I. Definition of Mathematical Model At present, there is no unified and accurate definition of mathematical model, because different angles have different definitions. However, we can give such a definition: "A mathematical model is an abstract simplified structure about a part of the real world and used for special purposes." Specifically, the mathematical model is for a certain purpose. Equations or inequalities established by mathematical symbols such as letters and mathematics, and mathematical structural expressions such as charts, images and block diagrams that describe the characteristics of objective things and their internal relations. Generally speaking, the process of mathematical modeling can be represented by the following block diagram: mathematics is produced in the demand of practical application, and mathematical models need to be established 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. In particular, new technologies and new processes are developing vigorously, and computers are popularized and widely used. Mathematics plays a key role in many high technologies. Therefore, mathematical modeling has been endowed with more important significance by the times. Second, the methods and steps of establishing a mathematical model 1. Model preparation should understand the actual background of the problem, clarify the modeling purpose, collect all necessary information, and try to understand the characteristics of the object. 2. Make modeling assumptions according to the characteristics of the object and the purpose of modeling, make necessary and reasonable simplification of the problem, and make assumptions with accurate language. This is a crucial step in modeling. If all the factors of the problem are taken into account, it is undoubtedly a courageous but clumsy act. Therefore, a superb modeler can give full play to his imagination, insight and judgment, and is good at distinguishing priorities. In order to make the handling method simple, he should try his best to linearize and homogenize the problem. 3. Model composition According to the assumptions made, the causal relationship of the object is analyzed, and the internal laws of the object and appropriate mathematical tools are used. 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. However, we should remember that the purpose of establishing mathematical models is to make people understand and understand. So the simpler the tool, the more valuable it is. 4. The model can be solved by various traditional and modern mathematical methods, such as solving equations, drawing, proving theorems, logical operations, numerical operations and so on. , especially computer technology. The solution of a practical problem often needs complicated calculation, and it often needs a computer to simulate the operation of the system. Therefore, programming ability and familiarity with mathematical software packages are very important. 5. Model analysis carries out mathematical analysis on the model solution. 3. The guiding ideology of mathematics competition is that traditional mathematics competition generally emphasizes theoretical knowledge. The content to be investigated is single, and the data is simple and clear, which is not allowed to be completed by a calculator. In this regard, the mathematical model competition is a "discipline", which mostly comes from the actual production or scientific research process. 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 some reasonable assumptions, so it can only be done. Not the only one), the result of the report is a series of "papers". It can be seen that the "mathematical model competition" focuses on application, and it is a comprehensive ability competition with mathematical knowledge as the guide and article writing ability as the supplement. 4. There are three basic components in the common contest questions: 1. The background of realistic problems involves a wide range-society, economy, management and life. New problems in modern science, etc. Generally, there is an exact practical problem. 2.-@/v 1e+[. h2d4n &； The assumptions of A0A 1W are 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 some parameters or graphs; 4) There are some supplementary assumptions that can be maneuvered and exerted, or contestants can generate data according to their own collection or simulation .3.2N9U8] # b; The questions that U $ 0z asks to answer often have several questions, which are generally not the only answers. Generally, it includes the following two parts: 1) Compare the definite answer (basic answer); 2) More detailed or higher-level discussion results (often discussing the formulation and results of the optimal scheme). V. What is the basic content and format of submitting a paper? Submit a paper, the basic content and format are roughly divided into three parts: 1. (H4 \ m-t，O-[，u ' g &； ~ title, the abstract part of the topic-write a more accurate topic (not just questions A and B). -200-300 words, including the main features, modeling methods and main results of the model. When there are many contents, it is best to have a catalogue. 2. The central part of the problem is 1), analyzing the problem. 2) Model establishment: ① Supplementary assumptions. ② Model form (there can be multiple models); ③ model solving; ④ Model properties; 3) Design and computer realization of calculation method. 4) Analysis and test of the results. 5) Discussion-advantages and disadvantages of the model, improvement direction and popularization of new ideas. 6) References-pay attention to the format. 3.