Popularizing Mathematical Methods in the People’s Republic of China: Some Personal Experiences

225 Pages·1989·6.409 MB·English

by  Loo-Keng Hua, Yuan Wang (auth.), J. G. C. Heijmans (eds.)

MATHEMATICAL
MODELING
No.2
Edited by
William F. Lucas, Claremont Graduate School
Maynard Thompson, Indiana University
Hua Loo-Keng Wang Yuan
lbpularizing Mathematical
Methods in the
People's Republic of China
Some Personal Experiences
Revised and Edited by J.G.C. Heijmans
Birkhauser
Boston . Basel . Berlin
Hua Loo-Keng Wang Yuan J.G.C. Heijmans
(1911-1985) Academia Sinica Department of Mathematics
Institute of Mathematics University of Texas at Arlington
Beijing Arlington, TX 76019
People's Republic of China U.S.A.
Library of Congress Cataloging-in-Publication Data
Hua, Loo-keng, 1910-
Popularizing mathematical methods in the People's Republic of China:
Some personal experiences I by Hua Loo-keng and Wang Yuan;
revised and edited by l.G.C. Heijmans.
p. cm.-(Mathematical modeling; 2)
Includes bibliographies.
ISBN-13: 978-1-4684-6759-8 (alk. paper)
I. Mathematical optimization. 2. Industrial engineering
Mathematics. 3. Operations research. I. Wang, Yiian, fl. 1963-
II. Heijmans, l.G.C. III. Title. IV. Series: Mathematical
modeling ; no. 2.
QA402.5.H79 1989
519-dc19 88-39804
CIP
Printed on acid-free paper.
© Birkhiiuser Boston, 1989
Softcover reprint of the hardcover 1s t edition 1989
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner.
ISBN-l3: 978-1-4684-6759-8 e-ISBN-13: 978-1-4684-6757-4
001: 10.10071978-1-4684-6757-4
Text provided in camera-ready form by the editor.
9 8 7 6 5 4 3 2 1
Contents
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Preface by the editor.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
XVll
An Obituary of Loo-Keng Bua, by B. Balberstam ................ xix
CBAPTERO
INTRODUCTION
§ 0.1 Three principles.............................................. 1
§ 0.2 Looking for problems in the literature.. . . . . . . . . . . . . . . . . . . . . 3
§ 0.3 Looking for problems in the workshop......... . . . . . . . . . . . . . 5
§ 0.4 Optimum seeking methods (O.S.M) .......................... 8
§ 0.5 The Fibonacci search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
§ 0.6 The golden number and numerical integration..... . . . . . . . . 11
§ 0.7 Overall planning methods................................... 13
§ 0.8 On the use of statistics. .. .. .. .. .. .. .. . .. .. . .. .. .. .. .. .. .. .. . 18
§ 0.9 Conduding remarks. ................ ........................ 23
CHAPTER 1
ON THE CALCULATION OF MINERAL RESERVES AND HILLSIDE
AREAS ON CONTOUR MAPS
§ 1.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 25
§ 1.2 Calculation of mineral reserves. .... .. .. .. . .. .. .. .. .. . . .. .. 26
§ 1.3 Calculation of hillside areas... .. .. .. .. . .. .. . .. .. .. .. .. .. .. . 32
References. ......................................................... 40
v
vi
CHAPTER 2
THE MESHING GEAR-PAIR PROBLEM
§ 2.1 Introduction................................................. 41
§ 2.2 Simple continued fractions...................... . . . . . . . . . . . . 42
§ 2.3 Farey series.................................................. 47
§ 2.4 An algorithm for the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
§ 2.5 The solution to the meshing gear-pair problem. .......... 51
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 55
CHAPTER 3
OPTIMUM SEEKING METHODS (single variable)
§ 3.1 Introduction................................................. 57
§ 3.2 Unimodal functions... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
§ 3.3 Method of trials by shifting to and fro. .... . . . . . . . . . . . . . . . 60
§ 3.4 The golden section method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
§ 3.5 The proof of Theorem 3.1.................................. 63
§ 3.6 The Fibonacci search....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
§ 3.7 The proof of Theorem 3.2... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
§ 3.8 The bisection method. ..................................... 74
§ 3.9 The parabola method. ..................................... 76
References. ......................................................... 78
CHAPTER 4
OPTIMUM SEEKING METHODS (several variables)
§ 4.1 Introduction................................................ 79
§ 4.2 Unimodal functions (several variables).................... 79
§ 4.3 The bisection method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
§ 4.4 The successive approximation method. .. . . . . . . . . . . . . . . . . . 85
§ 4.5 The parallel line method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
§ 4.6 The discrete case with two factors.. . . . . . . . . . . . . . . . . . . . . . . . 89
vii
54.7 The equilateral triangle method............................ 91
54 .8 The gradient method........................................ 94
5 4.9 The paraboloid method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
54.10 Convex bodies............................................... 99
