1 understanding of the mathematical model and mathematical modeling. Fundamentals of mathematical models. Preparing to ODE or EDI in Mathematics
Як систему рівнянь, або арифметичних співвідношень, або геометричних фігур, або комбінацію того й іншого, дослідження яких засобами математики має відповісти на поставлені питання про властивості деякої сукупності властивостей об'єкта реального світу, як сукупність математичних співвідношень, рівнянь, нерівностей, що описують основні regularity, power in the following process, object or system.
At automated systems the control of the mathematical model is based on the algorithm of the controller's functioning. Whose algorithm is chosen, how to change piercing infusion in the fallow, the type of change is set in order for the management to be reached.
Classification of models
Formal classification of models
The formal classification of models is based on the classification of victorious mathematical methods. Often found in forms of dichotomies. For example, one of the popular sets of dichotomies:
and so far. The model was induced by the skin in a linear number, non-linear, deterministic, purely stochastic, ... Naturally, it is possible to change the type: in one case, the zoning (with a wide range of parameters), in the other, the division of the model is thin.
Classification according to the method of presenting the object
The order of the formal classification of the model depends on the way the object is presented:
- Structural and functional models
Models-hypotheses in science cannot be brought to light once and for all, we can only talk about them being unrecorded as a result of the experiment.
As the model of the first type has been induced, it means that it is timely confessed for the truth and it is possible to concentrate on other problems. However, it cannot be a dot in progress, but rather an hour-long pause: the status of a model of the first type can be more than an hour.
Phenomenological model
Another type is the phenomenological model ( "Let's act like this, nibi ..."), to avenge a mechanism for describing a phenomenon, if this mechanism is not enough reconciliations, it cannot be sufficiently confirmed by obvious data, otherwise it is nasty to use obvious theories and the accumulation of knowledge about the object. That is why the phenomenological models determine the status of timchasov decisions. It is important that the evidence is still unknown, and it is necessary to continue the search for “correct mechanisms”. For example, the caloric model and the quark model of elementary particles are considered to be another type of Peierls.
The role of the model in the study may change from time to time, it may be that new data and theories confirm the phenomenological model and will be promoted to the status of a hypothesis. Similarly, new knowledge can step by step become superficial with models-hypotheses of the first type, and they can be translated into another. Thus, the quark model is transformed step by step into a category of hypotheses; atomism in physics vinik as a temporal solution, but with the passage of the history of transitions in the first type. And the axis of the model of the ether went through a path from type 1 to type 2, and at the same time it is known by science.
The idea of forgiveness is even more popular with budding models. Ale forgiveness bovaє reznim. Payerls sees three types of problems with modelling.
Closeness
The third type of models is proximity ( “We respect the great chi even small”). Although it is possible to be inspired to describe the completed system, it does not mean that it can be found with the help of a computer. Zagalnopriynyat priyom razі - vykoristannya podblizhenya (models type 3). Among them models of linear guidance. Rivnyannya are replaced by linear ones. Standard butt - Ohm's law.
Dumkov experiment
m x ¨ = − k x (\displaystyle m(\ddot(x))=-kx),de x ¨ (\displaystyle (\ddot (x))) means to a friend x (\displaystyle x) by the hour: x ¨ = d 2 x d t 2 (\displaystyle (\ddot (x))=(\frac (d^(2)x)(dt^(2)))).
Otrimane is equal to the mathematical model of the examined physical system. This model is called "harmonic oscillator".
For the formal classification, the model is linear, deterministic, dynamic, sedentary, uninterrupted. In the process of її, they made me impersonal allowance (about the daytime of the callous forces, the daytime of rubbing, the troubles of respiration, etc.), as if they really can’t win.
In terms of reality, the most common model is type 4 forgiveness(“It is omitted for the sake of clarity of the details”), the omissions are omitted from the deacons of the sutti in the universal singularity (for example, disipation). For someone close (say, while the vіdkhilenny vіdhіnі vіd vіd іvnovagi іn small, іn а small terti, іn thе rіght іn nοt thе great hour і іn dοtrimannі іnshih minds), such a model іѕ tο well describe thе real mechanical system, thе oscillа іdkinіtі factors mаyut znіkuїlі їїї ї ї ї ї ї ї ї ї ї ї ї . However, the model can be refined by taking into account any of these factors. Tse brought to a new model, with a wider (if I want to re-surface) area of stoking.
Vtіm, with a refined model, the folding and її mathematical elaboration can be significant in terms of maturity and maturity, the model is practical. In most cases, the simplest model makes it possible to more briefly and more accurately extend the real system, less folding (and, formally, “correct”).
If you want to bring the model of the harmonic oscillator to the objects, distant types of physics, the change status may be different. For example, with the addition of this model to biological populations, it should be recognized, better for everything, up to type 6 analogy(“Vrahuemo is less than deyaki speciality”).
Short and soft models
The harmonic oscillator is an example of the so-called “hard” model. Vaughn is taken away by the strong idealization of a real physical system. The dominance of the harmonic oscillator is clearly changed by small fluctuations. For example, to add to the right side of the small dodanok − ε x ˙ (\displaystyle -\varepsilon (\dot (x)))(rubbing) ( ε > 0 (\displaystyle \varepsilon >0)- deaky small parameter), then exponentially fading colivanya, so change the sign of the additional addendum (ε x ˙) (\displaystyle (\varepsilon (\dot (x)))) then tertya transform into pumping and the amplitude of the injection exponentially increases.
In order to improve the nutrition about the stagnation of the zhorstkoy model, it is necessary to understand, on the basis of the facts and factors, with which we were opposed. It is necessary to follow the soft models, as if they look like a small burrowed zhorstkoy. For a harmonic oscillator, the stench can be set, for example, to the coming equals:
m x ¨ = − k x + ε f(x , x ˙).Here f (x , x ˙) (\displaystyle f(x,(\dot (x))))- deak function, in which case the force can be reversed by losing the coeficient of the hardness of the spring in the form of stretching. Explicit form of the function f (\displaystyle f) don't tease us at once.
As we know, the behavior of the soft model is fundamentally not affected by the behavior of the hard model (independently of the explicit mind of the factors that make you feel bad, like the stink of dosit little), the task is to follow the hard model. Otherwise, stosuvannya results, otrimanih schodo zhorstkoї model, instead of additional results.
If the system saves its own behavior in case of a small cloudiness, then it seems that it is structurally stable. The harmonic oscillator is an example of a structurally unstable (rough) system. Prote, this model can be vikoristovuvatime vyvchennya vyvchennya protsessiv on obrazhenih intervals of the hour.
Universality of models
The most important mathematical models sound like an important power universality: fundamentally different real phenomena can be described by one and the same mathematical model. Let's say that the harmonic oscillator describes not only the behavior of the vantage on the springs, but also other oscillatory processes, which often may be similar to our nature: small oscillating of the pendulum, oscillating of equal parts U (\displaystyle U)- similar to the vessel or change the strength of the struma in the kolivalny circuit. In this manner, cultivating one mathematical model, we cultivate a whole class of phenomena described by it. The very isomorphism of laws, which is manifested by mathematical models in different segments of scientific knowledge, is the feat of Ludwig von Bertalanff to create "the ignorant theory of systems."
Direct turnaround of mathematical modeling
Іsnuє impersonal tasks related to mathematical modeling. First, you need to come up with a basic scheme of the object that is being modeled, to practice yoga within the framework of the idealization of this science. Так, вагон поїзда перетворюється на систему пластин і складніших тіл з різних матеріалів, кожен матеріал задається як його стандартна механічна ідеалізація (щільність, модулі пружності, стандартні характеристики міцності), після чого складаються рівняння, по дорозі якісь деталі відкидаються як несуттєві, виробляються розрахунки, compare with the models, the model is being specified, and so on. Proto-development of technologies for mathematical modeling of the basic development of the process on the main warehouse elements.
Traditionally, there are two main classes of tasks associated with mathematical models: direct and reverse.
