Table Of ContentENCYCLOPAEDIA OF MATHEMATICS
Volume 7
ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
Scieniijic Board
S. Albeverio, J. B. Alblas, S. A. Amitsur, I. J. Bakelman, G. Bakker, J. W. de Bakker, C.
Bardos, H. Bart, H. Bass, A. Bensoussan, M. Bercovier, L. Berkovitz, M. Berger, E. A.
Bergshoeff, E. Bertin, F. Beukers, A. Beutelspacher, H. P. Boas, J. Bochnak, H. J. M. Bos, B.
L. J. Braaksma, T. P. Branson, D. S. Bridges, A. E. Brouwer, M. G. de Bruin, R. G. Bums, H.
Capel, P. Cartier, C. Cercignani, J. M. C. Clark, Ph. Clement, A. M. Cohen, J. W. Cohen, P.
Conrad, H. S. M. Coxeter, R. F. Curtain, M. H. A. Davis, M. V. Dekster, C. Dellacherie, G.
van Dijk, H. C. Doets, I. Dolgachev, A. Dress, J. J. Duistermaat, D. van Dulst, H. van Duyn,
H. Dym, A. Dynin, M. L. Eaton, W. Eckhaus, P. van Emde Boas, H. Engl, G. Ewald, V. I.
Fabrikant, A. Fasano, M. Fliess, R. M. Fossum, B. Fuchssteiner, G. B. M. van der Geer, R. D.
Gill, V. V. Goldberg, J. de Graaf, J. Grasman, P. A. Griffith, A. W. Grootendorst, L. Gross, P.
Gruber, K. P. Hart, G. Heckman, A. J. Hermans, W. H. Hesselink, C. C. Heyde, M. W. Hirsch,
K. H. Hofmann, A. T. de Hoop, P. J. van der Houwen, N. M. Hugenholtz, J. R. Isbell, A.
Isidori, E. M. de Jager, D. Johnson, P. T. Johnstone, D. Jungnickel, M. A. Kaashoek, V. Kac,
W. L. J. van der Kallen, D. Kanevsky, Y. Kannai, H. Kaul, E. A. de Kerf, W. Klingenberg, T.
Kloek, J. A. C. Kolk, G. Komen, T. H. Koomwinder, L. Krop, B. Kuperschmidt, H. A.
Lauwerier, J. van Leeuwen, H. W. Lenstra Jr., J. K. Lenstra, H. Lenz, M. Levi, J. Lin
denstrauss, J. H. van Lint, F. Linton, A. Liulevicius, M. Livshits, W. A. J. Luxemburg, R. M.
M. Mattheij, L. G. T. Meertens, I. Moerdijk, J. P. Murre, H. Neunzert, G. Y. Nieuwland, G. J.
Olsder, B. 0rsted, F. van Oystaeyen, B. Pareigis, K. R. Parthasarathy, K. R. Parthasarathy, I. I.
Piatetskil-Shapiro, H. G. J. Pijls, N. U. Prabhu, E. Primrose, A. Ramm, C. M. Ringel, J. B. T.
M. Roerdink, K. W. Roggenkamp, G. Rozenberg, W. Rudin, S. N. M. Ruysenaars, A. Salam,
A. Salomaa, P. Saunders, J. P. M. Schalkwijk, C. L. Scheffer, R. Schneider, J. A. Schouten, F.
Schurer, J. J. Seidel, A. Shenitzer, V. Snaith, T. A. Springer, J. H. M. Steenbrink, J. D.
Stegeman, F. W. Steutel, P. Stevenhagen, I. Stewart, R. Stong, L. Streit, K. Stromberg, L. G.
Suttorp, D. Tabak, F. Takens, R. J. Takens, N. M. Temme, S. H. Tijs, B. Trakhtenbrot, N. S.
Trudinger, L. N. Vaserstein, M. L. J. van de Vel, F. D. Veldkamp, W. Vervaat, P. M. B.
Vitanyi, N. J. Vlaar, H. A. van der Vorst, J. de Vries, F. Waldhausen, B. Wegner, J. J. O. O.