5 4.11 Qie Kuai Fa................................................. 104
54.12 The 0-1 variable method •................................... 106
R.eferences ........................................................... 109
CHAPTERS
THE GOLDEN NUMBER AND NUMERICAL INTEGRATION
5 5.1 Introduction ................................................. III
5 5.2 Lemmas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113
5 5.3 Error estimation for the quadrature formula. ............. 117
5 5.4 A result for {} and a lower bound for the quadrature
formula. ..................................................... 120
5 5.5 R.emarks.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 122
R.eferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 124
CHAPTER 6
OVERALL PLANNING METHODS
5 6.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . .. ... . . . . . . . . . . . . . . . . .. 125
5 6.2 Critical Path Method....................................... 125
5 6.3 Float.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 127
5 6.4 Parallel operations and overlapping operations. .......... 130
5 6.5 Manpower scheduling.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 134
R.eferences ........................................................... 136
CHAPTER 7
PROGRAM EVALUATION AND REVIEW TECHNIQUE (PERT)
S7 .1 Introduction................................................. 137
viii
S7 .2 Estimation ofthe probability........................... . . .. 138
S7 .3 Computation process ........................................ 142
S 7.4 An elementary approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145
S7 .5 R.emarks •.................................................... 147
R.eferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 149
CHAPTER 8
MACHINE SCHEDULING
S8 .1 Introduction. ................................................ 151
S8 .2 Two-machine problem ...................................... 151
S8 .3 A lemma..................................................... 154
§ 8.4 Proof of Theorem 8.1. ...................................... 156
R.eferences ........................................................... 158
CHAPTER 9
THE TRANSPORTATION PROBLEM (GRAPHICAL METHOD)
§ 9.1 Introduction •................................................ 159
§ 9.2 One cycle. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
§ 9.3 Proof of Theorem 9.1....................................... 170
R.eferences... . . . . . . .. . . . . . . . .. .. . . . .. . . . . . .. . . .. . . . . . . . . . . . . . . . . .. . .. 172
CHAPTER 10
THE TRANSPORTATION PROBLEM (SIMPLEX METHOD)
§ 10.1 Introduction. .............................................. 173
§ 10.2 Eliminated unknowns and feasible solutions. . . . . . . . . . . .. 179
§ 10.3 Criterion numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 182
§ 10.4 A criterion for optimality ................................. 188
§ 10.5 Characteristic numbers.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. 189
§ 10.6 Substitution ................................................ 193
§ 10.7 Linear programming ...................................... 197
R.eferences.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198
ix
CHAPTER 11
THE POSTMAN PROBLEM
S 11.1 Introduction •............................................... 201
S 11.2 Euler paths. ................................................ 204
S 11.3 A necessary and sufficient criterion for an optimum
solution. ................................... . . . . . . . . . . . . . . .. 205
R.eference&.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 208
Acknowledgements
I would like to express my gratitude first of all to Dr. B.O. Pollak
of Bell Laboratories in the United States, for so kindly inviting me to
present my report entitled "Some Personal Experiences in Popularizing
Mathematical Methods in the People's Republic of China", both at the
Fourth International Conference on Mathematical Education and at Bell
Laboratories!. Dr. Pollak wrote to me on several occasions urging me to
make additions to my report and collate the material for publication as a
book with Birkhi user Boston, Inc. The fact that this has now been done
is due to his kind recommendation and sincere concern.
My special thanks go to all colleagues, workers and professionals
who have been engaged in the popularization of mathematical methods
over the past 20 years and have done a great deal of creative work in this
field. Because of the vast number of people involved, I am unable to list
each name individually, but I would like to make particular mention of
Messrs. Chen De Quan and Ji Lei, who have been with me since the
inception of my work up to the present time.
A part of the book has been written by Professor Wang Yuan,
based on my report. In view of my own heavy work load and poor
health, this book could not have been completed so quickly had he not
participated in the writing of this book.
Finally, I must also thank Drs. K. Peters and W. Klump of
Birkhiuser Boston, Inc., for their help during the production of this
book.
December, 1982
Bua Loo.Keng
xi