Straight ahead: the structure of the model and all parameters are taken into account, main task- Carry out the follow-up model for the acquisition of core knowledge about the object. How statically navantazhennya vytrimaє mist? As a reaguvatime on a dynamic urge (for example, on a march of a company of soldiers, or on a train passing on a different flight), as a lighter sound barrier, so as not to fall apart in the flutter, - the axis of a typical butt is applied directly. The setting of the correct direct task (the task of proper nutrition) requires special mastery. If you don’t set the right nutrition, then the place can collapse, so it was necessary to create a model for yoga behavior. So, in 1879. near Great Britain, a metal bridge collapsed across the Firth of Tay, the designers of which inspired a model of the bridge, built it up for a 20-fold reserve of minerality for the purpose of the corynes, but they forgot about the wind, which is constantly cloudy in quiet places. I through the second time the rocks of the wines were called.
In the simplest way (one equal oscillator, for example) it’s even easier to get straight to the point of obvious perfection of that equal.
Zvorotne zavdannya: to see anonymous possible models, you need to select a specific model on the basis of additional data about the object. Most often, the structure of the house model and it is necessary to assign some unknown parameters. Additional information can be applied to additional empirical data, or else to the object ( project manager). Additional data can be found independently in the process of completing the final task ( passive watchfulness) or be the result of a specially planned experiment during the course of the decision ( active watchfulness).
One of the first applications of the virtuosic accomplishment of a pivotal task, with the most recent and most accessible data of Newton's inspirations, is the method of reinforcing forces by rubbing against dying out colivans.
As another example, you can bring mathematical statistics. The head of the scientific center - the development of registration methods, describe and analyze these warnings and experiments with the method of prompting imovirnіsnyh models of mass vipadical manifestations. That impersonal possible models are surrounded by imovirnіsnymi models. For specific tasks, a lot of models are more heavily marked.
Computer systems and modeling
To support the mathematical modeling of the expansion of the system of computer mathematics, for example, Maple, Mathematica, Mathcad, MATLAB, VisSim and other. They allow you to create formal and block models, both simple and folding processes and attachments, and easily change the parameters of models during the course of modeling. block models represented by blocks (mostly graphical ones), the collection of which is given by the diagram of the model.
Appendices butts
Malthus model
In line with the model propagated by Malthus, the rate of growth is proportional to the flow rate of the population, which is described by differential equations:
x ˙ = α x (\displaystyle (\dot(x))=\alpha x),de α (\displaystyle \alpha)- a certain parameter, which is determined by the difference between the people and the death rate. Decisions on which equal is the exponential function x(t) = x 0 e α t (\displaystyle x(t)=x_(0)e^(\alpha t)). Like the people overturn death ( α > 0 (\displaystyle \alpha >0)), the expansion of the population is unfenced and even slightly growing. Indeed, what cannot be obtained through the exchange of resources. With the reach of a certain critical population commitment, the model ceases to be adequate, and the shards of the resource exchange. The refined model of Malthus can be a logistic model, as described by the differential equations of Verhulst:
x ˙ = α (1 − x x s) x (\displaystyle (\dot (x))=\alpha \left(1-(\frac(x)(x_(s)))\right)x),de - "Equally important" expansion of the population, with which the population is exactly compensated for by mortality. Population expansion in such a model is of equal importance x s (\displaystyle x_(s)), moreover, such a behavior is structurally stable.
Hijack-victim system
It is acceptable that two kinds of creatures live on the deakіy territory: rabbits (eating roslins) and foxes (eating rabbits). Give me a bunch of rabbits x (\displaystyle x), the number of foxes y (\displaystyle y). Vikoristovuyuchi model of Malthus with the necessary amendments, scho vrakhovuyut podїdannya rabbits foxes, it comes to the offensive system, as may be Trays - Volterra:
( x ˙ = (α − c y) x y ˙ = (− β + d x) y (\displaystyle (\begin(cases)(\dot (x))=(\alpha -cy)x\\(\dot (y ))=(-\beta +dx)y\end(cases)))The behavior of this system is not structurally stable: a small change in the parameters of the model (for example, what is the cost of resources needed by rabbits) can lead to a significant change in behavior.
With certain values of the parameters, the system can become equally important if the number of rabbits and foxes is constant. Vіdhilennya vіd tsogo I'll bring to the stepwise fading coliving of the number of rabbits and foxes.
The situation is possible and protilezhna, if there is any small change in the situation of the equal, it will lead to catastrophic consequences, right up to the total extinction of one of the sights. For information about those, which of these scenarios are implemented, the Volterri model - Trays are not given: here you need additional follow-up.
Div. also
Notes
- "A mathematical representation of reality" (Encyclopaedia Britanica)
- Novik I. B., About philosophical nutrition of cybernetic modeling. M., Knowledge, 1964.
- Rad B. Ya., Yakovlev S. A., Modeling systems: Navch. for universities - 3rd type., revised. that dod. - M: Vishch. school, 2001. - 343 p. ISBN 5-06-003860-2
- Samarsky A. A., Mikhailov A. P. Mathematical modeling. Ideas. Methods. Apply. - 2nd species., Vipr. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X.
- Mishkis A. D. Elements of the theory of mathematical models. - 3rd species, Vipr. - M: KomKniga, 2007. - 192 s ISBN 978-5-484-00953-4
- Sevostyanov, A. G. Modeling of technological processes: assistant / A. G. Sevostyanov, P. A. Sevostyanov. - M .: Easy that Kharchova promislovist, 1984. - 344 p.
- Rotach V.Ya. The theory of automatic curing. - 1st. - M.: ZAT " vidavnichy booth MEI", 2008. - S. 333. - 9 p. - ISBN 978-5-383-00326-8.
- Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena(English). Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII + 562 pp. ISBN 3-540-35885-4. The date of completion is 18 March 2013. Archived on March 18, 2013.
- “The theory is respected by a linear chi non-linear fallow in that, which is a linear chi non-linear - mathematical apparatus, yak - linear chi non-linear - mathematical models out of vicorist. ... without listing the rest. Modern physicist, let me re-create the designation of such an important essence, like non-linearity, more for everything, introducing bi іnakshe, i, giving the priority of non-linearity as the more important and widening of the two prolongations, denoting the “non-linear” Danilov Yu. A., Lectures on non-linear dynamics. Elementary request. Series "Synergetics: from the past to the present". View.2. – M.: URSS, 2006. – 208 p. ISBN 5-484-00183-8
- “Dynamic systems, which are modeled by the last number of sizable differential levels, are called fixed or point systems. The stench is described with the aid of the final phase space and is characterized by the final number of steps of freedom. One and the same system in different minds can be seen either as a serendipity, or as a divided one. Mathematical models of separate subdivisions of systems - ce differential equivalence of private inferior ones, integral equivalence or primal equivalence from behind argument. The number of steps of freedom of the divided system is inexhaustible, and an inexhaustible amount of data is required for the purpose of becoming.
Anishchenko V. S., Dynamic systems, Sorosievskiy osvitniy zhurnal, 1997 № 11, p. 77-84. - “Depending on the nature of the subsequent processes in the system S, all types of modeling can be subdivided into deterministic and stochastic, static and dynamic, discrete, interruption-free and discrete-interruption-free. Deterministic modeling in the form of deterministic processes, so that processes, in which the day-to-day occurrence of vague infusions is transferred; stochastic modeling of imaginative processes and processes. … Static modeling is used to describe the behavior of the object at the hour, and dynamic modeling is used to describe the behavior of the object at the hour. Discrete modeling is used to describe processes, as they are transferred to discrete ones, in such a way that continuous modeling allows you to visualize continuous processes in systems, and discretely-uninterrupted modeling is used to describe the processes, if you want to see the presence of discrete, so and interruptless processes.
Rad B. Ya., Yakovlev S. A., Modeling systems: Navch. for universities - 3rd type., revised. that dod. - M: Vishch. school, 2001. - 343 p. ISBN 5-06-003860-2 - The structure (appendices) of the modeled object, the essence of the method of investigating the quality and interrelationship of the components of the object; such a model is called structural. Well, the model only looks like those, like the object is functioning - for example, like the wine reacts to the outward infusions, it is called a functional or, figuratively, a black box. Possible models of combined type. Mishkis A. D. Elements of the theory of mathematical models. - 3rd species, Vipr. - M: KomKniga, 2007. - 192 p.
