Geometrical fractals, defined

Here, Amici is the microscopic area which is defined within the fractal limits y, the dimensionless geometrical parameter X0, the spatial outer (upper) cutoff /fmacr, the macroscopic area which is defined beyond the fractal limits (= Aex in Eq. 15) and Xi represents the spatial inner (lower) cutoff. 

The notion fractal means an object possessing a complex structure the form of elements of which repeats on all levels of measurement—from macroscopic to microscopic. Fractal is defined as a geometrical figure consisting of identical motif repeating itself on an ever-reduced scale [1,2]. 

Surfaces of most materials, including natural and synthetic, porous and non-porous, and amorphous and crystalline, are fractal on a molecular scale. Mandelbrot defines that a fractal object has a dimension D which is greater than the geometric or physical dimension (0 for a set of disconnected points, 1 for a curve, 2 for a surface, and 3 for a solid volume), but less than or equal to the embedding dimension in an enclosed space (embedding Euclidean space dimension is usually 3). Various methods, each with its own advantages and disadvantages, are available to obtain... 

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 ... 

This study shows that small and well defined molecules behave in a complex manner when chemically linked to the surface of a solid. Their behaviour is strongly dependent on the characteristics of the graft and the surface, and on geometrical factors like the fractality of the surface. Obviously, the grafting ratio, which determines the intensity of interactions that adjacent grafted molecules experience is also important. 

For arbitrary dimension d of Euclidean space in which the fractal set Qf is embedded and arbitrary form of the initial element, the mass (the number of bonds) can be defined by singling out the factorial geometric coefficient F ... 

Finally, we define a strange attractor to be an attractor that exhibits sensitive dependence on initial conditions. Strange attractors were originally called strange because they are often fractal sets. Nowadays this geometric property is regarded as less important than the dynamical property of sensitive dependence on initial conditions. The terms chaotic attractor and fractal attractor are used when one wishes to emphasize one or the other of those aspects. 

However, the frontier between transport and temporal transfer is not always clearly defined. In some systems where geometry is not simple, being described by fractal geometry for instance, the proportion of transport and temporal transfer fluctuates according to the geometric dimensions of the system (Le Mehaute and Crepy 1983). A value for p between 0 and 1 but different from 1/2 characterizes the transfer. 

---