Table Of ContentENCYCLOPAEDIA OF MATHEMATICS
Volume 8
ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
Scientific Board
S. Albeverio, J. B. Alblas, S. A. Amitsur, 1. 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, 1. 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. 1.
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. Kupershmidt, 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, M. Livshits, W. A. J. Luxemburg, R. M. M. Mattheij, L.
G. T. Meertens, 1. Moerdijk, J. P. Murre, H. Neunzert, G. Y. Nieuwland, G. J. Olsder, B.
0rsted, F. van Oystaeyen, B. Pareigis, K. R. Parthasarathy, I. 1. Piatetskil-Shapiro, H. G. J.
Pijls, N. U. Prabhu, E. Primrose, A. Ramm, C. M. Ringel, J. B. T. M. Roerdink, K. W. Rog
genkamp, G. Rozenberg, W. Rudin, S. N. M. Ruysenaars, A. Salam, A. Salomaa, 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,
1. 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, L. N. Vaserstein, M. L. J. van de Vel, F. D.
Veldkamp, 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 8
Reaction-Diffusion Equation - Stirling Interpolation Fonnula
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 978-90-481-8237-4 ISBN 978-94-015-1235-0 (eBook)
DOl 10.1007/978-94-015-1235-0
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
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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. Il'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. UI'yanov, S. V. Yablonski!
Translation Arrangements Committee
V. I. Bityutskov, R. V. Gamkrelidze, Yu. V. Prokhorov
'Soviet Encyclopaedia' Publishing House
PREFACE
This ENCYCLOPAEDIA OF MATHEMATICS 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 or'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. 93DI5). 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 60J
(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 come out
as Bemshteln.
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-Iess well defined level
Vlll
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-Iess 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 try out 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 CW!, Amster
dam. The technical (mark-up-Ianguage) 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
REAcnON-DiFFUSION EQUATION system, in which specific natural effects are modelled, will
have solutions which reflect some known natural
Editorial comments. A system of partial differential
phenomenon of interest whose causes are incompletely
equations of the form
known. Then one looks for the existence and the stability of
aaut
= Ot.u+f(u), solutions of the system in question which have properties
analogous to the phenomenon in question.
where u = u (x, t) = (u 1, ... , Un), t. is the Laplace operator
Regarding the initial value problem i), the theory of ana
in the spatial variables x, 0 is a non-negative non-zero diag
lytic semi-groups, which in this context relies on the opera
onal matrix, and f is a function from a domain in Rn into Rn.
tor Ot. being sectorial, has developed as one of the most
Many generalizations of these equations have also been stu
commonly used approaches to existence and uniqueness
died, such as result when f depends also on the first-order
[A2]. The study of steady solutions ii) has used a variety of
x-derivatives of u, when the operator t. is replaced by other,
methods, such as recasting the problem as a fixed-point
possibly non-linear, operators, or when the matrix 0 is not
problem for a mapping in some suitable function space and
diagonal. If extra first-order terms appear in the system as a using topological-degree methods. In the case when n = 1
model for convective transport effects, the system is some
or the system has certain monotonicity properties, methods
times termed reaction-advection-diffusion equation.
based on upper-and-Iower solutions (el. Upper-and-Iower
Such equations arise as models of diverse natural
functions method) provide easier alternatives (see [A 1] and
phenomena [A1], but their most natural roots lie in the study
[A3], e.g.).
of chemical systems: the components of the vector u may
The travelling-wave problem iii) can be viewed as seeking
then represent concentrations of chemical species which
a parameter c for which there exists a connection between
are present, the term Ot.u represents the diffusive transport
two critical points for the system of ordinary differential
of those species, possibly through a chemical solution, and
equations resulting from the substitution u = U(x - ct). One
f (u) represents the production or destruction of species
of the principal tools in this connection has been the power
resulting from reactions among them (if the rates of all such
ful Conley index [A4] , [A3]. Below some other methods
reactions are known, as functions of u, then the explicit
which have been devised more recently are mentioned.
form of f can be written down).
