Applied quantitative analysis and modeling of international relations. Mathematical methods in international relations. Meaning of the method problem

Improvement of computer technology, further development of the mathematical apparatus increases the range of

E. G. Baranovsky, N. N., Vladislavleva
changes in exact methods in the humanities, including in international relations. The use of mathematical methods in the conduct of political research makes it possible to expand the traditional methods of qualitative analysis and improve the accuracy of predictive estimates. International relations are a sphere of public activity with a huge number of factors, events and relationships of the most diverse nature, therefore, on the one hand, this area of knowledge is very difficult to formalize, but on the other hand, for a complete and systematic analysis, it is necessary to introduce common concepts and a certain unified language: “Politics, dealing with problems of fantastic complexity needs a common language... There is a need for a coherent and universal logic and precise methods for evaluating the impact of a particular policy on the achievement of goals. You need to learn to visualize complex structures clearly in order to make the right decisions. .
Mathematical tools used today in the study of international relations, in the vast majority of cases, were borrowed from the related social sciences, which in turn drew them from the natural sciences. It is customary to single out the following types of mathematical tools: 1) means of mathematical statistics; 2) apparatus of algebraic and differential equations; 3) game theory, computer simulation, information and logic systems, "non-quantitative sections" of mathematics.
Mathematical approaches in the analysis of international relations are used in two ways - to solve tactical (local) issues and to analyze strategic (global) problems. Mathematics also acts as a useful tool for building a model of international relations of various levels of complexity. At the same time, it should be taken into account that “the application of quantitative methods in the social sciences is based on the creation of such models, which in their essence depend not so much on the absolute values of the numbers, but on their order. Such models are not designed to obtain numerical
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results, but rather to answer questions about whether or not some property, for example, stability, takes place.
When constructing formalized models and applying mathematical methods, the following conditions must be taken into account.
1) Conceptual models should allow formalizing the existing information array into quantitatively measurable indicators. 2) When building forecasts based on the use of formalized methods, it should be taken into account that they are able to calculate a limited number of options in strictly defined areas of application.
The main steps in building a formal model include:
1. Development of hypotheses and development of a system of categories.
2. The choice of methods for obtaining conclusions and the logic of transforming theoretical knowledge into practical consequences.
3. Choice of mathematical display, adequately applied theory.
It should be noted that the problems that arise when constructing a system of hypotheses and categories are the most difficult to resolve. A hypothesis should be such a theoretical construction that, on the one hand, would adequately reflect the qualitative aspects of the object of study, and on the other hand, would provide for the division of the object into formalizable and measured units or isolating a system of indicators that adequately reflect the state of the object and the changes that occur in it.
There are also special requirements for the categories used in the formalization process. They must correspond not only to theoretical approaches and a system of hypotheses, but also to the criteria of mathematical clarity, that is, to be operational. The best option seems to be the construction of a categorical apparatus according to the “pyramid” principle, so that the content of the most generalized categories is gradually revealed by categories covering specific phenomena, and would be reduced to categories that go to quantitatively measurable indicators.

Methods for analyzing international conflicts
The formalization of political science categories and systems of hypotheses, the construction on this basis of a model of a conflict situation and process suggest that within the framework of a formal description it is necessary to state the largest possible number of ideas in the most capacious form. At this stage, the important points are the generalization and simplification of international processes and phenomena. The greatest difficulty is the translation of qualitative categories into a quantitative (measurable) form, which essentially boils down to assessing the significance of each category ... For this, the scaling method is used.
The mathematical tools used in the field of applied analysis of international relations include the following methods.
I. Extrapolation. The technique is an extrapolation of events and phenomena of the past to future period, for which data is collected in accordance with selected indicators at certain time intervals. As a rule, extrapolation is done only in relation to small time intervals in the future, since the probability of error increases significantly with a longer period. This is called the forecast lead depth. To determine it, you can use the dimensionless indicator of the depth (range) of forecasting proposed by V. Belokon: ? =?t/tx, ?t absolute lead time; tX is the value of the evolutionary nickname of the predicted object. Formalized methods are effective, if the magnitude of the lead time? " 1.
The basis of extrapolation methods is the study of time series, which are time-ordered sets of measurements of certain characteristics of the object or process under study. The time series can be represented in the following form:
уt = Xt + ?t where
Xt is a deterministic non-random component of the process; 136

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international conflicts
?t - stochastic random component of the process.
If the deterministic component (trend) хt characterizes the existing dynamics of the development of the process as a whole, then the stochastic component еt reflects random fluctuations or noises of the process. Both components of the process are determined by some functional mechanism that characterizes their behavior in time. The task of forecasting is to determine the type of extrapolating functions хt, еt based on the initial empirical data. To estimate the parameters of the selected extrapolation function, the least squares method, the exponential smoothing method, the probabilistic modeling method, and the adaptive smoothing method are used.
2. Correlation and regression analysis. This method allows you to identify the presence or absence of relationships between variables, as well as to determine the nature of such relationships, that is, to find out what is the cause (independent variable) and what is the effect (dependent variable).
For the linear case, the multiple regression model is written as:
Y = X x? + ?, where
Y - vector of function values (dependent variable); X - vector of values of independent variables;
? - vector of coefficient values;
? is the vector of random errors.
3. Factor analysis. A systematic approach to forecasting complex objects means the maximum possible consideration of the totality of variables that characterize the object, and the relationships between them. Factor analysis makes it possible to make such an account and at the same time reduce the dimension of system studies. The main idea of the method is that variables (indicators) that are closely correlated with each other indicate the same reason. Among the available indicators, their groups are searched for, which have a high level (value) of correlation, and on their basis, the so-called complex variables are created, which are combined by

N, G. Baranovsky, N. N. Vladislavleva
Methods of analysis of international conflicts
correlation coefficient. On the basis of indicators,
factors.
1. Spectral analysis. This method allows you to accurately describe processes whose dynamics contain oscillatory or harmonic components. The process under study can be represented as:
х(t) = х1(t) + х2(t) + х3(t) + ?(t), where
х1(t) - secular level;
x2(t) - seasonal fluctuations with a twelve-month period; х3(t) - fluctuations with a period greater than the seasonal ones, but shorter than those of the corresponding secular level fluctuations;
?(t) - random fluctuations with a wide range of periods, but with a small intensity.
Spectral analysis makes it possible to identify the main vibrations in complex structures and calculate the frequency and duration of the phase. The basis of the method is the selection of the structure of the oscillatory process and the construction of a graph of sinusoidal oscillations. To do this, chronological data is collected, an equation of oscillation is compiled, cycles are calculated, on the basis of which graphs are built.
5. Game theory. One of the main methods for analyzing conflict situations is game theory, which was initiated by the work of von Neumann in the 1920s and 1940s. After a period of rapid prosperity and an excessive abundance of research from the 50s to the early 70s, a noticeable decline occurred in the development of game theory. In part, the disappointment in game theory is explained by the fact that, despite the many mathematical results and proven theorems, researchers have not been able to make significant progress in solving the problem they set themselves: to create a model of human behavior in society and learn how to predict the possible outcomes of conflict situations. However, the efforts expended were not in vain. It turned out that of the concepts developed in game theory, they are very convenient for describing all kinds of problems that arise in the study of conflict situations.

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international conflicts
Game theory allows you to: structure the problem, present it in a foreseeable form, find areas of quantitative assessments, orderings, preferences and uncertainty, identify dominant strategies, if they exist; fully solve the problems that are described by stochastic models: identify the possibility of reaching an agreement and explore the behavior of systems capable of agreement (cooperation), that is, the interaction area near the saddle point, equilibrium point or Pareto agreement. However, many questions remain behind the possibilities provided by game theory. Game theory proceeds from the principle of average risk, which is far from always true for the behavior of participants in a real conflict. Game theory does not take into account the presence of random variables that describe the behavior of the conflicting parties, does not provide a quantitative description of the structural components of the conflict situation, does not take into account the degree of awareness of the parties, the ability of the parties to quickly change goals, etc. However, this does not detract from the advantages that the application of game theory gives to problem solving at certain stages of the conflict. It should be noted that for a systematic study of conflicts, there are two ways: 1. To describe the interaction of systems in a fairly general form, taking into account all significant factors and based on systemography, to detect and investigate the possible nature of the interaction of the conflicting parties, the causes of the conflict, mechanisms, course, outcomes, etc. Such models turn out to be large-scale, requiring large computational resources, but at the same time they give a multifaceted, sufficiently reliable result. 2. Assume that the parties, the causes and nature of the conflict are known, identify the main factors, build simple calculation models to assess the weight of the a priori factor and the results of the conflict. The path is rather narrow, but economical and efficient, giving concrete results for the parameters of interest in a short period of time. Both methods are used depending on the nature of the research objectives. For strategic research aimed at identifying

E. G. Baranovsky, N. N. Vladislavleva
Methods for analyzing international conflicts
potential conflicts, influence on the entire system of international relations, the formation of a long-term strategy for the behavior of the state in relation to a possible conflict situation, the degree of influence directly on the interests of the state, etc., of course, the first method of organizing research is preferable. To solve short-term tasks of a tactical nature, the second of the described methods is used.
In addition to such a division, it is proposed to consider the use of various mathematical methods depending on the stage of the conflict and the set of specific structural components of the conflict situation or process that need to be assessed. For example, to develop and describe a strategy for the behavior of a participant at a stage when the conflict has not yet escalated into an armed phase and there is an opportunity to negotiate a mutually acceptable agreement, it is proposed to consider the possibility of using game theory. Within the framework of the theory of cooperative agreements, the issue of sustainability will be considered, an agreement has already been reached, which is important point post-conflict settlement. To assess the "acceptable damage" and "pain threshold" we will use quantitative analysis. As mentioned earlier, one of the most important structural components of a conflict situation is potential, in particular, an indicator of the intensity of the conflict. To construct a tension curve, it is proposed to use factor analysis, methods of mathematical statistics and probability theory. Let's take a closer look at the proposed methods.
The resolution of this or that conflict means the achievement of a mutually acceptable agreement between the parties to the conflict. Politicians instinctively choose the best among the worst outcomes as the starting point from which they begin to develop a cooperative position. The minimax principle, game theory and the procedure for coordinating the interests of the parties in cooperative games formalize this practice.
Negotiations and agreement on the positions of the parties contribute to the achievement of compromises, which may be the desired solution to the conflict. At the same time, the parties involved in the conflict

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may use a variety of basic behavioral strategies. By forming alliances with each other, blocs of states can improve their negotiating positions and secure a greater degree of cooperation from their partners. Sophisticated methods of using threats, sanctions and even the use of force are used by states to force other states to cooperate with them. The threat of non-cooperation may bring less benefits to both parties. A small state may persuade a larger state to cooperate with it in such a way that each of them, acting together, will receive a greater gain. On the other hand, a larger state may impose cooperation on a smaller one, because the latter may be in dire need of the gains possible as a result of such cooperation.
Before proceeding to a formalized presentation of the basic concepts of game theory, it is necessary to dwell on two important conditions for the application this method: awareness of the participants about the situation and the formation of their goals. In game-theoretic modeling of conflict situations, it is usually assumed that the entire situation of the conflict is known to all participants, in any case, each participant clearly represents his interests, opportunities and goals. Of course, in real conditions, the refinement of ideas occurs right up to the very end of negotiations on the choice of a joint solution. However, the idealization adopted in game theory seems to be justified, at least as an initial stage of scientific analysis.
The process of forming the goals of the participants is most clearly described in the work of Yu.B. Germeier. .
Any solution can be represented as a result
striving to achieve some goal in the considered
process.
Any process from the point of view of making a decision or forming goals is quite adequately described by a finite set of some quantities (1
E. G. Baranovsky, N. N. Vladislavleva
Methods for analyzing international conflicts

3. The goal of the decision maker can be expressed in terms of
in the form of certain strivings for the values of Wi and only for them. In the general case, there may be several participants (n) in the process pursuing different goals.
4. Goals should be formulated as clearly as possible and not changed during the time of the process considered in the decision. The variability of the goal over time entails the impossibility of making clear rational decisions.
5. Goals can be set, inspired and educated.
6. The process of setting goals should be careful, clear and stable over time. Goals should be structurally simplified as the dimension of the process increases. To form goals; only the most general and rough characteristics of the change set XV should be used. To facilitate the process of forming goals, an orienting analysis of the methods of forming goals and a language for describing these methods are necessary.
A well-defined goal can be expressed as
the desire to increase some single scalar efficiency criterion w0, defined as a function of only the vector W: w0 = Ф(W)
Basically, the following types of elementary methods for the formation of common criteria (convolution of criteria) are used in practice:

b) lexicographic convolution of the criteria, when the maximum of the criterion Wi is first searched, then on the set

a) the choice of one (for example, the first) as a single criterion when imposing restrictions of the form Wi > Аi (i>1) on the rest or, in general, only imposing restrictions Wi > Аi on all criteria. In the latter case, a single criterion can be
present in the form:

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Methods for building and analyzing models of international conflicts

criterion W2 is maximized, and so on. until all criteria are exhausted or at the next iteration the maximum is reached at a single point;
c) summation with weights or economic convolution:

where?i are some positive numbers, usually normalized by the condition

d) convolution of the minimum type (Germeier convolution):

Here, in principle, Wio is any constant, but it is most natural to take the minimum value of the i-th criterion as Wio, and the maximum (desirable) value as Wim.
Economic convolution is used if the deterioration in the value of one of the criteria can, in principle, be compensated by an improvement in the value of any other. In Germeierian convolution, the criteria are not interchangeable. When modeling conflict situations, the second method of convolution is more often used, since it is believed that it is impossible to negotiate if it is assumed that any increase in the risk of a conflict escalating into an armed stage can be offset by some other advantages.
sustainable agreements. Let us dwell on a systematic exposition of the main questions of the theory of cooperative agreements. We will adhere to the generally accepted idea of cooperation as a certain association of subjects (persons, organizations, countries) that satisfies three conditions: 1) all subjects participate in cooperation voluntarily; 2) all subjects can dispose of their resources at will; 3) it is beneficial for all subjects to participate in cooperation.

E. G. Baranovsky, N. N. Vladislavleva
Methods for analyzing international conflicts
Cooperative agreements (institutions of consent) are the basis of the modern theory of conflicts as a set of mathematical methods that make it possible to study those informal connections that arise between the participants in the conflict and help to find a solution to the conflict on the paths of building consent institutions.
Let there be n participants in the conflict, they are assigned numbers i= = 1, ... , n and they form a set N = (1, ... , n). All actions that participant number 1 can take to achieve his goals are limited by the set Xi. The elements xi of this set are usually called strategies. The complete set х = (х1, ... , хn) of strategies of all participants is called the outcome of the conflict situation.
In order to set the interests and aspirations of each participant, it is necessary to describe which of the possible outcomes of the conflict situation are most preferable for him, which are less. A very general and technically convenient way of such a description is related to the objective functions or payoff functions of the participants. Suppose that for each participant i(i = 1, ..., m) the function fi (x) = fi (x1, ..., xn) is given on the set of all possible outcomes, that is, the value of fi depends not only on own strategy xi. Outcome x is more preferable to participant i than outcome y if and only if fi(x) > fi(y). In the future, we will conditionally call the values of fi (x) the “gains” of the corresponding participants.
Let the participants in the conflict situation gather to jointly choose their strategies (in practice, these are political negotiations between the participants in the conflict). In principle, they can agree on the implementation of any outcome of the conflict. But since each participant strives for the greatest possible value of his “winning” and cannot but reckon with the similar desire of partners, some outcomes will certainly not be realized, and different versions of agreements have different degrees of “viability”.
Let one of the participants (participant 1) give up any relationship with partners altogether and decide to act independently.

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Methods for building and analyzing models of international conflicts
independently, If participant i chooses some strategy хi of his own, then the “payoff” he receives will, in any case, not be less than the minimum of the objective function fi (х) = fi (х1, ..., хn), for all possible values variables x1 ... , xn, except for xi. Having chosen his strategy xi in such a way as to maximize this minimum, participant i will be able to count on winning

Therefore, the offer of a variant that barks participant i "win" less than the guaranteed result? i has no chance of getting his consent. Therefore, we will assume that only outcomes x satisfying the inequalities fi(x) > ?i are discussed as possible options for a joint decision; for all iєN. The set of such outcomes will be denoted by IR - the set of individually rational outcomes. Note that it is necessarily non-empty: if each participant applies his own guaranteeing strategy, then the outcome from the set IR is realized.
The question of the sustainability of a possible agreement is very important. The option under discussion may be advantageous when compared with a guaranteed result?i, but not advantageous when compared with a unilateral breach of the agreement.
Let the participants agree on a joint choice of some outcome x. For the stability of this agreement, it is necessary that the violation of it by any participant is not beneficial to the violator. If there are two participants (N = (1, 2)), then this condition is written as the fulfillment of two systems of inequalities:

for all y1єX1 , y2єX2, or as a fulfillment of the system of equations

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E. G. Baranovsky, N. N. Vladislavleva
Methods for analyzing international conflicts
For an arbitrary number of participants, we introduce the notation
x ¦¦ yi - the outcome of the conflict, in which participant i applies the strategy yi, and all other participants use the strategy хj. Then the conditions for the stability of the agreement on choosing the outcome x = (x1, ..., xn) consist in the fulfillment of the inequalities fi(x) > fi (x II yi) for all i є N , yiєxi, or in the fulfillment of the equalities:

these conditions were first formulated by J. Nash in 1950. Outcomes that satisfy them are called Nash equilibrium, as well as equilibrium points or simply equilibria. The set of outcomes will be denoted by NE.
From the definition of equilibrium, it does not follow at all that equilibrium outcomes should exist at all. Indeed, it is not difficult to construct examples of conflict situations that do not have equilibrium outcomes at all. All that theory can offer to participants in such situations is to expand the set of outcomes (that is, the set of collective strategies) either by finding unaccounted for strategic opportunities or by deliberately introducing additional possibilities. As a general way of such expansion, it can be pointed out that, firstly, taking into account the natural dynamics of a violation, which is beneficial from the point of view of short-term interests, may turn out to be disadvantageous if more remote consequences are taken into account; secondly, an increase in the mutual awareness of the participants - if the parties to the conflict manage to organize an effective system of mutual control, then the potential violator of the agreement will have to take into account the possibility of an unfavorable reaction of partners to his deviation from the strategy stipulated by the agreement, which will nullify the benefit from violating the agreement.
However, the existence of equilibrium outcomes does not mean that it will be easy for the participants to enter into a cooperative agreement. Consider an example called the Prisoner's Dilemma. Two participants have two strategies "peacefulness" and "aggressiveness". The preferences of the participants on the set of four outcomes are as follows. In the most

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Methods for building and analyzing models of international conflicts
the participant who has chosen the strategy of aggressiveness against a peaceful partner turns out to be in a better position. In second place is the outcome in which both participants are peaceful. Next comes the outcome in which both are aggressive, and, finally, the worst thing is to be peaceful, against an aggressive partner. Assigning conditional numerical values of the “payoff” functions to these outcomes, we obtain the following payoff matrix:
(5, 5) (0,10) (10,0) (1, 1).
As is customary in game theory, we assume that the strategies of participant 1 correspond to the rows of the matrix, the strategies of participant 2 correspond to columns (the first row (column) is a peaceful strategy, the second is aggressive), the first number in brackets is “winning” of participant 1 in the corresponding outcome, the second is “winning » of participant 2. It is easy to check that for any partner’s strategy it is more profitable for each participant to be aggressive, therefore the only equilibrium outcome is the use of aggressive strategies by both participants, which gives each participant a “payoff” equal to 1. However, this approach is not very attractive for participants, because By applying strategies of peacefulness, they could both increase their "payoff". Thus, we see that the fulfillment of the Nash conditions is by no means the only requirement that it makes sense to impose on a potential agreement.
In order to formulate in general terms another natural requirement suggested by the considered example, let us imagine that in a general situation two versions of the agreement are discussed: to realize the outcome x and to realize the outcome y. Generally speaking, some participants benefit from outcome x, others
outcome at. If, however, it happens that the outcome x is more beneficial for someone than y, and the outcome y is not better for everyone than x, then there seems to be no point for the participants to agree on the implementation of outcome y. In this case, outcome x is said to be Pareto-dominant outcome y.

