Proceedings of the International Mathematical Conference

203 Pages·1982·3.574 MB·iii-xv, 3-202\203

by  Louis H.Y. Chen, Tze Beng Ng and M.J. Wicks (Eds.)

NORTH-HOLLAND
MATHEMATICS STUDIES 74
Proceedings of the
International Mathematical Conference,
Singapore 198 1
Edited by
LOUIS H. Y. CHEN
TZE BENG NG
and
M. J. WICKS
MathematicsD epartment
National University of Singapore
1982
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK . OXFORD
North-Holland Publishing Company, 1982
AN 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 otherwbe, without the prior permission of the copyright owner.
ISBN: 0 444 86510 I
Publishers:
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PRINTED IN THE NETHERLANDS
V
PREFACE
The present volume contains the Proceedings of a conference and work-
shop held at the National University of Singapore in June 1981. The following
contributions to the conference are not included in these proceedings.
D. L. Burkholder, Topics in martingale theory and Brownian motion.
D. L. Burkholder, Martingale transforms and the geometry of Banach
spaces.
M. Fukushima, On absolute continuity of reversible diffusion processes.
B. H. Neumann, Not quite inner automorphisms of groups.
We would like to thank the many people, especially Drs. Arjen Sevenster,
Mathematics editor of North-Holland Publishing company who cooperated to make
this volume possible.
L.H.Y. CHEN
T.B. NG
M.J. WICKS
ix
INTRODUCTION
The International Mathematical Conference was held at the National
University of Singapore from 1 to 13 June, 1981. It was jointly organized by
the Singapore Mathematical Society and the Department of Mathematics of the
National University of Singapore. The sponsors of the conference were UNESCO,
the International Mathematical Union (IMU), the Mathematical Society of Japan
(MSJ) and the Southeast Asian Mathematical Society (SEAMS). UNESCO made a
grant for the financial support of participants from within the Southeast
Asian region, while IMU contributed to the travel expenses of some of the
invited speakers. The contribution of MSJ is witnessed by the presence of
six invited speakers from Japan.
The initiative for the conference came from SEAMS which adopted it as
one of its major activities for 1981. There were 132 participants from 16
countries. A conference of this size would not have been possible without
adequate financing. Among the contributions the largest came from the Lee
Foundation. The conference also received financial support from Shell Eastern
Petroleum (Pte) Ltd and National Kap (Pte) Ltd. The facilities for the
conference were provided by the National University of Singapore.
In the planning of the conference the Organizing Committee received
valuable advice and assistance from J. W. S. Cassels, S. S. Chern, B. H.Neumann
(IMU), K. It;, Y. Kawada (MSJ) and W. S. Tan (SEAMS).
The Organizing Committee would like to express its deep gratitude to
the above organizations and individuals and to the participants for making
the conference one of the most stimulating and fruitEul mathematical meetings
in Southeast Asia.
Peng Tsu Ann
Chairman
Organizing Committee
xi
ORGANIZING COMMITTEE
Chairman : PENG Tsu Ann
Co-Chairman : TEH Hoon Heng
Secretary : Louis H. Y. CHEN
Assistant Secretaries : LA1 Choy Heng
LEONG Yu Kiang
Treasurer : KOH Khee Meng
Members : CHEW Kim Lin
CHONG Chi Tat
LEE Peng Yee
Malcolm WICKS
SUBCOMMITTEES
Accommodation : CHENG Kai Nah CHAN Gin Hor
LEE Peng Yee
NG Peng Nung
Finance : TEH Hoon Heng PENG Tsu Ann
Malcolm WICKS
Hospital it y : Malcolm WICKS CHOW Yoong Ming
LAM Lay Yong
WAN Fook Sun
Programnes : Stephen T.L. CHOY CHEN Chum Chong
LEONG Yu Kiang
R. SUBRAMANIAN
WOO Kai Yuen
Pub1 ications : LEE Peng Yee CHONG Chi Tat
LING Kiong Doong
NG Tze Beng
Public Relations : Louis H.Y. CHEN KOH Khee Meng
Leonard Y .H. YAP
Registration : TAN Sie Keng CHANG Boon Chong
LA1 Choy Heng
LIM Chee Seng
Transport : SHEE Sze Chin CHANG Kai Meng
Peter C.T. RING
S. J. WILSON
YAP Kim Yew
xii
LIST OF WORKSHOP LECTURES