-J3ZL+w'm)t9a, U Appendix calculation program, block diagram, various calculation processes, intermediate calculation results and various charts. There is no need to learn a lot of mathematical knowledge systematically, which is not allowed by time and energy. The genius of many excellent papers lies not in how much mathematical knowledge they use, but in their comprehensive, practical, problem-solving or innovative thinking. Sometimes, they may encounter some knowledge that they haven't learned. Why? The mathematical knowledge used in excellent papers is most likely to be used in mathematical modeling competitions. Of course, you need to double it. Specifically, there are the following three aspects: the first aspect: the application ability of mathematical knowledge can be summarized into the following categories: 1) probability and mathematical statistics 2) overall planning and spool planning 3) differential equation; There is also knowledge that intersects with computer knowledge: computer simulation. Some students have never learned the above content at all, and some students have only learned a little probability and mathematical statistics. What about the knowledge of differential equations? As for the word "self-study", I Excel heard Fan Yi, the teacher in charge of correcting mathematical models, say that "it is a better answer sheet to solve the answer sheet that others can solve with advanced theory by the simplest and easiest mathematical method". Using "Mathematica" software, language ability is preferred. Most of this knowledge is learned by students themselves in their spare time. The third aspect: the writing ability of the paper has been said before. The full text of the test paper is essay-style, and the writing of the article has a strict format. It is not easy to express your ideas clearly. Sometimes one question is not clearly stated before another. Teachers who grade papers all have * * * knowledge. If an article is still not interesting after reading 10 minutes, then it is likely to be ignored. Seven. How to divide the work in the group? The traditional standard answers are-mathematics, programming and writing. In fact, the division of labor is not so clear, but there is a premise that everyone has a good relationship. Otherwise, it is easy to produce contradictions. The division of labor is too clear, which will make people dependent on ideas and unwilling to use their brains. The ideal division of labor is this: everyone in the mathematical modeling competition group can be competent for the work of others, even if she (he) is the only one left in the group. Mathematical modeling competition can still be solved. The division of labor in the competition is only to improve work efficiency and make better achievements. The specific suggestions are as follows: there must be a person with a lively brain and good thinking ability, who does not want to belong to mathematics; There must be someone who can program and implement some algorithms. In addition, you need to have a well-written paper, but it doesn't matter if you don't write well. Read more excellent papers of others, and use word, Word and Visio several times to become it. First, the importance of writing a mathematical model answer sheet is 1. Evaluate the performance, level, award-winning level and mathematical model answer sheet of the team. This is the only basis. The answer sheet is the written form of the competition result. 3. The training of writing answer sheet is the basic training of scientific writing. 2. The basic contents of the answer sheet and the problems needing attention. 1. Marks the rationality of hypothesis, creativity of modeling, rationality of results and clarity of expression. 2. The article structure of the answer sheet 1) abstract. Background analysis, etc. 3) Hypothesis and symbolic description of the model (table). 4) Model establishment (problem analysis, formula derivation, basic model, final or simplified model, etc.). ) .5) Design or selection of calculation methods for model solving; Algorithm design or selection, algorithm ideological basis, steps and implementation, calculation block diagram; The name of the software used; Quote or establish necessary mathematical propositions and theorems; Solutions and processes. 6) Result representation, analysis and test, error analysis and model test. 7) Model evaluation, characteristics, advantages and disadvantages, improvement methods and popularization. 8) References. 9) Appendix, calculation block diagram and detailed drawing. 3. 1) Summarize the problems needing attention. Including: a. Mathematical classification of models (in mathematics, b. Thinking of modeling); C. algorithm idea (solution idea); D modeling features (model advantages, modeling ideas or methods, algorithm features, result test, sensitivity analysis, model test, etc.). ); E. Main results (numerical results and conclusions; Answer all the "questions" asked by the topic. ▲ Pay attention to expression: accurate, concise, clear-cut, grammatical, neat and beautiful fonts; It is best to print, but it must conform to the article format. Be sure to proofread carefully. 2) Restatement of the problem. 3) Model assumptions. According to the scoring principle determined by the National Organizing Committee, the rationality of basic assumptions is very important. A. make assumptions according to the conditions in the topic. B. make assumptions according to the requirements in the topic. Key assumptions are indispensable; Assumption should conform to the meaning of the question. 