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)
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.…
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.
Fractal vector optical fields.
Pan, Yue; Gao, Xu-Zhen; Cai, Meng-Qiang; Zhang, Guan-Lin; Li, Yongnan; Tu, Chenghou; Wang, Hui-Tian
2016-07-15
We introduce the concept of a fractal, which provides an alternative approach for flexibly engineering the optical fields and their focal fields. We propose, design, and create a new family of optical fields-fractal vector optical fields, which build a bridge between the fractal and vector optical fields. The fractal vector optical fields have polarization states exhibiting fractal geometry, and may also involve the phase and/or amplitude simultaneously. The results reveal that the focal fields exhibit self-similarity, and the hierarchy of the fractal has the "weeding" role. The fractal can be used to engineer the focal field. PMID:27420485
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
ERIC Educational Resources Information Center
Barton, Ray
1990-01-01
Presented is an educational game called "The Chaos Game" which produces complicated fractal images. Two basic computer programs are included. The production of fractal images by the Sierpinski gasket and the Chaos Game programs is discussed. (CW)
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.…
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)
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.
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. PMID:24978815
Electromagnetic fields in fractal continua
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Mena, Baltasar; Patiño, Julián; Morales, Daniel
2013-04-01
Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum ΦD3⊂E3 with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space Fα accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed.
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)
Persistence intervals of fractals
NASA Astrophysics Data System (ADS)
Máté, Gabriell; Heermann, Dieter W.
2014-07-01
Objects and structures presenting fractal like behavior are abundant in the world surrounding us. Fractal theory provides a great deal of tools for the analysis of the scaling properties of these objects. We would like to contribute to the field by analyzing and applying a particular case of the theory behind the P.H. dimension, a concept introduced by MacPherson and Schweinhart, to seek an intuitive explanation for the relation of this dimension and the fractality of certain objects. The approach is based on recently elaborated computational topology methods and it proves to be very useful for investigating scaling hidden in dimensions lower than the “native” dimension in which the investigated object is embedded. We demonstrate the applicability of the method with two examples: the Sierpinski gasket-a traditional fractal-and a two dimensional object composed of short segments arranged according to a circular structure.
Fractal funcitons and multiwavelets
Massopust, P.R.
1997-04-01
This paper reviews how elements from the theory of fractal functions are employed to construct scaling vectors and multiwavelets. Emphasis is placed on the one-dimensional case, however extensions to IR{sup m} are indicated.
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)
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)
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.
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
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, i&tacute;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 (Muuḱil Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoiŕ models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mes&hacute; cells was designed and calibrated in the studied area. The statistically sound power law relations were
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.
Entanglement entropy on fractals
NASA Astrophysics Data System (ADS)
Faraji Astaneh, Amin
2016-03-01
We use the heat kernel method to calculate the entanglement entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cutoff parameter is (generally) a fractional number, which, indeed, is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log-periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the holographic calculation in a certain hyperscaling-violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyperscaling-violating theory with the spectral dimension of the fractal. We provide additional support by comparing the behavior of the thermal entropy in terms of the temperature, computed for two geometries, the fractal geometry and the hyperscaling-violating background.
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).
Skin Depth of Electromagnetic Wave through Fractal Crustal Rocks
NASA Astrophysics Data System (ADS)
Takahara, Kazutaka; Muto, Jun; Nagahama, Hiroyuki
Skin depth of electromagnetic (EM) wave depends on frequency of EM wave ν and electrical properties of rocks and minerals. Previous studies have theoretically assumed that the skin depth Lα(ν) can be expressed as a function of frequency ν by Lα(ν) ∝ ν -φ and φ = 1 at high frequency or φ = 1/2 at low frequency. Based on fractal theory of rocks, we point out that the frequency exponent φ reflects internal fractal structures (i.e., occupancy, distribution and connectivity) of dielectric/conductive matrices of rocks such as pores, cracks, grain boundaries, inclusions and various fluids. Laboratory measurements of dielectric constant and conductivity of granite and previous studies on various rocks as a function of frequency show that φ is an exponent ranging from 1/4 to 1. By extrapolation of the skin depth by laboratory measurements at a given frequency into at other frequencies, the skin depth with variation in φ becomes longer or shorter than that by previous studies. Moreover, at a given frequency, the skin depth decreases with increasing a fractal dimension of fracture systems (decreasing φ). Thus, the skin depth of EM wave through the crust for detecting seismo-EM radiations and through rock salt domes for detecting ultra-high energy neutrinos depends on fractal structures of dielectric/conductive matrices in heterogeneous crust.
Fractal dimensions of sinkholes
NASA Astrophysics Data System (ADS)
Reams, Max W.
1992-05-01
Sinkhole perimeters are probably fractals ( D=1.209-1.558) for sinkholes with areas larger than 10,000 m 2, based on area-perimeter plots of digitized data from karst surfaces developed on six geologic units in the United States. The sites in Florida, Kentucky, Indiana and Missouri were studied using maps with a scale of 1:24, 000. Size-number distributions of sinkhole perimeters and areas may also be fractal, although data for small sinkholes is needed for verification. Studies based on small-scale maps are needed to evaluate the number and roughness of small sinkhole populations.
FRACTAL DIMENSION 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
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)
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.…
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.
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.
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 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...
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!
NASA Astrophysics Data System (ADS)
Burdzy, Krzysztof; Hołyst, Robert; Pruski, Łukasz
2013-05-01
We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n×n with periodic boundary conditions. A fraction p⩽20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (nlnn)2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by Mγ where γ=-1.18 for p=0.1, decreasing to γ=-1.28 for p=0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a=1.56 for p=0.20 to a=1.42 for p=0.001 (for n=4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.
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.
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.
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.
Electromagnetism on anisotropic fractal media
NASA Astrophysics Data System (ADS)
Ostoja-Starzewski, Martin
2013-04-01
Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.
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.)
NASA Astrophysics Data System (ADS)
Brothers, Harlan J.
2015-03-01
Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.
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.
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.
Map of fluid flow in fractal porous medium into fractal continuum flow.
Balankin, Alexander S; Elizarraraz, Benjamin Espinoza
2012-05-01
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided. PMID:23004869
Map of fluid flow in fractal porous medium into fractal continuum flow
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Elizarraraz, Benjamin Espinoza
2012-05-01
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow ds is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.
NASA Astrophysics Data System (ADS)
Eliazar, Iddo; Klafter, Joseph
2008-09-01
The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results - extending them from affine to general nonlinear scaling schemes.
Fractality of light's darkness.
O'Holleran, Kevin; Dennis, Mark R; Flossmann, Florian; Padgett, Miles J
2008-02-01
Natural light fields are threaded by lines of darkness. For monochromatic light, the phenomenon is familiar in laser speckle, i.e., the black points that appear in the scattered light. These black points are optical vortices that extend as lines throughout the volume of the field. We establish by numerical simulations, supported by experiments, that these vortex lines have the fractal properties of a Brownian random walk. Approximately 73% of the lines percolate through the optical beam, the remainder forming closed loops. Our statistical results are similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior. PMID:18352372
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. PMID:16393505
Anomalous Diffusion in Fractal Globules
NASA Astrophysics Data System (ADS)
Tamm, M. V.; Nazarov, L. I.; Gavrilov, A. A.; Chertovich, A. V.
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X2(t )⟩˜tαF with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.
Fractal electronic devices: simulation and implementation.
Fairbanks, M S; McCarthy, D N; Scott, S A; Brown, S A; Taylor, R P
2011-09-01
Many natural structures have fractal geometries that exhibit useful functional properties. These properties, which exploit the recurrence of patterns at increasingly small scales, are often desirable in applications and, consequently, fractal geometry is increasingly employed in diverse technologies ranging from radio antennae to storm barriers. In this paper, we explore the application of fractal geometry to electrical devices. First, we lay the foundations for the implementation of fractal devices by considering diffusion-limited aggregation (DLA) of atomic clusters. Under appropriate growth conditions, atomic clusters of various elements form fractal patterns driven by DLA. We perform a fractal analysis of both simulated and physical devices to determine their spatial scaling properties and demonstrate their potential as fractal circuit elements. Finally, we simulate conduction through idealized and DLA fractal devices and show that their fractal scaling properties generate novel, nonlinear conduction properties in response to depletion by electrostatic gates. PMID:21841218
Fractal electronic devices: simulation and implementation
NASA Astrophysics Data System (ADS)
Fairbanks, M. S.; McCarthy, D. N.; Scott, S. A.; Brown, S. A.; Taylor, R. P.
