Changes in fractal dimension during aggregation.
Chakraborti, Rajat K; Gardner, Kevin H; Atkinson, Joseph F; Van Benschoten, John E
2003-02-01
Experiments were performed to evaluate temporal changes in the fractal dimension of aggregates formed during flocculation of an initially monodisperse suspension of latex microspheres. Particle size distributions and aggregate geometrical information at different mixing times were obtained using a non-intrusive optical sampling and digital image analysis technique, under variable conditions of mixing speed, coagulant (alum) dose and particle concentration. Pixel resolution required to determine aggregate size and geometric measures including the fractal dimension is discussed and a quantitative measure of accuracy is developed. The two-dimensional fractal dimension was found to range from 1.94 to 1.48, corresponding to aggregates that are either relatively compact or loosely structured, respectively. Changes in fractal dimension are explained using a conceptual model, which describes changes in fractal dimension associated with aggregate growth and changes in aggregate structure. For aggregation of an initially monodisperse suspension, the fractal dimension was found to decrease over time in the initial stages of floc formation.
Fractal dimension of alumina aggregates grown in two dimensions
NASA Technical Reports Server (NTRS)
Larosa, Judith L.; Cawley, James D.
1992-01-01
The concepts of fractal geometry are applied to the analysis of 0.4-micron alumina constrained to agglomerate in two dimensions. Particles were trapped at the bottom surface of a drop of a dilute suspension, and the agglomeration process was directly observed, using an inverted optical microscope. Photographs were digitized and analyzed, using three distinct approaches. The results indicate that the agglomerates are fractal, having a dimension of approximately 1.5, which agrees well with the predictions of the diffusion-limited cluster-cluster aggregation model.
Fractal dimension and mechanism of aggregation of apple juice particles.
Benítez, E I; Lozano, J E; Genovese, D B
2010-04-01
Turbidity of freshly squeezed apple juice is produced by a polydisperse suspension of particles coming from the cellular tissue. After precipitation of coarse particles by gravity, only fine-colloidal particles remain in suspension. Aggregation of colloidal particles leads to the formation of fractal structures. The fractal dimension is a measure of the internal density of these aggregates and depends on their mechanism of aggregation. Digitized images of primary particles and aggregates of depectinized, diafiltered cloudy apple juice were obtained by scanning electron microscopy (SEM). Average radius of the primary particles was found to be a = 40 ± 11 nm. Maximum radius of the aggregates, R(L), ranged between 250 and 7750 nm. Fractal dimension of the aggregates was determined by analyzing SEM images with the variogram method, obtaining an average value of D(f) = 2.3 ± 0.1. This value is typical of aggregates formed by rapid flocculation or diffusion limited aggregation. Diafiltration process was found to reduce the average size and polydispersity of the aggregates, determined by photon correlation spectroscopy. Average gyration radius of the aggregates before juice diafiltration was found to be R(g) = 629 ± 87 nm. Average number of primary particles per aggregate was calculated to be N = 1174. PMID:21339133
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
The fractal dimension of the protein aggregates can be estimated by dynamic mechanical methods when the particle aggregates are imbedded in a polymer matrix. Nanocomposites were formed by mixing hydrolyzed soy protein isolate (HSPI) nanoparticle aggregates with styrene-butadiene (SB) latex, followe...
Aggregation of liposomes in presence of La3+: a study of the fractal dimension.
Sabín, Juan; Prieto, Gerardo; Ruso, Juan M; Messina, Paula; Sarmiento, Félix
2007-07-01
A study of the fractal dimension of the aggregation of three different types of large unilamellar vesicles, formed by egg yolk phosphatidylcholine (EYPC), dimyristoyl-phosphocholine (DMPC), and dipalmitoyl-phosphocholine (DPPC), in the presence of La3+, is presented. Aggregate liposome fractal dimensions were calculated by two methods, aggregation kinetics, using the approaches diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA) and angle-scattering light dispersion. Electrophoretic measurements show a similar variation of the zeta potential (zeta potential) for EYPC and DPPC, with a small increase of initial positive values. However, the zeta potential of DMPC changes from a initial negative value to near zero with increasing La3+ concentration. The evolution of the aggregate sizes was followed by light scattering. DPPC and DMPC show a RLCA regimen growth at low La3+ concentrations and a DLCA regimen at higher concentrations. In the case of EYPC, the final size of aggregation strongly depends on La3+ concentration. The calculated fractal dimension is in the range 1.8 to 2.1.
Fractal dimension of soil aggregates as an index of soil erodibility
NASA Astrophysics Data System (ADS)
Ahmadi, Abbas; Neyshabouri, Mohammad-Reza; Rouhipour, Hassan; Asadi, Hossein
2011-04-01
SummaryAggregate stability is an influential factor governing soil erodibility. The fractal dimension of soil aggregates has been related to their size distributions and stabilities. Several fractal models have been proposed for estimating fractal dimension of soil aggregates. This study was conducted to investigate how closely the soil interrill erodibility factor in WEPP model can be correlated to and predicted from soil aggregate size distribution or from their fractal dimensions. Samples from 36 soil series with contrasting properties were collected from northwest of Iran. The fractal dimensions of soil aggregates were calculated from Rieu and Sposito ( D n), Tyler and Wheatcraft ( D mT), and Young and Crawford ( D mY) models using aggregate size distribution (ASD) data. A rainfall simulator with drainable tilting flume (1 × 0.5 m) at slope of 9% was employed and total interrill erosion ( TIE), total splashed soil ( TS) and interrill erodibility factor ( K i) were calculated for 20, 37, and 47 mm h -1 rainfall intensities. Results showed that both D n and D mT estimated from aggregate wet-sieving data characterized ASD of the examined soils and significantly ( p < 0.01) correlated to TS, TIE and K i. Values of D n and D mT estimated from dry-sieving data only correlated to TS but not to TIE and K i. Using air-dried aggregates of 4.75-8 mm size range, instead of aggregates <4.75 mm, in wet-sieving was better for estimating D n as an index for the predication of TIE, TS and K i. Correction of ASD for the particle fraction greater than lower sieve mesh size in each size class decreased the correlation coefficient between TIE, TS or K i and D n or D mT. The values of D mY were not correlated to TS, TIE and K i. The correlation coefficient TIE and K i with D n and D mT derived from wet-sieving data, were higher than those with wet-aggregate stability (WAS), mean weight diameter (MWD) and geometric mean diameter (GMD), implying that D n and D mT may be better
NASA Astrophysics Data System (ADS)
Braga, F. L.; Mattos, O. A.; Amorin, V. S.; Souza, A. B.
2015-07-01
Clusters formation models have been extensively studied in literature, and one of the main task of this research area is the analysis of the particle aggregation processes. Some work support that the main characteristics of this processes are strictly correlated to the cluster morphology, for example in DLA. It is expected that in the DLA clusters formation with particles containing different sizes the modification of the aggregation processes can be responsible for changes in the DLA morphology. The present article is going to analyze the formation of DLA clusters of particles with different sizes and show that the aggregates obtained by this approach generate an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affect the fractal dimension of the clusters.
Meng, Zhiyong; Hashmi, Sara M; Elimelech, Menachem
2013-02-15
The time-evolutions of nanoparticle hydrodynamic radius and aggregate fractal dimension during the aggregation of fullerene (C(60)) nanoparticles (FNPs) were measured via simultaneous multiangle static and dynamic light scattering. The FNP aggregation behavior was determined as a function of monovalent (NaCl) and divalent (CaCl(2)) electrolyte concentration, and the impact of addition of dissolved natural organic matter (humic acid) to the solution was also investigated. In the absence of humic acid, the fractal dimension decreased over time with monovalent and divalent salts, suggesting that aggregates become slightly more open and less compact as they grow. Although the aggregates become slightly more open, the magnitude of the fractal dimension suggests intermediate aggregation between the diffusion- and reaction-limited regimes. We observed different aggregation behavior with monovalent and divalent salts upon the addition of humic acid to the solution. For NaCl-induced aggregation, the introduction of humic acid significantly suppressed the aggregation rate of FNPs at NaCl concentrations lower than 150mM. In this case, the aggregation was intermediate or reaction-limited even at NaCl concentrations as high as 500mM, giving rise to aggregates with a fractal dimension of 2.0. For CaCl(2)-induced aggregation, the introduction of humic acid enhanced the aggregation of FNPs at CaCl(2) concentrations greater than about 5mM due to calcium complexation and bridging effects. Humic acid also had an impact on the FNP aggregate structure in the presence of CaCl(2), resulting in a fractal dimension of 1.6 for the diffusion-limited aggregation regime. Our results with CaCl(2) indicate that in the presence of humic acid, FNP aggregates have a more open and loose structure than in the absence of humic acid. The aggregation results presented in this paper have important implications for the transport, chemical reactivity, and toxicity of engineered nanoparticles in aquatic
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.
Fractal aggregates in Titan's atmosphere
NASA Astrophysics Data System (ADS)
Cabane, M.; Rannou, P.; Chassefiere, E.; Israel, G.
1993-04-01
The cluster structure of Titan's atmosphere was modeled by using an Eulerian microphysical model with the specific formulation of microphysical laws applying to fractal particles. The growth of aggregates in the settling phase was treated by introducing the fractal dimension as a parameter of the model. The model was used to obtain a vertical distribution of size and number density of the aggregates for different production altitudes. Results confirm previous estimates of the formation altitude of photochemical aerosols. The vertical profile of the effective radius of aggregates was calculated as a function of the visible optical depth.
Fractal aggregates in tennis ball systems
NASA Astrophysics Data System (ADS)
Sabin, J.; Bandín, M.; Prieto, G.; Sarmiento, F.
2009-09-01
We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the images of the cluster of balls, following Forrest and Witten's pioneering studies on the aggregation of smoke particles, to estimate their fractal dimension.
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.
Classical Liquids in Fractal Dimension.
Heinen, Marco; Schnyder, Simon K; Brady, John F; Löwen, Hartmut
2015-08-28
We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension dl. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here, we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semianalytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
Soy protein isolate (SPI) is obtained from soybeans by removing soybean oil and soy carbohydrates. Soy protein nanoparticles were prepared by alkaline hydrolysis of SPI and centrifugal separation process. Structurally, SPI is a globular protein and its aggregates in water consist of sphere-like pr...
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.
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.
Fractality à la carte: a general particle aggregation model
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-19
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
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.
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
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates. PMID:19257067
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates.
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.
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.
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.
Emergence of fractals in aggregation with stochastic self-replication.
Hassan, Md Kamrul; Hassan, Md Zahedul; Islam, Nabila
2013-10-01
We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse. Furthermore, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given, and the relationship between this fractal dimension and a conserved quantity is pointed out.
Fractal 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
Structural Interpretations of Static Light Scattering Patterns of Fractal Aggregates.
Lambert; Thill; Ginestet; Audic; Bottero
2000-08-15
A method based on static light scattering by fractal aggregates is introduced to extract structural information. In this study, we determine the scattered intensity by a fractal aggregate calculating the Structure and the Form factors noted, respectively, S(q) and F(q). We use the approximation of the mean field Mie scattering by fractal aggregates (R. Botet, P. Rannou, and M. Cabane, appl. opt. 36, 8791, 1997). This approximation is validated by a comparison of the scattering and extinction cross sections values calculated using, on the one hand, Mie theory with a mean optical index n) and, on the other hand, the mean field approximation. Scattering and extinction cross sections values differ by about 5%. We show that the mean environment of primary scatterers characterized by the optical index n(s) must be taken into account to interpret accurately the scattering pattern from fractal aggregates. Numerical simulations were done to evaluate the influence of the fractal dimension values (D(f)>2) and of the radius of gyration or the number of primary particles within the aggregates (N=50 to 250) on the scatterers' mean optical contrast (n(s)/n). This last parameter plays a major role in determining the Form factor F(q) which corresponds to the primary particles' scattering. In associating the mean optical index (n) to structural characteristics, this work provides a theoretical framework to be used to provide additional structural information from the scattering pattern of a fractal aggregate (cf. Part II). Copyright 2000 Academic Press.
BOX DIMENSIONS OF α-FRACTAL FUNCTIONS
NASA Astrophysics Data System (ADS)
Akhtar, Md. Nasim; Prasad, M. Guru Prem; Navascués, M. A.
2016-08-01
The box dimension of the graph of non-affine, continuous, nowhere differentiable function fα which is a fractal analogue of a continuous function f corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of α-fractal function fα for equally spaced as well as arbitrary data sets are found.
Pre-Service Teachers' Concept Images on Fractal Dimension
ERIC Educational Resources Information Center
Karakus, Fatih
2016-01-01
The analysis of pre-service teachers' concept images can provide information about their mental schema of fractal dimension. There is limited research on students' understanding of fractal and fractal dimension. Therefore, this study aimed to investigate the pre-service teachers' understandings of fractal dimension based on concept image. The…
Aggregates and Superaggregates of Soot with Four Distinct Fractal Morphologies
NASA Technical Reports Server (NTRS)
Sorensen, C. M.; Kim, W.; Fry, D.; Chakrabarti, A.
2004-01-01
Soot formed in laminar diffusion flames of heavily sooting fuels evolves through four distinct growth stages which give rise to four distinct aggregate fractal morphologies. These results were inferred from large and small angle static light scattering from the flames, microphotography of the flames, and analysis of soot sampled from the flames. The growth stages occur approximately over four successive orders of magnitude in aggregate size. Comparison to computer simulations suggests that these four growth stages involve either diffusion limited cluster aggregation or percolation in either three or two dimensions.
NASA Astrophysics Data System (ADS)
Wei, Wei; Cai, Jianchao; Hu, Xiangyun; Han, Qi; Liu, Shuang; Zhou, Yingfang
2016-08-01
A theoretical effective thermal conductivity model for nanofluids is derived based on fractal distribution characteristics of nanoparticle aggregation. Considering two different mechanisms of heat conduction including particle aggregation and convention, the model is expressed as a function of the fractal dimension and concentration. In the model, the change of fractal dimension is related to the variation of aggregation shape. The theoretical computations of the developed model provide a good agreement with the experimental results, which may serve as an effective approach for quantitatively estimating the effective thermal conductivity of nanofluids.
Fractal Dimension Analysis of Putative Martian Coastlines
NASA Astrophysics Data System (ADS)
Gianelli, G. A.
2005-08-01
Prior research is equivocal on the existence and location of Martian coastlines. This study proposes a novel method of analyzing putative coastlines; fractal dimensions provide a quantitative measurement of the complexity and nature of a fractal. Geological evidence points to a coastline at the elevation of -3790 meters, called the Deuteronilus contact. It is hypothesized that the fractal dimensions of this putative Martian coastline will be comparable to those of Earth shorelines. A topographic map with a contour line at -3790 meters was obtained from the U. S. Geological Survey, reflecting the most recent Mars Orbiter Laser Altimeter data. The map was cropped into sixty and twenty degree segments, and the putative coastline was isolated from extraneous features. A program which used the box-counting method calculated the fractal dimensions of the putative shorelines. The 22 results were tabulated and compared to 17 fractal dimensions of Earth shorelines, collected from published articles. Ranges were 1.07 to 1.66 for Earth and 1.141 to 1.436 for Mars. The mean was 1.28 for the Mars data and 1.22 for the Earth data, a slight difference that asteroid craters could account for. An unpaired t-test could not prove that the two data sets were significantly different. Although the past existence of a coastline at the Deuteronilus contact cannot be definitively proven without on site investigations, the hypothesis that the fractal dimensions of the putative Martian coastline would be comparable to those of Earth's was accepted, thereby substantiating the claims for the existence of a large northern ocean.
Application of fractal dimensions to study the structure of flocs formed in lime softening process.
Vahedi, Arman; Gorczyca, Beata
2011-01-01
The use of fractal dimensions to study the internal structure and settling of flocs formed in lime softening process was investigated. Fractal dimensions of flocs were measured directly on floc images and indirectly from their settling velocity. An optical microscope with a motorized stage was used to measure the fractal dimensions of lime softening flocs directly on their images in 2 and 3D space. The directly determined fractal dimensions of the lime softening flocs were 1.11-1.25 for floc boundary, 1.82-1.99 for cross-sectional area and 2.6-2.99 for floc volume. The fractal dimension determined indirectly from the flocs settling rates was 1.87 that was different from the 3D fractal dimension determined directly on floc images. This discrepancy is due to the following incorrect assumptions used for fractal dimensions determined from floc settling rates: linear relationship between square settling velocity and floc size (Stokes' Law), Euclidean relationship between floc size and volume, constant fractal dimensions and one primary particle size describing entire population of flocs. Floc settling model incorporating variable floc fractal dimensions as well as variable primary particle size was found to describe the settling velocity of large (>50 μm) lime softening flocs better than Stokes' Law. Settling velocities of smaller flocs (<50 μm) could still be quite well predicted by Stokes' Law. The variation of fractal dimensions with lime floc size in this study indicated that two mechanisms are involved in the formation of these flocs: cluster-cluster aggregation for small flocs (<50 μm) and diffusion-limited aggregation for large flocs (>50 μm). Therefore, the relationship between the floc fractal dimension and floc size appears to be determined by floc formation mechanisms. PMID:20937512
Vahedi, Arman; Gorczyca, Beata
2012-09-01
Here we introduce a distribution of floc fractal dimensions as opposed to a single fractal dimension value into the floc settling velocity model developed in earlier studies. The distribution of fractal dimensions for a single floc size was assumed to cover a range from 1.9 to 3.0. This range was selected based on the theoretically determined fractal dimensions for diffusion-limited and cluster-cluster aggregation. These two aggregation mechanisms are involved in the formation of the lime softening flocs analyzed in this study. Fractal dimensions were generated under the assumption that a floc can have any value of normally distributed fractal dimensions ranging from 1.9-3.0. A range of settling velocities for a single floc size was calculated based on the distribution of fractal dimensions. The assumption of multiple fractal dimensions for a single floc size resulted in a non-unique relationship between the floc size and the floc settling velocity, i.e., several different settling velocities were calculated for one floc size. The settling velocities calculated according to the model ranged from 0 to 10 mm/s (average 2.22 mm/s) for the majority of flocs in the size range of 1-250 μm (average 125 μm). The experimentally measured settling velocities of flocs ranged from 0.1 to 7.1 mm/s (average 2.37 mm/s) for the flocs with equivalent diameters from 10 μm to 260 μm (average 124 μm). Experimentally determined floc settling velocities were predicted well by the floc settling model incorporating distributions of floc fractal dimensions calculated based on the knowledge of the mechanisms of aggregation, i.e., cluster-cluster aggregation and diffusion-limited aggregation. PMID:22673348
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.
Box-covering algorithm for fractal dimension of weighted networks
NASA Astrophysics Data System (ADS)
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-10-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.
Fractal dimension and nonlinear dynamical processes
NASA Astrophysics Data System (ADS)
McCarty, Robert C.; Lindley, John P.
1993-11-01
Mandelbrot, Falconer and others have demonstrated the existence of dimensionally invariant geometrical properties of non-linear dynamical processes known as fractals. Barnsley defines fractal geometry as an extension of classical geometry. Such an extension, however, is not mathematically trivial Of specific interest to those engaged in signal processing is the potential use of fractal geometry to facilitate the analysis of non-linear signal processes often referred to as non-linear time series. Fractal geometry has been used in the modeling of non- linear time series represented by radar signals in the presence of ground clutter or interference generated by spatially distributed reflections around the target or a radar system. It was recognized by Mandelbrot that the fractal geometries represented by man-made objects had different dimensions than the geometries of the familiar objects that abound in nature such as leaves, clouds, ferns, trees, etc. The invariant dimensional property of non-linear processes suggests that in the case of acoustic signals (active or passive) generated within a dispersive medium such as the ocean environment, there exists much rich structure that will aid in the detection and classification of various objects, man-made or natural, within the medium.
Laser light scattering as a probe of fractal colloid aggregates
NASA Technical Reports Server (NTRS)
Weitz, David A.; Lin, M. Y.
1989-01-01
The extensive use of laser light scattering is reviewed, both static and dynamic, in the study of colloid aggregation. Static light scattering enables the study of the fractal structure of the aggregates, while dynamic light scattering enables the study of aggregation kinetics. In addition, both techniques can be combined to demonstrate the universality of the aggregation process. Colloidal aggregates are now well understood and therefore represent an excellent experimental system to use in the study of the physical properties of fractal objects. However, the ultimate size of fractal aggregates is fundamentally limited by gravitational acceleration which will destroy the fractal structure as the size of the aggregates increases. This represents a great opportunity for spaceborne experimentation, where the reduced g will enable the growth of fractal structures of sufficient size for many interesting studies of their physical properties.
Fractal dimensions of flocs between clay particles and HAB organisms
NASA Astrophysics Data System (ADS)
Wang, Hongliang; Yu, Zhiming; Cao, Xihua; Song, Xiuxian
2011-05-01
The impact of harmful algal blooms (HABs) on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency (RE) was influenced by many factors, including clay type and concentration, algal growth stage, and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema costatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay. For the original clay, the two-dimensional fractal dimension ( D 2) was 1.92 and three-dimensional fractal dimension ( D 3) 2.81, while for the modified clay, D 2 was 1.84 and D 3 was 2.50. The addition of polyaluminum chloride (PACl) lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D 2 and D 3. Due to the decrease of D 3, and the increase of the effective sticking coefficient, the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D 3 and D 2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D 3 and D 2 of flocs were the lowest.
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.
Fractal dimension in nonhyperbolic chaotic scattering
NASA Technical Reports Server (NTRS)
Lau, Yun-Tung; Finn, John M.; Ott, Edward
1991-01-01
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
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.
Permeability of collapsed cakes formed by deposition of fractal aggregates upon membrane filtration.
Park, Pyung-Kyu; Lee, Chung-Hak; Lee, Sangho
2006-04-15
We have investigated, theoretically, the physical properties of cake layers formed from aggregates to obtain a better understanding of membrane systems used in conjunction with coagulation/flocculation pretreatment. We developed a model based on fractal theory and incorporated a cake collapse effect to predict the porosity and permeability of the cake layers. The floc size, fractal dimension, and transmembrane pressure were main parameters that we used in these model calculations. We performed experiments using a batch cell device and a confocal laser-scanning microscope to verify the predicted specific cake resistances and porosities under various conditions. Based on the results of the model, the reduction in inter-aggregate porosity is more important than that in intra-aggregate porosity during the cake collapsing process. The specific cake resistance decreases upon increasing the aggregate size and decreasing the fractal dimensions. The modeled porosities and specific cake resistances of the collapsed cake layer agreed reasonably well with those obtained experimentally.
Trabecular Bone Mechanical Properties and Fractal Dimension
NASA Technical Reports Server (NTRS)
Hogan, Harry A.
1996-01-01
Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again
Surface characterization of proteins using multi-fractal property of heat-denatured aggregates
Lahiri, Tapobrata; Mishra, Hrishikesh; Sarkar, Subrata; Misra, Krishna
2008-01-01
Multi-fractal property of heat-denatured protein aggregates (HDPA) is characteristic of its individual form. The visual similarity between digitally generated microscopic images of HDPA with that of surface-image of its individual X-ray structures in protein databank (PDB) displayed using Visual Molecular Dynamics (VMD) viewer is the basis of the study. We deigned experiments to view the fractal nature of proteins at different aggregate scales. Intensity based multi-fractal dimensions (ILMFD) extracted from various planes of digital microscopic images of protein aggregates were used to characterize HDPA into different classes. Moreover, the ILMFD parameters extracted from aggregates show similar classification pattern to digital images of protein surface displayed by VMD viewer using PDB entry. We discuss the use of irregular patterns of heat-denatured aggregate proteins to understand various surface properties in native proteins. PMID:18795110
Pyramidal fractal dimension for high resolution images.
Mayrhofer-Reinhartshuber, Michael; Ahammer, Helmut
2016-07-01
Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥1024×1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.
Pyramidal fractal dimension for high resolution images
NASA Astrophysics Data System (ADS)
Mayrhofer-Reinhartshuber, Michael; Ahammer, Helmut
2016-07-01
Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥1024 ×1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.
Pyramidal fractal dimension for high resolution images.
Mayrhofer-Reinhartshuber, Michael; Ahammer, Helmut
2016-07-01
Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥1024×1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images. PMID:27475069
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 .
Estimating fractal dimension of medical images
NASA Astrophysics Data System (ADS)
Penn, Alan I.; Loew, Murray H.
1996-04-01
Box counting (BC) is widely used to estimate the fractal dimension (fd) of medical images on the basis of a finite set of pixel data. The fd is then used as a feature to discriminate between healthy and unhealthy conditions. We show that BC is ineffective when used on small data sets and give examples of published studies in which researchers have obtained contradictory and flawed results by using BC to estimate the fd of data-limited medical images. We present a new method for estimating fd of data-limited medical images. In the new method, fractal interpolation functions (FIFs) are used to generate self-affine models of the underlying image; each model, upon discretization, approximates the original data points. The fd of each FIF is analytically evaluated. The mean of the fds of the FIFs is the estimate of the fd of the original data. The standard deviation of the fds of the FIFs is a confidence measure of the estimate. The goodness-of-fit of the discretized models to the original data is a measure of self-affinity of the original data. In a test case, the new method generated a stable estimate of fd of a rib edge in a standard chest x-ray; box counting failed to generate a meaningful estimate of the same image.
NASA Astrophysics Data System (ADS)
Chang, Kuo-En; Lin, Tang-Huang; Lien, Wei-Hung
2015-04-01
Anthropogenic pollutants or smoke from biomass burning contribute significantly to global particle aggregation emissions, yet their aggregate formation and resulting ensemble optical properties are poorly understood and parameterized in climate models. Particle aggregation refers to formation of clusters in a colloidal suspension. In clustering algorithms, many parameters, such as fractal dimension, number of monomers, radius of monomer, and refractive index real part and image part, will alter the geometries and characteristics of the fractal aggregation and change ensemble optical properties further. The cluster-cluster aggregation algorithm (CCA) is used to specify the geometries of soot and haze particles. In addition, the Generalized Multi-particle Mie (GMM) method is utilized to compute the Mie solution from a single particle to the multi particle case. This computer code for the calculation of the scattering by an aggregate of spheres in a fixed orientation and the experimental data have been made publicly available. This study for the model inputs of optical determination of the monomer radius, the number of monomers per cluster, and the fractal dimension is presented. The main aim in this study is to analyze and contrast several parameters of cluster aggregation aforementioned which demonstrate significant differences of optical properties using the GMM method finally. Keywords: optical properties, fractal aggregation, GMM, CCA
Fractal 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 dimension and unscreened angles measured for radial viscous fingering.
Praud, Olivier; Swinney, Harry L
2005-07-01
We have examined fractal patterns formed by the injection of air into oil in a thin (0.127 mm) layer contained between two cylindrical glass plates of 288 mm diameter (a Hele-Shaw cell), for pressure differences in the range 0.25 < or = DeltaP < or = 1.75 atm. We find that an asymptotic structure is reached at large values of the ratio r/b, where r is the pattern radius and b the gap between the plates. Both the driving force and the size of the pattern, which reaches r/b = 900, are far larger than in past experiments. The fractal dimension D0 of the pattern for large r/b is 1.70 +/- 0.02. Further, the generalized dimensions D(q) of the pattern are independent of q , D(q) approximately 1.70 for the range examined, -11 < q < 17; thus the pattern is self-similar within the experimental uncertainty. The results for D(q) agree well with recent calculations for diffusion-limited aggregation (DLA) clusters. We have also measured the probability distribution of unscreened angles. At late times, the distribution approaches a universal (i.e., forcing and size-independent) asymptotic form that has mean 145 degrees Celsius and standard deviation 36 degrees Celsius. These results indicate that the distribution function for the unscreened angle is an invariant property of the growth process. PMID:16089960
Fractal dimension of mesospheric radar backscatter at 2. 75 MHz
Hall, C.; Armstrong, R.J.; La Hoz, C. )
1991-04-01
The authors identified the fractal dimension of radar returns from the mesopause region at 2.75 MHz. The input dataset was a time series of echo amplitude at a discrete height obtained from a partial reflection radar operating at Ramfjordmoen in northern Norway. Two different algorithms both of which yield approximations to the fractal dimension have been employed and give almost identical results. The radar echo dataset in question exhibits a dimension of between 7 and 8.
Fractal dimension of cerebral surfaces using magnetic resonance images
Majumdar, S.; Prasad, R.R.
1988-11-01
The calculation of the fractal dimension of the surface bounded by the grey matter in the normal human brain using axial, sagittal, and coronal cross-sectional magnetic resonance (MR) images is presented. The fractal dimension in this case is a measure of the convolutedness of this cerebral surface. It is proposed that the fractal dimension, a feature that may be extracted from MR images, may potentially be used for image analysis, quantitative tissue characterization, and as a feature to monitor and identify cerebral abnormalities and developmental changes.
Fractal aggregation growth and the surrounding diffusion field
NASA Astrophysics Data System (ADS)
Miyashita, Satoru; Saito, Yukio; Uwaha, Makio
2005-10-01
Silver metal trees grow and form a forest at the edge of a Cu plate in the AgNO3 water solution in a two-dimensional ( d=2) cell. The local structure of the forest is similar to that of the diffusion-limited aggregation (DLA), but the whole pattern approaches a uniform structure. Its growth dynamics is characterized by the fractal dimension Df of DLA. Time-dependence of the tip height is found to satisfy the scaling relation with the solute concentration c, and the asymptotic growth velocity V is consistent with the power law V˜c expected from the theory. The thickness ξc of the diffusion boundary layer is measured by the Michelson interferometry, and the scaling relation is also confirmed.
An extended fractal growth regime in the diffusion limited aggregation including edge diffusion
NASA Astrophysics Data System (ADS)
Ghosh, Aritra; Batabyal, R.; Das, G. P.; Dev, B. N.
2016-01-01
We have investigated on-lattice diffusion limited aggregation (DLA) involving edge diffusion and compared the results with the standard DLA model. For both cases, we observe the existence of a crossover from the fractal to the compact regime as a function of sticking coefficient. However, our modified DLA model including edge diffusion shows an extended fractal growth regime like an earlier theoretical result using realistic growth models and physical parameters [Zhang et al., Phys. Rev. Lett. 73 (1994) 1829]. While the results of Zhang et al. showed the existence of the extended fractal growth regime only on triangular but not on square lattices, we find its existence on the square lattice. There is experimental evidence of this growth regime on a square lattice. The standard DLA model cannot characterize fractal morphology as the fractal dimension (Hausdorff dimension, DH) is insensitive to morphology. It also predicts DH = DP (the perimeter dimension). For the usual fractal structures, observed in growth experiments on surfaces, the perimeter dimension can differ significantly (DH ≠ DP) depending on the morphology. Our modified DLA model shows minor sensitivity to this difference.
Fractal aggregates induced by liposome-liposome interaction in the presence of Ca2+.
Sabín, J; Prieto, G; Ruso, J M; Sarmiento, F
2007-10-01
We present a study of the fractal dimension of clusters of large unilamellar vesicles (LUVs) formed by egg yolk phosphatidylcholine (EYPC), dimyristoylphosphocholine (DMPC) and dipalmitoylphosphocholine (DPPC) induced by Ca2+ . Fractal dimensions were calculated by application of two methods, measuring the angular dependency of the light scattered by the clusters and following the evolution of the cluster size. In all cases, the fractal dimensions fell in the range from 2.1 to 1.8, corresponding to two regimes: diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA). Whereas DMPC clusters showed a typical transition from the RLCA to the DLCA aggregation, EYPC exhibited an unusual behaviour, since the aggregation was limited for a higher concentration than the critical aggregation concentration. The behaviour of DPPC was intermediate, with a transition from the RLCA to the DLCA regimes with cluster sizes depending on Ca2+ concentration. Studies on the reversibility of the aggregates show that EYPC and DPPC clusters can be re-dispersed by dilution with water. DMPC does not present reversibility. Reversibility is evidence of the existence of secondary minima in the DLVO potential between two liposomes. To predict these secondary minima, a correction of the DLVO model was necessary taking into account a repulsive force of hydration.
Fractal dimensions of rampart impact craters on Mars
NASA Technical Reports Server (NTRS)
Ching, Delwyn; Taylor, G. Jeffrey; Mouginis-Mark, Peter; Bruno, Barbara C.
1993-01-01
Ejecta blanket morphologies of Martian rampart craters may yield important clues to the atmospheric densities during impact, and the nature of target materials (e.g., hard rock, fine-grained sediments, presence of volatiles). In general, the morphologies of such craters suggest emplacement by a fluidized, ground hugging flow instead of ballistic emplacement by dry ejecta. We have quantitatively characterized the shape of the margins of the ejecta blankets of 15 rampart craters using fractal geometry. Our preliminary results suggest that the craters are fractals and are self-similar over scales of approximately 0.1 km to 30 km. Fractal dimensions (a measure of the extent to which a line fills a plane) range from 1.06 to 1.31. No correlations of fractal dimension with target type, elevation, or crater size were observed, though the data base is small. The range in fractal dimension and lack of correlation may be due to a complex interplay of target properties (grain size, volatile content), atmospheric pressure, and crater size. The mere fact that the ejecta margins are fractals, however, indicates that viscosity and yield strength of the ejecta were at least as low as those of basalts, because silicic lava flows are not generally fractals.
Fractal structure and the dynamics of aggregation of synthetic melanin in low pH aqueous solutions
Huang, J.S.; Sung, J.; Eisner, M.; Moss, S.C.; Gallas, J.
1989-01-01
We have used static and dynamic light scattering to study the dynamics of aggregation of synthetic melanin, an amorphous biopolymeric substance, in low pH aqueous solution. We have found that, depending on the final pH value of the solutions, there existed two regimes of the aggregation kinetics, one corresponding to diffusion limited aggregation (DLA), and the other corresponding to reaction limited aggregation (RLA). The precipitates formed in these two regimes can be characterized by fractal structures. We have found fractal dimensions of d/sub f/ = 1.8 for the DLA clusters and d/sub f/ = 2.2 for the RLA clusters. These results agree well with the proposed limits of the fractal dimensions of the gold aggregates formed in aqueous solutions by Weitz et al.
A simple method for estimating the fractal dimension from digital images: The compression dimension
NASA Astrophysics Data System (ADS)
Chamorro-Posada, Pedro
2016-10-01
The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.
Size dependent fractal aggregation mediated through surfactant in silica nanoparticle solution
NASA Astrophysics Data System (ADS)
Kumar, Sugam; Aswal, V. K.; Kohlbrecher, J.
2012-06-01
Small-angle neutron scattering (SANS) has been used to study aggregation of anionic silica nanoparticles in presence of cationic surfactant (DTAB) in aqueous solution. The measurements were carried out for different sizes of nanoparticles (8.2, 16.4 and 26.4 nm) at fixed (1 wt%) nanoparticles and surfactant concentration. It is found that the adsorption of surfactant micelles on the silica nanoparticles leads to the aggregation of nanoparticles, which is characterized by a fractal structure. The number of adsorbed micelles on nanoparticle increases from 7 to 152 with the increase in the size of the nanoparticle from 8.2 to 26.4 nm, whereas interestingly the fractal dimension remains same. The aggregate morphology in these systems is expected to be governed by the diffusion limited aggregation.
Roth, Eric J; Gilbert, Benjamin; Mays, David C
2015-10-20
Experiments reveal a wide discrepancy between the permeability of porous media containing colloid deposits and the available predictive equations. Evidence suggests that this discrepancy results, in part, from the predictive equations failing to account for colloid deposit morphology. This article reports a series of experiments using static light scattering (SLS) to characterize colloid deposit morphology within refractive index matched (RIM) porous media during flow through a column. Real time measurements of permeability, specific deposit, deposit fractal dimension, and deposit radius of gyration, at different vertical positions, were conducted with initially clean porous media at various ionic strengths and fluid velocities. Decreased permeability (i.e., increased clogging) corresponded with higher specific deposit, lower fractal dimension, and smaller radius of gyration. During deposition, fractal dimension, radius of gyration, and permeability decreased with increasing specific deposit. During flushing with colloid-free fluid, these trends reversed, with increased fractal dimension, radius of gyration, and permeability. These observations suggest a deposition scenario in which large and uniform aggregates become deposits, which reduce porosity, lead to higher fluid shear forces, which then decompose the deposits, filling the pore space with small and dendritic fragments of aggregate.
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.
Fractal Aggregates in Tennis Ball Systems
ERIC Educational Resources Information Center
Sabin, J.; Bandin, M.; Prieto, G.; Sarmiento, F.
2009-01-01
We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the…
Relationship between Fractal Dimension and Agreeability of Facial Imagery
NASA Astrophysics Data System (ADS)
Oyama-Higa, Mayumi; Miao, Tiejun; Ito, Tasuo
2007-11-01
Why do people feel happy and good or equivalently empathize more, with smiling face imageries than with ones of expressionless face? To understand what the essential factors are underlying imageries in relating to the feelings, we conducted an experiment by 84 subjects asked to estimate the degree of agreeability about expressionless and smiling facial images taken from 23 young persons to whom the subjects were no any pre-acquired knowledge. Images were presented one at a time to each subject who was asked to rank agreeability on a scale from 1 to 10. Fractal dimensions of facial images were obtained in order to characterize the complexity of the imageries by using of two types of fractal analysis methods, i.e., planar and cubic analysis methods, respectively. The results show a significant difference in the fractal dimension values between expressionless faces and smiling ones. Furthermore, we found a well correlation between the degree of agreeability and fractal dimensions, implying that the fractal dimension optically obtained in relation to complexity in imagery information is useful to characterize the psychological processes of cognition and awareness.
EEG signal features extraction based on fractal dimension.
Finotello, Francesca; Scarpa, Fabio; Zanon, Mattia
2015-08-01
The spread of electroencephalography (EEG) in countless applications has fostered the development of new techniques for extracting synthetic and informative features from EEG signals. However, the definition of an effective feature set depends on the specific problem to be addressed and is currently an active field of research. In this work, we investigated the application of features based on fractal dimension to a problem of sleep identification from EEG data. We demonstrated that features based on fractal dimension, including two novel indices defined in this work, add valuable information to standard EEG features and significantly improve sleep identification performance. PMID:26737209
Comparison of Two Numerical Methods for Computing Fractal Dimensions
NASA Astrophysics Data System (ADS)
Shiozawa, Yui; Miller, Bruce; Rouet, Jean-Louis
2012-10-01
From cosmology to economics, the examples of fractals can be found virtually everywhere. However, since few fractals permit the analytical evaluation of generalized fractal dimensions or R'enyi dimensions, the search for effective numerical methods is inevitable. In this project two promising numerical methods for obtaining generalized fractal dimensions, based on the distribution of distances within a set, are examined. They can be applied, in principle, to any set even if no closed-form expression is available. The biggest advantage of these methods is their ability to generate a spectrum of generalized dimensions almost simultaneously. It should be noted that this feature is essential to the analysis of multifractals. As a test of their effectiveness, here the methods were applied to the generalized Cantor set and the multiplicative binomial process. The generalized dimensions of both sets can be readily derived analytically, thus enabling the accuracy of the numerical methods to be verified. Here we will present a comparison of the analytical results and the predictions of the methods. We will show that, while they are effective, care must be taken in their interpretation.
Fractal dimension of microbead assemblies used for protein detection
Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos
2014-01-01
We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1μm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70–1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles. PMID:25195559
Fractal dimension of microbead assemblies used for protein detection.
Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos; Kopelman, Raoul
2014-11-10
We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 μm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles.
Measuring border irregularities of skin lesions using fractal dimensions
NASA Astrophysics Data System (ADS)
Ng, Vincent T. Y.; Lee, Tim K.
1996-09-01
Malignant melanoma is the most common cancer in people less than 35 years of age and incident rates are increasing by approximately 5 percent per annum in many white populations, including British Columbia, Canada. In 1994, a clinical study has been established to digitize melanocytic lesions under a controlled environment. Lesions are digitized from patients who are referred to the Colored Pigment Lesion Clinic in the University of British Columbia. In this paper, we investigate how to use fractal dimensions (FDs) in measuring the irregularity of a skin lesion. In a previous project, we have experimented with 6 different methods to calculate fractal dimensions on a small number of images of skin lesions, and the simple box-counting method performed the best. However, the method did not exploit the intensity information of the images. With the new set of images which are digitized under the controlled environment, we utilize the differential box counting method to exploit such information. Four FD measures, including the direct FD, the horizontal and the vertical smoothing FDs, and the multi- fractal dimension of order two, are calculated based on the original color images. In addition, these 4 FD features are repeatedly calculate for the blue band of the images. This paper reports the different features through the calculations of the fractal dimensions and compares their differentiation power in the use of diagnosis of images of skin lesions.
An image retrieval system based on fractal dimension.
Yao, Min; Yi, Wen-Sheng; Shen, Bin; Dai, Hong-Hua
2003-01-01
This paper presents a new kind of image retrieval system which obtains the feature vectors of images by estimating their fractal dimension; and at the same time establishes a tree-structure image database. After preprocessing and feature extracting, a given image is matched with the standard images in the image database using a hierarchical method of image indexing.
Pandey, Apoorva; Chakrabarty, Rajan K; Liu, Li; Mishchenko, Michael I
2015-11-30
Soot aggregates (SAs)-fractal clusters of small, spherical carbonaceous monomers-modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension Df = 1.8) and aged (Df = 2.6) SAs at 550 nm wavelength using the numerically-exact superposition T-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships. PMID:26698786
NASA Technical Reports Server (NTRS)
Pandey, Apoorva; Chakrabarty, Rajan K.; Liu, Li; Mishchenko, Michael I.
2015-01-01
Soot aggregates (SAs)-fractal clusters of small, spherical carbonaceous monomers-modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension Df = 1.8) and aged (Df = 2.6) SAs at 550 nm wavelength using the numericallyexact superposition T-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships.
Bałazy, Anna; Podgórski, Albert
2007-07-15
Nonspherical particles, such as fractal-like aggregates emitted by diesel engines, are commonly met in the ambient air. Some of them are believed to be carcinogenic to humans, thus their efficient removal is of crucial practical importance. A fibrous filter is the device commonly used for aerosol purification but the literature lacks experimental data concerning aggregates filtration. Effect of aggregates' parameters (fractal dimension, primary particle radius) as well as fiber diameter and air velocity on the filtration efficiency is investigated theoretically using the modified Brownian dynamics method. Three different expressions for the friction coefficient evaluation for the aggregates were examined. The results obtained indicate that structure of an aggregate, filter structure and process conditions strongly influence the aggregates deposition efficiency, which significantly differs from the values determined for mass-equivalent spherical particles. The results determined using the Brownian dynamics approach were compared with the values calculated using classical single fiber theory and noticeable discrepancy was observed for the most penetrating particles, while both approaches agree for the limiting cases of small or large particles. Peclet number based on the mobility radius and the interception parameter based on the outer radius are the proper criteria to describe diffusional and deterministic deposition of aggregates.
Liver ultrasound image classification by using fractal dimension of edge
NASA Astrophysics Data System (ADS)
Moldovanu, Simona; Bibicu, Dorin; Moraru, Luminita
2012-08-01
Medical ultrasound image edge detection is an important component in increasing the number of application of segmentation, and hence it has been subject of many studies in the literature. In this study, we have classified the liver ultrasound images (US) combining Canny and Sobel edge detectors with fractal analysis in order to provide an indicator about of the US images roughness. We intend to provide a classification rule of the focal liver lesions as: cirrhotic liver, liver hemangioma and healthy liver. For edges detection the Canny and Sobel operators were used. Fractal analyses have been applied for texture analysis and classification of focal liver lesions according to fractal dimension (FD) determined by using the Box Counting method. To assess the performance and accuracy rate of the proposed method the contrast-to-noise (CNR) is analyzed.