According to the assistant of Radov and Yakovlev: “the model (Latin modulus - the world) is the object-protector of the object-original, which ensures the transfer of certain powers to the original”. (S. 6) “Replacing one object with another, with the method of removing information about the most important power of the original object for the additional object-model, is called a model”. (p. 6) “Before mathematical modeling, it is reasonable to understand the process of establishing the validity of a given real object of a certain mathematical object, called a mathematical model, and following this model, which allows us to take the characteristics of a real object, which is considered. The type of mathematical model to be deposited as in the nature of a real object, so the task of verifying the object and the necessary reliability and accuracy of the development of this task.
Nareshti, the most concise description of the mathematical model: “Rivnyannya, who expresses the idea».
Classification of models
Formal classification of models
The formal classification of models is based on the classification of victorious mathematical methods. Often found in forms of dichotomies. For example, one of the popular sets of dichotomies:
and so far. The model was induced by the skin in a linear number, non-linear, deterministic, purely stochastic, ... Naturally, it is possible to change the type: in one case, the zoning (with a wide range of parameters), in the other, the division of the model is thin.
Classification according to the method of presenting the object
The order of the formal classification of the model depends on the way the object is presented:
- Structural and functional models
Structural models represent the object as a system with its attachment and mechanism of functioning. Functional models do not win such manifestations and show that the behavior (functioning) of the object is accepted. In their borderline expression, stinks are also called “black box” models. It is also possible to combine types of models, which are sometimes called models. orphan screenshots».
Changes and formal models
May all the authors, who describe the process of mathematical modeling, show that, in the future, there will be a special ideal design, replacement model. There is no tired terminology here, other authors name this ideal object conceptual model , smart model or front model. Why is the final mathematical construction called formal model or simply a mathematical model taken after the formalization of the given replacement model (before the model). Pobudova change models can be developed for an additional set of ready-made idealizations, like in mechanics, de ideal springs, hard bodies, ideal pendulums, spring centers, and then give ready-made structural elements of a change model. However, in the circles of knowledge, where there is no complete formalization of theories (the leading edge of physics, biology, economics, sociology, psychology, and most other areas), the creation of change models is sharply reduced.
Zmistovna classification of models
The same hypothesis in science does not happen once and for all. More clearly stated by Richard Feynman:
“We always have the ability to prostrate a theory, but, to show respect, we can’t at all prove that it’s right. It is acceptable that you hung the hypothesis in the distance, razrahuvali, to what extent you know, and explained that these findings are experimentally confirmed. What does it mean that your theory is correct? Hі, just tse means that you didn’t get far enough to її prostuvati.
As the model of the first type has been induced, it means that it is timely confessed for the truth and it is possible to concentrate on other problems. However, it cannot be a dot in progress, but rather an hour-long pause: the status of a model of the first type can be more than an hour.
Type 2: Phenomenological model (let's act like this, nibi yakby…)
Phenomenological model to replace the mechanism of the description of the phenomenon. However, this mechanism is not enough reconciliation, it cannot be sufficiently confirmed by evidence, otherwise it is nasty to use evidence theories and accumulated knowledge about the object. That is why the phenomenological models determine the status of timchasov decisions. It is important that it is still unknown and it is necessary to continue the search for “correct mechanisms”. For example, the caloric model and the quark model of elementary particles are considered to be another type of Peierls.
The role of the model in the study may change from time to time, it may be that new data and theories confirm the phenomenological model and will be promoted to the status of a hypothesis. Similarly, new knowledge can step by step become superficial with models-hypotheses of the first type and can be translated into another. Thus, the quark model is transformed step by step into a category of hypotheses; atomism in physics vinik as a temporal solution, but with the passage of the history of transitions in the first type. And the axis of the model of ether, passed the way from type 1 to type 2, and at the same time it is known by science.
The idea of forgiveness is even more popular with budding models. Ale forgiveness bovaє reznim. Payerls sees three types of problems with modelling.
Type 3: Closeness (we respect the great chi even the smallest)
Although it is possible to be inspired to describe the completed system, it does not mean that it can be found with the help of a computer. Zagalnopriynyat priyom razі - vykoristannya podblizhenya (models type 3). Among them models of linear guidance. Rivnyannya are replaced by linear ones. Standard butt - Ohm's law.
A axis i type 8, extensions in mathematical models of biological systems.
Type 8: Demonstration of ability (smut - show the inner non-superability of ability)
Tsezh uyavnі experiment with obvious essences, yakі demonstrate that peredbachuvane apparition uzgodzhuєtsya with the basic principles that internally is not superb. In this case, the main type of models is type 7, yakі rozkryvayut prihovanі protirіchchya.
One of the most famous such experiments is Lobachevsky's geometry (Lobachevsky called it "manifest geometry"). The second example is the mass production of formally kinetic models of chemical and biological colivans, auto-curing and other. The paradox of Einstein - Podilsky - Rosen was conceived as a model of type 7, to demonstrate the super-smartness of quantum mechanics. By an absolutely unplanned rank, he changed into a type 8 model - a demonstration of the possibility of quantum teleportation of information.
butt
Let's take a look at the mechanical system, which is made up of springs, fixed from one end, that vantage by the mass, attached to the free end of the spring. Vvazhatimemo, that the vantage can collapse only into the straight axis of the spring (for example, ruh vіdbuvaєtsya vdovzh shear). Let's have a mathematical model of the whole system. Describe the system's rise to the center of vantage to the first position of equality. Let's describe the interplay of springs and vantage for help Hooke's law() after which we speed up another Newton's law, so that we can say yoga in the form of differential alignment:
de means to a friend at a later time: .
Otrimane is equal to the mathematical model of the examined physical system. This model is called "harmonic oscillator".
Behind the formal classification, the model is linear, deterministic, dynamic, sedentary, uninterrupted. In the process of її, they made me impersonal allowance (about the daytime of the callous forces, the daytime of rubbing, the troubles of respiration, etc.), as if they really can’t win.
In terms of reality, the most common model is type 4 forgiveness(“It is omitted for the sake of clarity of the details”), the omissions are omitted from the deacons of the sutti in the universal singularity (for example, disipation). For someone close (say, while the vіdkhilenny vіdhіnі vіd vіd іvnovagi іn small, іn а small terti, іn thе rіght іn nοt thе great hour і іn dοtrimannі іnshih minds), such a model іѕ tο well describe thе real mechanical system, thе oscillа іdkinіtі factors mаyut znіkuїlі їїї ї ї ї ї ї ї ї ї ї ї ї . However, the model can be refined by taking into account any of these factors. Tse bred to a new model, with a larger wide (even newly fringed) area of zastosuvannya.
Vtіm, with a refined model, the folding and її mathematical elaboration can be significant in terms of maturity and maturity, the model is practical. In most cases, the simplest model makes it possible to more briefly and more accurately extend the real system, less folding (and, formally, “correct”).
If you want to bring the model of the harmonic oscillator to the objects, distant types of physics, the change status may be different. For example, with the addition of this model to biological populations, it should be recognized, better for everything, up to type 6 analogy(“Vrahuemo is less than deyaki speciality”).
Short and soft models
The harmonic oscillator is an example of the so-called “hard” model. Vaughn is taken away by the strong idealization of a real physical system. In order to improve nutrition about її zastosuvannya, it is necessary to understand, how many suttєvimi є factors, which mi znehtuvali. In other words, it is necessary to finish the "m'yaku" model, so that the small "zhorstkoy" will go out. You can ask yourself, for example, we will attack the equals:
Here - a deuce function, for which the strength can be reversed, the fallowing coefficient of the hardness of the spring in the form of stretching, is a small parameter. The explicit form of the function doesn't fool us all at once. As we know, the behavior of the soft model is fundamentally not affected by the behavior of the hard model (independently of the explicit mind of the factors that make you feel bad, like the stink of dosit little), the task is to follow the hard model. Otherwise, stosuvannya results, otrimanih schodo zhorstkoї model, instead of additional results. For example, the solution of the harmonic oscillator is equal to the function of the mind, so that the constant amplitude of the oscillation. Why is it so obvious that a real oscillator is constantly changing for a long time with a constant amplitude? Hі, oskіlki looking at the system zі sіlki zavgodno small thirds (zavzhdi present in the real system), we omit quenching colivanya. The behavior of the system has clearly changed.
If the system saves its own behavior in case of a small cloudiness, then it seems that it is structurally stable. The harmonic oscillator is an example of a structurally unstable (rough) system. Prote, this model can be vikoristovuvatime vyvchennya vyvchennya protsessiv on obrazhenih intervals of the hour.
