"System approach in modeling" - Process - dynamic change of the system in time. System - a set of interrelated elements that form integrity or unity. Peter Ferdinand Drucker. Systems approach in organizations. A systematic approach as the basis for the introduction of specialized education. The founders of the system approach: Structure is a way of interaction between the elements of the system through certain connections.

"ISO 20022" - Elements of the methodology of the international standard. Comparison of composition and properties. Appointment. Modeling process. Features of the methodology. Simulation results. openness and development. Migration. Title of the International Standard. Aspects of universality. Tools. Activity. The composition of documents.

"The concept of model and modeling" - Types of models by branches of knowledge. Types of models. Basic concepts. Types of models depending on the time. Types of models depending on the external dimensions. Model adequacy. Figurative-sign models. The need to create models. Modeling. Models modeling.

"Models and Modeling" - Changing the size and proportions. A mathematical model is a model presented in the language of mathematical relations. A block diagram is one of the special varieties of a graph. Analysis of an object. Structural model - representation of the information sign model in the form of a structure. Real phenomenon. Abstract. Verbal.

"Model development steps" - Descriptive information models are usually built using natural languages ​​and drawings. Building a descriptive information model. The main stages of development and research of models on the computer. Stage 4. Stage 1. Stage 5 Model of the solar system. Practical task. Stage 3. Stage 2.

"Modeling as a method of cognition" - In biology - the classification of the animal world. Definitions. Definition. In physics, it is an information model of simple mechanisms. Modeling as a method of cognition. Forms of representation of information models. Tabular model. The process of building information models using formal languages ​​is called formalization.

There are 18 presentations in total in the topic

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A mathematical model is a mathematical representation of reality, one of the variants of a model, as a system, the study of which allows obtaining information about some other system. The process of building and studying mathematical models is called mathematical modeling. All natural and social sciences that use the mathematical apparatus are, in fact, engaged in mathematical modeling: they replace the object of study with its mathematical model and then study the latter. The connection of a mathematical model with reality is carried out with the help of a chain of hypotheses, idealizations and simplifications. With the help of mathematical methods, as a rule, an ideal object is described, built at the stage of meaningful modeling. General information

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No definition can fully cover the real-life activity of mathematical modeling. Despite this, definitions are useful in that they attempt to highlight the most significant features. According to Lyapunov, mathematical modeling is an indirect practical or theoretical study of an object, in which not the object of interest to us is directly studied, but some auxiliary artificial or natural system (model) that is in some objective correspondence with the object being known, capable of replacing it in certain respects. and giving, in its study, ultimately, information about the modeled object itself. In other versions, the mathematical model is defined as an object-substitute of the original object, providing the study of some properties of the original, as "an" equivalent "of the object, reflecting in mathematical form its most important properties - the laws to which it obeys, the connections inherent in its constituent parts", as a system of equations, or arithmetic relations, or geometric figures, or a combination of both, the study of which by means of mathematics should answer the questions posed about the properties of a certain set of properties of a real world object, as a set of mathematical relations, equations, inequalities that describe the basic patterns inherent the process, object or system being studied. Definitions

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The formal classification of models is based on the classification of the mathematical tools used. Often built in the form of dichotomies. For example, one of the popular sets of dichotomies is: Linear versus non-linear models; Concentrated or distributed systems; Deterministic or stochastic; Static or dynamic; Discrete or continuous and so on. Each constructed model is linear or non-linear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed models in another, etc. Formal classification of models

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Along with the formal classification, the models differ in the way they represent the object: Structural or functional models. Structural models represent an object as a system with its own device and functioning mechanism. Functional models do not use such representations and reflect only the externally perceived behavior (functioning) of the object. In their extreme expression, they are also called "black box" models. Combined types of models are also possible, sometimes referred to as gray box models. Mathematical models of complex systems can be divided into three types: Black box models (phenomenological), Gray box models (a mixture of phenomenological and mechanistic models), White box models (mechanistic, axiomatic). Schematic representation of black box, gray box and white box models

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Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction, a meaningful model, is built. There is no established terminology here, and other authors call this ideal object a conceptual model, a speculative model, or a premodel. In this case, the final mathematical construction is called a formal model or simply a mathematical model obtained as a result of the formalization of this content model (pre-model). A meaningful model can be built using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. provide ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other areas), the creation of meaningful models becomes much more complicated. Content and formal models