Ferretti, R.; Zhang, J.; Buffle, J.
1998-12-15
The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (M{sub w} = 1.36 {times} 10{sup 6}, M{sub w}/M{sub n} = 1.53; M{sub w} = 3.69 {times} 10{sup 4}, M{sub w}/M{sub n} = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, D{sub f} = 1.84 {+-} 0.02 and 1.73 {+-} 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension D{sub f} {approx} 1.8. Moreover, for all three regimes, the D{sub f} values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure.
Segmentation of magnetic resonance image using fractal dimension
NASA Astrophysics Data System (ADS)
Yau, Joseph K. K.; Wong, Sau-hoi; Chan, Kwok-Leung
1997-04-01
In recent years, much research has been conducted in the three-dimensional visualization of medical image. This requires a good segmentation technique. Many early works use first-order and second-order statistics. First-order statistical parameters can be calculated quickly but their effectiveness is influenced by many factors such as illumination, contrast and random noise of the image. Second-order statistical parameters, such as spatial gray level co-occurrence matrices statistics, take longer time to compute but can extract the textural information. In this investigating, two different parameters, namely the entropy and the fractal dimension, are employed to perform segmentation of the magnetic resonance images of the head of a male cadaver. The entropy is calculated from the spatial gray level co-occurrence matrices. The fractal dimension is calculated by the reticular cell counting method. Several regions of the human head are chosen for analysis. They are the bone, gyrus and lobe. Results show that the parameters are able to segment different types of tissue. The entropy gives very good result but it requires very long computation time and large amount of memory. The performance of the fractal dimension is comparable with the entropy. It is simple to estimate and demands lesser memory space.
Fractal dimension analysis of malignant and benign endobronchial ultrasound nodes
2014-01-01
Background Endobronchial ultrasonography (EBUS) has been applied as a routine procedure for the diagnostic of hiliar and mediastinal nodes. The authors assessed the relationship between the echographic appearance of mediastinal nodes, based on endobronchial ultrasound images, and the likelihood of malignancy. Methods The images of twelve malignant and eleven benign nodes were evaluated. A previous processing method was applied to improve the quality of the images and to enhance the details. Texture and morphology parameters analyzed were: the image texture of the echographies and a fractal dimension that expressed the relationship between area and perimeter of the structures that appear in the image, and characterizes the convoluted inner structure of the hiliar and mediastinal nodes. Results Processed images showed that relationship between log perimeter and log area of hilar nodes was lineal (i.e. perimeter vs. area follow a power law). Fractal dimension was lower in the malignant nodes compared with non-malignant nodes (1.47(0.09), 1.53(0.10) mean(SD), Mann–Whitney U test p < 0.05)). Conclusion Fractal dimension of ultrasonographic images of mediastinal nodes obtained through endobronchial ultrasound differ in malignant nodes from non-malignant. This parameter could differentiate malignat and non-malignat mediastinic and hiliar nodes. PMID:24920158
Fractal Dimension in Eeg Signals during Muscle Fatigue
NASA Astrophysics Data System (ADS)
Huang, Haibin; Yao, Bin; Yue, Guang; Brown, Robert; Jing, Liu
2003-10-01
Fractal dimension (FD) has been successfully used to characterize signals in the format of time series. In this study, we calculated FD of EEG signals recorded during human muscle fatigue as a measure of changes in the EEG signal complexity along fatigue. Subjects performed 200 intermittent handgrip contractions at 100contraction level. Each contraction lasted 2 s, followed by a 5-s rest. EEG data were recorded from the scalp along with handgrip force and muscle EMG signals. The FD computation was based on measurements of the length (Lk) of the signal at 6 different temporal resolutions (k = 1, 2, ¡, 6). FD was determined from the relationship between Lk and k using the least square fit. The results showed that: (1) EEG fractal dimension associated with the motor performance was significantly higher than that during the rest period; (2) changes in the fractal dimension along the process of fatigue showed a significant correlation with the decline in force and EMG signals.
Fractal dimension of particle showers measured in a highly granular calorimeter.
Ruan, Manqi; Jeans, Daniel; Boudry, Vincent; Brient, Jean-Claude; Videau, Henri
2014-01-10
We explore the fractal nature of particle showers using Monte Carlo simulation. We define the fractal dimension of showers measured in a high granularity calorimeter designed for a future lepton collider. The shower fractal dimension reveals detailed information of the spatial configuration of the shower. It is found to be characteristic of the type of interaction and highly sensitive to the nature of the incident particle. Using the shower fractal dimension, we demonstrate a particle identification algorithm that can efficiently separate electromagnetic showers, hadronic showers, and nonshowering tracks. We also find a logarithmic dependence of the shower fractal dimension on the particle energy.
Fractal model of consolidation of weakly aggregated colloidal dispersions
NASA Astrophysics Data System (ADS)
Potanin, A. A.; Russel, W. B.
1996-04-01
A microrheological model for the consolidation of aggregated colloidal dispersion is presented. Interparticle forces are responsible for the formation of the fractal-like network of particles chains. Aggregation is supposed to be weak in that the chains are gradually created and annihilated. In the framework of this model a constitutive equation is proposed which includes linear dependence of the normal stress upon consolidation rate at low stresses and yieldlike behavior as the stress reaches critical value. Our constitutive equation thus generalizes earlier models based on the compressive yield stress concept by taking into account the deformation in the low-compression limit (analog to the Newtonian flow in the low-shear limit). The constitutive equation is employed to calculate numerically concentration profiles of consolidating samples in better agreement with the experimental data for weakly aggregated alumina dispersions.
Surface evaluation by estimation of fractal dimension and statistical tools.
Hotar, Vlastimil; Salac, Petr
2014-01-01
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380
Surface Evaluation by Estimation of Fractal Dimension and Statistical Tools
Salac, Petr
2014-01-01
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380
NASA Astrophysics Data System (ADS)
Li, Jian-Hua; Yu, Bo-Ming; Zou, Ming-Qing
2009-11-01
We report a model for the fractal dimension Ds of rough surfaces based on the fractal distribution of roughness elements on surfaces and the fractal character of surface profiles. The proposed model for the fractal dimension Ds is expressed as a function of the fractal dimensions D for conic roughness diameter/height and Dp for surface profile, maximum roughness base diameter λmax, the ratio β of conic roughness height to its base radius as well as the ratio λminλmax of the minimum to the maximal base diameter.
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.
Multiparticle sintering dynamics: from fractal-like aggregates to compact structures.
Eggersdorfer, Max L; Kadau, Dirk; Herrmann, Hans J; Pratsinis, Sotiris E
2011-05-17
Multiparticle sintering is encountered in almost all high temperature processes for material synthesis (titania, silica, and nickel) and energy generation (e.g., fly ash formation) resulting in aggregates of primary particles (hard- or sinter-bonded agglomerates). This mechanism of particle growth is investigated quantitatively by mass and energy balances during viscous sintering of amorphous aerosol materials (e.g., SiO(2) and polymers) that typically have a distribution of sizes and complex morphology. This model is validated at limited cases of sintering between two (equally or unequally sized) particles, and chains of particles. The evolution of morphology, surface area and radii of gyration of multiparticle aggregates are elucidated for various sizes and initial fractal dimension. For each of these structures that had been generated by diffusion limited (DLA), cluster-cluster (DLCA), and ballistic particle-cluster agglomeration (BPCA) the surface area evolution is monitored and found to scale differently than that of the radius of gyration (moment of inertia). Expressions are proposed for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering. These expressions are important in design of aerosol processes with population balance equations (PBE) and/or fluid dynamic simulations for material synthesis or minimization and even suppression of particle formation. PMID:21488641
Speech Emotion Recognition Based on Parametric Filter and Fractal Dimension
NASA Astrophysics Data System (ADS)
Mao, Xia; Chen, Lijiang
In this paper, we propose a new method that employs two novel features, correlation density (Cd) and fractal dimension (Fd), to recognize emotional states contained in speech. The former feature obtained by a list of parametric filters reflects the broad frequency components and the fine structure of lower frequency components, contributed by unvoiced phones and voiced phones, respectively; the latter feature indicates the non-linearity and self-similarity of a speech signal. Comparative experiments based on Hidden Markov Model and K Nearest Neighbor methods are carried out. The results show that Cd and Fd are much more closely related with emotional expression than the features commonly used.
Multiorder boundaries among discrete domains: relative fractal dimension and others.
Xuan, Qi; Du, Fang; Wu, Tie-Jun
2010-03-01
In nature and society, most of competitions take place on the boundaries among a group of domains where different individuals or colonies share common resources; therefore, it is widely believed that domain boundaries play important roles in the evolution of many complex systems. Here, we first give a definition for multiorder boundaries among discrete domains and then propose a general method to calculate their relative fractal dimension, i.e., the ratio of the fractal dimension of the boundaries versus that of the domains themselves. Through analyzing three types of real-world discrete domains, several interesting results are revealed. For example, the limitation on the number of domains that an individual can join in may produce longer boundaries indicating more cruel competitions among the domains. Besides, the individuals with more social links are always considered more important in social networks, and it is found that these individuals as valuable resources of social domains are always centralized on the boundaries of higher order. PMID:20370288
Zhang, Wei; Li, Cai-Ting; Wei, Xian-Xun; Gao, Hong-Liang; Wen, Qing-Bo; Fan, Xiao-Peng; Shu, Xin; Zeng, Guang-Ming; Wei, Wei; Zhai, Yun-Bo; He, Yi-De; Li, Shan-Hong
2011-05-15
A cake collapse model was developed by taking the combined effects of fractal dimension, relaxation ratio, coordination number, and aggregate diameter into consideration. The cake porosity including intraaggregate and interaggregate porosities was modeled successively by three typical coordination numbers (n = 6, 8, and 12). Accordingly, an inversion method made it possible to deduce the coordination number using the measured cake porosities, and the reverse-calculated value with minimum error and the corresponding relaxation ratios were applied as the parameters for the model. As a result, the profiles of intraaggregate and interaggregate porosities and cake porosity were respectively predicted in contrast to the integrated variation of the relaxation ratio and the fractal dimension. Furthermore, a comparison between the model predictions of the cake pressure drop gradients with and without aggregate compression was conducted to validate the presence of cake collapse. The results show that the predictions based on the proposed collapse model are in agreement with the experiments, and the coordination number is one of the key factors that must be incorporated into the cake collapse models.
Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2014-09-01
The correlation between the void ratio of swelled montmorillonite and the vertical overburden pressure can be expressed as {e}{ m} = Kp{ s}{D{ s}-3}. The surface fractal dimension Ds of five bentonites were estimated from the swelling deformation tests according to this fractal correlation. The reliability of surface fractal dimension obtained from the swelling deformation test was confirmed by nitrogen adsorption test, with identical values of surface fractal dimension obtained from both tests. The surface fractal dimension can also be used to estimate the swelling deformation of bentonite, after calculating the swelling coefficient K from the parameters of diffuse double layer (DDL) model in the osmotic swelling phase. Comparison of the model predictions with a number of experimental results on swelling deformation of both Na dominant and Ca dominant bentonites suggests that the surface fractal model works excellent in the cases tested.
NASA Astrophysics Data System (ADS)
Maggi, Federico
2015-09-01
A comprehensive set of experiments was carried out to investigate the effect of the fractal architecture of granular aggregates on the free-fall acceleration through a still water column. Test aggregates were first generated numerically with a method that allowed to control the fractal dimension d and, next, three stochastic replicates were lithographically fabricated for each of six values of d ranging between 1.9 and 2.7. The recorded position, velocity and acceleration served to analyze their dynamics in the Reynolds and Galilei number space, and to calculate the momentum rate of change and the intensity of drag (viscous and impact) and inertial forces (added mass and Basset-Bousinnesq). Analysis of these forces highlighted a strong dependence on d; additionally, integration of these forces in the particle momentum equation allowed to identify an additional resistance Rx that showed a strong correlation with d. A correlation analysis of Rx with various scaling laws combining velocity and acceleration suggested that Rx could be described by a nonlinear drag force and a force intermediate between drag and inertia. It was therefore concluded that irregular granular fractal aggregates accelerating in water are subject to highly complex and nonlinear hydrodynamic effects caused by surface roughness and volume porosity, and that these effects have tight connection with the internal and external fractal characteristics of the aggregates.
Fractal aggregation of ZnO nanoparticles under different aqueous solution chemistries
NASA Astrophysics Data System (ADS)
Zhou, D.; Keller, A. A.
2009-12-01
The aggregation of ZnO nanoparticles influences not only their environmental transport but also their toxicity. In natural aquatic systems, the ubiquitous presence of natural organic matter (NOM) can lead to interactions with released ZnO nanoparticles (NPs) and influence their transport. In this study, the aggregation kinetics of ZnO with and without NOM under different ionic strength and pH were examined by both time-resolved dynamic light scattering and sedimentation experiments. ZnO aggregates faster as the pH approaches its point of zero charge. The role of ionic strength role was examined by determining the reaction-limited cluster regime (RLCR) and diffusion-limited cluster regime (DLCR). The critical coagulation concentration (CCC) was determined as 25mM using NaCl as the electrolyte at pH 8. A higher initial ZnO concentration leads to faster aggregation. DLVO calculations agreed well with the experimental data. At high NOM concentration, ZnO aggregation was significantly slowed. However, at low NOM concentration, bridging effect was observed. Finally, the fractal dimensions of ZnO aggregates under different conditions were determined by static light scattering (SLS).
Fractal dimension, walk dimension and conductivity exponent of karst networks around Tulum.
NASA Astrophysics Data System (ADS)
Hendrick, Martin; Renard, Philippe
2016-06-01
Understanding the complex structure of karst networks is a challenge. In this work, we characterize the fractal properties of some of the largest coastal karst network systems in the world. They are located near the town of Tulum (Quintana Roo, Mexico). Their fractal dimension d_f, conductivity exponent tilde{mu} and walk dimension d_w are estimated using real space renormalization and numerical simulations. We obtain the following values for these exponents: d_f≈ 1.5, d_w≈ 2.4, tilde{mu}≈ 0.9. We observe that the Einstein relation holds for these structures tilde{mu} ≈ -d_f + d_w. These results indicate that coastal karst networks can be considered as critical systems and this provides some foundations to model them within this framework.
Analysis of fractal dimensions of rat bones from film and digital images
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.
2001-01-01
OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.
Fractal Dimensions of In Vitro Tumor Cell Proliferation
Lambrou, George I.
2015-01-01
Biological systems are characterized by their potential for dynamic adaptation. One of the challenges for systems biology approaches is their contribution towards the understanding of the dynamics of a growing cell population. Conceptualizing these dynamics in tumor models could help us understand the steps leading to the initiation of the disease and its progression. In vitro models are useful in answering this question by providing information over the spatiotemporal nature of such dynamics. In the present work, we used physical quantities such as growth rate, velocity, and acceleration for the cellular proliferation and identified the fractal structures in tumor cell proliferation dynamics. We provide evidence that the rate of cellular proliferation is of nonlinear nature and exhibits oscillatory behavior. We also calculated the fractal dimensions of our cellular system. Our results show that the temporal transitions from one state to the other also follow nonlinear dynamics. Furthermore, we calculated self-similarity in cellular proliferation, providing the basis for further investigation in this topic. Such systems biology approaches are very useful in understanding the nature of cellular proliferation and growth. From a clinical point of view, our results may be applicable not only to primary tumors but also to tumor metastases. PMID:25883653
Nonlinear analysis of EEG in major depression with fractal dimensions.
Akar, Saime A; Kara, Sadik; Agambayev, Sumeyra; Bilgic, Vedat
2015-08-01
Major depressive disorder (MDD) is a psychiatric mood disorder characterized by cognitive and functional impairments in attention, concentration, learning and memory. In order to investigate and understand its underlying neural activities and pathophysiology, EEG methodologies can be used. In this study, we estimated the nonlinearity features of EEG in MDD patients to assess the dynamical properties underlying the frontal and parietal brain activity. EEG data were obtained from 16 patients and 15 matched healthy controls. A wavelet-chaos methodology was used for data analysis. First, EEGs of subjects were decomposed into 5 EEG sub-bands by discrete wavelet transform. Then, both the Katz's and Higuchi's fractal dimensions (KFD and HFD) were calculated as complexity measures for full-band and sub-bands EEGs. Last, two-way analyses of variances were used to test EEG complexity differences on each fractality measures. As a result, a significantly increased complexity was found in both parietal and frontal regions of MDD patients. This significantly increased complexity was observed not only in full-band activity but also in beta and gamma sub-bands of EEG. The findings of the present study indicate the possibility of using the wavelet-chaos methodology to discriminate the EEGs of MDD patients from healthy controls. PMID:26738004
Fractal dimension analysis of cerebellum in Chiari Malformation type I.
Akar, Engin; Kara, Sadık; Akdemir, Hidayet; Kırış, Adem
2015-09-01
Chiari Malformation type I (CM-I) is a serious neurological disorder that is characterized by hindbrain herniation. Our aim was to evaluate the usefulness of fractal analysis in CM-I patients. To examine the morphological complexity features of this disorder, fractal dimension (FD) of cerebellar regions were estimated from magnetic resonance images (MRI) of 17 patients with CM-I and 16 healthy control subjects in this study. The areas of white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF) were calculated and the corresponding FD values were computed using a 2D box-counting method in both groups. The results indicated that CM-I patients had significantly higher (p<0.05) FD values of GM, WM and CSF tissues compared to control group. According to the results of correlation analysis between FD values and the corresponding area values, FD and area values of GM tissues in the patients group were found to be correlated. The results of the present study suggest that FD values of cerebellar regions may be a discriminative feature and a useful marker for investigation of abnormalities in the cerebellum of CM-I patients. Further studies to explore the changes in cerebellar regions with the help of 3D FD analysis and volumetric calculations should be performed as a future work.
Entanglement entropy of highly degenerate States and fractal dimensions.
Castro-Alvaredo, Olalla A; Doyon, Benjamin
2012-03-23
We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-1/2 representation, pointing in various directions, form a sphere. We show that for subsystems with a large number m of local degrees of freedom, the entanglement entropy diverges as d/2 logm, where d is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing d as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder.
Wang Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.
2010-12-15
The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski-Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM.
Electroencephalographic Fractal Dimension in Healthy Ageing and Alzheimer's Disease.
Smits, Fenne Margreeth; Porcaro, Camillo; Cottone, Carlo; Cancelli, Andrea; Rossini, Paolo Maria; Tecchio, Franca
2016-01-01
Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi's fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20-89 years) and 67 Alzheimer's Disease (AD) patients (age: 50-88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called "free" copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging. PMID:26872349
Electroencephalographic Fractal Dimension in Healthy Ageing and Alzheimer's Disease.
Smits, Fenne Margreeth; Porcaro, Camillo; Cottone, Carlo; Cancelli, Andrea; Rossini, Paolo Maria; Tecchio, Franca
2016-01-01
Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi's fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20-89 years) and 67 Alzheimer's Disease (AD) patients (age: 50-88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called "free" copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging.
Structure and flow of dense suspensions of protein fractal aggregates in comparison with microgels.
Inthavong, Walailuk; Kharlamova, Anna; Chassenieux, Christophe; Nicolai, Taco
2016-03-14
Solutions of the globular whey protein β-lactoglobulin (β-lg) were heated at different protein concentrations leading to the formation of polydisperse fractal aggregates with different average sizes. The structure of the solutions was analyzed with light scattering as a function of the protein concentration. The osmotic compressibility and the dynamic correlation length decreased with increasing concentration and became independent of the aggregate size in dense suspensions. The results obtained for different aggregate sizes could be superimposed after normalizing the concentration with the overlap concentration. Dense suspensions of fractal protein aggregates are strongly interpenetrated and can be visualized as an ensemble of fractal 'blobs'. The viscosity of the heated β-lg solutions increased extremely sharply above 80 g L(-1) and diverged at 98 g L(-1), mainly due to the sharply increasing aggregate size. At a fixed aggregate size, the viscosity increased initially exponentially with increasing concentration and then diverged. The increase was stronger when the aggregates were larger, but the dependence of the viscosity on the aggregate size was weaker than that of the osmotic compressibility and the dynamic correlation length. The concentration dependence of the viscosity of solutions of fractal β-lg aggregates is much stronger than that of homogeneous β-lg microgels. The behavior of fractal aggregates formed by whey protein isolates was similar. PMID:26864954
NASA Technical Reports Server (NTRS)
Bazell, David; Dwek, Eli
1990-01-01
Mathis and Whiffen (1989) have recently suggested that interstellar dust particles are fluffy aggregates of submicron-size particles composed of various astronomical minerals. These dust particles should exhibit optical properties that are quite different from standard dust, characterized by spherical particles of various homogeneous mineral composition. In this paper, the discrete dipole approximation (DDA) method is used to examine the effects of chemical inhomogeneities and spatial structure on the optical properties of interstellar Mathis-Whiffen-type dust particles. The spatial structure of the dust is represented by its fractal dimension, and the chemical inhomogeneities are simulated by randomly assigning the composition of the occupied sites in the structure to be either carbon or silicate. It is found that compositional inhomogeneities are the dominant parameter affecting the shape of the 9.7 and 18 micron silicate bands. Some bands-shape variations can be attributed to the fractal dimension of the dust. The results derived here can be used to explain or constrain variations in these parameters among various astronomical objects.
NASA Astrophysics Data System (ADS)
Lasue, Jeremie; Levasseur-Regourd, Anny-Chantal; Hadamcik, Edith; Botet, Robert; Renard, Jean-Baptiste
In situ missions have shown that cometary dust particles have low densities and are easily fragmenting aggregates [1]. The linear polarization of the solar light scattered by cometary dust corresponds to bell-shaped (with a small negative branch and a maximum below 30%) phase curves with a quasi-linear increase with the wavelength between 30° and 50° phase angle [2]. Such physical properties of the cometary material are reconciled by a fractal model of cometary dust and comet nuclei as formed by aggregation in reduced gravity as studied by laboratory experiments and numerical simulations. Reduced gravity light scattering experiments: The CODAG-LSU experiment (1999) gave the first indication of the light scattering properties transition between single particles and low dimensions fractal aggregates (D 1.3) [3, 4]. Such studies will be pursued on board the ISS with the ICAPS precursor experiment. The PROGRA2 experiment is designed to study the light scattering properties of particles levitated during dedicated microgravity flights or with ground-based configurations [5]. The material properties are chosen so as to be relevant in the context of cosmic dust from cometary and asteroidal origins. It is especially useful to disentangle the effects of varying albedos of constitutive materials [6], shape and size of constitutive grains [7]. Some of the results are interpreted in terms of fractal aggregates growth. Light scattering numerical simulations Based on numerical simulations and in coherence with the experimental results, a model of cometary coma by a mixture of fractal aggregates of up to 256 sub-micron sized spheroidal grains and compact spheroidal particles is shown to reproduce the polarimetric observations of comets such as 1P/Halley or C/1995 O1 Hale-Bopp [8]. Physical parameters of the size distribution of particles (minimum and maximum size, shape of the size distribution and quantity and location of absorbing and non-absorbing particles) can be retrieved
Hentschel, H George E; Levermann, Anders; Procaccia, Itamar
2002-07-01
We study the fractal and multifractal properties (i.e., the generalized dimensions of the harmonic measure) of a two-parameter family of growth patterns that result from a growth model that interpolates between diffusion-limited aggregation (DLA) and Laplacian growth patterns in two dimensions. The two parameters are beta that determines the size of particles accreted to the interface, and C that measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian growth are obtained at beta=0, C=0 and beta=2, C=1, respectively. The main purpose of this paper is to show that there exists a line in the beta-C phase diagram that separates fractal (D<2) from nonfractal (D=2) growth patterns. Moreover, Laplacian growth is argued to lie in the nonfractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for beta=0, C>0, and derive for them a scaling relation D=2D(3). We then propose that this family has growth patterns for which D=2 for some C>C(cr), where C(cr) may be zero. Next we consider the whole beta-C phase diagram and define a line that separates two-dimensional growth patterns from fractal patterns with D<2. We explain that Laplacian growth lies in the region belonging to two-dimensional growth patterns, motivating the main conjecture of this paper, i.e., that Laplacian growth patterns are two dimensional. The meaning of this result is that the branches of Laplacian growth patterns have finite (and growing) area on scales much larger than any ultraviolet cutoff length. PMID:12241482
Fractal Dimension Analysis of Transient Visual Evoked Potentials: Optimisation and Applications
Boon, Mei Ying; Henry, Bruce Ian; Chu, Byoung Sun; Basahi, Nour; Suttle, Catherine May; Luu, Chi; Leung, Harry; Hing, Stephen
2016-01-01
Purpose The visual evoked potential (VEP) provides a time series signal response to an external visual stimulus at the location of the visual cortex. The major VEP signal components, peak latency and amplitude, may be affected by disease processes. Additionally, the VEP contains fine detailed and non-periodic structure, of presently unclear relevance to normal function, which may be quantified using the fractal dimension. The purpose of this study is to provide a systematic investigation of the key parameters in the measurement of the fractal dimension of VEPs, to develop an optimal analysis protocol for application. Methods VEP time series were mathematically transformed using delay time, τ, and embedding dimension, m, parameters. The fractal dimension of the transformed data was obtained from a scaling analysis based on straight line fits to the numbers of pairs of points with separation less than r versus log(r) in the transformed space. Optimal τ, m, and scaling analysis were obtained by comparing the consistency of results using different sampling frequencies. The optimised method was then piloted on samples of normal and abnormal VEPs. Results Consistent fractal dimension estimates were obtained using τ = 4 ms, designating the fractal dimension = D2 of the time series based on embedding dimension m = 7 (for 3606 Hz and 5000 Hz), m = 6 (for 1803 Hz) and m = 5 (for 1000Hz), and estimating D2 for each embedding dimension as the steepest slope of the linear scaling region in the plot of log(C(r)) vs log(r) provided the scaling region occurred within the middle third of the plot. Piloting revealed that fractal dimensions were higher from the sampled abnormal than normal achromatic VEPs in adults (p = 0.02). Variances of fractal dimension were higher from the abnormal than normal chromatic VEPs in children (p = 0.01). Conclusions A useful analysis protocol to assess the fractal dimension of transformed VEPs has been developed. PMID:27598422
A Fractal Model for the Capacitance of Lunar Dust and Lunar Dust Aggregates
NASA Technical Reports Server (NTRS)
Collier, Michael R.; Stubbs, Timothy J.; Keller, John W.; Farrell, William M.; Marshall, John; Richard, Denis Thomas
2011-01-01
Lunar dust grains and dust aggregates exhibit clumping, with an uneven mass distribution, as well as features that span many spatial scales. It has been observed that these aggregates display an almost fractal repetition of geometry with scale. Furthermore, lunar dust grains typically have sharp protrusions and jagged features that result from the lack of aeolian weathering (as opposed to space weathering) on the Moon. A perfectly spherical geometry, frequently used as a model for lunar dust grains, has none of these characteristics (although a sphere may be a reasonable proxy for the very smallest grains and some glasses). We present a fractal model for a lunar dust grain or aggregate of grains that reproduces (1) the irregular clumpy nature of lunar dust, (2) the presence of sharp points, and (3) dust features that span multiple scale lengths. We calculate the capacitance of the fractal lunar dust analytically assuming fixed dust mass (i.e. volume) for an arbitrary number of fractal levels and compare the capacitance to that of a non-fractal object with the same volume, surface area, and characteristic width. The fractal capacitance is larger than that of the equivalent non-fractal object suggesting that for a given potential, electrostatic forces on lunar dust grains and aggregates are greater than one might infer from assuming dust grains are sphericaL Consequently, electrostatic transport of lunar dust grains, for example lofting, appears more plausible than might be inferred by calculations based on less realistic assumptions about dust shape and associated capacitance.
Fractal Dimension Analysis of Gustatory Electroencephalograms in Humans
NASA Astrophysics Data System (ADS)
Igasaki, Tomohiko; Murayama, Nobuki
To quantify the neural dynamics of the brain responsible for gustatory recognition and discrimination, fractal dimensions (FDs) of electroencephalograms (EEGs), which were measured under resting and three gustatory stimulation states, were investigated. The seven normal subjects sat on a chair with the chin resting on a frame made of plaster bandage and eyes closed. Distilled water (DW), high concentrated taste (HCT) solution (300 mM NaCl, 1 mM quinine-HCl, 40 mM acetic acid and 500 mM sucrose) and low concentrated taste (LCT) solution (51 mM NaCl, 0.026 mM quinine-HCl, 3 mM acetic acid and 14 mM sucrose) were randomly delivered to the anterior region of the tongue which was protruded slightly out of the mouth. FDs of EEGs from Cz in the resting and in the DW stimulation state were 5.43±1.01 and 4.94±1.03, respectively. In the HCT stimulation state, FD significantly decreased to 4.20±1.08 as compared with that in the resting (P<0.001). While, in the LCT stimulation state, FD significantly increased to 5.77±1.02 as compared with that in the HCT stimulation state (P<0.001). These results suggest that information processing of the brain is relatively simple when easily recognized tastes are applied.
Heterogeneities Analysis Using the Generalized Fractal Dimension and Continuous Wavelet Transform
NASA Astrophysics Data System (ADS)
Ouadfeul, S.; Aliouane, L.; Boudella, A.
2012-04-01
The main goal of this work is analyze heterogeneities from well-logs data using the wavelet transform modulus maxima lines (WTMM). Firstly, the continuous wavelet transform (CWT) with sliding window is calculated. The next step consists to calculate the maxima of the modulus of the CWT and estimate the spectrum of exponents. The three generalized fractal dimensions D0, D1 and D2 are then estimated. Application of the proposed idea at the synthetic and real well-logs data of a borehole located in the Algerian Sahara shows that the fractal dimensions are very sensitive to lithological variations. The generalized fractal dimensions are a very robust tool than can be used for petroleum reservoir characterization. Keywrods: reservoir, Heterogeneities, WTMM, fractal dimension.
NASA Astrophysics Data System (ADS)
Arefiev, K.; Nesterenko, V.; Daneykina, N.
2016-06-01
The results of communication research between the wear resistance of the K applicability hard-alloy cutting tools and the fractal dimension of the wear surface, which is formed on a back side of the cutting edge when processing the materials showing high adhesive activity are presented in the paper. It has been established that the wear resistance of tested cutting tools samples increases according to a fractal dimension increase of their wear surface.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2011-03-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2010-10-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia
Sehatpour, Pejman; Long, Jun; Gui, Weihua; Qiao, Jianping; Javitt, Daniel C.; Wang, Zhishun
2016-01-01
A failure of adaptive inference—misinterpreting available sensory information for appropriate perception and action—is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7–54.3 and healthy controls, 24.9–51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07–2.18 vs. median: 2.1730, range: 2.15–2.23, p<0.001; Cohen’s effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05–2.19 vs. median: 2.1760, range: 2.12–2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40–2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM
Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia.
Zhao, Guihu; Denisova, Kristina; Sehatpour, Pejman; Long, Jun; Gui, Weihua; Qiao, Jianping; Javitt, Daniel C; Wang, Zhishun
2016-01-01
A failure of adaptive inference-misinterpreting available sensory information for appropriate perception and action-is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7-54.3 and healthy controls, 24.9-51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07-2.18 vs. median: 2.1730, range: 2.15-2.23, p<0.001; Cohen's effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05-2.19 vs. median: 2.1760, range: 2.12-2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40-2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM surface detected in
The fractal dimensions of the spatial distribution of young open clusters in the solar neighbourhood
NASA Astrophysics Data System (ADS)
de La Fuente Marcos, R.; de La Fuente Marcos, C.
2006-06-01
Context: .Fractals are geometric objects with dimensionalities that are not integers. They play a fundamental role in the dynamics of chaotic systems. Observation of fractal structure in both the gas and the star-forming sites in galaxies suggests that the spatial distribution of young open clusters should follow a fractal pattern, too. Aims: .Here we investigate the fractal pattern of the distribution of young open clusters in the Solar Neighbourhood using a volume-limited sample from WEBDA and a multifractal analysis. By counting the number of objects inside spheres of different radii centred on clusters, we study the homogeneity of the distribution. Methods: .The fractal dimension D of the spatial distribution of a volume-limited sample of young open clusters is determined by analysing different moments of the count-in-cells. The spectrum of the Minkowski-Bouligand dimension of the distribution is studied as a function of the parameter q. The sample is corrected for dynamical effects. Results: .The Minkowski-Bouligand dimension varies with q in the range 0.71-1.77, therefore the distribution of young open clusters is fractal. We estimate that the average value of the fractal dimension is < D> = 1.7± 0.2 for the distribution of young open clusters studied. Conclusions: .The spatial distribution of young open clusters in the Solar Neighbourhood exhibits multifractal structure. The fractal dimension is time-dependent, increasing over time. The values found are consistent with the fractal dimension of star-forming sites in other spiral galaxies.
Archaeon and archaeal virus diversity classification via sequence entropy and fractal dimension
NASA Astrophysics Data System (ADS)
Tremberger, George, Jr.; Gallardo, Victor; Espinoza, Carola; Holden, Todd; Gadura, N.; Cheung, E.; Schneider, P.; Lieberman, D.; Cheung, T.
2010-09-01
Archaea are important potential candidates in astrobiology as their metabolism includes solar, inorganic and organic energy sources. Archaeal viruses would also be expected to be present in a sustainable archaeal exobiological community. Genetic sequence Shannon entropy and fractal dimension can be used to establish a two-dimensional measure for classification and phylogenetic study of these organisms. A sequence fractal dimension can be calculated from a numerical series consisting of the atomic numbers of each nucleotide. Archaeal 16S and 23S ribosomal RNA sequences were studied. Outliers in the 16S rRNA fractal dimension and entropy plot were found to be halophilic archaea. Positive correlation (R-square ~ 0.75, N = 18) was observed between fractal dimension and entropy across the studied species. The 16S ribosomal RNA sequence entropy correlates with the 23S ribosomal RNA sequence entropy across species with R-square 0.93, N = 18. Entropy values correspond positively with branch lengths of a published phylogeny. The studied archaeal virus sequences have high fractal dimensions of 2.02 or more. A comparison of selected extremophile sequences with archaeal sequences from the Humboldt Marine Ecosystem database (Wood-Hull Oceanography Institute, MIT) suggests the presence of continuous sequence expression as inferred from distributions of entropy and fractal dimension, consistent with the diversity expected in an exobiological archaeal community.
Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.
Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter
2014-11-20
An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants.
Simplifying the calculation of light scattering properties for black carbon fractal aggregates
NASA Astrophysics Data System (ADS)
Smith, A. J. A.; Grainger, R. G.
2014-08-01
Black carbon fractal aggregates have complicated shapes that make the calculation of their optical properties particularly computationally expensive. Here, a method is presented to estimate fractal aggregate light scattering properties by optimising simplified models to full light scattering calculations. It is found that there are no possible spherical models (at any size or refractive index) that well represent the light scattering in the visible or near-thermal infrared. As such, parameterisations of the light scattering as a function of the number of aggregate particles is presented as the most pragmatic choice for modelling distributions of black carbon when the large computational overheads of rigorous scattering calculations cannot be justified. This parameterisation can be analytically integrated to provide light scattering properties for lognormal distributions of black carbon fractal aggregates and return extinction cross sections with 0.1% accuracy for typical black carbon size distributions. Scattering cross sections and the asymmetry parameter can be obtained to within 3%.
Simplifying the calculation of light scattering properties for black carbon fractal aggregates
NASA Astrophysics Data System (ADS)
Smith, A. J. A.; Grainger, R. G.
2014-02-01
Black carbon fractal aggregates have complicated shapes that make the calculation of their optical properties particularly computationally expensive. Here, a method is presented to estimate fractal aggregate light scattering properties by optimising simplified models to full light scattering calculations. It is found that there are no possible spherical models (at any size or refractive index) that well represent the light scattering in the visible, or near-thermal infrared. As such, parameterisations of the light scattering as a function of the number of aggregate particles is presented as the most pragmatic choice for modelling distributions of black carbon when the large computational overheads of rigorous scattering calculations cannot be justified. This parameterisation can be analytically integrated to provide light scattering properties for log-normal distributions of black carbon fractal aggregates and return extinction cross-sections with 0.1% accuracy for typical black carbon size distributions. Scattering cross-sections and the asymmetry parameter can be obtained to within 3%.
Local fractal dimension based approaches for colonic polyp classification.
Häfner, Michael; Tamaki, Toru; Tanaka, Shinji; Uhl, Andreas; Wimmer, Georg; Yoshida, Shigeto
2015-12-01
This work introduces texture analysis methods that are based on computing the local fractal dimension (LFD; or also called the local density function) and applies them for colonic polyp classification. The methods are tested on 8 HD-endoscopic image databases, where each database is acquired using different imaging modalities (Pentax's i-Scan technology combined with or without staining the mucosa) and on a zoom-endoscopic image database using narrow band imaging. In this paper, we present three novel extensions to a LFD based approach. These extensions additionally extract shape and/or gradient information of the image to enhance the discriminativity of the original approach. To compare the results of the LFD based approaches with the results of other approaches, five state of the art approaches for colonic polyp classification are applied to the employed databases. Experiments show that LFD based approaches are well suited for colonic polyp classification, especially the three proposed extensions. The three proposed extensions are the best performing methods or at least among the best performing methods for each of the employed databases. The methods are additionally tested by means of a public texture image database, the UIUCtex database. With this database, the viewpoint invariance of the methods is assessed, an important features for the employed endoscopic image databases. Results imply that most of the LFD based methods are more viewpoint invariant than the other methods. However, the shape, size and orientation adapted LFD approaches (which are especially designed to enhance the viewpoint invariance) are in general not more viewpoint invariant than the other LFD based approaches.
Fractal Dimension Characterization of in-vivo Laser Doppler Flowmetry signals
NASA Astrophysics Data System (ADS)
Srinivasan, Gayathri; Sujatha, N.
Laser Doppler Blood Flow meter uses tissue backscattered light to non-invasively assess the blood flow rate. qualitatively. As there is large spatial variability and the temporal heterogeneity in tissue microvasculature, the measured blood flow rate is expressed in relative units. A non-linear approach in order to understand the dynamics of the microcirculation led to the fractal characterization of the blood flow signals. The study presented in the paper aims to analyze the fractal behavior of Laser Doppler Flow (LDF) signals and to quantitatively estimate the fractal dimension of waveforms using Box-Counting method. The measured Fractal dimension is an estimate of temporal variability of tissue perfusion. The rate at which fractal dimension varies as a function of location between individuals, exhibits a weak correlation with time. Further studies with a larger number of subjects are necessary to test the generality of the findings and if changes in dimension are reproducible in given individuals. In conclusion, the fractal dimension determined by Box-counting method may be useful for characterizing LDF time series signals. Future experiments evaluating whether the technique can be used to quantify microvascular dysfunction, as commonly occurring in conditions such as Diabetes, Raynaud's phenomenon, Erythromelalgia and Achenbach syndrome needs to be evaluated.
Black and brown carbon fractal aggregates from combustion of two fuels widely used in Asian rituals
NASA Astrophysics Data System (ADS)
Chakrabarty, Rajan K.; Arnold, Ian J.; Francisco, Dianna M.; Hatchett, Benjamin; Hosseinpour, Farnaz; Loria, Marcela; Pokharel, Ashok; Woody, Brian M.
2013-06-01
Incense sticks and mustard oil are the two most popular combustion fuels during rituals and social ceremonies in Asian countries. Given their widespread use in both closed and open burning activities, it is important to quantify the spectral radiative properties of aerosols emitted from the combustion of both fuels. This information is needed by climate models to assess the impact of these aerosols on radiative forcing. In this study, we used a 3-wavelength integrated photoacoustic-nephelometer - operating simultaneously at 405, 532 and 781nm - to measure the optical coefficients of aerosols emitted from the laboratory combustion of mustard oil lamp and two types of incense sticks. From the measured optical coefficients at three wavelengths, time-varying single scattering albedo (SSA), absorption Ångström exponent (AAE), and scattering Ångström exponent (SAE) were calculated. For incense smoke particles, the time-averaged mean AAE values were found to be as high as 8.32 (between 405 and 532nm) and 6.48 (between 532 and 781nm). This spectrally-varying characteristic of AAE indicates that brown carbon - a class of organic carbon which strongly absorbs solar radiation in the blue and near ultraviolet - is the primary component of incense smoke aerosols. For aerosols emitted from the burning of mustard oil lamp, the time-averaged mean AAE values were ˜1.3 (between 405 and 781nm) indicating that black carbon (BC) is the primary constituent. Scanning electron microscopy combined with image processing revealed the morphology of incense smoke aerosols to be non-coalescing and weakly-bound aggregates with a mean two-dimensional (2-d) fractal dimension (Df)=1.9±0.07, while the mustard oil smoke aerosols had typical fractal-like BC aggregate morphology with a mean 2-d Df=1.85±0.09.
Fractal dimension-bound spatio-temporal analysis of digital mammograms
NASA Astrophysics Data System (ADS)
Shanmugavadivu, P.; Sivakumar, V.; Sudhir, Rashmi
2016-02-01
A new Fractal Dimension-based diagnosis technique for the change detection and time-series analysis of masses in the temporal digital mammogram is presented in this paper. As the digital mammograms are confirmed as a reliable source for the prognosis of breast cancer, the demand for the development of precise computer aided detection techniques is constantly on the increase. This formed the basis for the development of this method using Fractal geometry, which is an efficient mathematical approach that deals with self-similar and irregular geometric objects called fractals. This work comprises of the detection of spatial masses using Fractal Hurst bound enhancement and segmentation of those temporal masses using Fractal Thresholding. The consultant radiologist's assessment of mass lesions forms the baseline for comparison and validation of the detected masses. Further, this research work performs temporal analysis of mass lesions, detected from the mammograms of the current and the respective prior view using the principle of Fractal Dimension. The precision of Fractal-dimension based temporal texture analysis of malignant masses of digital mammograms subsequently attributes to their characterization.
NASA Astrophysics Data System (ADS)
Boness, D. A.; Terrell-Martinez, B.
2010-12-01
As part of an ongoing undergraduate research project of light scattering calculations involving fractal carbonaceous soot aggregates relevant to current anthropogenic and natural sources in Earth's atmosphere, we have read with interest a recent paper [E.T. Wolf and O.B Toon,Science 328, 1266 (2010)] claiming that the Faint Young Sun paradox discussed four decades ago by Carl Sagan and others can be resolved without invoking heavy CO2 concentrations as a greenhouse gas warming the early Earth enough to sustain liquid water and hence allow the origin of life. Wolf and Toon report that a Titan-like Archean Earth haze, with a fractal haze aggregate nature due to nitrogen-methane photochemistry at high altitudes, should block enough UV light to protect the warming greenhouse gas NH3 while allowing enough visible light to reach the surface of the Earth. To test this hypothesis, we have employed a rigorous T-Matrix arbitrary-particle light scattering technique, to avoid the simplifications inherent in Mie-sphere scattering, on haze fractal aggregates at UV and visible wavelenths of incident light. We generate these model aggregates using diffusion-limited cluster aggregation (DLCA) algorithms, which much more closely fit actual haze fractal aggregates than do diffusion-limited aggregation (DLA) algorithms.
