Shannon, an American scholar, established Shannon's positive theorem with his engineering intuition in his groundbreaking masterpiece. A. X. инчин, a famous mathematician in the former Soviet Union, made a strict mathematical arrangement of Shannon's work, wrote a monograph and listed the mathematical proofs of Shannon's positive theorem. In the conclusion of the monograph, he said: "The sufficient condition on which Shannon's positive definite theorem is based is too strong and should be weakened, and pointed out that it is not easy to weaken the inverse theorem to the necessary and sufficient condition, and a new concept in essence must be introduced." Hu pioneered the study of Shannon's inverse theorem for the first time. Based on the difficulty of the subject, he first discussed two aspects of communication mode, namely source and channel. In his thesis "Information Stability of Channel Sequences", he introduced the new concept of "ε-fan" and successfully found out the necessary and sufficient conditions related to the channel for the first time. This is the "Hu Theorem" first mentioned at Berkeley International Conference. Later, he studied information sources, and introduced the new concept of "ε-compressibility" of information sources for the first time in his paper "Three Inverse Theorems of Shannon Theory in Information Theory", and once again successfully found the necessary and sufficient conditions related to information sources. After overcoming the above two key difficulties, on this basis, combining the related properties of information source and channel, a 47-page creative article Shannon's Positive and Negative Theorem of Abstract Variables in Communication System is completed in 196 1. In this paper, he completely and thoroughly solved the basic problem of Shannon, that is, he found the necessary and sufficient conditions of Shannon's basic proposition. When Hu delivered a speech at the International Conference on Information Theory, he received a great response on the spot. (2) Information quantity is a basic concept in information theory. His other basic work on information theory is On Information Quantity. Shannon introduced one-dimensional and two-dimensional information into his basic work and gave the quantitative relationship between them. Later, the school of the former Soviet Union introduced some information of three variables and found several variables, such as A.H. Colmo Golo (Koлмогоpов) and H.M. Gelfond (гелв). This theorem is often included in textbooks of modern information theory. (3) Subsequently, Hu discussed the basic problems of Shannon under different standards in three articles, and obtained the necessary and sufficient conditions for the establishment of Shannon's basic proposition.
(4) Communication has developed from one-way communication (one transmission and one reception) in 1950s to satellite multi-channel communication (multi-radio transmitting and receiving) after 1960s. If only one or two variables were used in one-way communication in the 1950s, and the information of any limited number of variables introduced in his paper was invisible at that time, it would be of practical use in satellite multi-channel communication. This is also the reason why modern information theory documents and books cite his papers more. After 1970s, Hu overcame many unique difficulties in multi-channel communication on the basis of Shannon's positive and inverse theorem and combined with the relationship between any number of variables, and further successfully extended the results of Shannon's positive and inverse theorem to the general occasions of multi-channel communication models (see three articles published in 1980s), and continued to be highly praised by the international information theory circle. (5) For the most important evaluation of Hu's information theory work abroad, please refer to the book Comprehensive Evaluation of New Achievements in Information Theory by Professor Kotz. The book has 83 pages, covering the main works of information theory in various countries in the world, of which 9 pages are devoted to Hu, and many are full pages. As far as the proportion of individuals in the full text is concerned, he is the most. This can roughly reflect the foreign evaluation of his international status in information theory. Why does Hu occupy a prominent position in the above Coates' works? This is mainly because the content discussed in his works is relatively basic, which has attracted the attention of the information theory circle. (6) Over the years, Hu has trained a number of excellent doctoral students, master students, advanced students and undergraduates in information theory in Nankai, some in theoretical research departments and more in military departments. International conferences on information theory are often held now, and many of our participants are students from Nankai or have studied in Nankai. A stochastic process is a probability space with many events as a curve. After understanding the nature of "simple" stochastic process, people can approximately understand the nature of "complex" stochastic process by using the approximation of "simple" stochastic process to "complex" stochastic process. In order to really calculate the degree of approximation between them, under what conditions can the topological structure in the functional space of all stochastic processes be "quantified", which has become a very important subject in this discipline. On the basis of probability theory and measure theory, Hu used many sharp tools in functional topological space to write σ-topology and measure in topological space. There are two achievements in this paper: (1) The relationship between measures in σ-additive topological space and measures in general topological space is studied, and the necessary and sufficient conditions for their consistency are found.
(2) On the basis of (1), the necessary and sufficient conditions for quantifying the above topological structure are found. This article was published in major magazines in the former Soviet Union, and has been repeatedly quoted since then. (1) In computer science, there is no general definition of "computer" except the simplest so-called "Turing machine". In his article Computer Mathematical Model, he introduced the general constructive definition of "computer"-a computer is a discrete calculator that realizes infinite transformation (i.e. "computable function") through finite transformation (i.e. "program") composed of a limited number of instructions, or a "computer" is a discrete calculator that generates infinite transformation through finite transformation, which is a kind of "finite" discrete calculator.
The article on parallel computer and its software research is an application of the above article. (2) At the beginning of the 20th century, the third mathematical crisis caused by the paradox in Cantor's infinite set theory has not reached the same conclusion as the first and second crises. D Hilbert set aside Cantor's infinite set theory and established strict formal mathematics on the basis of new mathematics with the help of finite mapping principle. After K. Gdel's incompleteness theorem was published in 193 1, the discussion about the third crisis of mathematics was relatively calm. Two basic problems of mathematical foundation are only discussed in the field of mathematical philosophy. These two problems are: first, the truth of mathematics. On the one hand, Godel's incompleteness theorem declares that "at least some theorems in formal arithmetic cannot be proved". On the other hand, ordinary mathematicians always think that "any theorem in mathematics can be proved"; Second, the object of mathematics: On the one hand, when ordinary mathematicians study general mathematics, they think that the object of study is some objective reality, but after reading Hilbert's formal mathematics theory, they regard the object of mathematics as only a limited transformation of a series of symbols. The article "On General Mathematics and Formal Mathematics" attempts to answer the above questions with strict mathematical methods rather than from the perspective of mathematical philosophy. This paper holds that there are essential differences between general mathematics and formal mathematics. The former freely uses Cantor's infinite set theory, while the latter strictly follows the principle of finiteness without Cantor's infinite set theory. According to the general principle of computer, Hilbert's formal mathematics is just the simulation of ordinary mathematics by computer, so formal mathematics can also be called machine mathematics. Following the principle of finiteness, with the help of finiteness, some infinity can be produced, but no infinity can be produced, which is the essential limitation of computer function. So with the help of computers, ordinary mathematics can only simulate a small part, not all. In this way, formal mathematics (machine mathematics) is similar to its simulated ordinary mathematics, but it is essentially different after all; First, theorems in general mathematics can always be proved by deduction of inference rules including arbitrary infinite sets, but they do not necessarily follow limited formal (machine) inference rules, which is the root of why any general mathematics is complete (theorems can be proved) and formal arithmetic is incomplete (theorems cannot be proved); Second, the object of general mathematics is some objective reality, but the object of formal mathematics (machine mathematics), that is, the object directly processed by computer, is only a limited transformation of a series of symbols, but because formal mathematics (machine mathematics) is the simulation of general mathematics, in the final analysis, a series of symbols are still a reflection of some objective reality.