4) Establishment of the model. A. Basic model: i) First, there must be a mathematical model: mathematical formula, scheme, etc. Ⅱ) Basic model, complete, correct and concise; B. Simplify the model: Ⅰ) Clearly explain the simplification idea and basis; Ii) Simplify the model and give it as completely as possible; C. the model should be practical and effective, and the principle is to solve problems effectively. Mathematical modeling faces practical problems and does not pursue the high (level), deep (engraving) and difficult (degree) of mathematics. I) If it can be solved by elementary methods, there is no need for advanced methods; Ii) If it can be solved by simple methods, there is no need for complicated methods; Iii) If you can use a method that can be understood and understood by others, you don't need a method that can only be understood and understood by a few people. D encourage innovation, but be pragmatic, and don't digress and do new things. Mathematical model innovation can appear in: ▲ modeling, model itself, good methods of simplification, good strategies, etc. ; ▲ The model is being solved; ▲ Results representation, analysis, testing and model testing; ▲ Promotion part. E. Problems needing attention in the process of problem analysis and deduction: Ⅰ) Analysis: pertinent and accurate; Terminology: professional and expert; 3) Principles and basis: correct and clear; Ⅳ) Description: concise and to the point, listing the key steps; ⅴ) Avoid: layman's words, unclear technical terms, confusing and lengthy expressions. 5) model solving. A. When a mathematical proposition needs to be established, the description of the proposition should conform to the norms of the mathematical proposition and be as rigorous as possible. B it is necessary to explain the principle, thinking, basis and steps of the calculation method or algorithm. If the existing software is used, explain the reasons for using the software. C. calculation process, software name. 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 or improve the algorithm, calculation method or model. C. The questions, numerical results and conclusions required to be answered in the topic must 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. The result representation: centralized, clear at a glance, intuitive and convenient for comparative analysis. ▲ Numerical result representation: a well-designed table; If possible, in the form of a chart. ▲ It is better to use a graphic solution. 7) If necessary, answer questions and conduct qualitative or regular discussions. The final conclusion should be clear. 8) The advantages of model evaluation are outstanding and the disadvantages are inevitable. Here you can change the original requirements and re-model. Don't play with new mathematical terms when popularizing or improving the direction. 9) Refer to 10) Appendix for details. It is best not to list errors. The main results and data should be listed in the text, not afraid of repetition. Check the three key points of the answer sheet through three levels: a. the correctness, rationality and innovation of the model; B. correctness and rationality of the results; C. the text is clear, the analysis is incisive, and the summary is wonderful. 3. What questions need to be answered in thinking and work planning before writing the answer sheet-what problems need to be solved in modeling; How to answer questions ―― how to express the results; What key data should be listed for each question-what key data should be calculated for modeling; For each quantity, list one or more sets of numbers-whether to calculate one or more sets of numbers. 4. The principle required for the answer sheet is 1. Accuracy-science; 2. Organizational logic; 3. Simplicity-the beauty of mathematics; 4. Innovation-one of the goals of research and application is also the need of talent training; 5. Practicality-modeling and practical problem requirements. 5. Modeling concept 1. The application of consciousness should solve practical problems, and 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 To solve problems by mathematical methods, there must be a mathematical model; The mathematical abstraction of the problem model is universal and scientific, and is not limited to the solution of this specific problem. 3. Innovation consciousness modeling has its own characteristics, which is more reasonable, scientific, effective and practical. More universal application significance; Innovation is not just for innovation. 1. Time allocation is also very important in the competition between time and physical strength. If the assignment is not good, the paper may not be finished, so it is necessary to make general arrangements from the beginning, such as making the first small question on the first day and the second small question on the second day, which will be stressful. If you are not busy writing at first, you can record some key points of group discussion, and don't be too neat and casual. It's not too late to start writing papers on the third day. Another thing to say is that physical strength should keep up. Generally, you can only sleep less than 10 hours for three days. It is recommended to stay up late for programming several times before the game, but don't stay up until the day before the game. Ha ha. Teamwork is the key to winning the prize. In the three-day competition, team communication may take up more than half of the time. When there are differences, how to solve them is very important, and even directly determines whether you can win the prize. My suggestion is "compromise". Don't always think that your point of view is correct. Listen to other people's views and seek common ground between them. Before the competition, we should cultivate cooperation, such as working on a project together and making full use of everyone's advantages. You can also prepare graph theory for