphotese d'un modele de la structure de la matiere, Revista Matematica y Fisica Teorica, Serie A, Volumen XY, No.1, y.2, 1964 3) I. Gottlieb, Some hypotheses on space, time and gravitation, Studies in Gravitation Theory, CIP Press, Bucharest, 1988, pp.227-234 as well as some recent papers (published in the coauthorship with his disciples): 4)M. Agop, Gottlieb speace-time. A fractal axiomatic model of the Universe. in Particles and Fields, Editors: M.Agop and P.D. Ioannou, Athens University Press, 2005, pp. 59-141 5) I. Gottlieb, M.Agop and V.Enache, Games with Cantor's dust. Chaos, Solitons and Fractals, vol.40 (2009) pp. 940-945 6) I. Gottlieb, My picture over the World, Bull. of the Polytechnic Institute of Iasi. Tom LVI)LX, Fasc. 1, 2010, pp. 1-18. The book contains also a dedication to father Vasile Gottlieb and wife Cleopatra
NASA Astrophysics Data System (ADS)
Frederiksen, Richard D.; Dahm, Werner J. A.; Dowling, David R.
1997-05-01
Results from an earlier experimental assessment of fractal scale similarity in one-dimensional spatial and temporal intersections in turbulent flows are here extended to two- and three-dimensional spatial intersections. Over 25000 two-dimensional (2562) intersections and nearly 40 three-dimensional (2563) intersections, collectively representing more than 2.3 billion data points, were analysed using objective statistical methods to determine which intersections were as fractal as stochastically scale-similar fractal gauge sets having the same record length. Results for the geometry of Sc [dbl greater-than sign]1 scalar isosurfaces and the scalar dissipation support span the range of lengthscales between the scalar and viscous diffusion scales [lambda]D and [lambda][nu]. The present study finds clear evidence for stochastic fractal scale similarity in the dissipation support. With increasing intersection dimension n, the data show a decrease in the fraction of intersections satisfying the criteria for fractal scale similarity, consistent with the presence of localized non-fractal inclusions. Local scale similarity analyses on three-dimensional (643) intersections directly show such intermittent non-fractal inclusions with characteristic lengthscale comparable to [lambda][nu]. These inclusions lead to failure of the relation among codimensions Dn[identical with]D[minus sign](3[minus sign]n) when applied to simple average dimensions, which has formed the basis for most previous assessments of fractal scale-similarity. Unlike the dissipation support geometry, scalar isosurface geometries from the same data were found not to be as fractal as fractional Brownian motion gauge sets over the range of scales examined.
Fractal fluctuations in cardiac time series
NASA Technical Reports Server (NTRS)
West, B. J.; Zhang, R.; Sanders, A. W.; Miniyar, S.; Zuckerman, J. H.; Levine, B. D.; Blomqvist, C. G. (Principal Investigator)
1999-01-01
Human heart rate, controlled by complex feedback mechanisms, is a vital index of systematic circulation. However, it has been shown that beat-to-beat values of heart rate fluctuate continually over a wide range of time scales. Herein we use the relative dispersion, the ratio of the standard deviation to the mean, to show, by systematically aggregating the data, that the correlation in the beat-to-beat cardiac time series is a modulated inverse power law. This scaling property indicates the existence of long-time memory in the underlying cardiac control process and supports the conclusion that heart rate variability is a temporal fractal. We argue that the cardiac control system has allometric properties that enable it to respond to a dynamical environment through scaling.
Fractal power law in literary English
NASA Astrophysics Data System (ADS)
Gonçalves, L. L.; Gonçalves, L. B.
2006-02-01
We present in this paper a numerical investigation of literary texts by various well-known English writers, covering the first half of the twentieth century, based upon the results obtained through corpus analysis of the texts. A fractal power law is obtained for the lexical wealth defined as the ratio between the number of different words and the total number of words of a given text. By considering as a signature of each author the exponent and the amplitude of the power law, and the standard deviation of the lexical wealth, it is possible to discriminate works of different genres and writers and show that each writer has a very distinct signature, either considered among other literary writers or compared with writers of non-literary texts. It is also shown that, for a given author, the signature is able to discriminate between short stories and novels.
Fractal Analysis of Drainage Basins on Mars
NASA Technical Reports Server (NTRS)
Stepinski, T. F.; Marinova, M. M.; McGovern, P. J.; Clifford, S. M.
2002-01-01
We used statistical properties of drainage networks on Mars as a measure of martian landscape morphology and an indicator of landscape evolution processes. We utilize the Mars Orbiter Laser Altimeter (MOLA) data to construct digital elevation maps (DEMs) of several, mostly ancient, martian terrains. Drainage basins and channel networks are computationally extracted from DEMs and their structures are analyzed and compared to drainage networks extracted from terrestrial and lunar DEMs. We show that martian networks are self-affine statistical fractals with planar properties similar to terrestrial networks, but vertical properties similar to lunar networks. The uniformity of martian drainage density is between those for terrestrial and lunar landscapes. Our results are consistent with the roughening of ancient martian terrains by combination of rainfall-fed erosion and impacts, although roughening by other fluvial processes cannot be excluded. The notion of sustained rainfall in recent Mars history is inconsistent with our findings.
A ``fractal'' modification of Torricelli's formula
NASA Astrophysics Data System (ADS)
Maramathas, Athanasios J.; Boudouvis, Andreas G.
2010-03-01
A modification is proposed of Torricelli’s (1608-1647) formula for the velocity of water discharging from a small hole at the bottom of a large tank filled with fractal solid material. The new formula takes proper account of the mechanical energy losses due to flow in the solid matrix, thus expanding the area of validity of the classical Torricelli’s formula. Moreover, it offers a convenient alternative to Darcy’s law for estimating the discharge rate from an aquifer. The new formula was derived from laboratory experiments, with a low-Reynolds number discharge flow (Darcian flow). It was tested in a natural karst aquifer where the flow is non-Darcian, at Almiros spring on the island of Crete (Greece). In both cases, the predictive capability of the modified formula is established.
Capillary Condensation in a Fractal Porous Medium
Broseta, Daniel; Barre, Loic; Vizika, Olga; Shahidzadeh, Noushine; Guilbaud, Jean-Pierre; Lyonnard, Sandrine
2001-06-04
Small-angle x-ray and neutron scattering are used to characterize the surface roughness and porosity of a natural rock which are described over three decades in length scales and over nine decades in scattered intensities by a surface fractal dimension D=2.68{+-}0.03 . When this porous medium is exposed to a vapor of a contrast-matched water, neutron scattering reveals that surface roughness disappears at small scales, where a Porod behavior typical of smooth interfaces is observed instead. Water-sorption measurements confirm that such interface smoothing is due predominantly to the water condensing in the most strongly curved asperities rather than covering the surface with a wetting film of uniform thickness.
Stochastic and fractal analysis of fracture trajectories
NASA Technical Reports Server (NTRS)
Bessendorf, Michael H.
1987-01-01
Analyses of fracture trajectories are used to investigate structures that fall between 'micro' and 'macro' scales. It was shown that fracture trajectories belong to the class of nonstationary processes. It was also found that correlation distance, which may be related to a characteristic size of a fracture process, increases with crack length. An assemblage of crack trajectory processes may be considered as a diffusive process. Chudnovsky (1981-1985) introduced a 'crack diffusion coefficient' d which reflects the ability of the material to deviate the crack trajectory from the most energetically efficient path and thus links the material toughness to its structure. For the set of fracture trajectories in AISI 304 steel, d was found to be equal to 1.04 microns. The fractal dimension D for the same set of trajectories was found to be 1.133.
From Fractal Trees to Deltaic Networks
NASA Astrophysics Data System (ADS)
Cazanacli, D.; Wolinsky, M. A.; Sylvester, Z.; Cantelli, A.; Paola, C.
2013-12-01
Geometric networks that capture many aspects of natural deltas can be constructed from simple concepts from graph theory and normal probability distributions. Fractal trees with symmetrical geometries are the result of replicating two simple geometric elements, line segments whose lengths decrease and bifurcation angles that are commonly held constant. Branches could also have a thickness, which in the case of natural distributary systems is the equivalent of channel width. In river- or wave-dominated natural deltas, the channel width is a function of discharge. When normal variations around the mean values for length, bifurcating angles, and discharge are applied, along with either pruning of 'clashing' branches or merging (equivalent to channel confluence), fractal trees start resembling natural deltaic networks, except that the resulting channels are unnaturally straight. Introducing a bifurcation probability fewer, naturally curved channels are obtained. If there is no bifurcation, the direction of each new segment depends on the direction the previous segment upstream (correlated random walk) and, to a lesser extent, on a general direction of growth (directional bias). When bifurcation occurs, the resulting two directions also depend on the bifurcation angle and the discharge split proportions, with the dominant branch following the direction of the upstream parent channel closely. The bifurcation probability controls the channel density and, in conjunction with the variability of the directional angles, the overall curvature of the channels. The growth of the network in effect is associated with net delta progradation. The overall shape and shape evolution of the delta depend mainly on the bifurcation angle average size and angle variability coupled with the degree of dominant direction dependency (bias). The proposed algorithm demonstrates how, based on only a few simple rules, a wide variety of channel networks resembling natural deltas, can be replicated
Fractals and Forecasting in Earthquakes and Finance
NASA Astrophysics Data System (ADS)
Rundle, J. B.; Holliday, J. R.; Turcotte, D. L.
2011-12-01
It is now recognized that Benoit Mandelbrot's fractals play a critical role in describing a vast range of physical and social phenomena. Here we focus on two systems, earthquakes and finance. Since 1942, earthquakes have been characterized by the Gutenberg-Richter magnitude-frequency relation, which in more recent times is often written as a moment-frequency power law. A similar relation can be shown to hold for financial markets. Moreover, a recent New York Times article, titled "A Richter Scale for the Markets" [1] summarized the emerging viewpoint that stock market crashes can be described with similar ideas as large and great earthquakes. The idea that stock market crashes can be related in any way to earthquake phenomena has its roots in Mandelbrot's 1963 work on speculative prices in commodities markets such as cotton [2]. He pointed out that Gaussian statistics did not account for the excessive number of booms and busts that characterize such markets. Here we show that both earthquakes and financial crashes can both be described by a common Landau-Ginzburg-type free energy model, involving the presence of a classical limit of stability, or spinodal. These metastable systems are characterized by fractal statistics near the spinodal. For earthquakes, the independent ("order") parameter is the slip deficit along a fault, whereas for the financial markets, it is financial leverage in place. For financial markets, asset values play the role of a free energy. In both systems, a common set of techniques can be used to compute the probabilities of future earthquakes or crashes. In the case of financial models, the probabilities are closely related to implied volatility, an important component of Black-Scholes models for stock valuations. [2] B. Mandelbrot, The variation of certain speculative prices, J. Business, 36, 294 (1963)
Wetting characteristics of 3-dimensional nanostructured fractal surfaces
NASA Astrophysics Data System (ADS)
Davis, Ethan; Liu, Ying; Jiang, Lijia; Lu, Yongfeng; Ndao, Sidy
2017-01-01
This article reports the fabrication and wetting characteristics of 3-dimensional nanostructured fractal surfaces (3DNFS). Three distinct 3DNFS surfaces, namely cubic, Romanesco broccoli, and sphereflake were fabricated using two-photon direct laser writing. Contact angle measurements were performed on the multiscale fractal surfaces to characterize their wetting properties. Average contact angles ranged from 66.8° for the smooth control surface to 0° for one of the fractal surfaces. The change in wetting behavior was attributed to modification of the interfacial surface properties due to the inclusion of 3-dimensional hierarchical fractal nanostructures. However, this behavior does not exactly obey existing surface wetting models in the literature. Potential applications for these types of surfaces in physical and biological sciences are also discussed.
Fractal Particles: Titan's Thermal Structure and IR Opacity
NASA Technical Reports Server (NTRS)
McKay, C. P.; Rannou, P.; Guez, L.; Young, E. F.; DeVincenzi, Donald (Technical Monitor)
1998-01-01
Titan's haze particles are the principle opacity at solar wavelengths. Most past work in modeling these particles has assumed spherical particles. However, observational evidence strongly favors fractal shapes for the haze particles. We consider the implications of fractal particles for the thermal structure and near infrared opacity of Titan's atmosphere. We find that assuming fractal particles with the optical properties based on laboratory tholin material and with a production rate that allows for a match to the geometric albedo results in warmer troposphere and surface temperatures compared to spherical particles. In the near infrared (1-3 microns) the predicted opacity of the fractal particles is up to a factor of two less than for spherical particles. This has implications for the ability of Cassini to image Titan's surface at 1 micron.
Controlled growth of polyaniline fractals on HOPG through potentiodynamic electropolymerization.
Bhattacharjya, Dhrubajyoti; Mukhopadhyay, Indrajit
2012-04-10
Polyaniline (PANI) in fractal dimension has been electrodeposited reproducibly on highly oriented pyrolytic graphite (HOPG) from 0.2 M aniline in 1 M aqueous HCl solution by potentiodynamic sweeping in the range of -0.2 to 0.76 V vs Ag/AgCl at room temperature. Fractal growth of PANI dendrimers is affected by diffusion limited polymerization (DLP) at a sweep rate of 15 mV s(-1) for 43 min. This type of PANI dendrimer is prepared for the first time on such large area HOPG substrate by electrochemical technique using rather simple cell setup. The fractal dimension has been determined by chronoamperometry (CA) and box counting technique and is found to vary from 1.4 to 1.9 with the duration of electropolymerization. The sweep rate, terminal oxidation potential, and the diverse surface anisotropy of the HOPG surface are found to be crucial factors in controlling the growth of such PANI fractals.
Evaluation of Two Fractal Methods for Magnetogram Image Analysis
NASA Technical Reports Server (NTRS)
Stark, B.; Adams, M.; Hathaway, D. H.; Hagyard, M. J.
1997-01-01
Fractal and multifractal techniques have been applied to various types of solar data to study the fractal properties of sunspots as well as the distribution of photospheric magnetic fields and the role of random motions on the solar surface in this distribution. Other research includes the investigation of changes in the fractal dimension as an indicator for solar flares. Here we evaluate the efficacy of two methods for determining the fractal dimension of an image data set: the Differential Box Counting scheme and a new method, the Jaenisch scheme. To determine the sensitivity of the techniques to changes in image complexity, various types of constructed images are analyzed. In addition, we apply this method to solar magnetogram data from Marshall Space Flight Centers vector magnetograph.
Fractal analysis of circulating platelets in type 2 diabetic patients.
Bianciardi, G; Tanganelli, I
2015-01-01
This paper investigates the use of computerized fractal analysis for objective characterization by means of transmission electron microscopy of the complexity of circulating platelets collected from healthy individuals and from type 2 diabetic patients, a pathologic condition in which platelet hyperreactivity has been described. Platelet boundaries were extracted by means of automatically image analysis. Local fractal dimension by box counting (measure of geometric complexity) was automatically calculated. The results showed that the platelet boundary observed by electron microscopy is fractal and that the shape of the circulating platelets is significantly more complex in the diabetic patients in comparison to healthy subjects (p < 0.01), with 100% correct classification. In vitro activated platelets from healthy subjects show an analogous increase of geometric complexity. Computerized fractal analysis of platelet shape by transmission electron microscopy can provide accurate, quantitative, data to study platelet activation in diabetes mellitus.
Hands-On Fractals and the Unexpected in Mathematics
ERIC Educational Resources Information Center
Gluchoff, Alan
2006-01-01
This article describes a hands-on project in which unusual fractal images are produced using only a photocopy machine and office supplies. The resulting images are an example of the contraction mapping principle.
Applications of fractal geometry to dynamical evolution of sunspots
Milovanov, A.V.; Zelenyi, L.M. )
1993-07-01
A fractal model for sunspot dynamics is presented. Formation of a sunspot in the solar photosphere is considered from the viewpoint of aggregation of magnetic flux tubes on a fractal geometry. Fine structure of the magnetic flux tubes is analyzed for a broad class of non-Maxwellian plasma distribution functions. The sunspot fractal dimension is proved to depend on the parameters of the plasma distribution function, enabling one to investigate intrinsic properties of the solar plasma by means of powerful geometrical methods. Magnetic field dissipation in the tubes is shown to result in effective sunspot decay. Sunspot formation and decay times as well as the diffusion constant [ital K] deduced by using the fractal model, are in a good agreement with observational data. Disappearance of umbras in decaying sunspots is interpreted as a second-order phase transition reminiscent of the transition through the Curie point in ferromagnetics.
Is fractal geometry useful in medicine and biomedical sciences?
Heymans, O; Fissette, J; Vico, P; Blacher, S; Masset, D; Brouers, F
2000-03-01
Fractal geometry has become very useful in the understanding of many phenomena in various fields such as astrophysics, economy or agriculture and recently in medicine. After a brief intuitive introduction to the basis of fractal geometry, the clue is made about the correlation between Df and the complexity or the irregularity of a structure. However, fractal analysis must be applied with certain caution in natural objects such as bio-medical ones. The cardio-vascular system remains one of the most important fields of application of these kinds of approach. Spectral analysis of the R-R interval, morphology of the distal coronary arteries constitute two examples. Other very interesting applications are founded in bacteriology, medical imaging or ophthalmology. In our institution, we apply fractal analysis in order to quantitate angiogenesis and other vascular processes.
Fractal characterization and wettability of ion treated silicon surfaces
NASA Astrophysics Data System (ADS)
Yadav, R. P.; Kumar, Tanuj; Baranwal, V.; Vandana, Kumar, Manvendra; Priya, P. K.; Pandey, S. N.; Mittal, A. K.
2017-02-01
Fractal characterization of surface morphology can be useful as a tool for tailoring the wetting properties of solid surfaces. In this work, rippled surfaces of Si (100) are grown using 200 keV Ar+ ion beam irradiation at different ion doses. Relationship between fractal and wetting properties of these surfaces are explored. The height-height correlation function extracted from atomic force microscopic images, demonstrates an increase in roughness exponent with an increase in ion doses. A steep variation in contact angle values is found for low fractal dimensions. Roughness exponent and fractal dimensions are found correlated with the static water contact angle measurement. It is observed that after a crossover of the roughness exponent, the surface morphology has a rippled structure. Larger values of interface width indicate the larger ripples on the surface. The contact angle of water drops on such surfaces is observed to be lowest. Autocorrelation function is used for the measurement of ripple wavelength.
