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Author: anonymous
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1, the origin and history of mathematical model competition
Mathematical model competition is a college student competition activity initiated by American Institute of Industrial and Applied Mathematics on 1985. Its purpose is to stimulate students' enthusiasm for learning mathematics, improve their comprehensive ability to use computer technology to build mathematical models and solve practical problems, encourage students to actively participate in extracurricular scientific and technological activities, broaden their knowledge, cultivate innovative spirit and cooperative consciousness, and promote the reform of college mathematics teaching system, teaching content and teaching methods. The National Mathematical Modeling Competition for College Students is an annual communication competition for colleges and universities nationwide sponsored by the Ministry of Education of People's Republic of China (PRC) and the People's Republic of China (PRC) Mathematical Society. Its purpose is: innovation.
Knowledge, team spirit, focus on participation and fair competition. 1992 was founded in China. Since its establishment, it has been strongly supported and cared by the Ministry of Education of People's Republic of China (PRC) and China Industrial and Applied Mathematics Association, showing a rapid development momentum. As far as 2003 is concerned, the registration stage was definitely affected by SARS, but there were 5406 teams from 637 universities in 30 provinces (municipalities and autonomous regions) and Hong Kong, and the number of vocational and technical colleges increased even faster. It can be said that mathematical modeling has become the largest extracurricular scientific and technological activity in colleges and universities in China.
2. What is mathematical modeling?
Mathematical modeling (mathematics
Modelling) is a mathematical thinking method, that is, "through mental activities, often visual or symbolic, construct a representation of realistic phenomena that can grasp its important and useful characteristics." From the perspective of science, engineering, economy and management, mathematical modeling is a powerful mathematical tool, which can approximate and "solve" practical problems through abstraction and simplification with mathematical language and methods. As the name implies, the word modeling means "shaping art" in English, so it can be understood that when we look at problems from different sides and angles, there will be endless mathematical models, so as to establish mathematical models.
Creation has certain artistic characteristics. The most important feature of mathematical modeling is to accept the test of practice and modify the process of model improvement many times.
3, the content of the competition
Competition topics generally come from practical problems that have been properly simplified in engineering technology and management science. Participants are not required to master in-depth professional knowledge in advance, but only need to have studied mathematics courses in ordinary universities. The topic has great flexibility, allowing participants to exert their creative ability. Participants should complete a paper (answer sheet) including model hypothesis, establishment and solution, design and computer realization of calculation method, analysis and test of results, improvement of model, etc. Competition awards are based on the rationality of assumptions, the creativity of modeling, the correctness of results and the clarity of text expression.
4, the steps of competition
Modeling is a very complicated creative labor. There are all kinds of things in the real world, and it is impossible to use some rules and regulations.
Box specifies how to build various models, and here is just a summary of the general steps and principles of modeling:
1) model preparation: First of all, we should understand the actual background of the problem, clarify the requirements of the topic, and collect all kinds of necessary information.
2) Model assumption: In order to use mathematical methods, it is usually necessary to make necessary and reasonable assumptions about the problem, so as to highlight the main characteristics of the problem and ignore the secondary aspects of the problem.
3) Model composition: according to the assumptions made and the relationship between things, the relationship between various quantities is constructed to solve the problem.
4) Model solving: Solving the mathematical problems obtained in the previous step by using known mathematical methods often requires further simplification or assumption. For mathematical problems, pay attention to using simple mathematical tools as much as possible.
5) Model analysis: analyze the obtained solution, and pay special attention to whether the obtained result is stable when the data changes.
6) Model test: analyze the actual meaning of the obtained results and compare them with the actual situation to see if they are in line with the reality. If it is not ideal, it is necessary to modify and supplement the hypothesis, or re-model and constantly improve it.
7) Model application: The established model must be used in practical application to generate benefits, and it will be continuously improved and perfected in application.
5. Classification of models
According to the application field of the model.
Biological mathematical model
Medical mathematical model
Geological mathematical model
Quantitative economic model
Mathematical sociological model
Depending on whether random factors are considered.
deterministic model
Stochastic model
Depending on whether the change of model is considered.
static model
Dynamic model
Depending on the applied discrete method or continuous method
discrete model
Continuous model
According to the mathematical method of establishing the model.
Geometric model
Differential equation model
Graph theory model
Planning theoretical model
Markov chain model
According to people's understanding of the development process of things
White box model:
Refers to those models with clear internal laws. Such as mechanics, heat, electricity and related engineering and technical problems.
Grey box model:
It refers to those problems whose internal laws are not very clear, and there is still a lot of work to be done to establish and improve the model in different degrees.
Such as meteorology and ecological economics.
Black box model:
Refers to some phenomena whose internal laws are still unknown. Such as life science and social science. However, due to many influencing factors and complicated relationships, it can also be simplified as a grey box model to study.
6. The application of mathematical modeling
Today, mathematical modeling has very specific applications in the following aspects of national economy and social activities.
Analysis and design
For example, describe the changing law of drug concentration in human body to analyze the curative effect of drugs; The mathematical model of transonic airflow and shock wave is established, and a new type of aircraft airfoil is designed by numerical simulation.
Forecasting and decision-making
In the production process, there must be prediction models such as product quality index prediction, meteorological prediction, population prediction and economic growth prediction. The price strategy to maximize economic benefits and the equipment maintenance scheme to minimize costs are examples of decision-making models.
Control and optimization
The optimal control of electric power and chemical production process and parameter optimization in part design should be based on mathematical models. It is an urgent and arduous task to establish a mathematical model for control and optimization of large-scale systems.
Planning and management
Production planning, resource allocation, transportation network planning, reservoir optimal operation, queuing strategy and material management can all be solved by operational research model.
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