2011-09-01
Many natural structures have fractal geometries that exhibit useful functional properties. These properties, which exploit the recurrence of patterns at increasingly small scales, are often desirable in applications and, consequently, fractal geometry is increasingly employed in diverse technologies ranging from radio antennae to storm barriers. In this paper, we explore the application of fractal geometry to electrical devices. First, we lay the foundations for the implementation of fractal devices by considering diffusion-limited aggregation (DLA) of atomic clusters. Under appropriate growth conditions, atomic clusters of various elements form fractal patterns driven by DLA. We perform a fractal analysis of both simulated and physical devices to determine their spatial scaling properties and demonstrate their potential as fractal circuit elements. Finally, we simulate conduction through idealized and DLA fractal devices and show that their fractal scaling properties generate novel, nonlinear conduction properties in response to depletion by electrostatic gates.
Fractal dynamics of bioconvective patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1991-01-01
Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.
Diffusion, Dispersion, and Uncertainty in Anisotropic Fractal Porous Media
NASA Astrophysics Data System (ADS)
Monnig, N. D.; Benson, D. A.
2007-12-01
Motivated by field measurements of aquifer hydraulic conductivity (K), recent techniques were developed to construct anisotropic fractal random fields, in which the scaling, or self-similarity parameter, varies with direction and is defined by a matrix. Ensemble numerical results are analyzed for solute transport through these 2-D "operator-scaling" fractional Brownian motion (fBm) ln(K) fields. Contrary to some analytic stochastic theories for monofractal K fields, the plume growth rates never exceed Mercado's (1967) purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance. Apparent super-stratified growth must be the result of other demonstrable factors, such as initial plume size. The addition of large local dispersion and diffusion does not significantly change the effective longitudinal dispersivity of the plumes. In the presence of significant local dispersion or diffusion, the concentration coefficient of variation CV={σc}/{\\langle c \\rangle} remains large at the leading edge of the plumes. This indicates that even with considerable mixing due to dispersion or diffusion, there is still substantial uncertainty in the leading edge of a plume moving in fractal porous media.
Small-angle scattering from fat fractals
NASA Astrophysics Data System (ADS)
Anitas, Eugen M.
2014-06-01
A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.
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 kinetics in drug release from finite fractal matrices
NASA Astrophysics Data System (ADS)
Kosmidis, Kosmas; Argyrakis, Panos; Macheras, Panos
2003-09-01
We have re-examined the random release of particles from fractal polymer matrices using Monte Carlo simulations, a problem originally studied by Bunde et al. [J. Chem. Phys. 83, 5909 (1985)]. A certain population of particles diffuses on a fractal structure, and as particles reach the boundaries of the structure they are removed from the system. We find that the number of particles that escape from the matrix as a function of time can be approximated by a Weibull (stretched exponential) function, similar to the case of release from Euclidean matrices. The earlier result that fractal release rates are described by power laws is correct only at the initial stage of the release, but it has to be modified if one is to describe in one picture the entire process for a finite system. These results pertain to the release of drugs, chemicals, agrochemicals, etc., from delivery systems.
The fractal aggregation of asphaltenes.
Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott
2013-07-16
This paper discusses time-resolved small-angle neutron scattering results that were used to investigate asphaltene structure and stability with and without a precipitant added in both crude oil and model oil. A novel approach was used to isolate the scattering from asphaltenes that are insoluble and in the process of aggregating from those that are soluble. It was found that both soluble and insoluble asphaltenes form fractal clusters in crude oil and the fractal dimension of the insoluble asphaltene clusters is higher than that of the soluble clusters. Adding heptane also increases the size of soluble asphaltene clusters without modifying the fractal dimension. Understanding the process of insoluble asphaltenes forming fractals with higher fractal dimensions will potentially reveal the microscopic asphaltene destabilization mechanism (i.e., how a precipitant modifies asphaltene-asphaltene interactions). It was concluded that because of the polydisperse nature of asphaltenes, no well-defined asphaltene phase stability envelope exists and small amounts of asphaltenes precipitated even at dilute precipitant concentrations. Asphaltenes that are stable in a crude oil-precipitant mixture are dispersed on the nanometer length scale. An asphaltene precipitation mechanism is proposed that is consistent with the experimental findings. Additionally, it was found that the heptane-insoluble asphaltene fraction is the dominant source of small-angle scattering in crude oil and the previously unobtainable asphaltene solubility at low heptane concentrations was measured. PMID:23808932
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 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.
Fractal Electronic Circuits Assembled From Nanoclusters
NASA Astrophysics Data System (ADS)
Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.
2009-07-01
Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.
[Fractal analysis of liver fibrosis].
Soda, G; Nardoni, S; Bosco, D; Grizzi, F; Dioguardi, N; Melis, M
2003-04-01
This study realized by two different study groups use of Fractal geometry to quantify the complex collagen deposition during chronic liver disease. Thirty standard needle liver biopsy specimens were obtained from patients with chronic HCV-related disease. Three mu-thick sections were cut and stained by means of Picrosirius stain, in order to visualise collagen matrix. The degree of fibrosis was measured using a quantitative scoring system based on the computer-assisted evaluation of the fractal dimension of the deposited collagen surface. The obtained results by both study groups, show that the proposed method is reproducible, rapid and inexpensive. The complex distribution of its collagenous components can be quantified using a single numerical score. This study demonstrated that it is possible to quantify the collagen's irregularity in an objective manner, and that the study of the fractal properties of the collagen shapes is likely to reveal more about its structure and the complex behaviour of its development. PMID:12768879
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.
Texture Analysis In Cytology Using Fractals
NASA Astrophysics Data System (ADS)
Basu, Santanu; Barba, Joseph; Chan, K. S.
1990-01-01
We present some preliminary results of a study aimed to assess the actual effectiveness of fractal theory in the area of medical image analysis for texture description. The specific goal of this research is to utilize "fractal dimension" to discriminate between normal and cancerous human cells. In particular, we have considered four types of cells namely, breast, bronchial, ovarian and uterine. A method based on fractal Brownian motion theory is employed to compute the "fractal dimension" of cells. Experiments with real images reveal that the range of scales over which the cells exhibit fractal property can be used as the discriminatory feature to identify cancerous cells.
Roughness Perception of Haptically Displayed Fractal Surfaces
NASA Technical Reports Server (NTRS)
Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)
2000-01-01
Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.
Fractal Feature Analysis Of Beef Marblingpatterns
NASA Astrophysics Data System (ADS)
Chen, Kunjie; Qin, Chunfang
The purpose of this study is to investigate fractal behavior of beef marbling patterns and to explore relationships between fractal dimensions and marbling scores. Authors firstly extracted marbling images from beef rib-eye crosssection images using computer image processing technologies and then implemented the fractal analysis on these marbling images based on the pixel covering method. Finally box-counting fractal dimension (BFD) and informational fractal dimension (IFD) of one hundred and thirty-five beef marbling images were calculated and plotted against the beef marbling scores. The results showed that all beef marbling images exhibit fractal behavior over the limited range of scales accessible to analysis. Furthermore, their BFD and IFD are closely related to the score of beef marbling, suggesting that fractal analyses can provide us a potential tool to calibrate the score of beef marbling.
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.
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…
Fractal statistics of cloud fields
NASA Technical Reports Server (NTRS)
Cahalan, Robert F.; Joseph, Joachim H.
1989-01-01
Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data, with 80 and 30 m spatial resolution, respectively, have been employed to study the spatial structure of boundary-layer and intertropical convergence zone (ITCZ) clouds. The probability distributions of cloud areas and cloud perimeters are found to approximately follow a power-law, with a different power (i.e., fractal dimension) for each cloud type. They are better approximated by a double power-law behavior, indicating a change in the fractal dimension at a characteristic size which depends upon cloud type. The fractal dimension also changes with threshold. The more intense cloud areas are found to have a higher perimeter fractal dimension, perhaps indicative of the increased turbulence at cloud top. A detailed picture of the inhomogeneous spatial structure of various cloud types will contribute to a better understanding of basic cloud processes, and also has implications for the remote sensing of clouds, for their effects on remote sensing of other parameters, and for the parameterization of clouds in general circulation models, all of which rely upon plane-parallel radiative transfer algorithms.
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…
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.
