Application of mathematical methods in international relations literature. Methods in the theory of international relations. Method Problem Significance

Having decided on the answer to the question of what the science of international relations studies, another one should be posed: how do we get knowledge? This question involves thinking about research methods. The problem of the method is one of the most important for any science, because we are talking about how to get new knowledge and how to apply it in practice .

In its most general meaning, a method can be defined as a way to achieve a goal(from the Greek "way to something"). Methods of scientific knowledge is a certain sequence of actions, operations, techniques, the implementation of which is necessary to solve cognitive, theoretical and practical problems in science; the application of methods leads either to the achievement of the set goal, or brings it closer to it. According to I.P. Pavlov, "the method holds the fate of research in its hands," in other words, the results of scientific activity largely depend on how adequate the set of research methods will be.

The research method turns out to be fruitful - that is, contributing to the disclosure of the essential properties and regular connections of the object - only when it is adequate to the nature of the object under study and corresponds to a certain stage of its study. "Since the fruitfulness of the scientific method is determined by how much it corresponds to the nature of the object, the researcher must have preliminary knowledge about the object, on the basis of which he will develop research methods and their system," they note Russian philosophers V.S. Stepin and A.N. Elsukov... - This means that the correct scientific method, being a necessary prerequisite for true knowledge, itself follows and is determined by the already existing knowledge about the object. Such knowledge must contain the essential characteristics of the object, and therefore it has the character of theoretical knowledge. Thus, a close relationship is established between theory and method. "In other words, the scientific method is the practical application of theory," theory in action. "

Methods can be classified on several grounds, for example, by levels of knowledge (methods of empirical and theoretical research); by the accuracy of predictions (deterministic and stochastic, or probabilistic-statistical); by the functions they perform in cognition (systematization, explanation and prediction); by subject area (methods used in physics, biology, sociology, political science, etc.).

Another possible option is classification of research methods by research levels to which they correspond. According to this classification, methods are divided into general, general scientific and private (specific scientific).

Highest level- general methods (level of methodology) - combines the general principles of cognition and the categorical structure of science as a whole. At this level, the general direction of research is set, the fundamental principles of approach to the object of study, "a system of guidelines for cognitive activity" ... These methods highlight universal principles and give knowledge about the universal laws of the development of nature, society and thinking, which are at the same time the laws of the world cognition.

In modern scientific knowledge, the so-called general scientific approaches , which set a certain focus scientific research, fix a certain aspect of it, although they do not strictly indicate the specifics of specific research tools. This allows us to consider them as "methodological orientation" and refer to this methodological level of scientific research tools.

As a similar approach to the study of international relations, one should include systemic , adopted by almost all, with a few exceptions, theoretical directions and schools in modern TMO. The systems approach is often considered as a concretization of the dialectical principle of universal connection. The systems approach is based on the study of objects as systems. It is characterized by a holistic consideration of a certain set of objects - material or ideal. In this case, the integrity of the object implies that the relationship between the set of objects under consideration and their interaction lead to the emergence of new integrative properties systems that are absent from its constituent objects. The specificity of the systems approach is the orientation towards the study of factors that ensure the integrity of the object as a system. ... The main problematic within the framework of the systemic approach is formed by the identification of the diverse so-called "system-forming" connections, which are primarily "responsible for the integrity of the phenomenon or object under study."

The use of a systematic approach contributes to the creation of such theoretical constructions, which can be, "on the one hand, so meaningful as to fully reflect reality, and on the other, so formal that when they are mutually correlated, general patterns can be found that allow not only to reflect that and to streamline the material under study and the process of research itself. "

The application of a systematic approach makes it possible to present the object of study in its unity and integrity... Its focus on identifying correlations (interdependence) between the interacting elements helps to find the "rules" of such interaction, or patterns of functioning of the system. This is the advantage of a systems approach. However, it should be borne in mind that any advantages can be continued in the form of disadvantages. With regard to the systematic approach, the latter include excessive formalization that can lead to an impoverishment of our understanding of international relations.

A systematic approach to research (and in particular the study of international relations) is implemented in several versions, among them: structural and functional, by the type of cybernetic model. As for the first , then he orients the researcher on the study of the internal structure of the system, on the identification of regularities in the processes of ordering elements in the system, on the analysis of the specifics and nature of the connections between the elements, on the one hand, and on the identification of the features of the functioning of systems, abstracting from their substrate-structural basis, on the other .

An approach according to the principle of the cybernetic model presupposes consideration of the system as a whole and its constituent elements as flexibly reacting to changes in the system under the influence of external or internal influences, or the environment of the system ... Moreover, the influence of the environment can be so significant that the evolution of the system is considered as co-evolution with the environment. This version of the systems approach emphasizes the stability of the system against external influences and its "behavior" in response to requirements or support from the environment. This approach is often identified with the "black box" technique, which abstracts from the content of the "black box", focusing on the problem of detecting functional dependencies between the input and output parameters of the system.

Specificity of general scientific methods, as well as general scientific categories on which they are based is determined "relative indifference to specific types of subject matter and, at the same time, an appeal to some common features" ... In other words, they are independent of the type of scientific problems being solved and can be used in various subject areas. General scientific methods are developed within the framework of formal and dialectical logics. These include such as observation, experiment, modeling, analysis and synthesis, induction and deduction, analogy, comparison, etc. ...

At the level of general scientific methods the systems approach is implemented in the form of general systems theory (OTS), which is the concretization and expression of the principles of the systems approach... One of the founders of general systems theory is Austrian theoretical biologist who immigrated to the United States, Ludwig von Bertalanffy (1901-1972). In the late 1940s. he put forward a program for the construction of a general theory of systems, providing for the formulation of general principles and laws of behavior of systems, regardless of their type and nature of their constituent elements and relations between them. The systems theory also performs the tasks of describing systems and its constituent elements, explaining the interaction between the system and the environment, as well as intrasystem processes, under the influence of which the system changes and / or destruction. Within the framework of the system theory, general scientific categories are developed, such as element, subsystem, structure, environment.

The elements - these are the smallest units within any system, from which, in turn, its individual parts can be formed (as a rule, in hierarchically organized systems - biological, social) - subsystems. The latter are relatively self-contained, smaller systems."Since they participate in the implementation of a single goal of the entire system, then their functioning and activities are subordinated to the tasks of the general system and are controlled by it." At the same time, subsystems carry out their special functions within the system and therefore have relative independence. The study of the elements of the system allows you to determine its structure. However, the more important category of systems analysis is the structure of the system. In the broadest sense, the latter is understood as connection and interrelation between elements, thanks to which new integrative properties of the system arise .

The third group of scientific methods are private (private) - methods of a particular science. Highlighting them assumes that their application is limited to only one area. Moreover, the presence of such methods is considered one of the conditions for recognizing the autonomy of a particular discipline. However, this requirement is far from always applicable to the social sciences. As a rule, social sciences do not have their own specific method, inherent only to them. They "borrow" the general scientific methods and methods of other sciences (both social and natural science), refracting them in relation to their object of research.

To assess how the discipline we are considering has developed, perhaps more important is another division of research methods - into "traditional" and "scientific". This juxtaposition emerged as a result of the "behaviorist revolution" of the 1950s. and was at the center of the second "big controversy" within the TMO. " The modernist "or" scientific "direction insisted on transferring the methods of the exact and natural sciences to social disciplines, emphasizing that only in this case, studies of the sphere of social relations can claim the status of" science. " "Scientific" methods formed an operational-applied, analytical-predictive approach associated with "formalization, data calculation (quantification), verifiability (or falsifiability) of conclusions, etc." ... This approach, new to the discipline, was contrasted with "traditional" historical-descriptive, or intuitive-logical. The latter until the middle of the twentieth century. was the only basis for the study of international relations. The traditional approach was based largely on history, philosophy and law, with an emphasis on the singular, unique in the historical, and in particular political, process. Proponents of the traditional approach emphasized the inadequacy of "scientific" quantitative methods, the groundlessness of their claims to universality ... Thus, one of the most prominent representatives of the traditional approach and the founder of the school of political realism G. Morgenthau noted that such a phenomenon as power, so important for understanding the essence of international relations, "represents the quality of interpersonal relations, which can be checked, evaluated, guessed, but which cannot be measured quantitatively... Of course, it is possible and necessary to determine how many votes can be given to a politician, how many divisions or nuclear warheads disposes of the government; but if I need to understand how much power a politician or government has, then I will have to put aside the computer and the calculating machine and start thinking about historical and, certainly, qualitative indicators. "

“The essence of political phenomena,” notes P.A. Tsygankov, “cannot be investigated in any way fully using only applied methods. V public relations in general, and in international relations in particular, stochastic processes dominate, which 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 are quite legitimate here, combining observation and reflection, comparison and intuition, knowledge of facts and imagination. Their usefulness and effectiveness is confirmed by both modern research and fruitful intellectual traditions " ... In other words, the opposition "modernist" methods "traditional The feeling of their dichotomy appeared due to the fact that they were introduced into the study of international relations historically consistently. However, it should be admitted that they complement each other and without such an integrated approach to the choice of research instruments, any of our theoretical constructions are doomed to failure. In this sense, one should probably consider the statements that the main drawback of our discipline is that the process of turning the science of international relations into an applied one should be considered too categorical. " The process of development of science is not linear, but rather reciprocal, writes P.A. Tsygankov. - There is no transformation of it from a historical-descriptive into an applied one, but the refinement and correction of theoretical positions through applied research (which, in fact, is possible only at a certain, rather high stage of its development) and the "return of debt" to "applied workers" in the form of a more durable and operational theoretical and methodological basis.

The introduction of "scientific" into international relations research methods represented "the assimilation of many relevant results and methods of sociology, psychology, formal logic, and natural and mathematical sciences." All this made the research toolkit much wider and gave rise to a kind of "methodological explosion" . At the same time, in the formation of modern ideas about the nature of international relations, an increasingly prominent role began to play applied projects. "The advancement of applied research" to the forefront "of the study of international relations, - notes K.P. Borishpolets, - led to the appeal of a wide range of specialists to special scientific instruments focused on the collection of empirical information, quantitative methods of its processing, preparation of analytical conclusions in the form of prognostic assumptions ". The scientific turnover of international relations research organically includes interdisciplinary applied analysis techniques ... The latter presuppose, first of all, the sum of procedures for collecting and processing empirical material. In the analysis of international relations such methods of sociological and political science data collection, such as polling and interviewing; they took a fairly strong place methods of content analysis, event analysis and cognitive mapping .

First developments content analysis is associated with the name of G. Lasswell and the works of his school at Stanford University ... In its most general form, this technique is viewed as a systematic study of the content of the text, the identification and assessment of the characteristics of the text material "in order to answer the question of what the author wants to emphasize (hide)." There are several stages in the application of this technique: text structuring, processing of the information array using matrix tables, quantification of information material. The most common way to assess the content of the text under study is calculation of the frequency of use of a semantic unit of analysis Is a quantitative, or frequency, version of content analysis. There is also a qualitative type of content analysis, which is focused not on the direct quantitative measurement of the semantic units of the information array, but on " taking into account the combination of qualitative and quantitative indicators ", characteristic of them.

Event Analysis , or event analysis, is one of the most common methods of applied analysis of international relations. It is based "on monitoring the course and intensity of events and the purpose of determining the main trends in the evolution of the situation in individual countries and in the international arena." The essence of the technique can be expressed by the formula: "who says or does what, in relation to whom and when." The application of the technique includes: compiling an information databank, dividing this array into separate observation units and their coding, correlating the selected facts and phenomena with the sorting system adopted in connection with the project's tasks.

Cognitive mapping technique aims at analyzing the perception of the international situation by decision-makers. This technique originated in the framework of cognitive psychology, which concentrates its attention "on the peculiarities of the organization, dynamics and formation of a person's knowledge about the world around him." The central concept of cognitive psychology is a "scheme" (map), which is a "graphical representation of a plan (strategy) in the mind of a person for collecting, processing and storing information", which is the basis of his ideas about the past, present and probable future. The use of cognitive mapping techniques involves identification of the basic concepts used by the decision-maker; establishing causal relationships between them, as well as assessing the significance and "density" of these relationships " .

