<p>It is with great pleasure that I am writing the preface for my little book, "Starting with the Unit Circle", in the office of Springer Verlag in Heidel berg. This is symbolic of the fact that I have once again joined in the main stream of scientific exchange between East and West. Since the establishment of the People's Republic of China, I have written "An Introduction to Number Theory" for the young people studying Number Theory: for the young people studying algebra, Prof. Wan Zhe-xian (Wan Che-hsien) and I have written "Classical Groups"; for those studying the theory of functions of several complex variables, I have written "Har monic Analysis of Functions of Several Complex Variables in the Classical Domains", * and for university students I have written "Introduction to Higher Mathematics". The present volume had been written for those who were beginning to engage in research at the Chinese University of Science and Technology and at the Guangdong Zhongshan University. Its purpose is none other than to make the students see the crucial ideas in their simplest manifestations, so that when they go on to the more complex parts of modem mathematics, they will not be without guidance. For example, in the first chapter when I point out that the Poisson kernel is just the Jacobian of some transformation, I am merely revealing the source of one of the main tools in my work on harmonic analysis in the classical domains.</p>

Starting with the Unit Circle: Background to Higher Analysis

Crafts, Hobbies & Home

Loo Keng Hua

Gardening & Landscape Design

Springer

Owing to the developments and applications of computer science, ma thematicians began to take a serious interest in the applications of number theory to numerical analysis about twenty years ago. The progress achieved has been both important practically as well as satisfactory from the theoretical view point. It'or example, from the seventeenth century till now, a great deal of effort was made in developing methods for approximating single integrals and there were only a few works on multiple quadrature until the 1950's. But in the past twenty years, a number of new methods have been devised of which the number theoretic method is an effective one. The number theoretic method may be described as follows. We use num ber theory to construct a sequence of uniformly distributed sets in the s dimensional unit cube G , where s ~ 2. Then we use the sequence to s reduce a difficult analytic problem to an arithmetic problem which may be calculated by computer. For example, we may use the arithmetic mean of the values of integrand in a given uniformly distributed set of G to ap s proximate the definite integral over G such that the principal order of the s error term is shown to be of the best possible kind, if the integrand satis fies certain conditions.

Applications of Number Theory to Numerical Analysis

Computers & Technology

Y. Wang

Computer Science

Introduction to Number Theory

Science & Math

Mathematics

Springer, Verlag

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