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Showing papers on "Fractal dimension published in 2000"


Journal ArticleDOI
TL;DR: In this article, the behavior of the small fractal Koch monopole is numerically and experimentally analyzed, and it is shown that as the number of iterations on the small Koch monopoles are increased, the Q of the antenna approaches the fundamental limit for small antennas.
Abstract: Fractal objects have some unique geometrical properties. One of them is the possibility to enclose in a finite area an infinitely long curve. The resulting curve is highly convoluted being nowhere differentiable. One such curve is the Koch curve. In this paper, the behavior the Koch monopole is numerically and experimentally analyzed. The results show that as the number of iterations on the small fractal Koch monopole are increased, the Q of the antenna approaches the fundamental limit for small antennas.

457 citations


Journal ArticleDOI
TL;DR: A new method using signal summation conversion (SSC) greatly improves the classification and the reliability of Ĥ, the estimates of H, for the times series, and suggests that the flow signal is the summation of a set of local velocities from neighboring vessels that are negatively correlated, as if induced by local resistance fluctuations.
Abstract: Many physiological signals appear fractal, in having self-similarity over a large range of their power spectral densities. They are analogous to one of two classes of discretely sampled pure fractal time signals, fractional Gaussian noise (fGn) or fractional Brownian motion (fBm). The fGn series are the successive differences between elements of a fBm series; they are stationary and are completely characterized by two parameters, σ2, the variance, and H, the Hurst coefficient. Such efficient characterization of physiological signals is valuable since H defines the autocorrelation and the fractal dimension of the time series. Estimation of H from Fourier analysis is inaccurate, so more robust methods are needed. Dispersional analysis (Disp) is good for noise signals while bridge detrended scaled windowed variance analysis (bdSWV) is good for motion signals. Signals whose slopes of their power spectral densities lie near the border between fGn and fBm are difficult to classify. A new method using signal summation conversion (SSC), wherein an fGn is converted to an fBm or an fBm to a summed fBm and bdSWV then applied, greatly improves the classification and the reliability of Ĥ, the estimates of H, for the times series. Applying these methods to laser-Doppler blood cell perfusion signals obtained from the brain cortex of anesthetized rats gave Ĥ of 0.24±0.02 (SD, n=8) and defined the signal as a fractional Brownian motion. The implication is that the flow signal is the summation (motion) of a set of local velocities from neighboring vessels that are negatively correlated, as if induced by local resistance fluctuations.

340 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a contact problem in which an elastic half-plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series.
Abstract: A contact problem is considered in which an elastic half–plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n − 1 and that at scale n . Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n , except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n , tending to a power–law behaviour at large n which corresponds to a limiting fractal dimension of (2 − D ), where D is the fractal dimension of the surface profile. However, it is not a ‘simple’ fractal, in the sense that it deviates from the power–law form at low n , at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n . † The authors dedicate this paper to the memory of Dr J. F. Archard, 1918–1989.

226 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present an analysis of the development of the Tel Aviv metropolis by using the concept of fractals, and the fractal dimension of the entire metropolis, and of its parts, was estimated as a function of time, from 1935 onwards.
Abstract: We present here an analysis of the development of the Tel Aviv metropolis by using the concept of fractals. The fractal dimension of the entire metropolis, and of its parts, was estimated as a function of time, from 1935 onwards. The central part and the northern tier are fractal at all times. Their fractal dimension increased with time. However, the metropolis as a whole can be said to be fractal only after 1985. There is a general tendency towards fractality, in the sense that the fractal dimension of the different parts converge towards the same value.

195 citations


Journal ArticleDOI
TL;DR: This paper summarizes the state of the art and introduces several updated developments in analysis and description of patch patterns and patch dynamics by means of Mandelbrot’s fractal analysis, with an emphasis on current research results and a personal view.

194 citations


Journal ArticleDOI
TL;DR: Shih et al. as discussed by the authors defined the structure of particulate colloidal protein gels using salt-induced cold gelation of heat-denatured whey protein isolate (WPI) as a model.

