Spatial self-similarity

The q-dependence of the intensity scattered per monomer reflects the mass self-similarity(x) and spatial self-similarity >) of the clusters. In the case of a monodisperse sample (M = Af, R = R) the intensity scattered at q R > leads to the fractal dimension ... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

It should be stressed here that the specific power dependences from the above self-affine fractal interfaces are maintained even during the relatively long time (or number of random jumps) interval. This implies that the morphology of the self-affine fractal interfaces tested is possibly characterized by the self-similar fractal dimension within a relatively wide spatial cutoff range. 

Fritton, S.P., McLeod, KJ. and Rubin, C.T. (2000) Quantifying the strain history of bone spatial uniformity and self-similarity of low magnitude strains. Journal of Biomechanic 33 317-325... 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

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. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

Inside the Debye sphere, strong electron acceleration takes place. The electrical field that surrounds the ion current channel accelerates the electrons toward the filaments where they are deflected on the induced magnetic field. The scenario is depicted in fig. 2. It have previously been shown that the ion filaments are generated in a self-similar coalescence process (Medvedev et al., 2004) which implies that a spatial Fourier decomposition exhibits power law behavior. As a result, the electrons are accelerated to a power law distribution function (fig. 3) as shown by Hededal et al., 2004. 

Molecule Crystal lattice Homogeneous surface Lattice defect Irregular structure Rotation/reflection Spatial translation Surface translation Homotopy Dilation (self-similarity) Molecular point group Space group 2-dimensional unit cell Burgers vector Fractal dimension Spectroscopy X-ray analysis Adsorption studies Crysttd properties Scaling laws... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

Processes in nature often result in fractal-like forms that differ from the mathematical fractals such as the Koch curve in two ways (a) the self-similarity is not exact but is a congruence in a statistical sense and (b) the number of repeated splittings is finite and random fractals have an upper and a lower cutoff length. A spatial example of a random fractal is the colloidal gold particle agglomerate shown earlier in Figure 7.4. 

FIG. 11 Spatially periodic suspensions of fractal aggregates. The aggregate in (a) contains 1024 cubic particles of size a- it was built with the hierarchical model with linear trajectories. The deterministic self-similar flake in (b) is at the third-generation stage with b = 5. 

Fractal theory was subsequently extended to irregular or quasi-random surfaces lacking well-defined self-similarity [323, 326, 329-331]. Pajkossy and Nyikos [332] carried out simulations of blocking electrodes with a self-similar spatial capacitance distribution and found that the calculated impedances exhibited CPE behavior. 

Cross-section sets for Assembly 6A were generated in the manner described in Ref. 3 using ENDF/B Version 1 i.e., MC homogeneous cross sections, corrected tor resonance spatial self-shielding in U and D, and wei ited with the unit-cell fluxes. The keff of the unitcell calculated with 1-D, Sn (n > 16) and the anisotropic option (Pt only) was 1% greater than that calculated In the same manner with the transport approximation. This difference disappeared, however, when the transverse dimension was made Infinite this indicated that the decrease in leakage associated with the use of S total Instead of S transport was responsible for this difference. Similar difference in keff between the two methods was found for the as-built system. 

The distribution of the distance between two points in the freely-jointed chain is also well approximated by a Gaussian distribution, but with the chain length Njc in Eq. (8) replaced by the number of statistical segments between the points. This makes a freely-jointed chain self-similar on various length scales. If we look at it with different spatial resolutions it looks the same, as long as the resolution is much larger than and less than N bK-... 

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