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. PMID:26169924
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
NASA Astrophysics Data System (ADS)
Velde, B.; Dubois, J.; Moore, D.; Touchard, G.
1991-05-01
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) 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) 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) 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.
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 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.
Fuzzy fractals, chaos, and noise
Zardecki, A.
1997-05-01
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.
The topology of fractal universes
NASA Technical Reports Server (NTRS)
Hamilton, A. J. S.
1988-01-01
It is shown how the genus per unit volume of isodensity surfaces in general nonlinear universes is related to the entire hierarchy of correlation functions. The general relation between the correlation function, the probability distribution of densities at several points, and the probability distributions of density and its derivatives at a point are given. Formulas for the area and genus per unit volume of isodensity surfaces are presented. As an application, after first reviewing the case of Gaussian fields, analytic results are reported for one particular example of a thoroughly nonlinear universe, Mandelbrot's Rayleigh-Levy random-walk fractal. While this fractal bears little resemblance to the real universe of galaxies, it possesses the singular and theoretically interesting property that in it cluster-cluster correlations are identically equal to galaxy-galaxy correlations to all orders.
Fractal calculus involving gauge function
NASA Astrophysics Data System (ADS)
Golmankhaneh, Alireza K.; Baleanu, Dumitru
2016-08-01
Henstock-Kurzweil integral or gauge integral is the generalization of the Riemann integral. The functions which are not integrable because of singularity in the senses of Lebesgue or Riemann are gauge integrable. In this manuscript, we have generalized Fα-calculus using the gauge integral method for the integrating of the functions on fractal set subset of real-line where they have singularities. The suggested new method leads to the wider class of functions on the fractal subset of real-line that are *Fα-integrable. Using gauge function we define *Fα-derivative of functions their Fα-derivative is not exist. The reported results can be used for generalizing the fundamental theorem of Fα-calculus.
Fractals, malware, and data models
NASA Astrophysics Data System (ADS)
Jaenisch, Holger M.; Potter, Andrew N.; Williams, Deborah; Handley, James W.
2012-06-01
We examine the hypothesis that the decision boundary between malware and non-malware is fractal. We introduce a novel encoding method derived from text mining for converting disassembled programs first into opstrings and then filter these into a reduced opcode alphabet. These opcodes are enumerated and encoded into real floating point number format and used for characterizing frequency of occurrence and distribution properties of malware functions to compare with non-malware functions. We use the concept of invariant moments to characterize the highly non-Gaussian structure of the opcode distributions. We then derive Data Model based classifiers from identified features and interpolate and extrapolate the parameter sample space for the derived Data Models. This is done to examine the nature of the parameter space classification boundary between families of malware and the general non-malware category. Preliminary results strongly support the fractal boundary hypothesis, and a summary of our methods and results are presented here.
Ghosh, Subrata; Dutta, Mrinal; Ray, Kanad; Fujita, Daisuke; Bandyopadhyay, Anirban
2016-06-01
We introduce a new class of fractal reaction kinetics wherein two or more distinct fractal structures are synthesized as parts of a singular cascade reaction in a single chemical beaker. Two examples: sphere ↔ spiral & triangle ↔ square fractals, grow 10(6) orders from a single dendrimer (8 nm) to the visible scale. PMID:27166589
Electrodynamic properties of fractal clusters
NASA Astrophysics Data System (ADS)
Maksimenko, V. V.; Zagaynov, V. A.; Agranovski, I. E.
2014-07-01
An influence of interference on a character of light interaction both with individual fractal cluster (FC) consisting of nanoparticles and with agglomerates of such clusters is investigated. Using methods of the multiple scattering theory, effective dielectric permeability of a micron-size FC composed of non-absorbing nanoparticles is calculated. The cluster could be characterized by a set of effective dielectric permeabilities. Their number coincides with the number of particles, where space arrangement in the cluster is correlated. If the fractal dimension is less than some critical value and frequency corresponds to the frequency of the visible spectrum, then the absolute value of effective dielectric permeability becomes very large. This results in strong renormalization (decrease) of the incident radiation wavelength inside the cluster. The renormalized photons are cycled or trapped inside the system of multi-scaled cavities inside the cluster. A lifetime of a photon localized inside an agglomerate of FCs is a macroscopic value allowing to observe the stimulated emission of the localized light. The latter opens up a possibility for creation of lasers without inverse population of energy levels. Moreover, this allows to reconsider problems of optical cloaking of macroscopic objects. One more feature of fractal structures is a possibility of unimpeded propagation of light when any resistance associated with scattering disappears.
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 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.
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.
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.
Some problems in fractal differential equations
NASA Astrophysics Data System (ADS)
Su, Weiyi
2016-06-01
Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.
Fractal signatures in the aperiodic Fibonacci grating.
Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam
2014-05-01
The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies. PMID:24784044
Order-fractal transitions in abstract paintings
NASA Astrophysics Data System (ADS)
de la Calleja, E. M.; Cervantes, F.; de la Calleja, J.
2016-08-01
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.
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.
Diffraction from fractal grating Cantor sets
NASA Astrophysics Data System (ADS)
Golmankhaneh, Alireza K.; Baleanu, D.
2016-08-01
In this paper, we have generalized the Fα-calculus by suggesting Fourier and Laplace transformations of the function with support of the fractals set which are the subset of the real line. Using this generalization, we have found the diffraction fringes from the fractal grating Cantor sets.
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. PMID:25727720
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…
A fractal-like resistive network
NASA Astrophysics Data System (ADS)
Saggese, A.; De Luca, R.
2014-11-01
The equivalent resistance of a fractal-like network is calculated by means of approaches similar to those employed in defining the equivalent resistance of an infinite ladder. Starting from an elementary triangular circuit, a fractal-like network, named after Saggese, is developed. The equivalent resistance of finite approximations of this network is measured, and the didactical implications of the model are highlighted.
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…
Fractals and cosmological large-scale structure
NASA Technical Reports Server (NTRS)
Luo, Xiaochun; Schramm, David N.
1992-01-01
Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a 'limited' fractal with dimension D of about 1.2. This is not a 'pure' fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process.
Dimensionally Frustrated Diffusion towards Fractal Adsorbers
NASA Astrophysics Data System (ADS)
Nair, Pradeep R.; Alam, Muhammad A.
2007-12-01
Diffusion towards a fractal adsorber is a well-researched problem with many applications. While the steady-state flux towards such adsorbers is known to be characterized by the fractal dimension (DF) of the surface, the more general problem of time-dependent adsorption kinetics of fractal surfaces remains poorly understood. In this Letter, we show that the time-dependent flux to fractal adsorbers (1
Randomness in fractals, connectivity dimensions, and percolation
NASA Astrophysics Data System (ADS)
Perreau, M.; Levy, J. C. S.
1989-10-01
The structural properties of random fractals embedded in a d-dimensional Euclidean space are studied by means of transfer-matrix formalism of fractal sets. For d=1, both global and local approaches have been investigated, leading to the definition of a subdimension that is different from the fractal dimension and depends on the probability distribution. This subdimension is shown to be identical for the global and local approaches; then, the scaling corrections involved in this subdimension are the same for both these approaches. For d>1, only the local approach can be generalized, characterizing the connectivity properties of these structures. There are exactly d subdimensions called connectivity dimensions that prove to be useful to describe percolation properties of these fractals. Several percolation thresholds are shown, and the fractal dimension of the sets at the percolation threshold are related to the connectivity dimensions.
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.
Random sequential adsorption on fractals
NASA Astrophysics Data System (ADS)
Ciesla, Michal; Barbasz, Jakub
2012-07-01
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions.
Random sequential adsorption on fractals.
Ciesla, Michal; Barbasz, Jakub
2012-07-28
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions. PMID:22852643
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.
Fractal Dimensions of Macromolecular Structures
Todoroff, Nickolay; Kunze, Jens; Schreuder, Herman; Hessler, Gerhard; Baringhaus, Karl-Heinz; Schneider, Gisbert
2014-01-01
Quantifying the properties of macromolecules is a prerequisite for understanding their roles in biochemical processes. One of the less-explored geometric features of macromolecules is molecular surface irregularity, or ‘roughness’, which can be measured in terms of fractal dimension (D). In this study, we demonstrate that surface roughness correlates with ligand binding potential. We quantified the surface roughnesses of biological macromolecules in a large-scale survey that revealed D values between 2.0 and 2.4. The results of our study imply that surface patches involved in molecular interactions, such as ligand-binding pockets and protein-protein interfaces, exhibit greater local fluctuations in their fractal dimensions than ‘inert’ surface areas. We expect approximately 22 % of a protein’s surface outside of the crystallographically known ligand binding sites to be ligandable. These findings provide a fresh perspective on macromolecular structure and have considerable implications for drug design as well as chemical and systems biology. PMID:26213587
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. PMID:21230135
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.
