Fig. 5.31. A demonstrative visual representation of the reign lengths of dynasties a and b as graphs.

Fig. 5.32. A density function demonstrating the distribution of points pertinent to the set vir(D).

Fig. 5.32. A density function demonstrating the distribution of points pertinent to the set vir(D).

dependent pairs of dynasties. In other words, such distances between a priori dependent dynasties of annals, and those between a priori independent ones, prove to be comparable to each other. They appear to have "the same order of magnitude".

Moreover, it is impossible to determine the "similarity" or "dissimilarity" of two dynasties, or, to be more precise, graphs of their rule, "at a glance". Visual similarity of two graphs can indicate nothing. It is possible to give examples of a priori independent dynasties, the graphs of rule of which prove to be "very similar", although there will be no actual dependence. It turns out that visual proximity can easily lead to confusion in this problem. A reliable quantitative estimation is necessary, one that would eliminate unsteady subjective considerations like "similar" or "not similar".

Thus, the aim is to explain whether such a natural measure of proximity does exist in general within a set of all virtual dynasties, which would make it possible to confidently separate dependent dynasties from independent ones, or make the "distance" between a priori dependent dynasties "small", and the "distance" between a priori independent dynasties "large". Moreover, these "small" and "large" values should be essentially different from one another, for example, by one or several orders of magnitude.

Such a measure of proximity, or "distance between dynasties", appears to actually exist. We will now turn to the description of this coefficient c(a, b).

Thus, we plotted a set of dynasties D in space R15. Two most typical errors usually committed by chroniclers were simulated. Each dynasty of the set D was subjected to disturbances of types (1) and (2). In this case, each point from D multiplied into several points, which led to the increase of the set. We designated the set obtained as vir(D). The set vir(D) turned out to consist of approximately 15X1011 points.

We will consider "dynastic vector a" to be a random vector in Rk, passing through the set vir(D). Then, on the basis of the set vir(D) we can create a probability density function z. With this aim in mind, the entire space R15 was divided into standard cubes of sufficiently small size, so that no point of the set vir(D) would fall on the boundary of any cube. If x is an internal point of a cube, then we may assume that z(x) =

the number of points from the set vir(D) falling into the cube the total quantity of points in the set vir(D) '

It is clear that for a point which lies on a boundary of any cube, it is possible to consider z(x) = 0. Function z(x) reaches its maximum in the area of especially high concentration of dynasties from the set vir(D), and it drops to zero where there are no points of set (D), fig. 5.32. Thus, the graph of function z(x) clearly shows how the set of virtual dynasties vir(D) is distributed within space Rk, - in other words, where this set is "thick", "dense", and where it is rarefied.

Now we are given two dynasties a = (av ... ak) and b = (bv ... ,bk), and we want to estimate how close or distant they are. Let us plot a /c-dimensional parallelepiped P'(a, b) with its center in point a, which has as diagonal vector a-by fig. 5.33. If we project the parallelepiped P'(ay b) on the ¿-coordinate axis, we will obtain a segment with the ends

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