Showing papers on "Fractal dimension published in 1991"

Fractals and Disordered Systems

Abstract: 1 Fractals and Multifractals: The Interplay of Physics and Geometry (With 30 Figures).- 1.1 Introduction.- 1.2 Nonrandom Fractals.- 1.3 Random Fractals: The Unbiased Random Walk.- 1.4 The Concept of a Characteristic Length.- 1.5 Functional Equations and Fractal Dimension.- 1.6 An Archetype: Diffusion Limited Aggregation.- 1.7 DLA: Fractal Properties.- 1.8 DLA: Multifractal Properties.- 1.8.1 General Considerations.- 1.8.2 "Phase Transition" in 2d DLA.- 1.8.3 The Void-Channel Model of 2d DLA Growth.- 1.8.4 Multifractal Scaling of 3d DLA.- 1.9 Scaling Properties of the Perimeter of 2d DLA: The "Glove" Algorithm.- 1.9.1 Determination of the l Perimeter.- 1.9.2 The l Gloves.- 1.9.3 Necks and Lagoons.- 1.10 Multiscaling.- 1.11 The DLA Skeleton.- 1.12 Applications of DLA to Fluid Mechanics.- 1.12.1 Archetype 1: The Ising Model and Its Variants.- 1.12.2 Archetype 2: Random Percolation and Its Variants.- 1.12.3 Archetype 3: The Laplace Equation and Its Variants.- 1.13 Applications of DLA to Dendritic Growth.- 1.13.1 Fluid Models of Dendritic Growth.- 1.13.2 Noise Reduction.- 1.13.3 Dendritic Solid Patterns: "Snow Crystals".- 1.13.4 Dendritic Solid Patterns: Growth of NH4Br.- 1.14 Other Fractal Dimensions.- 1.14.1 The Fractal Dimension dw of a Random Walk.- 1.14.2 The Fractal Dimension dmin ? 1/?? of the Minimum Path.- 1.14.3 Fractal Geometry of the Critical Path: "Volatile Fractals".- 1.15 Surfaces and Interfaces.- 1.15.1 Self-Similar Structures.- 1.15.2 Self-Affine Structures.- 1.A Appendix: Analogies Between Thermodynamics and Multifractal Scaling.- References.- 2 Percolation I (With 24 Figures).- 2.1 Introduction.- 2.2 Percolation as a Critical Phenomenon.- 2.3 Structural Properties.- 2.4 Exact Results.- 2.4.1 One-Dimensional Systems.- 2.4.2 The Cayley Tree.- 2.5 Scaling Theory.- 2.5.1 Scaling in the Infinite Lattice.- 2.5.2 Crossover Phenomena.- 2.5.3 Finite-Size Effects.- 2.6 Related Percolation Problems.- 2.6.1 Epidemics and Forest Fires.- 2.6.2 Kinetic Gelation.- 2.6.3 Branched Polymers.- 2.6.4 Invasion Percolation.- 2.6.5 Directed Percolation.- 2.7 Numerical Approaches.- 2.7.1 Hoshen-Kopelman Method.- 2.7.2 Leath Method.- 2.7.3 Ziff Method.- 2.8 Theoretical Approaches.- 2.8.1 Deterministic Fractal Models.- 2.8.2 Series Expansion.- 2.8.3 Small-Cell Renormalization.- 2.8.4 Potts Model, Field Theory, and ? Expansion.- 2.A Appendix: The Generating Function Method.- References.- 3 Percolation II (With 20 Figures).- 3.1 Introduction.- 3.2 Anomalous Transport in Fractals.- 3.2.1 Normal Transport in Ordinary Lattices.- 3.2.2 Transport in Fractal Substrates.- 3.3 Transport in Percolation Clusters.- 3.3.1 Diffusion in the Infinite Cluster.- 3.3.2 Diffusion in the Percolation System.- 3.3.3 Conductivity in the Percolation System.- 3.3.4 Transport in Two-Component Systems.- 3.3.5 Elasticity in Two-Component Systems.- 3.4 Fractons.- 3.4.1 Elasticity.- 3.4.2 Vibrations of the Infinite Cluster.- 3.4.3 Vibrations in the Percolation System.- 3.4.4 Quantum Percolation.- 3.5 ac Transport.- 3.5.1 Lattice-Gas Model.- 3.5.2 Equivalent Circuit Model.- 3.6 Dynamical Exponents.- 3.6.1 Rigorous Bounds.- 3.6.2 Numerical Methods.- 3.6.3 Series Expansion and Renormalization Methods.- 3.6.4 Continuum Percolation.- 3.6.5 Summary of Transport Exponents.- 3.7 Multifractals.- 3.7.1 Voltage Distribution.- 3.7.2 Random Walks on Percolation.- 3.8 Related Transport Problems.- 3.8.1 Biased Diffusion.- 3.8.2 Dynamic Percolation.- 3.8.3 The Dynamic Structure Model of Ionic Glasses.- 3.8.4 Trapping and Diffusion Controlled Reactions.- References.- 4 Fractal Growth (With 4 Figures).- 4.1 Introduction.- 4.2 Fractals and Multifractals.- 4.3 Growth Models.- 4.3.1 Eden Model.- 4.3.2 Percolation.- 4.3.3 Invasion Percolation.- 4.4 Laplacian Growth Model.- 4.4.1 Diffusion Limited Aggregation.- 4.4.2 Dielectric Breakdown Model.- 4.4.3 Viscous Fingering.- 4.4.4 Biological Growth Phenomena.- 4.5 Aggregation in Percolating Systems.- 4.5.1 Computer Simulations.- 4.5.2 Viscous Fingers Experiments.- 4.5.3 Exact Results on Model Fractals.