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Bory-Reyes, Juan; Shapiro, Michael
2016-02-01
One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.
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. PMID:24025993
Fractal-like structures in colloid science.
Lazzari, S; Nicoud, L; Jaquet, B; Lattuada, M; Morbidelli, M
2016-09-01
The present work aims at reviewing our current understanding of fractal structures in the frame of colloid aggregation as well as the possibility they offer to produce novel structured materials. In particular, the existing techniques to measure and compute the fractal dimension df are critically discussed based on the cases of organic/inorganic particles and proteins. Then the aggregation conditions affecting df are thoroughly analyzed, pointing out the most recent literature findings and the limitations of our current understanding. Finally, the importance of the fractal dimension in applications is discussed along with possible directions for the production of new structured materials. PMID:27233526
Fractal-based wideband invisibility cloak
NASA Astrophysics Data System (ADS)
Cohen, Nathan; Okoro, Obinna; Earle, Dan; Salkind, Phil; Unger, Barry; Yen, Sean; McHugh, Daniel; Polterzycki, Stefan; Shelman-Cohen, A. J.
2015-03-01
A wideband invisibility cloak (IC) at microwave frequencies is described. Using fractal resonators in closely spaced (sub wavelength) arrays as a minimal number of cylindrical layers (rings), the IC demonstrates that it is physically possible to attain a `see through' cloaking device with: (a) wideband coverage; (b) simple and attainable fabrication; (c) high fidelity emulation of the free path; (d) minimal side scattering; (d) a near absence of shadowing in the scattering. Although not a practical device, this fractal-enabled technology demonstrator opens up new opportunities for diverted-image (DI) technology and use of fractals in wideband optical, infrared, and microwave applications.
Formation of fractal islands on nonlattice substrates
NASA Astrophysics Data System (ADS)
Luo, Meng-Bo; Ye, Gao-Xiang; Xia, A.-Gen; Jin, Jin-Sheng; Yang, Bo; Xu, Jian-Min
1999-01-01
A Monte Carlo study on the formation of fractal islands on nonlattice substrates is presented. The islands, including disc aggregates and single discs, perform two-dimensional diffusion along four directions with different diffusion step lengths and rigid rotation about their centers of mass on a nonlattice square with periodic boundary conditions. It is found that the fractal dimension of the ramified islands is almost independent of the diffusion step length, rigid rotation angle, and disc size. However, the fractal dimension increases linearly with the surface coverage. Our simulation results are in good agreement with the previous experimental findings of the aggregation of the silver atomic islands on silicone oil surfaces.
Sporadically Fractal Basin Boundaries of Chaotic Systems
Hunt, B.R.; Ott, E.; Rosa, E. Jr.
1999-05-01
We demonstrate a new type of basin boundary for typical chaotic dynamical systems. For the case of a two dimensional map, this boundary has the character of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary being smooth {open_quotes}almost everywhere,{close_quotes} its fractal dimension exceeds one (implying degradation of one{close_quote}s ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a boundary {ital sporadically fractal}. {copyright} {ital 1999} {ital The American Physical Society}
NASA Astrophysics Data System (ADS)
Suzuki, Hiroki; Nagata, Kouji; Sakai, Yasuhiko; Hasegawa, Yutaka
2013-07-01
The fractal geometry of turbulent mixing of high-Schmidt-number scalars in multiscale, fractal-generated turbulence (FGT) is experimentally investigated. The difference between the fractal geometry in FGT and that in classical grid turbulence (CGT) generated by a biplane, single-scale grid is also investigated. Nondimensional concentration fields are measured by a planar laser-induced fluorescence technique whose accuracy has recently been improved by our research group, and the fractal dimensions are calculated by using the box-counting method. The mesh Reynolds number is 2500 for both CGT and FGT. The Schmidt number is about 2100. It is found that the threshold width ΔCth, when applying the box-counting method, does not affect the evaluation of the fractal dimension at large scales; therefore, the fractal dimensions at large scales have been investigated in this study. The results show that the fractal dimension in FGT is larger than that in CGT. In addition, the fractal dimension in FGT monotonically increases with the onset of time (or with the downstream direction), whereas that in CGT is almost constant with time. The investigation of the number of counted boxes in a unit area, together with the above results, suggests that turbulent mixing is more enhanced in FGT from the viewpoints of fractal geometry and expansion of the mixing interface.
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 cartography of urban areas
NASA Astrophysics Data System (ADS)
Encarnação, Sara; Gaudiano, Marcos; Santos, Francisco C.; Tenedório, José A.; Pacheco, Jorge M.
2012-07-01
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 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
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
Riemann zeros, prime numbers, and fractal potentials.
van Zyl, Brandon P; Hutchinson, David A W
2003-06-01
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels. PMID:16241330
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)
Structural investigations of fat fractals using small-angle scattering
NASA Astrophysics Data System (ADS)
Anitas, Eugen M.
2015-01-01
Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.
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.
A fractal circular polarized RFID tag antenna
NASA Astrophysics Data System (ADS)
Chaouki, Guesmi; Ferchichi, Abdelhak; Gharsallah, Ali
2013-09-01
In this paper, we present a novel fractal antenna for radiofrequency identification (RFID) tags. The proposed antenna has a resonant frequency equal to 2.45GHz and circular polarization. The fractal technique was very useful to obtain a miniaturization of antenna size by more than 30%. The gain and directivity of the antenna are acceptable for the desired RFID application. All the results are obtained using CST Microwave simulation tool.
Multiplexing of encrypted data using fractal masks.
Barrera, John F; Tebaldi, Myrian; Amaya, Dafne; Furlan, Walter D; Monsoriu, Juan A; Bolognini, Néstor; Torroba, Roberto
2012-07-15
In this Letter, we present to the best of our knowledge a new all-optical technique for multiple-image encryption and multiplexing, based on fractal encrypting masks. The optical architecture is a joint transform correlator. The multiplexed encrypted data are stored in a photorefractive crystal. The fractal parameters of the key can be easily tuned to lead to a multiplexing operation without cross talk effects. Experimental results that support the potential of the method are presented. PMID:22825170
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 Analysis of DNA Sequence Data
NASA Astrophysics Data System (ADS)
Berthelsen, Cheryl Lynn
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the "sandbox method." Analysis of 164 human DNA sequences compared to three types of control sequences (random, base -content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than do invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Fractal scaling of microbial colonies affects growth
NASA Astrophysics Data System (ADS)
Károlyi, György
2005-03-01
The growth dynamics of filamentary microbial colonies is investigated. Fractality of the fungal or actinomycetes colonies is shown both theoretically and in numerical experiments to play an important role. The growth observed in real colonies is described by the assumption of time-dependent fractality related to the different ages of various parts of the colony. The theoretical results are compared to a simulation based on branching random walks.
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 analysis of yeast cell optical speckle
NASA Astrophysics Data System (ADS)
Flamholz, A.; Schneider, P. S.; Subramaniam, R.; Wong, P. K.; Lieberman, D. H.; Cheung, T. D.; Burgos, J.; Leon, K.; Romero, J.
2006-02-01
Steady state laser light propagation in diffuse media such as biological cells generally provide bulk parameter information, such as the mean free path and absorption, via the transmission profile. The accompanying optical speckle can be analyzed as a random spatial data series and its fractal dimension can be used to further classify biological media that show similar mean free path and absorption properties, such as those obtained from a single population. A population of yeast cells can be separated into different portions by centrifuge, and microscope analysis can be used to provide the population statistics. Fractal analysis of the speckle suggests that lower fractal dimension is associated with higher cell packing density. The spatial intensity correlation revealed that the higher cell packing gives rise to higher refractive index. A calibration sample system that behaves similar as the yeast samples in fractal dimension, spatial intensity correlation and diffusion was selected. Porous silicate slabs with different refractive index values controlled by water content were used for system calibration. The porous glass as well as the yeast random spatial data series fractal dimension was found to depend on the imaging resolution. The fractal method was also applied to fission yeast single cell fluorescent data as well as aging yeast optical data; and consistency was demonstrated. It is concluded that fractal analysis can be a high sensitivity tool for relative comparison of cell structure but that additional diffusion measurements are necessary for determining the optimal image resolution. Practical application to dental plaque bio-film and cam-pill endoscope images was also demonstrated.