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Peierls' work gives a classification of mathematical models used in physics and, more broadly, in the natural sciences. In the book by A. N. Gorban and R. G. Khlebopros, this classification is analyzed and expanded. This classification is focused primarily on the stage of constructing a meaningful model. Hypothesis Models of the first type - hypotheses ("this could be"), "represent a trial description of the phenomenon, and the author either believes in its possibility, or even considers it to be true." According to Peierls, these are, for example, the Ptolemy model of the solar system and the Copernican model (improved by Kepler), the Rutherford model of the atom and the Big Bang model. Model-hypotheses in science cannot be proven once and for all, one can only talk about their refutation or non-refutation as a result of the experiment. If a model of the first type is built, then this means that it is temporarily recognized as true and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of the model of the first type can only be temporary. Phenomenological model The second type, the phenomenological model (“behave as if…”), contains a mechanism for describing the phenomenon, although this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or is poorly consistent with the available theories and accumulated knowledge about the object. . Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown, and the search for "true mechanisms" must continue. Peierls refers, for example, the caloric model and the quark model of elementary particles to the second type. The role of the model in research may change over time, it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Similarly, new knowledge can gradually come into conflict with models-hypotheses of the first type, and they can be transferred to the second. Meaningful classification of models

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Thus, the quark model is gradually moving into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history it passed into the first type. But the ether models have gone from type 1 to type 2, and now they are outside of science. The idea of ​​simplification is very popular when building models. But simplification is different. Peierls distinguishes three types of simplifications in modeling. Approximation The third type of models is approximations (“we consider something very large or very small”). If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (models of type 3). Among them are linear response models. The equations are replaced by linear ones. The standard example is Ohm's law. If we use the ideal gas model to describe sufficiently rarefied gases, then this is a type 3 model (approximation). At higher gas densities, it is also useful to imagine a simpler situation with an ideal gas for qualitative understanding and evaluation, but then this is already type 4. Simplification noticeable and not always controllable effect on the result. The same equations can serve as a model of type 3 (approximation) or 4 (we omit some details for clarity) - this depends on the phenomenon for which the model is used to study. So, if linear response models are used in the absence of more complex models (that is, non-linear equations are not linearized, but linear equations describing the object are simply searched), then these are already phenomenological linear models, and they belong to the following type 4 (all non-linear details " omitted for clarity). Examples: application of an ideal gas model to a non-ideal one, the van der Waals equation of state, most models of solid state, liquid and nuclear physics. The path from microdescription to the properties of bodies (or media) consisting of a large number of particles, Meaningful classification of models (continued)

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very long. Many details have to be left out. This leads to models of the fourth type. Heuristic model The fifth type is the heuristic model (“there is no quantitative confirmation, but the model contributes to a deeper insight into the essence of the matter”), such a model retains only a qualitative similarity to reality and gives predictions only “in order of magnitude”. A typical example is the mean free path approximation in kinetic theory. It gives simple formulas for the coefficients of viscosity, diffusion, thermal conductivity, consistent with reality in order of magnitude. But when building a new physics, it is far from immediately obtained a model that gives at least a qualitative description of an object - a model of the fifth type. In this case, a model is often used by analogy, reflecting reality at least in some way. Analogy The sixth type is an analogy model (“let's take into account only some features”). Peierls gives a history of the use of analogies in Heisenberg's first paper on the nature of nuclear forces. Thought experiment The seventh type of models is the thought experiment (“the main thing is to refute the possibility”). This type of simulation was often used by Einstein, in particular, one of these experiments led to the construction of the special theory of relativity. Suppose that in classical physics we follow a light wave at the speed of light. We will observe an electromagnetic field periodically changing in space and constant in time. According to Maxwell's equations, this cannot be. From here, Einstein concluded: either the laws of nature change when the frame of reference changes, or the speed of light does not depend on the frame of reference, and chose the second option. Possibility demonstration The eighth type is the possibility demonstration (“the main thing is to show the internal consistency of the possibility”), such models are also thought experiments with imaginary entities, demonstrating that the alleged phenomenon is consistent with the basic principles and Meaningful classification of models (continued)

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internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions. One of the most famous such experiments is Lobachevsky's geometry. (Lobachevsky called it "imaginary geometry".) Another example is the mass production of formal kinetic models of chemical and biological vibrations, autowaves. The Einstein - Podolsky - Rosen paradox was conceived as a thought experiment to demonstrate the inconsistency of quantum mechanics, but in an unplanned way over time turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information. The substantive classification is based on the stages preceding mathematical analysis and calculations. Eight types of models according to Peierls are eight types of research positions in modeling. Meaningful classification of models (continued)

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actually useless. Often, a simpler model allows you to better and deeper explore the real system than a more complex (and, formally, “more correct”) one. If we apply the harmonic oscillator model to objects that are far from physics, its meaningful status may be different. For example, when applying this model to biological populations, it should most likely be attributed to type 6 analogy (“let's take into account only some features”). Example (continued)

