## Ui M

Fig. 5.30. "Globular clusters" vir(M) and vir(N) corresponding to two a priori independent and different dynasties M and N that are separated by a considerable distance.

(2) Either q{ = piy or q{ coincides with the number Pi +pi+i.

It is clear that each such vector (dynasty) q may be considered as a dynasty of annals, resulting from an actual dynasty p by a "reproduction thereof" due to errors (1) and (2) made by chroniclers. In other words, we take each real dynasty p = (ply p2>..., pk) from the list D and apply "disturbances" (1) and (2) to it, which means that we either swap places of two adjacent numbers pi and pi+ly or substitute a certain number pt by the sum pt + pi+ly or sum p^ + pt. For each number z, we use the above operations just once, that is, we do not consider "long iterations" of operations at the same place i. As a result, we obtain a certain number of virtual dynasties {q = vir(p)} from one dynasty p. The quantity of such virtual dynasties is easy to calculate.

Thus, each "point" from set D is "multiplied" and generates a certain set of "virtual points" surround ing it, a "surrounding cloud", or "globular cluster", fig. 5.29. We may come across some of the obtained virtual dynasties in a certain chronicle (in this case they will be dynasties of annals), while others remain just "theoretically possible", or "virtual".

By uniting all virtual dynasties obtained from all actual dynasties p, which compose our list of dynasties D, we obtain a certain set vir(D)y i. e., "a cloaking cloud" for the initial set of dynasties D.

Thus, for each actual dynasty M the set of dynasties of annals describing it can be pictured as a "globular cluster" vir(M). Let us now consider the two actual dynasties M and N. If the small distortions principle formulated by us is accurate, then the globular clusters vir(M) and vir(N) corresponding to two a priori independent, different actual dynasties M and N do not intersect in space Rky which means that they must be arranged sufficiently far from each other, q.v. in fig. 5.30.

Now let a and b stand for two certain dynasties from set vir(D)y for example, two dynasties of annals, q.v. in fig. 5.31. We would like to introduce a certain quantitative measure of proximity between two dynasties, or "measure the distance between them" - estimate how distant they are from each other, in other words, the easiest method would be as follows. Regarding both dynasties as vectors in space Rky it would be possible just to take the Euclidean distance between them, or calculate the number r(ayb)y the square of which assumes the form of

However, numeric experiments with specific dynasties of annals show that this distance does not make it possible to confidently separate dependent and in