Box-covering algorithm for fractal dimension of weighted networks.
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-01-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the "Sierpinski" weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed. PMID:24157896
Box-covering algorithm for fractal dimension of weighted networks
NASA Astrophysics Data System (ADS)
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-10-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U∝l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area. PMID:25836036
Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique
NASA Technical Reports Server (NTRS)
Qiu, Hong-Lie; Lam, Nina Siu-Ngan; Quattrochi, Dale
1997-01-01
With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.
Fractal Analysis of Cervical Intraepithelial Neoplasia
Fabrizii, Markus; Moinfar, Farid; Jelinek, Herbert F.; Karperien, Audrey; Ahammer, Helmut
2014-01-01
Introduction Cervical intraepithelial neoplasias (CIN) represent precursor lesions of cervical cancer. These neoplastic lesions are traditionally subdivided into three categories CIN 1, CIN 2, and CIN 3, using microscopical criteria. The relation between grades of cervical intraepithelial neoplasia (CIN) and its fractal dimension was investigated to establish a basis for an objective diagnosis using the method proposed. Methods Classical evaluation of the tissue samples was performed by an experienced gynecologic pathologist. Tissue samples were scanned and saved as digital images using Aperio scanner and software. After image segmentation the box counting method as well as multifractal methods were applied to determine the relation between fractal dimension and grades of CIN. A total of 46 images were used to compare the pathologist's neoplasia grades with the predicted groups obtained by fractal methods. Results Significant or highly significant differences between all grades of CIN could be found. The confusion matrix, comparing between pathologist's grading and predicted group by fractal methods showed a match of 87.1%. Multifractal spectra were able to differentiate between normal epithelium and low grade as well as high grade neoplasia. Conclusion Fractal dimension can be considered to be an objective parameter to grade cervical intraepithelial neoplasia. PMID:25302712
On the stability of fractal globules.
Schram, Raoul D; Barkema, Gerard T; Schiessel, Helmut
2013-06-14
The fractal globule, a self-similar compact polymer conformation where the chain is spatially segregated on all length scales, has been proposed to result from a sudden polymer collapse. This state has gained renewed interest as one of the prime candidates for the non-entangled states of DNA molecules inside cell nuclei. Here, we present Monte Carlo simulations of collapsing polymers. We find through studying polymers of lengths between 500 and 8000 that a chain collapses into a globule, which is neither fractal, nor as entangled as an equilibrium globule. To demonstrate that the non-fractalness of the conformation is not just the result of the collapse dynamics, we study in addition the dynamics of polymers that start from fractal globule configurations. Also in this case the chain moves quickly to the weakly entangled globule where the polymer is well mixed. After a much longer time the chain entangles reach its equilibrium conformation, the molten globule. We find that the fractal globule is a highly unstable conformation that only exists in the presence of extra constraints such as cross-links. PMID:23781815
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.
Evolution of Fractal Patterns during a Classical-Quantum Transition
Micolich, A. P.; Taylor, R. P.; Davies, A. G.; Bird, J. P.; Newbury, R.; Fromhold, T. M.; Ehlert, A.; Linke, H.; Macks, L. D.; Tribe, W. R.
2001-07-16
We investigate how fractals evolve into nonfractal behavior as the generation process is gradually suppressed. Fractals observed in the conductance of semiconductor billiards are of particular interest because the generation process is semiclassical and can be suppressed by transitions towards either fully classical or fully quantum-mechanical conduction. Investigating a range of billiards, we identify a ''universal'' behavior in the changeover from fractal to nonfractal conductance, which is described by a smooth evolution rather than deterioration in the fractal scaling properties.
Construction of fractal nanostructures based on Kepler-Shubnikov nets
NASA Astrophysics Data System (ADS)
Ivanov, V. V.; Talanov, V. M.
2013-05-01
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 2 D 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.
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.
a New Construction of the Fractal Interpolation Surface
NASA Astrophysics Data System (ADS)
Ri, Songil
2015-10-01
In this paper, we introduce a new construction of the fractal interpolation surface (FIS) using an even more general iterated function systems (IFS) which can generate self-affine and non self-affine fractal surfaces. Here we present the general types of fractal surfaces that are based on nonlinear IFSs.
a Type of Fractal Interpolation Functions and Their Fractional Calculus
NASA Astrophysics Data System (ADS)
Liang, Yong-Shun; Zhang, Qi
2016-05-01
Combine Chebyshev systems with fractal interpolation, certain continuous functions have been approximated by fractal interpolation functions unanimously. Local structure of these fractal interpolation functions (FIF) has been discussed. The relationship between order of Riemann-Liouville fractional calculus and Box dimension of FIF has been investigated.
Multiscale differential fractal feature with application to target detection
NASA Astrophysics Data System (ADS)
Shi, Zelin; Wei, Ying; Huang, Shabai
2004-07-01
A multiscale differential fractal feature of an image is proposed and a small target detection method from complex nature clutter is presented. Considering the speciality that the fractal features of man-made objects change much more violently than that of nature's when the scale is varied, fractal features at multiple scales used for distinguishing man-made target from nature clutter should have more advantages over standard fractal dimensions. Multiscale differential fractal dimensions are deduced from typical fractal model and standard covering-blanket method is improved and used to estimate multiscale fractal dimensions. A multiscale differential fractal feature is defined as the variation of fractal dimensions between two scales at a rational scale range. It can stand out the fractal feature of man-made object from natural clutters much better than the fractal dimension by standard covering-blanket method. Meanwhile, the calculation and the storage amount are reduced greatly, they are 4/M and 2/M that of the standard covering-blanket method respectively (M is scale). In the image of multiscale differential fractal feature, local gray histogram statistical method is used for target detection. Experiment results indicate that this method is suitable for both kinds background of land and sea. It also can be appropriate in both kinds of infrared and TV images, and can detect small targets from a single frame correctly. This method is with high speed and is easy to be implemented.
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…
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-01
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades. PMID:26305773
Fractal materials, beams, and fracture mechanics
NASA Astrophysics Data System (ADS)
Ostoja-Starzewski, Martin; Li, Jun
2009-11-01
Continuing in the vein of a recently developed generalization of continuum thermomechanics, in this paper we extend fracture mechanics and beam mechanics to materials described by fractional integrals involving D, d and R. By introducing a product measure instead of a Riesz measure, so as to ensure that the mechanical approach to continuum mechanics is consistent with the energetic approach, specific forms of continuum-type equations are derived. On this basis we study the energy aspects of fracture and, as an example, a Timoshenko beam made of a fractal material; the local form of elastodynamic equations of that beam is derived. In particular, we review the crack driving force G stemming from the Griffith fracture criterion in fractal media, considering either dead-load or fixed-grip conditions and the effects of ensemble averaging over random fractal materials.
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. PMID:22461844
``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.
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.
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 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.
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.
Edges of Saturn's rings are fractal.
Li, Jun; Ostoja-Starzewski, Martin
2015-01-01
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn's rings are fractal. PMID:25883885
Spontaneous emission from a fractal vacuum
NASA Astrophysics Data System (ADS)
Akkermans, Eric; Gurevich, Evgeni
2013-08-01
Spontaneous emission of a quantum emitter coupled to a QED vacuum with a deterministic fractal structure of its spectrum is considered. We show that the decay probability does not follow a Wigner-Weisskopf exponential decrease but rather an overall power law behavior with a rich oscillatory structure, both depending on the local fractal properties of the vacuum spectrum. These results are obtained by giving first a general perturbative derivation for short times. Then we propose a simplified model which retains the main features of a fractal spectrum to establish analytic expressions valid for all time scales. Finally, we discuss the case of a Fibonacci cavity and its experimental relevance to observe these results.
Fractal structures in casting films from chlorophyll
NASA Astrophysics Data System (ADS)
Pedro, G. C.; Gorza, F. D. S.; de Souza, N. C.; Silva, J. R.
2014-04-01
Chlorophyll (Chl) molecules are important because they can act as natural light-harvesting devices during the photosynthesis. In addition, they have potential for application as component of solar cell. In this work, we have prepared casting films from chlorophyll (Chl) and demonstrated the occurrence of fractal structures when the films were submitted to different concentrations. By using optical microscopy and the box-count method, we have found that the fractal dimension is Df = 1.55. This value is close to predicted by the diffusion-limited aggregation (DLA) model. This suggests that the major mechanism - which determines the growth of the fractal structures from Chl molecules - is the molecular diffusion. Since the efficiencies of solar cells depend on the morphology of their interfaces, these finds can be useful to improve this kind of device.
Role of Fractals in Solid Earth Geophysics
NASA Astrophysics Data System (ADS)
Dimri, V. P.
2007-12-01
Various studies carried out across the globe reveal that many of the Earth's processes satisfy fractal statistics, where examples range from the frequency-size statistics of earthquakes to the time series of the Earth's magnetic field. The scaling property of fractal signal is very much appealing for descriptions of many geological features. It is observed from the German Continental Deep Drilling Programme (KTB) and many other deep bore wells around the world that the source distribution of density, magnetic susceptibility, electrical conductivity, acoustic impedance etc. follows power-law, hence they are fractal in nature. This finding has been incorporated in various geophysical techniques to better understand the non-linear processes in Earth systems. Theoretical relation between source and potential fields is established and based on that techniques for gravity and magnetic interpretation methods have been reformulated. A new scaling power spectral method is developed to understand source behaviour and parameters of the Earth's interior. Further, fractal concept of tessellation has been used to model the complex geometrical object, which was hitherto unaddressed. An entirely new technique has been proposed to generate the complex geometrical structures with desired physical property variation for forward and inverse modeling of the geophysical data. Further, the concept of fractal distribution of frequency and magnitude of earthquakes is exploited in aftershock study of the major earthquakes such as, Uttarkashi (1991), Latur (1993), Jabalpur (1997), Chamoli (1999), Bhuj (2001) and Muzzafarabad (2005). This study revealed that the Himalayan earthquakes follow multifractal distribution however, shield earthquakes follow monofractal distribution. This finding has been used to explain the earthquake mechanism in Himalayan and shield areas. The fractal study was extended to sea earthquakes and wave propagation modeling is done to understand the effect of Tsunami
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.
Wave interactions with continuous fractal layers
NASA Technical Reports Server (NTRS)
Kim, Y.; Jaggard, D. L.
1991-01-01
Many natural structures possess self-similar multiscales which can be characterized by power law spectra. Under appropriate conditions, knowledge of the strength of these scale sizes provides information on the physical processes which formed these objects. In this paper, we investigate wave interactions with continuous fractal layers which model geological and variegated structures. Since fractal characteristics of the layers are embedded in the scattered field, they can be retrieved under appropriate conditions. This inversion can be performed in either the frequency or the time domain as desired.
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.
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. PMID:23985488
Fractal boundaries in magnetotail particle dynamics
NASA Astrophysics Data System (ADS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-07-01
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.
Problems of Geophysics that Inspired Fractal Geometry
NASA Astrophysics Data System (ADS)
Mandelbrot, B. B.
2001-12-01
Fractal geometry arose when the speaker used then esoteric mathematics and the concept of invariance as a tool to understand diverse ``down-to-earth'' practical needs. The first step consisted in using discontinuous functions to represent the variation of speculative prices. The next several steps consisted in introducing infinite-range (global) dependence to handle data from geophysics, beginning with hydrology (and also again in finance). This talk will detail the speaker's debt and gratitude toward several specialists from diverse areas of geophysics who had the greatest impact on fractal geometry in its formative period.
Communities and classes in symmetric fractals
NASA Astrophysics Data System (ADS)
Krawczyk, Małgorzata J.
2015-07-01
Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analyzed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class.
Is a chaotic multi-fractal approach for rainfall possible?
NASA Astrophysics Data System (ADS)
Sivakumar, Bellie
2001-04-01
Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono-fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi-fractal approaches. The assumption behind multi-fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi-fractals. However, it has been observed recently that multi-fractal approaches might provide positive evidence of a multi-fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi-fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi-fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi-fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi-fractal approach for
NASA Astrophysics Data System (ADS)
Li, Jun; Ostoja-Starzewski, Martin
2013-11-01
In two recent papers [Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.85.025302 85, 025302(R) (2012) and Phys. Rev. E10.1103/PhysRevE.85.056314 85, 056314 (2012)], the authors proposed fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. While in general providing a certain advancement of continuum mechanics modeling of fractal media to fluid flows, some results and statements to previous works need clarification. We first show that the nonlocal character those authors alleged in our paper [Proc. R. Soc. A1364-502110.1098/rspa.2009.0101 465, 2521 (2009)] actually does not exist; instead, all those works are in the same general representation of derivative operators differing by specific forms of the line coefficient c1. Next, the claimed generalization of the volumetric coefficient c3 is, in fact, equivalent to previously proposed product measures when considering together the separate decomposition of c3 on each coordinate. Furthermore, the modified Jacobian proposed in the two commented papers does not relate the volume element between the current and initial configurations, which henceforth leads to a correction of the Reynolds’ transport theorem. Finally, we point out that the asymmetry of the Cauchy stress tensor resulting from the conservation of the angular momentum must not be ignored; this aspect motivates a more complete formulation of fractal continuum models within a micropolar framework.
Li, Jun; Ostoja-Starzewski, Martin
2013-11-01
In two recent papers [Phys. Rev. E 85, 025302(R) (2012) and Phys. Rev. E 85, 056314 (2012)], the authors proposed fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. While in general providing a certain advancement of continuum mechanics modeling of fractal media to fluid flows, some results and statements to previous works need clarification. We first show that the nonlocal character those authors alleged in our paper [Proc. R. Soc. A 465, 2521 (2009)] actually does not exist; instead, all those works are in the same general representation of derivative operators differing by specific forms of the line coefficient c(1). Next, the claimed generalization of the volumetric coefficient c(3) is, in fact, equivalent to previously proposed product measures when considering together the separate decomposition of c(3) on each coordinate. Furthermore, the modified Jacobian proposed in the two commented papers does not relate the volume element between the current and initial configurations, which henceforth leads to a correction of the Reynolds' transport theorem. Finally, we point out that the asymmetry of the Cauchy stress tensor resulting from the conservation of the angular momentum must not be ignored; this aspect motivates a more complete formulation of fractal continuum models within a micropolar framework. PMID:24329394
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
Fractal electrodynamics via non-integer dimensional space approach
NASA Astrophysics Data System (ADS)
Tarasov, Vasily E.
2015-09-01
Using the recently suggested vector calculus for non-integer dimensional space, we consider electrodynamics problems in isotropic case. This calculus allows us to describe fractal media in the framework of continuum models with non-integer dimensional space. We consider electric and magnetic fields of fractal media with charges and currents in the framework of continuum models with non-integer dimensional spaces. An application of the fractal Gauss's law, the fractal Ampere's circuital law, the fractal Poisson equation for electric potential, and equation for fractal stream of charges are suggested. Lorentz invariance and speed of light in fractal electrodynamics are discussed. An expression for effective refractive index of non-integer dimensional space is suggested.
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 Structure in Human Cerebellum Measured by MRI
NASA Astrophysics Data System (ADS)
Zhang, Luduan; Yue, Guang; Brown, Robert; Liu, Jingzhi
2003-10-01
Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men, 12 women). CB images were resampled to a series of image sets with different three-dimensional resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57+/-0.01. No significant difference in the CB fractal dimension was observed between men and women. Fractal dimension may serve as a quantitative index for structural complexity of the CB at its developmental, degenerative, or evolutionary stages.
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
Lightning and the Heart: Fractal Behavior in Cardiac Function
BASSINGTHWAIGHTE, JAMES B.; van BEEK, J. H. G. M.
2010-01-01
Physical systems, from galactic clusters to diffusing molecules, often show fractal behavior. Likewise, living systems might often be well described by fractal algorithms. Such fractal descriptions in space and time imply that there is order in chaos, or put the other way around, chaotic dynamical systems in biology are more constrained and orderly than seen at first glance. The vascular network, the syncytium of cells, the processes of diffusion and transmembrane transport might be fractal features of the heart. These fractal features provide a basis which enables one to understand certain aspects of more global behavior such as atrial or ventricular fibrillation and perfusion heterogeneity. The heart might be regarded as a prototypical organ from these points of view. A particular example of the use of fractal geometry is in explaining myocardial flow heterogeneity via delivery of blood through an asymmetrical fractal branching network. PMID:21938081
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
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.
Fractal Dimensions and Entropies of Meragi Songs
NASA Astrophysics Data System (ADS)
Aydemir, Adnan; Gündüz, Güngör
Melodies can be treated as time series
A fractal model for crustal deformation
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1986-01-01
It is hypothesized that crustal deformation occurs on a scale-invariant matrix of faults. For simplicity, a two-dimensional pattern of hexagons on which strike-slip faulting occurs is considered. The behavior of the system is controlled by a single parameter, the fractal dimension. Deformation occurs on all scales of faults. The fractal dimension determines the fraction of the total displacement that occurs on the first-order or primary faults. The value of the fractal dimension can be obtained from the frequency-magnitude relation for earthquakes. The results are applied to the San Andreas fault system in central California. Earthquake studies give D = 1.90. The main strand of the San Andreas fault is associated with the primary faults of the fractal system. It is predicted that the relative velocity across the main strand is 2.93 cm/yr. The remainder of the relative velocity of 5.5 cm/yr between the Pacific and North American plates occurs on higher-order faults. The predicted value is in reasonably good agreement with the value 3.39 + or - 0.29 cm/yr obtained from geological studies.
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.
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,…
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.
NASA Astrophysics Data System (ADS)
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.
Turbulence on a Fractal Fourier Set.
Lanotte, Alessandra S; Benzi, Roberto; Malapaka, Shiva K; Toschi, Federico; Biferale, Luca
2015-12-31
A novel investigation of the nature of intermittency in incompressible, homogeneous, and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension D from the original three dimensional case to a strongly decimated system with D=2.5, where only about 3% of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set E(k)∼k(-5/3+3-D) explains the results. At small scales, the intermittency of the vorticity field is observed to be quasisingular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at D∼2.98. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism. PMID:26764993
Fractional transport equation on random fractals
NASA Astrophysics Data System (ADS)
Zeng, Qiuhua; Li, Houqiang; Fang, Yaquan
1998-12-01
According to the ways of H.E. Roman and M. Giona with the constitutive equation of diffusive particles in isotropic and homogeneous three dimensions and the Laplace transform we derive the multiscaling fractional transport equation in disordered fractal media, whose solution is consistent with literature results.
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. PMID:26651668
Light Scattering From Fractal Titania Aggregates
NASA Astrophysics Data System (ADS)
Pande, Rajiv; Sorensen, Christopher M.
1996-03-01
We studied the fractal morphology of titania aggregates by light scattering. Titanium dioxide particles were generated by the thermal decomposition of titanium tetra-isopropoxide(TTIP) in a glass furnace at various temperatures in the range of 100 - 500^o C. We scattered vertically polarized He-Ne laser (λ = 6328Ålight from a laminar aerosol stream of particles and measured the optical structure factor. This structure factor shows Rayleigh, Guinier, fractal and Porod regimes. The radius of gyration Rg was determined from the Guinier analysis. The data were then fit to the Fisher-Burford form to determine the fractal dimension of about 2.0. This fit also delineated the crossover from the fractal to Porod regime, which can be used to determine the monomer particle size of about 0.1 μm. These optical measurements will be compared to electron microscope analysis of aggregates collected from the aerosol. This work was supported by NSF grant CTS-9908153.
Scaling of Hamiltonian walks on fractal lattices.
Elezović-Hadzić, Suncica; Marcetić, Dusanka; Maletić, Slobodan
2007-07-01
We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs Z(N), for the lattice with N>1 sites, varies as omega(N)}N(gamma). We explicitly calculate the exponent gamma for several members of GM and MSG families, as well as for n-simplices with n=3, 5, and 7. For n-simplex fractals with even n we find different scaling form: Z(N) approximately omega(N)mu(N1/d(f), where d(f) is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers. PMID:17677410
Fractal surfaces from simple arithmetic operations
NASA Astrophysics Data System (ADS)
García-Morales, Vladimir
2016-04-01
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent H that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.
Dimension of a fractal streamer structure
NASA Astrophysics Data System (ADS)
Lehtinen, Nikolai G.; Østgaard, Nikolai
2015-04-01
Streamer corona plays an important role in formation of leader steps in lightning. In order to understand its dynamics, the streamer front velocity is calculated in a 1D model with curvature. We concentrate on the role of photoionization mechanism in the propagation of the streamer ionization front, the other important mechanisms being electron drift and electron diffusion. The results indicate, in particular, that the effect of photoionization on the streamer velocity for both positive and negative streamers is mostly determined by the photoionization length, with a weaker dependence on the amount of photoionization, and that the velocity is decreased for positive curvature, i.e., convex fronts. These results are used in a fractal model in which the front propagation velocity is simulated as the cluster growth probability [Niemeyer et al, 1984, doi:10.1103/PhysRevLett.52.1033]. Monte Carlo simulations of the cluster growth for various ratios of background electric field E to the breakdown field Eb show that the emerging transverse size of the streamers is of the order of the photoionization length, and at the larger scale the streamer structure is a fractal similar to the one obtained in a diffusion-limited aggregation (DLA) system. In the absence of electron attachment (Eb = 0), the fractal dimension is the same (D ˜ 1.67) as in the DLA model, and is reduced, i.e., the fractal has less branching, for Eb > 0.
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)
Fractal analysis of narwhal space use patterns.
Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R
2004-01-01
Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice. PMID:16351924
Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions
Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander
2010-01-21
A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/r{sup D}propor tor{sup -D} where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis.The fractal coefficients analysis provides some important advantages in examining earthquake spatial
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
Novel optical password security technique based on optical fractal synthesizer
NASA Astrophysics Data System (ADS)
Wu, Kenan; Hu, Jiasheng; Wu, Xu
2009-06-01
A novel optical security technique for safeguarding user passwords based on an optical fractal synthesizer is proposed. A validating experiment has been carried out. In the proposed technique, a user password is protected by being converted to a fractal image. When a user sets up a new password, the password is transformed into a fractal pattern, and the fractal pattern is stored in authority. If the user is online-validated, his or her password is converted to a fractal pattern again to compare with the previous stored fractal pattern. The converting process is called the fractal encoding procedure, which consists of two steps. First, the password is nonlinearly transformed to get the parameters for the optical fractal synthesizer. Then the optical fractal synthesizer is operated to generate the output fractal image. The experimental result proves the validity of our method. The proposed technique bridges the gap between digital security systems and optical security systems and has many advantages, such as high security level, convenience, flexibility, hyper extensibility, etc. This provides an interesting optical security technique for the protection of digital passwords.
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.
Multispectral image fusion based on fractal features
NASA Astrophysics Data System (ADS)
Tian, Jie; Chen, Jie; Zhang, Chunhua
2004-01-01
Imagery sensors have been one indispensable part of the detection and recognition systems. They are widely used to the field of surveillance, navigation, control and guide, et. However, different imagery sensors depend on diverse imaging mechanisms, and work within diverse range of spectrum. They also perform diverse functions and have diverse circumstance requires. So it is unpractical to accomplish the task of detection or recognition with a single imagery sensor under the conditions of different circumstances, different backgrounds and different targets. Fortunately, the multi-sensor image fusion technique emerged as important route to solve this problem. So image fusion has been one of the main technical routines used to detect and recognize objects from images. While, loss of information is unavoidable during fusion process, so it is always a very important content of image fusion how to preserve the useful information to the utmost. That is to say, it should be taken into account before designing the fusion schemes how to avoid the loss of useful information or how to preserve the features helpful to the detection. In consideration of these issues and the fact that most detection problems are actually to distinguish man-made objects from natural background, a fractal-based multi-spectral fusion algorithm has been proposed in this paper aiming at the recognition of battlefield targets in the complicated backgrounds. According to this algorithm, source images are firstly orthogonally decomposed according to wavelet transform theories, and then fractal-based detection is held to each decomposed image. At this step, natural background and man-made targets are distinguished by use of fractal models that can well imitate natural objects. Special fusion operators are employed during the fusion of area that contains man-made targets so that useful information could be preserved and features of targets could be extruded. The final fused image is reconstructed from the
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.
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'.
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.
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
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.
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 energy carpets in non-Hermitian Hofstadter quantum mechanics.
Chernodub, Maxim N; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion. PMID:26565163
Local Earth's gravity field in view of fractal dimension
NASA Astrophysics Data System (ADS)
Mészárosová, Katarína; Minarechová, Zuzana; Janák, Juraj
2013-04-01
The poster presents the relative roughness of chosen characteristics of the Earth's gravity field in several small regions in area of Slovakia (e.g. free-air anomaly, Bouguer anomaly, gravity disturbance...) using the values of fractal dimension. In this approach, a three dimensional box counting method and the Hurst analysis method are applied to estimate the values of fractal dimensions. Then the computed fractal dimension values are used to compare all 3D models of all chosen characteristics.
Fractal 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 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.
Fractal tiles associated with shift radix systems.
Berthé, Valérie; Siegel, Anne; Steiner, Wolfgang; Surer, Paul; Thuswaldner, Jörg M
2011-01-15
Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine). PMID:24068835
Fractal property of eye movements in schizophrenia.
Yokoyama, H; Niwa, S; Itoh, K; Mazuka, R
1996-08-01
On the basis of a temporal model of animal behavior we conducted temporal analysis of eye movements in schizophrenic subjects (n = 10) and normal controls (n = 10). We found a fractal property in schizophrenic subjects, the fixation time of eye movement during reading ambiguous and difficult sentences showing a clear inverse power law distribution. An exponential distribution of a nonfractal nature was found in normal controls. PMID:8855352
Fractal dimension based corneal fungal infection diagnosis
NASA Astrophysics Data System (ADS)
Balasubramanian, Madhusudhanan; Perkins, A. Louise; Beuerman, Roger W.; Iyengar, S. Sitharama
2006-08-01
We present a fractal measure based pattern classification algorithm for automatic feature extraction and identification of fungus associated with an infection of the cornea of the eye. A white-light confocal microscope image of suspected fungus exhibited locally linear and branching structures. The pixel intensity variation across the width of a fungal element was gaussian. Linear features were extracted using a set of 2D directional matched gaussian-filters. Portions of fungus profiles that were not in the same focal plane appeared relatively blurred. We use gaussian filters of standard deviation slightly larger than the width of a fungus to reduce discontinuities. Cell nuclei of cornea and nerves also exhibited locally linear structure. Cell nuclei were excluded by their relatively shorter lengths. Nerves in the cornea exhibited less branching compared with the fungus. Fractal dimensions of the locally linear features were computed using a box-counting method. A set of corneal images with fungal infection was used to generate class-conditional fractal measure distributions of fungus and nerves. The a priori class-conditional densities were built using an adaptive-mixtures method to reflect the true nature of the feature distributions and improve the classification accuracy. A maximum-likelihood classifier was used to classify the linear features extracted from test corneal images as 'normal' or 'with fungal infiltrates', using the a priori fractal measure distributions. We demonstrate the algorithm on the corneal images with culture-positive fungal infiltrates. The algorithm is fully automatic and will help diagnose fungal keratitis by generating a diagnostic mask of locations of the fungal infiltrates.
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.
Aero-acoustic performance of Fractal Spoilers
NASA Astrophysics Data System (ADS)
Nedic, J.; Ganapathisubramani, B.; Vassilicos, C.; Boree, J.; Brizzi, L.; Spohn, A.
2010-11-01
One of the major environmental problems facing the aviation industry is that of aircraft noise. The work presented in this paper, done as part of the OPENAIR Project, looks at reducing spoiler noise through means of large-scale fractal porosity. It is hypothesised that the highly turbulent flow generated by these grids, which have multi-length-scales, would remove the re-circulation region and with it, the low frequency noise it generates. In its place, a higher frequency noise is introduced which is susceptible to atmospheric attenuation, and would be deemed less offensive to the human ear. A total of nine laboratory scaled spoilers were looked at, seven of which had a fractal design, one conventionally porous and one solid for reference. All of the spoilers were mounted on a flat plate and inclined at 30^o to the horizontal. Far-field, microphone array and PIV measurements were taken in an anechoic chamber to determine the acoustic performance and to study the flow coming through the spoilers. A significant reduction in sound pressure level is recorded and is found to be very sensitive to small changes in fractal grid parameters. Wake and drag force measurements indicated that the spoilers increase the drag whilst having minimal effect on the lift.
The contact mechanics of fractal surfaces
NASA Astrophysics Data System (ADS)
Buzio, Renato; Boragno, Corrado; Biscarini, Fabio; Buatier de Mongeot, Francesco; Valbusa, Ugo
2003-04-01
The role of surface roughness in contact mechanics is relevant to processes ranging from adhesion to friction, wear and lubrication. It also promises to have a deep impact on applied science, including coatings technology and design of microelectromechanical systems. Despite the considerable results achieved by indentation experiments, particularly in the measurement of bulk hardness on nanometre scales, the contact behaviour of realistic surfaces, showing random multiscale roughness, remains largely unknown. Here we report experimental results concerning the mechanical response of self-affine thin films indented by a micrometric flat probe. The specimens, made of cluster-assembled carbon or of sexithienyl, an organic molecular material, were chosen as prototype systems for the broad class of self-affine fractal interfaces, today including surfaces grown under non-equilibrium conditions, fractures, manufactured metal surfaces and solidified liquid fronts. We observe that a regime exists in which roughness drives the contact mechanics: in this range surface stiffness varies by a few orders of magnitude on small but significant changes of fractal parameters. As a consequence, we demonstrate that soft solid interfaces can be appreciably strengthened by reducing both fractal dimension and surface roughness. This indicates a general route for tailoring the mechanical properties of solid bodies.
Fractality of pulsatile flow in speckle images
NASA Astrophysics Data System (ADS)
Nemati, M.; Kenjeres, S.; Urbach, H. P.; Bhattacharya, N.
2016-05-01
The scattering of coherent light from a system with underlying flow can be used to yield essential information about dynamics of the process. In the case of pulsatile flow, there is a rapid change in the properties of the speckle images. This can be studied using the standard laser speckle contrast and also the fractality of images. In this paper, we report the results of experiments performed to study pulsatile flow with speckle images, under different experimental configurations to verify the robustness of the techniques for applications. In order to study flow under various levels of complexity, the measurements were done for three in-vitro phantoms and two in-vivo situations. The pumping mechanisms were varied ranging from mechanical pumps to the human heart for the in vivo case. The speckle images were analyzed using the techniques of fractal dimension and speckle contrast analysis. The results of these techniques for the various experimental scenarios were compared. The fractal dimension is a more sensitive measure to capture the complexity of the signal though it was observed that it is also extremely sensitive to the properties of the scattering medium and cannot recover the signal for thicker diffusers in comparison to speckle contrast.
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 dimension analyses of lava surfaces and flow boundaries
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.
1993-01-01
An improved method of estimating fractal surface dimensions has been developed. The accuracy of this method is illustrated using artificially generated fractal surfaces. A slightly different from usual concept of linear dimension is developed, allowing a direct link between that and the corresponding surface dimension estimate. These methods are applied to a series of images of lava flows, representing a variety of physical and chemical conditions. These include lavas from California, Idaho, and Hawaii, as well as some extraterrestrial flows. The fractal surface dimension estimations are presented, as well as the fractal line dimensions where appropriate.
FAST TRACK COMMUNICATION: Weyl law for fat fractals
NASA Astrophysics Data System (ADS)
Spina, María E.; García-Mata, Ignacio; Saraceno, Marcos
2010-10-01
It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Elizarraraz, Benjamin Espinoza
2013-11-01
The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. EPLEEE81063-651X10.1103/PhysRevE.85.025302 85, 025302(R) (2012); PLEEE81063-651X10.1103/PhysRevE.85.056314 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds’ transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.
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.
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.
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.
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 energy measurement and the singularity energy spectrum analysis
NASA Astrophysics Data System (ADS)
Xiong, Gang; Zhang, Shuning; Yang, Xiaoniu
2012-12-01
The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.
Fractal 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...
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.
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.
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
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.
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 .
A fractal approach to probabilistic seismic hazard assessment
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1989-01-01
The definition of a fractal distribution is that the number of objects (events) N with a characteristic size greater than r satisfies the relation N proportional to r exp - D is the fractal dimension. The applicability of a fractal relation implies that the underlying physical process is scale-invariant over the range of applicability of the relation. The empirical frequency-magnitude relation for earthquakes defining a b-value is a fractal relation with D = 2b. Accepting the fractal distribution, the level of regional seismicity can be related to the rate of regional strain and the magnitude of the largest characteristic earthquake. High levels of seismic activity indicate either a large regional strain or a low-magnitude maximum characteristic earthquake (or both). If the regional seismicity has a weak time dependence, the approach can be used to make probabilistic seismic hazard assessments.
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.