Fern leaves and cauliflower curds are not fractals.
Lev-Yadun, Simcha
2012-05-01
The popular demonstration of drawing a mature fern leaf as expressed by Barnsley's fractal method is mathematically and visually very attractive but anatomically and developmentally misleading, and thus has limited, if any, biological significance. The same is true for the fractal demonstration of the external features of cauliflower curds. Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Moreover, fern leaves and cauliflower curds develop from the inside out through a process totally different from fractal drawing procedures. The above cases demonstrate a general problem of using mathematical tools to investigate or illustrate biological phenomena in an irrelevant manner. A realistic set of mathematical equations to describe fern leaf or cauliflower curd development is needed.
Fractal dynamics of body motion in patients with Parkinson's disease.
Sekine, Masaki; Akay, Metin; Tamura, Toshiyo; Higashi, Yuji; Fujimoto, Toshiro
2004-03-01
In this paper, we assess the complexity (fractal measure) of body motion during walking in patients with Parkinson's disease. The body motion of 11 patients with Parkinson's disease and 10 healthy elderly subjects was recorded using a triaxial accelerometry technique. A triaxial accelerometer was attached to the lumbar region. An assessment of the complexity of body motion was made using a maximum-likelihood-estimator-based fractal analysis method. Our data suggest that the fractal measures of the body motion of patients with Parkinson's disease are higher than those of healthy elderly subjects. These results were statistically different in the X (anteroposterior), Y (lateral) and Z (vertical) directions of body motion between patients with Parkinson's disease and the healthy elderly subjects (p < 0.01 in X and Z directions and p < 0.05 in Y direction). The complexity (fractal measure) of body motion can be useful to assess and monitor the output from the motor system during walking in clinical practice.
Monitoring the Depth of Anaesthesia Using Fractal Complexity Method
NASA Astrophysics Data System (ADS)
Klonowski, W.; Olejarczyk, E.; Stepien, R.; Jalowiecki, P.; Rudner, R.
We propose a simple and effective method of characterizing complexity of EEG-signals for monitoring the depth of anaesthesia using Higuchi's fractal dimension method. We demonstrate that the proposed method may compete with the widely used BIS monitoring method.
FRACTAL DIMENSION RESULTS FOR CONTINUOUS TIME RANDOM WALKS
Meerschaert, Mark M.; Nane, Erkan; Xiao, Yimin
2013-01-01
Continuous time random walks impose random waiting times between particle jumps. This paper computes the fractal dimensions of their process limits, which represent particle traces in anomalous diffusion. PMID:23482421
Derivation of Archie's' law based on a fractal pore volume
NASA Astrophysics Data System (ADS)
Wang, Hongtao; Liu, Tangyan
2017-03-01
The geometrical mechanism behind Archie's law has been extensively investigated for many years, but is as yet inadequately understood. In this research, we present a straight-forward theoretical derivation revealing that the geometrical mechanism behind Archie's law is well represented in terms of a fractal pore volume. This representation is verified by the results of numerical simulations of the electrical conductivity obtained from deterministic fractal models. The derivation naturally suggests some new physical interpretations for Archie's parameters. It is revealed that the fractal building process determines the values of the cementation exponent and the prefactor. In addition, the prefactor is also determined by the initial states of the porosity and formation factor, which together define the initial state of the fractal building process.
Fractal based curves in musical creativity: A critical annotation
NASA Astrophysics Data System (ADS)
Georgaki, Anastasia; Tsolakis, Christos
In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.
Vortex-ring-fractal Structure of Atom and Molecule
Osmera, Pavel
2010-06-17
This chapter is an attempt to attain a new and profound model of the nature's structure using a vortex-ring-fractal theory (VRFT). Scientists have been trying to explain some phenomena in Nature that have not been explained so far. The aim of this paper is the vortex-ring-fractal modeling of elements in the Mendeleev's periodic table, which is not in contradiction to the known laws of nature. We would like to find some acceptable structure model of the hydrogen as a vortex-fractal-coil structure of the proton and a vortex-fractal-ring structure of the electron. It is known that planetary model of the hydrogen atom is not right, the classical quantum model is too abstract. Our imagination is that the hydrogen is a levitation system of the proton and the electron. Structures of helium, oxygen, and carbon atoms and a hydrogen molecule are presented too.
The Classification of HEp-2 Cell Patterns Using Fractal Descriptor.
Xu, Rudan; Sun, Yuanyuan; Yang, Zhihao; Song, Bo; Hu, Xiaopeng
2015-07-01
Indirect immunofluorescence (IIF) with HEp-2 cells is considered as a powerful, sensitive and comprehensive technique for analyzing antinuclear autoantibodies (ANAs). The automatic classification of the HEp-2 cell images from IIF has played an important role in diagnosis. Fractal dimension can be used on the analysis of image representing and also on the property quantification like texture complexity and spatial occupation. In this study, we apply the fractal theory in the application of HEp-2 cell staining pattern classification, utilizing fractal descriptor firstly in the HEp-2 cell pattern classification with the help of morphological descriptor and pixel difference descriptor. The method is applied to the data set of MIVIA and uses the support vector machine (SVM) classifier. Experimental results show that the fractal descriptor combining with morphological descriptor and pixel difference descriptor makes the precisions of six patterns more stable, all above 50%, achieving 67.17% overall accuracy at best with relatively simple feature vectors.
Facilitated diffusion of proteins through crumpled fractal DNA globules
NASA Astrophysics Data System (ADS)
Smrek, Jan; Grosberg, Alexander Y.
2015-07-01
We explore how the specific fractal globule conformation, found for the chromatin fiber of higher eukaryotes and topologically constrained dense polymers, affects the facilitated diffusion of proteins in this environment. Using scaling arguments and supporting Monte Carlo simulations, we relate DNA looping probability distribution, fractal dimension, and protein nonspecific affinity for the DNA to the effective diffusion parameters of the proteins. We explicitly consider correlations between subsequent readsorption events of the proteins, and we find that facilitated diffusion is faster for the crumpled globule conformation with high intersegmental surface dimension than in the case of dense fractal conformations with smooth surfaces. As a byproduct, we obtain an expression for the macroscopic conductivity of a hypothetic material consisting of conducting fractal nanowires immersed in a weakly conducting medium.
[Dimensional fractal of post-paddy wheat root architecture].
Chen, Xin-xin; Ding, Qi-shuo; Li, Yi-nian; Xue, Jin-lin; Lu, Ming-zhou; Qiu, Wei
2015-06-01
To evaluate whether crop rooting system was directionally dependent, a field digitizer was used to measure post-paddy wheat root architectures. The acquired data was transferred to Pro-E, in which virtual root architecture was reconstructed and projected to a series of planes each separated in 10° apart. Fractal dimension and fractal abundance of root projections in all the 18 planes were calculated, revealing a distinctive architectural distribution of wheat root in each direction. This strongly proved that post-paddy wheat root architecture was directionally dependent. From seedling to turning green stage, fractal dimension of the 18 projections fluctuated significantly, illustrating a dynamical root developing process in the period. At the jointing stage, however, fractal indices of wheat root architecture resumed its regularity in each dimension. This wheat root architecture recovered its dimensional distinctness. The proposed method was applicable for precision modeling field state root distribution in soil.
Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects
NASA Astrophysics Data System (ADS)
Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.
2013-04-01
This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to
Fractal geometry of some Martian lava flow margins: Alba Patera
NASA Technical Reports Server (NTRS)
Kauhanen, K.
1993-01-01
Fractal dimension for a few lava flow margins on the gently sloping flanks of Alba Patera were measured using the structured walk method. Fractal behavior was observed at scales ranging from 20 to 100 pixels. The upper limit of the linear part of log(margin length) vs. log(scale) profile correlated well to the margin length. The lower limit depended on resolution and flow properties.
Fractal characterization of a fractured chalk reservoir - The Laegerdorf case
Stoelum, H.H.; Koestler, A.G.; Feder, J.; Joessang, T.; Aharony, A.
1991-03-01
What is the matrix block size distribution of a fractured reservoir In order to answer this question and assess the potential of fractal geometry as a method of characterization of fracture networks, a pilot study has been done of the fractured chalk quarry in Laegerdorf. The fractures seen on the quarry walls were traced in the field for a total area of {approximately}200 {times} 45 m. The digitized pictures have been analyzed by a standard box-counting method. This analysis gave a fractal dimension of similarity varying from 1.33 for fractured areas between faults, to 1.43 for the fault zone, and 1.53 for the highly deformed fault gouge. The amplitude showed a similar trend. The fractal dimension for the whole system of fractures is {approximately}1.55. In other words, fracture networks in chalk have a nonlinear, fractal geometry, and so matrix block size is a scaling property of chalk reservoirs. In terms of rock mechanics, the authors interpret the variation of the fractal dimension as follows: A small fractal dimension and amplitude are associated with brittle deformation in the elastic regime, while a large fractal dimension and amplitude are associated with predominantly ductile, strain softening deformation in the plastic regime. The interaction between the two regimes of deformation in the rock body is a key element of successful characterization and may be approached by seeing the rock as a non-Newtonian viscoelastic medium. The fractal dimension for the whole is close to a material independent limit that constrains the development of fractures.
Discrimination of walking patterns using wavelet-based fractal analysis.
Sekine, Masaki; Tamura, Toshiyo; Akay, Metin; Fujimoto, Toshiro; Togawa, Tatsuo; Fukui, Yasuhiro
2002-09-01
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.
Tissue as a self-organizing system with fractal dynamics.
Waliszewski, P; Konarski, J
2001-01-01
Cell is a supramolecular dynamic network. Screening of tissue-specific cDNA library and results of Relative RT-PCR indicate that the relationship between genotype, (i.e., dynamic network of genes and their protein regulatory elements) and phenotype is non-bijective, and mendelian inheritance is a special case only. This implies non-linearity, complexity, and quasi-determinism, (i.e., co-existence of deterministic and non-deterministic events) of dynamic cellular network; prerequisite conditions for the existence of fractal structure. Indeed, the box counting method reveals that morphological patterns of the higher order, such as gland-like structures or populations of differentiating cancer cells possess fractal dimension and self-similarity. Since fractal space is not filled out randomly, a variety of morphological patterns of functional states arises. The expansion coefficient characterizes evolution of fractal dynamics. The coefficient indicates what kind of interactions occurs between cells, and how far from the limiting integer dimension of the Euclidean space the expanding population of cells is. We conclude that cellular phenomena occur in the fractal space; aggregation of cells is a supracollective phenomenon (expansion coefficient > 0), and differentiation is a collective one (expansion coefficient < 0). Fractal dimension or self-similarity are lost during tumor progression. The existence of fractal structure in a complex tissue system denotes that dynamic cellular phenomena generate an attractor with the appropriate organization of space-time. And vice versa, this attractor sets up physical limits for cellular phenomena during their interactions with various fields. This relationship can help to understand the emergence of extraterrestial forms of life. Although those forms can be composed of non-carbon molecules, fractal structure appears to be the common feature of all interactive biosystems.
New 5-adic Cantor sets and fractal string.
Kumar, Ashish; Rani, Mamta; Chugh, Renu
2013-01-01
In the year (1879-1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5-adic Cantor one-fifth set as an example of fractal string have been introduced. Moreover, the applications of 5-adic Cantor one-fifth set in string theory have also been studied.
Intrinsic half-metallicity in fractal carbon nitride honeycomb lattices.
Wang, Aizhu; Zhao, Mingwen
2015-09-14
Fractals are natural phenomena that exhibit a repeating pattern "exactly the same at every scale or nearly the same at different scales". Defect-free molecular fractals were assembled successfully in a recent work [Shang et al., Nature Chem., 2015, 7, 389-393]. Here, we adopted the feature of a repeating pattern in searching two-dimensional (2D) materials with intrinsic half-metallicity and high stability that are desirable for spintronics applications. Using first-principles calculations, we demonstrate that the electronic properties of fractal frameworks of carbon nitrides have stable ferromagnetism accompanied by half-metallicity, which are highly dependent on the fractal structure. The ferromagnetism increases gradually with the increase of fractal order. The Curie temperature of these metal-free systems estimated from Monte Carlo simulations is considerably higher than room temperature. The stable ferromagnetism, intrinsic half-metallicity, and fractal characteristics of spin distribution in the carbon nitride frameworks open an avenue for the design of metal-free magnetic materials with exotic properties.
Exploring the relationship between fractal features and bacterial essential genes
NASA Astrophysics Data System (ADS)
Yong-Ming, Yu; Li-Cai, Yang; Qian, Zhou; Lu-Lu, Zhao; Zhi-Ping, Liu
2016-06-01
Essential genes are indispensable for the survival of an organism in optimal conditions. Rapid and accurate identifications of new essential genes are of great theoretical and practical significance. Exploring features with predictive power is fundamental for this. Here, we calculate six fractal features from primary gene and protein sequences and then explore their relationship with gene essentiality by statistical analysis and machine learning-based methods. The models are applied to all the currently available identified genes in 27 bacteria from the database of essential genes (DEG). It is found that the fractal features of essential genes generally differ from those of non-essential genes. The fractal features are used to ascertain the parameters of two machine learning classifiers: Naïve Bayes and Random Forest. The area under the curve (AUC) of both classifiers show that each fractal feature is satisfactorily discriminative between essential genes and non-essential genes individually. And, although significant correlations exist among fractal features, gene essentiality can also be reliably predicted by various combinations of them. Thus, the fractal features analyzed in our study can be used not only to construct a good essentiality classifier alone, but also to be significant contributors for computational tools identifying essential genes. Project supported by the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2014FM022).
Fractal continuum model for tracer transport in a porous medium.
Herrera-Hernández, E C; Coronado, M; Hernández-Coronado, H
2013-12-01
A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.
Modeling of Aperiodic Fractal Waveguide Structures for Multifrequency Light Transport
NASA Astrophysics Data System (ADS)
Hiltunen, Marianne; Dal Negro, Luca; Feng, Ning-Ning; Kimerling, Lionel C.; Michel, Jurgen
2007-07-01
In this paper, we present the design of a novel waveguide structure capable of multifrequency transmission bands with strongly enhanced electric field states. The concept of the structure is based on aperiodic and quasi-periodic fractal ordering of scattering subunits combined within a traditional channel-waveguide scheme. The resulting 3-D fractal waveguides are characterized by complex transmission spectra and sustain quasi-localized field modes with strong enhancement effects due to the lack of translational symmetry. In this paper, we will describe how it is possible to accurately model these complex waveguide structures within a simple 1-D model. We will explore the formation of photonic band gaps and the character of the quasi-localized states in fractal waveguide structures generated according to different deterministic rules, such as Fibonacci, Thue Morse, and Rudin Shapiro sequences. Furthermore, we will qualitatively compare the characteristics of the optical gaps and field states in periodic, fractal, and aperiodic waveguides. The results of our comparative study will show that fractal waveguides based on aperiodic order exhibit the richest transmission spectra with field-enhancement effects occurring at multiple frequencies. The proposed fractal waveguide design can provide an attractive route toward the fabrication of optically active devices for multiwavelength operation.
Experimental control of scaling behavior: what is not fractal?
Likens, Aaron D; Fine, Justin M; Amazeen, Eric L; Amazeen, Polemnia G
2015-10-01
The list of psychological processes thought to exhibit fractal behavior is growing. Although some might argue that the seeming ubiquity of fractal patterns illustrates their significance, unchecked growth of that list jeopardizes their relevance. It is important to identify when a single behavior is and is not fractal in order to make meaningful conclusions about the processes underlying those patterns. The hypothesis tested in the present experiment is that fractal patterns reflect the enactment of control. Participants performed two steering tasks: steering on a straight track and steering on a circular track. Although each task could be accomplished by holding the steering wheel at a constant angle, steering around a curve may require more constant control, at least from a psychological standpoint. Results showed that evidence for fractal behavior was strongest for the circular track; straight tracks showed evidence of two scaling regions. We argue from those results that, going forward, the goal of the fractal literature should be to bring scaling behavior under experimental control.
Fractal analysis of lumbar vertebral cancellous bone architecture.
Feltrin, G P; Macchi, V; Saccavini, C; Tosi, E; Dus, C; Fassina, A; Parenti, A; De Caro, R
2001-11-01
Osteoporosis is characterized by bone mineral density (BMD) decreasing and spongy bone rearrangement with consequent loss of elasticity and increased bone fragility. Quantitative computed tomography (QCT) quantifies bone mineral content but does not describe spongy architecture. Analysis of trabecular pattern may provide additional information to evaluate osteoporosis. The aim of this study was to determine whether the fractal analysis of the microradiography of lumbar vertebrae provides a reliable assessment of bone texture, which correlates with the BMD. The lumbar segment of the spine was removed from 22 cadavers with no history of back pain and examined with standard x-ray, traditional tomography, and quantitative computed tomography to measure BMD. The fractal dimension, which quantifies the image fractal complexity, was calculated on microradiographs of axial sections of the fourth lumbar vertebra to determine its characteristic spongy network. The relationship between the values of the BMD and those of the fractal dimension was evaluated by linear regression and a statistically significant correlation (R = 0.96) was found. These findings suggest that the application of fractal analysis to radiological analyses can provide valuable information on the trabecular pattern of vertebrae. Thus, fractal dimensions of trabecular bone structure should be considered as a supplement to BMD evaluation in the assessment of osteoporosis.
Tan, Wanyu; Li, Yongmei; Tan, Kaixuan; Duan, Xianzhe; Liu, Dong; Liu, Zehua
2016-12-01
Radon diffusion and transport through different media is a complex process affected by many factors. In this study, the fractal theories and field covering experiments were used to study the fractal characteristics of particle size distribution (PSD) of six kinds of geotechnical materials (e.g., waste rock, sand, laterite, kaolin, mixture of sand and laterite, and mixture of waste rock and laterite) and their effects on radon diffusion. In addition, the radon diffusion coefficient and diffusion length were calculated. Moreover, new formulas for estimating diffusion coefficient and diffusion length functional of fractal dimension d of PSD were proposed. These results demonstrate the following points: (1) the fractal dimension d of the PSD can be used to characterize the property of soils and rocks in the studies of radon diffusion behavior; (2) the diffusion coefficient and diffusion length decrease with increasing fractal dimension of PSD; and (3) the effectiveness of final covers in reducing radon exhalation of uranium tailings impoundments can be evaluated on the basis of the fractal dimension of PSD of materials.
Poosapadi Arjunan, Sridhar; Kumar, Dinesh Kant
2014-01-01
This research study investigates the fractal properties of surface Electromyogram (sEMG) to estimate the force levels of contraction of three muscles with different cross-sectional areas (CSA): m. quadriceps--vastus lateralis, m. biceps brachii, andm. flexor digitorum superficialis. The fractal features were computed based on the fractal analysis of sEMG, signal recorded while performing sustained muscle contraction at different force levels. A comparison was performed between the fractal features and five other features reported in the literature. Linear regression analysis was carried out to determine the relationship between the force of contraction (20-100%) and features of sEMG. The results from the coefficients of regression r² show that the new fractal feature, maximum fractal length of the signal has highest correlation (range 0.88-0.90) when compared with other features which ranges from 0.34 to 0.74 for the three different muscles. This study suggests that the estimation of various levels of sustained contraction of muscles with varied CSA will provide a better insight into the biomechanics model that involves muscle properties and muscle activation.
Fractal Hearts are Healthy Hearts---Are Fractal Companies Healthy Companies?
NASA Astrophysics Data System (ADS)
Argyle, Bronson; Hart, Gus
2008-03-01
Fractal analyses of cardiac rhythms have implied that healthy individuals have complex cardiac behavior whereas aged or unhealthy individuals show either more random or more periodic behavior. Does this marker of ``complexity = health'' show up elsewhere? Can this technique be used in other fields as well? Specifically, does a Detrended Fluctuation Analysis of S&P 500 bid prices reveal long range correlations in volatility, similar to those observed in cardiac interbeat intervals? If such correlations exist, measures of market complexity could be compared with individual securities to assess corporate strength and vitality.
Menger sponge-like fractal body created by a novel template method.
Mayama, H; Tsujii, K
2006-09-28
We have established experimental strategies on how to create a Menger sponge-like fractal body and how to control its fractal dimension. The essence was to utilize alkylketene dimer (AKD), which spontaneously forms super-water-repellent fractal surface. We prepared "fractal AKD particles" with fractal surface structure as templates of pores in fractal body. The fractal body was synthesized by filling the remained space between the packed template particles with a tetramethyl orthosilicate solution, solidifying it by the sol-gel process, and removing the template by calcinations. We have succeeded in systematically creating fractal bodies of silica with different cross-sectional fractal dimensions D(cs)=1.87, 1.84, and 1.80 using "fractal template particles" compressed under the ratio=1.0, 2.0, and 3.0, respectively. We also discussed the possibilities of their fractal geometries in comparison with mathematical models. We concluded that the created fractal bodies were close to a Menger sponge and its modified one. Our experimental strategy allows us to design fractality of porous materials.
Scaling relations in the diffusive infiltration in fractals
NASA Astrophysics Data System (ADS)
Aarão Reis, F. D. A.
2016-11-01
In a recent work on fluid infiltration in a Hele-Shaw cell with the pore-block geometry of Sierpinski carpets (SCs), the area filled by the invading fluid was shown to scale as F ˜tn , with n <1 /2 , thus providing a macroscopic realization of anomalous diffusion [Filipovitch et al., Water Resour. Res. 52, 5167 (2016), 10.1002/2016WR018667]. The results agree with simulations of a diffusion equation with constant pressure at one of the borders of those fractals, but the exponent n is very different from the anomalous exponent ν =1 /DW of single-particle diffusion in the same fractals (DW is the random-walk dimension). Here we use a scaling approach to show that those exponents are related as n =ν (DF-DB) , where DF and DB are the fractal dimensions of the bulk and the border from which diffusing particles come, respectively. This relation is supported by accurate numerical estimates in two SCs and in two generalized Menger sponges (MSs), in which we performed simulations of single-particle random walks (RWs) with a rigid impermeable border and of a diffusive infiltration model in which that border is permanently filled with diffusing particles. This study includes one MS whose external border is also fractal. The exponent relation is also consistent with the recent simulational and experimental results on fluid infiltration in SCs, and explains the approximate quadratic dependence of n on DF in these fractals. We also show that the mean-square displacement of single-particle RWs has log-periodic oscillations, whose periods are similar for fractals with the same scaling factor in the generator (even with different embedding dimensions), which is consistent with the discrete scale invariance scenario. The roughness of a diffusion front defined in the infiltration problem also shows this type of oscillation, which is enhanced in fractals with narrow channels between large lacunas.
Persistent Fluctuations in Stride Intervals under Fractal Auditory Stimulation
Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J.; Daffertshofer, Andreas
2014-01-01
Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals. PMID:24651455
Persistent fluctuations in stride intervals under fractal auditory stimulation.
Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J; Daffertshofer, Andreas
2014-01-01
Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals.
Multiple rate-determining steps for nonideal and fractal kinetics.
Vlad, Marcel O; Popa, Vlad T; Segal, E; Ross, John
2005-02-17
We show that the kinetic model of a single rate-determining step in a reaction mechanism can be extended to systems with multiple overall reactions for which the elementary reactions obey nonideal or fractal kinetics. The following assumptions are necessary: (1) The system studied is either closed or open, but no constraints exist preventing the evolution toward equilibrium. (2) Elementary reactions occur in pairs of forward and backward steps. (3) The kinetics of the elementary steps are either nonideal or fractal and are compatible with equilibrium thermodynamics. (4) The number of reaction routes is identical with the number of rate-determining steps. If these hypotheses are valid, then the overall reaction rates can be explicitly evaluated: they have a form similar to the kinetic equations for the elementary reactions and the apparent reaction orders and fractal coefficients can be expressed analytically in terms of the kinetic parameters of the elementary reactions. We derive a set of relationships which connect the equilibrium constants of the reaction routes, the corresponding overall rate coefficients, and the stoichiometric numbers of the rate-determining steps. We also derive a set of generalized Boreskov relations among the apparent activation energies of the forward and backward overall processes, the corresponding reaction enthalpies, and the stoichiometric coefficients of the rate-determining steps. If the elementary reactions obey fractal kinetics, the same is true for the rate-determining steps. The fractal exponents of the forward and backward overall reactions are linear combinations of the fractal exponents of the fractal elementary reactions. Similar to the theory of single rate-determining steps, our approach can be used for selecting suitable reaction mechanisms from experimental data.
Bioinspired fractal electrodes for solar energy storages.
Thekkekara, Litty V; Gu, Min
2017-03-31
Solar energy storage is an emerging technology which can promote the solar energy as the primary source of electricity. Recent development of laser scribed graphene electrodes exhibiting a high electrical conductivity have enabled a green technology platform for supercapacitor-based energy storage, resulting in cost-effective, environment-friendly features, and consequent readiness for on-chip integration. Due to the limitation of the ion-accessible active porous surface area, the energy densities of these supercapacitors are restricted below ~3 × 10(-3) Whcm(-3). In this paper, we demonstrate a new design of biomimetic laser scribed graphene electrodes for solar energy storage, which embraces the structure of Fern leaves characterized by the geometric family of space filling curves of fractals. This new conceptual design removes the limit of the conventional planar supercapacitors by significantly increasing the ratio of active surface area to volume of the new electrodes and reducing the electrolyte ionic path. The attained energy density is thus significantly increased to ~10(-1) Whcm(-3)- more than 30 times higher than that achievable by the planar electrodes with ~95% coulombic efficiency of the solar energy storage. The energy storages with these novel electrodes open the prospects of efficient self-powered and solar-powered wearable, flexible and portable applications.
Complexity, fractals, disease time, and cancer.
Spillman, W B; Robertson, J L; Huckle, W R; Govindan, B S; Meissner, K E
2004-12-01
Despite many years of research, a method to precisely and quantitatively determine cancer disease state remains elusive. Current practice for characterizing solid tumors involves the use of varying systems of tumor grading and staging and thus leaves diagnosis and clinical staging dependent on the experience and skill of the physicians involved. Although numerous disease markers have been identified, no combination of them has yet been found that produces a quantifiable and reliable measure of disease state. Newly developed genomic markers and other measures based on the developing sciences of complexity offer promise that this situation may soon be changed for the better. In this paper, we examine the potential of two measures of complexity, fractal dimension and percolation, for use as components of a yet to be determined "disease time" vector that more accurately quantifies disease state. The measures are applied to a set of micrographs of progressive rat hepatoma and analyzed in terms of their correlation with cell differentiation, ratio of tumor weight to rat body weight and tumor growth time. The results provide some support for the idea that measures of complexity could be important elements of any future cancer "disease time" vector.
Fractal and Euclidean descriptors of platelet shape.
Kraus, Max-Joseph; Neeb, Heiko; Strasser, Erwin F
2014-01-01
Platelet shape change is a dynamic membrane surface process that exhibits remarkable morphological heterogeneity. Once the outline of an irregular shape is identified and segmented from a digital image, several mathematical descriptors can be applied to numerical characterize the irregularity of the shapes surface. 13072 platelet outlines (PLO) were segmented automatically from 1928 microscopic images using a newly developed algorithm for the software product Matlab R2012b. The fractal dimension (FD), circularity, eccentricity, area and perimeter of each PLO were determined. 972 PLO were randomly assigned for computer-assisted manual measurement of platelet diameter as well as number, width and length of filopodia per platelet. FD can be used as a surrogate parameter for determining the roughness of the PLO and circularity can be used as a surrogate to estimate the number and length of filopodia. The relationship between FD and perimeter of the PLO reveals the existence of distinct groups of platelets with significant structural differences which may be caused by platelet activation. This new method allows for the standardized continuous numerical classification of platelet shape and its dynamic change, which is useful for the analysis of altered platelet activity (e.g. inflammatory diseases, contact activation, drug testing).
Analysis of Texture Using the Fractal Model
NASA Technical Reports Server (NTRS)
Navas, William; Espinosa, Ramon Vasquez
1997-01-01
Properties such as the fractal dimension (FD) can be used for feature extraction and classification of regions within an image. The FD measures the degree of roughness of a surface, so this number is used to characterize a particular region, in order to differentiate it from another. There are two basic approaches discussed in the literature to measure FD: the blanket method, and the box counting method. Both attempt to measure FD by estimating the change in surface area with respect to the change in resolution. We tested both methods but box counting resulted computationally faster and gave better results. Differential Box Counting (DBC) was used to segment a collage containing three textures. The FD is independent of directionality and brightness so five features were used derived from the original image to account for directionality and gray level biases. FD can not be measured on a point, so we use a window that slides across the image giving values of FD to the pixel on the center of the window. Windowing blurs the boundaries of adjacent classes, so an edge-preserving, feature-smoothing algorithm is used to improve classification within segments and to make the boundaries sharper. Segmentation using DBC was 90.8910 accurate.
Bioinspired fractal electrodes for solar energy storages
Thekkekara, Litty V.; Gu, Min
2017-01-01
Solar energy storage is an emerging technology which can promote the solar energy as the primary source of electricity. Recent development of laser scribed graphene electrodes exhibiting a high electrical conductivity have enabled a green technology platform for supercapacitor-based energy storage, resulting in cost-effective, environment-friendly features, and consequent readiness for on-chip integration. Due to the limitation of the ion-accessible active porous surface area, the energy densities of these supercapacitors are restricted below ~3 × 10−3 Whcm−3. In this paper, we demonstrate a new design of biomimetic laser scribed graphene electrodes for solar energy storage, which embraces the structure of Fern leaves characterized by the geometric family of space filling curves of fractals. This new conceptual design removes the limit of the conventional planar supercapacitors by significantly increasing the ratio of active surface area to volume of the new electrodes and reducing the electrolyte ionic path. The attained energy density is thus significantly increased to ~10−1 Whcm−3- more than 30 times higher than that achievable by the planar electrodes with ~95% coulombic efficiency of the solar energy storage. The energy storages with these novel electrodes open the prospects of efficient self-powered and solar-powered wearable, flexible and portable applications. PMID:28361924
Suboptimal fractal coding scheme using iterative transformation
NASA Astrophysics Data System (ADS)
Kang, Hyun-Soo; Chung, Jae-won
2001-05-01
This paper presents a new fractal coding scheme to find a suboptimal transformation by performing an iterative encoding process. The optimal transformation can be defined as the transformation generating the closest attractor to an original image. Unfortunately, it is impossible in practice to find the optimal transformation, due to the heavy computational burden. In this paper, however, by means of some new theorems related with contractive transformations and attractors. It is shown that for some specific cases the optimal or suboptimal transformations can be obtained. The proposed method obtains a suboptimal transformation by performing iterative processes as is done in decoding. Thus, it requires more computation than the conventional method, but it improves the image quality. For a simple case where the optimal transformation can actually be found, the proposed method is experimentally evaluated against both the optimal method and the conventional method. For a general case where the optimal transformation in unavailable due to heavy computational complexity, the proposed method is also evaluated in comparison with the conventional method.
NASA Technical Reports Server (NTRS)
Garneau, S.; Plaut, J. J.
2000-01-01
The surface roughness of the Vastitas Borealis Formation on Mars was analyzed with fractal statistics. Root mean square slopes and fractal dimensions were calculated for 74 topographic profiles. Results have implications for radar scattering models.
Hierarchical socioeconomic fractality: The rich, the poor, and the middle-class
NASA Astrophysics Data System (ADS)
Eliazar, Iddo; Cohen, Morrel H.
2014-05-01
Since the seminal work of the Italian economist Vilfredo Pareto, the study of wealth and income has been a topic of active scientific exploration engaging researches ranging from economics and political science to econophysics and complex systems. This paper investigates the intrinsic fractality of wealth and income. To that end we introduce and characterize three forms of socioeconomic scale-invariance-poor fractality, rich fractality, and middle-class fractality-and construct hierarchical fractal approximations of general wealth and income distributions, based on the stitching of these three forms of fractality. Intertwining the theoretical results with real-world empirical data we then establish that the three forms of socioeconomic fractality-amalgamated into a composite hierarchical structure-underlie the distributions of wealth and income in human societies. We further establish that the hierarchical socioeconomic fractality of wealth and income is also displayed by empirical rank distributions observed across the sciences.
Broadband light-scattering spectroscopy on fractal and non-fractal relaxors
NASA Astrophysics Data System (ADS)
Koreeda, Akitoshi; Ogawa, Tomohiro; Katayama, Daisuke; Fujii, Yasuhiro; Tachibana, Makoto
2016-10-01
We show the quasi-elastic light scattering (QELS) spectra of two groups of relaxors: the first group includes relaxors that exhibit glasslike low-temperature thermal conductivity and heat capacity, namely, Pb(Mg1/3Nb2/3)O3 (PMN), (1 - x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT), Pb(Zr1/3Nb2/3)O3 (PZN), and (Na1/2Bi1/2)TiO3 (NBT). The other group consists of relaxors exhibiting a normal (crystal) temperature dependence of the thermal conductivity and heat capacity, namely, K1- x Li x TaO3 (KLT) and KTa1- x Nb x O3 (KTN). The crystals of the first group yielded self-similar (power-law) QELS spectra, indicating the existence of fractal networks/clusters of polar nanoregions, while those of the second group did not show any self-similarity in the QELS spectra. These results imply that the glasslike low-temperature thermal conductivity and heat capacity in relaxors can be attributed to the vibrational modes specific to fractal networks/clusters formed by polar nanoregions.
Scalings and decay of fractal-generated turbulence
NASA Astrophysics Data System (ADS)
Hurst, D.; Vassilicos, J. C.
2007-03-01
A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence. The resulting turbulent flows have been studied using hot wire anemometry. Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax/tmin. With relatively small blockage ratios, as low as σ =25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage ratio classical grids in similar wind tunnels and wind speeds U. The scalings and decay of the turbulence intensity u'/U in the x direction along the tunnel's center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v '/U and w'/U): (i) for fractal cross grids (Df=2), (u'/U)2=tr2CΔPfct (x/Meff); (ii) for fractal I grids, (u'/U)2=tr(T/Lmax)2CΔPfct(x/Meff), where T is the tunnel width and Lmax is the maximum bar length on the grid; (iii) for space-filling (Df=2) fractal square grids, the turbulence intensity builds up as the turbulence is convected downstream until a distance xpeak from the grid is reached where the turbulence intensity peaks and then decays exponentially, u'2=upeak'2exp [-(x-xpeak)/lturb], where upeak'2 increases linearly with tr, xpeak∝tminT/Lmin (Lmin being the minimum bar length on the grid), and lturb∝λ2U /ν (ν being the kinematic viscosity of the air and λ being the Taylor microscale); λ remains approximately constant during decay at x ≫xpeak. The longitudinal and lateral integral length scales also remain approximately constant during decay at x ≫xpeak.
Fractal symmetry of protein interior: what have we learned?
Banerji, Anirban; Ghosh, Indira
2011-08-01
The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic
Controlling Molecular Growth between Fractals and Crystals on Surfaces.
Zhang, Xue; Li, Na; Gu, Gao-Chen; Wang, Hao; Nieckarz, Damian; Szabelski, Paweł; He, Yang; Wang, Yu; Xie, Chao; Shen, Zi-Yong; Lü, Jing-Tao; Tang, Hao; Peng, Lian-Mao; Hou, Shi-Min; Wu, Kai; Wang, Yong-Feng
2015-12-22
Recent studies demonstrate that simple functional molecules, which usually form two-dimensional (2D) crystal structures when adsorbed on solid substrates, are also able to self-assemble into ordered openwork fractal aggregates. To direct and control the growth of such fractal supramolecules, it is necessary to explore the conditions under which both fractal and crystalline patterns develop and coexist. In this contribution, we study the coexistence of Sierpiński triangle (ST) fractals and 2D molecular crystals that were formed by 4,4″-dihydroxy-1,1':3',1″-terphenyl molecules on Au(111) in ultrahigh vacuum. Growth competition between the STs and 2D crystals was realized by tuning substrate and molecular surface coverage and changing the functional groups of the molecular building block. Density functional theory calculations and Monte Carlo simulations are used to characterize the process. Both experimental and theoretical results demonstrate the possibility of steering the surface self-assembly to generate fractal and nonfractal structures made up of the same molecular building block.
