Graphs, fractal dimension

FIGURE 17.4 USAXS and SALS results for samples Al, A2, and A3 described in the text. Each sample shows four structural levels with the i g for some of the levels indicated in the graph. The power-law value for the second level, corresponding to the mass-fractal dimension, d(, is also indicated. 

Hence Fig 19 represents, in fact five correlations Other many more relations are available [64] Here, in an analogy with self-similarity the concept of fractal dimensions has been extended to graphs by... 

It is considered that is determined by the properties of the supermolecular structure of epoxy polymers, namely, by the degree of local order in them. This assumption arises due to the well-known succession of relationships between the diffusion process and d [28], between the diffusion process and the fluctuation free volume [119], and between and the cluster structure [120]. Therefore, the fractal dimension of the cluster structure of crosslinked polymers [105, 110], estimated from Equation (11.27), is taken to be d. Since the D value is determined for an Euclidean space with d = 2, it was also assumed in Equation (11.27) that d = 2. The d value was found for a three-dimensional space therefore, a graph [121] for converting d from three- to two-dimensional space was used in subsequent calculations. The D values were further calculated from Equation (11.8) in terms of the AO and AS hypotheses. Figure 11.13 shows the dependences of D on calculated from Equations (11.8) and (11.39). The dependences of D on and the absolute magnitudes of D are in good agreement, especially when the AO hypothesis is used. 

The principle of the box counting method mainly involves an iteration operation to an initial square, whose area is supposed to be 1 and which covers the entire graph. The initial square is divided into four sub-squares and so on. After the n times operations, the number of sub-squares, which contain the discrete points of the profile graph are counted and the length L of the profile is approximately obtained. Then the fractal dimension is calculated as D=l+log L/(n.log2). 

Figure 13.2 Determination of the fractal dimension of a surface using the box-coimting method. The graph shows the number of boxes of size r necessary to approximate the measured surface profile. A planar surface would be precisely described independent of box size and the graph would be linear with slope -1. On a rough stuface, the profile would be sufficiently resolved only within certain boundaries, which are denoted I = loer. andL = logr. The fractal dimension is derived from the slope within these boundaries.

Figure 9.8. Data generated by mercury intrusion porosimetry can be reinterpreted using the concepts of fractal geometry, by replotting the data on log-log scales. The slopes of the linear regions of the resulting graph can be viewed as fractal dimensions in data space, a) Traditional presentation of mercury intrusion porosimetry data, b) Data of (a) plotted on log-log scales.

Fractal-Like Dimensions of Graphs 3-2 Correlation with Physical Properties 3-3 Ordering of Structures ... 

Second, fractals have noninteger dimensions. This means that they are entirely different from the graphs of fines and conic sections that we have in fundamental Euclidean geometry. The... 

In case of reaction elapsion in the Euclidean spaces the value D is equal to dimension of this space d and for fractal spaces D is accepted equal to spectral dimension ds [6], By graphing p.4=(l-0 (where Q is conversion degree) as a function of t in double logarithmic coordinates the value D from the slope of these graphs can be determined. It was found, that the mentioned graphs fall apart on two linear parts at f<100 min with small slope and at f>100 min the slope essentially increases. In this case the value varies within the limits 0.069-3.06. Since the considered reactions are elapsed in Euclidean space, that is pointed by a linearity of kinetic curves Q-t, this means, that the reetherification reaction elapses in specific medium with Euclidean dimension d, but with connectivity degree, characterized by spectral dimension c4, typical for fractal spaces [5]. 

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