Fractal boundary

Cantor strings (or Cantor harps) are one-dimensional drums with a Cantor set as a fractal boundary. They have been studied extensively and an important relationship to the Riemann conjecture was shown [6]. Cantor chains are their discrete counterpart, introduced by the author in [7],... 

Multiple oscillations, fractal boundaries and final state sensitivity... 

Surface fractals have a fractal boundary but a compact, nonfractal bulk. Common to all methods of fractal analysis is that yardsticks of different size probe the surface. Depending on the method, the probes may be molecules of different size. 

Kaye, B.H., 1986. The description of two-dimensional rugged boundaries in fine particle science by means of fractal dimensions. Powder Technology, 46, 245-254. 

In order to examine the current response to the imposition of the potential step on the self-affine fractal interface, the current transients were calculated theoretically by random walk simulation.153 The simulation cell was taken as the square area bottom boundary which is replaced by one of the self-affine fractal profiles in Figure 7. The details of the simulation condition were described in their publication.43... 

The investigation of the behaviour of Z(r, t) in a finite system is a difficult problem due to the fact that the boundary conditions for (7.3.3) are unknown for fractal systems although one can use some scaling arguments based on the knowledge of the properties of the infinite system. In the case of the independent production of different particles we obtain... 

In the context of particle aggregation, the particular fractal structures called mass fractals are usually considered. Mass fractals correlate the location of mass with radius, as opposed to other types of fractal constructs such as boundary fractals, surface fractals, and pore fractals (see Ref. 17). The power law which correlates the size of an aggregate with the number of primary particles in the aggregate is given by... 

The release problem can be seen as a study of the kinetic reaction A+B —> B where the A particles are mobile, the B particles are static, and the scheme describes the well-known trapping problem [88]. For the case of a Euclidean matrix the entire boundary (i.e., the periphery) is made of the trap sites, while for the present case of a fractal matrix only the portions of the boundary that are part of the fractal cluster constitute the trap sites, Figure 4.11. The difference between the release problem and the general trapping problem is that in release, the traps are not randomly distributed inside the medium but are located only at the medium boundaries. This difference has an important impact in real problems for two reasons ... 

Figure 4.11 A percolation fractal embedded on a 2-dimensional square lattice of size 50 x 50. Cyclic boundary conditions were used. We observe, especially on the boundaries, that there are some small isolated clusters, but these are not isolated since they are actually part of the largest cluster because of the cyclic boundary conditions. Exits (release sites) are marked in dark gray, while all lighter grey areas are blocked areas. Reprinted from [87] with permission from American Institute of Physics.

The drug molecules move inside the fractal matrix by the mechanism of diffusion, assuming excluded volume interactions between the particles. The matrix can leak at the intersection of the percolation fractal with the boundaries of the square box where it is embedded, Figure 4.11. 

It has been stated that heterogeneous reactions taking place at interfaces, membrane boundaries, or within a complex medium like a fractal, when the reactants are spatially constrained on the microscopic level, culminate in deviant reaction rate coefficients that appear to have a sort of temporal memory. Fractal kinetic theory suggested the adoption of a time-dependent rate constant , with power-law form, determined by the spectral dimension. This time-dependency could also be revealed from empirical models. 

There are a number of different fractal dimensions commonly used to describe a specific property of a system. These fractal dimensions and methods to obtain them are explained in detail in Boundary and Surface Fractal Dimensions and Mass Fractal Dimension. Fractal dimensions used in specific applications will be shown also in the related section. 

In a simple technique to determine fractal dimension of a rugged boundary, a series of polygons of side length 1 are constructed on the image of the object. The perimeter of the polygon, P, then becomes the approximation of the perimeter at resolution jS. The boundary fractal dimension, Dl, which varies from... 

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