All the methods discussed above are aimed at developing predictive capabilities in the framework of the science of international relations and thereby strengthening its applied nature. ... Often these techniques have an independent meaning, however, they can be combined with various mathematical tools and system modeling. The essence of the latter lies in the fact that it is a way of operating with an object, which consists in replacing the original with a model that is in a certain objective relationship with a directly cognizable object. ... Usually, three sequential stages of modeling are distinguished: logical-intuitive analysis, formalization and quantification. "Accordingly, three classes of models are distinguished: meaningful, formalized and quantified." The first stage of modeling is essentially a traditional research practice, when a scientist uses his knowledge, logic and intuition to create a model for studying an international phenomenon. At the second stage, the meaningful model is formalized - the transition from a predominantly descriptive to a predominantly matrix-graphic one. The solution to the problem of identifying trends in changing international situations is possible at the third stage of modeling - quantification.

Doubts about the possibility of strict formalization and quantification of the phenomena of international life have always existed. However, at the present stage of development of the science of international relations, the prospects for modeling are assessed "with moderate optimism." Perhaps, now no one will categorically insist on the conclusion of N. Wiener that "the humanities are a wretched field for new mathematical methods." The use of mathematical tools in the applied analysis of international relations is an independent problem.

Consideration of applied methods of analysis of international relations pushes for the separation of research methods depending on at what stage of research they are used (methods of collecting material, processing and ordering, theoretical justification, proof, or otherwise, methods used at the stage of empirical, theoretical research and the stage of building a scientific theory).

Special attention should be paid to the decision method. , which presupposes the concentration of the researcher's attention on studying the process of making foreign policy decisions. Now this method, originally developed for the analysis of processes in foreign policy, is widely used in political science. As applied to the study of international relations, it is focused on studying the process of developing and implementing foreign policy decisions and is designed to help identify its essence. For any researcher, the starting point of analysis is a foreign policy decision, and it is important to determine which variables caused its adoption. The application of the decisional method can be compared to the "decomposition" of multi-stage situations that make up the decision-making process. In the process of implementing the method, the researcher must focus on four "key points": decision-making centers, decision-making process, the political decision itself, and, finally, its implementation. ... The application of the decisional method involves identifying the circle of key "players" or decision-makers, as well as assessing the role of each of them. If we are talking about important foreign policy decisions, then attention will be paid to the top political leadership of the country. (the head of state and his advisers, ministers of foreign affairs, defense, etc.). It should also be borne in mind that each of the designated persons has his own staff of assistants involved in the process of receiving and processing information. The analysis of the circle of decision-makers also requires the researcher to pay attention to their personal and role characteristics.

Based on a common approach, several models for analyzing the process of making foreign policy decisions . The first model is based on rational choice - there is an understanding of the decision-making process as rational, i.e. assuming the maximization of goals while minimizing the funds spent. The model assumes that the process of foreign policy goal-setting is based on objective and unshakable national interests, and the decision-maker has all the necessary set of information to assess all possible alternatives of action and is able to choose the best option for action. In practice, the implementation of such a model is impossible.

In the "behavioral model "analysis of the process of making foreign policy decisions, the emphasis is on the individual characteristics of the cognitive process of decision-makers, it is emphasized that the behavior of politicians largely depends on their vision of reality. The results of such a study are used to predict the behavior of decision-makers in a given situation.

Another model assigns a key role to bureaucracy (the so-called bureaucratic model of politics ). Foreign policy decisions, according to this model, it is the result of bargaining and "confrontation" between various bureaucratic structures seeking to realize their interests. In this case, all other "players", including parliamentary institutions and the public, are nothing more than extras.

"Pluralistic model" proceeds from the fact that the decision-making process is largely chaotic. The public could have a much greater influence on him, but its influence is realized through the struggle of organized "interest groups". Society is heterogeneous, and a conflict of different interests within society is inevitable. At the same time, it is emphasized that only a small number of individuals and institutions are involved in the process of developing the most important decisions, while the public is for the most part an "outside observer". The final political decision is the result of a "struggle" between different "interest groups".

Organizational Behavior Model assumes that decisions are made by various government entities operating in accordance with their well-established decision-making routines (standard operating procedures). The latter include procedures for collecting, processing and transmitting information and allow you to standardize the solution of complex but repetitive routine issues. We can say that this allows you to cope with problems without making a decision in each specific single case - this decision is "programmed" by standard operating procedures. In other words, the life of each "organization" (government structure) has its own logic. The decision-making process turns out to be fragmented, and the final decision is the result of the interaction of structures of different possibilities to influence.

All of the above models focus on the internal state mechanism for making foreign policy decisions. However, we must not forget that the process of developing a foreign policy course is always "placed" in a certain external context, and the influence of external factors is just as strong. The "transnational model" of foreign policy analysis involves taking into account the influence of the external environment - the global economic, social and cultural context of the foreign policy of any state. Other models have also become widespread: such as, for example, model of elitism, democratic politics and etc. .

Another fairly common method of studying the decision-making process in the framework of the science of international relations is associated with game theory ... The latter is based on the theory of probability and extends the concept of "play" to all types of human activity. Game theory is the construction of models to analyze or predict various types of actor behavior. Canadian researcher J.-R. 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" ... Within the framework of this model, the behavior of a decision-maker is analyzed in his relationship with other "players" pursuing the same goal. "Wherein the task is not in describing the behavior of players or their reaction to information about the behavior of the enemy, but in finding the best of possible options decisions for each of them in the face of the predicted decision of the enemy " .

Keywords

MIZHNARODNi VIDNOSINI / POLITICAL ANALYSIS/ FORECAST / CONTENT-ANALYSIS / ANALYSIS OF DOCUMENTS/ INTERPRETATION / INTERNATIONAL RELATIONSHIPS / POLITICAL ANALYSIS / FORECASTING/ CONTENT ANALYSIS / ANALYSIS OF DOCUMENTS/ INTERPRETATION / INTERNATIONAL RELATIONS / POLITICAL ANALYSIS / FORECASTING / CONTENT ANALYSIS / DOCUMENT ANALYSIS / INTERPRETATION

annotation scientific article on political science, the author of the scientific work - Dzera M.M., Pasichny R.Ya.

International relationships as a sphere of human coexistence encompass political, economic, diplomatic, cultural and other ties and relations between actors acting in the international arena. The presence of such a large number of subjects and the importance of their relationships are the reason for the need to analyze this area in order to determine the trends of their development and mutual influence between them. For studying international relations they apply most of the general scientific methods, however, along with them, they also use special methodological approaches, due to the fact that world political processes have their own specifics, differ from the political processes unfolding within individual states. An important place in the study of world politics and international relations belongs to the methods of instrumental observation, in particular content analysis, analysis of documents, a method of observing the reflection of political reality in the media. With the help of the above methods, it becomes possible to fix and observe an event, with subsequent assessment and establishment of cause-and-effect relationships.

The text of the scientific work on the topic "Modern Methods of Research in International Relations"

Scientific Vknik ЛНУВМБТ iMeHi S.Z. Gzhitskogo, 2017, vol. 19, no. 76

HayKoBHH BiCHUK ^ BBiBctKoro HanjoHantHoro ymBepcureTy BeTepHHapHoi MegunuHH Ta 6ioTexHonoriH iMeHi C.3. IxuntKoro Scientific Messenger of Lviv National University of Veterinary Medicine and Biotechnologies named after S.Z. Gzhytskyj

ISSN 2519-2701 print ISSN 2518-1327 online

http://nvlvet.com.ua/

Suchasha methods of doslvdzhen international vidnosin

MM. Dzera1, R. Ya. Pasichny2 [email protected]

1Lviv National University of Veterinary Medicine and Byotechnologist imeni S.Z. Gzhitsky,

vul. Pekarska, 50, m. Lviv, 79010, Ukraine;

2Lviv National University "Lvivska nolytechnika" st. Stepana Bandery, 12, Lviv, 79013, Ukraine

M1zhnarodt vgdnosini yak the sphere of people’s attitudes to the state of affairs spheres with a viznachennya tendency of 1x rosette and vzaemovplivu mgzh them.

For vivchennya mzhnarodnyh vgdnosin to be stashed by the state of the scientific methods, protest at once with them viko-christovoy to g special methodology It is more important to follow the methods of instrumental caution, content-analgzu zokrem, ana-lgzu dokumentgv, the method of caution in the way of thinking politically I dshsnostg in the mas-medg. Behind the help of the most important methods, in a flock of mobile fgksatzgya that guarded go, with the further otstyuvannya and the establishment of causal symptoms of ulcers.

Key words: mgnarng vgdnosini, poltichny analgs, forecasting, content-analgs, analgs documentgv, tterp-pulling.

Modern methods of research of international relations

MM. Dzera1, R. Ya. Pasichny2 [email protected]

1 Lviv National University Veterinary Medicine and Biotechnology named after S.Z. Gzhitsky,

st. Pekarskaya, 50, Lviv, 79010, Ukraine;

2Lviv National University "Lviv Polytechnic", st. Stepana Bandera, 12, Lviv, 79013, Ukraine

International relations as a sphere of human coexistence encompass political, economic, diplomatic, cultural and other ties and relations between actors acting in the international arena. The presence of such a large number of subjects and the importance of their relationships are the reason for the need to analyze this area in order to determine the trends of their development and mutual influence between them.

To study international relations, most general scientific methods are used, however, along with them, special methodological approaches are used, due to the fact that world political processes have their own specifics, differ from political processes unfolding within individual states. An important place in the study of world politics and international relations belongs to the methods of instrumental observation, in particular, content analysis, analysis of documents, the method of observing the reflection of political reality in the media. With the help of the above methods, it becomes possible to fix and observe an event, with subsequent assessment and establishment of cause-and-effect relationships.

Key words: international relations, political analysis, forecasting, content analysis, document analysis, interpretation.

Dzera, M.M., Pasichnyy, R.Y. (2017). Modern research methods International Relations. Scientific Messenger LNUVMBT named after S.Z. Gzhytskyj, 19 (76), 144-146.

HayKoBHH BicHHK .HHyBMET iMeHi C.3. iKHibKoro, 2017, t 19, No. 76

Modern research methods International Relations

M.M. Dzera1, R.Y. Pasichnyy2 [email protected]

1Lviv National University of Veterinary Medicine and Biotechnologies named after S.Z. Gzhytskyi,

Pekarska Str., 50, Lviv, 79010, Ukraine;

2Lviv National Polytechnic University "Lviv Polytechnic", Stepan Bandera Str., 12, Lviv 79013, Ukraine

To study international relations most used scientific methods, but both of them and use special methodological approaches, due to the fact that world political processes have their own specifics are different from political processes unfolding within individual states. An important role in the study of world politics and international relations belongs instrumental observation techniques, including content analysis, document analysis, observation method reflected the political reality in the media. Using the above named methods becomes possible fixing and monitoring developments with further evaluation and establishing causation.

Select individual research methods in the analysis of international relations, determined by the characteristics of the task, so with the aim of research training public perception ofpowerful solutions in the political arena, pay attention to methods such as analysis of documents and their content, method of illumination and interpretation in the media.

Political analysis involves a systematic assessment of political reality and viability of alternative policies, which tend to have a form of political documents.

The study of relevant documents gives researchers important information about the foreign policy of countries and their development trends, the reasons for acceptance of foreign policy decisions in a given international situation. However, study of topical international problems, this method has several disadvantages. As part of the documents may be closed nature, due to the state secret, a researcher operating with only open sources and not having all the information about the scenario of the international situation, can make the wrong conclusion.

Key words: international relations, political analysis, forecasting, content analysis, document analysis, interpretation.

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HayKOBHH BicHHK HHyBMET iMeHi C.3. I ^ H ^ Koro, 2017, t 19, no. 76

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npu aHagi3i Mi®HapogHHx BigHocuH 3HaxoguTb 3a-crocyBaHHa i MeTog iBeHT-aHagi3y (aHagi3 nogiH), 3acHO-BaHHH Ha cnocTepe®eHHi 3a guHaMiKoro nogiH Ha Mi®-HapogHin apeHi 3 MeToro BH3HaneHHa ochobhhx TeHgeH- ^ h po3BHTKy nogiTHHHoi "crnya ^ ib OKpeMux KpaiHax, perioHax ib cBiTi b ^ goMy. 3rigHo 3 gaHHMH gocgi-g®eHb, 3a gonoMororo iBeHT-aHagi3y MO®Ha ycnimHo BHBnaTH Mi®HapogHi neperoBopu. y tsbOmy BunagKy b tsentpi yBaru nepe6yBae guHaMiKa noBegiHKH ynacHHKiB neperoBopHoro ppotsecy, iHTeHcuBHicTb BucyBaHHa npo -position, guHaMiKa B3aeMHux nocTynoK i Tg

Ei6.iorpa $ iHni iiocii. lanim

Popova, O.V. (2011). Politicheskij analiz i prognoziro-

in economics, social psychology, sociology and demography.

Sociological theories of O. Comte and E. Durkheim, starting from XIX. in., nourished the idea of transferring them from sociology to other social sciences. The decisive influence on the formation of new directions in the study of international relations was exerted by the almost coinciding in time and interconnected appearance of the general theory of systems, the principles of which were set forth in the 1930s by L. von Bertalanffy, and cybernetics.

They gave a powerful impetus to behavioristics (from the English word behavior or behavior) 36, i.e.

researching behavior at the individual, collective and social levels by measuring it. The prerequisites for the rapid development of behaviorism in the 50s, the so-called "behaviorist revolution" in the social sciences, were laid by American psychologists (C. Merriam, G. Lasswell) in the 1920s-30s, when they substantiated the idea

studying political behavior as the main subject of political research

science 37.

Based on general systems theory, information theory and cybernetics, the behaviorist direction

became dominant among the "modernist" in the study of international relations. And in the very

behaviorist direction can be conditionally distinguished groups of researchers: 1) operated

non-mathematical concepts, in particular, based on the theory of structural and functional analysis by T.

Parsons and D. Easton's method of systems analysis of politics; 2) who used quantitative methods and such

mathematical theories like J. von Neumann's game theory or N. Wiener's and W. Ross Ashby's information theory

(K. Deutsch, L. Singer, D. Modelsky, A. Rapoport).

We emphasize once again that one should beware of a rigid classification of `` modernist '' directions: it was a flow of various variations, a fusion of ideas and methods of exact and humanitarian knowledge, a shift of efforts from the development of a universal theory based on historical and philosophical knowledge to the theory of systems and at the same time to empirical research based on the measurement of observable data outside their ideological or philosophical significance.

However, the very rejection of philosophical views as a theoretical basis for the study of international relations, as many Soviet international experts believed, in fact could mean an appeal to the philosophy of "neopositivism." One way or another, "modernism" was sharply different from traditional trends in its striving for accurate, empirical evidence.

One of the most prominent "modernists", who was the president of the American Association for Political Science, K. Deutsch thus motivated the appeal to empirical methods: '"Modern methods of storing and returning information, electronic computers make it possible to handle large amounts of data if we know we want to do with them, and if we have an adequate political theory that can help formulate questions and interpret the findings. Computers cannot be used as a substitute for thinking, just as data cannot be used as a substitute for judgment. But computers can help us carry out the analysis that offers new thinking to the theory ... The availability of large masses of relevant data and computer processing methods open up broad and deeper foundations for political theory, while at the same time it differs from theory in broader and more complex problems ”38 ...

Most of the supporters of traditional approaches led by G. Morgenthau rejected or skeptically

referred to the application of methods adopted from economics in the study of international relations,

sociology and psychology. Although earlier in the Soviet scientific literature the difference in

methodology between American "traditionalists" and "modernists", it was essential and at the first

pores reflected opposite approaches.

In our opinion, M. Merle correctly spoke about the advantages and disadvantages of the new methods. Noting about the rejection of them by “political realists” that “it would be absurd to justify the intellectual tradition of the lack of research tools” that expands these methods, he expressed doubts about the possibility of quantifying data on international relations due to the lack of many statistical indicators or the unreliability of statistics in many countries of the immense scale and complexity of the international sphere39.

Let us try to extract from the long dispute between “traditionalists” and “modernists” the most

significant arguments of both: (see Table 1) There is no doubt that the arguments of the supporters of the old and new

approaches on each side contained a grain of truth. But on the rejection of "modernism" by traditionalists

an important objective circumstance affected: the views of the "realists" who became the leading school

traditional directions, were confirmed by the practice of US foreign policy, because in essence their own views

she was inspired. Therefore, their reaction to the seemingly heavy heaps in the methodology

was quite understandable. Another thing is that this reaction contradicted the objective tendency towards integration.

sciences, expanding the possibilities of humanitarian research by the achievements of natural sciences, their theories and

Arguments of the "traditionalists" Arguments of the ‘modernists"

1. Quantitative and other methods, taken mainly from economic science, are alien to the science of international relations, in which there is no hierarchy and organization inherent in relations within the state (social

economic or political). 1. Traditional approaches have unreliable scientific instruments, assessment criteria are speculative, concepts and terms are vague.

2. In international relations, in addition to material, and non-material factors (national feelings, the will of political leaders) are manifested, which are difficult to systematize, their combination is unique and lends itself only to qualitative assessments 2. The analysis of modern international relations is based on outdated ideas.

3. The distinction between nations (national spirit, traditions, culture) is also qualitative in nature.

3. Inapplicability of the theories of traditionalists, in particular

"Realists" for quantification.

4. The foreign policy of the state acts as a historically conditioned integrity that cannot be quantified, just like strength (power). 4. Limited predictive power of the concepts of traditionalists, their generalizations are unverifiable.

So, let us trace briefly the most significant stages in the formation of American "modernism". Describing new, "modernist" approaches in the study of international relations, experts

it is often said that their essence is focused in the behavioral methods, which ”have already been mentioned and which meant the use of methods for analyzing empirical data, the construction of various models based on system concepts.

2. "FIELD THEORY" QUINCY Wright

One of the pioneers of "modernist" approaches was the renowned historian and sociologist Quincy Wright, who published the two-volume Study of War in 1942. Specializing in the study of war, K. Wright began with the systematization of all data on wars that occurred in the history of mankind. Then, based on the structural-functional method of analysis, he proposed an interdisciplinary approach to the study of international relations, which would combine the consideration of empirical data, their generalization and the development of a general theory, a model that can be verified by an application to reality. K. Wright was puzzled by the creation of a general theory of international relations. He listed 16 disciplines necessary from his point of view to create a scientific theory, the so-called "field theory" of international relations: 1) international politics, 2) military art, 3) the art of diplomacy, 4) foreign policy of the state, 5) colonial government , 6) international organizations, 7) international law, 8) world economy, 9) international communications, 10) international education, 11) political geography, 12) political demography, 13) technocracy, 14) sociology, 15) psychology, 16) ethics of international relations.

K. Wright considered one of the goals of such an “integrated” science to be the ability to foresee the future. He was a sincere pacifist, opposed “ cold war”, Criticized US foreign policy, in particular the Vietnam war.

3. SYSTEM APPROACH OF MORTON A. KAPLAN

The next notable milestone in the formation of "modernism" after the publication of K. Wright's book in 1955 was M. Kaplan's work "System and Process in International Politics" 40 (1957). It is believed that it was in this work that a systematic approach to the study of international

relations based on the general theory of systems, or rather, its version set out in the book

W. Ross Ashby's "The Construction of the Brain" 41 (1952). M. Kaplan's work has been widely known for a long time,

but the evolution that has been taking place in international relations since the late 1980s, all the more revives interest in his hypotheses, allowing them to test their predictive capabilities.

M. Kaplan's book is also remarkable for the fact that it reveals the connection, the continuity between the new approach and traditional "realism", since the initial for the author is the fundamental concept

"Classical" theory - "balance of power". M. Kaplan suggested that from a certain historical time (approximately from the 18th century) in international relations, global systems were formed, which,

while changing, they retained their main quality - “ultrastability”. Using a concept from cybernetics (“input

Exit ”), he tried more accurately than the“ classics ”to define the basic rules for the optimal behavior of states (“ actors ”) in the“ balance of power ”system that had existed since the 18th century. before World War II. He described six rules for the normal, from his point of view, functioning of the system, in which there should be at least 5

actors. So, each of them had to be guided by the following rules:

1) build up force, but, if possible, prefer negotiations to the conduct of hostilities;

2) it is better to go to war than to miss the chance to increase strength;

3) it is better to end the war than to exclude from the system the main national actor (against whom force was used),

4) discourage any coalition or actor seeking to dominate the international system;

5) deter actors who apply supranational principles of organization and behavior;

6) allow defeated or weakened primary actors to take their place in the system as partners and help secondary actors to raise their status.

The system that emerged as a result of World War II is the second global international system

v history, according to M. Kaplan, was defined by him as “free (or“ weakly connected ”) bipolar system”,

v which bipolarity was limited by the action of the UN and the power of actors who remained neutral. In addition to two real historical systems, M. Kaplan imagined 4 hypothetical ones that can

to form from a “free bipolar system”:

1) a rigid bipolar system, where all actors are drawn into one or another block, and the neutral position is excluded (the system is less stable than “free bipolarity”);

2) the universal international system of the confederal type;

3) a hierarchical system dominated by one bloc, where nation states would find themselves in the position of autonomous,

4) system of "veto" or multipolar system, in which the number of powers that have nuclear weapons and providing nuclear deterrence.

Later M. Kaplan supplemented these models with 4 variations:

1) A very free bipolar system, where the degree of nuclear equilibrium would increase, the blocs weakened, and nuclear weapons partially spread.

2) A system of relaxed tension (or detente), which presupposed evolution in the superpowers (“liberalization” of the USSR and democratization of US foreign policy), which made it possible to limit arms to a minimum level.

3) An “unstable bloc system” where the arms race will continue and tensions will rise.

4) Incomplete nuclear proliferation system(15-20 countries). It is similar to the previous system, but in it the nuclear potentials of the superpowers do not reach the level of the ability to deliver the first crushing blow, and in it, coalitions between the superpowers and small nuclear countries are possible, which would further increase the likelihood of war.

“Realists” criticized M. Kaplan for the abstractness of his models. The Australian scientist H. Bull, who worked at the London Institute for Strategic Studies, reproached M. Kaplan for the fact that his models “are divorced from reality and are unable to develop any understanding of the dynamics of international politics or

moral dilemmas generated by these dynamics ”42.

While admitting a certain amount of fairness in such criticism, for the sake of fairness, we recall that

M. Kaplan himself did not at all pretend to be a biblical prophet and quite realistically considered

the possibilities of scientific foresight using system modeling. Emphasizing the failure of any

theory of international relations to predict the future in its specific manifestations, he limited

the predictive value of their hypothetical models by knowing: 1) the conditions for the invariability of the system, 2) the conditions

changes in the system, 3) the nature of these changes.

M. Kaplan's methodology still had a certain cognitive value, helping to imagine the likely evolution of international relations. And if none of his proposed 8 hypotheses (apart from the real free bipolar system) has not been fully realized, then some of them are still partially confirmed by the trends of modern development. In the Soviet scientific literature until the second half of the 80s, when the principles of “new thinking” were formulated, M. Kaplan's position on the evolution of the USSR was sharply criticized as “unacceptable”, as “completely contrary to reality” or “directed

between countries ”. The process of "perestroika" and the destruction of the USSR, however, prove that today it is impossible not to recognize the scientific significance of M. Kaplan's scenario forecasts.

4. CHARACTERISTIC FEATURES OF "MODERNIST" RESEARCH IN THE LATE 50s - 60s

Since the late 1950s, a real boom in international relations research began in the United States based on

new methods. Thousands of works have appeared, university schools have been formed, which are distinguished not only by methodological criteria, but also by research subjects. Several attempts at classifications have been made in the United States. The most detailed classification of works in English was proposed by the prominent American international specialist Bruce Russet, who compiled a sociometric table of the citation index of more than 70 authors. Having chosen for this publication 1968-1986, he conditionally divided all scientists into 12 groups according to the criteria of methodology or object of research, and of them 15 authors were simultaneously assigned to two groups, 9 - to three groups. The largest group was made up of scientists from Yale University or collaborators, mainly engaged in “international integration” (16 people) 43.

Another detailed classification was given by the American international expert F. Burgess, who singled out seven

directions ("cognitive rationalism", the study of behavior in terms of its goals, reasons, etc.)

etc.), “theory of strength”, the study of the decision-making process, theory of strategy, theory of communications, theory

fields (see above for a summary of the method proposed by K, Wright), systems theory (M. Kaplan and his followers) 44.

extremely time consuming. (This work was largely done in the already mentioned

innovations introduced into the science of international relations by "modernists", and then we will consider the main theoretical directions of "modernism" and present a number of specific examples of the application of these methods, in particular, in determining the power of states.

5. APPLICATION OF THE SYSTEM APPROACH

The use of a systems approach meant a major shift in both theory and methodology of studying international relations - a departure from the "state-centric" views on international relations as the "sum" of foreign policies of states.