174 citations


Proceedings ArticleDOI
01 Aug 2000
TL;DR: A new clustering algorithm, based in the fractal properties of the data sets, which is capable of recognizing clusters of arbitrary shape and which places points incrementally in the cluster for whic h the change in the Fractal dimension after adding the point is the least.
Abstract: Clustering is a widely used knowledge discovery technique. It helps uncovering structures in data that were not previously known. The clustering of large data sets has received a lot of atten tion in recen t years, how ever,clustering is a still a challenging task since many published algorithms fail to do well in scaling with the size of the data set and the number of dimensions that describe the points, or in nding arbitrary shapes of clusters, or dealing e ectively with the presence of noise. In this paper, we presen t a new clustering algorithm, based in the fractal properties of the data sets. The new algorithm which w e call F ractal Clustering (F C) places poin ts incrementally in the cluster for whic h the change in the fractal dimension after adding the point is the least. This is a very natural way of clustering points, since points in the same cluster have a great degree of selfsimilarity among them (and m uch lessself-similarity with respect to poin tsin other clusters). F C requires one scan of the data, is suspendable at will, pro viding the best answ er possible at that point, and is incremental. We sho w via experiments that F C e ectively deals with large data sets, high-dimensionality and noise and is capable of recognizing clusters of arbitrary shape.

156 citations


Journal ArticleDOI
TL;DR: The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms and avoids possible underestimates of fractal dimension.

138 citations


Journal ArticleDOI
TL;DR: In this article, a novel partial discharge (PD) defect identification method is described, where a suitable set of parameters are determined and then used as input variables to a neural network for the purpose of identifying the defects within the insulation.
Abstract: A novel partial discharge (PD) defect identification method is described. Starting with PD data on different families of specimens, a suitable set of parameters are determined and then used as input variables to a neural network for the purpose of identifying the defects within the insulation. In this procedure the statistical Weibull analysis is performed on PD pulse amplitude histograms to obtain the scale parameter /spl alpha/ and the shape parameter /spl beta/. Thereafter, the two statistical operators (skewness and kurtosis) and two fractal parameters (fractal dimension and lacunarity) are evaluated from the PD phase on the discharge epoch histogram and from the 3 dimensional (pulse amplitude/phase/discharge rate) histogram, respectively. Following the exposition of the basic mathematical concepts regarding the above parameters, experimental results are reported on the recognition capability of the method in defining the defect category in a number of different specimens.

127 citations


Journal ArticleDOI
TL;DR: In this article, the morphology of structures formed during two-dimensional aggregation of Ag particles in three different solvents was studied by transmission electron microsocopy (TEM), and the patterns were dendrite, fractal, and fractal with the fractal dimension of 1.81, 1.73, and 1.70 corresponding to the aggregating structure from deionized water, ethanol, and 0.01 M C12H25NaSO4 solutions, respectively.
Abstract: With γ-irradiation on AgNO3 dissolved in deionized water, ethanol, and 0.01 M C12H25NaSO4, metallic Ag with different dispersed states and particle sizes was obtained. In the former two solvents, metallic Ag particles precipitated from the solutions and the Ag particle size was about 100 and 20 nm, respectively. In the latter solvent, Ag colloid was produced. The morphology of structures formed during two-dimensional aggregation of Ag particles in three different solvents was studied by transmission electron microsocopy (TEM). The patterns were dendrite, fractal (with center), and fractal (no center) with the fractal dimension of 1.81, 1.73, and 1.70 corresponding to the aggregating structure from deionized water, ethanol, and 0.01 M C12H25NaSO4 solutions, respectively. The formation mechanism of different patterns was illustrated from different aspects.

