Combinatorial mathematics is also called discrete mathematics, but sometimes people add combinatorial mathematics and graph theory together to calculate discrete mathematics. Combinatorial mathematics is a branch of mathematics that has developed rapidly since the emergence of computers. Computer science is the science of algorithm, and the object of computer processing is discrete data, so the processing of discrete objects has become the core of computer science, and the science of studying discrete objects is only combinatorial mathematics. The development of combinatorial mathematics has changed the dominant position of analysis and algebra in traditional mathematics. Modern mathematics can be divided into two categories: one is to study continuous objects, such as analysis and equations, and the other is to study combinatorial mathematics of discrete objects. Combinatorial mathematics not only plays an extremely important role in basic mathematics research, but also has important applications in other disciplines, such as computer science, coding and cryptography, physics, chemistry, biology and so on. The development of calculus and modern mathematics laid the foundation for the modern industrial revolution. The development of combinatorial mathematics is the foundation of computer revolution in this century. A computer can be called a computer because it is programmed, and the program is an algorithm. In most cases, computer algorithms are aimed at discrete objects, rather than numerical calculations. It is because of the combinatorial algorithm that people feel that computers seem to have thinking.
Combinatorial mathematics has important application value not only in software technology, but also in enterprise management, traffic planning, war command, financial analysis and other fields. In America, there is a company named after Combinatorial Mathematics. They use combinatorial mathematics to improve the efficiency of enterprise management. This company is very successful. In addition, experimental design is also a subject with great application value, and its mathematical principle is combinatorial design. Combinatorial design is used to solve the experimental design problems in industry, and there are specialized companies in the United States to develop software in this field. Recently, a famous combinatorial mathematician in Germany used combinatorial mathematics to study the structure of drugs, which saved a lot of money for pharmaceutical companies and attracted the attention of pharmaceutical industry.
At the inaugural meeting of the Combinatorial Mathematics Research Center of Nankai University in June, 1997, 1 1, Academician Wu Wenjun pointed out that each era has its own special requirements, which makes mathematics have a new look and some new branches of mathematics appear. Combinatorial mathematics, a new branch, also came into being under the requirements of the times. Recently, Academician Wu Wenjun pointed out that information technology will probably bring a fundamental change to mathematics itself, and combinatorial mathematics will show its important role. Academician Yang Le also pointed out that combinatorial mathematics is playing an increasingly important role in both application and theory, and its future development is very vital and promising. China should advocate research in this field. Academician Wan Zhexian even gave an example to show that the mathematicians of the older generation in China, such as Hua, Xu and Wu Wenjun, not only attached importance to combinatorial mathematics, but also made great contributions to some basic problems of combinatorial mathematics. Due to the needs of the development of combinatorial mathematics in China and the development of information industry in China, it is urgent to develop combinatorial mathematics in China.
2. Combination of mathematics and computer software
With the development of computer network, the use of computers has affected people's work, life, study, social activities and business activities, and the application of computers is basically realized by software. I heard a saying in America that the economic strength of a country in the future can be directly reflected from the software industry. It may not be easy to tell the root cause of China's backwardness in software. In addition to technical and scientific reasons, it may also be related to our culture, management level, education level, ideological quality and many other factors. In addition to these humanistic factors, one of the most fundamental reasons is that the mathematical foundation of information technology in China is very weak. If this problem is not solved, it will be difficult for us to become a software power. However, the problem is by no means so simple. The development of information technology has involved deep mathematical knowledge, and mathematics itself has also developed to a deep and wide extent. It's not just a question that a few clever minds think about, but more importantly, it needs collective cooperation and strength, just as software development needs the cooperation of many people. The key reason why American software can lead is that they are strong and have many excellent talents on the basis of mathematics. Most people may think that mathematics is a pure basic science, and the solution of 1+ 1 may have no practical significance. If so, it doesn't matter if the development of a pure subject lags behind for several years or even ten years. However, the development of China's software industry has put forward an urgent demand for mathematical foundation: network algorithm and analysis, information compression, network security, coding technology, system software, parallel algorithm, mathematical mechanization and computer reasoning, etc. In addition, there are many algorithms that need mathematical foundation, such as logistics planning, financial engineering, computer-aided design and so on. If our software industry still focuses on application software and secondary development, then we will also let foreign enterprises grab a big market in the field of application software. If we vigorously support and invest on the mathematical basis of information technology now, it is better to do it late than never; As long as we can find the mathematical foundation of information technology, then we may turn the tables or even turn the tables in the competition of software industry. The research on mathematical mechanization initiated and led by Academician Wu Wenjun has occupied an important position in the field of information technology for China. With a solid mathematical foundation, we naturally have the competitiveness of software development. A few more such jobs will open up a new prospect for our software industry. It is worth noting that India has a good foundation in statistics and combinatorial mathematics, which may also be the reason for the great development of Indian software industry in recent years.