Universality of models
The most important mathematical models sound like an important power universality: fundamentally different real phenomena can be described by one and the same mathematical model. Let's say that the harmonic oscillator describes not only the behavior of the vantage on the springs, but also other coliving processes, which often seem to be similar to our nature: small swaying of the pendulum, swaying of the rod in the underside of the vessel, or changing the strength of the strum in the strum circuit. In this manner, cultivating one mathematical model, we cultivate a whole class of phenomena described by it. The very isomorphism of laws, which is manifested by mathematical models in different segments of scientific knowledge, is the feat of Ludwig von Bertalanff on the creation of the “Zahalny Theory of Systems”.
Direct turnaround of mathematical modeling
Іsnuє impersonal tasks related to mathematical modeling. First, you need to come up with a basic scheme of the object that is being modeled, to practice yoga within the framework of the idealization of this science. Так, вагон поїзда перетворюється на систему пластин і складніших тіл з різних матеріалів, кожен матеріал задається як його стандартна механічна ідеалізація (щільність, модулі пружності, стандартні характеристики міцності), після чого складаються рівняння, по дорозі якісь деталі відкидаються, як несуттєві , Виробляються розрахунки , compare with the models, the model is being specified, and so on. Proto-development of technologies for mathematical modeling of the basic development of the process on the main warehouse elements.
Traditionally, there are two main classes of tasks associated with mathematical models: direct and reverse.
Straight ahead: the structure of the model and її parameters are taken into account, the main task is to carry out the follow-up of the model for the acquisition of the basic knowledge of the object. How statically navantazhennya vytrimaє mist? As a reaguvatime on a dynamic urge (for example, on a march of a company of soldiers, or on a train passing on a different flight), as a lighter sound barrier, so as not to fall apart in the flutter, - the axis of a typical butt is applied directly. The setting of the correct direct task (the task of proper nutrition) requires special mastery. If you don’t set the right nutrition, then the place can collapse, so it was necessary to create a model for yoga behavior. So, in 1879. in Great Britain, a metal bridge collapsed across the river Tey, the designers of which inspired a model of the bridge, roared it for a 20-fold supply of capital per day of corynes, and then forgot about the wind, which is constantly cloudy in quiet places. I through the second time the rocks of the wines were called.
In the simplest way (one equal oscillator, for example) it’s even easier to get straight to the point of obvious perfection of that equal.
Zvorotne zavdannya: to see anonymous possible models, you need to select a specific model on the basis of additional data about the object. Most often, the structure of the house model, and it is necessary to assign some unknown parameters. Additional information can be applied to additional empirical data, or else to the object ( project manager). Additional data can be found independently in the process of completing the final task ( passive watchfulness) or be the result of a specially planned experiment during the course of the decision ( active watchfulness).
One of the first examples of the virtuosic accomplishment of a pivotal task with the highest possible number of available inspirations I. Newton's method of reinforcing forces is rubbing against the guarding fading coils.
As another example, you can bring mathematical statistics. The head of the scientific center - the development of registration methods, describe and analyze these warnings and experiments with the method of prompting imovirnіsnyh models of mass vipadical manifestations. Tobto. impersonal possible models are surrounded by imovirnіsnymi models. For specific tasks, a lot of models are more heavily marked.
Computer systems and modeling
To support the mathematical modeling of the expansion of the system of computer mathematics, for example, Maple, Mathematica, Mathcad, MATLAB, VisSim and other. They allow you to create formal and block models, both simple and folding processes and attachments, and easily change the parameters of models during the course of modeling. block models represented by blocks (mostly graphical ones), the collection of which is given by the diagram of the model.
Appendices butts
Malthus model
The speed of growth is proportional to the streaming expansion of the population. Vaughn is described by differential equals
de - deaky parameter, which is determined by the difference between the people and death. Decisions of which equal is the exponential function. As a nation outweighs mortality (), the expansion of the population is unfettered and even more rapidly growing. It dawned on me that you really can’t get through the exchange of resources. With the reach of a certain critical population commitment, the model ceases to be adequate, and the shards of the resource exchange. The refined model of Malthus can be a logistic model, as described by the differential equations of Verhulst
de - "Equally important" expansion of the population, with which the population is exactly compensated for by mortality. Population expansion in such a model is of equal importance, and such a behavior is structurally stable.
Hijack-victim system
It is acceptable that two kinds of creatures live on the deakіy territory: rabbits (eating roslins) and foxes (eating rabbits). Let me know the number of rabbits, the number of foxes. Vikoristovuyuchi model of Malthus with the necessary amendments, scho vrakhovuyut podїdannya rabbits foxes, it comes to the offensive system, as may be Trays - Volterra:
Tsya system may be equally important, if the number of rabbits and foxes is constant. Whenever I begin, I will bring up the number of rabbits and foxes, similar to those of the harmonic oscillator. Like a harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, that the security of the resources necessary for rabbits) can lead to a significant change in behavior. For example, an equally important camp can become stable, and the number of numbers will fade. The situation is possible and protilezhna, if there were any small change in the situation of the equal, it would lead to catastrophic consequences, right up to the total extinction of one of the sights. For information about those, which of these scenarios are implemented, the Volterra model - Trays are not given: here you need additional follow-up.
Notes
- "A mathematical representation of reality" (Encyclopaedia Britanica)
- Novik I. B., About philosophical nutrition of cybernetic modeling. M., Knowledge, 1964.
- Rad B. Ya., Yakovlev S. A., Modeling systems: Navch. for universities - 3rd type., revised. that dod. - M: Vishch. school, 2001. - 343 p. ISBN 5-06-003860-2
- Samarsky A. A., Mikhailov A. P. Mathematical modeling. Ideas. Methods. Apply. - 2nd species., Vipr. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X
- Mishkis A. D. Elements of the theory of mathematical models. - 3rd species, Vipr. - M: KomKniga, 2007. - 192 s ISBN 978-5-484-00953-4
- Sevostyanov, A.G. Modeling of technological processes: assistant / A.G. Sevostyanov, P.A. Sevostyanov. - M .: Easy that Kharchova promislovist, 1984. - 344 p.
- Wiktionary: mathematical models
- CliffsNotes.com. Earth Science Glossary. 20 Sep 2010
- Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4
- “The theory is respected by a linear chi non-linear fallow in that, which is a linear chi non-linear - mathematical apparatus, yak - linear chi non-linear - mathematical models out of vicorist. ... without listing the rest. Modern physicist, let me re-create the designation of such an important essence, like non-linearity, more for everything, introducing bi іnakshe, i, giving the priority of non-linearity as the more important and widening of the two prolongations, denoting the “non-linear” Danilov Yu. A., Lectures on non-linear dynamics. Elementary request. Series "Synergetics: from the past to the present". View.2. – M.: URSS, 2006. – 208 p. ISBN 5-484-00183-8
- “Dynamic systems, which are modeled by the last number of sizable differential levels, are called fixed or point systems. The stench is described with the aid of the final phase space and is characterized by the final number of steps of freedom. One and the same system in different minds can be seen either as a serendipity, or as a divided one. Mathematical models of separate subdivisions of systems - ce differential equivalence of private inferior ones, integral equivalence or primal equivalence from behind argument. The number of steps of freedom of the divided system is inexhaustible, and an inexhaustible amount of data is required for the purpose of becoming. Anishchenko V. S., Dynamic systems, Sorosievskiy osvitniy zhurnal, 1997 № 11, p. 77-84.