The theory of reaction-diffusion systems can be viewed as
The variable x is often confined to a domain Q with boun
incorporating all of the theory of autonomous ordinary dif
dary aQ, and then solutions are sought which satisfy
ferential systems duldt=f(u) (cf. Autonomous system),
specific boundary conditions on aQ. These are generally of
since when homogeneous Neumann boundary conditions
the form
are imposed, solutions of the latter system automatically
au, constitute x-independent solutions of the corresponding
a'a;;+b,u,= = h" xEaQ, i=1, ... ,n,
reaction-diffusion systems. But, of course, solutions with
striking spatial characteristics arise as well; and ,in fact, the
where al av is the derivative normal to aQ, a, and b are not
i possible spatial structure of solutions is one of the most
both zero (unless u, does not 'diffuse'), and h, is a given
often investigated aspects.
function. Again, generalizations, such as to non-linear boun
Some of the best-studied examples of reaction-diffusion
dary conditions, abound.
systems are the following.
The specific problems of interest are: i) the initial value
a) The scalar Fisher equation
problem, in which u(x, 0) is given and u(x, t) is sought for
t ;;.0; ii) the steady problem, in which solutions independent aaut =
t.u+f(u),
of t are sought; and iii) the travelling-wave problem, in
which Q = R and solutions are sought of the special form [A5], [A6], where f has exactly two zeros. This equation ori
=
u(x, t) U(x - ct). ginally arose in connection with population genetics.
Because of their strong connections with the applied sci b) The scalar bistable diffusion equation, [A7], [A6], [AS],
ences and the limited number of important properties com of the same form but where f has exactly three simple zeros
mon to all members of this unwieldy class of systems, the and is negative between the first two. This equation also has
research impetus in this field comes more from viewing the connections with population genetics, but knowledge of its
systems as models of specific natural phenomena, rather properties is even more important in connection with the
than from interest in them for their own sake. A typical role it plays as a component part of more complicated sys
motivation, for example, may be to ask whether a certain tems.
REACTION-DIFFUSION EQUATION
c) The FitzHugh-Nagumo system their mathematical foundation is still rudimentary. Analogous
au phenomena exist in three dimensions: structures which
-at = ~u+ f(u)-v, rotate about curves in space, called filaments, which them
av selves migrate according to certain approximate laws. An
-at = au-bv' important challenge for the future is to better understand
(and provide a firm mathematical foundation for) the con
where f has the properties given in b) (see the references in
nections between such dynamic spatial patterns and their
[A3] and [A9] for this and generalizations). It is a simplifica
laws of motion, on the one hand, and the underlying partial
tion of higher-order systems such as the Hodgkin-Huxley
system, which arise as models of signal transmission on differential equations on the other.
nerve axons and in cardiac tissue. References
d) The thermal-diffusion model in chemical reactor theory [A 1] FIFE, P.c.: Mathematical aspects of reacting and diffusing
and combustion [A10], [A11]. In this model, systems, Lecture notes in biomathematics, 28, Springer,
1979.
u =(uo, ... ,un), Uo represents temperature, the other
[A2] HENRY, D.: Geometric theory of semilinear parabolic equa
components of u represent concentrations of chemical
tions, Lecture notes in mathematics, 840, Springer, 1981.
species, and the components of f are given by [A3] SMOLLER, 1.: Shock waves and reaction-diffusion equations,
Springer, 1983.
flu) = "2>A(uo)m/(u),
[A4] CONLEY, c.: Isolated invariant sets and the Morse index,
/
Amer. Math. Soc., 1978.
the summation being over all reactions occurring in the
[A5] KOLMOGOROV, A.N., PETROVSKII, l.G. and PISKUNOV, N.S.: 'A
material (these reactions are indexed by 0. Here, m/ is a study of the diffusion equation with increase in the quantity
(mass action) monomial in U1, ... ,un appropriate to the 1- of matter, and its application to a biological problem', 8yull.
th reaction, b/ is the 'reaction constant' for that reaction, Moskov. Gas. Univ. 17 (1937),1-72 (in Russian).
and the numbers aj/ are 'stoichiometric parameters', specify [A6] ARONSON, D.G. and WEINBERGER, H.F.: 'Multidimensional
ing the amount of species j (or heat, in case j = 0) pro nonlinear diffusion arising in population genetics', Adv. in
Math. 30 (1978), 33-76.
duced or consumed in reaction I. [A7] KANEL', YA.I.: 'On the stability of solutions of the Cauchy
In all of the above examples, the existence and the stabil problem for the equations arising in combustion theory', Mat.
ity of travelling-wave solutions is of paramount importance; Sb. 59 (1962), 245-288 (in Russian).
and in case d), other spatially or temporally ordered solu [A8] FIFE, P.c. and McLEOD, J.B.: 'The approach of solutions of
nonlinear diffusion equations to travelling front solutions',
tions are important as well.
Arch. Rational Mech. Anal. 65 (1977), 335-361.
In many applications, solutions with frontal or interfacial
[A9] FIFE, P.c.: Dynamics of internal layers and diffusive inter
properties arise [A9]. For example, a moving surface in 3- faces, CBMS-NSF Reg. Conf. Ser. Appl. Math., 53, SIAM,
space may exist near which some components of u experi 1988.
ence dramatic changes. These changes form an interior [A10] ARIS, R.: The mathematical theory of diffusion and reaction
in permeable catalysts, 1-2, Clarendon Press, 1975.
layer at the surface in question. They have been studied in
[A 11] BUCKMASTER, J. and LUDFORD, G.S.S.: Theory of laminar
the context of phase-field equations (a reaction-diffusion
flow, Cambridge Univ. Press, 1982.
system with non-diagonal D), in which they represent phase [A12] JONES, c.: 'Stability of the travelling wave solution of the
interfaces, of the bistable equation, and of the FitzHugh - Nagumo system', Trans. Amer. Math. Soc. 286
FitzHugh - Nagumo equations and their generalizations, in (1984), 431-469.
[A13] ALEXANDER, J., GARDNER, R. and JONES, c.: 'A topological
which they may represent phase changes, changes in the
invariant arising in the stability analysis of travelling waves',
electrochemical properties of neural or cardiac tissue, or
J. Reine Angew. Math. (To appear).
changes in the chemical state of a medium. [A14] NISHIURA, Y., MIMURA, M., IKEDA, H. and FUJII. H.: 'Singu
The stability of waves in one space dimension for systems lar limit analysis of stability of travelling wave solutions in
with n > 1 is a considerably more difficult field of investiga bistable reaction-diffusion systems', SIAM J. Math. Anal. 21
(1990). 85-122.
tion than is their existence. Much of the work here has been P.e. Fife
done for the FitzHugh - Nagumo equations [A 12]. Recently
a new technique, the stability index of J. Alexander, R. AMS 1980 Subject Classification: 35K57, 80A25,
Gardner and C. Jones [A 13], was developed and applied to 80A30, 80A32, 92A09, 92A15, 92A17
a number of travelling-wave problems.
For travelling-wave and stationary solutions with inter REAL ALGEBRAIC VARIETY - The set A = X(R) of
faces (see above), a technique called the SLEP method has
real points of an algebraic variety X defined over the
been developed to study stability questions (see [A4] and
field R of real numbers. A real algebraic variety is said
the references therein).
to be non-Singular if X is non-singular. In such a case A
For FitzHugh - Nagumo and related systems, the most
is a smooth variety. and its dimension dimA is equal to
Important patterned solutions in two space dimensions are
the dimension of the complex variety CA = X(C); the
rotating spirals, which are extremely prevalent and
latter is known as the complexification of the variety A.
apparently extremely stable structures for this and many
other models for excitable media. Despite the great interest Non-singular regular complete intersections have
in these rotating solutions and the large number of papers been most thoroughly studied. These are varieties X in
the concept has generated (see the references in [A9]), the projective space RplJ which are non-singular regular
2