3. The situation of combinatorial mathematics abroad
Looking at the world software industry, it is easy to see a strange phenomenon: the United States is in an absolute monopoly position. A fundamental reason for this phenomenon is the rapid development of computer science in the United States. Many of the most authoritative people in computer science today come from combinatorial mathematics. The most important computer science departments in the United States (MIT, Princeton, Stanford, Harvard, Yale, ...) have first-class combinatorial mathematicians. It is an indisputable fact that computer science has brought great benefits by promoting the software industry. Combinatorial mathematics has long been a very important subject abroad, and it can even be said to be the basis of computer science. Some big companies, such as IBM, AT & amp； T has the strongest joint research center in the world. Bill Gates of Microsoft recently advocated and supported the basic research of computer science. For example, the realization of linear programming algorithm in Bell Laboratories and the algorithm of computer network are obviously not made public because of their obvious commercial value. There is a trend in the United States that software related to new algorithms can be patented. If we follow this trend, the investment and competition in combinatorial mathematics and computer algorithms will become more and more fierce all over the world. The U.S. government has also established the Center for Discrete Mathematics and Theoretical Computer Science, DIMACS (with Princeton University, Rutgers University, AT & amp； T co-founded, located at Rutgers University), the center has always been an important research position of combinatorial mathematics theory computer science. The National Institute of Mathematical Sciences (founded by Mr. Chen Shengshen) chose combinatorial mathematics as the research topic in 1997 and organized a one-year research activity. NEC Corporation of Japan has also set up a research center in the United States. Theoretical computer science and combinatorial mathematics have become their important research topics, and R. Tarjan, director of the center, is the authority of combinatorial mathematics. I am familiar with the important national laboratory in the United States (Los Alamos National Laboratory, famous for building the world's first atomic bomb), which has always attached importance to the research of applied mathematics including combinatorial mathematics since the Manhattan Project. The computer simulation project of combinatorial mathematics that I came into contact with cost 30 million dollars. Not only that, the laboratory has been actively enriching its research strength in combinatorial mathematics recently. Sandia National Laboratory, another important national laboratory in the United States, has an institution specializing in combinatorial mathematics and computer science, mainly engaged in the research of combinatorial coding theory and cryptography, and has a high position in the American government and international academic circles. Because the structure and biological phenomena of DNA in biology are closely related to combinatorial mathematics, all countries attach great importance to the research of bioinformatics, which is also an important field where combinatorial mathematics can play a role. The Xiangshan Conference held in Beijing not long ago showed that the country attached great importance to bioinformatics. It is said that IBM will also set up a bioinformatics research center. Because DNA is the sequence structure in combinatorial mathematics, Professor Rota, an academician of American Academy of Sciences and founder of modern combinatorial mathematics, predicted that combinatorial problems in biology would become the frontier field of combinatorial mathematics.
There are many research centers of combinatorial mathematics in American universities, national research institutions, industrial, military and intelligence departments, and a lot of money has been invested in research. But their income far exceeds their investment, and more importantly, they gather the best talents in the field of combinatorial mathematics in the world. Combinatorial mathematics can be seen everywhere in advanced software products, or more precisely combinatorial algorithms. Traditional computer algorithms can be divided into two categories, one is combinatorial algorithm, and the other is numerical algorithm (including computational mathematics and informatics related to processing various information and data). In my humble opinion, there is another computer algorithm in recent years: symbolic calculation algorithm. The machine proof method initiated by Academician Wu Wenjun belongs to symbolic computation, which has aroused high evaluation in the world and is called Wu method. There is also a special journal of symbolic computation in the world. Symbolic algorithm and Wu method are also closely related to algebraic combinatorics. Combinatorial mathematics, numerical calculation (including computational mathematics, scientific calculation, nonlinear science and informatics related to processing all kinds of information and data) and statistics may be the most widely used branches of mathematics, and the value of combinatorial mathematics is even less than that of statistics and numerical calculation. Due to the development of mathematical mechanization in recent years and its importance in computer science, combining mathematical mechanization, scientific calculation and combinatorial mathematics can be said to be the foundation of China's information industry. Combinatorial mathematicians H. Wilf and D. Zeilberger 1998 won the Steele Prize of American Mathematical Society 1998 for their achievements in the mechanical proof of combinatorial identities.
