Encyclopedic Dictionary of Mathematics: A-N

2171 Pages·1987·177.64 MB·English

by  Mathematical Society of Japan

Encyclopedic
Dictionary
of
Mathematics
Second Edition

Encyclopedic
Dictionary
Of
Mathematics
Second Edition
by the
Mathematical Society
of Japan
edited by
Kiyosi Itô
Volume I
A-N
The MIT Press
Cambridge, Massachusetts,
and London, England
First MIT Press paperback edition, 1993
Originally published in Japanese in 1954 by
Iwanami Shoten, Publishers, Tokyo, under the title
Iwanami Süyaku Ziten. Copyright ‘(2 1954, revised
and augmented edition 0 1960, second edition
c 1968, third edition 0 1985 by Nihon Sugakkai
(Mathematical Society of Japan).
English translation of the third edition CC; 1987 by
The Massachusetts Institute of Technology.
Al1 rights reserved. No part of this book may be
reproduced in any form or by any means, electronic
or mechanical, including photocopying, recording,
or by any information storage and retrieval system,
without permission in writing from The MIT Press.
This book was set in Monolaser Times Roman by
Asco Trade Typesetting Ltd., Hong Kong, and
printed and bound by Arcata Graphies, Kingsport in
the United States of America.
Library of Congres Cataloging-in-Publication Data
Iwanami sügaku jiten (ziten). English.
Encyclopedic dictionary of mathematics.
Translation of Iwanami sugaku jiten.
Includes bibliographies and indexes.
1. Mathematics-Dictionaries. 1. Itô, Kiyosi,
1915- II. Nihon Sügakkai. III. Title.
QA5.18313’ 1986 510’.3’21 86-21092
ISBN 0-262-09026-O (HB), o-262-59020-4 (PB)
Contents Forewords vii
Prefaces ix
Introduction xix
Encyclopedic Dictionary of Mathematics
Appendix A. Tables of Formulas 1421
Appendix B. Numerical Tables 1820
Statistical Tables for Reference 1837
Numerical Tables for Reference 1838
Journals 1840
Publishers 1848
Special Notation 1850
Systematic List of Articles 1857
Alphabetical List of Articles 1863
Contributors 1867
Translators 1873
Name Index 1875
Subject Index 1917

Foreword to the The Mathematical Society of Japan takes
Second English Edition pleasure in presenting this second edition of
our Encyclopedic Dictionary $Mathematics
to every researcher and user of mathematics.
It is intended to be a compact, up-to-date
source of information comprising, as com-
pletely as possible, all significant results in ah
fields of our Science, pure and applied, from
the elementary to the advanced level. The
success of the tïrst edition owed much to the
kind assistance given by the American Mathe-
matical Society. As described in the preface,
the members of our Society have taken re-
sponsibility for compiling this new edition.
We hope that it Will be as useful to the mathe-
maticians of today as the first edition was
to the mathematicians of yesterday. We also
hope that this edition Will be followed in
years to corne by subsequent ones incorporat-
ing the future development of our Science.
Hikosaburo Komatsu
President 198551987
Mathematical Society of Japan
Foreword to the The American Mathematical Society welcomes
First English Edition the publication of the Encyclopedic Dictionaqj
of‘ Mathematics. For many years we have been
fascinated by the publication in Japanese,
Iwanami Sügaku Ziten, because we saw that
this was an encyclopedia that contained effec-
tive and penetrating information about all
the tïelds of advanced mathematical research.
We were also frustrated because we could not
read Japanese and SO we could not really
reach out to this expert and effective source
of information. We now welcome the fact that
the second Japanese edition has been trans-
lated into English and we look forward to the
fascination which we cari now have in getting
at this rich mine of information.
Saunders MacLane
President 197331974
American Mathematical Society