1 June - 5 June
M. Nagata Connnutative algebra and algebraic geometry
H. Matsumura HomotogicaZ methods in comtctive algebra
D. L. Burkholder Topics in martingale theory and Bromian motion
T. Hida Calculus of Brownian functionals
M. Fukushima Markov processes and functional analysis
LIST OF INVITED LECTURES
8 June M. Nagata Applications of theory of valuation rings
J. W. S. Cassels Rational quadratic foms
B. H. Neumann Not quite inner automorphisms of groups
9 June D. L. Burkholder Martingale transforms and the geometry of Banach
spaces
N. Iwahori Some topics on Weyl groups and Coxeter groups
H. Nagao SevemZ correspondences in the representation theory
of finite groups
10 June T. Hida White noise analysis and its applicatiovrs
B. Hartley Powers of the augmentation ideal
11 June P. Erd& Some of my favowite problems which recently have
been solved
H. Matsumura Recent developments in the theory of excellent rings
12 June E. C. Milner Some recent results in the theory of partially
ordered sets
M. Fukushima On absolute continuity of reversible diffusion
processes
LIST OF SHORT TALKS
9 June Z. T. Hou Closure of renmaZ sequences for circle-operation
(read by 2. S. Liang)
H. H. Teh A generalization of S. EiZenberg's theoq on noml
foms in the studg of format Zangwrges
List of Lectures and Short Talks Xiii
C. T. Chong Global and local addssibility : II. Major subsets
and automorphisms
S. Darwis Nuees dynamiques : a method for classification
G. D. Hu
On the optimum approximation solution in the order
and
for principal oomponent analysis (read by 2. Wang)
Y. C. Chang
Z. Soejoeti Joint asymptotic normality of the least squares
estimators and the mean square error in the usual
linear regression model
M. F. Chen On construction problem for weakly symnetrizable
&-processes (read by S. J. Yan)
S. Nualtaranee On the maximum principle for a subhmonic function
in a cone and in a cylinder
K. F. Tien Relationships between analytical regularization and
regularization by surgery
S. Yodintra The cardinal number of the set of &percomplex
nwnbers
G. H. Chan A conjecture involving generalized matrix functions
10 June K. M. Chong A stu& of nmrtingales using spectral Orders
S. J. Yan
Potentiality and reversibitity for general speed
M. 'F. Chen
fmctions
and
W. D. Ding
L. H. Y. Chen Martingale convergence d a the square +ction
S. M. Nababan Oscillation criteria for second order nonlinear
inequalities
A. Buraczewski A generalization of detsnninant theory of Fredholm
operators in Banach spaces
S. T. L. Choy Integml representation of multiplicative, involution
presedng opemtors in L(C,(S,A) ,B)
C. C. Chen
On the strong p&i$ of the lattice of sublattices
and
of a finite distributive lattice
K. M. Koh
C. C. Chen
On the lmer length of the Zattice of subkttices
and
of a finite distributiv~p lanar lattice
K. M. Koh
C. C. Chen
K. M. Koh On the length of the lattices of subalgebras of a
and Stone algebm
K. L. Teo
11 June Z. S. Liang On Delphic-semigroups in stochastic point processes
xiv List of Lectures and Short Talks
M. F. Chen
Uniqueness criterion for &-processes
and
(read by S. J. Yan)
X. G. Zhen
M. 2. Shaharir A derivation of the Schroedinger equation via the
Fresne 1 stochastic int egra 1
C. R. Hwang Limit problem for some linear global constraint
2. Wang Last exit distributions and m a x i m excursion for
Brownian motion
Z. T. Hou
Markov processes and fietd theoq
and
(read by S. J. Yan)
M. F. Chen
D. Avis Cornbinatorial aspects of computational geometry
G. L. Chia
and Some results on superconpact graphs
C. K. Lim
R. B. Eggleton
and Covering lattice points with discs
R. J. Armstrong
C. H. Kan Norrri-attaining vectors of operators on L spaces
P
xv
LIST OF PARTICIPANTS
ABDUL RAZAK BIN SALLEH, Universiti Kebangsaan Malaysia, Malaysia.
ABU OSMAN BIN MD TAP, Universiti Kebangsaan Malaysia, Malaysia.
ARIFIN, ACHMAD, Institut Teknologi Bandung, Indonesia.
AVIS, DAVID, McGill University, Canada.
BURKHOLDER, D.L., University of Illinois at Urbana-Champaign, U.S.A.