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)
LFN, QPO and fractal dimension of X-ray light curves from black hole binaries
NASA Astrophysics Data System (ADS)
Prosvetov, Art; Grebenev, Sergey
The origin of the low frequency noise (LFN) and quasi-periodic oscillations (QPO) observed in X-ray flux of Galactic black hole binaries is still not recognized in spite of multiple studies and attempts to model this phenomenon. There are known correlations between the QPO frequency, X-ray power density, X-ray flux and spectral state of the system, but there is no model that can do these dependences understandable. For the low frequency (~1 Hz) QPO we still have no even an idea capable to explain their production and don't know even what part of an accretion disc is responsible for them. Here we attempted to measure the fractal dimension of X-ray light curves of several black hole X-ray binaries and to study its correlation with the frequency of quasi periodic oscillations observed in their X-ray light-curves. The fractal dimension is a measure of the space-filling capacity of the light curves' profile. To measure the fractal dimension we used R/S method, which is fast enough and has good reputation in financial analytic and materials sciences. We found that if no QPO were observed in X-ray flux from the particular source, the fractal dimension is equal to the unique value which is independent on the source, its luminosity or its spectral state. On the other hand if QPO were detected in the flux, the fractal dimension deviated from its usual value. Also, we found a clear correlation between the QPO frequency and the fractal dimension of the emission. The relationship between these two parameters is solid but nonlinear. We believe that the analysis of X-ray light curves of black hole binaries using the fractal dimension has a good scientific potential and may provide an addition information on the geometry of accretion flow and fundamental physical parameters of the system.
Age-related reduction of chromatin fractal dimension in toluidine blue - stained hepatocytes.
Pantic, Igor; Petrovic, Danica; Paunovic, Jovana; Vucevic, Danijela; Radosavljevic, Tatjana; Pantic, Senka
2016-07-01
In this study, we proposed a hypothesis that chromatin of mouse hepatocytes exhibits age-related reduction of fractal dimension. This hypothesis was based on previously published works demonstrating that complexity of biological systems such as tissues, decreases during the process of physiological aging. Liver tissue was obtained from 24 male mice divided into 3 age groups: 10-days-old (young, juvenile), 210-days-old (adult) and 390-days-old. The tissue was stained using a modification of toluidine blue (nucleic acid - specific) staining method. A total of 480 chromatin structures (20 for each animal) were analyzed. For each structure, the values of fractal dimension, lacunarity, textural angular second moment and inverse difference moment were calculated using ImageJ software and its plugins. The results indicated the age-related reduction in fractal dimension and increase in lacunarity (p<0.01). Fractal dimension is a potentially good indicator of age associated changes in chromatin structure. To our knowledge, this is the first study to show that fractal complexity of hepatocyte chromatin decreases during the process of physiological aging. Fractal analysis as a method could be useful in detection of small age-related changes in chromatin distribution not otherwise visible with naked eye on conventional tissue micrographs. PMID:27412950
Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.
2001-01-01
Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.
Fractal dimension analysis for spike detection in low SNR extracellular signals
NASA Astrophysics Data System (ADS)
Salmasi, Mehrdad; Büttner, Ulrich; Glasauer, Stefan
2016-06-01
Objective. Many algorithms have been suggested for detection and sorting of spikes in extracellular recording. Nevertheless, it is still challenging to detect spikes in low signal-to-noise ratios (SNR). We propose a spike detection algorithm that is based on the fractal properties of extracellular signals and can detect spikes in low SNR regimes. Semi-intact spikes are low-amplitude spikes whose shapes are almost preserved. The detection of these spikes can significantly enhance the performance of multi-electrode recording systems. Approach. Semi-intact spikes are simulated by adding three noise components to a spike train: thermal noise, inter-spike noise, and spike-level noise. We show that simulated signals have fractal properties which make them proper candidates for fractal analysis. Then we use fractal dimension as the main core of our spike detection algorithm and call it fractal detector. The performance of the fractal detector is compared with three frequently used spike detectors. Main results. We demonstrate that in low SNR, the fractal detector has the best performance and results in the highest detection probability. It is shown that, in contrast to the other three detectors, the performance of the fractal detector is independent of inter-spike noise power and that variations in spike shape do not alter its performance. Finally, we use the fractal detector for spike detection in experimental data and similar to simulations, it is shown that the fractal detector has the best performance in low SNR regimes. Significance. The detection of low-amplitude spikes provides more information about the neural activity in the vicinity of the recording electrodes. Our results suggest using the fractal detector as a reliable and robust method for detecting semi-intact spikes in low SNR extracellular signals.
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.
2004-01-01
The accuracy of traditional multispectral maximum-likelihood image classification is limited by the skewed statistical distributions of reflectances from the complex heterogenous mixture of land cover types in urban areas. This work examines the utility of local variance, fractal dimension and Moran's I index of spatial autocorrelation in segmenting multispectral satellite imagery. Tools available in the Image Characterization and Modeling System (ICAMS) were used to analyze Landsat 7 imagery of Atlanta, Georgia. Although segmentation of panchromatic images is possible using indicators of spatial complexity, different land covers often yield similar values of these indices. Better results are obtained when a surface of local fractal dimension or spatial autocorrelation is combined as an additional layer in a supervised maximum-likelihood multispectral classification. The addition of fractal dimension measures is particularly effective at resolving land cover classes within urbanized areas, as compared to per-pixel spectral classification techniques.
NASA Astrophysics Data System (ADS)
Gao, Wei; Zakharov, Valery P.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Artemyev, Dmitry N.; Kornilin, Dmitry V.
2015-07-01
Optical coherence tomography (OCT) is usually employed for the measurement of retinal thickness characterizing the structural changes of tissue. However, fractal dimension (FD) could also character the structural changes of tissue. Therefore, fractal dimension changes may provide further information regarding cellular layers and early damage in ocular diseases. We investigated the possibility of OCT in detecting changes in fractal dimension from layered retinal structures. OCT images were obtained from diabetic patients without retinopathy (DM, n = 38 eyes) or mild diabetic retinopathy (MDR, n = 43 eyes) and normal healthy subjects (Controls, n = 74 eyes). Fractal dimension was calculated using the differentiate box counting methodology. We evaluated the usefulness of quantifying fractal dimension of layered structures in the detection of retinal damage. Generalized estimating equations considering within-subject intereye relations were used to test for differences between the groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of fractal dimension to discriminate between the eyes of DM, MDR and healthy eyes. Significant decreases of fractal dimension were observed in all layers in the MDR eyes compared with controls except in the inner nuclear layer (INL). Significant decreases of fractal dimension were also observed in all layers in the MDR eyes compared with DM eyes. The highest area under receiver operating characteristic curve (AUROC) values estimated for fractal dimension were observed for the outer plexiform layer (OPL) and outer segment photoreceptors (OS) when comparing MDR eyes with controls. The highest AUROC value estimated for fractal dimension were also observed for the retinal nerve fiber layer (RNFL) and OS when comparing MDR eyes with DM eyes. Our results suggest that fractal dimension of the intraretinal layers may provide useful
Mukherjee, Anika; Chan, Adrian D C; Keating, Sarah; Redline, Raymond W; Fritsch, Michael K; Machin, Geoffrey A; Cornejo-Palma, Daniel; de Nanassy, Joseph; El-Demellawy, Dina; von Dadelszen, Peter; Benton, Samantha J; Grynspan, David
2016-01-01
The distal villous hypoplasia (DVH) pattern is a placental correlate of fetal growth restriction. Because the pattern seems to involve less complexity than do appropriately developed placental villi, we postulated that it may be associated with lower fractal dimension-a mathematical measure of complexity. Our study objectives were to evaluate interobserver agreement related to the DVH pattern among expert pathologists and to determine whether pathologist classification of DVH correlates with fractal dimension. A study set of 30 images of placental parenchyma at ×4 magnification was created by a single pathologist from a digital slide archive. The images were graded for the DVH pattern according to pre-specified definitions and included 10 images graded as "no DVH" (grade = 0), 10 with mild to moderate DVH (grade = 1), and 10 with severe DVH (grade = 2). The images were randomly sorted and shown to a panel of 4 international experts who similarly graded the images for DVH. Weighted kappas were calculated. For each image, fractal dimension was calculated by the Box Counting method. The correlation coefficient between (1) the averaged DVH scores obtained by the 5 pathologists and (2) fractal dimension was calculated. The mean weighted kappa score among the observers was 0.59 (range: 0.42-0.70). The correlation coefficient between fractal dimension and the averaged DVH score was -0.915 (P < 0.001). Expert pathologists achieve fair to substantial agreement in grading DVH, indicating consensus on the definition of DVH. Distal villous hypoplasia correlates extremely well with fractal dimension and represents an objective measure for DVH.
NASA Astrophysics Data System (ADS)
Huber, Franz J. T.; Will, Stefan; Daun, Kyle J.
2016-11-01
Inferring the size distribution of aerosolized fractal aggregates from the angular distribution of elastically scattered light is a mathematically ill-posed problem. This paper presents a procedure for analyzing Wide-Angle Light Scattering (WALS) data using Bayesian inference. The outcome is probability densities for the recovered size distribution and aggregate morphology parameters. This technique is applied to both synthetic data and experimental data collected on soot-laden aerosols, using a measurement equation derived from Rayleigh-Debye-Gans fractal aggregate (RDG-FA) theory. In the case of experimental data, the recovered aggregate size distribution parameters are generally consistent with TEM-derived values, but the accuracy is impaired by the well-known limited accuracy of RDG-FA theory. Finally, we show how this bias could potentially be avoided using the approximation error technique.
Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation
NASA Astrophysics Data System (ADS)
Santos, Andrés; de Haro, Mariano López
2016-06-01
Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1 ≤d ≤3 ) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015), 10.1103/PhysRevLett.115.097801], a good agreement being observed.
Salinas-Nolasco, Manlio Favio; Méndez-Vivar, Juan
2010-03-16
Among several analysis techniques applied to the study of surface passivation using dicarboxylic acids, small angle X-ray scattering (SAXS) has proved to be relevant in the physicochemical interpretation of the surface association resulting between calcium carbonate and the molecular structure of malonic acid. It is possible to establish chemical affinity principles through bidimensional geometric analysis in terms of the fractal dimension obtained experimentally by SAXS. In this Article, we present results about the adsorption of malonic acid on calcite, using theoretical and mathematical principles of the fractal dimension.
Smith, R.L. Mecholsky, J.J.
2011-05-15
Fractal analysis has been used as a method to study fracture surfaces of brittle materials. However, it has not been determined if the fractal characteristics of brittle materials is consistent throughout the fracture surface. Therefore, the fractal dimensional increment of the mirror, mist, and hackle regions of the fracture surface of silica glass was determined using atomic force microscopy. The fractal dimensional increment of the mirror region (0.17-0.26) was determined to be statistically greater than that for the mist (0.08-0.12) and hackle (0.08-0.13) regions. It is thought that the increase in the fractal dimensional increment is caused by a greater tortuosity in the mirror region due to, most likely, the slower crack velocity of the propagating crack in that region and that there is a point between the mirror and mist region at which the fractal dimension decreases and becomes constant. - Research Highlights: {yields} The fracture surface of silica glass does not have a constant fractal dimension. {yields} Mirror region has greater fractal dimension than mist or hackle region. {yields} Fractal dimension decreases between mirror and mist region. {yields} Greater fractal dimension could be due to slower crack velocity in mirror region.
Electrochemical Growth of Ag Junctions and Diffusion Limited Aggregate (DLA) Fractal Simulation
NASA Astrophysics Data System (ADS)
Olson, Zak; Tuppan, Sam; Kim, Woo-Joong; Seattle University Team
2015-03-01
We attempt construction of a single atom connection between two copper wires. By applying a DC voltage across the wires when immersed in a silver nitrate solution, we deposit silver until a junction is formed. The deposited silver forms a fractal structure that can be simulated with a diffusion limited aggregation model.
Wang, Jing-Yan; Hu, Ting-Xing; Gong, Wei; Gong, Yuan-Bo; Luo, Cheng-De
2010-06-01
By using fractal model, this paper studied the fractal dimension of soil aggregate structure (D) in the slope farmland (CK), its 5-year de-farmed Neosinocalamus affinis plantation (NAP), Bambusa pervariabilis x Dendrocalamopsis oldhami plantation (BDP), Alnus crenastogyne + Neosinocalamus affinis plantation (ANP), and abandoned farmland (AFL) in south Sichuan Province of China, and analyzed the relationships between the D and soil physical and chemical properties. In the de-farmed plantations and abandoned farmland, the contents of > 0.25 mm soil aggregates and water-stable aggregates were increased significantly, compared with those in the slope farmland. The D was 1.377-2.826, being in the order of NAP < BDP < ANP < AFL < CK, and decreased with the increasing contents of > 0.25 mm soil aggregates and water-stable aggregates. Comparing with CK, de-farming increased the soil natural water content, capillary porosity, and contents of soil organic matter, total N, alkali-hydrolysable N, total P, and total K, and decreased soil bulk density, non-capillary porosity, and aeration porosity. There were close relationships between the fractal dimension of soil aggregate structure and the soil physical and chemical properties. All the results suggested that the de-farming of slope farmland was beneficial to the increase of the contents of > 0.25 mm soil aggregates and water-stable aggregates, and the enhancement of soil structure stability. The D could be used as an ideal index to evaluate soil fertility, and planting Neosinocalamus affinis on the de-farming slope farmland was a good measure for the improvement of soil fertility in the research area.
Fractal dimensions: A new paradigm to assess spatial memory and learning using Morris water maze.
Singh, Surjeet; Kaur, Harpreet; Sandhir, Rajat
2016-02-15
Morris water maze has been widely used for analysis of cognitive functions and relies on the time taken by animal to find the platform i.e. escape latency as a parameter to quantify spatial memory and learning. However, escape latency is confounded by swimming speed which is not necessarily a cognitive factor. Rather, path length may be a more appropriate and reliable parameter to assess spatial learning. This paper presents fractal dimension as a new paradigm to assess spatial memory and learning in animals. Male wistar rats were administrated with pentylenetetrazole and scopolamine to induce chronic epilepsy and dementia respectively. Fractal dimension of the random path followed by the animals on Morris water maze was analyzed and statistically compared among different experimental groups; the results suggest that fractal dimension is more reliable and accurate parameter to assess cognitive deficits compared to escape latency. Thus, the present study suggests that fractal dimensions could be used as an independent parameter to assess spatial memory and learning in animals using Morris water maze.
NASA Astrophysics Data System (ADS)
Langton, C. M.; Whitehead, M. A.; Haire, T. J.; Hodgskinson, R.
1998-02-01
There has been considerable debate on the relative dependence of broadband ultrasound attenuation (nBUA, ) upon the density and structure of cancellous bone. A nonlinear relationship between nBUA and porosity has recently been demonstrated using stereolithography models, indicating a high structural dependence for nBUA. We report here on the measurement of trabecular perimeter and fractal dimension on the two-dimensional images used to create the stereolithography models. Adjusted coefficients of determination with nBUA were 94.4% and 98.4% for trabecular perimeter and fractal dimension respectively. The feature of fractal dimension representing both the porosity and connectivity of a given structure is most exciting. Further work is required to determine the relationship between broadband ultrasound attenuation and fractal dimension in complex three-dimensional cancellous bone structures.
Structural and Fractal Dimensions are Reliable Determinants of Grain Yield in Soybean
Technology Transfer Automated Retrieval System (TEKTRAN)
Reliable models are needed to describe plants with complex geometric structures, quantify the impact of management strategies on the plant’s geometric distribution in space and time, and predict yield as a function of fractal dimension. We measured growth and development variables on single soybean ...
Estimation of Fractal Dimension in Differential Diagnosis of Pigmented Skin Lesions
NASA Astrophysics Data System (ADS)
Aralica, Gorana; Milošević, Danko; Konjevoda, Paško; Seiwerth, Sven; Štambuk, Nikola
Medical differential diagnosis is a method of identifying the presence of a particular entity (disease) within a set of multiple possible alternatives. The significant problem in dermatology and pathology is the differential diagnosis of malignant melanoma and other pigmented skin lesions, especially of dysplastic nevi. Malignant melanoma is the most malignant skin neoplasma, with increasing incidence in various parts of the world. It is hoped that the methods of quantitative pathology, i.e. morphometry, can help objectification of the diagnostic process, since early discovery of melanoma results in 10-year survival rate of 90%. The aim of the study was to use fractal dimension calculated from the perimeter-area relation of the cell nuclei as a tool for the differential diagnosis of pigmented skin lesions. We analyzed hemalaun-eosin stained pathohistological slides of pigmented skin lesions: intradermal naevi (n = 45), dysplastic naevi (n = 47), and malignant melanoma (n = 50). It was found that fractal dimension of malignant melanoma cell nuclei differs significantly from the intradermal and dysplastic naevi (p ≤ 0. 001, Steel-Dwass Multiple Comparison Test). Additionaly, ROC analysis confirmed the value of fractal dimension based evaluation. It is suggested that the estimation of fractal dimension from the perimeter-area relation of the cell nuclei may be a potentially useful morphometric parameter in the medical differential diagnosis of pigmented skin lesions.
NASA Astrophysics Data System (ADS)
Ouadfeul, S.-A.; Aliouane, L.; Tourtchine, V.
2013-09-01
In this paper, we use the so-called the Wavelet Transform Modulus Maxima lines (WTMM) technique for estimation of the capacity, the information and the correlation fractal dimensions of the Intermagnet Observatories time series. Analysis of Hermanus, Baker-Lake, Kakioka, Albibag and Wingst observatories data shows that the correlation and the information dimensions can be used a supplementary indexes for geomagnetic disturbances identification.
Kalauzi, Aleksandar; Bojić, Tijana; Vuckovic, Aleksandra
2012-07-01
The exact mathematical relationship between FFT spectrum and fractal dimension (FD) of an experimentally recorded signal is not known. In this work, we tried to calculate signal FD directly from its Fourier amplitudes. First, dependence of Higuchi's FD of mathematical sinusoids on their individual frequencies was modeled with a two-parameter exponential function. Next, FD of a finite sum of sinusoids was found to be a weighted average of their FDs, weighting factors being their Fourier amplitudes raised to a fractal degree. Exponent dependence on frequency was modeled with exponential, power and logarithmic functions. A set of 280 EEG signals and Weierstrass functions were analyzed. Cross-validation was done within EEG signals and between them and Weierstrass functions. Exponential dependence of fractal exponents on frequency was found to be the most accurate. In this work, signal FD was for the first time expressed as a fractal weighted average of FD values of its Fourier components, also allowing researchers to perform direct estimation of signal fractal dimension from its FFT spectrum.
Fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-01-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, have not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension, 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with fractal dimension 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas. PMID:23629025
NASA Astrophysics Data System (ADS)
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-06-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.
Pacagnelli, Francis Lopes; Sabela, Ana Karênina Dias de Almeida; Mariano, Thaoan Bruno; Ozaki, Guilherme Akio Tamura; Castoldi, Robson Chacon; do Carmo, Edna Maria; Carvalho, Robson Francisco; Tomasi, Loreta Casquel; Okoshi, Katashi; Vanderlei, Luiz Carlos Marques
2016-01-01
Background Right-sided heart failure has high morbidity and mortality, and may be caused by pulmonary arterial hypertension. Fractal dimension is a differentiated and innovative method used in histological evaluations that allows the characterization of irregular and complex structures and the quantification of structural tissue changes. Objective To assess the use of fractal dimension in cardiomyocytes of rats with monocrotaline-induced pulmonary arterial hypertension, in addition to providing histological and functional analysis. Methods Male Wistar rats were divided into 2 groups: control (C; n = 8) and monocrotaline-induced pulmonary arterial hypertension (M; n = 8). Five weeks after pulmonary arterial hypertension induction with monocrotaline, echocardiography was performed and the animals were euthanized. The heart was dissected, the ventricles weighed to assess anatomical parameters, and histological slides were prepared and stained with hematoxylin/eosin for fractal dimension analysis, performed using box-counting method. Data normality was tested (Shapiro-Wilk test), and the groups were compared with non-paired Student t test or Mann Whitney test (p < 0.05). Results Higher fractal dimension values were observed in group M as compared to group C (1.39 ± 0.05 vs. 1.37 ± 0.04; p < 0.05). Echocardiography showed lower pulmonary artery flow velocity, pulmonary acceleration time and ejection time values in group M, suggesting function worsening in those animals. Conclusion The changes observed confirm pulmonary-arterial-hypertension-induced cardiac dysfunction, and point to fractal dimension as an effective method to evaluate cardiac morphological changes induced by ventricular dysfunction. PMID:27223643
NASA Astrophysics Data System (ADS)
Wei, Wei; Cai, Jianchao; Hu, Xiangyun; Fan, Ping; Han, Qi; Lu, Jinge; Cheng, Chu-Lin; Zhou, Feng
2015-03-01
The fractal dimension of random walker (FDRW) is an important parameter for description of electrical conductivity in porous media. However, it is somewhat empirical in nature to calculate FDRW. In this paper, a simple relation between FDRW and tortuosity fractal dimension (TFD) of current streamlines is derived, and a novel method of computing TFD for different generations of two-dimensional Sierpinski carpet and three-dimensional Sierpinski sponge models is presented through the finite element method, then the FDRW can be accordingly predicted; the proposed relation clearly shows that there exists a linear relation between pore fractal dimension (PFD) and TFD, which may have great potential in analysis of transport properties in fractal porous media.
Are fractal dimensions of the spatial distribution of mineral deposits meaningful?
Raines, G.L.
2008-01-01
It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the
Effect of mobile phone radiation on brain using EEG analysis by Higuichi's fractal dimension method
NASA Astrophysics Data System (ADS)
Smitha, C. K.; Narayanan, N. K.
2013-01-01
venient window on the mind, revealing synaptic action that is moderately to strongly co-relate with brain state. Fractal dimension, measure of signal complexity can be used to characterize the physiological conditions of the brain. As the EEG signal is non linear, non stationary and noisy, non linear methods will be suitable for the analysis. In this paper Higuichi's fractal method is applied to find the fractal dimension. EEGs of 5 volunteers were recorded at rest and on exposure to radiofrequency (RF) emissions from mobile phones having different SAR values. Mobiles were positioned near the ears and then near the cz position. Fractal dimensions for all conditions are calculated using Higuich's FD estimation algorithm. The result shows that there are some changes in the FD while using mobile phone. The change in FD of the signal varies from person to person. The changes in FD show the variations in EEG signal while using mobile phone, which demonstrate transformation in the activities of brain due to radiation.
NASA Astrophysics Data System (ADS)
Tijera, Manuel; Maqueda, Gregorio; Cano, José L.; López, Pilar; Yagüe, Carlos
2010-05-01
The wind velocity series of the atmospheric turbulent flow in the planetary boundary layer (PBL), in spite of being highly erratic, present a self-similarity structure (Frisch, 1995; Peitgen et., 2004; Falkovich et., 2006). So, the wind velocity can be seen as a fractal magnitude. We calculate the fractal dimension (Komolgorov capacity or box-counting dimension) of the wind perturbation series (u' = u- ) in the physical spaces (namely velocity-time). It has been studied the time evolution of the fractal dimension along different days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The data analysed was recorded in the experimental campaign SABLES-98 (Cuxart et al., 2000) at the Research Centre for the Lower Atmosphere (CIBA) located in Valladolid (Spain). In this work the u, v and w components of wind velocity series have been measured by sonic anemometers (20 Hz sampling rate). The fractal dimension versus the integral length scales of the mean wind series have been studied, as well as the influence of different turbulent parameters. A method for estimating these integral scales is developed using the normalized autocorrelation function and a Gaussian fit. Finally, it will be analysed the variation of the fractal dimension versus stability parameters (as Richardson number) in order to explain some of the dominant features which are likely immersed in the fractal nature of these turbulent flows. References - Cuxart J, Yagüe C, Morales G, Terradellas E, Orbe J, Calvo J, Fernández A, Soler MR, Infante C, Buenestado P, Espinalt A, Joergensen HE, Rees JM, Vilá J, Redondo JM, Cantalapiedra IR and Conangla L (2000) Stable atmospheric boundary-layer experiment in Spain (SABLES98): a report. Boundary- Layer Meteorol 96:337-370 - Falkovich G and Kattepalli R. Sreenivasan (2006) Lessons from Hidrodynamic Turbulence. Physics Today 59: 43-49 - Frisch U (1995) Turbulence the legacy of A.N. Kolmogorov Cambridge University Press 269pp - Peitgen H, Jürgens H and
Fractal Dimensions for Radioisotope Pollution Patterns by Nuclear Power Plant Accidents
NASA Astrophysics Data System (ADS)
Saito, K.; Ogawa, S.
2015-04-01
The radioisotope pollution shows two types of patterns: dry and wet deposits for nuclear power plant accidents. Two surface pollution patterns were analysed by fractal. In Fukushima nuclear power plant accident, surface pollution by wet deposits was estimated to occur. However, actually it was no rain and white crystals were observed on the surface. Then, fractal analysis was carried out for the spatial distribution patterns of radio isotopes on the surface to judge the types of deposits. As a reference, Chernobyl nuclear power plant accident was checked for the spatial distribution patterns of radioisotopes on the surface. The objective patterns by fractal analysis were the surface pollution maps in Fukushima and Chernobyl, Abukuma river watershed map, and NOAA/AVHRR. The calculation of fractal dimensions was carried out with the box counting for binarized images. Fractal analysis results suggested the next conclusions. The radioisotope pollution in Fukushima might occur in both dry and wet deposits. The dry deposit might make the pollution pattern similar to the watershed, while the wet deposit might make the pollution pattern similar to cloud images. Moreover, most radioisotope contaminants might flow on the road in the forest valley and deposit on forest with and without rainfall in Fukushima.
Zone Specific Fractal Dimension of Retinal Images as Predictor of Stroke Incidence
Kumar, Dinesh Kant; Hao, Hao; Unnikrishnan, Premith; Kawasaki, Ryo; Mitchell, Paul
2014-01-01
Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, α = 0.05) compared with SFD (H = 0.51, P = 0.475, α = 0.05) and BC (H = 0.41, P = 0.520, α = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed. PMID:25485298
Nuclear fractal dimension as a prognostic factor in oral squamous cell carcinoma.
Goutzanis, L; Papadogeorgakis, N; Pavlopoulos, P M; Katti, K; Petsinis, V; Plochoras, I; Pantelidaki, C; Kavantzas, N; Patsouris, E; Alexandridis, C
2008-04-01
Strong theoretical reasons exist for using fractal geometry in measurements of natural objects, including most objects studied in pathology. Indeed, fractal dimension provides a more precise and theoretically more appropriate approximation of their structure properties and especially their shape complexity. The aim of our study was to evaluate the nuclear fractal dimension (FD) in tissue specimens from patients with oral cavity carcinomas in order to assess its potential value as prognostic factor. Relationships between FD and other factors including clinicopathologic characteristics were also investigated. Histological sections from 48 oral squamous cell carcinomas as well as from 17 non-malignant mucosa specimens were stained with Hematoxylin-Eosin for pathological examination and with Feulgen for nuclear complexity evaluation. The sections were evaluated by image analysis using fractal analysis software to quantify nuclear FD by the box-counting method. Carcinomas presented higher mean values of FD compared to normal mucosa. Well differentiated neoplasms had lower FD values than poorly differentiated ones. FD was significantly correlated with the nuclear size. Patients with FD lower than the median value of the sample had statistically significant higher survival rates. Within the sample of patients studied, FD was proved to be an independent prognostic factor of survival in oral cancer patients. In addition this study provides evidence that there are several statistically significant correlations between FD and other morphometric characteristics or clinicopathologic factors in oral squamous cell carcinomas. PMID:17692559
Fractal Dimension and Vessel Complexity in Patients with Cerebral Arteriovenous Malformations
Reishofer, Gernot; Koschutnig, Karl; Enzinger, Christian; Ebner, Franz; Ahammer, Helmut
2012-01-01
The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs. PMID:22815946
NASA Astrophysics Data System (ADS)
Zhang, Chen; Ni, Zhiwei; Ni, Liping; Tang, Na
2016-10-01
Feature selection is an important method of data preprocessing in data mining. In this paper, a novel feature selection method based on multi-fractal dimension and harmony search algorithm is proposed. Multi-fractal dimension is adopted as the evaluation criterion of feature subset, which can determine the number of selected features. An improved harmony search algorithm is used as the search strategy to improve the efficiency of feature selection. The performance of the proposed method is compared with that of other feature selection algorithms on UCI data-sets. Besides, the proposed method is also used to predict the daily average concentration of PM2.5 in China. Experimental results show that the proposed method can obtain competitive results in terms of both prediction accuracy and the number of selected features.
Study of fractal dimension in chest images using normal and interstitial lung disease cases
NASA Astrophysics Data System (ADS)
Tucker, Douglas M.; Correa, Jose L.; Souto, Miguel; Malagari, Katerina S.
1993-09-01
A quantitative computerized method which provides accurate discrimination between chest radiographs with positive findings of interstitial disease patterns and normal chest radiographs may increase the efficacy of radiologic screening of the chest and the utility of digital radiographic systems. This report is a comparison of fractal dimension measured in normal chest radiographs and in radiographs with abnormal lungs having reticular, nodular, reticulonodular and linear patterns of interstitial disease. Six regions of interest (ROI's) from each of 33 normal chest radiographs and 33 radiographs with positive findings of interstitial disease were studied. Results indicate that there is a statistically significant difference between the distribution of the fractal dimension in normal radiographs and radiographs where disease is present.
Oscillations in the evaluation of fractal dimension of RR intervals time series.
Muñoz Diosdado, A; Gálvez Coyt, G; Pérez Uribe, B M
2010-01-01
Previously, we have reported the presence of oscillations in the graphs we have used to evaluate the Higuchi's fractal dimension in RR intervals time series of congestive heart failure (CHF) patients in the sleep phase but these oscillations hardly appear in all the six hours of the awake phase. In this paper we report the same analysis for heart rate time series for different groups of healthy subjects; we are looking for the presence of this kind of oscillations in other situations. We analyzed all the time series in the Exaggerated Heart Rate Oscillations database of Physionet during two meditation techniques: volunteers with spontaneous breathing, subjects in meditation, volunteers in a metronomic breathing group and elite athletes. We have found oscillations in the graphs of the Higuchi's fractal dimension in the heart rate time series of subjects in meditation and metronomic breathing and this fact coincides with previous reported results.
Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation.
Santos, Andrés; de Haro, Mariano López
2016-06-01
Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1≤d≤3) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.097801], a good agreement being observed.
Klatt, Jan; Gerich, Carola E; Gröbe, Alexander; Opitz, Jörg; Schreiber, Jürgen; Hanken, Henning; Salomon, Georg; Heiland, Max; Kluwe, Lan; Blessmann, Marco
2014-09-01
Early detection and complete resection of oral carcinomas is of crucial importance for patient survival. This could be significantly improved by developing a non-invasive, sensitive and real-time detection technique. Time-resolved autofluorescence measurement is state-of-the-art technology originally developed for non-destructive inspection of material. In this study, we measured time-resolved autofluorescence in tumours and healthy tissues of the oral cavity ex vivo and calculated the corresponding fractal dimension which was significantly higher in tumours than in healthy tissues (1.8 vs. 1.6, P < 0.001, unpaired t-test) with non-overlapping 95% confidential intervals 1.88-1.84 and 1.57-1.69, respectively. Very high specificity (86%) could be reached at 100% sensitivity. The area under the curve was 99%, further suggesting the superior prediction potential of fractal dimension based on time-resolved autofluorescence spectra.
Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation.
Santos, Andrés; de Haro, Mariano López
2016-06-01
Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1≤d≤3) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.097801], a good agreement being observed. PMID:27415227
Estimating the level of dynamical noise in time series by using fractal dimensions
NASA Astrophysics Data System (ADS)
Sase, Takumi; Ramírez, Jonatán Peña; Kitajo, Keiichi; Aihara, Kazuyuki; Hirata, Yoshito
2016-03-01
We present a method for estimating the dynamical noise level of a 'short' time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise.
NASA Astrophysics Data System (ADS)
Kharchenko, D. O.
For the system with colored multiplicative noise the nonlinearity of the synergetic potential like φ^{2+m} model in Langevin equation was shown to be capable of providing the expanse of the stochastic system phase space. The concrete system of the population dynamics with the noise correlation time τ_cto∞ is examined. The fractal dimension of that kind of a system is defined as D=m, in contrast to the system with a white noise were D=0.
NASA Astrophysics Data System (ADS)
Kovalnogov, Vladislav N.; Fedorov, Ruslan V.; Generalov, Dmitry A.; Khakhalev, Yury A.; Zolotov, Aleksandr N.
2016-06-01
The results of numerical and experimental studies of the structure and the frictional resistance of the turbulent flow with the effects obtained by using a modified model of the mixing Prandtl with the fractal dimension of pressure fluctuations. A construction of a cooled turbine blade was designed based on the results. The construction comprises a combined cooling and cylindrical cavity on the blade surface and the inner surface of the cooling channels.
Pancheliuga, V A; Pancheliuga, M S
2013-01-01
In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the "all possible combination" method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes. PMID:23755565
Banerji, Anirban; Ghosh, Indira
2009-01-01
A robust marker to describe mass, hydrophobicity and polarizability distribution holds the key to deciphering structural and folding constraints within proteins. Since each of these distributions is inhomogeneous in nature, the construct should be sensitive in describing the patterns therein. We show, for the first time, that the hydrophobicity and polarizability distributions in protein interior follow fractal scaling. It is found that (barring ‘all-α’) all the major structural classes of proteins have an amount of unused hydrophobicity left in them. This amount of untapped hydrophobicity is observed to be greater in thermophilic proteins, than that in their (structurally aligned) mesophilic counterparts. ‘All-β’(thermophilic, mesophilic alike) proteins are found to have maximum amount of unused hydrophobicity, while ‘all-α’ proteins have been found to have minimum polarizability. A non-trivial dependency is observed between dielectric constant and hydrophobicity distributions within (α+β) and ‘all-α’ proteins, whereas absolutely no dependency is found between them in the ‘all-β’ class. This study proves that proteins are not as optimally packed as they are supposed to be. It is also proved that origin of α-helices are possibly not hydrophobic but electrostatic; whereas β-sheets are predominantly hydrophobic in nature. Significance of this study lies in protein engineering studies; because it quantifies the extent of packing that ensures protein functionality. It shows that myths regarding protein interior organization might obfuscate our knowledge of actual reality. However, if the later is studied with a robust marker of strong mathematical basis, unknown correlations can still be unearthed; which help us to understand the nature of hydrophobicity, causality behind protein folding, and the importance of anisotropic electrostatics in stabilizing a highly complex structure named ‘proteins’. PMID:19834622
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-01-01
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus. PMID:27138171
NASA Astrophysics Data System (ADS)
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-05-01
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus.
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-01-01
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus. PMID:27138171
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-05-03
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus.
Fractal dimension analysis of aluminum oxide particle for sandblasting dental use.
Oshida, Y; Munoz, C A; Winkler, M M; Hashem, A; Itoh, M
1993-01-01
Aluminum oxide particles are commonly used as a sandblasting media, particularly in dentistry, for multiple purposes including divesting the casting investment materials and increasing effective surface area for enhancing the mechanical retention strengths of succeedingly applied fired porcelain or luting cements. Usually fine aluminum oxide particles are recycled within the sandblasting machine. Ceramics such as aluminum oxides are brittle, therefore, some portions of recycling aluminum oxide particles might be brittle fractured. If fractured sandblasting particles are involved in the recycling media, it might result in irregularity metallic materials surface as well as the recycling sandblasting media itself be contaminated. Hence, it is necessary from both clinical and practical reasons to monitor the particle conditions in terms of size/shape and effectiveness of sandblasting, so that sandblasting dental prostheses can be fabricated in optimum and acceptable conditions. In the present study, the effect of recycling aluminum oxide particles on the surface texture of metallic materials was evaluated by Fractal Dimension Analysis (FDA). Every week the alumina powder was sampled and analyzed for weight fraction and contaminants. Surface texture of sandblasted standard samples was also characterized by FDA. Results indicate very little change in particle size, while the fractal dimension increased. Fractal dimension analysis showed that the aluminum oxide particle as a sandblasting media should be replaced after 30 or 40 min of total accumulated operation time.
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.
Cluster-Cluster Aggregation Calculations of Fractal Haze Particles: Titan and the Early Earth
NASA Astrophysics Data System (ADS)
Terrell-Martinez, Bernice; Boness, David
2010-10-01
The atmosphere of the Archean Earth (3.8 to 2.5 billion years ago) is thought to have been dominated by a thick hydrocarbon haze similar to that of Titan's current atmosphere. To understand radiative transport in the atmospheres of the early Earth and of Titan, it is necessary to compute light scattering in UV, visible, and IR wavelength ranges for realistic fractal aggregate hydrocarbon aerosol particles. We report preliminary work on MATLAB, True BASIC, and Fortran programs to simulate the growth of fractal aggregate aerosols through diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) physical processes. The results of these computations are being used with a T-Matrix light scattering program to test recently published, widely-reported conclusions about the early Earth and the faint young Sun paradox [E. T. Wolf and O. B. Toon, Science 328, 1266 (2010)]. This modeling is also relevant to understanding atmospheric carbonaceous soot aerosol anthropogenic and natural effects on climate change of Earth today.
NASA Astrophysics Data System (ADS)
Ul'yanov, A. S.; Lyapina, A. M.; Ulianova, O. V.; Fedorova, V. A.; Uianov, S. S.
2011-04-01
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated.
Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S
2011-04-30
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated. (optical technologies in biophysics and medicine)
NASA Astrophysics Data System (ADS)
Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun
1995-05-01
This paper describes the quantitative measurement of the amount of fibrosis in the rat liver using the fractal dimension of the shape of power spectrum. The shape of the power spectrum of the scattered echo from biotissues is strongly affected by its internal structure. The fractal dimension, which is one of the important parameters of the fractal theory, is useful to express the complexity of shape of figures such as the power spectrum. From in vitro experiments using rat liver, it was found that this method can be used to quantitatively measure the amount of fibrosis in the liver, and has the possibility for use in the diagnosis of human liver cirrhosis.
Modified box dimension and average weighted receiving time on the weighted fractal networks
Dai, Meifeng; Sun, Yanqiu; Shao, Shuxiang; Xi, Lifeng; Su, Weiyi
2015-01-01
In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is. PMID:26666355
Modified box dimension and average weighted receiving time on the weighted fractal networks
NASA Astrophysics Data System (ADS)
Dai, Meifeng; Sun, Yanqiu; Shao, Shuxiang; Xi, Lifeng; Su, Weiyi
2015-12-01
In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.
a New Method for Calculating the Fractal Dimension of Surface Topography
NASA Astrophysics Data System (ADS)
Zuo, Xue; Zhu, Hua; Zhou, Yuankai; Li, Yan
2015-06-01
A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass-Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.
Nuclear Fractal Dimensions as a Tool for Prognostication of Oral Squamous Cell Carcinoma
Yinti, Shanmukha Raviteja; Boaz, Karen; Lewis, Amitha J; Ashokkumar, Pandya Jay; Kapila, Supriya Nikita
2015-01-01
Background Carcinogenesis follows complex molecular alterations, which are triggered by subtle chromatin architectural changes that are imperceptible to the human eye. As the treatment decisions in Oral Squamous Cell Carcinoma (OSCC) are hindered by the imprecise clinical stage determination and inter-observer variability in histological grading, focus in recent years has shifted to discovering identifiers related to neoplastic cell morphology studied through computer-aided image analysis. One such approach is the assessment of fractal geometry, a technique first described by Mandelbrot, which aids in precise assessment of architecture of natural objects. Assessment and quantification of degree of complexity of these fractal objects (self-similarities in structural complexity at different magnifying scales) is described as fractal dimension (FD). Aim To evaluate the nuclear fractal dimension (NFD) in OSCC using computer-aided image analysis. Materials and Methods Histological sections of 14 selected cases of Oral Squamous Cell Carcinoma (OSCC) and 6 samples of normal buccal mucosa (as control) were stained with Haematoxylin-Eosin and Feulgen stain for histopathological examination and evaluation of nuclear complexity respectively. Fifteen HPF at Invasive Tumour Front (ITF) and Tumour Proper (TP) of Feulgen-stained sections were selected and photographed in test and control samples. At ITF, TP and normal buccal mucosa 200 nuclei each were selected and analyzed using Image J software to quantify FD. The test and control groups were compared statistically using Independent sample t-test and One-way ANOVA. Results Nuclear FD increased progressively towards worst tumour staging as compared to normal buccal mucosa. Conclusion Nuclear FD can be considered for quantification of nuclear architectural changes as a prognostic indicator in OSCC. PMID:26674013
Lu, Xiao-long; Zheng, Qin; Yin, Xian-zhen; Xiao, Guang-qing; Liao, Zu-hua; Yang, Ming; Zhang, Ji-wen
2015-06-01
The shape and structure of granules are controlled by the granulation process, which is one of the main factors to determine the nature of the solid dosage forms. In this article, three kinds of granules of a traditional Chinese medicine for improving appetite and promoting digestion, namely, Jianwei Granules, were prepared using granulation technologies as pendular granulation, high speed stirring granulation, and fluidized bed granulation and the powder properties of them were investigated. Meanwhile, synchrotron radiation X-ray computed micro tomography (SR-µCT) was applied to quantitatively determine the irregular internal structures of the granules. The three-dimensional (3D) structure models were obtained by 3D reconstruction, which were more accurately to characterize the three-dimensional structures of the particles through the quantitative data. The models were also used to quantitatively compare the structural differences of granules prepared by different granulation processes with the same formula, so as to characterize how the production process plays a role in the pharmaceutical behaviors of the granules. To focus on the irregularity of the particle structure, the box counting method was used to calculate the fractal dimensions of the granules. The results showed that the fractal dimension is more sensitive to reflect the minor differences in the structure features than the conventional parameters, and capable to specifically distinct granules in structure. It is proved that the fractal dimension could quantitatively characterize the structural information of irregular granules. It is the first time suggested by our research that the fractal dimension difference (Df,c) between two fractal dimension parameters, namely, the volume matrix fractal dimension and the surface matrix fractal dimension, is a new index to characterize granules with irregular structures and evaluate the effects of production processes on the structures of granules as a new
Lu, Xiao-long; Zheng, Qin; Yin, Xian-zhen; Xiao, Guang-qing; Liao, Zu-hua; Yang, Ming; Zhang, Ji-wen
2015-06-01
The shape and structure of granules are controlled by the granulation process, which is one of the main factors to determine the nature of the solid dosage forms. In this article, three kinds of granules of a traditional Chinese medicine for improving appetite and promoting digestion, namely, Jianwei Granules, were prepared using granulation technologies as pendular granulation, high speed stirring granulation, and fluidized bed granulation and the powder properties of them were investigated. Meanwhile, synchrotron radiation X-ray computed micro tomography (SR-µCT) was applied to quantitatively determine the irregular internal structures of the granules. The three-dimensional (3D) structure models were obtained by 3D reconstruction, which were more accurately to characterize the three-dimensional structures of the particles through the quantitative data. The models were also used to quantitatively compare the structural differences of granules prepared by different granulation processes with the same formula, so as to characterize how the production process plays a role in the pharmaceutical behaviors of the granules. To focus on the irregularity of the particle structure, the box counting method was used to calculate the fractal dimensions of the granules. The results showed that the fractal dimension is more sensitive to reflect the minor differences in the structure features than the conventional parameters, and capable to specifically distinct granules in structure. It is proved that the fractal dimension could quantitatively characterize the structural information of irregular granules. It is the first time suggested by our research that the fractal dimension difference (Df,c) between two fractal dimension parameters, namely, the volume matrix fractal dimension and the surface matrix fractal dimension, is a new index to characterize granules with irregular structures and evaluate the effects of production processes on the structures of granules as a new
NASA Astrophysics Data System (ADS)
Jing, Juntao; Feng, Pingfa; Wei, Shiliang; Zhao, Hong; Liu, Yunfeng
2016-11-01
Rotary ultrasonic grinding machining (RUGM) has been employed in Si3N4 ceramics parts machining widely, and the surface morphology is related with surface friction and wear properties directly. It is necessary to investigation on the surface morphology characterization to improve surface quality. Surface morphology of Si3N4 ceramics for rotary ultrasonic grinding machining was investigated based on fractal theory in the paper. The fractal features of surface morphology have been proved with qualitative and quantitative investigation. Differential box-counting method and peleg-blanket method was applied to calculate fractal dimension, but low calculation accuracy was found. So a new fractal dimension calculation method called perimeter-volume method has been proposed. The results show that the calculation deviation rate of morphology fractal dimension is only 2.5%. Meanwhile, the influence of spindle speed, cutting depth, feed rate and cutting force on fractal dimension has also been investigated. The investigation results provide the support for surface morphology optimization.
