docx - Mathematical Cybernetics. Mathematical cybernetics.docx - mathematical cybernetics Cybernetics in the ussr

He called it the science of effective organization, and Gordon Pask expanded the definition to include flows of information “from any source,” from the stars to the brain.

According to another definition of cybernetics, proposed in 1956 by L. Kuffignal (eng.), one of the pioneers of cybernetics, cybernetics is "the art of ensuring the effectiveness of action."

Another definition suggested by Lewis Kaufman (eng.): "Cybernetics is the study of systems and processes that interact with themselves and reproduce themselves."

Cybernetic methods are used to study the case when the action of the system in the environment causes some change in the environment, and this change is manifested on the system through feedback, which causes changes in the way the system behaves. The study of these "feedback loops" is the essence of the methods of cybernetics.

Modern cybernetics was born, including research in various areas of control systems, the theory of electrical circuits, mechanical engineering, mathematical modeling, mathematical logic, evolutionary biology, neurology, anthropology. These studies appeared in 1940, mainly in the works of scientists on the so-called. Macy conferences (eng.).

Other areas of research that influenced the development of cybernetics or were influenced by it: control theory, game theory, systems theory (a mathematical analogue of cybernetics), psychology (especially neuropsychology, behaviorism, cognitive psychology) and philosophy.

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Sphere of cybernetics

All controlled systems are the object of cybernetics. Systems that cannot be controlled, in principle, are not objects of study in cybernetics. Cybernetics introduces concepts such as the cybernetic approach, the cybernetic system. Cybernetic systems are considered abstractly, regardless of their material nature. Examples of cybernetic systems are automatic controllers in technology, computers, the human brain, biological populations, and human society. Each such system is a set of interconnected objects (system elements) capable of perceiving, memorizing and processing information, as well as exchanging it. Cybernetics develops general principles for creating control systems and systems for the automation of mental work. The main technical means for solving problems of cybernetics are computers. Therefore, the emergence of cybernetics as an independent science (N. Wiener, 1948) is associated with the creation of these machines in the 40s of the XX century, and the development of cybernetics in theoretical and practical aspects - with the progress of electronic computing technology.

Complex systems theory

Complex systems theory analyzes the nature of complex systems and the reasons underlying their unusual properties.

A method for modeling a complex adaptive system

In computing

In computing, cybernetics is used to control devices and analyze information.

In engineering

Cybernetics in engineering is used to analyze system failures, where small errors and flaws can cause an entire system to fail.

In economics and management

In mathematics

In psychology

In sociology

History

In ancient Greece, the term "cybernetics", originally denoting the art of the helmsman, began to be used figuratively to refer to the art of the statesman who ruled the city. In this sense, he, in particular, is used by Plato in the "Laws".

James Watt

The first artificial automatic regulating system, the water clock, was invented by the ancient Greek mechanic Ctesibius. In his water clock, water flowed from a source, such as a stabilizing tank, into a pool, then from the pool onto the clock mechanisms. The Ktesibius device used a cone-shaped flow to control the water level in its tank and adjust the water flow rate accordingly to maintain a constant water level in the tank so that it was neither overfilled nor drained. It was the first truly automatic self-adjusting artificial device that did not require any external interference between feedback and control mechanisms. Although they naturally did not refer to this concept as a science of cybernetics (they considered it a field of engineering), Ctesibius and other masters of antiquity, such as Heron of Alexandria or the Chinese scientist Su Song, are considered among the first to study cybernetic principles. Research on mechanisms in corrective feedback machines dates back to the late 18th century, when James Watt's steam engine was equipped with a control device, a centrifugal feedback controller, in order to control the speed of the engine. A. Wallace described feedback as "essential to the principle of evolution" in his famous 1858 work. In 1868, the great physicist J. Maxwell published a theoretical article on control devices, one of the first to consider and improve the principles of self-regulating devices. J. Ikskul applied the feedback mechanism in his model of the functional cycle (Funktionskreis) to explain the behavior of animals.

XX century

Modern cybernetics began in the 1940s as an interdisciplinary field of research, combining control systems, electrical circuit theory, mechanical engineering, logical modeling, evolutionary biology, and neurology. Electronic control systems date back to the work of Bell Labs engineer Harold Black in 1927 using negative feedback to control amplifiers. The ideas are also related to the biological work of Ludwig von Bertalanffy in general systems theory.

Cybernetics as a scientific discipline was based on the work of Wiener, McCulloch, and others such as W.R. Ashby and W.G. Walter.

Walter was one of the first to build autonomous robots to aid in animal behavior research. Along with Great Britain and the United States, France was an important geographic location for early cybernetics.

Norbert Wiener

During this stay in France, Wiener received an offer to write an essay on the unification of this part of applied mathematics, which is found in the study of Brownian motion (the so-called Wiener process) and in the theory of telecommunications. The following summer, in the United States, he used the term "cybernetics" as the title of a scientific theory. This title was intended to describe the study of "purposeful mechanisms" and was popularized in the book Cybernetics, or Control and Communication in the Animal and Machine (Hermann & Cie, Paris, 1948). In Great Britain, the Ratio Club was formed around this in 1949 (eng.).