BURACZEWSKI, ADAM, University of Papua New Guinea, Papua New Guinea.
CARRILLO, LUIS D., Iligan Institute of Technology, Philippines.
CASSELS, J.W.S., University of Cambridge, U.K.
CASTILLO, ZANITA C., Philippines.
CHAN, GIN HOR, National University of Singapore, Singapore.
CHAN, KAI MENG, National University of Singapore, Singapore.
CHANG, KOK WAH, University of Calgary, Canada.
CHEN, CHUAN CHONG, National University of Singapore, Singapore.
CHEN, LOUIS H.Y., National University of Singapore, Singapore.
CHENG, KAI NAH, National University of Singapore, Singapore.
CHIA, GEK LING, University of Malaya, Malaysia.
CHONG, CHI TAT, National University of Singapore, Singapore.
CHONG, KONG MING, University of Malaya, Malaysia.
CHOW, YOONG MING, National University of Singapore, Singapore.
CHOY, STEPHEN T.L., National University of Singapore, Singapore.
CHUA, SENG KEE, National University of Singapore, Singapore.
DARWIS, SUTAWANIR, Institut Teknologi Bandung, Indonesia.
DEFENSOR, CORAZON G., University of the Philippines, Philippines.
DEL ROSARIO, ESTELITA B., Polytechnic University of the Philippines, Philippines.
EGGLETON, R.B., University of Newcastle, Australia.
ELEAZAR, MARIA N., Philippines.
ERDOS, PAUL, Hungarian Academy of Sciences, Hungary.
FUKUSHIMA, MASATOSHI, Osaka University, Japan.
HADIWIDJOJO, MOEHARTI, IKIP, Yogyakarta, Indonesia.
HANG, KIM HOO, National University of Singapore, Singapore.
HANTINGAN, ARMAND, Universitas Palangka Raya, Indonesia.
HARAHAP, ACHMAD SAID, Universitas Sumatera Utara, Indonesia.
HARAHAP, MARWAN, Universitas Sumatera Utara, Indonesia.
HARTLEY, B., University of Manchester, U.K.
HEE, JUAY GUAN, National University of Singapore, Singapore.
HENG, SIEW JOO, National University of Singapore, Singapore.
HIDA, TAKEYUKI, Nagoya University, Japan.
HUANG, HUE1 CHUEN, National University of Singapore, Singapore.
HWANG, CHII-RUEY, Academia Sinica, Taipei.
Roceedinm of the
International Mathematical Conference
L.H. Y. Chen, T.B. Ng,M J. wlcks (eds.)
@ North-Hdlandhblishing Compmy. 1982 3
APPLICATIONS OF THE THEORY OF VALUATION RINGS
Masayoshi Nagata
Department of Mathematics
Kyoto University
Kyoto
Japan
The theory of valuation rings was founded in connection with number theory. The
valuations concerned are multiplicative valuations having values in non-negative
real numbers. Then W. Krull [l ]d efined the notion of an additive valuation
(called an "Allgemeine Bewertung") having values in an ordered additive group.
This is a generalization of a p-adic valuation which Krull employed to study
commutative rings.
We shall review first some of the well known and important applications of the
theory of valuations (81). Then we prove a generalization of a well known exis-
tence theorem of valuation rings (82). In the last section, we give a theorem
which gives a sufficient condition for a field to be finitely generated, as an
application of the theory of valuation rings. Furthermore we add some modifica-
tion of the theorem,
In this article, by a ring, we mean a commutative ring with identity.
1. REVIEWS ON WELL KNOWN AND IMPORTANT APPLICATIONS,
]
Krull [2 applied the theory of valuation rings especially to study normal
rings, and obtained the theory of Krull rings (called "Diskrete Hauptordnung").
The notion of Krull ring is a generalization of that of noetherian normal ring
and plays an important r$le in studying noetherian rings. Furthermore, Krull
made use of the generalized valuations in many other situations. One basic fact
is that a normal ring is the intersection of valuation rings containing it. This
was given by Krull [l 1.
0. Zariski [l] employed valuation rings in order to study correspondences
between algebraic varieties. A basic fact for this may be expressed that a valu-
tion ring of the function field, k(V), of a variety V over a field k characterizes
a full set of simultaneous specializations of corresponding points (in the family
of birationally equivalent varieties) to a given generic point of V.
In the case of multiplicative valuations, their completions play an important