Anisotropic diffusion limited aggregation in three dimensions: universality and nonuniversality.
Goold, Nicholas R; Somfai, Ellák; Ball, Robin C
2005-09-01
We explore the macroscopic consequences of lattice anisotropy for diffusion limited aggregation (DLA) in three dimensions. Simple cubic and bcc lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an fcc lattice, our data suggest at least two stable fixed points of anisotropy, one matching the bcc case. Hexagonal growths, favoring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resembles real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the DLA model. The algorithm maintains a hierarchy of sphere coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions. PMID:16241431
Anisotropic diffusion limited aggregation in three dimensions: Universality and nonuniversality
NASA Astrophysics Data System (ADS)
Goold, Nicholas R.; Somfai, Ellák; Ball, Robin C.
2005-09-01
We explore the macroscopic consequences of lattice anisotropy for diffusion limited aggregation (DLA) in three dimensions. Simple cubic and bcc lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an fcc lattice, our data suggest at least two stable fixed points of anisotropy, one matching the bcc case. Hexagonal growths, favoring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resembles real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the DLA model. The algorithm maintains a hierarchy of sphere coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions.
THE FRACTAL DIMENSION OF STAR-FORMING REGIONS AT DIFFERENT SPATIAL SCALES IN M33
Sanchez, Nestor; Alfaro, Emilio J.; Anez, Neyda; Odekon, Mary Crone
2010-09-01
We study the distribution of stars, H II regions, molecular gas, and individual giant molecular clouds in M33 over a wide range of spatial scales. The clustering strength of these components is systematically estimated through the fractal dimension. We find scale-free behavior at small spatial scales and a transition to a larger correlation dimension (consistent with a nearly uniform distribution) at larger scales. The transition region lies in the range {approx}500-1000 pc. This transition defines a characteristic size that separates the regime of small-scale turbulent motion from that of large-scale galactic dynamics. At small spatial scales, bright young stars and molecular gas are distributed with nearly the same three-dimensional fractal dimension (D {sub f,3D} {approx}< 1.9), whereas fainter stars and H II regions exhibit higher values, D {sub f,3D} {approx_equal} 2.2-2.5. Our results indicate that the interstellar medium in M33 is on average more fragmented and irregular than in the Milky Way.
Fractal dimension analysis of landscape scale variability in greenhouse gas production potentials
NASA Astrophysics Data System (ADS)
da Silva Bicalho, Elton; Spokas, Kurt; La Scala, Newton, Jr.
2015-04-01
Soil greenhouse gas emission is influenced by tillage and management practices that modify soil attributes directly related to the dynamics of soil carbon in the agricultural environment. The aim of this study was to assess the soil CO2 and N2O production potentials and their spatial variability characterized by fractal dimension in different scales, in addition to their correlation with other soil attributes. The quantification of soil CO2 and N2O production was carried out from dry soil samples collected in a grid of 50 × 50 m containing 133 points arranged symmetrically on a sugarcane area under green residue management in southern Brazil. Laboratory incubations were used to analyze greenhouse gas dynamics by gas chromatography. Soil CO2 and N2O production were correlated significantly (P < 0.05) with microbial biomass, silt and clay content, pH, available phosphorus, sum of metal cations (bases), and cation exchange capacity. Similarly, these soil attributes also were correlated with microbial biomass, supporting their role in soil microbial activity and greenhouse gas production. Furthermore, variations in the fractal dimension over the scale indicate that the pattern of the spatial variability structure of soil CO2 production potential was correlated to that observed for microbial biomass, pH, available phosphorus, sum of bases, and cation exchange capacity. On the other hand, only the spatial structure of the clay content, pH and the sum of bases were correlated with the soil N2O production. Therefore, examining the fractal dimension enables the spatially visualization of altering processes across a landscape at different scales, which highlights properties that influence greenhouse gas production and emission in agricultural areas.
ERIC Educational Resources Information Center
McCartney, M.; Myers, D.; Sun, Y.
2008-01-01
The divider dimensions of a range of maps of Ireland dating from 1567 to 1893 are evaluated, and it is shown that for maps produced before 1650 the fractal dimension of the map can be correlated to its date of publication. Various classroom uses and extensions are discussed. (Contains 2 figures.)
NASA Astrophysics Data System (ADS)
Ahammer, Helmut; DeVaney, Trevor T. J.
2004-03-01
The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.
Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy
NASA Astrophysics Data System (ADS)
Kristoufek, Ladislav; Vosvrda, Miloslav
2014-07-01
We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [L. Kristoufek, M. Vosvrda, Physica A 392, 184 (2013)]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock market indices across the world, we find that the most efficient markets are situated in the Eurozone (the Netherlands, France and Germany) and the least efficient ones in the Latin America (Venezuela and Chile).
Berntson, G. M.; Stoll, P.
1997-01-01
Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales. Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly
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)
Monthus, Cécile
2015-09-01
For Gaussian Spin-Glasses in low dimensions, we introduce a simple Strong Disorder renormalization at zero temperature in order to construct ground states for Periodic and Anti-Periodic boundary conditions. The numerical study in dimensions d = 2 (up to sizes 20482) and d = 3 (up to sizes 1283) yields that Domain Walls are fractal of dimensions ds(d = 2) ≃ 1.27 and ds(d = 3) ≃ 2.55, respectively.
Chappard, D; Legrand, E; Haettich, B; Chalès, G; Auvinet, B; Eschard, J P; Hamelin, J P; Baslé, M F; Audran, M
2001-11-01
Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.
Chappard, D; Legrand, E; Haettich, B; Chalès, G; Auvinet, B; Eschard, J P; Hamelin, J P; Baslé, M F; Audran, M
2001-11-01
Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods. PMID:11745685
NASA Astrophysics Data System (ADS)
Willie, Robert
2016-09-01
In this paper, we study a model system of equations of the time dependent Ginzburg-Landau equations of superconductivity in a Lorentz gauge, in scale of Hilbert spaces E^{α } with initial data in E^{β } satisfying 3α + β ≥ N/2, where N=2,3 is such that the spatial domain of the equations [InlineEquation not available: see fulltext.]. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor {{U}}subset E^{α } compact in E^{β } if α >β , uniform differentiability of orbits on the attractor in E0\\cong L2, and the existence of an explicit finite bounding estimate on the fractal dimension of the attractor yielding that its Hausdorff dimension is as well finite. Uniform boundedness in (0,∞ )× Ω of solutions in E^{1/2}\\cong H1(Ω ) is in addition investigated.
A Fractal Dimension and Wavelet Transform Based Method for Protein Sequence Similarity Analysis.
Yang, Lina; Tang, Yuan Yan; Lu, Yang; Luo, Huiwu
2015-01-01
One of the key tasks related to proteins is the similarity comparison of protein sequences in the area of bioinformatics and molecular biology, which helps the prediction and classification of protein structure and function. It is a significant and open issue to find similar proteins from a large scale of protein database efficiently. This paper presents a new distance based protein similarity analysis using a new encoding method of protein sequence which is based on fractal dimension. The protein sequences are first represented into the 1-dimensional feature vectors by their biochemical quantities. A series of Hybrid method involving discrete Wavelet transform, Fractal dimension calculation (HWF) with sliding window are then applied to form the feature vector. At last, through the similarity calculation, we can obtain the distance matrix, by which, the phylogenic tree can be constructed. We apply this approach by analyzing the ND5 (NADH dehydrogenase subunit 5) protein cluster data set. The experimental results show that the proposed model is more accurate than the existing ones such as Su's model, Zhang's model, Yao's model and MEGA software, and it is consistent with some known biological facts. PMID:26357222
A Fractal Dimension and Wavelet Transform Based Method for Protein Sequence Similarity Analysis.
Yang, Lina; Tang, Yuan Yan; Lu, Yang; Luo, Huiwu
2015-01-01
One of the key tasks related to proteins is the similarity comparison of protein sequences in the area of bioinformatics and molecular biology, which helps the prediction and classification of protein structure and function. It is a significant and open issue to find similar proteins from a large scale of protein database efficiently. This paper presents a new distance based protein similarity analysis using a new encoding method of protein sequence which is based on fractal dimension. The protein sequences are first represented into the 1-dimensional feature vectors by their biochemical quantities. A series of Hybrid method involving discrete Wavelet transform, Fractal dimension calculation (HWF) with sliding window are then applied to form the feature vector. At last, through the similarity calculation, we can obtain the distance matrix, by which, the phylogenic tree can be constructed. We apply this approach by analyzing the ND5 (NADH dehydrogenase subunit 5) protein cluster data set. The experimental results show that the proposed model is more accurate than the existing ones such as Su's model, Zhang's model, Yao's model and MEGA software, and it is consistent with some known biological facts.
Nieckarz, Zenon; Tatoń, Grzegorz; Kozerska, Magdalena; Skrzat, Janusz; Sioma, Andrzej
2015-01-01
We presented a novel approach to studies of the vascular grooves located on the inner surface of the cranial vault. A three-dimensional vision system that acquired the endocranial surface topography was used for this purpose. The acquired data were used to generate images showing the branching pattern of the middle meningeal artery. Fractal dimension was used to characterize and analyze branching pattern complexity. We discussed the usefulness of the latter method and indicated difficulties and potential errors connected to the fractal dimension application. The technique introduced for recording traits of the object surface appears to be helpful in anatomical study of morphological variation of dural vascularization. It may also be applicable in paleoneurological research based on analysis of the cranial remnants. Fractal dimension should be used carefully as a method sensitive to many aspects of data acquisition and processing. PMID:25807002
NASA Astrophysics Data System (ADS)
Guo, Long; Cai, XU
2009-08-01
It is shown that many real complex networks share distinctive features, such as the small-world effect and the heterogeneous property of connectivity of vertices, which are different from random networks and regular lattices. Although these features capture the important characteristics of complex networks, their applicability depends on the style of networks. To unravel the universal characteristics many complex networks have in common, we study the fractal dimensions of complex networks using the method introduced by Shanker. We find that the average 'density' (ρ(r)) of complex networks follows a better power-law function as a function of distance r with the exponent df, which is defined as the fractal dimension, in some real complex networks. Furthermore, we study the relation between df and the shortcuts Nadd in small-world networks and the size N in regular lattices. Our present work provides a new perspective to understand the dependence of the fractal dimension df on the complex network structure.
NASA Astrophysics Data System (ADS)
Inclan, Rosa Maria
2016-04-01
Knowledge on three dimensional soil pore architecture is important to improve our understanding of the factors that control a number of critical soil processes as it controls biological, chemical and physical processes at various scales. Computed Tomography (CT) images provide increasingly reliable information about the geometry of pores and solids in soils at very small scale with the benefit that is a non-invasive technique. Fractal formalism has revealed as a useful tool in these cases where highly complex and heterogeneous meda are studied. One of these quantifications is mass dimension (Dm) and spectral dimension (d) applied to describe the water and gas diffusion coefficients in soils (Tarquis et al., 2012). In this work, intact soil samples were collected from the first three horizons of La Herreria soil. This station is located in the lowland mountain area of Sierra de Guadarrama (Santolaria et al., 2015) and it represents a highly degraded type of site as a result of the livestock keeping. The 3D images, of 45.1 micro-m resolution (256x256x256 voxels), were obtained and then binarized following the singularity-CA method (Martín-Sotoca et al. 2016). Based on these images Dm and d were estimated. The results showed an statistical difference in porosity, Dm and d for each horizon. This fact has a direct implication in diffusion parameters for a pore network modeling based on both fractal dimensions. These soil parameters will constitute a basis for site characterization for further studies regarding soil degradation; determining the interaction between soil, plant and atmosphere with respect to human induced activities as well as the basis for several nitrogen and carbon cycles modeling. References Martin Sotoca; J.J. Ana M. Tarquis, Antonio Saa Requejo, and Juan B. Grau (2016). Pore detection in Computed Tomography (CT) soil 3D images using singularity map analysis. Geophysical Research Abstracts, 18, EGU2016-829. Santolaria-Canales, Edmundo and the Gu
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
A new version of Visual tool for estimating the fractal dimension of images
NASA Astrophysics Data System (ADS)
Grossu, I. V.; Felea, D.; Besliu, C.; Jipa, Al.; Bordeianu, C. C.; Stan, E.; Esanu, T.
2010-04-01
This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images (Grossu et al., 2009 [1]). The earlier version was limited to bi-dimensional sets of points, stored in bitmap files. The application was extended for working also with comma separated values files and three-dimensional images. New version program summaryProgram title: Fractal Analysis v02 Catalogue identifier: AEEG_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 9999 No. of bytes in distributed program, including test data, etc.: 4 366 783 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 30 M Classification: 14 Catalogue identifier of previous version: AEEG_v1_0 Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 1999 Does the new version supersede the previous version?: Yes Nature of problem: Estimating the fractal dimension of 2D and 3D images. Solution method: Optimized implementation of the box-counting algorithm. Reasons for new version:The previous version was limited to bitmap image files. The new application was extended in order to work with objects stored in comma separated values (csv) files. The main advantages are: Easier integration with other applications (csv is a widely used, simple text file format); Less resources consumed and improved performance (only the information of interest, the "black points", are stored); Higher resolution (the points coordinates are loaded into Visual Basic double variables [2]); Possibility of storing three-dimensional objects (e.g. the 3D Sierpinski gasket). In this version the optimized box-counting algorithm [1] was extended to the three
NASA Astrophysics Data System (ADS)
Liu, Changjun; Dong, Leiyun; Jiang, Xiaodong
2012-07-01
The fractal dimension (FD) of surfaces has been widely used to characterize the properties of materials. However, most of the previous researches were concentrated on the correlation between the FD of surfaces and mechanical properties of materials, such as impact energy and fracture toughness, etc. The aim of this paper is to characterize the spheroidization grade and strength of 15CrMo steel through determination of FD of cementite phase on the basis of two-dimension microstructural image. Two methods, namely slit-island method (SIM) and box-counting method (BCM), are used to determine the value of FD. It is found that the FD value evaluated by using BCM is generally higher than that evaluated by SIM. This phenomenon may be due to the difference in the principles used in different methods. Whether SIM or BCM is used, the spheroidization grade in 15CrMo steel linearly increases with decreasing the value of FD. The relationship between the FD value, D, and spheroidization grade, S g, can be approximately expressed as D≌-0.11 S g+ A, where A is a constant value which is depended on the evaluation method. Both the ultimate strength and the yielding strength of 15CrMo steel increase with increasing FD of cementite phase. There may be a common relationship between the FD of cementite phase and strength of 15CrMo steel. When the FD of cementite phase in 15CrMo steel is determined, the strength of this steel can be evaluated. The present paper can provide a novel method to evaluate the strength and spheroidization grade of carbon steel through determination of fractal dimension (FD) of cementite phase.
Hadjidemetriou, Marilena; Pippa, Natassa; Pispas, Stergios; Demetzos, Costas
2013-06-01
Dimethoxycurcumin (DMC) is a lipophilic analog of curcumin found in Curcuma longa Linn., which is known to possess significant activity against various cancer cell lines. The purpose of this study was to develop suitable liposomal formulations in order to overcome DMC's poor water solubility and to study the aggregation kinetic profile using the fractal analysis. DMC was incorporated into liposomal formulations composed of DPPC, DPPC:DPPG:chol (9:1:1 molar ratio) and DPPC:DODAP:chol (9:1:1 molar ratio) liposomes. Light scattering techniques were used to elucidate the physicochemical parameters of the liposomal formulations with and without DMC. The structural characteristics of the incorporated molecule were found to be crucial and promote the aggregation mechanism depending also on the liposomes' composition. The results of our study contribute to the overall scientific efforts to prepare efficient carriers for DMC and could be a useful tool in order to study more efficiently the kinetics of the aggregation process of the liposomal carriers. PMID:23294394
Amino acid induced fractal aggregation of gold nanoparticles: Why and how.
Doyen, Matthieu; Goole, Jonathan; Bartik, Kristin; Bruylants, Gilles
2016-02-15
Gold colloids are the object of many studies as they are reported to have potential biological sensing, imaging and drug delivery applications. In the presence of certain amino acids the aggregation of the gold nanoparticles into linear structures is observed, as highlighted by the appearance of a second plasmon band in the UV-Vis spectra of the colloid. The mechanism behind this phenomenon is still under debate. In order to help elucidate this issue, the interaction between gold colloids and different amino acids, modified amino acids and molecules mimicking their side-chain was monitored by UV-Vis absorption, DLS and TEM. The results show that phenomenon can be rationalized in terms of the Diffusion Limited Colloid Aggregation (DLCA) model which gives rise to the fractal aggregation colloids. The global charge of the compound, which influences the ionic strength of the solution, and the ease with which the compound can interact with the GNPs and affect their surface potential, are, the two parameters which control the DLCA regime. Calculations based on the Derjaguin, Landau, Verwey and Overbeek (DLVO) theory confirm all the experimental observations. PMID:26613335
Amino acid induced fractal aggregation of gold nanoparticles: Why and how.
Doyen, Matthieu; Goole, Jonathan; Bartik, Kristin; Bruylants, Gilles
2016-02-15
Gold colloids are the object of many studies as they are reported to have potential biological sensing, imaging and drug delivery applications. In the presence of certain amino acids the aggregation of the gold nanoparticles into linear structures is observed, as highlighted by the appearance of a second plasmon band in the UV-Vis spectra of the colloid. The mechanism behind this phenomenon is still under debate. In order to help elucidate this issue, the interaction between gold colloids and different amino acids, modified amino acids and molecules mimicking their side-chain was monitored by UV-Vis absorption, DLS and TEM. The results show that phenomenon can be rationalized in terms of the Diffusion Limited Colloid Aggregation (DLCA) model which gives rise to the fractal aggregation colloids. The global charge of the compound, which influences the ionic strength of the solution, and the ease with which the compound can interact with the GNPs and affect their surface potential, are, the two parameters which control the DLCA regime. Calculations based on the Derjaguin, Landau, Verwey and Overbeek (DLVO) theory confirm all the experimental observations.
Accretion of Fine Particles: Sticking Probability Estimated by Optical Sizing of Fractal Aggregates
NASA Astrophysics Data System (ADS)
Sugiura, N.; Higuchi, Y.
1993-07-01
Sticking probability of fine particles is an important parameter that determines (1) the settling of fine particles to the equatorial plane of the solar nebula and hence the formation of planetesimals, and (2) the thermal structure of the nebula, which is dependent on the particle size through opacity. It is generally agreed that the sticking probability is 1 for submicrometer particles, but at sizes larger than 1 micrometer, there exist almost no data on the sticking probability. A recent study [1] showed that aggregates (with radius from 0.2 to 2 mm) did not stick when collided at a speed of 0.15 to 4 m/s. Therefore, somewhere between 1 micrometer and 200 micrometers, sticking probabilities of fine particles change from nearly 1 to nearly 0. We have been studying [2,3] sticking probabilities of dust aggregates in this size range using an optical sizing method. The optical sizing method has been well established for spherical particles. This method utilizes the fact that the smaller the size, the larger the angle of the scattered light. For spheres with various sizes, the size distribution is determined by solving Y(i) = M(i,j)X(j), where Y(i) is the scattered light intensity at angle i, X(j) is the number density of spheres with size j, and M(i,j) is the scattering matrix, which is determined by Mie theory. Dust aggregates, which we expect to be present in the early solar nebula, are not solid spheres, but probably have a porous fractal structure. For such aggregates the scattering matrix M(i,j) must be determined by taking account of all the interaction among constituent particles (discrete dipole approximation). Such calculation is possible only for very small aggregates, and for larger aggregates we estimate the scattering matrix by extrapolation, assuming that the fractal nature of the aggregates allows such extrapolation. In the experiments using magnesium oxide fine particles floating in a chamber at ambient pressure, the size distribution (determined by
Ferer, M; Bromhal, Grant S; Smith, Duane H
2009-07-01
Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact /linear flow at a characteristic crossover time, tau , in three dimensions to systems with as many as a 10(6) pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, ttau. Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, ttau, the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, beta=1.0+/-0.1. We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Lévy flights, the height variations across the interface appear to be random with occasional large height variations. PMID:19658710
Ferer, M.; Bromhal, Grant S.; Smith, Duane H.
2009-07-01
Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact/linear flow at a characteristic crossover time, τ, in three dimensions to systems with as many as a 10^{6} pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, t >> τ. Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, t >> τ, the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, β=1.0±0.1. We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Levy flights, the height variations across the interface appear to be random with occasional large height variations.
Phulari, Rashmi G S; Rathore, Rajendrasinh S; Talegaon, Trupti Pramod
2016-01-01
Background: Various clinical and histological factors have helped in predicting the survival of patients with oral squamous cell carcinoma (OSCC). However, there has been a need for more specialized diagnostic and prognostic factors to avoid subjective variation among opinion. Thus, fractal dimension (FD) can be used as an index of the morphological changes that the epithelial cells undergo during their transformation into neoplastic cell. In oral cancer study, nuclear FD (NFD) can be used as a quantitative index to discriminate between normal, dysplastic and neoplastic oral mucosa. Aim: To use nuclear fractal geometry to compare the morphometric complexity in the normal, epithelial dysplasia and OSCC cases and to verify the difference among the various histological grades of dysplasia and OSCC. It was fulfilled by estimating the FDs of the nuclear surface. Materials and Methods: Histopathologically diagnosed cases of epithelial dysplasia and OSCC were taken from the archives. Photomicrographs were captured with the help of Lawrence and Mayo research microscope. The images were then subjected to image analysis using the Image J software with FracLac plugin java 1.6 to obtain FDs. FD of ten selected nuclei was calculated using the box-counting algorithm. Statistical Analysis: was done using descriptive analysis, ANOVA and Tukey's honest significant difference post hoc tests with STATAIC-13 software. Results and Conclusion: NFD can provide valuable information to discriminate between normal mucosa, dysplasia and carcinoma objectively without subjective discrimination. PMID:27721604
NASA Astrophysics Data System (ADS)
Donadio, Carlo; Magdaleno, Fernando; Mazzarella, Adriano; Mathias Kondolf, G.
2015-07-01
By applying fractal geometry analysis to the drainage network of three large watercourses in America and Europe, we have calculated for the first time their fractal dimension. The aim is to interpret the geomorphologic characteristics to better understand the morphoevolutionary processes of these fluvial morphotypes; to identify and discriminate geomorphic phenomena responsible for any difference or convergence of a fractal dimension; to classify hydrographic patterns, and finally to compare the fractal degree with some geomorphic-quantitative indexes. The analyzed catchment of Russian (California, USA), Ebro (Spain), and Volturno (Italy) rivers are situated in Mediterranean-climate regions sensu Köppen, but with different geologic context and tectonic styles. Results show fractal dimensions ranging from 1.08 to 1.50. According to the geological setting and geomorphic indexes of these basins, the lower fractal degree indicates a prevailing tectonics, active or not, while the higher degree indicates the stronger erosion processes on inherited landscapes.
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)
SU-D-BRA-04: Fractal Dimension Analysis of Edge-Detected Rectal Cancer CTs for Outcome Prediction
Zhong, H; Wang, J; Hu, W; Shen, L; Wan, J; Zhou, Z; Zhang, Z
2015-06-15
Purpose: To extract the fractal dimension features from edge-detected rectal cancer CTs, and to examine the predictability of fractal dimensions to outcomes of primary rectal cancer patients. Methods: Ninety-seven rectal cancer patients treated with neo-adjuvant chemoradiation were enrolled in this study. CT images were obtained before chemoradiotherapy. The primary lesions of the rectal cancer were delineated by experienced radiation oncologists. These images were extracted and filtered by six different Laplacian of Gaussian (LoG) filters with different filter values (0.5–3.0: from fine to coarse) to achieve primary lesions in different anatomical scales. Edges of the original images were found at zero-crossings of the filtered images. Three different fractal dimensions (box-counting dimension, Minkowski dimension, mass dimension) were calculated upon the image slice with the largest cross-section of the primary lesion. The significance of these fractal dimensions in survival, recurrence and metastasis were examined by Student’s t-test. Results: For a follow-up time of two years, 18 of 97 patients had experienced recurrence, 24 had metastasis, and 18 were dead. Minkowski dimensions under large filter values (2.0, 2.5, 3.0) were significantly larger (p=0.014, 0.006, 0.015) in patients with recurrence than those without. For metastasis, only box-counting dimensions under a single filter value (2.5) showed differences (p=0.016) between patients with and without. For overall survival, box-counting dimensions (filter values = 0.5, 1.0, 1.5), Minkowski dimensions (filter values = 0.5, 1.5, 2.0, 2,5) and mass dimensions (filter values = 1.5, 2.0) were all significant (p<0.05). Conclusion: It is feasible to extract shape information by edge detection and fractal dimensions analysis in neo-adjuvant rectal cancer patients. This information can be used to prognosis prediction.
Topological Vulnerability Evaluation Model Based on Fractal Dimension of Complex Networks.
Gou, Li; Wei, Bo; Sadiq, Rehan; Sadiq, Yong; Deng, Yong
2016-01-01
With an increasing emphasis on network security, much more attentions have been attracted to the vulnerability of complex networks. In this paper, the fractal dimension, which can reflect space-filling capacity of networks, is redefined as the origin moment of the edge betweenness to obtain a more reasonable evaluation of vulnerability. The proposed model combining multiple evaluation indexes not only overcomes the shortage of average edge betweenness's failing to evaluate vulnerability of some special networks, but also characterizes the topological structure and highlights the space-filling capacity of networks. The applications to six US airline networks illustrate the practicality and effectiveness of our proposed method, and the comparisons with three other commonly used methods further validate the superiority of our proposed method.
Topological Vulnerability Evaluation Model Based on Fractal Dimension of Complex Networks
Gou, Li; Wei, Bo; Sadiq, Rehan; Sadiq, Yong; Deng, Yong
2016-01-01
With an increasing emphasis on network security, much more attentions have been attracted to the vulnerability of complex networks. In this paper, the fractal dimension, which can reflect space-filling capacity of networks, is redefined as the origin moment of the edge betweenness to obtain a more reasonable evaluation of vulnerability. The proposed model combining multiple evaluation indexes not only overcomes the shortage of average edge betweenness’s failing to evaluate vulnerability of some special networks, but also characterizes the topological structure and highlights the space-filling capacity of networks. The applications to six US airline networks illustrate the practicality and effectiveness of our proposed method, and the comparisons with three other commonly used methods further validate the superiority of our proposed method. PMID:26751371
Zhang, Jiong; Bekkers, Erik; Abbasi-Sureshjani, Samaneh
2016-01-01
The retinal fractal dimension (FD) is a measure of vasculature branching pattern complexity. FD has been considered as a potential biomarker for the detection of several diseases like diabetes and hypertension. However, conflicting findings were found in the reported literature regarding the association between this biomarker and diseases. In this paper, we examine the stability of the FD measurement with respect to (1) different vessel annotations obtained from human observers, (2) automatic segmentation methods, (3) various regions of interest, (4) accuracy of vessel segmentation methods, and (5) different imaging modalities. Our results demonstrate that the relative errors for the measurement of FD are significant and FD varies considerably according to the image quality, modality, and the technique used for measuring it. Automated and semiautomated methods for the measurement of FD are not stable enough, which makes FD a deceptive biomarker in quantitative clinical applications. PMID:27703803
Topological Vulnerability Evaluation Model Based on Fractal Dimension of Complex Networks.
Gou, Li; Wei, Bo; Sadiq, Rehan; Sadiq, Yong; Deng, Yong
2016-01-01
With an increasing emphasis on network security, much more attentions have been attracted to the vulnerability of complex networks. In this paper, the fractal dimension, which can reflect space-filling capacity of networks, is redefined as the origin moment of the edge betweenness to obtain a more reasonable evaluation of vulnerability. The proposed model combining multiple evaluation indexes not only overcomes the shortage of average edge betweenness's failing to evaluate vulnerability of some special networks, but also characterizes the topological structure and highlights the space-filling capacity of networks. The applications to six US airline networks illustrate the practicality and effectiveness of our proposed method, and the comparisons with three other commonly used methods further validate the superiority of our proposed method. PMID:26751371
Electroencephalographic Fractal Dimension in Healthy Ageing and Alzheimer’s Disease
Cottone, Carlo; Cancelli, Andrea; Rossini, Paolo Maria; Tecchio, Franca
2016-01-01
Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi’s fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20–89 years) and 67 Alzheimer’s Disease (AD) patients (age: 50–88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called “free” copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging. PMID:26872349
NASA Astrophysics Data System (ADS)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to
Puškaš, Nela; Zaletel, Ivan; Stefanović, Bratislav D; Ristanović, Dušan
2015-03-01
Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones.
Kopelman, R
1988-09-23
Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds.
NASA Astrophysics Data System (ADS)
Skorupski, Krzysztof; Mroczka, Janusz; Wriedt, Thomas; Riefler, Norbert
2014-06-01
In many branches of science experiments are expensive, require specialist equipment or are very time consuming. Studying the light scattering phenomenon by fractal aggregates can serve as an example. Light scattering simulations can overcome these problems and provide us with theoretical, additional data which complete our study. For this reason a fractal-like aggregate model as well as fast aggregation codes are needed. Until now various computer models, that try to mimic the physics behind this phenomenon, have been developed. However, their implementations are mostly based on a trial-and-error procedure. Such approach is very time consuming and the morphological parameters of resulting aggregates are not exact because the postconditions (e.g. the position error) cannot be very strict. In this paper we present a very fast and accurate implementation of a tunable aggregation algorithm based on the work of Filippov et al. (2000). Randomization is reduced to its necessary minimum (our technique can be more than 1000 times faster than standard algorithms) and the position of a new particle, or a cluster, is calculated with algebraic methods. Therefore, the postconditions can be extremely strict and the resulting errors negligible (e.g. the position error can be recognized as non-existent). In our paper two different methods, which are based on the particle-cluster (PC) and the cluster-cluster (CC) aggregation processes, are presented.
NASA Astrophysics Data System (ADS)
Meir, Yigal; Aharony, Amnon
1989-05-01
We investigate the problem of flow in porous media near the percolation threshold by studying the generelized model of Viscous Fingering (VF) on fractal structures. We obtain analytic expressions for the fractal dimensions of the resulting structures, which are in excellent agreement with existing experimental results, and exact relations for the exponent Dt, which describes the scaling of the time it takes the fluid to cross the sample, with the sample size, in terms of geometrical exponents for various experimental situations. Lastly, we discuss the relation between the continuous viscous fingers model and stochastic processes such as dielectric breakdown model (DBM) and diffusion limited aggregation (DLA).
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 dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; Dou, Jie; Wang, Cheng
2015-08-12
The morphological properties of kaolin flocs were investigated in a Couette-flow experiment at the steady state under seven shear flow conditions (shear rates of 5.36, 9.17, 14, 24, 31, 41 and 53 s^{-1}). These properties include a one-dimensional (1-D) fractal dimension (D_{1}), a two-dimensional (2-D) fractal dimension (D_{2}), a perimeter-based fractal dimension (D_{pf}) and an aspect ratio (AR). They were calculated based on the projected area (A), equivalent size, perimeter (P) and length (L) of the major axis of the floc determined through sample observation and an image analysis system. The parameter D_{2}, which characterizes the relationship between the projected area and the length of the major axis using a power function, A ∝ L^{D2}, increased from 1.73 ± 0.03, 1.72 ± 0.03, and 1.75 ± 0.04 in the low shear rate group (G = 5.36, 9.17, and 14 s^{-1}) to 1.92 ± 0.03, 1.82 ± 0.02, 1.85 ± 0.02, and 1.81 ± 0.02 in the high shear rate group (24, 31, 41 and 53 s^{-1}), respectively. The parameter D_{1} characterizes the relationship between the perimeter and length of the major axis by the function P ∝ L^{D1} and decreased from 1.52 ± 0.02, 1.48 ± 0.02, 1.55 ± 0.02, and 1.63 ± 0.02 in the low shear group (5.36, 9.17, 14 and 24 s^{-1}) to 1.45 ± 0.02, 1.39 ± 0.02, and 1.39 ± 0.02 in the high shear group (31, 41 and 53 s^{-1}), respectively. The results indicate that with increasing shear rates, the flocs become less elongated and that their boundary lines become tighter and more regular, caused by more breakages and possible restructurings of the flocs. The parameter D_{pf}, which is related to the perimeter and the projected area through the function , decreased as the shear rate increased almost linearly. The parameter AR, which is the ratio of the length of the major axis and equivalent diameter, decreased from 1.56, 1
Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; Dou, Jie; Wang, Cheng
2015-08-12
The morphological properties of kaolin flocs were investigated in a Couette-flow experiment at the steady state under seven shear flow conditions (shear rates of 5.36, 9.17, 14, 24, 31, 41 and 53 s-1). These properties include a one-dimensional (1-D) fractal dimension (D1), a two-dimensional (2-D) fractal dimension (D2), a perimeter-based fractal dimension (Dpf) and an aspect ratio (AR). They were calculated based on the projected area (A), equivalent size, perimeter (P) and length (L) of the major axis of the floc determined through sample observation and an image analysis system. The parameter D2, which characterizes the relationship between the projectedmore » area and the length of the major axis using a power function, A ∝ LD2, increased from 1.73 ± 0.03, 1.72 ± 0.03, and 1.75 ± 0.04 in the low shear rate group (G = 5.36, 9.17, and 14 s-1) to 1.92 ± 0.03, 1.82 ± 0.02, 1.85 ± 0.02, and 1.81 ± 0.02 in the high shear rate group (24, 31, 41 and 53 s-1), respectively. The parameter D1 characterizes the relationship between the perimeter and length of the major axis by the function P ∝ LD1 and decreased from 1.52 ± 0.02, 1.48 ± 0.02, 1.55 ± 0.02, and 1.63 ± 0.02 in the low shear group (5.36, 9.17, 14 and 24 s-1) to 1.45 ± 0.02, 1.39 ± 0.02, and 1.39 ± 0.02 in the high shear group (31, 41 and 53 s-1), respectively. The results indicate that with increasing shear rates, the flocs become less elongated and that their boundary lines become tighter and more regular, caused by more breakages and possible restructurings of the flocs. The parameter Dpf, which is related to the perimeter and the projected area through the function , decreased as the shear rate increased almost linearly. The parameter AR, which is the ratio of the length of the major axis and equivalent diameter, decreased from 1.56, 1.59, 1.53 and 1.51 in the low shear rate group to 1.43, 1.47 and 1.48 in the high shear rate group. These changes in Dpf and AR show that the flocs become
Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Chen, Falin
2014-01-01
Background This study aimed to investigate the correlation between quantitative retinal vascular parameters such as central retinal arteriolar equivalent (CRAE) and retinal vascular fractal dimension (D(f)), and cardiovascular risk factors in the Chinese Han population residing in the in islands of southeast China. Methodology/Principle Findings In this cross-sectional study, fundus photographs were collected and semi-automated analysis software was used to analyze retinal vessel diameters and fractal dimensions. Cardiovascular risk factors such as relevant medical history, blood pressure (BP), lipids, and blood glucose data were collected. Subjects had a mean age of 51.9±12.0 years and included 812 (37.4%) males and 1,357 (62.6%) females. Of the subjects, 726 (33.5%) were overweight, 226 (10.4%) were obese, 272 (12.5%) had diabetes, 738 (34.0%) had hypertension, and 1,156 (53.3%) had metabolic syndrome. After controlling for the effects of potential confounders, multivariate analyses found that age (β = 0.06, P = 0.008), sex (β = 1.33, P = 0.015), mean arterial blood pressure (β = −0.12, P<0.001), high-sensitivity C-reactive protein (β = −0.22, P = 0.008), and CRVE (β = 0.23, P<0.001) were significantly associated with CRAE. Age (β = −0.0012, P<0.001), BP classification (prehypertension: β = −0.0075, P = 0.014; hypertension: β = −0.0131, P = 0.002), and hypertension history (β = −0.0007, P = 0.009) were significantly associated with D(f). Conclusions/Significance D(f) exhibits a stronger association with BP than CRAE. Thus, D(f) may become a useful indicator of cardiovascular risk. PMID:25188273
NASA Astrophysics Data System (ADS)
Alonso, C.; Benito, R. M.; Tarquis, A. M.
2012-04-01
such complexities from remote sensing images and will applied in this study to see the scaling behavior for each sensor in generalized fractal dimensions. The studied area is located in the provinces of Caceres and Salamanca (east of Iberia Peninsula) with an extension of 32 x 32 km2. The altitude in the area varies from 1,560 to 320 m, comprising natural vegetation in the mountain area (forest and bushes) and agricultural crops in the valleys. Scaling analysis were applied to Landsat-5 and MODIS TERRA to the normalized derived vegetation index (NDVI) on the same region with one day of difference, 13 and 12 of July 2003 respectively. From these images the area of interest was selected obtaining 1024 x 1024 pixels for Landsat image and 128 x 128 pixels for MODIS image. This implies that the resolution for MODIS is 250x250 m. and for Landsat is 30x30 m. From the reflectance data obtained from NIR and RED bands, NDVI was calculated for each image focusing this study on 0.2 to 0.5 ranges of values. Once that both NDVI fields were obtained several fractal dimensions were estimated in each one segmenting the values in 0.20-0.25, 0.25-0.30 and so on to rich 0.45-0.50. In all the scaling analysis the scale size length was expressed in meters, and not in pixels, to make the comparison between both sensors possible. Results are discussed. Acknowledgements This work has been supported by the Spanish MEC under Projects No. AGL2010-21501/AGR, MTM2009-14621 and i-MATH No. CSD2006-00032
NASA Astrophysics Data System (ADS)
Kou, Jian-Long; Lu, Hang-Jun; Wu, Feng-Min; Xu, You-Sheng
2008-05-01
A tumour vascular network, characterized as an irregularly stochastic growth, is different from the normal vascular network. We systematically analyse the dependence of the branching. It is found that anastomosis of tumour on time is according to a number of tumour images, and both the fractal dimensions and multifractal spectra of the tumours are obtained. In the cases studied, the fractal dimensions of the tumour vascular network increase with time and the multifractal spectrum not only rises entirely but also shifts right. In addition, the best drug delivery stage is discussed according to the difference of the singularity exponent δα(δα = αmax — αmin), which shows some change in the growth process of the tumour vascular network. A common underlying principle is obtained from our analysis along with previous results.
Manera, M; Dezfuli, B S; Borreca, C; Giari, L
2014-11-01
Fractal analysis is a reliable method for describing, summarizing object complexity and heterogeneity and has been widely used in biology and medicine to deal with scale, size and shape management problems. The aim of present survey was to use fractal analysis as a complexity measure to characterize mast cells (MCs) degranulation in a rainbow trout ex vivo model (isolated organ bath). Compound 48/80, a condensation product of N-methyl-p-methoxyphenethylamine with formaldehyde, was adopted as MCs degranulation agent in trout intestinal strips. Fractal dimension (D), as a measure of complexity, 'roughness' and lacunarity (λ), as a measure of rotational and translational invariance, heterogeneity, in other words, of the texture, were compared in MCs images taken from intestinal strips before and after compound 48/80 addition to evaluate if and how they were affected by degranulation. Such measures were also adopted to evaluate their discrimination efficacy between compound 48/80 degranulated group and not degranulated group and the results were compared with previously reported data obtained with conventional texture analysis (image histogram, run-length matrix, co-occurrence matrix, autoregressive model, wavelet transform) on the same experimental material. Outlines, skeletons and original greyscale images were fractal analysed to evaluate possible significant differences in the measures values according to the analysed feature. In particular, and considering outline and skeleton as analysed features, fractal dimensions from compound 48/80 treated intestinal strips were significantly higher than the corresponding untreated ones (paired t and Wilcoxon test, p < 0.05), whereas corresponding lacunarity values were significantly lower (paired Wilcoxon test, p < 0.05) but only for outline as analysed feature. Outlines roughness increase is consistent with an increased granular mediators interface, favourable for their biological action; while lacunarity (image
Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension.
Backes, André Ricardo; Gerhardinger, Leandro Cavaleri; Batista Neto, João do Espírito Santo; Bruno, Odemir Martinez
2015-02-01
Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered.
Reljin, Natasa; Reyes, Bersain A; Chon, Ki H
2015-04-27
In this paper, we propose the use of blanket fractal dimension (BFD) to estimate the tidal volume from smartphone-acquired tracheal sounds. We collected tracheal sounds with a Samsung Galaxy S4 smartphone, from five (N = 5) healthy volunteers. Each volunteer performed the experiment six times; first to obtain linear and exponential fitting models, and then to fit new data onto the existing models. Thus, the total number of recordings was 30. The estimated volumes were compared to the true values, obtained with a Respitrace system, which was considered as a reference. Since Shannon entropy (SE) is frequently used as a feature in tracheal sound analyses, we estimated the tidal volume from the same sounds by using SE as well. The evaluation of the performed estimation, using BFD and SE methods, was quantified by the normalized root-mean-squared error (NRMSE). The results show that the BFD outperformed the SE (at least twice smaller NRMSE was obtained). The smallest NRMSE error of 15.877% ± 9.246% (mean ± standard deviation) was obtained with the BFD and exponential model. In addition, it was shown that the fitting curves calculated during the first day of experiments could be successfully used for at least the five following days.
Reljin, Natasa; Reyes, Bersain A.; Chon, Ki H.