Another important merit of the "systemists" was that they expanded their ideas about the participants (actors) of the international system, considering as such, in addition to the main actors - states, international organizations, non-state political forces (for example, parties), religious organizations and economic forces mainly transnational corporations. David Singer of the University of Michigan put forward in 1961 in a well-known article the idea of “levels of analysis” that unite two areas - international systems and the nation state. D. Singer singled out the main border in the search for phenomena that affect international politics: 1) internal phenomena occurring within the borders of the state, 2) external phenomena occurring outside the state border45.

The application of the principle of general systems theory not only expanded the concept of "actors"

international relations (and, in essence, changed the understanding of their structure), but also led international relations to

the formation of the concept of "environment". Let's reproduce the simplest scheme, which is given in many

foreign textbooks and monographs, graphically depicting a systematic approach to the study of political

a sphere that presupposes the existence of an “external environment” (Fig. l):

Picture 1

Often this approach to the analysis of political systems is called the method of D. Iston, which is described in his work “System Analysis political life”*. With regard to international relations, the concept “ environment'' Gets complicated. It seems to be quite simple for a state, rather specific for groups of states or coalitions; finally, one can imagine a more complex “external environment *” for the entire system of interstate relations, which can be considered international relations as a whole. But what is the “external environment” for the global system of international relations, if we accept the assumption of its existence? There is no unambiguous answer to this question in the scientific literature.

In the 60s, a number of works appeared in the United States aimed at studying the foreign policy of the state, considered "surrounded by the environment." Several interesting publications on this topic belong to the spouses G. and M. Spraugs *. They proposed the concept of "ecological triad" (the term "ecology" is used here in a broad sense): 1) personality of a certain character (statesman), 2) conditions that surround it (environment), 3) interaction of personality and conditions. G. and M. Sprout distinguish 3 types of interaction:

The first type is environmental possibilism, i.e. opportunities representing the conditions in which the decision-maker operates. These conditions change historically. For example, they say. Napoleon could not threaten Moscow with a nuclear bombardment (neither could the Germans in 1914, although they could reach Moscow faster with the help of railways than Napoleon could do it), the Romans could not move their legions from Italy to Britain for hours or even days, Theodore Roosevelt in 1905 could not raise American prestige by sending a man to the moon (he decided to send the American flag around the world travel), the Persian king Darius could not use the phone to clarify disagreements with Alexander before the Macedonian campaign in Asia; the Spaniards in the Middle Ages could not rely on the resources of the New World to repel the Islamic invasion of the Iberian Peninsula, etc.

The main idea of G. and M. Spraugov is that the decision-makers are limited by the opportunities provided by the world around them.

The second type of interaction is environmental probabilism, i.e. the probability with which this or that event will occur. In other words, assuming that states interact, the authors focus on the likelihood that an individual will act in a certain way in a “certain environment”. For example, what was the likelihood that the United States and the USSR would become rivals as two superpowers after World War II? Or what is the possibility of interaction between Burma and Bolivia, small states in different regions of the world, separated by thousands of miles?

The third type of interaction is cognitive behavio sm, i.e. behavior of a person making a decision based on knowledge of the environment. Such a person interacts with the surrounding world through the images of this surrounding world. She acts on the basis of how she perceives this world. This perception can be very different from reality.

6. USE OF CYBERNETIC CIRCUITS IN A SYSTEM APPROACH

A powerful impetus to the systems approach is given by the theory of communication and the means of cybernetics. As a result of their application, the concept of states, nations, political regimes has been formed as cybernetic systems with “input” and “output”, controlled by a feedback mechanism (“stimulus” - “reaction”). The pioneer and the most major representative The "cybernetic" approach was the patriarch of American political science K. Deutsch.

Subsequently, American colleagues, French international experts, recognizing the positive use of cybernetic tools for analyzing such a complex system as the state, criticized K. Deutsch, believing that his methodology overestimates the rational nature of decision-making by the center of the political system and that it is closer to physics than social sciences.

K. Deutsch, explaining the "cybernetic approach" to foreign policy, compared the decision-making process with playing electric billiards. The player sets the initial speed to the ball, it moves, colliding with obstacles that change the trajectory of its movement. The point of falling or stopping depends simultaneously on the initial impulse, subsequent maneuvers of the player and the impact of obstacles.

Criticizing K. Loich, the French internationalists P.-F. Gonidek and R. Sharven draw attention to the fact that

v Unlike physics, obstacles in the international sphere represent not only explicit, but also hidden influences, intersections of interests * (ie, “obstacles” themselves in motion). Therefore, K. Deutsch's "cybernetic" method is more suitable for the analysis of military strategies than politics, since in military area the behavior of states is more rigid and mutually determined.

Nevertheless, there is no doubt that computers have dramatically expanded the use of mathematical tools in the study of international relations, making it possible to go, in addition to the already used methods of mathematical statistics, algebraic and differential equations, to new methods: computer modeling, solving information-logical problems. But first of all, the capabilities of computers stimulated research on proven methods in mathematical statistics aimed at formalizing qualitative characteristics, attempts to measure “strength *”, “power,” “solidarity,” “integration,” “aggressiveness,” and so on. Let us clarify that, although a number of methods were specially developed by him for the study of international relations, their development for political science as a whole was of more significant importance.

V monographs by S. V. Melikhov contain significant reference data on the use of quantitative methods in American political science, mainly factor analysis (as well as multivariate correlation, regression, analysis of variance and time series analysis) * ”.

Famous international scholars who applied mathematical methods in the 50s - 60s * in the USA were A. Rapoport, K. Deutsch, D. Singer, G. Getzkov, O. Holsti, B. Russet, R. Rammel, D. Tsinnes and others. But the extreme popularity of mathematics at that time involved the so-called * quantitative ”research

v social sciences of many amateurs who did not have a professional mathematics, flaunting some separately "snatched" methods and concepts from the mathematical arsenal.

from about the 70s, when the great, or better to say, exaggerated hopes were not justified. Soviet international experts from the NMEMO expressed the following opinion on this score: “On the whole, the scarcity of results from the use of mathematics in the“ interdisciplinary ”study of international relations is associated with the underdevelopment of mathematics itself, which may be suitable for this specificity. Apparently, the branch of mathematics that would correspond to the subject of research under consideration has not yet been developed. Attempts to borrow mathematical tools from other branches of science, which were created specifically for the needs of these branches, turned out to be unsuccessful. ”

7. DIFFICULTIES OF APPLICATION OF MATHEMATICAL KINDIES IN POLITICAL SCIENCE

In our opinion, some of the difficulties in the reliable application of mathematical methods in the study of politics and history at a theoretical level are as follows:

1. It is difficult to quantify the spiritual sphere, consciousness, movement of ideas and mentality, individual qualities of those who make decisions. With logical thinking, a person is subject to

and the sphere of subconscious drives, emotions, passions affecting rational thinking, which in the behavior of state and political leaders often makes decisions difficult to predict.

Although, in theory, the system or “environment” should impose restrictions on their deviation from the most rational choice, history shows that the role of the state leader often turns out to be decisive, while he himself, making a decision, becomes immune to objective information, and acts on the basis of the subjective , to a large extent intuitively, understanding the political process and the intentions of opponents and other actors. As an example, let us recall the behavior of J. Stalin on the eve of Hitler's aggression against the USSR.

2. The second difficulty is related to the first one, but it covers the social sphere as a whole, where many influences, interests, factors intersect, and it seems impossible to establish and measure them relative to each other. Again, history shows that an insignificant, according to visible signs, or large, but formerly unchanged parameter can dramatically change its value and have a decisive impact.

An example from the relatively recent past is the four to fivefold rise in the price of oil in 1973, which caused the global energy crisis in the short term and, in the long term, caused the restructuring of the world economy. The same factor in the short term had a beneficial effect on the foreign trade of the USSR, and in the long term contributed to * the maturing crisis of the Soviet economy and the decline of the Soviet system as a whole. Meanwhile, the most significant change in the international economic system of the 70s. was not predicted in the models. So, in the well-known forecast of world development "Goal 2000", published on the eve of the 1973-1974 energy crisis. by the famous American futurologist G. Kahn, the oil factor did not figure among the variables at all ”*. those. many large, but suddenly developed processes in the economic, social and political spheres turn out to be unpredictable, which, of course, is not an indisputable proof of their unpredictability.

3. Finally, some processes seem to be random, stochastic, because the causes that cause them are invisible (at a given time). If we figuratively compare the social sphere with a bioslope organism, then the reasons for this are similar to a virus that does not show activity for a long time. due to the lack of favorable environmental conditions or their unknown internal "clockwork". With regard to international relations, it is important not to lose sight of the historical aspect, since the origins of some processes not observed by contemporaries are fixed in national traditions, national consciousness. In contrast to the evolution of nature (excluding anthropogenic impact and cataclysms), in which the length of time on a scale human history is minimal, in the world social sphere the complexity of systems in space is interconnected with strong, historically accelerating mutations.

As if summing up the results of behavioristic studies of international relations in the 50s and 60s, the English internationalist L. Reynalls spoke about the revealed methodological difficulties: “We are talking about the problems of inadequacy of intellectual tools. The human mind is completely incapable of creating a system that includes the entire ensemble of constituent elements and interactions on a worldwide scale. Such a system should be simplified.

But as soon as simplification is allowed, reality is immediately falsified, and simplification is nothing more than an abstraction of reality '' **.

One of the leading American behaviorists D. Singer argued the opposite point of view: “We cannot build a global system as a complex of very flexible, mobile co-optations, territorial

and others, including smaller formations, which can now be linked not only through governments, but be internal or extra-national as well as national as in the sphere

In this dispute, the skepticism of traditionalists is understandable, but he can hardly convince a serious researcher that the methods of the exact sciences are a priori unsuitable in the study of international relations. Naturally, these methods first began to be used in demography, economics, which, according to the subject of research

are, as it were, intermediate between the exact and “purely” humanities, where with the expansion of such a subject of research as the sphere of consciousness, the most adequate forms of cognition (figurative-metaphorical thinking, intuitive-experiential assessments, etc.) also expand. It is no coincidence that the qualitative and other methods of mathematics, biology, physics, transferred through the `` intermediate '' sciences into political science, international relations, by the way, gave the most noticeable results in those studies, the subject of which also turned out to be closer to physics or cybernetics than to purely humanities ...

8. EXAMPLES OF APPLICATION OF MATHEMATICAL MEANS IN MODELING MILITARY CONFLICTS AND ARMS RACE (L. RICHARDSON'S MODEL)

These examples primarily relate to the military-strategic area, where the criteria for the behavior of states, as well as the behavior itself, are tightened, and the significance of various influences and interests is assessed in a single dimension of the balance of forces and potentials, i.e. one way or another, the number of factors that are subject to quantification decreases.

Back in the 1930s, the Scottish mathematician L. Richardson began to create a mathematical model of war and international conflict. According to A. Rapoport, L. Richardson viewed international relations as a “physical system”. In the 50s, his method attracted the attention of American authors, but L. Richardson, improving it, retained the priority and achieved widespread recognition in the West of his model as a classic in the field of military-strategic research using mathematics, which can be seen from its citation index in foreign countries. literature. L. Richardson proposed a system of differential equations:

dx / dt = ky - α х + g
				βy

where x and y are the levels of armaments of the two countries, k and l are the “defense coefficients” (the government's idea of the enemy's strategy); α and β - coefficients of the "cost" of military efforts; g and h - coefficients of "aggressiveness" 262 (the degree of militarism or peacefulness of foreign policy).

Another quantitative analysis technique that has found widespread use in

foreign studies, contained in the project "Correlation of War", developed

under the leadership of D. Singer *. It is based on the paired correction technique. D. Singer set the task of establishing, on the one hand, the correlation between the number of wars and the military potential of European states from the Congress of Vienna 1815 to 1965, on the other hand, between several parameters of wars (occurrence, intensity, duration)

and parameters characterizing the international system (the number and strength of alliances, the number

international organizations).

In the project, using factor analysis, six indicators of military strength were identified: 1) the total population, 2) the population in cities over 20,000 thousand inhabitants; 3) the amount of energy consumed; 4) production of steel and iron;

5) the level of military spending; 6) the size of the armed forces. One project output

states that long-term equilibrium in the European system of the XIX century. hindered the intensity of wars and, conversely, wars of the XX century. caused by changes in the balance of power in favor of one power or coalition. Another less obvious conclusion is

the fact that the intensification of the process of the formation of unions in the XIX century. increased the likelihood

the emergence of wars, while in the international system of 1900-1945. strengthening alliances

game models (G. Gettskov, R. Brody). Game theory originated in the 40s. Since the late 1950s, games in the field of international relations have been simulated without and with the help of computers (O-Benson, J. Crand). The Soviet international experts who analyzed them believe that the use of logical and mathematical methods and computer modeling opened up a promising direction, but were restrained by the "insufficiency of the existing mathematical means, and, above all, game theory".