127 citations


Journal ArticleDOI
TL;DR: In this paper, the fractal dimensions derived from laser-induced fluorescence (LIF) of OH and Mie scattering images were determined by analysis of the flame front images using the caliper technique.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated changes to the ground state when a weak perturbation is applied to the bulk of the spin glass system and showed that replica symmetry breaking is possible.
Abstract: The nature of the spin glass state is investigated by studying changes to the ground state when a weak perturbation is applied to the bulk of the system. We consider short range models in three and four dimensions and the infinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. Our results for the SK and Viana-Bray models agree with the replica symmetry breaking picture. The data for the short range models fit naturally a picture in which there are large scale excitations which cost a finite energy but whose surface has a fractal dimension, ${d}_{s}$, less than the space dimension $d$. We also discuss the possible crossover to other behavior at larger length scales than the sizes studied.

Journal ArticleDOI
Ruikang K. Wang1
TL;DR: In this article, a method for modelling the complicated soft tissue, based on the fractal approach, permitting numerical evaluation of the phase functions and four optical properties of tissue (scattering coefficient, reduced scattering coefficient, backscattering coefficient, and anisotropy factor) by the use of the Mie scattering theory.
Abstract: A knowledge of the local refractive index variations and size distribution of scatterers in biological tissue is required to understand the physical processes involved in light-tissue interaction. This paper describes a method for modelling the complicated soft tissue, based on the fractal approach, permitting numerical evaluation of the phase functions and four optical properties of tissue—scattering coefficient, reduced scattering coefficient, backscatter-ing coefficient, and anisotropy factor—by the use of the Mie scattering theory. A key assumption of the model is that refractive index variations caused by microscopic tissue elements can be treated as particles with size distribution according to the power law. The model parameters, such as refractive index, incident wavelength, and fractal dimension, that are likely to affect the predictions of optical properties are investigated. The results suggest that the fractal dimension used to describe how biological tissue can be approximated by par...

Proceedings ArticleDOI
01 Feb 2000
TL;DR: The theoretical and empirical results show that previous worst-case analysis of nearest neighbor search in high dimensions are over-pessimistic, to the point of being unrealistic, and the performance depends critically on the intrinsic ("fractal") dimensionality as opposed to the embedding dimension that the uniformity assumption incorrectly implies.
Abstract: Nearest neighbor queries are important in many settings, including spatial databases (find the k closet cities) and multimedia databases (find the k most similar images). Previous analyses have concluded that nearest neighbor search is hopeless in high dimensions, due to the notorious "curse of dimensionality". However, their precise analysis over real data sets is still an open problem. The typical and often implicit assumption in previous studies is that the data is uniformly distributed, with independence between attributes. However, real data sets overwhelmingly disobey these assumptions; rather, they typically are skewed and exhibit intrinsic ("fractal") dimensionalities that are much lower than their embedding dimension, e.g., due to subtle dependencies between attributes. We show how the Hausdorff and correlation fractal dimensions of a data set can yield extremely accurate formulas that can predict I/O performance to within one standard deviation. The practical contributions of this work are our accurate formulas which can be used for query optimization in spatial and multimedia databases. The theoretical contribution is the 'deflation' of the dimensionality curse. Our theoretical and empirical results show that previous worst-case analysis of nearest neighbor search in high dimensions are over-pessimistic, to the point of being unrealistic. The performance depends critically on the intrinsic ("fractal") dimensionality as opposed to the embedding dimension that the uniformity assumption incorrectly implies.

Journal ArticleDOI
TL;DR: In this paper, the Stokes equation was adopted to model the fluid external to the aggregate and the Brinkman equation for the internal flow, and the results were summarised in terms of three parameters: drag coefficient, fluid collection efficiency and settling factor Z.

Journal ArticleDOI
TL;DR: The applications of fractal analysis of bone in medical imaging in general and dental radiographs in particular are introduced and it is remembered that all stages in the analytical chain have an impact on the results.
Abstract: OBJECTIVE To introduce the applications of fractal analysis of bone in medical imaging in general and dental radiographs in particular. RESULTS Various methods for measuring fractal dimension have been used to compare normal with osteoporotic bone with contradictory results. This disparity may be attributed to differences in the anatomical sites studied and differences in methods used to obtain the 2D images. However, differences in methods used to measure fractal dimension can also be responsible. CONCLUSION When fractal dimension is used to study bone it must be remembered that all stages in the analytical chain have an impact on the results. Thus, to obtain more conclusive results, studies on the fractal dimension of bone should be carefully designed and individual methods thoroughly evaluated.

