this confirms the peak correlation principle for the volume functions of dependent texts.

This dependence of texts can be expressed numerically. Let us introduce the following "distance" between volume functions volX(t) and vol Y(t) for the two texts X and 7, each divided into clusters of separate per annum fragments X(t) and Y(t), respectively. Let us recall that the fragments X(t) and Y(t) describe the events of just one year t.

Let parameter t vary within the time interval from year A to year B. Let us designate by t(Xy 1), t(X, 2), ..., t(X, N) the years in which such peaks, or local maxima, occur on volume graph vol X( t). Accordingly, let us designate the peaks of the volume graph vol Y(t) by t(Yy 1), t(Yy2)y...y t(YyM).

For each point t(Xy z), let us find the point nearest to it in the sequence t(Yy 1), t(Yy 2),..., t(Yy M). Let it be a certain point t(Yy k). Let p(i) designate the distance between them in years, i.e. the absolute difference value t(Xy i) - t(Yy k). In other words, we shall find out which local maximum of Y is the nearest to the selected local maximum of X.

In a perfectly similar manner, swapping the roles of X and Yy for each point t(Yyj) let us find the nearest point to it in the sequence t(Xy 1), t(Xy 2),..., t(Xy N). Let it be a certain point t(X, s). Let q(j) designate the distance between them in years, or the absolute value of difference t(Yyj) - t(X, s).

Finally, we assume the following sum as "the distance between X and Y":

R(XyY)=p(l)+p(2)+ ... + p(N)+ q(l)+ q(2)+ ... + q(M).

The meaning of the distance R(Xy Y) is completely clear. For each local maximum of function vol X(t) we find the nearest local maximum of function vol Y(t)y determine the distance between them in years, and sum up the numbers obtained. Then we repeat this operation after swapping the positions of chronicles X and Y. Summing up the numbers obtained, we obtain R(X, Y). It is clear that R(X, Y) =R(YyX).

If distance R(X, Y) equals zero for a certain pair of texts X and Y, consequently, their volume function graphs peak simultaneously. The greater this distance, the worse the correlation between their local maxima points. It is also possible to examine the asymmetrical distance from X to Y, assuming that

Likewise, the asymmetrical distance from Y to X is determined, namely, q(Y, X) = q(l)-\- g(2)+...+ q(M).

Let us numerically estimate a degree of dependence between the historical texts 1-22 listed above, for which we shall calculate a 22X22 square matrix of two-by-two distances R(X, Y), where X and Y pass through all texts 1-22, independently from each other. Let us then calculate a frequency histogram, for which we shall consider the horizontal axis, on which we shall mark the integer points: 0, 1, 2, 3, ... and plot the following graph. Let us calculate the number of zeroes in the matrix {R(X, Y)} obtained earlier. The number obtained will be plotted on the vertical axis at the point of which horizontal coordinate is equal to zero. Then we shall calculate the number of unities in the matrix {R(X, Y)}, plot the obtained number on the vertical axis at the point of which horizontal coordinate is equal to 1, and so on. We shall come up with a graph called frequency histogram. What can a study of the obtained histogram tell us?

If the chronicles selected for the analysis are de-

pendent, then the majority of two-by-two distances between the chronicles must be expressed in small numbers, which is to say, the chronicles must be close to each other, meaning that the majority of matrix elements {R(X, Y)} must be "small" or close to zero. In that case, however, the absolute maximum of the frequency histogram must be shifted to the left, that is, there should be a large set of small frequencies. On the contrary, if there are many independent texts among those under investigation, then the maximum of the frequency histogram is shifted to the right, q.v. in fig. 5.26. The share of "large" and "medium" two-by-two distances between the chronicles should therefore increase.

This observation makes it possible to evaluate the degree of dependence or independence for a group of chronicles by plotting an appropriate frequency histogram based on matrix {R(X, Y)}. Namely, a shift of the maximum to the left indicates a possible dependence of chronicles, while a shift of the maximum to the right indicates a possible independence.

This idea was used to evaluate the degree of dependence of historical texts 1-22 enumerated above. Fig. 5.27 shows the experimental histogram of the matrix {R(X, Y)} for texts 1-22. This matrix proved to possess many small numbers, therefore the maximum of the histogram is visibly shifted to the left. This indicates the dependence of historical texts 1-22.

For comparison, let us plot a histogram for independent texts. To present an example, we decided to compare three chronicles A, B, C mentioned below with the preceding texts 1-22. The three additional chronicles are:

A: Povesf Vremennykh Let, allegedly 850-1110 a.d., B: Akademicheskaya letopis\ allegedly 1336-1446 a.d., C: Nikiforovskaya letopis\ allegedly 850-1430 a.d.

For each of them, a volume function was calculated and all local maxima found. Let us calculate all two-by-two distances of {R(Xy Y)}, where X passes through the three chronicles A, By C, and Y passes through the historical texts 1-22. As a result, we obtain a rectangular 3 X22 matrix {R(Xy Y)}. Then a frequency histogram was calculated, with its result shown in fig. 5.28. A totally different nature of this histogram is distinctly visible - its maximum moved to the right. This indicates independence of two groups of texts: {A, By C} and {texts 1-22}. Each of these groups can certainly contain dependent texts.

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