Dimension topological

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

In some publications (e.g., [208]), the E parameter is considered as a topological dimension and E+1 as die dimension of the corresponding Euclidean space. 

One is accustomed to associating topological dimensions with special objects dimension 1 with a curve, dimension 2 with a square, and dimension 3 with a cube. Because there are severe difficulties for the definition of the topological dimension dt, it is convenient to associate the topological dimension of an object with its cover dimension da. 

In the simplest case, a system is called Euclidean or nonfractal if its topological dimension dt is identical to the fractal dimension df. This means dt = df = 1 for a curve, dt = df = 2 for a surface, and dt = df = 3 for a solid. The following relationship holds for the three expressions of dimensionality... 

Fig. 1 (A) Profiles of fumed pigment fine particles where the fractal dimension, 5, describes the ruggedness of the profile. (B) Four lines with identical topological dimensions with varying degrees of ruggedness as seen from their corresponding fractal dimensions. (From Ref. l)...

Devreux [146] has shown that transport properties of perchlorate doped polypyrrole can be analysed com-plementarily by spin dynamics (using NMR measurements) and microwave measurements. The approach based on anomalous diffusion theory involves two parameters of dimensionality, d (which is the topological dimension) and d (which is the spectral dimension i.e. the exponent of the power law describing the increase with time of the number of sites visited by a random walker). 

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... 

C curve, the topological dimension is equal to one and the Hausdorff-Besicovic dimension is equal to two and that is why it is considered fractal, despite the fact that neither of these dimensions is itself fractional. In most cases, however, a fractal has a fractional dimension. 

Figure 1 Topological dimension p of various molecular models. A zero-dimensional (OD) model corresponds to the nuclear geometry, a ID model is specified by the nuclear geometry and bond connectivity, a 2D model is a molecular surface (a contour surface), and a 3D model is a continuum of molecular surfaces.

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

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]. 

As it has been shown above, polymers macromolecular coils in solution are fractal objects, i.e., self-similar objects, having dimension, which differs from their topological dimension. The coil fractal dimension Dp characterizing its structure (a coil elements distribution in space), can be determined according to the Eq. (4). The exponent ax values for polyarylate Ph-2 solutions in three solvents (tetrachloroethane, tetrahydrofuran and 1,4-dioxane) are adduced in [36]. The values ar] for the same polyarylate are also given in paper [37]. This allows to use the Eq. (4) for the macromolecular coil of Ph-2 Devalue estimation in the indicated solvents. The estimations showed D variation from 1.55 in tetrachloroethane (good solvent for Ph-2) up to 1.78 in chloroform. As it is known [38],... 

At present there are several methods of filler structure (distribution) determination in polymer matrix, both experimental [10, 35] and theoretical [4]. All the indicated methods describe this distribution by fractal dimension of filler particles network. However, correct determination of any object fractal (Hausdorff) dimension includes three obligatory conditions. The first from them is the indicated above determination of fiiactal dimension numerical magnitude, which should not be equal to object topological dimension. As it is known [36], any real (physical) fractal possesses fiiactal properties within a certain scales range. Therefore, the second condition is the evidence of object self-similarity in this scales range [37]. And at last, the third condition is the correct choice of measurement scales range itself As it has been shown in Refs. [38, 39], the minimum range should exceed at any rate one self-similarity iteration. 

If, in Figure 2.4, one takes the part of I4 that corresponds to the interval [0,1/3] of the initiator, and scales this part up horizontally and vertically by three, one obtains h. In the limit n -> 00, however, this same scaling up of any segment of / would reproduce the von Koch curve itself. Furthermore, like the Peano-Hilbert space-filling curves, the von Koch curve is nowhere differentiable. All of these curves have a topological dimension equal to unity a simple stretching operation (or rectification ) transforms them into an infinite straight line. 

Section 2.2 introduced a large array of monstrous mathematical beings that exhibit pathological properties defying the traditional concept of dimension. In Section 2.3, we saw that various alternatives to the traditional topological dimension have been devised by mathematicians. In spite of their multiplicity, these dimensions have tended to make the mathematical monsters somewhat less terrifying. 

Mandelbrot beUeved initially that one would do better without a precise definition of fractals. His original essay [1] contains none. By 1977, however, he saw the need to produce at least a tentative definition. It is the now classical statement that a fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension [4, 5,10]. For example, the Cantor set is a fractal, according to this viewpoint, since Dh = 0.631 > Z)r=0. 

F often has some form of self-similarity, perhaps approximate or statistical usually, the fractal dimension of F (defined in some way) is greater than its topological dimension ... 

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