Zhang San and optimize methods for Li Si, and then communicate with you in a few days. All these will destroy the relationship between teams. Don't copy some words from the topic, but go straight to the point. Write down how you analyze the problem and how to solve it. The most important thing is to know what the conclusion is. If you don't write a conclusion in the domestic competition, you won't get the prize. It takes at least two hours to ponder, so don't underestimate its importance. It is necessary to read more about how the abstracts of excellent papers are written. And it should be one of the topics prepared before the game. 4. Writing papers should be formal. Papers must be written in the form of abstract, restatement of problems, model hypothesis, symbolic explanation, problem analysis, (establishing, analyzing and solving models), ……, references, appendices, etc. Generally, some articles with structural failure will be eliminated first in the initial evaluation. Without the structure of the paper, the content is useless no matter how good it is. The structure in front of the paper generally does not change. You can arrange your own structure according to the actual situation in the future. The omitted parts can include result description, sensitivity analysis, other models, model expansion, advantages and disadvantages analysis, etc. Read more excellent papers and you will know what other forms there are. Appendices can post some algorithm flowcharts or relatively large results or charts. 5. Model hypothesis and model establishment. After the judges read the abstract, they immediately looked at the model hypothesis. There is a general method, that is, copying a few sentences, which can be used as a hypothesis in the topic and will leave a good impression on people. After all, it means that you have already examined the questions. But we can't copy them all. You should add some assumptions to your paper, preferably not too specific. Some important parameters should not be fixed, and only certain values will make people feel that the limitations of the paper are very strong. The establishment of the model is based on your analysis of the problem, and the mathematical symbols proposed should be close to the establishment of the model. It is best to let the judges see the symbols on the same page, and the mathematical formula should be rigorous and the derivation should be rigorous, which all reflect a person's mathematical quality and ability. Even if your deduction is wrong, others will mistake you for being right when they see your battle. 6. Illustrated pictures can add color. I heard that an inaccurate news is that the judges like to program papers with Matlab. I don't know if this is the case. However, this shows that teachers need to read a paper with charts or tables. If the paper is written like a book, it is estimated that no one will be interested, especially scientific papers. The reason why Matlab programming is favored is because the graphics processing ability provided by Matlab is very powerful and the illustrations are particularly strong. If there is a lot of data in the conclusion, it is best to explain it in the form of a chart, which will make your paper more convincing. It will also be well received by the judges. 1. Ten algorithms that should be mastered in mathematical modeling competition 1. Monte Carlo algorithm, also called stochastic simulation algorithm, is an algorithm to solve problems through computer simulation, and at the same time, it can test the correctness of its own model through simulation, which is an essential method in the competition. 2. Data processing algorithms such as data fitting, parameter estimation and interpolation usually encounter a lot of data to be processed in competition. The key to data processing lies in these algorithms, and Matlab is usually used as a tool. 3. In the modeling competition of linear programming, integer programming, multivariate programming and quadratic programming, most of the problems belong to optimization problems. Many times, these problems can be described by mathematical programming algorithms, which are usually realized by Lindo and Lingo software. 4. Graph theory algorithms can be divided into many kinds. Including the shortest path, network flow, bipartite graph and other algorithms. Problems related to graph theory can be solved by these methods and need careful preparation. 5. Computer algorithms commonly used in algorithm design include dynamic programming, backtracking search, divide-and-conquer algorithm, branch and bound, etc. Many occasions can be used for competitions. 6. Three non-classical algorithms of optimization theory: simulated annealing method, neural network and genetic algorithm are used to solve some difficult optimization problems, which are very helpful to some problems, but they are difficult to realize and need to be used carefully. 7. Grid algorithm and exhaustive method Grid algorithm and exhaustive method are the best algorithms for violent search, and they are applied in many competition problems. When focusing on the model itself and ignoring the algorithm, this violent scheme can be used, and it is best to use some high-level languages as programming tools. 8. Some problems of the continuous discretization method are practical, the data can be continuous, and the computer only recognizes discrete data. Therefore, it is very important to discretize it, and then replace differential with difference, so as to seek sum instead of integral. 