Cayley trees

One example of a tree-based separator system is shown below in Fig. 2.8 where the Bethe lattice or Cayley tree is shown (Wilson, 1996). This graph can be expanded to any number of levels and can function with dilferent types of columns and electrophoretic elements. This is not the only graph that can function as a complex multidimensional separator system. But it is an example of something with multiple... 

FIGURE 2.8 Separator systems cascaded to form a Bethe lattice or Cayley tree where the point of introduction is the graph vertex 0 and solute can be sampled from any of the outward nodes at position 1, 2, 3, 4, and so on. The sample loops and valves are not shown. 

A useful toy theoretical model which captures the essential features of self-entangled dendritic polymers is the monodisperse Cayley tree in which each chain segment branches with a fixed functionality z at each of its ends, except those at the extremity of the molecule (see Eig. 13). Smaller versions of these structures, too low in molecular weight to be entangled, have been synthesised and are usually referred-to as dendrimers [47]. 

The result of treating the Flory-Stockmayer ensemble in this way and at the same level of approximation as that of the Cayley tree sketched in the previous section, is that a very similar relaxation spectrum to Eq. (36) is predicted, but with a value for the exponent 0 of 4 rather then 2. A disentanglement transition is also found, as well as the appeafing feature that at each seniority a fixed fraction of the molecules still entangled actually renormalise into linear polymers (like the comb topology) and relax by reptation rather than by further fluctuation [49]. 

Figure 3. Robust super high-spin organic macromolecule with a dendritic architecture based on a three-coordinated Cayley tree.

Our choice was the two series of dendritic polymers 5 and 6, depicted in Figure 4, which have all their open-shell centers (or trivalent carbon atoms) sterically shielded by an encapsulation with six bulky chlorine atoms in order to increase their life expectancies and thermal and chemical stabilities. Indeed, it is very well known that the monoradical counterpart of both series of polyradicals, the perchlo-rotriphenyl methyl radical, shows an astonishing thermal and chemical stability for which the term of inert free radical was coined. The series of dendrimer polymers 5 and 6 differ in the nature and multiplicity (or branching) of their central core unit, N, as well as in their branch-juncture multiplicities, N Thus, series 5 has a hyperbranched topology with = 3 and = 4, while dendrimer series 6 has a lower level of branching with = 3 and = 2, and the topology of a three-coordinated Cayley tree. 

Network structures are still determined by nodes and strands when long chains are crosslinked at random, but the segmental spacing between two consecutive crosslinks, along one chain, is not uniform in these systems which are currently described within the framework of bond percolation, considered within the mean field approximation. The percolation process is supposed to be developed on a Cayley tree [15, 16]. Polymer chains are considered as percolation units that will be linked to one another to form a gel. Chains bear chemical functions that can react with functions located on crosslinkers. The functionality of percolation units is determined by the mean number f of chemical functions per chain and the gelation (percolation) threshold is given by pc = (f-1)"1. The... 

The procedure of fractal set construction can be shown using the Cayley tree so that each fractal set has its own Cayley tree [25,26]. We show a Cayley tree with branch characteristic j = 4. 

The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. 

The following constitutes the definition of the distance between two points in an ultrametric space. The points in an ultrametric space on a given hierarchical level are the ends of the Cayley tree branches (Fig. 13). The number of points on the rath level of the Cayley tree is equal to Nn = j". Each point on the nth level can be numbered ... 

The points of the discrete ultrametric space (Cayley tree junctions) on the nth level, namely, N , are divided into clusters (groups). Each cluster contains j points the distance between which is / = 1 and has its progenitor on the (.n — l)th level. The number of such clusters is N /j = jn l ... 

Representation (134) corresponds to Cayley tree division / into n groups, each of them consisting of clusters. Each of the clusters of the group is... 

This approximate equation means that the ultrametric space has a logarithmic metric. Thus, when constructing a fractal set, each element corresponds to a point of the ultrametric space with geometric image represented by the Cayley tree. 

The hierarchical chain of changes from the initial state (t = 0) to the final one (t —> oo) can be compared to a Cayley tree [25,26] (see Fig. 64). Here, the knots of the Cayley tree will correspond to static ensembles a and p which correspond to the dots in ultrametric space divided by the distance lap. 

Figure 64. Schematic of a self-similar structure potential energy landscape and of the Cayley tree.

The value of lap is defined by the number of steps over the levels of the Cayley tree up to the mutual knot in Fig. 64 and it yields the extent of a hierarchical link. Therefore, both the barrier height, Qap, and the relaxation time, xap, are connected with functions of the distance lap in ultrametric space, that is,... 

The parallel action of different relaxation channels is only possible under conditions of hierarchical co-subordination of the corresponding collection of static ensembles. The hierarchical co-subordination means that the parallel net of channels of the next level having relaxation time tW does not act until channels with the given relaxation time X 1-1 > tW act. Thus, the fastest processes take place first they correspond to surmounting the barriers of minimum height Gap- Here, static ensembles merge with each other, and the system attains a higher hierarchical level of the Cayley tree. 

We remark that the self-similar potential energy landscape (Fig. 68) resembles a Cayley tree (Fig. 68), provided each minimum at a certain self-... 

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