Mathematical Modeling of introduction

Modeling background

 

Mathematical techniques

For nearly half a century, with the rapid development of computer technology, the application of mathematics to play not only in engineering, natural sciences and other fields increasingly important role, but with unprecedented breadth and depth to economics, management, finance, new areas of biology, medicine, environment, geology, population, transportation and other penetration, so-called mathematical technology has become an important part of contemporary high-tech.

Mathematical model (Mathematical Model) is an analog, with mathematical symbols, mathematical formulas, programs, graphics, abstract yet simple portrayal of the actual nature of the subject property, which may explain some objective phenomenon, or can predict the future the law of development, or to provide optimal strategy or tactics better in some sense to control the development of certain phenomena. Direct replica of the real problem is not the general mathematical model, it often requires both people to establish further detailed observation and analysis of practical problems, we need flexible people skillfully use a variety of mathematical knowledge. This abstract from the actual application of knowledge in the subject, to extract the mathematical model of the process is called mathematical modeling (Mathematical Modeling).

No matter what kind of practical problems to solve in the field of science and technology and production mathematically, or combined with other disciplines to form interdisciplinary primary and crucial step is the study of the mathematical model and are calculated solving (usually by means of a computer); mathematical modeling and computer technology in the role of knowledge economy era can be described as even more powerful.

 

Modeling applications

Mathematics is the study of the relationship between the real world and the amount of space in the form of science, in the long history of its emergence and development, it has been a wide variety of applications and issues closely related. Mathematical features not only abstract rigor, logic concept, clarity and integrity of the conclusion of the system, but also in the breadth of its application. Since the 20th century, with the growing popularity of the rapid development of science and technology and computers, people on various issues more and more accurate, making mathematics more widely and deeply, especially in this era of knowledge economy of the 21st century , the status of mathematical sciences will change dramatically, it is come to the forefront from the reserve of the national economy and science and technology. Economic development of globalization, the rapid development of computers, expanding the theory and methods of mathematics, making mathematics has become an important contemporary high-tech components and think tanks, mathematics has become a common technique capable of implementation. Awareness of students of applied mathematics and capacity has become an important aspect of mathematics teaching.

 

Modeling process

Model preparation

Background to understand the actual problem, clarify its practical significance, to master a variety of information objects. To mathematical thinking essence to accommodate the problem, mathematical thinking through the whole process of the problem, and then use mathematical language to describe the problem. Requirements in line with mathematical theory, in line with the mathematical habits, clear and accurate.

 

The model assumes

The object and features of the actual object modeling, necessary to simplify the problem and make some assumptions with the appropriate precise language.

 

Model

On the basis of assumptions, using an appropriate mathematical tools to characterize the mathematical relationship between the variables constant, the corresponding mathematical structure (as far as possible with simple mathematical tools).

 

Model Solution

Using the data acquired to make computing (or approximated) on all parameters of the model.

 

Model Analysis

To build on the idea of ​​the model are explained, and the results obtained were analyzed mathematically.

 

Model Checking

The model results were compared with the actual situation, in order to verify the accuracy of the model, rationality and applicability. If the model is in good agreement with the actual, will have to give its actual meaning of the results, and interpretation. If the model is consistent with the actual poor, you should modify the assumptions, modeling process is repeated again.

Model application and promotion

Application methods differ for the purpose and nature of the problem varies modeling, and to promote the model is based on the existing model of the model have a more comprehensive consideration, the establishment of model more in line with reality.

 

Modeling significance

the way of thinking

Mathematical modeling is a mathematical way of thinking, is the use of mathematical language and methods, by abstracting and simplifying the establishment and can be approximated portray "solve" a powerful mathematical tool practical problems.

Mathematical modeling is the process of real-world phenomena using mathematical language to describe. Here both the actual phenomenon of natural phenomena such as the inclusion of specific free-fall phenomenon, also contains an abstract phenomenon such as customers tend to value a commodity taken. Described herein include not only external morphological description internal mechanism, but also includes content prediction, testing and interpretation of the actual phenomena.