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The most important mathematical models usually have an important property of universality: fundamentally different real phenomena can be described by the same mathematical model. For example, a harmonic oscillator describes not only the behavior of a load on a spring, but also other oscillatory processes, often of a completely different nature: small oscillations of a pendulum, fluctuations in the liquid level in a U-shaped vessel, or a change in the current strength in an oscillatory circuit. Thus, studying one mathematical model, we study at once a whole class of phenomena described by it. It is this isomorphism of the laws expressed by mathematical models in various segments of scientific knowledge that led Ludwig von Bertalanffy to create a “general systems theory”. Universality of models

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There are many problems associated with mathematical modeling. First, it is necessary to come up with the basic scheme of the object being modeled, to reproduce it within the framework of the idealizations of this science. So, a train car turns into a system of plates and more complex bodies made of different materials, each material is given as its standard mechanical idealization (density, moduli of elasticity, standard strength characteristics), after which equations are drawn up, along the way some details are discarded as insignificant, calculations are made, compared with measurements, the model is refined, and so on. However, for the development of mathematical modeling technologies, it is useful to disassemble this process into its main constituent elements. Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse. Direct task: the structure of the model and all its parameters are considered known, the main task is to study the model in order to extract useful knowledge about the object. What static load can the bridge withstand? How it will react to a dynamic load (for example, to the march of a company of soldiers, or to the passage of a train at different speeds), how the plane will overcome the sound barrier, whether it will fall apart from flutter - these are typical examples of a direct task. Setting the correct direct problem (asking the correct question) requires special skill. If the right questions are not asked, the bridge can collapse, even if a good model has been built for its behavior. So, in 1879 in the UK, a metal railway bridge across the River Tey collapsed, the designers of which built a model of the bridge, calculated it for a 20-fold margin of safety against the payload, but forgot about the winds constantly blowing in those places. And after a year and a half it collapsed. In the simplest case (one oscillator equation, for example), the direct problem is very simple and reduces to an explicit solution of this equation. Inverse problem: many possible models are known, it is necessary to choose a specific model based on additional data Direct and inverse problems of mathematical modeling

Algorithm drawing up a mathematical model:

• Make a brief statement of the problem statement:

A) find out how many quantities are involved in the task;

B) identify the relationship between these quantities.

2. Make a drawing for the problem (in problems of movement or problems of geometric content) or a table.

3. Designate one of the values ​​​​for X (better, a smaller value).

4. Taking into account the connections, make a mathematical model.

Problem 1. (No. 86 (1)).

The apartment consists of 3 rooms with a total area of ​​42 sq.m. The first room is 2 times smaller than the second, and the second is 3 square meters. m more than a third. What is the area of ​​each room in this apartment?

Task 2. (No. 86 (2)).

Sasha paid 11200 rubles for the book, pen and notebook. A pen is 3 times more expensive than a notebook and 700 r. cheaper than a book. How much does a notebook cost?

Problem 3. (No. 86 (3)).

A motorcyclist traveled a distance between two cities equal to

980 km, in 4 days. On the first day he covered 80 km less than on the second day, on the third day he covered half the distance covered in the first two days, and on the fourth day he covered the remaining 140 km. How far did the motorcyclist travel on the third day?

Problem 4. (No. 86 (4))

The perimeter of a quadrilateral is 46 in. Its first side is 2 times smaller than the second and 3 times smaller than the third side, and the fourth side is 4 cm larger than the first side. What are the lengths of the sides of this quadrilateral?

Problem 5. (No. 87)

One of the numbers is 17 less than the second, and their sum is 75. Find the largest of these numbers.

Problem 6. (No. 99)

20 participants performed in three parts of the concert. In the second section there were 3 times fewer participants than in the first, and in the third section - 5 participants more than in the second. How many participants in the concert performed in each section?

I can (or not):

Skills

Points

0 or 1

Reveal the number of quantities involved in the task

Reveal relationships between quantities

I understand what it means

B) "everything"

I can make a mathematical model

I can create a new problem for a given mathematical model

Homework:

1) № 87 (2, 3, 4), № 102 (1);

2) Compose a problem for the mathematical model of the problem

Mathematical model- this is a set of mathematical objects and relationships between them, adequately reflecting the properties and behavior of the object under study.

Mathematics in the most general sense deals with the definition and use of symbolic models. A mathematical model covers a class of undefined (abstract, symbolic) mathematical objects such as numbers or vectors, and the relationships between these objects.