- 4.5.4 Crossover to Homogeneous Behavior.- 4.6 Crossover in Dielectric Breakdown with Cutoffs.- 4.7 Is Growth Multifractal?.- 4.8 Conclusion.- References.- 5 Fractures (With 18 Figures).- 5.1 Introduction.- 5.2 Some Basic Notions of Elasticity and Fracture.- 5.2.1 Phenomenological Description.- 5.2.2 Elastic Equations of Motion.- 5.3 Fracture as a Growth Model.- 5.3.1 Formulation as a Moving Boundary Condition Problem.- 5.3.2 Linear Stability Analysis.- 5.4 Modelisation of Fracture on a Lattice.- 5.4.1 Lattice Models.- 5.4.2 Equations and Their Boundary Conditions.- 5.4.3 Connectivity.- 5.4.4 The Breaking Rule.- 5.4.5 The Breaking of a Bond.- 5.4.6 Summary.- 5.5 Deterministic Growth of a Fractal Crack.- 5.6 Scaling Laws of the Fracture of Heterogeneous Media.- 5.7 Hydraulic Fracture.- 5.8 Conclusion.- References.- 6 Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts (With 8 Figures).- 6.1 Introduction.- 6.2 The Electrode Problem and the Constant Phase Angle Conjecture.- 6.3 The Diffusion Impedance and the Measurement of the Minkowski-Bouligand Exterior Dimension.- 6.4 The Generalized Modified Sierpinski Electrode.- 6.5 A General Formulation of Laplacian Transfer Across Irregular Surfaces.- 6.6 Electrodes, Roots, Lungs,.- 6.7 Fractal Catalysts.- 6.8 Summary.- References.- 7 Fractal Surfaces and Interfaces (With 27 Figures).- 7.1 Introduction.- 7.2 Rough Surfaces of Solids.- 7.2.1 Self-Affine Description of Rough Surfaces.- 7.2.2 Growing Rough Surfaces: The Dynamic Scaling Hypothesis.- 7.2.3 Deposition and Deposition Models.- 7.2.4 Fractures.- 7.3 Diffusion Fronts: Natural Fractal Interfaces in Solids.- 7.3.1 Diffusion Fronts of Noninteracting Particles.- 7.3.2 Diffusion Fronts in d = 3.- 7.3.3 Diffusion Fronts of Interacting Particles.- 7.3.4 Fluctuations in Diffusion Fronts.- 7.4 Fractal Fluid-Fluid Interfaces.- 7.4.1 Viscous Fingering.- 7.4.2 Multiphase Flow in Porous Media.- 7.5 Membranes and Tethered Surfaces.- 7.6 Conclusions.- References.- 8 Fractals and Experiments (With 18 Figures).- 8.1 Introduction.- 8.2 Growth Experiments: How to Make a Fractal.- 8.2.1 The Generic DLA Model.- 8.2.2 Dielectric Breakdown.- 8.2.3 Electrodeposition.- 8.2.4 Viscous Fingering.- 8.2.5 Invasion Percolation.- 8.2.6 Colloidal Aggregation.- 8.3 Structure Experiments: How to Determine the Fractal Dimension.- 8.3.1 Image Analysis.- 8.3.2 Scattering Experiments.- 8.3.3 Sacttering Formalism.- 8.4 Physical Properties.- 8.4.1 Mechanical Properties.- 8.4.2 Thermal Properties.- 8.5 Outlook.- References.- 9 Cellular Automata (With 6 Figures).- 9.1 Introduction.- 9.2 A Simple Example.- 9.3 The Kauffman Model.- 9.4 Classification of Cellular Automata.- 9.5 Recent Biologically Motivated Developments.- 9.A Appendix.- 9.A.1 Q2R Approximation for Ising Models.- 9.A.2 Immunologically Motivated Cellular Automata.- 9.A.3 Hydrodynamic Cellular Automata.- References.- 10 Exactly Self-similar Left-sided Multifractals with new Appendices B and C by Rudolf H. Riedi and Benoit B. Mandelbrot (With 10 Figures).- 10.1 Introduction.- 10.1.1 Two Distinct Meanings of Multifractality.- 10.1.2 "Anomalies".- 10.2 Nonrandom Multifractals with an Infinite Base.- 10.3 Left-sided Multifractality with Exponential Decay of Smallest Probability.- 10.4 A Gradual Crossover from Restricted to Left-sided Multifractals.- 10.5 Pre-asymptotics.- 10.5.1 Sampling of Multiplicatively Generated Measures by a Random Walk.- 10.5.2 An "Effective" f(?).- 10.6 Miscellaneous Remarks.- 10.7 Summary.- 10.A Details of Calculations and Further Discussions.- 10.A.1 Solution of (10.2).- 10.A.2 The Case ?min = 0.- 10.B Multifractal Formalism for Infinite Multinomial Measures, by R.H. Riedi and B.B. Mandelbrot.- 10.C The Minkowski Measure and Its Left-sided f(?), by B.B. Mandelbrot.- 10.C.1 The Minkowski Measure on the Interval [0,1].- 10.C.2 The Functions f(?) and f?(?) of the Minkowski Measure.- 10.C.3 Remark: On Continuous Models as Approximations, and on "Thermodynamics".- 10.C.4 Remark on the Role of the Minkowski Measure in the Study of Dynamical Systems. Parabolic Versus Hyperbolic Systems.- 10.C.5 In Lieu of Conclusion.- References.

