Ramification infinite

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

When it is added that open-chain structures also include endless ramifications of branched chains (p. 176) that hybrid chain-ring structures exist in profuse abundance and that a large variety of so-called typical groups (—COOH, —CHO, —NH —CO NHa, etc.) may be substituted for a hydrogen atom at almost any position of a chain or rii carrying such an atom, some idea may be gained of the infinite variety of the organic molecular world. 

Of course there are many unsolved problems, and possible directions for further research in this area. The most interesting problem would be to try to extend these exact solutions to some fractals with infinite ramification index. There are some studies of statistical physics models of interacting degrees of freedom on Sierpinski carpets, using Monte Carlo simulations, or approximate renormalization group using bond-moving, or other ad-hoc approximations. An exactly soluble case would be very instructive here. 

In the case of fractal substrates, one has to distinguish between two main subclasses of structures, namely deterministic and random fractals. Within the class of deterministic fractals, one additionally has a subdivision in finitely and infinitely ramified fractals. Here, (either finite or infinite) ramification refers to the number of cut operations which are required to disconnect any given subset of the structure, the upper limit of which is independent of the chosen subset [7,8]. An example of a finitely ramified structure is the Sierpinski triangular lattice, whereas the Sierpinski square lattice is an example of an infinitely ramified structure. See Figs. 2(a) and 6 in Section 4 for the respective sketches of these structures in d = 2. 

This Subsection deals with the Sierpinski carpet (d = 2) and its corresponding sponge (d = 3), further on called Sierpinski square lattices (an example in d = 2 is shown in Fig. 6). Note that in this Subsection, we use the identical notation as in the previous Subsection discussing Sierpinski triangular lattices, avoiding a reiteration of Eqs. (17) to (22). Nonetheless, a major difference between the two types of Sierpinski structures is the ramification, as Sierpinski triangular lattices are finitely ramified, whereas Sierpinski square lattices are infinitely ramified. The infinitely ramification renders the application of RG technique elusive and has drastic consequences on the actual values obtained numerically for the exponents. 

It should be noted that the Hertz model assumes the contact of elastic half-spaces. In other words, the bodies in contact are assumed to be infinitely thick. However, when studying thin films, this may not necessarily be tme. The ramifications and prescribed solution to this problem were worked out by Akhremitchev and Walker. ... 

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