A mathematical relation is a hypothetical rule relating two or more symbolic objects. Many relationships can be described using mathematical operations that relate one or more objects to another object or to a set of objects (the result of an operation). The abstract model, with its objects of an arbitrary nature, relations and operations, is defined by a consistent set of rules that introduce operations that can be used and establish general relationships between their results. The constructive definition introduces a new mathematical model, using already known mathematical concepts (for example, the definition of addition and multiplication of matrices in terms of addition and multiplication of numbers).

A mathematical model will reproduce appropriately selected aspects of a physical situation if a correspondence rule can be established relating specific physical objects and relations to certain mathematical objects and relations. It can also be instructive and/or interesting to build mathematical models for which there are no analogues in the physical world. The most commonly known mathematical models are systems of integers and real numbers and Euclidean geometry; the defining properties of these models are more or less direct abstractions of physical processes (counting, ordering, comparison, measurement).

Objects and operations of more general mathematical models are often associated with sets of real numbers, which can be correlated with the results of physical measurements.

Mathematical modeling is a method of qualitative and (or) quantitative description of a process using the so-called mathematical model, in the construction of which a real process or phenomenon is described using one or another adequate mathematical apparatus. Mathematical modeling is an integral part of modern research.

Mathematical modeling is a typical discipline located, as it is now often said, at the "junction" of several sciences. An adequate mathematical model cannot be built without deep knowledge of the object that is “served” by the mathematical model. Sometimes an illusory hope is expressed that a mathematical model can be created jointly by a mathematician who does not know the object of modeling, and a specialist in the “object” who does not know mathematics. For successful activity in the field of mathematical modeling, it is necessary to know both mathematical methods and the object of modeling. This is connected, for example, with the presence of such a specialty as a theoretical physicist, whose main activity is mathematical modeling in physics. The division of specialists into theoreticians and experimenters, which has been established in physics, will undoubtedly occur in other sciences, both fundamental and applied.

Due to the variety of applied mathematical models, their general classification is difficult. In the literature, classifications are usually given, which are based on various approaches. One of these approaches is related to the nature of the process being modeled, when deterministic and probabilistic models are distinguished. Along with such a widespread classification of mathematical models, there are others.

Classification of mathematical models based on the features of the applied mathematical apparatus . It includes the following varieties.

Usually, such models are used to describe the dynamics of systems consisting of discrete elements. From the mathematical side, these are systems of ordinary linear or non-linear differential equations.

Mathematical models with lumped parameters are widely used to describe systems consisting of discrete objects or sets of identical objects. For example, the dynamic model of a semiconductor laser is widely used. In this model, two dynamic variables appear - the concentrations of minor charge carriers and photons in the active zone of the laser.

In the case of complex systems, the number of dynamic variables and, consequently, differential equations can be large (up to 102 ... 103). In these cases, various methods of system reduction are useful, based on the temporal hierarchy of processes, assessing the influence of various factors and neglecting the insignificant among them, etc.

The method of successive extension of the model can lead to the creation of an adequate model of a complex system.

Models of this type describe the processes of diffusion, heat conduction, propagation of waves of various nature, etc. These processes can be not only of a physical nature. Mathematical models with distributed parameters are widely used in biology, physiology and other sciences. Most often, the equations of mathematical physics, including nonlinear ones, are used as the basis of a mathematical model.

The fundamental role of the principle of greatest action in physics is well known. For example, all known systems of equations describing physical processes can be derived from extremal principles. However, in other sciences extremal principles play an essential role.

The extremal principle is used when approximating empirical dependencies by an analytical expression. The graphic representation of such a dependence and the specific form of the analytical expression describing this dependence is determined using the extremal principle, called the least squares method (Gauss method), the essence of which is as follows.

Let an experiment be carried out, the purpose of which is to study the dependence of some physical quantity Y from physical quantity x. It is assumed that the values x and y linked by functional dependency

The form of this dependence needs to be determined from experience. Let us assume that as a result of the experiment, we obtained a number of experimental points and built a dependence graph at from X. Usually, the experimental points on such a graph are not located quite correctly, they give some spread, i.e., they reveal random deviations from the visible general pattern. These deviations are associated with measurement errors that are inevitable in any experiment. Then the problem of smoothing the experimental dependence, which is typical for practice, arises.

To solve this problem, a calculation method is usually used, known as the method of least squares (or the Gauss method).

Of course, the listed varieties of mathematical models do not exhaust the entire mathematical apparatus used in mathematical modeling. The mathematical apparatus of theoretical physics and, in particular, its most important section, elementary particle physics, is especially diverse.

The areas of their application are often used as the main principle of the classification of mathematical models. With this approach, the following areas of application are distinguished:

physical processes;

technical applications, including controlled systems, artificial intelligence;

life processes (biology, physiology, medicine);

large systems associated with the interaction of people (social, economic, environmental);

humanities (linguistics, art).