Characterizing the lacunarity of random and deterministic fractal sets

Abstract: The notion of lacunarity makes it possible to distinguish sets that have the same fractal dimension but different textures. In this paper we define the lacunarity of a set from the fluctuations of the mass distribution function, which is found using an algorithm we call the gliding-box method. We apply this definition to characterize the geometry of random and deterministic fractal sets. In the case of self-similar sets, lacunarity follows particular scaling properties that are established and discussed in relation to other geometrical analyses.

Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. Theory

Abstract: Recent efforts to characterize soil water properties in terms of porosity and particle-size distribution have turned to the possibility that a fractal representation of soil structure may be especially apt. In this paper, we develop a fully self-consistent fractal model of aggregate and pore-space properties for structured soils. The concept underlying the model is the representation of a soil as a fragmented fractal porous medium. This concept involves four essential components: the mathematical partitioning of a bulk soil volume into self-similar pore- and aggregate-size classes, each of which is identified with a successive fragmentation step; the definition of a uniform probability for incomplete fragmentation in each size class; the definition of fractal dimensions for both completely and incompletely fragmented porous media; and the definition of a domain of length scales across which fractal behavior occurs. Model results include a number of equations that can be tested experimentally: (i) a fractal dimension ≤3; (ii) a decrease in aggregate bulk density (or an increase in porosity) with increasing aggregate size; (iii) a power-law aggregate-size-distribution function; (iv) a water potential that scales as an integer power of a similarity ratio; (v) a power-law expression for the water-retention curve; and (vi) an expression for hydraulic conductivity in terms of the conductivities of single-size arrangements of fractures embedded in a regular fractal network. Future research should provide experimental data with which to evaluate these predictions in detail.