Cybernetics in the USSR

Dutch sociologists Geyer and Van der Zouven in 1978 identified a number of features of the emerging new cybernetics. “One of the features of the new cybernetics is that it considers information as built and restored by a person interacting with the environment. This provides the epistemological foundation of science when viewed from the point of view of the observer. Another feature of the new cybernetics is its contribution to overcoming the problem of reduction (contradictions between macro- and microanalysis). Thus, it connects the individual to the society. " Geyer and Van der Zouven also noted that “the transition from classical cybernetics to new cybernetics leads to a transition from classical problems to new ones. These changes in thinking include, among others, changes from an emphasis on the governed system to the governing one and the factor that directs governing decisions. And a new emphasis on communication between multiple systems that try to control each other. "

Famous teachers

• L. A. Petrosyan - Doctor of Physics and Mathematics, Professor, Professor of the Department of Mathematical Game Theory and Static Solutions. Research Area: Mathematical Game Theory and Its Applications
• A. Yu. Aleksandrov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Biomedical Systems Management. Scientific supervision: qualitative methods of the theory of dynamical systems, stability theory, control theory, theory of nonlinear oscillations, mathematical modeling
• SN Andrianov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Computer Modeling and Multiprocessor Systems. Scientific direction: mathematical and computer modeling of complex dynamic systems with control
• LK Babadzhanyants - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Mechanics of Controlled Motion. Area of \u200b\u200bscientific leadership: mathematical problems of analytical and celestial mechanics, space dynamics, the theory of existence and continuity of the solution of the Cauchy problem for ordinary differential equations, stability theory and controlled motion, numerical methods for solving ill-posed problems, creating packages of applied programs
• VM Bure - Doctor of Technical Sciences, Associate Professor, Professor of the Department of Mathematical Game Theory and Static Solutions. Scientific leadership: probabilistic and statistical modeling, data analysis
• E. Yu. Butyrskiy - Doctor of Physics and Mathematics, Professor, Professor of the Department of Control Theory, St. Petersburg State University. Area of \u200b\u200bAcademic Leadership: Management Theory
• EI Veremey - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Computer Technologies and Systems. Scientific supervision: development of mathematical methods and computational algorithms for optimizing control systems and methods of their computer modeling
• EV Gromova - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Game Theory and Statistical Decisions. Research area: game theory, differential games, cooperative game theory, applications of game theory in management, economics and ecology, mathematical statistics, statistical analysis in medicine and biology
• OI Drivotin - Doctor of Physical and Mathematical Sciences, Senior Researcher, Professor of the Department of Theory of Control Systems for Electrophysical Equipment. Scientific leadership: modeling and optimization of the dynamics of beams of charged particles, theoretical and mathematical problems of classical field theory, some problems of mathematical physics, computer technologies in physical problems
• NV Egorov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Modeling Electromechanical and Computer Systems. Scientific leadership: information-expert and intelligent systems, mathematical, physical and natural modeling of the structural elements of computing devices and electromechanical systems, diagnostic systems based on electron and ion beams, emission electronics and physical aspects of methods for monitoring and controlling the properties of a solid surface
• A. P. Zhabko - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Control Theory. Scientific supervision: differential-difference systems, robust stability, analysis and synthesis of plasma control systems
• V. V. Zakharov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Mathematical Modeling of Energy Systems. Scientific leadership: optimal control, game theory and applications, operations research, applied mathematical (intelligent) logistics, traffic theory
• NA Zenkevich - Associate Professor of the Department of Mathematical Game Theory and Statistical Decisions. Research area: game theory and its applications in management, theory of conflict-controlled processes, quantitative methods of decision-making, mathematical modeling of economic and business processes
• A. V. Zubov - Doctor of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Theory of Microprocessor Control Systems. Research Area: Database Management and Optimization
• AM Kamachkin - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Higher Mathematics. Scientific supervision: qualitative methods of the theory of dynamic systems, theory of nonlinear oscillations, mathematical modeling of nonlinear dynamic processes, theory of nonlinear automatic control systems
• VV Karelin - candidate of physical and mathematical sciences, associate professor, associate professor of the department of mathematical theory of modeling control systems. Scientific direction: identification methods; nonsmooth analysis; observability; adaptive control
• A. N. Kvitko - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Information Systems. Scientific direction: boundary value problems for controlled systems; stabilization, methods of optimization of program movements, motion control of aerospace complexes and other technical objects, development of algorithms for computer-aided design of intelligent control systems
• V. V. Kolbin - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Mathematical Theory of Economic Decisions. Scientific direction: mathematical
• VV Kornikov - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Biomedical Systems Management. Scientific leadership: stochastic modeling in biology, medicine and ecology, multivariate statistical analysis, development of mathematical methods for multi-criteria assessment and decision-making under uncertainty, decision-making systems in financial management problems, mathematical methods for analyzing non-numerical and incomplete information, Bayesian models of uncertainty and risk
• ED Kotina - Doctor of Physical and Mathematical Sciences, Associate Professor, Professor of the Department of Control Theory. Scientific leadership: differential equations, control theory, mathematical modeling, optimization methods, analysis and formation of the dynamics of charged particle beams, mathematical and computer modeling in nuclear medicine
• D. V. Kuzyutin - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Game Theory and Statistical Decisions. Scientific direction: mathematical game theory, optimal control, mathematical methods and models in economics and management
• GI Kurbatova - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Modeling Electromechanical and Computer Systems. Scientific leadership: nonequilibrium processes in the mechanics of inhomogeneous media; computer fluid dynamics in the Maple environment, problems of gradient optics, problems of modeling the transportation of gas mixtures through offshore pipelines
• OA Malafeev - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Modeling of Socio-Economic Systems. Area of \u200b\u200bscientific leadership: modeling competitive processes in the socio-economic sphere, research of nonlinear dynamic conflict-controlled systems
• SE Mikheev - Doctor of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Theory of Modeling Control Systems, St. Petersburg State University. Area of \u200b\u200bscientific leadership: nonlinear programming, acceleration of convergence of numerical methods, simulation of oscillations and sound perception by the human ear, differential games, management of economic processes
• VD Nogin - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Control Theory. Area of \u200b\u200bscientific leadership: theoretical, algorithmic and applied issues of the theory of decision making in the presence of several criteria
• A. D. Ovsyannikov - Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Programming Technology. Scientific supervision: computer modeling, computational methods, modeling and optimization of the dynamics of charged particles in accelerators, modeling and optimization of plasma parameters in tokamaks
• DA Ovsyannikov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Theory of Control Systems for Electrophysical Equipment. Scientific leadership: control of charged particle beams, control under uncertainty conditions, mathematical methods for optimizing accelerating and focusing structures, mathematical methods for controlling electrophysical equipment
• IV Olemskoy - Doctor of Physics and Mathematics, Associate Professor, Professor of the Department of Information Systems. Research area: numerical methods for solving ordinary differential equations
• A. A. Pechnikov - Doctor of Technical Sciences, Associate Professor, Professor of the Department of Programming Technology. Scientific leadership: webometrics, problem-oriented systems based on web technologies, multimedia information systems, discrete mathematics and mathematical cybernetics, software systems and models, mathematical modeling of social and economic processes
• LN Polyakova - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Mathematical Theory of Modeling Control Systems. Scientific guidance: nonsmooth analysis, convex analysis, numerical methods for solving nonsmooth optimization problems (minimization of the maximum function, the difference of convex functions), the theory of multivalued mappings
• AV Prasolov - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Modeling Economic Systems. Scientific leadership: mathematical modeling of economic systems, statistical forecasting methods, differential equations with aftereffect
• S. L. Sergeev - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Programming Technology. Scientific leadership: integration and application of modern information technologies, automated control, computer modeling
• MA Skopina - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Higher Mathematics. Scientific leadership: wavelet theory, harmonic analysis, function approximation theory
• G. Sh. Tamasyan - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Theory of Modeling Control Systems. Scientific leadership: nonsmooth analysis, nondifferentiable optimization, convex analysis, numerical methods for solving nonsmooth optimization problems, calculus of variations, control theory, computational geometry
• SI Tarashnina - Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematical Game Theory and Statistical Decisions. Research Area: Mathematical Game Theory, Cooperative Games, Pursuit Games, Statistical Data Analysis
• I.B.Tokin - Doctor of Biological Sciences, Professor, Professor of the Departments of Management of Medical and Biological Systems. Scientific leadership: modeling the effect of radiation on mammalian cells; analysis of metastable states of cells, processes of autoregulation and repair of damaged cells, mechanisms of restoration of tissue systems under external influences; human ecology
• A. Yu. Uteshev - Doctor of Physics and Mathematics, Professor, Professor of the Department of Biomedical Systems Management. Scientific guidance: symbolic (analytical) algorithms for systems of polynomial equations and inequalities; computational geometry; computational aspects of number theory, coding, encryption; qualitative theory of differential equations; tasks of the optimal location of enterprises (facility location)
• V.L. Kharitonov - Doctor of Physical and Mathematical Sciences, Professor of the Department of Control Theory. Scientific leadership: control theory, equations with lagging argument, stability and robust stability
• S. V. Chistyakov - Doctor of Physics and Mathematics, Professor of the Department of Mathematical Game Theory and Statistical Decisions of St. Petersburg State University. Research area: optimal control theory, game theory, mathematical methods in economics
• V.I.Shishkin - Doctor of Medical Sciences, Professor, Professor of the Department of Diagnostics of Functional Systems. Field of scientific leadership: mathematical modeling in biology and medicine, the use of mathematical models for the development of diagnostic methods and prognosis of diseases, computer support in medicine, mathematical modeling of technological processes for the production of element base for medical diagnostics devices
• AS Shmyrov - Doctor of Physics and Mathematics, Professor, Professor of the Department of Mechanics of Controlled Motion, St. Petersburg State University. Scientific leadership: optimization methods in space dynamics, qualitative methods in Hamiltonian systems, approximation of distribution functions, methods of countering comet-asteroid hazard

