Introduction to Mathematical Modeling Knowledge for Xiaobai

1. Definition of Mathematical Model

Now the mathematical model does not have a unified and accurate definition, because there can be different definitions from different angles. However, we can give the following definition: "A mathematical model is an abstract, simplified structure about part of the real world and made for a special purpose." Specifically, a mathematical model is for a certain purpose, using letters, mathematics Equations or inequalities established with mathematical symbols and other mathematical symbols, as well as mathematical structural expressions such as diagrams, images, and block diagrams that describe the characteristics of objective things and their internal connections.

Mathematics is produced in the needs of practical applications, and mathematical models must be established to solve practical problems. In this sense, mathematical modeling has the same ancient history as mathematics. For example, Euclidean geometry is an ancient mathematical model, and Newton's law of universal gravitation is a shining example of mathematical modeling. Today, mathematics has penetrated into other fields of science and technology with unprecedented breadth and depth. Fields where mathematics was rarely used in the past are now rapidly becoming quantitative and quantitative, requiring the establishment of a large number of mathematical models. Especially with the booming of new technologies and new processes, the popularization and wide application of computers, mathematics plays a key role in many high and new technologies. Therefore, mathematical modeling is endowed with more important meaning by the times.

2. Methods and steps for building mathematical models

1. Model preparation

To understand the actual background of the problem, clarify the purpose of modeling, collect the necessary information, and try to clarify the characteristics of the object.

2. Model assumptions

According to the characteristics of the object and the purpose of modeling, necessary and reasonable simplification of the problem and making assumptions in precise language are crucial steps in modeling. If all the factors of the problem are considered, it is undoubtedly a courageous but poor method. Therefore, a superb modeler can give full play to imagination, insight and judgment, and is good at distinguishing primary and secondary, and in order to make the treatment method Simple, the problem should be linearized and homogenized as much as possible.

3. Model composition

Analyze the causal relationship of the object according to the assumptions made, use the internal laws of the object and appropriate mathematical tools to construct the equation relationship between various quantities or other mathematical structures. At this time, we will enter a vast world of applied mathematics. There are many lovely children under the knees of old people with high mathematics and probability. They are graph theory, queuing theory, linear programming, game theory and many others. Great country, there are holes in the sky. However, we should keep in mind that the establishment of mathematical models is to allow more people to understand and apply them, so the simpler the tool, the more valuable it is.

4. Model solving

Various traditional and modern mathematical methods, such as solving equations, drawing graphics, proving theorems, logic operations, and numerical operations, especially computer technology, can be used. The solution of a practical problem often requires complicated calculations, and in many cases, the operation of the system must be simulated by a computer. Therefore, the ability to program and be familiar with mathematical software packages is very important.

5. Model analysis

Perform a mathematical analysis of the model solution. "Looking horizontally, you can see mountains and peaks, and the distances and distances are different." Whether you can make a detailed and precise analysis of the model results determines whether your model can reach a higher level. Also remember that no matter what the situation is, error analysis and data stability analysis are required.

Example: There are several chickens and rabbits in a cage. It is known that they have 8 heads and 22 feet in total. How many chickens and rabbits are there in the cage?

Solution: Assuming that there are x chickens and y rabbits in the cage, from the known conditions
x+y=8
2x+4y=22

After solving the above binary equation, the solution x=5, y=3, that is, there are 5 chickens and 3 rabbits in the cage. Substituting this result into the original question for verification shows that the desired result is correct.

According to the example, the following mathematical modeling steps can be obtained:

1. Make assumptions based on the background of the problem and the purpose of modeling (this question implicitly assumes that chickens and rabbits are normal, except for deformed chickens and rabbits)
2. Indicate the required unknown with a letter
3. List mathematical formulas or graphs based on known common sense (the common sense in this question is that chickens and rabbits have one head and chickens have 2 legs, and rabbits have 4 legs)
4. find solutions to mathematical expressions
5. Verify the correctness of the obtained results

This is the general procedure for mathematical modeling

3. The guiding ideology of the questions in the digital simulation competition

Traditional mathematics competitions generally focus on theoretical knowledge. The content to be tested is single, the data is simple and clear, and calculators are not allowed to complete. In this regard, the math-analog competition question is a "topic", most of which originate from the actual production or the process of scientific research. It is a comprehensive problem with huge data and needs to be completed by computer. 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 certain reasonable assumptions, so it can only be better, not unique), and the reported results is a thesis. It can be seen that the "mathematical and analog competition" focuses on application. It is a competition of comprehensive ability based on mathematical knowledge to guide computer application ability and supplemented by article writing ability.

4. Common question types in the competition

There are three basic components in the question structure of the competition:

1. Practical problem background

It involves a wide range of topics - including society, economy, management, life, environment, natural phenomena, engineering technology, new problems emerging in modern science, etc. Generally, there is a more precise practical problem.

2. some assumptions

There are several situations as follows:

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 several parameters or graphs;
4. Contains certain flexible and playable supplementary assumptions, or contestants can collect or simulate data according to themselves.

3. Questions to answer

There are often several questions, and generally not a single answer. Generally includes the following two parts:

1. More deterministic answers (basic answers);
2. More detailed or higher-level discussion results (often discussing formulations and results of optimal solutions).

5. To submit a paper, what is the basic content and format?

To submit a paper, the basic content and format can be roughly divided into three parts:

1. Title, abstract section

Topic - Write a more precise topic (you can't just write A and B questions).
Abstract – 200-300 words, including main features of the model, modeling methodology and main results.
It is better to have a directory when there are many contents.

