Fractals definition

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] ... 

As it is known [1], the following relationship is one from the fractal definitions in reference to a macromolecular coil ... 

Nevertheless, the Eqs. (1) and (2) are valid for different objects. If Flory equation is correct for arbitraiy coils, then the fractal Eq. (1) — for only semi-similar ones (by the fractal definition [3]). 

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. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

At definite conditions the value H is defined by dimension Df (Euclidean or fractal) of reaction product (heptylbenzoate molecule) only [8] ... 

There are some very special characteristics that must be considered as regards colloidal particle behavior size and shape, surface area, and surface charge density. The Brownian motion of particles is a much-studied field. The fractal nature of surface roughness has recently been shown to be of importance (Birdi, 1993). Recent applications have been reported where nanocolloids have been employed. Therefore, some terms are needed to be defined at this stage. The definitions generally employed are as follows. Surface is a term used when one considers the dividing phase between... 

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. 

However, the same difficulty that observers meet in defining a cluster exist for theorists to define clusters in a numerical simulation typical numerical simulations handled several millions dark matter particle and a similar number of gas particle when hydro-dynamical processes are taken into account the actual distribution of dark matter, at least on non linear scales is very much like a fractal, for which the definition of an object is somewhat conventional Different algorithms are commonly used to define clusters. Friend of friend is commonly used because of its simplicity, however its relevance to observations is very questionable, especially for low mass systems. On the analytical side... 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

Avnir et al. llbl have examined the classical definitions and terminology of chirality and subsequently determined that they are too restrictive to describe complex objects such as large random supermolecular structures and spiral diffusion-limited aggregates (DLAs). Architecturally, these structures resemble chiral (and fractal) dendrimers therefore, new insights into chiral concepts and nomenclature are introduced that have a direct bearing on the nature of dendritic macromolecular assemblies, for example, continuous chirality measure44 and virtual enantiomers. ... 

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... 

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