E. G. Baranovsky, N. N. Vladislavleva
Methods for the analysis of international conflicts
Conflict outcomes that are not dominated by any others, that is, cannot be rejected on the basis of these considerations, are called Pareto optimal or efficient. Let us give a precise definition: the outcome x is Pareto optimal if and only if, for any outcome y, the inequality fi(y) > fi (x) for at least one i єN implies the existence of jєN for which fj(y) > fj (х ). Indeed, the above condition means exactly that if there is a participant interested in discussing outcome y instead of outcome x, then there is a participant interested in the opposite. The set of Pareto-optimal outcomes will be denoted as RO.
In game theory, the set IR P RO, that is, the set of Pareto optimal individually rational outcomes, is usually called the negotiation set, as if assuming that with reasonable behavior of the participants, negotiations on a joint decision will end from this set.
Along with the advantages that mathematical methods provide, there are a number of difficulties that limit the possibilities of their application to the analysis of international conflicts. The first such difficulty is related to taking into account the human factor, which plays a significant role in the decision-making process. Possessing logical thinking, a person is also subject to the sphere of subconscious drives, emotions, passions that affect rational thinking, which in the behavior of state and political leaders often makes decisions difficult to predict. Although theoretically the system or environment should impose restrictions on their deviations from the most rational choice, history shows that the role of the state leader often turns out to be decisive, while he himself, when making a decision, becomes immune to objective information, and acts on the basis of subjectively established, in largely intuitive, understanding the political process and the intentions of opponents and other actors.
Another difficulty is related to the fact that some processes seem to be random, stochastic, because at the time of the study their causes are invisible. If figuratively

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compare a political song with a biological organism, then the causes of this are like a virus that long time does not show activity due to the lack of favorable environmental conditions. With regard to international relations and conflicts, it is important not to lose sight of the historical aspect, since the origins of some of the processes observed by contemporaries are enshrined in national traditions, national consciousness.
Of course, mathematical models themselves cannot answer the question of how to resolve existing contradictions, they cannot become a panacea for all conflicts, but they greatly facilitate the management of conflict processes, reduce the level of resources expended, help choose the most optimal behavior strategy, which reduces the number of losses. , including human ones.
To date, applied modeling of international relations is being carried out in many institutions of industry. developed countries. But, of course, the palm among them belongs to such centers as Stanford, Chicago, California universities, the Massachusetts Institute of Technology, the International Center for Peacekeeping in Canada.
In the next chapter, we will look at some examples of international conflict prayers.

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MATHEMATICAL METHODS IN INTERNATIONAL RELATIONS. MATHEMATICAL AND APPLIED CALCULATIONS OF REPEATING THE REVOLUTIONARY POSSIBILITIES OF "COLOR SCENARIOS" IN THE COMMONWEALTH OF INDEPENDENT STATES

International relationships - component science, including diplomatic history, international law, the world economy, military strategy and many other disciplines that study various aspects of a single object for them. Of particular importance for her is the "theory of international relations", which, in this case, is understood as a set of multiple conceptual generalizations presented by theoretical schools polemicizing with each other and constituting the subject field of a relatively autonomous discipline. In this sense, the "theory of international relations" is both very old and very young. Already in ancient times, political philosophy and history raised questions about the causes of conflicts and wars, about the means and methods for achieving order and peace between peoples, about the rules for their interaction, etc. - and therefore it is old. But at the same time, it is also young - as a systematic study of observed phenomena, designed to identify the main determinants, explain behavior, reveal typical, recurring in the interaction of international factors. Tsygankov P.A. Theory of international relations: textbook / P.A. Tsygankov. - 2nd ed., corrected. and additional - M.: Gardariki, 2007. - 557 p.

The sphere of international relations is mobile and constantly changing. Now, in the period of world globalization, integration and, at the same time, regionalization, the number and diversity of participants in international relations has increased significantly. Transnational actors have appeared: intergovernmental organizations, transnational corporations, international non-governmental organizations, religious organizations and movements, domestic political regions, international criminal and terrorist organizations. As a result, international relations have become more complex, have become even more unpredictable, it has become more difficult to determine the true, real goals and interests of their participants, to develop a state strategy and formulate state interests. Therefore, at present it is important to be able to analyze and evaluate events in the field of international relations, to see the goals of their participants, and to set priorities. To do this, you need to study international relations. In the process of studying, study methods, their advantages and disadvantages play a significant role. Therefore, the topic is “Mathematical Methods in International Relations. Mathematical and applied calculations of the revolutionary possibilities of the “color scenario” in the Commonwealth of Independent States” is relevant and modern.

In this work, a prognostic method was applied, which largely helped to build a chain of logically complete conclusions from a study of the likelihood of a repeat of "color revolutions" in the CIS countries. Therefore, it is advisable to begin with the consideration and definition of the concept of this method.

In international relations, there are both relatively simple and more complex predictive methods. The first group can include such methods as, for example, conclusions by analogy, the method of simple extrapolation, the Delphi method, scenario building, etc. To the second - the analysis of determinants and variables, a systematic approach, modeling, analysis of chronological series (ARIMA), spectral analysis, computer simulation, etc. The Delphi method implies a systematic and controlled discussion of the problem by several experts. Experts submit their assessments of this or that international event to the central body, which conducts their generalization and systematization, after which it returns to the experts again. Being carried out several times, such an operation makes it possible to state more or less serious discrepancies in the indicated estimates. Taking into account the generalization carried out, the experts either amend their initial assessments, or strengthen their opinion and continue to insist on it. The study of the causes of discrepancies in expert assessments makes it possible to identify previously unnoticed aspects of the problem and fix attention both on the most (in case of coincidence of expert assessments) and the least (in case of their discrepancy) probable consequences of the development of the analyzed problem or situation. In accordance with this, the final assessment is developed and practical advice. Scenario building - this method consists in building ideal (i.e. mental) models of the likely development of events. Based on the analysis of the current situation, hypotheses are put forward - which are simple assumptions and are not subject to any verification in this case - about its further evolution and consequences. At the first stage, the analysis and selection of the main factors determining, according to the researcher, further development situations. The number of such factors should not be excessive (as a rule, no more than six elements are distinguished) in order to provide a holistic vision of the whole set of future options arising from them. At the second stage, hypotheses are put forward (based on simple "common sense") about the proposed phases of evolution of the selected factors over the next 10, 15 and 20 years. At the third stage, the selected factors are compared and, on their basis, a number of hypotheses (scenarios) corresponding to each of them are put forward and described in more or less detail. This takes into account the consequences of interactions between the identified factors and imaginary options for their development. Finally, in the fourth step, an attempt is made to create indicators of the relative likelihood of the scenarios described above, which are classified (quite arbitrarily) according to their degree of probability for this purpose.3. Khrustalev M.A. System modeling of international relations. Abstract for the competition degree doctor of political sciences. - M., 1992, p. 8, 9. The concept of a system (system approach) is widely used by representatives of various theoretical trends and schools in the science of international relations. Its generally recognized advantage is that it makes it possible to present the object of study in its unity and integrity, and, therefore, contributing to finding correlations between interacting elements, helps to identify the "rules" of such interaction, or, in other words, the laws of the functioning of the international system. On the basis of a systematic approach, a number of authors distinguish international relations from international politics: if the constituent parts of international relations are represented by their participants (actors) and “factors” (“independent variables” or “resources”) that make up the “potential” of participants, then the elements of international politics are only actors. Modeling - the method is associated with the construction of artificial, ideal, imaginary objects, situations, which are systems, the elements and relations of which correspond to the elements and relations of real international phenomena and processes. Let us consider such a type of this method as - complex modeling. In the same place - the construction of a formalized theoretical model, which is a trinary synthesis of methodological (philosophical theory of consciousness), general scientific (general systems theory) and particular scientific (international relations theory) approaches. The construction is carried out in three stages. At the first stage, “pre-model tasks” are formulated, which are combined into two blocks: “evaluative” and “operational”. In this regard, such concepts as "situations" and "processes" (and their types), as well as the level of information, are analyzed. Based on them, a matrix is built, which is a kind of "map", designed to provide the researcher with the choice of an object, taking into account the level of information security.

As for the operational block, the main thing here is to single out the nature (type) of models (conceptual, theoretical and concrete) and their forms (verbal or content, formalized and quantified) on the basis of the triad “general-special-singular”. The selected models are also presented in the form of a matrix, which is a theoretical model of modeling, reflecting its main stages (form), stages (character) and their relationship.

At the second stage, we are talking about building a meaningful conceptual model as the starting point for solving the general research problem. Based on two groups of concepts - "analytical" (essence-phenomenon, content-form, quantity-quality) and "synthetic" (matter, movement, space, time), presented in the form of a matrix, a "universal cognitive construction - configurator" is built, setting the general framework of the study. Further, on the basis of the selection of the above logical levels of study of any system, the noted concepts are subjected to reduction, as a result of which the “analytical” (essential, content, structural, behavioral) and “synthetic” (substrate, dynamic, spatial and temporal) characteristics of the object are distinguished. Based on the “system oriented matrix configurator” structured in this way, the author traces specific features and some trends in the evolution of the system of international relations.

At the third stage, a more detailed analysis of the composition and internal structure of international relations is carried out, i.e. construction of its expanded model. Here, the composition and structure (elements, subsystems, connections, processes) are distinguished, as well as the “programs” of the system of international relations (interests, resources, goals, mode of action, balance of interests, balance of forces, relations). Interests, resources, goals, course of action are elements of the "program" of subsystems or elements. Resources, characterized as a "non-system-forming element", are subdivided by the author into resources of means (material-energy and information) and resources of conditions (space and time).

The "program of the system of international relations" is a derivative in relation to the "programs" of elements and subsystems. Its backbone element is the "correlation of interests" of various elements and subsystems with each other. The non-system-forming element is the concept of “balance of forces”, which could be more accurately expressed by the term “balance of means” or “correlation of potentials”. The third derived element of this "program" is the "relationship" understood by the author as a kind of evaluative representation of the system about itself and about the environment.

At the same time, it would be wrong to exaggerate the importance of a systematic approach and modeling for science, to ignore their weaknesses and shortcomings. Paradoxical as it may seem, the main one is the fact that no model - even the most flawless in its logical foundations - gives confidence in the correctness of the conclusions drawn on its basis. This, however, is acknowledged by the author of the work discussed above, when he speaks of the impossibility of constructing an absolutely objective model of the system of international relations. We add that there is always a certain gap between the model constructed by this or that author and the actual sources of the conclusions that he formulates about the object under study. And the more abstract (that is, the more strictly logically justified) the model is, and also the more adequate to reality its author strives to make his conclusions, the wider the indicated gap. In other words, there is a serious suspicion that when formulating conclusions, the author relies not so much on the model construction he has built, but on the initial assumptions, the “building material” of this model, as well as on others not related to it, including “intuitive logical" methods. Hence the question, which is very unpleasant for “uncompromising” supporters of formal methods: could those (or similar) conclusions that appeared as a result of a model study be formulated without a model? A significant discrepancy between the novelty of such results and the efforts made by researchers on the basis of system modeling makes us think that an affirmative answer to this question looks very reasonable.

As for the systematic approach as a whole, its shortcomings are a continuation of its merits. Indeed, the advantages of the concept of "international system" are so obvious that it is used, with few exceptions, by representatives of all theoretical trends and schools in the science of international relations. However, as the French political scientist M. Girard rightly noted, few people know exactly what it really means. It continues to retain a more or less rigorous meaning for functionalists, structuralists, and systemists. For the rest, it is most often nothing more than a beautiful scientific epithet, convenient for decorating an ill-defined political object. As a result, this concept turned out to be oversaturated and devalued, which makes it difficult to use it creatively.

Agreeing with the negative assessment of the arbitrary interpretation of the concept of "system", we emphasize once again that this does not at all mean doubts about the fruitfulness of applying both the systematic approach and its specific incarnations - system theory and system analysis - to the study of international relations.

The role of prognostic methods of international relations can hardly be overestimated: after all, in the final analysis, both analysis and explanation of facts are needed not by themselves, but for the sake of making forecasts of the possible development of events in the future. In turn, forecasts are made in order to make an adequate international political decision. An important role in this is called upon to play an analysis of the decision-making process of a partner (or opponent).

Thus, in my work, an analysis was made of the possibility of repeating the “color scenario” in the CIS countries by constructing a tabular matrix, which, in turn, presents the criteria for situations in this moment in this CIS state. It should be noted that the situation criteria assessment score was 5, since in the countries of the former Soviet Union the tendency of comparison according to the system above 5 points remains unchanged, in connection with which, the author proposed a 5-point scale; criteria) by Internet ( social media: Facebook, Odnoklassniki, etc.).

The table presents 7 criteria that can most affect the likelihood of a repetition of revolutions in a given region: the weakness of the state, the weakness of law enforcement agencies, the split of the elites, the spread of anti-government utopia, external pressure, confrontational agitation and propaganda, and the activity of the masses. Commonwealth members proposed Independent States on an individual basis, as well as on a regional basis, the average score of the highest probability of repetition was calculated.

As can be seen from the table, Ukraine has a score close to the maximum - 4, in which the situation with the problem of the weakness of the political system remains acute to this day, as a result of which the ideas of anti-government utopia are close to 4 points, which confirms the deplorable situation in this state. Speaking about external pressure, the participants of the social survey gave the maximum score - 5, which is a complete lack of self-determination, dependence on external influence and helplessness of this state from foreign interventions and infusions of financial investments by it. The split of the elites is also an important problem in this zone, since according to the schedule, 5 points were noted, i.e. at the moment, Ukraine is divided into several parts, the split elites dictate their ideas for conducting politics, which undoubtedly puts the state in one of the poorest countries in the world today. The average risk score for a repeat of the "color revolutions" was 4.

Further on consideration are the problems of our country - Kyrgyzstan, for which the survey participants determined the maximum score - 5 among all CIS countries, when compared with neighboring Tajikistan, our state has military-economic, political and economic weaknesses that prevent our country from being one step ahead neighboring republics. Despite the confrontational agitation and propaganda close to the minimum score - 2, the rest of the criteria are mostly close to - 4, it turns out that at the moment the situation after two revolutions did not give any lessons and the consequences were meaningless. The average probability score for the repetition of revolutions in our republic was 3.6.

However, paradoxically, the situation in Tajikistan remains not the best, when compared with the same Georgia, which also suffered two “color revolutions”, Tajikistan has socio-economic, political weaknesses, an overshooting unemployment rate demoscope.ru/weekly /2015/0629/barom07.php in this country forces citizens to leave to work in Russia (including the problem of drug trafficking, criminal activities of extremist groups, the danger of religious extremism, clannishness). In Tajikistan, the average score was - 3, 4.

Turkmenistan is one of the "closed" countries of the former USSR, today it is in last place, the average score for repeating the "color scenario" of which was only 1.7. Does this result indicate that the state is classified in its economic, political and military matters, or in fact, this state is one of the most prosperous in given time, everyone decides for himself. Even comparing the same Uzbekistan (3 points) on foreign assistance, Turkmenistan has 2 points, confirming that this country exists to the greatest extent “on its own”, providing its people and statehood with its own efforts. Thus, taking the last place in this list.

international color revolution state

The work will include a graph of the average repetition rate of "color revolutions" in the CIS countries on an individual basis, i.e. if the tabular matrix shows how the evaluation work was performed according to certain criteria, then the graph allows you to see the whole situation of this problem, where there is the highest coefficient of repetition of the "color scenario", and where - the smallest. From which it follows that the highest probability of repetition (on an individual basis) in Ukraine is 4 points, and the lowest in Turkmenistan and Uzbekistan is about 2 points.

However, if the great danger Ukraine has a repetition of revolutions (4 points), then by division into regional characteristics, the countries of the so-called Transcaucasia (Azerbaijan, Georgia, Armenia) have the highest average score - 2.9, compared with Eastern Europe, which has 2.8 points, Central Asia has - 2.7 points, which puts our region in last place in terms of the possibility of repeating the "color scenario", despite the difference of 0.1 points compared to other regions of the CIS.

The totality of economic (unemployment, low wages, low labor productivity, uncompetitiveness of the industry), socio-medical (disability, old age, high morbidity), demographic (single-parent families, a large number of dependents in the family), educational and qualification (low level of education, insufficient professional training), political (military conflicts, forced migration), regional-geographical (unequal development of regions), religious-philosophical and psychological (austerity as a way of life, foolishness) causes the countries of Transcaucasia to take the first place in terms of the level of backwardness and poverty of the regions of the CIS countries, which inevitably leads to the likelihood of a repetition of revolutionary situations in this region. The dissatisfaction of civil society, despite the dictatorship of some states of the Central Asian region (Uzbekistan, Turkmenistan), can spill out through careful external sponsorship and investment influences and specially trained youth opposition, despite excessive democracy, according to the author, in countries such as Kyrgyzstan, Ukraine, the likelihood of a repetition of revolutions is really high, since the consequences of the past “color revolutions” are not justified in any way and the results did not lead to any significant changes, except that only the “tops” of power changed.

Summing up, this section helped in many ways to reveal the essence of the topic “General and specific features of “color revolutions” in the CIS countries”, the method of applied and mathematical analysis carried out led to the conclusion that the likelihood of a repeat of “color revolutions” is not taken if measures are not taken to prevent these conflicts. situations and fundamentally change the issues of poverty in Eastern Europe, resolve conflicts at the interethnic level in Azerbaijan, Armenia and Georgia, and end the problem of clans and nepotism in Central Asia.

