FRACTAL PROPERTIES OF HYPERBOLIC CURVES

In this paper, we study the fractal properties of the hyperbolic curve introduced by J. Belair ([Be]). We obtain some conditions of nowhere-differentiability of this kind of curves and its Bouligand dimension, and find a class of curves which are almost everywhere differertiable and have Bouligand dimensions being greater than one simultaneously.

Effects of Microwave Heating on Pore Fractal Properties of Blast Furnace Sludge

Effects of microwave heating on the pore fractal properties of blast furnace sludge （BFS） and relative mechanism were studied. The results show that the morphology features of iron bearing sinter and coke particles, which are the main constituents of the BFS, were remarkably changed by microwave heating. The porosity, surface roughness and specific surface area of modified particle surface all increased obviously. Combining with fractal meth-od called Sierpinski model, the fractal dimensions of sinter, coke and others increased from 2.35, 2.24 and 2.58 to 2.65, 2.44 and 2.61 respectively, after modification by microwave heating. The results predicted that the reaction mechanism of microwave heating for BFS is related to two aspects. Different mineral phases existed in BFS particles incline to dissociate each other due to their different microwave absorbability~ some recombination or reconstruction of matters or structure leads to structure defects, which have great influences on the surface morphology characteris-tics and chemical properties. The research indicated that fractal dimension can be used as an effective factor for quan-titative analysis of the pore changes in morphology of the sludge. Furthermore, it is helpful for separation and ex- traction of valuable constituent from BFS.

Fractal characterization and frequency properties of near-fault ground motions

This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T. Meanwhile, the box dimension of the velocity histories Dye is negatively correlated with the characteristic period T and improved characteristic period Tgi.

Fractal analysis of canopy architectures of Acacia angustissima, Gliricidia sepium, and Leucaena collinsii for estimation of aboveground biomass in a short rotation forest in eastern Zambia

A study was conducted at Msekera Regional Agricultural Research Station in eastern Zambia to （1） describe canopy branching properties of Acacia angustissima, Gliricidia sepium and Leucaena collinsii in short rotation forests, （2） test the existence of self similarity from repeated iteration of a structural unit in tree canopies, （3） examined intra-specifie relationships between functional branching characteristics, and （4） determine whether allometric equations for relating aboveground tree biomass to fractal properties could accurately predict aboveground biomass. Measurements of basal diameter （Din0） at 10em aboveground and total height （H）, and aboveground biomass of 27 trees were taken, but only nine trees representative of variability of the stand and the three species were processed for functional branching analyses （FBA） of the shoot systems. For each species, fractal properties of three trees, includ- ing fractal dimension （Dfract）, bifurcation ratios （p） and proportionality ratios （q） of branching points were assessed. The slope of the linear re- gression ofp on proximal diameter was not significantly different （P 〈 0.01） from zero and hence the assumption that p is independent of scale, a pre-requisite for use of fraetal branching rules to describe a fraetal tree canopy, was fulfilled at branching orders with link diameters 〉1.5 cm. The proportionality ration q for branching patterns of all tree species was constant at all scales. The proportion of q values 〉0.9 （fq） was 0.8 for all species. Mean fraetal dimension （Df^ct） values （1.5-1.7） for all species showed that branching patterns had an increasing magnitude of intricacy. Since Dfraet values were 〉1.5, branching patterns within species were self similar. Basal diameter （D10）, proximal diameter and Dfraet described most of variations in aboveground biomass, suggesting that allometric equa- tions for relating aboveground tree biomass to fractal properties could accurately predict aboveground biomass. Thus, assessed Acacia, Gliri- cidia and Leucaena trees were fractals and their branching propertiescould be used to describe variability in size and aboveground biomass.

HOLDER PROPERTY OF FRACTAL INTERPOLATION FUNCTION

The purpose of this paper is to prove a Holder property about the fractal interpolation function L(x), ω(L,δ)=O(δ~α), and an approximate estimate |f-L|≤2{α(h)+||f||/1-h^(2-D)·h^(2-D)}, where D is a fractal dimension of L(x).

STUDY ON IMAGE EDGE PROPERTY LOCATION BASED ON FRACTAL THEORY

A novel approach of printed circuit board(PCB)image locating is presented Based on the rectangle mark image edge of PCB,the featur es is used to describe the image edge and the fractal properby of image edge is analyzed It is proved that the rectangle mark image edge of PCB has some fractal features A method of deleting unordinary curve noise and compensating the length of the fractal curve is put forward,which can get the fractal dimension value from one complex edge fractal property curve The relation between the dim ension of the fractal curve and the turning angle of image can be acquired from an equation,as a result,the angle value of the PCB image is got exactly A real image edge analysis result confirms that the method based on the fractal theory is a new powerful tool for angle locating on PCB and related image area.

Lyapunov exponent and dimension of the strange attractor of elastic frictional system

In this paper, we have calculated the spectrum of Lyapunov exponent of the strange attractor for a single degree of freedom in elastic system with a two-state variable friction law via the method advanced by Wolf. The system is expressed by the following dimensionless equation:where,and f are dimensionless state variable, logarithm slip velocity and frictional stress, respectively;β1,β2,ρ,and K are dimensionless system parameters.The state of this system is chaotic when dimensionless parameters are β1=1. 00, β2=0. 84, ρ=0. 048, =0. 198 85, K=0. 0685.The Lyapunov exponent spectrum of its strange attractor has been calculated as follows:λ1=0. 0179, λ2=0, λ3=-0. 1578The dimension of this strange attractor has also been calculated as DL=D0=2.11where DL and D0 denote Lyapunov dimension and Kolmogorov dimension respectively.