- “Depending on the nature of the subsequent processes in the system S, all types of modeling can be subdivided into deterministic and stochastic, static and dynamic, discrete, interruption-free and discrete-interruption-free. Deterministic modeling in the form of deterministic processes, so that processes, in which the day-to-day occurrence of vague infusions is transferred; stochastic modeling of imaginative processes and processes. … Static modeling is used to describe the behavior of the object at the hour, and dynamic modeling is used to describe the behavior of the object at the hour. Discrete modeling is used to describe processes, as they are transferred to discrete ones, in such a way that continuous modeling allows you to visualize continuous processes in systems, and discretely-uninterrupted modeling is used to describe the processes, if you want to see the presence of discrete, so and interruptless processes. Rad B. Ya., Yakovlev S. A. ISBN 5-06-003860-2
- The structure (appendices) of the modeled object, the essence of the method of investigating the quality and interrelationship of the components of the object; such a model is called structural. Well, the model only looks like those, like the object is functioning - for example, like the wine reacts to the outward infusions, it is called a functional or, figuratively, a black box. Possible models of combined type. Mishkis A. D. ISBN 978-5-484-00953-4
- “Obvious, but the most important stage of the first step in the choice of a mathematical model is the elimination of the possibility of a clear statement about the object that is being modeled, and the refinement of this design model, based on informal discussions. It is not possible to waste time on that zusil at this stage, in the light of the new significant world to lay down the success of all the success. More than once it happened that the practice was significant, it was stained on the top of a mathematical task, it turned out to be ineffective, or to instill a stained glass through lack of respect to the right side. Mishkis A. D. Elements of the theory of mathematical models. - 3rd species, Vipr. - M: KomKniga, 2007. - 192 s ISBN 978-5-484-00953-4, p. 35.
- « Description of the conceptual model of the system. On the basis of the following details of the system model: a) the conceptual model M is described in abstract terms and concepts; b) a description of the model is given based on the selection of typical mathematical schemes; c) a residual hypothesis is accepted; d) the selection of the procedure for approximating real processes with a prompt model is being developed. Rad B. Ya., Yakovlev S. A., Modeling systems: Navch. for universities - 3rd type., revised. that dod. - M: Vishch. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.
- Blehman I. I., Mishkis A. D.,
There is still no standardized terminology and it is unlikely to appear, but the shards for the history of mathematical modeling and many scientists have dealt with this topic.
Mathematical modeling stagnates in various spheres of human life. Such as, for example: mathematics, biochemistry, medicine, etc.
Designation of the mathematical model, given by A.D. Mishkis.
Let me calculate the total value S of the powers of the object A (object: system, situation, phenomenon, process, and so on). Navismo my future mathematical object A ”- arithmetic spіvvіdnosheniya, geometrical figure, the system of equalities toshcho, following which the methods of mathematics can give in the form of supplying the power of S. U. to this particular type the mathematical object A" is called the mathematical model of the object A along with the totality of powers S. The designated ones give intelligibility not only to those that objects A and A "might differ in nature, but those that A" signify not only the original itself A, a, and sskunistya yogo doslіfezhuvans of the power of S. Yakshcho, two doslizlizni of the one об об qdodnikh pellets s1 і s2 yogo power, vidpovіdnі math "a1 a2 power of mathematical models - їх multiplicity Apparently, what is at stake here is not only the multiplicity of models, due to їх іеrarchіchnostі, and th result of generations of the need for further expansion of different systems, ... S1 S2 yogo authorities.
For example, one and the same mass of dome gloom can be seen as from the point of view of the low winds generated by it, which spread far along the surface of the earth and we see it as a wind blow in front of a cob of strong rain, so it is a zone of high electrical activity of the atmosphere. All of the development of the object to become a high place for the protection of the visited ships. Skhіdnі unsafe streams at the stages of zlotu - landing, through a significant change in the magnitude of the underground force of the wing of the wing of the vessel (a sharp change directly in the wind speed from the sustrichny on the way). Strong electric fields, which are blamed on such a gloom, can create a discharge of atmospheric electricity (bliskavka), as a result of the injection of some kind on an inspected vessel, it may become a new or frequent failure of the radio-electronic equipment on board an inspected vessel. It was understood that in the first case for the model, the aerodynamic dynamics are equal and the field of winding flow fluctuations is maintained (the mathematical model of the totality of the symbol S1). In another way, the electrical structure of the cloud is twisting and it will be an electrodynamic model (showing the sign of S2).
Another, most important power is the unity of mathematical models. The obvious fact is those that different real systems or their replacement models can form one and the same mathematical model.
Vagomim in the theory of mathematical modeling is the constant improvement of all aspects of the model for the purposes of follow-up. To that, it is visible to the foreground the deakі suttєvі for the sake of the specialness mechanical systems that process.
In the first place, factors, which signify such objects, are characterized as the world's values - parameters.
In another way, such models are based on a level that describes the fundamental laws of nature (mechanics), which do not require revision and clarification. Navіt ready private models of okremih vyschi, shcho vykoristovuyutsya when folded more zagalnyh, well-formulated and described from the point of view of minds and areas of congestion.
Thirdly, the magnitude of the change in the development of models of mechanical systems and processes represents a description of the inaccurate characteristics of the object, both functional and numerical.
Fourthly, none of these models lead to the need to pay a lot of factors that are added to the behavior of the object, not only those that are due to the laws of nature. All these features lead to the fact that the models of mechanical systems and processes are considered mainly to the class of mathematical ones.
Mathematical models are based on the mathematical description of the object. The mathematical description, of course, before we think, includes the interrelationship of the parameters of the object, which characterizes its particularity of functioning. Such links can be given at the sight:
Malyunok 2.1.1 - Relationship of object parameters
The first few of the designated species may have a common name: analytical deposits.
Mathematically describe the revenge of the individual on the interrelationship of the elements and parameters of the object (laws and laws), and the latest set of functional and numerical data of the object (characteristics; parameters of the model. This mathematical description is the totality of functions, methods, and calculation data, which allows you to take the result.
However, the mathematical model may include a part of the mathematical inventory (mostly deaks of the data), and, moreover, the descriptions of all the allowances can be spared, the selections should be made, and the algorithms will transfer the current and current data from the model to the original.
Malyunok 2.1.2 - Mathematical description of the model
As an addition to the classification of the mathematical model, depending on the nature of the object, the development of tasks and zastosovuvannyh methods, they can be introduced by the following types:
- Rozrahunkov (algorithms, nomograms, formulas, graphs, tables);
– vіdpovіdnі (butt: model in a wind tunnel and real flight of the aircraft in the atmosphere);
– similar (proportional similar parameters and the same mathematical inventory);
- non-linear and linear (described by functions that can only measure the main parameters in steps 0 and 1, or be types of functions),
- Non-stationary and stationary (deposit or independent per hour),
- discrete or uninterrupted,
- stochastic or deterministic (imovirnіsnі, unambiguously exact: models of mass service, imitatsіyni and іn.),
- fuzzy and fuzzy (apply fuzzy multipliers: close to 10; deep chi dribno; good bad).
Vihodyachi z historical backgrounds it turned out so that under the mathematical model for an hour there is only one particular type of models, which only single-valued direct mathematical description can be found in visually enumerative algorithms, or analytical deposits - that the mathematical model is determined, for the help of which, for some other things, it is impossible to one and the same result. There is a wide range of deterministic models that establish links with the parameters of the original for additional coefficients of proportionality, all of which are equal to one hour at a time. Mathematically describe, vikoristovuvane such a model, naturally look like a description without intermediary to the original - it is true: the model has the same original mathematical description. In the minds of such simplicity of misunderstandings, the engineer understands that the model is no longer like a model, but like an original. However, such a mathematical model is only a model with a lot of simplifications, cleverness, abstractions, omissions, and underpinnings. It is necessary to "forgive" the process of good modeling, which seems to be impossible, because the model either follows the original, or else it does not. Nedbale stavlennya up to tsgogo to bring to faceless pardons in applied studies, and taking away the results do not correspond to the real state of speeches.
As an antipode of deterministic models, the simulation models are presented.
Imitation models (stochastic) - mathematical models of such originals, including elements of such daily analytical type of mathematical inventory. Mathematically describe the imitation models to find in your own case the descriptions of vipadkovyh processes (stochastic). In the capacity of such a description, various forms of laws have been subdivided, which can be put on the basis of a statistical analysis of the results of caution for the original.
Mathematical description of simulation models vipadic values, How to describe the phenomenon, it can include a description of the interrelationships of variable values (for example, for the help of models of the theory of mass service), as well as an algorithm for statistical testing (Monte Carlo method for the implementation of vipadkovy elementary pods). It is clear that the simulation models of vicorists are the mathematical apparatus of the theory of intelligence: mathematical statistics, the theory of mass service, and the method of statistical testing.
The concept of the model and the modeling.
A model for a wide range of minds- whether it be an image, an analogue of manifestations or installations of an image, a description, a diagram, an armchair, a map of something, be it an obligation, a process, or a manifestation that is victorious like a yoga substitute or a representative. The object itself, the process, is called the original of this model.