2. Center part

1) Ask questions and analyze them.
2) Model establishment:
① Supplementary assumptions, clarification of concepts, and introduction of parameters;
② Model form (multiple forms of models are possible);
③ Model solution;
④ Model properties;
3) Calculation method design and computer implementation.
4) Result analysis and inspection.
5) Discussion - the advantages and disadvantages of the model, the direction of improvement, and the promotion of new ideas.
6) References - There is also a specific format.

3. Appendices

Calculation program, block diagram.
Various solving calculus processes and calculation of intermediate results.
Various graphs and tables.
(The thesis has its strict format, here is just a brief statement, the detailed content is reserved for the next issue, please watch)

6. Do I need to learn a lot of knowledge to participate in the mathematical modeling competition?

There is no need to learn a lot of mathematical knowledge systematically, which is not allowed by time and energy. The brilliance of many excellent papers is not the amount of mathematical knowledge used, but the comprehensive thinking, practicality, problem solving or innovation. Sometimes, you may encounter some knowledge that you have not learned in the paper, what should you do? Use what you have learned now, and the mathematical knowledge used in excellent papers is most likely to be used in mathematical modeling competitions. Of course, it is necessary for you to look it up.
Specifically, there are probably the following three aspects:

1. Ability to apply mathematical knowledge

In summary, there are generally the following categories:
1) Probability and mathematical statistics
2) Overall planning and axis programming
3) Differential equations;
related basic knowledge of mathematics includes
1, linear programming 6, optimization theory
2, nonlinear programming 7, management operations Science
3, Discrete Mathematics 8, Difference Equation
4, Probability and Statistics 9, Analysis Hierarchy
Process 5, Ordinary Differential Equations
There is also knowledge intersecting with computer knowledge: computer simulation.
Some students have never learned the above content, and some students have only learned a little probability and mathematical statistics. What about the knowledge of differential equations? One word "self-study", I remember the teacher in charge of digital and analog marking once said that "the answer sheet that can solve the answer sheet that others can solve with advanced theory is a better answer sheet with the simplest and most simple mathematical method."

2. Computer skills

Generally speaking, all students who have participated in the mathematics competition can skillfully apply the word processing software "Word", master the use of the electronic form "Excel"; the use of "Mathematica" software, it is best to have language skills. Most of this knowledge is learned by the students themselves in their spare time.

3. Thesis writing skills

As mentioned above, the full text of the examination paper is in the form of a thesis, and the writing of the article has a relatively strict format. It is not easy to express one's thoughts clearly. Sometimes one question is not clearly stated before another question is said. The teachers who graded the papers had a consensus that if an article was read for more than 10 minutes and still did not arouse interest, the article is likely to be sidelined this time.

7. How to learn problem-solving methods from modeling examples

When looking at the sample questions, it depends on how the sample questions are done, that is, how to cut in, how to choose reasonable assumptions, how to analyze the established model, etc. Common mathematical modeling methods are:

1. Mechanistic analysis derives a model from fundamental physical laws and structural data of the system.

1. Proportional Analysis – The most basic and most commonly used method of establishing functional relationships between variables.
2. Algebraic methods – the main method for solving discrete problems (discrete data, symbols, graphs).
3. Logical method is an important method in the study of mathematical theory. It is widely used in decision-making and countermeasures for practical problems in the fields of sociology and economics.
4. Ordinary Differential Equation – To solve the law of change between two variables, the key is to establish the expression of "instantaneous rate of change".
5. Partial Differential Equations – Solve the law of variation between a dependent variable and more than two independent variables.

2. Data analysis method uses statistical methods to establish mathematical models from a large amount of observation data

1. Regression analysis method – used for a set of observations (xi, fi) i=1,2,…,n of the function f(x), to determine the expression of the function, because it deals with static independent data, it is called Mathematical statistics methods.
2. Time series analysis method – dealing with dynamic related data, also known as process statistics method.

3. Simulation and other methods

1. Computer simulation (simulation) – essentially a statistical estimation method, equivalent to a sample test. ① Discrete System Simulation – There is a set of state variables. ② Continuous system simulation - with analytical expressions or system structure diagrams.
2. Factor test method – do local tests on the system, and then carry out continuous analysis and modification according to the test results to obtain the required model structure.

8. How should the work be divided in the group?

The traditional standard answers are - math, programming, writing

In fact, the division of labor does not need to be so clear, but there is a premise that everyone has a good relationship. Otherwise, conflicts will easily arise. The division of labor is too clear, which will make people feel dependent and unwilling to use their brains. The ideal division of labor is like this: everyone in the Mathematical Modeling Contest team can do the work of others, even if there is only her (him) left in the team, they can still handle the Mathematical Modeling Contest. The division of labor in the competition is only to improve work efficiency and make better results.

Specific suggestions are as follows:

There must be a person with a more active mind and good at thinking about problems. This person can barely be attributed to mathematics; there must be a person who can program and realize some algorithms. In addition, there needs to be a paper that is well written, but it doesn't matter if it is not well written. Just read more excellent papers from others and use Office a few more times.

Mathematical modeling is a kind of scientific research work, which requires a team thinking mode of research and discussion. To analyze, argue, inspire each other, and brainstorm broadly. Every student must actively participate and think positively. If the three people do not cooperate well, it will reduce the efficiency and lead to the failure of the entire modeling study.