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Tsygankov P. Political sociology of international relations

Chapter IV. The problem of method in the sociology of international relations

The main purpose of this chapter is to introduce the most widely used methods, techniques and techniques for studying international relations and foreign policy. It does not pose such a rather complicated and independent task of how to teach how to use them. However, its solution would be impossible, since this requires, firstly, detailed description tech or other methods, illustrated by examples of their specific application in research work in the analysis of a certain object of international relations, and, secondly (and this is the main thing), practical participation in this or that scientific-theoretical or scientific-applied project, since, as you know, one cannot learn to swim without entering the water.

At the same time, it should be borne in mind that each researcher (or research team) usually uses his favorite method (or a group of them), corrected, supplemented and enriched taking into account the existing conditions and tools. It is also important to keep in mind that the application of one method or another depends on the object and objectives of the study, as well as (which is very important) on the available material resources.

Unfortunately, we have to note the fact that the specialized literature devoted to the problem of methods and especially applied methods of analyzing international relations is very scarce (especially in Russian) and therefore difficult to access.

1. Significance of the method problem

The problem of the method is one of the most important problems of science, since ultimately it is about teaching, obtaining new knowledge, how to apply it in practice. At the same time, this is one of the most difficult problems, which precedes the study of its object by science, and is the result of such study. It precedes the study of the object because the researcher from the very beginning must have a certain amount of techniques and means to achieve new knowledge. It is the result of the study, because the knowledge obtained as a result of it concerns not only the object itself, but also the methods of its study, as well as the application of the results obtained in practical activities. Moreover, the researcher is faced with the problem of method already when analyzing the literature and the need to classify and evaluate it.

Hence the ambiguity in understanding the content of the very term "method". It means both the sum of techniques, means and procedures for the study of its subject by science, and the totality of already existing knowledge. This means that the problem of method, while having an independent meaning, is at the same time closely connected with the analytical and practical role of theory, which also plays the role of method.

The widespread opinion that each science has its own method is only partly true: most of the social sciences do not have their own specific, only inherent method. Therefore, in one way or another, in relation to their object, they refract general scientific methods and methods of other (both social and natural sciences) disciplines. In this regard, it is generally accepted that the methodological approaches of political science (including the political sociology of international relations) are built around three aspects: the strictest possible separation of the research position from moral value judgments or personal views; the use of analytical techniques and procedures that are common to all social sciences, which plays decisive role in the establishment and subsequent examination of the facts; striving for systematization, or, in other words, for the development of common approaches and the construction of models that facilitate the discovery of "laws" 1 .

And although it is emphasized that what has been said does not mean the need for “complete exclusion” from science of value judgments or personal positions of the researcher, nevertheless, he inevitably faces the problem of a broader nature, the problem of the relationship between science and ideology. In principle, this or that ideology, understood in a broad sense as a conscious or unconscious choice of a preferred point of view, always exists. It is impossible to avoid this, to “de-ideologize” in this sense. Interpretation of facts, even the choice of "observation angle", etc. inevitably conditioned by the point of view of the researcher. Therefore, the objectivity of the research suggests that the researcher must constantly remember about the "ideological presence" and strive to control it, to see the relativity of any conclusions, taking into account such a "presence", strive to avoid one-sided vision. The most fruitful results in science can be achieved not with the rejection of ideology (this is at best a delusion, and at worst a conscious cunning), but under the condition of ideological tolerance, ideological pluralism and "ideological control" (but not in the sense of the control of official political ideology in relation to science, and vice versa in the sense of the control of science over any ideology). As for the problem of values, it would not be an exaggeration to say that the difficulties that Russian sociology is experiencing today are connected precisely with the lack of a value principle. The environment of severe political pressure that dominated the country for many years led to the development of Soviet sociology of the herd within the American behavioral tradition, giving preference to operational, instrumental approaches and methods. This allowed her, as it were, to "get rid" of ideology: Soviet sociologists were among the first among domestic social scientists who stopped believing in ideological myths. But, on the other hand, having not accepted in its time the traditions of theoretical sociology, for example, the French school with its Durkheimian traditions, or the German phenomenological sociology of Max Scheller, etc., Soviet (and post-Soviet) sociology, which inherited it, has not yet managed to adapt to the new , a post-non-classical trend in world social (including sociological, political, and any other) science, where there is a renaissance of values, an anthropological approach, attention to socio-cultural specifics, etc.

The foregoing also applies to the so-called methodological dichotomy, which, however, is often observed not only in domestic, but also in Western science of international relations. We are talking about opposing the so-called traditional historical-descriptive, or intuitive-logical approach to operational-applied, or analytical - predictive, associated with the use of exact science methods, formalization, data calculation (quantification), verifiability (or falsification) of conclusions, etc. In this regard, for example, it is argued that the main shortcoming of the science of international relations is the protracted process of its transformation into applied science 2. Such statements suffer from excessive categoricalness. The process of development of science is not linear, but rather mutual: it does not turn from historical descriptive into applied science, but refinement and correction of theoretical positions through applied research (which, indeed, are possible only at a certain, fairly high stage of its development) and “return debt" to "applicants" in the form of a more solid and operational theoretical and methodological basis.

Indeed, in the world (primarily American) science of international relations, since the early 1950s, many relevant results and methods of sociology, psychology, formal logic, as well as natural and mathematical sciences have been mastered. At the same time, the accelerated development of analytical concepts, models and methods, progress towards a comparative study of data, and the systematic use of the potential of electronic computing technology begin. All this contributed to the significant progress of the science of international relations, bringing it closer to the needs of practical regulation and forecasting of world politics and international relations. At the same time, this by no means led to the displacement of the former, “classical” methods and concepts.

For example, the operational nature of the historical sociological approach to international relations and its predictive capabilities were demonstrated by R. Aron. One of the most prominent representatives of the "traditional", "historical-descriptive" approach, G. Morgenthau, pointing out the insufficiency of quantitative methods, wrote, not without reason, that they could hardly claim to be universal. Such an important phenomenon for understanding international relations as, for example, power, “is the quality interpersonal relationships which can be checked, evaluated, guessed, but which cannot be measured quantitatively... Of course, it is possible and necessary to determine how many votes a politician can have, how many divisions or nuclear warheads the government has; but if I need to understand how much power a politician or government has, then I will have to put aside the computer and adding machine and start thinking about historical and certainly qualitative indicators.

Indeed, the essence of political phenomena cannot be investigated in any complete way with the help of applied methods alone. Social relations in general, and international relations in particular, are dominated by stochastic processes that defy deterministic explanations. Therefore, the conclusions of the social sciences, including the science of international relations, can never be finally verified or falsified. In this regard, the methods of "high" theory, combining observation and reflection, comparison and intuition, knowledge of facts and imagination, are quite legitimate here. Their usefulness and effectiveness are confirmed by both modern research and fruitful intellectual traditions.

At the same time, as M. Merl rightly noted about the controversy between supporters of "traditional" and "modernist" approaches in the science of international relations, it would be absurd to insist on intellectual traditions where exact correlations between the collected facts are necessary. Everything that can be quantified must be quantified 4 . We will return to the controversy between "traditionalists" and "modernists".

Here it is important to note the illegitimacy of the opposition between "traditional" and "scientific" methods, the falsity of their dichotomy. In fact, they complement each other. Therefore, it is quite legitimate to conclude that both approaches “are on an equal footing, and the analysis of the same problem is carried out independently by different researchers” (see note 4, p. 8). Moreover, within the framework of both approaches, the same discipline can use, albeit in different proportions, different methods: general scientific, analytical, and concrete empirical (however, the difference between them, especially between general scientific and analytical, is also rather arbitrary). In this respect, the political sociology of international relations is no exception. Turning to a more detailed consideration of these methods, it is worth once again emphasizing the conditionality, the relativity of the boundaries between them, their ability to “flow” into each other.

2. General scientific methods

General scientific methods constitute the starting point, the foundation of any discipline, however far from high theory it may be. However, considering the use of general scientific methods in the sociology of international relations, it makes no sense to dwell on the description of such theoretical and philosophical methods as historical and logical, analysis and synthesis, the principle of priority, ascent from the abstract to the concrete, etc. All of them have a place, but to seek and demonstrate their application in a given discipline, as experience already available in this respect5 shows, is an unproductive exercise. On the other hand, it seems much more productive to consider those methods that, with all the variety of methodological approaches, are used most often in the science of international relations and give concrete research results. In this sense, the sociology of international relations in its object is characterized by the generalization and systematization of facts based on the study of historical, analytical and other documents, rigorous scientific observations and comparative analysis. This implies a refusal to be locked into the boundaries of a particular discipline, an attempt to comprehend the object of study in integrity and, as far as possible, in unity, opening up the prospect of discovering trends and patterns of its functioning and evolution. Hence the importance attached in the study of international relations to the systems approach and the method of modeling closely related to it. .Let's consider these methods in more detail.

Systems approach

The concept of a system (it will be discussed in more detail below) is widely used by representatives of various theoretical trends and schools in the science of international relations. Its generally recognized advantage is that it makes it possible to present the object of study in its unity and integrity, and, therefore, by helping to find correlations between interacting elements, it helps to identify the "rules" of such interaction, or, in other words, the laws of the functioning of the international system. On the basis of a systematic approach, a number of authors distinguish international relations from international politics: if the components of international relations are represented by their participants (authors) and "factors" ("independent variables" or "resources") that make up the "potential" of participants, then the elements of international politics only the authors of 6,7,8 speak.

The systems approach should be distinguished from its specific incarnations of systems theory and systems analysis. System theory fulfills the tasks of constructing, describing and explaining systems and their constituent elements, the interaction of the system and the environment, as well as intrasystem processes, under the influence of which the system is changed and/or destroyed 9 . As for system analysis, it solves more specific problems; representing a set of practical techniques, techniques, methods, procedures, thanks to which a certain ordering is introduced into the study of an object (in this case, international relations) (see: note 9, p. 17; note 10, p. 100).

From the point of view of R. Aron, "the international system consists of political units that maintain regular relations with each other and which can be drawn into a general war" 11 . Since the main (and in fact the only) political units of interaction in the international system for Aron are states, at first glance one might get the impression that he identifies international relations with world politics. However, limiting, in fact, international relations to a system of interstate interactions, R. Aron at the same time not only paid great attention to the assessment of resources, the potential of states that determine their actions in the international arena, but also considered such an assessment to be the main task and content of the sociology of international relations. At the same time, he represented the potential (or power) of the state as an aggregate consisting of a broad geographical environment, material and human resources, and the ability of collective action (see note 11, p. 65). Thus, proceeding from a systematic approach, Aron outlines, in essence, three levels of consideration of international (interstate) relations: the level of the interstate system, the level of the state and the level of its power (potential).

D. Rosenau proposed in 1971 another scheme, including six levels of analysis: 1) individuals-“creators” of policy and their characteristics; 2) their positions and roles; 3) the structure of the government in which they operate; 4) the society in which they live and govern; 5) the system of relations between the nation state and other participants in international relations; 6) world system 12 . Describing the systematic approach represented by different levels of analysis, B. Russett and H. Starr emphasize that the choice of one or another level is determined by the availability of data and the theoretical approach, but by no means by the whim of the researcher. Therefore, in each case of applying this method, it is necessary to find and define several different levels. At the same time, explanations at different levels do not have to be mutually exclusive, they can be complementary, thereby deepening our understanding.

Serious attention is paid to the systematic approach in the domestic science of international relations. The works published by the researchers of IMEMO, MGIMO, ISKAN, IVAN and other academic and university centers testify to the significant progress Russian science in both systems theory 13,14 and systems analysis 15,16 . Thus, the authors of the textbook "Fundamentals of the Theory of International Relations" believe that "the method of the theory of international relations is a systematic analysis of the movement and development of international events, processes, problems, situations, carried out with the help of existing knowledge, foreign policy data and information, special methods and techniques of research » (see note 15, p. 68). The starting point of such an analysis is, from their point of view, three levels of study of any system: 1) the level of composition of the set of elements that form it; 2) the level of internal structure is a set of regular relationships between elements; 3) the level of the external structure is the totality of the relationship of the system as a whole with the environment (note 15, p. 70).

Let us consider the method of system analysis in its static and dynamic dimensions in relation to the study of the state's foreign policy.

Static measurement includes the analysis of "determinants", "factors" and "variables".

One of the followers of Aron, R. Bosk, in his work "Sociology of the World" presents the potential of the state as a set of resources that it has to achieve its goals, consisting of two types of factors: physical and spiritual.

Physical (or directly tangible) factors include the following elements:

1.1 Space (geographical position, its advantages and disadvantages).

1.2 Population (demographic power).

1.3 The economy in such manifestations as: a) economic resources; b) industrial and agricultural potential; c) military power.

In turn, the spiritual (or moral, or social, not directly tangible) factors include:

2.1 Type of political regime and its ideology.

2.2 The level of general and technical education of the population.

2.3 "National morality", the moral tone of society.

2.4 Strategic position in the international system (for example, within a community, union, etc.).

These factors constitute a set of independent variables that affect the foreign policy of states, by studying which it is possible to predict its changes 17 .

Graphically, this concept can be represented as the following diagram:

The diagram gives a visual representation of both the advantages and disadvantages of this concept. The advantages include its operationality, the possibility of further classification of factors based on the database, their measurement and analysis using computer technology. As for the shortcomings, then, apparently, the most significant of them is the actual absence in this scheme (with the exception of paragraph 2.4) of factors external environment that have a significant (sometimes decisive) impact on the foreign policy of states.

In this regard, the concept of F. Bryar and M.-R. Jalili 18 looks much more complete, which can also be presented in the form of a diagram (see Fig. 2).

Conventions

Physical factors

A.1 - Geographic location
A.2 - Natural resources
A.3 - Demographic situation

Structural factors

B.1 - Political institutions
B.2 - Economic institutions
B.3 - Ability to use physical and social environment; technological, economic and human potential
B.4 - Political parties
B.5 - Pressure groups
B.6 - Ethnic groups
B.7 - Faith groups
B.8 - Language groups
B.9 - Social mobility
B.10 - Territorial structure; share of urban and rural population
B.11 - Level of national agreement

Cultural and human factors

B.1 (Culture):
B.1.1 Value system
B.1.2 Language
B.1.3 Religion
B.2 (Ideology):
B.2.1 Authority's self-assessment of its role
B.2.2 Her self-perception
B.2.3 Her perception of the world
B.2.4 Primary means of pressure
B.3 (Collective mentality):
B.3.1 Historical memory
B.3.2 The image of the "other"
B.3.3 Conduct in the field of international obligations
B.3.4 Particular sensitivity to a national security issue
B.3.5 Messianic traditions
B.4 Qualities of decision makers (decision makers):
B.4.1 Perception of one's environment
B.4.2 Perception of the world
B.4.3 Physical qualities
B.4.4 Moral qualities

As can be seen from the diagram, this concept, having all the advantages of the previous one, overcomes its main drawback. Its main idea is the close relationship of internal and external factors, their mutual influence and interdependence in influencing the foreign policy of the state. In addition, within the framework of internal independent variables, these factors are presented here much more fully, which significantly reduces the possibility of missing any important nuance in the analysis. At the same time, the scheme reveals that what has been said applies much less to external independent variables, which are only marked on it, but not structured in any way. This circumstance testifies that with all the "equality" of internal and external factors, the authors still clearly prefer the former.

It should be emphasized that in both cases the authors by no means absolutize the importance of factors in influencing foreign policy. As R. Bosk shows, having entered the war against France in 1954, Algeria did not possess most of these factors, and yet it managed to achieve its goal.

Indeed, attempts at a naive-deterministic description of the course of history in the spirit of the Laplace paradigm as a movement from the past through the present to a predetermined future reveal their failure with particular force precisely in the sphere of international relations, where stochastic processes dominate. The foregoing is especially characteristic of the current transitional stage in the evolution of the world order, which is characterized by increased instability and is a kind of bifurcation point that contains many alternative paths of development and, therefore, does not guarantee any predetermination.

Such a statement does not mean at all that no forecasts in the sphere of international relations are possible in principle. It is about seeing the limits, the relativity, the ambivalence of the predictive possibilities of science. This also applies to such a specific process as the process of making foreign policy decisions.

Decision Process Analysis (DPR) is a dynamic measurement system analysis of international politics and, at the same time, one of the central problems of social science in general and the science of international relations in particular. Studying the determinants of foreign policy without taking into account this process can turn out to be either a waste of time, from the point of view of predictive capabilities, or a dangerous delusion, because this process is the “filter” through which the totality of factors influencing foreign policy is “sifted” by a person (persons) decision maker (DM).

The classical approach to the analysis of SPR, reflecting the "methodological individualism" characteristic of the Weberian tradition, includes two main stages of research 19 . At the first stage, the main decision-makers are identified (for example, the head of state and his advisers, ministers: foreign affairs, defense, security, etc.), and the role of each of them is described. This takes into account that each of them has a staff of advisers with the authority to request any information they need in a particular government department.

The next step is to analyze political preferences decision makers, taking into account their worldview, wholesale, political views, leadership style, etc. An important role in this respect was played by the works of R. Snyder, X . Brook 20, B. Sapan and R. Jervis.

F. Briar and M.R. Jalili, summarizing the methods of analysis of PPR, distinguish four main approaches.

The first of these can be called the rational choice model, in which the decision is made by a single and rationally thinking leader based on the national interest. It is assumed that: a) the decision maker acts with regard to the integrity and hierarchy of values, about which he has a fairly stable idea; b) he systematically possible consequences of his choice; c) The PPR is open to any new information that could influence the decision.

The second approach assumes that the decision is made under the influence of a set of government structures acting in accordance with established routine procedures. The decision turns out to be divided into separate fragments, and the fragmentation of government structures, the peculiarities of their selection of information, the complexity of mutual relations with each other, differences in the degree of influence and authority, etc. are an obstacle to PPR, based on a systematic assessment of the consequences of a particular choice.

In the third model, the decision is viewed as the result of bargaining in a complex game between members of the bureaucratic hierarchy, government apparatus, etc. each representative of which has its own interests, its own positions, its own ideas about the priorities of the state's foreign policy.

Finally, the fourth approach draws attention to the fact that in many cases decision makers are in a complex environment and have incomplete, limited information. Besides. they are unable to assess the consequences of one choice or another. In such a setting, they have to break down problems by reducing the information used to a small number of variables.

In the analysis of PPR, the researcher must avoid the temptation to use one or another of these approaches "in pure form". IN real life the processes they describe vary in a wide variety of combinations, the study of which should show which of them in each specific case it should be based on and with what others it should be connected (see note 18, p. 71-74).

Decision-making analysis is often used to predict the possible evolution of a particular international situation, such as an interstate conflict. At the same time, not only the factors related “directly” to the PPR are taken into account, but also the potential (a set of resources) that the person or authority that makes the decision has. An interesting technique in this regard, which includes elements of quantitative formalization and is based on various PPR models. proposed in the article by Sh.Z.

Sultanov "Analysis of decision making and the conceptual scheme of forecasting" (see note 10, pp. 71-82).

Modeling

This method is associated with the construction of artificial, ideal, imaginary objects, situations, which are systems, the elements and relations of which correspond to the elements and relations of real international phenomena and processes.

One of the common types of modeling that has become widespread in the science of international relations is related to game theory. Game theory is a theory of decision making in a specific social context, where the concept of "game" extends to all types of human activity. It is based on the theory of probability and is the construction of models for analyzing or predicting various types of behavior of actors in special situations. Classical game theory was developed by mathematician D. von Poymann and economist O. Morgenstern in their joint work “Game Theory and economic behavior published by Princeton University Press in 1947. In the analysis of the behavior of international actors, it found application in the classic works of A. Rapoport, who studied its epistemological possibilities 21 , and T. Schelling, who extended it to the study of such international phenomena as conflicts, negotiations, arms control, deterrence strategy, etc. P. 22. A Canadian specialist in the sociology of international relations, J.-P. Derriennik, considers game theory as a theory of decision-making in a risky situation, or, in other words, as an area of application of the model of subjectively rational action in a situation where all events are unpredictable. If we are talking about a game with several players, then we are dealing with the theory of interdependent decisions, where the risk situation is common, and unpredictability follows for each player from the actions of another. A risky situation finds its solution if its risky nature is eliminated. In a two-player game, when one of the players makes a bad decision, the other gets an extra payoff. If both play well (that is, they act rationally), then neither has a chance to improve his payoff beyond what the rules of the game allow.

In game theory, therefore, the behavior of decision makers in their mutual relations associated with the pursuit of the same goal. In this case, the task is not to describe the behavior of the players or their reaction to information about the behavior of the enemy, but to find the best possible solution for each of them in the face of the predicted decision of the enemy. Game theory shows that the number of types of situation in which players can find themselves is finite. Moreover, it can be reduced to a small number of game models that differ in the nature of the goals, the possibilities of mutual communication and the number of players.

There are games with different number of players: one, two or many. For example, the dilemma of whether or not to take an umbrella with you in unsettled weather is a one-player game (for nature takes no account of man's decisions), which will cease to be such when meteorology becomes an exact science (see note 23, p. thirty).

In a two-player game, such as the famous Prisoner's Dilemma, the players are unable to communicate with each other, so each makes a decision based on the rational behavior of the other. The rules of the game are likened to the rules of a situation in which two people (A and B), who have committed a joint crime and have fallen into the hands of justice, receive from his representatives an offer of voluntary confession (that is, of betrayal in relation to his accomplice). At the same time, everyone is warned about the following: I. If A is recognized (P), B is not recognized (N), then A receives freedom (C), B the maximum punishment (C); 2. If A is not recognized (N), B is recognized (P), then A receives the maximum punishment (C), B freedom (C); 3. If both A and B confess, then both receive severe, although not the maximum punishment (T); 4. If both do not confess, then both receive the minimum punishment (Y).

Graphically, the prisoner's dilemma is presented in the form of such a scheme (Fig. 3):

Ideally, for each of the accomplices, freedom is better than the minimum punishment, the minimum punishment is better than the severe one, and the latter is better than the maximum: S>U>T>B. Therefore, for both, the most profitable option would be N, N. In fact, deprived of the opportunity to communicate with another, not trusting him, everyone expects betrayal from the accomplice (for A this is: N, P) and, trying to avoid B, decides to betray, considering it the least risky, As a result, both choose betrayal ( P, P), and both receive severe punishment.

In terms of symbolic logic, the situation can be represented as follows:

1.( P(A)& P(B)) (S(A)&C(B))

2. ( P(A) & P(B)) ( V(A) & S(B))

3. ( P(A) & P(B)) (T(A) & T(V))

4. (P(A)&P(B)) (U(A)&U(B))

This model was applied to the analysis of many international situations: for example, the foreign policy of Nazi Germany, or the arms race of the period of the 1950s and 1970s. In the latter case, the situation for the two superpowers was based on the severity of the mutual risk presented by nuclear weapons, and the desire of both to avoid mutual destruction. The result was an arms race that did not benefit either side.

Game theory allows you to find (or predict) a solution in some situations: that is, to indicate the best possible solution for each participant, to calculate the most rational way of behavior in various types of circumstances. Nevertheless, it would be a mistake to exaggerate its importance as a method for studying international relations, and even more so as a practical method for developing a strategy and tactics of behavior on the world stage. As we have already seen, decisions taken in international relations are by no means always of a rational nature. In addition, for example, the Prisoners' Dilemma does not take into account that in the field of international relations there are mutual obligations and agreements, and there is also the possibility of communication between participants even during the most intense conflicts.

Consider another type of complex modeling using the example of the work of M.A. Khrustalev "System Modeling of International Relations" (see note 2).

The author sets the task of building a formalized theoretical model representing the trinary methodological (philosophical theory of consciousness), general scientific (general systems theory) and particular scientific (international relations theory) approaches. The construction is carried out in three stages. At the first stage, “pre-model tasks” are formulated, which are combined into two blocks: “evaluative” and “operational”. In this regard, the author analyzes such concepts as "situations" and "processes" (and their types), as well as the level of information. Based on them, a matrix is built, which is a kind of "map", designed to provide the researcher with the choice of an object, taking into account the level of information security.

As for the operational block, the main thing here is to single out the nature (type) of models (conceptual, theoretical and concrete) and their forms (verbal or content, formalized into quantified) on the basis of the triad “general-special-singular”. The selected models are also presented in the form of a matrix, which is a theoretical model of modeling, reflecting its main stages (form), stages (character) and their relationship.

At the second stage, we are talking about building a meaningful conceptual model as a starting point for solving the general research problem. On the basis of two groups of concepts "analytical" (essence-phenomenon, content-form, quantity-quality) and "synthetic" (matter, movement, space, time), presented in the form of a matrix, a "universal cognitive construction configurator" is built, which sets general research framework. Further, based on the selection of the above logical levels of study of any system, the noted concepts are subjected to reduction, as a result of which “analytical” (essential, content, structural, behavioral) and “synthetic” (substrate, dynamic, spatial and temporal) characteristics of the object are distinguished. Based on the “system oriented matrix configurator” structured in this way, the author traces specific features and some trends in the evolution of the system of international relations.

At the third stage, a more detailed analysis of the composition and internal structure of international relations is carried out, that is, the construction of its detailed model. Here, the composition and structure (elements, subsystems, connections, processes) are distinguished, as well as the “programs” of the system of international relations (interests, resources, goals, mode of action, balance of interests, balance of forces, relations). Interests, resources, goals, course of action are elements of the "program" of subsystems or elements. Resources, characterized as a "non-system-forming element", are subdivided by the author into resources of means (material-energy and information) and resources of conditions (space and time).

The "program of the system of international relations" is a derivative in relation to the "programs" of elements and subsystems. Its backbone element is the "correlation of interests" of various elements and subsystems with each other. The non-system-forming element is the concept of “balance of forces”, which could be more accurately expressed by the term “balance of means” or “correlation of potentials”. The third derived element of this "program" is the "relationship", understood by the author as a kind of evaluative representation of the system about itself and about the environment.

Based on the theoretical model constructed in this way, M.A. Khrustalev analyzes the real processes characteristic of modern stage world developed. He notes that if key factor, which determined the evolution of the system of international relations throughout its history, was interstate conflict interaction within the framework of stable confrontational axes, then by the 90s of the XX century. there are prerequisites for the transition of the system to a different qualitative state. It is characterized not only by the breaking of the global confrontational axis, but also by the gradual formation of stable axes of comprehensive cooperation between the developed countries of the world. As a result, an informal subsystem of developed states appears in the form of a world economic complex, the core of which has become the “seven” of leading developed countries, which has objectively turned into a control center that regulates the process of developing the system of international relations. The fundamental difference between such a “control center” and the League of Nation or the UN is that it is the result of self-organization, and not the product of “social engineering” with its characteristic static completeness and poor adequacy to dynamic changes in the environment. As the governing center, the G7 solves two important tasks for the functioning of the system of international relations: first, the elimination of existing and preventing the emergence of regional confrontational military-political axes in the future; secondly, stimulating the democratization of countries with authoritarian regimes (the creation of a single world political space). Highlighting, taking into account the model he proposed, also other trends in the development of the system of international relations, M.A. Khrustalev considers the emergence and consolidation of the concept of “world community” and the identification of the idea of a “new world order” to be very symptomatic, emphasizing at the same time that the current state of the system of international relations as a whole does not yet meet the modern needs of the development of human civilization.

Such a detailed consideration of the method of system modeling as applied to the analysis of international relations allows us to see both the advantages and disadvantages of both this method itself and the system approach as a whole. The advantages include the above-mentioned generalizing, synthesizing nature of the systematic approach. It allows you to discover both the integrity of the object under study and the diversity of its constituent elements (subsystems), which can be participants in international interactions, relations between them, spatio-temporal factors, political, economic, religious characteristics, etc. A systematic approach makes it possible not only to fix certain changes in the functioning of international relations, but also to discover the causal relationships of such changes with the evolution of the international system, to identify the determinants that affect the behavior of states. System modeling gives the science of international relations those opportunities for theoretical experimentation, which, in its absence, it is practically deprived of. It also provides an opportunity for the complex application of applied methods and techniques of analysis in their most diverse combination, thereby expanding the prospects for research and their practical benefits for explaining and predicting international relations and world politics.

At the same time, it would be wrong to exaggerate the importance of a systematic approach and modeling for science, to ignore their weaknesses and shortcomings. Paradoxical as it may seem, the main one is the fact that no model, even the most flawless in its logical foundations, gives confidence in the correctness of the conclusions drawn on its basis. This, however, is recognized by the author of the work considered above, when he speaks of the impossibility of constructing an absolutely objective model of the system of international relations (see note 2, p. 22). We add that there is always a certain gap between the model constructed by this or that author and the actual sources of the conclusions that he formulates about the object under study. And the more abstract (that is, the more strictly logically justified) the model is, and also the more adequate to reality its author strives to make his conclusions, the wider the indicated gap. In other words, there is a serious suspicion that when formulating conclusions, the author relies not so much on the model construction he has built, but on the initial assumptions, the “building material” of this model, as well as on others not related to it, including “intuitive logical" methods. Hence the question, which is very unpleasant for “uncompromising” supporters of formal methods: could those (or similar) conclusions that appeared as a result of a model study be formulated without a model? A significant discrepancy between the novelty of such results and the efforts made by researchers on the basis of system modeling makes us think that an affirmative answer to this question looks very reasonable. As B. Russett and H. Starr emphasize in a similar connection: “to a certain extent, the proportion of each contribution can be determined using data collection and analysis methods typical of modern social sciences. But in all other respects we remain in the realm of conjecture, intuition and informed wisdom” (see note 12, p. 37).

As for the systematic approach as a whole, its shortcomings are a continuation of its merits. Indeed, the advantages of the concept of "international system" are so obvious that it is used, with few exceptions, by representatives of all theoretical trends and schools in the science of international relations. However, as the French political scientist M. Girard rightly swept, few people know exactly what it means in reality. It continues to retain a more or less rigorous meaning for functionalists, structuralists, and systemists. For the rest, it is most often nothing more than a beautiful scientific epithet, convenient for decorating an ill-defined political object. As a result, this concept turned out to be oversaturated and devalued, which hinders its creative use 24 .

Agreeing with the negative assessment of the arbitrary interpretation of the concept of "system", we emphasize once again that this does not at all mean doubts about the fruitfulness of applying both the systems approach and its specific incarnations of system theory and system analysis to the study of international relations.

System analysis and modeling are the most common of the analytical methods, which are a set of complex research methods, procedures and techniques of an interdisciplinary nature related to the processing, classification, interpretation and description of data. On their basis and with their use, many other analytical methods of a more particular nature have appeared and become widespread, to the consideration of which we will now proceed.

3. Other analytical methods

The most common of them are content analysis, event analysis, the method of cognitive mapping and their numerous varieties (see: note 2; 10; 16).

Cotpent analysis in political science was first applied by the American researcher G. Lasswell and his collaborators in the study of the propaganda orientation of political texts and described by them in 1949. 25 . In its most general form, this method can be represented as a systematic study of the content of a written or oral text with the fixation of the most frequently repeated phrases or plots in it. Further, the frequency of these phrases or plots is compared with their frequency in other written or oral messages, known as neutral, on the basis of which a conclusion is made about the political orientation of the content of the text under study. Describing this method, M .A. Xpy stalev and K.P. Borishpolets distinguish such stages of its application as: text structuring associated with primary processing information material; processing of the information array using matrix tables; quantification of information material, allowing to continue its analysis with the help of electronic computers (see note 16, pp. 86-94).

The degree of rigor and operationality of the method depends on the correctness of the selection of primary units of analysis (terms, phrases, semantic blocks, topics, etc.) and units of measurement (for example, a word, phrase, section, page, etc.).

Event analysis (or event data analysis) is aimed at processing public information showing "who says or does what, in relation to whom and when." The systematization and processing of the relevant data is carried out according to the following criteria: 1) the initiating subject (who); 2) plot or "issue - area" (what); 3) the target subject (in relation to whom) and 4) the date of the event (when) (see note 8, pp. 260-261). Events systematized in this way are summarized in matrix tables, ranked and measured using a computer. The effectiveness of this method requires the presence of a significant data bank. Scientific and applied projects using event analysis differ in the type of behavior studied, the number of politicians, according to the studied time parameters, the number of sources used, the typology of matrix tables, etc.

As for the method of cognitive mapping, it is aimed at analyzing how one or another politician perceives a certain political problem.

American scientists R. Snyder, H. Brook and B. Sapin showed in 1954 that political leaders' decisions can be based not only and not so much on the reality that surrounds them, but on how they perceive it. In 1976, R. Jervis in his work "Perception and misperception (misperception) in international politics" showed that in addition to emotional factors the decision made by a particular leader is influenced by cognitive factors. From this point of view, the information received by decision makers is assimilated and ordered by them “with correction” for their own views on the outside world. Hence the tendency to underestimate any information that contradicts their system of values and the image of the enemy, or, on the contrary, to give an exaggerated role to insignificant events. An analysis of cognitive factors makes it possible to understand, for example, that the relative constancy of the state's foreign policy is explained, along with other reasons, by the constancy of the views of the respective leaders.

The method of cognitive mapping solves the problem of identifying the basic concepts used by a politician and finding the cause-and-effect relationships between them. “As a result, the researcher receives a map-scheme, on which, based on the study of the speeches and speeches of a political figure, his perception of the political situation or individual problems in it is reflected” (see note 4, p. 6).

In applying the described methods, which have a number of undoubted advantages, the possibility of obtaining new information based on the systematization of already known documents and facts, increasing the level of objectivity, the possibility of measuring, etc., the researcher also faces serious problems. This is the problem of information sources and its reliability. , availability and completeness of databases, etc. But the main problem is the problem of the costs that require research using content analysis, event analysis and the method of cognitive mapping. Database compilation, their encoding, programming, etc. take a significant amount of time, require expensive equipment, require the involvement of appropriate specialists, which ultimately translates into significant amounts.

Taking into account these problems, Professor B. Korani of the University of Montreal proposed a methodology with a limited number of indicators of the behavior of an international author, which are considered as key (most characteristic) (see: note 8, p. 263265). There are only four such indicators: the method of diplomatic representation, economic transactions, interstate visits and agreements (contracts). These indicators are classified according to their type (for example, agreements can be diplomatic, military, cultural, or economic) and their level of significance. Then a matrix table is compiled, giving a visual representation of the object under study. So, the table reflecting the exchange of visits looks like this:

As for the methods of diplomatic representation, their classification is based on their level (ambassador level or lower level) and whether it is a direct representation or through the intermediary of another country (resident or non-resident). The combination of these data can be represented as follows:

On the basis of such data, conclusions are drawn regarding the way the international author behaves in time and space: with whom does he maintain the most intense interactions, in what period and in what area do they occur, etc.

Using this technique, B. Korani established that almost all military-political relations that, for example, Algeria had in the 70s, he maintained with the USSR, while the level of economic relations with the entire socialist camp was rather weak. In fact, most of Algeria's economic relations were directed towards cooperation with the West, and especially with the United States, "the main imperialist power." As B. Korani writes, “such a conclusion, contrary to “common sense” and first impressions [we recall that Algeria belonged in these years to the countries of the “socialist orientation”, adhering to the course of “anti-imperialist struggle and all-round cooperation with the countries of socialism” P.Ts. ], could not be done, and it could not be believed without the use of a rigorous methodology, supported by the systematization of data ”(see note 8, p. 264). Perhaps this is a somewhat exaggerated estimate. But in any case, this technique is quite effective, fairly evidence-based and not too expensive.

However, its limitations should also be emphasized, which, however, is common to all the above methods. As its author himself admits, it cannot (or can only partially) answer the question about the causes of certain phenomena. Such methods and techniques are much more useful at the level of description rather than explanation. They give, as it were, a photograph, a general view of the situation, they show what is happening, but without clarifying why. But this is precisely what their purpose is to perform a diagnostic role in the analysis of certain events, situations and problems of international relations. However, for this they need primary material, the availability of data that are subject to further processing and the accumulation of which is carried out on the basis of private methods.

4. Private methods

Private methods are understood as the sum of interdisciplinary procedures used for the accumulation and primary systematization of empirical material (“data”). Therefore, sometimes they are also called "research techniques". To date, more than a thousand such techniques are known, from the simplest (for example, observation) to quite complex ones (such as situational games approaching one of the stages of system modeling). The most famous of them are questionnaires, interviews, expert surveys, and expert meetings. A variation of the latter is, for example, the "Delphian technique" when independent experts submit their assessments of an international event to the central body, which generalizes and systematizes them, and then returns them to the experts. Taking into account the generalization carried out, the experts either amend their initial assessments, or strengthen their opinion and continue to insist on it. In accordance with this, the final assessment is developed and practical recommendations are given.

Consider the most common analytical methods: observation, study of documents, comparison, experiment.

Observation

As is known, the elements of this method are the subject of observation, the object and means of observation. There are different types of observations. So, for example, direct observation, in contrast to indirect (instrumental) observation, does not involve the use of any technical equipment or tools (television, radio, etc.). It can be external (similar to that carried out, for example, by parliamentary journalists or special correspondents in foreign countries) and included (when the observer is a direct participant in an international event: diplomatic negotiations, a joint project or an armed conflict). In turn, direct observation differs from indirect observation, which is carried out on the basis of information obtained through interviews, questionnaires, etc. In the science of international relations, indirect and instrumental observation is mainly possible. The main disadvantage of this method of data collection is the large role of subjective factors associated with the activity of the subject, his (or primary observers) ideological preferences, imperfection or deformity of the means of observation, etc. (see note 5, pp. 57-58).

