Fig. 5.21. Function graphs F(x) and g(x) = 1 - F(x).
Fig. 5.21. Function graphs F(x) and g(x) = 1 - F(x).
non-decreasing graph with values increasing from 0 to 1, character of this increase differing for various chronicles.
Let us consider a new function g(x) - I- F(x). See fig. 5.21. Its graph does not increase. Omitting mathematical precision, we shall formulate the next model.
The function g(x) - I- F(x) should behave in the poor, early zone of the chronicle as function exp(-Xxa).
In mathematical statistics, distributions of such kind are called the Weibull-Gnedenko distributions which are used in mathematical statistics for the description of similar processes.
Therefore, we have two degrees of freedom at our disposal: the parameter X and the parameter a, swapping which, we can try to approximate the function 1 - F(x). If we manage to do it for specific chronicles, this will prove our theoretical model.
The statistical experiment that we performed with actual chronicles demonstrated that the decrease of j i
Fig. 5.22. Depiction of the two parameters - the shape and the volume of the chronicle in question - with a point on a plane.
the graph 1 - F(x) is indeed fairly well approximated by the function exp(-Xxa), given a suitable choice of values for X and a.
As a result, we can juxtapose over each chronicle - or rather over its beginning - the poor zone thereof, of the two numbers X and a reflecting the character of the chronicle's volume function behaviour. We shall call X the parameter of the chronicle's volume, and a the parameter of the chronicle's form.
The parameter a turns out to be more important to us since, as statistical experiments have demonstrated, it is this parameter that has a better sense of the distribution character of individual scarce peaks of volume graphs within the poor zone of a chronicle. The parameter a will be the first to indicate whether chronicles are dependent or independent. The parameter X is rather responsible for the chronicle's volume, it demonstrates how rich or poor the chronicle is.
So, our hypothesis, or the statistical model may now be formulated in the following way.
a) If chronicles X and Fare dependent, then their pairs of corresponding parameters (ax, Xx) and (ay, Xy) should be similar, stipulating that they are calculated for the poor zones of the chronicles.
b) If the chronicles X and Fare independent, then their pairs of corresponding parameters (ax, Xx) and (GCy, XY) should be at some distance from each other.
It is convenient to picture the pair of numbers (a, X) as a point on an ordinary plane with Cartesian coordinates a and X. See fig. 5.22.
2.7. The hypothesis about the increase of the "form" parameter of a chronicle in the course of time
We shall now consider two different historical epochs: one with a poor primary information fund, and one with a rich primary fund. In the latter case, we assume the volume of this fund to be more or less constant for each year. Then, it can be demonstrated (omitting mathematical details) that the value a in the first, poor case should be less than the value of a in the second, rich case (, ). See also articles 2.13 - 2.15. In other words, poor primary funds are characterized by small values of a, and the rich primary information funds by large values of a.
But the closer historical epoch (A, B) is to our time, the better do the primary information funds survive. Today, for instance, written information is, by and large, on the average kept better than in the distant past. Therefore, the value of the parameter a should "on the average" increase, as we shift the time period (A, B) under study from left to right on the time axis, i.e., closer to us.
2.8. The list and characteristics of the Russian chronicles we investigated
1) Povesf vremennykh let (Story of Years of Time). See Literary Memorials of the Ancient Rus\ The Beginning of the Russian Literature. Moscow, 1978.
This famous chronicle covers the events in the history of Russia, allegedly between the IX and XII century a.d. The main part of the chronicle describes the epoch of the alleged years 850-1110 a.d. in the consensual chronology. The chronicle begins with a poor zone approximately one hundred years long, starting allegedly in 850 a.d. and ending in the alleged year 940 a.d. The next part of the chronicle, beyond 1050-1110 a.d., is fairly rich.
2) Nikiforovskaya letopisy (The Nikiforov Chronicle), of the Byelorussian-Lithuanian group of chronicles. See The Complete Russian Chronicles, Volume 35, Moscow, 1980. The period of 650 between the alleged years 850 a.d. and 1450 a.d. has been taken for our research work.
3) SuprasVskaya letopis* (The SuprasT Chronicle), of the Byelorussian-Lithuanian group of chronicles. See
The Complete Russian Chronicles (CRC for short), volume 35, Moscow, 1980. The period for which this chronicle provides the dates is allegedly 850-1450 a.d. This chronicle, as well as the Nikiforov one, can be rather ranked among poor texts in comparison with the richer Povesf vremennykh let.
4) Akademicheskaya letopis'(The Academy Chronicle). See CRC, volume 35, Moscow, 1980. We have researched the period of 1338-1378 a.d. This chronicle is intermediate between poor and rich texts.
5) Kholmogorskaya letopis' (TheKholmogory Chronicle). See CRC, volume 33, St. Petersburg, 1977. It covers the period of the alleged years 850-1560 a.d. This chronicle contains both rich and poor zones.
6) Dvinskoy letopisets (The Dvina Book of Chronicles). Short and full editions. See CRCy volume 33, St. Petersburg, 1977. It covers the period of 1390-1750 a.d. This chronicle contains both rich and poor zones.
All these chronicles begin with poor zones, which comes as no surprise. A. T. Fomenko calculated the volume functions. See Chroni, Appendix 5.1. Among the listed chronicles, there are a priori dependent and a priori independent ones. For instance, among the a priori dependent are:
a) Nikiforovskaya letopis' and SuprasVskaya letopis b) Povesf vremennykh let and Nikiforovskaya leto-pis\ therefore SuprasVskaya letopis\ too.
c) Short and full versions of Dvinskoy letopisets.
A priori independenty for instance, are the part of
Dvinskoy letopisets covering the XIV century a.d., and the next one covering the XV century a.d.
The fact of dependence or independence of the listed chronicles has been confirmed in  and  on the basis of the maxima correlation principle, q.v. above.
All listed chronicles were divided into pieces covering approximately 100 years, each one examined with the method stated above. As a result, the parameters ax and Xx, and the correlation coefficient r indicating how well the corresponding graph exp(-fop) approximates the decreasing graph 1 - F(x), were calculated (see table 5.1).
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