Controlling the efficiency of trapping in treelike fractals.
Wu, Bin; Zhang, Zhongzhi
2013-07-14
Efficiently controlling the trapping process, especially the trapping efficiency, is central in the study of trap problem in complex systems, since it is a fundamental mechanism for diverse other dynamic processes. Thus, it is of theoretical and practical significance to study the control technique for trapping problem. In this paper, we study the trapping problem in a family of proposed directed fractals with a deep trap at a central node. The directed fractals are a generalization of previous undirected fractals by introducing the directed edge weights dominated by a parameter. We characterize all the eigenvalues and their degeneracies for an associated matrix governing the trapping process. The eigenvalues are provided through an exact recursive relation deduced from the self-similar structure of the fractals. We also obtain the expressions for the smallest eigenvalue and the mean first-passage time (MFPT) as a measure of trapping efficiency, which is the expected time for the walker to first visit the trap. The MFPT is evaluated according to the proved fact that it is approximately equal to reciprocal of the smallest eigenvalue. We show that the MFPT is controlled by the weight parameter by modifying which the MFPT can scale superlinealy, linearly, or sublinearly with the system size. Thus, this work paves a way to delicately controlling the trapping process in the fractals.
Fractal image perception provides novel insights into hierarchical cognition.
Martins, M J; Fischmeister, F P; Puig-Waldmüller, E; Oh, J; Geissler, A; Robinson, S; Fitch, W T; Beisteiner, R
2014-08-01
Hierarchical structures play a central role in many aspects of human cognition, prominently including both language and music. In this study we addressed hierarchy in the visual domain, using a novel paradigm based on fractal images. Fractals are self-similar patterns generated by repeating the same simple rule at multiple hierarchical levels. Our hypothesis was that the brain uses different resources for processing hierarchies depending on whether it applies a "fractal" or a "non-fractal" cognitive strategy. We analyzed the neural circuits activated by these complex hierarchical patterns in an event-related fMRI study of 40 healthy subjects. Brain activation was compared across three different tasks: a similarity task, and two hierarchical tasks in which subjects were asked to recognize the repetition of a rule operating transformations either within an existing hierarchical level, or generating new hierarchical levels. Similar hierarchical images were generated by both rules and target images were identical. We found that when processing visual hierarchies, engagement in both hierarchical tasks activated the visual dorsal stream (occipito-parietal cortex, intraparietal sulcus and dorsolateral prefrontal cortex). In addition, the level-generating task specifically activated circuits related to the integration of spatial and categorical information, and with the integration of items in contexts (posterior cingulate cortex, retrosplenial cortex, and medial, ventral and anterior regions of temporal cortex). These findings provide interesting new clues about the cognitive mechanisms involved in the generation of new hierarchical levels as required for fractals.
Gene essentiality prediction based on fractal features and machine learning.
Yu, Yongming; Yang, Licai; Liu, Zhiping; Zhu, Chuansheng
2017-02-28
Essential genes are required for the viability of an organism. Accurate and rapid identification of new essential genes is of substantial theoretical interest to synthetic biology and has practical applications in biomedicine. Fractals provide facilitated access to genetic structure analysis on a different scale. In this study, machine learning-based methods using solely fractal features are presented and the problem of predicting essential genes in bacterial genomes is evaluated. Six fractal features were investigated to learn the parameters of five supervised classification methods for the binary classification task. The optimal parameters of these classifiers are determined via grid-based searching technique. All the currently available identified genes from the database of essential genes were utilized to build the classifiers. The fractal features were proven to be more robust and powerful in the prediction performance. In a statistical sense, the ELM method shows superiority in predicting the essential genes. Non-parameter tests of the average AUC and ACC showed that the fractal feature is much better than other five compared features sets. Our approach is promising and convenient to identify new bacterial essential genes.
Fractal variability: An emergent property of complex dissipative systems
NASA Astrophysics Data System (ADS)
Seely, Andrew J. E.; Macklem, Peter
2012-03-01
The patterns of variation of physiologic parameters, such as heart and respiratory rate, and their alteration with age and illness have long been under investigation; however, the origin and significance of scale-invariant fractal temporal structures that characterize healthy biologic variability remain unknown. Quite independently, atmospheric and planetary scientists have led breakthroughs in the science of non-equilibrium thermodynamics. In this paper, we aim to provide two novel hypotheses regarding the origin and etiology of both the degree of variability and its fractal properties. In a complex dissipative system, we hypothesize that the degree of variability reflects the adaptability of the system and is proportional to maximum work output possible divided by resting work output. Reductions in maximal work output (and oxygen consumption) or elevation in resting work output (or oxygen consumption) will thus reduce overall degree of variability. Second, we hypothesize that the fractal nature of variability is a self-organizing emergent property of complex dissipative systems, precisely because it enables the system's ability to optimally dissipate energy gradients and maximize entropy production. In physiologic terms, fractal patterns in space (e.g., fractal vasculature) or time (e.g., cardiopulmonary variability) optimize the ability to deliver oxygen and clear carbon dioxide and waste. Examples of falsifiability are discussed, along with the need to further define necessary boundary conditions. Last, as our focus is bedside utility, potential clinical applications of this understanding are briefly discussed. The hypotheses are clinically relevant and have potential widespread scientific relevance.
Wideband fractal antennas for holographic imaging and rectenna applications
NASA Astrophysics Data System (ADS)
Bunch, Kyle J.; McMakin, Douglas L.; Sheen, David M.
2008-04-01
At Pacific Northwest National Laboratory, wideband antenna arrays have been successfully used to reconstruct three-dimensional images at microwave and millimeter-wave frequencies. Applications of this technology have included portal monitoring, through-wall imaging, and weapons detection. Fractal antennas have been shown to have wideband characteristics due to their self-similar nature (that is, their geometry is replicated at different scales). They further have advantages in providing good characteristics in a compact configuration. We discuss the application of fractal antennas for holographic imaging. Simulation results will be presented. Rectennas are a specific class of antennas in which a received signal drives a nonlinear junction and is retransmitted at either a harmonic frequency or a demodulated frequency. Applications include tagging and tracking objects with a uniquely-responding antenna. It is of interest to consider fractal rectenna because the self-similarity of fractal antennas tends to make them have similar resonance behavior at multiples of the primary resonance. Thus, fractal antennas can be suited for applications in which a signal is reradiated at a harmonic frequency. Simulations will be discussed with this application in mind.
Wideband Fractal Antennas for Holographic Imaging and Rectenna Applications
Bunch, Kyle J.; McMakin, Douglas L.; Sheen, David M.
2008-04-18
At Pacific Northwest National Laboratory, wideband antenna arrays have been successfully used to reconstruct three-dimensional images at microwave and millimeter-wave frequencies. Applications of this technology have included portal monitoring, through-wall imaging, and weapons detection. Fractal antennas have been shown to have wideband characteristics due to their self-similar nature (that is, their geometry is replicated at different scales). They further have advantages in providing good characteristics in a compact configuration. We discuss the application of fractal antennas for holographic imaging. Simulation results will be presented. Rectennas are a specific class of antennas in which a received signal drives a nonlinear junction and is retransmitted at either a harmonic frequency or a demodulated frequency. Applications include tagging and tracking objects with a uniquely-responding antenna. It is of interest to consider fractal rectenna because the self-similarity of fractal antennas tends to make them have similar resonance behavior at multiples of the primary resonance. Thus, fractal antennas can be suited for applications in which a signal is reradiated at a harmonic frequency. Simulations will be discussed with this application in mind.
The concept of fractals in the pharmaceutical sciences.
Koch, H P
1993-09-01
An old problem is sometimes better understood when it is viewed from a new perspective than by investigating it repeatedly with classical methods. Such a new perspective is provided by the concept of fractals. Briefly, this term defines a spread class of geometric shapes, whose subunits replicate the structure of the larger unit in accordance with the formalization of the idea of self-similarity. This concept was introduced by Mandel-brot in respond to the need for a more sophisticated explanation of numerous phenomena commonly encountered in nature. In this review we will provide a short introduction into the concept of fractals and a concise overview of those studies where fractal geometry has been employed so far in pharmaceutical research. We believe that a review of the most recent findings in the application of fractal geometry to problems encountered in the pharmaceutical sciences might be helpful also to those people who are involved in the search for novel drug substances, as well as in the investigation of their performance in vitro and in vivo. A few suggestions will be made where this concept probably could be helpful in the future and instructions will be given on how the fractal dimension of rugged objects can be estimated practically.
Fractal markets: Liquidity and investors on different time horizons
NASA Astrophysics Data System (ADS)
Li, Da-Ye; Nishimura, Yusaku; Men, Ming
2014-08-01
In this paper, we propose a new agent-based model to study the source of liquidity and the “emergent” phenomenon in financial market with fractal structure. The model rests on fractal market hypothesis and agents with different time horizons of investments. What is interesting is that though the agent-based model reveals that the interaction between these heterogeneous agents affects the stability and liquidity of the financial market the real world market lacks detailed data to bring it to light since it is difficult to identify and distinguish the investors with different time horizons in the empirical approach. results show that in a relatively short period of time fractal market provides liquidity from investors with different horizons and the market gains stability when the market structure changes from uniformity to diversification. In the real world the fractal structure with the finite of horizons can only stabilize the market within limits. With the finite maximum horizons, the greater diversity of the investors and the fractal structure will not necessarily bring more stability to the market which might come with greater fluctuation in large time scale.
Fractal approach to the description of the auroral region
Chernyshov, A. A. Mogilevsky, M. M.; Kozelov, B. V.
2013-07-15
The plasma of the auroral region, where energetic particles precipitate from the magnetosphere into the ionosphere, is highly inhomogeneous and nonstationary. In this case, traditional methods of classical plasma physics turn out to be inapplicable. In order to correctly describe the dynamic regimes, transition processes, fluctuations, and self-similar scalings in this region, nonlinear dynamics methods based of the concepts of fractal geometry and percolation theory can be used. In this work, the fractal geometry and percolation theory are used to describe the spatial structure of the ionospheric conductivity. The topological properties, fractal dimensions, and connective indices characterizing the structure of the Pedersen and Hall conductivities on the nightside auroral zone are investigated theoretically. The restrictions imposed on the fractal estimates by the condition of ionospheric current percolation are analyzed. It is shown that the fluctuation scalings of the electric fields and auroral glow observed in the auroral zone fit well the restrictions imposed by the critical condition on the percolation of the Pedersen current. Thus, it is demonstrated that the fractal approach is a promising and convenient method for studying the properties of the ionosphere.
Analysis of fractal electrodes for efficient neural stimulation
Golestanirad, Laleh; Elahi, Behzad; Molina, Alberto; Mosig, Juan R.; Pollo, Claudio; Chen, Robert; Graham, Simon J.
2013-01-01
Planar electrodes are increasingly used in therapeutic neural stimulation techniques such as functional electrical stimulation, epidural spinal cord stimulation (ESCS), and cortical stimulation. Recently, optimized electrode geometries have been shown to increase the efficiency of neural stimulation by increasing the variation of current density on the electrode surface. In the present work, a new family of modified fractal electrode geometries is developed to enhance the efficiency of neural stimulation. It is shown that a promising approach in increasing the neural activation function is to increase the “edginess” of the electrode surface, a concept that is explained and quantified by fractal mathematics. Rigorous finite element simulations were performed to compute electric potential produced by proposed modified fractal geometries. The activation of 256 model axons positioned around the electrodes was then quantified, showing that modified fractal geometries required a 22% less input power while maintaining the same level of neural activation. Preliminary in vivo experiments investigating muscle evoked potentials due to median nerve stimulation showed encouraging results, supporting the feasibility of increasing neural stimulation efficiency using modified fractal geometries. PMID:23874290
Relationship between Fractal Dimension and Agreeability of Facial Imagery
NASA Astrophysics Data System (ADS)
Oyama-Higa, Mayumi; Miao, Tiejun; Ito, Tasuo
2007-11-01
Why do people feel happy and good or equivalently empathize more, with smiling face imageries than with ones of expressionless face? To understand what the essential factors are underlying imageries in relating to the feelings, we conducted an experiment by 84 subjects asked to estimate the degree of agreeability about expressionless and smiling facial images taken from 23 young persons to whom the subjects were no any pre-acquired knowledge. Images were presented one at a time to each subject who was asked to rank agreeability on a scale from 1 to 10. Fractal dimensions of facial images were obtained in order to characterize the complexity of the imageries by using of two types of fractal analysis methods, i.e., planar and cubic analysis methods, respectively. The results show a significant difference in the fractal dimension values between expressionless faces and smiling ones. Furthermore, we found a well correlation between the degree of agreeability and fractal dimensions, implying that the fractal dimension optically obtained in relation to complexity in imagery information is useful to characterize the psychological processes of cognition and awareness.
Fractal geometry in an expanding, one-dimensional, Newtonian universe.
Miller, Bruce N; Rouet, Jean-Louis; Le Guirriec, Emmanuel
2007-09-01
Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the mu (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure.
Fractal dynamics in the ionization of helium Rydberg atoms
NASA Astrophysics Data System (ADS)
Xu, Xiulan; Zhang, Yanhui; Cai, Xiangji; Zhao, Guopeng; Kang, Lisha
2016-11-01
We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory. By introducing a fractal approach to describe the chaotic dynamical behavior of the ionization, we identify the fractal self-similarity structure of the escape time versus the distribution of the initial launch angles of electrons, and find that the self-similarity region shifts toward larger initial launch angles with a decrease in the scaled energy. We connect the fractal structure of the escape time plot to the escape dynamics of ionized electrons. Of particular note is that the fractal dimensions are sensitively controlled by the scaled energy and magnetic field, and exhibit excellent agreement with the chaotic extent of the ionization systems for both helium and hydrogen Rydberg atoms. It is shown that, besides the electric and magnetic fields, core scattering is a primary factor in the fractal dynamics. Project supported by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2014AM030).
Modified fractal model and rheological properties of colloidal networks.
Tang, Dongming; Marangoni, Alejandro G
2008-02-15
The scaling relationship between the storage modulus (G(')) and the volume fraction of solids (Phi) in fat crystal networks has been explained by the fractal model developed by our group. However, many experimental results and simulation studies suggest that the stress distribution within a colloidal network is dramatically heterogeneous, which means that a small part of the network carries most of the stress, while the other part of the network does not contribute much to the elastic properties of the system. This concept was introduced into a modified fractal model. The volume fraction of solids term (Phi) in the original fractal model was replaced by Phi(e), the effective volume fraction of solids, in the modified fractal model, which represents the volume fraction of stress-carrying solids. A proposed expression for Phi(e) is given and a modified expression for the scaling relationship between G(') and Phi is obtained. The modified fractal model fits the experiment data well and successfully explains the sometimes observed nonlinear log-log behavior between the storage modulus of colloidal networks and their volume fraction of solids.
Energy and Laplacian on Hanoi-type fractal quantum graphs
NASA Astrophysics Data System (ADS)
Alonso-Ruiz, Patricia; Kelleher, Daniel J.; Teplyaev, Alexander
2016-04-01
This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of one-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. Classical quantum graphs and fractal spaces such as the Hanoi attractor are included among them. We begin with proving the existence of a resistance form on the Hanoi attractor, and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of Laplacians corresponding to weakly self-similar measures on the Hanoi attractor. These estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal. We then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph. Finally, we extend our spectral results to a large class of weakly self-similar fractal quantum graphs.
Synthesis of Cobalt Oxides Thin Films Fractal Structures by Laser Chemical Vapor Deposition
Haniam, P.; Kunsombat, C.; Chiangga, S.; Songsasen, A.
2014-01-01
Thin films of cobalt oxides (CoO and Co3O4) fractal structures have been synthesized by using laser chemical vapor deposition at room temperature and atmospheric pressure. Various factors which affect the density and crystallization of cobalt oxides fractal shapes have been examined. We show that the fractal structures can be described by diffusion-limited aggregation model and discuss a new possibility to control the fractal structures. PMID:24672354
Catastrophes in the multi-fractal dynamics of social-economic systems
NASA Astrophysics Data System (ADS)
Kudinov, A. N.; Tsvetkov, V. P.; Tsvetkov, I. V.
2011-06-01
In the present paper, the concept of multi-fractal dynamics is developed. The problem concerning catastrophes in this dynamics is studied in detail. In the framework of the concept of fractal curve as a thick curve, it is proved that the cell approach to measuring the fractal dimension D is equivalent to measuring the dependence of the length L of the line on the scope δ. The introduction of a fractal scale of temperatures T f is suggested.
Deterministic fractals: extracting additional information from small-angle scattering data.
Cherny, A Yu; Anitas, E M; Osipov, V A; Kuklin, A I
2011-09-01
The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.
Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance
ERIC Educational Resources Information Center
Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy
2011-01-01
Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…
Fourier series analysis of fractal lenses: theory and experiments with a liquid-crystal display.
Davis, Jeffrey A; Sigarlaki, Sean P; Craven, Julia M; Calvo, María Luisa
2006-02-20
We report on a Fourier series approach that predicts the focal points and intensities produced by fractal zone plate lenses. This approach allows us to separate the effects of the fractal order from those of the lens aperture. We implement these fractal lenses onto a liquid-crystal display and show experimental verification of our theory.
Hagerhall, C M; Laike, T; Küller, M; Marcheschi, E; Boydston, C; Taylor, R P
2015-01-01
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.
Form in the Natural Environment: Fractal Computer Graphics and Wassily Kandinsky.