Fractal growth of two-dimensional islands: Au on Ru(0001).

Abstract: The two-dimensional growth of Au on Ru(0001) in the submonolayer range has been investigated with scanning tunneling microscopy. Upon deposition at room temperature, highly dendritic islands of one layer thickness grow on large Ru terraces. These irregular island shapes are removed upon annealing to 650 K. The dendritic islands exhibit a fractal character, and a dimensional analysis yields a fractal dimension of 1.72\ifmmode\pm\else\textpm\fi{}0.07. The results are in quantitative agreement with a two-dimensional diffusion-limited-aggregation growth mechanism.

Fractal dimensions of aggregates determined from steady-state size distributions

Abstract: Aggregates formed by Brownian motion, shear coagulation, and differential sedimentation have fractal geometries. In order to model coagulation of fractal aggregates, we have derived a set of collision functions containing a fractal dimension for use in a general coagulation equation. These collision functions predict greater collision frequencies than models based on aggregates with Euclidean properties. Assuming only one mechanism of aggregate formation is dominant for a range of particle sizes, we also incorporated a fractal dimension in a dimensional analysis of steady-state particle-size distributions

Applications of fractal analysis to physiology

Abstract: This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.

Random Fractals: characterization and measurement

Abstract: Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A brief mathematical characterization of random fractals is presented with emphasis on variations of Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized.

Structure and self‐similarity in silty and sandy soils: the fractal approach

Abstract: SUMMARY Soil structure was studied using the concept of fractals and related to soil texture and aggregate properties such as surface charges and aggregate stability. The mass and porosity fractal dimensions (Dm and Dp) of silty and sandy soils were determined on in situ soils using a variety of soil sections (thin, very-thin and ultra-thin), by image analysis on a continuous scale from m to 10−9 to 10−1m. Surface fractal dimensions (Ds) of these soils were determined on < 2 mm air-dried samples using mercury porosimetry and the fractal cube generator model. The results suggest that soils are not pore fractals but mass and surface fractals with Dm= 1.1 Ds when the dimension of the embedding Euclidean space d is 3. The soil structures could possibly be described by fractal diffusion-limited aggregation with complex interconnected aggregates or by fractal cluster–cluster aggregation models. As a preliminary conclusion, the fractal approach appears to be a potentially useful tool for understanding the underlying mechanisms in the creation or destruction of soil structure.

Multifractal wave functions at the Anderson transition

Abstract: Electronic wave functions in disordered systems are studied within the Anderson model of localization. At the critical disorder in 3D we diagonalize very large (103 823\ifmmode\times\else\texttimes\fi{}103 823) secular matrices by means of the Lanczos algorithm. On all length scales the obtained strong spatial fluctuations of the amplitude of the eigenstates display a multifractal character, reflected in the set of generalized fractal dimensions and the singularity spectrum of the fractal measure. An analysis of 1D systems shows multifractality too, in contrast to previous claims.

Theory and numerical simulation of optical properties of fractal clusters.

Abstract: : Fractals, as introduced by Benoit Mandelbrot over ten years ago, are scale self-similar mathematical objects possessing nontrivial geometrical properties. There exist various physical realizations of fractals, and here we shall consider what we believe to be one of the most important such realizations, namely, fractal; clusters. Attention will be paid mainly to their optical properties. A fractal cluster is a system of interacting material particles called monomers. Theory of linear optical properties of fractal clusters is developed in this report. The theory is based upon the exact properties of dipole polarizability and assumption of the existence of scaling for the dipole excitations (eigenstates) of the fractal. This assumption is self-consistently validated by the results of the theory and is also confirmed by numerical stimulation in the framework of the Monte-Carlo method. Using exact relations and the scaling requirements, it is shown that the fractal absorption and density of eigenstates scale with the same exponent d sub o-1.