Academic partners

• Institute of Mathematics and Mechanics named after N. N. Krasovsky, Ural Branch of the Russian Academy of Sciences (Yekaterinburg)
• V. A. Trapeznikov Institute of Control Sciences RAS (Moscow)
• Institute for Applied Mathematical Research of the Karelian Scientific Center of the Russian Academy of Sciences (Petrozavodsk)

Projects and grants

Implemented within the program

• rFBR grant 16-01-20400 "Project for the organization of the Tenth International Conference" Game Theory and Management "(GTM2016)", 2016. Leader - L. A. Petrosyan
• sPbSU grant 9.38.245.2014 "Optimality principles in dynamic and differential games with a fixed and variable coalition structure", 2014–2016. Leader - L.A. Petrosyan
• sPbSU grant 9.38.205.2014 "New constructive approaches in nonsmooth analysis and nondifferentiable optimization and their applications", 2014–2016. Leader - V. F. Demyanov, L. N. Polyakova
• sPbSU grant 9.37.345.2015 “Control of the orbital motion of celestial bodies in order to counteract the comet-asteroid hazard”, 2015–2017. Leader - L.A. Petrosyan
• rFBR grant No. 14-01-31521_mol_a "Inhomogeneous approximations of nonsmooth functions and their applications", 2014–2015. Head - G. Sh. Tamasyan

Implemented with partner universities

• jointly with Qingdao University (China) - 17-51-53030 "Rationality and Sustainability in Network Games", from 2017 to the present. Leader - L.A. Petrosyan