By analogy with war games, “hard” imitations are distinguished, where certain conditions of behavior are set, and “free” ones. The former, as a rule, were used in attempts to model at the global level, the latter - for specific problems (most often for modeling conflicts). It seems that the experience of these models deserves more careful analysis by mathematicians for the possible use of valuable elements. Note that game, simulation models, as well as correlation, static ones, also concerned mainly the military-strategic area.

MAIN THEORETICAL DIRECTIONS OF "MODERNIST" RESEARCH

The conventionality of dividing the directions of "modernist" (behavioristic) studies of international relations according to two criteria - methodology and theory - is quite obvious. The existing theory itself is the methodological basis of cognition. For example, studies of the process of making foreign policy decisions can be viewed as a methodological principle in the analysis of foreign policy and, at the same time, as a theoretical direction. Nevertheless, theoretical constructions differ from methodology in that they have a specific subject of research. The "classical" approach to the study of international relations in American and Western European science was focused on a universal general theory. And since many “modernist” approaches proceeded from opposite, empirical attitudes, their result was a rejection of the search for a global theory and the formation of a number of private theories of international relations.

Abroad, there are many private theories and methods in the study of international relations. According to some estimates, only by the beginning of the 60s there were up to three dozen of them. However, several main ones stand out among them: theory international conflicts, the theory of integration, the theory of making foreign policy decisions, and in a broader sense - the theory of foreign policy. Finally, there is such a separate area as the study of peace problems (Peace research), which stood out from the study of international conflicts.

So, let's look at a number of examples of the characteristic features of particular theories of international relations.

1. GENERAL THEORY OF CONFLICT

The largest of these in terms of the number of studies and publications was the theory of international conflicts. Actually, conflict management is a broader branch of international research that considers conflict as a social phenomenon and behavior in all social spheres. In the United States and other Western countries, there is the so-called “general theory of conflict”, the dominant methodology of which is the systemic, structural-functional approaches in combination with behavioral-cybernetic techniques. The behavioral trend was reflected in the publications of the American journal “Journal of Conflict Resolution” founded in 1957. International conflicts turned out to be central theme on the pages of the journal, which has essentially become a priority scientific publication not only in the field of conflict studies, but to a large extent in the study of international relations in the United States as a whole. One of its most famous representatives is the conflict expert Kenneth Boulding.

The behavior of participants in an international conflict is considered by behaviorists approximately according to the scheme that is presented in the well-known work on quantitative methods, published under the editorship of D. Singer (see Fig. 2).

Picture 2

S - incentives caused by the behavior of states R - behavior of each state

r - stimulus score

s - intentions expressed depending on perception.

International conflicts is a topic that in the 70s and 80s, perhaps, became a priority for Soviet international scholars as well. In any case, in terms of the number of monographs in comparison with other subjects in the theory of international relations. The authors of foreign and domestic works emphasized that the main trends in the development and contradictions of the international sphere are focused in international conflicts, and if we take into account that the global problem of war was interpreted by many Western scientists as component conflictology, then it is logical to consider the theory of international conflicts in its approach to the level of the general theory of international relations. It is the vastness and significance of the subject that explains why the main direction in research on the general theory of conflict has taken the study of international conflicts.

The study of international conflicts in most cases pursues applied goals. Therefore, in

foreign conflictology from an applied point of view most often at the beginning two levels of analysis were distinguished: 1) analysis of the causes, structure and dynamics of conflicts, 2) "therapy", i.e. development of a methodology for their settlement (UN, international court in The Hague, negotiations, application of international legal norms, force). Then the third level emerged - the prevention of international conflicts. In particular, the idea of the possibility of conflict prevention and the need to develop appropriate means for this was formulated by the director of the Center for the Study of Conflicts at London University Cottage J. Burton.

2. THEORY OF INTEGRATION

Among the studies on the theory of international integration in Anglo-American literature, the works of K. Deutsch “Political community at the international level. Definition and Measurement Problems ”,“ Political Community and the North Atlantic Space. International Organization in the Light of a Historical Experiment ”, as well as“ Nationalism and Social Communication ”and a number of other works.

Considering that there can be no universal law according to which cooperation and integration processes develop, K. Deutsch named several conditions necessary for this. Among them, he first of all highlighted the commonality of political values and such psychological factors as knowledge of partners, the development of trade, the intensity of cultural exchange and exchange of ideas. K. Deutsch put forward a hypothesis about the predominance of communication factors in the formation of political communities and in maintaining their internal unity, cohesion, considering linguistic communication primarily from the point of view of information exchange. Each nation, people has special communicative means, which are expressed in the consolidated collective memory, symbols, habits, traditions.

Two American authors, R. Cobb and C. Elder, conducted a study on the basis of correlation analysis in order to determine the factors that determine rapprochement and cooperation in international relations, comparing the relationship between the selected fifty states of the world and the relationship within the North Atlantic community. As a result, two factors turned out to be predominant: 1) previous cooperation, 2) economic power, as can be seen from the following diagram (the significance of a number of factors was not revealed) (see Table 2 in the Appendix).

If we take into account that “previous cooperation” is itself the result of the action of other factors, then there remain two leading factors in terms of the level of correlation (economic and military power).

Other authors emphasize the predominance of the factor of the leading political force, the "hotbed" of integration. From these positions, the Belgian internationalist J. Barrea considered the history of the USA, Canada, Australia and South Africa, who believes that integration tends to develop around the "core area", representing one (possibly more) more powerful state, attracting into its orbit the surrounding area.

3. THEORY OF FOREIGN POLITICAL DECISION MAKING

Publications on this topic can be divided into “purely scientific”, in which real processes are analyzed, and scientific and applied, in which methods are developed to optimize decision-making. In Anglo-American studies, there are several approaches to assessing the process of foreign policy decisions.

One of the most popular in the 40-50s was the socio-psychological approach, in particular, the so-called "method of operational cipher" or "code". It was used by the sociologist N. Leits, who, on the basis of an analysis of Russian literature and the works of the Bolsheviks, tried to reconstruct the system of values (beliefs) of Soviet leaders and open their perception of the outside world. His goal was to create a collective image of the “Bolshevik perception” of reality, in order to try to understand the behavior of leaders based on this. Modified, this approach then turned into a psychological test of 10 questions, asked in order to find out a politician's view of the world. Philosophical questions were also clarified, for example, "is the political universe in essence some kind of harmony or collision?" "Is the future in politics predictable?" In addition, the list includes "instrumental" questions that clarify someone's behavior in the world of politics: "What is the best way to choose, goals or objects of political action?"

By the mid-1950s, a socio-psychological interpretation of the motives for decision-making was given by R. Snyder based on the ideas of M. Weber and the structural and functional analysis of T. Parsons. His method assumed the greatest possible consideration of factors, but their consideration through the prism of perception by those who make decisions. (In the early 60s, R. Snyder took up the problem of rationalizing foreign policy decisions).

V further in the USA, as well as in the UK, two approaches were most widespread

To evaluating decision making: behavioral, combining socio-psychological aspects with cybernetic concepts; and rational decision theory based on game theory.

The behaviorist approach using cybernetic means in the analysis of foreign policy decisions and actions of the state was one of the first to be applied by the professor of Washington University J. foreign policy decisions).

Let us reproduce the explanation of the decision-making process, which was developed by the American internationalist O. Holsti, who defended his dissertation on this topic at Stanford University. In his opinion, three phases should be distinguished in an ideal decision process. The first is a kind of push from the external environment. Perception of the impact of the external environment - the second phase, the process by which the decision maker selects, sorts, evaluates the information received regarding the surrounding world. Interpreting a deliberate "push" - the third phase. Both perception and interpretation depend on those images that already exist (are embedded) in the consciousness of the person making the decision. O. Holsty gave the following schematic description of perception and its relationship with images from the outside world and the system of values of the person making the decision (Fig. 3):

Even if we accept O. Holst's scheme as adequately describing the behavior of a political leader intending to make a certain decision, it cannot reflect the real process of its adoption. As a rule, many factors operate in it, for example, the structure of power, within which decisions are made. In the 60s and 70s, the concept of a bureaucratic process of making foreign policy decisions (G. Allison, M. Halperin, and others), in which foreign policy actions are presented as a product of interaction between various state structures, a compromise of interests, became widespread in the United States. Emphasizing the special role of bureaucracy, the supporters of this concept chose the main object of the analysis of the decision-making process (and absolutized the meaning of this object) those factors that are underestimated in the socio-psychological interpretation of O. Holsty.

A more complex model of the process of foreign policy decisions was developed by the English internationalist J. Burton, who is also a supporter of structural and functional analysis using the cybernetic stimulus-response scheme. The peculiarity of his approach lies in the development of the concept of “vectors of change”, acting from the outside on the state. J. Burton divides changes into primary and secondary. Primary factors are changes in the environment (geography, geology, biosphere), secondary factors are the result of social interaction of human societies. Let us present a diagram of the decision-making process according to J. Burton, given in his book "System, States, Diplomacy and Rules".

		Table 5
	The factor of changes in the external environment
“Entry of State A		State B ... N
reaction of social groups		government reaction
perception	Perception	perception
		classification and storage of information
	classification and storage of information
	decision process	politics
	execution
domestic law	international action	"Exit" of each state B ... N

internal coercion (police)		external compulsion
affected groups			factors of change
states whose interests are affected	The "entrance" of each state

The main purpose of this chapter is to acquaint with the most widely used methods, techniques and techniques in the study of International Relations and Foreign Policy. It does not pose such a rather complex and independent task as teaching how to use them. However, its solution would be impossible, since this requires, firstly, a detailed description of certain 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 one or another scientific-theoretical or scientific-applied project, since, as you know, you cannot learn to swim without entering the water.

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

Unfortunately, it is necessary to note the fact that the special literature devoted to the problem of methods and especially applied methods of analyzing international relations is very few (especially in Russian) and therefore difficult to access.

1. Significance of the problem of the method

The problem of the method is one of the most important problems of any science, since ultimately it is about teaching how to get 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 a study. It precedes the study of an object because the researcher must from the very beginning possess a certain amount of techniques and means of achieving 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 activity. Moreover, the researcher is faced with the problem of the method already when analyzing the literature and the need for its classification and evaluation.

Hence the ambiguity in understanding the content of the term "method" itself. 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 a method, while having an independent meaning, is at the same time closely related to the analytical and practical role of theory, which also plays the role of a method.

The widespread belief that each science has its own method is only partially true: most social sciences do not have their own specific, only inherent method. Therefore, in one way or another, they refract, in relation to their object, the general scientific methods and methods of other (both social and natural science) disciplines. In this regard, it is generally accepted that the methodological approaches of political science (including international relations) are built around three aspects:

Separation of the research position from moral value judgments or personal views as strictly as possible;

The use of analytical techniques and procedures that are common to all social sciences, which play a decisive role in establishing and subsequent consideration of facts;

The desire to systematize, or, in other words, to develop common approaches and build models that facilitate the discovery of "laws" (1).

And although at the same time it is emphasized that this remark does not mean the need for "complete expulsion" from science of value

judgments or personal positions of the researcher, nevertheless, he inevitably faces a 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, "de-ideologize" in this sense. Interpretation of facts, even the choice of "viewing angle", etc. are inevitably conditioned by the point of view of the researcher. Therefore, the objectivity of research assumes that the researcher must constantly remember about the "ideological presence" and strive to control it, see the relativity of any conclusions, given this "presence", strive to avoid one-sided vision. The most fruitful results in science can be achieved not with the denial of ideology (this is, at best, delusion, and at worst, deliberate cunning), but under the condition of ideological tolerance, ideological pluralism and "ideological control" (but not in the sense of what we are accustomed to recently the past of the control of the official political ideology in relation to science, and vice versa - in the sense of the control of science over any ideology).

This also applies to the so-called methodological dichotomy, which is often observed in international relations. We are talking about the opposition of the so-called traditional historical-descriptive, or intuitive-logical approach to the operational-applied, or analytical-prognostic, associated with the use of methods of exact sciences, formalization, data calculation (quantification), verifiability (or falsifiability) of conclusions, etc. ... In this regard, for example, it is argued that the main drawback of the science of international relations is the protracted process of its transformation into an applied science (2). Such statements are too categorical. The process of the development of science is not linear, but rather reciprocal: it does not turn from a historical-descriptive to an applied one, but the refinement and correction of theoretical positions through applied research (which, indeed, are possible only at a certain, sufficiently high stage of its development) and "Debt repayment" to "applied specialists" in the form of a more solid and operational theoretical and methodological basis.

Indeed, in the world (first of all, American) science of international relations, since the early fifties of the XX century, the assimilation of many relevant results and

methods of sociology, psychology, formal logic, as well as natural and mathematical sciences. At the same time, the accelerated development of analytical concepts, models and methods begins, progress towards the comparative study of data, the systematic use of the potential of electronic computing technology. 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 old, "classical" methods and concepts.

Thus, 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 far from claim to be universal. A phenomenon so important for understanding international relations as, for example, power - “represents the quality of interpersonal relations, which can be checked, evaluated, guessed, but which cannot be quantified ... Of course, it is possible and necessary to determine how many votes can be given to politics, 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 the calculating machine and start thinking about historical and, certainly, qualitative indicators ”(3).

Indeed, the essence of political phenomena cannot be investigated in any complete way using only applied methods. In social relations in general, and in international relations in particular, stochastic processes dominate, which do not lend themselves to 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 are quite legitimate here, combining observation and reflection, comparison and intuition, knowledge of facts and imagination. Their usefulness and effectiveness are confirmed by both modern research and fruitful intellectual traditions.

At the same time, as M. Merle correctly 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 accurate correlations between collected facts are needed. Anything that can be quantified must be quantified (4). We will return to the controversy between "traditionalists" and "modernists" later. Here it is important to note the illegality of the opposition of "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 "act on equal grounds, and the analysis of the same problem is carried out independently of each other by different researchers" (see: ibid., P. 8). Moreover, within the framework of both approaches, the same discipline can use - albeit in different proportions - various methods: general scientific, analytical and specific empirical. However, the difference between them, especially between general scientific and analytical ones, is also rather arbitrary, therefore, one must bear in mind the conventionality, the relativity of the boundaries between them, their ability to "flow" into each other. This statement is also true for International Relations. At the same time, we must not forget that the main purpose of science is to serve practice and, ultimately, to create the basis for making decisions that are most likely to contribute to the achievement of the goal.

In this regard, based on the conclusions of R. Aron, we can say that, in fundamental terms, the study of international relations requires a combination of approaches that are based on the theory (study of the essence, specificity and main driving forces of this special kind of social relations); sociology (search for determinants and patterns that determine its changes and evolution); history (the actual development of international relations in the process of changing eras and generations, which makes it possible to find analogies and exceptions) and praxeology (analysis of the process of preparation, adoption and implementation of an international political decision). In applied terms, we are talking about the study of facts (analysis of the set of available information); explanation the current situation (search for reasons designed to avoid the undesirable and ensure the desired development of events); forecasting further evolution of the situation (study of the likelihood of its possible consequences); preparing

solutions (drawing up a list of available means of influencing the situation, evaluating various alternatives) and, finally, accepting solutions (which should also not exclude the need for an immediate response to possible changes in the situation) (5).

It is not difficult to notice the similarity of methodological approaches and even the intersection of methods inherent in both levels of the study of international relations. This is also true in the sense that in both cases, some of the methods used meet all the goals set, while others are effective only for one or another of them. Let us consider in some detail some of the methods used at the applied level of International Relations.

INTRODUCTION

CHAPTER I. MATHEMATICAL METHODS AND INTERNATIONAL RELATIONS

§ 1. Modeling of socio-economic processes

policy analysis toolkit

§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. Functional spaces and the problem of representing dependencies as a superposition of elementary

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

§6. Basic approaches to using indicator systems

to analyze foreign policy processes

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

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

§1. Information countermeasures to strategic

intelligence

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

and foreign experience

§3. Methodology for an individual assessment of the consequences of the classification of foreign policy information

§4. The use of models of national, regional and world development for the classification of information .. 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 many foreign policy

indicators

§2. Lacunar series as a toolkit in the problem of characterizing political processes (trigonometric case)

§3. Lacunar series as a toolkit in the problem of characterizing political processes (the case of the system

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

lacunar systems

§5. Application of the technique of lacunar analysis to the problems of the representability of the political process as measurable

functions on multiple indicators

CONCLUSION (summary)

APPLICATION

1. The 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

information support systems of the UN Secretariat

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

5. Solving the problem of U. Rudin of characterizing the density of lacunary sets (political indicators)

LITERATURE

Recommended list of dissertations

Development of information technologies in the foreign policy 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 for making management decisions 2007, Doctor of Technical Sciences Ryzhov, Alexander Pavlovich
Theoretical and methodological problems of the formation of the strategy of Russia's foreign policy in the conditions of the formation of the global information space 1999, Doctor of Political Science Vladimir Rostislavovich Medinsky
Mechanisms for optimizing the foreign policy of the Russian Federation in the post-Soviet space 2006, candidate of political sciences Vorozhtsova, Elena Aleksandrovna
Information Processes as a Factor in the Development of Contemporary International Relations: A Political Analysis of the Developing World 2009, Doctor of Political Science Seyidov, Shakhrutdin Gadzhialievich

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

INTRODUCTION

The mathematization of modern science is a logical and natural process. If the differentiation of scientific knowledge leads to the emergence of new branches of science, then the integration processes in the knowledge 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 proclaimed 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 in his concept of a priorism.1 While in natural science, mathematics quickly and firmly took leading positions in social sciences, its successes were more modest. The use of mathematical methods turned out to be justified where concepts are stable and the task of establishing a connection between these concepts becomes meaningful, and not an endless redefinition of the concepts themselves. While recognizing determinism in the social sphere, one should thus 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 formalized it is, can and should be the subject of the application of mathematical methods. Politicians, practitioners of foreign policy departments, international scientists, sociologists, psychologists, geographers, military men, and others are extremely interested in scientific methods of studying international relations. Empiricism in international studies, i.e. the current associated with the study of statistical information in international relations brought many different and heterogeneous methods and algorithms to the theory. There was a need for systematization and a unified approach to statistical data. International information

mation as a special type of information required specialized processing methods. In the context of the dynamic development of events in the country, the secrecy regime that has been in effect since the end of World War II turned out to be an extreme anachronism. Back in 1989, preparatory work began to create a new, more advanced information mode. The first research stage 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 the classification of foreign policy information. In the complex of the 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 security of the country. Thus, the problem of correctly describing and predicting information flows in the MFA system turned out to be among the strategic ones that are especially important for the state.

International relations, as you know, include the entire set of relations between countries, including political, economic, military, scientific, cultural, etc. Modeling is an effective toolkit for explaining and predicting the observed object under study. Representatives of the exact (natural) and humanitarian sciences put a different meaning in the concept of a model, there is a so-called methodological dichotomy, when the historical-descriptive (or intuitive-logical) approach of representatives of the humanities is opposed to the analytical and prognostic approach associated with the use of methods of exact sciences.

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

1. Formation of the law connecting the main objects of the model, which requires knowledge of the facts and phenomena related to the studied phenomena - this stage ends with the recording in mathematical terms of the formulated qualitative ideas about the connections between the objects of the model;

2. Investigation 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 conformity of the constructed model to the criterion of practice. In the event that 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 associated with the analysis of the model and its modernization in connection with the accumulation of empirical data.

There is a widespread opinion that social sciences do not have their own specific, only inherent method; therefore, in one way or another, they refract, in relation to their object, general scientific methods and methods of other sciences. The 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 de-deologize their results.

Models of economic relations between states and regions seem to us to be a fairly well-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. The description of the political behavior of states in the international arena is a poorly structured, multi-factor task that does not lend itself well to formalization. In attempts to theoretically substantiate foreign policy since the beginning of the 20th century, various ideas have been put forward, the origins of which have their origins in political life. antique greece and the Roman movement within the framework of the historical-philosophical, moral-ethical and legal approaches was called "political idealism", the synonyms of which are also the names "moralism", "normativeism", "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", the leader of which was Professor of the University of Chicago G. Morgen-tau. In an effort to get away from ideology, realists increasingly began to turn to the study of empirical data using mathematical methods. This is how the trend of "modernists" appeared, who often absolutized mathematical methods in politics as the only reliable ones. The most balanced approach was the work

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

The ideas of content analysis (analysis of the content of the text) as a method of analyzing the most frequently encountered combinations in political texts were introduced into politics by the American researcher G. LasSuele6. Event analysis (analysis of event data) assumes the presence of an extensive database with a certain systematization and processing of data matrices. The cognitive mapping method 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 constitute only a preparatory part of quantitative data processing. The first mathematical model developed for purely political science is the well-known model of the dynamics of weapons of the Scottish mathematician and meteorologist L. Richardson, first published in 19397 L. Richardson suggested that the change in the total size of the weapons of a side participating in the arms race is proportional to the available weapons of the opposite side, and the restraining factor is its own economy, which cannot withstand the endless burden of armaments. These simple considerations translated

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

TA-pVch ^ (0.

By calculating the coefficients k, 1, m, n, L. Richardson obtained surprisingly accurate agreement of the calculated data with the empirical ones on the example of World War I, 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 assess 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 X (H) be the share of mathematical methods in the total volume of research on international topics at time 1; Assuming that the growth in international relations theory research using mathematical methods is proportional to their cash share, as well as the degree of distance from saturation A, we have the differential equation:

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

The greatest successes in international studies have been achieved by methods that allow statistically processing the aggregate of foreign policy information data. Factorial 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 "A Cross-Polity Survey" project was carried out under the leadership of A.Banks and R. Textor at the Massachusetts Institute of Technology. The project "Correlates of War Project: 1918-1965", which was headed by D. Singer, is devoted to the statistical processing of voluminous information about 144 nations and 93 wars for the period 1818-1965. In the project "Dimentions of Nations", which was developed at Northwestern University, computer implementations of the methods of factor analysis of the computing centers of Indiana, Chicago and Yale Universities, etc. were used. Practical tasks for the development of analytical methods for specific situations have been repeatedly set by the US Department of State for research centers. For example, D. Kirkpatrick, US Permanent Representative to the Security Council, asked to develop a methodology by which American aid to developing countries would be clearly correlated with the results of voting at the UN General Assembly of these countries in comparison with the US position. The US Department of State also attempted, by analyzing data from an expert survey, to assess the likelihood of the seizure of the American embassy in Tehran during the well-known events. Enough full reviews on the application of mathematical methods in the theory of international relations were 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

Demiy of the Ministry of Foreign Affairs of Russia has been held since 1987. The author has constructed 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" during the development of 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 main idea of managing the flows of foreign policy information on the basis of a synthetic criterion of state power goes back to the early works of G. Morgenthau10. The indicators of the power of the state, cited 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 Ministry of Foreign Affairs of Russia A.K. Subbotin to create a model for managing information resources. The construction of mathematically correct models for managing the flows of foreign policy information using synthetic criteria seems to be a difficult task. On the one hand, the convolution of a set of single indicators into a single universal indicator, even satisfying the necessary conditions of invariance, obviously leads to the 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 can be the author's approach using the apparatus of functional 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 there may be, in particular, 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 basic indicators, these synthetic indicators are transformed covariantly, in contrast to the base ones, which are transformed contravariantly. Thus, the proposed method essentially contains a tensor approach in general systems theory, which comes from the American researcher G. Crohn.

The system of single indicators (indicators) characterizing the state or political process is the main information base for making foreign policy decisions. Making decisions on different systems of indicators leads, generally speaking, to inconsistent, if not to say 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 remedy this situation, procedures for assessing the measure of consistency of indicator samples should be developed. In the absence of such algorithms, not only the possibility of any adequate mathematical modeling in the system of international relations is questioned, 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:

what is taken in this moment we are interested in and to what it is impossible to apply any coherent theory, generalizations or unify methods? " ), describing the system of international relations, as a kind of initially existing set, which is obviously 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 is considered by us as actual or complete, or extended or existential. The infinite set is considered as existing in the form of a complete set, before and independently of any process of its generation or construction by a person, as if it completely lay before us for our review. "According to the abstraction of actual infinity in an infinite set, each of its elements can , but in fact, it is fundamentally impossible to fix and describe each element of an infinite set. Abstraction of actual infinity is an abstraction from this impossibility, "... relying on the abstraction of actual infinity, we get the opportunity to stop movement, individualize each element of an infinite set." 14 Abstraction actual infinity in mathematics has its supporters and opponents.The opposite point of view of the constructivists - the abstraction of potential infinity is based on the strict mathematical concept of the algorithm: the existence of only those objects that can but build as a result of some procedure.

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 the state standardization bodies.15 Within the framework of the task of developing procedures for agreeing the results obtained from various samples of the indicator system, the problem of space arises in the categories of which the corresponding mathematical model is built. 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 to obtain various quantitative characteristics. The spaces introduced in fact turn out to be linear normed spaces with norms of the same name, 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 in relation to linear transformations of the space itself. So, the main idea of factorial data analysis, which is most widespread in international studies, is the search for a suitable orthogonal transformation that transforms the initial set of observation vectors into another, the interpretation of the properties of which is a simpler and more visual task. Is it easy to see that the orthogonal transformation is 1? do not preserve the metric in the Minkowski spaces bp for the case p> 2, therefore the question is natural on what subspaces of the metric 1? and]> are equivalent. The problem acquires the correct formulation in the case of specific orthogonal transformations. Statement of a similar problem for a special orthogonal transformation - discrete transformation

Fourier - allows you to understand the complexity and depth of the problem. Meanwhile, it is the Fourier transform that is widely used 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 nonharmonic oscillations arising in electronic systems (Hertz dipole, microphone) require other, non-trigonometric orthogonal systems for their study, for example, the system of Walsh functions16. 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 another language, its spectral decomposition. The problem of 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 for the function to be "homogeneous" on the set of selected indicators. With a clear definition of the concept of "homogeneity" or "monogeneity", various mathematical problems arise. In particular, the correct formulation of the above-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 f (x) e b2 this function belongs to the space bp for some p> 2. For reasons of generality, one should not limit ourselves to considering only discrete Fourier transforms, since the emerging problems are common for the continual 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 presented in the following sections. A classical example of an orthogonal transformation for the case of a discrete Fourier transform is a transformation with a Hadamard matrix, therefore

the Fourier transform for the orthogonal Walsh system is also 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 means and methods for describing and defining a certain formal theory, as well as studying its properties. Metatheory is an essential part of the formalization method." In the work, in particular, it is proposed as a metatheory for the study of the system of international relations, the apparatus of finite functions and lacunar series.

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

Chapter I (Mathematical Methods and International Relations) is introductory. Section 1 gives a description of the subject area - the system of international relations and that part of it that belongs to the sphere of political relations. The article provides an overview of the development of political science and the emergence of mathematical methods in political research. The main trends in the science of international relations are considered - political idealism, political realism, empiricism, behaviorism, 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 the foreign ministries of foreign countries and Russia. §3 of the work is devoted to a critical analysis of the state of affairs with existing mathematical

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

In paragraph 7 of Chapter I, the main tasks and problems of the metatheory of international relations related to decision-making based on indicators are considered.

Chapter 2 (Models of information classification in the information resource management system in foreign policy) is devoted to the use of quantitative methods in structuring the flows of foreign policy information used in the process of making a foreign policy decision. With regard 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 gives the optimum to the power of the state. The conceptual approach to choosing the structure of indicators goes back to the works of the American

Rican researcher D.Kh. Smith as a combination of political, scientific, economic, technological and humanitarian factors. The article also examines the 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 the 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 the classification of foreign policy information. A system of models for processing expert information by 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 different types information on the individual components of the power of the state. Such estimates are based on the approach of G. Grenevsky and M. Kem-pisti about the separation of two streams - material and informational, despite the fact that Information system in politics is not only a system of movement and transformation of messages, but also a regulatory system. The power of the state acts as an object of regulation.

In Chapter III of the work (Spectral characteristics in mathematical models of the system of international relations), the metric characteristics of the target functions of the models are investigated using the apparatus of spectral analysis.

Problems. The specificity of systems of models in the theory of international relations is the use of various systems of indicators, or, in mathematical language, finite functions. Finiteness in the broad sense presupposes that the function vanishes (disappears) outside some set, the measure of which is small in relation 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 also called spectrum lacunarity. So, the lacunarity of a sound 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 a single space of political indicators) functions and their metric properties. Existing models spectral analysis using the entire spectral range is initially inaccurate, since in the real world, the spectrum of an object is lacunar. Taking into account the 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 the class of models based on the system of political indicators. The system of indicators can be associated with a formal series for the selected 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 can be investigated 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: do different formal series for a fixed system of indicators represent different functions? In the second case (the dual problem), the subject of study is the subsets on which the metrics in bp (p> 2) are equivalent to the metric br. It is obvious that the entire conceivable system of indicators is in a certain sense "overflowing" - there are many mutually dependent indicators among the indicators. 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 = 1.2, .....

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

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

Most general concept the concept of a lacunary system of order p, or simply a system, appeared in the works of S. Sidon and S. Banach. A rigorous theory of lacunary systems based on

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

Let us give the necessary definitions.

DEFINITION 1. Let an orthonormal system of functions (^ (x)) be given on a finite segment [a, b]. The system (^ (x)) is said to be a Br-system for some p> 2 if for every polynomial L (x) = X akk (x) the following estimate is valid:

(|| Ы (x) I Рех) "Р< 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 every polynomial H (x) = I a] A (x) the estimate

(/ I I (x) 12s1x) 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 hereinafter be called lacunary systems. Within the framework of considering subsystems of a fixed complete orthonomised system (Tsx)), we will adhere to the notation (nk) eA (p), or (nk) eL (2) if (nk) 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)). There is a well-known construction by U. Rudin, which 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 A (p) -set (p> 2) in any segment of length N contains at most CG \ Γ2 / p points, where the constant C> 0 does not depend on H, that is, has a density of zero power order. For the sets A (1), U. Rudin only succeeded in showing that the indicated sets do not contain arbitrarily long arithmetic progressions, therefore U. Rudin posed the question of whether the A (p) -sets have density zero for any p> 018. In 1975 the Hungarian mathematician E. Szemerédi19 gave an extremely difficult proof of the fact that sequences that do not contain arbitrarily long arithmetic progressions have a density of zero, but the density of such sequences turned out to be not of power order. In addition, both the question of estimating the density of A (p) -sets for the case of arbitrary p> 0 and the question of constructing specific dense sets that do not contain progressions or other, in a sense, regular sets, remained open. In the presented work, the hypothesis of U. Rudin found its complete solution. For the proof, we introduced the concept of a return 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 return segment, but not every return segment is a segment of an arithmetic progression, as follows from the definition:

DEFINITION 2. Let integers r, pi, wr, ..., ti be given; b> 2 such that ttz> 0, mk> nc + m2 + tz + ... + Shk-1.

Then the set of all points of the form r + lice + 821112, + .... + e5m5, where r) = 0 or 1, is called a return 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 (nk) does not contain recurrent segments of length 2П, then for any segment In of length N the following inequality holds:

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

card ((nk) n In) 0 do not depend on N. Moreover, all the 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, since the density of prime numbers has a different order from the power order. A sequence of primes has a special place in mathematics, and therefore any new result on 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 - U. Rudin showed that (k2) e A (4), but it is not clear how the situation is for other pe (0,4].

THEOREM 3. Let integers p, n> 2, and also integers

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

Then the set of all tuples 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 2П.

Using the construction used in the proof of Theorem 3, it is possible to construct sets that do not contain arithmetic progressions of length 3 - the most interesting case of sequences that do not contain a progression. The results of F. Berend20 are known in

this direction, however, they were obtained in a non-constructive way. There is also L. Moser's infinite construction21 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 applies to all indices k, n not exceeding some number N.

The trigonometric system (e> nx) is multiplicative, 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, the system of Walsh functions occupies a special place. This system is a natural completion of the well-known Rademacher system and is defined (in Paley's numbering) as follows:

w0 ^, \ ¥ n (x) = [rk + 1 (x)] ak, xe, in the case when n> 1 has the form n = where ak take the values 0 or 1, and rk (x) = sign zm (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: if A1 =] C ak2k, nz = Xbk2k, where the numbers ak, bk are equal to 0 or 1, then ns = A1 © m = X ak-bk 2k. Then for any n, w the relation \ Yn (x) "\ Ym (x) =" \ Yn © m (x) is true. It is easy to see that M2n (x) = Γn + 1 (x), n = 0,1,2 ..., but it is natural to consider other lacunary subsystems of the system of Walsh functions.

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

forms were studied by the French researcher A. Bonami22, 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 we used in the proof of Theorem 1 allows to carry over Bonami's estimates obtained by her only for the case p> 2 to the case of any p> 0. Namely,

THEOREM 4. The sets A (p), p> 0 for the Walsh-Paley system have a density of order zero, that is, the estimate is valid card ((nk) n In) 0 and its (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, each element except zero is invertible, that is, 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

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

The following is true

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 in a segment of length N (or manifolds of cardinality N) contain more than 1/2 N1 / 2 points. 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 № / 4).

The main result of Chapter 3 is 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 finite trigonometric Vinogradov sums (respectively, Walsh sums), or that the same, 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

Ux) = where e ^ are 0 or 1 and Xe ^ B

the inequality is true:

I I Щ2п pc / r) | 2<С вр^/р) 8/(8+2) (*)

k, 0< пк<р 12

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

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

(nk) eA (2 + v-p) for any p, 0< р< 2+8.

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

E8] = B, 8j are 0 or 1

the inequality is true

S | R (nk / p) | 2

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

О or 1 and Ssj-s the estimate (**) holds, then the sequence

(nk) eA (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 O or 1 (i.e., an idempotent polynomial) is directly related to the problems of coding theory. As is known, a linear (n, k) - code (к< 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 be given R (x) = EsjWj (x), where Sj are 0 or 1 and Ssj = s. To each point x of the space En we associate 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 coordinatewise multiplication. In this case, the formula R (x) = s-2 (the number of minus ones in the codeword) 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 is replaced by 1 during the addition modulo 2, then we come to the standard form of a binary code with a standard weighting function. In this case, go-

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

The following result allows one to estimate the distributions of values of idempotent Walsh polynomials using entropy estimates.

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

Then W2 (])> d22K-% 9

where Ha = - (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 for the weight of a binary code, which refine the well-known bound by S. Johnson.24

The main point that determines the interest in lacunary systems is the fact that the behavior of a lacunary series on a set of positive measure determines the behavior of a series over the entire interval of definition. In particular, there is no nontrivial lacunary (in the sense of 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 statement remains valid for any trigonometric BR-system (p> 2). At the moment it is

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

THEOREM 10. Suppose that (nk) eA (2 + e), s> 0 and the set E c is such that u.E> O. Then there exists a positive number X such that

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

for any finite polynomial H (x) = Eake "nkx.

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

THEOREM 11. Suppose that (nk) eA (2 + e), e> 0, and the set E c is such that pE> 0. In addition, let the sequence (nk) have the property nk © w - »ω for k> 1> 0. Then, for any λ> 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 goes 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 k © 1 - »ω 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 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 is carried out under much more general assumptions. The first results in this direction belong to the famous American mathematicians N. Wiener and

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. Quasianalyticity of lacunary representations, i.e. a property close to the properties of analytic functions (as is known, if a power series is equal to zero on a set with a limit point, then all its coefficients are equal to zero), manifests itself in terms of the smoothness of the functions.

DEFINITION 3. A function f (x), defined on some interval [a, b], is said to belong to the class Lip a with some cce (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 C> 0 does not depend on choice x, y... If the function f (x) satisfies the estimate:

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

sits on y, then the function f (x) is said to belong to the class Lip (2, a).

We have installed

THEOREM 12. Let the set of functions (cos nkx, sin Pkx) be an Sp-system for some p> 2 and the function f (x) e Lip (2, a) for some a> 0. 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 Hadamard series is lacunar and the function f (x) e Lip a, a> 0, then the series everywhere converges 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 IM, On rows with gaps, Matematicheskii sbornik-, nickname, 1975, vol. 98, N 4, pp. 538-563;

2. Mikheev IM, Lacunar subsystems of the Walsh function system, Siberian Mathematical Journal, 1979, No. 1, pp. 109-118;

3. Mikheev IM, On methods of optimization of the structure of technological processes, (co-author Martynov GK), Reliability and quality control, 1979, N.5;

4. Mikheev IM, Methodology for choosing the optimal version of the technological process of the production line by the method of random search using a computer, (co-author Martynov GK), Standards Publishing House, 1981

5. Mikheev IM, Methodology for estimating the parameters of nonlinear regression models of technological processes, (co-author Martynov GK), Standards Publishing House, 1981;

6. Mikheev IM, Technique for optimizing the parameters of technological systems in their design, (co-author Martynov GK), Standards Publishing House, 1981;

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

8. Mikheev IM, 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 VNIIS, issue 49, 1983, pp. 65-68;

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

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

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

13. Mikheev IM, Model of maintaining the strategic balance of forces between the USSR and the United States in the conditions of phased disarmament, In Sat. 1 "Management and informatics in foreign policy", DA USSR Ministry of Foreign Affairs, 1990, (ed. Anikin VI, Mikheev IM), pp. 40-45;

14. Mikheev IM, Methodology for predicting the results of voting in the UN, In Sat. "Management and informatics in foreign policy", DA USSR Ministry of Foreign Affairs 1990 (ed. Anikin VI, Mikheev IM), pp. 45-52;

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

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

17. Mikheev IM, Internal factors hindering the development of foreign economic relations of the USSR, (co-authors Subbotin AK, Shestakova IV, Vakhidov A.B.), Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

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

19. Mikheev IM, Using quantitative methods in forecasting world development, Moscow, Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1990;

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

21. Mikheev I.M. and others, Problems of information resources management in the USSR, (a team of authors, ed. Subbotin A.K.), Diplomatic Academy of the USSR Ministry of Foreign Affairs, 1991

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

23. Mikheev I.M., Cluster analysis methodology for assessing and making 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 IM, Research of information support of international relations using functional spaces, Materials of the 4th international conference "Informatization of security systems ISB-95" of the International Forum of Informatization, Moscow, November 17, 1995, pp. 20-22;

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

26. Mikheev IM, Mathematics of border studies, Collection of articles of the Department of border studies of the International Academy of Informatization, vol. 2, M., Department of Borderology of the Moscow Aviation Institute, 1996, pp. 116-119

The total volume of the thesis, 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 support of the UN Secretariat ( Appendix 3). There are also listings of programs for processing voting results in the UN (Appendix 4) and the solution of U. Rudin's problem on the density of lacunary sets (Appendix 5).

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

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Conclusion of the thesis 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 above results 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 the group structure by the operation of symmetric difference, which made it possible to apply the theory of characters of Abelian groups and linear transformations (first of all, the discrete Fourier transform with the Hadamard matrix). This method, in contrast to the traditional methods of convolution (averaging) of single criteria, does not lead to the loss of the original information.

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

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

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

List of dissertation research literature Doctor of physico-mathematical sciences Mikheev, Igor Mikhailovich, 1997

LITERATURE

1 see N.A. Kiseleva, Mathematics and Reality, M., Moscow State University, 1967, p. 107

2 A.N. Tikhonov, Mathematical model, see Encyclopedia of Mathematics, vol. 3, pp. 574-575

3 see O. Holsti, An Adaptation of the "General Inquier" for Systematic Análisis of Political Documents, Behavior Science, 1964, v. nine

4 see C. Mc. Clelland, The Management and Analysis of International Event Date: A Computerised System for Monitoring and Projecting Event Flows. University of Southern California, Los Angeles, 1971; Ph. Burgess, Indicators of International Behavior: an Assessment of Events Date Research, L., 1972

5 see M. Bonham, M. Shapiro, Cognitive Processes and Political Decision-Making, International Studies Quarterly, 1973, v. 47, p. 147-174

6 H. Lasswell, N. Leites, The Language of Politics: Studies in Quantitative Semantics, N.Y., 1949

7 L. Richardson, Generalized Forein Politics, British Journal of Psychology: Monograf Supplement, vol. 23, Cambridge, 1939; see also A. Rappoport, F. Levis, Richardsons Mathematical Theory of War, The Journal of Conflict Resolution, September 1957, N.l

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

9 M. Ward, (ed.), Theories, Models and Simulations in International Relations, N.Y., 1985

10 H. Morgenthau, Politics Among Nations: The Strugle for Power, 4th .. ed., N.Y., 1967

11 D.H. Smith, Values of Transnational Associations, Intern. Trans. Assoc., 1980, No. 5, 245-258; N. 6-7, 302-309

12 M. Kaplan, Is International Relations a Discipline?, The Journal of Politics, 1961, v. 23, No. 3

13 S. Kleene, Introduction to Metamathematics, M.b I.L., 1957, p. 49

14 P.S. Novikov, Elements of mathematical logic, M., Fizmatgiz, 1950, p. 80

15 cm. Selection of the nomenclature of indicators of the quality of industrial products, GOST 22851-77; Selection and standardization of reliability indicators, GOST 230003-83

16 cm. H.F. Harmut, Information transfer by orthogonal functions, M., 1975

17 A.G. Dragalin, Metatheory, Encyclopedia of Mathematics, 1982, vol. 3, p. 651

18 W. Rudin, Trigonometric series with gaps, Journal of Mathematics and Mechanics, vol. 9, No. 2 (1960), p. 217

19 E. Szemeredi, On sets of integers containing no k-elements of arithmetic progression, Acta Arith., 27 (1975), 199-245

20 F.A. Berend, On sets of integers which contain no three terms in arithmetic progression, Proc. Nat. Acad. Sci. USA 32 (1946) 331-332

21L. Moser, On non -averaging sets of integers, Canad. J. of Math., 5 (1953), 245-252

22 A. Bonami, Ensemles A (p) dans le dual de D °°, Ann. Inst. Fourier, Grenoble, 18, 2 (1968), 193-204; 20.2 (1970), 335-402

23 Ph. Delsart, Weight of linear codes and strongly regular normed spased, Disc. Math. , 3 (1972), 47-64

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25 A. Zigmund, Trigonometric series, Cambridge University Press, 1959, v. 1,2

26 see J.-P. Kahane, Lacunary Taylor and Fourier Series, Bull. Amer. Math. Soc., 70, N. 2, (1964), 199-213

27 P.B. Kennedy, On the coefficient in certain Fourier series, J. London Math. Soc., 33 (1958), p. 206

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 Science 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. Prospects, problems, methods, M., 1972, p. 204

35 History of political doctrines, M., 1994, 139 p.

36 Ibid., P. 4

37 Ibid, p. 14

38 Political Science Dictionary, M., 1994, part 2, p. 73

39 P.A. Tsygankov, Political Sociology of International Relations, M., Radix, 1994, p. 72

40 S.V. Melikhov, Quantitative methods in American Political Science, M., Science, 1979, p. 3

41 Ibid., P. 4

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

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

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

48 N. Wiener, I am a mathematician, M., Nauka, 1964, pp. 29-30

49 A. D. Aleksandrov, General view of mathematics, collection of articles. "Mathematics, its content, method and meaning", v.1, Ed. AN SSSR, 1956, pp. 59, 68

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., Science, 1979

61 V.M. Zhukovskaya, I.B. Muchnik, Factor analysis in socio-economic research, M., Statistics, 1976

62 Quantitative methods in the study of political processes, comp. Sergiev A.V., M., Progress, 1972

63 Questions of foreign policy forecasting, ref. collection, M., INION, 1980

64 Modern Western Theories of International Relations, ref. collection, M., INION, 1982

65 G.A. Satarov, Multidimensional scaling, Interpretation and analysis of data in sociological research, M., Nauka, 1987

66 G.A. Satarov, S.B. Stankevich, Ideological Delimitation in the US Congress, Sociological Research, 1982, N 2

67 S.I. Lobanov, Practical experience of quantitative analysis (using a computer) of the voting results of the UN member states: methodological aspects, in collection of articles. "A systematic approach: analysis and forecasting of international relations, M., MGIMO, 1991, pp. 33-50

68 V.P. Akimov, Modeling and mathematical methods in the study of international relations, in the book. "Political sciences and scientific and technological revolution", M., Science, 1987, pp. 193-205

69 M.A. Khrustalev, Systemic modeling of international relations, abstract for the degree of Doctor of Political Science, M., MGIMO, 1991

70 International Research, Scientific Information Bulletin, No. 3, otv. ed. E.I. Skakunov, 1990

71 Quantitative Methods in Soviet and American Historiography, M. Nauka, 1983 (ed. By I. Kovalchenko)

72 Quantitative methods in foreign historical science (historiography 70-80 years). Scientific and analytical review, M., INION, 1988

73 Problems of information resource management in the USSR, a team of authors, otv. ed. Subbotin A.K., M., 1991

74 M. Ward, (ed.) Theories, models and simulation 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., P. 14,15

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

79 see, for example, Thomas L. Saati, Mathematical Models of 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 processes, 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, B. 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 thery of social action with application to conflict within nation, Genaral Systems Yearbook, 1965, v. ten

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., Radix, 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., Science, 1976, pp. 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., P. 314, 308

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

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

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

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

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

105 A.A. Kokoshin, Interuniversity Consortium for Policy Studies, United States of America, No. 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, No. 6, pp. 12-17

108 B.C. Kretov, I.E. Vlasov, B.JI. Dudikhin, I. V. Frolov, Some aspects of creating a system of information support for decision-making by operational-diplomatic officers of the Ministry of Foreign Affairs of the Russian Federation, Scientific and technical information, ser. 1, 1994, No. 6, pp. 18-22

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

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

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

112 A.B. Grishin, Fundamental problems creation of "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.B.), M., Progress, 1972

114 A. Dutta, Reasoning with imprecite knowlage in expert systems, Inf. Sei. (USA), 1985, v. 37, N. 1-3, p. 3-34

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

116 Courant and Robbins, "What is Mathematics", M., 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, "Architecture of Mathematics", in the book "N. Bourbaki, Essays on the History of Mathematics, Moscow, IL, 1963

125 A.A. Lyapunov, "On the foundation and style of modern mathematics", Mathematical education, 1960, N 5

K.E. 126 Plohotnikov, 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, R. 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, Massachusets, 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., Science, 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, Concept of creating an intellectual system for monitoring political conflicts, M., Scientific Research Center of the Ministry of Foreign Affairs of the Russian Federation,

138 B.B. Dudikhin, I.P. Belyaev, Application of modern information technologies to analyze 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

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

142 cm. Methods for assessing the level of quality 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 cm.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, Moscow, Nauka, 1979

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

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

152 Ibid., P. 58

153 Problems of Information Resources Management 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 information resources management 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 cm, for example, Fujii Haruo, Nikonno kokka kimitsu (Japanese state secret), Tokyo, 1972; Kimitsu Hogo To Gendai (Defense of Secrets and Modernity), Tokyo, 1983.

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

170 R. Wynn, C. Holden, Introduction to Applied Econometric Analysis, M., 1971

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

173 See EZ 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. Wynn, K. Holden, Introduction to Applied Econometric Analysis, M., 1981, p. 112

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

175 A. D. Ursul, Informatization of Society (Introduction to Social Informatics), Tutorial, 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.BC Dadayan), M., Economics, 1984

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

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

183 R. Hillsman, Strategic Intelligence and Political Decisions, M., IL, 1959, p. 7

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

185 R. Hillsman, Strategic Intelligence and Political Decisions, M., IL, 1959, pp. 19-20

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

187 see 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 standart, 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; Utilitas Math. , 1 (1972), 121-140

202I. Okun, Factor analysis, M., 1974, p. 112 203 Agaev, N. Ya. Vilenkin, G.M. Jafarli, A.I. Rubinstein, 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 fonktions 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, no. 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, 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 V. F. Gaposhkin, On lacunary series in multiplicative systems of functions, Siberian Mathematical Journal, 1971, 12, number 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 and G. Steingauz, Theory of orthogonal series, Moscow, Fiz-matgiz, 1958

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

226 A. Bonami, Ensemles L (p) danse le dual de D00, Ann. Inst. Fourier, 18 (1969), # 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. Steingauz, Theory of orthogonal series, Moscow, Fiz-matgiz, 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. fiz.-mat.-nauk, 3 (1961), 35-41

234 P.L. Ul'yanov, Solved and unsolved problems in the theory of trigonometric and orthogonal series, UMN, 19 (1964), no. 1, 3-69

235 G. Polia and G. Szege, 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!

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

259 P. Erdos, P. Turan, On some sequences of integers, J. London Math. Soc., 11 (1936), 261-264

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241 E. Szemeredi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sei. Hungar., 20 (1969), 89-104

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244 F.A. Behrend, On sets of integers which contain no three terms in arithmetical progressions, Proc. Nat. Acad. Sei., USA, 32 (1946), 331-332

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247 W. Rudin, Trigonometric series with gaps, J. Math. Mech., 9 (1960), 203227

249 I.M. Mikheev, On rows with gaps, Mat. collection, 98 (1975), 537-563

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annotation scientific article on political science, the author of the scientific work - Dzera M.M., Pasichny R.Ya.

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