We can also intuitively understand this concept: Let mathematical modeling is a pure mathematician (refers only to study mathematics and applied mathematics regardless of mathematicians in practice) become physicists, biologists, economists and even psychologists, etc. process.

General mathematical model is a mathematical simplified practical things. It is often close to the real thing there is a sense of abstract form, but the real thing, and it is essentially different. To describe a real phenomenon can have a variety of ways, such as audio, video, metaphor, rumors and so on. In order to describe more scientific, logical, objectivity and reproducibility, people widely believed to adopt a more stringent language to describe a variety of phenomena, this language is mathematics. Use mathematical language to describe something called a mathematical model. Sometimes we need to do some experiments, but these experiments are often out of abstract mathematical models instead of the actual object and the corresponding experiments, the experiment itself is also a theory practical alternative.

 

Mathematical model

The application of mathematics to solve various practical problems, the mathematical model is a critical step, but also a very difficult step. Mathematical model of the process, is to simplify the complex practical problems of abstract mathematical structure is reasonable process. Through the survey, collecting data, observation and study of the inherent characteristics of the actual object and the inherent law, seize the main contradiction, and establish relationships reflect the number of practical problems, and then use mathematical theories and methods to analyze and solve problems. This requires deep solid mathematical foundation, keen insight and imagination, interest in practical problems and extensive knowledge. Mathematical modeling is linked with the practical problems of mathematical bridge, the media is widely used in various fields of mathematics, and science and technology is the main way mathematical transformation, the important role of mathematical modeling in the development of science and technology more and more mathematics and engineering world widespread attention, it has become a modern science and technology workers have the key capabilities of a . In order to meet the needs of the development of science and technology and culture of high-quality, high-level scientific and technological personnel, mathematical modeling has gradually carried out in university education, more and more domestic and mathematical modeling teaching university courses and ongoing participation in an open mathematical modeling contest, the mathematical modeling contest as a teaching and teaching reform of institutions of higher learning and training high-level scientific and technological talents of an important aspect of many colleges and universities are mathematical modeling and teaching reform combined efforts to explore more effective mathematics modeling Approach nurture talent and new ideas for the 21st century, compared with other mathematical courses in our colleges and universities, mathematical modeling has great difficulty, involving a wide range, flexible form, teachers and students require high mathematical construction teaching mode itself is a constantly explore, continue to innovate and improve and enhance the process. In order to change the traditional teacher-centered teaching model to the classroom based teaching, imparting knowledge-based past, mathematical modeling courses guiding principle is: laboratory-based, student-centered, problem-based line, in order to cultivate ability to target organized teaching. By teaching the students to understand the whole process of using mathematical theories and methods to analyze and solve problems, and improve their ability to analyze and solve problems; increase their interest in mathematics and applied mathematics awareness and ability, so that they work in the future can regularly expect to use mathematics to solve problems, they try to raise awareness of the use of computer software and modern high-tech achievements, can mathematics, computer organically combine to solve practical problems. Mathematical modeling were students, teachers use some good questions in advance design inspire and guide students to actively access to literature and learning new knowledge, students are encouraged to Active discussion and debate, students take the initiative to explore and strive to forge ahead style of study, cultivate students' ability to engage in preliminary research work, students cultivate the spirit of unity and cooperation, form a lively environment and atmosphere, the focus of the teaching process is to create an environment in which to induce students' desire to learn, develop their self-learning ability, enhance their mathematical qualities and ability to innovate, improve their quality of math, the emphasis is acquiring new knowledge, it is the process of problem-solving, rather than knowledge and results. Most accepted to participate in Mathematical Contest in Modeling training before students need to learn, such as mathematical statistics, optimization, graph theory, differential equations, computational methods, neural networks, analytic hierarchy process, fuzzy mathematics, mathematical software packages, etc. "Short Course "(or lectures), with much of the hours, most inspiring about some of the basic concepts and methods, mainly by the students themselves to learn, and fully mobilize the enthusiasm of the students, give full play to the potential of the students. Seminar way training is widely used, the students themselves report, discussions, debates, teachers, mainly from the questioning, answering, the role of counseling, competition must be using a computer and appropriate software, such as Spss, Lingo, Maple, Mathematica, Matlab even typesetting software.

