Fractal dimension from

Several algorithms have been used to determine surface fractal dimension from SPM images. The most popular ones are... 

Several simple relations have been proposed for the determination of the fractal dimensions from the results of such experiments as gas molecular probe method, transmission electron microscopy (TEM), small-angle X-ray scattering, neutron scattering, and laser light scattering.63,66,113 116 Among those techniques, gas molecular probe method and image analysis method have been widely used for the calculation of the surface... 

Therefore, it appears likely that the approach considering the solid volume of primary aggregates, as evaluated from the two-dimensional cross-section area by Eq. (12), leads to an overestimation of the mass fractal dimension. A more realistic estimate is obtained with Eq. (13). By referring to Eq. (12), the data obtained by Herd et al. [108] show a successively increasing value of the mass fractal dimension from to df 2.8 with increas-... 

Box counting method is commonly used to obtain mass fractal dimension from an aggregate s projected area. 

Fig. 12 The variation of apparent permeability coefficient with boundary fractal dimension. (From Ref. " f)...

Fig. 1 (A) Profiles of fumed pigment fine particles where the fractal dimension, 5, describes the ruggedness of the profile. (B) Four lines with identical topological dimensions with varying degrees of ruggedness as seen from their corresponding fractal dimensions. (From Ref. l)...

Fractal dimensions from decay time measurements. 

Perfect, E. 1999. Estimating soil mass fractal dimension from water retention curves. Geoderma 88 221-232. 

Nevertheless, the determination of the fractal dimension from a data set thought to be chaotic is often of interest. A number of different dimensions exist in the literature, including the Hausdorff dimension, the information dimension, the correlation dimension, and the Lyapunov dimension. Which of these is the true fractal dimension Of the ones in this list, the information dimension, Di, has the most basic and fundamental definition, so we often think of it as the true fractal dimension. Because the information dimension is impractical to calculate directly, however, most investigators have taken to finding the correlation dimension, Dq, as an estimate of the fractal dimension. Grassberger and Procaccia published a straightforward and widely used algorithm for the calculation of the correlation dimension. On the other hand, the Lyapunov... 

Calculation of fractal dimensions from sorptometry and porosimetry data... 

Figure 3.2 Example aggregates drawn from classes having mass fractal dimension (from top to bottom) of 1.74, 2.11 and 2.41. Each aggregate contains 2000 primary particles. Note the increasing compactness as the mass fractal dimension increases.

This relationship was used to calculate aggregate fractal dimensions from previous settling rate studies. However, this model was not found to describe the settling behaviour of fractal aggregates adequately due to the perhaps unwarranted assumption that and / are the only variables affecting U, which implies that the drag coefficient will be the same for fractals as for impermeable spheres in the laminar flow region 1). 

However, the porosimetric method does not directly measure the initial area So. To evaluate So, Kozak et al. [40] modified Neimark s approach based on the application of Equation (6.14). In addition to Neimark s approach, other alternative approaches have been developed that are aimed at evaluating the fractal dimension from mercury intrusion data [47],... 

In case of clustered events, the fractal dimension of the events is used to find linear, planar, or spatial clusters. The analysis of clusters aims at finding planar clusters, because planar clusters are the most hazardous structures compared with linear or spatial ones. A drop of the fractal dimension from 3 to 2 within a certain time period indicates a concentration and focusing of an initially diffuse spatial event distribution in a plane which forms the nucleus of a fracture. 

Fractal curves generally have a dimension, D, between 1 and 2, fractal surfaces between 2 and 3. The value of D can be worked out knowing the mathematical procedme by which the fractal is generated. It is unlikely that such a procedure will be known for a fractal observed in nature. The evaluation of fractal dimension from experimental measurements is discussed in Fractals and surface roughness. 

Fig. 8.8 Examples of von Koch curves for three different fractal dimensions (From ref. [317], copyright (1985), with permission from Elsevier)...

Fig. 4.6 Fractal dimension from growth of Xg = 2 x Rg, from fragmentation, from box-counting, and from pair-correlation versus aggregation number, each time averaged from 100 aggregates left DLCA aggregates, right RLCA aggregates...

Mean values from four measurements of profile lines were used to calculate the fractal dimension from formula (10.29). Specimens of dolomite and gravel concretes were selected for fractal analysis from specimens with different values of... 

In Figure 5-3, the suscephbility /(< )/( -I-1) is reported. Therefore, from equation (5-7) and from the hypothesis that C(independent measurement of the fractal dimension from SANS experiments, D= 2.4 (Vacher, 1988), and of the spectral dimension by INS experiments, d= 1.3 0.1 (Vacher, 1989, 1990 Courtens, 1990), aUowed the determination of the scaling exponent of C(Different models have been proposed for the calculahon of C(ft>), based on the scaling properties ofthe local strain in fractals (Boukenter,... 

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