Structure self-similar

Weitz D A and Huang J S 1984 Self-similar structures and the kinetics of aggregation of gold colloids Kinetics of Aggregation and Geiationed F Family and D P Landau (Amsterdam North-Holland) pp 19-28... 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

Power law relaxation is no guarantee for a gel point. It should be noted that, besides materials near LST, there exist materials which show the very simple power law relaxation behavior over quite extended time windows. Such behavior has been termed self-similar or scale invariant since it is the same at any time scale of observation (within the given time window). Self-similar relaxation has been associated with self-similar structures on the molecular and super-molecular level and, for suspensions and emulsions, on particulate level. Such self-similar relaxation is only found over a finite range of relaxation times, i.e. between a lower and an upper cut-off, and 2U. The exponent may adopt negative or positive values, however, with different consequences and... 

In media of fractal structure, non-integer d values have been found (Dewey, 1992). However, it should be emphasized that a good fit of donor fluorescence decay curves with a stretched exponential leading to non-integer d values have been in some cases improperly interpreted in terms of fractal structure. An apparent fractal dimension may not be due to an actual self-similar structure, but to the effect of restricted geometries (see Section 9.3.3). Another cause of non-integer values is a non-random distribution of acceptors. 

Fractals are mathematically defined self-similar structures (Fig. 1.11) [26]. The scaffold of cascade or dendritic molecules is fractal if the atoms are considered to be points and the bonds to be strictly one-dimensional lines. Self-similarity... 

Observations of the liver reveal an anatomically unique and complicated structure, over a range of length scales, dominating the space where metabolism takes place. Consequently, the liver was considered as a fractal object by several authors [4,248] because of its self-similar structure. In fact, Javanaud [275], using ultrasonic wave scattering, has measured the fractal dimension of the liver as approximately df 2 over a wavelength domain of 0.15-1.5mm. 

Fig. 30a behaves similarly to that of the NBR/N220-samples shown in Fig. 29, i.e., above a critical frequency it increases according to a power law with an exponent n significantly smaller than one. In particular, just below the percolation threshold for 0=0.15 the slope of the regression line in Fig. 30a equals 0.98, while above the percolation threshold for 0=0.2 it yields n= 0.65. This transition of the scaling behavior of the a.c.-conductivity at the percolation threshold results from the formation of a conducting carbon black network with a self-similar structure on mesoscopic length scales. 

Due to the characteristic self-similar structure of the CCA-clusters with fractal dimension df 1.8 [3-8, 12], the cluster growth in a space-filling configuration above the gel point O is limited by the solid fraction Oa of the clusters. The cluster size is determined by a space-filling condition, stating that, up to a geometrical factor, the local solid fraction Oa equals the overall solid concentration O ... 

Some of the intriguing properties of the golden ratio include (i) its connection with self-similar structures, evidenced by its definition as a continued fraction ... 

Pouzot et al. (2004) reported applicability of the fractal model to )6-lactoglobulin gels prepared by heating at 80 C and pH 7 and O.IM NaCl. They suggested that the gels may be considered as collections of randomly close packed blobs with a self-similar structure characterized by a fractal dimension Z)f 2.0 0.1. 

For f=3.7% (and above), D-2, indicating that the network is sufficiently dense and uniform that the blend can be considered an effective medium-, i.e. the fractal dimensionality is the same as the spatial dimensionality. As f is decreased toward the percolation threshold, D becomes less than the spatial dimensionality indicating a self-similar structure with holes on every length scale. At f f,., the analysis of the mass density distribution yielded D = 1.5. 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

If Eq. (14) holds then anomalous diffusion may appear only for D = 0 and very strong Lagrangian velocity correlations. The latter condition can be realized— for example, in time periodic velocity fields where the Lagrangian phase space has a complicated self-similar structure of islands and cantori [30]. Here superdiffusion is due to the almost trapping, for arbitrarily long time, of the ballistic trajectories close to the cantori, which are organized in complicated selfsimilar structures. 

Conclusion. Calculations of the dependence of the conductivity and the relative permittivity of chaotic hierarchical self-similar structures of composites were performed using a fractal model in the entire range of concentrations of inhomogeneities at various frequencies of an external field. The metal-insulator transition was shown to occur not only near the percolation threshold. It was also shown that the transition depends on the concentration of the metallic phase and the frequency of the external field. 

Figure 64. Schematic of a self-similar structure potential energy landscape and of the Cayley tree.

Figure 69. Schematic of the self-similar structure of a dielectric medium.

Of course, this volume is finite in the case of compact structures in space. Conversely, it may vanish for certain homogeneous (i.e. self-similar) structures. This happens for structures with fractal dimensions and in particular for Brownian chains. For such structures... 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic structure of the chemical repeat units [4, 5 and 6]. The self-similar structure [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

Fig. 4.14. Self-similar structure of the attraction basins of the two limit cycles in the case of final state sensitivity described in fig. 4.13. The initial values of p and y being fixed, the initial value of a is varied in a continuous manner. A vertical line is traced when the system evolves towards limit cycle LC2. The white zones correspond to values for which the system evolves towards limit cycle LCl. Successive enlargements of the domains of variation of a illustrate the self-similarity of the random alternation between the two limit cycles, as a function of the initial substrate concentration (Decroly Goldbeter, 1984b).

The parameter C increases with the molar mass and is related to the branching probability p. For self-similar structures, one expects that the particle topology does not vary with molar mass, that is, the form factor should be independent of molar mass. These nonrandomly branched... 

Certain structures, when examined on different scales from small to large, always appear exactly the same. Such structures are said to be self-similar or to be endowed with the symmetry of self-similarity. Self-similar structures are found to be invariably associated with the geometrical relationship known as the golden ratio, or what Johannes Kepler (1571 - 1630) referred to as the "divine proportion", adding ... 

Kepler s obsession with the golden ratio kept him occupied in an effort to demonstrate the self-similar structure of the solar system and the music of the spheres. His biggest problem probably was insufficient data. 

West and Goldberger, 1987). Many plants have the same self-similar structure, as do shorelines, clouds, and mountains. 

Self-similar structures do not have a single length scale. The property of interest (say the diameters of the respiratory airways, for instance) will then be dependent on the scale of interest, usually decreasing in value as the scale becomes smaller (West and Goldberger, 1987). The result is that instead of an exponential function where the ratio of larger scale to smaller scale is a constant, the larger scale is related to the smaller scale through a power law relationship. 

This property may be used as lead-lag components in controlling the information transfer over the network. Closed-loop feedback is utilized to control the signals transferred over the network. Intermediate self-similar structures or switches may modulate feedback signals and control the system behavior. 

---