 

Modeling origin

West case

Mathematical modeling is entering some Western countries universities in the 1960s and 1970s, the Chinese are also several universities in the early 1980s the mathematical modeling into the classroom. After 20 years of development, the vast majority of colleges and many colleges have set up various forms of mathematical modeling courses and seminars for the training of students' ability to analyze the use of mathematical methods to solve practical problems has opened up an effective way.

Mathematical Contest in Modeling was first appeared in 1985 in the United States in 1989 under several teachers in the organization and promotion of education in mathematical modeling, students at several universities in China began to participate in the US competition, and increasing enthusiasm in recent years the number of participating schools, the number of teams accounted for a large proportion. It can be said, Mathematical Contest in Modeling is born in the United States, flowering in China, results.

 

China case

In 1992 he held a mathematical model of the league of 10 cities by the Chinese Students Society for Industrial and Applied Mathematics organization, 314 teams of 74 universities participated. The Ministry of Education to discover and foster, nurture this new thing, the decision since 1994, co-sponsored by the National Mathematical Contest in Modeling Department of Higher Education and the China Society for Industrial and Applied Mathematics, the annual session. Over the past decade the scale of this competition to more than 25% average annual growth rate of development.

In 2009 the country has 33 provinces / municipalities / autonomous regions (including Hong Kong and Macao Special Administrative Region) 1 137 schools, 15,046 teams (Group A where 12276 teams, 2770 Group B team), more than 45,000 students from all majors contest, is the largest number of entries over the years (which is the first entry in Tibet and Macau).  [2] 

 

Modeling Competition

National Mathematical Contest in Modeling

National Mathematical Contest in Modeling by the State Department of Higher Education

And China Society for Industrial and Applied Mathematics co-sponsored. Competition title comes from the general aspects of engineering and management science appropriately simplify the process of practical problems, does not require participants to master the pre-depth expertise, only learned of college mathematics curriculum includes a complete model assumptions, the establishment and solving, and computer calculation methods of design implementation, test papers and the results of the analysis and improvement of aspects of the model (ie respondents). Competition awards based on the assumption of rationality, creative modeling, correctness and clarity of the textual representation of the results of the main criteria.  [2] 

Unified national title contest, take way communication contest, carried out in a relatively concentrated form; competition usually held in three days in early September each year (as little as possible in order to ensure everyone's delayed course, it is generally two days including weekends); students with the team for participating units, 3 people per team and a teacher as counseling, professional limited.

 

 

Competition Statutes (2008)

General Provisions Article

National Mathematical Contest in Modeling (hereinafter referred to as competition) Ministry of Education and Higher Education Division of China Industrial and Applied Mathematics Institute co-sponsored mass scientific and technological activities for the National College with the aim of enthusiasm motivate students to learn mathematics, students develop a mathematical model to improve and comprehensive ability to use computer technology to solve practical problems, to encourage the students to actively participate in extracurricular science and technology activities, to develop their knowledge, develop creative spirit and sense of cooperation and promote the reform of mathematics teaching system, teaching content and methods.

The second contest content

Topic greater flexibility for participants to play their creativity. Participants should be based on questions asked, assuming that a complete model, including the establishment and solution, calculation methods and computer design implementation, analysis and testing, improvement model aspects of the paper results (ie respondents).

Article in the form of competition, rules and disciplines

1. Unified national title contest, take way communication contest, carried out in a relatively concentrated form.

2. Contest held every year, usually held within three days before and after one weekend.

3. Students participating in a team as a unit, each team three people (must belong to the same school), professional limited. This competition undergraduate, specialist two groups, undergraduate students participate in undergraduate group competition, college students to participate in competitions specialist group (group competition may also participate in undergraduate), graduate students may not participate. Each team can set up a guidance teacher (or teacher groups), engaged in counseling and organizational work before the competition, but team members must be avoided during the competition, shall not participate in discussions or guidance, or else the disciplinary process.