Wiegerinck, J. C. Willems, J. M. Wills, B. de Wit, S. A. Wouthuysen, S. Yuzvinskil, L.
Zalcman
ENCYCLOPAEDIA
OF
MATHEMATICS
Volume 7
Orbit - Rayleigh Equation
An updated and annotated translation of the Soviet
'Mathematical Encyclopaedia'
KLUWER ACADEMIC PUBLISHERS
Dordrecht / Boston / London
Library of Congress Cataloging-in-Publication Data
Matematicheskaia entsiklopediia. English.
Encyclopaedia of mathematics.
1. Mathematics--Dictionaries. I. Hazewinkel, Michiel. II. Title.
QA5.M3713 1987 510'.3'21 87-26437
ISBN 978-90-481-8236-7
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
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ISBN 978-90-481-8236-7 ISBN 978-94-015-1237-4 (eBook)
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SOVIET MATHEMATICAL ENCYCLOPAEDIA
Editor-in-Chief
I. M. Vinogradov
Editorial Board
S. 1. Adyan, P. S. Aleksandrov, N. S. Bakhvalov, A. V. Bitsadze, V. 1. Bityutskov (Deputy
Editor-in-Chief), L. N. Bol'shev, A. A. Gonchar, N. V. Efimov, V. A. II'in, A. A. Karatsuba,
L. D. Kudryavtsev, B. M. Levitan, K. K. Mardzhanishvili, E. F. Mishchenko, S. P. Novikov,
E. G. Poznyak, Yu. V. Prokhorov (Deputy Editor-in-Chief), A. 1. Shirshov, A. G. Sveshnikov,
A. N. Tikhonov, P. L. VI'yanov, S. V. Yablonskii
Translation Arrangements Committee
V. I. Bityutskov, R. V. Gamkrelidze, Yu. V. Prokhorov
'Soviet Encyclopaedia' Publishing House
PREFACE
This ENCYCLOPAEDIA OF MA THEMA TICS aims to be a reference work for all parts of mathe
matics. It is a translation with updates and editorial comments of the Soviet Mathematical
Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in
1977-1985. The annotated translation consists of ten volumes including a special index
volume.
There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type
articles dealing with the various main directions in mathematics (where a rather fine subdivi
sion has been used). The main requirement for these articles has been that they should give a
reasonably complete up-to-date account of the current state of affairs in these areas and that
they should be maximally accessible. On the whole, these articles should be understandable to
mathematics students in their first specialization years, to graduates from other mathematical
areas and, depending on the specific subject, to specialists in other domains of science, en
gineers and teachers of mathematics. These articles treat their material at a fairly general level
and aim to give an idea of the kind of problems, techniques and concepts involved in the area
in question. They also contain background and motivation rather than precise statements of
precise theorems with detailed definitions and technical details on how to carry out proofs and
constructions.
The second kind of article, of medium length, contains more detailed concrete problems,
results and techniques. These are aimed at a smaller group of readers and require more back
ground expertise. Often these articles contain more precise and refined accounts of topics and
results touched upon in a general way in the first kind of article.
Finally, there is a third kind of article: short (reference) definitions.
Practically all articles (all except a few of the third kind) contain a list of references by
means of which more details and more material on the topic can be found. Most articles were
specially written for the encyclopaedia and in such cases the names of the original Soviet
authors are mentioned. Some articles have another origin such as the Great Soviet Ency
clopaedia (Bol'shaya Sovetskaya Entsiklopediya or BSE).
Communication between mathematicians in various parts of the world has certainly greatly
improved in the last decennia. However, this does not mean that there are so-to-speak 'one-to
one onto' translations of the terminology, concepts and tools used by one mathematical school
to those of another. There also are varying traditions of which questions are important and
which not, and what is considered a central problem in one tradition may well be besides the
point from the point of view of another. Even for well-established areas of mathematical
inquiry, terminology varies across languages and even within a given language domain. Fur
ther, a concept, theorem, algorithm, ... , which is associated with one proper name within one
tradition may well have another one in another, especially if the result or idea in question was
indeed discovered independently and more-or-Iess simultaneously. Finally, mathematics is a
very dynamic science and much has happened since the original articles were finalized (mostly
around 1977). This made updates desirable (when needed). All this, as well as providing
vii
PREFACE
additional references to Western literature when needed, meant an enormous amount of work
for the board of experts as a whole; some indeed have done a truly impressive amount of work.