(The areas of application are listed in descending order according to the level of adequacy of the models).

Types of mathematical models: deterministic and probabilistic, theoretical and experimental factorial. Linear and non-linear, dynamic and static. continuous and discrete, functional and structural.

Classification of mathematical models (TO - technical object)

The structure of a model is an ordered set of elements and their relationships. A parameter is a value that characterizes a property or mode of operation of an object. The output parameters characterize the properties of the technical object, and the internal parameters characterize the properties of its elements. External parameters are the parameters of the external environment that affect the functioning of the technical object.

Mathematical models are subject to the requirements of adequacy, economy, universality. These claims are contradictory.

Depending on the degree of abstraction in describing the physical properties of a technical system, three main hierarchical levels are distinguished: the upper or metalevel, the middle or macrolevel, and the lower or microlevel.

The meta level corresponds to the initial stages of design, at which scientific and technical1 search and forecasting, development of a concept and technical solution, and development of a technical proposal are carried out. To build mathematical models of the metalevel, methods of morphological synthesis, graph theory, mathematical logic, automatic control theory, queuing theory, and finite automaton theory are used.

At the macrolevel, an object is considered as a dynamic system with lumped parameters. Mathematical models of the macrolevel are systems of ordinary differential equations. These models are used in determining the parameters of a technical object and its functional elements.

At the micro level, an object is represented as a continuous medium with distributed parameters. To describe the processes of functioning of such objects, partial differential equations are used. At the micro level, elements of a technical system that are indivisible in terms of functional characteristics, called basic elements, are designed. In this case, the base element is considered as a system consisting of a set of similar functional elements of the same physical nature, interacting with each other and being influenced by the external environment and other elements of the technical object, which are the external environment in relation to the base element.

According to the form of representation of mathematical models, invariant, algorithmic, analytical and graphical models of the design object are distinguished.

AT invariant form, the mathematical model is represented by a system of equations without regard to the method of solving these equations.

AT algorithmic in the form of model relations are associated with the chosen numerical solution method and are written in the form of an algorithm - a sequence of calculations. Algorithmic models include imitation, models designed to simulate the physical and information processes occurring in the object during its operation under the influence of various environmental factors.

Analytical the model represents the explicit dependences of the desired variables on the given values ​​(usually, the dependences of the output parameters of the object on internal and external parameters). Such models are obtained on the basis of physical laws, or as a result of direct integration of the original differential equations. Analytical mathematical models make it easy and simple to solve the problem of determining the optimal parameters. Therefore, if it is possible to obtain a model in this form, it is always advisable to implement it, even if it requires performing a number of auxiliary procedures. Such models are usually obtained by experimental design (computational or physical).

Graphic(circuit) model is represented in the form of graphs, equivalent circuits, dynamic models, diagrams, etc. To use graphical models, there must be a rule of one-to-one correspondence between conditional images of graphical elements and components of invariant mathematical models.

The division of mathematical models into functional and structural ones is determined by the nature of the displayed properties of the technical object.

Structural models display only the structure of objects and are used only in solving problems of structural synthesis. Parameters of structural models are signs of functional or structural elements that make up a technical object and in which one version of the object structure differs from another. These parameters are called morphological variables. Structural models take the form of tables, matrices and graphs. The most promising is the use of tree-like graphs of the AND-OR-tree type. Such models are widely used at the meta level when choosing a technical solution.

Functional models describe the processes of functioning of technical objects and have the form of systems of equations. They take into account the structural and functional properties of the object and allow solving problems of both parametric and structural synthesis. They are widely used at all levels of design. At the meta level, functional tasks allow solving forecasting problems, at the macro level - choosing the structure and optimizing the internal parameters of a technical object, at the micro level - optimizing the parameters of basic elements.

According to the methods of obtaining functional mathematical models are divided into theoretical and experimental.

Theoretical models are obtained based on the description of the physical processes of the object's functioning, and experimental- based on the behavior of the object in the external environment, considering it as a "black box". Experiments in this case can be physical (on a technical object or its physical model) or computational (on a theoretical mathematical model).

When constructing theoretical models, physical and formal approaches are used.

The physical approach is reduced to the direct application of physical laws to describe objects, for example, the laws of Newton, Hooke, Kirchhoff, etc.

The formal approach uses general mathematical principles and is used in the construction of both theoretical and experimental models. Experimental models are formal. They do not take into account the entire complex of physical properties of the elements of the technical system under study, but only establish the relationship between the individual parameters of the system that can be varied and (or) measured during the experiment. Such models provide an adequate description of the processes under study only in a limited region of the parameter space, in which the parameters were varied in the experiment. Therefore, experimental mathematical models are of a particular nature, while physical laws reflect the general patterns of phenomena and processes occurring both in the entire technical system and in each of its elements separately. Consequently, experimental mathematical models cannot be accepted as physical laws. However, the methods used to build these models are widely used in testing scientific hypotheses.