2015-01-01
In this paper, we propose the use of blanket fractal dimension (BFD) to estimate the tidal volume from smartphone-acquired tracheal sounds. We collected tracheal sounds with a Samsung Galaxy S4 smartphone, from five (N = 5) healthy volunteers. Each volunteer performed the experiment six times; first to obtain linear and exponential fitting models, and then to fit new data onto the existing models. Thus, the total number of recordings was 30. The estimated volumes were compared to the true values, obtained with a Respitrace system, which was considered as a reference. Since Shannon entropy (SE) is frequently used as a feature in tracheal sound analyses, we estimated the tidal volume from the same sounds by using SE as well. The evaluation of the performed estimation, using BFD and SE methods, was quantified by the normalized root-mean-squared error (NRMSE). The results show that the BFD outperformed the SE (at least twice smaller NRMSE was obtained). The smallest NRMSE error of 15.877% ± 9.246% (mean ± standard deviation) was obtained with the BFD and exponential model. In addition, it was shown that the fitting curves calculated during the first day of experiments could be successfully used for at least the five following days. PMID:25923929
Takeda, T; Sakata, A; Matsuoka, T
1999-08-01
1. Occurrence of miniature endplate potentials (MEPP) in the sartorius muscle of Rana catesbiana in high Mg2+ Ringer solution were observed in standard intracellular recording. Intervals and amplitudes of sequentially occurring MEPP were registered and analyzed. 2. Interval histograms of a time series of MEPP showed exponential-like pattern as reported in the classical study by Fatt and Katz (1952). The cumulative distribution of the intervals plotted in logarithmic axes showed two distinct phases. In shorter intervals (< 1s), curve along exponential decay was observed, and in longer intervals (> or = 1s) linear decay can be seen. The latter power-law relation gave dimensions of 4.111 +/- 0.812 (mean and S.D.). Self-similarity in longer range implies a time-scale invariant nature and may suggest fractal nature in restoration process of synaptic vesicles, while exponential decay in the short time interval range implies random release of transmitter packet from the readily releasable pool. 3. Fluctuation of amplitudes in sequentially occurred MEPP were analyzed according to Higuchi's cumulative route-length analysis. The estimates for sequential amplitude curve showed the power-law relation in a logarithmic plot whose inclination (= D) estimated with linear regression analysis was 1.996 +/- 0.007 (mean and S.D.). This results indicate that fluctuation in the amplitude of MEPP shows possible maximum complexity as a graphic curve in 2-D plane. Similar result was obtained for fluctuation of intervals of successively occurring MEPP. PMID:10621955
A new way of describing meiosis that uses fractal dimension to predict metaphase I
2005-01-01
Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465
NASA Astrophysics Data System (ADS)
Fernandes, Maurício Anderson; Ribeiro Rosa, Edvaldo Antônio; Johann, Aline Cristina Batista Rodrigues; Grégio, Ana Maria Trindade; Trevilatto, Paula Cristina; Azevedo-Alanis, Luciana Reis
2016-01-01
Objectives: To test the capacity of the digital tool, fractal dimension (FD) analysis, in identifying subtle differences in bone pattern in patients with renal osteodystrophy (RO), correlated with the time of hemodialysis, in different regions of interest, delineated on panoramic and periapical radiographs. Study design: A total of 34 patients with chronic renal disease undergoing hemodialysis were submitted to panoramic and periapical radiographs. Different regions of interest were delineated on the mandibular body and ramus. FD was analyzed by means of the software program ImageJ and correlated with the time of hemodialysis. Results: The sample consisted of 34 subjects. The time of hemodialysis varied from 1 to 286 months. There was significant correlation between the time of hemodialysis and the FD values in the region delineated in the mandibular angle (r = 0.498; p = 0.003) and this was shown in the periapical radiographs as well (r = -0.349; p = 0.043). Conclusions: FD analysis was a useful tool in detecting alterations caused by RO in bone pattern, in panoramic and periapical radiographs.
Earthquake frequency-magnitude distribution and fractal dimension in mainland Southeast Asia
NASA Astrophysics Data System (ADS)
Pailoplee, Santi; Choowong, Montri
2014-12-01
The 2004 Sumatra and 2011 Tohoku earthquakes highlighted the need for a more accurate understanding of earthquake characteristics in both regions. In this study, both the a and b values of the frequency-magnitude distribution (FMD) and the fractal dimension ( D C ) were investigated simultaneously from 13 seismic source zones recognized in mainland Southeast Asia (MLSEA). By using the completeness earthquake dataset, the calculated values of b and D C were found to imply variations in seismotectonic stress. The relationships of D C -b and D C -( a/ b) were investigated to categorize the level of earthquake hazards of individual seismic source zones, where the calibration curves illustrate a negative correlation between the D C and b values ( D c = 2.80 - 1.22 b) and a positive correlation between the D C and a/ b ratios ( D c = 0.27( a/ b) - 0.01) with similar regression coefficients ( R 2 = 0.65 to 0.68) for both regressions. According to the obtained relationships, the Hsenwi-Nanting and Red River fault zones revealed low-stress accumulations. Conversely, the Sumatra-Andaman interplate and intraslab, the Andaman Basin, and the Sumatra fault zone were defined as high-tectonic stress regions that may pose risks of generating large earthquakes in the future.
Assessing severity of obstructive sleep apnea by fractal dimension sequence analysis of sleep EEG
NASA Astrophysics Data System (ADS)
Zhang, J.; Yang, X. C.; Luo, L.; Shao, J.; Zhang, C.; Ma, J.; Wang, G. F.; Liu, Y.; Peng, C.-K.; Fang, J.
2009-10-01
Different sleep stages are associated with distinct dynamical patterns in EEG signals. In this article, we explored the relationship between the sleep architecture and fractal dimension (FD) of sleep EEG. In particular, we applied the FD analysis to the sleep EEG of patients with obstructive sleep apnea-hypopnea syndrome (OSAHS), which is characterized by recurrent oxyhemoglobin desaturation and arousals from sleep, a disease which received increasing public attention due to its significant potential impact on health. We showed that the variation of FD reflects the macrostructure of sleep. Furthermore, the fast fluctuation of FD, as measured by the zero-crossing rate of detrended FD (zDFD), is a useful indicator of sleep disturbance, and therefore, correlates with apnea-hypopnea index (AHI), and hourly number of blood oxygen saturation (SpO 2) decreases greater than 4%, as obstructive apnea/hypopnea disturbs sleep architecture. For practical purpose, a modified index combining zDFD of EEG and body mass index (BMI) may be useful for evaluating the severity of OSAHS symptoms.
NASA Astrophysics Data System (ADS)
Neves, L. A.; Oliveira, F. R.; Peres, F. A.; Moreira, R. D.; Moriel, A. R.; de Godoy, M. F.; Murta Junior, L. O.
2011-03-01
This paper presents a method for the quantification of cellular rejection in endomyocardial biopsies of patients submitted to heart transplant. The model is based on automatic multilevel thresholding, which employs histogram quantification techniques, histogram slope percentage analysis and the calculation of maximum entropy. The structures were quantified with the aid of the multi-scale fractal dimension and lacunarity for the identification of behavior patterns in myocardial cellular rejection in order to determine the most adequate treatment for each case.
2013-01-01
Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522
Fuss, Franz Konstantin
2013-01-01
Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522
Early stage fractal growth in thin films below the percolation limit
NASA Astrophysics Data System (ADS)
Batabyal, R.; Mahato, J. C.; Das, Debolina; Dev, B. N.
2013-02-01
We demonstrate the fractal growth of epitaxial Ag thin films on Si(111) surfaces using scanning tunneling microscopy (STM). The initial stage growth of Ag thin films provides islands of compact shape. These compact-shaped two-dimensional (2D) islands follow the Euclidian dimension 2. As the islands grow they become fractal in nature. The fractal (Hausdorff) dimension of the islands depends on the coverage of the Ag thin films. The mechanism responsible for this fractal nature of the Ag nanostructures varies from diffusion limited aggregation (DLA) to diffusion limited cluster aggregation (DLCA).
Mossotti, Victor G.; Eldeeb, A. Raouf
2000-01-01
Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.
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.
2010-01-01
Background Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony. Methods Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization. Results A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total. Conclusions The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process. PMID:20846449
Technology Transfer Automated Retrieval System (TEKTRAN)
The aggregate structure of phthalic anhydride (PA) modified soy protein isolate (SPI) was investigated by estimating its fractal dimension from the equilibrated dynamic strain sweep experiments. The estimated fractal dimensions of the filler aggregates were less than 2, indicating that these partic...
Local connected fractal dimension analysis in gill of fish experimentally exposed to toxicants.
Manera, Maurizio; Giari, Luisa; De Pasquale, Joseph A; Sayyaf Dezfuli, Bahram
2016-06-01
An operator-neutral method was implemented to objectively assess European seabass, Dicentrarchus labrax (Linnaeus, 1758) gill pathology after experimental exposure to cadmium (Cd) and terbuthylazine (TBA) for 24 and 48h. An algorithm-derived local connected fractal dimension (LCFD) frequency measure was used in this comparative analysis. Canonical variates (CVA) and linear discriminant analysis (LDA) were used to evaluate the discrimination power of the method among exposure classes (unexposed, Cd exposed, TBA exposed). Misclassification, sensitivity and specificity, both with original and cross-validated cases, were determined. LCFDs frequencies enhanced the differences among classes which were visually selected after their means, respective variances and the differences between Cd and TBA exposed means, with respect to unexposed mean, were analyzed by scatter plots. Selected frequencies were then scanned by means of LDA, stepwise analysis, and Mahalanobis distance to detect the most discriminative frequencies out of ten originally selected. Discrimination resulted in 91.7% of cross-validated cases correctly classified (22 out of 24 total cases), with sensitivity and specificity, respectively, of 95.5% (1 false negative with respect to 21 really positive cases) and 75% (1 false positive with respect to 3 really negative cases). CVA with convex hull polygons ensured prompt, visually intuitive discrimination among exposure classes and graphically supported the false positive case. The combined use of semithin sections, which enhanced the visual evaluation of the overall lamellar structure; of LCFD analysis, which objectively detected local variation in complexity, without the possible bias connected to human personnel; and of CVA/LDA, could be an objective, sensitive and specific approach to study fish gill lamellar pathology. Furthermore this approach enabled discrimination with sufficient confidence between exposure classes or pathological states and avoided
Comparison of different fractal dimension measuring algorithms for RE-TM M-O films
NASA Technical Reports Server (NTRS)
Bernacki, Bruce E.; Mansuripur, M.
1991-01-01
Noise in magneto-optical recording devices is discussed. In general, it appears that either the divider technique or amplitude spectrum technique may be used interchangeably to measure the fractal dimension (D) in the domain wall structure of ideal images. However, some caveats must be observed for best results. The divider technique is attractive for its simplicity and relatively modest computation requirements. However, it is sensitive to noise, in that noise pixels that touch the domain boundary are interpreted as being part of the boundary, skewing the measurement. Also, it is not useful in measuring nucleation-dominated films or domains that have significant amounts of structure within the interior of the domain wall. The amplitude spectrum method is more complex, and less intuitive than the divider method, and somewhat more expensive to implement computationally. However, since the camera noise tends to be white, the noise can be avoided in the measurement of D by avoiding that portion of the curve that is flat (due to the white noise) when the least squares line is fit to the plot. Also, many image processing software packages include a Fast Fourier Transformation (FFT) facility, while the user will most likely have to write his own edge extraction routine for the divider method. The amplitude spectrum method is a true two dimensional technique that probes the interior of the domain wall, and in fact, can measure arbitrary clusters of domains. It can also be used to measure grey-level images, further reducing processing steps needed to threshold the image.
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
NASA Astrophysics Data System (ADS)
Mehan, Sumit; Kumar, Sugam; Aswal, V. K.
2014-04-01
Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
Mehan, Sumit Kumar, Sugam Aswal, V. K.
2014-04-24
Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.
Zhang, Lihui; Duan, Feng; Huang, Yaji; Chyang, Chiensong
2015-12-01
The changes in pore structure characteristics of sewage sludge particles under effect of calcium magnesium acetate (CMA) during combustion were investigated, the samples were characterized by N2 isothermal absorption method, and the data were used to analyze the fractal properties of the obtained samples. Results show that reaction time and the mole ratio of calcium to sulfur (Ca/S ratio) have notable impact on the pore structure and morphology of solid sample. The Brunauer-Emmett-Teller (BET) specific surface area (SBET) of sample increases with Ca/S ratio, while significant decreases with reaction time. The fractal dimension D has the similar trend with that of SBET, indicating that the surface roughness of sludge increases under the effect of CMA adding, resulting in improved the sludge combustion and the desulfurization process.
NASA Astrophysics Data System (ADS)
Flores-Marquez, E. L.; Galvez-Coyt, G.; Cifuentes-Nava, G.
2012-12-01
Fractal analysis of the total magnetic field (TMF) time series from 1997 to 2003 at Popocatépetl Volcano is performed and compared with the TMF-series of the Teoloyucan Magnetic Observatory, 100 km away. Using Higuchi's fractal dimension method (D). The D changes over time for both series were computed. It was observed, when the time windows used to compute D increase in length, both series show nearly the same behavior. Some criteria of comparison were employed to discriminate the local effects inherent to volcano-magnetism. The simultaneous maximum in D (1.8) of the TMF series at Popocatépetl Volcano and the recovered volcanic activity indicates a scaling relation of the TMF at Popocatépetl Volcano and demonstrates a link between the magnetic field and volcanic activity.
Crystal, Howard A.; Holman, Susan; Lui, Yvonne W.; Baird, Alison E.; Yu, Hua; Klein, Ronald; Rojas-Soto, Diana Marcella; Gustafson, Deborah R.; Stebbins, Glenn T.
2016-01-01
Objective The fractal dimension of retinal arteries and veins is a measure of the complexity of the vascular tree. We hypothesized that retinal fractal dimension would be associated with brain volume and white matter integrity in HIV-infected women. Design Nested case-control within longitudinal cohort study. Methods Women were recruited from the Brooklyn site of the Women’s Interagency HIV study (WIHS); 34 HIV-infected and 21 HIV-uninfected women with analyzable MRIs and retinal photographs were included. Fractal dimension was determined using the SIVA software program on skeletonized retinal images. The relationship between predictors (retinal vascular measures) and outcomes (quantitative MRI measures) were analyzed with linear regression models. All models included age, intracranial volume, and both arterial and venous fractal dimension. Some models were adjusted for blood pressure, race/ethnicity, and HIV-infection. Results The women were 45.6 ± 7.3 years of age. Higher arterial dimension was associated with larger cortical volumes, but higher venous dimension was associated with smaller cortical volumes. In fully adjusted models, venous dimension was significantly associated with fractional anisotropy (standardized β = -0.41, p = 0.009) and total gray matter volume (β = -0.24, p = 0.03), and arterial dimension with mean diffusivity (β = -0.33,.p = 0.04) and fractional anisotropy (β = 0.34, p = 0.03). HIV-infection was not associated with any retinal or MRI measure. Conclusions Higher venous fractal dimension was associated with smaller cortical volumes and lower fractional anisotropy, whereas higher arterial fractal dimension was associated with the opposite patterns. Longitudinal studies are needed to validate this finding. PMID:27158911
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)
Lin, Fan; Zhu, Pengli; Huang, Feng; Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Yu, Peng; Lin, Jing; Chen, Falin
2015-05-01
The objective of this study was to evaluate the association of the central retinal arteriolar equivalent (CRAE) and the retinal vascular fractal dimension, two quantitative parameters that reflect microcirculation, with aortic stiffness. In this cross-sectional study, we identified the cardiovascular risk factors in 2169 subjects using a health questionnaire, physical examinations and laboratory examinations. We evaluated the aortic stiffness using noninvasive brachial-ankle pulse wave velocity (baPWV) and assessed the microcirculatory alterations with CRAE and retinal vascular fractal dimension, which were measured using fundus photography and semiautomatic quantitative software, respectively. The increase in baPWV (Q1-Q4) correlated with an increased likelihood of the central retinal artery narrowing and a reduction in the retinal vascular fractal dimension. Further adjustment of the cardiovascular risk factors diminished the association between baPWV and CRAE, but increased the association between baPWV and retinal vascular fractal dimension. Elevated baPWV correlates with reduced CRAE and retinal vascular fractal dimension. Such a finding supports macrocirculation- and microcirculation-associated hypotheses.
Fractal signatures in analogs of interplanetary dust particles
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Banerjee, Varsha; Puri, Sanjay
2014-10-01
Interplanetary dust particles (IDPs) are an important constituent of the earths stratosphere, interstellar and interplanetary medium, cometary comae and tails, etc. Their physical and optical characteristics are significantly influenced by the morphology of silicate aggregates which form the core in IDPs. In this paper we reinterpret scattering data from laboratory analogs of cosmic silicate aggregates created by Volten et al. (2007) [1] to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension dm≃1.75. The same fractal dimension also characterizes clusters obtained from diffusion limited aggregation (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically transported silicate particles.
NASA Astrophysics Data System (ADS)
Wu, Yu; Cheng, Tianhai; Zheng, Lijuan; Chen, Hao
2016-10-01
Light absorption enhancement of aged soot aerosols is highly sensitive to the morphologies and mixing states of soot aggregates and their non-absorbing coatings, such as organic materials. The quantification of these effects on the optical properties of thinly coated soot aerosols is simulated using an effective model with fixed volume fractions. Fractal aggregated soot was simulated using the diffusion limited aggregation (DLA) algorithm and discretized into soot dipoles. The dipoles of non-absorbing aerosols, whose number was fixed by the volume fraction, were further generated from the neighboring random edge dipoles. Their optical properties were calculated using the discrete dipole approximation (DDA) method and were compared with other commonly used models. The optical properties of thinly coated soot calculated using the fixed volume fraction model are close to (less than ~10% difference) the results of the fixed coating thickness model, except their asymmetry parameters (up to ~25% difference). In the optical simulations of thinly coated soot aerosols, this relative difference of asymmetry parameters and phase functions between these realistic models may be notable. The realizations of the fixed volume fraction model may introduce smaller variation of optical results than those of the fixed coating thickness model. Moreover, the core-shell monomers model and homogeneous aggregated spheres model with the Maxwell-Garnett (MG) theory may underestimate (up to ~20%) the cross sections of thinly coated soot aggregates. The single core-shell sphere model may largely overestimate (up to ~150%) the cross sections and single scattering albedo of thinly coated soot aggregates, and it underestimated (up to ~60%) their asymmetry parameters. It is suggested that the widely used single core-shell sphere approximation may not be suitable for the single scattering calculations of thinly coated soot aerosols.
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.
NASA Astrophysics Data System (ADS)
Cymberknop, L.; Legnani, W.; Pessana, F.; Bia, D.; Zócalo, Y.; Armentano, R. L.
2011-12-01
The advent of vascular diseases, such as hypertension and atherosclerosis, is associated to significant alterations in the physical properties of arterial vessels. Evaluation of arterial biomechanical behaviour is related to the assessment of three representative indices: arterial compliance, arterial distensibility and arterial stiffness index. Elasticity is the most important mechanical property of the arterial wall, whose natures is strictly non-linear. Intervention of elastin and collagen fibres, passive constituent elements of the arterial wall, is related to the applied wall stress level. Concerning this, appropriate tools are required to analyse the temporal dynamics of the signals involved, in order to characterize the whole phenomenon. Fractal geometry can be mentioned as one of those techniques. The aim of this study consisted on arterial pressure and diameter signals processing, by means of nonlinear techniques based on fractal geometry. Time series morphology was related to different arterial stiffness states, generated by means of blood flow variations, during experiences performed in vitro.
Self-organized one-atom thick fractal nanoclusters via field-induced atomic transport
NASA Astrophysics Data System (ADS)
Batabyal, R.; Mahato, J. C.; Das, Debolina; Roy, Anupam; Dev, B. N.
2013-08-01
We report on the growth of a monolayer thick fractal nanostructures of Ag on flat-top Ag islands, grown on Si(111). Upon application of a voltage pulse at an edge of the flat-top Ag island from a scanning tunneling microscope tip, Ag atoms climb from the edge onto the top of the island. These atoms aggregate to form precisely one-atom thick nanostructures of fractal nature. The fractal (Hausdorff) dimension, DH = 1.75 ± 0.05, of this nanostructure has been determined by analyzing the morphology of the growing nanocluster, imaged by scanning tunneling microscopy, following the application of the voltage pulse. This value of the fractal dimension is consistent with the diffusion limited aggregation (DLA) model. We also determined two other fractal dimensions based on perimeter-radius-of-gyration (DP) and perimeter-area (D'P) relationship. Simulations of the DLA process, with varying sticking probability, lead to different cluster morphologies [P. Meakin, Phys. Rev. A 27, 1495 (1983)]; however, the value of DH is insensitive to this difference in morphology. We suggest that the morphology can be characterized by additional fractal dimension(s) DP and/or D'P, besides DH. We also show that within the DLA process DP = DH [C. Amitrano et al., Phys. Rev. A 40, 1713 (1989)] is only a special case; in general, DP and DH can be unequal. Characterization of fractal morphology is important for fractals in nanoelectronics, as fractal morphology would determine the electron transport behavior.
Wagenseil, R.
1991-01-01
There are persistent difficulties in monitoring nonpoint source pollution and in the related field of hydrology. The problems stem from variations in spatial distribution which are poorly understood and difficult to model with established methods. Two recent developments may offer a solution, if they are combined with care. The first development is the increasing capability of computer mapping, called geographic information systems (GIS). These systems can store, retrieve, and manipulate data with an explicit spatial structure. The second development is the field of fractal mathematics. Fractal mathematics includes geometric sets which have simple descriptions, despite complex appearances. One family of such fractal sets are the Brownian surfaces, which capture many of the qualities of natural land surfaces in a simple statistical model. Up until now, the Brownian models have been constrained by the assumption that the same statistical relationship holds over the entire surface. This is called the constraint of stationarity. The need to study how the landscape differs by location leads to relaxing the constraint of stationarity. This, in turn, causes some profound changes in the model. A special computer program applies the new model to a set of three-dimensional digital maps of natural terrain (DEMs). The model performs well, and highlights differences in landforms. This suggests several new approaches to spatial variation.
Taylor, Adele M.; MacGillivray, Thomas J.; Henderson, Ross D.; Ilzina, Lasma; Dhillon, Baljean; Starr, John M.; Deary, Ian J.
2015-01-01
Purpose Cerebral microvascular disease is associated with dementia. Differences in the topography of the retinal vascular network may be a marker for cerebrovascular disease. The association between cerebral microvascular state and non-pathological cognitive ageing is less clear, particularly because studies are rarely able to adjust for pre-morbid cognitive ability level. We measured retinal vascular fractal dimension (Df) as a potential marker of cerebral microvascular disease. We examined the extent to which it contributes to differences in non-pathological cognitive ability in old age, after adjusting for childhood mental ability. Methods Participants from the Lothian Birth Cohort 1936 Study (LBC1936) had cognitive ability assessments and retinal photographs taken of both eyes aged around 73 years (n = 648). IQ scores were available from childhood. Retinal vascular Df was calculated with monofractal and multifractal analysis, performed on custom-written software. Multiple regression models were applied to determine associations between retinal vascular Df and general cognitive ability (g), processing speed, and memory. Results Only three out of 24 comparisons (two eyes × four Df parameters × three cognitive measures) were found to be significant. This is little more than would be expected by chance. No single association was verified by an equivalent association in the contralateral eye. Conclusions The results show little evidence that fractal measures of retinal vascular differences are associated with non-pathological cognitive ageing. PMID:25816017
Mincione, Gabriella; Di Nicola, Marta; Di Marcantonio, Maria Carmela; Muraro, Raffaella; Piattelli, Adriano; Rubini, Corrado; Penitente, Enrico; Piccirilli, Marcello; Aprile, Giuseppe; Perrotti, Vittoria; Artese, Luciano
2015-10-01
Fractal dimension (FD) in tissue specimens from patients with oral squamous cell carcinoma (OSCC) was evaluated. FD values in different stages of OSCC, and the correlations with clinicopathological variables and patient survival were investigated. Histological sections from OSCC and control non-neoplastic mucosa specimens were stained with hematoxylin-eosin for pathological analysis and with Feulgen for nuclear evaluation. FD in OSCC groups vs. controls revealed statistically significant differences (P < 0.001). In addition, a progressive increase of FD from stage I and II lesions and stage III and IV lesions was observed, with statistically significant differences (P = 0.003). Moreover, different degrees of tumor differentiation showed a significant difference in the average nuclear FD values (P = 0.001). A relationship between FD and patients' survival was also detected with lower FD values associated to longer survival time and higher FD values with shorter survival time (P = 0.034). These data showed that FD significantly increased during OSCC progression. Thus, FD could represent a novel prognostic tool for OSCC, as FD values significantly correlated with patient survival. Fractal geometry could give insights into tumor morphology and could become an useful tool for analyzing irregular tumor growth patterns.
Light Scattering by Fractal Dust Aggregates. I. Angular Dependence of Scattering
NASA Astrophysics Data System (ADS)
Tazaki, Ryo; Tanaka, Hidekazu; Okuzumi, Satoshi; Kataoka, Akimasa; Nomura, Hideko
2016-06-01
In protoplanetary disks, micron-sized dust grains coagulate to form highly porous dust aggregates. Because the optical properties of these aggregates are not completely understood, it is important to investigate how porous dust aggregates scatter light. In this study, the light scattering properties of porous dust aggregates were calculated using a rigorous method, the T-matrix method, and the results were then compared with those obtained using the Rayleigh-Gans-Debye (RGD) theory and Mie theory with the effective medium approximation (EMT). The RGD theory is applicable to moderately large aggregates made of nearly transparent monomers. This study considered two types of porous dust aggregates—ballistic cluster-cluster agglomerates (BCCAs) and ballistic particle-cluster agglomerates. First, the angular dependence of the scattered intensity was shown to reflect the hierarchical structure of dust aggregates; the large-scale structure of the aggregates is responsible for the intensity at small scattering angles, and their small-scale structure determines the intensity at large scattering angles. Second, it was determined that the EMT underestimates the backward scattering intensity by multiple orders of magnitude, especially in BCCAs, because the EMT averages the structure within the size of the aggregates. It was concluded that the RGD theory is a very useful method for calculating the optical properties of BCCAs.
Light Scattering by Fractal Dust Aggregates. I. Angular Dependence of Scattering
NASA Astrophysics Data System (ADS)
Tazaki, Ryo; Tanaka, Hidekazu; Okuzumi, Satoshi; Kataoka, Akimasa; Nomura, Hideko
2016-06-01
In protoplanetary disks, micron-sized dust grains coagulate to form highly porous dust aggregates. Because the optical properties of these aggregates are not completely understood, it is important to investigate how porous dust aggregates scatter light. In this study, the light scattering properties of porous dust aggregates were calculated using a rigorous method, the T-matrix method, and the results were then compared with those obtained using the Rayleigh–Gans–Debye (RGD) theory and Mie theory with the effective medium approximation (EMT). The RGD theory is applicable to moderately large aggregates made of nearly transparent monomers. This study considered two types of porous dust aggregates—ballistic cluster–cluster agglomerates (BCCAs) and ballistic particle–cluster agglomerates. First, the angular dependence of the scattered intensity was shown to reflect the hierarchical structure of dust aggregates; the large-scale structure of the aggregates is responsible for the intensity at small scattering angles, and their small-scale structure determines the intensity at large scattering angles. Second, it was determined that the EMT underestimates the backward scattering intensity by multiple orders of magnitude, especially in BCCAs, because the EMT averages the structure within the size of the aggregates. It was concluded that the RGD theory is a very useful method for calculating the optical properties of BCCAs.
NASA Astrophysics Data System (ADS)
Koshiro, Yoko; Watanabe, Manabu; Takai, Rikuo; Hagiwara, Tomoaki; Suzuki, Toru
Size and shape of ice crystals in frozen food materials are very important because they affect not only quality of foods but also the viability of industrial processing such as freeze-drying of concentration. In this study, 30%wt sucrose solution is used as test samples. For examining the effect of stabilizerspectine and xantan gum is added to the sucrose solution. They are frozen on the cold stage of microscope to be observed their growing ice crystals under the circumstance of -10°C. Their size and shape are measured and quantitatively evaluated by applying fractal analysis. lce crystal of complicated shape has large fractal dimension, and vice versa. It successflly categorized the ice crystals into two groups; one is a group of large size and complicated shape, and the other is a group of small size and plain shape. The critical crystal size between the two groups is found to become larger with increasing holding time. It suggests a phenomenological model for metamorphoses process of ice crystals. Further, it is indicated that xantan gum is able to suppress the smoothing of ice crystals.
Frozen Fractals All Around: Aggregate Particles in the Plumes of Enceladus
NASA Astrophysics Data System (ADS)
Gao, P.; Kopparla, P.; Zhang, X.; Ingersoll, A. P.
2015-12-01
Estimates of the total particulate mass of the plumes of Enceladus are important to constrain theories of particle formation and transport at the surface and interior of the satellite. We revisit the calculations of Ingersoll & Ewald (2011), who estimated the particulate mass of the Enceladus plumes from strongly forward scattered light in Cassini ISS images. We model the plume as a combination of spherical particles and irregular aggregates resulting from the coagulation of spherical monomers, the latter of which allows for plumes of lower particulate mass. Though a continuum of solutions are permitted by the model, the best fits to the ISS data consist either of low mass plumes composed entirely of small aggregates or high mass plumes composed of large aggregates and spheres. The high mass plumes can be divided into a population of large aggregates with total particulate mass of 116 ± 12 × 103 kg, and a mixed population of spheres and aggregates consisting of a few large monomers that has a total plume particulate mass of 166 ± 42 × 103 kg, consistent with the results of Ingersoll & Ewald (2011). Meanwhile, the low particulate mass aggregate plumes have masses of 25 ± 4 × 103 kg, leading to a solid to vapor mass ratio of 0.07 ± 0.01 for the plume. If indeed the plumes are made of such aggregates, then a vapor-based origin for the plume particles is possible. The process of aggregate formation by the coagulation of monomers, which depends on the bulk monomer number density inside the plume vents, requires a total plume vent cross sectional area of at most 1800 m2 to allow for the aggregates to form before the monomers are ejected into space. Differentiation between the high mass and low mass solutions may be possible if forward scattering observations are taken at scattering angles <2°, or else an independent plume particulate mass measurement becomes available.
NASA Astrophysics Data System (ADS)
Raupov, Dmitry S.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Kornilin, Dmitry V.; Zakharov, Valery P.; Khramov, Alexander G.
2016-04-01
Optical coherence tomography (OCT) is usually employed for the measurement of tumor topology, which reflects structural changes of a tissue. We investigated the possibility of OCT in detecting changes using a computer texture analysis method based on Haralick texture features, fractal dimension and the complex directional field method from different tissues. These features were used to identify special spatial characteristics, which differ healthy tissue from various skin cancers in cross-section OCT images (B-scans). Speckle reduction is an important pre-processing stage for OCT image processing. In this paper, an interval type-II fuzzy anisotropic diffusion algorithm for speckle noise reduction in OCT images was used. The Haralick texture feature set includes contrast, correlation, energy, and homogeneity evaluated in different directions. A box-counting method is applied to compute fractal dimension of investigated tissues. Additionally, we used the complex directional field calculated by the local gradient methodology to increase of the assessment quality of the diagnosis method. The complex directional field (as well as the "classical" directional field) can help describe an image as set of directions. Considering to a fact that malignant tissue grows anisotropically, some principal grooves may be observed on dermoscopic images, which mean possible existence of principal directions on OCT images. Our results suggest that described texture features may provide useful information to differentiate pathological from healthy patients. The problem of recognition melanoma from nevi is decided in this work due to the big quantity of experimental data (143 OCT-images include tumors as Basal Cell Carcinoma (BCC), Malignant Melanoma (MM) and Nevi). We have sensitivity about 90% and specificity about 85%. Further research is warranted to determine how this approach may be used to select the regions of interest automatically.
Berry, Hugues
2002-01-01
Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes. PMID:12324410
NASA Astrophysics Data System (ADS)
Chadee, X. T.
2007-05-01
The fractal dimension, Lyapunov-exponent spectrum, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily wind speeds over the Caribbean region. It can be shown that this dimension is greater than 8. However, the number of data points may be too small to obtain a reliable estimate of the Grassberger-Procaccia (1983a) correlation dimension because of the limitations discussed by Ruelle (1990). These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia (1983b) are sensitive to the selection of the time delay and are thus unreliable. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. (1991). Using this method, it is found that the error-doubling time is 2-3 days for time series over the Caribbean region. This is comparable to the predictability time found by Waelbrock (1995) for a single station in Mexico. The predictability time over land is slightly less than that over ocean which tends to have higher climatic signal-to-noise ratio. This analysis impacts on the selection of prediction tools (deterministic chaotic linear and non-linear maps or linear stochastic modeling) for wind speeds in the short term for wind energy farm resource planning and management. We conclude that short term wind predictions in the Caribbean region, for a few days ahead, may be best done with a stochastic model instead of a deterministic chaotic model. References Grassberger, P., and I. Procaccia. 1983a. Measuring the strangeness of attractors. Physica D 9: 189-208. Grassberger, P., and I. Procaccia. 1983b. Estimating the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28
Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain
NASA Astrophysics Data System (ADS)
Macé, Nicolas; Jagannathan, Anuradha; Piéchon, Frédéric
2016-05-01
We present a theoretical framework for understanding the wave functions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a symmetry of wave functions under permutation of site and energy indices. We compute the wave-function renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wave functions, as well as their global averages. The multifractality of wave functions is seen to appear at next-to-leading order in ρ . Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of ρ well outside the domain of applicability of perturbation theory.
Beretta-Piccoli, Matteo; D’Antona, Giuseppe; Barbero, Marco; Fisher, Beth; Dieli-Conwright, Christina M.; Clijsen, Ron; Cescon, Corrado
2015-01-01
Purpose Over the past decade, linear and non-linear surface electromyography descriptors for central and peripheral components of fatigue have been developed. In the current study, we tested fractal dimension (FD) and conduction velocity (CV) as myoelectric descriptors of central and peripheral fatigue, respectively. To this aim, we analyzed FD and CV slopes during sustained fatiguing contractions of the quadriceps femoris in healthy humans. Methods A total of 29 recreationally active women (mean age±standard deviation: 24±4 years) and two female elite athletes (one power athlete, age 24 and one endurance athlete, age 30 years) performed two knee extensions: (1) at 20% maximal voluntary contraction (MVC) for 30 s, and (2) at 60% MVC held until exhaustion. Surface EMG signals were detected from the vastus lateralis and vastus medialis using bidimensional arrays. Results Central and peripheral fatigue were described as decreases in FD and CV, respectively. A positive correlation between FD and CV (R=0.51, p<0.01) was found during the sustained 60% MVC, probably as a result of simultaneous motor unit synchronization and a decrease in muscle fiber CV during the fatiguing task. Conclusions Central and peripheral fatigue can be described as changes in FD and CV, at least in young, healthy women. The significant correlation between FD and CV observed at 60% MVC suggests that a mutual interaction between central and peripheral fatigue can arise during submaximal isometric contractions. PMID:25880369
Clinical relevance of the fractal dimension of F0 perturbations computed by the box-counting method.
Boek, W; Wieneke, G H; Dejonckere, P H
1997-12-01
The box-counting method for determining the fractal dimension (Dfj) was applied to the fundamental frequency (F0) perturbations in normal and pathological voices in order to assess its clinical value. The upper limit of these Dfj values was similar for both groups, but the distribution for pathological voices extended to lower values than for the normal voices. However, these lower values were most probably the result of one or a few outlying frequency points due to incorrect determination of the vocal period. The Dfj of normal voices were within the range of values found for randomly varying F0 values. It was concluded, that the vocal perturbations in pathological voices are also more or less randomly distributed. So, the Dfj, at that least determined with the box-counting method, do not contain clinically relevant information in addition to the traditional measures for the extent of the vocal period perturbations. An exception is special perturbation types like diplophonia. The result of the computation is very sensitive for voice breaks and vibrato and depends on the number of periods. PMID:9422278
Boccia, Gennaro; Dardanello, Davide; Beretta-Piccoli, Matteo; Cescon, Corrado; Coratella, Giuseppe; Rinaldo, Nicoletta; Barbero, Marco; Lanza, Massimo; Schena, Federico; Rainoldi, Alberto
2016-01-01
Over the past decade, linear and nonlinear surface electromyography (EMG) variables highlighting different components of fatigue have been developed. In this study, we tested fractal dimension (FD) and conduction velocity (CV) rate of changes as descriptors, respectively, of motor unit synchronization and peripheral manifestations of fatigue. Sixteen elderly (69 ± 4 years) and seventeen young (23 ± 2 years) physically active men (almost 3-5 h of physical activity per week) executed one knee extensor contraction at 70% of a maximal voluntary contraction for 30 s. Muscle fiber CV and FD were calculated from the multichannel surface EMG signal recorded from the vastus lateralis and medialis muscles. The main findings were that the two groups showed a similar rate of change of CV, whereas FD rate of change was higher in the young than in the elderly group. The trends were the same for both muscles. CV findings highlighted a non-different extent of peripheral manifestations of fatigue between groups. Nevertheless, FD rate of change was found to be steeper in the elderly than in the young, suggesting a greater increase in motor unit synchronization with ageing. These findings suggest that FD analysis could be used as a complementary variable providing further information on central mechanisms with respect to CV in fatiguing contractions.
Davies, Nia Anne; Harrison, Nicholas Kim; Morris, Roger H Keith; Noble, Simon; Lawrence, Matthew James; D'Silva, Lindsay Antonio; Broome, Laura; Brown, Martin Rowan; Hawkins, Karl M; Williams, Phylip Rhodri; Davidson, Simon; Evans, Phillip Adrian
2015-11-25
Venous thromboembolism (VTE) is common in cancer patients, and is the second commonest cause of death associated with the disease. Patients with chronic inflammation, such as cancer, have been shown to have pathological clot structures with modulated mechanical properties. Fractal dimension (df) is a new technique which has been shown to act as a marker of the microstructure and mechanical properties of blood clots, and can be performed more readily than current methods such as scanning electron microscopy (SEM). We measured df in 87 consecutive patients with newly diagnosed lung cancer prior to treatment and 47 matched-controls. Mean group values were compared for all patients with lung cancer vs controls and for limited disease vs extensive disease. Results were compared with conventional markers of coagulation, fibrinolysis and SEM images. Significantly higher values of df were observed in lung cancer patients compared with controls and patients with extensive disease had higher values than those with limited disease (p< 0.05), whilst conventional markers failed to distinguish between these groups. The relationship between df of the incipient clot and mature clot microstructure was confirmed by SEM and computational modelling: higher df was associated with highly dense clots formed of smaller fibrin fibres in lung cancer patients compared to controls. This study demonstrates that df is a sensitive technique which quantifies the structure and mechanical properties of blood clots in patients with lung cancer. Our data suggests that df has the potential to identify patients with an abnormal clot microstructure and greatest VTE risk.
Jurczyszyn, Kamil; Osiecka, Beata J; Ziółkowski, Piotr
2012-01-01
Fractal dimension analysis (FDA) is modern mathematical method widely used to describing of complex and chaotic shapes when classic methods fail. The main aim of this study was evaluating the influence of photodynamic therapy (PDT) with cystein proteases inhibitors (CPI) on the number and morphology of blood vessels inside tumor and on increase of effectiveness of combined therapy in contrast to PDT and CPI used separately. Animals were divided into four groups: control, treated using only PDT, treated using only CPI and treated using combined therapy, PDT and CPI. Results showed that time of animal survival and depth of necrosis inside tumor were significantly higher in CPI+PDT group in contrast to other groups. The higher value of fractal dimension (FD) was observed in control group, while the lowest value was found in the group which was treated by cystein protease inhibitors. The differences between FD were observed in CPI group and PDT+CPI group in comparison to control group. Our results revealed that fractal dimension analysis is a very useful tool in estimating differences between irregular shapes like blood vessels in PDT treated tumors. Thus, the implementation of FDA algorithms could be useful method in evaluating the efficacy of PDT.
NASA Astrophysics Data System (ADS)
Yan, Kun
2007-04-01
In this paper, by discussing the basic hypotheses about the continuous orbit and discrete orbit in two research directions of the background medium theory for celestial body motion, the concrete equation forms and their summary of the theoretic frame of celestial body motion are introduced. Future more, by discussing the general form of Binet's equation of celestial body motion orbit and it's solution of the advance of the perihelion of planets, the relations and differences between the continuous orbit theory and Newton's gravitation theory and Einstein's general relativity are given. And by discussing the fractional-dimension expanded equation for the celestial body motion orbits, the concrete equations and the prophesy data of discrete orbit or stable orbits of celestial bodies which included the planets in the Solar system, satellites in the Uranian system, satellites in the Earth system and satellites obtaining the Moon obtaining from discrete orbit theory are given too. Especially, as the preliminary exploration and inference to the gravitation curve of celestial bodies in broadly range, the concept for the ideal black hole with trend to infinite in mass density difficult to be formed by gravitation only is explored. By discussing the position hypothesis of fractional-dimension derivative about general function and the formula form the hypothesis of fractional-dimension derivative about power function, the concrete equation formulas of fractional-dimension derivative, differential and integral are described distinctly further, and the difference between the fractional-dimension derivative and the fractional-order derivative are given too. Subsequently, the concrete forms of measure calculation equations of self-similar fractal obtaining by based on the definition of form in fractional-dimension calculus about general fractal measure are discussed again, and the differences with Hausdorff measure method or the covering method at present are given. By applying
Observation of a crossover in kinetic aggregation of Palladium colloids
NASA Astrophysics Data System (ADS)
Ghafari, M.; Ranjbar, M.; Rouhani, S.
2015-10-01
We use field emission scanning electron microscope (FE-SEM) to investigate the growth of palladium colloids over the surface of thin films of WO3/glass. The film is prepared by Pulsed Laser Deposition (PLD) at different temperatures. A PdCl2 (aq) droplet is injected on the surface and in the presence of steam hydrogen the droplet is dried through a reduction reaction process. Two distinct aggregation regimes of palladium colloids are observed over the substrates. We argue that the change in aggregation dynamics emerges when the measured water drop Contact Angel (CA) for the WO3/glass thin films passes a certain threshold value, namely CA ≈ 46°, where a crossover in kinetic aggregation of palladium colloids occurs. Our results suggest that the mass fractal dimension of palladium aggregates follows a power-law behavior. The fractal dimension (Df) in the fast aggregation regime, where the measured CA values vary from 27° up to 46° according to different substrate deposition temperatures, is Df = 1.75(± 0.02) - the value of Df is in excellent agreement with kinetic aggregation of other colloidal systems in fast aggregation regime. Whereas for the slow aggregation regime, with CA = 58°, the fractal dimension changes abruptly to Df = 1.92(± 0.03). We have also used a modified Box-Counting method to calculate fractal dimension of gray-level images and observe that the crossover at around CA ≈ 46° remains unchanged.
Diffusion-limited aggregation on curved surfaces
NASA Astrophysics Data System (ADS)
Choi, J.; Crowdy, D.; Bazant, M. Z.
2010-08-01
We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusion-limited aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere (K>0) and the pseudo-sphere (K<0), which approximate "bumps" and "saddles" in smooth surfaces, respectively. Although the curvature affects the global morphology of the aggregates, the fractal dimension (in the curved metric) is remarkably insensitive to curvature, as long as the particle size is much smaller than the radius of curvature. We conjecture that all aggregates grown by conformally invariant transport on curved surfaces have the same fractal dimension as DLA in the plane. Our simulations suggest, however, that the multifractal dimensions increase from hyperbolic (K<0) to elliptic (K>0) geometry, which we attribute to curvature-dependent screening of tip branching.