Key points

• The program consists of educational and research components. The educational component includes the study of academic disciplines, including methods of mathematical cybernetics, discrete mathematics, control system theory, mathematical programming, mathematical theory of operations research and game theory, mathematical theory of recognition and classification, mathematical theory of optimal control, and teaching practice. The curriculum provides for a set of optional disciplines, allowing graduate students to form an individual training schedule. The task of the research component of training is to obtain results, the scientific value and novelty of which allows publication in scientific journals included in the scientometric bases of the RSCI, WoS and Scopus
• The mission of this educational program is to train highly qualified personnel capable of critical analysis and assessment of modern scientific achievements, generating new ideas when solving research and practical problems, including in interdisciplinary areas.
• Graduates who have mastered the program:
  • know how to design and carry out complex research, including interdisciplinary, based on a holistic systemic scientific worldview
  • ready to participate in the work of Russian and international research teams to solve urgent scientific and scientific and educational problems and to use modern methods and technologies of scientific communication in the state and foreign languages
  • are able to plan and solve problems of their own professional and personal development, independently carry out research activities in the relevant professional field using modern research methods and information and communication technologies, as well as be ready for teaching activities in the main educational programs of higher education

None N / A

The collection continues (since 1988) the mathematical direction of the world famous series "Problems of Cybernetics". The collection includes original and review articles on the main directions of world science, containing the latest results of fundamental research.

The authors of the collection are mainly well-known specialists, some of the articles were written by young scientists who have recently received bright new results. Among the directions presented in the collection are the theory of synthesis and complexity of control systems; problems of expressibility and completeness in the theory of functional systems related to multivalued logics and automata; fundamental issues of discrete optimization and recognition; problems of extremal problems for discrete functions (problems of Fejer, Turan, Delsarte on a finite cyclic group); research of mathematical models of information transmission in communication networks; a number of other branches of mathematical cybernetics are also presented.

Special mention should be made of the review article by O.B Lupanov “A. N. Kolmogorov and the theory of circuit complexity ”. Issue 16 - 2007 For specialists, graduate students, students interested in the current state of mathematical cybernetics and its applications.

Information storage and retrieval theory

Valery Kudryavtsev Educational literature Absent

A new type of database representation is introduced, called the information-graph data model, generalizing the previously known models. The main types of problems of information retrieval in databases are considered and problems of the complexity of solving these problems are investigated in relation to the information-graph model.

A mathematical apparatus has been developed for solving these problems, based on the methods of the complexity theory of control systems, probability theory, as well as on the original methods of characteristic carriers of a graph, optimal decomposition and dimensionality reduction.

The book is intended for specialists in the field of discrete mathematics, mathematical cybernetics, recognition theory and algorithmic complexity.

Test recognition theory

Valery Kudryavtsev Educational literature Absent

A logical approach to pattern recognition is described. Its main concept is a test. Analysis of a set of tests allows one to build functionals that characterize the image and procedures for calculating their values. Qualitative and metric properties of tests, functionals and recognition procedures are indicated.

The results of solving specific problems are presented. The book can be recommended to mathematicians, cybernetics, computer scientists and engineers as a scientific monograph and as a new technological apparatus, as well as a textbook for undergraduate and graduate students specializing in mathematical cybernetics, discrete mathematics and mathematical informatics.

Problems in set theory, mathematical logic and theory of algorithms

Igor Lavrov Educational literature Absent No data

In the book, in the form of tasks, the foundations of set theory, mathematical logic and the theory of algorithms are systematically presented. The book is intended for the active study of mathematical logic and related sciences. Consists of three parts: "Set theory", "Mathematical logic" and "Theory of algorithms".

Tasks are provided with instructions and answers. All the necessary definitions are formulated in brief theoretical introductions to each paragraph. The third edition of the book was published in 1995. The collection can be used as a textbook for mathematical departments of universities, pedagogical institutes, as well as in technical universities in the study of cybernetics and informatics.

For mathematicians - algebraists, logicians and cybernetics.

Foundations of the theory of boolean functions

Sergey Marchenkov Technical literature Absent No data

The book contains an extensive introduction to the theory of Boolean functions. The basic properties of Boolean functions are stated and a criterion for functional completeness is proved. A description of all closed classes of Boolean functions (Post classes) is given and a new proof of their finite geneability is given.

The definition of Post classes in terms of some standard predicates is considered. Foundations of Galois theory for Post classes are presented. Two “strong” closure operators are introduced and investigated: parametric and positive. Partial Boolean functions are considered and a criterion for functional completeness is proved for the class of partial Boolean functions.

The complexity of the implementation of Boolean functions by circuits of functional elements is investigated. For undergraduate, graduate students and high school teachers studying and teaching discrete mathematics and mathematical cybernetics. Approved by UMO for classical university education as a textbook for students of higher educational institutions studying in the areas of HPE 010400 "Applied Mathematics and Informatics" and 010300 "Fundamental Informatics and Information Technologies".

Numerical optimization methods 3rd ed., Rev. and add. Academic Bachelor's textbook and workshop

Alexander Vasilievich Timokhov Educational literature Bachelor. Academic course

The textbook is written on the basis of courses of lectures on optimization, which were given by the authors for a number of years at the Faculty of Computational Mathematics and Cybernetics of the Lomonosov Moscow State University. The main attention is paid to methods of minimizing functions of a finite number of variables.

The publication includes theory and numerical methods for solving optimization problems, as well as examples of applied models, which are reduced to this type of mathematical problems. The appendix contains all the necessary information from mathematical analysis and linear algebra.

Physics. Practical course for university applicants

V. A. Makarov Educational literature Absent

The manual is intended for students in graduating classes of secondary schools with advanced study of physics and mathematics. It is based on problems in physics that have been offered to applicants at the Faculty of Computational Mathematics and Cybernetics of Moscow State University over the past 20 years.

M.V. Lomonosov. The material is divided into topics in accordance with the program of entrance examinations in physics for applicants to Moscow State University. Each topic is preceded by a short summary of the basic theoretical information that is necessary to solve the problems and will be useful in preparing for the entrance exams.

In total, the collection includes about 600 problems, over half of them are supplied with detailed solutions and methodological instructions. For students preparing to enter the physics and mathematics departments of universities.

