**1. INTRODUCTION. SHORT HISTORY OF THE PROBLEM.**

In the literature, history, linguistics very often appears the problem - who is the real author of some text? Or, are several different texts written by one and the same author? For example, is it true that Plato's dialogs were written by one and the same author? Is true that the Shakespeare's plays are creations of one person? Or, they were written by different people? The special interest have the problems connected with possible plagiarism.

Let us recall some approaches to the solution of these problems.

For example, in the paper of W.Fucks [1] was analyzed the problem of authorship for some ancients texts on the basis of statistical analysis of different grammar structures of their language.

Many papers were devoted to the discovery of quantitative rules and parameters which allow to distinguish different literary genres, namely, poetry, dramaturgy, et cetera [2]. The attempt of using of exact mathematical methods for the solving of plagiarism problem is represented, for example, in the book [10].

The problem of discovering of author's invariants is discussed in many scientific papers. For example, the linguistic structure of different authors was analyzed with the help of individual words, in particular, - preposition "v" (engl. in), particle "ne" (engl. not) [3], or with the help of the length of the words and sentences [4]. But, as was discovered during the experiments, the linguistic spectrums of individual words do not allow to find the stable author's invariants. This fact was formulated in 1916 by academician A.A.Markov [5], who noted that for the sample of large volume the results of such a kind should "oscillate near some middle value, which is specific for a given language", and do not distinguish different authors. In other words, very often the parameters for different authors oscillate near some common constant, which is the characteristics of the language, but not the authors.

Useful approach was demonstrated in the works of W.Fucks. He suggested to correspond to each author the following numerical characteristics: mean quantity of the syllables (in the words) and mean quantity of the words in the sentences. This idea allows us to represent each text (each author) by some point on 2-plane (in case of two parameters), or by some point in multidimensional space (in case of many analogous parameters).

Let us note interesting research in Russian linguistics. See, for example, [6], [7], [8], [9].

**2. WHAT IS AUTHOR'S INVARIANT?**

We suggest "dynamic approach" instead of "static approach". Here we use the term "static approach", when each author is "represented" by some set of "static numbers". For example: (mean length of the words in the whole text, mean length of the sentences in the whole text,...). We will divide the text of each individual author on the set of "subtexts", "chapters", "samples" and will calculate the evolution of some fixed parameter through the whole text. In other words, we calculate some value for each sample, and because the set of samples is ordered, we obtain some graph showing the oscillation of chosen parameter inside the text of the author.

We call AUTHOR'S INVARIANT the quantitative characteristics of literary texts (i.e. some numerical parameter), which satisfy to following conditions:

1) The graph (the "curve") of this dynamic parameter is "constant" (in other words, oscillate very close to some fixed constant) through all the texts of each author.

2) The dynamic parameter takes different values at least for several groups of authors. (Of course, there are some authors with close values of this dynamic parameter). In other words, there are authors with considerable different values of dynamic parameter.

We will try to find the dynamic parameter such that the number of such "different authors groups" is more or less large. It is clear that the discovery of such invariant is very complicated problem because of very mixed character of grammar structures used in literary texts. The simple numerical experiments show that the finding of the invariants separating some authors, is nontrivial task. The reason is as follows. When author write the book, he is under the influence not only unconscious factors, but also under conscious factors. For example, the frequency of using the rare and foreign words in the book is some indicator of author's stile and his erudition. But this parameter can be easily controlled by the author of the conscious level, because rare and foreign words are inserted in the text sufficiently rare. And each time the author "says" something like: "here I specially insert some specific rare of foreign word". As a result, the frequency (of rare and foreign words) cannot serve as "author's invariant". This remark is confirmed by concrete numerical experiments with concrete texts. Because author control this frequency, he can easily change it from one his book to another. In this case we obtain the jump of this numerical characteristics.

Let us formulate the more detailed necessary conditions for author's invariant.

1) The invariant (dynamic parameter) should be sufficiently integral, "massive", and should be stable, not under conscious control of the author. In other words, this author's characteristics should be his "unconscious parameter", located so "deep", that author even not think about it. If this parameter suddenly changes, this event should be "local", "very short in time", and after several small fluctuations, the parameter should return to its previous value, which is typical for a given author.

2) The parameter should be "constant" through all the texts (books) of the author. In other words, the dynamic parameter must have only small fluctuations (around some mean value) along the all consecutive samples through all the books of a given author. Exactly this property allows us to call this parameter "author's invariant".

3) Finally, the dynamic parameter should distinguish different groups of the authors. In other words, should be sufficiently large number of different groups of the authors characterizing by considerable different values of this invariant. The values of invariant for the authors inside the same group can be very close.

The third condition is important. Really, there are dynamic parameters which have small fluctuations inside the books of different authors, but the values of the parameter for these authors are very close, practically identical. All different authors have the same value of this "author's parameter". It is clear that in this case the parameter cannot be considered as "author's invariant".

ONLY THE COMBINATION OF ALL THREE CONDITIONS MENTIONED ABOVE ALLOW US TO SPEAK THAT WE FIND THE AUTHOR'S INVARIANT.

**3. OUR APPROACH. SAMPLES AND STEPS.**

Let us consider the texts (books) of some author and let us enumerate them in chronological order, i.e. in the order they were written. We call this sequence of the texts by the TEXT OF A GIVEN AUTHOR. Thus, in our approach, text of the author can consist of several his books - novels, stories et cetera. Then we extract from the text the consecutive individual fragments consisting of the same fixed number of words. We call these fragments - samples. The number of the words in the sample we call the "volume of the sample".

These samples we extract from the text step by step. The distance between the samples is the same and is equal to some fixed constant, which we call "value of step" (Fig.1). The volume of the sample and the value of step we can vary.

Thus, we go through the texts of the author and after each 10 pages, for example, of standard book text, we extract the samples of the same volume. We started from the volume of the sample equal to 2000 words. If the text is large, then we can choose many samples. For short texts the number of the samples is evidently small. In this case the results of statistical analysis can be unstable.

Let us choose some concrete linguistic parameter, for example the frequency of using by the author the preposition "v" (engl.in). We extract the consecutive samples and for each sample calculate the value of this parameter. As a result, we obtain some number for each sample. For different samples this value can be different. Let us construct the graph showing the evolution of the parameter. We mark on the horizontal line the numbers 1,2,3..., which enumerate the consecutive samples. Then, we put on the vertical lines the values of the parameter. As a result, the behavior of the parameter is represented by some curve on the plane.

Thus, we represent each author not by a point on the plane (or in the space), as it was done in the papers [1], [2], but some curve, some graph. The curve visually represents the evolution of a given parameter along all the texts of a given author. It turns out, that such a graphs are very useful for the problem of finding the author's invariants. Really, now the problem can be reformulated as follows.

It is necessary to find such a linguistic parameter and such optimal value of samples, that the corresponding graphs become (for all authors) PRACTICALLY HORIZONTAL LINES, "straight lines", i.e. their oscillations should be "very small".

If we find such a parameter, then its values have small deviation from some mean value through all the texts of each individual author. This effect - the smoothing of the curve and its transformation into the "horizontal straight line" - we call STABILIZATION of the linguistic parameter.

But, let us recall, the fact of stabilization is insufficient to claim that we really have discovered the author's invariant. It is necessary that for different groups of the authors the obtained "horizontal straight lines" should have sufficiently different position, they should be on different "heights". Let us recall once more, that sometimes the "horizontal lines" representing different authors can be very close one to another, In this case the value of author's invariants are very close. We will collect all authors with close invariants in one group. If the values of invariant for some two texts are close, we cannot conclude that these texts belong to one author.