ERIC Educational Resources Information Center
Geake, John; Porter, Jim
1992-01-01
Reports on study of use of fractal geometry in a computer graphics program to improve the perception of intermediate grade level students in their paintings. Finds that students are more likely to use changing shapes and colors after viewing slides of fractal computer graphics. Concludes that fractal computer graphics would make highly engaging…
Studying fractal geometry on submicron length scales by small-angle scattering
Wong, P.; Lin, J.
1988-08-01
Recent studies have shown that internal surfaces of porous geological materials, such as rocks and lignite coals, can be described by fractals down to atomic length scales. In this paper, the basic properties of self-similar and self-affine fractals are reviewed and how fractal dimensions can be measured by small-angle scattering experiments are discussed.
Fractal Dimension of Certain Continuous Functions of Unbounded Variation
NASA Astrophysics Data System (ADS)
Liang, Y. S.; Su, W. Y.
Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.
Temperature induced smoothing of initially fractal grain boundaries
Streitenberger, P.; Foerster, D.; Kolbe, G.; Veit, P.
1996-01-01
Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fractal geometry has attracted great attention. It has been reported that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. In the present paper, for the first time, measurements of the change in the roughness of initially fractal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially fractal dimension.
Fractal properties of macrophage membrane studied by AFM.
Bitler, A; Dover, R; Shai, Y
2012-12-01
Complexity of cell membrane poses difficulties to quantify corresponding morphology changes during cell proliferation and damage. We suggest using fractal dimension of the cell membrane to quantify its complexity and track changes produced by various treatments. Glutaraldehyde fixed mouse RAW 264.7 macrophage membranes were chosen as model system and imaged in PeakForce QNM (quantitative nanomechanics) mode of AFM (atomic force microscope). The morphology of the membranes was characterized by fractal dimension. The parameter was calculated for set of AFM images by three different methods. The same calculations were done for the AFM images of macrophages treated with colchicine, an inhibitor of the microtubule polymerization, and microtubule stabilizing agent taxol. We conclude that fractal dimension can be additional and useful parameter to characterize the cell membrane complexity and track the morphology changes produced by different treatments.
Preliminary evidence for a theory of the fractal city.
Batty, M; Xie, Y
1996-10-01
"In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States...."
A fractal-based approach to lake size-distributions
NASA Astrophysics Data System (ADS)
Seekell, David A.; Pace, Michael L.; Tranvik, Lars J.; Verpoorter, Charles
2013-02-01
The abundance and size distribution of lakes is critical to assessing the role of lakes in regional and global biogeochemical processes. Lakes are fractal but do not always conform to the power law size-distribution typically associated with fractal geographical features. Here, we evaluate the fractal geometry of lakes with the goal of explaining apparently inconsistent observations of power law and non-power law lake size-distributions. The power law size-distribution is a special case for lakes near the mean elevation. Lakes in flat regions are power law distributed, while lakes in mountainous regions deviate from power law distributions. Empirical analyses of lake size data sets from the Adirondack Mountains in New York and the flat island of Gotland in Sweden support this finding. Our approach provides a unifying framework for lake size-distributions, indicates that small lakes cannot dominate total lake surface area, and underscores the importance of regional hypsometry in influencing lake size-distributions.
Trabecular architecture analysis in femur radiographic images using fractals.
Udhayakumar, G; Sujatha, C M; Ramakrishnan, S
2013-04-01
Trabecular bone is a highly complex anisotropic material that exhibits varying magnitudes of strength in compression and tension. Analysis of the trabecular architectural alteration that manifest as loss of trabecular plates and connection has been shown to yield better estimation of bone strength. In this work, an attempt has been made toward the development of an automated system for investigation of trabecular femur bone architecture using fractal analysis. Conventional radiographic femur bone images recorded using standard protocols are used in this study. The compressive and tensile regions in the images are delineated using preprocessing procedures. The delineated images are analyzed using Higuchi's fractal method to quantify pattern heterogeneity and anisotropy of trabecular bone structure. The results show that the extracted fractal features are distinct for compressive and tensile regions of normal and abnormal human femur bone. As the strength of the bone depends on architectural variation in addition to bone mass, this study seems to be clinically useful.
Broadband fractal acoustic metamaterials for low-frequency sound attenuation
NASA Astrophysics Data System (ADS)
Song, Gang Yong; Cheng, Qiang; Huang, Bei; Dong, Hui Yuan; Cui, Tie Jun
2016-09-01
We fabricate and experimentally characterize a broadband fractal acoustic metamaterial that can serve to attenuate the low-frequency sounds at selective frequencies ranging from 225 to 1175 Hz. The proposed metamaterials are constructed by the periodic Hilbert fractal elements made of photosensitive resin via 3D printing. In analogy to electromagnetic fractal structures, it is shown that multiple resonances can also be excited in the acoustic counterpart due to their self-similar properties, which help to attenuate the acoustic energy in a wide spectrum. The confinement of sound waves in such subwavelength element is evidenced by both numerical and experimental results. The proposed metamaterial may provide possible alternative for various applications such as the noise attenuation and the anechoic materials.
Fractal structure of sequential behaviour patterns: an indicator of stress
Alados, C.L.; Escos, J.M; Emlen, J.M.
1996-01-01
The detection of stress arising from parasitic infection bySarcoptes scabieisand from pregnancy is explored, using a fractal analysis of head lifting behaviour and feeding–non-feeding activity sequences in female Spanish ibex,Capra pyrenaica, under natural conditions. Because organisms under stress increase their metabolic rate and, in consequence, energy consumption, it follows that stress will, generally, lead to a reduction in complexity (fractal dimension) of exploratory behaviour. In the present study the fractal dimension of the three measures of complexity used declined with stress, both from pregnancy and from parasitic infection. This observation provides a new and effective way to assess the general state of animals’ health in the field, without the need for capture and handling.
Power-law hereditariness of hierarchical fractal bones.
Deseri, Luca; Di Paola, Mario; Zingales, Massimiliano; Pollaci, Pietro
2013-12-01
In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ⩽ β ⩽1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power law.
Chaotic transients and fractal structures governing coupled swing dynamics
Ueda, Y.; Enomoto, T. ); Stewart, H.B. )
1990-01-01
Numerical simulations are used to study coupled swing equations modeling the dynamics of two electric generators connected to an infinite bus by a simple transmission network. In particular, the effect of varying parameters corresponding to the input power supplied to each generator is studied. In addition to stable steady operating conditions, which should correspond to synchronized, normal operation, the coupled swing model has other stable states of large amplitude oscillations which, if realized, would represent non-synchronized motions: the phase space boundary separating their basins of attraction is fractal, corresponding to chaotic transient motions. These fractal structures in phase space and the associated fractal structures in parameter space will be of primary concern to engineers in predicting system behavior.
Fractal structure of the liver: effect on drug elimination.
Weiss, Michael
2013-02-01
Liver modeling in pharmacokinetics has been based on outflow curves of extracellular tracers obtained in single-pass perfused rat livers. These reference curves represent the hepatic transit time densities (TTD) of tracers. Since the fractal structure of the sinusoidal network implies a TTD with power-law tail, the question is whether the use of conventional empirical TTDs with exponential tail may lead to biased estimates of hepatic clearance. A simulation study using a novel TTD model that accounts for fractal heterogeneity of hepatic flow shows that the bias is less than about 5%. Using this approach to determine the influence of hepatic flow dispersion on drug extraction, only a minor effect was found. The results demonstrate that there is no need for specific fractal models of hepatic drug elimination.
Fractal signatures in analogs of interplanetary dust particles
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Banerjee, Varsha; Puri, Sanjay
2014-10-01
Interplanetary dust particles (IDPs) are an important constituent of the earths stratosphere, interstellar and interplanetary medium, cometary comae and tails, etc. Their physical and optical characteristics are significantly influenced by the morphology of silicate aggregates which form the core in IDPs. In this paper we reinterpret scattering data from laboratory analogs of cosmic silicate aggregates created by Volten et al. (2007) [1] to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension dm≃1.75. The same fractal dimension also characterizes clusters obtained from diffusion limited aggregation (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically transported silicate particles.
Liver ultrasound image classification by using fractal dimension of edge
NASA Astrophysics Data System (ADS)
Moldovanu, Simona; Bibicu, Dorin; Moraru, Luminita
2012-08-01
Medical ultrasound image edge detection is an important component in increasing the number of application of segmentation, and hence it has been subject of many studies in the literature. In this study, we have classified the liver ultrasound images (US) combining Canny and Sobel edge detectors with fractal analysis in order to provide an indicator about of the US images roughness. We intend to provide a classification rule of the focal liver lesions as: cirrhotic liver, liver hemangioma and healthy liver. For edges detection the Canny and Sobel operators were used. Fractal analyses have been applied for texture analysis and classification of focal liver lesions according to fractal dimension (FD) determined by using the Box Counting method. To assess the performance and accuracy rate of the proposed method the contrast-to-noise (CNR) is analyzed.
Fractal calibration in size-exclusion chromatography I. An introduction.
García-Lopera, Rosa; Irurzun, Isabel; Abad, Concepción; Campos, Agustín
2003-05-09
The elution behaviour of different polymer-solvent systems in three types of organic columns for SEC has been compared and interpreted. The experimental data show that the classical universal calibration is not accomplished. Deviations from a unique curve are observed due to the binary and ternary interactions between the components of the system (solvent, polymer and gel) which results on secondary mechanisms accompanying the main pure or "ideal" SEC separation mechanism. Both, enthalpic and entropic effects are interpreted in terms of the swelling and crosslinking degrees of the gel packings, and are also related with the fractal characteristics of their surfaces, such as the fractal dimension and the available pore size. Moreover, a relationship between the fractal dimension of the pore surface and the chromatographic distribution coefficient is proposed.
Synthetic turbulence, fractal interpolation, and large-eddy simulation.
Basu, Sukanta; Foufoula-Georgiou, Efi; Porté-Agel, Fernando
2004-08-01
Fractal interpolation has been proposed in the literature as an efficient way to construct closure models for the numerical solution of coarse-grained Navier-Stokes equations. It is based on synthetically generating a scale-invariant subgrid-scale field and analytically evaluating its effects on large resolved scales. In this paper, we propose an extension of previous work by developing a multiaffine fractal interpolation scheme and demonstrate that it preserves not only the fractal dimension but also the higher-order structure functions and the non-Gaussian probability density function of the velocity increments. Extensive a priori analyses of atmospheric boundary layer measurements further reveal that this multiaffine closure model has the potential for satisfactory performance in large-eddy simulations. The pertinence of this newly proposed methodology in the case of passive scalars is also discussed.
Voronoi cells, fractal dimensions and fibre composites.
Summerscales, J.; Guild, F. J.; Pearce, N. R. L.; Russell, P. M.
2001-02-01
The use of fibre-reinforced polymer matrix composite materials is growing at a faster rate than the gross domestic product (GDP) in many countries. An improved understanding of their processing and mechanical behaviour would extend the potential applications of these materials. For unidirectional composites, it is predicted that localized absence of fibres is related to longitudinal compression failure. The use of woven reinforcements permits more effective manufacture than for unidirectional fibres. It has been demonstrated experimentally that compression strengths of woven composites are reduced when fibres are clustered. Summerscales predicted that clustering of fibres would increase the permeability of the reinforcement and hence expedite the processing of these materials. Commercial fabrics are available which employ this concept using flow-enhancing bound tows. The net effect of clustering fibres is to enhance processability whilst reducing the mechanical properties. The effects reported above were qualitative correlations. To improve the design tools for reinforcement fabrics we have sought to quantify the changes in the micro/meso-structure of woven reinforcement fabrics. Gross differences in the appearance of laminate sections are apparent for different weave styles. The use of automated image analysis is essential for the quantification of subtle changes in fabric architecture. This paper considers Voronoi tessellation and fractal dimensions for the quantification of the microstructures of woven fibre-reinforced composites. It reviews our studies in the last decade of the process-property-structure relationships for commercial and experimental fabric reinforcements in an attempt to resolve the processing vs. properties dilemma. A new flow-enhancement concept has been developed which has a reduced impact on laminate mechanical properties.
Fractal Theory for Permeability Prediction, Venezuelan and USA Wells
NASA Astrophysics Data System (ADS)
Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana
2014-05-01
Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.
A New Fractal Model of Chromosome and DNA Processes
NASA Astrophysics Data System (ADS)
Bouallegue, K.
Dynamic chromosome structure remains unknown. Can fractals and chaos be used as new tools to model, identify and generate a structure of chromosomes?Fractals and chaos offer a rich environment for exploring and modeling the complexity of nature. In a sense, fractal geometry is used to describe, model, and analyze the complex forms found in nature. Fractals have also been widely not only in biology but also in medicine. To this effect, a fractal is considered an object that displays self-similarity under magnification and can be constructed using a simple motif (an image repeated on ever-reduced scales).It is worth noting that the problem of identifying a chromosome has become a challenge to find out which one of the models it belongs to. Nevertheless, the several different models (a hierarchical coiling, a folded fiber, and radial loop) have been proposed for mitotic chromosome but have not reached a dynamic model yet.This paper is an attempt to solve topological problems involved in the model of chromosome and DNA processes. By combining the fractal Julia process and the numerical dynamical system, we have finally found out four main points. First, we have developed not only a model of chromosome but also a model of mitosis and one of meiosis. Equally important, we have identified the centromere position through the numerical model captured below. More importantly, in this paper, we have discovered the processes of the cell divisions of both mitosis and meiosis. All in all, the results show that this work could have a strong impact on the welfare of humanity and can lead to a cure of genetic diseases.
Fractal analysis of bone architecture at distal radius.
Tomomitsu, Tatsushi; Mimura, Hiroaki; Murase, Kenya; Sone, Teruki; Fukunaga, Masao
2005-12-20
Bone strength depends on bone quality (architecture, turnover, damage accumulation, and mineralization) as well as bone mass. In this study, human bone architecture was analyzed using fractal image analysis, and the clinical relevance of this method was evaluated. The subjects were 12 healthy female controls and 16 female patients suspected of having osteoporosis (age range, 22-70 years; mean age, 49.1 years). High-resolution CT images of the distal radius were acquired and analyzed using a peripheral quantitative computed tomography (pQCT) system. On the same day, bone mineral densities of the lumbar spine (L-BMD), proximal femur (F-BMD), and distal radius (R-BMD) were measured by dual-energy X-ray absorptiometry (DXA). We examined the correlation between the fractal dimension and six bone mass indices. Subjects diagnosed with osteopenia or osteoporosis were divided into two groups (with and without vertebral fracture), and we compared measured values between these two groups. The fractal dimension correlated most closely with L-BMD (r=0.744). The coefficient of correlation between the fractal dimension and L-BMD was very similar to the coefficient of correlation between L-BMD and F-BMD (r=0.783) and the coefficient of correlation between L-BMD and R-BMD (r=0.742). The fractal dimension was the only measured value that differed significantly between both the osteopenic and the osteoporotic subjects with and without vertebral fracture. The present results suggest that the fractal dimension of the distal radius can be reliably used as a bone strength index that reflects bone architecture as well as bone mass.
Testing Fractal Methods on Observed and Simulated Solar Magnetograms
NASA Technical Reports Server (NTRS)
Adams, M.; Falconer, D. A.; Lee, J. K.; Jones, C.
2003-01-01
The term "magnetic complexity" has not been sufficiently quantified. To accomplish this, we must understand the relationship between the observed magnetic field of solar active regions and fractal dimension measurements. Using data from the Marshall Space Flight Center's vector magnetograph ranging from December 1991 to July 2001, we compare the results of several methods of calculating a fractal dimension, e.g., Hurst coefficient, the Higuchi method, power spectrum, and 2-D Wavelet Packet Analysis. In addition, we apply these methods to synthetic data, beginning with representations of very simple dipole regions, ending with regions that are magnetically complex.
Extended Vicsek fractals: Laplacian spectra and their applications.
Dolgushev, Maxim; Liu, Hongxiao; Zhang, Zhongzhi
2016-11-01
Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.
Fractal scaling and fluid flow in fracture networks in rock
Barton, C.C.
1996-12-31
Recovery of oil and gas resources and injection of toxic waste materials requires quantitative models to describe and predict the movement of fluids in rock. Existing models based on pore-space flow are inappropriate for study of the more rapid process of fluid flow through fracture networks. This type of flow is not a simple function of the fracture characteristics at any particular scale, but rather the integration of fracture contributions at all scales. The mathematical constructs of fractal geometry are well suited to quantify and model relationships within complex systems that are statistically self-similar over a wide range of scales. Analyses show that fracture traces mapped on two-dimensional slices through three-dimensional nature fracture networks in rock follow a fractal scaling law over six orders of magnitude. Detailed measurements of 17 two-dimensional samples of fracture networks (at diverse scales in rocks of dissimilar age, lithology, and tectonic setting) show similar fractal dimensions in the range 1.3-1.7. The range in fractal dimension implies that a single physical process of rock fracturing operates over a wide range of scales, from microscopic cracks to large, regional fault systems. The knowledge that rock-fracture networks are fractal allows the use of data from a one-dimensional drill-hole sample to predict the two- and three-dimensional scaling of the fracture system. The spacing of fractures in drill holes is a fractal Cantor distribution, and the range of fractal dimension is 0.4-0.6, which is an integer dimension less than that of fracture-trace patterns exposed on two-dimensional, planar sections. A reconstruction of the fracture history at the point of initial connectivity across the network (percolation) has a fractal dimension of 1.35 as compared to a dimension of 1.9 for the percolation cluster in a two-dimensional model. Paleo flow was mapped based on the deposition of aqueous minerals on the fracture surface.
Fractal dimension of alumina aggregates grown in two dimensions
NASA Technical Reports Server (NTRS)
Larosa, Judith L.; Cawley, James D.
1992-01-01
The concepts of fractal geometry are applied to the analysis of 0.4-micron alumina constrained to agglomerate in two dimensions. Particles were trapped at the bottom surface of a drop of a dilute suspension, and the agglomeration process was directly observed, using an inverted optical microscope. Photographs were digitized and analyzed, using three distinct approaches. The results indicate that the agglomerates are fractal, having a dimension of approximately 1.5, which agrees well with the predictions of the diffusion-limited cluster-cluster aggregation model.