Optimization methods 3rd ed., Rev. and add. Academic Baccalaureate Textbook and Workshop

Vyacheslav Vasilievich Fedorov Educational literature Bachelor and Master. Academic course

The textbook is written on the basis of courses of lectures on optimization, which for a number of years were read by the authors at the Faculty of Computational Mathematics and Cybernetics of the Moscow State University. M.V. Lomonosov. The main attention is paid to methods of minimizing functions of a finite number of variables.

The edition includes tasks. The appendix contains all the necessary information from mathematical analysis and linear algebra.

Intelligent systems. Theory of storage and retrieval of information, 2nd ed., Rev. and add. Tutorial for tank

The main types of problems of information retrieval in databases are considered, problems of the complexity of solving these problems are investigated in relation to the information-graph model.

Analytic geometry

V. A. Ilyin Educational literature Absent No data

The textbook was written based on the teaching experience of the authors at the Moscow State University. M.V. Lomonosov. The first edition was published in 1968, the second (1971) and third (1981) stereotyped editions, the fourth edition (1988) was supplemented with material devoted to linear and projective transformations.

Mathematical game theory is an integral part of the vast branch of mathematics - operations research. Game theory methods are widely used in ecology, psychology, cybernetics, biology - wherever many participants pursue different (often opposite) goals in joint activities.

But the main area of \u200b\u200bapplication of this discipline is economics and social sciences. The textbook includes topics that are basic and required in the training of economists. It presents the classic sections of game theory, such as matrix, bimatrix noncooperative and statistical games, and modern developments, for example, games with incomplete and imperfect information, cooperative and dynamic games.

The theoretical material in the book is widely illustrated with examples and provided with assignments for individual work, as well as tests.

CYBERNETICS, the science of management, which studies mainly by mathematical methods the general laws of receiving, storing, transferring and transforming information in complex control systems. There are other, slightly different definitions of cybernetics. Some are based on the information aspect, others - algorithmic, in others the concept of feedback is highlighted, as expressing the specificity of cybernetics. In all definitions, however, the task of studying by mathematical methods of control systems and processes and information processes is necessarily indicated. A complex control system in cybernetics is understood as any technical, biological, administrative, social, ecological or economic system. Cybernetics is based on the similarity of control and communication processes in machines, living organisms and their populations.

The main task of cybernetics is the study of general laws underlying control processes in various environments, conditions, and areas. These are, first of all, the processes of transmission, storage and processing of information. At the same time, control processes take place in complex dynamic systems - objects that have variability and the ability to develop.

Historical sketch... It is believed that the word "cybernetics" was first used by Plato in the dialogue "Laws" (4th century BC) to mean "management of people" [from the Greek ϰυβερνητιϰή - the art of managing, from here come the Latin words gubernare (to govern) and gubernator (governor) ]. In 1834 A. Ampere in his classification of sciences used this term to denote "the practice of government." The term was introduced into modern science by N. Wiener (1947).

The cybernetic principle of automatic control based on feedback was implemented in automatic devices by Ctesibius (circa 2nd - 1st century BC; float water clock) and Heron of Alexandria (around 1st century AD). During the Middle Ages, many automatic and semi-automatic devices were created, used in watch and navigation mechanisms, as well as in watermills. Systematic work on the creation of teleological mechanisms, that is, machines demonstrating expedient behavior, equipped with corrective feedback, began in the 18th century in connection with the need to regulate the operation of steam engines. In 1784, J. Watt patented a steam engine with an automatic regulator, which played an important role in the transition to industrial production. The beginning of the development of the theory of automatic control is considered to be an article by J.C. Maxwell on regulators (1868). IA Vyshnegradskiy is considered to be the founder of the theory of automatic regulation. In the 1930s, in the works of I.P. Pavlov, a comparison of the brain and electrical switching circuits was outlined. PK Anokhin studied the activity of the organism on the basis of the theory of functional systems developed by him, in 1935 he proposed the so-called method of reverse afferentation - a physiological analogue of feedback in controlling the behavior of the organism. The finally necessary prerequisites for the development of mathematical cybernetics were created in the 1930s by the works of A.N. Kolmogorov, V.A.Kotelnikov, E.L. Post, A.M. Turing, A. Church.

The need to create a science dedicated to the description of control and communication in complex technical systems in terms of information processes and providing the possibility of their automation was realized by scientists and engineers during the Second World War. Complex systems of weapons and other technical means, command and control of troops and their supply in theaters of military operations have increased attention to the problems of automation of command and control and communications. The complexity and variety of automated systems, the need to combine various means of control and communication in them, the new possibilities created by computers have led to the creation of a unified, general theory of control and communication, a general theory of information transfer and transformation. These tasks, to one degree or another, required the description of the studied processes in terms of collecting, storing, processing, analyzing and evaluating information and obtaining a management or predictive decision.

Since the beginning of the war, N. Wiener (together with the American designer W. Bush) participated in the development of computing devices. In 1943, he began developing a computer together with J. von Neumann. In this regard, meetings were held at the Princeton Institute for Advanced Study (USA) in 1943-44 with the participation of representatives of various specialties - mathematicians, physicists, engineers, physiologists, and neurologists. Here the Wiener-von Neumann group was finally formed, which included scientists W. McCulloch (USA) and A. Rosenbluth (Mexico); the work of this group made it possible to formulate and develop cybernetic ideas in relation to real technical and medical problems. Wiener summed up these studies in his book Cybernetics, published in 1948.

A significant contribution to the development of cybernetics was made by N. M. Amosov, P. K. Anokhin, A. I. Berg, E. S. Bir, V. M. Glushkov, Yu. V. Gulyaev, S. V. Emelyanov, Yu. I. Zhuravlev, A. N. Kolmogorov, V. A. Kotelnikov, N. A. Kuznetsov, O. I. Larichev, O. B. Lupanov, A. A. Lyapunov, A. A. Markov, J. von Neumann , B. N. Petrov, E. L. Post, A. M. Turing, Ya. Z. Tsypkin, N. Chomsky, A. Church, K. Shannon, S. V. Yablonsky, as well as domestic scientists M. A Aizerman, V. M. Akhutin, B. V. Biryukov, A. I. Kitov, A. Ya. Lerner, Viach. Viach. Petrov, Ukrainian scientist A.G. Ivakhnenko.

The development of cybernetics was accompanied by its absorption of individual sciences, scientific directions and their sections and, in turn, the emergence in cybernetics and the subsequent separation from it of new sciences, many of which formed functional and applied sections of informatics (in particular, pattern recognition, image analysis, artificial intelligence). Cybernetics has a rather complex structure, and in the scientific community there is no full agreement on the directions and sections that are its integral parts. The interpretation proposed in this article is based on the traditions of Russian schools of informatics, mathematics and cybernetics and on provisions that do not cause serious disagreements between leading scientists and specialists, most of whom agree that cybernetics is devoted to information, the practice of its processing and technology related to information systems; studies the structure, behavior and interaction of natural and artificial systems that store, process and transmit information; develops his own conceptual and theoretical foundations; has computational, cognitive and social aspects, including the social significance of information technology, since computers, individuals and organizations process information.

Since the 1980s, there has been a slight decline in interest in cybernetics. It is associated with two main factors: 1) during the formation of cybernetics, the creation of artificial intelligence seemed to many to be a simpler task than it was in reality, and the prospect of its solution related to the foreseeable future; 2) on the basis of cybernetics, having inherited its basic methods, in particular mathematical ones, and almost completely absorbing cybernetics, a new science arose - informatics.

The most important research methods and communication with other sciences. Cybernetics is an interdisciplinary science. It arose at the intersection of mathematics, the theory of automatic regulation, logic, semiotics, physiology, biology and sociology. The formation of cybernetics took place under the influence of trends in the development of mathematics itself, the mathematization of various fields of science, the penetration of mathematical methods into many areas of practical activity, and the rapid progress of computer technology. The process of mathematization was accompanied by the emergence of a number of new mathematical disciplines, such as algorithmic theory, information theory, operations research, game theory, which constitute an essential part of the apparatus of mathematical cybernetics. On the basis of the problems of the theory of control systems, combinatorial analysis, graph theory, and coding theory, discrete mathematics arose, which is also one of the main mathematical tools of cybernetics. In the early 1970s, cybernetics emerged as a physical and mathematical science with its own subject of research - the so-called cybernetic systems. A cybernetic system consists of elements; in the simplest case, it can also consist of one element. A cybernetic system receives an input signal (representing the input signals of its elements), has internal states (that is, sets of internal states of elements are defined); By processing the input signal, the system converts the internal state and produces an output signal. The structure of a cybernetic system is set by a set of relationships connecting input and output signals of elements.

In cybernetics, the tasks of analysis and synthesis of cybernetic systems are essential. The task of the analysis is to find the properties of information transformation carried out by the system. The task of synthesis is to build a system according to the description of the transformation that it must carry out; the class of elements that the system can consist of is fixed. Of great importance is the problem of finding cybernetic systems defining the same transformation, that is, the problem of the equivalence of cybernetic systems. If we set the functional of the quality of the work of cybernetic systems, then the problems arise of finding the best system in the class of equivalent cybernetic systems, that is, the system with the maximum value of the quality functional. In cybernetics, problems of the reliability of cybernetic systems are also considered, the solution of which is aimed at increasing the reliability of the functioning of systems by improving their structure.

For fairly simple systems, the listed problems can usually be solved by classical means of mathematics. Difficulties are caused by the analysis and synthesis of complex systems, which in cybernetics mean systems that do not have simple descriptions. These are usually the cybernetic systems studied in biology. The direction of research, for which the name "the theory of large (complex) systems" is fixed, has been developing in cybernetics since the 1950s. In addition to complex systems in living nature, complex systems of production automation, economic planning systems, administrative and economic systems, and military systems are studied. Research methods of complex control systems form the basis of systems analysis and operations research.

To study complex systems in cybernetics, both an approach using mathematical methods and an experimental approach using various experiments are used either with the object under study or with its real physical model. The main methods of cybernetics include algorithmization, the use of feedback, the method of machine experiment, the method of "black box", the systems approach, formalization. One of the most important achievements of cybernetics is the development of a new approach - a mathematical modeling method. It consists in the fact that experiments are carried out not with a real physical model, but with a computer implementation of the model of the object under study, built according to its description. This computer model, including programs that implement changes in the parameters of an object in accordance with its description, is implemented on a computer, which makes it possible to carry out various experiments with the model, register its behavior under various conditions, change certain structures of the model, etc.

The theoretical basis of cybernetics is mathematical cybernetics, devoted to the methods of studying wide classes of cybernetic systems. Mathematical cybernetics uses a number of branches of mathematics, such as mathematical logic, discrete mathematics, probability theory, computational mathematics, information theory, coding theory, number theory, automata theory, complexity theory, and mathematical modeling and programming.

Depending on the field of application in cybernetics, there are: technical cybernetics, including the automation of technological processes, the theory of automatic control systems, computer technology, the theory of computers, automatic design systems, the theory of reliability; economic cybernetics; biological cybernetics, including bionics, mathematical and machine models of biosystems, neurocybernetics, bioengineering; medical cybernetics, which deals with the management process in medicine and health care, the development of simulation and mathematical models of diseases, the automation of diagnostics and treatment planning; psychological cybernetics, including the study and modeling of mental functions based on the study of human behavior; physiological cybernetics, including the study and modeling of the functions of cells, organs and systems in conditions of norm and pathology for the purposes of medicine; linguistic cybernetics, including the development of machine translation and communication with computers in natural language, as well as structural models for processing, analyzing and evaluating information. One of the most important achievements of cybernetics is the identification and formulation of the problem of modeling human thinking processes.