Modeling - tse doslіdzhennya kakogos ob'єkta chi system ob'єktіv way pobudovi that vyvchennya їх models. The choice of models for the designation or clarification of characteristics and rationalization of methods for stimulating objects that are newly constructed.
Any method of scientific research is based on the idea of modeling, with which in theoretical methods there are different signs, abstract models, in experimental ones - object models.
In case of further folding, the real phenomenon is replaced by a simple copy or a scheme, sometimes such a copy will serve only for the purpose of remembering and in the event of an attack, knowing about the need for the appearance. Sometimes a scheme has been suggested to show the nature of the rice, allowing it to expand into the mechanism of the appearance, giving the possibility of transferring it to change. One and the same thing can be confirmed by different models.
The task of the doslidnik is to convey the nature of the phenomenon and the interruption of the process.
Sometimes, what is an accessible object, but experiments with it are expensive or lead to serious environmental consequences. Knowledge about such processes is taken for the help of models.
An important point is that the very nature of science transfers the discovery of one particular phenomenon, but to a wide class of native phenomena. Beforehand, the need to formulate some kind of blatant categorical assertions, as they are called laws. Naturally, with such a formula, there is no need to go into detail. In order to more clearly reveal the regularity, one should go for roughness, idealization, schematism, so that one does not show the very thing, but more accurately, a copy of the model. Muster the laws of the law about the model, and there is nothing surprising in the fact that sometimes the deacons of scientific theories are recognized as inapplicable. Tse not to lead to the collapse of science, shards one model was replaced by another. more than today.
I emphasize the role of science to play mathematical models, everyday material and tools of these models - mathematical understanding. The stench piled up and drank for thousands of years. Modern mathematics is given exclusively and universally possible to achieve. Practically understanding mathematics, skin mathematical object, starting with understanding numbers, mathematical model. When prompted by a mathematical model, the object, which is being developed, otherwise the phenomena see those particularities, drawings and details, like from one side to cover more or less information about the object, and otherwise allow mathematical formalization. Mathematical formalization means that the features and details of an object can be put in the context of adequate mathematical understanding: numbers, functions, matrices, etc. The same links and vowels, vyyavleni і perebachuvanі in ob'єkti, scho vychaєєєєєєєєєєєєєєєєєєєєєєєі між ехмімій гого details і warehouse parts can be written for additional mathematical vydnosin: evenness, unevenness, equalities. The result will have a mathematical description of the completed process, which is a manifestation, so that it is a mathematical model.
The development of a mathematical model is always tied to the actual rules of action on objects that are being developed. These rules reflect links between causes and consequences.
Pobudov's mathematical model is the central stage in the further development of the design of any system. According to the quality of the model, to deposit the entire analysis of the object. Pobudova model - the procedure is not formal. To lay heavily in the sight of the past, I will bring that relish, always rely on the singing of the material. The model can be accurate, adequate, and can be handy for sampling.
Mathematical modeling.
Classification of mathematical models.
Mathematical models can bedeterminations і stochastic .
Determination models i- tse models, in which it is established mutually-uniquely the difference between the changes to describe the object of the appearance.
Such a hypothesis is based on the known mechanism of the functioning of objects. The object, which is often modeled, is foldable and the decoding of this mechanism can be more laborious and long in an hour. In some way it is necessary to go in this order: to carry out experiments on the original, to process and reject the results, without delving into the mechanism and theory of the object, which is modeled after additional methods of mathematical statistics and . Have a vipadka to gainstochastic Model . At stochastic models of links between them change to have a vipadical character, but in principle. Having poured in a majestic number of factors, each day they bring to a vipadkovy set of significant objects that describe any manifestation. Behind the nature of the regimes, the model isstatistical і dynamic.
statisticalModelincludes a description of the links between the main changes in the object that is being modeled, in the installed mode without improving the change in the parameters of the hour.
At dynamicmodelsdescribes the links between the main changes in the object, which are modeled during the transition from one mode to another.
Models are running discreteі uninterrupted, as well as mixed type. At uninterrupted changes take the value of the current interval,discretechange the insulated value.
Linear models- all functions and blue lines that describe the modelnot linearin a different direction.
Mathematical modeling.
Wimogi , which are presented to models.
1. Universality- characterizes the extent to which the model of the doslidzhuvanih powers of the real object.
- Adequacy - zdatnіst vіdbіvati nebhіdnі vіdnі vіlnostі ob'єkta z pohibkoi not vіdshe zadії.
- Accuracy - the value of the characteristics of a real object and the values of these characteristics, taken away for the help of models, are estimated by the step of zbіgu.
- Economy - Signed by the resources of EOM memory that hour for implementation and operation.
Mathematical modeling.
Main stages of modeling.
1. Statement of the problem.
The purpose is to analyze and analyze that path and achieve the goal of shaping the wild approach to the end of the problem. At this stage, it is necessary to thoroughly understand the essence of the assigned task. Sometimes it is correct to put the task not less smoothly, lower or lower. Staging is not a formal process, wild rules no.
2. The development of theoretical foundations and the selection of information about the object of the original.
At what stage is it possible to choose or develop a different theory. As if nothing, causal-hereditary connections are established between changing descriptive objects. Entry and exit dates are recognized, allowances are accepted.
3. Formalization.
Polyagaє at the choice of the system of mental meanings and with the help of writing down the words between the warehouse objects like mathematical expressions. The class is set up to which one can see the otriman's mathematical model of the object. The values of these parameters at this stage may not be specified.
4. Select the solution method.
At this stage, the residual parameters of the models are restored to improve the functioning of the object. For otrimano ї mathematical problem, either a method of development is chosen, or a special method is developed. When choosing a method, the knowledge of the coristuvach, the value, as well as the value of the retailer, are forfeited.
5. Implementation of the model.
Having developed the algorithm, the program is written, so that it is improved, tested and the solution of the required task comes out.
6. Analysis of the taken information.
There is a decision to cancel the transfer of the decision, to control the error of the model.
7. Rechecking the adequacy of the real object.
Results, subtracted for the model, will be submittedor with explicit object information, or an experiment is carried out and the results are shown with rozrahunkovimi.
The modeling process is iterative. At times of unfavorable results of the stages 6. or 7. it is possible to turn to one of the early stages, which could lead to the development of a recent model. This stage and all steps are being specified, and as the model is refined, the results will not be taken away.
Mathematical model - tse approximations describe whether there is a class of phenomena or objects in the real world of my mathematics. The main meta-modeling is to follow up the objects and convey the results of future warnings. However, modeling is the only method of knowing the necessary light, which allows me to cherish it.
Mathematical modeling and connection with him, the computer experiment is indispensable in quiet situations, if the natural experiment is impossible or difficulties for quiet reasons. For example, it is not possible to set up a natural experiment of history, to distort, “what would be b, yakby ...” It is impossible to distort the correctness of this other cosmological theory. In principle, it is possible, but hardly sensible, to set up an experiment with a wider range of ailments, for example, plague, or create a nuclear vibe, in order to recover this legacy. However, all the whole can be worked on a computer, having in advance the mathematical models of the phenomena that are being developed.
1.1.2 2. Main stages of mathematical modeling
1) Pobudova model. At this stage, some kind of “non-mathematical” object appears - a natural phenomenon, a construction, an economic plan, a manufacturing process, etc. At this stage, as a rule, a clear description of the situation is difficult. On the back of the head, the main features of the phenomenon and the link between them on the Yakish line are revealed. Then the knowledge of some of the deposits is formulated by my mathematics, so that a mathematical model will be formed. The most important stage of modeling.
2) Derivation of the mathematical task, to which point the model. At this stage, great attention is paid to the development of algorithms and numerical methods for solving problems on the EOM, for the help of which results can be obtained with the necessary accuracy in an allowable hour.
3) Interpretation of the contents of the observations from the mathematical model.The findings, derived from the model of my mathematics, are interpreted by mine, adopted by my gallery.
4) Revalidation of the adequacy of the model.At what stage it is necessary to determine which results of the experiment with the theoretical implications of the model in terms of singing accuracy are used.
5) Model modification.At this stage, either an aggravated model is considered, so that it will be adequately effective, or else it should be simplified in order to reach a practically acceptable solution.