Before Professor Ji 'an-carlo rota died last year, he specifically asked me to appeal to the relevant departments and leaders in China that combinatorial mathematics is the foundation of the computer software industry, and China will eventually become a big software country, but the breakthrough to achieve this goal is to develop combinatorial mathematics. China lags far behind the United States in software technology and behind the United States and Europe in combinatorial mathematics. If China only wants to follow the West in software technology and doesn't work hard on combinatorial mathematics, then China's software will always be in a backward state. He particularly emphasized the role of combinatorial mathematics in computer science, and strengthened the teaching and personnel training of combinatorial mathematics in university computer department.
Recently, an electronic publication "Discrete Mathematics and Theoretical Computer Science" founded by Thomson Scientific Company is a good illustration. Its content involves many aspects of discrete mathematics and computer science. Due to the promotion and demand of computer software, combinatorial mathematics has become a broad and profound subject, which requires a deep mathematical foundation and gradually becomes the mainstream branch of mathematics. Gail Fonder, a great mathematician recognized in this century, predicts that combinatorial mathematics and geometry will be the frontier of mathematical research in the next century. This view has been endorsed not only by the international mathematics community, but also by the mathematics community in China.
Canadian Experimental Mathematics Research Center was established in Montreal. Their ideas may be similar to those of Academician Wu Wenjun's Mathematical Mechanization Research Center. Mechanization and algorithmic mathematics will not only make mathematics serve computer science, but also make computers serve mathematical research. Academician Wu Wenjun pointed out that China's traditional mathematics itself has strong algorithmic ideas.
In the future, computers will develop in a more intelligent direction, and the way out is mathematical algorithms and mathematical mechanization. Another convincing phenomenon is that combinatorial mathematicians can always find a good job in the computer department or computer company of a university, and an excellent combinatorial mathematician is naturally an excellent computer scientist. On the contrary, all computer departments in American universities have courses in combinatorial mathematics.
Besides the above, Europe is also actively developing combinatorial mathematics. Britain, France, Germany, the Netherlands, Denmark, Austria, Sweden, Italy, Spain and other countries have established various forms of combinatorial mathematics research centers. In recent years, South American countries are also actively promoting the research of combinatorial mathematics. Australia and New Zealand have also established powerful combinatorial mathematics research institutions. It is worth mentioning that developed countries in Asia also attach great importance to the study of combinatorial mathematics. Japan has a Combinatorial Mathematics Research Center and imported talents from the United States, which not only supports domestic research in Japan, but also funds research on related topics in the United States, thus making the development of Combinatorial Mathematics in Japan extremely rapid in recent years. Taiwan Province Province and Hongkong also introduced talents from the United States to develop combinatorial mathematics. Singapore, South Korea and Malaysia are also actively promoting the research and personnel training of combinatorial mathematics. The Mathematics Research Center of Taiwan Province Province is also considering the development of combinatorial mathematics as the key direction. There is obviously a reason why the world loves combinatorial mathematics so much, that is, without combinatorial mathematics, there would be no computer science and no computer software.
4. Combinatorial mathematics tidbits
* * In our daily life, we often encounter the problem of combinatorial mathematics. If you carefully observe a map of the world, you will find that if a country is colored with one color, then only four colors are needed to ensure that the colors of every two neighboring countries are different. This coloring effect can be clearly displayed in every country. But in order to prove that this conclusion is indeed a famous world problem, it was finally solved with the help of computers, and recently people found a simpler proof.
* * The third-order Rubik's Cube is recorded on the ancient China Heluotu, that is, nine numbers from one to nine are arranged in three rows and three columns, so that the sum of the three numbers on each row, column and diagonal line is fifteen. There are many ingenious structures similar to magic squares in combinatorial mathematics. 1977, American voyager spacecraft 1 and 2 brought the Rubik's cube as a signal of human wisdom.
