Ultrametric space

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 ... 

Thus, each point on the nth level of an ultrametric space corresponds to an n-digit number in the /-digit system of calculation (Fig. 13) ... 

The distance between two points in the ultrametric space is defined by the number of steps from these points to the common limit. For example, the distance between points 00 and 03 equals 1, and the distance between points 02 and 12 equals 2 (Fig. 13). Thus, the distance between two points in the ultrametric space with coordinates given by w-digit numbers in the /-digit system of calculation only depends on which digit these numbers first differ and does not depend on the specific values of this difference. 

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 ... 

The unity of clusters corresponds to an arbitrary distance l between points in the ultrametric space. All points are united in j subclusters with distance l — 1 and having/ points. Thus, the group of clusters formed on the hierarchical level n — 1 corresponds to an arbitrary distance l. If the limit transition is made when n —> oo, then the number of the points attaining level n approaches infinity (Nn —y oo) that is, the intervals between points Ax = 1/Nn become infinitely small, and the ultrametric space itself becomes continuous. On the Cayley free, the transition to a continuous ultrametric space indicates a condensation of the hierarchical levels. 

The distance between two points in an ultrametric space in the conventional Euclidean sense can be defined as... 

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. 

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,... 

There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Onc complex numbers were equally outlandish, but th frequently proved the shortest path between real results. Similady, the first two topics named have already provided a number of wormhole paths. There is no telling where all this is leading -fortunatdy. 

The ultrametricity of the tree is a direct consequence of the non-intersecting property of the locally optimal paths. In ultrametric space, any three points A, A2 and A3 satisfy the inequality [22] ... 

Looking at overlaps between three valleys (Mezard et al. 1984) it turned out to one s surprise that there are restrictions on the values these overlaps can take there is no probability associated with all three overlaps different. Such restrictions characterize an ultrametric space. Physically it arises from a hierarchical structure of valleys within valleys within.. . . 

Zumofen,G., Blumen, A., and Klafter, J., Reaction Kinetics on Ultrametric Spaces, J.Chem.Phys. 84 6679(1986). 

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