Studying documents

As applied to international relations, it has the peculiarity that an “unofficial” researcher often does not have free access to sources of objective information (unlike, for example, staff analysts, experts from international agencies or security officials). The ideas of this or that regime about state secrets and security play a big role in this. In the USSR, for example, the volume of oil production, the level of industrial production, etc., remained a subject of state secrets for a long time; there was a huge array of documents and literature intended only "for official use", the ban on the free circulation of foreign publications remained, a huge number of institutions and institutions were closed to "outsiders". There is another problem that makes it difficult to use this method, which is one of the initial, basic for any research in the field of social and political sciences: this is the problem of financial resources necessary for the acquisition, processing and storage of documents, payment of labor costs associated with this, and so on. It is understandable, therefore, that the more developed a state is and the more democratic its political regime, the more favorable opportunities there are for research in the social and political sciences. Unfortunately, for modern Russia, both of these problems are very relevant. And the aggravation of the economic crisis, combined with the turn of the value priorities of the mass consciousness towards mercantilism, associated with the loss of many spiritual guidelines, unusually exacerbates the difficulties of research work in general and in the field of international relations in particular.

The most accessible are official documents: messages from the press services of diplomatic and military departments, information about visits of statesmen, statutory documents and statements of the most influential intergovernmental organizations, declarations and messages from power structures, political parties and public associations, etc. At the same time, unofficial written, audio and audiovisual sources are also widely used, which in one way or another can contribute to an increase in information about the events of international life: records of the opinions of individuals, family archives, unpublished diaries. Memories of direct participants in certain international events, wars, diplomatic negotiations, official visits can be of great importance. This also applies to the forms of such memories written or oral, direct or restored, etc. An important role in the collection of data is played by the so-called iconographic documents: paintings, photographs, films, exhibitions, slogans. Thus, in the conditions of the closeness prevailing in the USSR, increased secrecy and, consequently, the practical inaccessibility of unofficial information, American Sovietologists paid great attention to the study of iconographic documents, for example, reports from festive demonstrations and parades. The features of the design of the columns, the content of slogans and posters, the number and personal composition of officials present on the podium, and, of course, the types of demonstrations were studied. military equipment and armaments 26 .

Comparison

It is also a method that is common to many disciplines. According to B. Russet and H. Starr, it began to be applied in the science of international relations only from the mid-60s, when the incessant growth in the number of states and other international actors made it both possible and absolutely necessary (see note 12, p. 46). The main advantage of this method is that it aims at the search for common, recurring in the field of international relations. The need to compare states and their individual characteristics (territory, population, level of economic development, military potential, length of borders, etc.) stimulated the development of quantitative methods in the science of international relations and, in particular, measurement. Thus, if there is a hypothesis that large states are more prone to unleashing war than all others, then there is a need to measure the size of states in order to determine which of them is large and which is small, and by what criteria. In addition to this, the “spatial”, aspect of measurement, there is a need to measure “in time”, that is, to find out in historical retrospect what size of the state enhances its “inclination” to war (see note 12, p. 4748).

At the same time, comparative analysis makes it possible to obtain scientifically significant conclusions based on the dissimilarity of phenomena and the uniqueness of the situation. Thus, comparing iconographic documents (in particular, photographs and newsreels) reflecting the departure of French soldiers to the army in 1914 and in 1939, M. Ferro discovered an impressive difference in their behavior. The smiles, the dances, the atmosphere of general rejoicing that prevailed at the Gare de l'Est in Paris in 1914 contrasted sharply with the picture of despondency, hopelessness, and a clear unwillingness to go to the front, observed at the same station in 1939. Since these situations could not have developed under the influence of the pacifist movement (according to written sources, it was never as strong as on the eve of 1914, and, on the contrary, almost did not manifest itself at all before 1939), a hypothesis was put forward according to which one from the explanation of the contrast described above, it must be that in 1914, unlike in 1939, there was no doubt about who the enemy was: the enemy was known and identified. The proof of this hypothesis became one of the ideas of a very interesting and original study devoted to understanding the First World War 27 .

Experiment

The method of experiment as the creation of an artificial situation in order to test theoretical hypotheses, conclusions and positions is one of the main ones in the natural sciences. In the social sciences, the most widespread form of it is simulation games, which are a kind of laboratory experiment (as opposed to a field experiment). There are two types of simulation games: without the use of electronic computers and with its use. In the first case, we are talking about individual or group actions related to the performance of certain roles (for example, states, governments, politicians or international organizations) in accordance with a pre-designed scenario. At the same time, the participants must strictly observe the formal conditions of the game controlled by its leaders: for example, in the case of an imitation of an interstate conflict, all parameters of the state whose role is played by the participant must be taken into account, economic and military potential, participation in unions, stability of the ruling regime, etc. Otherwise, such a game can turn into mere entertainment and a waste of time in terms of cognitive results. Computer-assisted simulation games offer much broader exploratory perspectives. Based on the relevant databases, they make it possible, for example, to reproduce a model of diplomatic history. Starting with the simplest and most plausible model for explaining the current events of crises, conflicts, the creation of intergovernmental organizations, etc., it is further explored how it fits the previously selected historical examples. Through trial and error, changing the parameters of the original model, adding previously omitted variables in it, taking into account cultural and historical values, shifts in the dominant mentality, etc., one can gradually move towards achieving its greater correspondence with the reproduced model of diplomatic history and on the basis of comparison these two models to put forward reasonable hypotheses about the possible development of current events in the future.

Concluding our discussion of the methods used in the science of international relations, we summarize the main conclusions concerning our discipline.

Firstly, the absence of "own" methods in the sociology of international relations does not deprive it of the right to exist and is not a reason for pessimism: not only social, but also many "natural sciences" are successfully developing using "interdisciplinary" methods common with other sciences. and procedures for studying your object. Moreover, interdisciplinarity is increasingly becoming one of the important conditions scientific progress in any branch of knowledge. We emphasize once again that each science uses general theoretical (characteristic of all sciences) and general scientific (characteristic of a group of sciences) methods of cognition.

Secondly, the most common in the sociology of international relations are such general scientific methods as observation, study of documents, a systematic approach (system theory and system analysis), and modeling. Applied interdisciplinary methods (content analysis, event analysis, etc.), which are developing on the basis of general scientific approaches, are widely used in it, as well as private methods for collecting and primary data processing. At the same time, all of them are modified taking into account the object and goals of the study and acquire new specific features here, being fixed as “their own” methods of this discipline. We note in passing that the difference between analytical, applied and particular methods is quite relative: the same methods can act both as general scientific approaches and as specific methods (for example, observation).

Thirdly, like any other discipline, the sociology of international relations in its entirety, as a certain set of theoretical knowledge, acts simultaneously as a method of knowing its object. Hence the attention paid in this work to the basic concepts of this discipline: each of them, reflecting one or another side of international realities, in epistemological terms, carries a methodological load, or, in other words, plays the role of a guideline for further study of its content only the deepening and expansion of knowledge, but to their concretization in relation to the needs of practice.

Finally, it should be emphasized again that the best result is achieved with complex use various research methods and techniques. Only in this case, the researcher can hope to discover recurrence in a chain of disparate facts, situations and events, that is, a kind of regularity (respectively, deviant) of international relations.

Notes

Braud Ph. La science politic. Paris, 1992, p.3.
Khrustalev M.A.. System modeling of international relations Abstract for the degree of doctor of political sciences M., 1992, p.89.
Tsygankov A.P.. Hans Morgenthau: a look at foreign policy // Power and Democracy. Digest of articles. Ed. P.A. Tsygankov a. M., 1992, p.171.
Lebedeva M.M.., Tyulin IG Applied interdisciplinary political science: opportunities and prospects / / System approach: analysis and forecasting of international relations (experience in applied research). Collection of scientific papers. Ed. Doctor of Political Sciences I.G. Tyulin. M., 1991.
Chrysalis E. Problems of the theory of international relations (translated from Polish). M., 1980, p.52-56; 60-61.
Hoffmann S. Theorie et relations internationales. Paris, 1965, p.428.
Merle M. Les acteurs dans les relations internationales. Paris, 1986.
Korany B. et colL Analyze des relations internationales. Approches, concepts et donnees. Montreal. 1987.
Braillard Ph. Philosophy e et relations internationales. Paris, 1965.
IN AND. Lenin and dialectics of modern international relations. Collection of scientific papers. Ed. Ashina G.K., Tyulina I.G. M., 1982.
Aron R. Paix et Guerre entre les nations., p., 1984, p.l03.
Rassettb., Starr H. World Politics. Menu for Choice. San-Francisco, 1981.
Pozdnyakov E.A.. System approach and international relations. M., 1976.
System, structure and process of development of international relations / Ed. ed. V.I.Gantman. M., 1984.
Antyukhina-Moskovchenko V.I.., Zlobin A.A., Khrustalev M.A. Fundamentals of the theory of international relations. M., 1988.
Analytical methods in the study of international relations. Collection of scientific papers. Ed. Tyulina I.G., Kozhemtsova A.S., Khrusgaleva MA. M., 1982.
Bosc R. Sociologie de la paix. Paris, 1965, p.47-48.
Braillard Ph., Djalili M.-R. Les relations internationales. Paris, 1988, p.65-71.
Senarklens P.de. La politiqoe intemationale. Paris, 1992, p.44-47.
Rapoport A. N-Person Game T h eo ri e, Concepts and Applications. Un. of Michigan Press, 1970.
SnyderR.C. , Bruck H. W , Sapin B. Decision-Making as an Approach to the Study of International Politics. 1954.
SchellingT. The Strategy of Conflict Oxford, 1971.
Derriennic J.-P. Esquisse de problematique pour un e sociologie des relations internationales. Grenoble. 1977, p.29-33.
Girard M. Turbulence dans la theorie politique intemationale ou James Rosenau inventeur// Revue francaise de science politique. Vol. 42, no. 4, out 1992, p.642.
LasswellH. & Leites N. The Language of Politics: Studies in Quantitative Semantics. N.Y., 1949.
Batalov E.A. What is applied political science// Conflicts and consensus. 1991. WE.

Ferro M. Penser la Premiere Guerre Mondiale. In: Penser le XX-e siecle. Bruxelles, 1990.

INTRODUCTION

CHAPTER I. MATHEMATICAL METHODS AND INTERNATIONAL RELATIONS

§ 1. Modeling of socio-economic processes-

political analysis tools

§2. New information technologies and their role in modeling international politics

§3. The need to build mathematical models

new generation on a single methodological basis

§4. Function spaces and the problem of representing dependencies as superpositions of elementary ones

§5. Combinatorial models of political behavior, ..,

§6. Main approaches to using indicator systems

for the analysis of foreign policy processes

§7. The space of indicators in the system of international relations - the main tasks of metatheory

CHAPTER II. MODELS OF CLASSIFICATION OF INFORMATION IN THE SYSTEM OF MANAGEMENT OF INFORMATION RESOURCES IN THE FOREIGN POLICY SPHERE

§1. Information counteraction to strategic

intelligence

§2. Classification of information as an element of the information resource management system - domestic

and foreign experience

§3. Methodology for Individual Assessment of the Consequences of Classifying Foreign Policy Information

§4. Use of models of national, regional and world development for information classification. 163 §5. Coding as a way to protect information from unauthorized access - mathematical models

CHAPTER III. SPECTRAL CHARACTERISTICS IN MATHEMATICAL MODELS OF THE SYSTEM

INTERNATIONAL RELATIONS

§ 1. The group structure of the set of foreign policy

indicators

§2. Lacunary series as tools in the problem of characterization of political processes (trigonometric case)

§3. Lacunar series as tools in the problem of characterization of political processes (the case of the system

§4. Solution of P. Kennedy's problem of spectrum characterization

lacunar systems

§5. Applying the Lacunary Analysis Technique to Problems of the Representability of the Political Process as Measurable

functions on a set of indicators

CONCLUSION (summary)

APPLICATION

1. Main political indicators used in studies of the system of international relations

2. Tables of proximity measures used in mathematical models and in the processing of empirical data

3. About the experience of functioning of the automated

UN Secretariat information systems

4. Listings of programs for quantitative processing of voting results at the UN General Assembly

5. Solution of U. Rudin's problem of characterization of the density of lacunar sets (political indicators)

LITERATURE

Recommended list of dissertations

Development of information technologies in the foreign policy activities of the Russian Federation: problems and prospects 2005, candidate of political sciences Glebova, Irina Sergeevna
Methods and Algorithms for Processing Fuzzy Information in Intelligent Support Systems in Making Management Decisions 2007, Doctor of Technical Sciences Ryzhov, Alexander Pavlovich
Theoretical and Methodological Problems of Forming the Strategy of Russia's Foreign Policy in the Conditions of the Formation of the Global Information Space 1999, doctor of political sciences Medinsky, Vladimir Rostislavovich
Mechanisms for optimizing the foreign policy activities of the Russian Federation in the post-Soviet space 2006, candidate of political sciences Vorozhtsova, Elena Alexandrovna
Information processes as a factor in the development of modern international relations: political analysis of the developing world 2009, doctor of political sciences Seidov, Shakhrutdin Gadzhialievich

Introduction to the thesis (part of the abstract) on the topic "Application of mathematical methods in the study of the system of international relations using functional spaces"

INTRODUCTION

Mathematization modern science is a regular and natural process. If the differentiation of scientific knowledge leads to the emergence of new branches of science, then the integration processes in the cognition of the world lead to a kind of diffusion of scientific ideas from one area to another. In the 18th century, Immanuel Kant not only proclaims the slogan "every science is a science insofar as it is mathematics", but also puts the ideas of the axiomatic construction of Euclid's geometry into his concept of apriorism.1 While in natural science mathematics quickly and firmly took a leading position, in the field social sciences, its successes were more modest. The use of mathematical methods turned out to be justified where the concepts are of a stable nature and the task of establishing a connection between these concepts becomes meaningful, and not an endless redefinition of the concepts themselves. Recognizing determinism in the social sphere, one should thereby recognize the existence of a scientific basis in the theory of international relations. Therefore, the system of international relations, no matter how complex and poorly formalizable it may be, can and should be the subject of application of mathematical methods. Politicians, practitioners of foreign policy departments, international scientists, sociologists, psychologists, geographers, military men, etc. are extremely interested in scientific methods of studying international relations. Empiricism in international studies, i.e. The trend associated with the study of statistical information in international relations has introduced many different and heterogeneous methods and algorithms into the theory. There was a need for systematization and a unified approach to statistical data. International Information

Macia as a special type of information needed specialized processing methods. In the context of the dynamic development of events in the country, the regime of secrecy that has been in force since the end of the Second World War turned out to be an extreme anachronism. Back in 1989, preparatory work began to create a new, more advanced information regime. The first research phase of the work covered the period from 1988 to 1990 and included the development of a draft law on state secrets and the protection of classified information, as well as the search for a concept to prevent damage from incorrect classification of information. The Ministry of Foreign Affairs was entrusted with the task of searching for legal and procedural norms for classifying foreign policy information. In the complex of problems that have arisen, the leading place was taken by the problem of constructing a mathematical model of the impact of information classification on the country's security. Thus, the problem of correct description and forecasting of information flows in the system of the Ministry of Foreign Affairs turned out to be among the strategic ones, which are especially important for the state.

International relations, as you know, include the totality of relations between countries, including political, economic, military, scientific, cultural, etc. Modeling is an effective toolkit that allows you to explain and predict the observed object under study. Representatives of the exact (natural) and humanities put different meanings into the concept of a model; there is a so-called methodological dichotomy when the historical-descriptive (or intuitive-logical) approach of the representatives of the humanities is contrasted with the analytical and prognostic approach associated with the application of methods of the exact sciences.

As A.N. Tikhonov 2 "A mathematical model is an approximate description of any class of phenomena of the external world, expressed with the help of mathematical symbols." Mathematical modeling is usually understood as the study of a phenomenon with the help of its mathematical model. In the cited article by A.N. Tikhonov subdivides the process of mathematical modeling into 4 stages -

1. Formation of a law that links the main objects of the model, which requires knowledge of facts and phenomena related to the phenomena under study - this stage ends with a record in mathematical terms of formulated qualitative ideas about the relationships between the objects of the model;

2. The study of mathematical problems to which the mathematical model leads. The main question of this stage is the solution of the direct problem, i.e. obtaining through the model of the output data of the described object - typical mathematical problems are considered here as an independent object;

3. The third stage is connected with checking the consistency of the constructed model with the criterion of practice. If it is required to determine the parameters of the model to ensure its consistency with practice, such problems are called inverse;

4. Finally, the last stage is related to the analysis of the model and its modernization in connection with the accumulation of empirical data.

There is a widespread opinion that the social sciences do not have their own specific, only inherent method; therefore, they somehow refract general scientific methods and methods of other sciences in relation to their object. Mathematization of social science is due to the desire to clothe their positions and ideas in

precise, abstract mathematical forms and models, the desire to dei-deologize their results.

Models of economic relations between states and regions seem to us to be a fairly developed area - the science of applying quantitative methods in economic research is called econometrics. The peak of research in this area is apparently associated with the well-known work of D. Forrester "World Dynamics", which describes a model of global development implemented in a special machine language "DINAMO". Less well known are the results of mathematical modeling of political processes. Description of the political behavior of states in the international arena is a poorly structured, difficult to formalize multi-factorial task. In attempts to theoretically substantiate foreign policy since the beginning of the 20th century, various ideas have been put forward, the beginning of which has its origins in the political life of ancient Greece and Rome; the names "moralism", "normativism", "legalism". The practical experience of the pre-war crisis and the Second World War put forward new ideas of pragmatism, which would make it possible to link the theory and practice of foreign policy with the realities of the 20th century. These ideas served as the basis for the creation of the school of "political realism", whose leader was Professor G. Morgenthau of the University of Chicago. In an effort to get away from ideology, realists increasingly began to turn to the study of empirical data by mathematical methods. This is how the current of "modernists" appeared, who often absolutized mathematical methods in politics as the only reliable ones. The most balanced approach differed works

D. Singer, K. Deutsch, who saw effective tools in mathematical methods, but did not exclude a person from the decision-making system. The well-known mathematician J. von Neumann believed that politics should develop its own mathematics; of the existing mathematical disciplines, he considered game theory to be the most applicable in political research. In the variety of formalized methods, the most common methods are content analysis,3 event analysis4 and the method of cognitive mapping.5

The ideas of content analysis (text content analysis) as a method of analyzing the most frequently occurring combinations in political texts were introduced into politics by the American researcher G. Lasuel 6 . Event analysis (analysis of event data) implies the existence of an extensive database with a certain systematization and processing of data matrices. The method of cognitive mapping was developed in the early 70s specifically for political research. Its essence lies in the construction of a combinatorial graph, in the nodes of which there are goals, and the edges define the characterization of possible connections between the goals. These methods still cannot be attributed to mathematical models, since they are aimed at presenting, structuring data and are only a preparatory part of quantitative data processing. The first mathematical model developed for purely political science is the well-known model of arms dynamics by the Scottish mathematician and meteorologist L. Richardson, first published in 1939. side, and the deterrent is their own economy, which cannot withstand the endless burden of armaments. These simple considerations, translated

translated into mathematical language, give a system of linear differential equations that can be integrated: 6A

TA-pWh^(0.

