Fractal dimension definition

Porous materials have attracted considerable attention in their application in electrochemistry due to their large surface area. As indicated in Section I, there are two conventional definitions concerning with the fractality of the porous material, i.e., surface fractal and pore fractal.9"11 The pore fractal dimension represents the pore size distribution irregularity the larger the value of the pore fractal dimension is, the narrower is the pore size distribution which exhibits a power law behavior. The pore fractal dimensions of 2 and 3 indicate the porous electrode with homogeneous pore size distribution and that electrode composed of the almost samesized pores, respectively. 

There are two conventional definitions in describing the fractality of porous material - the pore fractal dimension which represents the pore distribution irregularity56,59,62 and the surface fractal dimension which characterizes the pore surface irregularity.56,58,65 Since the geometry and structure of the pore surfaces are closely related to the electro-active surface area which plays a key role in the increases of capacity and rate capability in practical viewpoint, the microstructures of the pores have been quantitatively characterized by many researchers based upon the fractal theory. 

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

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]. 

Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. 

In the open flow case, the total amount of product within the mixing zone, accumulated on the unstable manifold of the chaotic saddle, can be calculated by recalling the definition of the fractal dimension Df, that states that the number of boxes of linear size w needed to cover the fractal set scales as w Df. Multiplying by the area of these... 

A fractal object such as a C curve may have some unusual properties. The properties are that it has a fractal dimension but this fractal dimension is not a fraction. In the case of the C curve it is equal to two. The reason this object is still a fractal relates to a definition of fractal dimension. First, one defines two concepts of dimensions the topological dimension, which corresponds to our usual concept of a dimension, and a so-called Hausdorff-Besicovic dimension. If for a given object the two dimensions defined are different, the object is said to have a fractal dimension. In the case of the... 

Once the reconstructed portrait is found, the same type of analyses as were described for the simulated data—that is, construction of Poincare sections and Poincare maps, calculation of the fractal dimension, etc.—can be carried out. Of these, the calculation of the fractal dimension is often of interest, although, as cautioned earlier, the knowledge of this number cannot, alone, distinguish chaotic data from nonchaotic data. An understanding of the route by which the suspected chaotic state arises is also necessary before a definitive statement can be made. 

Nevertheless, the determination of the fractal dimension from a data set thought to be chaotic is often of interest. A number of different dimensions exist in the literature, including the Hausdorff dimension, the information dimension, the correlation dimension, and the Lyapunov dimension. Which of these is the true fractal dimension Of the ones in this list, the information dimension, Di, has the most basic and fundamental definition, so we often think of it as the true fractal dimension. Because the information dimension is impractical to calculate directly, however, most investigators have taken to finding the correlation dimension, Dq, as an estimate of the fractal dimension. Grassberger and Procaccia published a straightforward and widely used algorithm for the calculation of the correlation dimension. On the other hand, the Lyapunov... 

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are ... 

As it has been noted above, the fractal dimension macromolecu-lar coil in solution is determined by two interactions groups interactions polymer-solvent and interactions of coil elements between themselves. Such definition allows to link between themselves the dimension and Flory-Huggins interaction parameter which was determined as follows [1] ... 

As it has been shown in Ref [72], the fractional exponent v coincides with the fractal dimension of Cantor s set and indicates a fraction of the system states, being preserved during the entire evolution time t. Let us remind that Cantor s set is considered in one-dimensional Euclidean space (d=l) and therefore its fractal dimension d < by virtue of the fractal definition [86]. For fractal objects in Euclidean spaces with higher dimensions d> ) as V one should accept fractional part or [76, 77] ... 

---