Ali, Zulfiqar; Elamvazuthi, Irraivan; Alsulaiman, Mansour; Muhammad, Ghulam
2016-01-01
Voice disorders are associated with irregular vibrations of vocal folds. Based on the source filter theory of speech production, these irregular vibrations can be detected in a non-invasive way by analyzing the speech signal. In this paper we present a multiband approach for the detection of voice disorders given that the voice source generally interacts with the vocal tract in a non-linear way. In normal phonation, and assuming sustained phonation of a vowel, the lower frequencies of speech are heavily source dependent due to the low frequency glottal formant, while the higher frequencies are less dependent on the source signal. During abnormal phonation, this is still a valid, but turbulent noise of source, because of the irregular vibration, affects also higher frequencies. Motivated by such a model, we suggest a multiband approach based on a three-level discrete wavelet transformation (DWT) and in each band the fractal dimension (FD) of the estimated power spectrum is estimated. The experiments suggest that frequency band 1-1562 Hz, lower frequencies after level 3, exhibits a significant difference in the spectrum of a normal and pathological subject. With this band, a detection rate of 91.28 % is obtained with one feature, and the obtained result is higher than all other frequency bands. Moreover, an accuracy of 92.45 % and an area under receiver operating characteristic curve (AUC) of 95.06 % is acquired when the FD of all levels is fused. Likewise, when the FD of all levels is combined with 22 Multi-Dimensional Voice Program (MDVP) parameters, an improvement of 2.26 % in accuracy and 1.45 % in AUC is observed.
Ali, Zulfiqar; Elamvazuthi, Irraivan; Alsulaiman, Mansour; Muhammad, Ghulam
2016-01-01
Voice disorders are associated with irregular vibrations of vocal folds. Based on the source filter theory of speech production, these irregular vibrations can be detected in a non-invasive way by analyzing the speech signal. In this paper we present a multiband approach for the detection of voice disorders given that the voice source generally interacts with the vocal tract in a non-linear way. In normal phonation, and assuming sustained phonation of a vowel, the lower frequencies of speech are heavily source dependent due to the low frequency glottal formant, while the higher frequencies are less dependent on the source signal. During abnormal phonation, this is still a valid, but turbulent noise of source, because of the irregular vibration, affects also higher frequencies. Motivated by such a model, we suggest a multiband approach based on a three-level discrete wavelet transformation (DWT) and in each band the fractal dimension (FD) of the estimated power spectrum is estimated. The experiments suggest that frequency band 1-1562 Hz, lower frequencies after level 3, exhibits a significant difference in the spectrum of a normal and pathological subject. With this band, a detection rate of 91.28 % is obtained with one feature, and the obtained result is higher than all other frequency bands. Moreover, an accuracy of 92.45 % and an area under receiver operating characteristic curve (AUC) of 95.06 % is acquired when the FD of all levels is fused. Likewise, when the FD of all levels is combined with 22 Multi-Dimensional Voice Program (MDVP) parameters, an improvement of 2.26 % in accuracy and 1.45 % in AUC is observed. PMID:26531753
Chowdhury, Indranil; Walker, Sharon L; Mylon, Steven E
2013-01-01
A systematic investigation was conducted to understand the role of aquatic conditions on the aggregate morphology of nano-TiO2, and the subsequent impact on their fate in the environment. In this study, three distinctly sized TiO2 nanoparticles (6, 13, and 23 nm) that had been synthesized with flame spray pyrolysis were employed. Nanoparticle aggregate morphology was measured using static light scattering (SLS) over a wide range of solution chemistry, and in the presence of natural organic matter (NOM). Results showed that primary nanoparticle size can significantly affect the fractal dimension of stable aggregates. A linear relationship was observed between surface areas of primary nanoparticles and fractal dimension indicating that smaller primary nanoparticles can form more compact aggregate in the aquatic environment. The pH, ionic strength, and ion valence also influenced the aggregate morphology of TNPs. Increased pH resulted a decrease in fractal dimension, whereas higher ionic strength resulted increased fractal dimension particularly for monovalent ions. When NOM was present, aggregate fractal dimension was also affected, which was also notably dependent on solution chemistry. Fractal dimension of aggregate increase for 6 nm system in the presence of NOM, whereas a drop in fractal dimension was observed for 13 nm and 23 nm aggregates. This effect was most profound for aggregates comprised of the smallest primary particles suggesting that interactions of NOM with smaller primary nanoparticles are more significant than those with larger ones. The findings from this study will be helpful for the prediction of nanoparticle aggregate fate in the aquatic environment. PMID:24592445
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1986-01-01
The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is observed that the models yield a fractal behavior for fragmentation; however, different values for the fractal dimension are produced . It is concluded that fragmentation is a scale invariant process and that the fractal dimension is a measure of the fragility of the fragmented material.
Attracted diffusion-limited aggregation.
Rahbari, S H Ebrahimnazhad; Saberi, A A
2012-07-01
In this paper we present results of extensive Monte Carlo simulations of diffusion-limited aggregation (DLA) with a seed placed on an attractive plane as a simple model in connection with the electrical double layers. We compute the fractal dimension of the aggregated patterns as a function of the attraction strength α. For the patterns grown in both two and three dimensions, the fractal dimension shows a significant dependence on the attraction strength for small values of α and approaches that of the ordinary two-dimensional (2D) DLA in the limit of large α. For the nonattracting case with α = 1, our results in three dimensions reproduce the patterns of 3D ordinary DLA, while in two dimensions our model leads to the formation of a compact cluster with dimension 2. For intermediate α, the 3D clusters have a quasi-2D structure with a fractal dimension very close to that of the ordinary 2D DLA. This allows one to control the morphology of a growing cluster by tuning a single external parameter α. PMID:23005417
Attracted diffusion-limited aggregation
NASA Astrophysics Data System (ADS)
Rahbari, S. H. Ebrahimnazhad; Saberi, A. A.
2012-07-01
In this paper we present results of extensive Monte Carlo simulations of diffusion-limited aggregation (DLA) with a seed placed on an attractive plane as a simple model in connection with the electrical double layers. We compute the fractal dimension of the aggregated patterns as a function of the attraction strength α. For the patterns grown in both two and three dimensions, the fractal dimension shows a significant dependence on the attraction strength for small values of α and approaches that of the ordinary two-dimensional (2D) DLA in the limit of large α. For the nonattracting case with α=1, our results in three dimensions reproduce the patterns of 3D ordinary DLA, while in two dimensions our model leads to the formation of a compact cluster with dimension 2. For intermediate α, the 3D clusters have a quasi-2D structure with a fractal dimension very close to that of the ordinary 2D DLA. This allows one to control the morphology of a growing cluster by tuning a single external parameter α.
Yao, Hiroshi; Morita, Yoshinobu; Kimura, Keisaku
2008-02-01
Morphological and spectroscopic properties of pseudoisocyanine (PIC) J aggregates produced at mica/solution interfaces have been characterized by absorption/fluorescence spectroscopy, fluorescence microscopy, and atomic force microscopy. Addition of organic solvents (1-propanol (PrOH) or 1,4-dioxane (Dox)) into aqueous solutions of the PIC dye induced a transition of the morphology of the interfacial J aggregates. The characteristic feature of this transition is the thickness (or height) change of the aggregate domain layers from three-dimensions to two-dimensions: The domain area of the J aggregates was dependent on the amount of the organic cosolvent, while the domain thickness was dependent on the type of the cosolvent. In pure aqueous solution, the J aggregates at the mica/water interface had a three-dimensional structure with the height of approximately 3 nm (multilayer structure). In mixed solvents of PrOH/water or Dox/water (5 or 10 vol%), the interfacial aggregates became a bilayer or monolayer structure, respectively, assuming that PIC molecules are adsorbed on their molecular plane perpendicular to the mica surface. Meanwhile, optical properties (band width and peak position) of the J band were invariant upon addition of the organic cosolvents, suggesting that molecular packing in the J aggregates is essentially unchanged. These results revealed that spectroscopic properties of the interfacial PIC J aggregates were determined only by the lateral (two-dimensional) interaction within the adsorbed monolayer of PIC molecules on mica, and interlayer interaction in the multilayered J aggregate was consequently small. PMID:17963780
NASA Astrophysics Data System (ADS)
Albert, Helena; Perugini, Diego; Martí, Joan
2014-05-01
The volcanic unit of Montaña Reventada is an example of magma mixing in Tenerife (Canary Islands, Spain). The eruptive process has been detonated by a basanite intruding into a phonolite magma chamber. This eruption started with a basanite followed by a phonolite. Montaña Reventada phonolite is characterized by the presence of mafic enclaves. These enclaves represent about the 2% of the outcrop and have been classified like basanites, phono-tephrite and tephri-phonolite. The enclaves have different morphologies, from rounded to complex fingers-like structures, and usually exhibit cuspate terminations. This study aims to provide a new perspective on the 1100 AD Montaña Reventada eruption quantifying the textural heterogeneities related to the enclaves generated by the mixing process. The textural study was carried out using a fractal geometry approach, and its results were used to calculate some parameters related to magma chamber dynamics. Photographs of 67 samples were taken normal to the surface of the enclaves with the aim of delineating the contact between the enclaves and the host rocks. The resulted pictures were processed with the NIH (National Institutes of Health) image analysis software to generate binary images in which enclaves and host rock were replaced by black and white pixels, respectively. The fractal dimension (Dbox) has been computed by using the box-counting method in order to quantify the complexity of the enclaves morphology. Viscosity ratio (μR) between the phonolite and the enclaves has been calculated as follows: log(μR) = 0.013e3.34Dbox PIC The viscosity of the enclaves has been calculated according to the μRvalue with the higher frequency and to the calculated viscosity of the phonolite between 900° and 1200° . We hypothesized that this value corresponds to the amount of mafic magma present in the system, while the other values represent different degrees of mingling and chemical diffusion. Viscosity of the basanite can be
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.
NASA Astrophysics Data System (ADS)
Karemore, Gopal; Nielsen, Mads
2009-02-01
Structural texture measures are used to address the aspect of breast cancer risk assessment in screening mammograms. The current study investigates whether texture properties characterized by local Fractal Dimension (FD) and Lacunarity contribute to asses breast cancer risk. FD represents the complexity while the Lacunarity characterize the gappiness of a fractal. Our cross-sectional case-control study includes mammograms of 50 patients diagnosed with breast cancer in the subsequent 2-4 years and 50 matched controls. The longitudinal double blind placebo controlled HRT study includes 39 placebo and 36 HRT treated volunteers for two years. ROIs with same dimension (250*150 pixels) were created behind the nipple region on these radiographs. Box counting method was used to calculate the fractal dimension (FD) and the Lacunarity. Paired t-test and Pearson correlation coefficient were calculated. It was found that there were no differences between cancer and control group for FD (P=0.8) and Lacunarity (P=0.8) in crosssectional study whereas earlier published heterogeneity examination of radiographs (BC-HER) breast cancer risk score separated groups (p=0.002). In the longitudinal study, FD decreased significantly (P<0.05) in the HRT treated population while Lacunarity remained insignificant (P=0.2). FD is negatively correlated to Lacunarity (-0.74, P<0.001), BIRADS (-0.34, P<0.001) and Percentage Density (-0.41, P<0.001). FD is invariant to the mammographic texture change from control to cancer population but marginally varying in HRT treated population. This study yields no evidence that lacunarity or FD are suitable surrogate markers of mammographic heterogeneity as they neither pick up breast cancer risk, nor show good sensitivity to HRT.
NASA Technical Reports Server (NTRS)
Wiscombe, W.
1999-01-01
The purpose of this paper is discuss the concept of fractal dimension; multifractal statistics as an extension of this; the use of simple multifractal statistics (power spectrum, structure function) to characterize cloud liquid water data; and to understand the use of multifractal cloud liquid water models based on real data as input to Monte Carlo radiation models of shortwave radiation transfer in 3D clouds, and the consequences of this in two areas: the design of aircraft field programs to measure cloud absorptance; and the explanation of the famous "Landsat scale break" in measured radiance.
Waliszewski, Przemyslaw
2016-01-01
Background: Tumor grading, PSA concentration, and stage determine a risk of prostate cancer patients with accuracy of about 70%. An approach based on the fractal geometrical model was proposed to eliminate subjectivity from the evaluation of tumor aggressiveness and to improve the prediction. This study was undertaken to validate classes of equivalence for the spatial distribution of cancer cell nuclei in a larger, independent set of prostate carcinomas. Methods: The global fractal capacity D0, information D1 and correlation D2 dimension, the local fractal dimension (LFD) and the local connected fractal dimension (LCFD), Shannon entropy H and lacunarity λ were measured using computer algorithms in digitalized images of both the reference set (n = 60) and the test set (n = 208) of prostate carcinomas. Results: Prostate carcinomas were re-stratified into seven classes of equivalence. The cut-off D0-values 1.5450, 1.5820, 1.6270, 1.6490, 1.6980, 1.7640 defined the classes from C1 to C7, respectively. The other measures but the D1 failed to define the same classes of equivalence. The pairs (D0, LFD), (D0, H), (D0, λ), (D1, LFD), (D1, H), (D1, λ) characterized the spatial distribution of cancer cell nuclei in each class. The co-application of those measures enabled the subordination of prostate carcinomas to one out of three clusters associated with different tumor aggressiveness. For D0 < 1.5820, LFD < 1.3, LCFD > 1.5, H < 0.7, and λ > 0.8, the class C1 or C2 contains low complexity low aggressive carcinomas exclusively. For D0 > 1.6980, LFD > 1.7644, LCFD > 1.7051, H > 0.9, and λ < 0.7, the class C6 or C7 contains high complexity high aggressive carcinomas. Conclusions: The cut-off D0-values defining the classes of equivalence were validated in this study. The cluster analysis suggested that the number of the subjective Gleason grades and the number of the objective classes of equivalence could be decreased from seven to three without a loss of clinically
Effect of Dust Coagulation Dynamics on the Geometry of Aggregates
NASA Technical Reports Server (NTRS)
Nakamura, R.
1996-01-01
Master equation gives a more fundamental description of stochastic coagulation processes rather than popular Smoluchowski's equation. In order to examine the effect of the dynamics on the geometry of resulting aggregates, we study Master equation with a rigorous Monte Carlo algorithm. It is found that Cluster-Cluster aggregation model is a good approximation of orderly growth and the aggregates have fluffy structures with a fractal dimension approx. 2. A scaling analysis of Smoluchowski's equation also supports this conclusion.
NASA Astrophysics Data System (ADS)
Yang, Yang; Wang, Ya Ping; Li, Chunyan; Gao, Shu; Shi, Benwei; Zhou, Liang; Wang, Dandan; Li, Gaocong; Dai, Chen
2016-04-01
Interactions between turbulence, suspended sediment concentration (SSC), settling velocity, effective density, fractal dimension, and floc size were studied on the tide-dominated, muddy coastal shelf of the southwestern Yellow Sea, China. The measurements were carried out in July 2013 at two sites located in water depths of 21.2 and 22.1 m. Negative correlations were observed between shear rate, SSC, effective density, and mean floc size, which supports the results of previous numerical, experimental, and field studies. A significant positive correlation was observed between near-bed SSC and shear rate, an indication that SSC variations are controlled by turbulence and re-suspension. In addition, significant linear relationships were found between settling velocity and other parameters (floc size, turbulence, SSC, effective density, and fractal dimension) at the two sites, indicating that the controlling factors on settling velocity are spatially variable. Principal component analysis was applied to determine the relative importance of turbulence, flocculation ability, and SSC as controls on floc size in situ. The relative contributions of turbulence, flocculation ability, and SSC to floc size (at both sites) were ~33.0%, 30.3%, and 29.7%, respectively, this being a new field-based quantitative analysis of the controls on floc size. The findings demonstrate that, in nature, flocculation ability affects floc size to the same degree as turbulence and SSC. Therefore, predictions of floc size in coastal marine environments require constraints not only on turbulence and SSC, but also on flocculation ability.
Dimensions, maximal growth sites, and optimization in the dielectric breakdown model.
Mathiesen, Joachim; Jensen, Mogens H; Bakke, Jan Oystein Haavig
2008-06-01
We study the growth of fractal clusters in the dielectric breakdown model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent alpha_{min} ) as a function of the growth exponent eta of the DBM model. We do not find evidence for a phase transition from fractal to nonfractal growth for a finite eta value. Simultaneously, we observe that the limit of nonfractal growth (D-->1) is consistent with alpha_{min}-->12 . Finally, using an optimization principle, we give a recipe on how to estimate the effective value of eta from temporal growth data of fractal aggregates.
Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert
1998-01-01
MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.
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.
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.
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 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.
Naik, Ganesh R; Arjunan, Sridhar; Kumar, Dinesh
2011-06-01
The surface electromyography (sEMG) signal separation and decphompositions has always been an interesting research topic in the field of rehabilitation and medical research. Subtle myoelectric control is an advanced technique concerned with the detection, processing, classification, and application of myoelectric signals to control human-assisting robots or rehabilitation devices. This paper reviews recent research and development in independent component analysis and Fractal dimensional analysis for sEMG pattern recognition, and presents state-of-the-art achievements in terms of their type, structure, and potential application. Directions for future research are also briefly outlined.
Fractal growth in the presence of a surface force field
NASA Astrophysics Data System (ADS)
Carlier, F.; Brion, E.; Akulin, V. M.
2012-05-01
We numerically simulate the dynamics of atomic clusters aggregation deposited on a surface interacting with the growing island. We make use of the well-known DLA model but replace the underlying diffusion equation by the Smoluchowski equation which results in a drifted DLA model and anisotropic jump probabilities. The shape of the structures resulting from their aggregation-limited random walk is affected by the presence of a Laplacian potential due to, for instance, the surface stress field. We characterize the morphologies we obtain by their Hausdorff fractal dimension as well as the so-called external fractal dimension. We compare our results to previously published experimental results for antimony and silver clusters deposited onto graphite surface.
NASA Astrophysics Data System (ADS)
Menshutin, A. Yu.; Shchur, L. N.
2011-09-01
Two-dimensional structures grown with Witten and Sander algorithm are investigated. We analyze clusters grown off-lattice and clusters grown with antenna method with N=3,4,5,6,7 and 8 allowed growth directions. With the help of variable probe particles technique we measure fractal dimension of such clusters D(N) as a function of their size N. We propose that in the thermodynamic limit of infinite cluster size the aggregates grown with high degree of anisotropy ( N=3,4,5) tend to have fractal dimension D equal to 3/2, while off-lattice aggregates and aggregates with lower anisotropy ( N>6) have D≈1.710. Noise-reduction procedure results in the change of universality class for DLA. For high enough noise-reduction value clusters with N⩾6 have fractal dimension going to 3/2 when N→∞.
Engineering images designed by fractal subdivision scheme.
Mustafa, Ghulam; Bari, Mehwish; Jamil, Saba
2016-01-01
This paper is concerned with the modeling of engineering images by the fractal properties of 6-point binary interpolating scheme. Association between the fractal behavior of the limit curve/surface and the parameter is obtained. The relationship between the subdivision parameter and the fractal dimension of the limit fractal curve of subdivision fractal is also presented. Numerical examples and visual demonstrations show that 6-point scheme is good choice for the generation of fractals for the modeling of fractal antennas, bearings, garari's and rock etc. PMID:27652066
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.}
NASA Astrophysics Data System (ADS)
Mezzenga, Raffaele; Fischer, Peter
2013-04-01
The aggregation of proteins is of fundamental relevance in a number of daily phenomena, as important and diverse as blood coagulation, medical diseases, or cooking an egg in the kitchen. Colloidal food systems, in particular, are examples that have great significance for protein aggregation, not only for their importance and implications, which touches on everyday life, but also because they allow the limits of the colloidal science analogy to be tested in a much broader window of conditions, such as pH, ionic strength, concentration and temperature. Thus, studying the aggregation and self-assembly of proteins in foods challenges our understanding of these complex systems from both the molecular and statistical physics perspectives. Last but not least, food offers a unique playground to study the aggregation of proteins in three, two and one dimensions, that is to say, in the bulk, at air/water and oil/water interfaces and in protein fibrillation phenomena. In this review we will tackle this very ambitious task in order to discuss the current understanding of protein aggregation in the framework of foods, which is possibly one of the broadest contexts, yet is of tremendous daily relevance.
Nanoparticle aggregation: principles and modeling.
Zhang, Wen
2014-01-01
The high surface area to volume ratio of nanoparticles usually results in highly reactive and colloidal instability compared to their bulk counterparts. Aggregation as well as many other transformations (e.g., dissolution) in the environment may alter the physiochemical properties, reactivity, fate, transport, and biological interactions (e.g., bioavailability and uptake) of nanoparticles. The unique properties pertinent to nanoparticles, such as shape, size, surface characteristics, composition, and electronic structures, greatly challenge the ability of colloid science to understand nanoparticle aggregation and its environmental impacts. This review briefly introduces fundamentals about aggregation, fractal dimensions, classic and extended Derjaguin-Landau-Verwey-Overbeak (DLVO) theories, aggregation kinetic modeling, experimental measurements, followed by detailed discussions on the major factors on aggregation and subsequent effects on nanomaterial transport and reactivity.
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 Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
Fractal processes in soil water retention
Tyler, S.W.; Wheatcraft, S.W. )
1990-05-01
The authors propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that they used for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar holes (or pores) over a wide range of scales. The authors evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. They relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet soil. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Their fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides paramters which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.
Aggregation of a hydrophobically modified poly(propylene imine) dendrimer
NASA Astrophysics Data System (ADS)
Tan, Susheng; Su, Aihua; Ford, Warren T.
2008-10-01
The poly(propylene imine) dendrimer DAB-dendr- (NH2)8 was hydrophobically modified with dodecanoyl end groups. The modified dendrimer was deposited onto mica by adsorption from solution and observed by atomic force microscopy. With the decrease of adsorption time, the modified dendrimer varied from continuous film to scattered islands. For the adsorption time of 20s the dendrimer formed a sub-monolayer thin film that contained many fractal aggregates of fractal dimension 1.80 that were > 1 μm in diameter and no more than 0.8nm thick. After 5 months at 1#1 , the initial fractal aggregates transformed into disks and other less-branched shapes with average heights of the domains of 0.6nm and 0.4nm, respectively. Formation of the fractal aggregates is explained by diffusion-limited aggregation. The slow reorganization of dendrimer molecules in the fractal aggregates occurs at a temperature well above the Tg of the dendrimer.
Multi-scale interactions in Dictyostelium discoideum aggregation
NASA Astrophysics Data System (ADS)
Dixon, James A.; Kelty-Stephen, Damian G.
2012-12-01
Cellular aggregation is essential for a wide range of phenomena in developmental biology, and a crucial event in the life-cycle of Dictyostelium discoideum. The current manuscript presents an analysis of multi-scale interactions involved in D. discoideum aggregation and non-aggregation events. The multi-scale fractal dimensions of a sequence of microscope images were used to estimate changing structure at different spatial scales. Three regions showing aggregation and three showing non-aggregation were considered. The results showed that both aggregation and non-aggregation regions were strongly multi-fractal. Analyses of the over-time relationships among nine scales of the generalized dimension, D(q), were conducted using vector autoregression and vector error-correction models. Both types of regions showed evidence that across-scale interactions serve to maintain the equilibrium of the system. Aggregation and non-aggregation regions also showed different patterns of effects of individual scales on other scales. Specifically, aggregation regions showed greater effects of both the smallest and largest scales on the smaller scale structures. The results suggest that multi-scale interactions are responsible for maintaining and altering the cellular structures during aggregation.
Characterization and modeling of thermal diffusion and aggregation in nanofluids.
Gharagozloo, Patricia E.; Goodson, Kenneth E.
2010-05-01
Fluids with higher thermal conductivities are sought for fluidic cooling systems in applications including microprocessors and high-power lasers. By adding high thermal conductivity nanoscale metal and metal oxide particles to a fluid the thermal conductivity of the fluid is enhanced. While particle aggregates play a central role in recent models for the thermal conductivity of nanofluids, the effect of particle diffusion in a temperature field on the aggregation and transport has yet to be studied in depth. The present work separates the effects of particle aggregation and diffusion using parallel plate experiments, infrared microscopy, light scattering, Monte Carlo simulations, and rate equations for particle and heat transport in a well dispersed nanofluid. Experimental data show non-uniform temporal increases in thermal conductivity above effective medium theory and can be well described through simulation of the combination of particle aggregation and diffusion. The simulation shows large concentration distributions due to thermal diffusion causing variations in aggregation, thermal conductivity and viscosity. Static light scattering shows aggregates form more quickly at higher concentrations and temperatures, which explains the increased enhancement with temperature reported by other research groups. The permanent aggregates in the nanofluid are found to have a fractal dimension of 2.4 and the aggregate formations that grow over time are found to have a fractal dimension of 1.8, which is consistent with diffusion limited aggregation. Calculations show as aggregates grow the viscosity increases at a faster rate than thermal conductivity making the highly aggregated nanofluids unfavorable, especially at the low fractal dimension of 1.8. An optimum nanoparticle diameter for these particular fluid properties is calculated to be 130 nm to optimize the fluid stability by reducing settling, thermal diffusion and aggregation.
NASA Astrophysics Data System (ADS)
Deng, Yan-Hong; Ye, Chao; Yuan, Yuan; Liu, Hui-Min; Cui, Jin
2011-04-01
We investigate the effect of silicone oil viscosity on the aggregation behavior of C:F clusters deposited on silicone oil liquid substrates with viscous coefficients of 100, 350 and 500mm2/s by C4F8 dual-frequency capacitively coupled plasma. The aggregated C:F clusters all exhibit a branch-like fractal structure. However, the fractal dimension decreases from 1.67 to 1.45 with the silicone oil viscous coefficient increasing from 100mm2/s to 500 mm2/s. Owing to the fractal dimension of 1.67 and 1.45, corresponding to the diffusion-limited-aggregation (DLA) model and the cluster-cluster-aggregation (CCA) model respectively, the results show that the increase of silicone oil viscosity can lead to the change of C:F clusters aggregating on a silicone oil liquid substrate from DLA to CCA growth.
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.
Applications of Fractal Analytical Techniques in the Estimation of Operational Scale
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Quattrochi, Dale A.
2000-01-01
resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.
Numerical Simulations of Single and Multiple Scattering by Fractal Ice Clusters
NASA Technical Reports Server (NTRS)
Dlugach, Janna M.; Mishchenko, Michael I.; Mackowski, Daniel W.
2011-01-01
We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4<=R<=10 cm and fractal dimensions D(sub f) 2.5 and 3. We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D(sub f) 2.5, but the differences can increase somewhat for D(sub f)3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.
Frankel, A.
1991-01-01
The high-frequency falloff ??-y of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R-D, D being the fractal dimension. In the model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R??, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (?? = 0) produces a main shock with an ??-2 falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to "islands' of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 - ??. Thus D = 2 corresponds to constant stress drop scaling (?? = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. -from Author
Characterization of anomalies by applying methods of fractal analysis
Sakuma, M.; Kozma, R.; Kitamura, M.
1996-01-01
Fractal analysis is applied in a variety of research fields to characterize nonstationary data. Here, fractal analysis is used as a tool of characterization in time series. The fractal dimension is calculated by Higuchi`s method, and the effect of small data size on accuracy is studied in detail. Three types of fractal-based anomaly indicators are adopted: (a) the fractal dimension, (b) the error of the fractal dimension, and (c) the chi-square value of the linear fitting of the fractal curve in the wave number domain. Fractal features of time series can be characterized by introducing these three measures. The proposed method is applied to various simulated fractal time series with ramp, random, and periodic noise anomalies and also to neutron detector signals acquired in a nuclear reactor. Fractal characterization can successfully supplement conventional signal analysis methods especially if nonstationary and non-Gaussian features of the signal become important.
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.
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.
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.
Fractal identification of supercell storms
NASA Astrophysics Data System (ADS)
Féral, Laurent; Sauvageot, Henri
2002-07-01
The most intense and violent form of convective storm is the supercell storm, usually associated with heavy rain, hail, and destructive gusty winds, downbursts, and tornadoes. Identifying a storm cell as a supercell storm is not easy. What is shown here, from radar data, is that when an ordinary, or multicell storm evolves towards the supercellular organization, its fractal dimension is modified. Whereas the fractal dimension of the ordinary convective storms, including multicell thunderstorms, is observed around 1.35, in agreement with previous results, the fractal dimension of supercell storms is found close to 1.07. This low value is due to the unicellular character of supercells. The present paper suggests that the fractal dimension is a parameter that should be considered to analyse the dynamical organization of a convective field and to detect and identify the supercell storms, either isolated or among a population of convective storms.
Robustness of the fractal regime for the multiple-scattering structure factor
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Botet, Robert; Puri, Sanjay
2016-08-01
In the single-scattering theory of electromagnetic radiation, the fractal regime is a definite range in the photon momentum-transfer q, which is characterized by the scaling-law behavior of the structure factor: S(q) ∝ 1 /q df. This allows a straightforward estimation of the fractal dimension df of aggregates in Small-Angle X-ray Scattering (SAXS) experiments. However, this behavior is not commonly studied in optical scattering experiments because of the lack of information on its domain of validity. In the present work, we propose a definition of the multiple-scattering structure factor, which naturally generalizes the single-scattering function S(q). We show that the mean-field theory of electromagnetic scattering provides an explicit condition to interpret the significance of multiple scattering. In this paper, we investigate and discuss electromagnetic scattering by three classes of fractal aggregates. The results obtained from the TMatrix method show that the fractal scaling range is divided into two domains: (1) a genuine fractal regime, which is robust; (2) a possible anomalous scaling regime, S(q) ∝ 1 /qδ, with exponent δ independent of df, and related to the way the scattering mechanism uses the local morphology of the scatterer. The recognition, and an analysis, of the latter domain is of importance because it may result in significant reduction of the fractal regime, and brings into question the proper mechanism in the build-up of multiple-scattering.
Phase transition in diffusion limited aggregation with patchy particles in two dimensions
NASA Astrophysics Data System (ADS)
Kartha, Moses J.; Sayeed, Ahmed
2016-08-01
The influence of patchy interactions on diffusion-limited aggregation (DLA) has been investigated by computer simulations. In this model, the adsorption of the particle is irreversible, but the adsorption occurs only when the 'sticky patch' makes contact with the sticky patch of a previously adsorbed particle. As we vary the patch size, growth rate of the cluster decreases, and below a well-defined critical patch size, pc the steady state growth rate goes to zero. The system reaches an absorbing phase producing a non-equilibrium continuous phase transition. The order parameter close to the critical value of the patch size shows a power law behavior ρ (∞) ∼(p -pc) β, where β = 0.2840. We have found that the value of the critical exponent convincingly shows that this transition in patchy DLA belongs to the directed percolation universality class.
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.
Conformal dynamics of fractal growth patterns without randomness
Davidovitch; Feigenbaum; Hentschel; Procaccia
2000-08-01
Many models of fractal growth patterns (such as diffusion limited aggregation and dielectric breakdown models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we introduce a wide class of fractal growth models with highly complex geometry but without any randomness in their growth rules. The models are defined in terms of deterministic itineraries of iterated conformal maps, generating the function Phi((n))(omega) which maps the exterior of the unit circle to the exterior of an n-particle growing aggregate. The complexity of the evolving interfaces is fully contained in the deterministic dynamics of the conformal map Phi((n))(omega). We focus attention on a class of growth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gained is used to introduce a scaling theory of the fractal growth patterns and to identify the exponent that determines the fractal dimension.
Brahma, Neil; Talbot, Jan B
2014-04-01
The aggregation rate and mechanism of 150 nm alumina particles in 1mM KNO3 with various additives used in chemical mechanical planarization of copper were investigated. The pH of each suspension was ∼8 such that the aggregation rate was slow enough to be measured and analyzed over ∼120 min. In general, an initial exponential growth was observed for most suspensions indicating reaction-limited aggregation. After aggregate sizes increase to >500 nm, the rate followed a power law suggesting diffusion-limited aggregation. Stability ratios and fractal dimension numbers were also calculated to further elucidate the aggregation mechanism. PMID:24491325
Fractal and spherulitic morphology of silicon nitride crystallized from amorphous films
Kahn, A.D.; Grabowski, K.S.; Donovan, E.P.; Carosella, C.A.; Hubler, G.K.
1988-01-01
Thin films of substoichiometric silicon nitride were grown by the use of ion beam assisted deposition. The amorphous films were annealed at high temperatures (1017-1200C) to produce crystalline alpha-Si3N4. Both highly symmetric spherulitic crystal morphologies and irregular fractal aggregates were seen. In the latter case, a fractal dimension of 1.2 was measured. These two macroscopically different forms possessed correspondingly different microstructures. The morphologies were found to be determined by the temperature of the anneal.
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.…
NASA Astrophysics Data System (ADS)
Bicalho, E. S.; Teixeira, D. B.; Panosso, A. R.; Perillo, L. I.; Iamaguti, J. L.; Pereira, G. T.; La Scala, N., Jr.
2012-04-01
Soil CO2 emission (FCO2) is influenced by chemical, physical and biological factors that affect the production of CO2 in the soil and its transport to the atmosphere, varying in time and space depending on environmental conditions, including the management of agricultural area. The aim of this study was to investigate the structure of spatial variability of FCO2 and soil properties by using fractal dimension (DF), derived from isotropic variograms at different scales, and construction of fractograms. The experimental area consisted of a regular grid of 60 × 60 m on sugarcane area under green management, containing 141 points spaced at minimum distances ranging from 0.5 to 10 m. Soil CO2 emission, soil temperature and soil moisture were evaluated over a period of 7 days, and soil chemical and physical properties were determined by sampling at a depth of 0.0 to 0.1 m. FCO2 showed an overall average of 1.51 µmol m-2 s-1, correlated significantly (p < 0.05) with soil physical factors such as soil bulk density, air-filled pore space, macroporosity and microporosity. Significant DF values were obtained in the characterization of FCO2 in medium and large scales (from 20 m). Variations in DF with the scale, which is the fractogram, indicate that the structure of FCO2 variability is similar to that observed for the soil temperature and total pore volume, and reverse for the other soil properties, except for macroporosity, sand content, soil organic matter, carbon stock, C/N ratio and CEC, which fractograms were not significantly correlated to the FCO2 fractogram. Thus, the structure of spatial variability for most soil properties, characterized by fractogram, presents a significant relationship with the structure of spatial variability of FCO2, generally with similar or dissimilar behavior, indicating the possibility of using the fractogram as tool to better observe the behavior of the spatial dependence of the variables along the scale.
NASA Astrophysics Data System (ADS)
Shi, Juanjuan; Liang, Ming; Guan, Yunpeng
2016-02-01
The conventional way for bearing fault diagnosis under variable rotational speed generally includes prefiltering, resampling based on shaft rotating frequency and order spectrum analysis. However, its application is confined by three major obstacles: a) knowledge-demanding parameter determination required by prefiltering, b) unavailable shaft rotating frequency for resampling as it is coupled with instantaneous fault characteristic frequency (IFCF) by a fault characteristic coefficient (FCC) which cannot be decided without knowing what fault actually exists, and c) complicated and error-prone resampling process. As such, we propose a new method to address these problems. The proposed method free from prefiltering and resampling mainly contains the following steps: a) extracting envelope by windowed fractal dimension (FD) transform, requiring no prefiltering, b) with the envelope signal, performing short time Fourier transform (STFT) to get a clear time frequency representation (TFR), from which the IFCF and the basic demodulator for generalized demodulation (GD) can be obtained, c) applying the generalized demodulation to the envelope signal with the current demodulator, converting the trajectory of the current time-frequency component into a linear path parallel to the time axis, d) frequency analyzing the demodulated signal, followed by searching the amplitude of the constant frequency where the linear path is situated. Updating demodulator via multiplying the basic demodulator by different real numbers (i.e., coefficient λ) and repeating the steps (c)-(d), the resampling-free order spectrum is then obtained. Based on the resulting spectrum, the final diagnosis decision can be made. The proposed method for its implementation on the example of simulated data is presented. Finally, experimental data are employed to validate the effectiveness of the proposed technique.
NASA Astrophysics Data System (ADS)
Takenaka, Mikihito; Kobayashi, Toshiaki; Saijo, Kenji; Tanaka, Hirokazu; Iwase, Naoki; Hashimoto, Takeji; Takahashi, Masaoki
2004-08-01
We investigated time evolution of shear moduli in the physical gelation process of 1,3:2,4-bis-O-(p-methylbenzylidene)-D-sorbitol in polystyrene melt. At the gel point, storage and loss shear moduli, G' and G″, were described by the power law of frequency ω, G'˜G″˜ωn, with the critical exponent n being nearly equal to 2/3, in agreement with the value predicted by the percolation theory. We also investigated the structure factor over two decades in length scale at gel point by using ultra-small-angle X-ray scattering, and small-angle X-ray scattering. We found the power-law behavior in low-q region, indicating that the gel network forms the self-similar structure with mass-fractal dimension. Comparison between the exponent of mass-fractal dimension from structure factor and that from viscoelasticity indicates that hydrodynamic interactions are completely screened out and the excluded volume effects are dominant in the gel. The gel strength was found to increase with the decrease in the lower limit length scale of fractality.
Xiao, Feng; Lam, Kit Ming; Li, Xiao-yan
2013-05-01
An advanced particle-tracking and flow-visualization technology, particle image velocimetry (PIV), was utilized to investigate the hydrodynamic properties of large aggregates in water. The laser-based PIV system was used together with a settling column to capture the streamlines around two types of aggregates: latex particle aggregates and activated sludge (AS) flocs. Both types of the aggregates were highly porous and fractal with fractal dimensions of 2.13±0.31 for the latex particle aggregates (1210-2144 μm) and 1.78±0.24 for the AS flocs (1265-3737 μm). The results show that PIV is a powerful flow visualization technique capable of determining flow field details at the micrometer scale around and through settling aggregates and flocs. The PIV streamlines provided direct experimental proof of internal flow through the aggregate interiors. According to the PIV images, fluid collection efficiency ranged from 0.052 to 0.174 for the latex particle aggregates and from 0.008 to 0.126 for AS flocs. AS flocs are apparently less permeable than the particle aggregates, probably due to the extracellular polymeric substances (EPSs) produced by bacteria clogging the pores within the flocs. The internal permeation of fractal aggregates and bio-flocs would enhance flocculation between particles and material transport into the aggregates.
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.
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.
Electromagnetic charges in aggregation phenomena.
NASA Astrophysics Data System (ADS)
Rioux, Claude; Slobodrian, R. J.
Introduction The mechanism of fine particles aggregation is of great importance in many areas of research, in particular environment sciences where the state of aggregation defines the removal speed of dust from the atmosphere. The study of this mechanism is also important to understand the first stage of planet formation from the solar nebula. The aggregates formed are generally fractals and, as mentioned in the literature [1], the fractal dimensions and the site growth probability measures of the resulting fractal structures strongly depend on the properties of the forces that cause the aggregation. Theory and experimental apparatus We began this study by the aggregation between two charged particles and we are now consid-ering the aggregation between two magnetized particles. The aggregations are produced in a gas at a pressure between 10 and 1000 mbar and by using the applicable simplifications; we find that the distance (r) between the particles as a function of time (t) is given by the following equations: r=Ce(tf -t)1/3 for the electrical attraction r=Cm(tf -t)1/5 for the magnetic dipoles aligned in an external magnetic field. The apparatus built for these measurements consists of an experimental cell from which two perpendicular views are combined via an optical system in one image recorded by a video camera. From the video, we can then measure the distance between the particles as a function of time and reconstruct the trajectories in 3-D. The horizontal and vertical resolutions are respectively 0.86 and 0.92 microns per pixel. With a depth of field of 250 microns, the usable volume for 3-D observation in then 250 microns x 250 microns x 443 microns. Results and discussion A first version of the apparatus was tested on an electrical force aggregation and the results [2] show that the corresponding equation is a good representation of the phenomenon. Preliminary results, from an experiment using iron particles, show that the magnetic force can be seen in
Interface fractal construction in Ni⧸KBr⧸Ni system
NASA Astrophysics Data System (ADS)
Shang, Chang He; Li, Heng De
1994-04-01
Non-equilibrium aggregation behavior on a surface has attracted increasing attention among researchers. It was found that at the early stage of film formation particles could coagulate into the form of diffusion-limited aggregation (DLA) on a free substrate surface. By using scanning tunnelling microscopy, Hwang, Schroder, Gunther and Behm [Phys. Rev. Lett. 67 (1991) 3279] recently showed that on the clean Ru surface, Au atoms grew into irregular islands of a fractal character as well. Similar monolayer growth was also found on a constrained surface, i.e., the interface between two lattice planes. In this paper, we report on the cluster aggregation behavior between two metallic layers. Samples were prepared by alternatively depositing pure constituent materials onto freshly cleaved NaCl single crystals in a high vacuum. Transmission electron microscopy was used to characterize the surface construction. It was found that KBr dendritic islands on the constrained surface also had a fractal geometry. The growth dynamics could be modified by adding interface impurities. Our experiment showed that Al addition onto the constrained surface could accelerate the growth process and degraded the fractal dimension. Detailed results will be presented, and some possible mechanisms will also be discussed.
Self-Similarity in Game-Locked Aggregation
NASA Astrophysics Data System (ADS)
Wang, Chao; Xiong, Wan-Ting; Wang, You-Gui
2012-12-01
A collective game is studied via agent-based modeling approach, where a group of adaptive learning players seek for their best positions on a vertical line. The movements of players are driven by benefits obtained from interactions. The game falls into an evolutionary stable state, at which aggregations of players on the line emerge. The pattern of these aggregates exhibits self-similarity at different scales with a fractal dimension of 0.58. The underlying mechanism of this aggregation is unique in that aggregates are resulted from mutual lock-in of players. This game-locked aggregation, in contrast with the diffusion limited aggregation, is applicable to a broader scope of aggregation processes.
Diffusion-Limited Aggregation with Polygon Particles
NASA Astrophysics Data System (ADS)
Deng, Li; Wang, Yan-Ting; Ou-Yang, Zhong-Can
2012-12-01
Diffusion-limited aggregation (DLA) assumes that particles perform pure random walk at a finite temperature and aggregate when they come close enough and stick together. Although it is well known that DLA in two dimensions results in a ramified fractal structure, how the particle shape influences the formed morphology is still unclear. In this work, we perform the off-lattice two-dimensional DLA simulations with different particle shapes of triangle, quadrangle, pentagon, hexagon, and octagon, respectively, and compare with the results for circular particles. Our results indicate that different particle shapes only change the local structure, but have no effects on the global structure of the formed fractal cluster. The local compactness decreases as the number of polygon edges increases.
