First, the definition of mathematical model

At present, there is no uniform and accurate definition of mathematical model, because different angles can have different definitions. However, we can give the following definition: "A mathematical model is an abstract and simplified structure about a part of the real world for a special purpose." Specifically, a mathematical model is an equation or inequality established by letters, mathematics and other mathematical symbols, and it is a mathematical structural expression that describes the characteristics of objective things and their internal relations, such as charts, images, block diagrams, etc. Generally speaking, the mathematical modeling process can be represented by the following block diagram:

Mathematics is produced in the demand of practical application, and it is necessary to establish a mathematical model to solve practical problems. In this sense, mathematical modeling has an ancient history like mathematics. For example, Euclid geometry is an ancient mathematical model, and Newton's law of universal gravitation is also a shining example of mathematical modeling. Today, mathematics has penetrated into other scientific and technological fields with unprecedented breadth and depth. In the past, the field of mathematics was rarely used, but now it is rapidly moving towards quantification and quantification, and a large number of mathematical models need to be established. Especially with the vigorous rise of new technologies and new processes and the popularization and wide application of computers, mathematics plays a very key role in many high and new technologies. Therefore, mathematical modeling has been endowed with more important significance by the times.

Second, the methods and steps of establishing mathematical model

1. Model preparation

It is necessary to understand the actual background of the problem, clarify the purpose of modeling, collect all kinds of necessary information, and try to understand the characteristics of the object.

2. Model assumptions

According to the characteristics of the object and the purpose of modeling, it is a crucial step to simplify the problem reasonably and make assumptions with accurate language. If all the factors of the problem are taken into account, it is undoubtedly a courageous act and the method is very poor. Therefore, a superb modeler can give full play to his imagination, insight and judgment, be good at distinguishing priorities, and linearize and homogenize problems as much as possible in order to simplify the handling methods.

3. Model composition

According to the assumptions made, the causal relationship of the object is analyzed, and the equation relationship between various quantities or other mathematical structures are constructed by using the internal laws of the object and appropriate mathematical tools. At this time, we will enter a vast world of applied mathematics, where there are many lovely children at the knees of the elderly with high numbers and probabilities. They are graph theory, queuing theory, linear programming, game theory and many other theories. They are really a great country with unique views. But we should remember that the mathematical model is established for more people to understand and apply, so the simpler the tool, the more valuable it is.

4. Model solving

We can use all kinds of traditional and modern mathematical methods, especially computer technology, such as solving equations, drawing pictures, proving theorems, logical operations, numerical operations and so on. Solving a practical problem often requires complicated calculation, and in many cases, the system operation has to be simulated by computer, so programming ability and familiarity with mathematical software packages are very important.

5. Model analysis

Mathematically analyze the model solution. "From the other side of the mountain, the distance is different." Whether you can analyze the model results carefully and accurately determines whether your model can reach a higher level. Also remember that in either case, error analysis and data stability analysis are needed.

Third, the guiding ideology of mathematical model competition

The traditional mathematics competition generally emphasizes theoretical knowledge, with simple content and clear data, which cannot be completed by a calculator. In this regard, mathematical model competition is a "discipline", which mostly comes from the process of production practice or scientific research. It is a comprehensive problem with a huge amount of data, which needs a computer to complete. The answer is often not unique (the mathematical model is an actual simulation, an approximate expression of the actual problem, and its completion is under reasonable assumptions, so it can only be excellent, not unique), and the reported result is a "paper". It can be seen that "Mathematical Model Competition" focuses on application, which is a comprehensive ability competition with mathematical knowledge as the guide and computer application ability and article writing ability as the supplement.

Four, the common problems in the competition

The structure of the competition question has three basic components:

1. Background of practical problems

It covers a wide range-including society, economy, management, life, environment, natural phenomena, engineering technology, new problems in modern science and so on. Generally, there are more precise practical problems.

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There are the following situations:

1) There are only qualitative assumptions such as processes and rules, but no specific quantitative data;

2) Give some measured or statistical data;

3) Give some parameters or graphs;

4) There are some supplementary assumptions that can be maneuvered and played, or contestants can generate data according to their own collection or simulation.

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There are often several questions, and they are generally not the only answers. Generally includes the following two parts:

1) Compare the definitive answer (basic answer);

2) More detailed or higher-level discussion results (often discussing the formulation and results of the optimal scheme).

4 model solving.

A. When a mathematical proposition needs to be established:

Proposition narrative should conform to the norms of mathematical propositions and be as rigorous as possible.

B need to explain the principle, thinking, basis and steps of the calculation method or algorithm.

If existing software is used, explain the reason for using the software and the name of the software.

C. In the calculation process, intermediate results are unnecessary and should not be listed.

D. Try to work out a reasonable numerical result.

5. Analysis and test of the results; Model checking and model modification; The results show that.

A. the correctness or rationality of the final numerical results is the first;

B, carrying out necessary tests on numerical results or simulation results;

When the result is incorrect, unreasonable or has a large error, analyze the reasons and modify and improve the algorithm, calculation method or model.

C. The questions, numerical results and conclusions required to be answered in the topic should be listed one by one;

D. List data: consider whether it is necessary to list multiple groups of data, or compare and analyze the data with additional data, so as to provide a basis for proposing various schemes;

E. Expression of results: it should be centralized, clear at a glance, intuitive and easy for comparative analysis.

Verb (abbreviation of verb) modeling concept

1. Application consciousness

To solve practical problems, the results and conclusions should be in line with reality;

The model, method and results should be easy to understand and convenient for practical application; Think and deal with problems from the standpoint of users.

2. Mathematical modeling

Solving problems by mathematical methods requires mathematical models;

The mathematical abstraction of the problem model is universal and scientific, and is not limited to the solution of this specific problem.

3. Innovative consciousness

Modeling has its own characteristics, which is more reasonable, scientific, effective and practical. More universal application significance; Not just for innovation.