I must stress though that I am totally responsible for what is finally included and what is not of
all the material provided by the members of the board of experts.
Many articles are thus provided with an editorial comment section in a different and some
what smaller typeface. In particular, these annotations contain additional material, amplifica
tions, alternative names, additional references, . . . . Modifications, updates and other extra
material provided by the original Soviet authors (not a rare occurrence) have been incorporated
in the articles themselves.
The final (lO-th) volume of the ENCYCLOPAEDIA OF MATHEMATICS will be an index volume.
This index will contain all the titles of the articles (some 6600) and in addition the names of all
the definitions, named theorems, algorithms, lemmas, scholia, constructions, ... , which occur
in the various articles. This includes, but is by no means limited to, all items which are printed
in bold or italic. Bold words or phrases, by the way, always refer to another article with
(precisely) that title.
All articles have been provided with one or more AMS classification numbers according to
the 1980 classification scheme (not, for various reasons, the 1985 revision), as have all items
occurring in the index. A phrase or word from an article which is included in the index always
inherits all the classification numbers of the article in question. In addition, it may have been
provided with its own classification numbers. In the index volume these numbers will be listed
with the phrase in question. Thus e.g. the Quillen - Suslin theorem of algebraic K-theory will
have its own main classification numbers (these are printed in bold; in this case that number is
18F25) as well as a number of others, often from totally different fields, pointing e.g. to parts
of mathematics where the theorem is applied, or where there occurs a problem related to it (in
this case e.g. 93D 15). The index volume will also contain the inversion of this list which will,
for each number, provide a list of words and phrases which may serve as an initial description
of the 'content' of that classification number (as far as this ENCYCLOPAEDIA is concerned). For
more details on the index volume, its structure and organisation, and what kind of things can
be done with it, cf. the (future) special preface to that volume.
Classifying articles is a subjective matter. Opinions vary greatly as to what belongs where
and thus this attempt will certainly reflect the tastes and opinions of those who did the clas
sification work. One feature of the present classification attempt is that the general basic
concepts and definitions of an area like e.g. 55N (Homology and Cohomology theories) or 601
(Markov processes) have been assigned classification numbers like 55NXX and 60JXX if there
was no finer classification number different from ... 99 to which it clearly completely belongs.
Different parts of mathematics tend to have differences in notation. As a rule, in this
ENCYCLOPAEDIA in a given article a notation is used which is traditional in the corresponding
field. Thus for example the (repeated index) summation convention is used in articles about
topics in fields where that is traditional (such as in certain parts of differential geometry (tensor
geometry» and it is not used in other articles (e.g. on summation of series). This pertains
especially to the more technical articles.
For proper names in Cyrillic the British Standards Institute transcription system has been
used (cf. Mathematical Reviews). This makes well known names like S. N. Bernstein corne out
as Bernshteln.
In such cases, especially in names of theorems and article titles, the traditional spelling has
been retained and the standard transcription version is given between brackets.
Ideally an encyclopaedia should be complete up to a certain more-or-less well defined level
VIII
PREFACE
of detail. In the present case I would like to aim at the completeness level whereby every
theorem, concept, definition, lemma, construction which has a more-or-less constant and
accepted name by which it is referred to by a recognizable group of mathematicians occurs
somewhere, and can be found via the index. It is unlikely that this completeness ideal will be
reached with this present ENCYCLOPAEDIA OF MATHEMATICS, but it certainly takes substantial
steps in this direction. Everyone who uses this ENCYCLOPAEDIA and finds items which are not
covered, which, he feels, should have been included, is invited to inform me about it. When
enough material has come in this way supplementary volumes will be put together.
