Fractal index

The term D is called the fractal index and represents the packing change with distance from the centre of the floe. Computer simulation and experiments allow the value of D to be related to the mechanism of aggregation. Typical values are for ... 

Given that there is a relationship between the yield stress and the concentration we might expect a power relationship between them reflecting the fractal index, i.e. m. 

Once interpenetration occurs the resistance to deformation increases markedly, so for example we would expect compaction of a sediment to become limited, as would further concentration in a filter press. It is worth emphasising the point that this is a simplistic approach, as prior to interpenetration the clusters undergo structural rearrangements changing their fractal index at a critical volume fraction. A typical data set for yield stress is shown in Figure 6.16.19... 

The fractality index a obtained from analysis of SANS curves correlates with the adsorption capacity of carbon materials for unconjugated bilirubin adsorbed from HSA solution in micro-column single-pass experiments (Table 29.2) [10]. 

Roughness. Characterized by the global fractal index Df of the molecular space curve or surface. 

Local geometrical features are essential for understanding binding properties, catalytic behavior, and molecular recognition. Many of the descriptors used for global analysis can be adapted to study local features. For instance, mean and Gaussian curvature distributions of a surface, curvature and torsion of a molecular space curves, and the variation of the fractal index Df(r) over a molecular model serve this purpose. 

The salient property of mathematical random fractals processes is the existence of long-time correlations, here measured by the correlation index r, which can be related to the fractal dimension by [21]... 

The scaling index or fractal dimension marks the system s response and can be used as an indicator of the system s state of health. Since the fractal dimension is also a measure of the level of complexity, the change in dimension with disease suggests a new definition of disease as a loss of complexity, rather than the loss of regularity [56], This observation was first made by Goldberger and West, see, for example, Ref. [110]. 

In this range, the connecting set is a fractal that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length and the P s of the connecting set (the upper index oo means that the limit / —> oo is taken) should scale with distance from the critical point (i.e., percolation threshold pc = p ) as... 

Probability functions Y lx,ly,p) for fractal ensembles grown on several lattices (of the generating cells lx x ly where 2 < lx < 4,1 < ly < 4) are presented in the Appendix, while calculated values of the percolation threshold Pc, fractal dimension of the ensemble at p = 1, d (lxIy ), mean fractal dimension at p = pc df), and critical indices p(/v, ly) and v(/v, ly) are listed in Table IX. The index ai in this table is calculated from... 

The coefficients of the series (606) depend both on the fractional derivative of gth order of the fractal function/(x) at the point x = x0 and on the branching index j of the fractal ensemble for which the function/(x) is specified. 

Figure 3. Comparison of the trajectories of a Gaussian (left) and a Levy (right) process, the latter with index a = 1.5. While both trajectories are statistically self-similar, the Levy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps ( 7000).

Note that power-law behaviour is prevalent at gelation. This has been proposed to be due to a fractal or self-similar character of the gel. Note that the exponent )f is termed the fractal dimension. For any three-dimensional structure D = 3) the exponent Df<3 (where Df < 3 indicates an open structure and Df= 3 indicates a dense strucmre). Also Muthu-kumar (Muthukumar and Winter, 1986, Muthukumar, 1989) and Takahashi et al. (1994) show explicitly the relationship between fractal dimension (Df) and power-law index of viscoelastic behaviour (n). Interestingly, more recent work (Altmann, 2002) has also shown a direct relationship between the power-law behaviour and the mobility of chain relaxations, which will be discussed further in Chapter 6. 

From a professor of mathematics comes Cynthia Lomus Fun Mathematics Lessons (www.math.rice.edu/ lanius/Lessons/index.html), a wonderful collection of creative, practical activities. These lessons address topics ranging from basic math up to calculus and fractals. Whether it s a problem that uses pizza to talk about fractions, or one that teaches math through mapmaking, you and your child will both appreciate the engaging, straightforward tone of the site. Notes for teachers and extensive links to online resources are also available. 

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. 

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