Self-similarity concept

Contrary to self-similar concepts, He et al. [204] and Wang et al. [205] argued that once fracture occurs at a certain scale, the discontinuity invalidates any selfsimilar notion. In other words, the fracture introduces a singularity within the selfsimilar material at that scale. A scale lower would then not be self-similar. 

Note that (1.6) relies on the self-similarity concept since the number of identical objects m and the scale factor r in (1.5) have been replaced by the number of balls m(oj) and the reciprocal of the size 1/w, respectively. The limit (uj > 0) is being used to indicate the estimation of dc at the highest possible resolution, i.e., as the ball size uj decreases continuously. 

Bellan et al. present the existence of self-similarity as an empirical observation resulting from the inspection of simulation results, and they do not provide a mathematical foundation to the method. Similar concentration curves may be a result of the existence of very different timescales, and the application of QSSA or partial equilibrium may result in linear relations between the concentrations. However, in these articles self-similarity was fotmd among long lifetime ( heavy ) species, and therefore, the existence of self-similarity seems to be a consequence of possible lumping relationships within the system variables. Although the self-similarity concept seems to be related to the lumping of species, it is not equivalent to it, since the derived hnear functions ccmtain the concentrations of all heavy species and not only a selection of them. 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

The treatment of mixing of immiscible fluids starts with a description of breakup and coalescence in homogeneous flows. Classical concepts are briefly reviewed and special attention is given to recent advances—satellite formation and self-similarity. A general model, capable of handling breakup and coalescence while taking into account stretching distributions and satellite formation, is described. 

Now we will introduce briefly the concept of self-similar and self-affine fractals by considering the assumption that fractals are sets of points embedded in Euclidean E-dimensional space. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

Thus, the concept of zero total momentum, and of the existence of a small region which does not obey the self-similar solution, resolves the paradoxes which arose when the solution was compared with the conservation laws. 

The existence of two types of self-similar solutions, explicitly formulated by Ya.B. for the first time, stimulated extensive studies to clarify the general character of the difference between them and to apply the concept of self-similar solutions of the second kind to various problems in mathematical physics. The present state of the problem can be found in a monograph by G. I. Barenblatt.6 We note also the existence of an exact analytic solution with a rational self-similarity exponent when the adiabatic index is equal to 7/5.7... 

Self-similarity is a mathematical concept in which fractals have been applied to predict or simulate some solid material behavior. A fractal is generally a rough or... 

Therefore, the power-law behavior itself is a self-similar phenomenon, i.e., doubling of the time is matched by a specific fractional reduction of the function, which is independent of the chosen starting time self-similarity, independent of scale is equivalent to a statement that the process is fractal. Although not all power-law relationships are due to fractals, the existence of such a relationship should alert the observer to seriously consider whether the system is self-similar. The dimensionless character of a is unique. It might be a reflection of the fractal nature of the body (both in terms of structure and function) and it can also be linked with species invariance. This means that a can be found to be similar in various species. Moreover, a could also be thought of as the reflection of a combination of structure of the body (capillaries plus eliminating organs) and function (diffusion characteristics plus clearance concepts). 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

Hence Fig 19 represents, in fact five correlations Other many more relations are available [64] Here, in an analogy with self-similarity the concept of fractal dimensions has been extended to graphs by... 

This method of selecting catalytic sites significantly depends on spontaneous processes, in contrast to the development of artificial enzymes and catalytic antibodies. The selection process is based on self-assembly, selforganization and self-optimization. Therefore, this selection approach bears the characteristics of supramolecular chemistry. A similar concept is used in natural evolution processes, resulting in the complicated life forms we see around us today. Therefore, it is clear that we can design the self-organizational processes used in supramolecular chemistry to proceed according to the concepts followed by this natural evolutionary process. 

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 ... 

Self-similarity is expected to be one of the important concepts to understand statistics and motion in Hamiltonian systems. However, the Poincare-Birkhoff theorem and the two models introduced by Aizawa et al. and Meiss et al. are based on the two-dimensionality of the phase spaces, and they cannot be directly applied to high-dimensional systems. As far as I know, existence of the self-similarity has not been clearly exhibited, since visualizing the selfsimilarity is not easy due to the high-dimensionality of Poincare sections, which has 2N — 2 dimension for systems with N degrees of freedom. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

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