Fractal local dimension

It is the local dimension that describes the irregularity of the self-affine fractal. The local dimension can be determined by such methods as the box-counting method1,61,62,65 and the dividerwalking method.61,66 The box dimension dEB is defined by the... 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

Fractal dimension of clusters formed by flow-induced aggregation is independent of local flow type. 

A random coil is clearly a three-dimensional object when looked at from long distance. Locally, however, it resembles more a one-dimensional thread. Therefore it is sensible to describe the coil by a fractal dimension that lies closer to 1 (for other architectures somewhere between 1 and 3). Such disordered objects are called fractals [101,102]. 

Due to the characteristic self-similar structure of the CCA-clusters with fractal dimension df 1.8 [3-8, 12], the cluster growth in a space-filling configuration above the gel point O is limited by the solid fraction Oa of the clusters. The cluster size is determined by a space-filling condition, stating that, up to a geometrical factor, the local solid fraction Oa equals the overall solid concentration O ... 

The choice of dimension D, depends on the value of relation dhldm [9], At dm<0,6dh interaction of diffusant molecules with walls of free volume microvoid is small and transport process is controlled by fractal dimension of structure (structural transport). At dm<0,6dh on transport processes has strong influence interaction of diffusant molecules with walls of free volume microvoid, which are polymeric macromolecules surface with dimension >/(/)/ is the dimension of excess energy localization regions) [10], In this case Dt=Df (molecular transport) [9] is adopted. 

Let s consider structural aspect of t50 change due to introduction Z. As it is known [3, 15], for compositions HDPE+Z is observed the extreme rise of relative fraction of local order regions (clusters) cpci, that results to decrease of fractal dimension of structure df according to the equation [15] ... 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

It has been shown in Ref. 170 that fractional derivatives can be obtained by assuming that a set of relaxation times has a fractal nature. The parameter a in Eqs. (376) and (377) is the fractal dimension of the fractal set of relaxation times and characterizes the localization (spread) of the relaxation spectrum [170],... 

The notion of pointwise dimension allows us to quantify the local variations in scaling. Given a multifractal A, let be the subset of A consisting of all points with pointwise dimension a. If a is a typical scaling factor on A, then it will be represented often, so will be a relatively large set if a is unusual, then will be a small set. To be more quantitative, wenotethateach is itself a fractal, so it makes sense to measure its size by its fractal dimension. Thus, let /(a) denote the dimension of. Then /(a) is called the multifractal spectrum of A or the spectrum of scaling indices (Halsey et al. 1986). 

One of the most important assumptions in MM kinetics is that the reaction in question wiU proceed in a three-dimensional vessel filled with a well-stirred fluid that obeys Pick s law for diffusion. This is rarely the case in a living cell, where many reactions are localized to membranes (two dimensions) or to small regions somewhere within the cell, creating an effectively one-dimensional environment with little or no diffusion. To circumvent this limitation, fractal kinetics have been developed which allow for the approximation of enzymatic reaction velocities in vivo [7]. Fractal kinetics can utilize MM-type kinetic constants to create a model of events in a spatially restricted environment. Briefly, as the dimensionality of a reaction is reduced from three dimensions to one, the kinetic order of a bimolec-ular reaction, for example, increases from 2 in a three-dimensional case, to 2.46 in a two-dimensional environment (e.g., membrane), to 3 in a one-dimensional channel, up to 50 for the case where fractal dimensions are less than 1. In simple terms, the kinetic order is the sum of all stoichiometric coefficients of the reactants in a balanced chemical reaction equation. Rearranging the familiar equation for MM kinetics... 

Suppose the geometry of a compartmentalized system is described by a lattice of integral or fractal dimension of given size and shape, and characterized by N discrete lattice points (sites) embedded in a Euclidean space of dimension d = de and local connectivity or valency v. At time t = 0, assume that the diffusing coreactant A is positioned at a certain site j with unit probability. For f > 0 the probability distribution function p(f) governing the fate of the diffusing particle is determined by the stochastic master equation... 

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