Self-similarity properties

Power-law expressions are found at all hierarchical levels of organization from the molecular level of elementary chemical reactions to the organismal level of growth and allometric morphogenesis. This recurrence of the power law at different levels of organization is reminiscent of fractal phenomena. In the case of fractal phenomena, it has been shown that this self-similar property is intimately associated with the power-law expression [28]. The reverse is also true if a power function of time describes the observed kinetic data or a reaction rate higher than 2 is revealed, the reaction takes place in fractal physical support. 

Increasing time of exposure of the recording film results in the appearance of more and more points throughout space in an eventually dense spacefilling array. All points satisfy the same symmetry and self-similarity properties. 

Apparently from the plots of log10 K (Fig. 47a) and log10 p (Fig. 47b) of the fractal ensemble versus the iteration step, number n, all these elastic properties behave like fractals before an eventual levelling off. The latter is obviously associated with the upper limit of fractal-like asymptotics, above which the elastic properties of a system are no longer p dependent on the scale—that is, on the iteration number (the loss of the self-similarity property occurs at iteration step n q = logc/log/o which defines the correlation length c at the given concentration, p). 

The essential novel ingredient of cluster science is that the objects under study should retain self-similar properties as their size is increased. This is the concept of stackability [658]. Although clusters are finite in size, they may be made to grow indefinitely by stacking one more atom of the element from which they are composed. This may be taken as the defining property of clusters and distinguishes them from ordinary molecules. 

These relationships in combination with the Eq. (8) follow from the macromolecular coil self-similarity property, which is obligatory one for any fractal [40], namely, two sizes of the same coil and are scaled with AZM change equally, that is, they have the same scaling exponent 1/... 

The particular feature of a surface fractal is a dimensionality which does not correspond to a whole number, for example 1.75 instead of 2. This is due to the extreme division of the surface. Surface fractals have the self-similarity property in the sense that their geometric features do not change if the surface is magnified. Mathematically, surface self-similarity... 

The self-similar property of the component induces interesting properties into the system. 

So far our description of the random-coil chain basically assumes a dilute solution and we have not yet defined the term dilute solution. It has been discovered that when the concentration increases to a certain point, interesting phenomena occur chain crossover and chain entanglement. Chain crossover refers to the transition in configuration from randomness to some kind of order, and chain entanglement refers to the new statistical discovery of the self-similar property of the random coil (e.g., supercritical conductance and percolation theory in physics). Such phenomena also occur to the chain near the theta temperature. In this section, we describe the concentration effect on chain configurations on the basis of the theories advanced by Edwards (1965) and de Gennes (1979). In the next section, we describe the temperature effect, which is parallel to the concentration effect. 

The fractal dimension of the chain part between clusters was proposed to use as such characteristic in Refs. [6, 7] that is due to the following reasons. Firstly, the mentioned chain part possesses self-similarity property and has the dimension, differing from its topological dimension, that is, it is fractal by definition [20]. Secondly, it has been shown [21], that the value within the range of values 1.0 < < 2.0 characterizes exactly molecu-... 

As an example, we may consider the approaches now used to model processes on the stock market (e.g. using elements from the theory of fractals). Many such models are very familiar to networking engineers, where e.g. the self-similar properties of data traffic have been identified a few years ago. 

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