9. If the numerical analysis algorithm is written in high-level language in the competition, some commonly used algorithms in numerical analysis, such as solving equations, matrix operation, function integration, etc., need to write additional library functions to call. 10. There is a kind of problem related to graphics in the image processing algorithm competition, even if it has nothing to do with graphics. There should be many pictures in the newspaper, too How to display these pictures and how to deal with them are the problems that need to be solved, usually with Matlab. Second, what are the main classifications of mathematical software? What are their characteristics? Mathematical software can be divided into general mathematical software packages and professional mathematical software packages according to their functions. The general mathematical software package has complete functions, including all kinds of mathematics, numerical calculation, rich mathematical functions, special functions, drawing functions, interactive functions of user graphics, interfaces with other software and languages, and a huge external function library mechanism (toolbox). Common general mathematical software packages are Matlab, Mathematica and Maple. Among them, Matlab is a high-performance scientific and technological computing software, which is widely used in mathematical calculation, modeling, simulation, data analysis and processing, engineering drawing and other fields. Mathematica is a representative software for numerical and symbolic calculation, and Maple is good at symbolic operation and formula derivation. Special mathematics packages include MatheCAD, Tecplot, IDL, Surfer, Origin, SmartDraw, DSP2000 and other drawing software), numerical calculation classes: (MATCOM, IDL, DataFit, S-Spline, Lindo, Lingo, O-Matrix, Scilab, Octave), and numerical calculation library (linpack/lapack). Fluent, Femlab, FlexPDE, Algor, COSMOS, ABAQUS, ADINA), computational chemistry (Gaussian 98, Spartan, ADF 2000, Chemofffice), mathematical statistics (Gauss, SPSS, SAS, Splus, Statistics, Minitab), Mathematical formula typesetting (MathType, Miktex, Scientific Workplace, Scientific Nootbook). Third, several good books about the mathematical model competition ▲ Jiang Qiyuan's Mathematical Model (Second Edition) Higher Education Press ▲ Jiang Qiyuan, Xie Jinxing, Alfred's Mathematical Modeling (Third Edition) Higher Education Press ▲ Xiao Shutie, etc. Higher Education Press ▲ Zhu Daoyuan's Selected Cases of Mathematical Modeling Science Press ▲ Lei, Lecture Notes on Mathematical Modeling Peking University Press ▲ Ye et al. Teaching materials for national college students' mathematical modeling competition (1) ~ (4) Hunan Education Press ▲ Jiang and Xin Peiqing, University of Electronic Science and Technology of China Press 《 Mathematical Model and Computer Simulation 》 ▲ Yang Qifan and Xin Peiqing. Mathematical Modeling and Mathematical Experiment, Higher Education Press, springer Press IV. Basic account 1. Mathematical analysis II. Advanced algebra iii. Probability and mathematical statistics. Optimization theory 5. Graph theory 6. Combinatorial mathematics. Stability analysis of differential equations. Queuing theory 5. Common websites and ftp▲/hmcm mathematical model website▲166438+06438+06438. W'h6B%V8C animal population management/KOC-0/985B problem 5]%]+%Y9' 6V strategic material storage management/KOC-0/986A problem waterway survey data/KOC-0/986B problem emergency facilities location/KOC-0/987A problem salt storage/KOC-0/. 988A problem to determine the location of smugglers 1988B problem to load two railway flat cars 65T8a, t1b # l $ _; Distribution of X drugs in brain/KOOC-0/990b problem/KOOC-0/99/KOOC-0/A problem/KOOC-0/99/KOOC-0/B problem/KOOC-0/B minimum cost minimum spanning tree/KOOC-0/992a problem/KOOC-0/. K5j*j$X5V6Z Emergency Power Supply Repair System 1993A Problem 7K0G8I2 {2 _&; O6Q Accelerates the Composting of Restaurant Leftovers 1993B Problem Coal Discharge Platform Operation Scheme 1994A Problem Construction Cost 1994B Problem 9? 4E2|-D4n4h, e-t Computer Transmission 1995A Problem Single Spiral 1995B Problem Teacher Salary Distribution 1996A Problem Submarine Exploration 1996B Problem1M2' * V (I-[2Y C2AD 7 8 [-Q meeting decided/KOOC-0/998A problem MRI scanner/KOOC-0/998B problem+W5ku/KOOC-0/D-WH/KOOC-0/S9 @ m; |"t small star impact 1999B problem * U3B5P4U ({8c8n; [$h illegal assembly 1999C topic land pollution 2000A topic air traffic control 2000C topic; The number of elephants in n00y0s "l 'f4x 't9v5h2002a question %D:q; P)d"r Wind and Fountain 2002B Problem (w%S-D 1_#I7S! B airline oversold 2003A. I3N！ t； R.x{8p stuntmen 2003B gamma knife treatment scheme 2004A 6V 1w$g, r6L fingerprint unique? 2004B Question 0y$? :i*^； Faster express delivery system II. Cumcm (National Mathematical Modeling Competition for College Students) 1993A! N% P & ampw nonlinear frequency design 1993 B problem 7Y6S+[! F7c6^ team ranking problem1994a; @(j，S%n 1[！ D clear the way on the mountain1994b-v * a.j # u+w6v6 * m+[:x1m lock the box and pack it1995a problem 5N$w:Y9e 1I%Y/X A flight management model/. Operation scheduling of crown block and smelting furnace 1996A optimal fishing strategy 1996B water N)H:F%\7E0Z8O investment income and risk 1998B problem 5B#c%L3h9p4X6v4a disaster patrol route 19999A problem% f O0}5D。 N drilling layout 2000 A topic DNA sequence classification 2000 B topic $ J: D "A.V. & A-S; j ' q(T； P steel pipe ordering and transportation 200 1 A topic Three-dimensional reconstruction of blood vessels 200 1 B topic Bus dispatching 2002 A topic & Optimal design of linear light source 2002 B topic & amps 5l/g.n&; I t x7] Lottery in mathematics 2003 a:SARS spread 2003b: 8w: j m $ u "h: open-pit mine vehicle arrangement 2004a-l1h: n0r $`; Me! W