Fractal scaling and fluid flow in fracture networks in rock
Barton, C.C. )
1996-01-01
Recovery of oil and gas resources and injection of toxic waste materials requires quantitative models to describe and predict the movement of fluids in rock. Existing models based on pore-space flow are inappropriate for study of the more rapid process of fluid flow through fracture networks. This type of flow is not a simple function of the fracture characteristics at any particular scale, but rather the integration of fracture contributions at all scales. The mathematical constructs of fractal geometry are well suited to quantify and model relationships within complex systems that are statistically self-similar over a wide range of scales. Analyses show that fracture traces mapped on two-dimensional slices through three-dimensional nature fracture networks in rock follow a fractal scaling law over six orders of magnitude. Detailed measurements of 17 two-dimensional samples of fracture networks (at diverse scales in rocks of dissimilar age, lithology, and tectonic setting) show similar fractal dimensions in the range 1.3-1.7. The range in fractal dimension implies that a single physical process of rock fracturing operates over a wide range of scales, from microscopic cracks to large, regional fault systems. The knowledge that rock-fracture networks are fractal allows the use of data from a one-dimensional drill-hole sample to predict the two- and three-dimensional scaling of the fracture system. The spacing of fractures in drill holes is a fractal Cantor distribution, and the range of fractal dimension is 0.4-0.6, which is an integer dimension less than that of fracture-trace patterns exposed on two-dimensional, planar sections. A reconstruction of the fracture history at the point of initial connectivity across the network (percolation) has a fractal dimension of 1.35 as compared to a dimension of 1.9 for the percolation cluster in a two-dimensional model. Paleo flow was mapped based on the deposition of aqueous minerals on the fracture surface.
The Concept of Fractal Cosmos: III. Present State
NASA Astrophysics Data System (ADS)
Grujic, P.
2011-06-01
This is the sequel to the previous accounts on the rise and development of the concept of fractal cosmos, up to year 2001 (Grujic 2001, 2002). Here we give an overview of the present-day state of art, with the emphasis on the latest developments and controversies concerning the model of hierarchical universe. We describe both the theoretical advances and the latest empirical evidence concerning the observation of the large-scale structure of the observable universe. Finally we address a number of epistemological points, putting the fractal paradigm into a broader cosmological frame.
Homogenization of spatially dispersive 1D fractal metamaterials
NASA Astrophysics Data System (ADS)
Moeini, Samaneh
2016-04-01
A layered uniaxial dielectric structure is considered. The layers in the structure are distributed according to a one-dimensional fractal set. The resulting fractal metamaterial is homogenized with an original source-driven homogenization approach which is suitable for both numerical and analytical calculations. Due to the fact that the considered metamaterial is nonmagnetic, the only effective parameter which needs to be calculated is the effective permittivity dyadic e(ω, k). The effective permittivity is obtained analytically (by using a transfer matrix approach) and numerically (by using a Finite-Difference Time-Domain solver).
EEG signal features extraction based on fractal dimension.
Finotello, Francesca; Scarpa, Fabio; Zanon, Mattia
2015-01-01
The spread of electroencephalography (EEG) in countless applications has fostered the development of new techniques for extracting synthetic and informative features from EEG signals. However, the definition of an effective feature set depends on the specific problem to be addressed and is currently an active field of research. In this work, we investigated the application of features based on fractal dimension to a problem of sleep identification from EEG data. We demonstrated that features based on fractal dimension, including two novel indices defined in this work, add valuable information to standard EEG features and significantly improve sleep identification performance.
Effect of Na+ on surface fractal dimension of compacted bentonite
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2015-05-01
Compacted Tsukinuno bentonite was immersed into NaCl solutions of different concentrations in oedometers, and the surface fractal dimension of bentonite-saline association was measured by nitrogen adsorption isotherms. The application of the Frenkel-Halsey-Hill equation and the Neimark thermodynamic method to nitrogen adsorption isotherms indicated that the surface roughness was greater for the bentonite-saline association. The surface fractal dimension of bentonite increased in the NaCl solution with low Na+ concentration, but decreased at high Na+ concentration. This process was accompanied by the same tendency in specific surface area and microporosity with the presence of Na+ coating in the clay particles.
Fractal mechanisms in the electrophysiology of the heart
NASA Technical Reports Server (NTRS)
Goldberger, A. L.
1992-01-01
The mathematical concept of fractals provides insights into complex anatomic branching structures that lack a characteristic (single) length scale, and certain complex physiologic processes, such as heart rate regulation, that lack a single time scale. Heart rate control is perturbed by alterations in neuro-autonomic function in a number of important clinical syndromes, including sudden cardiac death, congestive failure, cocaine intoxication, fetal distress, space sickness and physiologic aging. These conditions are associated with a loss of the normal fractal complexity of interbeat interval dynamics. Such changes, which may not be detectable using conventional statistics, can be quantified using new methods derived from "chaos theory.".
Nonlinear dynamics, fractals, cardiac physiology and sudden death
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.
1987-01-01
The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.
Analysis of Geographical Distribution Patterns in Plants Using Fractals
NASA Astrophysics Data System (ADS)
Bari, A.; Ayad, G.; Padulosi, S.; Hodgkin, T.; Martin, A.; Gonzalez-Andujar, J. L.; Brown, A. H. D.
Geographical distribution patterns in plants have been observed since primeval times and have been used by plant explorers to trace the origin of plants species. These patterns embody the effects of fundamental law-like processes. Diversity in plants has also been found to be proportionate with the area, and this scaling behavior is also known as fractal behavior. In the present study, we use fractal geometry to analyze the distribution patterns of wild taxa of cowpea with the objective to locate where their diversity would be the highest to aid in the planning of targeted explorations and conservation measures.
Counting spanning trees on fractal graphs and their asymptotic complexity
NASA Astrophysics Data System (ADS)
Anema, Jason A.; Tsougkas, Konstantinos
2016-09-01
Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem 3.4. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpiński gasket, a non-post critically finite analog of the Sierpiński gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.
Person identification using fractal analysis of retina images
NASA Astrophysics Data System (ADS)
Ungureanu, Constantin; Corniencu, Felicia
2004-10-01
Biometric is automated method of recognizing a person based on physiological or behavior characteristics. Among the features measured are retina scan, voice, and fingerprint. A retina-based biometric involves the analysis of the blood vessels situated at the back of the eye. In this paper we present a method, which uses the fractal analysis to characterize the retina images. The Fractal Dimension (FD) of retina vessels was measured for a number of 20 images and have been obtained different values of FD for each image. This algorithm provides a good accuracy is cheap and easy to implement.
NASA Astrophysics Data System (ADS)
Cheng, Qiuming
2016-04-01
Singularity theory states that extreme geo-processes result in anomalous amounts of energy release or material accumulation within a narrow spatial-temporal interval. The products (e.g. mass density and energy density) caused by extreme geo-processes depict singularity without the ordinary derivative and antiderivative (integration) properties. Based on the definition of fractal density, the density measured in fractal dimensional space, in the current paper the author is proposing several operations including fractal derivative and fractal integral to analyze singularity of fractal density. While the ordinary derivative including fractional derivatives as a fundamental tool measuring the sensitivity of change of function (quantity as dependent variable) with change of another quantity as independent variable, the changes are measured in the ordinary space with additive property, fractal derivative (antiderivative) measures the ratio of changes of two quantities measured in fractal space-fractal dimensional space. For example, if the limit of ratio of increment of quantity (Δf) over the associated increment of time (Δtα) measured in α - dimensional space approaches to a finite value, then the limit is referred a α-dimensional fractal derivative of function fand denoted as f' = lim Δf--= df- α Δt→0 Δtα dtα According to the definition of the fractal derivative the ordinary derivative becomes the special case if the space becomes non-fractal space with α value as an integer. In the rest of the paper we demonstrate that fractal density concept and fractal derivative can be applied in describing singularity property of products caused by extreme or avalanche events. The extreme earth-thermal processes such as hydrothermal mineralization occurred in the earth crust, heat flow over ocean ridges, igneous activities or juvenile crust grows, originated from cascade earth dynamics (mantle convection, plate tectonics, and continent crust grow etc.) were analyzed
An Introduction to Flow and Transport in Fractal Models of Porous Media: Part I
NASA Astrophysics Data System (ADS)
Cai, Jianchao; San José Martínez, Fernando; Martín, Miguel Angel; Perfect, Edmund
2014-09-01
This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal conductivity of fractal porous media and hydrophobicity and surface erosion properties of fractal structures.
Methods of nanoassembly of a fractal polymer and materials formed thereby
Newkome, George R; Moorefield, Charles N
2014-09-23
The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to (36) Ru and (6) Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.
Methods of nanoassembly of a fractal polymer and materials formed thereby
Newkome, George R [Medina, OH; Moorefield, Charles N [Akron, OH
2012-07-24
The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to 36 Ru and 6 Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.
NASA Astrophysics Data System (ADS)
Li, Jian-Hua; Yu, Bo-Ming; Zou, Ming-Qing
2009-11-01
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 λmax, the ratio β of conic roughness height to its base radius as well as the ratio λminλmax of the minimum to the maximal base diameter.
Fractal analysis of high-resolution CT images as a tool for quantification of lung diseases
Uppaluri, R.; Mitsa, T.; Galvin, J.R.
1995-12-31
Fractal geometry is increasingly being used to model complex naturally occurring phenomena. There are two types of fractals in nature-geometric fractals and stochastic fractals. The pulmonary branching structure is a geometric fractal and the intensity of its grey scale image is a stochastic fractal. In this paper, the authors attempt to quantify the texture of CT lung images using properties of both types of fractals. A simple algorithm for detecting of abnormality in human lungs, based on 2-D and 3-D fractal dimensions, is presented. This method involves calculating the local fractal dimensions, based on intensities, in the 2-D slice to air edge enhancement. Following this, grey level thresholding is performed and a global fractal dimension, based on structure, for the entire data is estimated in 2-D and 3-D. High Resolution CT images of normal and abnormal lungs were analyzed. Preliminary results showed that classification of normal and abnormal images could be obtained based on the differences between their global fractal dimensions.
NASA Astrophysics Data System (ADS)
Pan, Xuezai; Shang, Xudong; Wang, Minggang; Zuo-Fei
With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.
[Fractal theory and its application in the analysis of soil spatial variability: a review].
Zhang, Fa-Sheng; Liu, Zuo-Xin
2011-05-01
Soil has spatial variability in its attributes. The analysis of soil spatial variability is of significance for soil management. This paper summarized the fractal theory and its application in spatial analysis of soil variability, with the focus on the utilization of moment method in calculating the fractal dimension of soil attributes, the multi-fractal analysis of soil spatial variability, and the scaling up of soil attributes based on multi-fractal parameters. The studies on the application of fractal theory and multi-fractal method in the analysis of soil spatial variability were also reviewed. Fractal theory could be an important tool in quantifying the spatial variability and scaling up of soil attributes.
A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets
NASA Astrophysics Data System (ADS)
Satija, Indubala I.
2016-11-01
This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden D3 symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.
Basic principles and applications of fractal geometry in pathology: a review.
Dey, Pranab
2005-10-01
The basic principles and prospects of fractal geometry in pathology are promising. All articles found with a PubMed search with the keywords fractal dimension (FD) and related to pathology were reviewed. All fractal objects have FDs, commonly calculated with box counting. Fractal geometry has been applied to measure the irregularities of nuclear and glandular margins to distinguish malignant lesions from benign ones, to measure the infiltrative margin of a malignant tumor, to assess tumor angiogenesis and to measure the distribution of collagen in tissue. Fractal geometry has also been applied to assess the irregular distribution of chromatin in malignant cells. Biologic model formation is possible with fractal geometry. In the future, fractal geometry may help with the diagnosis, understanding of pathogenesis and management of lesions. It may also provide new insights into disease processes.
He, Chiquan; Zhao, Kuiyi
2003-04-01
By using the principles and methods of fractal geometry theory, the relationship between above ground biomass and plant length or sheath height of Carex lasiocarpa population was studied. The results showed that there was a good static fractal relationship between them, and the resulted fractal dimension was an efficient description of the accumulation of above ground biomass in each organ. The dynamic fractal relationship showed that during the whole growing season, the increase of above ground biomass had a self-similarity, being a fractal growth process, and the pattern of its increase was the fractal dimension D. Based on these results, a fractal growth model of Carex lasiocarpa population was established, which regarded the bigger grass as the result of the amplification of seedling growth.
Takehara, Takuma; Ochiai, Fumio; Watanabe, Hiroshi; Suzuki, Naoto
2013-01-01
There is currently substantial literature to suggest that facial emotion recognition is impaired when other-race or inverted faces are presented. This study examined circumplex structures for recognising facial emotions under these conditions, directly measured those structures using a fractal dimension, and examined the difference between fractal dimensions. Results established that emotion ratings for the emotion prototypes used were sufficiently accurate under all conditions. Fractal analyses showed that the fractal dimensions of the circumplexes were significantly higher for recognition of facial emotions in other races than in one's own when the faces were presented upright; the fractal dimensions of the circumplexes were also higher for recognition of emotions in inverted faces than in upright faces in the own-race condition. The results suggest that a lower level of facial emotion recognition is associated with higher fractal dimension and that an increase of fractal dimension may be characterised by lack of facial familiarity.
[Recent progress of research and applications of fractal and its theories in medicine].
Cai, Congbo; Wang, Ping
2014-10-01
Fractal, a mathematics concept, is used to describe an image of self-similarity and scale invariance. Some organisms have been discovered with the fractal characteristics, such as cerebral cortex surface, retinal vessel structure, cardiovascular network, and trabecular bone, etc. It has been preliminarily confirmed that the three-dimensional structure of cells cultured in vitro could be significantly enhanced by bionic fractal surface. Moreover, fractal theory in clinical research will help early diagnosis and treatment of diseases, reducing the patient's pain and suffering. The development process of diseases in the human body can be expressed by the fractal theories parameter. It is of considerable significance to retrospectively review the preparation and application of fractal surface and its diagnostic value in medicine. This paper gives an application of fractal and its theories in the medical science, based on the research achievements in our laboratory.
A user-friendly modified pore-solid fractal model
Ding, Dian-yuan; Zhao, Ying; Feng, Hao; Si, Bing-cheng; Hill, Robert Lee
2016-01-01
The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined. PMID:27996013
A user-friendly modified pore-solid fractal model.
Ding, Dian-Yuan; Zhao, Ying; Feng, Hao; Si, Bing-Cheng; Hill, Robert Lee
2016-12-20
The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined.
Assembling molecular Sierpiński triangle fractals.
Shang, Jian; Wang, Yongfeng; Chen, Min; Dai, Jingxin; Zhou, Xiong; Kuttner, Julian; Hilt, Gerhard; Shao, Xiang; Gottfried, J Michael; Wu, Kai
2015-05-01
Fractals, being "exactly the same at every scale or nearly the same at different scales" as defined by Benoit B. Mandelbrot, are complicated yet fascinating patterns that are important in aesthetics, mathematics, science and engineering. Extended molecular fractals formed by the self-assembly of small-molecule components have long been pursued but, to the best of our knowledge, not achieved. To tackle this challenge we designed and made two aromatic bromo compounds (4,4″-dibromo-1,1':3',1″-terphenyl and 4,4‴-dibromo-1,1':3',1″:4″,1‴-quaterphenyl) to serve as building blocks. The formation of synergistic halogen and hydrogen bonds between these molecules is the driving force to assemble successfully a whole series of defect-free molecular fractals, specifically Sierpiński triangles, on a Ag(111) surface below 80 K. Several critical points that govern the preparation of the molecular Sierpiński triangles were scrutinized experimentally and revealed explicitly. This new strategy may be applied to prepare and explore various planar molecular fractals at surfaces.
Tearing Plastic: A Laboratory Exercise on Fractals and Hyperbolic Geometry
ERIC Educational Resources Information Center
Taylor, Ron; Timberlake, Todd
2007-01-01
In this article we describe a hands-on activity for a liberal arts mathematics course that focuses on the beauty and unity of mathematics. The purpose of the activity is to tie together several topics in the context of a "real-world" situation. These topics include: fractals, non-Euclidean geometry, symmetry, and Platonic solids. This activity…
Turbulent premixed flames on fractal-grid-generated turbulence
NASA Astrophysics Data System (ADS)
Soulopoulos, N.; Kerl, J.; Sponfeldner, T.; Beyrau, F.; Hardalupas, Y.; Taylor, A. M. K. P.; Vassilicos, J. C.
2013-12-01
A space-filling, low blockage fractal grid is used as a novel turbulence generator in a premixed turbulent flame stabilized by a rod. The study compares the flame behaviour with a fractal grid to the behaviour when a standard square mesh grid with the same effective mesh size and solidity as the fractal grid is used. The isothermal gas flow turbulence characteristics, including mean flow velocity and rms of velocity fluctuations and Taylor length, were evaluated from hot-wire measurements. The behaviour of the flames was assessed with direct chemiluminescence emission from the flame and high-speed OH-laser-induced fluorescence. The characteristics of the two flames are considered in terms of turbulent flame thickness, local flame curvature and turbulent flame speed. It is found that, for the same flow rate and stoichiometry and at the same distance downstream of the location of the grid, fractal-grid-generated turbulence leads to a more turbulent flame with enhanced burning rate and increased flame surface area.
Fractal analysis of fracture surfaces in ceramic materials
Wasen, J.; Heier, E.; Hansson, T.
1998-02-13
Fracture processes in metallic and ceramic systems do generally have, on the microscale, a stochastic (irregular) character resulting in an irregular fracture surface geometry. This irregularity of the fracture surface reflects the multiplicity of glide and cleavage planes as well as the variations in grain orientations in polycrystalline materials. Considerable amounts of experimental data from fracture surfaces demonstrate this kind of fracture behavior in a clear manner. This fact has inspired several investigators in the literature to apply fractal analysis on fracture surface in order to be able to interpret the fracture surface geometry in microstructural terms. Further, more or less successful attempts to connect the fractal properties of the fracture surface to the mechanical behavior of materials have been made. The purpose of this paper is to demonstrate relations between geometrical parameters (the fractal dimension and the angular properties) of fracture surfaces in various ceramic materials and to study the relation between the fractal properties of the fracture surface and the fracture toughness of the materials.