Fractal patterns formed by growth of radial viscous fingers*
NASA Astrophysics Data System (ADS)
Praud, Olivier
2004-03-01
We examine fractal patterns formed by the injection of air into oil in a thin (0.13 mm) layer contained between two cylindrical glass plates of 288 mm diameter (a Hele-Shaw cell) [1]. The resultant radially grown patterns are similar to those formed in Diffusion Limited Aggregation (DLA), but the relation between the continuum limit of DLA and continuum (Laplacian) growth remains an open question. Our viscous fingering patterns in the limit of very high pressure difference reach an asymptotic state in which they exhibit a fractal dimension of 1.70± 0.02, in good agreement with a calculation of the fractal dimension of a DLA cluster, 1.713± 0.003 [2]. The generalized dimensions are also computed and show that the observed pattern is self-similar with Dq = 1.70 for all q. Further, the probability density function of shielding angles suggests the existence of a critical angle close to 75 degrees. This result is in accord with numerical and analytical evidence of a critical angle in DLA [3]. Thus fractal viscous fingering patterns and Diffusion Limited Aggregation clusters have a similar geometrical structure. *Work conducted in collaboration with H.L. Swinney, M.G. Moore and Eran Sharon [1] E. Sharon, M. G. Moore, W. D. McCormick, and H. L. Swinney, Phys. Rev. Lett. 91, 205504 (2003). [2] B.Davidovitch et A. Levermann and I. Procaccia, Phys. Rev. E 62, 5919 (2000). [3] D. A. Kessler et al., Phys. Rev. E 57, 6913 (1998).
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.
Rheological and fractal characteristics of unconditioned and conditioned water treatment residuals.
Dong, Y J; Wang, Y L; Feng, J
2011-07-01
The rheological and fractal characteristics of raw (unconditioned) and conditioned water treatment residuals (WTRs) were investigated in this study. Variations in morphology, size, and image fractal dimensions of the flocs/aggregates in these WTR systems with increasing polymer doses were analyzed. The results showed that when the raw WTRs were conditioned with the polymer CZ8688, the optimum polymer dosage was observed at 24 kg/ton dry sludge. The average diameter of irregularly shaped flocs/aggregates in the WTR suspensions increased from 42.54 μm to several hundred micrometers with increasing polymer doses. Furthermore, the aggregates in the conditioned WTR system displayed boundary/surface and mass fractals. At the optimum polymer dosage, the aggregates formed had a volumetric average diameter of about 820.7 μm, with a one-dimensional fractal dimension of 1.01 and a mass fractal dimension of 2.74 on the basis of the image analysis. Rheological tests indicated that the conditioned WTRs at the optimum polymer dosage showed higher levels of shear-thinning behavior than the raw WTRs. Variations in the limiting viscosity (η(∞)) of conditioned WTRs with sludge content could be described by a linear equation, which were different from the often-observed empirical exponential relationship for most municipal sludge. With increasing temperature, the η(∞) of the raw WTRs decreased more rapidly than that of the raw WTRs. Good fitting results for the relationships between lgη(∞)∼T using the Arrhenius equation indicate that the WTRs had a much higher activation energy for viscosity of about 17.86-26.91 J/mol compared with that of anaerobic granular sludge (2.51 J/mol) (Mu and Yu, 2006). In addition, the Bingham plastic model adequately described the rheological behavior of the conditioned WTRs, whereas the rheology of the raw WTRs fit the Herschel-Bulkley model well at only certain sludge contents. Considering the good power-law relationships between the
On the radiative properties of soot aggregates part 1: Necking and overlapping
NASA Astrophysics Data System (ADS)
Yon, J.; Bescond, A.; Liu, F.
2015-09-01
There is a strong interest in accurately modelling the radiative properties of soot aggregates (also known as black carbon particles) emitted from combustion systems and fires to gain improved understanding of the role of black carbon to global warming. This study conducted a systematic investigation of the effects of overlapping and necking between neighbouring primary particles on the radiative properties of soot aggregates using the discrete dipole approximation. The degrees of overlapping and necking are quantified by the overlapping and necking parameters. Realistic soot aggregates were generated numerically by constructing overlapping and necking to fractal aggregates formed by point-touch primary particles simulated using a diffusion-limited cluster aggregation algorithm. Radiative properties (differential scattering, absorption, total scattering, specific extinction, asymmetry factor and single scattering albedo) were calculated using the experimentally measured soot refractive index over the spectral range of 266-1064 nm for 9 combinations of the overlapping and necking parameters. Overlapping and necking affect significantly the absorption and scattering properties of soot aggregates, especially in the near UV spectrum due to the enhanced multiple scattering effects within an aggregate. By using correctly modified aggregate properties (fractal dimension, prefactor, primary particle radius, and the number of primary particle) and by accounting for the effects of multiple scattering, the simple Rayleigh-Debye-Gans theory for fractal aggregates can reproduce reasonably accurate radiative properties of realistic soot aggregates.
Time evolution of quantum fractals
Wojcik; Bialynicki-Birula; Zyczkowski
2000-12-11
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density P(t)(x) = |Psi(x,t)|(2) is shown not to change during the time evolution. We prove a universal relation D(t) = 1+Dx/2 linking the dimensions of space cross sections Dx and time cross sections D(t) of the fractal quantum carpets.
Fractal dynamics of earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.
Zhao, Qiang; An, Quanfu; Qian, Jinwen; Wang, Xuesan; Zhou, Yang
2011-12-22
Poly(diallyldimethylammonium chloride)-sodium carboxymethyl cellulose polyelectrolyte complexes (PDDA-CMCNa PECs) solids were prepared and dispersed in NaOH aqueous solution. Self-assembly of PECs nanoparticles during the solvent evaporation was examined by field emission electron microscopy (FESEM), atomic force microscopy (AFM), and fractal dimension analysis. It was found that tree-shaped fractal patterns formed after the solvent (water) was dried at ambient temperatures, and the fractal pattern is composed of needle-shaped PEC aggregate (PECA) nanoparticles. Time-dependent FESEM observation revealed that the fractal pattern started with the formation of initial nucleon and it is growing, during which the diffusion limited aggregation (DLA) mechanism revealed and made the pattern branched. Physical insight into the DLA mechanism was discussed in detail. Effects of PEC concentrations, PEC compositions, solvent evaporation temperatures, pH of PEC dispersion, and chemical structures of PECs on the formation of self-assembled fractal pattern were studied. Generally, it was found that the morphologies, charge characters of PEC particles, and the solvent evaporation conditions play important roles during the fractal self-assembly process. PMID:22098094
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.
Fractal patterns from chemical dissolution
NASA Astrophysics Data System (ADS)
Daccord, Gérard; Lenormand, Roland
1987-01-01
The highly ramified patterns1,2 produced by the flow of a reactive fluid through a soluble porous medium have never been quantitatively described. The theoretical understanding of this phenomenon is limited to very simple conditions (such as the flow of a liquid through a; capillary3) due to the complexity of the coupling between the chemical reaction and the fluid flow. We show here that the dissolution patterns (DP) obtained experimentally by injecting water through pure plaster are fractal, for different geometries of the samples. In two dimensions, these DP are remarkably'similar to patterns associated with diffusion-limited aggregation4-6 (DLA), that is, dielectric breakdown7, viscous fingering8,9 and diffusion-limited polymerization10. In three dimensions, we compare them with DLA clusters grown in the same boundary conditions and find a good qualitative and quantitative similarity. These results should be of interest in different areas where chemical dissolution of porous media by a flowing fluid occurs, for example, in nature (the formation of caves) and in industry (in the oil industry where acids are routinely injected into oil reservoirs).
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.
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.
ERIC Educational Resources Information Center
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
ERIC Educational Resources Information Center
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.…
Aggregation in charged nanoparticles solutions induced by different interactions
NASA Astrophysics Data System (ADS)
Abbas, S.; Kumar, Sugam; Aswal, V. K.; Kohlbrecher, J.
2016-05-01
Small-angle neutron scattering (SANS) has been used to study the aggregation of anionic silica nanoparticles as induced through different interactions. The nanoparticle aggregation is induced by addition of salt (NaCl), cationic protein (lysozyme) and non-ionic surfactant (C12E10) employing different kind of interactions. The results show that the interaction in presence of salt can be explained using DLVO theory whereas non-DLVO forces play important role for interaction of nanoparticles with protein and surfactant. The presence of salt screens the repulsion between charged nanoparticles giving rise to a net attraction in the DLVO potential. On the other hand, strong electrostatic attraction between nanoparticle and oppositely charged protein leads to protein-mediated nanoparticle aggregation. In case of non-ionic surfactant, the relatively long-range attractive depletion interaction is found to be responsible for the particle aggregation. Interestingly, the completely different interactions lead to similar kind of aggregate morphology. The nanoparticle aggregates formed are found to have mass fractal nature having a fractal dimension (~2.5) consistent with diffusion limited type of fractal morphology in all three cases.
Fractal-mound growth of pentacene thin films
NASA Astrophysics Data System (ADS)
Zorba, Serkan; Shapir, Yonathan; Gao, Yongli
2006-12-01
The growth mechanism of pentacene film formation on SiO2 substrate was investigated with a combination of atomic force microscopy measurements and numerical modeling. In addition to the diffusion-limited aggregation (DLA) that has already been shown to govern the growth of the ordered pentacene thin films, it is shown here that the Schwoebel barrier effect steps in and disrupts the desired epitaxial growth for the subsequent layers, leading to mound growth. The terraces of the growing mounds have a fractal dimension of 1.6, indicating a lateral DLA shape. This growth morphology thus combines horizontal DLA-like growth with vertical mound growth.
Shape characteristics of equilibrium and non-equilibrium fractal clusters.
Mansfield, Marc L; Douglas, Jack F
2013-07-28
It is often difficult in practice to discriminate between equilibrium and non-equilibrium nanoparticle or colloidal-particle clusters that form through aggregation in gas or solution phases. Scattering studies often permit the determination of an apparent fractal dimension, but both equilibrium and non-equilibrium clusters in three dimensions frequently have fractal dimensions near 2, so that it is often not possible to discriminate on the basis of this geometrical property. A survey of the anisotropy of a wide variety of polymeric structures (linear and ring random and self-avoiding random walks, percolation clusters, lattice animals, diffusion-limited aggregates, and Eden clusters) based on the principal components of both the radius of gyration and electric polarizability tensor indicates, perhaps counter-intuitively, that self-similar equilibrium clusters tend to be intrinsically anisotropic at all sizes, while non-equilibrium processes such as diffusion-limited aggregation or Eden growth tend to be isotropic in the large-mass limit, providing a potential means of discriminating these clusters experimentally if anisotropy could be determined along with the fractal dimension. Equilibrium polymer structures, such as flexible polymer chains, are normally self-similar due to the existence of only a single relevant length scale, and are thus anisotropic at all length scales, while non-equilibrium polymer structures that grow irreversibly in time eventually become isotropic if there is no difference in the average growth rates in different directions. There is apparently no proof of these general trends and little theoretical insight into what controls the universal anisotropy in equilibrium polymer structures of various kinds. This is an obvious topic of theoretical investigation, as well as a matter of practical interest. To address this general problem, we consider two experimentally accessible ratios, one between the hydrodynamic and gyration radii, the other
Shape characteristics of equilibrium and non-equilibrium fractal clusters.
Mansfield, Marc L; Douglas, Jack F
2013-07-28
It is often difficult in practice to discriminate between equilibrium and non-equilibrium nanoparticle or colloidal-particle clusters that form through aggregation in gas or solution phases. Scattering studies often permit the determination of an apparent fractal dimension, but both equilibrium and non-equilibrium clusters in three dimensions frequently have fractal dimensions near 2, so that it is often not possible to discriminate on the basis of this geometrical property. A survey of the anisotropy of a wide variety of polymeric structures (linear and ring random and self-avoiding random walks, percolation clusters, lattice animals, diffusion-limited aggregates, and Eden clusters) based on the principal components of both the radius of gyration and electric polarizability tensor indicates, perhaps counter-intuitively, that self-similar equilibrium clusters tend to be intrinsically anisotropic at all sizes, while non-equilibrium processes such as diffusion-limited aggregation or Eden growth tend to be isotropic in the large-mass limit, providing a potential means of discriminating these clusters experimentally if anisotropy could be determined along with the fractal dimension. Equilibrium polymer structures, such as flexible polymer chains, are normally self-similar due to the existence of only a single relevant length scale, and are thus anisotropic at all length scales, while non-equilibrium polymer structures that grow irreversibly in time eventually become isotropic if there is no difference in the average growth rates in different directions. There is apparently no proof of these general trends and little theoretical insight into what controls the universal anisotropy in equilibrium polymer structures of various kinds. This is an obvious topic of theoretical investigation, as well as a matter of practical interest. To address this general problem, we consider two experimentally accessible ratios, one between the hydrodynamic and gyration radii, the other
Shape characteristics of equilibrium and non-equilibrium fractal clusters
NASA Astrophysics Data System (ADS)
Mansfield, Marc L.; Douglas, Jack F.
2013-07-01
It is often difficult in practice to discriminate between equilibrium and non-equilibrium nanoparticle or colloidal-particle clusters that form through aggregation in gas or solution phases. Scattering studies often permit the determination of an apparent fractal dimension, but both equilibrium and non-equilibrium clusters in three dimensions frequently have fractal dimensions near 2, so that it is often not possible to discriminate on the basis of this geometrical property. A survey of the anisotropy of a wide variety of polymeric structures (linear and ring random and self-avoiding random walks, percolation clusters, lattice animals, diffusion-limited aggregates, and Eden clusters) based on the principal components of both the radius of gyration and electric polarizability tensor indicates, perhaps counter-intuitively, that self-similar equilibrium clusters tend to be intrinsically anisotropic at all sizes, while non-equilibrium processes such as diffusion-limited aggregation or Eden growth tend to be isotropic in the large-mass limit, providing a potential means of discriminating these clusters experimentally if anisotropy could be determined along with the fractal dimension. Equilibrium polymer structures, such as flexible polymer chains, are normally self-similar due to the existence of only a single relevant length scale, and are thus anisotropic at all length scales, while non-equilibrium polymer structures that grow irreversibly in time eventually become isotropic if there is no difference in the average growth rates in different directions. There is apparently no proof of these general trends and little theoretical insight into what controls the universal anisotropy in equilibrium polymer structures of various kinds. This is an obvious topic of theoretical investigation, as well as a matter of practical interest. To address this general problem, we consider two experimentally accessible ratios, one between the hydrodynamic and gyration radii, the other
A Fractal Nature for Polymerized Laminin
Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana
2014-01-01
Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system. PMID:25296244
Quantitative evaluation of midpalatal suture maturation via fractal analysis
Kwak, Kyoung Ho; Kim, Yong-Il; Kim, Yong-Deok
2016-01-01
Objective The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. Methods The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman's correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. Results The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235. Conclusions There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation. PMID:27668195
Quantitative evaluation of midpalatal suture maturation via fractal analysis
Kwak, Kyoung Ho; Kim, Yong-Il; Kim, Yong-Deok
2016-01-01
Objective The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. Methods The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman's correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. Results The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235. Conclusions There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation.
Robinson,, Gilpin R.; Brown, William M.
2002-01-01
The United States uses large quantities of natural aggregate to build and maintain a continuously expanding infrastructure. In recent years, per capita demand for aggregate in the United States has grown to about 9.7 metric tons (10.7 tons) per person per year. Over the next 25 years, the aggregate industry expects to mine quantities equivalent to all aggregate mined in the United States over the past 100 years. The issues surrounding supply and demand for aggregate in the mid-Atlantic states of Maryland, Pennsylvania, Virginia, and West Virginia illustrate competing requirements for industrial minerals and many simultaneous social and environmental objectives.
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.
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.
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
NASA Astrophysics Data System (ADS)
Dawar, Anit; Chandra, Amita
2013-04-01
Diffusion limited aggregation (DLA) is being reported in a polymer electrolyte composites PEO:NH4I + Al2O3 (acidic, basic and neutral). The aggregating specie is found to be I3- by using XRD and Raman spectroscopy. The involvement of NH4+ ion in the fractal pattern formation has also been explained via ion pairing with I3- (i.e., M+ + I3- → M+I3-), by making ammonium tri-iodide specie. Raman mapping has been done to analyze the fractal forming species in detail. A qualitative model has been proposed to explain the observations. The aggregates that get frozen in the host polymer matrix by diffusive growth form different patterns and are found to have fractal dimension that varies from 1.6 to 1.8 which is consistent with diffusion limited aggregation. The steady state of ion diffusion in polymer electrolyte is analyzed in the framework of the Nernst-Planck equation. The ion transport mechanism in the polymer electrolyte composite (with and without fractal growth), has been studied with the help of ac conductivity (σ) and ionic/electronic transference number (ti/te).
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.
ERIC Educational Resources Information Center
Jurgens, Hartmut; And Others
1990-01-01
The production and application of images based on fractal geometry are described. Discussed are fractal language groups, fractal image coding, and fractal dialects. Implications for these applications of geometry to mathematics education are suggested. (CW)
Fractal 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.
Random walk through fractal environments.
Isliker, H; Vlahos, L
2003-02-01
We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.
Anisotropic diffusion-limited aggregation.
Popescu, M N; Hentschel, H G E; Family, F
2004-06-01
Using stochastic conformal mappings, we study the effects of anisotropic perturbations on diffusion-limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map m preferred directions for growth to a distribution on the unit circle, which is a periodic function with m peaks in [-pi,pi) such that the angular width sigma of the peak defines the "strength" of anisotropy kappa= sigma(-1) along any of the m chosen directions. The two parameters (m,kappa) map out a parameter space of perturbations that allows a continuous transition from DLA (for small enough kappa ) to m needlelike fingers as kappa--> infinity. We show that at fixed m the effective fractal dimension of the clusters D(m,kappa) obtained from mass-radius scaling decreases with increasing kappa from D(DLA) approximately 1.71 to a value bounded from below by D(min) = 3 / 2. Scaling arguments suggest a specific form for the dependence of the fractal dimension D(m,kappa) on kappa for large kappa which compares favorably with numerical results. PMID:15244564
Bower, C; Washington, C; Purewal, T S
1996-04-01
The effect of increasing solid phase concentration on the morphology and flocculation rate of model aerosol suspensions has been investigated. Suspensions of micronized salbutamol sulphate and lactose in trichlorotrifluoroethane (P113) were studied under conditions of increasing shear stress. By use of image analysis techniques, measurement of aggregate size, fractal dimension and rate of aggregation was performed. The effect of the surfactant sorbitan monooleate on morphology and flocculation rate was also studied. Increased solid phase concentration caused an increase in the rate of aggregation and average aggregate size at a given value of shear stress. Surfactant addition retarded the aggregation rate, and caused a shift from a diffusion-limited cluster aggregation to a reaction-limited cluster aggregation mechanism. The aggregate profiles showed a corresponding change from rugged and crenellated without surfactant, to increasingly smooth and Euclidian with increasing surfactant concentration. The morphological changes were characterized by a decrease in the average boundary fractal dimension which also correlated well with the corresponding reduction in aggregation rate.
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.
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
Black carbon fractal morphology and short-wave radiative impact: a modelling study
NASA Astrophysics Data System (ADS)
Kahnert, M.; Devasthale, A.
2011-08-01
We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of differences in the optical cross sections and asymmetry parameters. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in
Black carbon fractal morphology and short-wave radiative impact: a modelling study
NASA Astrophysics Data System (ADS)
Kahnert, M.; Devasthale, A.
2011-11-01
We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of different error sources. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there
Long-term Differences in Tillage and Land Use Affect Intra-aggregate Pore Heterogeneity
Kravchenko, A.N.; Wang, A.N.W.; Smucker, A.J.M.; Rivers, M.L.
2012-10-25
Recent advances in computed tomography provide measurement tools to study internal structures of soil aggregates at micrometer resolutions and to improve our understanding of specific mechanisms of various soil processes. Fractal analysis is one of the data analysis tools that can be helpful in evaluating heterogeneity of the intra-aggregate internal structures. The goal of this study was to examine how long-term tillage and land use differences affect intra-aggregate pore heterogeneity. The specific objectives were: (i) to develop an approach to enhance utility of box-counting fractal dimension in characterizing intra-aggregate pore heterogeneity; (ii) to examine intra-aggregate pores in macro-aggregates (4-6 mm in size) using the computed tomography scanning and fractal analysis, and (iii) to compare heterogeneity of intra-aggregate pore space in aggregates from loamy Alfisol soil subjected to 20 yr of contrasting management practices, namely, conventional tillage (chisel plow) (CT), no-till (NT), and native succession vegetation (NS). Three-dimensional images of the intact aggregates were obtained with a resolution of 14.6 {micro}m at the Advanced Photon Source, Argonne National Laboratory, Argonne, IL. Proposed box-counting fractal dimension normalization was successfully implemented to estimate heterogeneity of pore voxel distributions without bias associated with different porosities in soil aggregates. The aggregates from all three studied treatments had higher porosity associated with large (>100 {micro}m) pores present in their centers than in their exteriors. Pores 15 to 60 {micro}m were equally abundant throughout entire aggregates but their distributions were more heterogeneous in aggregate interiors. The CT aggregates had greater numbers of pores 15 to 60 {micro}m than NT and NS. Distribution of pore voxels belonging to large pores was most heterogeneous in the aggregates from NS, followed by NT and by CT. This result was consistent with presence of
A Simple Experiment That Demonstrates Fractal Behavior.
ERIC Educational Resources Information Center
Ko, Raphael H.; Bean, Charles P.
1991-01-01
Described is how the crumpling of paper balls exhibits the concept of a topological dimension similar to fractals. The mass of the crumpled paper ball is found to be proportional to its diameter raised to a nonintegral power. (KR)
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)
Random-walk simulation of the response of irregular or fractal interfaces and membranes
NASA Astrophysics Data System (ADS)
Meakin, P.; Sapoval, B.
1991-03-01
A method of simulating the response of an irregular interface is investigated. It is based on an exact mapping between the Laplace equation and the steady-state diffusion equation with mixed boundary conditions. Simulations in two dimensions show that diffusion-limited aggregation (DLA) and other self-similar fractal electrodes exhibit the so-called constant-phase-angle (CPA) behavior. In the case of DLA electrodes the CPA exponent is found by this method to be close to the inverse of the fractal dimension. We show that this is directly related to the fact that in d=2 the admittance is proportional to the overall size of the self-similar electrode.
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.
NASA Technical Reports Server (NTRS)
Huang, J.; Turcotte, D. L.
1989-01-01
The concept of fractal mapping is introduced and applied to digitized topography of Arizona. It is shown that the fractal statistics satisfy the topography of the state to a good approximation. The fractal dimensions and roughness amplitudes from subregions are used to construct maps of these quantities. It is found that the fractal dimension of actual two-dimensional topography is not affected by the adding unity to the fractal dimension of one-dimensional topographic tracks. In addition, consideration is given to the production of fractal maps from synthetically derived topography.
Abnormal Stability in Growth of Diffusion-Limited Aggregation
NASA Astrophysics Data System (ADS)
Ohta, Shonosuke
2009-01-01
An abnormal and unsteady growth of an isotropic cluster in diffusion-limited aggregation (DLA) is observed in stability analyses. Macroscopic fluctuation due to the delay of transition from a dendritic tip to a tip-splitting growth induces the anisotropy of DLA. An asymptotic deformation factor \\varepsilon∞ = 0.0888 is obtained from large DLA clusters. A symmetric oval model proposed from the dual-stability growth of DLA gives an asymptotic fractal dimension of 1.7112 using \\varepsilon∞. The correspondence of this model to the box dimension is excellent.
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.
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.
Analysis of sludge aggregates produced during electrocoagulation of model wastewater.
Załęska-Chróst, B; Wardzyńska, R
2016-01-01
This paper presents the results of the study of sludge aggregates produced during electrocoagulation of model wastewater of a composition corresponding to the effluents from the cellulose and paper industry. Wastewater was electrocoagulated statically using aluminium electrodes with a current density of 31.25 A m(-2) and 62.50 A m(-2). In subsequent stages of the treatment, sludge flocs were collected, their size was studied and their floc settling velocity (30-520 μm s(-1)) and fractal dimension (D) were determined. The values of D ranged from 1.53 to 1.95 and were directly proportional to the degree of wastewater treatment. Higher values of D were determined for sludge with lower water content (after 24 hours' settling). Fractal dimension can therefore be used as an additional parameter of wastewater treatment control. PMID:26744947
Bhosale, Sheshanath V.; Kalyankar, Mohan B.; Nalage, Santosh V.; Lalander, Cecilia H.; Bhosale, Sidhanath V.; Langford, Steven J.; Oliver, Ruth F.
2011-01-01
Self-assembled nanostructures of zwitterionic octaphosphanatoporphyrin 1, of either nanoparticles or nanorods, depending on small changes in the pH, is demonstrated based on the J-aggregates. Porphyrin 1 self-assembled into nanosphere aggregates with a diameter of about 70–80 nm in the pH range 5–7, and nanorod aggregates were observed at pH 8.5. Hydrogen bonding, π-π stacking and hydrophilic interactions play important roles in the formation of this nanostructure morphology. Nanostructures were characterized by UV/Vis absorbance, fluorescence, atomic force microscopy (AFM) and transmission electron microscopy (TEM). This interesting pH dependent self-assembly phenomenon could provide a basis for development of novel biomaterials. PMID:21673901
Simulation model of the fractal patterns in ionic conducting polymer films
NASA Astrophysics Data System (ADS)
Amir, Shahizat; Mohamed, Nor; Hashim Ali, Siti
2010-02-01
Normally polymer electrolyte membranes are prepared and studied for applications in electrochemical devices. In this work, polymer electrolyte membranes have been used as the media to culture fractals. In order to simulate the growth patterns and stages of the fractals, a model has been identified based on the Brownian motion theory. A computer coding has been developed for the model to simulate and visualize the fractal growth. This computer program has been successful in simulating the growth of the fractal and in calculating the fractal dimension of each of the simulated fractal patterns. The fractal dimensions of the simulated fractals are comparable with the values obtained in the original fractals observed in the polymer electrolyte membrane. This indicates that the model developed in the present work is within acceptable conformity with the original fractal.
Chen, Jingjing; He, Binghui; Wang, Xiaoyan; Ma, Yun; Xi, Weimin
2013-10-01
To evaluate if hedgerows could improve the soil physicochemical properties and enhance the soil anti-scouring and anti-shearing capabilities, the effects of Herba Andrographitis hedgerows on soil erodibility and fractal features on sloping cropland in the Three Gorges Reservoir Area were investigated. Results showed that: (1) the clay particle accumulation around the hedgerows was significantly affected by the hedgerows; (2) the fractal dimension of soil particles was positively correlated with both silt and clay contents and had a negative linear correlation with sand content; (3) fine-grained content significantly influenced fractal dimension of the soil particles; (4) soil erodibility K was significantly and positively correlated with the sand content (correlation coefficient r=0.870), but significantly and negatively correlated with the silt content (r=-0.538), clay content (r=-0.739), organic carbon content (r=-0.603), the aggregation degree (r=-0.486), and soil fractal dimension (r=-0.538); and (5) the contents of organic matter and clay particles in the soil were found to be the effective indicators for soil erodibility at the Three Gorges Reservoir Area. The hedgerows may improve soil fractal features and decrease soil erodibility. The effective distance between hedgerows on a slope of 10° was less than 6 m. PMID:23608982
Agglomeration due to Brownian motion of fractal-structured combustion aerosols
Kaplan, C.H.
1987-01-01
A dynamic Monte-Carlo type lattice model has been developed to simulate the agglomeration of non-spherical chain-line aggregate combustion aerosols due to Brownian motion. Simulations are carried out in the free molecular and continuum regimes, for both initial monodisperse and initial log-normally distributed aerosols, with and without source mechanisms. Preservation of the chain-like structure of the aggregate is accomplished throughout the simulation by describing the agglomerate as fractal, that is, scale-invariant, self-similar with a noninteger dimensionality. Simulation results indicate that cluster growth is more rapid in the free molecular regime than in the continuum. Aerosols and log-normal distributions retain their log-normal characteristics even after long coagulation times. The effect of the clusters' fractal dimension on the cluster growth rate is determined; the rate of agglomeration increases when the structure of the agglomerate is more fragmented (lower fractal dimension). An analytical solution to the coagulation equation is obtained for log-normal aerosols by calculating moments of the distribution and solving sets of moment equations to determine the size distribution parameters. Condition numbers are employed to determine which moments should be calculated to most accurately determine these parameters. Excellent agreement is obtained between the simulations and the solution to the moment equations. Experimental measurements of soot particle velocity in a premixed methane/air flame are made using laser Doppler velocimetry.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
Fractal-based image processing for mine detection
NASA Astrophysics Data System (ADS)
Nelson, Susan R.; Tuovila, Susan M.
1995-06-01
A fractal-based analysis algorithm has been developed to perform the task of automated recognition of minelike targets in side scan sonar images. Because naturally occurring surfaces, such as the sea bottom, are characterized by irregular textures they are well suited to modeling as fractal surfaces. Manmade structures, including mines, are composed of Euclidean shapes, which makes fractal-based analysis highly appropriate for discrimination of mines from a natural background. To that end, a set of fractal features, including fractal dimension, was developed to classify image areas as minelike targets, nonmine areas, or clutter. Four different methods of fractal dimension calculation were compared and the Weierstrass function was used to study the effect of various signal processing procedures on the fractal qualities of an image. The difference in fractal dimension between different images depends not only on the physical features extant in the images but in the underlying statistical characteristics of the processing procedures applied to the images and the underlying mathematical assumptions of the fractal dimension calculation methods. For the image set studied, fractal-based analysis achieved a classification rate similar to human operators, and was very successful in identifying areas of clutter. The analysis technique presented here is applicable to any type of signal that may be configured as an image, making this technique suitable for multisensor systems.
NASA Technical Reports Server (NTRS)
Garneau, S.; Plaut, J. J.
2000-01-01
The surface roughness of the Vastitas Borealis Formation on Mars was analyzed with fractal statistics. Root mean square slopes and fractal dimensions were calculated for 74 topographic profiles. Results have implications for radar scattering models.
Marchetti, Riccardo; Taloni, Alessandro; Caglioti, Emanuele; Loreto, Vittorio; Pietronero, Luciano
2012-08-10
We prove that the harmonic measure is stationary, unique, and invariant on the interface of diffusion limited aggregation (DLA) growing on a cylinder surface. We provide a detailed theoretical analysis puzzling together multiscaling, multifractality, and conformal invariance, supported by extensive numerical simulations of clusters built using conformal mappings and on a lattice. The growth properties of the active and frozen zones are clearly elucidated. We show that the unique scaling exponent characterizing the stationary growth is the DLA fractal dimension. PMID:23006279
Stationary Growth and Unique Invariant Harmonic Measure of Cylindrical Diffusion Limited Aggregation
NASA Astrophysics Data System (ADS)
Marchetti, Riccardo; Taloni, Alessandro; Caglioti, Emanuele; Loreto, Vittorio; Pietronero, Luciano
2012-08-01
We prove that the harmonic measure is stationary, unique, and invariant on the interface of diffusion limited aggregation (DLA) growing on a cylinder surface. We provide a detailed theoretical analysis puzzling together multiscaling, multifractality, and conformal invariance, supported by extensive numerical simulations of clusters built using conformal mappings and on a lattice. The growth properties of the active and frozen zones are clearly elucidated. We show that the unique scaling exponent characterizing the stationary growth is the DLA fractal dimension.
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
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.
Diffusion-limited aggregates grown on nonuniform substrates
NASA Astrophysics Data System (ADS)
Cornette, V.; Centres, P. M.; Ramirez-Pastor, A. J.; Nieto, F.
2013-12-01
In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, p, from the percolation threshold, pc up to the jamming coverage, pj. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of p, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different k sizes. Correspondingly, the fractal dimension of the aggregates increases as p raises in the same range pc≤p≤pj. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.
In situ liquid-cell electron microscopy of colloid aggregation and growth dynamics.
Grogan, Joseph M; Rotkina, Lolita; Bau, Haim H
2011-06-01
We report on real-time observations of the aggregation of gold nanoparticles using a custom-made liquid cell that allows for in situ electron microscopy. Process kinetics and fractal dimension of the aggregates are consistent with three-dimensional cluster-cluster diffusion-limited aggregation, even for large aggregates, for which confinement effects are expected. This apparent paradox was resolved through in situ observations of the interactions between individual particles as well as clusters at various stages of the aggregation process that yielded the large aggregates. The liquid cell described herein facilitates real-time observations of various processes in liquid media with the high resolution of the electron microscope. PMID:21797362
Classification of impervious land cover using fractals
NASA Astrophysics Data System (ADS)
Quackenbush, Lindi J.
Runoff from urban areas is a leading source of nonpoint source pollution in estuaries, lakes, and streams. The extent and type of impervious land cover are considered to be critical factors in evaluating runoff amounts and the potential for environmental damage. Land cover information for watershed modeling is frequently derived using remote sensing techniques, and improvements in image classification are expected to enhance the reliability of runoff models. In order to understand potential pollutant loads there is a need to characterize impervious areas based on land use. However, distinguishing between impervious features such as roofs and roads using only spectral information is often challenging due to the similarity in construction materials. Since spectral information alone is often lacking, spatial complexity measured using fractal dimension was analyzed to determine its utility in performing detailed classification. Fractal dimension describes the complexity of curves and surfaces in non-integer dimensions. Statistical analysis demonstrated that fractal dimension varies between roofs, roads, and driveways. Analysis also observed the impact of scale by determining statistical differences in fractal dimension, based on the size of the window considered and the ground sampled distance of the pixels under consideration. The statistical differences in fractal dimension translated to minor improvements in classification accuracy when separating roofs, roads, and driveways.
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.
Image analysis of sludge aggregates obtained at preliminary treatment of sewage.
Smoczyński, L; Ratnaweera, H; Kosobucka, M; Kvaal, K; Smoczyński, M
2014-01-01
The results of wastewater treatment by Al and Fe salts and by electrocoagulation with aluminum electrodes are discussed and interpreted. Those processes used alone or combined with biological treatment, were analyzed for 50 and 90% removal of phosphates. Scanning electron microscopy (SEM) of the resulting sludge from three coagulation processes defined the perimeter P and area A of 129-142 differently sized objects in each contrast-enhanced image. Plots of lg A against lg P revealed that the analyzed sludge samples were made of self-similar aggregates-flocs with fractal characteristics. The slope of 'log plots' was used to determine surface fractal dimension Da, which was extrapolated to volumetric fractal dimension Dv. Dv was applied in a quantitative description of sludge aggregates-flocs. Aggregates-flocs of sludge obtained by Al ions (pre-polymerized Al and electrocoagulation) were characterized by higher values of Dv in comparison with sludge obtained by iron salts. The structure of {Al(OH)(3)} and {Fe(OH)(3)} aggregate-flocs was graphically simulated to determine the effect of size distribution and Dv on sweep flocculation and sludge separation and dehydration. Phosphate removal efficiency of 50% occurred at low ratios of Al:P and Fe:P. Adsorption-charge neutralization was suggested during coagulation with pre-polymerized coagulants, and sweep flow mechanism during electrocoagulation. PMID:25259494
Wang, Li; Li, Jun; Li, Juan; Bai, Wei-Xia
2014-03-01
A field experiment on effects of tillage rotation and fertilization on corn continuous cropping-practiced lands was carried out in Heyang of Shaanxi in 2007-2012. The tillage types included annual rotation of no-tillage and subsoiling (NT-ST), subsoiling and conventional tillage (ST-CT), or conventional tillage and no-tillage (CT-NT), and yearly practice of no tillage (NT-NT), subsoiling (ST-ST) or conventional tillage (CT-CT). The fertilization treatments included balanced fertilization, low-rate fertilization and conventional fertilization, which were separately practiced against the different tillage types. The experiment investigated compositions, mean mass diameters (MWD), geometrical mean diameters (GMD) and fraction dimension numbers (D) of soil aggregates in 0-40 cm soil and contents of organic carbon in 0-60 cm soil. The results indicated that: 1) The increased tillage intensity caused the reduced mechanical stability and content of soil aggregates and increased soil organic carbon loss. No-tillage or tillage rotation increased the MWD, GMD and contents of soil organic carbon and soil aggregates with diameters of more than 0.25 mm, but decreased D. Under the same fertilization treatment, the contents of soil aggregates with diameters of more than 0.25 mm were ranked in the order of NT-NT>NT-ST>NT-CT>ST-ST>CT-ST>CT-CT, and under the same tillage rotations, the soil aggregates were more stable with the balanced or low- rate fertilization than with the conventional fertilization. 2) Mathematical fractal dimension fitting of soil aggregates indicated that the fractal dimension numbers of soil aggregates ranged within 2.247-2.681 by dry sieving and 2.897-2.976 by wet sieving. In 0-30 cm soil, the fractal dimension numbers of soil aggregates were significantly lower under no-tillage or tillage rotation than under conventional tillage, and in 0-40 cm soil, the fractal dimensions of soil aggregates increased with soil depth, and tended to stabilize at the soil
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.
Fractal analysis of surface topography in ground monocrystal sapphire
NASA Astrophysics Data System (ADS)
Wang, Qiuyan; Liang, Zhiqiang; Wang, Xibin; Zhao, Wenxiang; Wu, Yongbo; Zhou, Tianfeng
2015-02-01
The surface characterization of ground monocrystal sapphire is investigated by fractal analysis method. A serial of ground sapphire surfaces in ductile removal and brittle removal mode are obtained by grinding experiments, and their three dimensional (3D) and two dimensional (2D) fractal dimensions are calculated and analyzed by box-counting methods. The 3D surface fractal dimension Ds shows a negative correlation with the surface roughness parameter Ra and is sensitive to the ground surface defects. For the ground surface with larger fractal dimension Ds, its micro-topography is more exquisite with minor defects. Once the fractal dimension Ds become smaller, deep cracks and pronounced defects are exhibited in ground surface. Moreover, the material removal mode can be implied from the distribution of 2D cross-sectional profile fractal dimension DL. The workpiece surface generated in ductile removal mode has high surface quality with high 2D and 3D fractal dimensions. This study indicates that the box-counting fractal analysis is an effective method to evaluate ground sapphire surface comprehensively.
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.
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.
An experimental study of dense aerosol aggregations
NASA Astrophysics Data System (ADS)
Dhaubhadel, Rajan
We demonstrated that an aerosol can gel. This gelation was then used for a one-step method to produce an ultralow density porous carbon or silica material. This material was named an aerosol gel because it was made via gelation of particles in the aerosol phase. The carbon and silica aerosol gels had high specific surface area (200--350 sq m2/g for carbon and 300--500 sq m2/g for silica) and an extremely low density (2.5--6.0 mg/cm3), properties similar to conventional aerogels. Key aspects to form a gel from an aerosol are large volume fraction, ca. 10-4 or greater, and small primary particle size, 50 nm or smaller, so that the gel time is fast compared to other characteristic times. Next we report the results of a study of the cluster morphology and kinetics of a dense aggregating aerosol system using the small angle light scattering technique. The soot particles started as individual monomers, ca. 38 nm radius, grew to bigger clusters with time and finally stopped evolving after spanning a network across the whole system volume. This spanning is aerosol gelation. The gelled system showed a hybrid morphology with a lower fractal dimension at length scales of a micron or smaller and a higher fractal dimension at length scales greater than a micron. The study of the kinetics of the aggregating system showed that when the system gelled, the aggregation kernel homogeneity lambda attained a value 0.4 or higher. The magnitude of the aggregation kernel showed an increase with increasing volume fraction. We also used image analysis technique to study the cluster morphology. From the digitized pictures of soot clusters the cluster morphology was determined by two different methods: structure factor and perimeter analysis. We find a hybrid, superaggregate morphology characterized by a fractal dimension of Df ≈ to 1.8 between the monomer size, ca. 50 nm, and 1 mum micron and Df ≈ to 2.6 at larger length scales up to ˜ 10 mum. The superaggregate morphology is 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
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
NASA Astrophysics Data System (ADS)
Coskun, Aycan; Sonmez, Harun; Ercin Kasapoglu, K.; Ozge Dinc, S.; Celal Tunusluoglu, M.
2010-05-01
dimensions. To determine fractal dimensions of more than hundred andesite blocks in cores, a computer program namely FRACRUN were developed. Fractal geometry has been used as practical and popular tool to define particularly irregular shaped bodies in literature since the theory of fractal was developed by Mandelbrot (1967) (Hyslip and Vallejo, 1997; Kruhl and Nega, 1996; Bagde etal., 2002; Gulbin and Evangulova, 2003; Pardini, 2003; Kolay and Kayabali, 2006; Hamdi, 2008; Zorlu, 2009 and Sezer, 2009). Although there are some methods to determine fractal dimensions, square grid-cell count method for 2D and segment count method for 1D were followed in the algorithm of FRACRUN. FRACRUN has capable of determine fractal dimensions of many closed polygons on a single surface. In the study, a database composed of uniaxial compressive strength, volumetric block proportion, fractal dimensions and number of blocks for each core was established. Finally, prediction models were developed by regression analyses and compared with the empirical equations proposed by Sonmez et al. (2006). Acknowledgement This study is a product of ongoing project supported by TUBITAK (The Scientific and Technological Research Council of Turkey - Project No: 108Y002). References Bagde, M.N., Raina, A.K., Chakraborty, A.K., Jethwa, J.L., 2002. Rock mass characterization by fractal dimension. Engineering Geology 63, 141-155. Gokceoglu, C., 2002. A fuzzy triangular chart to predict the uniaxial compressive strength of the Ankara agglomerates from their petrographic composition. Engineering Geology, 66 (1-2), 39-51. Gulbin, Y.L., Evangulova, E.B., 2003. Morphometry of quartz aggregates in granites: fractal images referring to nucleation and growth processes. Mathematical Geology 35 (7), 819-833 Hamdi, E., 2008. A fractal description of simulated 3D discontinuity networks. Rock Mechanics and Rock Engineering 41, 587-599. Hyslip, J.P., Vallejo, L.E., 1997. Fractals analysis of the roughness and size distribution
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.
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.
Aggregation and dendritic growth in a magnetic granular system
NASA Astrophysics Data System (ADS)
González-Gutiérrez, J.; Carrillo-Estrada, J. L.; Ruiz-Suárez, J. C.
2013-12-01
We experimentally study the aggregation of non-Brownian paramagnetic beads in a vibrofluidized system induced by an external magnetic dipole. A dendritic growth is observed in real time, particle by particle, and with the naked eye. Two aggregation stages are observed, where tip, tip-split and side-branching growths are differentiated. We found clusters morphologically similar to those generated by a diffusion limited aggregation algorithm (DLA). However, in our case, due to the finite range of the magnetic field, the clusters reach a finite size and their structures exhibit different rates of aggregation. These are revealed by the existence of two different scaling relations of the mass with the gyration radius, and the nature of the radial mass distribution function. The structures of the clusters are fractal objects with an effective mass fractal dimension of around 1.8. We found that an exponential function describes the aggregation phenomenon as a function of time. This exponential behavior is independent of the final state of the morphology (shape and length) of the agglomerates.
Thermodynamically reversible generalization of diffusion limited aggregation.