* * When you pack a box, you will find that it is not easy to make the box as full as possible. You often need to make some adjustments. Theoretically speaking, the packing problem is a difficult combinatorial mathematics problem, which is not easy to solve even by using a computer.
* * In the math game in primary and secondary schools, there is a problem that a boatman wants to transport a wolf, a sheep and a cabbage across the river. The problem is that when people are away, the wolf wants to eat sheep and the sheep wants to eat cabbage, and his boat can only carry one of them at a time. How can he transport all three people across the river? This is a typical and simple combinatorial mathematics problem.
* * We will also encounter more complicated schedule and arrangement problems. For example, the Manhattan project to produce atomic bombs involves many processes, many personnel arrangements and many components. How to arrange the work of all kinds of personnel and the connection between processes to make the whole construction period as short as possible? These are typical examples of combinatorial mathematics.
* * Airline scheduling and flight setting are also problems of combinatorial mathematics. How to ensure that each flight can meet the transfer needs of different passengers, and at the same time make the flight distribution of each airport reasonable. In addition, how to make the most reasonable adjustment under some special circumstances such as flight delay is a problem of combinatorial mathematics.
* * For urban traffic management, traffic planning, which places may be blocked, where one-way streets should be set up, where overpasses should be built, how to set traffic lights in the most reasonable way, etc., are all problems of combinatorial mathematics.
* * How should a postman choose the path to complete the street under his jurisdiction from the post office? This is the famous "China postal route problem", which was put forward by Professor Guan Meigu, a combinatorial mathematician in China, and the famous combinatorial mathematician J edmonds and his collaborators gave the answer.
* * What is the most economical layout of communication network? Both bell laboratory and IBM have world-class combinatorial mathematicians studying this problem, which is directly related to huge economic benefits.
* * It is said that in the management of Holiday Inn, relevant procedures are also strictly stipulated, such as what the cleaners should change in the first step, what to clean and what to do in the second step. In short, he should at least go in and out of the room. Since such a simple job needs to pay attention to the process, let alone a complex project.
* * The management of warehouse and transportation is also a typical combinatorial mathematics problem. How to arrange transportation to make the warehouse fully play its role, and further, where is the most convenient place to store the goods (for example, the goods stored for a short time should be placed in an easy-to-access place).
* * We know that a floor can be paved with square bricks of the same shape (regardless of the edges), but if a floor is paved with bricks of different shapes but not square, can it be paved? This is not only a practical problem, but also a deep combinatorial mathematics problem.
There is a famous question in combinatorial mathematics: Is there a problem of marriage stability? If we can find two pairs (such as Zhang (male)-Li (female)-Zhao (male)-Wang (female), if Zhang (male) prefers Wang (female) and Wang (female) prefers Zhang (male), then there may be potential instability. Combinatorial mathematics can find a way to arrange marriage, so there is no such instability (of course, this is only a theoretical conclusion). This method of combinatorial mathematics has a practical application: American hospitals will consider the priority of applicants' wishes when deciding the admission of residents, and will also rank the applications. Neither the hospital nor the applicant will regret the overall plan considered in this order. In fact, the final admission plan of college entrance examination students can also use this method.
* * Combinatorial mathematics can also be used for financial analysis, determination of investment scheme, and how to find a good investment portfolio to reduce investment risk. The Research Center of Combinatorial Mathematics of Nankai University has developed the "Jinsha Stock Market Risk Analysis System" and put it into the market, which provides an effective risk prevention tool for short-term investors.
In a word, combinatorial mathematics is everywhere, and its main application is to find the optimal solution in various complex relationships. Therefore, combinatorial mathematics can be regarded as a science of quantitative relations, a science of quantitative operation research and a science of quantitative management.
Comrade Hu Jintao pointed out in his speech on 1998 when he accepted the May 4th Youth Medal that combinatorial mathematics is different from a branch of traditional pure mathematics, and it is also an applied and interdisciplinary subject. He hoped that China's combinatorial mathematics research could serve the country's economic construction.
If 2 1 century is the century of information society, then 2 1 century is bound to be a promising century of combinatorial mathematics.