The fractal structure of soil aggregates: its measurement and interpretation

Abstract: SUMMARY The theory of fractal geometry is presented with reference to soil structure. Recent work on relating fractal structure to pore structure in soils is reviewed. It is suggested that the connection made in previous work between the fractal dimension and soil moisture retention curves is based on simplified assumptions that complicate the interpretation of results. A simple method for estimating the fractal dimension, D, of natural aggregates which circumvents some of these assumptions is presented. Preliminary results of aggregates from soils under different management systems show that, for the soils examined, D ranged from 2.75 to 2.93. The use of D to quantify heterogeneity in soil is explored.

Analysis of texture in macroradiographs of osteoarthritic knees using the fractal signature.

Abstract: Texture of regions of macroradiographs (x5) of six normal and five osteoarthritic knee joints, taken on a high resolution microfocal x-ray unit, are examined using mathematical morphology. Radiographs of bones are two-dimensional projections of attenuation coefficient through the three-dimensional (3D) joint structure. Visible texture represents the summation of the attenuation from numerous thin plates of bone. Where there is no organization in the trabeculae, resultant radiographs approximate a fractal surface. Varying structuring element size in mathematical morphology allows estimation of fractal dimension over a range of resolution. Variation of fractal dimension with resolution, the fractal signature, indicates how images deviate from fractal surfaces. By correct choice of structuring element, a texture analysis method using the fractal signature has been developed, tolerant to changes in image acquisition and digitization. Texture in regions of radiographs of normal tibia approximates a fractal surface with dimension 2.8 as does vertical structure in arthritic patients. In osteoarthritic knee joints, horizontal tibial trabeculae thicken. Horizontal structure in the tibia on radiographs of arthritic patients deviates from the fractal model. This is indicated by peaks in the fractal signature whose height and position match a visual assessment of the degree of arthritic change.

Fractal Theory Applied to Soil Aggregation

Abstract: Recent advances in fractal theory may be applicable to the characterization of soil structure. This study explored two such applications. Aggregates were assumed to be approximated by cubes of constant dry density. Under this condition, the fractal dimension, D, provides a measure of fragmentation. The value of D increases with increasing fragmentation. The influence of energy input (wet sieving) on the D of aggregates from a Conestogo silt loam soil (finesilty, mixed, mesic Aquic Eutrocrept) was investigated. Aggregates were obtained from five cropping treatments, ranging from 15 yr of continuous corn (Zea mays L.) to 15 yr of continuous bromegrass (Bromus inermis Leyss.). The value of D was estimated from cumulative size-frequency distribution data plotted on a log-log scale. Computed values of D ranged from 2.51 to 3.52. Energy input, cropping history, and their interaction all had a significant effect on D. Fragmentation increased with energy input and time under corn production. Relations between D and the breakdown of individual aggregates were investigated. A scale-invariant breakdown model was developed and tested. The model permits calculation of apparent probabilities of failure, P, as a function of aggregate size, x. Stepwise multiple regression analysis selected the rate of change in P with x as the best predictor of D following energy input. Fractal theory offers potential for modeling aggregate breakdown, as well as characterizing the degree of fragmentation.

Fractal Network Model for Contact Conductance

Abstract: The topography of rough surfaces strongly influences the conduction of heat and electricity between two surfaces in contact. Roughness measurements on a variety of surfaces have shown that their structure follows a fractal geometry whereby similar images of the surface appear under repeated magnification. Such a structure is characterized by the fractal dimension D, which lies between 2 and 3 for a surface and between 1 and 2 for a surface profile. This paper uses the fractal characterization of surface roughness to develop a new network model for analyzing heat conduction between two contacting rough surfaces. The analysis yields the simple result that the contact conductance h and the real area of contact A{sub t} are related as h {approximately} A{sub t}{sup D/2} where D is the fractal dimension of the surface profile. Contact mechanics of fractal surfaces has shown that A{sub t} varies with the load F as A{sub t} {approximately} F{sup {eta}} where {eta} ranges from 1 to 1.33 depending on the value of D. This proves that the ocnductance and load are related as h {approximately} F{sup {eta}D/2} and resolves the anomaly in previous investigations, which theoretically and experimentally obtained different values for the load exponent. Themore » analytical results agreed well with previous experiments although there is a tendency for overprediction.« less