The N-Simplex and Its Generalizations towards Fractals
ERIC Educational Resources Information Center
Kosi-Ulbl, Irena; Pagon, Dusan
2002-01-01
Nature is full of different crystals and many of them have shapes of regular geometric objects. Those in which the fractal structure of a geometric object can be recognized are especially unusual. In this paper a generalization of one of these shapes is described: a formation, based on an n-dimensional simplex. The construction of an n-dimensional…
The Analysis of Leaf Shape Using Fractal Geometry.
ERIC Educational Resources Information Center
Hartvigsen, Gregg
2000-01-01
Describes ways to examine leaf structure and shape using fractal geometry. Students can test hypotheses using the leaves of replicated plants to look for non-linear trends in leaf shape along the stems of plants, across species, and under different environmental growth conditions. (SAH)
Fractal Explorations in Secondary Mathematics, Science, and Computer Science.
ERIC Educational Resources Information Center
Egnatoff, William J.
1991-01-01
Fractal geometry is introduced through examples of computational exploration of coastlines, self-similar curves, random walks, and population growth. These explorations, which include the construction of algorithms and the subsequent development and application of simple computer programs, lend themselves to self-directed study and advanced…
Fractal analysis of microvascular networks in malignant brain tumors.
Di Ieva, Antonio
2012-01-01
Brain tumors are characterized by a microvascular network which differs from normal brain vascularity. Different tumors show individual angiogenic patterns. Microvascular heterogeneity can also be observed within a neoplastic histotype. It has been shown that quantification of neoplastic microvascular patterns could be used in combination with the histological grade for tumor characterization and to refine clinical prognoses, even if no objective parameters have yet been validated. To overcome the limits of the Euclidean approach, we employ fractal geometry to analyze the geometric complexity underlying the microangioarchitectural networks in brain tumors. We have developed a computer-aided fractal-based analysis for the quantification of the microvascular patterns in histological specimens and ultra-high-field (7-Tesla) magnetic resonance images. We demonstrate that the fractal parameters are valid estimators of microvascular geometrical complexity. Furthermore, our analysis allows us to demonstrate the high geometrical variability underlying the angioarchitecture of glioblastoma multiforme and to differentiate low-grade from malignant tumors in histological specimens and radiological images. Based on the results of this study, we speculate the existence of a gradient in the geometrical complexity of microvascular networks from those in the normal brain to those in malignant brain tumors. Here, we summarize a new methodology for the application of fractal analysis to the study of the microangioarchitecture of brain tumors; we further suggest this approach as a tool for quantifying and categorizing different neoplastic microvascular patterns and as a potential morphometric biomarker for use in clinical practice.
Spin correlations in percolating networks with fractal geometry
Ikeda, H.; Iwasa, K.; Fernandez-Baca, J.A.; Nicklow, R.M.
1994-07-28
Using neutron scattering techniques, the authors investigated the magnetic correlations in diluted antiferromagnets close to the percolation threshold in which the magnetic connectivity takes a fractal form. Recent experimental results concerning the self-similarity of the magnetic order, and magnetic excitations in two-dimensional Ising and three-dimensional Heisenberg antiferromagnets are presented.
Fractal geometry impact on nuclear relaxation in irregular pores.
Sapoval, B; Russ, S; Petit, D; Korb, J P
1996-01-01
We apply a fractal description of pore surface irregularity to study the nuclear relaxation of a confined liquid. From the introduction of a length characteristic of diffusive and surface relaxing properties we describe three different relaxation regimes. These regimes show that the nuclear relaxation can be drastically modified by pore surface irregularity.
A user-friendly modified pore-solid fractal model
NASA Astrophysics Data System (ADS)
Ding, Dian-Yuan; Zhao, Ying; Feng, Hao; Si, Bing-Cheng; Hill, Robert Lee
2016-12-01
The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined.
Application of fractal theory in analysis of human electroencephalographic signals.
Paramanathan, P; Uthayakumar, R
2008-03-01
In medical discipline, complexity measure is focused on the analysis of nonlinear patterns in processing waveform signals. The complexity measure of such waveform signals is well performed by fractal dimension technique, which is an index for measuring the complexity of an object. Its applications are found in diverse fields like medical, image and signal processing. Several algorithms have been suggested to compute the fractal dimension of waveforms. We have evaluated the performance of the two famous algorithms namely Higuchi and Katz. They contain some problems of determining the initial and final length of scaling factors and their performance with electroencephalogram (EEG) signals did not give better results. In this paper, fractal dimension is proposed as an effective tool for analyzing and measuring the complexity of nonlinear human EEG signals. We have developed an algorithm based on size measure relationship (SMR) method. The SMR algorithm can be used to detect the brain disorders and it locates the affected brain portions by analyzing the behavior of signals. The efficiency of the algorithm to locate the critical brain sites (recurrent seizure portion) is compared to other fractal dimension algorithms. The K-means clustering algorithm is used for grouping of electrode positions.
Sandwich type plasmonic platform for MEF using silver fractals.
Raut, Sangram L; Rich, Ryan; Shtoyko, Tanya; Bora, Ilkay; Laursen, Bo W; Sørensen, Thomas Just; Borejdo, Julian; Gryczynski, Zygmunt; Gryczynski, Ignacy
2015-11-14
In this report, we describe a plasmonic platform with silver fractals for metal enhanced fluorescence (MEF) measurements. When a dye containing surface was brought into contact with silver fractals, a significantly enhanced fluorescence signal from the dye was observed. Fluorescence enhancement was studied with the N-methyl-azadioxatriangulenium chloride salt (Me-ADOTA·Cl) in PVA films made from 0.2% PVA (w/v) solution spin-coated on a clean glass coverslip. The Plasmonic Platforms (PP) were assembled by pressing together silver fractals on one glass slide and a separate glass coverslip spin-coated with a uniform Me-ADOTA·Cl in PVA film. In addition, we also tested ADOTA labeled human serum albumin (HSA) deposited on a glass slide for potential PP bioassay applications. Using the new PP, we could achieve more than a 20-fold fluorescence enhancement (bright spots) accompanied by a decrease in the fluorescence lifetime. The experimental results were used to calculate the extinction (excitation) enhancement factor (GA) and fluorescence radiative rate enhancements factor (GF). No change in emission spectrum was observed for a dye with or without contact with fractals. Our studies indicate that this type of PP can be a convenient approach for constructing assays utilizing metal enhanced fluorescence (MEF) without the need for depositing the material directly on metal structures platforms.
Fractals of graphene quantum dots in photoluminescence of shungite
NASA Astrophysics Data System (ADS)
Razbirin, B. S.; Rozhkova, N. N.; Sheka, E. F.; Nelson, D. K.; Starukhin, A. N.
2014-05-01
Viewing shungite as loosely packed fractal nets of graphene-based (reduced graphene oxide, rGO) quantum dots (GQDs), we consider photoluminescence of the latter as a convincing proof of the structural concept as well as of the GQD attribution to individual rGO fragments. We study emission from shungite GQDs for colloidal dispersions in water, carbon tetrachloride, and toluene at both room and low temperatures. As expected, the photoluminescence of the GQD aqueous dispersions is quite similar to that of synthetic GQDs of the rGO origin. The morphological study of shungite dispersions shows a steady trend of GQDs to form fractals and to drastically change the colloid fractal structure caused by the solvent exchange. Spectral study reveals a dual character of the emitting centers: individual GQDs are responsible for the spectra position while the fractal structure of GQD colloids ensures high broadening of the spectra due to structural inhomogeneity, thus causing a peculiar dependence of the photoluminescence spectra on the excitation wavelength. For the first time, photoluminescence spectra of individual GQDs were observed in frozen toluene dispersions, which paves the way for a theoretical treatment of the GQD photonics.
Multiscale Fractal Characterization of Hierarchical Heterogeneity in Sandstone Reservoirs
NASA Astrophysics Data System (ADS)
Liu, Yanfeng; Liu, Yuetian; Sun, Lu; Liu, Jian
2016-07-01
Heterogeneities affecting reservoirs often develop at different scales. Previous studies have described these heterogeneities using different parameters depending on their size, and there is no one comprehensive method of reservoir evaluation that considers every scale. This paper introduces a multiscale fractal approach to quantify consistently the hierarchical heterogeneities of sandstone reservoirs. Materials taken from typical depositional pattern and aerial photography are used to represent three main types of sandstone reservoir: turbidite, braided, and meandering river system. Subsequent multiscale fractal dimension analysis using the Bouligand-Minkowski method characterizes well the hierarchical heterogeneity of the sandstone reservoirs. The multiscale fractal dimension provides a curve function that describes the heterogeneity at different scales. The heterogeneity of a reservoir’s internal structure decreases as the observational scale increases. The shape of a deposit’s facies is vital for quantitative determination of the sedimentation type, and thus enhanced oil recovery. Characterization of hierarchical heterogeneity by multiscale fractal dimension can assist reservoir evaluation, geological modeling, and even the design of well patterns.
[Modeling continuous scaling of NDVI based on fractal theory].
Luan, Hai-Jun; Tian, Qing-Jiu; Yu, Tao; Hu, Xin-Li; Huang, Yan; Du, Ling-Tong; Zhao, Li-Min; Wei, Xi; Han, Jie; Zhang, Zhou-Wei; Li, Shao-Peng
2013-07-01
Scale effect was one of the very important scientific problems of remote sensing. The scale effect of quantitative remote sensing can be used to study retrievals' relationship between different-resolution images, and its research became an effective way to confront the challenges, such as validation of quantitative remote sensing products et al. Traditional up-scaling methods cannot describe scale changing features of retrievals on entire series of scales; meanwhile, they are faced with serious parameters correction issues because of imaging parameters' variation of different sensors, such as geometrical correction, spectral correction, etc. Utilizing single sensor image, fractal methodology was utilized to solve these problems. Taking NDVI (computed by land surface radiance) as example and based on Enhanced Thematic Mapper Plus (ETM+) image, a scheme was proposed to model continuous scaling of retrievals. Then the experimental results indicated that: (a) For NDVI, scale effect existed, and it could be described by fractal model of continuous scaling; (2) The fractal method was suitable for validation of NDVI. All of these proved that fractal was an effective methodology of studying scaling of quantitative remote sensing.
Partially Blended Constrained Rational Cubic Trigonometric Fractal Interpolation Surfaces
NASA Astrophysics Data System (ADS)
Chand, A. K. B.; Tyada, K. R.
2016-08-01
Fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions pi,j(θ) qi,j(θ), where pi,j(θ) and qi,j(θ) are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane Π, and consequently the proposed partially blended RCTFIS lies above the plane Π. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.
Fractal mechanisms of light scattering in biological tissue and cells
NASA Astrophysics Data System (ADS)
Xu, M.; Alfano, R. R.
2005-11-01
We use fractal continuous random media to model visible and near-infrared light scattering by biological tissue and cell suspensions. The power law of the reduced scattering coefficient, the anisotropy factor of scattering, and the phase function are derived with good agreement with experimental results. Implications for spectroscopic tissue diagnosis are discussed.
The Fractal Dimension of the ρ Ophiucus Molecular Cloud Complex
NASA Astrophysics Data System (ADS)
Lee, Yongung; Yi, Di; Kim, Y. S.; Jung, J. H.; Kang, H. W.; Lee, C. H.; Yim, I. S.; Kim, H. G.
2016-12-01
We estimate the fractal dimension of the ρ Ophiuchus Molecular Cloud Complex, associated with star forming regions. We selected a cube (v, l, b) database, obtained with J=1-0 transition lines of \\coand tco at a resolution of 22'' using a multibeam receiver system on the 14-m telescope of the Five College Radio Astronomy Observatory. Using a code developed within IRAF, we identified slice-clouds with two threshold temperatures to estimate the fractal dimension. With threshold temperatures of 2.25 K (3σ) and 3.75 K (5σ), the fractal dimension of the target cloud is estimated to be D = 1.52-1.54, where P ∝ A^{D/2} , which is larger than previous results. We suggest that the sampling rate (spatial resolution) of observed data must be an important parameter when estimating the fractal dimension, and that narrower or wider dispersion around an arbitrary fit line and the intercepts at NP = 100 should be checked whether they relate to rms noise level or characteristic structure of the target cloud. This issue could be investigated by analysing several high resolution databases with different quality (low or moderate sensitivity).
Routes to fractality and entropy in Liesegang systems
Kalash, Leen; Sultan, Rabih
2014-06-01
Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF{sub 2} and PbI{sub 2} Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF{sub 2} and PbI{sub 2} Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.
Observing the Twinkling Fractal Nature of the Glass Transition
NASA Astrophysics Data System (ADS)
Stanzione, Joseph, III; Wool, Richard; Strawhecker, Kenneth
2010-03-01
The main idea underlying the Twinkling Fractal Theory (TFT) of the glass transition is the development of dynamic percolating solid fractal structures near Tg, which are in dynamic equilibrium with the surrounding liquid. Solid and liquid clusters interchange at a frequency φTF, which is controlled by the population of intermolecular oscillators in excited energy levels in accord with the Orbach vibrational density of states for a particular fractal cluster g(φ) ˜ φ^df-1, where the fracton dimension df = 4/3. To an observer, these clusters would appear to be ``twinkling.'' A time-lapse tapping-mode atomic force microscopy (AFM) technique has been developed in order to experimentally confirm such phenomena. The twinkling behavior of amorphous, atactic polystyrene with MW = 194,000 g/mol, PDI = 1.07 (GPC) and Tg = 375 K (DSC-heating rate of 3 K/min) has been captured above (383 K), below (358 K), and well below (298 K) its Tg. Two-dimensional space images reveal fractal dimensions consistent with the TFT. The twinkling behavior was analyzed using a statistical autocorrelation function in conjunction with the apparent stretched exponential Kohlrausch-Williams-Watts relaxation function.
Fractal and geostatistical methods for modeling of a fracture network
Chiles, J.P.
1988-08-01
The modeling of fracture networks is useful for fluid flow and rock mechanics studies. About 6600 fracture traces were recorded on drifts of a uranium mine in a granite massif. The traces have an extension of 0.20-20 m. The network was studied by fractal and by geostatistical methods but can be considered neither as a fractal with a constant dimension nor a set of purely randomly located fractures. Two kinds of generalization of conventional models can still provide more flexibility for the characterization of the network: (a) a nonscaling fractal model with variable similarity dimension (for a 2-D network of traces, the dimension varying from 2 for the 10-m scale to 1 for the centimeter scale, (b) a parent-daughter model with a regionalized density; the geostatistical study allows a 3-D model to be established where: fractures are assumed to be discs; fractures are grouped in clusters or swarms; and fracturation density is regionalized (with two ranges at about 30 and 300 m). The fractal model is easy to fit and to simulate along a line, but 2-D and 3-D simulations are more difficult. The geostatistical model is more complex, but easy to simulate, even in 3-D.
Sandwich type plasmonic platform for MEF using silver fractals
NASA Astrophysics Data System (ADS)
Raut, Sangram L.; Rich, Ryan; Shtoyko, Tanya; Bora, Ilkay; Laursen, Bo W.; Sørensen, Thomas Just; Borejdo, Julian; Gryczynski, Zygmunt; Gryczynski, Ignacy
2015-10-01
In this report, we describe a plasmonic platform with silver fractals for metal enhanced fluorescence (MEF) measurements. When a dye containing surface was brought into contact with silver fractals, a significantly enhanced fluorescence signal from the dye was observed. Fluorescence enhancement was studied with the N-methyl-azadioxatriangulenium chloride salt (Me-ADOTA.Cl) in PVA films made from 0.2% PVA (w/v) solution spin-coated on a clean glass coverslip. The Plasmonic Platforms (PP) were assembled by pressing together silver fractals on one glass slide and a separate glass coverslip spin-coated with a uniform Me-ADOTA.Cl in PVA film. In addition, we also tested ADOTA labeled human serum albumin (HSA) deposited on a glass slide for potential PP bioassay applications. Using the new PP, we could achieve more than a 20-fold fluorescence enhancement (bright spots) accompanied by a decrease in the fluorescence lifetime. The experimental results were used to calculate the extinction (excitation) enhancement factor (GA) and fluorescence radiative rate enhancements factor (GF). No change in emission spectrum was observed for a dye with or without contact with fractals. Our studies indicate that this type of PP can be a convenient approach for constructing assays utilizing metal enhanced fluorescence (MEF) without the need for depositing the material directly on metal structures platforms.
Fractal texture analysis of the healing process after bone loss.
Borowska, Marta; Szarmach, Janusz; Oczeretko, Edward
2015-12-01
Radiological assessment of treatment effectiveness of guided bone regeneration (GBR) method in postresectal and postcystal bone loss cases, observed for one year. Group of 25 patients (17 females and 8 males) who underwent root resection with cystectomy were evaluated. The following combination therapy of intraosseous deficits was used, consisting of bone augmentation with xenogenic material together with covering regenerative membranes and tight wound closure. The bone regeneration process was estimated, comparing the images taken on the day of the surgery and 12 months later, by means of Kodak RVG 6100 digital radiography set. The interpretation of the radiovisiographic image depends on the evaluation ability of the eye looking at it, which leaves a large margin of uncertainty. So, several texture analysis techniques were developed and used sequentially on the radiographic image. For each method, the results were the mean from the 25 images. These methods compute the fractal dimension (D), each one having its own theoretic basis. We used five techniques for calculating fractal dimension: power spectral density method, triangular prism surface area method, blanket method, intensity difference scaling method and variogram analysis. Our study showed a decrease of fractal dimension during the healing process after bone loss. We also found evidence that various methods of calculating fractal dimension give different results. During the healing process after bone loss, the surfaces of radiographic images became smooth. The result obtained show that our findings may be of great importance for diagnostic purpose.
Quantitating the subtleties of microglial morphology with fractal analysis
Karperien, Audrey; Ahammer, Helmut; Jelinek, Herbert F.
