Simply put, the mathematical model contest is a mathematical expression of practical problems. Specifically, a mathematical model is an abstract and simplified mathematical structure of some real worlds for a certain purpose. More precisely, a mathematical model is a mathematical structure obtained by simplifying some necessary assumptions and using appropriate mathematical tools according to its unique internal laws for specific objects and specific goals. Mathematical structures can be mathematical formulas, algorithms, tables, charts, etc. Mathematical modeling is to establish a mathematical model, and the process of establishing a mathematical model is the process of mathematical modeling (see the flow chart of mathematical modeling process). Mathematical modeling is a mathematical thinking method, and it is a powerful mathematical means to describe and "solve" practical problems by using mathematical language and methods through abstraction and simplification.
Two. origin
1985 In the United States, an annual mathematical model of college students named MCM appeared (1987 was called modeling mathematics competition, and 1988 was renamed modeling mathematics competition, both written by MCM). This is no accident. Before 1985, there was only one kind of mathematics competition for college students in the United States (William Lowell Putman Mathematics Competition for short), which was sponsored by the American Mathematical Association (MAA-abbreviation of American Mathematical Association) twice a year and held on the first Saturday of 12. It has a great influence in the world and has now become a famous competition for international college students. The competition is held in February or March every year.
China participated in this competition for the first time since 1989, and achieved excellent results in previous years. After several years' participation in American competitions, it shows that China university students are competitive and innovative in mathematical modeling. In order to make this competition more extensive, the Chinese Society of Industrial and Applied Mathematics and the State Education Commission jointly sponsored the National Mathematical Modeling Competition for College Students (CMCM) 1990, which is held every September. Mathematical model competition is different from ordinary mathematical competition because it comes from practical problems or has a clear practical background. Its purpose is to cultivate college students' consciousness and ability to solve practical problems by mathematical methods. The whole competition is to complete a paper including problem analysis, model hypothesis and establishment, calculation results and discussion. Through training and competition, students not only greatly improve their consciousness and ability to solve practical problems by mathematical methods, but also get a very beneficial exercise in unity and cooperation, giving full play to collective strength to tackle key problems and writing scientific and technological papers.
Three. Method introduction
Firstly, the mechanism analysis method deduces the model from the basic physical laws and structural data of the system.
1. Proportional analysis-the most basic and commonly used method to establish the functional relationship between variables.
2. Algebraic method-the main method to solve discrete problems (discrete data, symbols, graphics).
3. Logical method-an important method of mathematical theory research, is widely used in the decision-making and countermeasures of practical problems in sociology and economics.
4. Ordinary differential equation-The key to solving the change law between two variables is to establish the expression of "instantaneous change rate".
5. Partial differential equation-solving the change law between the dependent variable and more than two independent variables.
Secondly, the data analysis method establishes a mathematical model from a large number of observation data through statistical methods.
1. regression analysis method-function expression (xi, fi) i = 1, 2 ... n of function f(x) used to determine a set of observations. Because it deals with static independent data, it is called mathematical statistics method.
2. Time series analysis-dealing with dynamic related data, also known as process statistics.
Third, simulation and other methods
1. computer simulation-essentially a statistical estimation method, equivalent to sampling test.
(1) discrete system simulation-there is a set of state variables.
(2) Continuous system simulation-there are analytical expressions or system structure diagrams.
2. Factor test method-local test of the system, and then continuous analysis and modification according to the test results to get the required model structure.
3. Artificial reality method-based on the understanding of the past behavior and future goals of the system, and taking into account the possible changes in related factors of the system, artificially form a system.
(See: Qi Huan's Mathematical Model Method, Huazhong University of Science and Technology Press, 1996).
IV. Types of problems
The structure of the competition question has three basic components:
First, the background of practical problems
1. It covers a wide range-including society, economy, management, life, environment, natural phenomena, engineering technology, new problems in modern science and so on.
2. Generally, there are more exact practical problems.
Second, some assumptions are as follows:
1. There are only qualitative assumptions such as processes and rules, but no specific quantitative data;
2. Give some measurement or statistical data;
3. Give some parameters or graphs;
4. There are some supplementary assumptions that can be maneuvered and exerted, or participants can generate data according to their own collection or simulation.
Third, there are often several questions to answer (generally not the only answer):
1. Comparative deterministic answer (basic answer);
2. More detailed or higher-level discussion results (often discussing the formulation and results of the optimal scheme).
ⅴ. Postgraduate Mathematical Model Competition
Submit a paper, the basic content and format are roughly divided into three parts:
I. Title and summary:
1. Title-Write a more exact title (not just A and B).
2. Abstract -200-300 words, including the main features, modeling methods and main results of the model.
3. It is best to have a catalogue when there are many contents.
Second, the central part:
1. Ask questions and analyze them.
2. Model establishment: ① Supplementary assumptions, clear concepts and introduction of parameters; ② Model form (there can be multiple models); ③ model solving; ④ Model properties;
3. Design of calculation method and computer realization.
4. Result analysis and testing.
5. Discussion-the advantages and disadvantages of the model, improving the direction and popularizing new ideas.
6. References-pay attention to the format.
III. Appendix:
1. Calculation program, block diagram.
2. Various calculus solving processes and intermediate results.
3. Various charts.