1.1.3 3. Classification of models
Models can be classified according to different criteria. For example, the nature of the emerging problems of the model can be subdivided into functional and structural ones. For the first time, all the quantities that characterize the object and the manifestation are clearly pronounced. In this case, some of them are considered as independent changes, others - as functions of these quantities. The mathematical model sounds like a system of equals of a different type (differential, algebraic only. Bud.), establishes the quantity of fallows between the analyzed values. In another way, the model characterizes the structure of a folding object, which is composed of four parts, between which there are simple links. As a rule, qi zv'azki do not fit the kіlkіs vimіr. To inspire such models, it is necessary to use graph theory manually. A graph is a mathematical object, which is a number of points (vertices) on a square and space, a number of lines (ribs).
According to the nature of the output data, the results of the transfer of the model can be subdivided into deterministic and imovirnisno-statistical. Models of the first type give simple, unambiguous forecasts. Models of another type are grounded on statistical information, and the transfer, taken away for their help, may have an imaginative character.
MATHEMATICAL MODELING AND ALL COMPUTERIZATION OF ABO SIMULATION MODELS
At the same time, if in the country there is no blatant computerization, in the case of fakhivtsiv in various professions, a little bit is brought to light: "The axis can be carried out in one's own EOM, then all the tasks will be seen right away." Tsya thought zovsіm is not true, on their own EOM without mathematical models of quiet chi іnshih protsesіv nichogo robiti і about zagalnu kom'yuterizatsіyu can only dream.
On confirmation of what has been said above, we will try to ground the need for modeling, including mathematical, rozkriёmo yogo advance in the known and transformed people of the world of the world, we can see the lack of it and pidemo ... to the imitation of modeling, tobto. modeling with EOM vikoristannyam. Ale, everything is bad.
We are looking forward to the question: what is the model?
The model is the material idea of the object representations, which replaces the original in the process of recognition (development), taking the important factors for the given type of authority.
A good model was made available for follow-up - lower real object. For example, inadmissible experiments with the economy of the country with a knowledge method, here one cannot do without a model.
Summarizing what has been said, you can ask questions about the power supply: why do you need models? In order to
- understand, like a powerful object (yogo structure, authority, laws of development, mutual modality with the necessary light).
- learn how to figure out the object (process) and choose the best strategies
- predict the consequences of the object.
What is positive about any model? Vaughn allows you to take new knowledge about the object, but, unfortunately, it is not known to the other world.
Modelformulated by my mathematicians using various mathematical methods is called a mathematical model.
Vihіdnym item її pobudovi є deyak zavdannya, for example, ekonomіchna. Widely expanded both descriptive and mathematical optimizations that characterize differences economic processes that phenomenon, for example:
- expanded resources
- rational rozkrіy
- transportation
- business consolidation
- mezheve planning.
How is the mathematical model supposed to work?
- In the first place, the meta that subject is formulated.
- Secondly, the most important indications are seen, the most important ones.
- Thirdly, the interrelationships between the elements of the model are verbally described.
- Dalі vzaєmozv'yazok formalіzuєtsya.
- І to conduct a study of the mathematical model and analysis of the final solution.
Vikoristovuyuchi tsey algorithm can be virishiti whether-yaku optimization problem, okrema and rich criteria, tobto. that in which there is not only one, but a sprat of goals, super-clear zocrema.
Let's give an example. The theory of mass service - the problem of the establishment of the black. It is necessary to bring in two factors - a visit to the morning of the outbuildings and a visit to the house for a change. Inducing a formal description of the model to carry out surveys, vicarious analysis and calculation methods. If the model is good, then if the model is good, then if the model is good, then if the model is bad, then it will improve and replace it. The criterion of adequacy is practice.
OPITIMISINI MODOLAI, at that number of bagatocriterILNI, May SPILNU POWER - VIDOMA meta (Abo Kilka Tsilley) for reaching the right to the name of the reservoir systems, do not go about the virisennya, Skilki about the pre -trial what are they getting. And here we are stuck with the difficulties of implementing the colossal plan. The stench pogogayut at the offensive:
- foldable system
- the real system is succumbing to the influx of expansive factors, the appearance of their analytical path of impossibility
- Possibility of establishing the original with the model is only on the cob that after the delay of the mathematical apparatus, tk. intermediate results may be analogous to real systems.
At the link with the overridden difficulties, who blame the shodo folding systems, The practice of vimagala is a more flexible method, and it appeared - imitation modeling "Simujation modeling".
Sound under the simulation model to understand the complex of programs for the EOM, which describes the functioning of four blocks of systems and the rules for interfacing between them. To overcome the need to carry out experiments with a simulation system (on the EOM) and to carry out statistical analysis of the results obtained. To finish with a wider butt of vikoristannya imitation models є virishennya tasks of mass service by the MONTE CARLO method.
In this rank, the robot with a simulation system is an experiment, we are working on the EOM. Why do they have prevails?
-Great proximity to the real system, lower in mathematical models;
- The block principle gives the possibility to verify the skin block before it is included in the system;
– The variety of fallows of a folded nature, not described by simple mathematical expressions.
Changed advantages signify shortfalls
-To encourage the imitation model to be more, more important and more expensive;
- For work with a simulation system, it is necessary to have a valid presence for the EOM class;
- vzaєmodіya koristuvacha and imitation model (interface) can not be folded, handy and kind;
- Pobudova іmіtаtsіynoї model vіmagає bolsh vyvchennya real protsesu, nizhne mathematical simulivannya.
Question: what can the simulation model do to replace the optimization methods? Hі, ale manually add їх. The simulation model is a program that implements a simple algorithm, for optimization of control, the optimization task is violated earlier.
Otzhe, nі EOM, nі mathematical model, nі algorithm on її doslіdzhennya porously scho impromptu vyrishiti to complete the task smoothly. But at once the stench reveals the power that allows you to know navkolishniy svit slander him for the punishment of people.
1.2 Classification of models
1.2.1
|
1.2.2 Classification in the gallery of victoria (Makarova N.A.)
Initial- on the face of it assistants, trainers , about buchayuchi programs |
1.2.3 Classification according to the method of manifestation Makarov N.A.)
Material
models- otherwise
can be called objects. The stench takes on the geometrical and physical power of the original and is sure to be really inspired. |
1.2.4 Classification of models, induced by the book "Land of Informatics" (Gein A.G.))"... the axis is simple at the first glance of the task: how many hours will it take to turn over the desert of Karakumi?" Vidpovid, understood lie down in the way of transferring. Yakscho rise in price for camels, then one term is needed, the second one is like driving a car, the third one is like flying by plane. And the most important - for planning, the cost of different models is more expensive. For the first time, the necessary model can be found in the memoirs of famous past deserters: even here you can’t do without information about the oasis and camel stitches. For another, there is irreplaceable information that can be found in the atlas of automobile routes. For the third one, you can speed up the layout of the flights. |
1.2.5 The classification of models was given by the help of A.I.BochkinWays to classify supernaturally rich .Reduced less deyakі, most vіdomі pіdstavi ta signs: discretenessі continuity, matrix those scalar models, static and dynamic models, analytical and information models, subject and figurative-sign models, scale and non-scale... |
EOM mіtsno veiled into our lives, and practically there is no such gallery of human activity, de not zastosovuvaetsya b EOM. EOM is widely victorious at the same time in the process of creation and follow-up of new machines, new technological processes and the search for optimal options; at the hour of the ceremony of economic tasks, at the hour of the ceremony of the ceremony of planning and managing the production of various equals. The creation of great objects in rocket technology, aviation, shipbuilding, and design of rowing, bridges, and others. vzagali is impossible without zastosuvannya EOM.
For the selection of the EOM for the execution of applied tasks, the first for all applied tasks can be "transferred" to the formal mathematical language, then. for a real object, the process of the system may be inspired by a mathematical model.
The word "Model" resembles the Latin modus (copy, image, contour). Modeling is the replacement of the current object A by another object B. Object A, which is being replaced, is called the original or the object of modeling, and replacing B is the model. In other words, the model is the object-substitute for the original object, which ensures the transfer of certain powers to the original.
The method of modeling is otrimannya, processing, submission of that vikoristannya іnformatsiї about objects, yakі vzaєmodіyut among themselves that zvonіshnіshnіm sredovischem; and the model here stands as a recognition of the features and regularities of the behavior of the object.
Mathematical modeling - tse zasіb vyvchennya real object, process chi system way їх replace the mathematical model, zruchnіshoyu for experimental follow-up for additional EOM.
Mathematical modeling - the process of inducing and developing mathematical models of real processes and phenomena. All natural sciences and branches of science that victorious mathematical apparatus, in fact, are engaged in mathematical modeling: they replace the real object of the yoga model and then we continue the rest. As if in times of modeling, a mathematical model does not describe a phenomenon that is constantly being developed, and food about the stability of taking away such a rank of results is even more significant. A mathematical model is the only way to describe the reality for the help of mathematical understanding.
The mathematical model reflects the essence of the object and the process of my work and other mathematical problems. Vlasne, the very mathematics of the goiter is due to its own reasons for the fact that it is supposed to be imagined, tobto. to model, my own specific regularity of the current world.
At mathematical modeling The development of the object is based on a supplementary model, formulated by my mathematics with the help of other quiet mathematical methods.
The path of mathematical modeling in our hour is richer all-season, lower modeling of full-scale. The majestic development of mathematical modeling gave rise to EOM, although the method itself was born overnight from mathematics a thousand years ago.
Mathematical modeling, as such, does not rely on computer support. A leather fahivets, who is professionally engaged in mathematical modeling, to do everything possible for an analytical follow-up model. Analytical solutions (tobto represented by formulas that reflect the results obtained through external data) sound easier and more informative than numerical ones. The feasibility of analytical methods in the development of folding mathematical problems, however, is more common and, as a rule, these methods are richly collapsible for numerical ones.
Mathematical model for approximation of real objects, processes of systems, expression in mathematical terms and taking the essence of drawings from the original. Mathematical models in a calculus form, with additional logical and mathematical constructions, describe the main power of the object, the process of the system, its parameters, internal and external connections.
All models can be divided into two classes:
- speech,
- ideal.
You can divide your speech models into:
- nature,
- physical,
- mathematical.
Ideal models can be subdivided to:
- at first,
- signs,
- mathematical.
Speech natural models are real objects, processes and systems, over which the experiments of science, technology and virobnichie are vibrated.
Speech physical models- all mock-ups, models that create the physical power of the originals (kinematic, dynamic, hydraulic, thermal, electric, light models).
Speech mathematical - all analogue, structural, geometric, graphic, digital and cybernetic models.
Ideal scientific models - circuit diagrams, maps, armchairs, graphs, graphs, analogues, structural and geometric models.
Ideal sign models - all symbols, alphabet, movie programming, ordering of records, topological record, framing of appearance.
Ideal mathematical models - analytical, functional, simulation, combined models.
At the guidance of the classification, the existing models may be underwhelming clouding (for example, analog). All models, krіm natural, you can go up to one class of obvious models, tk. є a product of the abstract thought of a human being.
Elements of Gri theory
At the end of the day, it’s a good idea to finish the task, and the complexity of the task and the necessary decision to calculate it sharply increases in size. However, the problems are not of a principled nature and are caused only by a great obligation of rozrachunkiv, which in a number of cases may appear practically unimaginable. An important side to the method of asking for a solution is left for whatever one and the same.
Illustrated on the butt of gr. Damo їy geometrical іnpretatsіyu - vzhe prostorov. Our three strategies, represented by three dots on the plane ; persha lie on the cob of coordinates (Fig. 1). friend and third - on axes Ohі OU on vіdstanі 1 vіd cob.
Axes I-I, II-II and III-III, perpendicular to the plane, are drawn through the points. . On the axis I-I, there are wins for the strategies; on the axes II-II and III-III - the wins for the strategies. Skin strategy of the enemy to be depicted as a flat area that you can see on axes I-I, II-II and III-III
with different strategies, that strategy . Inducing, in such a rank, the strategy of the opponent, we take away the family of flats above the trikutnik (Fig. 2).
For this family, it is also possible to induce the lower boundary of the vigrash, as we fought at the fall, and to know on the border of the cordon the point N with the maximum height over the area . Tsya height and will be the price of gr.
The frequencies of the strategies in the optimal strategy will be indicated by the coordinates (x, y) points N, and itself:
However, such a geometrical urge to wake up for a change is not easy to achieve and will require a great deal of time and effort. In a wild temperament, it can be transferred to - a peaceful expanse and use it as if it were sharpness, although the introduction of geometric terminology in a row of vibrations can appear corny. With the improvement of Igor, it is practical to use not geometrical analogies, but rozrachunk analytical methods, moreover, with the improvement of the calculation machines, the methods and single appendages.
All these methods essentially lead to the completion of the task by the way of the last samples, but the ordering of the sequence of samples allows you to induce an algorithm that leads to the completion in the most economical way.
Here we briefly mention one of the Rozrakhan's method - on the so-called method of "linear programming".
For this lady, I’ll start by stating the problem about the significance of the solution to the problem. Come on dana gra s t engraving strategies BUTі n engraving strategies At and the payoff matrix is given
It is necessary to know the solution of gr, so that the two optimal changes in the strategy of gravity A and B
de (days of numbers can be equal to zero).
Our optimal strategy S*A is responsible for ensuring that we win, no less, for whatever the enemy’s behavior, and win, even, with his optimal behavior (strategy S*B). Similarly, strategy S*B is obliged to secure the enemy’s program, no greater, if our behavior is equal and equal to our optimal behavior (strategy S*A).
Rozmіr tsіni gri u razі us nevіdoma; we will respect that she is dear to deakom positive number. In this way, we do not destroy the sleepiness of the world; if bulo > 0, obviously, it is sufficient if all elements of the matrix are non-negative. What can be attained, adding to the elements to achieve a large positive value L; under which price gri will increase by L, but the decision will not change.
Let me choose your optimal strategy S&A. That is our average win with the opponent's strategy of dominance:
Our optimal strategy S*A Volodye tim vlastivistyu, scho be-yakіy behavior of the enemy will ensure the safety of winning no less, nizh; otzhe, whether it be s numbers may be less. We take low minds:
(1)
We divide the unevenness (1) into a positive value and it is significant:
Todі umova (1) sign up with the viewer
(2)
de - Invisible numbers. so yak 
magnitudes please the mind
We want to increase our guarantees to win as much as possible; Evidently, in its right, part of the equality (3) takes on the minimum value.
In this order, the task of znakhodzhennya solution gri is led to an offensive mathematical problem: calculate unknown quantities , what to please the minds (2), so, schob їx sum
was minimal.
Sound the hour of the day when the day is due, which, according to the extreme values (maximums and minimums), the function of differentiation and equate to zero. But such a trick for a given type of disrespect, for that function Ф, like necessary turn to the minimum, linearly, and її pokhіdnі for all the arguments, do it alone, so that nowhere do not go back to zero. Later, the maximum of the function is reached here on the inter-field of change of arguments, which is determined by the incomprehensibility of the arguments by the minds (2). By accepting the significance of extreme values for additional differentiation, it is unacceptable and in quiet moods, if the maximum of the lower (or the minimum of the upper) between is won, like mi. for example, they robbed at the cherry of Igor. Indeed, the lower boundary is folded from straight lines, and the maximum is reached not at the point where it is close to zero (there is no such point), but at the interval or at the point of the cross section of straight lines.
For the implementation of similar tasks, which are often used in practice, a special apparatus is developed in mathematics line programming.
The line programming manager is set in this way.
Given system linear rivers:
(4)
It is necessary to know the unknown values of the quantities that satisfy the minds (4) and at the same time to use at least a given uniform linear function of the quantities (linear form):
It is easy to perekonatisya, scho posed higher than the task of the theory of Igor є we will call the problem of linear programming with 
From the first glance, you can get away that your mind (2) is not equivalent to minds (4), shards replace the signs of equivalence and replace the signs of nervousness. However, in the face of signs of unevenness, it is easy to get lost, introducing new fictitious, invisible changes and writing down the mind (2) at the sight:
(5)
Form F
Linear programming device allows a small number of last samples to choose the size , what to satisfy we will put vimogs. For greater clarity, we will demonstrate the installation of this device directly on the materials of specific games.
Enthusiasm...