4. During the race team members can use a variety of books, computers and software, browse on the Internet, but not with anyone outside the team (including online) discussion.

5. After the competition starts, the title match will be announced at the specified URL for downloading teams, teams respondents completed within the specified time, and carry out an assignment on time.

6. Participating institutions should instruct the relevant departments responsible for the organization and discipline of the competition supervision to ensure the impartiality of the normative and the school competition.

Article IV form of organization

1. Competition Commission by the National Mathematical Contest in Modeling Organization (hereinafter referred to as the National Organizing Committee) chaired, responsible for launching the annual registration, preparation of title race, national excellent organization of respondents review and awards, printed award certificates, organized a national awards ceremony.

2. Sub-division competition organizations. In principle, a province (autonomous regions and municipalities) as a division, and each division should have at least 20 teams to participate in six institutions. Neighboring province may be combined to form a division. Each division established Organizing Committee (hereinafter referred to as Division Organizing Committee), responsible for the Division of propaganda and registration, supervision and discipline competitions organized reviewers respondents work. No set up provincial schools division of the teams directly to the National Organizing Committee entered the competition.

3. The establishment of the organization Excellence Award in recognition of excellent or outstanding progress in the work organization of the competitions in the achievement of the Division Organizing Committee, with the number of participating teams and the number of school levy questions of quantity and quality, no discipline phenomenon, quality reviewers work, combined with the Division the specific circumstances to carry out the work as well as the main criteria of the National Organizing committee and other cooperation creatively.

Article Awarding

1. Each Division Organizing Committee to hire experts reviewers Committee, named this division's first-class, second prize (also an additional prize), winning will not exceed one-third of the remaining qualified respondents who have completed successfully obtain entry certificate.

2. Each Division Organizing Committee according to the number of outstanding provisions of the National Organizing Committee will send the National Division of the respondents of this committee. National Organizing Committee to hire experts reviewers national committee, according to uniform standards selected the national first-class, second prize of excellent responses from all arenas sent in.

3. National Division with the first and second prize winners were awarded certificates.

4. For violating the rules of the team, once discovered disqualified from the competition, results invalid. Of where schools should be warned, informed, until the abolition of the next school year to qualify. Division awards for work in violation of the provisions of the National Organizing Committee does not recognize the results of its awards.

Article opposition Institutional

1. Within two weeks of the date of publication the country (or each division) winners from any individuals and units can be challenged by the National Organizing Committee (or each Division Organizing Committee) is responsible for receiving.

2. Focus on the admissibility objection is a violation of the Articles of Association of the competition, including competition during the teachers involved, players and others to discuss, injustice and other reviewers. For respondents require re-evaluation in order to improve the appeal of the award-winning level, in principle, admissible special circumstances can be the first Division Organizing Committee after each audit, by the Division Organizing Committee of the National Organizing Committee for verification.

3. Objection must be made in writing. Personal objections raised need to notify my real name, affiliation, mailing address (including telephone or e-mail address, etc.), and there's autograph; objections raised by the unit shall indicate the name of the contact, communication address (including telephone or e-mail address, etc.), and the official seal. National Organizing Committee and all Division Organizing Committee of the person or entity to object kept confidential.

4. The objection concerning the admissibility of school authorities, have a responsibility to assist the National Organizing Committee and all Division Organizing Committee for the objection to investigate, and put forward opinions. National Organizing Committee or each Division Organizing Committee shall within two months after the end of the opposition period, the complainant replied processing results.

Article VII funds

1. The team's school to pay the entry fee to host Division Organizing Committee.

2. Division Organizing Committee to pay a certain amount of funds to the National Organizing Committee.

3. Funded education authorities at all levels.

4. Funding community.

Article VIII, interpretations and adaptations

This constitution started in 2008, its interpretation and amendment vested in the National Organizing Committee.