Fractal relation of mainstream length to catchment area in river networks

Abstract: Institute of Hydraulics, University of Genoa, ItalyMandelbrot's (1982) hypothesis that river length is fractal has been recently substantiated byHjelmfelt (1988) using eight rivers in Missouri. The fractal dimension of river length, d, is derived herefrom the Horton's laws of network composition. This results in a simple function of stream length andstream area ratios, that is, d = max (1, 2 log RL/\og RA). Three case studies are reported showing thisestimate to be coherent with measurements of d obtained from map analysis. The scaling properties ofthe network as a whole are also investigated, showing the fractal dimension of river network, D, todepend upon bifurcation and stream area ratios according to D = min (2, 2 log RB/log RA). Theseresults provide a linkage between quantitative analysis of drainage network composition and scalingproperties of river networks.

From viscous fingering to viscoelastic fracturing in colloidal fluids.

Abstract: Fluid-displacement experiments in Hele-Shaw cells filled with a viscoelastic fluid show a novel transition between a viscous fingering (VF) regime producing fractal patterns of ``fingers'' and a viscoelastic facturing (VEF) regime producing fractal patterns of ``cracks.'' VEF patterns are characterized by branching angles of 90\ifmmode^\circ\else\textdegree\fi{} with respect to the main crack, behind the tip, and by a lower fractal dimension than VF. The transition is controlled by several parameters, including the Deborah number and the system deformability.

Multifractal analysis of spatial distribution of microearthquakes in the Kanto region

Abstract: SUMMARY The spatial distribution of earthquakes is a fractal, which is characterized by a fractal dimension. However, if a spatial distribution has a heterogeneous fractal structure, a single value of fractal dimension [e.g. Do (capacity dimension) or D2 (correlation dimension)] is not enough to characterize it. From a multifractal viewpoint, we analysed the spatial distribution of microearthquakes in the Kanto region by using a local density function. Generalized dimensions, D,, of the spatial distribution were calculated from the slopes of generalized correlation integrals, C,(r) versus distance r, on a log-log plot, examining the self-similarity of the spatial distribution of microearthquakes. Self-similar structures are held well at scales from 1.26 to 12.6 km. Our results suggest that the spatial distribution of microearthquakes in the Kanto region is not a homogeneous fractal structure but a heterogeneous one with generalized dimensions D2 = 2.2 h D3 2 - .? D, = 1.7. The value of D,, the lower limit of fractal dimension, is the fractal dimension of the most intensive clustering in the heterogeneous fractal set. The fractal dimension of the most intensive clustering of microearthquakes in the Kanto region is 1.7.

Complex fractal dimension of the bronchial tree.

Abstract: The architecture of the mammalian lung has been shown to be correctly described using a fractal model with a complex dimension, related to a Cantor set with random errors, for four different species. Here we provide an interpretation of that model that has implications for biological evolution. We argue that fractals are more error tolerant than other structures and therefore have an evolutionary advantage.

Some results on the behavior and estimation of the fractal dimensions of distributions on attractors

Abstract: The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimensionσ(x) and the generalized Renyi dimensionsD(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension ν and information dimensionσ. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of ν andσ we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure forσ which corrects for an important bias term (the local measure density) and provides confidence intervals forσ. The general applicability of this method is illustrated with various numerical examples.

Turbulent premixed combustion modelling using fractal geometry

Abstract: The fractal geometry concept has been applied to the problem of turbulent premixed flame propagation This approach has been based on the fact that turbulent surfaces are wrinkled over a wide range of scales, and experimental evidence shows that the scales are self-similar Self-similarity of the scales over a wide range is the distinctive characteristics of fractal objects A new model, derived using the fractal geometry, for the wrinkled flames (or flames-sheets) regime has been proposed The distinct feature of the present formulation is the adoption of a variable inner cutoff scale as a function of turbulent and molecular diffusivities With a fractal dimension of 7/3 and an outer cutoff approximating the integral length scale of turbulence, our analysis has yielded u t /u 1 ∞(u′/u 1 ) 1/2 Re L 1/4 This model suggests a wrinkled flame structure influenced by a turbulent length scale as well as the turbulence intensity Comparison of the predictions of the model and the experimental data from a variety of rigs shows excellent agreement despite the uncertainties involved in the measurement of the turbulence parameters, propagation velocities, and the intuitive assumptions made in the theories