Having calculated the coefficients k, 1, m, n, L. Richardson obtained surprisingly accurate agreement between the calculated data and the empirical data on the example of the 1st World War, when Austria-Hungary and Germany were on one side, and Russia and France on the other. The equations made it possible to explain the dynamics of the armaments of the conflicting parties.

It is mathematical methods that make it possible to explain the dynamics of population growth, to evaluate the characteristics of information flows and other phenomena in the social world. Let us give, for example, an assessment of the dynamics of the spread of mathematical methods in international studies. Let Х(Ч) be the share of mathematical methods in the total volume of research on international topics at the time 1;. Assuming that the increase in research on the theory of international relations using mathematical methods is proportional to their present share, as well as the degree of remoteness from saturation A, we have a differential equation:

KX(A-X), the solution of which is the logistic curve.

The greatest success in international studies has been achieved by methods that allow statistical processing of the totality of data of foreign policy information. Factor methods,

cluster and correlation analysis made it possible to explain, in particular, the nature of the behavior of states when voting in collective bodies (for example, in the US Congress or at the UN General Assembly). Fundamental results in this direction belong to American scientists. Thus, the project "A Cross-Polity Survey" was carried out under the leadership of A.Banks and R. Textor at the Massachusetts Institute of Technology. The Correlates of War Project: 1918-1965, headed by D. Singer, is devoted to the statistical processing of voluminous information on 144 nations and 93 wars for the period 1818-1965. In the "Dimentions of Nations" project, which was developed at Northwestern University, computer implementations of factor analysis methods were used at the computer centers of Indiana, Chicago and Yale universities, etc. Practical tasks for the development of analytical methods for specific situations have been repeatedly set by the US State Department for research centers. For example, D. Kirkpatrick, the US Permanent Representative to the Security Council, asked to develop a methodology by which US aid to developing countries would be put in a clear correlation dependence on the results of voting at the UN General Assembly of these countries in comparison with the US position. The US State Department also attempted to assess the probability of the capture of the American embassy in Tehran during the known events through the analysis of expert survey data. Sufficiently complete surveys on the application of mathematical methods in the theory of international relations have been compiled, for example, by M. Nicholson 8 , M. Ward 9 and others.

The study of modern international relations by quantitative (mathematical) methods in the Diplomatic Academy

The MFA of Russia has been held since 1987. The author has built models for structuring and predicting the results of voting at the UN General Assembly both using computer statistical packages and using his own algorithms for structural data processing. Fundamentally new models of structuring the flows of foreign policy information were developed by the author within the framework of the interdepartmental government program "Secret" when developing a draft of a new state information regime. The need to develop new algorithms for structural data processing is strongly dictated by the practical needs of the Ministry of Foreign Affairs: new high-speed and highly efficient computer technology does not allow such luxury as old and too general algorithms. The basic idea of managing the flow of foreign policy information on the basis of a synthetic criterion of state power goes back to early work G. Morgenthau10. The indicators of the power of the state, given in one of his works by the American researcher D. Smith11, were used by a working group led by Professor of the Diplomatic Academy of the Russian Foreign Ministry A.K. Subbotin to create an information resource management model. The construction of mathematically correct models for managing the flow of foreign policy information using synthetic criteria seems to be a difficult task. On the one hand, convolution of a set of single indicators into a single universal indicator, even if it satisfies the necessary invariance conditions, obviously leads to a loss of information. On the other hand, alternative methods such as Pareto-optimal criteria are not able to resolve the situation in the case of incomparable systems of indicators (maximum elements in a partially ordered set).

One of the approaches that resolve this situation may be the author's approach using the apparatus of function spaces. In particular, in the space of indicators (indicators, components) of the power of the state, a subset of synthetic indicators is distinguished: among which, in particular, there may be linear functions of the main (basic) indicators. In the case of a linear change of variables (i.e., a change of basis) in the space of base indicators, these synthetic indicators are transformed covariantly, in contrast to the base ones, which are transformed contravariantly. Thus, the proposed method essentially contains the tensor approach in general systems theory, coming from the American researcher G. Kron.

The system of single indicators (indicators) characterizing the state or the political process is the main information base for making a foreign policy decision. Making decisions on different systems of indicators leads, generally speaking, to inconsistent, if not directly opposite, conclusions. When such conclusions are drawn using quantitative procedures, it undermines the credibility of the use of mathematical methods in international research. To correct this situation, procedures should be developed to assess the degree of consistency of indicator samples. In the absence of such algorithms, not only the possibility of any adequate mathematical modeling in the system of international relations is called into question, but also the very existence of a scientific approach to this problem. The well-known American researcher Morton Kaplan expressed these doubts in his work 12: “Does the subject of international relations involve any kind of coherent research, or is it an ordinary bag from which you take out and

is it taken that at the moment we are interested and to which it is impossible to apply any coherent theory, generalizations or unify methods?". Elimination of contradictions in the conclusions obtained on the basis of processing the results of observations for different subsystems of indicators, the paper proposes to carry out as follows. It is natural to consider all conceivable indicators (indicators) that describe the system of international relations as a kind of initially existing set, which, obviously, is infinite. This set is supposed to be considered actually infinite as a complete, complete set of indicators available to our review. Following S. Kleene13 "this infinity by us regarded as actual or complete, or extended or existential. An infinite set is considered as existing in the form of a complete set, before and independently of any process of generation or construction of it by a person, as if it were completely lying before us for our review. "According to the abstraction of actual infinity in an infinite set, each of its elements can be distinguished , but in fact, it is fundamentally impossible to fix and describe each element of an infinite set.The abstraction of the actual infinity is a distraction from this impossibility, "... relying on the abstraction of the actual infinity, we get the opportunity to stop the movement, to individualize each element of the infinite totality"14. Actual infinity in mathematics has its supporters and opponents.The opposite point of view of constructivists - the abstraction of potential infinity is based on a strict mathematical concept of an algorithm: the existence of only those objects that can be constructed as a result of some procedure is recognized.

An example of such formalized approaches to the selection of the nomenclature of indicators of the object under study are, for example, the methods used in state standardization bodies. or, which is practically the same thing, the problem of metrics in the system of indicators. The most common metrics of Euclid, Minkowski, Hamming, being introduced on a set of indicators, determine the type of abstract space in which the desired mathematical model is built. Namely, the presence of a metric allows us to talk about the degree of proximity of states in relation to each other and obtain various quantitative characteristics. The introduced spaces actually turn out to be linear normed spaces with like-named norms, i.e., Banach spaces. The main method in the theory of linear spaces is the method of studying the properties of a system of vectors with respect to linear transformations of the space itself. Thus, the main idea of factor analysis of data, which is most widely used in international studies, is the search for a suitable orthogonal transformation that transfers the original set of observation vectors to another, the interpretation of the properties of which is a simpler and more visual task. It is easy to see that orthogonal transformations in 1? do not preserve the metric in the Minkowski spaces bp for the case p > 2, so the natural question is on which subspaces of the metric 1? and ]> are equivalent. The problem acquires a correct formulation in the case of specific orthogonal transformations. Statement of a similar problem for a special orthogonal transformation - a discrete transformation

Fourier - allows you to understand the complexity and depth of the problem. Meanwhile, it is the Fourier transform that finds wide application in the theory of information transmission. The idea of representing a signal as a superposition of individual harmonics of a simple form has become widespread in electrical engineering. It should be noted that non-harmonic oscillations arising in electronic systems (Hertz dipole, microphone) require other, non-trigonometric orthogonal systems, for example, the system of Walsh functions16 for their study. In many cases, the properties of a function (signal, system of indicators) can be understood on the basis of the properties of its Fourier transform, or, in other words, its spectral decomposition. The problem of the homogeneity of a system of indicators can be formulated in terms of the spectral function of such a system - what should be the structure of the spectrum in order for the function to be "homogeneous" on the set of selected indicators. With a clear definition of the concept of "homogeneity" or "monogenicity" various mathematical problems arise. In particular, the correct statement of the mentioned problem of choosing a subspace on which the metrics b2 and bp are equivalent takes the following form: for what degree of lacunarity of the spectrum of the function ]Γ(x)eb2 does this function belong to the space bp for some p > 2. For reasons of generality, one should not confine oneself to considering only discrete Fourier transforms, since the problems that arise are also general for the continuum case. Other cases of "homogeneity" of the system of indicators originate from one of the works of the famous mathematician S. Mandelbroit from 1936 and are given in the following sections. A classic example of an orthogonal transformation for the case of a discrete Fourier transform is a transformation with a Hadamard matrix, so

the Fourier transform for an orthogonal Walsh system is otherwise called the Hadamard transform.

According to A.G. Dragalin17 "The set of mathematical theories used in the study of formal theories is called metamathematics; metatheory is a set of tools and methods for describing and defining some formal theory, as well as studying its properties. Metatheory is the most important part of the formalization method." The work, in particular, proposes as a metatheory for studying the system of international relations, the apparatus of finite functions and lacunar series.

One of the goals of the work is to develop an effective mathematical apparatus for analyzing the system of indicators in the concept of "political strength" by G. Morgenthau in relation to the tasks of metric-functional analysis of the system of indicators of the state's power in classifying foreign policy information.

Chapter I (Mathematical Methods and International Relations) is introductory. Section 1 describes the subject area - the system of international relations and that part of it that relates to the sphere of political relations. An overview of the development of political science and the emergence of mathematical methods in political research is given. The main currents in the science of international relations are considered - political idealism, political realism, empiricism, behavioralism, modernism. An overview of the main domestic and foreign publications on mathematical modeling in international relations is given. Section 2 examines the role of new information technologies in modeling international relations and the use of computer technology in foreign affairs agencies. foreign countries and Russia. §3 of the work is devoted to a critical analysis of the state of affairs with existing mathematical

scientific models in the field of international relations and substantiates the need to build a new generation of mathematical models on a single methodological basis. The concept of building a universal model of political behavior and the functional of the quality of political management is given, and in a certain sense the uniqueness of the solution of the task is shown. In § 4, questions of the problem of representing functional dependencies as a superposition of elementary dependencies are studied. Section 5 considers combinatorial models of political behavior. Section 6 is devoted to an overview of the main methods and regulations on the use of methods of political comparison of different sets of indicators, as well as methods for determining weight coefficients in integral indicators of the power of the state. The main methods (N.V. Deryugin, N. Bystrov, R. Veksman) of using the system of indicators to build the functional of the power of the state are given. Ch. Taylor's approach to building a system of indicators for political, economic and social analysis is also discussed.

Section 7 of Chapter I discusses the main tasks and problems of the metatheory of international relations related to decision-making based on indicators.

Chapter 2 (Information Classification Models in the Information Resources Management System in the Foreign Policy Sphere) is devoted to the application of quantitative methods in structuring the flows of foreign policy information used in the process of making a foreign policy decision. In relation to the tasks of management, in accordance with the general idea of the power of the state, such regulation of the information regime is chosen that delivers the optimum to the power of the state. The conceptual approach to choosing the structure of indicators goes back to the works of

rican researcher D.Kh. Smith as a combination of political, scientific, economic, technological and humanitarian factors. We also study domestic and foreign experience in managing information resources, including the legislative aspects of the information sphere in the USA, Germany, and France. A comparative analysis of existing models of national, regional and world development and their role in the classification of information flows is given. The main result of this chapter is the construction of models for individual assessment of the consequences of classifying foreign policy information. A system of models for processing expert information on a multi-criteria choice is also considered. A specific example of the use of the developed models is the calculation of an assessment of the consequences of an incorrect classification of foreign policy information on the basis of archival documents of bilateral relations from the archives of the Ministry of Foreign Affairs of the Russian Federation and a quantitative expression of the degree of influence various kinds information on the individual components of the power of the state. This kind of assessment is based on the approach of G. Grenevsky and M. Kempisti on the allocation of two flows - material and information, despite the fact that the information system in politics is not only a system for the movement and transformation of messages, but also a regulatory system. The object of regulation is the power of the state.

In Chapter III of the work (Spectral Characteristics in Mathematical Models of the System of International Relations), the metric characteristics of the objective functions of the models are studied using the spectral analysis apparatus.

Problems. The specificity of model systems in the theory of international relations is the use of various systems of indicators, or, in mathematical terms, finite functions. Finiteness in a broad sense implies the vanishing of a function (disappearance) outside a certain set, the measure of which is small with respect to the measure of the entire space. Such a set can be, for example, a segment on the real axis or a set of measure (density) zero. Finiteness for spectral functions (i.e., for Fourier transforms) is otherwise called spectrum lacunarity. Thus, the lacunarity of an audio signal means that not all harmonics (fundamental tones) are present in it. The idea of harmonizing studies using different systems of indicators is to consider the properties of sets of finite (on single space political indicators) functions and their metric properties. Existing spectral analysis models that use the entire spectral range are inherently inaccurate, because in the real world, the spectrum of an object is lacunar. Accounting for lacunarity will reveal the specific, deep properties of political processes, only their inherent features. In addition, taking into account the lacunarity in the process of transmitting foreign policy information in the transmitter-----joder-> receiver system will optimize the process of exchanging foreign policy information.

Thereby. the theory of lacunar series acts as a metatheory in relation to the theory of mathematical modeling of international relations, if we consider a class of models based on a system of political indicators. The system of indicators can be associated with a formal series according to the chosen system of orthogonal functions, and this approach generates its own class of problems. On the contrary, the system of indicators can be considered as values

some function, the properties of which are studied through its linear transformations (in particular, the discrete Fourier transform with the Hadamard matrix). In the first case, the main problem is the problem of uniqueness: whether different formal series represent different functions according to a fixed system of indicators. In the second case (the dual problem), the subject of study is the subsets on which the metrics in Lp (p > 2) are equivalent to the metric Lr. Obviously, the entire conceivable system of indicators is, in a certain sense, "overcrowded" - among the indicators there are many mutually dependent ones. The correct formulation of such problems requires strict mathematical definitions.

The lacunarity of the spectrum of a political (or other object) is usually understood as the presence of a system of inequalities:

_> A> 1, k \u003d 1.2, .....

in the spectral decomposition of the corresponding function Γ(x)=Ea]A(x); ak=0 if k£(pc).

Such lacunarity is otherwise called strong lacunarity, or Hadamard lacunarity, in honor of the French researcher J. Hadamard, who studied the properties of the analytic continuation of power series beyond the boundary of the circle of convergence. Subsequently, this condition was repeatedly weakened by a number of authors, however, other natural conditions on the density or growth of the sequence (pc) did not ensure the preservation of those functional properties that were present in the Hadamard lacunarity.

Most general concept it turned out the concept of a lacunar system of order p, or simply a system that arose in the works of S. Sidon and S. Banach. A rigorous theory of lacunar systems based on

on the theory of the Lebesgue integral, is quite complex for political research. Nevertheless, for reasons of completeness of presentation and the requirements of mathematical rigor, in all cases, along with discrete realizations, appropriate formulations are also given for continual analogues of the results obtained.

Let us give the necessary definitions.

DEFINITION 1. Let an orthonormal system of functions (^(x)) be given on a finite interval [a, b]. It is said that the system (^(x)) is a Br-system for some p > 2 if for any polynomial N(x) = X akGk(x) the estimate is true:

(|| N(x) I Pex) "P< С {II Ы(х) I 2(1х} 1/2 ,

where the constant C>0 does not depend on the choice of the polynomial H(x).

If, however, for any polynomial H(x) = I a] A(x) the estimate

(/ I R (x) 12c1x) 1/2< С {/| Я(х) | йх} ,

with some constant C > 0 independent of the choice of the polynomial H(x), then such a system is called a Banach system.

Br-systems and Banach systems will henceforth be called lacunar systems. Within the limits of consideration of subsystems of a fixed complete orthogonal system (Ux)) we will adhere to the designations (pc)eA(p) , or (pc)eA(2), if (pc) is the set of indices of the Br-system (respectively, the Banach system). The trigonometric system, or the system of Walsh-Paley functions, will be considered as the initial system (^(x)) . A well-known construction by U. Rudin allows one to generalize the concept of an A(p)-set to the case of any p>0. In 1960 U. Rudin showed that for

of the trigonometric system, the A(p)-set (p > 2) in any segment of length N contains at most CG\r2/p points, where the constant C > 0 does not depend on H, i.e. has density zero of power order. For the sets L(1) U. Rudin managed to show only that these sets do not contain arbitrarily long arithmetic progressions, therefore U. Rudin raised the question of whether L(p)-sets have zero density in the case of any p>018. In 1975, the Hungarian mathematician E. Semeredi19 gave an extremely complicated proof of the fact that sequences that do not contain arbitrarily long arithmetic progressions have density zero, but the density of such sequences turned out to be of non-power order. In addition, both the question of estimating the very density of A(p)-sets for the case of an arbitrary p > 0 and the question of constructing specific dense sets that do not contain progressions or otherwise regular sets in some sense remained open. In the presented work, U. Rudin's hypothesis has found its complete solution. For proof, we introduced the concept of a recurrent segment of length 2П, which is a generalization of the concept of a segment of an arithmetic progression - any arithmetic progression of length 2П is a recurrent segment, but not every recurrent segment is a segment of an arithmetic progression, as follows from the definition:

DEFINITION 2. Let integers r, pi, wg, ..., ti be given; b>2 such that mts >0, mk> pts + m2 + mz + ... + Shk-1 .

Then the set of all points of the form r + lice + 821112, + .... + e5m5, where r) = 0 or 1, is called a recurrent segment of length

The next cycle of theorems completely solves the problem of U. Rudin.

Chapter 3 uses a different (double) numbering of theorems. Theorems!,2,3 are proved in Appendix 5.

THEOREM 1. If the sequence (pc) does not contain recurrent segments of length 2П, then for any segment In of length N the inequality

card ((nk) n In) 0 do not depend on N. THEOREM 2. Any set (pk)eL(p) , p > 0, has density zero; moreover, for any natural N and for any segment In of length N, the following inequality holds:

card((nk)n In) 0 do not depend on N. In addition, all sets A(p) , p > 0 do not contain arbitrarily long recurrent segments.