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Yu. I. Zhuravlev, I.B. Gurevich.

Mathematical modeling capabilities

For any object of modeling, qualitative and quantitative characteristics are inherent. Mathematical modeling gives preference to identifying quantitative features and patterns of systems development. This modeling is largely abstracted from the specific content of the system, but it necessarily takes it into account, trying to display the system through the apparatus of mathematics. The truth of mathematical modeling, like mathematics in general, is verified not by correlation with a specific empirical situation, but by the fact that it can be derived from other sentences.

Mathematical modeling is a vast area of \u200b\u200bintellectual activity. This is a rather complex process of creating a mathematical description of the model. It includes several stages. NP Buslenko identifies three main stages: the construction of a meaningful description, a formalized scheme and the creation of a mathematical model. In our opinion, mathematical modeling consists of four stages:

first - a meaningful description of an object or process, when the main components of the system, the laws of the system are identified. It includes the numerical values \u200b\u200bof known characteristics and parameters of the system;

second - the formulation of an applied task or the task of formalizing a meaningful description of the system. An applied problem contains a statement of research ideas, basic dependencies, as well as a formulation of a question, the solution of which is achieved by formalizing the system;

third - construction of a formalized scheme of an object or process, which implies the choice of the main characteristics and parameters that will be used in formalization;

fourth - transformation of a formalized scheme into a mathematical model, when the creation or selection of the corresponding mathematical functions is in progress.

An extremely important role in the process of creating a mathematical model of the system is played by formalization, which is understood as a specific method of research, the purpose of which is to clarify knowledge by identifying its form (method of organization, structure as a connection between content components). The formalization procedure involves the introduction of symbols. As A. K. Sukhotin notes: "To formalize a certain content area means to build an artificial language in which concepts are replaced by symbols, and utterances - by combinations of symbols (formulas). A calculus is created when one can get others from one symbolic combination according to fixed rules." At the same time, thanks to formalization, such information is revealed that is not captured at the levels of meaningful analysis. It is clear that formalization is difficult in relation to complex systems characterized by the richness and variety of connections.

After the creation of a mathematical model, its application begins to study some real process. In this case, the set of initial conditions and the required quantities is first determined. There are several ways to work with the model: its analytical study by means of special transformations and problem solving; the use of numerical methods of solution, for example, the method of statistical tests or the Monte Carlo method, methods of simulation of random processes, as well as through the use of computer technology for modeling.

In mathematical modeling of complex systems, the complexity of the system must be taken into account. As N.P. Buslenko rightly notes, a complex system is a multilevel structure of interacting elements, combined into subsystems of various levels. The mathematical model of a complex system consists of mathematical models of elements and mathematical models of the interaction of elements. The interaction of elements is usually considered as a result of the totality of the effects of each element on other elements. The impact, represented by a set of its characteristics, is called signal.Therefore, the interaction of elements of a complex system is studied within the framework of the signal exchange mechanism. Signals are transmitted through communication channels located between the elements of a complex system. They have inputs and outputs.

dy. When constructing a mathematical model of the system, its interaction with the external environment is taken into account. In this case, the external environment is usually presented in the form of a certain set of objects that affect the elements of the system under study. Significant difficulty is the solution of such problems as displaying quality transitions of elements and systems from one state to another, displaying transient processes.

According to N.P.Buslenko, the signal exchange mechanism as a formalized scheme of interaction of elements of a complex system with each other or with objects of the external environment includes the following components:

a process for generating an output signal by a signal generating element;

determination of the transmission address for each characteristic of the output signal;

passing signals through communication channels and arranging input signals for elements that receive signals;

the response of the element receiving the signal to the incoming signal.

Thus, through successive stages of formalization, "cutting" the original problem into parts, the process of constructing a mathematical model is carried out.

Features of cybernetic modeling

The foundations of cybernetics were laid by the famous American philosopher and mathematician, professor at the Massachusetts Institute of Technology Norbert Wiener (1894-1964) in the work "Cybernetics, or Control and Communication in an Animal and a Machine" (1948). The word "cybernetics" comes from the Greek word for "helmsman". The great merit of N. Wiener is that he established the commonality of the principles of management activity for fundamentally different objects of nature and society. Management is reduced to the transfer, storage and processing of information, i.e. to various signals, messages, information. The main merit of N. Wiener lies in the fact that he was the first to understand the fundamental importance of information in management processes. Nowadays, according to Academician A. N. Kolmogorov, cybernetics studies systems of any nature that are capable of perceiving, storing and processing information and using it for control and regulation.

There is a certain spread in the definition of cybernetics as a science, in the allocation of its object and subject. According to the position of Academician A. I. Berg, cybernetics is the science of managing complex dynamical systems. The categorical apparatus of cybernetics is based on such concepts as "model", "system", "control", "information". The ambiguity of the definitions of cybernetics is due to the fact that different authors place emphasis on one or another basic category. For example, the emphasis on the category "information" forces us to consider cybernetics as the science of the general laws of obtaining, storing, transferring and transforming information in complex controlled systems, and the preference for the category "control" - as the science of modeling the control of various systems.

Such ambiguity is quite legitimate, because it is due to the polyfunctionality of cybernetic science, its fulfillment of diverse roles in knowledge and practice. At the same time, the focusing of interests on one function or another makes one see the whole science in the light of this function. This flexibility of cybernetic science speaks of its high cognitive potential.

