Self-similar features

W for K = 1 is shown in Figure 31. This figure is known as a devil s staircase because of its self-similar features. Notice that the blow-up of one small region of the staircase looks exactly like the entire staircase. An even higher resolution blow-up of a region between two small steps in the insert would, as well, resemble the entire staircase. This is the essence of self-similarity and is associated with the fact that the staircase has a fractal dimension. (The name devil s staircase is an indication of the frustration one would feel in trying to climb... 

A difference between these two concepts can be illustrated in many ways. Consider, for example, a mathematical pendulum in this case the old concept of trajectories around a center holds. On the other hand, in the case of a wound clock at standstill, clearly it is immaterial whether the starting impulse is small or large (as long as it is sufficient for starting, the ultimate motion will be exactly the same). Electron tube circuits and other self-excited devices exhibit similar features their ultimate motion depends on the differential equation itself and not on the initial conditions. 

The simplest expression incorporating the basic features of self-similarity and cut-off for nearly critical gels has the spectrum of the critical gel altered by a cut-off at the longest time [19] ... 

The issue of scaling was touched upon briefly in the previous section. Here, the quantitative features of scaling expressed as scaling laws for fractal objects or processes are discussed. Self-similarity has an important effect on the characteristics of fractal objects measured either on a part of the object or on the entire object. Thus, if one measures the value of a characteristic 9 (cu) on the entire object at resolution cu, the corresponding value measured on a piece of the object at finer resolution 9 (rcu) with r < 1 will be proportional to 9 (cu) ... 

When proteins fold into their tertiary structures, there are often subdivisions within the protein, designated as domains, which are characterised by similar features or motifs. A protein domain is a part of the protein sequence and structure that can evolve, function and exist independently of the rest of the protein chain. Many proteins consist of several structural domains. One domain may appear in a variety of evolutionarily related proteins. Domains vary in length from about 25 up to 500 amino acids. The shortest domains, such as zinc fingers , are stabilised by metal ions or disulfide bridges. Domains often form functional units, such as the calcium-binding EF hand domain of calmodulin. As they are self-stable, domains can be swapped by genetic engineering between one protein and another, to make chimera proteins. 

Then, as the first step of approaching the study of self-similarity, we investigate two features that must appear if self-similarity exists power-type distribution function of momenta and anomalous diffusion. We are particularly... 

In a Hamiltonian system having mean field interaction, referred to as the HMF model, we have investigated two features that must reflect self-similar hierarchy of phase space power-type distribution and anomalous diffusion. They have been reported in the same model for one type of initial condition, and we used a different type of initial condition to check generality. 

The important feature of all geometric shapes examined in this chapter is their self-similarity, that is, scaling invariance. The dimensions of such geometric objects can be defined using the Hausdorff-Besicovitch measure. 

The main feature of the definition in Eq. (76) is that max ln —> 0. Whence in general the Hausdorff-Besicovitch dimension is a local characteristic. For deterministic self-similar sets, the local Hausdorff-Besicovitch dimension coincides with the dimension of the set itself. For statistically homogeneous sets, however, the local Hausdorff-Besicovitch dimension may not coincide with the dimension of the whole set. 

Roughly speaking, fractals are complex geometric shapes with fine structure at arbitrarily small scales. Usually they have some degree of self-similarity. In other words, if we magnify a tiny part of a fractal, we will see features reminiscent of the whole. Sometimes the similarity is exact more often it is only approximate or statistical. 

A specialized method for similarity-based visualization of high-dimensional data is formed by self-organizing feature maps (SOM). The data items are arranged on a two-dimensional plane with the aid of neural networks, especially Kohonen nets. Similarity between data items is represented by spacial closeness, while large distances indicate major dissimilarities [968]. At the authors department, a system called MIDAS had already been developed which combines strategies for the creation of feature maps with the supervised generation of fuzzy-terms from the maps [967]. 

Therefore we see that a self-similar form of the governing DE can be obtained. Furthermore, the rate of approach to the rupture singularity, and the rate at which the horizontal scale of the depression in film thickness varies with time to the rupture event, are completely determined by the local balance of capillary and van der Waals forces with viscous dissipation that is inherent in this equation. In particular, those features of the rupture dynamics are independent of initial conditions or of the film dynamics away from the rupture point. 

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