American Mathematical Contest in Modeling

American Mathematical Contest in Modeling (including interdisciplinary contest) is sponsored by the National Science Foundation and the American Mathematical Society and the Mathematical Association of application of Operations Research Society of America, Society for Industrial and Applied Mathematics, Mathematical Society and a number of international agencies co-organized CD eleven international modeling contest. Competition requires the following three undergraduate students have not graduated from social engineering to solve a specific problem within 4 days with mathematical modeling and other knowledge, to submit papers in English.

 

 

data set

Mathematical modeling involving large data sets for research personnel to test and demonstrate mathematical modeling algorithm, such as:

1.2008 National Graduate Mathematical Modeling Contest questions and data

2.2011 Higher Education Press Cup National Undergraduate Mathematical Modeling Contest title

3. The density can be modeled iris training data set

4. Applied Bayesian Modelling Dataset (Bayesian modeling dataset)

5. Worksheets Data for Multilevel modelling (multilevel modeling work table-format data), etc.

 

Domestic materials

1. mathematical modeling algorithms and applications, keep the Secretary Kui, Sun Yuxi Jing, eds., National Defense Industry Press (2012).

2. mathematical model, Jiang Qiyuan compiled Higher Education Press (first edition 1987, second edition 1993, third edition 2003, fourth edition, 2011; first edition in 1992, the State Board of Education held its second National excellent textbook Award won the "national outstanding teaching award").

3. Mathematical modeling and case, Zhang Wanlong, eds, National Defense Industry Press (2014).

4. Mathematical Modeling entry and improve, Li Hanlong, eds, National Defense Industry Press (2013).

5 mathematical models and computer simulations, Jiang Yu Zhao, Xin Pei love knitting, University of Electronic Science and Technology Press, (1989).

6. Mathematical model of election talk (to Mathematics from the book), Hua, Wang Yuan with, Wang Ke translation, Hunan Education Press; (1991).

7. Mathematical model, a single peak, Zhu Limei, National Defense Industry Press (2011).

8. Mathematical model, Chen Yihua ed, Chongqing University Press, (1995)

9. Mathematical modeling analysis, Cai Changfeng eds., Academic Press, (1995).

10. Mathematical Contest in Modeling tutorial, Li Shangzhi editor, Jiangsu Education Press, (1996).

11. Getting mathematical modeling, Xuquan Zhi, Yang Jinhao ed, Chengdu Electronic Science and Technology Press, (1996).

12. Mathematical modeling, Shen Ji, Shijiu Yu, Gao Zhen Bin, Zhang Xiaowei, ed., Harbin Engineering University Press, (1996).

13. The mathematical model, Wang Wo ed., University of Science and Technology of China Press, (1996).

14. Mathematical modeling approach, Qi Huan edited, Huazhong University Press, (1996).

15. Mathematical modeling and experimental, mathematical modeling and engineering colleges Industrial Mathematics compiled seminar in Nanjing, Hehai University Press, (1996).

16. Mathematical models and mathematical modeling, to Liu Fu, who compiled Arts, Beijing Normal University Press Du (1997).

17. Mathematical modeling, Yuan Zhendong, Hong Yuan, Lin Wuzhong, Jiang Lumin ed, East China Normal University Press.

18. Mathematical model, Tan Yongji, Yu Wen topiramate compiled Fudan University Press (1997).

19. Mathematical model of practical tutorials, training fee, the middle layer editor Cheng Yuan, Sichuan University Press, (1998).

20. Selected excellent example of mathematical modeling (base construction engineering mathematics books), Wang Guoqiang editor, South China University of Technology Press, (1998).

twenty one. Economic mathematical model (Second Edition) (Base Construction Engineering Mathematics Series), Hong Yi, He Dehua, edited by Chang Zhihua, South China University of Technology Press, (1999).

twenty two. Mathematical model handouts, Lei Yan Gong, ed., University Press (1999).

twenty three. Mathematical modeling of fine stories, edited by Zhu Tao Yuan, Southeast University Press, (1999),

twenty four. Mathematical modeling approach to solve the problem, Liu Lai Fu, who edited literary, Beijing Normal University Press, (1999).