Preface to the written in English. Such replacement was not
Second English Edition done in compiling EDM.
(3) Years of birth and death are included in
the Name Index of EDM2 as far as possible.
This second edition of the Encyclopedic Dic- This information was given in ISZ2 but was
tionary of Mathematics is in substance an removed in compiling EDM.
English version of the third edition of Iwanami (4) The Subject Index of EDM2 is SO de-
Sügaku Ziten (in Japanese). We shall explain signed that every concept consisting of two or
how these two versions are related to each more words cari be traced from each of these
other and how the present edition differs from component words. (This principle was adopted
the first English edition. For the sake of sim- in ISZ2 and EDM but was not rigidly fol-
plicity we abbreviate Encyclopedic Dictionary lowed in ISZ3 due to the lack of space.)
of Mathematics and Iwanami Sügaku Ziten to (5) While a11e ditions of ISZ were in one
EDM and ISZ, respectively, and indicate the volume and EDM was in two volumes, EDM2
numbers of editions beyond the lïrst by Arabie is in four volumes: 1. Forewords, prefaces,
numerals. introduction, text A-E; II. Text FFN; III.
The prefaces of the previous editions Will Text O-Z; IV. Appendices, indexes and other
clarify how ISZ, its augumented edition, ISZ2, backmatter. The Systematic List of Articles
ISZ3, and EDM came into existence in 1954, appears in volume IV, page 1857.
1960, 1968, 1985, and 1977, respectively. EDM, While EDM2 is more voluminous than
published nine years later than ISZ2, consisted ISZ3, we hope that, being written in English
of its English translation and some new mate- and printed with generous margins, it cari
rials added to update its contents. In retro- easily provide readers with information on
spect, it was felt that this procedure was not every signitïcant result of today’s mathematics
adequate to tope with the recent rapid pro- and SO Will be useful to a large number of
gress of mathematics, and a suggestion was mathematicians. 1 am responsible for any
raised in the Mathematical Society of Japan shortcomings that may exist despite all our
that ISZ3 and EDM2 be produced simulta- efforts, and 1 would appreciate any remarks
neously. The favorable reaction of the mathe- from the readers. I hope to have occasion to
matical public to EDM encouraged us greatly. remedy any such shortcomings in the future.
In 1978 an agreement was made among the Finally 1 would like to repeat my thanks to
Society, Iwanami Shoten, Publishers, and The a11 the collaborators named in my preface to
MIT Press for the publication of ISZ3 and ISZ3. Also 1 express my gratitude to Yuji Ito
EDM2. An editorial committee was estab- for his tremendous effort to polish the English
lished in the Society with the members named of all manuscripts, to Shigeru Iitaka for his
in the preface to ISZ3. Manuscripts were pre- laborious assistance in the final stage of the
pared simultaneously in Japanese and English work, to Mrs. M. Nawata for her excellent
by each contributor, with Yuji Ito acting as secretarial work, and to the staff of the De-
linguistic consultant. partment of Mathematics at Gakushuin Uni-
1 should mention that we benefïted greatly versity for providing me with an office for
from the kind comments on EDM by the compilation for eight years.
following mathematicians: J. F. Adams, M.
Atiyah, A. Borel, H. Cartan, K. Chandrase- Kiyosi Itô
kharan, S. S. Chern, J. Dieudonné, E. Hewitt, December 1986
F. Hirzebruch, 0. Lehto, J. L. Lions, L. Markus,
and J. P. Serre. In particular, we are deeply
grateful to J. Dieudonné for his many detailed
comments.
Compared with ISZ3 and EDM, EDM2 has
the following characteristics:
(1) EDM2 contains many more new mathe-
matical results than EDM. For the details we
refer the reader to the main points of revision
mentioned in the preface to ISZ3, because
EDM2 has the same mathematical content
as ISZ3 and the additions made to ISZ2 and
subsequently in EDM are of relatively minor
importance.
(2) The Japanese textbooks listed in the
article references of ISZ3 are replaced in
EDM2 by references to standard textbooks
X
Preface to the accordingly. TO each article a parenthetical
Third Japanese Edition number is also attached, giving the subject
area to which the article belongs and its place
in that list.
The tïrst edition of Iwanami Sùgaku Ziten was (3) On the Text: Sections within each arti-
published in 1954, and a revised and aug- cle are headed by letters A, B, C, ; the ini-