Functional mathematical models can be linear and non-linear. Linear models contain only linear functions of quantities characterizing the state of the object during its operation, and their derivatives. The characteristics of many elements of real objects are non-linear. Mathematical models of such objects include non-linear functions of these quantities and their derivatives and refer to non-linear .

If the modeling takes into account the inertial properties of the object and (or) the change in time of the object or the external environment, then the model is called dynamic. Otherwise the model is static. The mathematical representation of a dynamic model in the general case can be expressed by a system of differential equations, and a static one - by a system of algebraic equations.

If the impact of the environment on the object is of a random nature and is described by random functions. In this case, it is necessary to build probabilistic mathematical model. However, such a model is very complex and its use in the design of technical objects requires a lot of computer time. Therefore, it is used at the final stage of design.

Most design procedures are performed on deterministic models. A deterministic mathematical model is characterized by a one-to-one correspondence between an external influence on a dynamic system and its response to this influence. In a computational experiment, when designing, some standard typical actions on an object are usually set: step, impulse, harmonic, piecewise linear, exponential, etc. They are called test actions.

Continuation of the Table “Classification of mathematical models

Types of mathematical models of technical objects
By taking into account the physical properties of TO	By the ability to predict results
dynamic	deterministic
Static	Probabilistic
continuous
Discrete
Linear
At this stage, the following steps are performed. A plan is drawn up for creating and using a software model. As a rule, the model program is created using computer simulation automation tools. Therefore, the plan indicates: type of computer; simulation automation tool; approximate costs of computer memory for creating a model program and its working arrays; the cost of machine time for one cycle of the model; estimates of costs for programming and debugging the model program. Then the researcher starts programming the model. The description of the simulation model serves as a specification for programming. The specifics of model programming work depends on the modeling automation tools that are available to the researcher. There are no significant differences between the creation of a model program and the usual offline debugging of program modules of a large program or software package. In accordance with the text, the model is divided into blocks and subblocks. In contrast to the usual offline debugging of program modules, when debugging blocks and subblocks of a program model, the amount of work increases significantly, since for each module it is necessary to create and debug an external environment simulator. It is very important to verify the implementation of the module functions in the model time t and estimate the cost of computer time for one cycle of the model as a function of the values ​​of the model parameters. The work is completed during autonomous debugging of the model components by preparing the forms for representing the input and output data of the simulation. Next, proceed to the second verification of the reliability of the system model program. During this check, the correspondence of operations in the program and the description of the model is established. To do this, the program is translated back into the model scheme (manual "scrolling" allows you to find gross errors in the model statics). After eliminating gross errors, a number of blocks are combined and complex debugging of the model begins using tests. Debugging for tests starts with a few blocks, then an increasing number of model blocks are involved in this process. Note that the complex debugging of the model program is much more difficult than debugging application packages, since simulation dynamics errors in this case are much more difficult to find due to the quasi-parallel operation of various model components. Upon completion of the complex debugging of the model program, it is necessary to re-estimate the costs of computer time for one cycle of calculations on the model. In this case, it is useful to obtain an approximation of the simulation time for one simulation cycle. The next step is to draw up technical documentation for a model of a complex system. By the end of the complex debugging of the model program, the following documents should be the result of the stage: description of the simulation model; description of the model program indicating the programming system and accepted notation; full scheme of the model program; complete recording of the model program in the modeling language; proof of the reliability of the model program (the results of complex debugging of the model program); description of input and output values ​​with necessary explanations (dimensions, scales, ranges of values, symbols); evaluation of the cost of computer time for one simulation cycle; instructions for working with the model program. To check the adequacy of the model to the object of study, after compiling a formal description of the system, the researcher draws up a plan for conducting full-scale experiments with a system prototype. If there is no prototype of the system, then a system of nested IMs can be used, differing from each other in the degree of detail of imitation of the same phenomena. Then the more detailed model serves as a prototype for the generalized IM. If it is impossible to build such a sequence, either because of the lack of resources to perform this work, or because of insufficient information, then they do without checking the adequacy of the IM. According to this plan, in parallel with the debugging of the IM, a series of full-scale experiments on a real system is carried out, during which control results are accumulated. Having control results and MI test results at his disposal, the researcher checks the adequacy of the model to the object. If errors are found during the debugging phase that can only be corrected in the previous phases, a return to the previous phase can take place. In addition to the technical documentation, the results of the stage are accompanied by a machine implementation of the model (a program translated in the machine code of the computer on which the simulation will take place). This is an important step in the creation of the model. In this case, you must