Fractal fracture of single crystal silicon

Abstract: The quantitative description of surfaces that are created during the fracture process is one of the fundamental issues in materials science. In this study, single crystal silicon was selected as a model material in which to study the correlation of fracture surface features as characterized by their fractal dimension for two different orientations of fracture with the fracture toughness of the material as measured using the strength-indentation and fracture surface analysis techniques. The fracture toughness on the {110} fracture plane of single crystal silicon was determined to be 1.19 MPa m1/2 for the {100} tensile surface and 1.05 MPa m1/2 for the {110} tensile surface using the indentation-strength three-point bending method. The fracture surface features of these two orientations are correspondingly different. Within our limitations of measurements (1–100 μm), the fractal dimension appeared different in different regions of the fracture surface. It has a higher value in the branching region and a lower value in the pre-branching and post-branching regions. The fractal dimensions are about the same in the pre-branching regions and post-branching region for these two orientations (D = 1.01 ± 0.01), i.e., nearly Euclidean (smooth); but the fractal dimensions are higher in the branching region for these two orientations. The fractal dimension is 1.10 ±0.4 for the {100} tensile surface and is 1.04 ±0.3 for the {110} tensile surface. If we select the highest dimension on a surface to represent the dimensionality of the surface, then a material with a higher fracture toughness has a higher fractal dimension in the branching region.

Non-equilibrium critical behaviour on fractal lattices

Abstract: A simple one-component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1(Df(2 Results from time-dependent simulations show a change in the value of the critical exponents with Df Two of the exponents interpolate very nicely between the values in one and two dimensions

Adsorption isotherm on fractally porous materials

Abstract: The effects of fractal geometry on adsorption isotherms have been intensively researched in recent The fractal concept has been used in studies on rough surfaces, microporous solids, and mesoporous (or macroporous) materials. The concept of fractals has proven successful in applying to a wide variety of complex surfaces. The fractal model, when applied to isotherm adsorption, leads to simpler interpretations of experimental data than do traditional ideas based on isolated deviation from planar geometry. The manner in which adsorption isotherms depend on fractal dimension D of a fractal surface is a currently interesting question. This question may be traced to the Hill isotherm equation in the multilayer adsorption regime819

Wave interactions with generalized Cantor bar fractal multilayers

Abstract: The reflection and transmission properties of finely divided fractal layers are investigated and characterized. The results for electromagnetic or optical waves normally incident upon generalized Cantor bar fractal multilayers are found for various fractal dimensions and stages of growth. A new exact self‐similar algorithm is described which makes use of the self‐similarity of the structures to clearly display the underlying physics. This fractal computational scheme provides the reflection and transmission coefficients for fractally distributed layers with extreme economy when compared to traditional approaches. Finally, a method for extracting fractal descriptors from scattered data is discussed. High‐ and low‐ frequency regimes are examined.

The size, shape and dimension of urban settlements

Abstract: In this paper, we propose a scale theory of urban form and growth which enables us to consistently explain and estimate relationships between urban population size, area, field and boundary length for a system of settlements. Our approach is based on a synthesis of allometry and fractal growth theory, and the associated relationships are uniquely specified by dimensional parameters whose values vary from 1 to 2, from the line to the plane. The theory assumes that the form of settlements is tentacular and that the population density of these forms is constant with respect to their size. After the theory has been presented, four relationships - two allometric, relating populations and boundaries (or envelopes) to urban areas, and two fractal, relating the same variables to the urban field size - are estimated for some 70 settlements which compose the urban system in the English County of Norfolk. The hypothesized values of the dimensions characterizing these four relationships are confirmed by regression estimates and these results are given further strength when the same relations are re-estimated for various subsets of settlements in the Norfolk urban system. We conclude that the geometric form of the settlements system is consistent with the model we have adopted, that population density is constant at all scales, and that urban boundaries have a degree of irregularity measured by a fractal dimension similar to that conventionally assumed for coastlines. Finally, we suggest directions for further research.