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%-Z&WL
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Selected Wor k s d
%-Zm % Wen-Tsun Wu Academia Sinica,...
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Selected Works f
%-Z&WL
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Selected Wor k s d
%-Zm % Wen-Tsun Wu Academia Sinica, China
1; World Scientific N E W JERSEY
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LONDON
SINGAPORE
- BElJlNG - SHANGHAI
*
HONG KONG
*
TAIPEI
*
CHENNAI
Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK oflce: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
SELECTED WORKS OF WEN-TSUN WU Copyright 0 2008 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN-13 978-981-279-107-8 ISBN-10 98 1-279-107-8
Printed in Singapore by B & JO Enterprise
Foreword
The present “Selected Papers” may be considered as a brief survey of my scientific career in mathematical sciences. My researches in mathematical sciences are consisting of two stages. The researches in the first stage, started in 1947, are in pure mathematics, mainly in algebraic topology, occasionally also in algebraic geometry. This ended actually in 1965, the beginning of cultural revolution. See Nos. 1-5 of &‘SelectedPapers”. During the cultural revolution there were however some sporadic research works in pure mathematics, with papers published a little later. See Nos. 6, 7, 14, 15, 18 of “Selected Papers”. Such researches stopped completely at the end of cultural revolution, viz. the year 1976. The second stage of my mathematical researches took place during the cultural revolution. It took place owing to my learning of the history of our proper mathematics in ancient times. See No. 17 of “Selected Papers”. During the cultural revolution I was sent to some computer-manufacture company to learn and work with laborers. Being striken by the powerfulness of computers I began to consider of applying computers to the study of mathematics. It results in a method of proving geometry theorems by means of computers. Extending further the method it gave rise to the subject what I called the Mathematics Mechanization which had an immense varieties of applications in science and technology, besides the mathematics itself. See Nos. 16, 19-30 of the “Selected Papers”. For some general description of my scientific career one may refer to the book “The Road of WU Wen-tsun” , written in Chinese by Professors HU and SHI, published by Shanghai Science-Technology Press, year 2002.
Wen-tsun Wu Dec. 27, 2007
V
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Contents
Foreword
V
1. On the product of sphere bundles and the duality theorem modulo two
1
2. Classes caract6ristiques et i-carrks d’une variBtk
15
3. Les i-carrhs dans une vari6tk grassmannihne
19
4. On the realization of complexes in Euclidean spaces I
23
5. On the realization of complexes in Euclidean spaces I11
71
6. On universal invariant forms
85
7. Theory of I*-functor in algebraic topology - Effective calculation and axiomatization of I*-functor on complexes
99
8. On the decision problem and the mechanization of theorem-proving in elementary geometry
117
9. Toward mechanization of geometry - Some comments on Hilbert’s “Grundlagen der Geometrie’’
139
10. The out-in complementary principle
153
11. A constructive theory of differential algebraic geometry based on works of J. F. Ritt with particular applications to mechanical theorem-proving in differential geometries
177
12. Basic principles of mechanical theorem-proving in elementary geometries
195
13. On zeros of algebraic equations - An application of Ritt principle
225
14. On the planar imbedding of linear graphs I
231
vii
15. On the planar imbedding of linear graphs I1
245
16. A mechanization method of geometry and its applications I
259
17. Recent studies of the history of Chinese mathematics
273
18. On Chern numbers of algebraic varieties with arbitrary singularities
285
19. Mechanical derivation of Newton’s gravitational laws from Kepler’s laws
295
20. A mechanization method of geometry and its applications I1
303
21. A mechanization method of geometry and its applications I11
307
22. On the foundation of algebraic differential geometry
325
23. On the genetic zero and Chow basis of an irreducible ascending set
345
24. Mechanical theorem proving of differential geometries and some of its applications in mechanics
367
25. On a finiteness theorem about optimization problems
387
26. On surface-fitting problem in CAGD
401
27. Central configurations in planet motions and vortex motions
411
28. On algebraico-differential equations-solving
425
29. On the construction of Groebner basis of a polynomial ideal based on Riquier-Janet theory
437
30. On “good” bases of algebraico-differential ideals
455
viii
ON THE PRODUCT OF SPHERE BUNDLES AND THE DUALITY THEOREM MODULO TWO BY Wu \ji‘m-wux (Ibxivetl August 15, 1947)
Introduction* Given two spiierc brindles GI and S2over the base complexes f i i and fi, respectively, it is possible t,o define in a natural way a “product bundle” over the product complex ]