do the following. First, make sure that the dynamics of the development of the algorithm for modeling the object of study is correct in the course of simulating its functioning (to verify the model). Secondly, to determine the degree of adequacy of the model and the object of study. The adequacy of a software simulation model to a real object is understood as the coincidence with a given accuracy of the vectors of the characteristics of the behavior of the object and the model. In the absence of adequacy, the simulation model is calibrated (“correct” the characteristics of the model component algorithms). The presence of errors in the interaction of model components returns the researcher to the stage of creating a simulation model. It is possible that in the course of formalization, the researcher oversimplified the physical phenomena, excluded from consideration a number of important aspects of the functioning of the system, which led to the inadequacy of the model to the object. In this case, the researcher must return to the stage of system formalization. In cases where the choice of formalization method turned out to be unsuccessful, the researcher needs to repeat the stage of compiling a conceptual model, taking into account new information and experience. Finally, when the researcher has insufficient information about the object, he must return to the stage of compiling a meaningful description of the system and refine it, taking into account the results of testing the previous system model. At the same time, the accuracy of the simulation of phenomena, the stability of the simulation results, the sensitivity of the quality criteria to changes in the model parameters are evaluated. It is very difficult to obtain these estimates in some cases. However, without the successful results of this work, neither the developer nor the IM customer will have confidence in the model. Different researchers, depending on the type of IM, have developed different interpretations of the concepts of accuracy, stability, stationarity, sensitivity of IM. So far, there is no generally accepted theory of imitation of phenomena on a computer. Each researcher has to rely on his experience in organizing the simulation and on his understanding of the features of the simulation object. The simulation accuracy of phenomena is an assessment of the influence of stochastic elements on the functioning of a complex system model. The stability of the simulation results is characterized by the convergence of the controlled simulation parameter to a certain value with an increase in the simulation time of a variant of a complex system. The stationarity of the simulation mode characterizes a certain steady-state equilibrium of processes in the system model, when further simulation is meaningless, since the researcher will not receive new information from the model and continuing the simulation practically only leads to an increase in computer time. It is necessary to provide for such a possibility and develop a method for determining the moment when the stationary simulation mode is reached. The MI sensitivity is represented by the value of the minimum increment of the selected quality criterion, calculated from the simulation statistics, with sequential variation of the simulation parameters over the entire range of their changes. This stage begins with the design of an experiment that allows the researcher to obtain maximum information with minimum computational effort. Statistical substantiation of the experimental plan is required. Experiment planning is a procedure for choosing the number and conditions of experiments that are necessary and sufficient to solve the problem with the required accuracy. At the same time, the following is essential: striving to minimize the total number of experiments, ensuring the possibility of simultaneous variation of all variables; the use of a mathematical apparatus that formalizes many of the actions of experimenters; choosing a clear strategy that allows you to make informed decisions after each series of experiments on the model. Then the researcher proceeds to carry out working calculations on the model. This is a very time-consuming process that requires a large computer resource and an abundance of clerical work. It should be noted that already at the early stages of creating an IM, it is necessary to carefully consider the composition and volume of modeling information in order to significantly facilitate further analysis of the simulation results. The result of the work are the simulation results. This stage completes the technological chain of stages of creating and using simulation models. Having received the simulation results, the researcher proceeds to interpret the results. The following simulation cycles are possible here. In the first cycle of the simulation experiment in the IM, the choice of options for the system under study is provided in advance by setting the initial conditions for the simulation for the computer program of the model. In the second cycle of the simulation experiment, the model is modified in the modeling language, and therefore re-translation and editing of the program is required. It is possible that in the course of interpreting the results, the researcher found the presence of errors either when creating the model or when formalizing the modeling object. In these cases, a return is made to the stages of constructing a description of the simulation model or to compiling a conceptual model of the system, respectively. The result of the stage of interpretation of the simulation results are recommendations for the design of the system or its modification. With the recommendations at their disposal, the researchers begin to make design decisions. The interpretation of the simulation results is significantly influenced by the imaging capabilities of the computer used and the simulation system implemented on it. 1. How is the classification of mathematical models based on the features of the applied mathematical apparatus. Mathematics abstract Development of an economic and mathematical model for optimizing the sectoral structure of production in the agricultural sector

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Presentation on the topic: Mathematical models (Grade 7)

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§ 2.4. Mathematical models The main language of information modeling in science is the language of mathematics. Models built using mathematical concepts and formulas are called mathematical models. A mathematical model is an information model in which the parameters and dependencies between them are expressed in mathematical form.