A consequence of this theorem is, in particular, the fact that the set of primes (pj) is not the set A(p) for any p>0, because the density of prime numbers has a non-power order. The sequence of prime numbers has a special place in mathematics, and therefore any new result about its properties is certainly interesting. For comparison, we note that the validity of a similar statement for a sequence of squares of natural numbers is already unknown.

THEOREM 3. Let integers p, n > 2 be given, as well as integers

ki, k2,..., kn, 0< ki< р-1, a=a(ki,k2,...kn)= 2р2пЕЬ(2р)п-;+£ h2.

Then the set of all collections a=a(ki,k2,...kn) consists of pn elements, is contained in the interval [ 0, n2n+2pn+2] and does not contain recurrent segments of length 2n.

Using the construction used in the proof of Theorem 3, one can construct sets that do not contain arithmetic progressions of length 3 - the most interesting case of sequences that do not contain progressions. The results of F. Behrend20 are known

this direction, however, they are obtained in a non-constructive way. There is also an infinite construction by L. Moser21 based on another idea.

The paper also investigates the question of the densities of A(p)-sets p>0, on structures other than arithmetic progressions and recurrent segments. An example of such a structure is the set (2k + 2n) , where the summation extends to all indices k,p not exceeding some number N.

The trigonometric system (e>nx) has the property of multiplicativity, i.e. together with each pair of functions, it also contains their product. In the general theory of multiplicative systems, along with the trigonometric system, a special place is occupied by the system of Walsh functions. This system is a natural completion of the well-known Rademacher system and is defined (in Paley numbering) as follows:

sho^, \¥n(x)=P[rk+1(x)]ak, xe, in the case when n>1 has the form n= where ak takes the values 0 or 1, and rk(x)=sign s (2kt1; x) -

Rademacher functions. When studying the properties of a system of Walsh functions, it is convenient to introduce the following addition operation ® in the group of non-negative integers: 2k. Then for any n, w the relation It is easy to see that M2n(x)=Gn+1(x), n=0,1,2..., but it is natural to consider other lacunar subsystems of the system of Walsh functions.

An analogue of recurrent segments in the case of subsystems of the system of Walsh-Paley functions are linear manifolds in linear space over a field of two elements. Designs like this

types were studied by the French researcher A. Bonami,22 who, in particular, showed that all A(p)-sets, p > 0 for the Walsh system do not contain linear manifolds of arbitrarily large dimension. The construction used by us in the proof of Theorem 1 allows transfer A. Bonami's estimates obtained by her only for the case p > 2 to the case of any p > 0. Namely, we have

THEOREM 4. The sets A(p), p > 0 for the Walsh-Paley system have a zero density of power order, i.e. card ((nk) n In) 0 and ee(0,1) do not depend on n.

An analogue of Theorem 3 for the Walsh-Paley system requires the use of the property of a finite-dimensional linear space over a field of two elements to be a finite field (such a field is called a Galois field). In the linear space Ern every element except the zero one is invertible, i.e. along with the element ae Ern, the element a-"e Ern is defined. Let two isomorphic spaces Er" and F211 be given. Let two bases be chosen in Ern and F211, respectively: ei,e2,...en and fi,f2,...fn. to each

we assign to the element a=Xsj ej e Ern the element φ(a):= Ssj f]e F2n.

The following

THEOREM 5. The set of points of the direct sum of the spaces Ern and F2" of the form a+φ_1(a) (a > 0) has cardinality 2n-1, lies in the space Ern © F2" of cardinality 22n, and does not contain linear manifolds of dimension 2.

It follows from Theorem 5 that there are sets that do not contain linear manifolds of dimension 2 (the so-called B2 sets) and which contain more than 1/2 N1/2 points in a segment of length N (or a manifold of cardinality N). The result of Theorem 5 is stronger than that of

A.Bonami (A.Bonami constructed an example of a sequence that does not contain linear manifolds of dimension 2 and cardinality No./4).

The main results of Chapter 3 are Theorems 6 and 7 for the trigonometric system and the system of Walsh-Paley functions, which make it possible to reduce the study of A(p)-sets, p > 0, to the study of I. Vinogradov’s finite trigonometric sums (respectively, Walsh sums), or, which the same goes for studying the properties of discrete idempotent polynomials.

THEOREM 6. Let a sequence of integers (nk)eA(2+5),s>0 Then there exists a constant C=C((nk)>0 such that for any natural p and any polynomial

Wx) = where e^ are equal to 0 or 1 and Xe^B

the inequality is true:

I I<С вр^/р) 8/(8+2) (*)

k, 0< пк<р 12

Conversely, if for a sequence (pc) there exists a constant C > 0 such that for any polynomial ux) = X^-ech*, where Ej are equal to 0

or 1 and Here the estimate (*) is valid, then the sequence

(pc)eL(2+v-p) for any p, 0< р< 2+8.

THEOREM 7. Let the sequence Pk)eL(2+8),8>0 according to the Walsh-Paley system, then there exists a constant C>0 such that for any natural p=2" and any polynomial R(x)=X^yy /x), 0< ] <р,

E8]=B,8j are 0 or 1

the inequality

S | R(nk/p) |2

Conversely, if for a sequence (pc) there exists a constant С> 0 such that for any polynomial R(x)= XsjWj(x), where 8j are

0 or 1 and Ssj-s the estimate (**) is valid, then the sequence

(pc)eL(2+v-p) for any p, 0< р< 2+s.

The distribution of the values of a trigonometric polynomial (or a Walsh-Paley polynomial) whose coefficients are equal to 0 or 1 (ie, an idempotent polynomial) is directly related to problems in coding theory. As is known, the linear (n,k)-code (k< п) называется любое к-мерное подпространство линейного пространства размерности п над полем из двух элементов. Весом элемента кода называется число единиц в двоичном разложении элемента по базису.

Fair

THEOREM 8. Let an idempotent polynomial in the Walsh-Paley system R(x)= EsjWj(x) be given, where Sj are equal to 0 or 1 and Ssj=s. To each point x of the space En we assign a vector of length s from 1 and -1 of the form, the components of which are equal to the value of the corresponding Walsh function present in the representation of the polynomial at the point x. This mapping is a homomorphism of the space En into the linear space E "n czEs, where the addition operation is understood as a coordinate-wise multiplication. In this case, the formula R (x) \u003d s-2 (the number of minus ones in the code word) is valid.

Thus, the value of the Walsh polynomial is determined by the number of minus ones in the corresponding linear code. If we rename the words in the code so that 1 is replaced by 0, and -1 by 1 during the operation of addition modulo 2, then we come to the standard form of the binary code with the standard weight function. In this case, let's go

The potent Walsh polynomial corresponds to a binary code in which all columns of the generating matrix are different. Such codes are called projective codes, or Delsarte codes.23

The following result makes it possible to estimate the distributions of the values of idempotent Walsh polynomials using entropy estimates.

THEOREM 9. Let an idempotent polynomial H(x) = be given on En, where s] are equal to 0 or 1 and 2^=5, 0<а< 1. Пусть 3-1, 3.2, £ Еп таковы, что И.^) >b a where all w form a system of independent vectors in E1 (1<п).

Then

where Na \u003d - (1 + a) / 2 ^ 2 (((1 + a) / 2) - (1-a) / 2 log2 (((l-a) / 2) is the entropy of the distribution of a quantity that takes two values with probabilities (1+a)/2 and (1-a)/2, respectively.

The paper also obtained estimates for the upper bound on the weight of a binary code, which refine the well-known S. Johnson bound.24

The main point that causes interest in lacunary systems is the fact that the behavior of a lacunary series on a set of positive measure determines the behavior of the series over the entire interval of definition. In particular, there is no non-trivial lacunary (according to Hadamard) trigonometric series that vanishes on a set of positive measure. This classical result of the American researcher A. Zygmund25 has been substantially improved by us, namely, A. Zygmund's assertion remains valid for any trigonometric BR-system (p > 2). At the moment this is

the best known result. This result follows from the following theorem:

THEOREM 10. Let ( pc )eL(2+e), s>0 and the set E c be such that u.E> 0. Then there exists a positive number X such that

II EakeM 2ex>A, Eak2 (***)

for any finite polynomial R(x) = Eake "nx.

For the system of Walsh-Paley functions, we have proved a similar theorem in the following form:

THEOREM 11. Let (pc) eL(2+e), e > 0, and let the set Ε c be such that pE > 0. In addition, let the sequence (pc) have the property pc © w -> ω for k > 1 > 0. Then for any A > 1 and any set E of positive measure there exists a natural number N such that for any polynomial K(x) = ^akmin, k(x), where the summation is over the numbers k, k > N, the following inequality holds:

¡\ K(x)| 2c1x>(|uE/A,)Eak2 (****) £

A specific feature of the Walsh system is the fact that the condition Pk © P1 -> o for k> 1> 0 in Theorem 11 cannot be weakened (in comparison with Theorem 10 for the trigonometric system).

In inequalities (***) and (****), it is essential that the estimates are carried out for any set of positive Lebesgue measure. In the case when the set E is an interval, the proof of estimates of this kind is greatly simplified and carried out under much more general assumptions. The first results in this direction belong to the famous American mathematicians N. Wiener and

A. Zygmund26, however, the apparatus developed by them is insufficient for obtaining such estimates in the case of replacing the interval with an arbitrary set of positive Lebesgue measure. Quasi-analyticity of lacunar representations, i.e. a property close to the properties of analytic functions (as is known, if a power series vanishes on a set having a limit point, then all its coefficients vanish) manifests itself in terms of the smoothness of functions.

Definition 3. A function f(x) defined on some interval [a, b] is said to belong to the class Lip a with some ce(0,1) if

sup I f(x)-f(y) I<С 5а, где верхняя грань берется по всем числам х,у отрезка [а,Ь] , расстояние между которыми не превосходит 5>0, and the constant С>0 does not depend on choice x,y. If the estimate is valid for the function f(x):

J! f(x+y)-f(x)l 2dx 0 does not depend

s from y, then we say that the function f(x) belongs to the class Lip(2,a).

We have installed

THEOREM 12. Let the set of functions (cos nk x, sin Px) be an Sp-system for some p > 2 and let f(x)e Lip(2, oc) be a function for some a > 0. Then if the series Eakcosnkx+bksinnkx converges on a set of positive measure to a function f(x), then this series converges almost everywhere to some function g(x)e Lip(2, a) and is its Fourier series.

In addition, if in the previous condition the series is lacunar in the sense of Adamar and the function f(x)e Lip a, a>0, then the series converges everywhere to this function and is its Fourier series.

The latter result gives a positive answer to the problem posed by the American researcher P.B. Kennedy27 in 1958

The main results of the work are reflected in the following publications:

1. Mikheev I.M., On series with lacunae, Mathematical collection, 1975, v. 98, N 4, pp. 538-563;

2. Mikheev I.M., Lacunar subsystems of the system of Walsh functions, Siberian Mathematical Journal, 1979, N. 1, pp. 109-118;

3. Mikheev I.M., On methods for optimizing the structure of technological processes, (co-author Martynov G.K.), Reliability and quality control, 1979, N.5;

4. Mikheev I.M., Methodology for choosing the optimal variant of the technological process of a production line by random search using a computer, (co-author Martynov G.K.), Standards Publishing House, 1981

5. Mikheev I.M., Methods for estimating the parameters of nonlinear regression models of technological processes, (co-author Martynov G.K.), Publishing house of standards, 1981;

6. Mikheev I.M., Methodology for optimizing the parameters of technological systems in their design, (co-author Martynov G.K.), Standards Publishing House, 1981;

7. Mikheev I.M., Methods of synthesis of optimal production and technological systems and their elements, taking into account the requirements of reliability, (co-author Martynov G.K.), Standards Publishing House, 1981;

8. Mikheev I.M., Trigonometric series with gaps, Analysis Mathematica, vol. 9, part 1, 1983, pp. 43-55;

9. Mikheev I.M., On mathematical methods in the problems of assessing the scientific and technical level and product quality, Scientific works of VNIIS, issue 49, 1983, pp. 65-68;

10. Mikheev I.M. , Methodology for individual assessment of the consequences of classifying foreign policy information, (co-author Firsova ID), Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1989;

11. Mikheev I.M., On the place of mathematical modeling in modern political science, Proceedings of the scientific symposium "New political thinking: problems, theories, methodologies and modeling of international relations", Moscow, September 13-14, 1989, p. 99 -102;

12. Mikheev I.M., On the application of quantitative (mathematical) methods in the study of international relations, (co-author Anikin V.I.), Proceedings of the scientific symposium "New political thinking: problems of theory, methodology and modeling of international relations", Moscow, 13 - September 14, 1989, pp. 102-106;

13. Mikheev, I.M., A model for maintaining the strategic balance of power between the USSR and the United States under conditions of phased disarmament, in Sat. 1 "Management and informatics in foreign policy activity", DA MFA USSR, 1990, (ed. Anikin V.I., Mikheev I.M.), pp. 40-45;

14. Mikheev I.M., Methods for predicting the results of voting in the UN, In Sat. "Management and Informatics in Foreign Policy Activities", DA USSR Ministry of Foreign Affairs, 1990 (ed. Anikin V.I., Mikheev I.M.), pp. 45-52;

15. Mikheev I.M., Methodology of the approach to building a universal model of world development, Proceedings of the international seminar "Technical, psychological and pedagogical problems of using

16. Mikheev I.M., Using models of national, regional and world development for classifying information, Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

17. Mikheev I.M., Internal factors hindering the development of foreign economic relations of the USSR, (co-authors Subbotin A.K., Shestakova I.V., Vakhidov A.V.), Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

18. Mikheev I.M. , The concept of conversion in the conditions of perestroika, (co-authors Vakhidov A.V., Subbotin A.K., Shestakova I.V.), Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

19. Mikheev I.M., The use of quantitative methods in forecasting world development, Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

20. Mikheev I.M., Problems of capital export from the USSR in the 90s, (co-authors Vakhidov A.V., Subbotin A.K.), Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1991;

21. Mikheev I.M. et al., Problems of managing information resources in the USSR, (team of authors, ed. Subbotin A.K.), Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1991

22. Mikheev I.M., Modeling and development of an automated control system in foreign policy processes and training of diplomatic personnel, Proceedings of the scientific and practical conference for the 60th anniversary of the Diplomatic Academy of the Ministry of Foreign Affairs of Russia, Moscow, October 19, 1994;

23. Mikheev I.M., Methods of cluster analysis of evaluation and adoption of foreign policy decisions, (co-authors Anikin V.I., La-

rionova E.V.), Diplomatic Academy of the Ministry of Foreign Affairs of the Russian Federation, Department of Management and Informatics, textbook, 1994;

24. Mikheev I.M., Research of information support of international relations using functional spaces, Proceedings of the 4th international conference "Informatization of security systems ISB-95" of the International Informatization Forum, Moscow, November 17, 1995, pp. 20-22;

25. Mikheev I.M., Research of information support of political systems, Proceedings of the international scientific-practical conference "Systems analysis on the threshold of the XXI century: theory and practice", Moscow, February 27-29, 1996, vol. 1, p. 79 -80;

26. Mikheev I.M., Mathematics of borderology, Collection of articles of the Department of borderology of the International Academy of Informatization, vol. 2, M., Department of border studies of the MAI, 1996, pp. 116-119

The total volume of the dissertation, including the Appendix and bibliography (249 titles) - 310 pages. The Appendix contains the main political indicators used in various studies (Appendix 1), tables of proximity measures (Appendix 2), information on the functioning of the AIS provided by the UN Secretariat ( App 3). Listings of programs for processing the results of voting in the UN (Appendix 4) and the solution of U. Rudin's problem on the density of lacunar sets (Appendix 5) are also given.

Similar theses in the specialty "Application of computer technology, mathematical modeling and mathematical methods in scientific research (by branches of science)", 05.13.16 HAC code

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Dissertation conclusion on the topic "Application of computer technology, mathematical modeling and mathematical methods in scientific research (by branches of science)", Mikheev, Igor Mikhailovich

CONCLUSION (summary)

The results presented indicate that:

1. The development of mathematical modeling in the field of international relations has its own history and well-established mathematical tools - mainly methods of mathematical statistics, the theory of differential equations and game theory. The paper analyzes the main stages in the development of mathematical thought in relation to the social sphere and the theory of international relations, substantiates the need to create mathematical models of a new generation on a single methodological basis, and proposes new combinatorial constructions in relation to the system of international relations.

2. Within the framework of the theory of political empiricism, the paper proposes a method for analyzing systems of political indicators using a group structure according to the operation of a symmetric difference, which made it possible to apply the theory of characters of Abelian groups and linear transformations (primarily the discrete Fourier transform with the Hadamard matrix). This method, unlike the traditional methods of convolution (averaging) of single criteria, does not lead to the loss of the original information.

3. A fundamentally new problem of managing information resources in the foreign policy sphere has been solved and a methodology for assessing the damage from incorrect classification of foreign policy information has been proposed, which is used in the practical work of the Russian Foreign Ministry.

4. The tasks of studying the political process as a function on a set of political indicators using spectral methods are set and solved.

5. Fundamentally new results on discrete approximation of a number of metric problems are obtained and a structural characteristic of exceptional sets in the space of indicators is revealed.

