Dimension Hausdorff

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

Since Ruelle s function vanishes if its argument takes the unit value P(l) =0, the Hausdorff dimension can be expanded in powers of the wavenumber as... 

In systems with two degrees of freedom such as the two-dimensional Lorentz gases, there is a single positive Lyapunov exponent X and the partial Hausdorff dimension of the set of nonescaping trajectories can be estimated by the ratio of the Kolmogorov-Sinai entropy to the Lyapunov exponent [ 1, 38]... 

Correlation dimension. The correlation dimension is calculated by measuring the Hausdorff dimension according to the method of Grassberger [36,39]. The dimension of the system relates to the fewest number of independent variables necessary to specify a point in the state space [40]. With random data, the dimension increases with increase of the embedding space. In deterministic data sets, the dimension levels off, even though the presence of noise may yield a slow rise. 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Umberger, D.K., Mayer-Kress, G. and Jen, E. (1986). Hausdorff dimensions for sets with broken scaling symmetry, in Dimensions and Entropies in Chaotic Systems, ed. G. Mayer-Kress (Springer, New York). 

Falconer (1990) discusses other fractal dimensions, the most important of which is the Hausdorff dimension. It is more subtle than the box dimension. The main conceptual difference is that the Hausdorff dimension uses coverings by small sets of varying sizes, not just boxes of fixed size e. It has nicer mathematical properties than the box dimension, but unfortunately it is even harder to compute numerically. 

Grassberger, P. (1981) On the Hausdorff dimension of fractal attractors. J. Stat. Phys. 26, 173. 

B.R. Hunt. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Am. Math. Soc., 126 791, 1998. 

To get an expression for d, we need to explore the relationship between the cross-section of the islands in a given layer with film coverage. If 5 represents the area of a two-dimensional cluster and D its fractal (Hausdorff) dimension, then the island cross-section is proportional to sp, where Pi = HD. D = 2 corresponds to a compact island, but for pentacene that grows by diffusion limited aggregation (DLA), D 513. Noting from before that, prior to coalescence the island density, n remains constant and composed of identical islands, then the total cross-section of a layer becomes d = n SP. Since 9 = n S, then Qp. Similarly, if after coalescence the number of holes in the film also remains constant, then the perimeter of each layer can be related to its coverage as shown in Figure 5.1.12b and c by [29] ... 

The dimension which is defined here is called capacity the true Hausdorff dimension is defined in a slightly more complicated way 3 in the present case, these two notions coincide. 

Critical exponent for the swelling of a polymer chain in good solvent this exponent is the inverse of the Hausdorff dimension of a Kuhnian chain it is also the exponent v of the Land-au-Ginzburg model for n = 0... 

Certain real systems seem to be described by OLA, notably electrodeposition on a sharp point (39) and dielectric breakdown (33,40). The second class involves cluster formation by the homogeneous aggregation of a collection of two clusters of comparable size (37, ) (cluster-cluster aggregation, CA) and the resultant aggregate has a more open structure and lower fractal dimension, D = 1.4 ( d = 2) and 1.8 (d = 3). Real smoke ( ) and colloids (41) seem to have D = 1.8 this is a satisfying verification of the model. A process that has not, however, been included in the simulations is rearrangement within the clusters. This would lead to denser structures with higher Hausdorff dimensions ( ). 

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... 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

The difference between the dj and values found by Colvin and Stapleton [66] can also be easily explained. Indeed, d is a chemical dimension, as follows from the way of its determination [27], d is the Hausdorff dimension. These values do not necessarily coincide. By using Equation (11.10a) for a fractal swollen in a monomeric solvent, we obtain D 2.26 for d = 1.65 and d = 3, which is in good agreement with both experimental data and the limiting D value for branched polymers [9]. 

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]. 

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