D'Souza, R M; Margolus, N H
1999-07-01
We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in a closed system obeying a local, deterministic, microscopically reversible dynamics. This model roughly corresponds to placing the irreversible diffusion limited aggregation model (DLA) in contact with a heat bath. Particles release latent heat when aggregating, while singly connected cluster members can absorb heat and evaporate. The heat bath is initially empty, hence we observe the flow of entropy from the aggregating gas of particles into the heat bath, which is being populated by diffusing heat tokens. Before the population of the heat bath stabilizes, the cluster morphology (quantified by the fractal dimension) is similar to a standard DLA cluster. The cluster then gradually anneals, becoming more tenuous, until reaching configurational equilibrium when the cluster morphology resembles a quenched branched random polymer. As the microscopic dynamics is invertible, we can reverse the evolution, observe the inverse flow of heat and entropy, and recover the initial condition. This simple system provides an explicit example of how macroscopic dissipation and self-organization can result from an underlying microscopically reversible dynamics. We present a detailed description of the dynamics for the model, discuss the macroscopic limit, and give predictions for the equilibrium particle densities obtained in the mean field limit. Empirical results for the growth are then presented, including the observed equilibrium particle densities, the temperature of the system, the fractal dimension of the growth clusters, scaling behavior, finite size effects, and the approach to equilibrium. We pay particular attention to the temporal behavior of the growth process and show that the relaxation to the maximum entropy state is initially a rapid nonequilibrium process, then subsequently it is a quasistatic process with a well defined temperature. PMID:11969759
[Chaos and fractals and their applications in electrocardial signal research].
Jiao, Qing; Guo, Yongxin; Zhang, Zhengguo
2009-06-01
Chaos and fractals are ubiquitous phenomena of nature. A system with fractal structure usually behaves chaos. As a complicated nonlinear dynamics system, heart has fractals structure and behaves as chaos. The deeper inherent mechanism of heart can be opened out when the chaos and fractals theory is utilized in the research of the electrical activity of heart. Generally a time series of a system was used for describing the status of the strange attractor of the system. The indices include Poincare plot, fractals dimension, Lyapunov exponent, entropy, scaling exponent, Hurst index and so on. In this article, the basic concepts and the methods of chaos and fractals were introduced firstly. Then the applications of chaos and fractals theories in the study of electrocardial signal were expounded with example of how they are used for ventricular fibrillation.
Comprehensive fractal description of porosity of coal of different ranks.
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure.
Comprehensive fractal description of porosity of coal of different ranks.
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407
Comprehensive Fractal Description of Porosity of Coal of Different Ranks
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407
Fractal-like behaviour of the BCC/FCC phase separation in the iron-gold alloys.
Błachowski, A; Ruebenbauer, K; Rakowska, A; Kac, S
2010-03-01
Iron-gold alloys with compositions Fe(70)Au(30) and Fe(50)Au(50) were prepared by arc melting. The alloys were investigated by means of the high-resolution scanning electron microscopy (SEM-FEG) in the as-cast state and upon annealing in two steps, i.e. at 250 degrees C for 24 h and subsequently at 500 degrees C for 48 h. The alloys were composed of two phases, i.e. a BCC phase rich in iron and a FCC phase rich in gold. The single-phase regions have equivalent diameter of about 50 nm. SEM images show self-similar structure for the spatial distribution of the above phases on scales ranging from about 1 mm till about 100 nm. The roughness of the images has been used to estimate a fractal dimension of the phase mixture. For larger scales of the as-cast samples one finds fractal dimension of about 1.7 for Fe(70)Au(30) composition, i.e. very close to the dimension of typical diffusion limited aggregation (DLA) fractals. For annealed samples, dimension 1.1 was found. PMID:20500404
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
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.
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.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
NASA Astrophysics Data System (ADS)
Chernodub, Maxim N.; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics.
Chernodub, Maxim N; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal structure of eigenmodes of unstable-cavity lasers.
Karman, G P; Woerdman, J P
1998-12-15
We show that the eigenmodes of unstable-cavity lasers have fractal structure, in contrast with the well-known stable-cavity eigenmodes. As with all fractals, the dynamic range over which self-similarity holds is limited; in this case the range is set by diffraction, i.e., by the Fresnel number of the resonator. We determine the fractal dimension of the mode profiles and show that it is related to the aperture shape. PMID:18091952
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.
Synthesis of Cobalt Oxides Thin Films Fractal Structures by Laser Chemical Vapor Deposition
Haniam, P.; Kunsombat, C.; Chiangga, S.; Songsasen, A.
2014-01-01
Thin films of cobalt oxides (CoO and Co3O4) fractal structures have been synthesized by using laser chemical vapor deposition at room temperature and atmospheric pressure. Various factors which affect the density and crystallization of cobalt oxides fractal shapes have been examined. We show that the fractal structures can be described by diffusion-limited aggregation model and discuss a new possibility to control the fractal structures. PMID:24672354
Synthesis of cobalt oxides thin films fractal structures by laser chemical vapor deposition.
Haniam, P; Kunsombat, C; Chiangga, S; Songsasen, A
2014-01-01
Thin films of cobalt oxides (CoO and Co3O4) fractal structures have been synthesized by using laser chemical vapor deposition at room temperature and atmospheric pressure. Various factors which affect the density and crystallization of cobalt oxides fractal shapes have been examined. We show that the fractal structures can be described by diffusion-limited aggregation model and discuss a new possibility to control the fractal structures. PMID:24672354
Experimental Hints of Gravity in Large Extra Dimensions?
NASA Astrophysics Data System (ADS)
Sigurdsson, Steinn; Ahluwalia, D. V.
Recent conjectures suggest the universe may have large extra dimensions, through which gravity propagates. This implies gross departures from Newton's law of gravity at small length scales. Here I consider some implications for particle dynamics on scales comparable to the compactification radius, Rc<~1 mm. During planet formation, coalescence of micron sized dust grains to planetesimals is a rate critical step. Blum et al. (2000) found dust grain aggregates form low fractal dimension structures in microgravity, consistent with high angular momentum coalescence. I consider the effects of non-Newtonian gravity on dust aggregation on scales less than Rc and show they naturally coalesce into low-dimensional structures with high specific angular momentum. We infer Rc~80 microns.
Retinal fractals and acute lacunar stroke.
Cheung, Ning; Liew, Gerald; Lindley, Richard I; Liu, Erica Y; Wang, Jie Jin; Hand, Peter; Baker, Michelle; Mitchell, Paul; Wong, Tien Y
2010-07-01
This study aimed to determine whether retinal fractal dimension, a quantitative measure of microvascular branching complexity and density, is associated with lacunar stroke. A total of 392 patients presenting with acute ischemic stroke had retinal fractal dimension measured from digital photographs, and lacunar infarct ascertained from brain imaging. After adjusting for age, gender, and vascular risk factors, higher retinal fractal dimension (highest vs lowest quartile and per standard deviation increase) was independently and positively associated with lacunar stroke (odds ratio [OR], 4.27; 95% confidence interval [CI], 1.49-12.17 and OR, 1.85; 95% CI, 1.20-2.84, respectively). Increased retinal microvascular complexity and density is associated with lacunar stroke.
Fractal 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.
Fractality of eroded coastlines of correlated landscapes
NASA Astrophysics Data System (ADS)
Morais, P. A.; Oliveira, E. A.; Araújo, N. A. M.; Herrmann, H. J.; Andrade, J. S., Jr.
2011-07-01
Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.
Fractal analysis of multiscale spatial autocorrelation among point data
De Cola, L.
1991-01-01
The analysis of spatial autocorrelation among point-data quadrats is a well-developed technique that has made limited but intriguing use of the multiscale aspects of pattern. In this paper are presented theoretical and algorithmic approaches to the analysis of aggregations of quadrats at or above a given density, in which these sets are treated as multifractal regions whose fractal dimension, D, may vary with phenomenon intensity, scale, and location. The technique is illustrated with Matui's quadrat house-count data, which yield measurements consistent with a nonautocorrelated simulated Poisson process but not with an orthogonal unit-step random walk. The paper concludes with a discussion of the implications of such analysis for multiscale geographic analysis systems. -Author
Neurocognitive derivation of protein surface property from protein aggregate parameters
Mishra, Hrishikesh; Lahiri, Tapobrata
2011-01-01
Current work targeted to predicate parametric relationship between aggregate and individual property of a protein. In this approach, we considered individual property of a protein as its Surface Roughness Index (SRI) which was shown to have potential to classify SCOP protein families. The bulk property was however considered as Intensity Level based Multi-fractal Dimension (ILMFD) of ordinary microscopic images of heat denatured protein aggregates which was known to have potential to serve as protein marker. The protocol used multiple ILMFD inputs obtained for a protein to produce a set of mapped outputs as possible SRI candidates. The outputs were further clustered and largest cluster centre after normalization was found to be a close approximation of expected SRI that was calculated from known PDB structure. The outcome showed that faster derivation of individual protein’s surface property might be possible using its bulk form, heat denatured aggregates. PMID:21572883
A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes
NASA Technical Reports Server (NTRS)
Hsui, Albert T.; Rust, Kelly A.; Klein, George D.
1993-01-01
Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.
Aggregates, broccoli and cauliflower
NASA Astrophysics Data System (ADS)
Grey, Francois; Kjems, Jørgen K.
1989-09-01
Naturally grown structures with fractal characters like broccoli and cauliflower are discussed and compared with DLA-type aggregates. It is suggested that the branching density can be used to characterize the growth process and an experimental method to determine this parameter is proposed.
Fractal Spectrum Technique for Quantitative Analysis of Volcanic Particle Shapes
NASA Astrophysics Data System (ADS)
Maria, A. H.; Carey, S. N.
2001-12-01
The shapes of volcanic particles reflect numerous eruptive parameters (e.g. magma viscosity, volatile content, degree of interaction with water) and are useful for understanding fragmentation and transport processes associated with volcanic eruptions. However, quantitative analysis of volcanic particle shapes has proven difficult due to their morphological complexity and variability. Shape analysis based on fractal geometry has been successfully applied to a wide variety of particles and appears to be well suited for describing complex features. The technique developed and applied to volcanic particles in this study uses fractal data produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Quantitative comparisons are carried out using multivariate statistical techniques such as cluster and principal components analysis. Compared with previous fractal methods that express shape in terms of only one or two fractal dimensions, use of multiple fractal dimensions results in more effective discrimination between samples. In addition, the technique eliminates the subjectivity associated with selecting linear segments on Richardson plots for fractal dimension calculation, and allows direct comparison of particles as long as instantaneous dimensions used as input to multivariate analyses are selected at the same scales for each particle. Applications to samples from well documented eruptions (e.g. Mt. St. Helens, Tambora, Surtsey) indicate that the fractal spectrum technique provides a useful means of characterizing volcanic particles and can be helpful for identifying the products of specific fragmentation processes (volatile exsolution, phreatomagmatic, quench granulation) and modes of volcanic deposition (tephra fall
Studying fractal geometry on submicron length scales by small-angle scattering
Wong, P.; Lin, J.
1988-08-01
Recent studies have shown that internal surfaces of porous geological materials, such as rocks and lignite coals, can be described by fractals down to atomic length scales. In this paper, the basic properties of self-similar and self-affine fractals are reviewed and how fractal dimensions can be measured by small-angle scattering experiments are discussed.
Fractal 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.
Fractal image analysis - Application to the topography of Oregon and synthetic images.
NASA Technical Reports Server (NTRS)
Huang, Jie; Turcotte, Donald L.
1990-01-01
Digitized topography for the state of Oregon has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with the results for Oregon. This approach can be easily applied to any digitzed image that obeys fractal statistics.
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.
Dimension of chaotic attractors
Farmer, J.D.; Ott, E.; Yorke, J.A.
1982-09-01
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on probabilistic properties (that is, they depend on the frequency with which a typical trajectory visits different regions of the attractor). Both our example and the previous work that we review support the conclusion that all of the probabilistic dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
Kinetics of chain aggregation of poly(methyl methacrylate) in isoamyl acetate
NASA Astrophysics Data System (ADS)
Nakata, Mitsuo; Nakagawa, Tomohide; Nakamura, Yoshiki; Wakatsuki, Syogo
1999-02-01
For dilute solutions of poly(methyl methacrylate) in isoamyl acetate with the molecular weight Mw=2.35×106, the phase separation process was observed as an aggregation process of polymer chains by light-scattering measurements. The aggregation process was measured for a period of hours at four polymer concentrations and at about 15 K below the phase separation temperature. The light-scattering data analysis by Guinier plot yielded the average molecular weight
Fractal analysis of bone structure with applications to osteoporosis and microgravity effects
Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E.; LeBlanc, A.; Lin, C.; Shackelford, L.
1995-12-31
The authors characterize the trabecular structure with the aid of fractal dimension. The authors use Alternating Sequential filters to generate a nonlinear pyramid for fractal dimension computations. The authors do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of the scheme is the rudimentary definition of self similarity. This allows them the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, the authors have shown that the ASF methods outperform other existing methods for fractal dimension estimation. They have shown that the fractal dimension remains the same when computed with both the X-Ray images and the MRI images of the patella. They have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, the authors have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, they have shown that the subject`s prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.
Theoretical study of statistical fractal model with applications to mineral resource prediction
NASA Astrophysics Data System (ADS)
Wei, Shen; Pengda, Zhao
2002-04-01
The statistical estimation of fractal dimensions is an important topic of investigation. Current solutions emphsize visual straight-line fitting, but nonlinear statistical modeling has the potential of making valuable contributions in this field. In this paper, we present the concepts of generalized fractal models and generalized fractal dimension and conclude that many geological models are special cases of the generalized models. We show that the power-function distribution possesses the fractal property of scaling invariance under upper truncation, which may help in lead statistical fractal modeling. A new method is developed on the basis of nonlinear regression to estimate fractal parameters. This method has advantages with respect to the traditional method based on linear regression for estimating the fractal dimension. Finally, the new method is illustrated by means of application to a real data set.
Ion-specific aggregation of hydrophobic particles.
López-León, Teresa; Ortega-Vinuesa, Juan Luis; Bastos-González, Delfina
2012-06-18
This work shows that colloidal stability and aggregation kinetics of hydrophobic polystyrene (PS) nanospheres are extremely sensitive to the nature of the salt used to coagulate them. Three PS latices and four aggregating electrolytes, which all share the same cation (Na(+)) but have various anions located at different positions in the classical Hofmeister series depending on their kosmotropic or chaotropic character, are used. The present study focuses on analyzing different aggregating parameters, such as critical coagulation concentrations (CCC), cluster size distributions (CSD), initial kinetic constants K(11), and fractal dimensions of the aggregates d(f). While aggregation induced by SO(4)(2-) and Cl(-) behaved according to the predictions of the classical Derjaguin-Landau-Verwey-Overbeek theory, important discrepancies are found with NO(3)(-), which become dramatic when using SCN(-). These discrepancies among the anions were far more significant when they acted as counterions rather than as co-ions. While SO(4)(2-) and Cl(-) trigger fast diffusion-limited aggregation, SCN(-) gives rise to a stationary cluster size distribution in a few aggregation times when working with cationic PS particles. Clear differences are found among all analyzed parameters (CCC, CSD, K(11), and d(f)), and the experimental findings show that particles aggregate in potential wells whose depth is controlled by the chaotropic character of the anion. This paper presents new experimental evidence that may help to understand the microscopic origin of Hofmeister effects, as the observations are consistent with appealing theoretical models developed in the last few years. PMID:22556130
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
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
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)
Glucose induced fractal colony pattern of Bacillus thuringiensis.
Roy, Manas K; Banerjee, Paromita; Sengupta, Tapas K; Dattagupta, Sushanta
2010-08-01
Growing colonies of bacteria on the surface of thin agar plates exhibit fractal patterns as a result of nonlinear response to environmental conditions, such as nutrients, solidity of the agar medium and temperature. Here, we examine the effect of glucose on pattern formation by growing colonies of Bacillus thuringiensis isolate KPWP1. We also present the theoretical modeling of the colony growth of KPWP1 and the associated spatio-temporal patterns. Our experimental results are in excellent agreement with simulations based on a reaction-diffusion model that describes diffusion-limited aggregation and branching, in which individual cells move actively in the periphery, but become immotile in the inner regions of the growing colony. We obtain the Hausdorff fractal dimension of the colony patterns: D(H.Expt)=1.1969 and D(H, R.D.=)1.1965, for experiment and reaction-diffusion model, respectively. Results of our experiments and modeling clearly show how glucose at higher concentration can prove to be inhibitory for motility of growing colonies of B. thuringiensis cells on semisolid support and be responsible for changes in the growth pattern. PMID:20553734
Deterministic fractals: extracting additional information from small-angle scattering data.
Cherny, A Yu; Anitas, E M; Osipov, V A; Kuklin, A I
2011-09-01
The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.
Preliminary Study of 2D Fracture Upscaling of Geothermal Rock Using IFS Fractal Model
NASA Astrophysics Data System (ADS)
Tobing, Prana F. L.; Feranie, Selly; Latief, Fourier D. E.
2016-08-01
Fractured rock plays important role in reservoir production. In larger scale, fractures are more likely to be heterogeneous and considered to be fractal in its nature. One of the characteristics of fractal structure is the scale independence. An investigation of fractal properties on natural fractured rock is therefore needed for modelling larger fracture. We have investigated the possibilities of fractal upscaling method to produce a larger geothermal fracture model based on smaller fracture data. We generate Iterated Function System (IFS) fractal model using parameters e.g. scale factor, angle between branch, initial line direction, and branch thickness. All the model parameters are obtained from smaller fracture data. We generate higher iteration model to be compared with larger geothermal fracture. The similarity between the IFS fractal model and natural fracture is measured by 2D box counting fractal dimension (D). The fractal dimension of first to fourth generation fractal model is (1.86 ± 0.02). The fractal dimension of the reference geothermal site is (1.86 ± 0.04). Besides of D, we found significant similarity of fracture parameters there are intensity and density between fracture model and natural fracture. Based on these result, we conclude that fractal upscaling using IFS fractal model is potential to model larger scale of 2D fracture.
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 structure of the time distribution of microfracturing in rocks
NASA Astrophysics Data System (ADS)
Feng, Xia-Ting; Seto, Masahiro
1999-01-01
Using acoustic emission data obtained from laboratory double torsion tests, we have analysed the fractal nature of a series of 29 granite microfracturing processes in time. The data represent a wide variety of timescales, stress environments (increasing load with a constant displacement rate, relaxation, creep), soaking conditions [air, water, dodecyl trimethyl ammonium bromide (DTAB), polyethelene oxide (PEO)], and material anisotropy. We find that the time distribution of rock microfracturing displays fractal and multifractal properties. In some cases, it has a single fractal or a multifractal structure. In other cases, it changes from a single fractal structure into a multifractal structure as the system evolves dynamically. We suggest that the heterogeneity of the rock, the distribution of joints or weak planes, the stress level, and the nature of the microfracturing mechanism lead to these multifractal properties. Whatever the fractal structure of the system, a lower fractal dimension is generally produced at near-failure of the rock due to an increased clustering. This result concerning the fractal-dimension decrease is consistent with the conclusion drawn from the spatial distribution of rock microfracturing. Therefore, from the vantage point of observation of the time distribution of rock microfracturing, the decrease of the fractal dimension has a potential use as a rock failure predictor.
Dynamic structure factor of vibrating fractals.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-02-10
Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.
Fractal 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.
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.…
Modeling of fractal intermediates in the self-assembly of silicatein filaments
NASA Astrophysics Data System (ADS)
Murr, Meredith; Thakur, Gunjan; Mezic, Igor; Morse, Daniel
2006-03-01
Silicateins are proteins with catalytic, structure-directing activity that are responsible for silica biosynthesis in certain sponges, Self-assembly of the silicatein monomers and oligomers was previously shown experimentally (Murr and Morse 2005) to form fibrous structures through the formation of diffusion limited, fractally patterned aggregates on the path to filament formation. We present a diffusion-limited aggregation (DLA) based model that is capable of capturing the basic properties of this self-assembly process. The Silicatein oligomer is modeled with three sites of attachment. Rules of attachment are specified that allow for specific interaction between these sites when oligomers are in proximity. The process differs from a DLA process in the following: 1) The process of aggregation is continued dynamically, i.e. the growing structures are spatially distributed and keep diffusing as they are growing 2) The molecules are oriented. Thus rotational diffusion is important. 3) The attachment can happen at more than 1 site and the strength of the active sites can be varied. We show that the self-assembled structures have a good level of similarity with the in-vitro experimental results. We quantify this by comparing the fractal dimension of the experimental data and the model output.
Fractal model of anomalous diffusion.
Gmachowski, Lech
2015-12-01
An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion.
Fractal structure of films deposited in a tokamak
NASA Astrophysics Data System (ADS)
Budaev, V. P.; Khimchenko, L. N.
2007-04-01
The surface of amorphous films deposited in the T-10 tokamak was studied in a scanning tunnel microscope. The surface relief on a scale from 10 nm to 100 μm showed a stochastic surface topography and revealed a hierarchy of grains. The observed variety of irregular structures of the films was studied within the framework of the concept of scale invariance using the methods of fractal geometry and statistical physics. The experimental probability density distribution functions of the surface height variations are close in shape to the Cauchy distribution. The stochastic surface topography of the films is characterized by a Hurst parameter of H = 0.68-0.85, which is evidence of a nontrivial self-similarity of the film structure. The fractal character and porous structure of deposited irregular films must be considered as an important issue related to the accumulation of tritium in the ITER project. The process of film growth on the surface of tokamak components exposed to plasma has been treated within the framework of the general concept of inhomogeneous surface growth. A strong turbulence of the edge plasma in tokamaks can give rise to fluctuations in the incident flux of particles, which leads to the growth of fractal films with grain dimensions ranging from nano-to micrometer scale. The shape of the surface of some films found in the T-10 tokamak has been interpreted using a model of diffusion-limited aggregation (DLA). The growth of films according to the discrete DLA model was simulated using statistics of fluctuations observed in a turbulent edge plasma of the T-10 tokamak. The modified DLA model reproduces well the main features of the surface of some films deposited in tokamaks.
Investigation on effects of aggregate structure in water and wastewater treatment.
Chau, K W
2004-01-01
The fractal structure and particle size of flocs are generally recognized as the two most crucial physical properties having impact on the efficiency of operation of several unit processes in water and wastewater treatment. In this study, an experimental investigation is undertaken on the effect of aggregate structure in water and wastewater treatment in Hong Kong. The fractal dimension of the resulting aggregate is employed as a measure of the aggregate structure. Small angle light scattering technique is used here. Different amounts of polymers are mixed to bacterial suspensions and the resulting structures are examined. The addition of polymer may foster aggregate formation by neutralization of the bacterial surface charge and enhance inter-particle bridging. The aggregation behavior may affect the efficiency of certain water and wastewater treatment processes such as dewatering and coagulation. The impacts of aggregate structure on two representative processes, namely, ultra-filtration membrane fouling and pressure filter dewatering efficiency, are studied. It is found that the looser flocs yield a more porous cake and less tendency to foul whilst more porous filter cakes yield more ready biosolids dewatering.
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-01-01
Chaos and fractal dimension are emerging modalities for the research of electroencephalogram (EEG) signal processing. The capability of measuring non-linear characteristics of the fractal dimension enables new methodologies to identify distinct brain activities. Recent studies on the topic focus on utilizing various types of fractals as features in order to design better brain state classification system. However, we have little insight about the EEG signals projected in fractal dimension. In this paper, we investigate the relationship between the non-linear characteristics of ongoing EEG signals and event-related desynchronization (ERD) during motor imagery. We observed a considerable synchronization between ERD and fractal dimension. This finding suggests further usage of chaos and fractal theory in investigating brain activities. PMID:26737207
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-01-01
Chaos and fractal dimension are emerging modalities for the research of electroencephalogram (EEG) signal processing. The capability of measuring non-linear characteristics of the fractal dimension enables new methodologies to identify distinct brain activities. Recent studies on the topic focus on utilizing various types of fractals as features in order to design better brain state classification system. However, we have little insight about the EEG signals projected in fractal dimension. In this paper, we investigate the relationship between the non-linear characteristics of ongoing EEG signals and event-related desynchronization (ERD) during motor imagery. We observed a considerable synchronization between ERD and fractal dimension. This finding suggests further usage of chaos and fractal theory in investigating brain activities.
Wavelet and fractal analysis of ground-vehicle images
NASA Astrophysics Data System (ADS)
Gorsich, David J.; Tolle, Charles R.; Karlsen, Robert E.; Gerhart, Grant R.
1996-10-01
A large number of terrain images were taken at Aberdeen Proving Grounds, some containing ground vehicles. Is it possible to screen the images for possible targets in a short amount of time using the fractal dimension to detect texture variations. The fractal dimension is determined using the wavelet transform for these visual images. The vehicles are positioned within the grass and in different locations. Since it has been established that natural terrain exhibits a statistical l/f self-similarity property and the psychophysical perception of roughness can be quantified by the same self-similarity, fractal dimensions estimates should vary only at texture boundaries and breaks in the tree and grass patterns. Breaks in the patterns are found using contour plots of the dimension estimates and are considered as perceptual texture variations. Variation in the dimension estimate is considered more important than the accuracy of the actual dimensions number. Accurate variation estimates are found even with low resolution images.
Fractal Analysis of Stress Sensitivity of Permeability in Porous Media
NASA Astrophysics Data System (ADS)
Tan, Xiao-Hua; Li, Xiao-Ping; Liu, Jian-Yi; Zhang, Lie-Hui; Cai, Jianchao
2015-12-01
A permeability model for porous media considering the stress sensitivity is derived based on mechanics of materials and the fractal characteristics of solid cluster size distribution. The permeability of porous media considering the stress sensitivity is related to solid cluster fractal dimension, solid cluster fractal tortuosity dimension, solid cluster minimum diameter and solid cluster maximum diameter, Young's modulus, Poisson's ratio, as well as power index. Every parameter has clear physical meaning without the use of empirical constants. The model predictions of permeability show good agreement with those obtained by the available experimental expression. The proposed model may be conducible to a better understanding of the mechanism for flow in elastic porous media.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.
1998-01-01
Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and
Fractal Analysis on Morphology of Laser Irradiated Vanadium Surfaces Under Different Ambient
NASA Astrophysics Data System (ADS)
Szkiva, Zs.; Bálint, Á. M.; Füle, M.; Nánai, L.
2013-12-01
Pulsed laser irradiated vanadium surface morphology under different ambient has been prepared and characterized using fractal dimension analysis method on scanning electron microscopy (SEM) images. In presence of different ambient, self-periodic and self-similar surface patterns (e.g. dots, islands, and pins) were grown and appeared in different shapes. The fractal dimension (FD) of this developed vanadium nanostructure was calculated by fractal box count method (FBM). The calculated fractal dimension (FD, Df) shows dependence on the different type on ambient and the number of laser shots.
Colloidal Aggregate Structure under Shear by USANS
NASA Astrophysics Data System (ADS)
Chatterjee, Tirtha; van Dyk, Antony K.; Ginzburg, Valeriy V.; Nakatani, Alan I.
2015-03-01
Paints are complex formulations of polymeric binders, inorganic pigments, dispersants, surfactants, colorants, rheology modifiers, and other additives. A commercially successful paint exhibits a desired viscosity profile over a wide shear rate range from 10-5 s-1 for settling to >104 s-1 for rolling, and spray applications. Understanding paint formulation structure is critical as it governs the paint viscosity profile. However, probing paint formulation structure under shear is a challenging task due to the formulation complexity containing structures with different hierarchical length scales and their alterations under the influence of an external flow field. In this work mesoscale structures of paint formulations under shear are investigated using Ultra Small-Angle Neutron Scattering (rheo-USANS). Contrast match conditions were utilized to independently probe the structure of latex binder particle aggregates and the TiO2 pigment particle aggregates. Rheo-USANS data revealed that the aggregates are fractal in nature and their self-similarity dimensions and correlations lengths depend on the chemistry of the binder particles, the type of rheology modifier present and the shear stress imposed upon the formulation. These results can be explained in the framework of diffusion and reaction limited transient aggregates structure evolution under simple shear.
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.
Effect of the particle shape on the optical properties of black carbon aggregates
NASA Astrophysics Data System (ADS)
Skorupski, Krzysztof
2016-04-01
Small particles tend to connect to each other and create large geometries, namely aggregates. To simplify the light scattering simulation process, they are usually modelled as assemblies of spheres positioned in point contact. This is a rough approximation because connections between them always exist. In this work we present answers to the three following questions: which optical properties of fractal-like aggregates are strongly dependent on the particle shape, what is the magnitude of the relative extinction error σCext when non-spherical particles are modelled as spheres and whether the relative extinction error σCext is dependent on the aggregate size Np. The paper was aimed at tropospheric black carbon particles and their complex refractive index m was based on the work by Chang and Charalampopoulos. The incident wavelength λ varied from λ = 300nm to λ = 900nm. For the light scattering simulations the ADDA algorithm was used. The polarizability expression was IGT_SO (approximate Integration of Greens Tensor over the dipole) and each particle, regardless of its shape, was composed of ca. Nd ≍ 1000 volume elements (dipoles). In the study, fractal-like aggregates consisted of up to Np = 300 primary particles with the volume equivalent to the volume of a sphere with the radius rp = 15nm. The fractal dimension was Df = 1:8 and the fractal prefactor was kf = 1:3. Geometries were generated with the tunable CC (Cluster-Cluster) algorithm proposed by Filippov et al. The results show that when the extinction cross section σCext is considered, the changes caused by the particle shape, which are especially visible for longer wavelengths λ cannot be neglected. The most significant difference can be observed for the regular tetrahedron. The relative extinction error σCext diminishes slightly along with the number of primary particles Np. However, even when large fractal-like aggregates are studied, it should not be considered as non-existent. On the contrary
NASA Astrophysics Data System (ADS)
Ingersoll, A. P.; Nakajima, M.; Ewald, S.; Gao, P.
2015-12-01
Postberg et al (2009) argued that the observed plume activity requires large vapor chambers above the evaporating liquid (left figure). Here we argue that large vapor chambers are unnecessary, and that a liquid-filled crack 1 meter wide extending along the 500 km length of the tiger stripes would be an adequate source (right figure). We consider controlled boiling (companion paper by Nakajima and Ingersoll 2015AGU) regulated by friction between the gas and the walls. Postberg et al use formulas from Rayleigh-Benard convection, which we argue does not apply when bubbles are transferring their latent heat across the liquid-gas interface. We show that modest convection currents in the liquid (few cm/s) can supply energy to the boiling zone and prevent it from freezing. Hedman et al (2013) reported brightness variations with orbital phase, but they also reported that their 2005 observations were roughly 50% higher than the 2009 observations. Here we extend the observation period to 2015 (Ingersoll and Ewald 2015). Our analysis relies on ISS images whereas Hedman et al rely on VIMS near-IR images, which have 40 times lower resolution. We successfully separate the brightness of the plume from the E-ring background. Our earlier analysis of the particle size distribution (Ingersoll and Ewald 2011) allows us to correct for differences in scattering angle. We confirm a general decline in activity over the 10-year period, but we find hints of fluctuations on shorter time scales. Kempf (Cassini project science meeting, Jan 22, 2015) reported that the mass of particles in the plumes could be an order of magnitude less than that reported by Ingersoll and Ewald (2011). Kempf used in situ particle measurements by CDA, whereas I&E used brightness observations and the assumption that the particles are solid ice. Here we show (Gao et al 2015AGU) that fractal aggregates fit the brightness data just as well as solid ice, and are consistent with the lower mass reported by Kempf.
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.
[Dimensional fractal of post-paddy wheat root architecture].
Chen, Xin-xin; Ding, Qi-shuo; Li, Yi-nian; Xue, Jin-lin; Lu, Ming-zhou; Qiu, Wei
2015-06-01
To evaluate whether crop rooting system was directionally dependent, a field digitizer was used to measure post-paddy wheat root architectures. The acquired data was transferred to Pro-E, in which virtual root architecture was reconstructed and projected to a series of planes each separated in 10° apart. Fractal dimension and fractal abundance of root projections in all the 18 planes were calculated, revealing a distinctive architectural distribution of wheat root in each direction. This strongly proved that post-paddy wheat root architecture was directionally dependent. From seedling to turning green stage, fractal dimension of the 18 projections fluctuated significantly, illustrating a dynamical root developing process in the period. At the jointing stage, however, fractal indices of wheat root architecture resumed its regularity in each dimension. This wheat root architecture recovered its dimensional distinctness. The proposed method was applicable for precision modeling field state root distribution in soil.
Scaling in the Diffusion Limited Aggregation Model
NASA Astrophysics Data System (ADS)
Menshutin, Anton
2012-01-01
We present a self-consistent picture of diffusion limited aggregation (DLA) growth based on the assumption that the probability density P(r,N) for the next particle to be attached within the distance r to the center of the cluster is expressible in the scale-invariant form P[r/Rdep(N)]. It follows from this assumption that there is no multiscaling issue in DLA and there is only a single fractal dimension D for all length scales. We check our assumption self-consistently by calculating the particle-density distribution with a measured P(r/Rdep) function on an ensemble with 1000 clusters of 5×107 particles each. We also show that a nontrivial multiscaling function D(x) can be obtained only when small clusters (N<10000) are used to calculate D(x). Hence, multiscaling is a finite-size effect and is not intrinsic to DLA.
Linear relationship statistics in diffusion limited aggregation
NASA Astrophysics Data System (ADS)
Saberi, Abbas Ali
2009-11-01
We show that various surface parameters in two-dimensional diffusion limited aggregation (DLA) grow linearly with the number of particles. We find the ratio of the average length of the perimeter and the accessible perimeter of a DLA cluster together with its external perimeters to the cluster size, and define a microscopic schematic procedure for attachment of an incident new particle to the cluster. We measure the fractal dimension of the red sites (i.e., the sites such that cutting each of them splits the cluster) as equal to that of the DLA cluster. It is also shown that the average number of dead sites and the average number of red sites have linear relationships with the cluster size.
He, Chiquan; Zhao, Kuiyi
2003-04-01
By using the principles and methods of fractal geometry theory, the relationship between above ground biomass and plant length or sheath height of Carex lasiocarpa population was studied. The results showed that there was a good static fractal relationship between them, and the resulted fractal dimension was an efficient description of the accumulation of above ground biomass in each organ. The dynamic fractal relationship showed that during the whole growing season, the increase of above ground biomass had a self-similarity, being a fractal growth process, and the pattern of its increase was the fractal dimension D. Based on these results, a fractal growth model of Carex lasiocarpa population was established, which regarded the bigger grass as the result of the amplification of seedling growth.
Fractal Structures on Fe3O4 Ferrofluid: A Small-Angle Neutron Scattering Study
NASA Astrophysics Data System (ADS)
Giri Rachman Putra, Edy; Seong, Baek Seok; Shin, Eunjoo; Ikram, Abarrul; Ani, Sistin Ari; Darminto
2010-10-01
A small-angle neutron scattering (SANS) which is a powerful technique to reveal the large scale structures was applied to investigate the fractal structures of water-based Fe3O4ferrofluid, magnetic fluid. The natural magnetite Fe3O4 from iron sand of several rivers in East Java Province of Indonesia was extracted and purified using magnetic separator. Four different ferrofluid concentrations, i.e. 0.5, 1.0, 2.0 and 3.0 Molar (M) were synthesized through a co-precipitation method and then dispersed in tetramethyl ammonium hydroxide (TMAH) as surfactant. The fractal aggregates in ferrofluid samples were observed from their SANS scattering distributions confirming the correlations to their concentrations. The mass fractal dimension changed from about 3 to 2 as ferrofluid concentration increased showing a deviation slope at intermediate scattering vector q range. The size of primary magnetic particle as a building block was determined by fitting the scattering profiles with a log-normal sphere model calculation. The mean average size of those magnetic particles is about 60 - 100 Å in diameter with a particle size distribution σ = 0.5.
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.
3D simulation of the Cluster-Cluster Aggregation model
NASA Astrophysics Data System (ADS)
Li, Chao; Xiong, Hailing
2014-12-01
We write a program to implement the Cluster-Cluster Aggregation (CCA) model with java programming language. By using the simulation program, the fractal aggregation growth process can be displayed dynamically in the form of a three-dimensional (3D) figure. Meanwhile, the related kinetics data of aggregation simulation can be also recorded dynamically. Compared to the traditional programs, the program has better real-time performance and is more helpful to observe the fractal growth process, which contributes to the scientific study in fractal aggregation. Besides, because of adopting java programming language, the program has very good cross-platform performance.
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.
NASA Astrophysics Data System (ADS)
Tremberger, George, Jr.; Flamholz, A.; Cheung, E.; Sullivan, R.; Subramaniam, R.; Schneider, P.; Brathwaite, G.; Boteju, J.; Marchese, P.; Lieberman, D.; Cheung, T.; Holden, Todd
2007-09-01
The absorption effect of the back surface boundary of a diffuse layer was studied via laser generated reflection speckle pattern. The spatial speckle intensity provided by a laser beam was measured. The speckle data were analyzed in terms of fractal dimension (computed by NIH ImageJ software via the box counting fractal method) and weak localization theory based on Mie scattering. Bar code imaging was modeled as binary absorption contrast and scanning resolution in millimeter range was achieved for diffusive layers up to thirty transport mean free path thick. Samples included alumina, porous glass and chicken tissue. Computer simulation was used to study the effect of speckle spatial distribution and observed fractal dimension differences were ascribed to variance controlled speckle sizes. Fractal dimension suppressions were observed in samples that had thickness dimensions around ten transport mean free path. Computer simulation suggested a maximum fractal dimension of about 2 and that subtracting information could lower fractal dimension. The fractal dimension was shown to be sensitive to sample thickness up to about fifteen transport mean free paths, and embedded objects which modified 20% or more of the effective thickness was shown to be detectable. The box counting fractal method was supplemented with the Higuchi data series fractal method and application to architectural distortion mammograms was demonstrated. The use of fractals in diffusive analysis would provide a simple language for a dialog between optics experts and mammography radiologists, facilitating the applications of laser diagnostics in tissues.
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.
Scaling of transverse nuclear magnetic relaxation due to magnetic nanoparticle aggregation.
Brown, Keith A; Vassiliou, Christophoros C; Issadore, David; Berezovsky, Jesse; Cima, Michael J; Westervelt, R M
2010-10-01
The aggregation of superparamagnetic iron oxide (SPIO) nanoparticles decreases the transverse nuclear magnetic resonance (NMR) relaxation time T2CP of adjacent water molecules measured by a Carr-Purcell-Meiboom-Gill (CPMG) pulse-echo sequence. This effect is commonly used to measure the concentrations of a variety of small molecules. We perform extensive Monte Carlo simulations of water diffusing around SPIO nanoparticle aggregates to determine the relationship between T2CP and details of the aggregate. We find that in the motional averaging regime T2CP scales as a power law with the number N of nanoparticles in an aggregate. The specific scaling is dependent on the fractal dimension d of the aggregates. We find T2CP∝N-0.44 for aggregates with d = 2.2, a value typical of diffusion limited aggregation. We also find that in two-nanoparticle systems, T2CP is strongly dependent on the orientation of the two nanoparticles relative to the external magnetic field, which implies that it may be possible to sense the orientation of a two-nanoparticle aggregate. To optimize the sensitivity of SPIO nanoparticle sensors, we propose that it is best to have aggregates with few nanoparticles, close together, measured with long pulse-echo times. PMID:20689678
[Composition and stability of soil aggregates in hedgerow-crop slope land].
Pu, Yu-Lin; Lin, Chao-Wen; Xie, De-Ti; Wei, Chao-Fu; Ni, Jiu-Pai
2013-01-01
Based on a long-term experiment of using hedgerow to control soil and water loss, this paper studied the composition and stability of soil aggregates in a hedgerow-crop slope land. Compared with those under routine contour cropping, the contents of > 0.25 mm soil mechanical-stable and water-stable aggregates under the complex mode hedgerow-crop increased significantly by 13.3%-16.1% and 37.8% -55.6%, respectively. Under the complex mode, the contents of > 0.25 mm soil water-stable aggregates on each slope position increased obviously, and the status of > 0.25 mm soil water-stable aggregates being relatively rich at low slope and poor at top slope was improved. Planting hedgerow could significantly increase the mean mass diameter and geometric mean diameter of soil aggregates, decrease the fractal dimension of soil aggregates and the destruction rate of > 0.25 mm soil aggregates, and thus, increase the stability and erosion-resistance of soil aggregates in slope cropland. No significant effects of slope and hedgerow types were observed on the composition, stability and distribution of soil aggregates.
A Fractal Dimensional Analysis on the Cloud Shape Parameters of Cumulus over Land.
NASA Astrophysics Data System (ADS)
Gotoh, Kazuo; Fujii, Yasuhiko
1998-10-01
High-resolution Landsat thematic mapper image data were employed in the present study to estimate the influence of regional wind systems on the macrophysical properties of cumulus clouds, such as perimeter fractal dimension, orientation angle, and cloud-base height. As a case study, an image of cumulus clouds was chosen in which small- and large-sized clouds prevail over the land in the presence of local winds. Cloud extraction from backgrounds becomes difficult when the background comprises various brightnesses. Furthermore, the surface brightness of a cloud is not uniform when shadows appear on the surface. These effects contribute to errors in cloud extraction only using a simple threshold. To address these problems, technical improvements in processing, including the combination of clustering, dynamic threshold, and edge detection, are presented and shown to produce more accurate cloud outlines seen from the zenith direction.The clouds' statistical area-perimeter relation was shown to confirm that the fractal dimension of the cumuli follows the double power law: two different perimeter fractal dimensions exist for larger clouds and for smaller clouds. Correspondence of the bending point (around 0.7 km2) in the area-perimeter relation with the bending point in the area-major axis relation was also demonstrated. Analysis of cloud orientation showed that larger clouds (>0.7 km2) indicate the approximate direction of the local wind, whereas smaller clouds do not show clear tendencies in their directions, suggesting that there is a difference in formation between larger and smaller clouds. These findings suggest that in the presence of horizontal local wind, larger clouds are likely to be formed by aggregation of smaller cloud cells (unicellular clouds) during the process of alignment to become long- and roll-shaped clouds (multicellular clouds).Uneven distribution of cloud-base height was also demonstrated in this study. Clouds with higher base levels are found to
Fractal analysis methods for solid alkane monolayer domains at SiO2/air interfaces.
Knüfing, Lydia; Schollmeyer, Hauke; Riegler, Hans; Mecke, Klaus
2005-02-01
A systematic evaluation of various fractal analysis methods is essential for studying morphologies of finite and noisy experimental patterns such as domains of long chain alkanes at SiO(2)/air interfaces. The derivation of trustworthy fractal dimensions crucially relies on the definition of confidence intervals for the assumed scaling range. We demonstrate that the determination of the intervals can be improved largely by comparing the scaling behavior of different morphological measures (area, boundary, curvature). We show that the combination of area and boundary data from coarse-grained structures obtained with the box-counting method reveals clear confidence limits and thus credible morphological data. This also holds for the Minkowski density method. It also reveals the confidence range. Its main drawback, the larger swing-in period at the lower cutoff compared to the box-counting method, is compensated by more details on the scaling behavior of area, boundary, and curvature. The sandbox method is less recommendable. It essentially delivers the same data as box-counting, but it is more susceptible to finite size effects at the lower cutoff. It is found that the domain morphology depends on the surface coverage of alkanes. The individual domains at low surface coverage have a fractal dimension of approximately 1.7, whereas at coverages well above 50% the scaling dimension is 2 with a large margin of uncertainty at approximately 50% coverage. This change in morphology is attributed to a crossover from a growth regime dominated by diffusion-limited aggregation of individual domains to a regime where the growth is increasingly affected by annealing and the interaction of solid growth fronts which approach each other and thus compete for the alkane supply.
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)
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.
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
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.
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