Modern cybernetics is a heterogeneous science (Fig. 21). It combines a set of sciences that study control in systems of various natures from a formal standpoint.

As noted, cybernetic modeling is based on a formal mapping of systems and their components using the concepts of "input" and "output", which characterize the relationship of an element with the environment. Moreover, each element is characterized by a certain number of "inputs" and "outputs" (Fig. 22).

Figure: 22.Cybernetic representation of an element

In fig. 22 X 1 , X 2 , ... X M schematically shown: "inputs" of an element, Y 1 , Y 2 , ..., У Н - "outputs" of the element, and FROM 1 , С 2, ..., С К - his states. The flows of matter, energy, information affect the "inputs" of the element, form on its states and ensure functioning at the "outputs". The quantitative measure of the interaction of "input" and "output" is intensity, which is, respectively, the amount of matter, energy, information per unit of time. Moreover, this interaction is continuous or discrete. Now you can build mathematical functions that describe the behavior of an element.

Cybernetics regards the system as a unity of control and control elements. The managed items are called the managed object, and the controls are called the governing system. The structure of the control system is based on a hierarchical principle. The controlling system and the controlled (object) are interconnected by direct and feedback links (Fig. 23), and in addition, by communication channels. The control system through the direct communication channel acts on the controlled object, correcting the effects of the environment on it. This leads to a change in the state of the control object and it changes its impact on the environment. Note that the feedback can be external, as shown in Fig. 23, or internal, which ensures the internal functioning of the system, its interaction with the internal environment.

Cybernetic systems are a special kind of system. As L.A. Petrushenko notes, the cybernetic system

a theme satisfies at least three requirements: "1) it must have a certain level of organization and a special structure; 2) therefore be able to perceive, store, process and use information, that is, represent an information system; 3) have control according to the feedback principle. A cybernetic system is a dynamic system, which is a collection of communication channels and objects and has a structure that allows it to extract (perceive) information from its interaction with the environment or another system and use this information for self-management according to the feedback principle. "

A certain level of organization means:

integration into the cybernetic system of the controlled and control subsystems;

hierarchy of the control subsystem and the fundamental complexity of the controlled subsystem;

the presence of deviations of the controlled system from the goal or from equilibrium, which leads to a change in its entropy. This predetermines the need to develop a managerial impact on it from the control system.

Information is the basis of the cybernetic system, which perceives, processes and transmits it. Information is information, knowledge of the observer about the system, a reflection of its measure of diversity. It defines the connections between the elements of the system, its "input" and "output". The informational nature of the cybernetic system is due to:

The need to obtain information about the impact of the environment on the controlled system;

the importance of information about the behavior of the system;

the need for information about the structure of the system.

Various aspects of the nature of information have been studied N. Wiener, K. Shannon, W. R. Ashby, L. Brillouin, A. I. Berg, V. M. Glushkov, N. M. Amosov, A. N. Kolmogorov etc. Philosophical Encyclopedic Dictionary gives the following interpretation of the term "information": 1) message, awareness of the state of affairs, information about something transmitted by people; 2) reduced, removable uncertainty as a result of receiving a message; 3) a message inextricably linked with control, a signal in the unity of syntactic, semantic and pragmatic characteristics; 4) transmission, reflection of diversity in any objects and processes (inanimate and living nature).

The most important properties of information include:

adequacy, those. compliance with real processes and objects;

relevance, those. compliance with the tasks for which it is intended;

right, those. correspondence of the way of expressing information to its content;

accuracy, those. reflecting relevant phenomena with minimal distortion or minimal error;

relevance or timeliness, those. the possibility of its use when the need for it is especially great;

universality, those. independence from individual private changes;

degree of detail, those. detail of information.

Any cybernetic system consists of elements that are connected by information flows. It contains information resources, receives, processes and transfers information. The system exists in a certain information environment and is subject to information noise. Its most important problems include: prevention of distortion of information during transmission and reception (the problem of children playing in the "deaf phone"); creation of an information language that would be understandable to all participants in management relations (communication problem); effective search, receipt and use of information in management (problem of use). The complex of these problems acquires a certain originality and diversity in

depending on the specifics of control systems. Thus, in the information systems of government bodies, as noted by NR Nizhnik and OA Mashkov, there is a need to resolve the following problems: creation of a service for information resources of government and government bodies; creation of a legal basis for its functioning; formation of infrastructure; creation of an information monitoring system; creation of an information service system.

Feedback is a kind of connection of elements, when the connection between the input of an element and the output of the same element is carried out either directly or through other elements of the system. Feedbacks are internal and external (Fig. 24).

Feedback management is a complex process that includes:

constant monitoring of system functioning;

comparison of the current functioning of the system with the goals of the system;

development of impact on the system to bring it in line with the goal;

introduction of impact into the system.

Feedbacks can be positive and negative. In this case, positive feedback enhances the action of the input signal, has the same sign with it. Negative feedback attenuates the input signal. Positive feedback worsens the stability of the system because it throws it out of balance, and negative feedback helps to restore equilibrium in the system.

An important role in cybernetic modeling is played by the concept of "black", "gray" and "white" boxes. A "black box" refers to a cybernetic system (object, process, phenomenon), regarding the internal organization, structure and behavior of the elements of which the observer (researcher) has no information, but it is possible to influence the system through its inputs and register its reactions at the output. In the process of manipulating the input and fixing the results at the input, the observer draws up a test report, the analysis of which makes it possible to lighten the "black box", i.e. to get an idea of \u200b\u200bits structure and patterns of transformation of the "input" signal into the "output" signal. Such a clarified box is called a "gray box", which, however, does not give a complete picture of its contents. If the observer fully represents the content of the system, its structure and signal conversion mechanism, then it turns into a "white box".

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