# Volume Functions Of Historical Texts And The Amplitude Correlation Principle

This section contains quotations from works by A. T. Fomenko and S. T. Rachev.

(S. T. Rachev, doctor of physics and mathematics, Professor, specialist in the field of probability theory and mathematical statistics, Research Fellow of the Institute of Mathematics of the Bulgarian Academy of Sciences; currently works in the USA.)

2.1. Dependent and independent chronicles. Volume function maxima correlation

We shall describe the results published by the authors in [723] and [ 1140]. As above, we shall call two historical chronicles X and Y dependent if they can be traced back to a common original source and record approximately the same events on the same time interval (A, B) in the history of the same region.

On the contrary, we shall consider two chronicles independent if they record events of substantially different time intervals (A, B) and (C, D), or describe events in obviously different geographical regions. We shall consider two time intervals substantially different if their intersection on the time axis (i.e., their common part) does not exceed half of their length. Hereinafter, for the sake of simplicity, we shall assume that chronicles compared describe time intervals of the same length, i. e., B - A = D - C.

Let chronicle X describe events on the time interval (A, B), and parameter t run through the years from year A to year B. As above, we shall mark the part of the chronicle that describes the events in the year t as X(t). For the sake of brevity, we shall conventionally call fragments X(t) chapters. Let us calculate the volume of each fragment in certain units, for instance, in quantity of lines, or in pages. In the examples below, the volume of chapters is calculated in lines. However, the choice of measurement unit is not of great importance here. During statistical processing we have normalized the volume of chapters by dividing them by the total volume of the chronicle, thus levelling a possible difference in choice of volume measurement units. So, we obtain the function vol X(t) that we call the volume function of the chronicle.

The correlation principle for local maxima points of the volume graphs was formulated and experimentally tested by A. T. Fomenko in [884]. The main idea in the basis of the principle and the methods pertinent to it is as follows: dependence or independence of chronicles can in certain cases be established by comparing their volume functions. Generally speaking, local maxima points of volume graphs of dependent chronicles should "correlate" (in a proper precise sense, see above), while independent chronicles should not display any "correlation", fig.5.1.

In their work [357], A. T. Fomenko, V. V. Kalash-nikov and S. T. Rachev, applied the general idea of volume function correlation for dependent chronicles, and the absence of correlation for independent chronicles, to volume functions themselves, that is, considering their amplitudes. Since the research involved the amplitudes of graphs, this enhanced form of correlation principle should have been tested on specific

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chronicles, which were performed in [357] with participation of N. Y. Rives. Detection methods for dependent and independent chronicles as offered in [357], turned out to be fairly efficient when comparing chronicles of approximately the same volume. However, the picture was becoming "smudged" when chronicles of substantially different volumes were compared. The current work specifies a new class of chronicles, for which the enhanced form of the local maxima amplitude correlation principle is correct.

The maxima correlation principle discovered by A. T. Fomenko relied upon the fact that different chroniclers, describing the same historical epoch, would generally use the same volume or fund of information that survived until their time. That is why, as our statistical experiments prove, they would describe in greater detail only those years from which many texts survived, and in smaller detail all the rest of them.

We shall recall the notion of primary information volume for events of epoch (A, B). Let C(t) be the volume of all documents written by the contemporaries of year t about the events of that year, fig. 5.2. Now, let X and Ybe chroniclers who are not contemporaries of the epoch (A, B) but willing to write its history. Let M (respectively N) be the year in which chronicler X (respectively Y) creates the chronicle for the epoch (A,B).

We shall recall that CM(t) is the volume of documents that survived from the epoch (A, B) till the moment M, or the epoch of the chronicler X, - in other words, the remainder of primary texts survived till M. Graph CM(t) is the volume graph for the surviving information about the events of the epoch (A, B). CN(t) is defined similarly.

The maxima correlation principle ensues from the following principle. Each chronicler X, describing the epoch (A, 5), "on the average" talks in greater detail about years in which the graph CM(t) peaks - i. e., the more documents from the epoch (A, B) are available to the chronicler X, the more detailed is his description of that time, q.v. in fig.5.3.

### 2.2. Rich and poor chronicles and chronicle zones

The definition of a poor chronicle or a rich one is intuitively clear from fig. 5.16. We shall call the chronicle with the "majority" of volumes vol X(t) equalling zero poor, where most of the years haven't been described by a chronicler. On the contrary, we shall call the chronicle with the "majority" of volumes vol X(t) other than zero and fairly large rich, where a chronicler reports ample information about the epoch (A, B).

In fact, for actual examples it is sometimes difficult to categorize a chronicle as either poor or rich, therefore, the introduction of new definitions - poor zone and rich zone of a chronicle - would be practical. Fig. 5.17 presents a relative volume graph of a chronicle with a poor beginning and a rich ending. Our research experience for specific chronicles makes it clear that the beginning of a long chronicle is a poor zone,

Fig. 5.18. The rich and the poor zones may alternate within one and the same chronicle.

Fig. 5.17. Tne poor initial zone of a chronicle, and a richer zone following it.

Fig. 5.17. Tne poor initial zone of a chronicle, and a richer zone following it.

Fig. 5.18. The rich and the poor zones may alternate within one and the same chronicle.

and its ending is a rich zone, typically, although there are chronicles with a poor zone "in the middle", q.v. in fig. 5.18.

2.3. Significant and insignificant zeroes of volume functions

In our study of a specific chronicle we shall assume the first year for which vol X(A) differs from zero as the leftmost point A on the time axis, the year is described by a chronicler, in other words, we shall call the zero of a volume graph significant if it is located to the right from the first non-null value, fig.5.19. If the zero is to the left from the first non-null value of the graph, then we shall call it insignificant. An insignificant zero indicates that not only does the chronicler know nothing about that particular year, but also nothing of preceding years in general. A significant zero indicates that, although the chronicler knows nothing about that particular year, he knows at least something about some of the previous years.

From this moment on, we shall not normalize the volume function, since we want to consider the magnitude of amplitudes of local maxima in our research.

### 2.4.The information respect principle

Let us consider a certain historical epoch (A, B) and a chronicler X who lives in year M, where M is much bigger than B, fig.5.20. Describing the events of the epoch (A, 5), the chronicler X has to rely on the surviving information fund CM(f), still available in his time. Our idea is that the chronicler X treats poor and rich zones of the survived information fund differently.

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