25. Theory and practice of mathematical modeling, Wu Xiang, Ng Man Tat, Cheng Lai Chi edited, National Defense University Press, (1999).

26, mathematical modeling case studies, Baiqi Ling editor, Ocean Press, (2000, Beijing).

27. Mathematics Experiment (colleges and universities Textbook series), Sun Xie, Zhang let the editor, Science Press, (2000).

28. Math test, Fu Peng, Gong ribs, Qiong-sun, in what city, eds., Academic Press, (2000).

29. Mathematical modeling and mathematical experiments, Zhao Jing, but Qi compiled Higher Education Press (2000).

30. Mathematica timely basis in mathematical modeling application (2nd edition), edited by Li Hanlong, National Defense Industry Press, (2016).

 

Competition reference book

l, Chinese students mathematical modeling contest, Li Daqian editor, Higher Education Press (1998).

2. Mathematical Contest in Modeling resource materials, (a) (2) (3), Ye Qixiao editor, Hunan Education Press (1993, 1997).

3. Mathematical Modeling Education and the International Mathematical Contest in Modeling "Engineering Mathematics" album, Ye Qixiao editor, "Engineering Mathematics" magazine, 1994).

Mathematical Contest in Modeling Guide 4. Students, Xiaohua Yong editor, Electronic Industry Press (2015).
Questions study Mathematical Contest in Modeling 5. National College, WANG Ji Jian editor, National Defense Industry Press (2015).

 

Foreign reference books

(In translation)

1, Introduction to the mathematical model, E. A. Bender forward, Yao Zhu Chen, Xu Weixuan translation, Popular Science Press (1982).

2. Mathematical model, [door] with Jiro Kondo, government Rong Zhang, M., Machinery Industry Press, (1985).

3. Differential equation model (mathematical model applied Series Volume 1), [the United States] W. F. Lucas editor, Zhu Yumin, M., National Defense University Press, (1988).

4. Political and related models (Application of Mathematical Model Series Volume 2), [the United States W. F. Lucas editor, Wang Guoqiu translated, National Defense University Press, (1996).

5. Discrete and system model (Application of Mathematical Model Series Volume 3), [the United States w. F. Lucas edited, translated into Lai Chi, National Defense University Press, (1996).

6. Life sciences model (Application of Mathematical Model Series Vol. 4), [the United States 1W. F. Lucas editor, Zhai Xiaoyan, M., National Defense University Press, (1996).

7. Mathematical model - continuous power systems and discrete dynamical systems, [British 1H. B. Grif6ths and A. 01dknow with, Xiao ceremony, Zhang Zhijun compilation, Science Press, (1996).

8. Mathematical modeling - four cases from the United Kingdom industry research (applied mathematics Renditions No. 4), the British] D. Burglles waiting, Yeqi Xiao, WU Qing Bao translation, World Book Publishing Company, (1997)

 

Professional reference books

(Lot of this book, only a few columns for reference):

1. Complex network algorithms and applications, keep the Secretary Kui, Sun Yuxi Jing, eds., National Defense Industry Press, (2015).

2. Mathematical model of water environment, [Germany] W. KinZE1bach with, Yeung Jun, Liu Zhaochang such as editing, China Building Industry Press, (1987).

3. Mathematical model of science and technology project, worthy of Qi edited, Railway Publishing House (1988)

4. Biomedical mathematical model, green free school ed, Hunan Science and Technology Press (1990).

5. Crop Pest Management and application of mathematical models, Po Zhelong editor, Guangdong Science and Technology Publishing House (1990).

6. System Science mathematical model, edited by Ouyang Liang, Shandong University Press, (1995).

7. Mathematical modeling and study of population ecology, indebted to the horse, Anhui Education Press, (1996)

8. Modeling, transformation, optimization - Progress comprehensive approach to structure, Sui Yunkang the Dalian University of Technology Press, (1986)

9. Genetic model analysis method, with Zhu Jun, China Agriculture Press (1997). (Department of Mathematics, Sun Yat-sen Wang Shousong editor, April 2001)