The ENCYCLOPAEDIA is alphabetical. Many titles consist of several words. Thus the problem
arises how to order them. There are several systematic ways of doing this of course, for in
stance using the first noun. All are unsatisfactory in one way or another. Here an attempt has
been made to order things according to words or natural groups of words as they are daily used
in practice. Some sample titles may serve to illustrate this: Statistical mechanics, mathemati
cal problems in; Lie algebra; Free algebra; Associative algebra; Absolute continuity;
Abstract algebraic geometry; Boolean functions, normal forms of. Here again taste plays a
role (and usages vary). The index will contain all permutations. Meanwhile it will be advisable
for the reader to tryout an occasional transposition himself. Titles like K-theory are to be
found under K, more precisely its lexicographic place is identical with 'K theory', i.e. '-' =
'space' and comes before all other symbols. Greek letters come before the corresponding Latin
ones, using the standard transcriptions. Thus X2-distribution (chi-squared distribution) is at the
beginning of the letter C. A* as in C"-algebra and *~regular ring is ignored lexicographically.
Some titles involve Greek letters spelled out in Latin. These are of course ordered just like any
other' ordinary' title.
This volume has been computer typeset using the (Unix-based) system of the CWI, Amster
dam. The technical (mark-up-language) keyboarding was done by Rosemary Daniels,
Chahrzade van 't Hoff and Joke Pesch. To meet the data-base and typesetting requirements of
this ENCYCLOPAEDIA substantial amounts of additional programming had to be done. This was
done by Johan Wolleswinkel. Checking the translations against the original texts, and a lot of
desk editing and daily coordination was in the hands of Rob Hoksbergen. All these persons, the
members of the board of experts, and numerous others who provided information, remarks and
material for the editorial comments, I thank most cordially for their past and continuing efforts.
The original Soviet version had a printrun of 150,000 and is completely sold out. I hope that
this annotated and updated translation will tum out to be comparably useful.
Bussum, August 1987 MICHIEL HAZEWINKEL
ix
ORBIT of a point x relative to a group G acting on a If G is an algebraic group and X is an algebraic
set X (on the left) - The set variety over an algebraically closed field k, with regular
= action (see Algebraic group of transformations), then
G(x) {g(x): gEG}.
any orbit G(x) is a smooth algebraic variety, open in
The set
G = {gEG: g(x)=x} its closure G(x) (in the Zariski topology), while G(x)
x
always contains a closed orbit of the group G (see [5]).
is a subgroup in G and is called the stabilizer or station
ary subgroup of the point x relative to G. The mapping In this case the morphism G~G(x), gf4g(x), induces
gf4g(x), gEG, induces a bijection between GIGx and an isomorphism of the algebraic varieties G I Gx and
G(x) if and only if it is separable (this condition is
the orbit G(x). The orbits of any two points from X
always fulfilled if k is a field of characteristic zero, d.
either do not intersect or coincide; in other words, the
Separable mapping). The orbits of maximal dimension
orbits define a partition of the set X. The quotient by
form an open set in X.
the equivalence relation defined by this partition is
The description of the structure of an orbit for a
called the orbit space of X by G and is denoted by
given action usually reduces to giving in each orbit a
X I G. By assigning to each point its orbit, one defines a
unique representative x, the description of the stabilizer
canonical mapping 1T x.G: X ~X I G. The stabilizers of
the points from one orbit are conjugate in G, or, more Gx and the description of a suitable class of functions
which are constant on the orbit (invariants) and which
precisely, Gg(x) = gGxg -). If there is only one orbit in
separate various orbits; these functions enable one to
X, then X is a homogeneous space of the group G and G
describe the location of the orbits in X (orbits are inter
is also said to act transitively on X. If G is a topological
sections of their level sets). This program is usually
group, X is a topological space and the action is con
called the problem of orbit decomposition. Many classifi
tinuous, then X I G is usually given the topology in
cation problems can be reduced to this problem. Thus,
which a set U C X I G is open in X I G if and only if the
h( Example 2) is a classification problem of bilinear sym
set 1T X, U) is open in X.
metric forms up to equivalence; the invariants in this
Examples. I) Let G be the group of rotations of a
case - the rank and signature - are 'discrete', while
plane X around a fixed point a. Then the orbits are all
circles with centre at a (including the point a itself). the stabilizer Gi, where f is non-degenerate, is the
corresponding orthogonal group. The classical theory of
2) Let G be the group of all non-singular linear
the Jordan form of matrices (as well as the theory of
transformations of a finite-dimensional real vector
other normal forms of matrices, cf. Normal form) can
space V, let X be the set of all symmetric bilinear forms
also be incorporated in this scheme: The Jordan form
on V, and let the action of G on X be defined by
is a canonical representing element (defined, admit
(gf)(u, v) = j(g-'(u),g-'(v)) for any u, VEV. tedly, up to the order of Jordan blocks) in the orbit of
Then an orbit of G on X is the set of forms which have the general linear group GLn(C) on the space of all
a fixed rank and signature. complex (n X n )-matrices, for the conjugation action
Let G be a real Lie group acting smoothly on a dif Yf 4A YA -); the coefficients of the characteristic poly
ferentiable manifold X (see Lie transformation group). nomial of a matrix Yare important invariants (which,
For any point x EX, the orbit G(x) is an immersed however, do not separate any two orbits). The idea of
submanifold in X, diffeomorphic to GI Gx (the dif considering equivalent objects as orbits of a group is
feomorphism is induced by the mapping gf4g(x), actively used in various classification problems, for
g E G). This submanifold is not necessarily closed in X example, in algebraic moduli theory (see [10]).
(i.e., not necessarily imbedded). A classical example is If G and X are finite, then
the 'winding of a torus', i.e. an orbit of the action of
the additive group R on the torus 1X /G 1 = -~I 1 ~ 1F ixg I,
xt:.G
T2 = {(Z,.Z2): z,EC, 1 Z, 1 =1, i=I,2} where I Y I is the number of elements of the set Y, and
defined by the formula Fixg = {XEX: g(x)=x}.
I(Z"Z2) = (eU",e'OTz2), IER, If G is a compact Lie group acting smoothly on a
where a is an irrational real number; the closure of its connected smooth manifold X, then the orbit structure
orbit coincides with T2. If G is compact, then all orbits of X is locally finite, i.e. for any point x EX there is a
are imbedded submanifolds. neighbourhood U such that the number of conjugacy
ORBIT
classes of different stabilizers Gy, y E U, is finite. In par theory of representations of the group G; see Orbit
ticular, if X is compact, then the number of different method.
conjugacy classes of stabilizers Gy, y EX, is finite. For References
any subgroup H in G, each of the sets [I] PALAIS, R.: The classification of G-spaces, Amer. Math. Soc.,
X(H) = {XEX: Gx is conjugate to H in G} [2] H19A6R0A. RY, F.: Graph theory, Addison-Wesley, 1969.
is the intersection of an open and a closed G-invariant [3] LUNA, D.: 'Slices etales', Bull. Soc. Math. France. 33 (1973),
81-105.
subset in X. Investigation of X(H) in this case leads to [4] LUNA, D.: 'Adherence d'orbite et invariants', Invent. Math. 29,
the classification of actions (see [1 D. no. 3 (1975), 231-238.
Analogues of these results have been obtained in the [5] BOREL, A.: Linear algebraic groups, Benjamin, 1969.
[6] STEINBERG, R.: Conjugacy classes in algebraiC groups, Lecture
geometric theory of invariants (cf. Invariants, theory of)
notes in math., 366, Springer, 1974.
(see [3]). Let G be a reductive algebraic group acting [7] Popov, V.L.: 'Stability criteria for the action of a semisimple
regularly on an affine algebraic variety X (the base field group on a factorial manifold', Math. USSR Izv. 4 (1970),
527-535. (Izv. Akad. Nauk. SSSR Ser. Mat. 34 (1970),523-
k is algebraically closed and has characteristic zero).
531)
The closure of any orbit contains a unique closed orbit.