List of references for dissertation research Doctor of Physical and Mathematical Sciences Mikheev, Igor Mikhailovich, 1997

LITERATURE

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2 A.N. Tikhonov, Mathematical model, see Mathematical encyclopedia, v. 3, pp. 574-575

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8 M. Nicholson, Formal Theories in International Relations, Cambridge University Press, Cambridge , 1988

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28 L.P. Borisov, Political Science, M., 1966, p.3

29 Fundamentals of political science (ed. V.P. Pugachev), M., 1994, 4.1, p. 17

30 Ibid., p. 18

31 Political dictionary, M., 1994, part 2, p. 71

33 Fundamentals of political science (ed. Pugachev V.P.), M., 1994, 4.1, p. 20

34 American Sociology. Perspectives, problems, methods, M., 1972, p. 204

35 History political doctrines, M., 1994, 139 pp.

36 Ibid., p. 4

37 Ibid., p. 14

38 Political dictionary, M., 1994, part 2, p. 73

39 P.A. Tsygankov, Political sociology of international relations, M., Radiks, 1994, p. 72

40 S.V. Melikhov, Quantitative methods in American political science, M., Nauka, 1979, p. 3

41 Ibid., p. 4

43 Mathematical methods in the social sciences, Moscow, Progress, 1973, p. 340

44 S.V. Melikhov, Quantitative Methods in American Political Science, M., Nauka, 1979, p. 11

46 A.N. Kolmogorov, Mathematics, TSB, ed. 2, v. 26

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50 Quantitative methods in the study of political processes, comp. Sergiev A.V., Review of the American scientific press, M., Progress, 1972, p. 23

51 Modern bourgeois theories of international relations, M., Nauka, 1976, pp. 7-8

52 Ibid., p. 28

53 G. Morgenthou, Policy among Nation, N.Y. , 1960, p. 34

54 D. Singer, Empirical theory in international relations, N.Y., 1965

55 D. Singer, Quantitative international politics: Insights and Evidence, N.Y., 1968

56 K. Deutsch, On political theory and political action, American political science review, 1971, v. 65

57 K. Deutsch, The Nerves of Goverment: models of political communication and control, N.Y. 1963

58 K. Deutsch, Nationalism and its alternatives, N.Y., 1969, p. 142-143

59 Modern bourgeois theories of international relations, M., Nauka, 1976

60 S.V. Melikhov, Quantitative Methods in American Political Science, M., Nauka, 1979

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73 Problems of Information Resources Management in the USSR, team of authors, responsible. ed. Subbotin A.K., M., 1991

74 M. Ward, (ed.) Theories, models and simulations on international relation, N.Y., 1985

75 Indicator Systems for Political, Economic and Social Analysis, ed. Ch. L. Taylor, Cambridge, 1980

76 M. Nicholson, Formal theories in international relations, Cambridge University Press, 1989

77 Ibid., pp. 14,15

78 L. Richardson, Generalized Foreign Politics, British Journal of Psychology, v. 23, Cambridge, 1939

79 see e.g. Thomas L. Saaty, Mathematical models conflict situations, M., Sov. radio, 1977, p. 93

80 Murray Wolfson , A mathematical model of the Cold W, in Peace Research Society: Papers, IX, Cambridge Conference, 1968

81 W.L. Hollist, An analysis of arms process es, International Studies, Quarterly, 1977, v. 21, N. 3

82 R. Abelson, A Derivation of Richardson's Equations, The Journal of Conflict Resolution, 1963, v.7, N. 1

83 D. Zinnes, An Event Model of Conflict Interaction, 12th International Political Science Association, World Congress, Rio de Janeiro, 1982

84 Yu.N. Pavlovsky, Simulation systems and models, M., Knowledge, 1990

85 H. Alker, W. Russett, World Politics in General Assamly, New Haven, London, 1965

86 S. Brams, Transaction Flows in the International System, American Political Science Review, December, 1966, vol. 60, N. 4

87 R. Rammel, A Field theory of social action with application to conflict within nation, Genaral Systems Yearbook, 1965, v. 10

88 H. Lasswell, N. Leites, The Language of Politics; Statues in Quantitative Semantics, N. 9, 1949

89 Ph. Burgess, Indicators of international behavior: an assessment of event data research, L., 1972

90 P.A. Tsygankov, Political sociology of international relations, M., Radiks, 1994, p. 90

91 S.I. Lobanov, Application of event analysis in modern political science, Metological aspect, Political sciences and scientific and technological revolution, M., Nauka, 1987, pp. 220-226

92 Modern bourgeois theories of international relations, M., Nauka, 1976, line 314,417-419

93 Ibid., p. 320

94 Ibid., p. 323

95 J. von Neumann, O. Morgenstern, Game Theory and Economic Behavior, M., 1970

96 see, for example, Modern bourgeois theories of international relations, M., Nauka, 1976, p. 313

97 Ibid., pp. 314, 308

98 D. Sahal, Technical progress: concepts, models, estimates, M., Finance and statistics, 1985; V.M. Polterovich, G.M. Khenkin, Diffusion of technologies and economic growth, M., CEMI AN USSR, 1988

99 Political sciences and scientific and technological revolution, M., Nauka, 1987, p. 165

101 N.N. Moiseev, Socialism and Informatics, Political Literature Publishing House, M., 1988, pp. 82-83

103 International relations after the Second World War (ed. N.N. Inozemtsev), vol. 1, M., 1962

104 G.A. Lebedev, New York Times Information Bank, USA: Economics, Politics, Ideology, N2, 1975, pp. 118-121

105 A.A. Kokoshin, Interuniversity Policy Research Consortium, United States of America, N 10, 1973, pp. 187-196

106 D. Nikolaev, Information in the system of international relations, M., International relations, 1978, p. 86

107 I.V. Babynin, B.C. Kretov, The main directions of automation of information and analytical activities of the Ministry of Foreign Affairs of the Russian Federation, Scientific and technical information, ser. 1, 1994, N 6, pp. 12-17

108 B.C. Kretov, I.E. Vlasov, B.JI. Dudikhin, I.V. Frolov, Some aspects of creating an information support system for decision-making by operational and diplomatic employees of the Ministry of Foreign Affairs of the Russian Federation, Scientific and technical information, ser. 1, 1994, N 6, pp. 18-22

109 E.I. Skakunov, Methodological problems in the study of political stability, International Studies, 1992, N 6, pp. 5-42

110 see, for example, M.A. Khrustalev, System modeling of international relations, abstract of the dissertation for the degree of Doctor of Political Sciences, M., MGIMO, 1991

111 Yu.N. Pavlovsky, Simulation systems and models, M., Znanie, 1990

112 A.B. Grishin, Fundamental problems of creating "man-machine" systems in international relations and foreign policy, M., Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1979

113 Quantitative methods in the study of political processes (compiled by Sergiev A.V.), M., Progress, 1972

114 A. Dutta, Reasoning with imprecite knowlage in expert systems, Inf. Sei. (USA), 1985, v. 37, no. 1-3, p. 3-34

115 E.JI. Feinberg, Intellectual Revolution; on the way to the union of two cultures, Questions of Philosophy, 1986, N 8, pp. 33-45

116 Courant and Robbins, What is Mathematics, Moscow, Gostekhizdat, 1947, p. 20

118 N. Luzin, Op. , volume 3

120 A.B. Paplauskas, "Trigonometric series from Euler to Lebesgue"

121 R. Reiff, Geschichte der unendlichen Reihe, Tubungen, 1889, p. 131

122 H. Luzin, Works, volume 3

123 H.A. Kiseleva, "Mathematics and Reality", Moscow, Moscow State University, 1967

124 N. Bourbaki, "The Architecture of Mathematics", in the book "N. Bourbaki, Essays on the History of Mathematics", M., IL, 1963

125 A.A. Lyapunov, "On the Foundation and Style of Modern Mathematics", Mathematical Education, 1960, N 5

126 C.E. Plokhotnikov, Normative model of global history, M., \/ Moscow State University, 1996

127 V.I. Baranov, B.S. Stechkin, Extremal combinatorial problems and their applications, M., Nauka, 1989

128 P. Erdos, P. Turan, On a problem of Sidon in additive number theory, J.L.M.S., 16, (1941), p. 212-213

129 j. Rosenau, The Scientific Study of Foreign Policy, N.Y., 1971, p. 108

130 Ch. L. Taylor (ed.), Indicator Systems for Political, Economic and Social Analysis, International Institute for Comparative Social Research, Cambridge, Massachusetts, 1980

131 P. R. Beckman, World Politics in the Twentieth Century , Prentice-Hall, Englewood Cliffs, New Jersey

132 M. Kaplan, Macropolitics: Selected Essays on the Philosophy and Science of Politics, N.Y., 1962, p. 209-214

133 see Modern bourgeois theories of international relations, M., Nauka, 1976, pp. 222-223

134 N. Bystrov, Methodology for assessing the power of the state, Foreign Military Review, N. 9, 1981, pp. 12-15

136 see, for example, I.V. Babynin, B.C. Kretov, F.I. Potapenko, I.V. Vlasov, I.V. Frolov, The concept of creating an intelligent system for monitoring political conflicts, M., Research Center of the Ministry of Foreign Affairs of the Russian Federation,

138 B.B. Dudikhin, I.P. Belyaev, The use of modern information technologies for the analysis of the activities of municipal elected bodies, "Problems of Informatization", vol. 2, 1992, pp. 59-62

139 A.A. Goryachev, Problems of forecasting world commodity markets, M., 1981

140 see, for example, G.M. Fikhtengolts, Course of differential and integral calculus, M., 1969, v. 1, p. 263

141 A.I. Orlov, "General view on the statistics of non-numerical nature", Analysis of non-numerical information, M., Nauka, 1985, pp. 60-61

142 see Methods for assessing the quality level of industrial products, GOST 22732-77, M., 1979; Guidelines for assessing the technical level and quality of industrial products, RD 50-149-79, M., 1979, p. 61

144 see V.V. Podinovsky, V.D. Nogin, Pareto-optimal solutions of multicriteria problems, M., Nauka, 1982, p. 5

145 S.K. Kleene, Introduction to Metamathematics, M., IL, 1957, pp. 61-62

146 see Analysis of non-numerical information, M., Nauka, 1985

147 V.A. Trenogin, Functional Analysis, M., Nauka, 1980, p. 31

148 M.M. Postnikov, Linear algebra and differential geometry, M., Nauka, 1979

149 A.E. Petrov, Tensor methodology in systems theory, M., Radio and communication, 1985

150 V. Platt, Information work of strategic intelligence, M., IL, 1958, pp. 34-35

152 Ibid., p. 58

153 Information resource management problems in the USSR, (ed. A.K. Subbotin), Diplomatic Academy of the USSR Ministry of Foreign Affairs, Moscow, 1991

154 National Security Information, Executive order N 12356, April 2, 1982 (Compilation, p. 376-386)

155 Freedom of Information Act of 1967, as amended (Compilation, p. 159162)

156 National Security Information, Executive order N 12065, June 28, 1978 (Hearings, p. 292-316)

157 National Security Information, Executive order N 12356, April 2, 1982 (Compilation, p. 376-386)

158 see, for example, Executive Order on Security Classificatio. Hearings Before a Subcommitee on the Commitee on Goverment Operations, (House), Washington D.C., 1982, VI

159 Code of Federal Regulation, 1.1.1 Title 22. Foreign Relation, 1986, Washington D.C.

160 m. Frank, E. Wiesband, Secrecy and foreign Policy, N.Y., Oxford University Press, 1974

161 Le secret administratif dans les pays developpes. Cujas. 1977, p. 170-179

163 B.H. Chernega, M.Yu. Karpov, The problem of secrecy and management of information resources in France and Germany, M., Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990, pp. 6-8

166 Problems of managing information resources in the USSR, (ed. Subbotin A.K.) M., Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1991, p.166

167 Ibid., p. 169

168 see, for example, Fujii Haruo, Nikonno kokka kimitsu (Japanese State Secret), Tokyo, 1972; Kimitsu hogo to gendai (Protection of Secrets and Modernity), Tokyo, 1983.

169 I.M. Mikheev, I.D. Firsova, Methodology for an individual assessment of the consequences of classifying foreign policy information, M., Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1989

170 R. Winn, K. Holden, Introduction to Applied Econometric Analysis, M., 1971

171 V. Plyuta, Comparative multidimensional analysis in economic research, M., 1980

173 See E.Z. Maiminas, Planning processes in the economy: information aspect, M., 1977, pp. 33-43; D. Bartholomew, Stochastic models of social processes, M., 1985, p. 68; R. Winn, K. Holden, introduction to applied econometric analysis, M., 1981, p. 112

174 A. Peccei, Human qualities, M., Progress, 1980

175 A.D. Ursul, Informatization of society (Introduction to social informatics), Textbook, M., 1990, p. 14

176 J. Forrester, World Dynamics, M., Nauka, 1978

177 D.N. Meadows, D.L. Meadows, J. Randers., W.W. Behrens, The Limits to Growth., N.Y., Universe Books, Potamak associated book, 1972

178 M. Mesarovic, E. Pestel, Mankind at the turning point, Toronto, 1974

179 B.A. Gelovani, A.A. Piontkovsky, V.V. Yurchenko, Modeling of global systems, M., VNIISI, 1975

180 Modeling of global economic processes, (ed. B.C. Dadayan), M., Economics, 1984

181 Intersectoral balance in the study of the capitalist economy, M. Nauka, 1975

182 Modeling of global economic processes, (ed. B.C. Dadayan), M., Economics, 1984

183 R. Hilsman, Strategic intelligence and political decisions, M., IL, 1959, p.7

184 Bible, Old Testament Books, Fourth Book of Moses. Numbers, Chapter 13

185 R. Hilsman, Strategic intelligence and political decisions, M., IL, 1959, pp. 19-20

186 cm. D. Kahn, The Codebreakers, MacMillan, New York, 1967

187 cm. M.H. Arshinov, L.E. Sadovsky, Codes and Mathematics, M., Nauka, 1983, pp. 5,13,14

188 A. Akritas, Fundamentals of computer algebra with applications, M., Mir, 1994, p. 263

189 A. Sinkov, Elementary cryptanalysis - a mathematical approach. The New Mathematical Library, no 22, Mathematical Association of America, Washington, D.C. , 1968

190 M.H. Arshinov, L.E. Sadovsky, Codes and Mathematics, M., Nauka, 1983, p. 11

191 Ibid p. 17

192 D.Kahn, The Codesbreakers, MacMillan, New York, 1967, p. 236-237

193 F. Gass, Solving a Jules Verne cryptogramm, Mathematics Magasin, 59, 3-11, 1986

194 M.H. Arshinov, L.E. Sadovsky, Codes and Mathematics, M., Nauka, 1983, p.39

195 L.S. Hill, Concerning certain linear transformatoin apparatus of crytography. American Mathematical Monthly, 38, 135-154, 1931

196 R. Lidl, G. Pilz, Applied abstruct algebra, Springer-Verlag, New York, 1984

197 E.V. Krishnamurty, V. Ramachandran, A criptograthic system, based on finite field transform, Proceedings of the Indian Academy of Science, (Math. Csi.) 89(1980) ,75-93

198 see W. Diffie, M.E. Hellman, Exhaustive cryptanalysis of NBS date encryption standard, Computer, 10, 74-84, June, 1977

199 M.E. Hellman, The mathematics of public-key cryptograthy. Scientific American 241, 130-139, August, 1979

200 R.C. Mercle, M.E. Hellman, Hiding information and signatures in trapdoor knapsacs. IEEE Transaction on Information Theory IT-24, 525530,1978

201 S.M. Johnson, Upper bounds for constant weight error correction codes, Disc. Math. 3(1972), 109-124; Utility Math. , 1(1972), 121-140

202I. Okun, Factor analysis, M., 1974, p. 112 203G.N. Agaev, N.Ya. Vilenkin, G.M. Jafarli, A.I. Rubinshtein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, 1981, p. 67)

204 ibid., p. 57

205 K. Weierstrass, Uber continuirlische Functionen eines reelen Arguments, die fur keinen Werth des letzteren einen bestimmten Differentialquotienten bezitzen, Konigl. Acad. Wis. , Math. Werke, II, 1872, 71-74

206 G.H. Hardy, Weierstrass's nondifferentiable funktion, Tran. Amer. Math. Soc., 17(1916), 301-325

207 J. Adamard, Essai sur les l "etude des fondktions donees par leur développement de Taylor, J. Math., 8(1892), 101-186

208 F. Risz, Uber die Fourier Koeffizienten einer stetiger Funktion von beschranter Schankung, Math. Z., 2(1918), 312-315

209 A. Zigmund, On lacunary trigonometric series, Trans. amer. Math. Soc., 34(1932), 435-446

210 V.F. Gaposhkin, Lacunary series and independent functions, Uspekhi matematicheskikh nauk, XXI, vol. 6(132), 1966, 3-82

211 A. Zigmund, On a theorem of Hadamard, Ann. soc. Polon. Math. , 21, No 1, 1948, 52-68

2.2 A. Bonami, Y. Meyer, Propriétés de convergence de certaines series trigonometriques, C.R. Acad. Sei. Paris, 269, No 2, 1969, 68-70

213 I.M. Mikheev, On a uniqueness theorem for series with gaps, y"" Mat. notes, 17, no. 6, 1975, 825-838

214 W. Rudin, Trigonometrical series with gaps, J. Math, and Mech., 9, No 2, 1960, 203-227

215 J.-P. Kahane, Lacunary Taylor and Fourier series, Bull. amer. Math. Soc., 70, No. 2, 1964, 199-213

216 K.F. Roth, Sur quelques ensemble d" entriers, C.R. Acad. Sci. Paris, 234, No 4, 1952, 388-390

217 A. Khinchine, A. Kolmogoroff, Uber die convergenz der Reihen deren Glieder durch den Zuffall bestimmt werden, Mat. Sat. , 1925, 32, 668677

218 G.W. Morgenthaler, On Walsh-Fourier series, Trans. amer. Math. Soc., 1957, 84, No 2, 472-507

219 V.F. Gaposhkin, Lacunary series and independent functions, Uspekhi matematicheskikh nauk, 1966, no. 6, 3-82

220 w.f. Gaposhkin, On lacunar series in multiplicative systems of functions, Siberian Mathematical Journal, 1971, 12, no. 1.65-83

221 A. Zigmund, On a theorem of Hadamard, Ann. Soc., Polonaise Math. , 1948, 21, No 2, 52-69

222 A.E. Ingham, Some trigonometrical inequalities with application to the theory of series, Math. Z., 1936, No. 41, 367-379

223 N.I. Fine, On the Walsh-Fourier series, Trans. amer. Math. Soc. 65(1949), 372-419

224 S. Kachmazh, G. Steinhaus, Theory of orthogonal series, M., Fizmatgiz, 1958

225 A. Sigmund, Trigonometric series, Vol. 1, M., Mir, 1965

226 A. Bonami, Ensemles L(r) danse le dual de D00 , Ann. Inst. Fourier, 18(1969), No 2, 193-204

227 M.E. Noble, Coefficient properties of Fourier series with a gap condition, Math. Ann. 128(1954), 55-62

228 P.B. Kennedy, Fourier series with gaps, Quart. J Math. 7(1956), 224230

229 P.B. Kennedy, On the coefficients in certain Fourier series, J. London Math. soc. , 33(1958), 196-207

230 S. Kachmazh, G. Steinhaus, Theory of orthogonal series, Moscow, Fizmatgiz, 1958

231 A. Sigmund, Trigonometric series, vol. 1, M., Mir, 1965

232 N.K. Bari, Trigonometric series, M., Fizmatgiz, 1961

233 A.A. Talalyan, On the convergence of Fourier series to + oo, Izvestiya AN Arm. SSR, ser. Physics and Mathematics, 3(1961), 35-41

234 P.L. Ulyanov, Solved and unsolved problems in the theory of trigonometric and orthogonal series, Uspekhi Mat. Nauk, 19 (1964), no. 1, 3-69

235 G. Polia and G. Sege, Problems and theorems from analysis, vol. 2, Gostekhizdat, Moscow, 1956

236 H.G. Eggleston, Sets of fractional dimentions which occur in some problem of number theory, Proc. London Math. Soc., Ser. 2, 54, 19511952,42-93

237 w. Rudin, Trigonometric series with gaps, J. Math. Mech. 9(1960), 203!

sh B.L. Van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15(1928), 212-216

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