2013-01-01
It is well established that microglial form and function are inextricably linked. In recent years, the traditional view that microglial form ranges between “ramified resting” and “activated amoeboid” has been emphasized through advancing imaging techniques that point to microglial form being highly dynamic even within the currently accepted morphological categories. Moreover, microglia adopt meaningful intermediate forms between categories, with considerable crossover in function and varying morphologies as they cycle, migrate, wave, phagocytose, and extend and retract fine and gross processes. From a quantitative perspective, it is problematic to measure such variability using traditional methods, but one way of quantitating such detail is through fractal analysis. The techniques of fractal analysis have been used for quantitating microglial morphology, to categorize gross differences but also to differentiate subtle differences (e.g., amongst ramified cells). Multifractal analysis in particular is one technique of fractal analysis that may be useful for identifying intermediate forms. Here we review current trends and methods of fractal analysis, focusing on box counting analysis, including lacunarity and multifractal analysis, as applied to microglial morphology. PMID:23386810
Fractal Reference Signals in Pulse-Width Modulation
NASA Technical Reports Server (NTRS)
Lurie, Boris; Lurie, Helen
2005-01-01
A report proposes the use of waveforms having fractal shapes reminiscent of sawteeth (in contradistinction to conventional regular sawtooth waveforms) as reference signals for pulse-width modulation in control systems for thrusters of spacecraft flying in formation. Fractal reference signals may also be attractive in some terrestrial control systems - especially those in which pulse-width modulation is used for precise control of electric motors. The report asserts that the use of fractal reference signals would enable the synchronous control of several variables of a spacecraft formation, such that consumption of propellant would be minimized, intervals between thruster firings would be long (as preferred for performing scientific observations), and delays in controlling large-thrust maneuvers for retargeting would be minimized. The report further asserts that whereas different controllers would be needed for different modes of operation if conventional pulsewidth modulation were used, the use of fractal reference signals would enable the same controller to function nearly optimally in all regimes of operation, so that only this one controller would be needed.
The fractal organization of ultradian rhythms in avian behavior.
Guzmán, Diego A; Flesia, Ana G; Aon, Miguel A; Pellegrini, Stefania; Marin, Raúl H; Kembro, Jackelyn M
2017-04-06
Living systems exhibit non-randomly organized biochemical, physiological, and behavioral processes that follow distinctive patterns. In particular, animal behavior displays both fractal dynamics and periodic rhythms yet the relationship between these two dynamic regimens remain unexplored. Herein we studied locomotor time series of visually isolated Japanese quails sampled every 0.5 s during 6.5 days (>10(6) data points). These high-resolution, week-long, time series enabled simultaneous evaluation of ultradian rhythms as well as fractal organization according to six different analytical methods that included Power Spectrum, Enright, Empirical Mode Decomposition, Wavelet, and Detrended Fluctuation analyses. Time series analyses showed that all birds exhibit circadian rhythms. Although interindividual differences were detected, animals presented ultradian behavioral rhythms of 12, 8, 6, 4.8, 4 h and/or lower and, irrespective of visual isolation, synchronization between these ultradian rhythms was observed. Moreover, all birds presented similar overall fractal dynamics (for scales ∼30 s to >4.4 h). This is the first demonstration that avian behavior presents fractal organization that predominates at shorter time scales and coexists with synchronized ultradian rhythms. This chronobiological pattern is advantageous for keeping the organism's endogenous rhythms in phase with internal and environmental periodicities, notably the feeding, light-dark and sleep-wake cycles.
Fractal-based image texture analysis of trabecular bone architecture.
Jiang, C; Pitt, R E; Bertram, J E; Aneshansley, D J
1999-07-01
Fractal-based image analysis methods are investigated to extract textural features related to the anisotropic structure of trabecular bone from the X-ray images of cubic bone specimens. Three methods are used to quantify image textural features: power spectrum, Minkowski dimension and mean intercept length. The global fractal dimension is used to describe the overall roughness of the image texture. The anisotropic features formed by the trabeculae are characterised by a fabric ellipse, whose orientation and eccentricity reflect the textural anisotropy of the image. Tests of these methods with synthetic images of known fractal dimension show that the Minkowski dimension provides a more accurate and consistent estimation of global fractal dimension. Tests on bone x-ray (eccentricity range 0.25-0.80) images indicate that the Minkowski dimension is more sensitive to the changes in textural orientation. The results suggest that the Minkowski dimension is a better measure for characterising trabecular bone anisotropy in the x-ray images of thick specimens.
[Numerical calculation of coagulation kinetics incorporating fractal theory].
Jin, Peng-kang; Jing, Min-na; Wang, Xiao-chang
2008-08-01
Based on the Smoluchowski equation, a kinetic model was formulated by introducing the fractal dimension. In the kinetic model, fractal dimension at different time is calculated by considering of the void and primary particles contained in the flocs. Using the kinetic model, the coagulation kinetics was calculated by the method of finite difference. The calculation results showed that the characteristics of the structure and collision efficiency play an important role in particle size distribution. The higher of the fractal dimension and the collision efficiency, the broader of the particle size distribution will be obtained, which indicated the flocs with large size were formed. The results also revealed a tendency of decrease in the fractal dimension with the increase of floc size, which is resulted from the unproportionate growth between the floc size and the number of the primary particles contained in the flocs. The validity of the calculation was proved by a series of experiments using aluminum sulfate as coagulant for the flocculation of humic substances.
Scaling relations in the diffusive infiltration in fractals.
Aarão Reis, F D A
2016-11-01
In a recent work on fluid infiltration in a Hele-Shaw cell with the pore-block geometry of Sierpinski carpets (SCs), the area filled by the invading fluid was shown to scale as F∼t^{n}, with n<1/2, thus providing a macroscopic realization of anomalous diffusion [Filipovitch et al., Water Resour. Res. 52, 5167 (2016)WRERAQ0043-139710.1002/2016WR018667]. The results agree with simulations of a diffusion equation with constant pressure at one of the borders of those fractals, but the exponent n is very different from the anomalous exponent ν=1/D_{W} of single-particle diffusion in the same fractals (D_{W} is the random-walk dimension). Here we use a scaling approach to show that those exponents are related as n=ν(D_{F}-D_{B}), where D_{F} and D_{B} are the fractal dimensions of the bulk and the border from which diffusing particles come, respectively. This relation is supported by accurate numerical estimates in two SCs and in two generalized Menger sponges (MSs), in which we performed simulations of single-particle random walks (RWs) with a rigid impermeable border and of a diffusive infiltration model in which that border is permanently filled with diffusing particles. This study includes one MS whose external border is also fractal. The exponent relation is also consistent with the recent simulational and experimental results on fluid infiltration in SCs, and explains the approximate quadratic dependence of n on D_{F} in these fractals. We also show that the mean-square displacement of single-particle RWs has log-periodic oscillations, whose periods are similar for fractals with the same scaling factor in the generator (even with different embedding dimensions), which is consistent with the discrete scale invariance scenario. The roughness of a diffusion front defined in the infiltration problem also shows this type of oscillation, which is enhanced in fractals with narrow channels between large lacunas.
Broadband enhanced graphene photodetector with fractal metasurface (Conference Presentation)
NASA Astrophysics Data System (ADS)
Wang, Di; Fang, Jieran; DeVault, Clayton T.; Chung, Ting-Fung; Chen, Yong P.; Boltasseva, Alexandra; Shalaev, Vladimir M.; Kildishev, Alexander V.
2016-09-01
Graphene has been demonstrated to be a promising photo-detection material because of its ultra-broadband absorption, compatibility with CMOS technology, and dynamic tunability. There are multiple known photo-detection mechanisms in graphene, among which the photovoltaic effect has the fastest response time thus is the prioritized candidate for ultrafast photodetector. There have been numerous efforts to enhance the intrinsically low sensitivity in graphene photovoltaic detectors using metallic plasmonic structures, but such plasmonic enhancements are mostly narrowband and polarization dependent. In this work, we propose a gold Cayley-tree fractal metasurface design that has a multi-band resonance, to realize broadband and polarization-insensitive plasmonic enhancement in graphene photovoltaic detectors. When illuminated with visible light, the fractal metasurface exhibits multiple hotspots at the metal-graphene interface, where the electric field of the incident electromagnetic wave is enhanced and contributes to generating excess electron-hole pairs in graphene. The large metal-graphene interface length in the fractal metasurface also helps to harvest at a higher efficiency the electron-hole pairs by built-in electric field due to metal-graphene potential gradient. To demonstrate the concept, we carried out experiment using Ar-Kr CW laser, an optical chopper, and lock-in amplifier. We obtained experimentally an almost constant ten-fold enhancement of photocurrent generated on the fractal metasurface compared to that generated on the plain gold-graphene edge, at all tested wavelengths (488 nm, 514 nm, 568 nm, and 647 nm). We also observed an unchanged photoresponse with respect to incident light polarization angles, which is a result of the highly symmetric geometry of the fractal metasurface.
Hermann, Philipp; Mrkvička, Tomáš; Mattfeldt, Torsten; Minárová, Mária; Helisová, Kateřina; Nicolis, Orietta; Wartner, Fabian; Stehlík, Milan
2015-08-15
Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.
Multi-fractal scaling comparison of the Air Temperature and the Surface Temperature over China
NASA Astrophysics Data System (ADS)
Jiang, Lei; Zhang, Jiping; Liu, Xinwei; Li, Fei
2016-11-01
The spatial and temporal multi-scaling behaviors between the daily Air Temperature (AT) and the Surface Temperature (ST) over China are compared in about 60-yr observations by Multi-fractal Detrended Fluctuation Analysis (MF-DFA) method. The different fractal phenomena and diversity features in the geographic distribution are found for the AT and ST series using MF-DFA. There are more multi-fractal features for the AT records but less for ST. The respective geographic sites show important scaling differences when compared to the multi-fractal signatures of AT with ST. An interval threshold for 95% confidence level is obtained by shuffling the AT records and the ST records. For the AT records, 93% of all observed stations shows the strong multi-fractal behaviors. In addition, the multi-fractal characteristics decrease with increasing latitude in South China and are obviously strong along the coast. The multi-fractal behaviors of the AT records between the Yangtze River and Yellow River basin and in most regions of Northwest China seem to be weak and not significant, even single mono-fractal features. However, for the ST records, the geographical distributions of multi-fractal phenomenon seem to be in disorder which account for 81% of the stations. The weak multi-fractal behaviors of the ST records are concentrated in North China, most regions of Northeast China.
The Use of Fractals for the Study of the Psychology of Perception:
NASA Astrophysics Data System (ADS)
Mitina, Olga V.; Abraham, Frederick David
The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber-Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
NASA Astrophysics Data System (ADS)
Chakrabarty, R. K.; Moosmuller, H.; Garro, M. A.; Garro, B. A.; Chancellor, S.; Herald, C. M.
2010-12-01
The three dimensional (3-d) fractal dimension D is considered the key morphological parameter for black carbon (BC) aggregates as it influences BC optical, physical, and chemical properties. Using electron microscopy and image processing techniques, there are several approaches for extracting D and other structural parameters of BC aggregates, which are commonly unknown, from their two dimensional microscopy images. The nested squares method (NSM), the perimeter grid method (PGM), and the ensemble method (EM) have found wide use as image analysis techniques for determination of D for individual and ensemble BC aggregates. However, so far no study has quantified the errors involved in the measured D by these three analysis techniques. In this talk, we highlight the errors associated with using these three fractal analysis techniques for the determination of D and put forth a recommendation on the most reliable analysis technique for use in future research. The talk will conclude by discussing the results obtained by (i) applying the recommended image analysis technique to real-world, flame-generated BC aggregates for calculating their D and other morphological properties; (ii) further making use of the calculated aggregate morphological parameters to theoretically determine BC optical properties using three well-known theories, namely Rayleigh-Debye-Gans (RDG) approximation, volume-equivalent Mie theory, and integral equation formulation for scattering (IEFS); and (iii) comparison of the theoretically determined BC optical properties with experimental data.
Microtopographic Inspection and Fractal Analysis of Skin Neoplasia
NASA Astrophysics Data System (ADS)
Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón
2008-04-01
Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh
NASA Astrophysics Data System (ADS)
Maslovskaya, A. G.; Barabash, T. K.
2017-01-01
The article presents some results of fractal analysis of ferroelectric domain structure images visualized with scanning electron microscope (SEM) techniques. The fractal and multifractal characteristics were estimated to demonstrate self-similar organization of ferroelectric domain structure registered with static and dynamic contrast modes of SEM. Fractal methods as sensitive analytical tools were used to indicate degree of domain structure and domain boundary imperfections. The electron irradiation-induced erosion effect of ferroelectric domain boundaries in electron beam-stimulated polarization current mode of SEM is characterized by considerable raising of fractal dimension. For dynamic contrast mode of SEM there was revealed that complication of domain structure during its dynamics is specified by increase in fractal dimension of images and slight raising of boundary fractal dimension.
Biophilic fractals and the visual journey of organic screen-savers.
Taylor, R P; Sprott, J C
2008-01-01
Computers have led to the remarkable popularity of mathematically-generated fractal patterns. Fractals have also assumed a rapidly expanding role as an art form. Due to their growing impact on cultures around the world and their prevalence in nature, fractals constitute a central feature of our daily visual experiences throughout our lives. This intimate association raises a crucial question - does exposure to fractals have a positive impact on our mental and physical condition? This question raises the opportunity for readers of this journal to have some visual fun. Each year a different nonlinear inspired artist is featured on the front cover of the journal. This year, Scott Draves's fractal art works continues this tradition. In May 2007, we selected twenty of Draves's artworks and invited readers to vote for their favorites from this selection. The most popular images will feature on the front covers this year. In this article, we discuss fractal aesthetics and Draves's remarkable images.
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
Mehan, Sumit Kumar, Sugam Aswal, V. K.
2014-04-24
Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.
The Three-Point Sinuosity Method for Calculating the Fractal Dimension of Machined Surface Profile
NASA Astrophysics Data System (ADS)
Zhou, Yuankai; Li, Yan; Zhu, Hua; Zuo, Xue; Yang, Jianhua
2015-04-01
The three-point sinuosity (TPS) method is proposed to calculate the fractal dimension of surface profile accurately. In this method, a new measure, TPS is defined to present the structural complexity of fractal curves, and has been proved to follow the power law. Thus, the fractal dimension can be calculated through the slope of the fitted line in the log-log plot. The Weierstrass-Mandelbrot (W-M) fractal curves, as well as the real surface profiles obtained by grinding, sand blasting and turning, are used to validate the effectiveness of the proposed method. The calculation values are compared to those obtained from root-mean-square (RMS) method, box-counting (BC) method and variation method. The results show that the TPS method has the widest scaling region, the least fit error and the highest accuracy among the methods examined, which demonstrates that the fractal characteristics of the fractal curves can be well revealed by the proposed method.
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-01-01
Chaos and fractal dimension are emerging modalities for the research of electroencephalogram (EEG) signal processing. The capability of measuring non-linear characteristics of the fractal dimension enables new methodologies to identify distinct brain activities. Recent studies on the topic focus on utilizing various types of fractals as features in order to design better brain state classification system. However, we have little insight about the EEG signals projected in fractal dimension. In this paper, we investigate the relationship between the non-linear characteristics of ongoing EEG signals and event-related desynchronization (ERD) during motor imagery. We observed a considerable synchronization between ERD and fractal dimension. This finding suggests further usage of chaos and fractal theory in investigating brain activities.
Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2014-09-01
The correlation between the void ratio of swelled montmorillonite and the vertical overburden pressure can be expressed as {e}{ m} = Kp{ s}{D{ s}-3}. The surface fractal dimension Ds of five bentonites were estimated from the swelling deformation tests according to this fractal correlation. The reliability of surface fractal dimension obtained from the swelling deformation test was confirmed by nitrogen adsorption test, with identical values of surface fractal dimension obtained from both tests. The surface fractal dimension can also be used to estimate the swelling deformation of bentonite, after calculating the swelling coefficient K from the parameters of diffuse double layer (DDL) model in the osmotic swelling phase. Comparison of the model predictions with a number of experimental results on swelling deformation of both Na dominant and Ca dominant bentonites suggests that the surface fractal model works excellent in the cases tested.
Temporal fractals in seabird foraging behaviour: diving through the scales of time
NASA Astrophysics Data System (ADS)
Macintosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan
2013-05-01
Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Lévy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (216 data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments.
Analysis of Fractional Flow for Transient Two-Phase Flow in Fractal Porous Medium
NASA Astrophysics Data System (ADS)
Lu, Ting; Duan, Yonggang; Fang, Quantang; Dai, Xiaolu; Wu, Jinsui
2016-03-01
Prediction of fractional flow in fractal porous medium is important for reservoir engineering and chemical engineering as well as hydrology. A physical conceptual fractional flow model of transient two-phase flow is developed in fractal porous medium based on the fractal characteristics of pore-size distribution and on the approximation that porous medium consist of a bundle of tortuous capillaries. The analytical expression for fractional flow for wetting phase is presented, and the proposed expression is the function of structural parameters (such as tortuosity fractal dimension, pore fractal dimension, maximum and minimum diameters of capillaries) and fluid properties (such as contact angle, viscosity and interfacial tension) in fractal porous medium. The sensitive parameters that influence fractional flow and its derivative are formulated, and their impacts on fractional flow are discussed.
Fractals in the nervous system: conceptual implications for theoretical neuroscience.
Werner, Gerhard
2010-01-01
This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.
Fractal pattern formation in metallic ink sessile droplets
NASA Astrophysics Data System (ADS)
Hadj-Achour, Miloud; Brutin, David
2014-11-01
We report a fingering instability that occurs during the spreading and evaporation of a nanosuspension droplet. The patterns has a fractal structure similar to those reported by N. Shahidzadeh-Bonn et al. (2008) for salt crystallisation, during evaporation of saturated Na2SO4 on a hydrophilic surface. The fingering instability has been widely studied for both Newtonian and non-Newtonian fluids. However, we describe for the first time that a fingering instability is observed for the spreading of a nanosuspension sessile droplet. We demonstrate that in certain cases, the contact line evolves through different spreading regimes according to J. De Coninck et al. (2001) with an enhancement in the evaporation rate due the formation of the fractal patterns.