[8] Popov, A.M.: 'Irreducible semisimple linear Lie groups with
There exists a partition of X into a finite union of finite stationary subgroups of general position', Funct. Anal.
locally closed invariant non-intersecting subsets, App!. 12, no. 2 (1978), 154-155. (Funkts. Anal. i Prilozhen. 12,
X= U XIX' such that: a) if X,YEX and G(x) is no. 2 (1978), 91-92)
a a [9] ELASHVILI, A.G.: 'Stationary subalgebras of points of the com
closed, then the stabilizer Gy is conjugate in G to a sub mon state for irreducible Lie groups', Funct. Anal. App!. 6, no.
2 (1972), 139-148. (Funkts. Anal. i Prilozhen. 6, no. 2 (1972),
group in Gx, while if G(y) is also closed, then Gy is
65-78)
conjugate to Gx; b) if XEXa, YEX/3, a=l=f3, and G(x) [10] MUMFORD, D. and FOGARTY, J.: Geometric invariant theory,
and G(y) are closed, then Gx and Gy are not conjugate Springer, 1982.
[II] KOSTANT, B.: 'Lie group representations on polynomial rings',
in G. If X is a smooth algebraic variety (for example, in
Amer. J. Math. 85, no. 3 (1963), 327-404.
the important case of a rational linear representation of
[12] HUMPHREYS, J.: Linear algebraic groups, Springer, 1975.
G in a vector space V = X), then there is a non-empty v.L. Popov
open subset Q in X such that G, and Gr are conjugate Editorial comments.
in G for any x, y EQ. The latter result· is an assertion
References
about a property of points in general position in X, i.e. [A1] Popov, V.L.: 'Modern developments in invariant theory', in
points of a non-empty open subset; there are also a Proc. Internat. Congress Mathematicians Berkeley, 1986,
number of other assertions of this type. For example, Amer. Math. Soc., 1988, pp. 394-406.
[A2] KRAFT, H.: Geometrische Methoden in der Invarian
for a rational linear representation of a semi-simple
tentheorie, Vieweg, 1984.
group G in a vector space V, the orbits of the points in [A3] KRAFT, H., SLODOWY, P. and SPRINGER, T.A. (EDS.): Alge
general position are closed if and only if their stabiliz braische Transformationsgruppen und Invariantentheorie,
DMV Seminar, 13, Birkhauser, 1989.
ers are reductive (see [7D; when G is irreducible, an
AMS 1980 Subject Classification: 20FXX, 20G05,
explicit expression of the stabilizers of the points in
22E45, 54H15, 14M17, 14L30
general position has been found (see [8], [9D. The study
of orbit closures is important in this context. So, the set
of x E V the closure of whose orbits contains the ele ORBIT METHOD - A method for studying unitary
representations of Lie groups. The theory of unitary
ment 0 of V coincides with the variety of the zeros of
representations (cf. Unitary representation) of nilpotent
all non-constant invariant polynomials on V; in many
Lie groups was developed using the orbit method, and
cases, and especially in the applications of the theory of
it has been shown that this method can also be used for
invariants to the theory of moduli, this variety plays a
other groups (see [1 D.
vital part (see [10]). Any two different closed orbits can
The orbit method is based on the following 'experi
be separated by invariant polynomials. The orbit G (x)
mental' fact: A close connection exists between unitary
is closed if and only if the orbit of the point x relative
irreducible representations of a Lie group G and its
to the normalizer of G(x) in G is closed (see [4]). The
orbits in the coadjoint representation. The solution of
presence of non-closed orbits is connected with proper
basic problems in the theory of representations using
ties of G: if G is unipotent (and X is affine). then any
the orbit method is achieved in the following way (see
orbit is closed (sec [6]). One aspect of the theory of
invariants concerns the study of orbit decompositions [2)).
of different concrete actions (especially linear represen Construction and classification of irreducible lInitar~
tations). One of these - the adjoint representation of a representations. Let ~~ be an orbit of a real Lie group
reductive group G - has been studied in detail (see. G in the coadjoint representation. let F be a point of
for example, [11)). This study is connected wi til the this orbit (which is a linear functional on the Lie alge-
2