Journal ArticleDOI
TL;DR: Sontag as discussed by the authors gave a brief recap of risk stratification on Piazza, and then the vast majority of today's lecture we will be talking about a new topic-in particular, physiological time series modeling.
Abstract: DAVID SONTAG: So I'll begin today's lecture by giving a brief recap of risk stratification. We didn't get to finish talking survival modeling on Thursday, and so I'll go a little bit more into that, and I'll answer some of the questions that arose during our discussions and on Piazza since. And then the vast majority of today's lecture we'll be talking about a new topic-in particular, physiological time series modeling. I'll give two examples of physiological time series modeling-the first one coming from monitoring patients in intensive care units, and the second one asking a very different type of question-that of diagnosing patients' heart conditions using EKGs.

Journal ArticleDOI
TL;DR: In this paper, a modeling approach that simulates changes in particle size distribution (PSD) due to coagulation by incorporating recently proposed fractal mathematics and introducing a new conceptual framework called the coalesced fractal sphere (CFS) assumption was described.

Journal ArticleDOI
Markus Hütter1
TL;DR: Brownian dynamics simulation shows how the porosity is reflected in the distribution of nearest neighbors around the center particle, i.e., the very local conformation in the particle network, and no intermediate fractal regime exists.

Journal ArticleDOI
TL;DR: In this article, the fractal dimension of two geochemically linked groups of mafic microgranular enclaves (MME) from the Sithonia Plutonic Complex (Northern Greece) was estimated using distribution maps of chemical elements inside enclaves.

Journal ArticleDOI
01 Jan 2000-Carbon
TL;DR: In this article, the authors used the modified BET and fractal Frenkel-Halsey-Hill (FHH) models to estimate surface fractal dimensions of five different carbons (Sorbonorite 4, GAC 1240, and three amorphous carbons) from analysis of gas and liquid (phenanthrene) adsorption isotherm data.


Journal ArticleDOI
TL;DR: In this paper, the fractal dimension of cluster-cluster aggregates, Df, appears to be a well-established property, but it exhibits a large range of possible values.
Abstract: While the fractal dimension of cluster-cluster aggregates, Df , appears to be a well-established property, the fractal prefactor, kg (also known as structural coefficient), continues to exhibit a large range of possible values. In the present paper, an attempt is made to clarify this issue which leads to conclusive results concerning the value to adopt for the fractal prefactor of simulated aggregates. Starting from a large population of "free" numerically simulated aggregates (i.e., where no restrictions to aggregate formation were imposed) the fractal properties obtained were estimated both using morphological concepts as well as light scattering theories. Furthermore, studies for aggregates having predefined morphological kg values (namely, kg > 2 and ca. 1) were also performed in order to check the viability of these values. The results obtained for the three different populations of aggregates considered were used to infer, through a best-fit analysis, the fractal properties. Our best estimates are kg

Journal ArticleDOI
TL;DR: A sectional model for aggregate aerosol dynamics accounting for gas-phase chemical reaction, coagulation, and sintering at nonisothermal conditions is presented in this article, where the aggregate structure was considered by the implementation of a constant mass fractal dimension on the aggregate collision diameter and an average, but variable, aggregate area within a given volume section.
Abstract: A sectional model for aggregate aerosol dynamics accounting for gas-phase chemical reaction, coagulation, and sintering at nonisothermal conditions is presented. The aggregate structure was considered by the implementation of a constant mass fractal dimension on the aggregate collision diameter and an average, but variable, aggregate area within a given volume section. The model showed separately the evolution of the size distribution of the primary and aggregate particles. Model predictions with respect to average primary particle diameters and aggregate particle-size distributions were evaluated by comparing with experimental data for synthesis of titania by TiCl4 oxidation in a furnace aerosol flow reactor. The effects from sintering time and mass fractal dimension are also discussed.

