Self-similarity postulate

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

This space-time model is a conjecture that has been described in detail [28] and will be reconsidered in chapter 7. A new aspect thereof, which derives from number theory, is that the general curvature of this space-time manifold [26, 29] relates to the golden mean. This postulate is required to rationalize the self-similar growth pattern that occurs at many levels throughout the observable universe. 

Figure 4 shows the SCF results in spheroidal coordinates for the excited bending state (0,4) as a function of the coordinate parameter a. For each of the states, the result is compared with the energy given by SCF in polar (hyperspherical) coordinates. Also shown are the results of a bare-mode approximation, a crude model which assumes for the mode a potential = 0), and similarly postulates a separate potential F( cq, > ) for the q mode, without any self-consistency in the treatment of the two modes. It is evident from Fig. 4 that the physically motivated elliptical (spheroidal) SCF modes do better in this case than the hyperspherical coordinates. Also, the SCF correction gives an important improvement on the bare-mode results. Most important, coordinate optimization, that is, imposing condition, (25) yields a noticeably better result than the SCF energy in a spheroidal system that is not refined for the best a value. 

To initiate this computational algorithm it is necessary to know two values of the excess function, namely, 5f x g) and df x g), as functions of the bare coupling parameter g. The scale length x should, in principle, be selected equal to the shortest interval of the generahzed time beyond which the nonideal system is expected to display self-similarity. Once 3f v, g) and g) have been evaluated, construction of the excess quantity 3f t, g) for t X becomes a relatively easy task. This is the computational advantage that one gains by using the RG method of calculation based on the postulate of self-similarity. 

While the functional equations (5.61) and (5.65), are very well suited to the above mentioned, iterative method for determining the (generalized) time evolution of the dynamic system f(t, g,fo), most presentations of the RG method have instead used an approach based on differential equations. These differential equations can be written directly as equations of evolution for the object function Sf t g) and the related effective coupling function g(t g) [3-5,16] or as a pair of partial differential equations known as the Callan-Symanzik equations [3-5,17]. These three forms of the RG theory are essentially equivalent. However, we personally favor the functional equation approach, not only from a computational point of view but because it provides better insight into the workings of the postulates of the self-similarity based RG technique. 

The introductory chapter summarizes the contentious issues which stimulate cosmological debate, including the chirality and self-similarity of space-time, the dark-matter postulate and the physical meaning of mathematical singularities. It cautions against the uncritical reliance on the rhetoric of authority and outlines the fundamental considerations which dictate the structure of the work that follows. 

It has been postulated that after some initial transient time following a quench, H x,t) should change its shape self-similarly with time. In this section, we examine what comes out of this postulate. The self-similarity of H (x, t) is mathematically equivalent to the condition that there exists a single time-dependent correlation length (t) which satisfies the relation... 

There is considerable interest in the role of infectious agents in the development of autoimmune diseases. Some of this interest is based on the concept of molecular mimicry as a causal mechanism. Molecular mimicry refers to the possible pathologic role of cross-reactive antibodies or T cells to a self-antigen that is structurally similar to, and thus shares epitopes with, a viral or other infectious agent. For most autoimmune diseases, however, evidence of molecular mimicry leading to disease is not conclusive.1819 Viruses and other infections also have a less-specific immune effect, stimulating toll-like receptors and proinflammatory cytokine secretion, which is another mechanism that has been postulated to influence autoimmune disease risk.20... 

The discovery of self-splicing introns showed that RNA could catalyse chemical reactions. Yet, unlike proteins, RNA has no functional groups with pKa values and chemical properties similar to those considered to be important in protein-based enzymes. Steitz and Steitz (1993) postulated that two metal ions were essential for catalysis by ribozymes using a mechanism similar to DNA cleavage, in which a free 3 OH is produced. They proposed,... 

In contrast, metal and stractural analyses of the second group of FALS mutants show that these proteins are fully capable of binding metals and retain enzymatic activity, if they are properly folded. Thus, they have been termed wild-type-like. It is postulated that these mutants have higher prooxidant activity (possibly via bicarbonate-mediated inactivation, see Scheme 2). Self-inactivation could lead to metal loss and similar aggregation effects described for the metal-binding-region mutants. 

It is of interest to compare these results with those reported for the analogous carbon- and nitrogen-containing series. All the experimental [71] (EPR parameters) and theoretical [36, 66] studies suggest that the diazirinyl radicals 3N present a somewhat similar 7r -structure. However, based on the formation of nitriles and dinitrogen, Maeda and Ingold [71] postulated that the bimolecular self-reaction of diazirinyl radicals 3N proceeds via an N-N cr-dimer of type 28 the isolation of such compounds, which suffer from the anti-aromaticity of the two liT-diazirine moieties of type l N, is unlikely (Fig. 11). There is no mention in the literature of radicals of type 3 N. 

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