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Mathematical modeling The modeling method makes it possible to apply the mathematical apparatus to solving practical problems. The concepts of number, geometric figure, equation, are examples of mathematical models. The method of mathematical modeling in the educational process has to be resorted to when solving any problem with practical content. To solve such a problem by mathematical means, it must first be translated into the language of mathematics (to build a mathematical model).

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In mathematical modeling, the study of an object is carried out by studying a model formulated in the language of mathematics. Example: you need to determine the surface area of ​​\u200b\u200bthe table. Measure the length and width of the table, and then multiply the resulting numbers. This actually means that the real object - the surface of the table - is replaced by an abstract mathematical model of a rectangle. The area of ​​this rectangle is considered to be the required one. Of all the properties of the table, three were singled out: the shape of the surface (rectangle) and the lengths of the two sides. Neither the color of the table, nor the material from which it is made, nor how it is used is important. Assuming that the table surface is a rectangle, it is easy to specify the input data and the result. They are related by S=ab.

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Consider an example of bringing the solution of a specific problem to a mathematical model. Through the porthole of the sunken ship, you need to pull out the treasure chest. Some assumptions about the shape of the chest and windows of the porthole and the initial data for solving the problem are given. Assumptions: The porthole has the shape of a circle. The chest has the shape of a rectangular parallelepiped. Initial data: D - porthole diameter; x - chest length; y - chest width; z is the height of the chest. End result: Message: may or may not be pulled.

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The system analysis of the problem condition revealed the relationship between the size of the porthole and the size of the chest, taking into account their shapes. The information obtained as a result of the analysis was displayed in formulas and relationships between them, so a mathematical model arose. The mathematical model for solving this problem is the following relationships between the initial data and the result:

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Example 1: Calculate the amount of paint for a floor in a gym. To solve the problem, you need to know the area of ​​\u200b\u200bthe floor. To complete this task, measure the length, width of the floor and calculate its area. The real object - the floor of the hall - is occupied by a rectangle, for which the area is the product of length and width. When buying paint, they find out what area can be covered with the contents of one can, and calculate the required number of cans. Let A be the length of the floor, B - the width of the floor, S1 - the area that can be covered with the contents of one can, N is the number of cans. The floor area is calculated using the formula S=A×B, and the number of cans needed to paint the hall is N= A×B/S1.

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Example 2: Through the first pipe the pool is filled in 30 hours, through the second pipe in 20 hours. How many hours will it take to fill the pool through two pipes? Solution: Let's denote the time of filling the pool through the first and second pipes A and B, respectively. Let us take the entire volume of the pool as 1, denote the desired time by t. Since the pool is filled through the first pipe in A hours, then 1/A is the part of the pool filled with the first pipe in 1 hour; 1/B - part of the pool filled with the second pipe in 1 hour. Therefore, the speed of filling the pool with the first and second pipes together will be: 1/A + 1 / B. You can write: (1 / A + 1 / B) t \u003d 1. received a mathematical model describing the process of filling the pool of two pipes. The desired time can be calculated by the formula:

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Example 3: Points A and B are located on the highway, 20 km apart. The motorcyclist left point B in the direction opposite to A at a speed of 50 km/h. Let us make a mathematical model describing the position of the motorcyclist relative to point A in t hours. In t hours the motorcyclist will travel 50t km and will be at a distance of 50t km + 20 km from A . If we denote by the letter s the distance (in kilometers) of the motorcyclist to point A, then the dependence of this distance on the time of movement can be expressed by the formula: S = 50t + 20, where t> 0. The mathematical model for solving this problem is the following relationships between the initial data and the result: Misha had x marks; Andrei has 1.5x. Misha got x-8, Andrey got 1.5x+8. According to the condition of the problem, 1.5x + 8 = 2 (x-8).

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The mathematical model for solving this problem is the following relationship between the initial data and the result: Misha had x stamps; Andrei has 1.5x. Misha got x-8, Andrey got 1.5x+8. According to the condition of the problem, 1.5x + 8 = 2 (x-8). The mathematical model for solving this problem is the following dependencies between the initial data and the result: x people work in the second workshop, 4x in the first, and x + 50 in the third. x+4x+x+50=470. The mathematical model for solving this problem is the following dependencies between the initial data and the result: the first number x; second x + 2.5. According to the condition of the problem, x / 5 = (x + 2.5) / 4.

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Sources Informatics and ICT: textbook for grade 7Author: Bosova LL Publisher: BINOM. Knowledge Laboratory, 2009 Format: 60x90/16 (in lane), 229 pp., ISBN: 978-5-9963-0092-1http://www.lit.msu.ru/ru/new/study )http://images.yandex.ru (pictures)