Journal ArticleDOI
01 Jan 2000-Carbon
TL;DR: In this article, the Dubinin-Izotova and Dubin-Stoeckli equations were used for evaluation of micropore structural parameters of examined carbon samples.

Journal ArticleDOI
01 Mar 2000-Langmuir
TL;DR: In this article, a scaling relation between fractal dimension and stability ratio is demonstrated for a charged polystyrene colloid undergoing aggregation, where the stability ratio was systematically decreased by reducing the electrostatic barrier to aggregation, and fractal dimensions were measured by static light scattering, while stability ratios were evaluated from early-time aggregation kinetics using dynamic light scattering.
Abstract: A scaling relation between fractal dimension and stability ratio is demonstrated for a charged polystyrene colloid undergoing aggregation The stability ratio, ie, the inverse collision efficiency, or inverse sticking probability, is systematically decreased by reducing the electrostatic barrier to aggregation Fractal dimensions are measured by static light scattering, while stability ratios are evaluated from early-time aggregation kinetics (t < 10t1/2) using dynamic light scattering The fractal dimension for the model hydrophobic colloid investigated increases monotonically with stability ratio, reflecting a continuous increase in particle packing density as the sticking probability is reduced

Journal ArticleDOI
01 Sep 2000-Fuel
TL;DR: In this paper, the surface fractal dimension (D s ) of Witbank coal with heat treatment was studied using small angle X-ray scattering, and it was found that D s changes systematically depending on the temperature and the heating rate.

Journal ArticleDOI
TL;DR: An approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height, is developed and the scale-dependent curvature is defined, and it is demonstrated that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained.
Abstract: We develop an approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height. Given a height profile of the surface we perform independent measurements of the fractal dimension of contour loops, and the exponent that characterizes their size distribution. Scaling formulas are derived, and used to relate these two geometrical exponents to the roughness exponent of a self-affine surface, thus providing independent measurements of this important quantity. Furthermore, we define the scale-dependent curvature, and demonstrate that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained. These nonlinear measures are used to characterize the morphology of computer generated Gaussian rough surfaces, surfaces obtained in numerical simulations of a simple growth model, and surfaces observed by scanning-tunneling microscopes. For experimentally realized surfaces the self-affine scaling is cut off by a correlation length, and we generalize our theory of contour loops to take this into account.

Journal ArticleDOI
TL;DR: In this paper, the fractal dimension of different kinds of music was analyzed in keeping with the time domain, and it was shown that the high information quantity is obtained in the high frequency domain.
Abstract: The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some different aspects of music well, the high information quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically proven that fractal dimension could distinguish different kinds of music very well: musics could be classified by their fractal dimensions.

Journal ArticleDOI
TL;DR: This study indicates that the backbone of bond TIP is loopless and completely different from that of site TIP, a problem which is relevant to multiphase flow in field-scale porous media, such as oil reservoirs and groundwater aquifers, as well as flow in rock fractures.
Abstract: We present the results of extensive Monte Carlo simulations of the invasion percolation model with trapping (TIP) with long-range correlations, a problem which is relevant to multiphase flow in field-scale porous media, such as oil reservoirs and groundwater aquifers, as well as flow in rock fractures. The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. We employ a highly efficient algorithm for simulating TIP, and a novel method for identifying the backbone of TIP clusters. Both site and bond TIP are studied. Our study indicates that the backbone of bond TIP is loopless and completely different from that of site TIP. We obtain precise estimates for the fractal dimensions of the sample-spanning cluster (SSC), the minimal path, and the backbone of site and bond TIP, and analyze the size distribution of the trapped clusters, in order to identify all the possible universality classes of TIP with long-range correlations. For site TIP with $Hg1/2$ the SSC and its backbone are compact, indicating a first-order phase transition at the percolation threshold, while the minimal paths are essentially straigth lines. For $Hl1/2$ the SSC, its backbone, and the minimal paths are all fractal with fractal dimensions that depend on the Hurst exponent H. The fractal dimension of the loopless backbone for bond TIP is much less than that of site TIP for any H.