mented edition appeared in 1960. Extensive tial section of each article gives an outline
revision of the work was carried out sub- and is devoted to an introductory and general
sequently, and the result was published in 1968 account of the topic with which the article
as the second edition, which has retained its is concerned. In the actual exposition of
valuable and useful life for the past 17 years. the articles, every effort has been made to
The English translation of this second edition pay close attention to the interrelationships
published by The MIT Press has achieved amongst various lïelds of mathematics. Even
international recognition. In the meantime, when there was no need to revise the original
remarkable progress has been made in mathe- article, changes were made in the bibliography
matical sciences: recognition of the interrela- to list items that would be more accessible to
tionships amongst various branches within the readers.
mathematics has increased signitïcantly, and (4) On the Appendices: The appendices have
the formulation of mathematics as a synthetic been revised and augmented along with the
entity is in the making. Furthermore, advanced text. Numerical tables were deleted when-
mathematical theories have been utilized fre- ever it was felt that scientilïc calculators or
quently in the physical, biological, and social microcomputers could easily reproduce their
sciences, and expectations for mathematics contents.
to be the basis of a11 the sciences have been (5) On the Indexes: The listing of mathe-
increasing. In order to tope with these devel- matical terms in Japanese has been changed,
opments, it was decided that the second edition as in the case of the arrangement of articles,
of Sügaku Ziten should be further revised and from the alphabetical order to the order of
an updated version be published as the third the Japanese syllabary. However, terms in
edition. The English translation of this third Japanese that start with a European word
edition Will also be published by The MIT have been separated out, and, in order to
Press. facilitate fïnding, have been listed indepen-
The main points of the revision are as dently in an Index for Mathematical Terms in
follows: Japanese with European Headings. The loca-
(1) On the Size and Scope of the Encyclo- tion of a term listed in the indexes is indicated
pedia: The total number of articles has in- by the article number and the letter heading
creased only slightly from that of the second of the section in the article where the term
edition, from 436 to 450. However, in view appears.
of the rapid and extensive development in
mathematics in recent years, many old articles We here outline briefly how the process of
have been reorganized and unilïed, and much compilation of this edition was organized. In
new material has been added. Consequently, the summer of 1978 the Mathematical Society
the mathematical content of the encyclopedia of Japan decided, following a proposa1 by
has increased a great deal over that of the Professors Shôkichi Iyanaga and Yukiyosi
second edition, and this, together with the Kawada, the chief editors of the previous
authors’ efforts to render the articles more editions of Sügaku Ziten, to undertake the
readable than before, necessitated a 50% in- compilation of the third edition. Following
crease in the number of pages devoted to the a resolution by the members of the Soci-
main text. The account of computer science ety, Kunihiko Kodaira, Sigeru Mizohata,
has been taken out of the subject area of Itiro Tamura, Nagayoshi Iwahori, Tosihusa
Numerical Analysis, and a new subject area Kimura, and myself have joined Iyanaga and
has been created for it; thus mathematics is Kawada to form an editorial committee and
now classified into 21 subject areas altogether have formulated basic guidelines for the com-
instead of 20. pilation of the third edition. The work of
(2) On the Arrangement of the Articles: In editing and compiling articles in each subject
the second edition, the title of each article area was delegated to the editors designated
was spelled out in Romanized Japanese, and for the areas, and 1 was asked to undertake the
articles were arranged alphabetically. In the responsibility of putting the entire volume in
third edition, however, the titles of the articles order.
are arranged in the order of the Japanese syl- The subject areas and their editors are as
labary (katakana) and each article is numbered follows: