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ENCYCLOPAEDIA OF MATHEMATICS Supplement Volume III

ENCYCLOPAEDIA OF MATHEMATICS Supplement Volume III

kd

KLUWER ACADEMIC PUBLISHERS D O R D R E C H T / BOSTON / L O N D O N

Library of Congress Cataloging-in-PublicationsData

ISBN 1-4020-0198-3

Published by Kluwer Academic Publishers, RO. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 141 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, RO. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-.free paper

All Rights Reserved © 2001 KluwerAcademic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in the Netherlands.

E N C Y C L O P A E D I A OF M A T H E M A T I C S

Managing Editor M. H a z e w i n k e l

List of Authors S. S. Abhyankar, V. Abramov, A. Adem, L. Aizenberg, S. Albeverio, Lufs J. Alias, H. Andrdka, B. N. Apanasov, I. Assani, K. Atanassov, S. Axler, A. Bagchi, K. Balachandran, R. B. Bapat, C. Bardos, T. Bartsch, R W. Bates, E. S. Belinsky, A. Ben-Israel, R. D. Benguria, Ch. Berg, V. Bergelson, E Beukers, A. Bloch, D. L. Boley, C. de Boor, J.-E Brasselet, R. Brown W. Dale Brownawell, T. Brzezinski, M. Buhmann, A. Bultheel, D. Bump, S. Caenepeel, R. E. Caflisch. B. D. Calvert, R. Carroll, O. Chan, F. Clarke, Flfivio Ulhoa Coelho, D. J. Collins, A. K. Common S. C. Coutinho, C. Croke, G. Csordas, Ratil E. Curto, H. G. Dales, L. Debnath, M. Deistler. A. Derighetti, J. K. Deveney, U. Dieter, R Dr~ixler, V. Drensky, M. Dror, C. F. Dunkl, A. Duval. T. Ehrhardt, B. Eisenberg, S. Elaydi, E. Elizalde, K. Engel, E. Enochs, M. Eytan, Y. Fang, E. J. Farrell. A. Fernfindez L6pez, C. Foias, A. S. Fraenkel, M. Fukushima, T. Gannon, J. von zur Gathen. S. Gelbart, L. Gemignani, S. K. Ghosh, J. F. Glazebrook, R Goerss, J. E. Goodman, B. Brent Gordon S. Goto, H. Gottschalk, W. Govaerts, S. W. Graham, M. J. Grannell, T. S. Griggs, R. I. Grigorchuck, J. W. Grossman, M. H. Gutknecht, U. Hahn, D. Harbater, G. Harder, K. R Hart, R Haukkanen, D. R. Heath-Brown, G. F. Helminck, D. Hensley, N. J. Hitchin, E den Hollander, J. W. Hovenier, Y.-Z. Huang, I. D. Iliev, N. Immerman, M. Inuiguchi, G. Isac, S. V. Ivanov, W. Jaco, M. Jacobsen, K. Jarosz, Soon-M. Jung, D. Jungnickel, N. Kamiya, A. Kanamori, J. Kania-Bartoszyfiska, W. Kaup, Y. Kawamata, H. Kellay, R S. Kenderov, O. Kerner, E. Khmaladze, J. Klamka, M. Klin, M. A. Ktopotek, E. H. Knill, J. Knopfmacher, M. N. Kolountzakis, V. Komkov, J. G. Krzy2, S. H. Kulkarni, J. R S. Kung, Hui-H. Kuo, K. M. Kuperberg, M. L. Lapidus, R. D. Lazarov, J. Lepowsky, C. Heng Li, E. R. Liflyand, W. A. Light, J. Lukeg, U. Lumiste, V. Lychagin, J. X. Madarfisz, F. Marcellfin, H. Martini, J. Mawhin, R A. McCoy, W. McCune, G . McGuire, C. V. M. van der Mee, D. J. Melville, R W. Michor, M. Mihalik, C. Moro~anu, A. O. Morris, C. J. Mulvey, V. Mufioz, S. Naimpally, Wtadystaw Narkiewicz, R. B. Nelsen, I. N6meti, E Neuman, L. Newelski, G. A. Niblo, M. A. Nielsen, V. Nistor, R. Norberg, T. Nowicki, M. Oberguggenberger, D. Olivari, T. C. O'Neil, R J. Oonincx, E. L. Ortiz, G. Owen, E. Pap, V. Paulauskas, D. B. Pearson, G. K. Pedersen, R. B. Pelz, W. V. Petryshyn, A. N. Philippou, D. Pigozzi, A. Pinkus, Z. Piotrowski, R. Pollack, A. Prfistaro, Andrfis Pr6kopa, J. Przytycki, A. G. Ramm, T. M. Rassias, S. Reich, R. Reischuk, S. E. Rodabaugh, A. Rodffguez Palacios, J. Rosenberg, A. Rucifiski, J. Sfindor,

R Schmid, J. M. Schumacher, S. K. Sehgal, D. Shoikhet, B. Silbermann, D. Simson, A. Sitaram, H. de Snoo, A. Softer, E Sottile, J. Spencer, H. M. Srivastava, J. D. Stegeman, D. Stegenga, R. Steinberg, R. J. Stroeker, H. Sumida, L~iszl6 A. Sz6kely, F. Todor, E. Tsekanovski]', A. Turull, N. Tzanakis, L. Unger, H. Upmeier, R. S. Varga, W. Vasconcelos, R J. Vassiliou, V. Vinnikov, M. Vuorinen, M. Waldschmidt, N. Watt, G. R Wene, J. Wiegerinck, R. A. Wijsman, R. W. Wittenberg, S. A. Wolpert, S. Xiang, L. Zalcman, A. I. Zayed, S. Zlobec, S. Zucker

PREFACE TO THE THIRD S U P P L E M E N T V O L U M E

The present volume of the ENCYCLOPAEDIA OF MATHEMATICS is the third of several (planned are three) supplementary volumes. In the prefaces to the original first ten volumes I wrote: 'Ideally, an encyclopaedia should be complete up to a certain more-or-less well defined level of detail. In the present case I would like to aim at a 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.' With these three supplementary volumes we go some steps further in this direction. I will try to say a few words about how much further. The first source of (titles of) articles was the collective of users of the original 10 volume ENCYCLOPAEDIAOF MATHEMATICS. Many users transmitted suggestions for additional material to be covered. These suggestions were taken seriously and checked against the 3.5M keyword list of the FIZ/STN database MATH in Karlsruhe. If the hit rate was 10 or better, the suggestion was usually accepted. For the second source I checked the index of volumes 1-9 against that same key phrase list (normalized). Everything with a hit frequency in the normalized list of 40 or better was checked and, if not really present--a casual mention did not suffice--resulted in an invitation to an expert to contribute something on it. This 'top 40' supplementary list already involves more articles than would fit in a single volume alone and the simple expedient was followed of processing first what came in first (while being carefull about groups of articles that refer heavily to each other and other matters such as timelyness). However, the three supplementary volumes together will surely cover the whole 'top 40' and actually go one step deeper, roughly to the level of the 'top 20'. For the final (as far as I can see at the moment only electronic) version of the ENCYCLOPAEDIAOF MATHEMATICS (WEB and CDROM both) I hope and expect to go as far as the 'top 6'. This means an estimated 32000 articles and an 120K standard key phrase list, a four-fold increase over the printed 13-volume version. It should be noted that if one actually checks one of these 'top 6' standard key phrases in the database MATH, the number of hits is likely to be quite a bit higher; such a search will also pick mentions in title and abstract (and not only those in the key-phrase field). The present volume has its own index. This index is structured exactly like Volume 10, the index to Volumes 1-9. For details I refer to the Introduction to that index volume. The number of authors involved in this volume is substantial and in a sense this ENCYCLOPAEDIA is more and more a community effort of the whole mathematical world. These authors are listed collectively on one of the preliminary pages, and individually below their contributions in the main body

vii

PREFACE TO THE SUPPLEMENT VOLUME

of this volume. I thank all of t h e m most cordially for their considerable efforts. The final responsability for what to include and what not, etc., however, is mine. As is clear from the above, I have m a d e heavy use of that invaluable resource the FIZ/STN MATH database in Karlsruhe. I thank that institution, in particular Dr. Olaf N i n n e m a n n and the 'MATH group', for their assistance and the facilities put at m y disposal. As in the case of the original 10 volumes, this one would not have existed without the very considerable efforts of Rob Hoksbergen, w h o took care of all coordination and administration, and an awful lot of other detail work besides. Bussum, October 1999 PROE DR. MICHIEL HAZEWINKEL

email: [email protected] CWI RO.Box 94079 1090GB Amsterdam The Netherlands Telephone: +31 - 20 - 592 4204 Fax: +31 - 2o - 592 4199

°

,

,

Vlll

A *-AUTONOMOUS CATEGORY - Let C be a symmetric c l o s e d m o n o i d a l c a t e g o r y (cf. also C a t e g o r y ) . A f u n c t o r ( - ) * : C°p --+ C is a duality functor if there exists an isomorphism d(A, t3) : B A ~ A ' B * , natural in A and B, such that for all objects A, B, C C C the following diagram commutes:

(B A ® c B )

c(A,BfC)

cA

.kd( A,B )@d( B,C )

(A*)('*) ® (B*)(c*)

.~d( A,C )

c(C*,B*,A*)os

(A*)(c*)

where in the bottom arrow s = s((A*) (B*), (B*)(c*)). A category is *-autonomous if it is a symmetric monoidal closed category with a given duality functor. It so happens that *-autonomous categories have reallife applications: they are models of (at least the finite part of) linear logic [2] and have uses in modelling processes. An example of a .-autonomous category is the category 7~¢g of sets and relations; duality is given by S* = S. In fact, B A -~ (A* ® t3). From a given symmetric monoidal closed category and an object in it (that serves as a dualizing object) one can construct a *-autonomous category (the so-called Uhu construction, [3]). It can be viewed as a kind of generalized topology. References

[1] BARa, M.: *-Autonomous categories, Vol. 752 of Lecture Notes in Mathematics, Springer, 1979. [2] BARR, M., AND WELLS, C.: Category theory for computing science, Publ. CRM, 1990. [3] CHU, P.-H.: 'Constructing *-autonomous categories', in M. BARI~(ed.): *-Autonomous categories, Vol. 752 of Lecture Notes in Mathematics, Springer, 1979, p. Appendix. Michel Eytan MSC1991: 18D10, 18D15 ABSOLUTELY

CONTINUOUS

INVARIANT

MEA-

S U R E - A d y n a m i c a l s y s t e m , treated as a space X

with a mapping T : X ~ X or a family of mappings T, may have a Iarge number of invariant measures (cf. also I n v a r i a n t m e a s u r e ) . Among them there are invariant measures that are absolutely continuous with respect to some canonical measure on X (cf. also A b s o l u t e l y c o n t i n u o u s m e a s u r e s ) , such as L e b e s g u e m e a s u r e for X C R ~, H a a r m e a s u r e when X is a t o p o l o g i cal g r o u p , or a product m e a s u r e when X is a shift space (cf. Shift d y n a m i c a l s y s t e m ) . The importance of absolutely continuous invariant measures is due to a heuristic belief that canonical measures are the ones which represent physical objects. There is a natural procedure for finding an absolutely continuous invariant measure, by iterating the canonical measure #. First construct the images of # under the mapping #~ = # o T -'~, then take the averages ~ = ~ k = 0 # k / n and take some weak* a c c u m u l a t i o n p o i n t . Special properties of the mapping (e.g. its uniform expansion) may be reflected in the properties of the limit measure (absolute continuity). An alternative (dual) way is to iterate the density function with the transfer operator, and use the properties of T to prove a c o m p a c t n e s s property of a resulting sequence. The existence of an absolutely continuous invariant measure is not granted and is due in many cases to hyperbolic properties of the mapping, such as large derivatives on big sets of points. Once found, the absolutely continuous invariant measure serves via the e r g o d i c t h e o r e m to pronounce statements about typical (with respect to the canonical measure) behaviour of the system. The ergodic theorem says that the long-time behaviour of the system is asymptotically described by the behaviour on ergodic components of the space. The time averages of observables (measurable functions) are then equal to their space averages (integrals). An invariant measure is ergodic if there are no non-trivial invariant sets - - if T - 1 A = A then either #(A) = 0 or # ( X \ A) = 0. One can say, imprecisely, that any

ABSOLUTELY C O N T I N U O U S I N V A R I A N T M E A S U R E invariant measure is a combination of invariant ergodic measures. One calls an invariant measure a Sinai-Bowen-Ruelle measure, or SBR measure, when it is a limit point of the averages of Dirac measures (cf. also D i r a c d i s t r i b u t i o n ) on the trajectories of points from a set of positive Lebesgue measure: = lim E k=0

ldT~z n

for any x E A with positive measure. When an SBR measure is absolutely continuous with respect to some natural measure on the space (most often the Lebesgue or Haar measure), then it is said that the system is chaotic or stochastic. When, on the other hand, the SBR measure is concentrated on a finite number of points, then the system is called deterministic (with a periodic attractor). All other systems are commonly called strange or wild. It is widely believed that typically the systems are either stochastic or deterministic (or a combination of them), but there are known examples of strange limit behaviour. See also S t r a n g e a t t r a c t o r ; C h a o s . References [1] CORNFELD, I.P., FOMIN, S.V., AND SINAL YA.G.: Ergodic theory, Springer, 1982. [2] DEVANEY, R.L.: An introduction to chaotic dynamical systems, Benjamin/Cummings, 1986. [3] KaENCEL, U.: Ergodic theorems, de Gruyter, 1985. [4] NErMARK, YU.I., AND LANDA, P.S.: Stochastic and chaotic oscillations, Kluwer Acad. Publ., 1992, p. Chap. 2. [5] VRmS, J. DE: Elements of topological dynamics, Kluwer Acad. Publ., 1993.

T. Nowicki MSC 1991: 28Dxx, 5 8 F l l , 58F13, 54H20 ABSOLUTELY

CONTINUOUS

MEASURES

-

S u p p o s e that on the m e a s u r a b l e s p a c e (X, 34) there

are given two measures # and y (of. also M e a s u r e ) . One says that ~ is absolutely continuous with respect to # ( d e n o t e d , << #) if It(A) = 0 implies ~(A) = 0 for any set A E 34. One also says that # dominates y. If the measure ~ is finite (i.e. ~,(X) < ec), then , << # if and only if for any .c > 0 there exists a 5 > 0 such that ,(A) < 5 whenever #(A) < e. The R a d o n - N i k o d : ~ m t h e o r e m says that if # and are a-finite measures and u << #, then there exists a #-integrable non-negative function f (a density, cf. also I n t e g r a b l e f u n c t i o n ) , called the Radon-Nikod~m derivative, such that ~(A) = fA f d#. Two such densities f and g may differ only on a null set (see M e a s u r e ) , i.e. #({x: f ( x ) ~ g(x)}) = 0. An example of a density (with respect to the L e b e s g u e m e a s u r e on the interval, i.e. the length) is the function f ( x ) = ~ Ix - q~1-1/2 .2 -~,

where (q~)~>_l is the sequence of all rational numbers in this interval. The measure is a-finite if X is the union of a countable family of sets with finite measure. Given a reference measure p on (X, 34), any measure may be decomposed into a sum of uc and us with ~'c << # and Ys _1_#, i.e. an absolutely continuous and a singular part. This is called the Lebesgue decomposition. A set of non-zero measure that has no subsets of smaller, but still positive, measure is called an atom of the measure. It is a common mistake to claim that the singular part of a measure must be concentrated on points which are atoms. A singular measure may be atomless, as is shown by the measure concentrated on the standard C a n t o r set which puts zero on each gap of the set and 2 - n on the intersection of the set with the interval of generation n. When some canonical measure p on X is fixed (as the L e b e s g u e m e a s u r e on R n or its subsets or, more generally, the H a a r m e a s u r e on a t o p o l o g i c a l g r o u p ) , one says that L, is absolutely continuous on X, meaning that ~, << #. Two measures which are mutually absolutely continuous are called equivalent. See also A b s o l u t e c o n t i n u i t y . References [1] HEWITT, E., AND STROMBERG, K.: Real and abstract analysis, Springer, 1965. [2] ROYDEN, H.L.: Real analysis, Macmillan, 1968.

T. Nowicki MSC 1991: 28-XX ABSTRACT

ALGEBRAIC

L O G I C - The study of

logical equivalence, more precisely, the study of the relationship between logical equivalence and logical truth. Meta-logical investigations take on a different character when the emphasis is placed on logical equivalence, one that is very algebraic in character. But, in contrast to traditional algebraic logic, abstract algebraic logic focuses on the process by which a class of algebras is associated with a logical system rather than the algebras that are obtained in the process. The strength of the connection between logical equivalence and logical truth can vary greatly depending on the particular logical system under consideration. One of the main tasks of abstract algebraic logic is the classification of logical systems based on the strength of this connection. It is very strong in classical logic and this gives classical logic its distinctly algebraic character. The way in which the algebras arise from logic has traditionally followed two distinct paths. The first is based on semantical considerations. In this approach the algebras are abstracted directly from a primitive

A B S T R A C T ALGEBRAIC LOGIC intuitive notion of logical equivalence, and the assettional aspect of the logic (the notion of logical truth) is expressed in its terms. The development of classical propositional logic (cf. also P r o p o s i t i o n a l c a l c u l u s ) followed this path with Boolean algebras coming before the classical propositional calculus (cf. also B o o l e a n algebra). Relation algebras and the way they arose from the calculus of relations is the modern paradigm for the semantics-based method. In the logistic approach, or rule-based approach, the process is inverted. The assertional part comes first and logical equivalence and the associated algebras are then defined by means of the so-called Lindenbaum-Tarski process. The paradigm for the logistic method is the intuitionistic propositional calculus, where the class of Heyting algebras is constructed from Heyting's formalization of Brouwer's intuitionism by the Lindenbaum-Tarski process (cf. also H e y t i n g f o r m a l s y s t e m ) . Cylindric and polyadic algebras were obtained by applying the semantics-based method to first-order predicate logic, but, at least in the case of cylindric algebras, the influence of the logistic approach is strongly evident. The basis of the abstract form of logical equivalence is Frege's principle that sentences, like proper names, have a denotation and that this denotation is their truth value. Two sentences are logically equivalent if they have the same denotation in every possible situation. Thus, according to Frege's principle, they are logically equivalent if they are true in exactly the same interpretations of the underlying uninterpreted logic. For logistic systems this principle has the following technical ramifications. By a language type one means a set A of connectives or operation symbols (cf. also P r o p o s i t i o n a l c o n n e c tive), depending on whether one views them from a Iogical or algebraic perspective. Each connective has associated with it a natural number, called its rank or arity. The set Fm of formulas (terms in an algebraic context) is constructed from the connectives and a fixed, denumerable set of (formula) variable symbols in the usual way. The corresponding formula algebra is denoted by F m . This is the 'absolutely free' algebra of type A with an n-ary operation /~Fm: Fm n _+ Fm for each A E A of arity n such that AFro(p0,..., p ~ - l ) is the formula A ~ 0 , . . . , p ~ - l , in prefix notation, or (~0)~ ~1) when n = 2 and infix notation is used. The operation of simultaneously substituting fixed but arbitrary formulas for variables is identified with the unique endomorphism of F m it determines. A logistic or deductive system is a pair ~ = ( F m , ~-~), where ~-~, the consequence relation of ~ , is a binary relation between sets of formulas and individual formulas satisfying the following well-known conditions: For all F, A C Fm and F ~ Fro,

• F F-~ 9~ for all %D E F; • F ~-D %o and A ~-~ ¢ for every ¢ E F imply A ~-~ ~; • F ~-D 9~ implies F ~ ~-~ %o for some finite F ~ C F

(finiteness); • F ~-~) ~ implies or(F) ~-~ a(~) for every substitution a (substitution invariance). Substitution invariance is the technical counterpart of the idea that logical consequence depends on form and not substance. It plays a key role in abstract algebraic logic because it is an essential feature of e q u a t i o n a l logic. A formula ~ is a theorem of • if F-~ ~ (i.e., it is a consequence of the empty set of formulas). The set of theorems, which may be empty, is denoted by Thm:D. A set T of formulas is a theory of a deductive system if it is closed under consequence, i.e., ~ C T whenever F _C T and F F-~ p. Thin T~ is the smallest theory. The set of all theories of D is denoted by ThT). The theory axiomatized by an arbitrary set F of formulas is the set of all formulas ~ such that F ~-z~ ~. Deductive systems in this sense include all the familiar sentential logics (cf. also P r o p o s i t i o n a l c a l c u l u s ) together with their various fragments and refinements - - for example, the classical and intuitionistic propositional calculi CPC and IPC, the intermediate logics (cf. also I n t e r m e d i a t e logic), the various modal logics, including $4 and $5 (cf. also M o d a l logic), and the multiple-valued logics of J. Lukasiewicz and E. Post (cf. also M a n y - v a l u e d logic). The substructural logics, such as BCK logic, relevance logic and linear logic can also be formulated as deductive systems, although they are often formulated as Gentzen-type systems (cf. also G e n t z e n f o r m a l s y s t e m ) . Even first-order predicate logic can be formalized as a deductive system, although in its usual formulation it is not substitution-invariant. A (logical) matrix is a structure of the form 92 = (A, F}, where A is an algebra (of the same language type as :D), the underlying algebra of 92, and F C_A, the designated set of 92. An interpretation of ~ is a matrix 92 together with a homomorphism h : F m --+ A from the algebra of formulas F m into the underlying algebra of 92. h(~) is to be thought of as the 'sense' or 'meaning' of the formula ~ under the interpretation, and ~ is 'true' or 'false' depending on whether or not h(~) E F. By the Frege principle, this truth value is the denotation of p under the interpretation. Truth must be preserved under consequence in the sense that, if F F-~ ~ and each E F is true under the interpretation, then ~ must also be true. A set F of formulas is said to define a class K of interpretations if K is the class of all interpretations in which each formula of F is true. Because truth is preserved under consequence, the theory axiomatized by F

ABSTRACT

ALGEBRAIC

LOGIC

also defines K; it turns out that it is the unique theory with this property. The formulas ~ and ~p are equivalent over a given class K of interpretations (in the Fregean sense) if they have the same truth value, i.e., h(~) E F if and only if h(¢) E F for each (91, h) in K. T h e y are logically equivalent (with respect to l)) if they are equivalent over the class of all interpretations. The semantical and logistic approaches diverge at this point. In the former attention is restricted to a specific class of interpretations whose peculiar structure may play an important role in the meta-theory. In contrast, in the logistic method every interpretation is considered, and consequently only its representation as an abstract set with operations and a designated subset is significant. Deduction systems treated in this way are sometimes referred to as uninterpreted. The foundations of the logistic method in abstract algebraic logic can be found in [28], [37]; for more recent work, see [38]. For relation, polyadic and cylindric algebras, see [11], [22], [23]. For the origin of the notion of a deductive system, see [36]. L o g i s t i c a b s t r a c t a l g e b r a i c logic. The assumption that the deductive system :D is uninterpreted implies that one need consider only those matrices 92 with the property that (92, h) is an interpretation for every h: F m --+ A. Such a matrix is called a (matriz) model of l); the class of all models of :/P is denoted by Mod ~. Secondly, the properties of a given class of interpretations are for most purposes completely specified by the theory that defines it. So matters can be simplified by considering equivalence of formulas over theories rather than classes of interpretations. Two formulas ~ and ~p are equivalent over a theory T, or T-equivalent, in the Fregean sense, if, for every theory S that extends T, ~ C S if and only if ~ E S; the binary relation between formulas defined this way is called the Frege relation of T and denoted by AwT. and ~ are logically equivalent (with respect to ~ ) if they are equivalent over every theory, or, equivalently, if they are equivalent over Thm :D, the set of theorems. In terms of the consequence relation, ~ and ~ are logically equivalent over T if and only if ~ and ¢ are each consequences of the other over T; symbolically, ~ = (rood A r T ) if T, q¢ ~-v ~ and T, ~ ~-w qa. If, as in the case of the classical and intuitionistic propositional calculi, ~ has a binary implication connective --+ for which the deduction theorem holds, then ~a = ~P (modAz~T) if and only if ~ ~ ¢ ~ T and --+ ~ G T, or equivalently if there is a biconditional, ++ ¢ G T. So, under these rather weak conditions on ~P logical equivalence in the Fregean sense agrees with the familiar definition of the concept.

As a consequence of Frege's principle, the following

rule of replacement, or eompositionality, holds when is the classical propositional calculus CPC or the intuitionistic propositional calculus IPC. If in any formula a subformula ~p is replaced by an equivalent formula ~p', the resulting formula q¢' is equivalent to qa. In algebraic terms this says that for every theory T of CPC or IPC, A > T is a congruence relation on the algebra of formulas Fro. A deductive system :D for which T-equivalence is compositional for every theory T is called Fregean or extensional; non-Fregean systems are called intensional. If D is Fregean, it is possible to form the quotient matrices (Fm/Az~T, T / A > T } , where T ranges over all theories. These are called the Lindenbaum-Tarski models of Z). The construction of the Lindenbaum-Tarski models is more complicated in the case of intensional systems, where the Frege relation is not a congruence relation. For example, for every theory T of the strong ibrm of the modal system $5, with necessitation as a rule of inference (see below), one has ~ -- ¢ (rood AssT) if and only if T , ~ ~-s5 ~ and T , ¢ ~-s5 ~ if and only if [5~ -+ ~ C T and [7~ --+ ~ E T. But this relation is not, in general, compositional. A s s T includes however a largest compositional equivalence relation, called the Suszko congruence of T over $5 and denoted by ~ s s T . It can be shown that ~ --- ¢ (mod ~ s s T ) if and only if ~ ~ ¢ E T. So ~ s s T captures the usual notion of T-equivalence for $5. Suszko congruences can be defined for the theories of any deductive system, and consequently one can construct Lindenbaum-Tarski models of any deductive system. But it turns out to be more useful to consider the Lindenbaum-Tarski process in a broader context. A subset F of the underlying set of an algebra A, of the same language type as l?, is called a filter of ~P if (A, F) is a model of :D; thus F is a filter if and only if it is closed under consequence in the sense that, if P ~-z~ ~a, one has h(p) C F for every h: F m -~ A such that h(¢) C F for every ¢ E F. The set of filters of :D on A is denoted by Fig A. It is easy to see that the theories are the filters on the formula algebra. The Frege relation and the Suszko congruence generalize to filters on an arbitrary algebra A in a natural way. The Frege relation A > F is the set of all pairs (a, b) of elements of A such that, for every filter G on A that includes F, a E G if and only if b C G. The Suszko congruence ~ > F is the largest congruence relation on A that is included in A > F ; it always exists. The quotient matrix (A/~z~F, F / ~ F } is called the Suszko-reduction of 92 and is denoted by 9.1"s. It can be shown that 9 2 . s . s coincides with (or, more precisely, is isomorphic to) 92* s. A model 92 is Suszko-reduced if 92.s = 92, i.e., ~ F is the identity relation. The class of all Suszko-reduced models of 7) is denoted by Mod* s l).

A B S T R A C T ALGEBRAIC LOGIC The Suszko-reduced models of 79 are those for which two elements are equivalent in the Fregean sense if and only if they are identical. The class of underlying algebras of the Suszkoreduced models of i9 is denoted by AlgMod* s 79. The underlying algebras of CPC are the Boolean algebras. Each filter of CPC on a Boolean algebra A is completely determined by its Suszko congruence ~ F ; more precisely, it coincides with the set of all elements of A equivalent under ~ F to the unit element T of A. Moreover, every congruence relation on A is the Suszko congrnence of a filter of CPC. It follows that the consequence relation of CPC is completely determined by the equational logic of Boolean algebras, and in this way the class of Boolean algebras Alg Mod* s CPC constitutes a complete algebraic semantics for CPC. In a similar way, the class of Heyting algebras AlgMod*S IPC and the class of so-called monadic algebras Alg Mod* s $5 constitute complete algebraic semantics for IPC and $5, respectively, and this is the case for ahnost all the familiar deductive systems. But in general the connection between the consequence relation of a deductive system 79 and the equational logic of Alg Mod* s 79 is much weaker. A central problem for abstract algebraic logic is the characterization of those deductive systems for which this connection is as strong as for the traditional logics. Early work on the Suszko and the closely related Leibniz and Tarski congruences discussed below can be found in [28], [33]. [34] contains historical information on the Tarski-Lindenbaum process. The essential idea of Fregean logic as presented here originated with R. Suszko p0], [35].

Algebraizable logics. The basis of the abstract algebraic logic definition of an algebraizable deductive system is the notion of bisimulation between the consequence relation of 79 and the equational consequence relation of a class of algebras. Let 79 be a deductive system and K a class of algebras over the same language type. Let E(x,y) : {e/(x,y): i • I} be a (possibly infinite) non-empty system of formulas in two variables. E(x,y) is said to be a faithful interpretation of the equational logic of K in 79 if, for every equation ~ ~ ¢ and every set of equations F ~ A, r~

A bK~ ~¢

iff E ( F , A ) [ - g E ( 9 , ~ ) ,

whereF~A~K~¢meansthat AF~A--+P~'~ is a quasi-identity of K. Also, E(F, A) = {c~(V,d): ~ 6 C F ~ A, i 6 I} and E(~o,¢) = {ci(go,¢):i • I}, and E(F, A) F~ E(~, ¢) is shorthand for the system of

entailments E ( F , A ) [-z~ e i ( ~ , ¢ ) for every i E I; 'iff' stands for 'if and only if'. Let

K(x) ~ L(x) = {aj(x) ~ l j ( x ) : j e J} be a non-empty system of equations in one variable. K(x) ~ L(x) is a faithful interpretation of 79 in the equational logic of K if, for all F U {~} C_ Fm,

r ~-~ ~

iff K(F) ~ L(r) ~K K(~) ~ n(qo),

where K ( F ) ~ L(F) = {~j(¢) ~ t j ( ¢ ) : O • F, j • J} and K ( ~ ) ~ n(~) = {~j(~o) ~ ,~y(~o): j • J}. The two interpretations are mutual inverses if

x H~-z) E(K(x), L(x)) and x ~ y =l~K K(E(x,y)) ~ n(E(x,y)). A pair of mutually invertible interpretations E(x, y) and K(x) ~ L(x) such as these is called a bisimulation between 19 and the equational logic of K. A deductive system 79 is algebraizable if there is a bisimulation between 79 and the equational logic of some class K of algebras; it is finitely algebraizable if the interpretations are finite. If 79 is algebraizable, then Alg Mod* s 79 is the largest class K with the above properties; it is called the equivalent algebraic semantics of 79. In general, Alg Mod*S 79 is not elementary (i.e., definable by a set of sentences of the first-order predicate logic), and in fact it is an elementary class just in case :D is finitely algebraizable. In this case Alg Mod* s 79 is a quasi-variety (cf. also Q u a s i - v a r i e t y ) The classical and intuitionistic propositional calculi CPC and IPC are finitely algebraizable and their equivalent quasi-varieties are, respectively, the varieties BA of Boolean algebras and HA of Heyting algebras. In both cases the faithful interpretation of the equational logic of the equivalent quasi-variety in the deductive system is given by E(x, y) = {x ++ y}, where ++ is, respectively, the classical and intuitionistic biconditional, and the inverse faithful interpretation is K(x) ..~ L(x) = {x ~ T}. Most of the deductive systems of traditional algebraic logic are finitely algebraizable with similar interpretations. However, there are finitely algebraizable logics with non-standard interpretations, for example the entailment logic E. The bisimulation between a finitely algebraizable deductive system 79 and the equational logic of its equivalent quasi-variety AlgMod*S79 induces a correspondence between the meta-logical properties of 79 and the algebraic properties of AlgMod*S 79. This correspondence has been the focus of considerable attention in abstract algebraic logic. One of the most important aspects of traditional algebraic logic is the way in which

ABSTRACT ALGEBRAIC LOGIC it can be used to reformulate meta-logical properties of a particular logical system in algebraic terms. Abstract algebraic logic provides a framework for studying such correspondences in a general context. Known metalogical or algebraic results can then be applied to obtain new results in the other domain. Some correspondences of this kind that have been established are between: meta-logical interpolation and algebraic amalgamation, e.g., the Craig interpolation theorem of CPC and the amalgamation property of BA; • definability (in the sense of the Beth definability theorem of first-order predicate logic) and the property that every epimorphism (in the categorical sense) is surjective; • the deduction theorem and the equational definability of principal congruences. •

The deduction theorem is the formal expression of one of the most important and useful properties of classical logic: to prove that an implication holds between propositions it suffices to give a proof of the conclusion on the basis of the assumption of the antecedent. It is such a familiar part of ordinary logical argumentation that it is hardly recognizable as being something whose use might be problematic. But in fact it is not part of the usual formalizations of CPC and must be proved as a meta-theorem. Moreover, while the deduction theorem remains valid for intuitionistic logic, it is known to fail in other important logics, for instance, certain systems of modal logic. It turns out that there is a close connection between the deduction theorem and the universal algebraic notion of definable principal congruence relations. The development of abstract algebraic logic was motivated in part by a desire to provide the proper context in which to formalize this connection precisely. The ultimate goal was to be able to apply the extensive work on the definability of principal congruence relations in u n i v e r s a l a l g e b r a to answer some important questions about the validity of the deduction theorem in a variety of logical systems. Let :D be a deductive system. A finite non-empty set A(x, y) = {50(x, y ) , . . . , 5,~-1 (x, y)} of binary formulas is called a deduction-detachment system for Z) if the following equivalence holds for all F U {~, ~} C_ Fm: r , ~ ~-v ~ iff

r

¢).

A deductive system has the deduction-detachment theorem if it has a deduction-detachment system. The implication x -+ y forms a singleton deduction-detachment system for CPC and IPC, while the formula [Tx -+ y

forms a singleton deduction-detachment system for the modal systems $4 and $5. For an algebra A, let CoA be the set of all congruences of A. If Q is a quasi-variety, then a congruence O on an arbitrary algebra A (not necessarily in Q) is called a Q-congruence if A/O C Q. If Q is a variety (cf. also Algebraic systems, variety of) and A E Q, then every congruence is a Q-congruence. The principal Q-congruence generated by the pair a and b, OQ(a,b), is the smallest Qcongruence that identifies a and b. A quasi-variety Q has equationally definable principal relative congruences (EDPRC) if there is a finite system of equations

e.~-l,l(x, y, z, w), in four variables, such that, for every A E Q a n d a l l a , b,e, d E A , c = d(OQ(a,b))

iff eA i,0 (a, b, c, d) = cA1 (a, b, c, d) for all i < m, e A j ( a , b , c , d ) = h ( e i , j ( x , y , z , w ) ) , where h: F m -+ A is any homomorphism such that h ( x ) = a , . . . , h(w) = d.

Q-congruence generation in the formula algebra can be interpreted as the consequence relation of the equational logic of Q. Thus, if Q has EDPRC, the defining equations %0(x, y, z, w) ~ ei,l(X, y, z, w) can be interpreted collectively as an implication connective for the equational logic. This observation is reflected in the following theorem: Let :D be a finitely algebraizable deductive system and let Q = Alg Mod* s~D be its equivalent quasi-variety. Then Z~ has the deduction-detachment theorem if and only if Q has EDPRC. A correspondence of this kind makes it possible to infer new metalogical properties from known algebraic ones and vice versa. In this way it can be proved that orthomodular logic does not have the deductiondetachment theorem. Orthomodular logic can be viewed as a finitely algebraizable deductive system whose equivalent quasi-variety is the variety of orthomodular lattices. There are several different deductive systems that fit this description. It can be shown that, if a quasivariety Q has EDPRC, then it has the relative congruence extension property. The variety of orthomodular lattices does not have this property. Thus, no orthomodular logic of the kind described above can have the deduction-detachment theorem with respect to any system of binary formulas. A definition of an abstract class of deductive systems with the algebraic properties of the traditional logics was first proposed in [6]; earlier definitions, notably that in [31], were not truly abstract because they relied on the existence of connectives with special properties. The

A B S T R A C T ALGEBRAIC LOGIC notion of algebraizability was subsequently extended in several ways ([15], [17], [24], [25]). For entailment logic and its algebraizability, see [1], [6]. For the deduction theorem in abstract algebraic logic, see [9], [14]. The result on orthomodular logic is found in [7], [29].

The algebraic hierarchy. Finitely algebraizable deductive systems exhibit the strongest connection between the meta-logical properties of 79 and the algebraic properties of Alg Mod* s 79. But there are deductive systems where the connection is not as strong but which still have a clear algebraic character. One of the goals of abstract algebraic logic is the classification of deductive systems of this kind. This leads to a hierarchy of systems with the finitely algebraizable systems at the top. There are several ways of specifying it. The most natural, in view of the definition of algebraizability, is in terms of progressively weaker notions of bisimulation. The characteristic property of all deductive systems that make up the hierarchy is that equivalence, as expressed by the Suszko congruence, is explicitly definable in some way. The classification of the hierarchy is based on the nature of the definition. Let E(x,y) = {ei(x,y): i • I} be a (possibly infinite) set of formulas in two variables. E(x, y) is a protoequivalence system for a deductive system 79 if }-~ E(x,x)

and

x, E(x,y) F-v y

(E-detachment). A non-empty proto-equivalence system E(x, y) is an

equivalence system if Z(x, y) Fv E(y, x),

Z(x, y), Z(y, z) Fv E(x, z),

and

E(xo,Yo),...,E(xn-I,Yn-I)

~-z)

F-v E(~xo,..., xn-1, )~Yo,... , Yn-1) for all ), • A (n is the rank of ;~). The reflexivity and transitivity axioms are superfluous as they are provable from the remaining conditions. The connection between equivalence systems and the Suszko congruence is expressed in the following theorem: A non-empty system E(x, y) of formulas in two variables is an equivalence system for a deductive system 79 if and only if it defines Suszko congruences in the sense that, for every algebra A and F E F i r A, ~vF

=

= {(a,b> E A2: eA(a,b) C F f o r all e(x,y) • E(x,y)}. A deductive system is protoalgebraic if it has a protoequivalence system. Every proto-equivalence system includes a finite subset that is also a proto-equivalence system. A deductive system is (finitely) equivalential if it has a (finite) equivalence system.

The formulas that faithfully interpret in a (finitely) algebraizable deductive system the equational logic of its equivalent algebraic semantics form a (finite) equivalence system. This leads to a meta-logical characterization theorem of (finitely) algebraizable deductive systerns that is intrinsic in the sense that it does not require a priori knowledge of the equivalent algebraic semantics: A deductive system is (finitely) algebraizable if and only if it has a (finite) equivalence system for which there exists a finite system K(x) ,~ L(x) of equations in one variable, called a system of defining equations, such that x ~ - v E ( K ( x ) , L(x)). This last condition abstracts an important property of the biconditional +~ of both classical and intuitionistic propositional logic, namely, that x and the biconditional x ++ T are interderivable. The protoalgebraic, (finitely) equivalential and (finitely) algebraizable deductive systems constitute, along with the weakly algebralzable systems discussed shortly, the algebraic hierarchy. Natural deductive systems can be found to separate all levels of the hierarchy. Protoalgebraicity is a very weak condition and almost all known deductive systems have the property. There are some that do not however, for example the conjunction/disjunction fragment of CPC, subintuitionistic logics and Belnap's logic. Almost all the weak modal logics, without necessitation as a rule of inference, are protoalgebraic but not equivalential. There are also examples of algebraizable logics that are not finitely equivalential, and hence also of logics that are equivalential but not finitely equivalential. In addition to the syntactical characterizations considered above, each level of the hierarchy can be characterized by both algebraic and model-theoretic means. The algebraic characterization makes use of the Leibniz congruence, a more primitive but more manageable variant of the Suszko congruence. Given any algebra A and any subset F of A, there is a largest congruence relation [ t F on A compatible with F in the sense that F is a union of equivalence classes of f t F . [ t F is called the Leibniz congruence of F. The relationship between the Leibniz and Suszko congruences is straightforward: For every deductive system 79 and F E F i r A, 5 v F = A {~tG: F C_ G e F i v A } . ft and ~ can both be viewed as operators mapping the lattice of 79-filters of A to the lattice of congruences of A. Note that the Leibniz and Suszko congruences coincide on 79-filters just in case the Leibniz operator is order-preserving, i.e., F _C G implies ~ F ___ f t G for all F, G • F i v A .

ABSTRACT ALGEBRAIC LOGIC Let 2) be a deductive system. Then the following

characterizations hold: i) 2) is protoalgebraic if and only if the Leibniz operator Ft is order-preserving, i.e., if and only if the Leibniz and Suszko congruences coincide; ii) 2) is equivalential if and only if it is protoalgebraic and fl commutes with inverse homomorphic iraages; more precisely, fth-l(F) = h - l ( ~ ~ F ) for every homomorphism h: A -+ B and every F E Fiz~ B; iii) 2) is finitely equivalential if and only if it is protoalgebraic and fl commutes with directed unions; more precisely, fl [_J 5c = U F e ~ FtF for every iV C Fiz) A that is upward-directed by inclusion; iv) 2) is algebraizable if and only if it is equivalential and fl is injective; v) 2P is finitely algebraizable if and only if it is finitely equivalential and fl is injective. A deductive system 2) is said to be weakly algebraizable if it is protoalgebraie and the Leibniz operator fl is injective. A syntactical characterization of weak algebraizability is also known. Calculation of the Leibniz congruences can be a practical matter for some small algebras. This gives a way of verifying that a deductive system is not finitely algebraizable, or that a quasi-variety is not the equivalent algebraic semantics of any deductive system. This method has been used to show that the relevance logic R and the various formalizations of modal logic without the rule of necessitation are not finitely algebraizable. It has also been used to show that the variety of complemented distributive lattices is not the equivalent algebraic semantics of any deductive system. There is also a model-theoretic characterization of the algebraic hierarchy similar to the well-known modeltheoretic characterizations of equational and quasiequational classes by G. Birkhoff and A. Mal'cev. The Leibniz-reduction of a model of a deductive system is defined just like the Suszko-reduction, except that the Leibniz congruence is used in place of the Suszko congruence. Mod*Lg) denotes the class of all Leibniz-reduced models of 2). If 2) is protoalgebraic, then Mod* s2) = Mod *L 2); this equality in fact characterizes protoalgebraic systems. In general, the best one has is that Mod* s 2) coincides with the class of all matrices isomorphic to a subdirect product of matrices in Mod *L D, in symbols Mod *s 2) = PSD Mod *L 2). For any class K of matrices, SK, PK, PsDK, and P u K denote, respectively, the classes of isomorphic images of submatrices, direct products, subdirect products, and ultraproducts of members of K. Let 2) be a deductive system. Then tim following characterizations hold: 8

a) 2) is protoalgebraic if and only if Mod*L2) = PSD Mod*L 2), i.e., Mod *L 2) = Mod *s 2); b) 2) is equivalential if and only if Mod*L2) = S P Mod *b 2); c) 2) is finitely equivalential if and only if Mod *L 2) = S P P u Mod *L 2), i.e., Mod*L 2) is a quasi-variety in the sense of Mal'cev; d) 2) is algebraizable if and only if it is equivalential and F is the minimal 2)-filter of A for each {A, F) C Mod *L 2); e) ~D is finitely algebraizable if and only if it is finitely equivalential and F is the minimal 2)-fitter of A for each {A, F) C Mod *L 2).

For papers on the specific levels of the algebraic hierarchy, see [5], [6], [12], [17], [24], [25]. Two references of a more general nature are [8], [16].

Protoalgebraic logics. Within the context of the model theory of first-order logic, a deductive system can be viewed as a strict universal Horn theory with a single unary predicate and without equality. (cf. also H o r n clauses, t h e o r y of). It is an interesting question as to how much of the model theory of strict universal Horn logic with equality can be recovered by means of the abstract Lindenbaum-Tarski process. In the case of finitely algebraizable deductive systems it can be essentially completely recovered already in the algebraic reducts of the Leibniz-reduced models. The same is true for finitely equivalential systems where the finite equivalence systems give a strong representation of equality, but here the filter part of the Leibniz-reduced model is essential and cannot be discarded. But much can be recovered even in the case of protoalgebraic systems where the proto-equivalence systems give a much weaker representation of equality. Protoalgebraic systems turn out to be the largest class of deductive systems 2? whose Leibniz-reduced model class Mod*L2) is well behaved in the sense of strict Horn logic with equality, and the key to this is the following correspondence theorem for 2P-filters that mirrors the correspondence theorem for congruences in universal algebra: Let 2) be a protoalgebraic deductive system, and let A and B be algebras and h: A -+ B a surjective homomorphism. Finally, let F0 be the smallest 2)-filter on B. Then the mapping F ~ h - l ( F ) is a one-to-one correspondence between the 2)-filters on B and the 2)-filters on A that include

h-l(v0). When A is taken to be Fro, the algebra of formulas, this correspondence establishes a close connection between the meta-logical properties of 2) and the algebraic properties of the class Mod*L 2) of Leibniz-reduced models of 2P.

ABSTRACT ALGEBRAIC LOGIC Every class K of matrices over the same language type A defines a deductive system D(K) = ( F m , ~K} over A in the following way. ~ o , . . . , ~ - 1 ~K ~ if, for every ( A , F ) E K and every h o m o m o r p h i s m h: F m --+ A, h(~) E F whenever h ( ~ 0 ) , . . . , h(@n-1) C F. The following theorem is a generalization of Mal'cev's well-known characterization of the strict universal Horn class generated by an arbitrary class of matrices: Let K be a class of Leibniz-reduced matrices over the same language type; then • if D(K) is protoalgebraic, then Mod*LD(K) = (SPPuK)*L; • if D(K) is equivalential, then Mod*LD(K) = SPPuK. The following theorem is an analogue of the finite basis theorem of K. Baker for congruence-distributive varieties and of the corresponding result for relatively congruence-distributive quasi-varieties. It uses the notion of filter-distributive deductive system. A deductive system D is filter-distributive if Fig A is a distributive lattice for every algebra A. Let K be a finite set of matrices. If D(K) is protoalgebraic and filter-distributive, then D(K) has a presentation by a finite set of axioms and inference rules [30]. An important related axiomatizability result can be found in [13]. In analogy to the algebraic hierarchy there is a classification of deductive systems in terms of progressively weaker versions of a deductive-detachment systern. Again protoalgebraic systems lie at the lowest level, and filter-distributive systems constitute another level of hierarchy. See [14], [16]. The generalization of Mal'cev's theorem above is one of many model-theoretic theorems of this kind involving various levels of the algebraic hierarchy, and the scope of the theory has been broadened to include infinitary universal Horn logic without equality [8], [12], [18], [191,

[20]. Second-order algebraizable logics. There are deductive systems with clear algebraic counterparts that are not protoalgebraic and hence not amenable to the methods of abstract algebraic logic discussed so far. Many examples of this kind arise as fragments of more expressive deductive systems that are finitely algebraizable. A paradigm for deductive systems of this kind is the conjunction/disjunction fragment CPCAv of classical propositional logic. It has a natural algebraic semantics, the variety DL of distributive lattices. In order to extend the standard theory of algebraizability to a wider class of deductive systems, recent investigations in abstract

algebraic logic have switched focus from D-filters to certain special classes of D-filters and to a natural generalization of the Leibniz congruence that applies to classes of D-filters. The non-algebraizability of CPCAv is reflected in the fact that, for an arbitrary algebra A, the Leibniz operator does not give a one-one correspondence between (CPCAv)-filters and DL-congruences. The correspondence can in a sense be recovered by replacing single (CPCAv)-filters by sets of (CPCAv)-filters, each of which is of the form Cr, where Cr consists of all (CPCAv)-filters t h a t are compatible with each m e m b e r of a fixed but arbitrary class F of congruences on A. The set of congruences F is completely arbitrary, but it turns out t h a t there is always a single congruence • such that C{o} = Cr, and in fact a smallest one with this property, and it is necessarily a DL-congruence. Moreover, all Dk-congruences can be obtained this way. Considerations such as these lead to the following notion. A full second-order filter of D on an algebra A is the set of all D-filters F on A such t h a t F is compatible with a fixed but arbitrary set of congruences. The set of full second-order filters on A is denoted by FFi~ A. It is easy to check that every C C F F i ~ A is an algebraic closed-set system of the universe A of A. For each C E FFi~ A the Frege relation AC is the largest binary relation on A (necessarily an equivalence relation) that is compatible with each F E C, and the second-order Leibniz congruence, also called the Tarski congruence, ~C is the largest congruence of A included in AC. A set C of D-filters on A is a full second-order filter of D if and only if the set of quotient filters {F/~tC : F E C} coincides with the set of all D-filters on the quotient algebra A / f t C . A full second-order model o l d is a secondorder matrix 9.1 = (A, C) where C E F F i ~ A. ~ is Leibniz reduced if f~C is the identity relation. FMod D (respectively, FMod *L D) is the class of all (Leibniz-reduced) full second-order models of D. The following assertion generalizes iv) above, the lattice isomorphism characterization of algebraizable deductive systems, and applies to all deductive systems. For any deductive system D and any algebra A the second-order Leibniz operator ft is a dual orderisomorphism between F F i ~ A and CoalgFMod.Lz) A, both partially ordered by set inclusion. A full second-order model, and more generally, any second-order matrix (A,C) where C is an algebraic closed-set system on A, can be naturally thought of as a model of a Gentzen system. In the context of abstract algebraic logic a Gentzen system can be viewed as a finitary and snbstitution-invariant consequence relation between sequentsl a sequent is a syntactical expression of the form F o , . . . , P , ~ - I > P~, where ~ 0 , . . . , ~ , ~ - i , ~ n

ABSTRACT ALGEBRAIC LOGIC is any finite, non-empty sequence of formulas• Let 91 = {A,C> be a second-order matrix, and let C: P(A) --+ P(A) be the closure operator on A associated with the algebraic closed-set system C. 91 is a model of a Gentzen system G if the following holds. For every entaihnent

~00,

0

• ..,~gr~o_ 1 [>@0;...

m--1 ' ' ' ' , ~9m--1 T~m-l--1

;~9 0

1-9 0 0 , . . . , 0~_~

>

>

@rn--1 ~_

~,

and every homomorphism h: F m -+ A, if h(@i) E •" (~n~-i)}) for each i < m, then h(~) E

c({h(00),..., A deductive system ~D is said to have a fully adequate Gentzen system if the class of full second-order models of :D is the class of models of a Gentzen system. (Strictly speaking, this is the form the definition takes when :D has at least one theorem. The definition together with the formulation of some of the results stated below must be modified slightly if there are no theorems.) The notion of finite algebraizability for deductive systems can be extended to Gentzen systems in a straightforward way. Just as in the case of deductive systems, if a Gentzen system g is finitely algebraizable, there is a unique quasi-variety Q that is equivalent to ~ in the sense that there is a bisimulation between the consequence relation of g (between sequents) and the equational consequence relation of Q. In view of the above observations it is natural to take a deductive system :D to be second-order finitely algebraizable if it has a fully adequate Gentzen system ~ such that g is finitely algebraizable. In this case, Alg FMod *L :D turns out to coincide with the equivalent quasi-variety of 6, and the consequence relation of Z) is definable (as part of the consequence relation of g) in the equational consequence relation of Alg FMod *L ©, but not vice versa unless ~D is also finitely algebraizable. In the latter case Alg FMod*L :D coincides with the equivalent quasi-variety of :D. When :D is second-order finitely algebraizable, AlgFMod*L:D is called the second-order equivalent quasi-variety of D. Strictly speaking, second-order finite algebraizability does not generalize (first-order) finite algebraizability since there are deductive systems that are finitely algebraizable but do not have a fully adequate Gentzen system. However, this new notion of algebraizability goes a long way toward settling some important questions left open by the earlier theory. One of these deals with the notion of strong finite algebraizability. A finitely algebraizable deductive system is strongly finitely algebraizable if its equivalent quasivariety is a variety. All the familiar deductive systems of algebraic logic, including both the Fregean and intensional ones, turn out to be strongly finitely algebraizable, but the standard theory is unable to account for this. 10

Self-extensionality is a much weakened form of the property of being Fregean. A deductive system 7? is selfeztensional if A ~ Thin ~D is a congruence relation on the formula algebra• Let ~D be a self-extensional deductive system that has either conjunction or the deduction-detachment theorem with a single deduction-detachment formula. Then is second-order finitely algebraizable and its secondorder equivalent quasi-variety Alg FMod *L ~D is actually a variety• The conjunction/disjunction fragment CPCAv of classical propositional calculus is self-extensional (in fact Fregean) with conjunction• Hence it is finitely algebraizable in the second-order (but not the first-order) sense. Its second-order equivalent quasi-variety Alg FMod *L ~D is the variety DL of distributive lattices. The modal logic $5 can be formulated as a deductive system in two ways, both of which have the same set of theorenis. The first and more familiar one, the strong form, is denoted by $5 s and has the necessitation rule Dp as an inference rule (cf. also P e r m i s s i b l e law (infere n c e ) ) along with m o d u s p o n e n s p,~-+ ~ The weak form, $5 w, has modus ponens as its only rule of inference. $5 s is finitely algebraizable but not self-extensional• $5 w is not algebraizable, but it is self-extensional and has both conjunction and the deduction-detachment theorem with a single deductiondetachment formula. So $5 w is second-order finitely algebraizable. Moreover, its generalized equivalent quasivariety is a variety; this turns out to be the variety of monadic algebras, which is also the equivalent quasivariety of $5 s. The main source for this section is [21], where references to other relevant sources can be found• The generalization of algebraizability to Gentzen systems is found in [32]. S e m a n t i c s - b a s e d a b s t r a c t a l g e b r a i c logic. In this important branch of abstract algebraic logic the fine structure of the interpretations of a deductive system is taken into account. It also features a refinement of the notion of language• Let A be a language type, assumed to be fixed. For an arbitrary set P disjoint from A, let Fmp be the set of formulas built up from the elements of P, thought of as abstract atomic formulas, using the connectives of A; the associated formula algebra is denoted by F m p . For each set P of atomic formulas, let Sp = (P, Modsp, mngsp, ~ s p ) be a four-tuple, where Modsp is a class, called the class of models of Se;

A B S T R A C T A L G E B R A I C LOGIC mngsp is a function that assigns to each 9N ~ Mod&. a function m n g s p , ~ with domain F m p , called the meaning function for F.R; and ~&. is a binary relation between Modsp and Fmp, called the validity relation. Sp is a semantical system if the following conditions hold for every model 921l:

A) h is an isomorphism between Me~Lg)I and Me~L91 such that mng&,,~ = mngsp,~ t oh; and B) h preserves the truth set, i.e., h(Fspf0I *L) = Fsp 91,L.

• the meaning of a formula does not change if a subformula is replaced by one with the same meaning, i.e., mngsp,m ~ is a homomorphism; • the validity of a formula depends only on its meaning, i.e., if mngsp,~(qo ) = mngsm~t(~b), then 9)I ~ s p if and only if 9)I ~ s p ~b.

• Alg Mod *L 2?Sp, the algebraic semantics of the underlying deductive system of Sp; and • M e M o d S p = {Me&.ff~: 91/E M o d e . } , the class of meaning algebras of Sp.

The meaning algebra of gJ[, in symbols Mesp991, is the image of F m p under the meaning homomorphism mngsp,~ x. The final defining condition of a semantical system is the following: • every homomorphism from the formula algebra into the meaning algebra of 9)I is the meaning function of some model, i.e., if h: F m p -4 MesegJ[, then there is a 9I E Modsp such that h = m n g s , , ~ . 9)I is a model of a set I' of formulas if g)I ~ s p ~b for each ¢ E F. The class of all models of F is Modsp r . The consequence relation of S is the relation F ~ s p that holds between a set of formulas P and an individual formula if M o d e . F C_ Modsp {~o}. ~ s . satisfies all the conditions of a consequence relation of a deductive system except possibly finiteness; however, most of the familiar semantical systems are finitary. ( F m p , ~ s p ) is called the underlying deductive system of Sp and is denoted by DSp. The theory of a model 9)I of Sp, in symbols T h s e if2, is the set of all formulas valid in 9)I. The truth filter of Mesp~R, FspgJt, is the image of Thsp 9)I under mngse,~x. Because the validity of a formula depends only on its meaning, the meaning matrix (MespgJ[, Fspg)I) together with the meaning function mng&, ~x is an interpretation of the underlying deductive system of Sp. As before, the Leibniz reduction of the meaning matrix by the Leibniz congruence of the truth filter, (MespfOI/f~F&.gJ[, Fsp~JJt/f~Fspff2g), is denoted by <Me2Lgrt, F*Lg)I\Sp /. The model-theoretic properties of a large class of different logical systems can be studied algebraically in this context. Consider, for example, the relation of elementary equivalence. Two models g)I and 91 of Sp are elementarily equivalent if Thsp 9)I = Ths~ 91. Let Sp be a semantical system. Two models 9)I and 91 of Sp are elementarily equivalent if and only if there is an isomorphism h between the Leibniz-reduced meaning matrices that commutes with the meaning functions,

i,e.,

Two different classes of algebras are associated with each semantical system $p:

Sp is protoalgebraic, equivalential, finitely equivalential, algebraizable, or finitely algebraizable if its underlying deductive system DSp has the property and the meaning matrix of every model of $ p is Leibniz-reduced, i.e., if M e M o d S p C A l g M o d *L DSp. In this case it can be shown that Alg Mod *L DSp = ®g3Me Mod 8p. In general, for a deductive system D there are many different semantical systems with underlying deductive system D. A natural semantical system for classical propositional logic is obtained by considering only the interpretations of CPC whose underlying algebra is a Boolean algebra of sets. More precisely, a model is a pair (X, v), where X is a set and v assigns a subset of X to each atomic formula in P. The meaning function is the unique homomorphism from F m p to the Boolean algebra of subsets of X that extends v. ~ is valid in (X, v) if its meaning is X. A semantical system for $5 is obtained in a similar way. A model is a three-tuple (X, x, v), where X is a set, x C X, and v assigns subsets of X to atomic formulas. The meaning function is the unique homomorphism from F m p into the Boolean algebra of subsets of X extending v such that, for every formula ~, the meaning of [:]~ is X if the meaning of p is X; otherwise the meaning of D~ is the empty set. p is valid in (X, x, v) if x is contained in the meaning of p. (X, x, v) represents a so-called 'possible worlds' model for $5; X is the set of possible worlds and a formula is valid in the model if it is true at the distinguished 'real world' x. One of the standard semantical systems for the firstorder predicate logic has as its models structures of the form (X, v), where X is a non-empty set and v assigns a subset of X ~ to each atomic formula. It is assumed that the individual variable symbols are arranged in an w-sequence. The meaning function is the unique homomorphism from F m into the Boolean algebra of subsets of X ~ extending v such that, for each formula ~, the meaning of 3vi ~ is the 'cylinder' that is swept out by moving the meaning of ~ parallel to the ith coordinate. The meaning algebra is the subalgebra of the wdimensional cylindric set algebra over X generated by the w-ary relations that are the meanings of the atomic 11

ABSTRACT ALGEBRAIC LOGIC formulas. Elementary equivalence in first-order logic is essentially captured by the notion of elementary equivalence in the semantical systems of this kind. The characterization of elementary equivalence given by A) and B) provides a way of investigating elementary equivalence algebraically. The algebraic study of some model-theoretic notions, such as definability, require semantical systems over varying sets of atomic formulas. A system $ = ($p : P a set) of semantical systems is called a general semantical system if the Sp are compatible in the sense that, for all P and P', Sp and Sp, are isomorphic in the natural sense whenever P and P ' have the same cardinality, and, if P C_ P', then every model of Sp extends to a model of Sp, and every model of Sp, restricts to a model of Sp. A general semantical system $ is protoalgebraie, equivalential, finitely equivalential, algebraizable , or finitely algebraizable if each of its component semantical systems Sp has this property. For every general semantical system ,5, AlgMod *L D`5 = I.J{AlgMod *L:D`SP: P a set} and M e M o d S = I.J{MeModSp : P a set}. Let `5 be a general semantical system. Let P, R and R' be disjoint sets of atomic formulas, and let ' be a bijection between R and R'. Let E(P, R) _C F m p u n be a set of formulas whose atomic formulas are in P U R. Then: • E ( P , R ) defines R explicitly over P (in ,5) if for every r E R there exists a ~or E F m p such that, for every 9)I E M o d s p u n ( E ( P , R ) ) , mngspun,~x(r ) = • E ( P , R ) defines R implicitly over P (in ,5) if for every g)I E MOdsp.RuR, (E(P, R) U E(P,/~')) and every r E R, mngs,uRuw,~ot(r ) = mngspuRuR,,~x(r'); here E(P, R') denotes the set of formulas obtained from E(P, R) by replacing each r 6 R by r'. • E ( P , R ) is a strong implicit definition of R over P (in ,5) if it defines R implicitly over P and every 9Jl E Mods~ (Th Modsp~R (E(P, R)) n Fmp) has an extension 9l ¢ Mods~.~R(E(P, R)).

S has the (weak) Beth definability property if for all E, P and R as above, E defines R implicitly over P (in the strong sense), then E defines R explicitly over P. Explicit definability always implies implicit definability. This is the well-known method of A. Padoa formulated in abstract algebraic logic. The algebraic analogue of the property of Beth is surjectivity of epimorphisms. Let K be a class of algebras over the same language type. A homomorphism h: A --> B, where A , B E K, is called an epimorphism 12

over K if for any pair of homomorphisms 9, g' : B --+ C, if g o h = g' o h, then g : g'. Let Ko C_ K be classes of algebras over the same language type. A homomorphism h: A -+ B, where A, B C K, is said to be Ko-extensible over K if for any C 6 Ko and every surjection f : A -+ C there is a DEK0andg:B~DsuchthatC_CDandgoh=f. Let S be a finitely algebraizable generalized semantical system. Then: I) S has the Beth definability property if and only if every epimorphism over Alg Mod *L 79S is surjective; II) $ has the weak Beth definability property if and only if every (MeMod$)-extensible epimorphism of Alg Mod *L 2)3 is surjective. The algebraic characterization of the weak Beth property requires a semantics-based context, but the result on the ordinary Beth property can be reformulated within logistic abstract algebraic logic and extended to equivalence deductive systems. The main references for semantics-based abstract algebraic logic are [4], [2], [3]. For the results on definability, see [26] and [27]. References [1] ANDERSON, A.R., AND BELNAP, N.D.: Entailment. The logic

of relevance and necessity, Vol. I, Princeton Univ. Press, 1975. [2] ANDRI~KA, H., KURUCZ, A., NI~METI, I., AND SAIN, I.: 'Applying algebraic logic: A general methodology': Proc. Summer School of Algebraic Logic, Kluwer Acad. Pubi., to appear, Short version in: [4]. [3] ANDRI~KA, H., AND Nt~METI, I.: 'General algebraic logic: A perspective on "what is logic"', in D. GABBAY (ed.): What is a logical system?, Clarendon Press, 1994, pp. 485-569. [4] ANDRI~KA, H., NI~METI, I., AND SAIN, I.: 'Applying algebraic logic to logic', in M. NIVAT ET AL. (eds.): Algebraic Method-

ology and Software Techn. (AMAST'93, Proc. 3rd Internat. Conf. Algebraic Methodology and Software Techn.), Workshops in Computing, Springer, 1994, pp. 3-26. [5] BLOK, W.J., AND PIGOZZI, D.: 'Protoalgebraic logics', Studia Logiea 45 (1986), 337-369. [6] BLOK, W.J., AND PIGOZZI, D.: Algebraizable logics, Vol. 396 of Memoirs, Amer. Math. Soe., 1989. [7] BLOK, W.J., AND PIGOZZI, D.: 'Local deduction theorems in algebraic logic', in H. ANDRI~KA, J.D. MONK, AND I. NI~METI (eds.): Algebraic Logic (Proc. Conf. Budapest 1988), Voh 54 of Colloq. Math. Soc. Y. Bolyai, North-Holland, 1991, pp. 75109. [8] BLOK, W.J., AND PIGOZZI, D.: 'Algebraic semantics for universal Horn logic without equality', in A. ROMANOWSKA AND J.D.H. SMITH (eds.): Universal Algebra and Quasigroup Theory, Heldermann, 1992, pp. 1-56. [9] BLOK, W.J., AND PIGOZZI, D.: 'Abstract algebraic logic and the deduction theorem', Bull. Symbolic Logic (to appear). [10] BLOOM, S.L., AND SUSZKO, R.: 'Investigations into the sentential logic with identity', Notre Dame J. Formal Logic 13 (1972), 289 308.

A B S T R A C T ANALYTIC N U M B E R T H E O R Y

[11] CHIN, L.H., AND TARSI(I, A.: 'Distributive and modular laws in relation algebras', Univ. California Publ. Math. New Set. 1, no. 9 (1951), 341-384. [12] CZELAKOWSKI,J.: 'Equivalential logics I-IF, Studia Logica 40 (1981), 227-236; 355-372. [13] CZELAKOWSKI,J.: 'Filter-distributive logics', Studia Logica 43 (1984), 353-377. [14] CZELAKOWSKI,J.: 'Algebraic aspects of deduction theorems', Studia Logica 44 (1985), 369-387. [15] CZELAKOWSKI, J.: 'Consequence operations: Foundational studies', Tcchn. Rept. Inst. Philosophy and Sociology Polish Acad. Sci. (1992). [16] CZELAKOWSKI,J.: Protoalgebraic logics, Vol. 10 of Trends in Logic-Studia Logica Libr., Kluwer Acad. Publ., 2001. [17] CZELAKOWSKI, J., AND JANSANA, R.: 'Weakly algebraizable logics', Y. Symbolic Logic 65 (2000), 641-668. [18] DELLUNDE,P., AND JANSANA,R.: 'Some characterization theorems for infinitary universal horn logic without equality', J. Symbolic Logic 61 (1996), 1242 -1260. [19] ELGUETA, R.: 'Characterizing classes defined without equality', Studia Logica 58 (1997), 357-394. [20] ELGUETA, R.: 'Subdirect representation theory for classes without equality', Algebra Univ. 40 (1998), 201-246. [21] FONT, J.M., AND JANSANA, R.: A general algebraic semantics for sentential logics, Vol. 7 of Lecture Notes in Logic, Springer, 1996. [22] HALMOS, P.R.: 'Algebraic logic h Monadic Boolean algebras', Compositio Math. 12 (1955), 217-249. [23] HENKIN, L., MONK, J.D., AND TARSKI, A.: Cylindric algebras, Parts I-II, North-Holland, 1971/85. [24] HERRMANN, B.: 'Equivalential and algebraizable logics', Studia Logica 57 (1996), 419-436. [25] HERRMANN, B.: 'Characterizing equivalential and algebraizable logics', Studia Logica 58 (1997), 305-323. [26] HOOGLAND, E.: 'Algebraic characterization of various Beth definability properties', Studia Logica 65 (2000), 91-112. [27] HOOGLAND, E.: Definability and interpolation. Modeltheoretic investigations, ILLC Dissert. Ser. DS-2001-05. Inst. Language, Logic and Computation, Amsterdam, 2001. [28] Lo~, J.: 'O matrycach logicznych', Set. B. Travaux de la Soc. Sci. et des Lettres de Wroc~aw 19 (1949). [29] MALINOWSKI,J.: 'The deduction theorem for quantum logicsome negative results', J. Symbolic Logic 55 (1990), 615-625. [30] PALASII~SKA,K.: 'Deductive systems and finite axiomatizability properties', PhD Thesis Iowa State Univ. (1994). [31] RASIOWA, H.: A n algebraic approach to non-classical logics, North-Holland, 1974. [32] REBAGLIATO, J., AND VERDI~I, V.: 'On the algebraization of some Gentzen systems', Fundam. Inform. 18 (1993), 319338, Special Issue on Algebraic Logic and its Applications. [33] SMILEY, T.: 'The independence of connectives', J. Symbolic Logic 27 (1962), 426-436. [34] SURMA, S.J.: 'On the origin and subsequent applications of the concept of the Lindenbaum algebra': Logic, Methodology and Philosophy of Science VI (Hannover 1979), NorthHolland, 1982, pp. 719-734. [35] SUSZKO, R.: 'Abolition of the Fregean axiom': Logic Colloquium (Boston 1972/3), Vol. 453 of Lecture Notes in Mathematics, Springer, 1975, pp. 169-236. [36] TARSKI, A.: '0ber einige fundamentale Begriffe der Metamathematik', C.R. Soc. Sci. Lettr. Varsovie Cl. III 23 (1930), 22-29.

[37] TARSKI, A.: 'Grundzfige der Systemenkalkiils. Erster Teil', Fundam. Math. 25 (1935), 503-526. [38] WdJeICKI, R.: Theory of logical calculi. Basic theory of consequence operations, Vol. 199 of Synthese Library, Reidel, 1988. D. Pigozzi

M S C 1991: 03Gxx, 03G25, 06F35 ABSTRACT

06Exx, 03G15,

ANALYTIC

NUMBER

03G05, 03G10,

T H E O R Y - The

central concept in abstract analytic number theory is that of an arithmetical semi-group G (defined below). It turns out that the study of such semi-groups and of (real- or complex-valued) functions on them makes it possible on the one hand to apply methods of classical a n a l y t i c n u m b e r t h e o r y in a unified way to a variety of asymptotic enumeration questions for isomorphism classes of different kinds of explicit mathematical objects. On the other hand, these procedures also lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration questions about the

(mostly non-arithmetical) concrete types of mathematical objects just alluded to. A r i t h m e t i c a l s e m i - g r o u p s . An arithmetical semigroup is, by definition, a commutative s e m i - g r o u p G with identity element 1, which contains a countable subset P such that every element a # 1 in G admits a unique factorization into a finite product of powers of elements of P, together with a reM-valued mapping ]-I on G such that: i) [11 = l , [ p l > l f o r p e P ; ii) [ab[ = ]a]. [b[ for all a,b e G; iii) the total number of elements a with [a[ < x is finite, for each x > 0. The elements of P are called the primes of G, and [-] is called the norm mapping on G. It is obvious that, corresponding to any fixed c > 1, the definition cg(a) = log~ [a[ yields a mapping 0 on G such that: A) 0(1) = 0, 0(p) > 0 for p E P; B) O(ab) = O(a) + O(b) for all a,b E G; C) the total number of elements a with O(a) <_ x is finite, for each x > 0. Conversely, any real-valued mapping c9with the properties A)-C) yields a norm on G, if one defines lal = c°(a). In cases where such a mapping 0 is of primary interest, G together with 0 is called an additive arithmetical semi-group, and one refers to 0 as the degree mapping on G. In most concrete examples of interest, it turns out that the norm or degree mappings represent natural 'size' or 'dimension' measures which are integervalued. With an eye to applications to natural examples 13

ABSTRACT

ANALYTIC

NUMBER

THEORY

there is therefore little loss in henceforth restricting attention to either a single integer-valued norm mapping ]'], or a single integer-valued degree mapping 0, on G. Depending on which case is being considered, special interest then attaches to the basic counting functions (for

ncZ) G(n) = ~ { a e

G: la[ = n } ,

P(n) = # {p • P : Ip[ = ,~} (or G#(n) = # { a • G: O(a) = n}, P # ( n ) = :ff{p • P: O(p) = n}, in the additive case). The prototype of all arithmetical semi-groups is of course the multiplicative semi-group N of all positive integers {1, 2,...}, with its subset PN of all rational prime numbers {2, 3, 5, 7,...}. Here one may define the norm of an integer n to be In) = n, so that the number N(n) = 1 for n > 1. The asymptotic behaviour of 7r(x) = ~ n < x PN(n) for large x forms the content of the famous prime number theorem, which states that ~(x)

~

X - -

asx

log x

-~

(aft also de la V a l l ~ e - P o u s s i n t h e o r e m ) . A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semi-groups. For example, it is true for the multiplicative semi-group GK of all non-zero ideals in the r i n g R = R ( K ) of all algebraic integers in a given a l g e b r a i c n u m b e r field K , with III = card(R/I) for any non-zero ideal I in R. Here, the prime ideals act as prime elements of the semi-group GK. A simple but nevertheless interesting example of an additive arithmetical semi-group is provided by the multiplicative semi-group Gq of all monic polynomials in one indeterminate X over a f i n i t e field Fq with q elements, with O(a) = deg(a) and the set Pq of prime elements represented by the irreducible polynomials (cfi also Irr e d u c i b l e p o l y n o m i a l ) . Here, G#q(n) = qn, and it can be proved that

. ? ( n ) = ;1

(r)qO/r, rln

where # is the classical M S b i u s f u n c t i o n on N. Up to isomorphism, Gq is the simplest special case of the semi-group GR of all non-zero ideals in the ring R = R ( K ) of all integral functions in an algebraic function field K in one variable X over Fq.

Arithmetical categories of semi-groups. Many interesting examples of concrete, but non-classical, arithmetical semi-groups can be found by considering certain specific classes of mathematical objects, such as groups, rings, topological spaces, and so on, together with appropriate 'direct product' operations and isomorphism relations 14

for those classes. It is convenient, though admittedly not quite precise, to temporarily ignore the corresponding morphisms and refer to such classes of objects as 'categories' (cf. also C a t e g o r y ) . Now consider some category C which admits a direct 'product' (or 'sum') operation x on its objects. Suppose that this operation x preserves C-isomorphisms, is commutative and associative up to C-isomorphism, and that C contains a 'zero' object 0 (unique up to Cisomorphism) such that A x 0 = A for all objects A in U. Then suppose that a theorem of Krull-Schmidt type is valid for C, i.e., suppose that every object A ~ 0 can be expressed as a finite x-product A - P1 x ..- x Pm of objects Pi ~ 0 that are indecomposable with respect to x, in a way that is unique up to permutation of terms and C-isomorphism. In most natural situations at least, one may reformulate these conditions on C by stating that the various isomorphism classes A of objects A in C form a set Gc that is i) a commutative semi-group with identity with respect to the multiplication operation A x B = A x B; ii) a semi-group with the unique factorization property with respect to the isomorphism classes of the indecomposable objects in C. For this reason, one may call the C-isomorphism classes P of indecomposable objects P the 'primes' of C or Gc. In many interesting cases (some of which are illustrated below), the category C also admits a 'norm' function i'I on objects which is invariant under Cisomorphism and has the following properties: i) i01 = 1, [PI > 1 for every indecomposable object P; ii) ] A x B[ = [A[. [BI for all objects A, B; iii) the total number of C-isomorphism classes of objects A of norm IA[ _< x is finite, for each real x > 0. Obviously, in such circumstances, the definition IAI = IAI provides a norm function on Gc satisfying the required conditions for an arithmetical semi-group. For these reasons, a category C with such further properties may be called an arithmetical category. Now consider some concrete illustrations for the above concepts, taken from [2], [3]. a) (Finite Abelian groups; cf. A b e l i a n g r o u p . ) One of the simplest non-trivial examples of an arithmetical category is provided by the category A of all finite Abelian groups, together with the usual direct product operation and the norm function [A[ = card(A). Here, the Krull-Schmidt theorem reduces to the well-known

A B S T R A C T ANALYTIC N U M B E R T H E O R Y

fundamental theorem on finite Abelian groups, the indecomposable objects of this kind being simply the various cyclic groups Zp. of prime-power order pr (cf. also Cyclic group). b) The category of all semi-simple associative rings of finite cardinality (cf. also Associative rings a n d

Some explicit illustrations of zeta-functions and Euler products are given below.

The Riemann zeta-function. For the basic semi-group N of positive integers, the z e t a - f u n c t l o n is (X3

¢(z) = ~ n-z;

algebras).

n=l

c) The category of all semi-simple finite-dimensional associate algebras over a given field F (cf. also Asso-

ciative rings and algebras; Semi-simple ring). d) The category of all semi-simple finite-dimensional Lie algebras over a given field F (cf. also Lie algebra). e) The category of all compact simply-connected globally symmetric Riemannian manifolds (cf. also

Globally symmetric Riemannlan space). f) The category T of topological spaces of finite cardinality (ef. also T o p o l o g i c a l s p a c e ) with the property that a space Y lies in T if and only if each connected component of Y lies in T. Z e t a - f u n c t i o n s a n d enumeration problems. For a given arithmetical semi-group G, information on the basic counting functions G(n), P(n) can often be obtained, algebraically or with the aid of analysis, via a certain series-production relation called the Euler product formula for G. Indeed, ignoring questions of convergence for the moment, note that (by the unique factorization into prime elements of G) the series

it is called the Riemann zeta-function, and the classical Euler product formula reads:

¢(z) =

C~

eK(z) = ~

I±l-z = Z K(nln-~, n:l

IEGK

where K(n) denotes the total number of ideals of norm n in GK; it is known as the Dedekind zeta-function of K . (See also Z e t a - f u n c t i o n . )

Monic polynomials over a finite field. For the additive arithmetical semi-group Gq of all monic polynomials in one indeterminate X over Fq (see above), the generating function may be written as OO

Zq(y) = E qnyn = (1 - qy)-l, n=O

n=l

[al-Z =

aEG

~

Ip~~ • " ' p ;r~ l

-~

=

all products p[1 ...p~-~ with Pi E P, ri , m 6 N

= 1+ ~ = H

( 1 - p - O -1.

The Dedekind zeta-function. Let GK denote the (abovementioned) arithmetical semi-group of all non-zero 'integral' ideals in a given algebraic number field K . The zeta-function for GK is then

O(9

Ca(z) : E G(n)n-Z = E = 1+

II primes p E N

IPll - ' ~ z ' ' '

and the above-mentioned explicit formula for P f (n) can be deduced as an algebraic consequence of the Euler product for Ga.

Finite Abelian groups. For the category A of all finite Abelian groups, the zeta-function may be written as IP,~l - ~

(1 + lpl - z + I p l - ~ + ' " )

....

_-

OG

CA(z) -- Z a(n)n-<

=

n=l

p6P

H 0- l-0

pCP

=~

( 1 - ~ - z ) -~(m~

OO

rn=2

As a function of z, ~G(z) is called the zeta-function of G. If G is an additive arithmetical semi-group with ]a] = c°(a) for some integer c > 1, one may substitute the symbol y for c -z and obtain the modified Eulerprod-

uct formula:

Z a ~ ( n ) < = H (1 - y~)--~(~); n=O

where a(n) denotes the total number of isomorphism classes of Abelian groups of order n. The discussion of 'primes' in A given above shows that here the Euler product may be written as a double product

m:l

then Za(y) = Enc~__oO#(n)yn is called the modified zeta-function (or generating function) of G.

cA(z):

II

(1-p-r0-1:II

r>l, primes p E N

(rz), r:l

by the Euler product formula for the Riemann zetafunction. For the subcategory A(p) of all finite Abelian pgroups, where p is a fixed prime number (cf. also pg r o u p ) , it is natural to regard A(p) as an additive arithmetical category, with degree mapping defined by O(A) = log; card(A). 15

ABSTRACT ANALYTIC N U M B E R T H E O R Y In that case, A(p) has exactly one prime of degree r for each r = 1, 2,.... Therefore the Euler product formula implies that A(p) has the generating function o(3

oo

I'I( 1 yr)--I = E P ( n ) y n '

Z A(p)(y ) =

--

r=l

n=0

where p(n) = a(p ~) is the total number of isomorphism classes of Abelian groups of degree n in the above sense. In fact, for n > 0, p(n) equals the total number of ways of partitioning n into a sum of positive integers, which is also the number of pseudo-metrizable finite topological spaces of cardinality n (see f) above). Thus, the corresponding latter category 7) (say) has the same generating function as .4(p). T y p e s o f a r i t h m e t i c a l s e m i - g r o u p s . Bearing in mind the emphasis on concrete realizations of arithmetical semi-groups in a variety of areas of mathematics, it is reasonable to classify them and to base further investigations according to common features which may be exhibited by the initial enumeration theorems for particular sets of examples. In that way, further questions and enumeration problems may be investigated uniformly under suitable covering assumptions or 'axioms' appropriate for particular natural sets of examples. On this basis, a small number of special types of arithmetical semi-groups have so far (2000) been found to predominate amongst natural concrete examples. Classical and axiom-A type semi-groups. The strictly classical arithmetical semi-groups of analytic number theory are the multiplicative semi-group of all positive integers and the multiplicative semi-group of all nonzero ideals in the ring of all algebraic integers in a given algebraic number field (see above). For example, H. Weber and E. Landau proved theorems to the effect that

asx-

,

(1)

n<~J

where GK is the semi-group of all 'integral' ideals in a given algebraic number field K. Landau in particular used (1) in order to extend many asymptotic results about arithmetical functions on N to similar functions on GK. In quite a different direction, P. Erdhs and G. Szekeres proved in 1934 for the category A of all finite Abelian groups that

E a(n) = AlX + O ( v ~ )

as x --+ 0%

(2)

n<x

where A1 = 1-It<2 ~(r) = 2.29.... At a later stage, for the category $ of semi-simple finite rings, I.G. Connell and J. Knopfmacher independently proved that = rt<x

16

+

O(v

)

as x

(3)

where A2 = 1-Ir,~=>2 ~(rrn2) = 2.49-... Strong concrete motivation was available for unifying certain further developments under the umbrella of general studies of an abstract arithmetical semi-group G satisfying the so-called axiom A: There exist constants AG > 0, 5 > 0 and r/ < 6 (all depending on G), such that

E a(n) = Aax 5 +O(z')

asx--+oc.

n<x

Theorems based on the assumption of axiom A often simultaneously generalize earlier results for N, GK and GA, and provide additional asymptotic enumeration theorems for a variety of arithmetical categories like $ and many others. Axiom A # type semi-groups. Consideration of the examples of multiplicative semi-groups of monic polynomials in one indeterminate, and also of enumeration theorems for some infinite families of explicit additive arithmetical categories connected with rings of integral functions in algebraic function fields over Fq (cf. [5], [4]), provides a wealth of motivation for studying an abstract additive arithmetical semi-group G satisfying axiom A#: There exist constants AG > 0, q > 1 and u < 1 (all depending on G) such that

G#(n) = Acq ~ +O(q "~)

asn--+oc.

With this axiom as a basis instead of axiom A, problems similar to those outlined above may be investigated, with similar motivation to those stimulating the axiom-A type studies. It then turns out that the ensuing results and methods of proof sometimes but not always possess parallels to those subject to axiom A. A curious illustration of a non-parallel result arises with the abstract prime number theorem (or abstract prime element theorem) subject to axiom A #. In 1976, Knopfmacher derived such a theorem, on the initial foundation of some plausible-looking lemmas parallel to ones under axiom A. However, in 1989 and later, other authors independently found and then closed certain gaps in those lemmas. The combined efforts of various authors then led to a final theorem with two cases, depending on whether or not Z a ( - q -1) = 0; contributions to this were made by S.D. Cohen, K.-H. Indlekofer, E. Manstavi~ius, R. Warlimont and W.-B. Zhang (see e.g. [1], [5]). A strange point about this result is that the case Z c ( - q -1) ~ 0 holds for all the natural examples which initially motivated axiom A #. Although ingenious examples in which Z a ( - q -1) = 0 have also been constructed, those found up to now might be viewed as somewhat pathological or contrived. Therefore, in terms of the 'natural-example-based approach' to this subject outlined in the beginning, it would not be unreasonable

ABSTRACT PRIME NUMBER THEORY to continue the present (2000) direction of investigation under the combined assumption of axiom A # with the additional axiom

za(-q -1) ¢ o. In fact (see e.g. [3], [4]) many consequences of axiom A # are unrelated to the value of ZG(-q-1), and so the simplifying additional axiom would only sometimes become relevant (but nevertheless reasonable to then assume at such a stage). Axiom C. The examples listed earlier included many involving an additive arithmetical category g for which Gc# (n) and Pc# (n) have quite a different behaviour from that given by axiom A #. Here, although the objects in g may sometimes be rather complicated, the presently (as of 2000) known structure theorems for those objects often lead to a relatively simple estimation for Pc# (n) or Try(x) = ~ n < x PC#(n). Surprisingly perhaps, it turns out that sharp asymptotic information can then be deduced about G~(n) or NC#(x) = ~n<x G#c (n) by methods of classical-type arithmetical partition theory, which were initiated by G.H. Hardy and G. Ramanujan in 1917. These methods belong to a quite different branch of classical a n a l y t i c n u m b e r t h e o r y from those involved in the earlier discussion of axiom A. On the basis of these new types of examples as motivation, one is led to investigations of an additive arithmetical semi-group G satisfying axiom C: There exist constants C > 0, ~ > 0 and ~, (all depending on G) such that ~ ( x ) ~ C x ~(logx) ~ a s x - ~ . A simple example of axiom C is provided when g denotes either the category A(p) of finite Abelian pgroups (cf. also p - g r o u p ) , or the category 7) of pseudometrizable finite topological spaces (cf. also P s e u d o m e t r i c space). Similar formulas hold for the categories of compact simply-connected Lie groups, or semi-simple finitedimensional Lie algebras over an algebraically closed field F of characteristic zero. Asymptotic deductions about G#(n) or N~(x) = ~n<xG#(n), subject to axiom C, could perhaps be referred to as 'inverse additive abstract prime number theorems'. Based on methods of generalized arithmetical partition theory, various theorems of this kind can be derived, as well as results about 'average values' of arithmetical functions on G, and on asymptotic 'densities' of certain subsets of G, subject to axiom C.

Axiom G1. Yet another natural class of additive arithmetical semi-groups G is provided by those satisfying axiom GI: 'Almost all' elements of G are prime, in the sense that G#(n) > 0 for sufficiently large n, and ~ a#(n)

as

It is known that various classes F of finite graphs define arithmetical semi-groups with this slightly surprising property. It is also known that, when k > 1, the multiplicative semi-group Gk,q of all monic polynomials in k indeterminates X1, • • •, Xk over a finite field Fq has the property stipulated in axiom G1. See A b s t r a c t p r i m e n u m b e r t h e o r y for a further discussion of arithmetical semi-groups and their corresponding abstract prime number theorems. References

[1] INDLEKOFER,K.-H., MANSTAVICIUS,E., AND WARLIMONT,R.: 'On a certain class of infinite products with an application to arithmetical semigroups', Archiv Math. 56 (1991), 446-453. [2] KNOPFMACHER, J.: Abstract analytic number theory, NorthHolland, 1975, Reprinted: Dover, 1990. [3] KNOPFMACHER, J.: Analytic arithmetic of algebraic function fields, M. Dekker, 1979. [4] KNOPFMACHER, J., AND ZHANG, W.-B.: Number theory arising from finite fields, analytic and probabilistic theory, M. Dekker, 2001. [5] ZHANG, W.-B.: 'Elementary proofs of the abstract prime number theorem for algebraic function fields', Trans. Amer. Math. Soc. 332 (1992), 923-937.

John Knopfmacher MSC1991: 11N80, 11Nxx, 11N45, 11N32 In various branches of number theory, abstract algebra, combinatorics, and elsewhere in mathematics, it is sometimes possible and convenient to formulate a variety of enumeration or counting questions in a unified way in terms of the concept of an arithmetical semi-group G (cf. A b stract analytic number theory; Semi-group). Special interest then attaches to the basic counting functions (for n E Z): ABSTRACT

PRIME

NUMBER

THEORY -

a ( n ) = # {a e a : lal = n ) ,

P(n)= P G ( n ) = # {pe P: IPl =n} (here, P denotes the set of 'prime' elements in G). If one of the functions G(n), P(n) has a certain type of asymptotic behaviour, it may then be possible to deduce that of the other by a uniform method of derivation. Theorems of the latter kind may then be referred to as abstract prime number theorems within the context considered. Some concrete examples are given below. Types of arithmetical semi-groups. Axiom A. The prototype of all arithmetical semi-groups is of course the multiplicative semi-group N of all positive integers {1, 2,...}, with its subset PN of all rational prime numbers {2, 3, 5, 7,...}. Here one may define the norm of an integer n to be Inl = n, so that the number N(n) = 1 for n > 1. Although N(n) would be too trivial to mention if one were not interested in a wider arithmetical theory, 17

ABSTRACT PRIME N U M B E R T H E O R Y the corresponding function PN(n) remains mysterious to this day (as of 2000). The asymptotic behaviour of 7r(x) = ~,~<x PN (n) for large x forms the content of the famous prime number theorem, which states that x

7r(x) ~ logx

asx-+ec

(cf. also de la V a l l ~ e - P o u s s i n t h e o r e m ) . A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semigroups. For example, it is true for the multiplicative semi-group GK of all non-zero ideals in the ring R = R ( K ) of all algebraic integers in a given a l g e b r a i c n u m b e r field K, with [ I ] = card(R/I) for any non-zero i d e a l I in R. Here, the prime ideals act as prime elements of the semi-group GK, and both the corresponding functions GK (n), PK (n) are non-trivial to estimate in general. However, Landau's prime ideal theorem establishes that ( 7rK (x) =

x n) ,,~ log x

as x --+ ec,

n<:~x

while classical theorems of Weber and Landau yield

n<x

for explicit positive constants AK and r]K < 1. Similar types of asymptotic behaviour are displayed by many quite different types of concrete arithmetical semi-groups (cf. [4, Part II], where these are referred to as 'semi-groups satisfying axiom A').

Axiom A #. For a simple but nevertheless interesting example of an additive arithmetical semi-group, one may consider the multiplicative semi-group Gq of all monic polynomials in one indeterminate X over a fin i t e field Fq with q elements, with O(a) = deg(a) and set Pq of prime elements represented by the irreducible polynomials (cf. also I r r e d u c i b l e p o l y n o m i a l ) . Here, G#q(n) = q~, and it is a well-known theorem that

and

(5 ) where An is a positive constant. Similar examples arise if one considers the semi-group GK of all 'integral divisors' of K , instead of Gn. Related types of asymptotic behaviour are also displayed by many quite different kinds of concrete additive arithmetical semi-groups (cf. [6], [7], where these are referred to as 'semi-groups satisfying axiom A #').

Axioms ~1 and ~x. Yet another natural class of additive arithmetical semi-groups G is provided by those satisfying axiom GI: 'Almost all' elements of G are prime, in the sense that G # ( n ) > 0 for sufficiently large n, and P # (n) ~ G # (n) as n -+ ec, i.e.,

p#(n) nli% d e ( n ) - 1. Examples of this slightly unexpected property are provided by various classes r of finite graphs with the property that a g r a p h H lies in F if and only if each connected component of H lies in F. Let the disjoint union tad be used as follows to define a 'product' operation on the set G r of all unlabelled graphs (i.e., isomorphism classes H of graphs H) in F: H1 • H2 = H1 Ud H2. Also, let 0(H) = ~vertices in H. Then GF becomes an additive arithmetical semi-group. For some classes F, GF satisfies axiom gl, and this is also true for the quite different multiplicative semigroup Gq,k formed by all associate-classes of non-zero polynomials in k > 1 indeterminates X1, • • •, Xk over a finite field Fq (cf. [5]). Ignoring the corresponding limit zero which occurs under axiom A #, and also under axiom A (in a certain sense), R. Warlimont [11] raised the interesting question as to whether there are any 'natural' instances of additive arithmetical semi-groups G satisfying axiom G~: There exists a 0 < ~ < 1 with lim P # ( n ) _ ;~.

1Z =

n

"(r)qn/~'

rln

where p is the classical M S b i u s f u n c t i o n on N. Up to isomorphism, Gq is the simplest special case of the semi-group Gn of all non-zero ideals in the r i n g R = R ( K ) of all integral functions in an a l g e b r a i c f u n c t i o n field K in one variable X over Fq. Here, the set Pn of prime ideals in R acts as the set of prime elements, and the degree O(I) is defined by qO(Z) = card(R/I). The case K = Fq(x) leads back to Gq, and in general it can be proved that

G#R(n) = Anq ~ + 0 ( 1 ) 18

asn--+ ec,

G# (n) In the next section, a variety of 'natural' examples of semi-groups satisfying axiom Gx for various values of in (0, 1) will be given.

Axiom 02. The concrete examples described below provide a variety of natural illustrations of additive arithmetical semi-groups G with the following property (axiom • ): There exist real constants C > O, q > 1, c~ > 1, depending on G, such that P # (n) ~ Cqnn -~

as n --~ co.

Under these assumptions one has (cf. [3]) an abstract (inverse) prime number theorem: If G is an additive

ABUNDANT NUMBER arithmetical semi-group satisfying axiom ~, then

G#(n) ,,~ CZc(q-1)q~n -~

as n -+ oc,

where ZG(y) : E _0 G#(r)Y r. It follows that if G satisfies axiom ~, then G also satisfies axiom G~, for A = AG = 1/ZG(q-1). The set 5 of all unlabelled (i.e., isomorphism classes of) finite forests forms an additive arithmetical semigroup, whose prime elements are the unlabelled trees. A famous theorem of R. Otter [8] states that the total number T#(n) of unlabelled trees with n vertices satisfies

7"#(n) ~ Coq~n -5/2

as n -+ co,

where Co and q0 are explicitly described positive constants (q0 > 1). E.M. Palmer and A.J. Schwenk [9] estimated the corresponding total number /T#(n) of all unlabelled forests with n vertices. They showed that

Jr#(n) ,,, KoCoq~n -5/2

a s n --+ co,

where K0 > 1 is also an explicitly described constant. This and various other families of trees provide 'natural' examples of Warlimont's axiom G~ as well as axiom @. As considered by P. Hanlon [2], an interval graph is defined mathematically as a finite graph H whose vertices are in one-to-one correspondence with a set of real closed intervals in such a way that two vertices are joined by an edge in H if and only if their corresponding intervals intersect non-trivially. If all the intervals have length one, H is called a unit-interval graph; if H is connected, and no two adjacent vertices have exactly the same set of neighbouring vertices, H is called reduced. Based on the asymptotic estimates of Hanlon [2] one may then deduce that the semi-group 5[ corresponding to all unit-interval graphs satisfies axiom ~. Substantial advances have occurred in recent years (as of 2000) concerning the asymptotic enumeration of the non-isomorphic (combinatorially distinct) convex 3polyhedra (or 3-polytopes). Indeed, let 7)E#(n) denote the total number of combinatorially distinct convex 3-polyhedra with n edges (cf. also P o l y h e d r o n ) . L.B. Richmond and N.C. Wormald [10] showed that

1 4 n_7/2 468x/~

as n -+ cx~.

Soon after that, E.A. Bender and Wormald [1] considered the corresponding numbers 7)v#(n), :P~(n) when n now represents the number of vertices, respectively faces, and derived a similar asymptotic estimate. Let SE, Sv, SF denote the additive arithmetical semigroups which arise from the set S of all combinatorial

equivalence classes of the above special 3-dimensional simplicial complexes. Then it follows from the abstract inverse prime number theorem above that SE, S v and SF are further natural examples of semi-groups satisfying axiom ¢. References

[1] BENDER, E.A., AND WORMALD, N.C.: 'Almost all convex polyhedra are asymmetric', Canad. J. Math. 27 (1985), 854871. [2] HANLON, P.: 'Counting interval graphs', Trans. Amer. Math. Soc. 272 (1982), 383-426. [3] KNOPFMACHER, A., AND KNOPFMACHER, J.: 'Arithmetical semi-groups related to trees and polyhedra', J. Combin. Th. 86 (1999), 85-102. [4] KNOPFMACHER: J.: Abstract analytic number theory, NorthHolland, 1975, Reprinted: Dover, 1990. [5] KNOPFMACHER, J.: 'Arithmetical properties of finite graphs and polynomials', J. Combin. Th. 20 (1976), 205-215. [6] KNOPFMACHER, J.: Analytic arithmetic of algebraic function fields, M. Dekker, 1979. [7] KNOPFMACHER, J.: AND ZHANG, W.-B.: Number theory arising from finite fields, analytic and probabilistic theory, M. Dekker, 2001. [8] OTTER, R.: 'The number of trees', Ann. of Math. 49 (1948), 583-599. [9] PALMER, E.M., AND SCHWENK, A.J.: 'On the number of trees in a random forest', J. Combin. Th. B 27 (1979), 109-121. [10] RICHMOND, L.B., AND WORMALD, N.C.: 'The asymptotic number of convex polyhedra', Trans. Amer. Math. Soc. 2"/3 (1982), 721-735. [11] WARLIMONT, R.: 'A relationship between two sequences and arithmetical semi-groups', Math. Nachr. 164 (1993), 201217.

John Knopfmacher MSC1991: 11N32, 11N45, 11Nxx Let a(n) denote the sum of the distinct divisors of an integer n (cf. Divisor; N u m b e r o f divisors). The integer n is called abundant if e(n) > 2n; deficient if a(n) < 2n; and perfect if a(n) = 2n (cf. also P e r f e c t n u m b e r ) . Note that some authors call a number n abundant if a(n) > 2n. Clearly, these numbers are in fact perfect or abundant (i.e. 'nondeficient') numbers. In [7], L.E. Dickson gives details on the early history of abundant numbers. G. Nicomachus (about 100) separated the even numbers into abundant, deficient and perfect, and dwelled on the ethical importance of the three types. A.M.S. Boethius (around 500), in a Latin exposition of the arithmetic of Nicomaehus, stated that perfect numbers are rare, while abundant ('superfluous') and deficient ('diminutos') numbers are found to an unlimited extent. N. Jordanus (around 1236) stated that every multiple of a perfect or abundant number is abundant. He attempted to prove the erroneous statement that all abundant numbers are even. C. Bovillus (around 1509) corrected this statement, by citing ABUNDANT

NUMBER

-

19

ABUNDANT NUMBER 45045 = 5 • 7- 9 - 11 • 13 and its multiples. Bachet de M6ziriac (around 1600) gave a proof t h a t 2~p is perfect if p = 2 n+l - 1 is a p r i m e n u m b e r , and abundant if p is composite. He remarked t h a t the odd number 945 is abundant. J. Broscius (around 1652) showed t h a t there are only 21 abundant numbers between 10 and 100 and all of them are even; the only odd abundant number less than 1000 is 945. (The statement by E. Lucas (1891) that 33 • 5 • 79 is the smallest odd abundant number is probably a misprint for 945 = 33 -5.7.) Ch. de Neuveglise (1700) proved that the products 3 . 4 , . . . , 8.9 of two consecutive numbers are abundant, and all multiplies of 6 or an abundant number are abundant. J. Struve (1808) considered abundant numbers which are products abc of three distinct prime numbers in ascending order; for a = 2, b = 3, c = 5 or 7, and for a = 2, b = 5, c = 7, abcd is abundant for any prime number d > c. Of the numbers < 1000, 52 are abundant.

be the counting function of primitive a - a b u n d a n t numbers. Erdgs proved t h a t [11]

Dickson (1913, [6]) called a non-deficient number primitive abundant if it is not a multiple of a smaller non-deficient number. He proved that there are only a finite number of primitive non-deficient numbers having a given number of distinct odd prime factors and a given number of factors 2.

for all 1 < m < n. Let Q(x) be the counting function of superabundant numbers. For two consecutive superabundant numbers n, n' they prove that

There is no odd abundant number with fewer than three distinct prime factors, the primitive ones with three are

and this was sharpened to nl/n <_ 1 + 1 / l v ~ n for an infinity of n by J.-L. Nicolas [19]. Alaoglu and Erdgs showed that Q(x) >__ Clogxloglogx/(logloglogx) 2, while Erd6s and Nicolas [12] demonstrated that liminf~_+~logQ(x)/loglogx > 5/48. Alaoglu and Erd6s [1] introduced also the notion of highly abundant number, a number n with the property t h a t a(n) > or(m) for all m < n. For the counting function H(x) of these numbers one has H(x) > (1 - c)(logx) 2 for all e > 0 and large x; if n is highly abundant, then the largest prime factor of n is less than C log n(log log n) 3. Erd6s and Nicolas [12] call a number n cube-flee superabundant if m < n implies a°(m)/m < cr°(n)/n, where a°(p s) = cr(ps) for a < 2 and cr°(p s) = 0 for a > 3 (with p a prime number and a a positive integer). They prove that if n o and n ~° are two consecutive cube-free superabundant numbers, then limsupnl°/n ° >_ 21/4 ,,~ 1, 19. A non-deficient number is called weird by S.J. Benkovski and Erd6s [4] if it is not pseudo-perfect (cf. also P e r f e c t n u m b e r ) . They proved that the density of weird numbers is positive. V. Siva R a m a P r a s a d and D.R. Reddy [20] say that a number n is primitive unitary a-abundant if cr*(n) > a n but a* (d) < a d for all d I n, d < n (a > 2). Here, or*(n) denotes the sum of unitary divisors of n (for these functions, as well as related results, see also [17]). Let Us be the set of these numbers. Then

33.5.7, 32 • 52 • 11,

32.52.7,

35 • 52 • 13,

32.5.72 ,

34 • 52 • 132,

33 • 53 • 132.

He gave also a table of all even abundant numbers < 6232. Dickson's result was a starting point for much further research. In 1949 and 1968, H.N. Shapiro ([23], [24]) proved the following result. Let a be a rational number. A necessary and sufficient condition that there exist infinitely m a n y primitive a-abundant numbers (i.e. a(n)/n >_ a but a(d)/d < a for all d I n, d < n) with k distinct prime factors is that a has a representation a--

ba(a) a~(b)

with GCD(a, b) = 1, b > 1, where co(a)+co(b) < k. Here, is the Euler t o t i e n t f u n c t i o n and w(a) denotes the number of distinct prime factors of a. In 1933, F. Behrend, H. Davenport and S. Chowla [5] showed that the density of non-deficient numbers exists and is positive. This result follows also from a theorem of P. Erd6s [8] stating that the sum of reciprocals of primitive abundant numbers converges. Let

As(x) = card {n G x: n primitive a-abundant} 20

_- o and t h a t [9]

xexp(-8(logxloglogx) 1/2) < A2(x) <

0 This was sharpened successively by A. Ivid [15], with -(v/-6 + z) in place of - 8 and - ( 1 / x / ~ - e) in place of - 1 / 2 5 ; and by M.R. Avidon [2], who considered - ( x / ~ + e) in place of - ( v ~ + e), and - ( 1 - e) in place of - ( 1 / v / ~ - e). L. Alaoglu and Erd6s [1] call a number n superabundant if ~(n) ~(-~) n

n' -

-

n

m

(log log n) 2 log n

lim sup n-+~,nEUc,

,

C=const>0,

- a. n

ACCEPTANCE-REJECTION

Miscellaneous

r e s u l t s . Let a E R . A n u m b e r n is

called a-non-deficient if a ( n ) / n >_ a. B y s h a r p e n i n g a result of O. G r i i n [14], H. Sali6 [22] p r o v e d t h a t t h e least p r i m e factor of e v e r y a - n o n - d e f i c i e n t n u m b e r w i t h m p r i m e factors is less t h a n C ( m log n) 1/~. Ch.R. Wall [28] p r o v e d t h a t t h e r e exist infinitely m a n y a b u n d a n t integers n - a ( m o d b) (with a a n d b given). Let k be fixed. T h e n t h e r e exist k consecutive a b u n d a n t n u m b e r s . T h e r e exist infinitely m a n y sequences of five consecutive deficient n u m b e r s . (See [27].) See [16] for a t a b l e of o d d p r i m i t i v e a b u n d a n t n u m b e r s n with five d i s t i n c t p r i m e factors for which 2<

a(n) 2 <2+-n 1010"

If k _> 8, t h e n u m b e r n = 1 . 3 . 5 . . . see [25].

(2k - 1) is a b u n d a n t ,

For o t h e r s results on deficient, perfect, or r e l a t e d n u m b e r s , see [17], [9], [10], [26], [21]. L. Moser [18] p r o v e d t h a t every integer > 105 can be e x p r e s s e d as t h e s u m of two a b u n d a n t n u m b e r s . A c t u ally, this is valid for integers > 20162, see [3]. For a t a b l e of a b u n d a n t n u m b e r s less t h a n 104, see

[13]. References [1] ALAOCLU,L., AND EaDSS, P.: 'On highly composite and similar numbers', Trans. Amer. Math. Soc. 56 (1944), 448-469. [2] AVIDON, M.R.: 'On the distribution of primitive abundant numbers', Acta Arith. 77 (1996), 195-205. [3] BACH, E., AND SHALLIT, J.: Algorithmic number theory, MIT, 1996, p. 334. [4] BENKOVSKI,S.J., AND ERDOS, P.: 'On weird and pseudoperfect numbers', Math. Comput. 28 (1974), 617 623. [5] DAVENPORT, H.: '/fiber numeri abundantes', Preuss. Akad. Wiss. Sitzungsber 26/29 (1933), 830 837. [6] DICKSON,L.E.: 'Finiteness of odd perfect and primitive abundent numbers with n distinct prime factors', Amer. J. Math. 35 (1913), 413-422. [7] DICKSON, L.E.: History of the theory of numbers, Vol. I (Divisibility and primality), Chelsea, 1919, Reprint: AMS 1999. [8] ERDOS, P.: 'On the density of the abundant numbers', J. London Math. Soc. 9 (1934), 278-282. [9] ERD()S, P.: 'On primitive abundant numbers', J. London Math. Soc. 9 (1935), 49-58. [10] ERD6S, P.: 'Note on consecutive abundant numbers', J. London Math. Soc. 13 (1938), 128 131. [11] ERDOS, P.: 'Remarks on number theory I, On primitive c~abundant numbers', Acta Arith. 5 (1958), 25-33. [12] ERD(JS, P., AND NICOLAS,J.-L.: 'R@artition des nombres superabondantes', Bull. Soc. Math. France 103 (1975), 65-90. [13] GLAISER, J.W.L.: Number-Divisor Tables, British Assoc. Math. Tables, 1940. [14] GRiJN, O.: @ber ungerade vollkommene Zahlen', Math. Z. 55 (1952), 353-354. [15] IVld, A.: 'The distribution of primitive abundant numbers', Studia Sci. Math. Hung. 20 (1985), 183-187. [16] KISHORE, M.: 'Odd integers n with five distinct prime factors for which 2 - 10 -12 < a(n)/n < 2 + 10 -12', Math. Comput. 32 (1978), 303 309.

METHOD

[17] MITRINOVIC,D.S., AND SANDOR, J.: Handbook of number theory, Kluwer Acad. Publ., 1995, In coop. with B. Crstici. [18] MOSER, L.: 'Problem E848', Amer. Math. Monthly 56 (1949), 478. [19] NICOLAS, J.-L.: 'Ordre maximal d'un 616ment du groupe Sn des permutations et 'highly composite numbers", Bull. Soc. Math. France 97 (1969), 129-191. [20] PRASAD, V. SIVA RAMA, AND REDDY, D.R.: 'On primitive unitary abundant numbers', Indian 3. Pure Appl. Math. 21 (1990), 40-44. [21] RIELE, H.J.J. TE: 'A theoretical and computational study of generalized aliquot sequences', Math. Centrum, Amsterdam (1975). [22] SALIg, H.: @ber abundante Zahlen', Math. Nachr. 9 (1953), 217-220. [23] SHAPIaO, H.N.: 'Note on a theorem of Dickson', Bull. Amer. Math. Soc. 55 (1949), 450-452. [24] SHAPIRO,H.N.: 'On primitive abundant numbers', Commun. Pure Appl. Math. 21 (1968), 111-1t8. [25] SIERPINSKI,W.: Teoria liczb, Vol. II, Warsawa, 1959. [26] Si,NDOa, J.: 'On a method of Galambos and K£tai concerning the frequency of deficient numbers', Publ. Math. (Debreeen) 39 (1991), 155 157. [27] WALL, CH.R.: 'Problem, 6356', Amer. Math. Monthly 88 (1981), 623, Solution by L.L. Foster: 90 (1983), 215-216. [28] WALL, CH.R.: 'Problem E3002', Amer. Math. Monthly 90 (1983), 400, Solution by N.J. Fine: 93 (1986), 814. J. Sdndor M S C 1991: l l A x x

ACCEPTANCE-REJECTION Introduced

by J. von N e u m a n n

METHOD

-

[8], this is t h e m o s t

a d a p t a b l e m e t h o d for s a m p l i n g from c o m p l i c a t e d dist r i b u t i o n s (cf. also S a m p l e ; D i s t r i b u t i o n ) . It works as follows. Let f ( x ) be a given p r o b a b i l i t y d e n s i t y (cf. also D e n s i t y o f a p r o b a b i l i t y d i s t r i b u t i o n ) , a n d let h(x) be a f u n c t i o n such t h a t f ( x ) <_ h(x) w i t h i n t h e range of f ( x ) . If t h e i n t e g r a l of h ( z ) over this r a n g e is a finite n u m b e r a , t h e n g(x) = h ( x ) / a is a p r o b a b i l i t y d e n s i t y function, a n d t h e following p r o c e d u r e is valid: 1) Take a r a n d o m s a m p l e X from t h e d i s t r i b u t i o n w i t h p r o b a b i l i t y d e n s i t y g(x) = h ( x ) / a . 2) G e n e r a t e a u n i f o r m r a n d o m d e v i a t e U between zero a n d one. If U <_ f ( X ) / h ( X ) , a c c e p t X as a s a m p l e from t h e d i s t r i b u t i o n f ( x ) . O t h e r w i s e reject X a n d go b a c k to S t e p 1. T h e ease of t h e m e t h o d d e p e n d s on t h e following p r o p e r t i e s of t h e hat f u n c t i o n h(x): A) One has to select a h a t function h(x) from which it is easy to sample. E x a m p l e s are given below. B) T h e p a r a m e t e r s of t h e h a t function have to be det e r m i n e d in such a way t h a t t h e a r e a a below h(x) becomes m i n i m a l . O p t i m a l h a t functions can be c a l c u l a t e d by a n a l y t i cal m e t h o d s . A s s u m e t h a t t h e h a t function touches f ( x ) 21

ACCEPTANCE-REJECTION METHOD at two points L (left) and R (right), where L < R. Furthermore, suppose that the hat function depends on two parameters, called m and s. Thus

f(L) : ag(L; m, s),

f ( R ) = c~g(R; m, s),

(1)

and f(x) <_ ag(x; m, s) for all other x. Since L and R are local maxima of f(x)/g(x; m, s), one has the necessary conditions

f'(L) f(L)

- -

_

f ( L ; m, s) g(L;m,s)

(2)

f'(/~) _ g'(R; m, s)

g'(L; m, s) g(L; m, s) '

(3)

-f'(R) -< f(R)

g'(R; m, s) 9(R; m, s) "

f(L(s)) _ f(R(s)) g(n(s);m(s),s) g(R(s);m(s),s)' -

(4)

This leads to the necessary conditions d ]no~(s)=

ds

-:= . f(L) dL -ctL ln g(L; m,s) ds 4-

d lng(L;m,s)dm~ + ~dl n g ( L ; m , s ) d lna(s)=

ds +

(See also Laplace distribution.) Samples are obtained as X +- m + T s E , where E is a standard exponential deviate and T a random sign ±. The fundamental identity (5) leads to 2s = R - L. is given as

= ~ . f d( R ) -gl~lng(R;m,s)

lng(R;rn, s)-~s +

dR ds

= 0, 4-

l n g ( R ; m , s ) = O.

The first expression in each line is zero, by (2) and (3). Solving both equations for dm/ds, and comparing, yields the fundamental relation

dln g( L; m, s) dln g( R; m, s) = dm ds _ dlng(R;m,s) dlng(L;m,s) dm ds

t~(x) = C,,s (1 + (x--m)2~s2n ]-(~+1)/2 for - e c < x < ec, n > 1, where

(5)

(1), (2), (3) and (5) contain five conditions for finding candidates L, R, m, s and c~.

(See also S t u d e n t d i s t r i b u t i o n . ) The t-family contains the C a u c h y d i s t r i b u t i o n for n = 1 and the n o r m a l d i s t r i b u t i o n for n -+ cc as extreme cases. For n = 2 and n = 3 special sampling methods are available: If U denotes a (0, 1)-uniform randora variable, then X +-- ( U - 1 / 2 ) / ( v / ( U - U2)/2) samples from t2 and values from the t3-distribution can be generated efficiently by the ratio-of-uniforms method of A.J. Kinderman and J.F. Monahan [5], which will be discussed subsequently. The fundamental identity (5) yields s 2 = ( R - m ) ( m - L). As before, L, R, s, m, and a can be determined explicitly. The ratio-of-uniforms method was introduced by Kinderman and Monahan for sampling from a density f(x). First the table mountain-function is constructed: Let a, b and c be real numbers and let k be equal to 1, but k = 2 might be another possible choice for some densities. Then ((c ]k+l -forx E ( - o o , a - c],

= | ( b _ c ] k + 1 forx e [ a - c , a - c + b l ,

Triangular hat functions. i

- "~-~ • -m 82

if if

m-s<x<m, m<x<m+s.

Samples may be obtained as X +-- m + s(U1 + U2 - 1). The fundamental identity (5) leads to s = R - L. With its help all constants L, R, m, s, and a can be determined. 22

f o r x E [a - c + b, cx~].

k \x-a]

Samples from the area below the table mountainfunction are obtained by the transformation

X = a + - - bV - c U1/k ,

Examples.

g(x; m, s) =

n+l F (--~-)

1

Otherwise, the first derivative of ln(f(x)/g(x; m, s)) has to be discussed in detail. Equations (1), (2) and (3) are four equations for the determination of L, R, m, and s. Assuming that L, R and rn can be expressed as functions of s, one has to minimize

a(s) =

for x _< m, forx~m.

g(m m, s)

If L and R are uniquely determined, they should satisfy the sufficient conditions

-f'(L) -< f(n)

g(x;m,s)=

{ lexp (L~) l e x p ( ~ -~)

Student-t hat functions. Its probability density function

and

f(n)

Double exponential hat functions. The double exponential (or Laplace) distribution is given by

Y = U1/k .

The table mountain-function is taken as a hat function for f ( x ) / f , where f = m a x f ( x ) . In this case the fundamental identity leads to f ( L ) = f ( R ) for calculating optimal constants.

Squeeze functions. Step 2 of the acceptance-rejection method can be improved if some lower bound

b(x) <_

=

f(x) h(x)'

for all -

< x <

ACCESSIBILITY FOR GROUPS is known

and easy to calculate. Step 2 then changes to:

2') Generate a uniform random deviate U between zero and one. If U _< b(X), accept X as a sample from the target distribution f(x). If U >_ f(X)/h(X), reject .32 and go back to I. Otherwise accept X. Squeeze functions have been constructed procedures. See [4] for some theory.

in many

References [i] AHRENS, J.H., AND DIETER, U.: 'Computer methods for sampling from gamma, beta, Poisson and binomial distributions', Computing 12 (1974), 223-246. [2] DEVROYE, L.: Non-uniform random variate generation, Springer, 1986. [3] DIETER, U.: 'Optimal acceptance-rejection methods for sampling from various distributions', in P.R. NELSON (ed.): The Frontiers of Statistical Computation, Simulation, ~ Modeling, Amer. Ser. Math. Management Sci., 1987. [4] DIETER, U.: 'Mathematical aspects of various methods for sampling from classical distributions': Proc. 1989 Winter Simulation Conference, 1989, pp. 477-483. [5] KINDERMAN, A.J., AND MONAHAN, J.F.: 'Computer generation of random variables using the ratio of uniform deviates', A C M Trans. Math. Software 3 (1977), 257-260. [6] KNUTH, D.E.: The art of computer programming, third ed., Vol. 2: Seminumerical algorithms, Addison-Wesley, 1998. [7] LEYDOLD, J.: 'Automatic sampling with the ratio-of-uniform method', A C M Trans. Math. Software 26 (2000), 78-88. [8] NEUMANN, J. VON: 'Various techniques used in connection with random digits. Monte Carlo methods', Nat. Bureau Standards 12 (1951), 36-38. [9] STADLOBER, E.: 'Sampling from Poisson, binomial and hypergeometric distributions: Ratio of uniforms as a simple and fast alternative', Math.-Statist. Sekt. 303 Forschungsgesellschaft Yoanneum, Graz, Austria (1989). [10] STADLOBER, E.: 'The ratio of uniforms approach for generating discrete random varates', J. Comput. Appl. Math. 31, no. 1 (1990), 181-189. U. D i e t e r

M S C 1991:62D05 A C C E S S I B I L I T Y FOR GROUPS - Accessibility is concerned with bounding the complexity of decompositions as graphs of groups for a discrete group. In his proof that groups of c o h o m o l o g i c a l d i m e n s i o n one are necessarily free, J.R. Stallings [13] made use of Grushko's theorem [6], which asserts that if a g r o u p G is generated by a subset of cardinality d and decomposes as a non-trivial f r e e p r o d u c t G = Hi*... *Hk, then k < d. In attempting to generalize the above Stallings' theorem to pairs of relative cohomological dimension one, C.T.C. Wall [14] conjectured t h a t there was a similar bound on decompositions as a f i n i t e l y - g e n e r a t e d g r o u p as non-trivial a m a l g a m a t e d free products and HNN-extensions (cf. H N N - e x t e n s i o n ) over finite subgroups [14]. Specifically, Wall defined a group G to be accessible if there is a positive integer d such that any

reduced decomposition as a graph of groups for G with finite edge groups has at most d edges. He conjectured that all finitely-generated groups are accessible. In a torsion-free group (cf. also G r o u p w i t h o u t t o r sion), the only allowed decompositions are free products so it follows from Grushko's theorem that finitelygenerated torsion-free groups are accessible. Despite considerable interest in Wall's conjecture, which motivated much progress in the understanding of splittings of groups, it t o o k until the 1990s before M.J. Dunwoody found a counterexample [3]. In the meantime several i m p o r t a n t classes of groups were shown to be accessible. Using algebraic techniques, P.A. Linnell showed that any finitely-generated group with a bound on the order of its finite subgroups is accessible [8], and, using geometric methods, Dunwoody proved Wall's conjecture for the class of almost finitely-presented groups, which contains as a subclass all finitely-presented groups [2]. Much of the development of the theory of group splittings is parallel to the theory of incompressible embedded surfaces in three-dimensional manifolds (cf. also T h r e e - d l m e n s i o n a l m a n i f o l d ) ; for example, the analogue of Grushko's theorem is Kneser's theorem [7] concerning embedded 7r2-incompressible 2-spheres in compact orientable three-dimensional manifolds (cf. also Kneser theorem). Incompressible surfaces in three-dimensional manifolds are now (as of 2000) largely understood, using the theory of surfaces of least area [9]. Dunwoody's proof of accessibility for finitely-presented groups uses a combinatorial analogue of this idea, which applies to group actions on a simply connected 2-complex, or more generally any 2-complex X such that Hi(X, Z2) = 0. He showed that any splitting of a group G over a finite subgroup gives rise to the existence of a minimal track in any such complex on which G acts. Reduced decompositions as a graph of groups for G with finite edge groups give rise to non-parallel disjoint tracks in the 2complex via a method known as resolution. Dunwoody showed t h a t if G acts co-compactly on the 2-complex, then there is a bound on the number of such disjoint tracks, and this establishes the accessibility of G. Applying the method to the Cayley complex of a finitelypresented group proves accessibility in this case. There have been generalizations of the notion of accessibility to cover decompositions over other classes of subgroups. M. Bestvina and M. Feighn were able to show that for a finitely-presented group G there is a bound on the number of vertices in any reduced decomposition as a graph of groups for G with small edge groups [1]. The technical notion of a 'small subgroup' used here includes the case of splittings over Abelian subgroups. 23

ACCESSIBILITY

FOR

GROUPS

Another approach to accessibility was pioneered by Z. Sela [11]. His definition is most naturally understood using Bass-Serre theory, which gives a duality between decompositions as graphs of groups for a group G and actions of G on trees which have no global fixed point [12]. Sela defines a decomposition as a graph of groups to be k-acylindrical if it is reduced, and no element of G fixes a path of length k + 1 in the corresponding BassSerre tree. He remarks that this notion is satisfied in many natural situations, including small decompositions of torsion-free word hyperbolic groups (cf. also H y p e r b o l i c g r o u p ) . His main result is that there is a bound on the number of vertices in any reduced k-acylindrical splitting of a finitely-generated, torsion-free, freely indecomposable group. The bound depends only on the group and the integer k. He remarks that this notion of accessibility holds for certain groups that are not accessible in the classical sense, but that the bound is not given in terms of other algebraic invariants of the group, unlike the accessibility discussed by Dunwoody and Bestvina and Feighn. There is now (as of 2000) a much more more powerful structure theorem for decompositions of a finitelypresented group over virtually cyclic subgroups, given by the JSJ decomposition. This powerful theorem gives a natural decomposition of any finitely-presented group which encodes all the information about splittings of the group over virtually infinite cyclic subgroups, and generalizes the Jaeo-Shalen-Johansson decomposition theorem for embedded tori in three-dimensional manifolds. It was initially proved in [10] for the class of word hyperbolic groups, while [4] gives a geometric derivation of the JSJ decomposition for finitely-presented groups using tracks in 2-complexes. The theorem asserts that any finitely-presented group G admits a canonical decomposition as a graph of groups in which all edge groups are virtually infinite cyclic, and any virtually cyclic subgroup over which G splits lies in a subgroup isomorphic to a finite extension of a surface group and which is conjugate to a vertex group of the decomposition. References [1] BESTVIAN,M., AND FEIGHN, M.: 'Bounding the complexity of simplicial group actions on trees', Invent. Math. 103 (1991), 449-469. [2] DUNWOODY, M.J.: 'The accessibility of finitely presented groups', Invent. Math. 81 (1985), 449-457. [3] DUNWOODY, M.J.: 'An inaccessible group', in G.A. NmLO AND M.A. ROLLER (eds.): Geometric Group Theory I, Vol. 181 of London Math. Soc. Lecture Notes, 1993, pp. 7578. [4] DUNWOODY, M.J., AND SACEEV, M.E.: 'JSJ-splittings for finitely presented groups over slender groups', Invent. Math. 135, no. 1 (1999), 25-44.

24

[5] GROMOV, M.: 'Hyperbolic groups', in S.M. GERSTEN (ed.): Essays in Group Theory, Vol. 8 of Math. Sci. Res. Inst. Publ., Springer, 1987, pp. 75 263. [6] GRUSHKO, I.A.: @ber des Basen eines freien Produktes yon Gruppen', Mat. Sb. 8 (1940), 169-182. [7] KNESER, H.: 'Geschlossene Fl~chen in dreidimensionalen Mannigfaltigkeiten', Jahresber. Deutseh. Math. Verein. 38

(1929), 248-260. [8] LINNELL, P.A.: 'On accessibility of groups.', J. Pure Appl. Algebra 30 (1983), 39-46. [9] MEEKS, W., AND YAU, S.T.: 'Topology of 3-dimensional manifolds and the embedding problems in minimal surface theory', Ann. of Math. 112 (1980), 441-485. [10] RIPS, E., AND SELA, Z.: 'Cyclic splittings of finitely presented groups and the canonical JSJ decomposition', Ann. of Math. (2) 146, no. 1 (1997), 53-109. [11] SELA,Z.: 'Acylindrical accessibility for groups', Invent. Math. 129, no. 3 (1997), 527-565. [12] SERRE, J.-P.: Trees, Springer, 1980. (Translated from the French.) [13] STALLINGS,J.R.: 'On torsion free groups with infinitely many ends', Ann. of Math. 88 (1968), 312-334. [14] ~VALL,C.T.C.: 'Pairs of relative cohomological dimension 1', J. Pure Appl. Algebra 1 (1971), 141-154. Graham A. Niblo

MSC 1991: 20Jxx, 20E22, 57Mxx ACNODE - An older term, hardly used nowadays (2000), for an isolated point, or hermit point, of a plane algebraic curve (cf. also A l g e b r a i c c u r v e ) .

For instance, the point (0, 0) is an acnode of the curve X 3 + X 2 + Y 2 = 0 i n R 2. References [1] WALKER,R.J.: Algebraic curves, Princeton Univ. Press, 1950, Reprint: Dover 1962. M. Hazewinkel

MSC 1991: 14Hxx ADDITIVE BASIS for the natural numbers - A set B of non-negative integers is called an additive basis of order h if every non-negative integer may be written in at least one way as a sum bl + " " + bh, with all bi C B. One is usually only interested to represent in such a way all sufficiently large positive integers and speaks then of an asymptotic additive basis. For example, the set of

AFFINE

squares is an asymptotic additive basis of order 4 (Lagrange's theorem). Most of the results about a s y m p t o t i c additive bases deal with the existence of 'thin' bases, for various meanings of the word thin. For a given set B C_ N one writes rB,h(z) for the number of representations of the non-negative integer z as a sum of h terms from B, the order of addition being insignificant. A well-known theorem of P. ErdSs [1] claims that there is an asymptotic basis B with C1 l o g x ~ rB,2(x ) ~ C2 l o g x ,

(1)

when x is sufficiently large. It is an open problem (as of 2000) if one can have a basis with rB,2 (x) = O(1OgX). The famous Erd6s-Turdn conjecture states that, for any asymptotic basis B of order 2, the sequence rm2(x), x E N, cannot be bounded. Erd6s' construction is probabilistic. He shows that if one takes the integer x to be in B with probability K ( l o g ( x ) / z ) 1/2, for a sufficiently large positive constant K , then with probability 1 the set B is an asymptotic basis satisfying (1) for suitable constants C1 and C2, for all but finitely many positive integers x. See also [5] for a modified probabilistic construction which can be derandomized to yield a polynomial-time algorithm for the determination of such a basis up to any desired integer ?%.

ErdSs' result was subsequently extended [2] to bases of order h > 2, where the obstacle of having the random variables rg,h(X) no more independent for different x was removed with the help of Janson's inequality [3]. Janson's inequality has also been used by J. Spencer [6] to prove that one m a y take a small subset A of the squares, of size IA n [1, x]l _< Cx ~/4 logx which is still an asymptotic additive basis of order 4. Another measure for a basis being small is that of minimality. An asymptotic additive basis of order h is called minimal if it has no proper subset that is an asymptotic additive basis of the same order. Such bases are known to exist for all h. For example, in [4] it is proved that for every h and every ct < 1/h there exists a minimal asymptotic basis of order h and with counting function of the order x ~. References

[1] EaD6s, P.: 'Problems and results in additive number theory':

Colloque sur la Theorid des Nombres (CBRM, Bruxelles), G. Thone, 1955, pp. 255-259. [2] ERD6S, P., AND TETALI, P.: 'Representations of integers as the sum of k terms', Random Struct. Algor. 1, no. 3 (1990), 245-261. [3] JANSON, S.: 'Poisson approximation for large deviations', Random Struct. Algor. 1, no. 2 (1990), 221-229. [4] JIA, X.-D., AND NATHANSON, M.B.: 'A simple construction for minimal asymptotic bases', Acta Arith. 52, no. 2 (1989), 95-101.

DESIGN

[5] KOLOUNTZAKIS,M.: ~An effective additive basis for the integers', Discr. Math. 145 (1995), 1-3; 307-313. [6] SPENCER, J.: 'Four squares with few squares': Number Theory (New York, 1991-1995), Springer, 1996, pp. 295-297.

Mihail N. Kolountzakis M S C 1991: l l P x x AFFINE D E S I G N - Let 7P = (V,/3) be a resolvable t - (v, k, A)-design (see T a c t i c a l c o n f i g u r a t i o n ) , that is, the block set of 7P is partitioned into parallel classes each of which in turn partitions the point set V. 7) is called affine, or affine resolvable, if there exists a constant # such that any two non-parallel blocks intersect in exactly # points. For proofs of the results stated below, see [1]. The affine l-designs are precisely the nets, see N e t (in f i n i t e g e o m e t r y ) , and the affine 3-designs coincide with the Hadamard 3-designs, t h a t is, the 3 - ( 4 # , 2#, # 1)-designs, cf. T a c t i c a l c o n f i g u r a t i o n . There are no non-trivial affine t-designs with t _> 4. Thus, the most interesting case is t h a t of affine 2-designs, which are often simply called affine designs. Any affine l-design satisfies the inequality r _< ( s 2 # - 1 ) / ( # - 1), where r denotes the number of blocks through a point and where s denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the l-design is an (afi:ine) 2-design. Any resolvable 2-design satisfies the inequality r _> k + A, and equality holds if and only the design is affine. In this case, all parameters of D may be written in terms of the two parameters s and #, as follows:

k = s#,

v = s2#,

A-

s##-I

I

r'

s2#- I p-1

'

and the design is denoted by A~(s). The outstanding problem in this area is to characterize the possible pairs (s, #) for which an A~ (s) exists. The only known pairs to date (2001) are those with s = 2 and the pairs of the form (q, q~-2) for some prime power q and some integer d _> 2. The case s = 2 corresponds to Hadamard 2 -designs, i.e. 2 - ( 4 # - 1, 2 # - 1, # - 1)-designs; any such design extends uniquely to a H a d a m a r d 3design, and existence - - which is equivalent to that of an H a d a m a r d m a t r i x of order 4# - - is conjectured for all values of #. The classical examples for the second case are the affine designs AG~_I (d, q) formed by the points and hyperplanes of the d-dimensional finite affine spaces AG(d, q) over the G a l o i s field GF(q) of order q (so q is a prime power here; cf. also Affine s p a c e ) . As to the case d = 2, a design A1 (s) is just an affine plane of order s, see also P l a n e . In general, an affine design cannot be characterized just by its parameters. For instance, the number of non-isomorphic designs with the same parameters as 25

AFFINE DESIGN

AGd-1 (d, q) grows exponentially with a growth rate of at least e k'lnk, where k = qd-1. Hence, it is desirable to characterize the designs A G d - l ( d , q) among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design 7) with )~ > I and s > 2 in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some AGd-1 (d, q); the same conclusion holds if 7) admits an automorphism group which is transitive on ordered triples of non-collinear points. See [1, Sec. XII.3] for proofs and further characterizations. In particular, there is a wealth of results characterizing the classical afflne planes AG(2, q) and other interesting classes of affine planes; for example, a result of Y. Hiramine [2] states that any finite affine plane that admits a collineation group acting primitively on points is a translation plane (cf. P l a n e ; P r i m i t i v e g r o u p o f p e r m u t a t i o n s ) . Detailed studies of translation planes may be found in [3] and [4].

References [1] BETH, T., JUNONICKEL, D., AND LEHZ, H.: Design theory, second ed., Cambridge Univ. Press, 1999. [2] HIRAMINE, Y.: 'Affine planes with primitive eollineation groups', J. Algebra 128 (1990), 366 383. [3] KALLAHER, M.J.: Afflne planes with transitive collineation groups, North-Holland, 1981. [4] Li)NEBURG, H.: Translation planes, Springer, 1980.

Consider now the following polynomial expressions in the parameter z:

P=Poz+P1

:=

z+

,

(3)

O(n) := Qoz n + OlZn-1 ... On, where the Qi are slu-valued functions depending on the variables x, t = (tn), and Q0 = P0 and Q1 = P1- For these data the zero-curvature equations read

Otn

-

Q(n)

+ [P' Q(~)] = 0 ¢~

0 ] =0, ~ [0 ~_p,_g<_Q(n)

(4)

which is an infinite tower of equations extending the system (1). The system (4) generalizes from SL2(C) to a general simple complex Lie a l g e b r a g, a regular element P0 in a C a r t a n s u b a l g e b r a b of g and an element Q0 in ~, see [4] and [10]. Solutions of the equations (4) can be obtained by the Zakharov-Shabat dressing method (cf. also S o l i t o n ) . Namely, consider the function

¢(x,t,z)

=

xPoz+

= exp

(5)

Qoz r

g(z)"

rzl

Dieter Jungnickel • exp

MSC 1991:05B30 AKNS-tIIERARCHY, Ablowitz-Kaup-NeweUSegur hierarchy An infinite tower of non-linear evolution equations that derives its name from the simplest non-trivial system of equations contained in it, the

-xPoz-

Qoz ~ ,

with g(z) belonging to the loop group C ~ (S 1 , SL2 (C)). If this ¢ factorizes as ¢ = ¢ _ ¢ + , with

¢ _ ( x , t , z ) = exp

X,i(x,t)z -~

,

(6)

AKNS-equations i ° q ( x , t ) =iqt = - ½ q ~ +q2r, - qr 2. i-~r(x,t) = ir~ = ~rxx ~

then conjugating the trivial connections (ef. also C o n n e c t i o n s o n a m a n i f o l d ) ( O / O x ) - Poz and ( O / O t ~ ) Qoz ~ with ¢_ gives connections of the required form:

~-Poz

partial, discontinuous initial (boundary) condit i o n s ) could be solved with the inverse scattering transform (cf. also K o r t e w e g - d e Vries e q u a t i o n ) . To get a natural embedding of the AKNS-equations in a larger system, one rewrites (1) in zero-curvature form as 0 2 + [P1,02] = 0,

(2)

where

"~ = 26

'

Q~ = \ - g r ~

~q,')"

,

(S)

It were M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur who showed that the initial value problem of this system of equations (cf. also D i f f e r e n t i a l equation,

bTel -

j(x,t)z j

¢+ = exp

¢ - = o-7 -

(~--1 co

Ot---~ -

(9 Q°zn¢-

-

{~(n)

Ot~

Since flatness is preserved by this procedure, this leads to solutions of (4). If Q0 = To, as in the AKNS-case, one can take just as well 8/cOx = O/cOtl. It was observed by H. Flaschka, Newell and T. Ratiu [5] that the equations (4) for the SL2(C)-case could be captured in the system

~.Q

= [Q(")'Q]'

'~ -> ~'

(7)

AKNS-HIERARCHY for the single series

Q=

(cf. also K a c - M o o d y

Qj_j

Qj=

hj

fj

j=0

(S)

-hy

'

n Q(n) = E QJzn-J" j=0

They showed that these equations are commuting Hamiltonian flows (cf. also H a m i l t o n i a n s y s t e m ) on the Lie algebra ~ i ~ 0 Xi z-i, Xi E sl2(C), with respect to natural P o i s s o n b r a c k e t s . Further, they introduced the flux tensor Fjk by

= =

(j

-

r)Q Qk+j_

+

(9) - k)Q Qk+j_

and proved the local conservation laws of the system, namely 0 (10) Otk Fij = ~tiFJk. The left-hand side of (10) is, in fact, even symmetric under permutations of the indices i, j, k and this property made them introduce a potential ~- by 0

0

Fjk -- Otj Otk log(~-).

s=EcAi(o1

0I}®ECA-i(O

i~0

i~0

1 0l)®Cc,

where c is the central element of A~1) that is in the kernel of the projection of A~1) onto the loop algebra of SL2 (C). The A~l)-module L(Ao) decomposes with respect to the action of the homogeneous Heisenberg algebra s as a direct sum of irreducible s-modules C[t] = C[tl, t2,...] labelled by the root lattice. Thus, one can write each element of L(A0) as T(t) = (~'l(t))l~Z. The group orbit of the highest weight vector can then be characterized by a set of so-called Hirota bilinear relations for the components (Tz). By using the representation theory one constructs a series of elements (gl) in a suitable completion of the Kac-Moody group associated with A~1) such that the vacuum expectation value of gl is exactly ~-l. The Birkhoff decomposition of the (gl) in that group then enables one to construct solutions of the lattice zerocurvature equations, [2]. In particular, the operators p(5 from (12) obtained in this way can be expressed in the components (7-/) by

(11)

The equations (7) are called the Lax equations of the AKNS-hierarchy. As such, the AKNS-hierarchy is a natural reduction of the two-component KP-hierarchy (cf. also K P - e q u a t l o n ; [7]), a fact that enables a description in the Grassmannian of that hierarchy. It was shown by M.J. Bergvelt and A.P.E. ten Kroode [1] that it is natural to consider a system of zerocurvature relations (4) labelled by the root lattice of the Lie algebra, where the operators at different sites of the lattice are linked by Toda-type differential-difference equations. For example, for nearest neighbour sites there holds q(l+l) = _(q(l))2r(Z ) ~_ q(I) log(q(1)),

a l g e b r a ) . In A~1) one takes the

homogeneous Heisenberg algebra

r(l+l) =

1 q(Z)' (12)

where

This phenomenon is due to the fact that there is a natural lattice group that commutes with the commuting flows corresponding to the parameters tn. In the representation-theoretic approach to soliton equations (see [3], [8]), the soliton equations occur as the equations describing the group orbit of the highest weight vector. A similar description holds for these combined differential-difference equations. Let L(A0) be the basic representation of the Kac-Moody Lie algebra A~1)

q(5 = 2i Tl+l , Tl

r (0 = - 2 i T1----L Tl

(13)

By using the adjoint action of the Kac-Moody group, Bergvelt and ten Kroode also showed [2] that the 0

0

r(~) "- Otj Or, log(vl)

(14)

give exactly the flux tensor from (9), thus furnishing a representation-theoretic basis for the results in [5]. A geometric way to look at T-functions, see e.g. [9], is to consider a h o m o g e n e o u s space over the relevant loop group L, a holomorphic line b u n d l e £ over this space and its pull-back over the corresponding central extension [, of L. If this last line bundle has a global holomorphic section, then T-functions measure the failure of equivariance of partial liftings from L to L with respect to this section. With this point of view, one can also arrive at the formulas in (13) by lifting the discrete group of transformations that commute with the flows from (6) appropriately, see [11]. An important class of equations associated with the AKNS-hierarchy are the so-called stationary AKNSequations. These are the differential equations for the functions q and r from the first-order differential operator

+

,

(15)

27

AKNS-HIERARCHY resulting from the existence of a (2 x 2)-matrix-valued differential operator n+l ~n+l :

(d)

i

E ~i i=0

that commutes with L. Such a pair is naturally associated with a h y p e r - e l l i p t i c c u r v e of genus n and that is why one calls P1 an algebro-geometric A K N S - p o t e n t i a l . The elliptic algebro-geometric AKNS-potentials have been characterized in [6]. They correspond exactly to the potentials for which the equation L ( ¢ ) = z ¢ has a meromorphic f u n d a m e n t a l s y s t e m o f s o l u t i o n s with respect to x for all values of the spectral parameter z E C. References [I] BERGVELT, M.J., AND }(ROODE, A.P.E. TEN: 'Differentialdifference AKNS equations and homogeneous Heisenberg algebras', Y. Math. Phys. 28 (1987), 302-306. [2] BERGVELT, M.J., AND KROODE, A.P.E. TEN: 'T-functions and zero curvature equations of Tod~AKNS type', Y. Math. Phys. 29 (1988), 1308-1320. [3] DATE, E., JIMBO, M., KASHIWARA, M., AND MIWA, T.: 'Transformation groups for soliton equations': Non-linear Integrable Systems; Classical Theory and Quantum Theory (Proc. R I M S Symp.), World Sci., 1983, pp. 41-119. [4] DRINFEL'D, V.G., AND SOKOLOV, V.V.: 'Lie algebras and equations of Korteweg de Vries type', Itogi Nauki i Tekhn. Ser. Sovrem. Probl. Mat. 24 (1984), 81-180. (In Russian.) [5] FLASCHKA,H., NEWELL, A.C., AND RATIU, W.: 'Kac-Moody Lie algebras and soliton equations II. Lax equations associated with A~l)', Physica 9 D (1983), 300 323. [6] GESZTESY, F., AND WEIKARD, R.: 'A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy', Aeta Math. 181 (1998), 63-108. [7] HELMINCK, G.F., AND POST, G.F.: 'A convergent framework for the multicomponent KP-hierarchy', Trans. Amer. Math. Soc. 324, no. 1 (1991), 271-292. [8] KAC, V.G.: Infinite dimensional Lie algebras, third ed., Cambridge Univ. Press, 1989. [9] SEGAL, G., AND WILSON, G.: 'Loop groups and equations of KdV type', Publ. Math. I H E S 63 (1985), 1 64. [10] WILSON, G.: 'The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras', Ergo& Th. Dynam. Syst. 1 (1981), 361-380. [11] WILSON, G.: 'The v-functions of the AKNS equations': Integrable Systems: the Verdict Memorial Conf. (Actes Colloq. Internat. Luminy), Vol. 115 of Progress in Math., 1993, pp. 131-145.

G.F. Helminck

MSC 1991: 22E65, 22E70, 35Q53, 35Q58, 58F07

existence of a single preserved distance for some mapping f implies that f is an isometry (cf. [1]).

Even if X, Y are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking r = 1 is no loss of generality). For example, the following question has not been solved yet (as of 2000): Is a mapping f from R 2 to R 3 preserving unit distance necessarily an isometry (cf. [18])7 The discussion of Aleksandrov's problem under certain additional conditions on the given mapping preserving unit distance has led to several interesting and new problems (cf. [13], [15], [16], [17], [IS]). The Aleksandrov problem has been solved for Euclidean spaces X = Y = Rn: For 2 _~ n < ce, the answer is positive [2], while for n = 1, ee, the answer is negative [2], [5], [14]. In general normed vector spaces, the answer is positive for a mapping that is contractive, surjective and preserves the unit distance. The problem also has a positive solution when X and Y are strictly convex vector spaces, provided f is a h o m e o m o r p h i s m and the dimension of X is greater than 2 (cf. [12], [20]). The Aleksandrov problem has also been solved in some special cases of mappings f : X -+ Y which preserve two distances with an integer ratio greater than 1, with X and Y strictly convex vector spaces and the dimension of X greater than 1 (cf. [3], [16], [18]). Furthermore, when X, Y are Hilbert spaces and the dimension of X is greater than 1, a lot of work has been done (cf. [21], [22], [23]; for example, when f preserves the two distances 1 and v~; when dim X > 3 and f preserves 1 and v/-2, then f is an affine isometry, etc.). Problems connected with stability of isometrics as well as non-linear perturbations of isometrics have been extensively studied in [4], [5], [6], [7], [8], [10], [9], [11]. References

[1] ALEKSANDROV, A.D.: 'Mapping of families of sets', Soviet Math. Dokl. 11 (1970), 116-120. [2] BECKMAN, F.S., AND QUARLES, D.A.: 'On isometrics of Euclidean spaces', Proc. Amer. Math. Soc. 4 (1953), 810-815. [3] BENZ, W., AND BERENS, H.: 'A contribution to a theorem of Ulam-Mazur', Aquat. Math. 34 (1987), 61-63. [4] BOURCAIN, D.G.: 'Approximate isometrics', Bull. Amer. Math. Soc. 52 (1946), 704-714. [5] CIESIELSKI, K., AND RASSIAS, TH.M.: 'On some properties of isometric mappings', Facta Univ. Set. Math. Inform. 7

(1992), 107-115.

ALEKSANDROV MAPPINGS - Let X, spective distances dl, mapping f : X --+ Y

PROBLEM FOR ISOMETRIC Y be two metric spaces, with red2 (cf. also M e t r i c space). A is defined to be an i s o m e t r y if d 2 ( f ( x ) , f ( y ) ) = d l ( x , y ) for all x , y E X. A mapping f : X --+ Y is said to preserve the distance r if for all x , y e X with d l ( x , y ) = r one has d 2 ( f ( x ) , f ( y ) ) = r. A.D. Aleksandrov has posed the problem whether the 28

[6] DOLINAR, G.: 'Generalized stability of isometrics', J. Math. Anal. Appl. 202 (2000), 39-56. [7] GEVIRTZ, J.: 'Stability of isometrics on Banach spaces', Proc. Amer. Math. Soc. 89 (1983), 633-636. [8] GRUBER, P.M.: 'Stability of isometries', Trans. Amer. Math. Soe. 245 (1978), 263-277. [9] HYERS, D.H., AND MAZUR, S.M.: 'On approximate isometries', Bull. Amer. Math. Soc. 51 (1945), 288-292. [10] LINDENSTRAUSS,J., AND SZANKOWSKI, A.: 'Non linear perturbations of isometrics', Astdrisque 131 (1985), 357-371.

ALGEBRAIC HOMOTOPY [ii] MASTIR, S.M., AND ULAM, S.: 'Sur les transformations isom~triques d'espaces vectoriels norm~s', C.R. Acad. Sei. Paris 194 (1932), 946 948. [12] MIELNIE, B., AND RASSIAS, TH.M.: 'On the Aleksandrov problem of conservative distances', Proc. Amer. Math. Soc. 116 (1992), 1115-1118. [13] RASSIAS, TH.M.: 'Is a distance one preserving maping between metric space always an isometry?', Amer. Math. Monthly 90 (1983), 200. [14] RASSIAS, TH.M.: 'Some remarks on isometric mappings', Facta Univ. Set. Math. Inform. 2 (1987), 49-52. [15] RASSIAS, TH.M.: 'The stability of linear mappings and some problems on isometries': Proc. Internat. Conf. Math. Anal. Appl. Kuwait, 1985, Pergamon, 1988, pp. 175-184. [16] RASSIAS, TH.M.: 'Mappings t h a t preserve unit distance', Indian J. Math. 32 (1990), 275-278. [17] RASSIAS, TH.M.: 'Remarks and problems', Aequat. Math. 39 (1990), 304. [18] RASSIAS, TH.M.: 'Properties of isometries and approximate isometries': Recent Progress in Inequalities, Kluwer Acad. Publ., 1998, pp. 325-345. [19] RASSrAS, TH.M.: 'Remarks and problems', Aequat. Math. 56 (1998), 304-306. [20] RASSIAS, TH.M., AND SEMRL, P.: 'On the Masur-Ulam theorem and the Aleksandrov probiem for unit distance preserving mapping', Proc. Amer. Math. Soc. 118 (1993), 919-925. [21] RASSIAS, TH.M., AND XIANG, SHUHUANG: 'On mappings with conservative distance and the M a z u r - U l a m theorem', Publ. EPT. t o a p p e a r (2000). [22] XIANG, SHUHUANG: 'Aleksandrov problem and mappings which preserves distances': Funct. Equations and Inequalities, Kluwer Acad. Publ., 2000. [23] XIANG, SHUHUANG: 'Mappings of conservative distances and the Mazur-Ulam theorem', J. Math. Anal. Appl. 254 (2001), 262-274.

Shuhuang Xiang MSC 1991:54E35 ALEXANDER-CONWAY POLYNOMIAL - The normalized version of the Alexander polynomial (cf. also A l e x a n d e r invariants). It satisfies the Conway skein relation (cf. also C o n w a y skein triple) A L + -- A L _ ~ Z A L o

and the initial condition AT1 = 1, where T1 is the trivial knot (cf. also K n o t t h e o r y ) . For z = V ~ - 1/v/t one gets the original Alexander polynomial (defined only up to ± t i ) . References [1] ALEXANDER, J.W.: 'Topological invariants of knots and links', Trans. Amer. Math. Soc. 30 (1928), 275-306. [2] CONWAY, J.H.: ' A n enumeration of knots and links', in J. LEECH (ed.): Computational problems in abstract algebra, Pergamon, 1969, pp. 329-358. [3] KAUFFMAN, L.H.: 'The Conway polynomial', Topology 20, no. 1 (1981), 101-108.

Jozef Przytycki MSC 1991:57M25

ALEXANDER

T H E O R E M O N B R A I D S - Every link

has a closed braid presentation (cf. also B r a i d t h e o r y ; Link).

This result, published by J.W. Alexander in 1923, allows one to study knots and links using the theory of braids, [1] (cf. also K n o t t h e o r y ) . Alexander's theorem has its roots in Brunn's result (1897) that every knot has a projection with only one multiple point (it is usually not a regular projection) [2]. The smallest number of braid strings used in the presentation is called the braid index of the link. References [1] ALEXANDER, J.W.: 'A lemma on systems of knotted curves', Proc. Nat. Acad. Sci. USA 9 (1923), 93-95. [2] BRUNN, H.K.: 'l~lber verknotete Kurven': Verh. Math. Kongr. Ziirich, 1897, pp. 256-259.

Jozef Przytycki MSC 1991:57M25 A L G E B R A I C H O M O T O P Y - It was J.H.C. Whitehead who introduced the term 'algebraic homotopy' in his address to the 1950 International Congress of Mathematicians. For him the main problems were those of describing, understanding and calculating homotopy types of spaces and homotopy classes of mappings. His work shows he attacked of this problem through several special cases.

i) Studying spaces with cells in a small range of dimensions, for example r-connected n-dimensional spaces. ii) Stabilizing. iii) Seeking higher-dimensional analogues of methods of combinatorial group theory. Later workers would also see algebraic homotopy as including: iv) using general notions of homotopy in a variety of algebraic contexts. See [8] for examples of i), ii) and iv), of algebraic models of homotopy types, and of many modes of study, for example that of rational homotopy theory. A further aspect is that 'deformation' methods are essential in a variety of subjects for the purposes of classification, and the comparison of these deformations, or homotopies, in different subjects has led to new techniques and results [1]. Important aspects of the aims of algebraic homotopy, however, are nearer to that of iii). It was early found that groups give algebraic models of pointed connected CWcomplexes X with zero homotopy groups above dimension one (so-called I-co-connected spaces; cf. also C W complex). Whitehead found the concept of c r o s s e d m o d u l e and, with S. MacLane, that these describe 2co-connected spaces. Since crossed modules occur in a 29

ALGEBRAIC H O M O T O P Y variety of algebraic contexts, for example in the cohomology of groups, this suggested connections between homotopical ideas and wider contexts, and also that non-Abelian concepts lay behind homotopy theory. The generalization of crossed modules to crossed complexes (cf. also C r o s s e d c o m p l e x ) has also been important as giving a 'linear' model of a class of non-simply connected spaces, in which techniques analogous to those of the Abelian chain complexes can be applied. Whitehead also indicates the use of these crossed complexes in simple homotopy theory (cf. also H o m o t o p y ) , which is an exploration of the use of elementary moves, analogous to the Tietze transformations of group presentation theory, but in all dimensions (see [1] for a generalization of results in [11] using crossed complexes). H.-J. Baues has also developed quadratic versions of crossed modules (see [8, Baues' article]), while crossed n-cubes of groups [6], [10] are equivalent to catn-groups and so also give a general model of homotopy (n + 1)-types. J.-L. Loday, starting from the problem of the obstruction to excision in a l g e b r a i c K - t h e o r y , introduced in [9] the notion of a catn-group, which is also described as an n-fold g r o u p o l d internal to the category of groups, and showed that they describe all homotopy (n + 1)types. While these structures are related to simplicial groups [10], their use in homotopy theory can come under a general scheme, as follows. Here, II is a f u n c t o r defined geometrically in terms of homotopy classes of mappings, B and B are 'classifying space functors', and U is a forgetful functor. One seeks such schemes in which H preserves certain useful co-limits, so that some calculations are possible; II o B is naturally equivalent to 1, giving a match of the two forms of data; and some transformation from 1 to B o II showing how much the algebraic data models the topological data: topological data -~ algebraic data topological spaces There is a kind of algebraic model of all CW-spaces, namely simplicial groups and simplicial groupoids, and these form a very important tool in algebraic topology and algebraic homotopy. However it is often difficult to obtain from them specific calculations. In particular, the above scheme does not apply to simplicial groups as algebraic data, because of the lack of a functor H preserving certain useful co-limits. The scheme certainly applies to filtered spaces and crossed complexes, and to n-cubes of pointed spaces and cat~-groups. In both cases this has led to specific calculations of homotopy types. Possibly more importantly, calculations of special cases of co-limits of these structures have led to new algebraic 30

tools, such as a non-Abelian tensor product of groups [5]. In any case, this scheme is related to aims of the algebraic topology of the early 20th century, to find and use in geometric situations higher-dimensional analogues of the non-Abelian f u n d a m e n t a l g r o u p . Such an aim was long thought a mirage, but now (2000) one can see that functors to forms of multiple groupoids can play such a role. It is also interesting to see that cubical tools developed by R. Brown and P.J. Higgins in the late 1970s and related to crossed complexes are now being used in concurrency theory [7]. This shows interesting analogies between paths in spaces and processes in computer science, and the applicability of a range of homotopical notions. It follows from Loday's results that a surprisingly complete and simple to write down description of homotopy n-types is in terms of n-fold groupoids, described inductively as groupoids for n = 1 and otherwise as internal groupoids in the category of (n - 1)-fold groupoids. These have a c l a s s i f y i n g s p a c e obtained by using n-simplicial sets. The wider implications of this need much more study. Thus, tools developed specifically with the aim of problems in homotopy theory have been found relevant more widely. This is analogous to the way h o m o l o g ical a l g e b r a developed from the homology and cohomology of groups to a widespread tool. In this sense, algebraic homotopy is also seen as related to or even another term for non-Abelian homological algebra, yielding rather general algebraic structures available as coefficients, or for analyzing a wide range of geometric, combinatorial or algebraic situations. As an example of this, see [3]. In [2] it is shown how the non-Abelian methods of algebraic homotopy have some impact on generalized Galois theories, for example in relation to central extensions of groups. References [1] BAUES, H.-J.: Algebraic homotopy, Cambridge Univ. Press, 1989. [2] BOaCEUX, F., AND JANELIDZE, G.: Galois theories, Cambridge Univ. Press, 2001. [3] BREEN, L.: Braided n-Catgories and a-Structures in Higher Category Theory (Evanston, Ill. 1997), Amer. Math. Soc., 1998, pp. 59-81. [4] BROWN, R.: 'Higher dimensional group theory', web article (1998), www.bangor.ac.uk/~mas010/hdaweb2.ht mh [5] BROWN, R., AND LODAY, J.-L.: 'Van Kampen theorems for diagrams of spaces', Topology 26 (1987), 311-334. [6] ELLIS, G.J., AND STEINER, e.: 'Higher-dimensional crossed modules and the homotopy groups of (n + 1)-ads', J. Pure Appl. Algebra 46 (1987), 117-136.

ALGEBRAIC LOGIC [7] GAUCHER, P.: 'Homotopy theory of higher dimensional categories and concurrency in computer science', Math. Struct. Computer Sci. 10 (2000), 481-524. [8] JAMES, I.M. (ed.): Handbook of algebraic topology, Elsevier, 1995. [9] LODAY, J.-L.: 'Spaces with finitely many homotopy groups', J. Pure Appl. Algebra 24 (1982), 179 202. [10] PORTER, T.: 'n-types of simplicial groups and crossed ncubes', Topology 32 (1993), 5-24. [11] WHITEHEAD, J.H.C.: 'Simple homotopy type', Amer. J. Math. 72 (1950), 1-57. R. Brown

MSC1991: 55Pxx, 55P15, 55U35 ALGEBRAIC LOGIC - Algebraic logic can be divided into two major parts: abstract (or universal) algebraic logic and 'concrete' algebraic logic (or algebras of relations of various ranks). Both are discussed below. A b s t r a c t a l g e b r a i c logic. This branch of algebraic logic is built around a duality theory which associates, roughly speaking, quasi-varieties of algebras to logical systems (logics for short) and vice versa. After the duality theory is elaborated, characterization theorems follow, characterizing distinguished logical properties of a logic L in terms of natural algebraic properties of the algebraic counterpart Alg(L) of L. A logic is, usually, a tuple 12 = (Fme, ModE, ~ e , mngc, Fz:) , where Fm is the set of formulas of 12, Mod is the class of models of 12, ~ L C Mod x Fm is the validity relation, rang: Mod x Fm -+ Sets is the semantical meaning (or denotation) function of £, and ~- is the syntactical provability relation of 12. More generally, a general logic consists of a class VocL of vocabularies and then to each vocabulary w E Vocz;, g associates a logic, i.e. a 5-tuple 12(w) = (FmT, ModT, ~ r , mngr, Fr) as indicated above. As an example, first-order logic is a general logic in the sense that to any collection of predicate symbols it associates a concrete first-order language built up from those predicate symbols (i.e. that vocabulary; cf. also M a t h e m a t -

ical logic). Of course, there are some conditions which logics and general logics have to satisfy, otherwise any 'crazy' odd 5-tuple would count as a logic, which one wants to avoid. (E.g., one assumes that if F Fc ¢ and F _C A, then A F c ¢, for F, A C_ Fmc.) For such conditions on logics and general logics, see [5], [34], [26], or, for the case of logics without semantics (i.e. without ModE), [9]. These conditions go back to pioneering papers of A. Tarski, cf. [10]. To each logic and general logic there is associated a set Cnnc of logical connectives, specified in such a way that F m e or Fm~ becomes an absolutely free algebra

(cf. also F r e e a l g e b r a ) generated by the atomic formulas of ~- and 12 and using Cnnc as algebraic operations. Hence one can view Cnnc as the similarity type of the algebras Fm~. Using the algebras Fm~ and the provability relation F-~, one can associate a class AIg~ (12) of algebras to £. Each of these algebras corresponds to a syntactical theory of £. Using Fm~ together with mng~ and ~ , , one can associate a second class AIg~(£) of algebras to £. AIg~ (12) represents semantical aspects of 12, e.g. each model 99l E Mod~ corresponds to an algebra in Alga(12). Often, the members of AIg~(£) are called representable algebras or meaning algebras of 12. Under mild conditions on 12, one can prove that Alge(12) is a q u a s i - v a r i e t y and that Alga(12) _C AIg~(£). If the logic 12 is complete, then SP Alga(12) = SP Alge(12), of. e.g. [5].

Examples. If 12 is propositional logic (cf. also Mathematical

logic; Propositional calculus), then Alge(12) = Alga(12) is the class BA of Boolean algebras (cf. B o o l e a n algebra). Let n E w. For the nvariable fragment L~ of first-order logic, Alge(L~) is the class CAm of cylindric algebras of dimension n, while AIg~(L~) is the class NCAn of representable cylindric algebras. For a certain variant L~ of first-order logic, AIg~(L~) is the class RCA~ of representable CA~s. L~ is called the full restricted first-order language in [12, Vol. II], cf. also [5, §6] and [9, Appendix C]. For the algebraic counterparts of other logics (as well as other versions of first-order logic), see [5]. Now, take the logic L~ as an example. The algebraic counterparts of theories of L~ are exactly the algebras in CA,~ and the interpretations between theories correspond exactly to the homomorphisms between CAns. Further, axiomatizable classes of models of L~ correspond to RCA~s and (semantic) interpretations between such classes of models correspond to special homomorphisms, called base-homomorphisms, between RCA,~s, cf. [12, Vol. II, p. 170]. Individual models of Ln correspond to simple RCAns and elementary equivalence of models corresponds to isomorphism of RCAns. The elements of an RCA~ corresponding to a model 9)I are best thought of as the relations definable in 9)I. Of the duality theory between logics and their algebraic counterparts only the translation AIg : 'logics' ---+ 'pairs of classes of algebras' was discussed above. The other direction can also be elaborated (and then a two-sided duality like Stone duality between BAs and certain topological spaces can occur; cf. also S t o n e space)i see [9, p.21] for more on such a two-sided duality between logics and quasi-varieties of algebras. 31

ALGEBRAIC LOGIC

Some equivalence theorems. Using the duality theory outlined above, logical properties of/2 can be characterized by algebraic properties of Alga(/2), Alge(£) (under some mild assumptions on Z;). E.g. the deduction property of £ is equivalent with Alge(£) having equationally definable principal congruences. The Beth definability property for L: is equivalent with surjectiveness of all epimorphisms in AIg~(£). The various definability properties (weak Beth, local Beth, etc.) and interpolation properties are equivalent with distinguished versions of the amalgamation property and surjectiveness of epimorphisms, respectively, in Alge(£) or AIg~(£). A kind of completeness theorem for/2 is equivalent with finite axiomatizability of Alga(E). Compactness of g is equivalent with Alga(g) being closed under ultraproducts. The above (and further) equivalence theorems are elaborated in e.g. [5]. Further such results can be found in e.g. [12], [9], [22], [23], [35], [15], [24], [25], work of J. Czelakowski, L. Maksimova, and the references in [19]. A duality theory for algebraic logic is in [11]. An overview of duality theories is in [3, Chap. 6]. C o n c r e t e a l g e b r a i c logic. This branch investigates classes of algebras that arise in the algebraization of the most frequently used logics. Below, attention is restricted to algebras of classical quantifier logics, algebras of the finite variable fragments Ln of these logics, relativized versions of these logics, e.g. the guarded fragment, and logics of the dynamic trend, whose algebras are relation algebras or relativized relation algebras. See also A b s t r a c t a l g e b r a i c logic. The objective is to 'algebraize' logics which extend classical propositional logic. The algebras of this propositional logic are Boolean algebras (cf. also B o o l e a n algebra). Boolean algebras are natural algebras of unary relations. One expects the algebras of the extended logics to be extensions of Boolean algebras to algebras of relations of higher ranks. The elements of a Boolean algebra are sets of points; one expects the elements of the new algebras to be sets of sequences (since relations are sets of sequences). n-ary representable cylindric algebras (RCA,~s) are algebras of n-ary relations. They correspond to the nvariable fragment L~ of first-order logic. The new operations are cylindrifications ci (i < n). If R _C ~U is a relation defined by a formula F ( v 0 , . . . , v ~ - l ) , then ci(R) C_ nU is the relation defined by the formula 3 v i a ( v 0 , . . . , V~-l). (To be precise, one should write cU for ci.) Assume n = 2, R C_ U x U. Then co(R) = U x Rng(R) and Cl(R) = Dora(R) x U. This shows that ci is a natural and simple operation on n-ary relations: it simply abstracts from the ith argument of the relation. 32

Let i < n, R C_ ~U. Then ci(R) =

= { ( b 0 , . . . , b i - l , a , bi+l,...,b,~-l) : a CU

and

3bi: b = ( b 0 , . . . , b i - l , b i , b i + l , . . . , b n - 1 ) E R } . In other words, if ~i: ~U --+ (~-I)U is the canonical projection along the ith factor, then c~(R) = ~ - % ( ( R ) ) .

~(U) = (P(U), n, U, - ) denotes the Boolean algebra of all subsets of U. The algebra of n-ary relations over U is 9le[~(U) = (q3(~U), c o , . . . , en-~, Id) where the constant operation Id is the n-ary identity relation, Id = { ( a , . . . , a) : a C U} over U. E.g. the smallest subalgebra of 91ei2(U) has _< 2 atoms, while that of ffle[~(U) has < 2 (~2) atoms. The class RCA~ of n-ary representable cylindric algebras is defined as RCA~ = SP {9Ida(U): Uis a set}, where S and P are the operators on classes of algebras corresponding to taking isomorphs of subalgebras and direct products, respectively. Let n > 2. Then RCA~ is a discriminator variety, with an undecidable but recursively enumerable equational theory. RCA~ is not finitely axiomatizable, fails to have almost any form of the amalgamation property and has non-surjective epimorphisms. Almost all of these theorems remain true if one throws away the constant Id (from RCA~) and closes up under S to make it a universally axiomatizable class. These properties imply theorems about L~ via the duality theory between logics and classes of algebras elaborated in a b s t r a c t algebraic logic. Further, usual set t h e o r y can be built up in L3 (and even in the equational theory of CA3). Hence L3 (and CA3)have the 'GSdel incompleteness property', cf. [31] and also G S d e l i n c o m p l e t e n e s s t h e o r e m . For first-order logic L~ with infinitely many variables (cf. e.g. [9, Appendix C]), the algebraic counterpart is RCA~ (algebras of w-ary relations ). To generalize RCAn to RCA~, one needs only a single non-trivial step: one has to brake up the single constant Id to a set of constants Idij = {q E ~U: qi = qj}, with i , j E a~. Now RCA~ = SP { (V(~U), ci, Idij)i,jc~ : U is a set }. The definition of RCA~ with a an arbitrary o r d i n a l n u m b e r is practically the same. RCA~ is an arithmetical variety, not axiomatizable by any set E of formulas involving only finitely many individual variables. Most of the theorems about RCA~ mentioned above carry over to RCAa.

ALGEBRAIC LOGIC The greatest element of a 'generic' RCA~ was required to be a Cartesian space ~U. If one removes this condition and replaces aU with an arbitrary a-ary relation V C ~U in the definition, one obtains the important generalization Crs~ of RCA~. Many of the negative properties of RCAa disappear in Crsa. E.g., the equational theory is decidable, is a variety generated by its finite members, enjoys the super-amalgamation property (hence the strong amalgamation property (SAP), too), etc. Logic applications of Crs~ abound, cf. e.g. [1], [7], [16], [33], [27]. Since RCA~ is not finite schema axiomatizable, a finitely schematizable approximation CA~ D RCA~ was introduced by Tarski. There are theorems to the effect that CAs approximate RCAs well, cf. [12, Vol. II, §3.2],

[3O]. The above illustrates the flavour of the theory of algebras of relations; important kinds of algebras not mentioned include relation algebras and quasi-polyadic algebras, cf. e.g. [12, Vol. II], [37], [14], [5], [32], [17], [29]. The theory of the latter two is analogous with that of RCA~s. Common generalizations of CAs, Crss, relation algebras, polycyclic algebras, and their variants is the important class of Boolean algebras with operators, cf., e.g., [20], [11], [18], [10], [2], [21]. For category-theoretic approaches, see [5] and the references therein. There are many open problems in this area (cf. e.g. [32], [13], [4, pp. 727-745], [36]). To mention one (open as of 2000): is there a variety V C CA~ having the strong amalgamation property (SAP) but not the superamalgamation property? Application areas of algebraic logic range from logic and linguistics through cognitive science, to even relativity theory, cf., e.g., the work of the Amsterdam school [8], [6], [28], [7], and [3]. This work was supported by the Hungarian National Foundation for Scientific Research T30314, T35192.

References [1] ANDRI~KA, H., BENTHEM, J. VAN, AND NI~METI, I.: 'Modal languages and bounded fragments of predicate logic', d. Philos. Logic 27 (1998), 217-274. [2] ANDREKA, H., G1VANT, S., MIKULJ~S, SZ., NI~METI, I., AND SIMON, A.: 'Notions of density that imply representability in algebraic logic', Ann. Pure Appl. Logic 91 (1998), 93 190. [3] ANDREKA, H., MADARASZ,J.X., AND NEMETI, I.: On the logical structure of relativity theories, A. R6nyi Inst. Math., 2001. [4] ANDRI~KA,H., MONK, J.D., AND NI~METI, I. (eds.): Algebraic logic (Proc. Conf. Budapest 1988), Vol. 54 of Colloq. Math. Soc. J. Bolyai, North-Holland, 1991. [5] ANDRt~KA, H., N~METL I., AND SAIN, I.: 'Algebraic logic': Handbook of Philosophical Logic, Vol. 1, Kluwer Acad. Publ., 2001. [6] BENTHEM, J. VAN: Language in action (categories, lambdas and dynamic logic), Vol. 130 of Studies in Logic, NorthHolland, 1991.

[7] BENTHEM, J.A.F.K. VAN: Exploring logical dynamics, Studies in Logic, Language and Information. CSLI Publ., 1996. [8] BENTHEM, J. VAN, AND MEULEN, A. TER (eds.): Handbook of Logic and Language, Elsevier, 199"/. [9] BLOK, W.J., AND PIOOZZI, D.L.: 'Algebraizable logics', Mereoirs Amer. Math. Soc. 77, no. 396 (1989). [10] GIVANT, S.R., AND MCKENZIE, R.N. (eds.): Vol. 1-4, Birkh~user, 1986. [111 GOLDBLATT, R.; 'Algebraic polymodal logic: A survey', Logic J. IGPL 8, no. 4 (2000), 393 450. [12] HENKIN, L., MONK, J.D., AND TARSKI, A.: Cylindric algebras, Vol. I-II, North-Holland, 1971/85. [13] HENKIN, L., MONK, J.D., TARSKI, A., ANDR~KA, H., AND NI~METI, I.: Cylindric set algebras, Vol. 883 of Lecture Notes in Math., Springer, 1981. [14] HmSCH, R., AND HODKINSON, I.: Relation algebras by games, Kluwer Acad. Publ., to appear. [15] HOOGLAND, E.: 'Algebraic characterizations of various Beth definability properties', Studia Logica 65, no. 1 (2000), 91112. [16] HOOGLAND, E., AND MARX, M.: 'Interpolation in guarded fragments', Studia Logica (2000). [17] 'Special issue on Algebraic Logic', Logic d. IGPL 8, no. 4 (2000). [18] JIPSEN, P., JdNSSON, B., AND RAFTEa, J.: 'Adjoining units to residuated Boolean algebras', Algebra Univ. 34, no. 2 (1995), 118-127. [19] 'Special issue on abstract algebraic logic', Studia Logica 65, no. I (20o0).

[20] J6NSSON, B., AND TARSKI, A.: 'Boolean algebras with operators': Alfred Tarski Collected Papers, Vol. 3, Birkhguser, 1986. [21] KuRucz, A.: 'Decision problems in algebraic logic', PhD Diss., Budapest (1997). [22] MADAR~SZ, J.X.: 'Interpolation and amalgamation: Pushing the limits (I)', Stadia Logica 61, no. 3 (1998), 311-345. [23] MADARJ~SZ,J.X.: 'Interpolation and amalgamation: Pushing the limits (II)', Studia Logica 62, no. 1 (1999), 1-19. [24] MADARJSZ, J.X.: 'Surjectiveness of epimorphisms in varieties of algebraic logic', Preprint A. Rdnyi Inst. Math. (2000). [25] MADARASZ, J.X., AND SAYED-AHMED, T.: 'Amalgamation, interpolation and epimorphisms, solutions to all problems of Pigozzi's paper, and some more', A. Rdnyi Inst. Math. (2001). [26] MAaTI-OLmT, N., AND MESEGUER, J.: 'General logics and logical frameworks', in D.M. GABBAY (ed.): What is a Logical System, Clarendon Press, 1994, pp. 355 392. [27] MARX, M., P6LOS, L., AND MASUCH, M. (eds.): Arrow logic and multi-modal logic, CSLI Publ., 1996. [28] MARX, M., AND VENEMA, Y.: Multi-dimensional modal logic, Kluwer Acad. Publ., 1997. [29] M~KHALEV, R.A., AND Pmz, G.F. (eds.): Handbook on the heart of algebra, Kluwer Acad. Publ., to appear. [30] MONK, J.D.: 'An introduction to cylindric set algebras', Logic J. IGPL 8, no. 4 (2000), 451-506. [31] N~METI, I.: 'Logic with three variables has GSdel's incompleteness property - - thus free cylindric algebras are not atomic', Manuscript Math. Inst. Hangar. Acad. Sei., Budapest (1986). [32] NI~METI, I.: 'Algebraization of quantifier logics, an introductory overview', Studia Logica 50, no. 3/4 (1991), 485 570, Special issue devoted to Algebraic Logic, eds.: W.J. Blok and

33

ALGEBRAIC LOGIC D.L. Pigozzi. This is a preliminary, short version (without proofs, etc.) of www.math-inst.hu/pub/algebraic-logic. [33] NI~METI, I.: 'Fine-structure analysis of first order logic', in M. MARX, L. POLOS, AND M. MASUCH (eds.): Arrow Logic and Multi-Modal Logic, CSLI Publ., 1996, pp. 221-247. [34] N~METI, I., AND ANDREKA, H.: 'General algebraic logic: A perspective on what is logic', in D.M. GABBAY (ed.): What is a Logical System, Clarendon Press, 1994, pp. 393-444. [35] PIGOZZI, D.L.: 'Amalgamation, congruence-extensions, and interpolation properties in algebras', Algebra Univ. 1, no. 3

(1972), 269-349.

[36] SIMON, A.: 'Non-representable algebras of relations', PhD Diss., Budapest (1997). [37] TARSKI, A., AND GIVANT, S.: A formalization of set theory without variables, Vol. 41 of Colloq. Publ., Amer. Math. Soe., 1987.

H. Andrdka J.X. Madardsz I. Ndmeti

[2] PRZYTYCKI, J.H., AND TSUKAMOTO, T.: 'The fourth skein module and the Montesinos-Nakanishi conjecture for 3algebraic links', J. K n o t Th. Ramifications (2001).

Jozef Przytycki M S C 1991:57M25 The concept of a triple system, i.e. a v e c t o r s p a c e V over a field K together with a K-trilinear mapping V × V × V --+ V, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras (cf. also Lie a l g e b r a ; N o n - a s s o c i a t i v e r i n g s a n d a l g e b r a s ) . A m o d u l e V over a field of characteristic not equal to two or three together with a trilinear mapping ( x , y , z ) -+ (xyz) from V × V × V to V is said to be an Allison Hein triple system (or a J-ternary algebra) if ALLISON-HEIN

MSC1991: 03Gxx TANGLES

-

i) n-algebraic tangles is the smallest family of ntangles satisfying 1) any n-tangle with 0 or 1 crossing is n-algebraic; 2) if A and B are n-algebraic tangles, then r i (A) * r j (B) is n-algebraic for any integers i, j, where r denotes the rotation of a tangle by the angle 7r/n and * denotes (horizontal) composition of tangles. ii) If in condition 2) above, B is restricted to tangles with no more than k crossings, one obtains the family of (n, k )-algebraic tangles. iii) If an m-tangle, T, is obtained from an (n, k)algebraic tangle (respectively, an n-algebraic tangle) by closing 2 n - 2 m of its endpoints without a crossing, then T is called an (n, k)-algebraie m-tangle, respectively an n-algebraic m-tangle. For m = 0 one obtains an (n, k)algebraic link, respectively an n-algebraic link. 2-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arboreseent tangles). The 2-fold branched covering of S 3 with a 2-algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is 2-algebraic. It is an open problem (as of 2001) to find, for a given n, a link which is not n-algebraic. The smallest n for which a link L is n-algebraic is called the algebraic index of the link (it is bounded from above by the braid and bridge indices of the link). For example, the algebraic index of the 81s knot is equal to 3. References [1] CONWA¥, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): Computational Problems in Abstract Algebra, Pergamon Press, 1969, pp. 329-358. 34

SYSTEM

<xy

A family of tangles (cf. T a n g l e ) defined recursively for any n as follows: ALGEBRAIC

TRIPLE

=

w>

+

-

=
(1) +

( x y z ) - (zyx> = ( z x y ) - ( x z y )

(2)

for all x , y , z , u , v , w E V. From the identities (1) and (2) one deduces the relation

K((abc) , d) + K(c, (abd) ) + K(a, K(c, d)b) = O, where K(a, b)c = (acb) -(bca). Hence this triple system may be regarded as a variation of a F r e u d e n t h a l K a n t o r t r i p l e s y s t e m . In particular, it is important that the linear span {K(a, b)}span of the set K(a, b) is a Jordan subalgebra (cf. also J o r d a n a l g e b r a ) of (End V) + with respect to A o B = (AB + BA)/2. References [1] ALLmON, B.N.: 'A construction of Lie algebras from Jternary algebras', Amer. J. Math. 98 (1976), 285-294. [2] HEIN, W.: 'A construction of Lie algebras by triple systems', Trans. Amer. Math. Soc. 205 (1975), 79-95. [3] KAMIYA,N.: 'A structure theory of Freudenthal-Kantor triple systems II', Commun. Math. Univ. Sancti Pauli 38 (1989), 41-60. [4] YAMAGUTI,K.: 'On the metasymplectic geometry and triple systems', Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. 306 (1977), 55-92. (In Japanese.)

Noriaki Kamiya M S C 1991:17A40 A generic term used to describe any condition on a function f such that all continuous functions satisfy it; one can also use it if the original f is not necessarily continuous. The original condition of continuity by A. Cauchy [2] was cleared by K. Weierstrass (late 1850s) from the vagueness of its formulation as well as its dependence upon motion (cf. also C o n t i n u o u s f u n c t i o n ) . One of the first conditions of 'almost continuity' was the ALMOST

CONTINUITY

-

A L M O S T - S P L I T SEQUENCE

Lipschitz condition, introduced in 1864; Riemannintegrable functions were studied in 1867 (cf. also Riemann integral), while in 1870 H. Hankel introduced pointwise discontinuous functions (cf. Discontinuity point; Discontinuous function).

Let C be an indecomposable non-projective finitelygenerated left R-module. Then there exists a short exact sequence

Nowadays (2000), the term 'almost continuity' is used for various conditions weakening the (topological) condition of continuity that the inverse image of any open set is open. For example, V. Volterra noticed that all realvalued separately continuous functions from the plane have a certain ahnost continuity property, which was later termed quasi-continuity, where it is required that the inverse image of every open set is semi-open, i.e., is contained in the closure of its interior; quasi-continuity has been successfully used in recent proofs of 'deep' results in topological algebra (cf. also Separate and joint continuity), in particular in the proof that all Cech-complete semi-topological groups are topological (A. Bouziad, [1]). Another frequently used type of almost continuity is the notion of near continuity, introduced by B.J. Pettis; it is used in place of linearity in topological versions of the c l o s e d - g r a p h theorem, where the spaces under consideration are not necessarily assumed to be linear [4]. The papers [5] and [3] serve as good guides in this rapidly growing field.

in R Mod, the category of finitely-generated left Rmodules, with the following properties:

References [1] BOUZIAD, A.: 'Every Cech-analytic Baire semitopological group is a topological group', Proc. Amer. Math. Soe. 124 (1996), 953-959. [2] CAUCHY, A.L.: 'Cours d'analyse d'l~cole Royale Polytechnique, 1821': Oeuvres Compldtes d'Augustin Cauchy, H Ser., Voh III, Gauthier-Villars, 1897. [3] GAULD, D., GREENWOOD, S., AND P~EILLY, I.: 'On variations of continuity', Invited Contribution, Topology Atlas (2000),

http://at .yorku.ca/t/a/i/c/32.htm. [4] PIOTROWSKI, Z., AND SZYMANSKI, A.: 'Closed graph theorem: Topological approach', Rend. Circ. Mat. Palermo 37 (1988), 88-99. [5] PRZEMSKI, M.: 'On forms of continuity and cliquishness', Rend. Circ. Mat. Palermo 42 (1993), 417 452.

Z. Piotrowski MSC 1991:54C08 ALMOST-SPLIT SEQUENCE, Auslander-Reitcn sequence - Roughly speaking, almost-split sequences are minimal non-split short exact sequences. They were introduced by M. Auslander and I. Reiten in 1974 1975 and have become a central tool in the theory of representations of finite-dimensional algebras (cf. also R e p -

resentation of an associative algebra). Let R be an Artin algebra, i.e. R is an associative ring with unity that is finitely generated as a module over its centre Z(/~), which is a commutative Artinian ring.

O --+ A g B a C -+ O

(1)

i) A and C are indecomposable; ii) the sequence does not split, i.e. there is no section s: 6' -+ B of g (a homomorphism such that gs = id), or, equivalently, there is no retraction of f (a homomorphism r: B --+ A such that r f = id); iii) given any h: Z --+ 6" with Z indecomposable and h not an isomorphism, there is a I/ft of h to B (i.e. a homomorphism h : Z + B in n Mod such that g ~ = h); iv) given any j : A + X with X indecomposable and j not an isomorphism, there is a homomorphism J: B ~ X such that j f = j. Note that if iii) (or, equivalently, iv)) were to hold for all h, not just those h that are not isomorphisms, the sequence (1) would be split, whence 'almost split'. Moreover, a sequence (1) with these properties is uniquely determined (up to isomorphism) by 6,, and also by A. This is the basic Auslander Reiten theorem on almostsplit sequences, [2], [3], [4], [5], [6]. For convenience (things also work more generally), let now R be a finite-dimensional algebra over an algebraically closed field k. The category n Mod is a Krull-Schmidt category (Krull-Remak-Schmidt category), i.e. a C C n Mod is indecomposable if and only if R End(6,, 6"), the endomorphism ring of C, is a local ring and (hence) the decomposition of a module in R Mod into indecomposables is unique up to isomorphism. Let C be an indecomposable and consider the contravariant functor X ~ R Mod(X, C). The morphisms g: X ~ 6" that do not admit a section (i.e. an s: C --+ X such that gs = id) form a vector subspace E c ( X ) C R Mod(X, C). Let S c be the quotient functor S c = R Mod(?, C ) / E c . Then, for an indecomposable D, S o ( D ) = k if D is isomorphic to C and zero otherwise. So S c is a simple functor. (All fimctors R Mod(?, C), E c , S c are viewed as k-functors, i.e. functors that take their values in the category of vector k-spaces.) If C is indecomposable, then (the Auslander-Reiten theorem, [8, p.4]) the simple functor S c admits a minimal projective resolution of the form 0 + R Mod(?, A) --+ n Mod(?, B) -+ -+ R Mod(?, 6") -+ S c + 0. If C is projective, A is zero, otherwise A is indecomposable. 35

ALMOST-SPLIT SEQUENCE If C is not projective, the sequence 0 --+ A -+ B -+ C --+ 0 is exact and is the almost-split sequence determined by C. This functorial definition is used in [9] in the somewhat more general setting of exact categories. For a good introduction to the use of almost-split sequences, see [11]; see also [7], [9] for comprehensive treatments. See also R i e d t m a n n c l a s s i f i c a t i o n for the use of almost-split sequences and the Auslander-Reiten quiver in the classification of self-injective algebras. The Bautista-Brunner theorem says t h a t if R is of finite representation type and 0 -+ A --+ /3 --+ C --~ 0 is an almost-split sequence, then B has at most 4 terms in its decomposition into indecomposables; also, if there are indeed 4, then one of these is projective-injective. This can be generalized, [10]. References [1] AUSLANDER,M.: 'The what, where, and why of almost split sequences': Proc. ICM 1986, Berkeley, Vol. I, Amer. Math. Soc., 1987, pp. 338-345. [2] AUSLANDER,M., AND REITEN, I.: 'Stable equivalence of dualizing R-varieties I', Adv. Math. 12 (1974), 306-366. [3] AUSLANDER, M., AND REITEN, I.: 'Representation theory of Artin algebras III', Cornmun. Algebra 3 (1975), 239-294. [4] AUSLANDER, M., AND REITEN, I.: 'Representation theory of Artin algebras IV', Cornrnun. Algebra 5 (1977), 443-518. [5] AUSLANDER, M., AND REITEN, [.: 'Representation theory of Artin algebras V', Cornrnun. Algebra 5 (1977), 519-554. [6] AUSLANDER, M., AND REITEN, I.: 'Representation theory of Artin algebras VI', Cornrnun. Algebra 6 (1978), 257-300. [7] AUSLANDER, M., REITEN, I., AND SMALO, S.O.: Representation theory of Artin algebras, Cambridge Univ. Press, 1995. [8] GABRIEL, P.: 'Auslander-Reiten sequences and representation-finite

algebras', in V. DLAB

AND

M. Hazewinkel M S C 1991:16G70 ALTERNATING A L G O R I T H M - An algorithm first proposed by J. von Neumann in 1933 [6]. It gives a method for calculating the orthogonal projection Puny onto the intersection of two closed subspaces U and V in a H i l b e r t s p a c e H in terms of the orthogonal projections P : H --+ U and Q: H --+ V (cf. also O r t h o g o n a l p r o j e c t o r ) . The result is that

lim ( p Q ) n f = P v n w f

36

lim ( ( I - Q)(I - P))'~ f = (I - PFTF-)f

n--~ oo

(1)

for all f E H . Here, PFT~- is the orthogonal projection of H onto the subspace U + V. Since it was first proposed, this algorithm has undergone m a n y generalizations, mainly concerning the kind of spaces in which the algorithm can be located. It occurs in a large number of practical applications, such as domain decomposition methods for solving linear systems of equations, and certain multi-grid schemes used for solving elliptic partial differential equations. For a survey account of a wide collection of applications, see [2]. The algorithm easily admits a generalization to a finite number of subspaces of H . Let f be a m e m b e r of the Hilbert space H , and let Ui, i = 1 , . . . , n, be closed subspaces of H. Let U = FI~IUi, and let P be the orthogonal projection of H onto U. Let Pi : H --+ Ui be the orthogonal projection onto Ui, i = 1 , . . . ,n. Given f E H , define {ft}e~l by fi = ( P n " ' P 1 ) e f , for e = 1 , 2 , . . . . The elements fe are the iterates in the alternating algorithm and the analogous convergence result is that l i f t - Pfll ~ 0 as t ~ ee. Quite a while later, other authors became interested in the rate of convergence of this algorithm. It was verified by N. AronszaYn [1] in the case of only two subspaces t h a t the rate of convergence is usually linear. T h a t is, there is a constant e < 1 such that life - P u n v f l l <_ cNe-1]lfl] for all f in H . The number c depends on the notion of an angle between two subspaces of a Hilbert space. The angle a between U and V is given by

P. GABRIEL

(eds.): Representation Theory I. Proc. Ottawa 1979 Conf., Springer, 1980, pp. 1-71. [9] GABRmL, P., AND ROITER, A.V.: Representations of finitedimensional algebras, Springer, 1997, p. Sect. 9.3. [10] LIu, SHINING: 'Almost split sequenes for non-regular modules', Fundarn. Math. 143 (1993), 183-190. [11] REITEN, I.: 'The use of almost split sequences in the representation theory of Artin algebras', in M. AUSLANDERAND E. LLUIS (eds.): Representation of Algebras. Proc. Puebla 1978 Workshop, Springer, 1982, pp. 29-104.

n-+(N)

It can be equivalently formulated as

for all f C H.

cosa=sup

(u,v):

uEUMV± ' } v E V M U ±, • I]ui],]lv][ < 1

In the present context, c = c o s a . In [7], K.T. Smith, D.C. Solman and S.L. Wagner gave the analogous result for more t h a n two subspaces. The convergence constant c remains a function of the angles between the subspaces, but is only a little more complicated than the case of two subspaces. Because of the importance of the rate of convergence, later authors have given improved rates. To get a very good expression for the rate of convergence, one needs to be very much more careful over the angles specified in the Smith-Solman-Wagner result. The best result know today (2000) is given in [5]. The Lorch theorem states that the angle between two subspaces U and V of a Hilbert space H is strictly positive if and only if U + V is a closed subspace of H . So if the sum is not closed, the result of AronszaYn does not provide linear convergence, and it is of interest to enquire what takes place in this situation. A result of C. Franchetti and W.A. Light [4] shows that if U + V is not

ANOVA closed, then the rate of convergence of the alternating algorithm may be arbitrarily slow. Interesting questions arise when the algorithm is considered not as an algorithm in a Hilbert space, but in a Banach space with some fairly strong assumptions on the norm. The operators P~ and P are then of course the operators of best approximation, so one needs at least to guarantee that such operators exist. For example, the ambient space X can be a strictly convex and reflexive Banach space (cf. also Reflexive space; S m o o t h s p a c e ) . In fact, W.J. Stiles proved [8] that if the dimension of such a space is at least 3, the convergence of the algorithm for all f in X and for all pairs U and V of closed subspaces suffices to guarantee that X is a Hilbert space. Stiles also proved t h a t in a finite-dimensional, smooth and strictly convex space, the alternating algorithm as given in (1) is always effective. Several authors have given improvements to this results (see [2]). The results of Franchetti and Light [3] show that the convergence of the algorithm can still be linear in this case, with an appropriate modification to the definition of the angle between subspaces. Finally, there are a number of interesting extensions where the projections are not uniquely defined. Instead, one may work in a space X with subspaces U and V which do not guarantee unique best approximations. An example of this is the D i l i b e r t o - S t r a u s algorithm, where X = C(S x T), U = C(S) and V = C(T). In this algorithm, which is a realization of (1), a selection must be employed to define the mappings P and Q.

A N O V A , analysis of variance - Here, ANOVA will be understood in the wide sense, i.e., equated to the univariate linear model whose model equation is y = X/3 + e,

in which y is an n × 1 observable random vector, X is a known (n × m ) - m a t r i x (the 'design matrix'), is an (m × 1)-vector of unknown parameters, and e is an (n x 1)-vector of unobservable random variables e~ (the 'errors') t h a t are assumed to be independent and to have a normal distribution with mean 0 and unknown variance 0.2 (i.e., the ei are independent identically distributed N(0, or2)). It is assumed throughout t h a t n > m. Inference is desired on ¢~ and a 2. The ei may represent m e a s u r e m e n t error a n d / o r inherent variability in the experiment. The model equation (1) can also be expressed in words by: y has independent normal elements Yi with common, unknown variance and expectation E(y) = X ~ , in which X is known and fl is unknown. In most experimental situations the assumptions made on e should be regarded as an approximation, though often a good one. Studies on some of the effects of deviations from these assumptions can be found in [49, Chap. i0], and [52] discusses diagnostics and remedies for lack of fit in linear regression models. To a certain extent the ANOVA ideas have been carried over to discrete data, then called the log-linear model; see [6], and [10]. MANOVA (multivariate analysis of variance) is the multivariate generalization of ANOVA. Its model equation is obtained fl'om (1) by replacing the column vectors y, 13, e by matrices Y, B, E to obtain

References [1] ARONSZAJN, N.: 'Theory of reproducing kernels', Trans. Amer. Math. Soc. 68 (1950), 337-404. [2] DEUTSCH, F.: 'The method of alternating projections', in S.P. SINGH (ed.): Approximation Theory, Spline Functions and Applications, Kluwer Acad. Publ., 1992, pp. 105 121. [3] FRANCHETTI, C., AND LIGHT, W.A.: 'The alternating algorithm in uniformly convex spaces', J. London Math. Soc. 29, no. 2 (1984), 454-555. [4] FRANCHETTI, C., AND LIGHT, W.A.: 'On the von Neumann alternating algorithm in Hilbert Space', J. Math. Anal. Appl. 114 (1986), 305-314. [5] KAYLAR, S., AND WEINERT, H.L.: 'Error bounds for the method of alternating projections', Math. Control Signals Syst. 1 (1988), 43-59. [6] NEUMANN, J. VON: Functional operators H. The geometry of orthogonal spaces, Vol. 22 of Ann. of Math. Stud., Princeton Univ. Press, 1950. [7] SMITH, K.T., SOLMAN~ D.C., AND WAGNER, S.L.: 'Practical and mathematical aspects of the problem of reconstructing objects from radiographs', Bull. Amer. Math. Soe. 83 (1977), 1227-1270. [8] STILES, W.J.: 'Closest point maps and their products', Nieuw Archief voor Wiskunde 13, no. 3 (1965), 19-29.

W.A. Light M S C 1991: 46Cxx

(1)

Y = X B + E,

(2)

where Y and E are n x p, B is m x p, and X is as in (1). The assumption on E is that its n rows are independent identically distributed N(0, E), i.e., the common distribution of the independent rows is p-variate normal with 0 mean and p x p non-singular covariance matrix E.

GMANOVA (generalized multivariate analysis of variance) generalizes the model equation (2) of MANOVA to Y = X 1 B X 2 + E,

(3)

in which E is as in (2), X1 is as X in (2), B is m x s, and X2 is an s x p second design matrix. Logically, it would seem that it suffices to deal only with (3), since (2) is a special case of (3), and (1) of (2). This turns out to be impossible and it is necessary to treat the three topics in their own right. This will be done, below. For unexplained terms in the fields of estimation and testing hypotheses, see [31], [32] (and also

Statistical hypotheses, verification of; Statistical estimation). 37

ANOVA A N O V A . This field is very large, well-developed, and well-documented. Only a brief outline is given here; see the references for more detail. An excellent introduction to the essential elements of the field is [49] and a short history is given in [48, Sect. 2]. Brief descriptions are also given in [27, headings Anova; General Linear Model]. Other references are [50] [51], [45], [25], and [14]. A collection of survey articles on m a n y aspects of ANOVA (and of MANOVA and GMANOVA) can be found in

[29]. In (1) it is assumed t h a t the p a r a m e t e r v e c t o r / 3 is fixed (even though unknown). This is called a fixed effects model, or Model I. In some experimental situations it is more appropriate to consider/3 r a n d o m and inference is then about parameters in the distribution of ft. This is called a random effects model, or Model II. It is called a mixed model if some elements of/3 are fixed, others random. There are also various randomization models that are not described by (1). For reasons of space limitation, only the fixed effects model will be treated here. For the other models see [49, Chaps. 7, 8,

9]. The name 'analysis of variance' was coined by R.A. Fisher, who developed statistical techniques for dealing with agricultural experiments; see [49, Sect. 1.1: references to Fisher]. As a typical example, consider the twoway layout for the simultaneous study of two different factors, for convenience denoted by A and B, on the measurement of a certain quantity. Let A have levels i = 1 , . . . , I , and let B have levels j = 1 , . . . , J. For each (i, j) combination, measurements Yijk, k = 1 , . . . , K , are made. For instance, in a study of the effects of different varieties and different fertilizers on the yield of tomatoes, let Yijk be the weight of ripe tomatoes from plant k of variety i using fertilizer j. The model equation is Yijk : P ~- O~i -~- ~ j -~ 7ij -~- eijk,

(4)

and it is assumed that the eijk a r e independent identically distributed N(O, a2). This is of the form (1) after t h e Yijk a n d eijk are strung out to form the column vectors y and e of (1) with n = I J K ; similarly, the parameters on the right-hand side of (4) form an (m × l)-vector /3, with m = l+I+J+IJ; finally, X in (I) has one column for each of the rn parameters, and in row (i,j, k) of X there is a 1 in the columns for #, (~i,/3j, and 7ij, and 0s elsewhere. Some of the customary terminology is as follows. Each (i, j) combination is a cell. In the example (4), each cell has the same number K of observations (balanced design); in general, the cell numbers need not be equal. The p a r a m e t e r s on the right-hand side of (4) are called the effects: # is the general mean, the as are the main effects for factor A, t h e / % for B, and the 7s are the interactions. 38

The extension to more t h a n two factors is immediate. There are then potentially more types of interactions; e.g., in a three-way layout there are three types of two-factor interactions and one type of three-factor interactions. Layouts of this type are called factorial, and completely crossed if there is at least one observation in each cell. The latter may not always be feasible for practical reasons if the number of cells is large. In that case it m a y be necessary to restrict observations to only a fraction of the cells and assume certain interactions to be 0. The judicious choice of this is the subject of design of experiments; see [25], [14]. A different type of experiment involves regression. In the simplest case the measurement y of a certain quantity may be modelled as y = c~+ ~t + error, where a and are unknown real-valued p a r a m e t e r s and t is the value of some continuously measurable quantity such as time, temperature, distance, etc.. This is called linear regression (i.e., linear in t). More generally, there could be an arbitrary polynomial in t on the right-hand side. As an example, assume quadratic regression and suppose t denotes time. Let Yi be the measurement on y at time ti, i = 1 , . . . , n. The model equation is yi = a+/3ti+vt~+ei, which is of the form (1) with (a, fl, 3')' = / 3 of (1). The matrix X of (1) has three columns corresponding to a, /3, and 7; the ith row of X is (1, ti, t~). Functions of t other than polynomials are sometimes appropriate. Frequently, t is referred to as a regressor variable or independent variable, and y the dependent variable. Instead of one regressor variable there may be several (multiple

regression). Factors such as t above whose values can be measured on a continuous scale are called quantitative. In contrast, categorical variables (e.g., variety of tomato) are called qualitative. A quantitative factor t may be treated qualitatively if the experiment is conducted at several values, say tl, t 2 , . . . , but these are only regarded as levels i = 1, 2 , . . . of the factor whereas the actual values tl, t ~ , . . , are ignored. The name analysis of variance is often reserved for models t h a t have only factors t h a t are qualitative or treated qualitatively. In contrast, regression analysis has only quantitative factors. Analysis of covarianee covers models that have both kinds of factors. See [49, Chap. 6] for more detail. Another important distinction involving factors is between the notions of crossing and nesting. Two factors A and B are crossed if each level of A can occur with each level of B (completely crossed if there is at least one observation for each combination of levels, otherwise incompletely or partly crossed). For instance, in the tomato example of the two-way layout (4), the two factors are crossed since each variety i can be grown with any fertilizer j. In contrast, factor B is said to be

ANOVA

nested within factor A if every level of B can only occur with one level of A. For instance, suppose two different manufacturing processes (factor A) for the production of cords have to be compared. From each of the two processes several cords are chosen (factor B), each cord cut into several pieces and the breaking strength of each piece measured. Here each cord goes only with one of the processes so that B is nested within A. Nested factors should be treated more realistically as random. However, for the analysis it is necessary to analyze the corresponding fixed effects model first. See [49, Sect. 5.3] for more examples and detail. Estimation and testing hypotheses. The main interest is in inference on linear functions of the parameter vector of (1), called parametric functions, i.e., functions of the form ¢ = c~fl, with c of order m x 1. Usually one requires point estimators (cf. also P o i n t e s t i m a t o r ) of such Cs to be unbiased (cf. also U n b i a s e d e s t i m a t o r ) . Of particular interest are the elements of the vector ~. However, there is a complication arising from the fact that the design matrix X in (1) may be of less than maximal rank (the columns can be linearly dependent). This happens typically in analysis of variance models (but not usually in regression models). For instance, in the two-way layout (4) the sum of the columns for the c~i equals the column for p. If X is of less than full rank, then the elements of ¢~ are not identifiable in the sense that even if the error vector e in (1) were 0, so that X ~ is known, there is no unique solution for/3. A f o r t i o r i the elements of ~ do not possess unbiased estimators. Yet, there are parametric functions that do have an unbiased estimator; they are called estimable. It is easily shown t h a t c ~ is estimable if and only if c ~ is in the row space of X (see [49, Sect. 1.4]). In particular, if one sets E(yi) = ~i and takes c ~ to be the ith row of X, then c~¢~ = r]i is estimable. Thus, ~ is estimable if and only if it is a linear combination of the elements of r / = E(y). The complication presented by a design matrix X that is not of full rank m a y be handled in several ways. First, a re-parametrization with fewer parameters and fewer columns of X is possible. Second, a popular way is to impose side conditions on the parameters that make them unique. For instance, in the two-way layout (4) often-used side conditions are: ~ c ~ i = 0, or, equivalently, c~. = 0 (where dotting on a subscript means averaging over that subscript); similarly, p. = 0, and 7i. = 0 for all i, 7.j = 0 for all j. Then all parameters are estimable and (for instance) the hypothesis 7/1 that all main effects of factor A are 0 can be expressed by: All ai are equal to zero. A third way of dealing with an X of less t h a n full rank is to express all questions of inference in terms of estimable parametric functions. For instance, if in (4) one writes ~]ij = # + c~i +/3j + 7ij (: ~(Yijk)),

then all ?]ij are estimable and 7/A can be expressed by stating t h a t all Vi. are equal, or, equivalently, t h a t all r]i. - rj.. are equal to zero. Another type of estimator that always exists is a least-squares estimator (LSE; cf. also L e a s t s q u a r e s , m e t h o d of). A least-squares estimator of ~ is any vector b minimizing IlY- xNI 2. a minimizing b (unique if and only if X is of full rank) is denoted by ~ and satisfies the normal equations X'X~ = X'y.

(5)

If ~ = c'j3 is estimable, then ~ = e@ is unique (even when ~ is not) and is called the least-squares estimator of ~. By the G a u s s - M a r k o v theorem (cf. also L e a s t s q u a r e s , m e t h o d of), ¢ is the minimum variance unbiased estimator of ¢. See [49, Sect. 1.4]. A linear hypothesis 7/ consists of one or more linear restrictions on fl:

~t:

x3~=0

(6)

with X3 of order q x m and rank q. Then 7t is to be tested against the alternative X3fl ~ 0. Let rank(X) = r. The model (1) together with 7/ of (6) can be expressed in geometric language as follows: The mean vector r / = E(y) lies in a linear subspace ft of n-dimensional space, spanned by the columns of X, and 7t restricts r/ to a further subspace c~ of f~, where d i m ( f t) = r and dim(w) = r - q. Further analysis is simplified by a transformation to the canonical system, below.

Canonical form. There is a transformation z = ry, with r of order n x n and orthogonal, so that the model (1) together with the hypothesis (6) can be put in the following form (in which z l , . . . , z ~ are the elements of z and ~i = E(zi)): z l , . . . , z ~ are independent, normal, with common variance ¢2; 4r+1 . . . . . (~ = 0, and, additionally, 7/ specifies 41 . . . . . ~q = 0. Note that ( q + l , . - . , ¢~ are unrestricted throughout. Any estimable parametric function can be expressed in the ?, form ~ = ~ i = 1 di~i, with constants di, and the leastsquares estimator of ~ is ¢ = E i r = l diz i. TO estimate ¢2 n one forms the sum of squares for error SSe = E i = r + l z2 and divides by n - r (= degrees of freedom for the error) to form the mean square MSe = S S e / ( n - r). Then MSe is an unbiased estimator of ¢2. A test of the hy2 pothesis 7 / c a n be obtained by forming SSn = x-'q z_~i=l Zi, with degrees of freedom q, and MSn = S S n / q . Then, if 7/ is true, the test statistic 5c = MST//MS~ has an F-distribution with degrees of freedom ( q , n - r). For a test of 7/ of level of significance c~ one rejects 7t if 5c > F~;q,n-~ (= the upper c~-point of the F-distribution with degrees of freedom (q, n - r)). This is 'the' F-test; it can be derived as a likelihood-ratio test (LR test) or 39

ANOVA as a uniformly most powerful invariant test (UMP invariant test) and has several other optimum properties; see [49, Sect. 2.10]. For the power of the F-test, see [49, Sect. 2.8].

Simultaneous confidence intervals. Let L be the linear space of all parametric functions of the form ¢ = q ~ i = 1 di~i, i.e., all ¢ that are 0 if 7t is true. The F-test provides a way to obtain simultaneous confidence intervals for all ~b E L with confidence level 1 - a (cf. also C o n f i d e n c e i n t e r v a l ) . This is useful, for instance, in cases where 7-I is rejected. Then any ¢ E L whose confidence interval does not include 0 is said to be 'significantly different from 0' and can be held responsible for the rejection of 7{. Observe that q-1 ~ 1 (zi-~i)2/MSe has an F-distribution with degrees of freedom (q, n - r) (whether or not 7{ is true) so that this quantity is <_F~;q,~_~ with probability 1 - c~. This inequality can be converted into a family of double inequalities and leads to the simultaneous confidence intervals P(~-S~_<¢<_~+S~,V~bEL)

=l-c~,

(7)

in which S = (qFc~;q,~_~)1/2 and ~ = IldlI(MS~) ~/2 is the square root of the unbiased estimator of the variance Ildll2~r2 of ¢ = ~ i =q1 dizi. Thus, the confidence interval for ~ has endpoints ~ + S ~ , and all ¢ E L are covered by their confidence intervals simultaneously with probability 1 - a. Note that (7) is stated without needing the canonical system so that the confidence intervals can be evaluated directly in the original system. With help of (7) the F-test can also be expressed as follows: 7/ is accepted if and only if all confidence intervals with endpoints ~ 4- S ~ cover the value 0. More generally, it is convenient to make the following definition: a test of a hypothesis 7{ is exact with respect to a family of simultaneous confidence intervals for a family of parametric functions if 7{ is accepted if and only if the confidence interval of every ¢ in the family includes the value of ¢ specified by 7{; see [53], [54]. Thus, the F test is exact with respect to the simultaneous confidence intervals (7). The confidence intervals obtained in (7) are called Scheffd-type simultaneous confidence intervals. Shorter confidence intervals of Tukey-type within a smaller class of parametric functions are possible in some designs. This is applicable, for instance, in the two-way layout of (4) with equal cell numbers if only differences between the c~i are considered important rather than all parametric functions that are 0 under 7{a (so-called contrasts). See [49, Sect. 3.6]. The canonical system is very useful to derive formulas and prove properties in a unified way, but it is usually not advisable in any given linear model to carry out the transformation z = F y explicitly. Instead, the necessary 40

expressions can be derived in the original system. For instance, if ~a and ~ are the orthogonal projections of y on ft and on w, respectively, then SSe = IlY- hall 2 and SS~t = II~a - ~ IL2. These projections can be found by solving the normal equations (5) (and one gets, for instance, ~a = X ~ ) , or by minimizing quadratic forms. As an example of the latter: In the two-way layout (4), minimize }-~ijk (yijk -~ij) 2 over the ~]ij. This yields r]Aij = Yij., SO that SSe = ~ijk(Yijk --YiJ.) 2. If desired, formulas can be expressed in vector and matrix form. As an example, if X is of maximal rank, then (5) yields = ( X ' X ) - l X ' y and s e e = y'(I,~ - X ( X ' X ) - I X ' ) y . Similar expressions hold under 7t after replacing X by a matrix whose columns span w. If X is not of maximal rank, then a generalized inverse may be employed. See [45, Sect. 4a.3] and [46]. M A N O V A . There are several good textbooks on multivariate analysis that treat various aspects of MANOVA. Among the major ones are [1], [8], [18], [30], [37], [42], and [45, Chap. 8]. See also [27, headings Multivariate Analysis; Multivariate Analysis Of Variance], and [29]. The ideas involved in MANOVA are essentially the same as in ANOVA, but there is an added dimension in that the observations are now multivariate. For instance, if measurements are made on p different features of the same individual, then this should be regarded as one observation on a p-variate distribution. The MANOVA model is given by (2). A linear hypothesis on B analogous to (6) is 7{ : X a B = 0, (8) with X3 as in (6). Any ANOVA testing problem defined by the choice of X in (1) and X3 in (6) carries over to the same kind of problem given by (2) and (8). However, since B is a matrix, there are other ways than (8) of formulating a linear hypothesis. The most obvious extension of (8) is 7{ :

X 3 B X 4 -- 0,

(9)

in which X4 is a known (p x pl)-matrix of rank Pl. However, (9) can be reduced to (8) by making the transformation Z = YX4, of order n x pl, r ---- g x 4 , F = EX4; then the model is Z = x r + F, with the rows of F independent identically distributed N(0, ~ 1 ) , E1 = X~EX4, and 7 / : X3F = 0. Thus, the transformed problem is as (2), (8), with z , r , F replacing Y , B , E . This can be applied, for instance, to profile analysis; see [30, Sect. 5.4 (A5)], [37, Sects. 4.6, 5.6]. There is a canonical form of the MANOVA testing problem (2), (8) analogous to the ANOVA problem (1), (6), the difference being that the real-valued random variables zi of ANOVA are replaced by 1 x p random vectors. These vectors form the rows of three random matrices, Z1 of order q x p, Z2 of order (r - q) x p, and

ANOVA Z3 of order (n - r) x p, all of whose rows are assumed independent and p-variate normal with common nonsingular covariance matrix E; furthermore, E(Z3) = 0, E(Z2) is unspecified, and 7/specifies E(Z1) = 0. It is assumed that n - r _> p. P u t E(Z1) = O, so that Z1 is an unbiased estimator of O. For testing 7-/ : O = 0, Z2 is ignored and the sums of squares SSzt and SSe of ANOVA are replaced by the (p x p)-matrices M~t = Z~Z1 and ME = Z~Za, respectively. An application of sufficiency plus the principle of invariance restricts tests of 7{ to those that depend only on the positive characteristic roots of M z t M ~ -1 (= the positive characteristic roots of Z1M~-IZ~). The case q = 1, when Z1 is a row vector, deserves special attention. It arises, for instance, when testing for zero mean in a single multivariate population or testing the equality of means in two such populations. Then F = Z1M~-Iz~ is the only positive characteristic root; (n - r)F is called Hotelling's T 2, and p-1 (n - r - p + 1)F has an F-distribution with degrees of freedom (p, n - r - p + 1), central or non-central according as 7{ is true or false. Rejecting 7{ for large values of F is uniformly most powerful invariant. If q _> 2 there is no best way of combining the q characteristic roots, so t h a t there is no uniformly most powerful invariant test (unlike there is in ANOVA). The following tests have been proposed: • reject 7{ if

IMEJ/IM~t + M E I

< const (Wilks LR

test); • reject 7{ if the largest characteristic root of M n M E i exceeds a constant (Roy's test); • reject 7{ if t r ( M ~ M E 1) > const (Lawley-

a linear function of O, then f ( Z l ) is both an unbiased estimator and a maximum-likelihood estimator of f ( O ) . An unbiased estimator of E is ( n - r ) - l M E , whereas its maximum-likelihood estimator is n - I M E .

Confidence intervals and sets. There are several kinds of linear functions of O t h a t are of interest. The direct analogue of a linear function of ( 1 , . . . , ~q in ANOVA is a function of the form a ' O (with a of order q x 1), which is a (1 x p)-vector. This leads to a confidence set in p-space for d O , rather than an interval. Simultaneous confidence sets for all a ' O can be derived from any of the proposed tests for 7{, but it turns out t h a t only Roy's m a x i m u m root test is exact with respect to these confidence sets (and not, for instance, the LR test of Wilks); see [53], [54]. The same is true for simultaneous confidence sets for all O b , and confidence intervals for all d O b . Simultaneous confidence sets for all a ' O were given in [17]. In [47] simultaneous confidence intervals for all d O b are derived (called 'double linear compounds'). These are special cases of all (possibly matrixvalued) functions of the form A O B are treated in [11]. The most general linear functions of O are of the form t r ( N O ) . Simultaneous confidence intervals for all such functions as N runs through all (px q)-matrices are given in [38]. These are derived from a test defined in terms of a symmetric gauge function rather than from Roy's m a x i m u m root test. In [53], [54] a generalization of this is given if N has its rank restricted; for r a n k ( N ) _< 1 this reproduces the confidence intervals of [47].

For references, see [1, Sects. 8.3, 8.6] or [37, Chap. 5]. For distribution theory, see [1, Sects. 8.4, 8.6], [42, Sects. 10.4 10.6], [56, Sect. 10.3]. Tables and charts can be found in [1, Appendix] and [37, Appendix[. The problem of expressing the matrices M ~ and ME in terms of the original model given by (2), (8) is very similar to the situation in ANOVA. One way is to express M~t and ME explicitly in terms of X and X3. Another is to consider the ANOVA problem with the same X and X3; if explicit formulas exist for SSn and SSe, they can be converted to M n and ME. For instance, SSe = ~ i y k (Yijk - Yij.)2 in the ANOVA two-way layout (4) converts to ME = Y~,ijk(Yijk --Yij.)'(Yijk --Yij.) in the corresponding MANOVA problem, where now the Yijl~ are (1 x p)-vectors.

Step-down procedures. Partition B into its columns i l l , . - . , tip; then 7{ of (8) is the intersection of the component hypotheses 7{j : X3flj = 0. Also partition Y into its columns Y l , . . . ,Yp. Then for each j = 1 , . . . ,p, the hypothesis 7-lj is tested with a univariate ANOVA F - t e s t that depends only on YI, • • •, Yj. If any 7{j is rejected, then 7/ is rejected. The tests are independent, which permits easy determination of the overall level of significance in terms of the individual ones. For details, history of the subject and references, see [39] and [40, Sect. 3]. A variation, based on P-values, is presented in [41]. Step-down procedures are convenient, but it is shown in [35] that even in the simplest case when q = 1, a step-down test is not admissible. Furthermore, a stepdown test is not exact with respect to simultaneous confidence intervals or confidence sets derived from the test for various linear functions of B; see [54, Sect. 4.4]. A generalization of step-down procedures is proposed in [39] by grouping the column vectors of Y and B into blocks.

Point estimation. In the canonical system Z1 is an un-

Random effects models. Some references on this topic in

biased estimator and the maximum-likelihood estimator of O (cf. also M a x l m u m - l i k e l i h o o d m e t h o d ) . If f is

MANOVA are [2] and [36]; see also references quoted therein.

Hotelling test); • reject 7{ if t r ( M n ( M n

+ ME) -1)

>

const

( Bartlett-Nanda-Pillai test).

41

ANOVA

Missing data. Statistical experiments involving multivariate observations bring in an element t h a t is not present with univariate observations, such as in ANOVA. Above, it has been taken for granted that of every individual in a sample all p variates are observed. In practice this is not always true, for various reasons, in which case some of the observations have missing data. (This is not to be confused with the notion of e m p t y cells in ANOVA.) If t h a t happens, one can group all observations with complete data together as the complete sample and call the remaining observations an incomplete sample. From a slightly different point of view, the incomplete sample is sometimes considered extra data on some of the variates. The analysis of MANOVA problems is more complicated when there are missing data. In the simplest case, all missing d a t a are on the same variates. This is a special case of nested missing data patterns. In the latter case explicit expressions of maximum-likelihood estimators are possible; see [4] and the references therein. For more complicated missing data patterns explicit maximum-likelihood estimators are usually not available unless certain assumptions are made on the structure of the unknown covariance matrix X]; see [4], [5] and [3]. The situation is even worse for testing. For instance, even in the simplest case of testing the hypothesis that the mean of a multivariate population is 0, if in addition to a complete sample there is an incomplete one taken on a subset of the variates, then there is no locally (let alone uniformly) m o s t - p o w e r f u l t e s t ; see [9]. Several aspects of estimation and testing in the presence of various patterns of missing data can be found in [23], wherein also appear m a n y references to other papers in the field. G M A N O V A . This topic has not been recognized as a distinct entity within multivariate analysis until relatively recently. Consequently, most of today's (2000) knowledge of the subject is found in the research literature, rather than in textbooks. (There is an introduction to GMANOVA in [42, Problem 10.18], and a little can be found in [8, Sect. 9.6, second part].) A good exposition of testing aspects of GMANOVA, pointing to applications in various experimental settings, is given in [20]. The general GMANOVA model was first stated in [43], where the motivation was the modelling of experiments on the comparison of growth curves in different populations. Suppose such a growth curve can be represented by a polynomial in the time t, say f(t) = /30 + flit + ... +/~kt k. If measurements are made on an individual at times t~,... ,tp, then these p data are thought of as one observation on a p-variate population with population mean ( f ( t l ) , . . . , f(tp)) and covariance matrix 51, where the fls and 51 are unknown parameters. Suppose m populations are to be compared and 42

a sample of size ni is taken from the ith population, i = 1 , . . . ,m. In order to model this by (3), let the ith column of X1 (corresponding to the ith population) have n i l s , and 0s otherwise. Specifically, the first column has a 1 in positions 1 , . . . , n l , the second in positions n~ + 1 , . . . , nl + n2, etc.; then n = ~ ni. Let the growth curve in the ith population be flio + f l i l t + "'" + fliktk; then the matrix B has m rows, the ith row being (fli0,...,flik), so that s = k + 1 in (3); and X2 has p columns, the j t h one being ( 1 , t j , . . . , t j )k. I (In the example given in [43], measurements were taken at ages 8, 10, 12, and 14 in a group of girls and a group of boys; each measurement was of a certain distance between two points inside the head (with help of an X-ray picture) that is of interest in orthodontistry to monitor growth.) Linear hypotheses are in general of the form (9). For instance, suppose two growth curves are to be compared, both assumed to be straight lines (k = 1) so that rn = 2, s = 2. Suppose the hypothesis is f l l l = f l 2 1 (equal slope in the two populations). Then in (9) one can take X3 = (1, - 1 ) and X4 = (0, 1)'. Other examples of GMANOVA may be found in [20]. A canonical form for the GMANOVA model was derived in [13]; it can also be found in [20, Sect. 3.2]. It can be obtained from the canonical form of MANOVA by partitioning the matrices Zi columnwise into three blocks, resulting in 9 matrices Zij, i , j = 1,2,3. Invariance reduction eliminates all Zij except [Z12, Zla] and [Z32, Za3] (the latter is used for estimating the relevant portion of the unknown covariance matrix E). It is given that E(Z13) = 0 and E[Z32, Zaa] = 0; inference is desired on O = E(Z12), e.g., to test the hypothesis 7/ : O = 0. Further sufficiency reduction leads to two matrix-valued statistics T I and T2 ([19], [20]), of which T~ is the most important and is built-up from the following statistic: Zo = Z12 - Z 1 3 R ,

(10) !

in which R = V3-~V32 (with Vjj, = Z3jZ3f ) is the estimated regression of Z12 on Z13, the true regression being E3~5132. T h a t inference on O should be centred on Zo can be understood intuitively by realizing that if 51 were known, then Zi2 - ZlSE3-~E32 minimizes the variances among all linear combinations of Z12 and Z~3 whose mean is O, and provides therefore better inference than using only Z12. The unknown regression is then estimated by R , leading to Z0 of (10). The essential difference between GMANOVA and MANOVA lies in the presence of Z13, which is correlated with Z12 and has zero mean. Then Zls is used as a covariate for Z12; see, e.g., [34]. However, not all models that appear to be GMANOVA produce such a covariate. More precisely, if in (3) rank(X2) = p, then it

ANOVA

t u r n s out t h a t in t h e canonical form t h e r e are no m a t r i ces Zi3 a n d t h e m o d e l reduces essentially to M A N O V A . T h i s s i t u a t i o n was e n c o u n t e r e d p r e v i o u s l y when it was p o i n t e d out t h a t t h e M A N O V A m o d e l (2) t o g e t h e r w i t h t h e G M A N O V A - t y p e h y p o t h e s i s (9) was i m m e d i a t e l y reducible to s t r a i g h t M A N O V A . T h e s a m e conclusion would have been r e a c h e d after t r e a t i n g (2), (9) as a special case of G M A N O V A a n d i n s p e c t i n g t h e c a n o n i c a l form. For a ' t r u e ' G M A N O V A t h e existence of Z13 is essential. A t y p i c a l e x a m p l e of t r u e G M A N O V A , where t h e covariate d a t a are built into t h e e x p e r i m e n t , was given in [7]. Inference on O can p r o c e e d using only T1 (e.g., [26], a n d [13]), b u t is not necessarily t h e best possible. For t e s t i n g 7-{ an essentially c o m p l e t e class of t e s t s include those t h a t also involve T~ explicitly. One such test is t h e locally m o s t - p o w e r f u l t e s t d e r i v e d in [19]. For t h e d i s t r i b u t i o n t h e o r y of (T1, T2) see [20, Sect. 3.6] a n d [55, Sect. 6.5]. A d m i s s i b i l i t y a n d i n a d m i s s i b i l i t y results were o b t a i n e d in [33]; c o m p a r i s o n of various t e s t s can also be found there. A n a t u r a l e s t i m a t o r of O is Zo of (10); it is an u n b i a s e d e s t i m a t o r a n d in [21] it is shown to be b e s t e q u i v a r i a n t . O t h e r k i n d s of e s t i m a t o r s have also been considered, e.g., in [22], in which several references to earlier w o r k can be found. S i m u l t a n e o u s confidence intervals a n d sets have been t r e a t e d in [15], [16], [26], a n d [28]. Special s t r u c t u r e s of t h e covariance m a t r i x E have b e e n s t u d i e d in [44], where also references to earlier work on r e l a t e d topics can be found. A n a t u r a l g e n e r a l i z a t i o n of the G M A N O V A m o d e l is i n d i c a t e d in [13] by h a v i n g a furt h e r p a r t i t i o n i n g of t h e blocks of Z s in t h e canonical Generalizations.

form. T h i s is called e x t e n d e d G M A N O V A in [20] a n d exa m p l e s are given there. A n o t h e r g e n e r a l i z a t i o n involves some r e l a x a t i o n of t h e usual a s s u m p t i o n s of m u l t i v a r i a t e n o r m a l i t y , etc. See [24], [12], [16].

References [1] ANDERSON,m.w.: An introduction to multivariate statistical analysis, 2nd ed., Wiley, 1984. [2] ANDERSON, T.W.: 'The asymptotic distribution of characteristic roots and vectors in multivariate components of variance', in L.J. GLESER, M.D. PERLMAN, S.J. PRESS, AND A.R. SAMPSON (eds.): Contributions to Probability and Statistics; Essays in Honor of Ingrain Olkin, Springer, 1989, p. 177-196. [3] ANDERSSON,S.A., MARDEN, J.I., AND PERLMAN, M.D.: 'Totally ordered multivariate linear models', Sankhy5 A 55 (1993), 370-394. [4] ANDERSSON, S.A., AND PERLMAN, M.D.: 'Lattice-ordered conditional independence models for missing data', Statist. Prob. Lett. 12 (1991), 465-486. [5] ANDERSSON,S.A., AND PERLMAN, M.D.: 'Lattice models for conditional independence in a multivariate normal distribution', Ann. Statist. 21 (1993), 1318-1358. [6] BISHOP, Y.M.M., FIENBERG,S.E., AND HOLLAND,P.W.: Discrete multivariate analysis: Theory and practice, MIT, 1975.

[7] COCHRAN, W.C-., AND BLISS, C.I.: 'Discrimination functions with covariance', Ann. Statist. 19 (1948), 151-176. [8] EATON, M.L.: Multivariate statistics, a vector space approach, Wiley, 1983. [9] EATON, M.L., AND KARIYA, T.: 'Multivariate tests with incomplete data', Ann. Statist. 11 (1983), 654-665. [10] FIENBERG, S.E.: The analysis of cross-classified categorical data, 2nd ed., MIT, 1980. [11] GABamL, K.K.: 'Simultaneous test procedures in multivariate analysis of variance', Biometrika 55 (1968), 489-504. [12] GIRI, N., AND DAS, K.: 'On a robust test of the extended MANOVA problem in elliptically symmetric distributions', Sankhy~ A 50 (1988), 234-248. [131 GLESER, L.J., AND OLKIN, I.: 'Linear models in multivariate analysis', in R.C. BOSE (ed.): Essays in Probability and Statistics: In memory of S.N. Roy, Univ. North Carolina Press, 1970, p. 267-292. [14] HINKELMANN, K., AND KEMPTHORNE, O.: Design and analysis of experiments, Vol. I: Introduction to experimental design, Wiley, 1994. [15] HOOPER, P.M.: 'Simultaneous interval estimation in the general multivariate analysis of variance model', Ann. Statist. 11 (1983), 666-673, Correction in: 12 (1984), 785. [16] HOOPER, P.M., AND YAU, W.K.: 'Optimal confidence regions in GMANOVA', Canad. J. Statist. 14 (1986), 315-322. [171 JENSEN, D.R., AND MAYER, L.S.: 'Some variational results and their applications in multiple inference', Ann. Statist. 5 (1977), 922-931. [181 JOHNSON, P~.A., AND WICHERN, D.W.: Applied multivariate statistical analysis, 2nd ed., Prentice-Hail, 1988. [19] KAmYA, T.: 'The general MANOVA problem', Ann. Statist. 6 (1978), 200-214. [20] KARIYA,T.: Testing in the multivariate general linear model, Kinokuniya, 1985. [21] KARIYA,T.: 'Equivariant estimation in a model with an ancillary statistic', Ann. Statist. 17 (1989), 920 928. [22] KARIYA,T., KONNO, Y., AND STRAWDERMAN,W.E.: 'Double shrinkage estimators in the GMANOVA model', J. Multivar. Anal. 56 (1996), 245-258. [23] KARIYA, T., KRISHNAIAH, P.R., AND RAO, C.R.: 'Statistical inference from multivariate normal populations when some data is missing', in P.R. KRISHNAIAH(ed.): Developm. in Statist., Vol. 4, Acad. Press, 1983, p. 137-148. [24] KARIYA,T., AND SINHA, B.K.: Robustness of statistical tests, Acad. Press, 1989. [25] KEMPTHORNE, O.: The design and analysis of experiments, Wiley, 1952. [26] KHATR%C.G.: 'A note on a MANOVA model applied to problems in growth curves', Ann. Inst. Statist. Math. 18 (1966), 75 86. [27] KOTZ, S., AND JOHNSON, N.L. (eds.): Encyclopedia of Statistical Sciences, Wiley, 1982/88. [28] KRISHNAIAH,P.R.: 'Simultaneous test procedures under general MANOVA models', in P.R. KRISHNAIAH(ed.): Multivariate Analysis II, Acad. Press, 1969, p. 121-143. [29] KRISHNA;AH, P.R. (ed.): Analysis of Variance, Vol. 1 of Handbook of Statistics, North-Holland, 1980. [30] KSHIRSAGAR,A.M.: Multivariate analysis, M. Dekker, 1972. [31] LEHMANN,E.L.: Theory of point estimation, Wiley, 1983. [32] LEHMANN, E L.: Testing statistical hypotheses, 2nd ed., Wiley, 1986.

43

ANOVA [33] MARDEN, J.I.: 'Admissibility of invariant tests in the general multivariate analysis of variance problem', Ann. Statist. 11 (1983), 1086 1099. [34] MARDEN, J.I., AND PERLMAN, M.D.: 'Invariant tests for means with covariates', Ann. Statist. 8 (1980), 25-63. [35] MARDEN, J.I., AND PERLMAN, M.D.: 'On the inadmissibility of step-down procedures for the Hotelling T 2 problem', Ann. Statist. 18 (1990), 172-190. [36] MATH•W,T., NIYOGI, A., AND SINHA, B.K.: 'Improved nonnegative estimation of variance components in balanced multivariate mixed models', J. Multivar. Anal. 51 (1994), 83101. [37] MOaaISON, D.F.: Multivariate statistical methods, 2rid ed., McGraw-Hill, 1976. [38] MUDHOLKAR, G.S.: 'On confidence bounds associated with multivariate analysis of variance and non-independence between two sets of variates', Ann. Math. Statist. 37 (1966), 1736-1746. [39] MUDHOLKAR, G.S., AND SUBBAIAH, P.: 'A review of stepdown procedures for multivariate analysis of variance', in R.P. GUPTA (ed.): Multivariate Statistical Analysis, NorthHolland, 1980, p. 161-178. [40] MUDHOLKAR, G.S., AND SUBBAIAH, P.: 'Some simple optimum tests in multivariate analysis', in A.K. GUPTA (ed.): Advances in Multivariate Statistical Analysis, Reidel, 1987, p. 253-275. [41] MUDHOLKAR, G.S., AND SUBBAIAH, P.: 'On a Fisherian detour of the step-down procedure for MANOVA', Commun. Statist. Theory and Methods 17 (1988), 599-611. [42] MUIRI-IEAD, R.J.: Aspects of multivariate statistical theory, Wiley, 1982. [43] POTTHOFF, R.F., AND ROY, S.N.: 'A generalized multivariate analysis of variance model useful especially for growth curve models', Biometrika 51 (1964), 313-326. [44] RAO, C.R.: 'Least squares theory using an estimated dispersion matrix and its application to measurement of signals', in L.M. LE CAM AND J. NEYMAN (eds.): Fifth Berkeley Syrup. Math. Statist. Probab., Vol. 1, Univ. California Press, 1967, p. 355-372. [45] RAO, C.R.: Linear statistical inference and its applications, second ed., Wiley, 1973. [46] RAO, C.R., AND MITRA, S.K.: Generalized inverses of matrices and its applications, Wiley, 1971. [47] ROY, S.N., AND BOSE, R.C.: 'Simultaneous confidence interval estimation', Ann. Math. Statist. 24 (1953), 513-536. [48] SCHEFF~, H.: 'Alternative models for the analysis of variance', Ann. Math. Statist. 27 (1956), 251-271. [49] SeHE~Fi, H.: The analysis of variance, Wiley, 1959. [50] SEARLE, S.R.: Linear models, Wiley, 1971. [51] SEARLE, S.R.: Linear models for unbalanced data, Wiley, 1987. [52] WEISBERG, S.: Applied linear regression, 2rid ed., Wiley, 1985. [53] WIJSMAN, R.A.: 'Constructing all smallest simultaneous confidence sets in a given class, with applications to MANOVA', Ann. Statist. 7 (1979), 1003-1018. [54] WIJSMAN, R.A.: 'Smallest simultaneous confidence sets with applications in multivariate analysis', in P.R. KRISHNAIAH (ed.): Multivariate Analysis V, North-Holland, 1980, p. 483498. [55] WIJSMAN, R.A.: 'Global cross sections as a tooi for factorization of measures and distribution of maximal invariants', Sankhyg A 48 (1986), 1-42.

44

[56] WIJSMAN, R.A.: Invariant measures on groups and their use in statistics, Vol. 14 of Lecture Notes Monograph Ser., Inst. Math. Statist., 1990. Robert A. W i j s m a n

M S C 1991: 62Jxx ANTI-LIE T R I P L E S Y S T E M - A triple system is a v e c t o r s p a c e V over a field K together with a K trilinear m a p p i n g V x V x V --4 V. A triple system V satisfying

{ x y z } = {yxz},

{xyz} + {yzx} + {xy{zu

}} = {{xyz}

(1)

{zxy}

v} + {z{xyu}

= 0,

(2)

}+ (3)

for all x, y, z, u, v E V, is called an anti-Lie triple sys-

tem. If instead of (1) one has {xyz} = - { y x z } , a L i e t r i p l e s y s t e m is obtained. Assume that V is an anti-Lie triple system and that T) is the Lie a l g e b r a of derivations of V containing the inner derivation L defined by L(x, y)z = {xyz}. Consider /2 = L0 ® L1 with L0 = T) and L1 = V, and with product given by [al, a2] = n(al, a2) • L(V, V), - [ a l , D1] [ D l , a l ] = Dial, [D1,D2] = D1D2 - D2D1 • I? for ai • V, Di • l? (i = 1, 2). T h e n the definition of anti-Lie triple system implies t h a t £ is a Lie s u p e r a l g e b r a (cf. also Lie a l g e b r a ) . Hence L(V, V) (9 V is an ideal of the Lie superalgebra Z; = 77®V. One denotes L(V, V ) ® V by L(V) and calls it the standard embedding Lie superalgebra of V. This concept is useful to obtain a construction of Lie superalgebras as well as a construction of Lie algebras from Lie triple systems. ----

References [1] FAULKNER, J.R., AND FERRAR, J.C.: 'Simple anti-Jordan pairs', Commun. Algebra 8 (1980), 993-1013. [2] KAMIYA, N.: 'A construction of anti-Lie triple systems from a class of triple systems', Memoirs Fae. Sci. Shimane Univ. 22 (1988), 51-62. Noriaki K a m i y a

MSC1991: 17A40, 17B60 The Ap6ry numbers a,~, bn are defined by the finite sums APIARY NUMBERS

-

(n_l_k)2 (;)2

an=

k k=0

~

'

b, =

(

) (~)2

n+ k=0

for every integer n > 0. T h e y were introduced in 1978 by R. Apgry in his highly remarkable irrationality proofs of ~(3) and C(2) = rc2/6, respectively. In the case of ~(3), Ap6ry showed that there exists a sequence of rational numbers c~ with denominator dividing l c m ( 1 , . . . , n ) 3 such that 0 < la,{(3) - c,I < 1) 4" for all n > 0. Together with the fact t h a t l c m ( 1 , . . . , n ) > 3 ", this implies the irrationality of 4(3). For a very lively and

A P P R O X I M A T I O N SOLVABILITY amusing account of Ap5ry's discovery, see [4]. In 1979 F. Beukers [1] gave a very short irrationality proof of ¢(3), motivated by the shape of the Ap6ry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of ~(5) so far (2001). T. Rival [5] proved the very surprising result that ~(2n + 1) ¢ Q for infinitely many n. It did not take long before people noticed a large number of interesting congruence properties of Ap~ry numbers. For example, a,~p~ - a,~p~-i (rood p3~) for all positive integers m, r and all prime numbers p _> 5. Another congruence is a ( p _ l ) / 2 =_ ~p (mod p) for all prime numbers p > 5. Here, % denotes the coefficient of q~ in the q-expansion of a modular cusp form. For more details see [2], [3]. References [1] BEUKERS, F.: 'A note on the irrationality of ~(3)', Bull. London Math. Soc. 11 (1979), 268-272. [2] BEUKERS, F.: 'Some congruences for the Ap~ry numbers', J. Number Theory 21 (1985), 141-155. [3] BEUKERS, F.: ' A n o t h e r conguence for the Ap~ry numbers', J. Number Theory 25 (1987), 201-210. [4] POORTEN, A.J. VAN DER: 'A proof t h a t Euler missed... Ap~ry's proof of the irrationality of ¢(3)', Math. Intelligencer 1 (1979), 195-203. [5] RIVAL, T.: 'La fonction z~ta de R i e m a n n pren une infinit6 de valeurs irrationnelles aux entiers impairs', C.R. Acad. Sci. Paris 3 3 1 (2000), 267 270.

Frits Beukers MSC 1991: 11Axx, 11M06, 11J72 APPROXIMATION SOLVABILITY, A-solvability Let X and Y be Banach spaces (ef. also B a n a c h space), let T : X -+ Y be a, possibly non-linear, mapping (cf. also N o n - l i n e a r o p e r a t o r ) and let F = {X~, P~; Y~, Q . } be an admissible scheme for (X, Y), which, for simplicity, is assumed to be a complete projection scheme, i.e. {Xn} C X and {Yn} C Y are finitedimensional subspaces with dim X~ = dim Y~ for each n and P~: Y ~ X~ and Qn: Y --+ X~ are linear projections such that P~x --+ x and Q~y ~ y for x E X and y ¢ Y. Clearly, such schemes exist if both X and Y have a Schauder basis (cf. also Basis; B i o r t h o g o n a l s y s t e m ) . Consider the equation

Tx=f,

xGX,

fEY.

(1)

One of the basic problems in f u n c t i o n a l a n a l y s i s is to 'solve' (1). Here, 'solvability' of (1) can be understood in (at least) two manners: A) solvability in which a solution x E X of (1) is somehow established; or B) approximation solvability of (1) (with respect to F), in which a solution x ¢ X of (1) is obtained as the limit (or at least, a limit point) of solutions xn C X~ of

finite-dimensional approximate equations: T~(x~) = Q,~f, x. E x.,

Qnf e

(2)

Tn = (Q T)Ix ,

with T~: X~ --+ Y~ continuous for each n. If x~ and x are unique, then (1) is said to be uniquely A-solvable. Although the concepts A) and B) are distinct in their purpose, they are not independent. In fact, sometimes knowledge of A) is essential for B) to take place. If X and Y are Hilbert spaces (cf. H i l b e r t space), the projections P~ and Q~ are assumed to be orthogonal (cf. O r t h o g o n a l p r o j e c t o r ) . If, for example, {¢n} C X and {~Pn} C Y are orthogonal bases, then Xn -- s p a n { ¢ l , . . . , ¢ n } and Yn -- s p a n { f x , . . . , ~ } , and P~x = ~i~=x(x, ¢i)¢i and Qny = ~ i n l ( y , ¢i)¢i n for x e X, y ¢ Y. In this case, setting x~ = ~ i = 1 a~'¢i, the coefficients a ~ , . . . , a~ are determined by (2), which reduces to the system

(T(x~),fj)=(f,¢j),

j=l,...,n.

A - p r o p e r . In studying the A-solvability of (1) one may ask: For what type of linear or non-linear mapping T : X ~ Y is it possible to show that (1) is uniquely A-solvable? It turns out that the notion of an A-proper mapping is essential in answering this question. A mapping T : X --+ Y is called A-proper if and only if T~ : Xn ~ Yn is continuous for each n and such that if { x ~ : x~j E Xnj } is any bounded sequence satisfying T~j (x~j) --+ g for some g ¢ Y, then there exist a sub! ! sequence {x~j} and an x C X such that x~j --+ x as j ~ oe and T(x) = g, as was first shown in [1]. It was found (see [2]) that there are intimate relationships between (unique) A-solvability and A-properness of T, shown by the following results: R1) If T : X --~ Y is a continuous linear mapping, then (1) is uniquely A-solvable if and only if T is A-proper and one-to-one. This is the best possible result, which includes as a special case all earlier results for the Galerkin or PetrovGalerkin method (cf. also G a l e r k i n m e t h o d ) . R2) If T is non-linear and IIT~(x) - T~(Y)II > ¢(11x - YII)

(3)

for all x , y ¢ X~, n >_ No, where ¢ is a continuous function on R with ¢(0) = 0, ¢(t) > 0 for t > 0 and ¢(t) -+ oc as t --+ oc, then (1) is uniquely A-solvable for each f ¢ Y if and only if T is A-proper and one-to-one. If T is continuous, then R2) holds without the condition that T be one-to-one. The result R2) includes various results for strongly monotone or strongly accretive 45

A P P R O X I M A T I O N SOLVABILITY mappings (cf. also A c c r e t i v e m a p p i n g ) . If T is a continuous linear mapping, then (3) reduces to

IIT,~(x)II _> e Ilxll

(4)

which is half the t o t a l arc length of the lemniscate r e = cos(2qb) (cf. also L e m n i s c a t e s ) , is closely related to the Gauss constant. Taking a = 1, b = ( v ~ ) -1, so = 1/2 and setting

for all x E X~, n > No, and some e > 0. If, in addition, the scheme F = {Xn, Pn; Yn, Qn} is nested, i.e. Xn C X~+] and Y~ C Y~+I for all n, and Q*w --+ w in Y* for each w C Y*, then T is A-proper and one-to-one if and only if (4) holds. In particular, by R1), equation (1) is uniquely A-solvable for each f E Y. Without this extra condition on F, equation (1) is uniquely A-solvable if (1) is solvable for each f C Y, or if either X or Y is reflexive (cf. also R e f l e x i v e s p a c e ) .

one obtains a sequence po,Pl,.., that converges quadratically to % [2] (see P i ( n u m b e r ~)). This means that each iteration roughly doubles the number of correct digits. This algorithm is variously known as the Brent-Salamin algorithm, the Gauss-Salamin algorithrri, or Salamin-Brent algorithm. There are also corresponding cubic, quartic, etc. algorithms, [2].

References

References

[1] PETRYSHYN, W.V.: 'On projectional-solvability and Fredholm alternative for equations involving linear A-proper operators', Arch. Rat. Anal. 30 (1968), 270-284. [2] PETRYSHYN, W.V.: Approximation-solvability of nonlinear functional and differential equations, Vol. 171 of Monographs, M. Dekker, 1993. W . V. P e t r y s h y n

M S C 1991:47H17

ARITHMETIC-GEOMETRIC MEAN PROCESS, arithmetic-geometric mean method, A GM process, A GM method, Lagrange arithmetic-geometric mean algorithm - Given two real numbers a = a0 and b = bo, one can form the successive arithmetic and geometric means as follows: 1

an+] = -~(an + bn),

bn+l =

a~n~.

(Cf. also A r i t h m e t i c mean; Geometric mean.) The sequences a 0 , a l , . . , and bo,bl,.., rapidly converge to a common value, denoted agm(a,b) and called the arithmetic-geometric mean, or sometimes the arithmetic-geometric average, of a and b. Indeed,

1

[an+l - bn+l] < ~ Jan - b~l. This so-called AGM process is useful for computing the J a c o b i e l l i p t i c f u n c t i o n s sn(ulk), sn(ulk), cn(ulk), dn(ulk), the Jacobi theta-functions Oi(v) (cf. also T h e t a - f u n c t i o n ) , and the Jacobi zeta-function (see [1, pp. 571-598] and [5, Chap. 6; p. 663]). The number

--

(0

2

is known as the Gauss lemniscate constant, or Gauss constant, [4]. Here, F denotes the G a m m a - f u n c t i o n . The lemniscate constant

_ L = 1(2~)_1/2F_ 46

Ck = a2k -- b 2k,

Sk ~- Sk--1 _ 2 k e k ,

Pk = 2 s k l a 2 k

[1] ABRAMOWlTZ, M., AND STEOUN, J.A. (eds.): Handbook of mathematical functions, Nat. Bureau Standards, 1964, (Dover reprint 1965). [2] BAILEY, D.H., BORWEIN, J.M., BORWEIN, P.B., AND PLOUFFE, S.: 'The quest for pi', Math. Intelligencer 19, no. 1

(1997), 50-57. [3] BRENT, R.P.: 'Fast multiple-precision evaluation of elementary functions', J. Assoc. Comput. Mach. 23 (1976), 242-251. [4] FINCH, S.: 'Favorite mathematical constants', W E B : www.mathsoft, c o m / a s o l v e / c o n s t a n t / g a u s s / g a u s s . h t m l

(2000). [5] PRESS, W.H., FLANNERY, B.P., TEUKOLSKY, S.A., AND VETTERLING, W.T.: Numerical recipes, Cambridge Univ. Press, 1986. [6] SALAMIN,E.: 'Computation of pi using arithmetic-geometric mean', Math. Comput. 30 (1976), 565 570.

M. Hazewinkel M S C 1991: 65D20, 26Dxx

ATIYAH-FLOER C O N J E C T U R E - A conjecture relating the instanton Floer homology of suitable threedimensional manifolds with the symplectic Floer homology of moduli spaces of flat connections over surfaces, and hence with the q u a n t u m cohomology of such moduli spaces. It was originally stated by M.F. Atiyah for homology 3-spheres in [1]. The extension of the conjecture to the case of m a p p i n g cylinders was prompted by A. Floer and solved in this case by S. Dostoglou and D. Salamon in [4].

Instanton Floer homology for three-dimensional manifolds was introduced by Floer in [5]. Let (Y, Py) be a pair consisting of a closed oriented 3-dimensional manifold Y and an SO(3)-bundle Py --+ Y. If either Y is a homology 3-sphere or bl (Y) > 0 and the second S t i e f e l W h i t n e y class w2(Py) # O, then the instanton Floer homology H F i•n s t /ky , p y j is defined as the homology of the Morse-type complex constructed out of the C h e r n S i m o n s f u n c t i o n a l . The critical points are flat connections and the connecting orbits are anti-self-dual connections on P y x R --+ Y x R decaying exponentially to flat connections A ± when t --+ 4-0o.

ATIYAH F L O E R C O N J E C T U R E The symplectic Floer homology for Lagrangian intersections was introduced by Floer in [7]. Let (M,w) be a s y m p l e c t i c m a n i f o l d which is monotone and simply connected. Let L0 and Lz be Lagrangian submanifolds of M. Then there are Floer homology groups HF~.YmP(M, Lo, L1). Now the critical points are the intersection points z E L0 71 Lz and the connecting orbits are J-holomorphic strips u: [0, 1] x R .9 M with u(0, t) C L0, tt(1,t) E L1 and limt-++oou(s,t) = x i , where z ± E L0 5 Lz and J is an a l m o s t - c o m p l e x s t r u c t u r e compatible with the symplectic form. Let E be a closed oriented surface of genus g _> 1 and let P -+ E be the trivial SO(3)-bundle. Then the moduli space WI(P) of flat connections on P is symplectic and smooth except at the trivial connection. Now, let Y = Y0 U2 171 be a Heegaard splitting of a homology 3sphere and consider the trivial SO(3)-bundle P y on V. Then the flat connections on E which extend to Y0 define a Lagrangian subspace £0 C Ad(P), and analogously £1 C A//(P). Taking care of the singularity one may define UIT'symp{ AA(P), £0, £ i ) . The Atiyah-Floer conjec**~. w-, ture reads HF inst(Y Py) -~ Hvsymp( £4(P) £o,£1).

(1)

This was originally conjectured by Atiyah in [1]. An overview of the problem appears in [11]. The problem is still open (as of 2000). The symplectic Floer homology for a symplectic mapping was introduced by Floer in [6]. Let ( M , E ) be a symplectic manifold which is monotone and simply connected. Let ¢: M .9 M be a symplectomorphism. Then the symplectic Floer homology H F .syrup (M, 0) can be defined as the Morse-type theory where the critical points are the fixed points of (/) and the connecting orbits are J-holomorphic strips u: [0, 1] x R .9 M with u(1,t) = 0(u(0, t)) which converge to fixed points z ± of q5 as t -9 +oc. For q5 = id, Floer proved [6] that HI'SyruP(]1// id) = H*(M). Moreover, there is a natural ring structure for the symplectic Floer homology [11], and in [10] it is proved that there is an isomorphism of rings HFsymp(M, id) ~ QH*(2~I), where QH*(M) is the quantum cohomology of M. Let E be a closed oriented surface of genus g _> 1 and let Q -9 E be the non-trivial SO(3)-bundle. The moduli space of flat connections Ad(Q) is a smooth symplectic manifold. Consider the m a p p i n g c y l i n d e r Y/ of a diffeomorphism f : E -9 E. This Y/fibres over the circle S 1 with fibre E. Lift f to a bundle mapping f : Q -9 Q. This gives an SO(3)-bundle Q~- -9 Yr. On the other hand, f induces a mapping 0)~: Jtd(Q) -9 j~/(Q). The AtiyahFloer eonject~tre for" mapping cylinders was proposed by Floer [3] and reads: * k~V 1 f ' (~_) ~} H F .syrup ( M (Q), 0 9 . HF in~t

(2)

In [4], Dostoglou and Salamon prove the existence of an isomorphism between these two Floer homologies by constructing an isomorphism at the chain level and identifying the boundary operators. The idea is named adiabatic li'mit and consists of stretching Y/ in the direction orthogonal to E. A very important case is that of f = id. Then ~id = 2 X S 1 and ~ i d = Q × $1 --+ E × S 1 is the SO(3)-bundle with ~U2(Qid) = PD[S1]. Therefore,

HFinst(E, × S l , Q x S 1) ~ HFS.ylnp(2b'/(Q), id) ~-

(3)

Qn*(M(Q)). Both Floer homologies have natural product structures, introduced by S.K. Donaldson (see [11]). A stronger version of the Atiyah-Floer conjecture establishes that (3) is an isomorphism of rings. The existence of such an isomorphism has been proved by V. Mufioz in [9], [8] by giving an explicit presentation of both rings in terms of the natural generators of the cohomology of A//(Q) and using the relationship of instanton Floer homology of 3-manifolds with Donaldson invariants of 4-manifolds [2]. Also, in [12] Salamon proves that the adiabatic limit isomorphism is indeed a ring isomorphism.

References [1] ATIYAH, M.F.: 'New invariants of three and four dimensional manifolds', Prvc. Syrup. Pure Math. 48 (1988). [2] DONALDSON, S.K.: 'On the work of Andreas Floer', Jahresbet. Deutsch. Math. Verein. 95 (1993), 103-120. [3] DOSTOGLOU,S., AND SALAMON,D.: 'Instanton homology and symplectic fixed points', in D. SALAMON (ed.): Symplectic Geometry: Proc. Conf., Voh 192 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1993, pp. 57-94. [4] DOSTOOLOU, S., AND SALAMON, D.: 'Self-dual instantons and holomorphic curves', Ann. of Math. 139 (1994), 581-640. [5] FLOER, A.: 'An instanton invariant for 3-manifolds', Comm. Math. Phys. 118 (1988), 215-240. [6] FLOER, A.: 'Morse theory for the symplectic action', J. D/if. Geom. 28 (1988), 513 547. [7] FLOER, A.: 'Symplectic fixed points and holomorphic spheres', Comm. Math. Phys. 120 (1989), 575-611. [8] Mu~oz, V.: ' Q u a n t u m cohomology of the moduli space of stable bundles over a Riemann surface', Duke Math. J. 98 (1999), 525 540. [9] MuF
47

ATIYAH-FLOER C O N J E C T U R E

[12] SALAMON,I).: 'Quantum products for mapping tori and the Atiyah-Floer conjecture', Preprint ETH-Ziirich (1999).

Vicente Mur~oz MSC 1991: 57R57, 58D27, 53C15 AUTOMATIC

CONTINUITY

FOR BANACH

AL-

G E B R A S - The basic question in automatic continuity

theory is the following. Let A and B be Banach algebras (cf. B a n a c h a l g e b r a ) , and let 0: A -+ B be a h o m o m o r p h i s m . What algebraic conditions on A a n d / o r B ensure that the homomorphism 0 is automatically continuous? A variation of the question is the following. Let A be a Banach algebra, let E be a Banach A-bimodule, and let D: A -+ E be a derivation (cf. also D e r i v a t i o n in a ring). What algebraic conditions on A a n d / o r E ensure that D is automatically continuous? There are important generalizations of the latter question: for various purposes it is important to replace the derivation by the more general notion of an 'intertwining mapping'. A special case of the automatic continuity problem for homomorphisms is the uniqueness-of-norm problem, which asks which Banach algebras have a unique complete algebra norm. For a substantial recent (as of 2000) account of automatic continuity theory for Banach algebras, see [4]; all the terms that are used here are defined in [4]. The starting point for automatic continuity theory is the easily proved fact that every character on a Banach algebra A (i.e., every homomorphism from A onto the complex field C) is automatically continuous (cf. also C o n t i n u o u s f u n c t i o n ) . This was already stated in the seminal work of I.M. Gel'fand around 1940. Note that there is a deep related question. Let A be a F r ~ c h e t alg e b r a (so that the topology of A is given by a sequence of algebra semi-norms on A, and A is complete). Then it is an open question (as of 2000) whether or not every character on A is automatically continuous. This is called Michael's problem, because it was raised in [12]. A positive result is known in many cases; a striking sufficient condition, involving analytic functions of several complex variables, for the continuity of ai1 such characters is given in [7]. It follows easily from the continuity of characters that every homomorphism 0: A -+ B from a Banach algebra A into a commutative semi-simple Banach algebra B is continuous. A closely related result is Johnson's uniqueness-of-norm theorem: Every semi-simple Banach algebra has a unique complete algebra norm. For lovely alternative proofs of this theorem, see [1] and [13]. There are non-semi-simple commutative Banach algebras having a unique complete algebra norm. For example, this is true of the convolution algebras L 1(R +, ~), where cJ is a weight function on R + (see [4, §5.2]). On the other 48

hand, there are even commutative Banach algebras with a one-dimensional (Jacobson) radical which do not have a unique complete algebra norm (see [4, §5.1]). Nevertheless there are striking open questions in this area: it is not known (as of 2000) whether a commutative Banach algebra which is an integral domain necessarily has a unique complete algebra norm; the question is also open for Banach algebras with a finite-dimensional radical; for partial results, see [5]. The following question is also open. Let A and B be Banach algebras, and let 0: A -+ B be a homomorphism. Suppose that B is semi-simple and that 0(A) = B. Is 0 automatically continuous? The separating space ®(T) of a linear mapping T : E -+ F, where E and F are Banach spaces, is defined to be the set of elements y C F such that there is a sequence (x, 0 in E with x~ --+ 0 in E and Txn --+ y in F. Clearly, ~ ( T ) is a closed linear subspace of F and, by the c l o s e d - g r a p h t h e o r e m , G(T) = {0} if and only if T is continuous. A key result in automatic continuity theory is the stability lemma. One version of this is as follows; there are many variations. Let E and F be Banach spaces, let T: E --+ F be a linear mapping, let (E~ : n C Z +) be a sequence of Banach spaces with E0 = E, and let Rn E B(E,~,E~-I) (n C N). Then ( ® ( T R 1 . . . R ~ ) : n ~ N) is a nest in F that stabilizes. This leads to a proof [11] that all derivations from a semi-simple Banach algebra to itself are automatically continuous, and to many other results. Let A be a Banach algebra, let E and F be Banach A-bimodules, and let T: E ~ F be a linear mapping. The continuity ideal if(T) of T is defined to be {a e A: a. ®(T) = G ( T ) . a = {0}}. This is an ideal in A, and the 'bigger Z(T) is, the more continuous T is'. This leads to proofs that all derivations from various Banach algebras A into each Banach A-bimodule are automatically continuous; see [4, §5.3]. For example, this is true whenever A is a C * - a l g e b r a , [14]. Let 0: A -+ B be a homomorphism between Banach algebras. The main boundedness theorem of W.G. Bade and P.C. Curtis Jr. [2] asserts that, in the case where A has many idempotents, the continuity ideal Z(0) is necessarily 'large'. This leads to a proof that every homomorphism from many algebras, including the algebra B(E) of all bounded linear operators on a Banach space E for certain Banach spaces E, is automatically continuous. Let C(f~) be the commutative C*-algebra of all continuous functions on a compact space f~, taken with the uniform norm on fL The theory of Bade and Curtis shows that each h o m o m o r p h i s m from C(~) into a Banach algebra must be continuous on a dense subalgebra of C(f~); they left open the question of whether such a

AVERAGE-CASE COMPUTATIONAL COMPLEXITY homomorphism is necessarily continuous on the whole of C(f~). Eventually it was proved [3], [8] (see [4, ~5.7]) that this is not the case: For each infinite compact space ~t, there is a discontinuous homomorphism from C(f~) into certain commutative Banach algebras. Indeed, it is known just which Banach algebras arise in this situation. Note t h a t the proof of this theorem requires the assumption of the c o n t i n u u m h y p o t h e s i s CH; that some additional set-theoretic hypothesis is required is a remarkable result that is discussed in [6]. In fact, there is a discontinuous homomorphism from 'most', perhaps all, infinite-dimensional, commutative Banach algebras. It is an attractive result of J.R. Esterle [9] that all epimorphisms from C(ft) onto a Banach algebra are automatically continuous; the analogous question for C*algebras is open (as of 2000). Let G be a locally compact group, and let L I ( G ) be the corresponding group algebra. It is an active area of research to determine whether or not all derivations from L 1 (G) into a Banach L 1 (G)-bimodule are a u t o m a t ically continuous. T h a t this is true for m a n y such groups G is proved in [4, §5.6]; a key paper on which this work is based is [15]. It is a challenging open question (as of 2000) to determine whether or not this is true for all groups G, or even for all discrete groups, in which case the corresponding group algebra is denoted by (1 (G). References [1] AUPETIT, B.: 'The uniqueness of complete norm topology

in Banach algebras and Banach Jordan algebras', J. Funct. Anal. 47 (1982), 1-6. [2] BADE, W.G., AND CURTIS JR., P.C.: 'Homomorphisms of commutative Banach algebras', Amer. Y. Math. 82 (1960), 589-608. [3] DALES, H.G.: 'A discontinuous homomorphism from C ( X ) ' , Amer. J. Math. 101 (1979), 647-734. [4] DALES, H.G.: Banach algebras and automatic continuity, Vol. 24 of London Math. Soc. Monographs, Clarendon Press, 2001. [5] DALES, H.G., AND LOY, R.J.: 'Uniqueness of the norm topology for Banach algebras with finite-dimensional radical', Proc. London Math. Soc. (3) 74 (1997), 633-661. [6] DALES, H.G., AND WOODIN, W.H.: An introduction to independence for analysts, Vo]. 115 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1987. [7] DIXON, P.G., AND ESTERLE, J.R.: 'Michael's problem and the Poincar~-Bieberbach phenomenon', Bull. Amer. Math. Soc. 15 (1986), 127 187. [8] ESTERLE, J.R.: 'Sur l'existence d'un homomorphisme discontinu de C(K)', Proe. London Math. Soc. (3) 36 (1978), 46-58. [9] ESTERLE, J.R.: 'Theorems of Gelfand-Mazur type and continuity of epimorphisms from C(K)', J. Funct. Anal. 36 (1980), 273 286. [10] JOHNSON, B.E.: 'The uniqueness of the (complete) norm topology', Bull. Amer. Math. Soc. 73 (1967), 537 539. [11] JOHNSON, B.E., AND SINCLAIR, A.M.: 'Continuity of derivations and a problem of Kaplansky', Amer. J. Math. 90 (1968), 1067-1073.

[12] MICHAEL, E.A.: 'Locally multiplicatively-convex topological algebras', Memoirs Amer. Math. Soe. 11 (1952). [13] P~ANSFORD~ T.J.: 'A short proof of Johnson's uniqueness-ofnorm theorem', Bull. Amer. Math. Soc. 21 (1989), 487-488. [14] RINGROSE, JAR.: 'Automatic continuity of derivations of operator algebras', J. London Math. Soc. (2) 5 (1972), 432-438. [15] WILLIS, G.A.: 'The continuity of derivations from group algebras: factorizable and connected groups', J. Austral. Math. Soc. (Ser. A) 52 (1992), 185 204. H.G. Dales

M S C 1991:46H40 AVERAGE-CASE COMPUTATIONAL COMPLEXITY - The efficiency of an algorithm ~4 is measured by the amount of computational resources used, in the first place time (number of computation steps) and space (amount of m e m o r y cells). These values m a y depend on the individual inputs given to .4. Thus, in general it is infeasible to give a complete description of the efficiency of an algorithm, simply because the amount of d a t a grows exponentially with respect to the size of the inputs. Traditionally, one has considered the so-called worstcase complexity, which for each input size n determines the m a x i m u m amount of resources used by A on any input of t h a t size. This conservative estimation provides a guaranteed upper bound, but in specific cases m a y be much too pessimistic. The simplex algorithm is a wellknown example that is heavily used in practice since it tends to find a solution very fast [3]. In the worst case, however, it requires an exponential amount of time [13]. To estimate the average-case behaviour of an algorithm precisely is quite a difficult task in general. Only few examples are known where this has been achieved so far (as of 2000; compare [6]). Quick-sort is one of the most extensively investigated algorithms. To sort n data items, it shows a quadratic worst-case and a O(n log n) average-case time complexity [22]. For a more detailed analysis of its running time beyond the expectation, see [21]. These results have been shown for the uniform distribution, where every p e r m u t a t i o n of the input data is equally likely. For biased distributions, the analysis becomes significantly more complicated, and for arbitrary distributions the claim is no longer true. The averagecase behaviour of standard quick-sort decreases significantly if the input sequence tends to be partially presorted. See below for more details on the influence of the distribution. To determine the average-case complexity of an algorithmic problem Q, i.e. to find the 'best' algorithm for Q with respect to the average utilization of its resources, is even harder. The sorting problem is one of the rare positive exceptions, since a matching lower bound can be derived by an information-theoretic argument almost as simple as for the worst case. Again, the logn! ~ n l o g n 49

AVERAGE-CASE C O M P U T A T I O N A L C O M P L E X I T Y lower bound only holds for the uniform distribution; for arbitrary distributions it has to be replaced by their entropy. For most problems that are believed not to be efficiently solvable in the worst case, for example the dVPcomplete ones (cf. also C o m p u t a t i o n a l c o m p l e x i t y classes; C o m p l e x i t y t h e o r y ; H P ) , little is known about their average-case complexity. For example, for SAT, the satisfiability problem for Boolean formulas in conjunctive normal form, quite a large variation of behaviour has been observed for different satisfiability algorithms (for an overview, see [19]). For certain distributions they perform quite well, but the question arises what are 'reasonable' or 'typical' distributions. Similar behaviour occurs for combinatorial optimization problems based on graphs if the distribution is given according to the random graph model [5]. For example, if the edge probability is not extremely small, the dV7)complete problems '3-colourability' and 'Hamiltonian circuit' can easily be solved, simply because almost-all such random graphs contain a Hamiltonian circuit and also a 4-clique, thus cannot be 3-coloured [2], [8]. For S A T the relation between the number of clauses and the number of variables seems to be most important: with few clauses a random formula has a lot of satisfying assignments which can be found quickly. On the other hand, m a n y clauses are likely to make a random formula unsatisfiable and there is a good chance that a proof for this property can be deduced fast. For S A T there is no simple notion of a uniform distribution, but there exists a critical relation between the number of clauses and variables, for which a random formula fulfilling this relation is equally likely to be satisfiable or unsatisfiable. It has been conjectured that this boundary behaves like a phase transition with a numerical value around 4.2, like it is known for properties of random graphs [18], [12]. For distributions in this range the average-case complexity of S A T is still unresolved (as of 2000; see also [14]). Complexity estimations have to be based on a precise machine model. For static models like straight-line programs, all inputs of a given size are treated by the same sequence of operations, thus there is no difference in the running time. Boolean circuits seem to behave similarly if one measures the length of the computation simply by the circuit depth. However, a dynamic notion of delay has been established such that for different inputs the results of the computation are available at different times [10]. Thus, the average-case behaviour is not identical to the worst-case behaviour - - an average-case analysis makes sense. A precise complexity estimation has been given for m a n y Boolean functions, exhibiting in some cases a provable exponential speed-up of the 50

average-case complexity in comparison to the worst-case complexity [9]. This holds for a wide range of distributions. The complexity for generating the distributions has a direct influence on the average-case behaviour of circuits. If one bases the complexity analysis on machine models like the T u r i n g m a c h i n e or the r a n d o m access machine (RAM), the situation gets even more complicated. In contrast to the circuit model, where for each circuit the number of input bits it can handle is fixed - - thus a family of circuits is needed for different input sizes - - , now a single machine can treat inputs of arbitrary length. Hence, there are two kinds of degree to which one can average the behaviour of an algorithm. First, for each n one could measure the expected runtime given a probability distribution #n on the set of all inputs of size n. Thus, for each input size n, a corresponding distribution is required. One then requires that this expectation as a function of n is majorized by a desired complexity bound, like a linear or polynomial function. A more general approach would be to have a single probability distribution # defined over the whole input space. Then one could also weight different input sizes - - not all values of n have to be equally important. It has been observed t h a t there exists a universal distribution that makes the average-case complexity of any algorithm as bad as its worst-case complexity up to a constant factor [16]. Such distributions are called malign. Fortunately, the universal distribution is not computable, i.e. there does not exist a mechanical procedure to generate input instances according to this distribution. Thus, this universally bad distribution will probably not occur 'in real life'. From these results one can infer that an average-case analysis has to take the underlying distribution carefully into account and that a significantly better average-case behaviour can only be expected for computationally simple distributions. A simple notion for this purpose is polynomial-time computability [15]. It requires t h a t given a (natural) linear ordering X 1 , X 2 , . . . on the problem instances, the aggregated probability ~ of a distribution with density function # (cf. also D e n s i t y o f a p r o b a b i l i t y d i s t r i b u t i o n ) can be computed in polynomial time (~(Xi) = ~ x j <_x~ # (Xj) ). Finer differentiations are discussed in [1], [17], [20], [11]. From a global distribution # one can derive its normalized restriction #~ to inputs X of length n as # n ( X ) : = # ( X ) / ~ I Y I = ~ #(Y)" For most algorithmic problems, already the worstcase complexity has not yet been determined precisely. But a lot of relative classifications are known, comparing the complexity of one problem to that of another problem. This is obtained by reductions between problems. Useful here are many-one reductions, i.e. mappings p

AVERAGE-CASE COMPUTATIONAL COMPLEXITY from instances X1 for a problem Q1 to instances X2 of a problem Q2, such that from the instance p(X1) for Q2 and its solution with respect to Q2 one can easily deduce a solution for X1 with respect to QI. In the case of decision problems, for example, where for every instance the answer is either 'yes' or 'no', one simply requires that p(X1) has the same answer as X1. Thus, an algorithm solving Q2 together with a reduction from Q1 to Q2 can be used to solve Q1- In order to make this transformation meaningful, the reduction itself should be of lower complexity t h a n Q1 and Q2. Then one can infer that Q1 is at most as difficult as Q2. Polynomial-time many-one reductions require that the function p can be computed in polynomial time. They play a central role in complexity analysis, since the class 79, polynomial time complexity, is a very robust notion to distinguish algorithmic problems t h a t are feasibly solvable from the intractable ones. To investigate the complexity classes C above 79, a problem Q in C with the property that all problems in this class can be reduced to Q is called complete for C. Such a Q can be considered as a hardest problem in C - - an efficient solution of C would yield efficient solutions for all other problems in this class. Complete problems capture the intrinsic complexity of a class. This methodology has been established well for worstcase complexity. When trying to apply it to the average case, one faces several technical difficulties. Now one has to deal with pairs, consisting of a problem Q and a distribution # for Q, called distributional problems. A meaningful reduction between such problems (Q1,#1) and (Q2,#2) has to make sure t h a t likely instances of Q1 are m a p p e d to likely instances of Q2, a property called dominance [15]. Secondly, polynomial on average time complexity cannot simply be defined by taking the expectation, which would mean t h a t for an algorithm A with individual running times timeA(X) on inputs X the inequality

E #,~(X).timeA(X) Ixl=n

<

O(n k)

has to hold for some fixed exponent k (where #~(X) denotes the normalized restriction of # as defined above). This would destroy the closure properties of polynomial time reductions. Instead, the relation E x

# ( X ) ( t i m e A ( X ) ) 1/k Ix]

<~

is demanded for some number k, which is more robust against polynomial increase of running times and takes better into account the global probability distribution over the input space [15]. To establish a notion corresponding to the worst-case class 79, one can consider distributional problems (Q, #)

and require that there exists an algorithm solving Q in time polynomial on average with respect to this specific # - - this class is n a m e d A79, average-79. Alternatively, to allow a direct comparison to worstcase classes, one can restrict to computational problems Q and require t h a t they can be solved polynomially on average for all distributions up to a certain complexity, for example for all polynomial-time computable distributions. The hope that i m p o r t a n t worst-case intractable problems could be solved efficiently on average for polynomial-time computable distributions, in the case of N'79 called Distil79, is questionable. There exists a surprisingly simple relation to worst-case complexity, namely the deterministic and non-deterministic exponential-time classes 19$X79 and AfgX79. If they are different - - which is generally believed to be true, but a weaker condition t h a n 79 ¢ Y79 - - , then Dist N'79 is not contained in ¢t79; in other words, there exist Af79complete problems such t h a t their distributional counterparts cannot be solved by a deterministic algorithm efficiently on the average. An example of such a problem complete for the average case is the following so-called domino tiling problem [15]. Given a set of dominos and a square partially covered with some of them, it requires one to fill the square completely. Every domino on each of its four sides carries a m a r k and in a legal tiling adjacent dominos must have identical marks at neighbouring sides. However, the number of problems complete for the average case t h a t are known so far is significantly smaller than that for the worst case [7]. To obtain a measure for average-case complexity that allows a better separation t h a n simply polynomial or super-polynomial, these approaches have to be refined significantly. Given a time bound T, the class A v D T i m e ( T ) contains all distributional problems (Q, p) solvable by an algorithm A such t h a t its average-case behaviour obeys the following bound for all n:

E IXl_>~

#(x)T-l(timeA(N)) < E

IXl

#(X),

-ixt_>n

where T -1 denotes a suitably defined inverse function of T. Given also a complexity bound V on the distributions, the class AvDTimeDis(T, V) consists of all problems Q that for all distributions # of complexity at most V can be solved in #-average time T. This way, hierarchy results have been obtained for the average-case complexity similar to the worst case, showing t h a t any slight increase of computation time or any slight decrease of the distribution complexity allows one to solve more algorithmic problems [20], [4]. By diagonalization 51

AVERAGE-CASE COMPUTATIONAL

COMPLEXITY

a r g u m e n t s one can even exhibit p r o b l e m s with worstcase time complexity T t h a t c a n n o t be solved in average time T I for any T I <_o(T). For a detailed discussion of these concepts and further results, see [7], [1], [25], [20], [23], [24]. References [1] BEN-DAVID, S., CHOR, B., GOLDREICH, O., AND LUBY, M.: 'On the theory of average ease complexity', J. Comput. Syst. Sci. 44 (1992), 193-219. [2] BENDER, E., AND WILF, H.: 'A theoretical analysis of backtracking in the graph coloring problem', J. Algorithms 6 (1985), 275-282. [3] BORCWARDT, K.: 'The average n u m b e r of pivot steps required by the simplex-method is polynomial', Z. Operat. Res. 26 (1982), 157-177. [4] CAI, J., AND SELMAN, A.: 'Fine separation of average time compiexity classes': Proc. 13 Syrup. Theoret. Aspects of Computer Sci., Vol. 1046 of Lecture Notes in Compter Sci., Springer, 1996, pp. 331-343. [5] ERDOS, P., AND RENYI, A.: 'On the strength of connectedhess of a random graph', Acta Math. Acad. Sci. Hungar. 12 (1961), 261-267. [6] FLAJOLET, P., SALVY, B., AND ZIMMERMANN, P.: 'Automatic average-case analysis of algorithms', Theoret. Computer Sci. 79 (1991), 37-109. [7] GUREVICH, Y.: 'Average case completeness', J. Comput. Syst. Sci. 42 (1991), 346-398. [8] GUREVICH,Y., AND SHELAH, S.: 'Expected computation time for Hamiltonian path problem', S I A M J. Comput. 16 (1987), 486-502. [9] JAKOBY, A.: 'Die Komplexitgt yon Prgfixfunktionen bezfiglich ihres mittleren Zeitverhaltens', Diss. Univ. Liibeek, Shaker (1997). [10] JAKOBY,A., REISCHUK,R., AND SCHINDELHAUER, C.: 'Circuit complexity: From the worst case to the average case': Proc. 25th A C M Symp. Theory of Computing, 1994, pp. 58 67. [11] JAKOBY, A., REISCHUK, R., AND SCHINDELHAUER,C.: 'Malign distributions for average case circuit complexity', Inform. Comput. 150 (1999), 187-208. [12] KAMATH, A., MOTWANI, R., PALEM, K., AND SPIRAKIS, P.: 'Tail bounds for occupancy and the satisfiability threshold', Random Struct. Algor. 7 (1995), 59-80. [13] KLEE, V., AND MINTY, @.: 'How good is the simplex algorithm': Inequalities, Vol. III, Acad. Press, 1972, pp. 159-175. [14] KRATOCHVIL,J., SAVICKY, P., AND TUZA, Z.: 'One more occurence of variables makes satisfiability jump from trivial to NP-complete', S I A M J. Comput. 22 (1993), 203-210. [15] LEVIN, L.: 'Average case complete problems', S I A M J. Cornput. 15 (1986), 285-286. [16] LI, M., AND VITANYI, P.: 'Average case complexity under the universal distribution equals worst case complexity', Inform. Proc. Lett. 42 (1992), 145 149. [17] MILTERSEN, P.: 'The complexity of malign measures', S I A M J. Comput. 22 (1993), 147-156. [181 MITCHEL, D., SEAMAN, B., AND LEVESQUE, H.: 'Hard and easy distributions of SAT problems': Proc. lOth Amer. Assoc. Artificial Intelligence, AAAI/MIT, 1992, pp. 459-465. [19] PURDOM, P.: 'A survey of average time analysis of satisfiability algorithms', J. Inform. Proees. 13 (1990), 449-455. [20] REISCHUK, R., AND SCHINDELHAUER, C.: 'An average complexity measure that yields tight hierarchies', Comput. Complexity 6 (1996), 133-173.

52

[21] ROSLER, U.: 'A limit theorem for quicksort', Theoret. Inform. Appl. 25 (1991), 85-100. [22] SEDCEWICK, R.: 'The analysis of quicksort programs', Acta Inform. 7 (1977), 327-355. [23] WANG, J.: 'Average-case computational complexity theory', in A. SEAMAN AND L. HEMASPAANDRA(eds.): Complexity Theory Retrospective, VoI. II, Springer, 1997, pp. 295 328. [24] WANG, J.: 'Average-case intractable problems', in D. Du AND K. KO (eds.): Advances in Languages, Algorithms, and Complexity, Kluwer Acad. Publ., 1997, pp. 313-378. [25] WANG, J., AND BELANGER, J.: 'On average P vs. average NP', in K. AMBOS-SPIES, S. HOMER, AND U. SCHNING(eds.): Complexity Theory Current Research, Cambridge Univ. Press, 1993, pp. 47-68. R. Reischuk

MSC1991:68Q15 A V E R A G E S A M P L E N U M B E R , ASN - A t e r m occurring in the area of statistics called s e q u e n t i a l a n a l ysis. In sequential procedures for statistical estimation, hypothesis testing or decision theory, the n u m b e r of observations t a k e n (or sample size) is not pre-determined, but depends on the observations themselves. T h e expected value of the sample size of such a procedure is called the average sample number. This depends on the underlying distribution of the observations, which is often unknown. If the distribution is d e t e r m i n e d by the Value of some p a r a m e t e r , then the average sample number becomes a function of t h a t p a r a m e t e r . If N denotes the sample size and 0 is the unknown p a r a m e t e r , then the average sample n u m b e r is given by

Ee(N) = E n P e ( N n=l

= n) = E P e ( N

> n).

(1)

n=0

Some aspects of the average sample n u m b e r in the case where the observations are independent Bernoulli r a n d o m variables Xi with Eo(Xi) = Po(Xi = 1) = 0 = 1 Po (Xi = 0) are given below. Consider the case of the curtailed version of a test of the h y p o t h e s e s / t 0 : 0 = 0 versus H I : 0 > 0 t h a t takes n observations and decides H I if X1 + • • • + X~ > 0 (cf. also S t a t i s t i c a l h y p o t h e s e s , v e r i f i c a t i o n of). T h e curtailed version of this test is sequential and stops the first time t h a t Xk = 1 or at time n, whichever comes first. Then the average sample n u m b e r is given by n--1

Ee(N) = ~

n--1

Pe(N > k) = ~--~(I - 0) k =

k=0 -- 1 - ( 1 - 0 ) n 0

(2)

k=0 for0>0.

This formula is a special case of Wald's l e m m a (see [3] and W a l d i d e n t i t y ) , which is very useful in finding or a p p r o x i m a t i n g average sample numbers. Wald's lemma states t h a t if !/, !/1, I/'2, • • • are independent r a n d o m variables with c o m m o n e x p e c t e d value (cf. also R a n d o m v a r i a b l e ) and N is a s t o p p i n g t i m e (i.e., N = k is

AVERAGE SAMPLE NUMBER determined by the observations K x , . . . , Yk) with finite expected value, then letting Sn = Y1 + • • • + Y,, E(Y)E(N) = E(SN).

(3)

Thus, for E(Y) # O, one has E(N) = E(SN)(E(Y)) -1. In this example, !// = Xi, E(Y) = 0, and FO(SN) = Po(SN = 1) = 1 -- PO(Sn = 0) = 1 - (1 - 0)% The average sample number then follows from (3) and agrees with (2). See [1] for asymptotic properties of the average sample number for curtailed tests in general. In t e s t i n g H 0 : 0 = p v e r s u s H i : 0 = q = 1 - p , the logarithm of the likelihood ratio (cf. also L i k e l i h o o d r a t i o t e s t ) after n observations is easily seen to be of the form Sn log(q/p), where !/i = 2 X i - 1 . Thus, ifp < .5, the sequential probability ratio test stops the first time t h a t Sn = K and decides H i or the first time that Sn = - J and decides H0 for positive integers J and K . In this case $1, S 2 , . . . is a r a n d o m w a l k taking steps to the right with probability 0 and E(Y) = 20 - 1 in formula (3). Thus, if 0 ¢ 1/2, the average sample number is E0(N) = PO(SN = K)K20-_Po(SN1

= -J)J

(4)

Well-known formulas from the theory of r a n d o m walks show that PO(SN = K ) = (1 - rJ)(1 - r K + J ) -1, where r = (1 - 0)/0. If 0 = .5, this method fails. One must then use another result of A. Wald, stating t h a t if E(Y) = 0, but E(Y/2) = cr2 < c~, then

a2E(N) = E(S~r).

(5)

In this example, for 0 = .5, one has (72 = .25 and P(SN = B2) = J ( J + K ) -1. Then (5) yields E(N) = 4 J K . In order to decide which of two sequential tests is better, it is important to be able to express the average sample number in terms of the error probabilities of the test. Then the average sample numbers of different tests with the same error probabilities can be compared. In this example, the probability of a type-I error

a = Pp(SN = K ) and the probability of a type-II error (cf. also E r r o r ) . From this one sees

/3 = Pq(SN = - J ) that

K-~log (~-~)

( l o g q ) -1

and -1

In particular, it follows t h a t if 0 = p, then a l o g ( L : ~-) + ( 1 - a ) l o g (1_-~) Ep(N) =

(p - q)log(q/p)

(6)

This formula and the analogous one with 0 = q are often used to measure the efficiency of the sequential probability ratio test against other sequential tests with the same error probabilities Similar formulas can be found for sequential probability ratio tests for sequences of independent random variables with other distributions, but in general the formulas are approximations since the tests will stop when the logarithm of the likelihood ratio first crosses the boundary rather than when it first hits the boundary. Many studies (see [2]) consider the adequacy of approximations for the average sample number of sequential probability ratio tests and some other sequential tests whose stopping times are more complicated. In areas such as signal detection, average sample numbers are also studied for sequential tests where the observations are not independent, identically distributed random variables. References

[1] EISENBERG,B., AND GHOSH, B.K.: 'Curtailed and uniformly most powerful sequential tests', Ann. Statist. 8 (1980), 11231131. [2] SIEMUND, D.: Sequential analysis: Tests and confidence intervals, Springer, 1985. [3] WALD,A.: Sequential analysis, Wiley, 1947. Bennett Eisenberg MSC1991: 62Lxx

53

B BAILY-BOREL COMPACTIFICATION, SatakeBaily-Borel compactification Let G be a semi-simple l i n e a r a l g e b r a i c g r o u p (cf. also S e m i - s i m p l e algeb r a i c g r o u p ) defined over Q, meaning that G can be embedded as a subgroup of GL(m, C) such that each element is diagonalizable (cf. also D i a g o n a l i z a b l e alg e b r a i c g r o u p ) , and that the equations defining G as an a l g e b r a i c v a r i e t y have coefficients in Q (and that the group operation is an algebraic morphism). Further, suppose G contains a torus (cf. A l g e b r a i c t o r u s ) that splits over Q (i.e., G has Q-rank at least one), and G is of Hermitian type, so that X := K \ G(R) can be given a complex structure with which it becomes a symmetric domain, where G(R) denotes the real points of G and K is a maximal compact subgroup. Finally, let F be an arithmetic subgroup (cf. A r i t h m e t i c g r o u p ) of G(Q), commensurable with the integer points of G. Then the arithmetic quotient V := X / F is a n o r m a l a n a l y t i c s p a c e whose Baily-Borel compactification, also sometimes called the Satake-Baily Borel compactification, is a canonically determined projective normal a l g e b r a i c v a r i e t y V*, defined over C, in which V is Zariski-open (cf. also Zariski t o p o l o g y ) [6] [7] [4] [5]. To describe V* in the complex topology, first note that the Harish-Chandra realization [12] of X as a bounded symmetric domain may be compactified by taking its topological closure. Then a rational boundary component of X is a boundary component whose s t a b i l i z e r in G(R) is defined over Q; based on a detailed analysis of the R-roots and Q-roots of G, there is a natural bijection between the rational boundary components of X and the proper maximal parabolic subgroups of G defined over Q. Let X* denote the union of X with all its rational boundary components. Then (cf. [19]) there is a unique topology, the Satake topology, on X* such that the action of G(Q) extends continuously and V* = X * / F with its quotient topology compact and Hausdorff. It also follows from the construction that V*

is a finite disjoint union of the form

v* = V u v 1 u . . . u v t , where 17/ = Fi/Fi for some rational boundary component Fi of X*, and Fi is the intersection of F with the stabilizer of Fi. In addition, V and each V/has a natural structure as a n o r m a l a n a l y t i c space; the closure of any V/is the union of V/with some Vjs of strictly smaller dimension; and it can be proved that every point v E V* has a fundamental system of neighbourhoods {Us} such that Us N V is connected for every s. In order to describe the structure sheaf of V* (cf. also S c h e m e ) with which it becomes a normal analytic space and a projective variety, define an A-function on an open subset U C V* to be a continuous complexvalued function on U whose restriction to U A 17/ is analytic, 0 < i < t, where V0 = V. Then, associating to each open U the C-module of A-functions on U determines the s h e a f A of germs of A-functions. Further, for each i the sheaf of germs of restrictions of A-functions to Vi is the structure sheaf of V/. Ultimately it is proved [7] that (V*, ~4) is a normal analytic space which can be embedded in some complex projective space as a projective, normal algebraic variety. The proof of this last statement depends on exhibiting that in the collection of A-functions there are enough automorphic forms for F, more specifically, Poincard-Eisenstein series, which generalize both Poincar@ series and Eisenstein series (cf. also T h e t a - s e r i e s ) , to separate points on V* as well as to provide a projective embedding.

History and examples. The simplest example of a BailyBorel compactification is when G = SL(2, Q), and F = SL(2, Z), and X is the complex upper half-plane,

on hi h (a

in

d) - ] . (The bounded realization of X is a unit disc, to which the upper half-plane maps by z ~-~ (z-v/-Z-1)/(z+ x/L~).) The properly discontinuous action of F on X extends to X* = x U O U { o o } , and V* = X * / F is a smooth

BANACH-JORDAN ALGEBRA projective curve. Since G has Q-rank one, V* - V is a finite set of points, referred to as cusps. Historically the next significant example was for the Siegel modular group, with G = Sp(2n, Q), and F = Sp(2n, Z), and X = ~ n consisting of n × n symmetric complex matrices with positive-definite imaginary part; here ( C \

D)in

Sp(2n, R ) a c t s on Z C ~ n by

/

Z ~-+ (AZ + B ) ( C Z + D) -1. I. Satake [18] was the first to describe a compactification of V~ = ~,~/F as V* = V~ U .-. U Vo endowed with its Satake topology (cf. also S a t a k e c o m p a c t i f i c a t i o n ) . Then Satake, H. Caftan and others (in [9]) and W.L. Baily [2] further investigated and exhibited the analytic and algebraic structure of V*, using automorphic forms as mentioned above. Baily [3] also treated the Hilbert-Siegel modular group, where G(Q) = Sp(2n, F) for a totally real n u m b e r field F. In the meanwhile, under only some mild assumption about G, Satake [19] constructed V* with its Satake topology, while I.I. Piateckii-Shapiro [16] described a normal analytic compactification whose topology was apparently weaker than that of the Baily-Borel compactification. Later, P. Kiernan [13] showed that the topology defined by Piateckii-Shapiro is homeomorphic to the Satake topology used by Baily and Borel. Other compactifications. Other approaches to the compactification of arithmetic quotients of symmetric domains to which the Satake and Baily-Borel approach may be compared are the Borel-Serre compactification [8], see the discussion in [21], and the method of toroidal embeddings [1]. Cohomology. Zucker's conjecture [20] that the (middle perversity) intersection cohomology [11] (cf. also I n t e r s e c t i o n h o m o l o g y ) of the Baily-Borel compactification coincides with its L2-cohomology, has been given two independent proofs (see [15] and [17]); see also the discussion and bibliography in [10]. Arithmetic and moduli. In many cases V has an interpretation as the moduli space for some family of Abelian varieties (cf. also M o d u l i t h e o r y ) , usually with some additional structure; this leads to the subject of Shimura varieties (aft also S h i m u r a v a r i e t y ) , which also addresses arithmetic questions such as the field of definition of V and V*. Geometrically, the strata of V* - V parameterize different semi-Abelian varieties, i.e., semidirect products of algebraic tori with Abelian varieties, into which the Abelian varieties represented by points on V degenerate. For an example see [14], where this is thoroughly worked out for Q-forms of SU(n, 1), especially for n = 2.

References [1] ASH, A., MUMFORD, D., RAPOPORT, M., AND TAI, Y.: Smooth compactifications of locally symmetric varieties, Math. Sci. Press, 1975. [2] BA;LY, JR., W.L.: 'On Satake's compactification of Vn', Amer. J. Math. 80 (1958), 348-364. [3] BAILY, JR., W.L.: 'On the Hilbert-Siegel modular space', Amer. J. Math. 81 (1959), 846-874. [4] BAILY, JR., W.L.: 'On the orbit spaces of arithmetic groups': Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, 1965, pp. 4-10. [5] BAILY, JR., W.L.: 'On compactifications of orbit spaces of arithmetic discontinuous groups acting on bounded symmetric domains': Algebraic Groups and Discontinuous Subgroups, Vol. 9 of Proc. Syrup. Pure Math., Amer. Math. Soc., 1966, pp. 281-295. [6] BAILY, JR., W.L., AND BOREL, A.: 'On the compactification of arithmetically defined quotients of bounded symmetric domains', Bull. Amer. Math. Soc. 70 (1964), 588 593. [7] BAILY, JR., W.L., AND BOREL, A.: 'Compactification of arithmetic quotients of bounded symmetric domains', Ann. of Math. (2) 84 (1966), 442-528. [8] BOREL, A., AND SERRE, J.P.: 'Corners and arithmetic groups', Comment. Math. Helv. 48 (1973), 436-491. [9] Fonctions automorphes, Vol. 1-2 of Sdm. H. Cartan 10i~me ann. (1957/8), Secr. Math. Paris, 1958. [10] GORESKY, M.: 'L2-cohomology is intersection cohomology', in R.P. LANGLANDSAND D. RAMAKRISHNAN(eds.): The Zeta Functions of Picard Modular Surfaces, Publ. CRM, 1992, pp. 47-63. [11] GORESKY, M., AND MACPHERSON, R.: 'Intersection homology, II', Invent. Math. 72 (1983), 135-162. [12] HARmH-CRANDRA: 'Representations of semi-simple Lie groups. VI', Amer. J. Math. 78 (1956), 564-628. [13] KIERNAN, P.: 'On the compactifications of arithmetic quotients of symmetric spaces', Bull. Amer. Math. Soc. 80 (1974), 109-110. [14] LANGLANDS,R.P., AND RAMAKRISHNAN,D. (eds.): The zeta functions of Picard modular surfaces, Publ. CRM, 1992. [15] LOOUENGA, E.: 'L2-cohomology of locally symmetric varieties', Computers Math. 67 (1988), 3-20. [16] PIATECKII-SHAPIRO,I.I.: 'Arithmetic groups in complex domains', Russian Math. Surveys 19 (1964), 83-109. (Uspekhi Mat. Nauk. 19 (1964), 93-121.) [17] SAPER, L., AND STERN, M.: 'L2-cohomology of arithmetic varieties', Ann. of Math. 132, no. 2 (1990), 1-69. [18] SATAKE,I.: 'On the compactification of the Siegel space', Y. Indian Math. Soc. (N.S.) 20 (1956), 259-281. [19] SATAKE,I.: 'On compactifications of the quotient spaces for arithmetically defined discontinuous groups', Ann. of Math. 72, no. 2 (1960), 555-580. [20] ZUCKER, S.: 'L2-cohomology of warped products and arithmetic groups', Ann. of Math. 70 (1982), 169-218. [21] ZUCKER, S.: 'Satake compactifications', Comment. Math. Helv. 58 (1983), 312 343.

B. Brent Gordon MSC1991: l l F x x , 20Gxx, 22E46

BANACH-JORDAN ALGEBRA, Jordan-Banach algebra - A J o r d a n a l g e b r a over the field of real or complex numbers, endowed with a complete norm H'[] 55

BANACH JORDAN A L G E B R A satisfying ° yll _< 11 11 Ilyll

for all x, y in the algebra. Since an (associative) B a n a c h algebra is a B a n a c h - J o r d a n algebra under the Jordan product x o y := (xy + yx)/2, the theory of Banach-Jordan algebras can be regarded as a generalization of that of Banach algebras. For forerunners in this last theory, see B a n a c h a l g e b r a and [5]. Pioneering papers on Banacb-Jordan algebras are [4], [19] and [13]. A relatively complete panoramic view of the results on Banach Jordan algebras can be obtained by combining [16], [3] and [7]. Spectral methods in B a n a c h - J o r d a n algebras have been possible thanks to the concept of invertible element in a Jordan algebra with a unit, introduced by N. Jacobson and K. MeCrimmon (see [12] or J o r d a n algebra). From this concept, the spectrum sp(J, x) of an arbitrary element x of a Banach-Jordan algebra J is defined as in the associative case, and the spectral radius formula lim{llxntl ~/n} max{lAI : A c sp(J, x)} holds. In fact, Banach Jordan algebras are 'locally spectrally' associative. This means that each element in such an algebra J can be imbedded in some closed associative subalgebra J' of J satisfying sp(J, x) = sp(J', x) for every x E J'. Then, for a single element in a complex B a n a c h - J o r d a n algebra, a holomorphic functional calculus follows easily. A Jordan algebra is said to be semi-simple (or semiprimitive, as preferred by people working in pure algebra) whenever its Jacobson-type radical [11] is zero (cf. also J a c o b s o n r a d i c a l ) . Refining spectral methods, B. Aupetit [2] gave a Jacobson-representation-theoryfree proof of Johnson's uniqueness-of-norm theorem for semi-simple Banach algebras, and extended the result to semi-simple Banach Jordan algebras. The absence of representation theory in Aupetit's proof was relevant because, although semi-simple Jordan algebras can be expressed as subdirect products of Jordan algebras which are 'primitive' (in a peculiar Jordan sense), primitive Jordan algebras were not well-understood at that time. Aupetit's methods have shown also useful in extending fi'om Banach algebras to Banach-Jordan algebras many other relevant results (see again [3]), as well as in obtaining a general non-associative variant of Johnson's theorem [15]. Recently, using work of E.I. Zel'manov [22] on Jordan algebras without any finiteness condition, primitive Banach-Jordan algebras have been described in detail [8]. Such a description has allowed one to extend to Banach Jordan algebras the Johnson-Sinclair theorem, stating that derivations on semi-simple Banach algebras (cf. also D e r i v a t i o n in a ring) are automatically continuous [18]. JB-algebras are defined as the real Banach-Jordan algebras J satisfying Ilxll 2 _< II~2 +y211 for all x,y C J. =

56

The basic theory of JB-algebras, originally due to E.M. Alfsen, F.W. Shultz and E. Stormer [1], is fully treated in [10]. If A is a C*-algebra, then the self-adjoint part A~a of A is a JB-algebra under the Jordan product. Closed subalgebras of Asa, for some C*-algebra A, become relevant examples of JB-algebras, and are called JC-algebras. Through the consideration of J B W algebras (i.e., JB-algebras that are dual Banach spaces, cf. also B a n a c h space), JBW-factors (i.e., prime JBWalgebras), and factor representations of a given JBalgebra J (i.e., w*-dense range homomorphisms from J to JBW-factors), the knowledge of arbitrary JB-algebras is reasonably reduced to that of JC-algebras and the exceptional J B - a l g e b r a / / 3 ( 0 ) of all Hermitian (3 x 3)matrices over the alternative d i v i s i o n a l g e b r a O of real octonions. JB*-algebras are defined as complex Banach-Jordan algebras J endowed with a conjugate-linear algebra involution * satisfying IIVx(x*)H = ]lzlt 3 for every z E J. Here, for x E J, Ux denotes the operator on J defined by Ux(y) := 2 x o ( z o y ) - x 2 o y for e v e r y y C J. Every C*-algebra becomes a JB*-algebra under its Jordan product. JB*-algebras are closely related to JBalgebras. Indeed, JB-algebras are nothing but the seifadjoint parts of JB*-algebras [20]. The one-to-one categorical correspondence between JB-algebras and JB*algebras derived from the above result completely reduces the *-theory of JB*-algebras to the theory of JBalgebras. However, JB*-algebras are of interest on their own, mainly due to their connection with complex analysis (see [6], [17], and [21]). Using Zel'manov's prime theorem, the structure theory of JB- and JB*-algebras can be refined as follows (see [9]). A JB*-algebra J is primitive if and only if it is of one of the following types: • J is the unique JB*-algebra whose self-adjoint part is H 3 ( O ) .

• There exists a complex Hilbert space (H, ('[')) of dimension _> 3, with a conjugation a and a a-invariant norm-one element 1, such that J = H as complex vector spaces, whereas the product o, the involution *, and the norm 11"11of J are given by x o y := (xll)y

+ (vtl)x -

: = 2(11 Hxll 2 : = (

lx) +

)1 2

respectively. • There exists a primitive C*-algebra A such that d is a closed self-adjoint Jordan subalgebra of the C*algebra M(A), of multipliers of A, containing A. • There exists a primitive C*-algebra with a *involution ~- such that J is a closed self-adjoint Jordan

B A N A C H - J O R D A N PAIR

subalgebra of M ( A ) contained in the ~--Hermitian part of M ( A ) and containing the T-tIermitian part of A. From the point of view of analysis, the Jordan identity x o (y o x 2) = (x o y) o x 2 (which, together with the commntativity, is characteristic of Jordan algebras) can be regarded as a theorem instead of as an axiom. Indeed, if a unital complete normed non-associative complex algebra A is subjected to the geometric conditions that, through the Vidav Pahner theorem, characterize C*-algebras in the associative setting, then A under the product x o y := (xy + y x ) / 2 and a suitable involution becomes a JB*-algebra [14]. This article is dedicated to the memory of Eulalia GarcYa Rus.

References [1] ALFSEN, E . ~ . , SHULTZ, F.W., AND STORMER, E.: 'A Ge]fand-Neumark theorem for Jordan algebras', Adv. Math. 28 (1978), 11 56. [2] AUPETIT, B.: 'The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras', d. Funct. Anal. 47 (1982), 7-25. [3] AUPETIT, B.: 'Recent trends in the field of Jordan Banach algebras', in J. ZEMXNEK (ed.): Functional Analysis and Operator Theory, Vol. 30, Banach Center Publ., 1994, pp. 9 19. [4] BALACHANDRAN, V.K., AND REMA, P.S.: 'Uniqueness of the complete norm topology in certain Banach Jordan algebras', Publ. Ramanujan Inst. 1 (1969), 283 289. [5] BONSALL, F.F., AND DUNCAN, J.: Complete normed algebras, Springer, 1973. [6] BRAUN, R.B., KAUP, W., AND UPMEIER, H.: 'A holomorphic characterization of Jordan C*-algebras', Math. Z. 161 (1978), 277-290. [7] CABaERA, M., MORENO, A., AND RODRmUEZ, A.: 'Normed versions of the Zel'manov prime theorem: positive results and limits', in A. GHEONDEA, R.N. GOLOGAN, AND D. TIMOTIN (eds.): Operator Theory, Operator Algebras and Related Topics (16th Intcrnat. Conf. Operator Theory, Timisoaru (Romania) July, 2-10, 1996), The T h e t a Foundation, Bucharest, 1997, pp. 65-77. [8] CABaERA, M., MORENO, A., AND RODRI~UEZ, A.: 'Zel'manov's theorem for primitive Jordan-Banach algebras', Y. London Math. Soe. 57 (1998), 231-244. [9] FERNANDEZ, A., GAaC~A, E., AND RODRmUEZ, A.: 'A Zelm a n o r prime theorem for JB*-algebras', J. London Math. Soc. 46 (1992), 319 335. [10] HANCHE-OLSEN, H., AND STORMER, E.: Jordan operator algebras, Voh 21 of Monograph Stud. Math., Pitman, 1984. [11] HOGBEN, L., AND MCCRIMMON~ K.: 'Maximal modular inner ideals and the Jacobson radical of a Jordan algebra', J. Algebra 68 (1981), 155-169. [12] JACOBSON, N.: Structure and representations of Jordan algebras, Vol. 37 of Colloq. Publ., Amer. Math. Soe., 1968. [13] PUTTER, P.S., AND YOOD, B.: 'Banach Jordan .-algebras', Proc. London Math. Soc. 41 (1980), 21-44. [14] RODRIGUEZ, A.: 'Nonassociative normed algebras spanned by hermit±an elements', Proc. London Math. Soc. 47 (1983), 258-274.

[15] RODRIGUEZ, A.: 'The uniqueness of the complete algebra norm topology in complete normed nonassociative algebras', d. Funct. Anal. 60 (1985), 1 15. [16] RODaIGUEZ, A.: ~Jordan structures in analysis', in W. KAUP, K. McCRIMMON, AND H.P. PETERSSON (eds.): Jordan Algebras (Proc. Conf. Oberwolfaeh, Germany, August, 9-I5, 1992), de Gruyter, 1994, pp. 97--186. [17] UPMEIER, H.: S y m m e t r i c Banach manifolds and Jordan C*algebras, North-Holland, 1985. [18] VILLENA,A.R.: 'Continuity of derivations on Jordan-Banach algebras.', Studia Math. 118 (1996), 205-229. [19] VIOLA DEVAPAKKIAM, C.: 'Jordan algebras with continuous inverse', Math. Japon. 16 (1971), 115-125. [20] WRIGHT, J . D . M . : 'Jordan C*-algebras', Michigan Math. d. 24 (1977), 291 302. [21] YOUNGSON, M.A.: 'Non unitM Banach Jordan algebras and C*-triple systems', Proc. Edinburgh Math. Soc. 24 (1981), 19-31. [22] ZEL'MANOV, E.: 'On prime Jordan algebras II', Sib. Math. J. 24 (1983), 89-104.

Angel Rodr{guez Palacios MSC1991: 17C65, 46H70, 46L70

B A N A C H - J O R D A N PAIR - A real or complex Jordan pair V = ( V + , V - ) (i.e. a pair V = (V + , V - ) of real or complex vector spaces together with trilinear mappings V ± x V T x V - --+ V +, (x,y,z) ~-+ { x y z } , symmetric in the outer variables, which satisfy:

for all u , v , w E V 4-, x , y E V~:; cf. also V e c t o r space), where the spaces V ± are endowed with complete norms (cf. also N o r m ) making continuous the triple products { x y z } , x , z C V ±, y C V ~:. Banach-Jordan triple systems (cf. also J o r d a n t r i p l e s y s t e m ) can be viewed as Banach Jordan pairs V with V + = V - . For the general theory of Jordan pairs, see [16]. In the notation of quadratic Jordan theory, Q~y = { x y x } / 2 . A classic example of a Banach-Jordan pair is that given by (BL(X, Y), BL(Y, X)), the space of continuous linear operators between two Banach spaces X, Y, with {abe} = abe + cba (of. also B a n a c h space; Linear o p e r a t o r ) . Any B a n a c h - J o r d a n a l g e b r a J gives rise to a Banach-Jordan pair (J, J), by taking the usual Jordan triple product. Moreover, Banach-Jordan pairs are interesting in themselves by their connection with bounded symmetric domains [17]. Jordan-Banach triple systems with additional geometric requirements emerged in the work of W. Kaup. See [13], [26] or JB*t r i p l e for a general report on the theory of JB*-triples (prime JB*-triples have been classified in [23]); see [24] for the structure theory of Hilbert triples, Hermit±an Hilbert triples and atomic JBW*-triples, and see [6], [7] for so-called H*-triple theory. A general account of both the geometric and the non-geometric theory of JordanBanach triple systems can be found in [25]. 57

BANACH-JORDAN PAIR Given a Banach-Jordan pair V and an element y C V -~, c~ = +, one can define a B a n a c h - J o r d a n algebra on V °, called the y-homotope of V ~ and denoted by V ~(~), by taking x. z = {xyz}/2 as product, and the original norm I111 of v ~. o f course, v ~(y) can be given an equivalent norm I1'11' so that ]Ix. zll' ~ Ilxll'llzll'. The set Ker(y) = {x E V~: Qyx = 0} is a closed ideal of V ~(y), and so the quotient V~(Y)/Ker(y) is a B a n a c h Jordan algebra, called the local algebra of V at y and usually denoted by V~. Important notions in the theory of Jordan pairs, such as the J a c o b s o n radleal, are defined in terms of homotopes (the adverb 'properly' usually refers to properties that hold in all homotopes); on the other hand, local algebras have become a prominent tool in the study of Jordan pairs [9]. Local algebras play a fundamental role in proving the automatic continuity of derivations on Banach-Jordan pairs [12], thus extending the extension, by A.R. Villena [28], of the Johnson-Sinclair theorem to Banach-Jordan algebras.

Finiteness conditions in B a n a c h - J o r d a n pairs. A basic finiteness condition in Jordan pairs is having finite capacity, i.e. the ascending and descending chain conditions on principal inner ideals (cf. also Chain condition). For a non-degenerate Jordan pair V (QxV T = 0 implies x = 0, x c V+), the socle of V can be characterized as the ideal Soc(V) of all y E V -~ such that Vy has finite capacity [22] (cf. also Socle). Notice that for a Jordan pair of type (BL(X, Y), BL(Y, X)), its socle is the ideal (FBL(X, Y), FBL(Y, X)) of all continuous linear operators having finite rank (cf. also Linear operator). Moreover, it has finite capacity if and only if one of the two vector spaces is finite dimensional. It is well known that under certain assumptions (von Neumann regularity, finite spectrum, algebraicity or coincidence with the socle) a semi-primitive associative Banach algebra is finite dimensional. This is no longer true for Jordan-Banach algebras, but, under any of the above requirements (in the appropriate Jordan versions), semi-primitive Jordan-Banach algebras have finite capacity [2]. Using this result and the fact that semi-primitivity is inherited by local algebras, it can be shown (see [18] and [15]) that for a semi-primitive complex Banach-Jordan pair, the socle, the set of properly algebraic elements, the largest properly algebraic ideal, the largest properly spectrmn-finite ideal, and the largest von Neumann regular ideal all coincide. Related results can be found in [14], [20] and [21]. Since yon Neumann regular Banach-Jordan pairs are idempotentfinite, it follows that any von Neumann regular Banach Jordan pair has finite capacity ([19] and [10]). Loos' classification of simple Jordan pairs of finite capacity (any Jordan pair of finite capacity is a direct sum of 58

finitely many simple ones) can be refined in the complex Banach setting: A simple complex Banaeh Jordan pair having finite capacity is either finite dimensional, the Jordan pair defined by a continuous symmetric bilinear form on a complex Banach vector space, or a Jordan pair of continuous linear operators between two Banach pairings, one of them being finite dimensional ([10] and [11]). Using Zel'manov's structure theorem for strongly prime Jordan systems [29] (see also [8]), prime complex B a n a c h - J o r d a n .-triples with non-zero socle and without nilpotent elements were classified in [4], getting as a consequence, among other results, the Bunce-Chu structure theorem [5] for compact JB*-triples. An associative algebra is Noetherian if it satisfies the ascending chain condition on left ideals (eft also As-

sociative rings and algebras; N o e t h e r i a n ring). A.M. Sinclair and A.W. Tullo showed [27] that a complex Noetherian Banach associative algebra is finite dimensional. For a Jordan algebra or pair, the suitable Noetherian condition is the ascending chain condition on inner ideals. M. Benslimane and N. Boudi [1] proved that a complex Noetherian Banach-Jordan algebra is finite dimensional. This result has been extended, [3], to Banach-Jordan pairs. Actually, the following has been proved: i) the J a c o b s o n radical of a Noetherian BanachJordan pair is finite dimensional; ii) non-degenerate Noetherian Banach-Jordan pairs have finite capacity; and iii) complex Noetherian Banach-Jordan pairs are finite dimensional.

References [1] BENSLIMANE, M., AND BOUDI, N.: 'Noetherian Jordan Banach algebras are finite-dimensional', J. Algebra 213 (1998), 340-350. [2] BENSLIMANE, 1VII., FERNANDEZ, A., AND KAIDI, A.: 'Caract~risation des alg~bres de Jordan Banach de capacit~ finie', Bull. Sei. Math. 112, no. 2 (1988), 473-480. [3] BOUDI, N., FERN.4.NDEZ, A., GARCIA, E., MARHNINE, H., AND ZARHOUTI, C.: 'Noetherian Jordan Banach pairs', Math. Proc. Cambridge Philos. Soc. 130 (2001), 25-36. [4] BOUHYA, t(., AND FERNJ,NDEZ, A.: 'Jordan-*-triples with minimal inner ideals and compact JB*-triples', Proc. London Math. Soc. 68 (1994), 380-398. [5] BUNCE, L.J., AND CHU, C.H.: 'Compact operations, multipliers and the Radon Nikodym property in JB*-triples', Pacific Y. Math. 53 (1992), 249 265. [6] CASTELLdN, A., CUENCA, J.A., AND MARTIN, C.: 'Jordan H*-triple systems', in S. GONZJ,LEZ (ed.): Non-Associative Algebra and Its Applications, Kluwer Acad. Publ., 1994, pp. 66-72. [7] CASTELLdN, A., CUENCA, J.A., AND MARTIN, C.: 'Special Jordan H*-triple systems', Commun. Algebra 28 (2000), 4699-4706. [8] D'AMOUR, A.: 'Quadratic Jordan algebras of hermitian type', Y. Algebra 149 (1992), 197-233.

B A N A C H - M A Z U R GAME [9] D'AMOuR, A., AND MCCRIMMON,K.: 'The local algebra of a Jordan system', J. Algebra 177 (1995), 199-239. [10] FERNiNDEZ, A., GARCIA, E., AND S~NCHEZ, E.: 'Von Neumann regular Jordan Banach triple systems', J. London Math. Soc. 42 (1990), 32-48. [11] FERN~,NDEZ, A., GARCIA, E., S~.NCHEZ, E., AND SILES, M.: 'Strong regularity and hermitian Hilbert triples', Quart. J. Math. Oxford 45 (1994), 43 55. [12] FERNiNDEZ, A., MARHNINE, H., AND ZARHOUT1, C.: 'Derivations on Banach Jordan pairs', Quart. J. Math. Oxford (to appear). [13] FRIEDMAN, Y.: 'Bounded symmetric domanin and the J B * triple structure in physics', Proc. 1992 Oberwolfach Conf. Jordan algebras (1994), 61-82. [14] HESSENBERGER, G.: 'Inessential and Riesz elements in Banach Jordan systems', Quart. J. Math. Oxford 47 (1996), 337-347. [15] HESSENBERGER, G.: 'A spectral characterization of the socle of Banach Jordan systems', Math. Z. 223 (1996), 561-568. [16] LOOS, O.: Jordan pairs, Vol. 460 of Lecture Notes in Mathematics, Springer, 1975. [17] LOOS, O.: Bounded symmetric domains and Jordan pairs, Lecture Notes. Univ. California, 1977. [18] Loos, O.: 'Properly algebraic and spectrum-finite ideals in Jordan systems', Math. Proc. Cambridge Philos. Soc. 114 (1993), 149-161. [19] LOOs, O.: 'Recent results on finiteness conditions in Jordan pairs': Proc. 1992 Oberwolfach Conf. Jordan algebras, de Gruyter, 1994, pp. 83 95. [20] Loos, O.: 'Nuclear elements in Banach Jordan pairs': Proc, 199"7 Internat. Conf. Jordan structures, Dept. ~_lgebra, Geom. y Topol. Univ. M£1aga, 1999, pp. 116-117. [21] MAOUeHE, A.: 'Caract~risation spectrales du radical et socle d'une paire de Jordan Banach', Canad. Math. Bull. 40

(1997), 488 497. [22] MONTANER, F.: 'Local PI theory of Jordan systems', J. Algebra 216 (1999), 302-327. [23] MORENO, A., AND RODRmUEZ, A.: 'On the Zelmanovian classification of prime YB*-triples', J. Algebra 226 (2000), 577613. [24] NEHER, E.: Jordan triple systems by the grid approach, Vol. 1280 of Lecture Notes in Mathematics, Springer, 1980. [25] RODRIGUEZ PALACIOS, A.: 'Jordan structures in analysis': Proc. 1992 Oberwolfach Conf. Jordan Algebras, de Gruyter, 1994, pp. 97 186. [26] Russo, B.: 'Structure of JB*-triples': Proc. 1992 Oberwolfach Conf. Jordan algebras, de Gruyter, 1994, pp. 209-237. [27] SINCLAm, A.M., AND TULLO, A.W.: 'Noetherian Banach algebras are finite dimensional', Math. Ann. 211 (1974), 151153. [28] VILLENA, A.R.: 'Derivations on Jordan-Banach algebras', Studia Math. 118 (1996), 205-229. [29] ZEL'MANOV~E.: 'On prime Jordan triple systems III', Sib. Math. J. 26 (1985), 71-82.

Antonio Ferndndez L@ez MSC 1991: 17A40, 17C65, 46H70, 46L70 B A N A C H - M A Z U R GAME - A game that appeared in the famous Scottish Book [11], [6], where its initial version was formulated as Problem 43 by the Polish mathematician S. Mazur: Given the space of real numbers R and a non-empty subset E of it, two players

A and /3 play a game in the following way: A starts by choosing a non-empty interval -To of R and then B responds by choosing a non-empty subinterval /1 of Io. Then player A in turn selects a non-empty interval /2 C I1 and B continues by taking a non-empty subinterval/3 of/2. This procedure is iterated infinitely many times. The resulting infinite sequence of nested intervals {/i~}~_0 is called a play. By definition, the player A wins this play if the intersection N~__oI~ has a common point with E. Otherwise B wins. Mazur had observed two facts: a) if the complement of E in some interval of R is of the first Baire category in this interval (equivalently, if E is residual in some interval of R, cf. also C a t e g o r y o f a set; B a l r e classes), then player A has a winning strategy (see below for the definition); and b) if E itself is of the first Baire category in R, then B has a winning strategy. The question originally posed by Mazur in Problem 43 of the Scottish Book (with as prize a bottle of wine!) was whether the inverse implications in the above two assertions hold. On August 4, 1935, S. Banach wrote in the same book that 'Mazur's conjecture is true'. The proof of this statement of Banach however has never been published. The game subsequently became known as the Banach-Mazur game. More than 20 years later, in 1957, J. Oxtoby [8] published a proof for the validity of Mazur's conjecture. Oxtoby considered a much more general setting. The game was played in a general t o p o l o g i c a l s p a c e X with ~ E C X and the two players A and B were choosing alternatively sets W0 D W1 D ... from an a priori prescribed family of sets W which has the property that every element of 142 contains a non-empty open subset of X and every non-empty open subset of X contains an element of 14;. As above, A wins if (n~= 0 ,~) A E ~ 0, otherwise B wins. Oxtoby's theorem says that B has a winning strategy if and only if E is of the first Baire category in X; also, if X is a c o m p l e t e m e t r i c space, then A has a winning strategy exactly when E is residual in some non-empty open subset of X. Later, the game was subjected to different generalizations and modifications. G e n e r a l i z a t i o n s . Only the most popular modification of this game will be considered. It has turned out to be useful not only in set-theoretic topology, but also in the geometry of Banach spaces, non-linear analysis, number theory, descriptive set theory, well-posedness in optimization, etc. This modification is the following: Given a topological space X, two players (usually called a and ~) alternatively choose non-empty open sets U1 D V1 D U2 D V2 D ... (in this sequence the Un 59

BANACH-MAZUR GAME are the choices of fl and the V,~ are the choices of a; thus, it is player ~ who starts this game). Player a wins the play {Un, Vn}n°°=l if N~=IUn = NnC~_lV n ¢ ~, otherwise wins. To be completely consistent with the general scheme described above, one may think t h a t E = X and a starts by always choosing the whole space X. This game is often denoted by B M ( X ) . A strategy s for the player a is a mapping which assigns to every finite sequence (U1 D V1 D ... D U~) of legal moves in B M ( X ) a non-empty open subset V, of X included in the last move of/3 (i.e. V~ C U~). A stationary strategy (called also a tactics) for a is a strategy for this player which depends only on the last move of the opponent. A winning strategy (a stationary winning strategy) s for a is a strategy such t h a t a wins every play in which his/her moves are obtained by s. Similarly one defines the (winning) strategies for ~. A topological space X is called weakly c~-favourable if a has a winning strategy in B M ( X ) , while it is termed c~-favourable if there is a stationary winning strategy for a in BM(X). It can be derived from the work of Oxtoby [8] (see also [5], [7] and [9]) t h a t the space X is a B a l r e s p a c e exactly when player f? does not have a winning strategy in BM(X). Hence, every weakly a-favourable space is a Baire space. In the class of metric spaces X, a metric space is weakly a-favourable if and only if it contains a dense and completely metrizable subspace. One can use these two results to see t h a t the Banach Mazur game is 'not determined'. I.e. it could happen for some space X that neither a nor ~ has a winning strategy. For instance, the Bernstein set X in the real line (cf. also N o n - m e a s u r a b l e set) is a Baire space which does not contain a dense completely metrizable subspace (consequently X does not admit a winning strategy for either a or ¢~). The above characterization of weak a-favourability for metric spaces has been extended for some nonmetrizable spaces in [10]. A characterization of a-favourability of a given c o m p l e t e l y - r e g u l a r s p a c e X can be obtained by means of the space C(X) of all continuous and bounded real-valued functions on X equipped with the usual supnorm Ilfll~ := sup{[f(x)l : x • x } . The following statement holds [4]: The space X is weakly a-favourable if and only if the set

{ f E C(X) : f attains its m a x i m u m in X} is residual in C(X). In other words, X is weakly ctfavourable if and only if almost-all (from the Baire category point of view) of the functions in C(X) attain their m a x i m u m in X. The rich interplay between X , C(X) and B M ( X ) is excellently presented in [3]. 60

The class of a-favourable spaces (spaces which admit a-winning tactics) is strictly narrower than the class of weakly a-favourable spaces. G. Debs [2] has exhibited a completely-regular topological space X which admits a winning strategy for a in B M ( X ) , but does not admit any a-winning tactics in BM(X). Under the name 'espaces tamisables', the a-favourable spaces were introduced and studied also by G. Choquet [1]. [10] is an excellent survey paper about topological games (including B M ( X ) ) . References [1] CHOQUET, a.: 'Une cIasse r~guli~res d'espaces de Baire', C.R. Acad. Sci. Paris Sdr. I 246 (1958), 218-220. [2] DEBS, G.: 'Strategies gagnantes dans certain jeux topologiqne', Fundam. Math. 126 (1985), 93-105. [3] DEBS, G., AND SAINT-RAYMOND, J.: 'Topological games and optimization problems', Mathematika 41 (1994), 117-132. [4] KENDEROV, P.S., AND REVALSKI, J.P.: 'The Banach-Mazur game and generic existence of solutions to optimization problems', Proc. Amer. Math. Soc. 118 (1993), 911-917. [51 KaOM, M.R.: 'Infinite games and special Baire space extensions', Pacific J. Math. 55, no. 2 (1974), 483-487. [6] MAULDIN, R.D. (ed.): The Scottish Book: Mathematics from the Scottish Card, Birkhiiuser, 1981. [7] McCoY, R.A.: 'A Baire space extension', Proc. Amer. Math. Soc. 33 (1972), 199-202. [8] OXTOBY, J.: 'The Banach-Mazur game and the Banach category theorem': Contributions to the Theory of Games III, Vol. 39 of Ann. of Math. Stud., Princeton Univ. Press, 1957, pp. 159-163. [9] SAINT-RAYMOND, J.: 'Jeux topologiques et espaces de Namioka', Proc. Amer. Math. Soc. 87 (1983), 499-504. [10] TELC~aSKI, R.: 'Topological games: On the fifth anniversary of the Banach-Mazur game', Rocky Mount. J. Math. 17 (1987), 227-276. [11] ULAM, S.M.: The Scottish Book, second ed., A LASL monograph. Los Alamos Sci. Lab., 1977.

P.S. Kenderov M S C 1991:54E52 B A N A C H - S T O N E THEOREM, Stone-Banach theorem - For a compact H a u s d o r f f s p a c e X , let C(X) be the B a n a c h s p a c e of all continuous scalar-valued functions on X , with the usual sup-norm: Ilfll = s u p { l f ( x ) l : x ¢ x } . If ~ is a surjective h o m e o m o r p h i s m from a compact space Y onto a compact space X and X is a continuous and unimodular scalar-valued function on Y, then

T(f) = x . f o~,

I • Co(X),

(1)

defines a linear isometry from Co(X) onto Co(Y). The Banach-Stone theorem asserts that any linear surjective isometry T: C(X) -~ C(Y) is of the above form. Here, if X is not necessarily compact, then Co(X) is the space of continuous functions t h a t vanish at infinity (i.e. the functions f such t h a t for all e > 0 there is a compact set

BAUER-FIKE K such t h a t l/(x)l < for x C X \ K ) . A u n i m o d u l a r function is one for which I)dy)l = I for all y. T h e t h e o r e m was proved in the real case for c o m p a c t metric spaces by S. B a n a c h [2]; M.H. Stone [10] proved t h a t the a s s u m p t i o n of metrizability was superfluous. Subsequently the t h e o r e m was extended to spaces of b o t h real- or complex-valued functions defined on a locally c o m p a c t Hausdorff space. The t h e o r e m has been extended further into several directions: 1) a B a n a c h space E has the Banach Stone property if the B a n a c h space Co(X, E) of E - v a l u e d continuous functions is isometric with C 0 ( Y , E ) if and only if X and Y are homeomorphic; the class of B a n a c h spaces with this p r o p e r t y includes the strictly convex B a n a c h spaces and the Banach spaces with strictly convex dual (see e.g. [3], [8]); 2) if there is an i s o m o r p h i s m T : C(X) --+ C(Y) such t h a t []THllT-I[[ = 1 + e < 2, then X and Y are homeomorphic (and consequently C(X) and C(Y) are isometric) [1], [4], where 2 is the best b o u n d [5]; 3) the t h e o r e m holds for several classes of subspaces of the spaces C(X), the most i m p o r t a n t one being the class of uniform algebras (cf. also U n i f o r m a l g e b r a ) .

THEOREM

[8] JAROSZ, K.: 'Small i s o m o r p h i s m s of C ( X , E) spaces', Pacific J. Math. 138, no. 2 (1989), 295 315. [9] ROCHBERG, P~.: ' D e f o r m a t i o n of uniform algebras on R i e m a n n surfaces', Pacific J. Math. 121, no. 1 (1986), 135-181. [10] STONE, M.H.: 'Applications of t h e theory of Boolean rings to general topology', Trans. Amer. Math. Soc. 41 (1937), 375-481.

K. Jarosz

M S C 1991: 46Exx BAUER-FIKE THEOREM As popularized in most texts on c o m p u t a t i o n a l l i n e a r a l g e b r a or numerical methods, the B a u e r - F i k e t h e o r e m is a t h e o r e m on the p e r t m b a t i o n of eigenvalues of a diagonalizable matrix. However, it is actually just one t h e o r e m out of a small collection of t h e o r e m s on the localization of eigenvalues within small regions of the complex plane [1]. A vector norm is a f u n c t i o n a l I1"11 on a vector x which satisfies three properties: 1) [[x[[ >_ O, and Hx[[ = 0 if and only if x - O. for any scalar a. 3) Ilx + yll _< Ilxll + IIyll (the triangle inequality).

2) II~xll = I~lIIxll

A matriz norm ]IAII satisfies the above three properties plus one more:

4) [IABII _< IIAIIIIBII

(the submultiplicative inequal-

ity).

A joint a p p r o a c h to the last two extensions gave rise to the p e r t u r b a t i o n t h e o r y of uniform algebras. A Banach algebra B is an e-perturbation (or e-metric perturbation) of a B a n a c h a l g e b r a A if there is an isom o r p h i s m T : A -+ B such t h a t IITIIIIT-~II < 1 + s . In such a situation the algebras A, B must share several i m p o r t a n t properties [7], [9]; if the algebra B is forced to be isometrically isomorphic with A, the algebra A is called stable. E x a m p l e s of stable uniform algebras inelude C(X), A(D) and H°°(D), with D the unit disc.

An operator norm is a m a t r i x n o r m derived from a related vector norm:

Isometries of several other classes of B a n a c h spaces have been shown to follow the same general p a t t e r n , similar to (1); one often refers to such results also as Banach-Stone theorems (for a given class of spaces; see e.g. [6]).

Ilxll~ = m a x l x i [ ,

References [1] AMIR, D.: ' O n i s o m o r p h i s m s of continuous function spaces', Israel d. Math. 3 (1965), 205 210. [2] BANACH, S.: Thgorie des opdrations lindaires, P W N , 1932. [3] BEHRENDS, E.: M-structure and the Banach Stone theorem, Voi. 736 of Lecture Notes in Mathematics, Springer, 1979. [4] CAMBERN, M.: 'OI1 i s o m o r p h i s m s with small bound', Proc. Amer. Math. Soc. 18 (1967), 1062-1066. [5] COHEN, H.B.: 'A b o u n d - t w o isomorphism between C ( X ) Banach spaces', Proc. Amer. Math. Soc. 50 (1975), 215 217. [6] FLEMING, R.J., AND JAMISON, J.E.: 'Isometrics on Banach spaces: a survey': Analysis, Geometry and Groups: a Riemann Legacy Volume, Hadronic Press, 1993, pp. 52 123. [7] JAROSZ, K.: Perturbations of Banach algebras, Vol. 1120 of Lecture Notes in Mathematics, Springer, 1985.

5) IIAII =

maxx¢0

IlAxll/llxll = maxlrxl,=l IIAxII.

T h e most c o m m o n vector n o r m s are:

Ilxlll = ~

Ixil, i

Ilxl12 =

Ixil 2

,

where xi denotes the ith e n t r y of the vector x. T h e operator norms derived from these are, respectively: [[A[[ 1 = m a x E [aij[~ J []All 2 = largest singular value of A,

Ildlt~ = ~ x Z laijl, i

where aij denotes the (i,j)th entry of the m a t r i x A. In particular, [[A[[oo is the ' m a x i m u m absolute row s u m ' and I[A[I1 is the ' m a x i m u m absolute column sum'. For details, see, e.g., [2, Sec. 2.2-2.3]. Below, the n o t a t i o n [[-[[ will denote a vector n o r m when applied to a vector and an o p e r a t o r m a t r i x norm when applied to a matrix. Let A be a real or complex (n × n ) - m a t r i x t h a t is diagonalizable (i.e. it has a complete set of eigenvectors 61

BAUER-FIKE THEOREM v l , . . . , v ~ corresponding to n eigenvalues A1,...,An, which need not be distinct; cf. also E i g e n v e c t o r ; E i g e n value; C o m p l e t e set). Its eigenvectors make up an (n x n)-matrix V; let A = diag{A1,...,An} be the diagonal matrix. Then A = V A V -1. Let E be an arbitrary (n x n)-matrix. The popular B a u e r - F i k e theorem states that for such A and E, every eigenvalue # of the matrix A + E satisfies the inequality

I#- AI -< [IvII " ]lV-1l]" [IEII ,

(1)

where A is some arbitrary eigenvalue of A. If ]IEI] is small, this theorem states that every eigenvalue of A + E is close to some eigenvalue of A, and that the distance between eigenvalues of A + E and eigenvalues of A varies linearly with the perturbation E. But this is not an asymptotic result valid only as E --+ 0; this result holds for any E. In this sense, it is a very powerful localization theorem for eigenvalues. As an illustration, consider the following situation. Let B be an arbitrary (n x n)-matrix, let A = diag{bll,...,bn~} be the diagonal part of B, and define E = B - A to be the matrix of all off-diagonal entries. Let p be an arbitrary eigenvalue of B. The matrix of eigenvectors for A is just the identity matrix, so the Bauer-Fike theorem implies that lit - b i i [ < IIEll, for some diagonal entry bii. The oc-norm IIEI]~ is just the maximum absolute row sum over the off-diagonal entries of B. In this norm the assertion means that every eigenvalue of B lies in a circle centred at a diagonal entry of B with radius equal to the maximum absolute row sum over the off-diagonal entries. This is a weak form of a Gershgorin-type theorem, which localizes the eigenvalues to circles centred at the diagonal entries, but the radius here is not as tight as in the true Gershgorin theory [2, sec. 7.2.1] (cf. also G e r s h g o r i n t h e o r e m ) . Another simple consequence of the Bauer-Fike theorem applies to symmetric, Hermitian or normal matrices (cf. also S y m m e t r i c m a t r i x ; N o r m a l m a t r i x ; H e r m i t i a n m a t r i x ) . If A is symmetric, Hermitian or normal, then the matrix V of eigenvectors can be made unitary (orthogonal if real; cf. also U n i t a r y m a t r i x ) , which means that v H v = 1, where V H denotes the complex conjugate transpose of V. Then IIV[12 = NV-1H2 = 1, and (1) reduces to 1 # - A [ < [[EI]. Hence, for symmetric, Hermitian or normal matrices, the change to any eigenvalue is no larger than the norm of the change to the matrix itself. The Bauer-Fike theorem as stated above has some limitations. The first is that it only applies to diagonalizable matrices. The second is that it says nothing about the correspondence between an eigenvalue p of the perturbed matrix A + E and the eigenvalues of the unperturbed matrix A. In particular, as E increases from zero, 62

it is difficult to predict how each eigenvalue of A + E will move, and there is no way to predict which eigenvalue of the original A corresponds to any particular eigenvalue # of A + E, except in certain special cases such as symmetric or Hermitian matrices. However, in their original paper [1], F.L. Bauer and C.T. Fike presented several more general results to relieve the limitation to diagonalizable matrices. The most important of these is the following: Let A be an arbitrary (n x n)-matrix, and let p denote any eigenvalue of A + E . Then either p is also an eigenvalue of A or

Ell

1 <_ 1 1 ( # I - A) - 1 .

_< II(it±- A)-lll • IIEII,

(2)

This result follows from a simple matrix manipulation, using the norm properties given above: (A + Z ) x = Itx = (itZ)x ( i t I - A) -1 • E x = x =* [1(itI - A) - t . Ell • IIxll _> flxll •

Even though this may appear to be a rather technical result, it actually leads to a great many often-used results. For example, if A = V A V -1 is diagonalizable, it is easy to show that

II(it±- A)-lfl = Ilv(itI- A)-lv-lll -< _< IlVll. II(itI- A)-lll • Ilv-'ll, which leads immediately to the popular Bauer-Fike theorem (using the fact that the operator norm of a diagonal matrix is just its largest entry in absolute value). One can also repeat the construction, in which B is an arbitrary matrix, A is the matrix consisting of the diagonal entries of B, and E = B - A is the matrix of off-diagonal entries. Using the norm II'fl~, the leftmost inequality in (2) then reduces to 1 < max -

i

(

1 ) - - " E ]bij[ , lit-- bii] j¢i

leading immediately to the first G e r s h g o r i n t h e o r e m (see also [2, Sec. 7.2.1]. References [1] BAUEn, F.L., AND FIKn, C.T.: 'Norms and exclusion theorems', Numer. Math. 2 (1960), 137-141. [2] GOLUB,G.H., AND LOAN, C.F. VAN: Matrix computations, third ed., Johns Hopkins Univ. Press, 1996. D.L. Boley

MSC 1991:15A42 BAUMSLAG-SOLITAR GROUP, Solitar-Baumslag group - The Baumslag Solitar groups are a particular class of two-generator one-relator groups which have played a surprisingly useful role in combinatorial and, more recently (the 1990s), geometric group theory. In a number of situations they have provided examples which

BAUMSLAG-SOLITAR GROUP mark boundaries between different classes of groups and they often provide a testbed for theories and techniques. These groups have a deceptively simple definition. For each pair rn and n of non-zero integers, there is a corresponding group defined by the presentation B S ( m , n ) = (a,b la Zbma=b'~), where, as usual, this notation means that the group is the quotient of the f r e e g r o u p on the two generators a and b by the normal closure (cf. also N o r m a l subgroup) of the single element a-lb'~ab-% When Iml = Ir l = 1, US(m, n) is the fundamental group of the torus or Klein bottle (cf. also K l e i n s u r f a c e ) , both of which have been long familiar and well understood - and can therefore be regarded as standing somewhat apart from the remaining groups. The Baumslag-Solitar groups were introduced in [1] to provide some simple examples of so-called nonHopfian groups. A group is called Hopfian (or nowadays Hopf) if every e p i m o r p h i s m from the group to itself is an i s o m o r p h i s m . (The name is derived from the topologist H. Hopf and is thought to reflect the fact that whether fundamental groups of manifolds are 'Hopfian' is of interest; cf. also H o p f g r o u p ; N o n - H o p f g r o u p . ) The condition itself is an instance of what are often referred to as finiteness conditions, i.e. conditions satisfied by finite groups which may or may not hold for infinite groups (cf. also G r o u p w i t h a f i n i t e n e s s c o n d i t i o n ) . It is of interest to see what implications there are among different finiteness conditions within classes of infinite groups. The main result concerning finiteness conditions and the BaumsIag-Solitar groups is as follows: a) B S ( m , n ) is residually finite (i.e. the intersection of all its subgroups of finite index is trivial) if and only if I. 1 =

or I' 1 = I or InI = 1. See also R e s i d u a l l y -

finite group. b) BS(rn, n) is Hopfian if and only if it is residually finite or 7c(rn) = 7r(n), where ~r(nz) denotes the set of prime divisors of m. Note that a simple well-known argument shows that a finitely-generated residually-finite group is Hopfian. Most famously, BS(2, 3) = (a, b l a-lb2a = b3) is nonHopfian; the mapping 05: a ~+ a, b ~ b2, is an epimorphism that is not an isomorphism. It is elementary to show the first (and t h a t 05 is well-defined) and standard techniques provide an easy verification that b- 1a - 1bab- 1a - 1bah- 1 is a non-trivial element of the kernel. An illustration of the weakness of the Hopfian property as a finiteness condition is provided by BS(12, 18), which is Hopfian but has a (normal) subgroup of index 6, the normal closure of a, which is non-Hopfian.

(Note that there is a minor error in the actual statement in the original [I] (which is basically an announcement of results), since it asserts that BS(nz, n) is residually finite if one of ra, n divides the other. The observation of this error and its correction are due to S. Meskin

[17].) The above result has been extended and generalized in various ways, but it appears to be difficult to go beyond results t h a t are somewhat particular and limited in scope. Although the above theorem divides the Baumslag Solitar groups into three classes: those that are residually finite, those t h a t are Hopfian but not residually finite and those t h a t are non-Hopfian, the most marked contrast is between those Baumslag-Solitar groups where Iml = 1 or Jn] = 1 and those for which

I- l,

¢ 1.

For a group BS(1,n) = (a,b I a - l b a = b~) there is an obvious h o m o m o r p h i s m onto the infinite cyclic g r o u p , obtained by setting b = 1, and standard techniques show that the kernel is isomorphic to the additive group of n-adic rational numbers. Thus, such groups are meta-Abelian (cf. M e t a - A b e l i a n g r o u p ) and have strong structural properties; in particular, they do not contain a free subgroup of rank two. Moreover, these groups have a particularly simple normal form for their elements: Each element is uniquely represented by a word of the form aibka - j where i , j > 0 and, i f i , j > 0, k is not divisible by n. When Irnl, Inl ~ 1, there is still a strong normal form result since all Baumslag Solitar groups are examples, indeed are the simplest possible examples, of an H N N e x t e n s i o n (see also [3] or [15] for a definition). Hence the following result holds for Baumslag-Solitar groups: Let w be a freely reduced word of BS(m, n) which represents the identity element. Then w has a subword of the form either a - l b k a where m [ k, or abka -1 where

nl/c. This result shows, for example, that in BS(2, 3) the word b-Za-Zbab-la-lbab -1 does not represent the identity, and can also be used to show that if [rnl, in] # 1, then BS(rn, n) contains a free subgroup of rank two.

B a u m s l a g - S o l i t a r groups as e x a m p l e s and counterexamples. Below, a number of results concerning Baumslag-Solitar groups will illustrate their role as testbeds in combinatorial and geometric group theory.

Autornorphisrns. The group BS(2, 4), and more generally B S ( m , n ) when Irnl, Inl # 1 and one of rn, n divides the other, has an infinitely-generated automorphism group [4]. A generating set, no finite part of which will generate the automorphism group, consists of the 63

BAUMSLAG-SOLITAR G R O U P automorphisms r:

a~a,

~r : ~r :

b~-~b -1,

a ~+ ab, b ~-+ b,

a F-+ ar+lb2a-r,

r > 1,

plus the inner automorphisms associated to a and b. A different kind of failure of finite generation is illustrated by the group BS(1, 2), where the automorphism ~: a ~+ ab, b ~ b has fixed subgroup (akba - k I k >_ 1) that is not finitely generated. This is easily established using the normal form specified above. Subgroups. Groups defined by generators and relations arise from topological and geometric contexts. It is perhaps therefore not altogether surprising that the Baumslag-Solitar groups play a role in questions concerning groups defined by particular topological or geometric conditions. A reasonably classical illustration of this is the fact that only the residually finite BaumslagSolitar groups can be realized as fundamental groups of 'nice' 3-manifolds (cf. also Three-dimensional manifold; Fundamental group). An exceptionally strong result [13] shows that those Baumslag-Solitar groups that are not residually finite cannot even occur as subgroups of nice 3-manifolds, thereby providing a 'negativity test' for deciding when a group is a 3-manifold group. A contrasting result found in [14] is that the rectaAbelian Baumslag Solitar groups are the only finitelygenerated (non-cyclic) solvable subgroups of one-relator groups (cf. also Solvable group). In the torsion-free case (when the single relator is not a proper power), this result is a corollary of the theorem in [10] that the metaAbelian Baumslag Solitar groups are the only finitelygenerated solvable groups of cohomological dimension two. A further result [11], more akin to the exclusion result for 3-manifold groups, is that, for a torsion-free one-relator group G, all maximal Abelian subgroups are malnormal if and only if G excludes as subgroups both the direct product of a free group of rank two with the infinite cyclic group, and all the meta-Abelian groups BS(1,n), n • 1. Geometric group theory. Observations with a fiavour similar to that for 3-manifold groups hold. For example, no Baumslag-Solitar group can be a subgroup of a word hyperbolic group (cf. also H y p e r b o l i c group). For automatic groups, the position is that BS(m, n) is automatic if Irnl = In] but is otherwise not automatic. It is unknown whether 'subgroup exclusion' occurs whenever [m I ¢ Inl. A variant condition, weaker than straight automaticity and known as 'asynchronous automaticity' is, however, satisfied by all Baumslag-Solitar groups. 64

An alternative view of this area is provided by the concept of isoperimetric inequality. This concerns, for a given group presentation, the relationship between the length of an otherwise arbitrary word which represents the identity element and the number of relators required to express this fact. The relationship is linear if and only if the group is word hyperbolic, and for automatic groups it is quadratic in the sense that the number of relators needed is no more than a quadratic function of the length of the word. It is not known (as of 2000) whether the converse to this last statement is true or false, the Baumslag-Solitar groups having, on this occasion, failed to play their traditional role of providing discriminating examples by requiring a number of relators that can always be exponential in the length of the word. (See [9].) Normal forms. The study of normal forms for elements of a group goes back to the roots of combinatorial group theory and Dehn's introduction of the word problem (cf. also Identity problem). Two ideas relating to normal forms are of interest here. One is that of describing, in terms of the concept of a regular language (cf. also For-

mal languages and automata; Grammar, regular) properties of particular sets of normal forms and how effectively they can be computed; it was the essential impetus for the study of automatic groups. The second is the idea of a growth function for a group, which is the complex power series whose nth coefficient is the number of elements of (minimal) length n relative to some fixed generating set (cf. also Polynomial and exponential growth in groups and algebras). All automatic groups have a growth function which is rational, see [6], while the residually-finite Baumslag-Solitar groups have rational growth [2], [5] but, as noted earlier, are not in general automatic. Interestingly, the argument in [2] establishing this fact uses finite-state automata, although it is actually the case [12] that there is no regular set of length-minimal normal forms for the meta-Abelian Baumslag-Solitar groups. Rigidity and convexity. Baumslag-Solitar groups continue to be used as test beds for theories and techniques, in particular those derived from metric space concepts applied to the word metric for a group. Two illustrations are as follows: 1) An important classification tool for metric spaces is the concept of quasi-isometry: Two metric spaces are quasi-isometric if there are bijective mappings between them in which distortion of distances is uniformly linearly bounded (cf. also Q u a s i - i s o m e t r i c spaces). A considerable amount of successful classification has been done on groups which arise in specific geometric and topological settings and also on groups which have

BAXTER ALGEBRA a nilpotent subgroup of finite index (cf. also N i l p o t e n t g r o u p ) . It has been shown [7], [8] that for the meta-Abelian Baumslag-Solitar groups, the position is essentially rigid; namely that BS(1,m) and BS(1,n) are quasi-isometric if and only if they have isomorphic subgroups of finite index. Furthermore, if G is quasiisometric to some BS(1, n), then, ignoring a finite normal subgroup, G has an index-one or -two subgroup BS(1,m) quasi-isometric to BS(1,n). This work is of particular interest since the meta-Abelian BaumslagSolitar groups do not have a natural geometric setting nor do they satisfy any kind of nilpotency condition. 2) Another metric space concept that has applications to groups is that of 'almost convexity': A presentation of a group G is almost convex if, whenever g and g' lie within a specific distance K of the origin and are at most a given distance apart, then there is a path of uniformly bounded length which strays no further than distance K from the origin. The motivation is that almost convexity implies efl:icient computation of the C a y l e y g r a p h and this has been studied with successful but not always positive outcomes for the fundamental groups of closed 3-manifold groups with uniform geometries. In its standard presentation,the metaAbelian group BS(1,m), m # 1, is not almost convex [is].

F u r t h e r r e a d i n g . General reference texts in combingtorial and geometric group theory, such as [3], [6], [15], [16], provide background reading for non-specialists. References

[11]

[12] [13] [14] [15] [16] [17]

[18] MILLER III, C.F., AND SHAPIRO, M.: 'Solvable BaumslagSolitar groups are not almost convex', Geom. Dedicata 72, no. 2 (1998), 123-127.

D.J. Collins

MSC 1991: 20F32, 05C25, 20Fxx BAXTER ALGEBRA - Baxter algebras originated in the following problem in fluctuation theory: Find the distribution functions of the maxima max{0, $1, •. •, Sn} of the partial sums So = 0, $1 ---- X 1 , $2 ---- X1 -]X 2 , . . . , S ~ = X1 + ' " + X~ of a sequence Xi of independent identicMly-distributed random variables (cf. also R a n d o m v a r i a b l e ) . A central result in this area is the Spitzer identity

Z

[1] BAUMSLAG, G., AND SOLITAR, D.: 'Some two generator onerelator non-Hopfian groups', Bull. Amer. Math. Soe. 689 (1962), 199 201. [2] BRAZIL, M.: 'Growth functions for some nonautomatic Baumslag-Solitar groups', Trans. Amer. Math. Soc. 342 (1994), 137 154.

[3] COLLINS, D.J., GRIGORCHUK,

[10]

Algorithms and Classification in Combinatorial Group Theory, Springer, 1992. GILDENHUYS, D.: 'Classification of soluble groups of cohomological dimension two', Math. Z. 166 (1979), 21 25. GILDENHUYS, D., KHARLAMPOVICH, O., AND MYASNIKOV, A.: 'CSA-groups and separated free constructions', Bull. Austral. Math. Soc. 52 (1995), 63 84. GROVES, J.M.J.: 'Minimal length normal forms for some soluble groups', J. Pure Appl. Algebra 114 (1996), 51-58. JACO, W.H., AND SHALEN: P.B.: Seifert fibered spaces in 3manifolds, Vol. 192 of Memoirs, Amer. Math. Sou., 1979. KARRASS, A., AND SOLITAR, D.: 'Subgroups of HNN groups and one-relator groups', Canad. Math. J. 23 (1971), 627 643. LYNDON, R.C., AND SCHUPP~ P.E.: Combinatorial group theory, Ergebn. Math. Grenzgeb. Springer, 1977. MAGNUS, W., KARRASS, A., AND SOLITAR, D.: Combinatorial group theory, Wiley, 1966. MESKIN~ S.: 'Non-residually finite one-relator groups', Trans. Amer. Math. Soc. 64 (1972), 105-114.

R.I., KURCHANOV,

n

where pn(t) is max{0, S 1 , . . . , Sn} tion of max{0, Sk}. resemblance to the

P.F., AND

ZIESCHANG: H.: Combinatorial group theory and applications to geometry, Vol. 58 of Encyclopaedia Math. Sci., Springer, 1993. [4] COLLINS, D.J., AND LEVIN, F.: 'Automorphisms and Hopficity of certain Baumslag-Solitar groups', Archly Math. 40

(1983), 385-400. [5] EDJVET, M., AND JOHNSON, D.L.: 'The growth of certain amalgamated free products and HNN-extensions', J. Austral. Math. Soe. 52 (1992), 285-298. [6] EPSTmN,D.B.A., CANNON, J.W., HOLT, D.F., LEVY, S.V.F., PATERSON, M.S., AND THURSTON, W.P.: Word processing in groups, Jones & Bartlett, 1992. [7] FAHB, B., AND MOSHER, L.: 'A rigidity theorem for the solvable Baumslag Solitar groups (With an appendix by Daryl Cooper)', Invent. Math. 131 (1998), 419-451. [8] FARB, B., AND MOSHER, L.: 'Quasi-isometric rigidity for the solvable Banmslag-Solitar groups II', Invent. Math. 137 (1999), 613-649. [9] GERSTEN, S.M.: 'Dehn functions and /l-norms of finite presentations', in G. BAUMSLAO AND C.F. MILLER, III (eds.):

:

exp

k(t

,

n=0

the characteristic function of and ~k (t) is the characteristic funcSpitzer's identity bears an uncanny

Waring identity

oo

n = n=0

=exp

(--1)kpk(xl,x2,...

-

,

k=l

where en (xl, x 2 , . . . ) are elementary symmetric functions and pk(xl, x2,...) are power sum symmetric functions. The algebraic structure underlying both identities is a Baxter algebra. These algebras were defined by G.-C. Rota in [2], [3]. A Baxter operator P on an a l g e b r a A over a field k is a l i n e a r o p e r a t o r from ~4 to itself satisfying the identity

P(xPy) + P ( y P x ) = qP(xy) + (Px)(Py),

(1)

where q is a constant in k. A Baxter algebra is an algebra with a Baxter operator. 65

B A X T E R ALGEBRA An example is the algebra of real-valued continuous functions on the interval [0, 1] with the integration operator Pf(x) =

/oxf ( t ) dr.

The formula for i n t e g r a t i o n b y p a r t s is identity (1) with q = 0. Another example is the B a n a c h a l g e b r a of characteristic functions of distribution functions of random variables (cf. also C h a r a c t e r i s t i c f u n c t i o n ; R a n d o m v a r i a b l e ) with the Baxter operator P which sends the characteristic function of a random variable X to the characteristic function of max{0, X}. T h a t is, if ~=

/?

exp(itx) dF(x),

problem for Baxter algebras with more than one generator is solved in a similar way by P. Cartier. In particular, an identity amongst symmetric functions can be translated into an identity satisfied by all Baxter algebras on one generator. For example, writing Waring's identity in terms of Baxter operators, one obtains ~

P(xP(xP(... (xPx)...)))A s =

rtzO

= exp - P

( - 1 ) t -~-

When P is the Baxter operator given in (2), this identity is Spitzer's identity. When P is the q-integral, this identity becomes the Eulerian identity

C2~

tnqn(n+l)/2

then P~ =

exp(itx) dF(x). (2) + Given any e n d o m o r p h i s m E (that is, a linear operator satisfying E ( x y ) = ( E x ) ( E y ) ) on an algebra A, the operator E 1-E is a Baxter operator if the infinite series converges. In particular, the q-integral p=E+E2+

....

P f ( t ) = f(qt) + f(q2t) +

f(q3t)

= (0, U l , U l + u 2 , u l + u2 + u 3 , . . . ) .

The standard Baxter algebra 13 is the smallest subalgebra of Jt containing x, y , . . . and closed under P. Rota [2], [3] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also F r e e a l g e b r a ) . If x is the sequence (Xl,X2,...), then the (k + 1)st term in P ( x ~) is the power sum symmetric function x~ + . . . + x ~ and the kth term in P ( x P ( . . . ( x P x ) . . . ) ) , where there are n occurrences of P, is e ~ ( x l , . . . , X k ) . Hence, the free Baxter algebra on one generator x is isomorphic to the algebra of symmetric functions (cf. also S y m m e t r i c f u n c t i o n ) . Because the elementary symmetric functions are algebraically independent, the free Baxter algebra in one generator x is isomorphic to the algebra of polynomials in the variables x, P x , P ( x P x ) , P ( x P ( x P x ) ), P ( x P ( x P ( x P x ) ) ), . . .. This solves the word problem (cf. also I d e n t i t y p r o b l e m ) for Baxter algebras with one generator. The word 66

n:l

k=l

References [1] BAXTER, G.: 'An analytic problem whose solution follows from a simple algebraic identity', Pacific J. Math. 10 (1960), 731-742. [2] ROTA, G.-C.: 'Baxter algebras and combinatorial identities I-IF, Bull. A m e r . Math. Soc. 75 (1969), 325-334. [3] ROTA, G.-C.: 'Baxter algebras: an introduction', in J.P.S. KUNG (ed.): Gian-Carlo Rota on Combinatorics, Birkh~user, 1995, pp. 504 512.

Joseph P.S. Kung

+...

is a Baxter operator. The standard Baxter algebra over a field F with generators x, y , . . . is defined in the following way. Let x = (Xz,X2,...), y = (Yl,Y2,...), ... be sequences such that the terms x~, x2, • • •, y~, Y2, • • • are algebraically independent. On the F-algebra ..4 with coordinate-wise addition and multiplication generated by x, y , . . . , define the Baxter operator P by P(Ul,U2,U3,...)

= ~(l+qkt).

MSC 1991: 05E05, 60G50

BENJAMIN-BONA-MAHONY EQUATION, B B M equation, regularized long wave equation - The model equation ut + ux + uux - Ux,t = 0, (1) where u(x,t): R × R --+ R and the subscripts denote partial derivatives with respect to time t and the position coordinate x. The Benjamin-Bona-Mahony equation serves as an approximate model in studying the dynamics of small-amplitude surface water waves propagating unidirectionally, while suffering non-linear and dispersive effects. (1) was introduced in [5] as an alternative of the famous K o r t e w e g - d e V r i e s e q u a t i o n ut + uz + uux + Uxxx = 0.

(2)

Unlike the Korteweg-de Vries equation, the BenjaminB o n ~ M a h o n y equation is not integrable by the inverse scattering method [10], [12]. As indicated by several numerical experiments, (1) has no multi-soliton solutions. It has been proved by A.C. Bryan and A.E.G. Stuart [8] that (1) has no analytic two-soliton solution. The equation has three independent invariants (conservation laws): • D(u) = f R u d x ; • E ( u ) = fR( •

2+ dx; a n d + 1 3)

BEREZIN T R A N S F O R M These quantities are time-independent during the time evolution of the solution u. The correctness of the initial value problem u(x,O) = g(x) (the C a u c h y p r o b l e m ) for (1) in Sobolev spaces We*(R~ ) = HS(R~), s > 1 (cf. also S o b o l e v space), was investigated in [5]. Equation (1) has a solitary wave solution u(x,t) = ¢ ( x - v t - e ) , where ¢(3) = 3 ( v 1) s e c h 2 { ~ V / ( v - 1 ) / ( 4 v ) ) (cf. also S o l i t o n ) , provided that the wave velocity v satisfies v ¢ [0,1]. The non-linear stability of the wave ¢ with respect to the p s e u d o - m e t r i c d ( u , ¢ ) ( t ) = i n f { ] ] u - ¢(x - v t - c)]]l: e E R} was established in [3] and [7] by a clever modification of Lyapunov's direct method in combination with a spectral decomposition technique. Here, ]l']tl is the norm in the Sobolev space H I ( R x ) . This means that the form of the solitary wave is stable under small perturbations in the form of the initial wave. G e n e r a l i z a t i o n s . The generalized B e n j a m i n - B o n a Mahony equation is an equation of the form ut + a(u)~ - u~,t = 0,

(3)

where a : R --+ R is a differentiable function. (3) allows two types of solitary waves: kink-shaped and bellshaped ones. Depending on the concrete form of the nonlinearity, these solitary waves can be stable or unstable with respect to the metric d(u, ¢). For more concrete resuits concerning (3), see [11, Chap. 4]. The generalized Benjamin-Bona-Mahony equation in higher dimensions reads ut - A u t + div ~(u) = 0,

(4)

where A is the L a p l a c e o p e r a t o r in R ~ and p C C I ( R ; R n ) . The uniqueness and global existence of a solution in Sobolev spaces to the initial boundary value problem for (4) in f~ x [0, T], with Dirichlet (or more general) boundary conditions, was proved in [2], [9]. Here, ft C R ~ is a hounded domain with smooth boundary. The Cauchy problem for (4) is studied in [1]. Non-local generalizations of the Benjamin-BonaMahony equation can be obtained after one writes (1) in the form M u t + u~ + u u , = 0 .

Here, M is a p s e u d o - d i f f e r e n t i a l o p e r a t o r (in fact, a Fourier multiplier operator), acting as M u ( ~ ) = m({)g(~), w h e r e ^ d e n o t e s the F o u r i e r t r a n s f o r m in the space variable. For the original Benjamin-BonaMahony equation one has m(~) = 1 + ~2. In general, one takes for m({) a positive even function such that its negative power m({) -1 is monotone decreasing on (0, oo) and belongs to L 1(R). See [4], [5] and the references therein for more details.

The variable-coefficient equation

Benjamin-Bona-Mahony

ut + a(t)ux + b(t)uVux - u ~ t = 0

describes the propagation of long weakly non-linear water waves in a channel of variable depth. This equation was studied in [6]. References [1] AVRIN, J.: 'The generalized Benjamin Bona-Mahony equation in R n with singular initial data', Nonlin. Anal. Th. Meth. Appl. 11 (1987), 139-147. [2] AVRIN, J., AND GOLDSTEIN, J.A.: 'Global existence for the Benjamin Bona-Mahony equation in arbitrary dimensions', Nonlin. Anal. Th. Meth. Appl. 9 (1985), 861-865. [3] BENJAMIN, T.B.: 'The stability of solitary waves', Proc. Royal Soc. London A 328 (1972), 153-183. [4] BENJAMIN, T.B.: 'Lectures on nonlinear wave motion', in A.C. NEWELL (ed.): Nonlinear Wave Motion, Vol. 15 of Lectures in Applied Math., Amer. Math. Soc., 1974, pp. 3-47. [5] BENJAMIN, T.B., BONA, J.L., AND MAHONY, J.J.: 'Model equations for long waves in nonlinear dispersive systems', Philos. Trans. Royal Soc. London A 272 (1972), 47-78. [6] BISOGNIN, V., AND PERLA MENZALA, G.: 'Asymptotic behaviour of nonlinear dispersive models with variable coefficients', Ann. Mat. Pura Appl. 168 (1995), 219-235. [7] BONA, J.L.: 'On the stability theory of solitary waves', Proc. Royal Soc. London A 344 (1975), 363-374. [8] BaYAN, A.C., AND STUART, A.E.G.: 'Solitons and the regularized long wave equation: a nonexistence theorem', Chaos, Solitons, Fraetals 7 (1996), 1881-1886. [9] CALVERT, B.: 'The equation A ( t , u ( t ) ) ' + B(t,u(t)) = 0', Math. Proc. Cambridge Philos. Soc. 79 (1976), 545-561. [10] GARDNER, C.S., GREENE, J.M., KRUSKAL, M.D., AND MIURA, R.M.: 'Method for solving the Korteweg-de Vries equation', Phys. Rev. Lett. 19 (1967), 1095-1097. [11] ILIEV, I.D., KHRISTOV, E., AND KIRCHEV, K.P.: Spectral methods in soliton equations, Vol. 73 of Pitman Monographs and Surveys Pure Appl. Math., Longman, 1994. [12] LAX, P.D.: 'Integrals of nonlinear equations of evolution and solitary waves', Commun. Pure Appl. Math. 21 (1968), 467490.

Iliya D. Iliev

MSC 1991: 76B15, 35Q53 BEREZIN TRANSFORM, Berezin transformation The Berezin transform associates smooth functions with operators on Hilbert spaces of analytic functions. The usual setting involves an open set ~ C C n and a H i l b e r t s p a c e H of analytic functions on ~ (cf. also A n a l y t i c f u n c t i o n ) . It is assumed that, for each z C ~, the point evaluation at z is a continuous l i n e a r funct i o n a l on H. Thus, for each z C ~, there exists a Ks C H such that f ( z ) = (f, K z ) for every f E H. Because Kz reproduces the value of functions in H at z, it is called the reproducing kernel. The normalized reproducing kernel kz is defined by kz = K~/]IK~II. For T a bounded operator on H , the Berezin transf o r m of T, denoted by T, is the complex-valued function 67

BEREZIN TRANSFORM on ft defined by

T(z) = . For each bounded operator T on H , the Berezin transform T is a bounded real-analytic function on ft. Properties of the operator T are often reflected in properties of the Berezin transform 2r. The Berezin transform is named in honour of F. Berezin, who introduced this concept in [4]. The Berezin transform has been useful in several contexts, ranging from the Hardy space (see, for example, [8]) to the Bargmann-Segal space (see, for example, [5]), with major connections to the Bloch space and functions of bounded mean oscillation (see, for example, [9]). However, the Berezin transform has been most successful as a tool to study operators on the Bergman space. For concreteness and simplicity, attention below is restricted to the latter setting. The Bergman space L~(D) (cf. also B e r g m a n spaces) consists of the analytic functions f on the unit disc D C C such that fD Ill 2 dA < co (here, dA denotes area measure, normalized so that the area of D equals 1). The normalized reproducing kernel is then given by the formula kz(w) = (1 - Izl2)/(1 - ~w) 2. For ~ E L°°(D, dA), the Toeplitz operator with symbol %¢ is the operator T~ on L~a(D) defined by T~f = P ( ~ f ) , where P is the orthogonal projection of L2(D, dA) onto L~(D) (cf. also T o e p l i t z o p e r a t o r ) . The Berezin transform of the function ~, denoted by ~, is defined to be the Berezin transform of the Toeplitz operator T~. This definition easily leads to the formula ~(z) = (1 - I z l 2 ) 2 D { l ~(w) : ~ w ] 4 dA(w). If p is a bounded h a r m o n i c f u n c t i o n on D, then the mean-value property can be used to show that = ~. The converse was proved by M. Engli~ [6]: if E L°°(D, dA) and ~ = ~, then ~ is harmonic on D. P. Ahern, M. Flores and W. Rudin [1] extended this result to functions ~ C L 1(D, dA) (the formula above for makes sense in this case) and showed that the higherdimensional analogue is valid up to dimension 11 but fails in dimensions 12 and beyond. The normalized reproducing kernel kz tends weakly to 0 as z -+ OD. This implies that if T is a c o m p a c t o p e r a t o r on the Bergman space L~, then T(z) --+ 0 as z --+ OD. Unfortunately, the converse fails. For example, if T is the operator on L~ defined by ( T f ) ( z ) = f ( - z ) , then T(z) = (1 - Iz12)2/(1 + Izl2) 2. Thus, in this case T(z) --~ 0 as z -+ OD, but T is not compact (in fact, this operator T is unitary, cf. also U n i t a r y o p e r a t o r ) . However, the situation is much nicer for Toeplitz operators, and even, more generally, for finite sums of finite products of Toeplitz operators. S. Axler and D. Zheng 68

[3] proved that such an operator is compact if and only if its Berezin transform tends to 0 at OD. The Berezin transform also makes an appearance in the decomposition of the Toeplitz algebra 7- generated by the Toeplitz operators with analytic symbol. Specifically, G. McDonald and C. Sundberg [7] proved that if T E 7-, then T can be written in the form T = T~ + C, where ~ is in the closed algebra generated by the bounded harmonic functions on the unit disc and C is in the commutator ideal of 7-. The choice of ~ is not unique, but taking ~ to be the Berezin transform of T always works (see [2]). References [1] AHERN, P., FLORES, M., AND RUDIN, W.: 'An invariant volume-mean-value property', J. Funct. Anal. 111 (1993), 380-397. [2] AXLER, S., AND ZHENG, D.: 'The Berezin transform on the ToepIitz algebra', Studia Math. 127 (1998), 113-136. [3] AXLER, S., AND ZHENO, D.: 'Compact operators via the Berezin transform', Indiana Univ. Math. J. 47 (1998), 387400. [4] BEREZIN, F.: 'Covariant and contravariant symbols of operators', Izv. Akad. Nauk. S S S R Ser. Mat. 36 (1972), 1134 1167. (In Russian.) [5] BERGER, C., AND COBURN, L.: 'Toeplitz operators and quanturn mechanics', J. Funct. Anal. 68 (1986), 273-299. [6] ENGLIg, M.: 'Functions invariant under the Berezin transform', J. Funct. Anal. 121 (1994), 233-254. [7] MCDONALD, G., AND SUNDBERG, C.: 'Toeplitz operators on the disc', Indiana Univ. Math. J. 28 (1979), 595-611. [8] STROETHOFF, K.: 'Algebraic properties of Toeplitz operators on the Hardy space via the Berezin transform': Function Spaces (Edwardsville, IL, 1998), Contemp. Math. 232, Amer. Math. Soc., 1999, pp. 313-319. [9J ZHU, K.: 'VMO, ESV, and Toeplitz operators on the Bergman space', Trans. A m e r . Math. Soc. 302 (1987), 617-646.

Sheldon Axler MSC 1991: 47B35, 46Cxx

BERNSTEIN-BEZIER FORM, Bernstein form, Bdzier polynomial - The Bernstein polynomial of order n for a function f , defined on the closed interval [0, 1], is given by the formula

j=0

with

The polynomial was introduced in 1912 (see, e.g., [3]) by S.N. Bernstein (S.N. BernshteYn) and shown to converge, uniformly on the interval [0, 1] as n ~ oc, to f in case f is continuous, thus providing a wonderfully short, probability-theory based, constructive proof of the Weierstrass approximation theorem (cf. W e i e r strass theorem).

BEURLING ALGEBRA The Bernstein polynomial Bnf is of degree < n and agrees with f in case f is a polynomial of degree < 1. It depends linearly on f and is positive on [0, 1] in case f is positive there, and so has served as the starting point of the theory concerned with the approximation of continuous functions by positive linear operators (see, e.g., [1] and A p p r o x i m a t i o n o f f u n c t i o n s , l i n e a r m e t h o d s ) , with the Bernstein operator, Bn, the prime example. See also B e r n s t e i n p o l y n o m i a l s . The (n + 1)-sequence {b~: j = 0 , . . . , n} is evidently linearly independent, hence a basis for the (n + 1)dimensional linear space II~ of all polynomials of degree < n which contains it. It is called the Bernstein-Bdzier basis, or just the Bernstein basis, and the corresponding representation

is the ruth Fourier coefficient of f (cf. also F o u r i e r coefficients). The Beurling algebra is defined as

The space M* was introduced by A. Beurling for establishing contraction properties of functions [2]: Let

f(t) = Z

la±nl _< a~, n >_ 1, where { a ; } is a non-increasing sequence of numbers with a finite sum. Then if

g(t) ~ ~ n =

is called the Bernstein-Bdzier form, or just the Bernstein form, for p E II~. Thanks to the fundamental work of P. B6zier and P. de Casteljau, this form has become the standard way in computer-aided geometric design (see, e.g., [2]) for representing a polynomial curve, that is, the image {p(t) : 0 < t < 1} of the interval [0, 1] under a vector-valued polynomial p. The coefficients aj in that form readily provide information about the value of p and its derivatives at both endpoints of the interval [0, 1], hence facilitate the concatenation of polynomial curve pieces into a more or less smooth curve. Somewhat confusingly, the term 'Bernstein polynomial' is at times applied to the polynomial b~t, the term 'B~zier polynomial' is often used to refer to the Bernstein-B~zier form of a polynomial, and, in the same vein, the term 'B6zier curve' is often used for a curve that is representable by a polynomial, as well as for the Bernstein B~zier form of such a representation.

bnei~t, --

bo=O,

00

is a contraction of f(t) (that is, for any pair of arguments tl, t2 the inequality t 9 ( t l ) - g ( t 2 ) l _< If(tl)-f(t2)l holds), then the Fourier series of g(t) also converges absolutely, and

rb f _< 10 }2 a:. n=--OG

n=l

A similar result was proved in [2] for the F o u r i e r t r a n s f o r m , whence integrability of the monotone majorant on the real line is considered. These two spaces coincide locally, hence have much in common. Subsequently, A* appeared in some other papers either in explicit or implicit form. See [1] for a detailed survey of the history and properties of .4*. It turned out that the consideration of summability of Fourier series by linear methods at Lebesgue points leads to the same space of functions. Let f be a 2~r-periodic integrable function with Fourier series ~ k ]'(k) eikz" Let be a continuous function on R = ( - o c , ec), representable as follows:

References [1] DEVORE, R.A.: The approximation of continuous functions by positive linear operators, Springer, 1972. [2] FARIN, G.: Curves and surfaces for computer aided geometric design, third ed., Acad. Press, 1993. [3] LORENTZ, G.G.: Bernstein polynomials, Univ. Toronto Press, 1953.

ao = O,

be an absolutely convergent F o u r i e r series such that

?Z

j=0

ane int,

a(x) = L e-ix~d~(t), where # is a finite B o r e l m e a s u r e satisfying f a 1. Consider the means

(f*d#)N:=lim/afh(~ff~)h-+o

dp(t) =

d#(u),

C. de Boor where

MSC 1991: 41A10, 41A15, 68U05 BEURLING ALGEBRA of Fourier series with summable majorant of coefficients An algebra closely refated to the Wiener algebra

A=

f : IlfllA=

~

f(m)<~

fh(x) = h - 1 / R p ( h ) f ( x - t) dt. Here, F(t) is infinitely differentiable, equal to 1 for ]t] < 1, vanishing for It] > 2, and such that f a p(t) dt= 1. For f sufficiently smooth one has

,

m=--o0

k

where ?(rr~) = (2w) - 1

//

f ( u ) e -imu du 7r

and these are the linear means of the F o u r i e r series generated by )~. 69

BEURLING ALGEBRA The linear means (f*dp)N(X) converge to f(x) as N -+ oo at all the Lebesgue points of each integrable f if and only if the measure # is absolutely continuous (cf. A b s o l u t e c o n t i n u i t y ) with respect to the L e b e s g u e m e a s u r e and 0 ° esssup I#'(x)I du < (x). u
1) A* is a B a n a c h a l g e b r a with the local property. 2) The space of maximal ideals (cf. also M a x i m a l ideal) of .4* coincides with [-7~, ~]. 3) A* is a regular Banach algebra with trivial radical. 4) If f • .4* and F(z) is defined and analytic on a neighbourhood of the set of values of the function f , then F o f • .4" (in particular, if f does not vanish anywhere, then 1/f • A*). oo The space 7)34 * of all sequences d = {d k}k=-~o with finite norm

-

1 g;--f

Idkl k = - n

is the dual space of .4*. The space 7)34 * is not separable (cf. also S e p a r a b l e space) and thus the space A*, like A, is not reflexive (cf. also R e f l e x i v e space). S p e c t r a l p r o p e r t i e s o f A*. The following two results are analogues for A* of the Herz theorem and of the Wiener-Ditkin theorem, respectively. Let fN be a function coinciding with f at each point 2rck/N, where k, N > 0 are integers, k < N, and fN is linear on intervals. Suppose f • A*. Then lim IIf - fN[IA* ---- 0.

N--+o~

Let At(t) = (1 - I t l / e ) + for It[ __

and

¢

• (0,7r/2),

At(t + 2~r) = A t(t), and V~ = 2A2t - At. Let f • A* and f(0) = 0. Then lira I l f v t l [ ~ . = 0.

t-+0

S y n t h e s i s p r o b l e m s . Writing f • S if f admits synthesis in the norm of.4* (cf. also S y n t h e s i s p r o b l e m s ) , for f • .4* the following statements hold. a) If f • Lip 1, then f • S (Lip(i/2) for .4). b) If f is absolutely continuous and in Lip ~, ~ > 1/2, then f • S (for f • ,4, f has to be of bounded variation). (Here, analogous conditions for .4 are given in parentheses; these assertions make up the Beurling Pollard

theorem.) The following statements describe structural properties of functions in A*. In the sequel, w(g, .)p stands for 70

i) B 1,1 1 C A* C ~2,1 ~1/2 , and the imbeddings into these Besov spaces (cf. also Nikol'ski~ s p a c e and H a n k e l o p e r a t o r ) are both continuous. ii) If f is absolutely continuous and 1

.4* possesses many properties which are similar to those of A:

[]dl[p~. = sup

the modulus of continuity, in the norm of L p, of a function g (cf. also C o n t i n u i t y , m o d u l u s of).

(~)-l/p'

o W(f'; t)p in

t -1 dt < oc

for some p C [1, 2], p' = p/(p - 1), then f E .4*. iii) There exists a continuously differentiable function f ¢ .4* for which

w(f';t)~--O((lnl)-l/2). The second inclusion in i) is sharp; indeed, for each ¢ > 0 R e + l / 2 such that f • A*. there exists a function f ig ~2,oo As for ii), it is sharp for p = 1. The question is open for p > 1 (as of 2000). There exists a function f • A* that is not of bounded variation (cf. also F u n c t i o n o f b o u n d e d v a r i a t i o n ) . The following condition, although being very simple, is surprisingly sharp. If f~ • A, then f • A*. If the Fourier series of f is lacunary in the sense of Hadamard (cf. also L a c u n a r y s e q u e n c e ) , then the converse statement holds. Note that the central problem of s p e c t r a l s y n t h e sis, that is, existence of sets that are not of synthesis as well as existence of functions not admitting synthesis, is still open (as of 2000) for A*. Another open question is connected with Beurling's initial result. It would be interesting to know whether the following statement, converse to that given above, is true or not: If for every f • A* one has F o f • .4, then F • Lip 1. Some of the results known for .4* have been generalized to the multi-dimensional case. References

[1] BELINSKY, E., LIFLYAND, E., AND TRIGUB, R.: 'The Banach algebra A* and its properties', J. Fourier Anal. Appl. 3 (1997), 103-129. [2] BEURLING, A.: 'On the spectral synthesis of bounded functions', Acta Math. 81 (1949), 225-238. [3] KAHANE, J.-P.: Sdries de Fourier absolument convergentes, Springer, 1970.

E.S. Belinsky E.R. Liflyand MSC1991: 42A28, 42A24, 42A16 Let f be a function in the Hardy class H 2 (cf. also H a r d y classes). The vector space spanned by the functions einOf, n >_0, is dense in H 2 if and only if f is an outer function (cf. also H a r d y classes). BEURLING

THEOREM

-

B I R K H O F F - R O T T EQUATION This follows from the characterization of closed shiftinvariant subspaces in H 2 as being of the form g H 2 with g an inner function. See B e u r l i n g - L a x t h e o r e m for further developmeats. References [1] BEAUZAMY,B.: Introduction to operator theory and invariant subspaces, North-Holland, 1988, p. 194. [2] MLAK, W.: Hilbert spaces and operator theory, Kluwer Acad. Publ., 1991, p. 188; 190. [3] Sz.-NACY, B., AND FOIAS, C.: Harmonic analysis of operatots on Hilbert spaces, North-Holland, 1970, p. 104.

M. Hazewinkel

MSC 1991: 46J15, 30D55, 47A15 BEZOUT DOMAIN -

See B e z o u t

ring.

MSC 1991: 13Fxx B I R K H O F F - R O T T EQUATION - A planar vortex sheet is a curve in a two-dimensional inviscid incompressible flow across which the tangential velocity is discontinuous (cf. also V o n K f i r m f i n v o r t e x s h e d d i n g ) . The vortex sheet is described by its complex position z(F, t) = x + iy. For simplicity, assume that the vorticity on the sheet is all positive and that the flow outside the sheet is irrotational. The sheet is parameterized by a real variable F which represents the circulation, i.e. 7 = tOz/OF] -1 is the vorticity density along the sheet. Vortex sheet evolution is then described by the B i r k h o f f Rott equation [1], [15]: Otz(F,t) = (27ci)-1PV

/_~

dU

~ z(F,t)~z(Fi,t).

(1)

Because of the singularity of the integral at U = F, the integral in (1) is understood as a Cauchy principal value integral (cf. also C a u e h y integral). Perturbations of a flat sheet of uniform strength grow due to the linear Kelvin-Helmholtz instability and at some time later the sheet begins to roll-up. D. Moore [12], [13] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting from smooth initial data. The singularity found by Moore has the form zr = O(F -1/2) in which z(F) = x + iy is the position and F is the circulation variable. This singularity form was later found to be generic [7]. Exact singular solutions of the non-linear Birkhoff Rott equation, corresponding to Moore's singularity, have been constructed in [6], [8]. Numerical simulations of the vortex sheet problem [10], [11], [16] have produced singular solutions which are in agreement with Moore's theory. Krasny's method [10] used a non-linear filter to remove the numerical noise generated by the physical instability, the convergence of which was proved in [3] for analytic initial data.

R. Krasny [9] also computed roll-up of a sheet, using a desingularized equation, and found that the sheet begins to roll-up immediately after the appearance of the first singularity. A general set of similarity solutions for a rolled-up vortex sheet were constructed numerically in [14]. Existence results almost up to the singularity time have been proved [5], [17], using the abstract C a u e h y K o v a l e v s k a y a t h e o r e m . The results for existence and for singularity formation use an extension of the Birkhoff-Rott equation (1) into the complex F-plane for analytic initial data. Since the linearization of (1) is elliptic in F and t (el. also E l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n ) , it is hyperbolic in the imaginary F direction (cf. also H y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n ) . Singularities in the initial data at complex values of F travel towards the real axis at a finite speed. The Birkhoff Rott equation has been extended to three-dimensional sheets in [4]. Short-time existence theory for the three-dimensional equations has been established in [17]. A computational method for the threedimensional equations was implemented in [2]. Open questions as of 2000 include the well-posedness for continuation after Moore's singularity and the form of singularities in three dimensions. References [1] BIRKHOFF, G.: Helmholtz and Taylor instability, Vol. XII of Proe. Syrup. Appl. Math., Amer. Math. Soc., 1962, pp. 55-76. [2] BRADY, M., LEONARD, A., AND PULLIN, D.I.: 'Regularized vortex sheet evolution in three dimensions', J. Comput. Phys. 146 (1998), 520-45. [3] CAFLISeH, R.E., HOU, T.Y., AND LOWENGRUB, J.: 'Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering', Math. Comput. 68 (1999), 1465-1496. [4] CAFLISCH, R.E., AND LI, X.: 'Lagrangian theory for 3D vortex sheets with axial or helical symmetry', Transport Th. Statist. Phys. 21 (1992), 559-578. [5] CAFLISCH, R.E., AND ORELLANA, O.F.: 'Long time existence for a slightly perturbed vortex sheet', Commun. Pure Appl. Math. 39 (1986), 807-838. [6] CAFLISCH, R.E., AND ORELLANA, O.F.: 'Singularity formulation and ill-posedness for vortex sheets', SIAM J. Math. Anal. 20 (1989), 293-307. [7] COWLEY, S.J., BAKER, G.R., AND TANVEER, S.: 'On the formation of Moore curvature singularities in vortex sheets', J. Fluid Mech. 378 (1999), 233-267. [8] DUCHON, J., AND ROBERT, R.: 'Global vortex sheet solutions of Euler equations in the plane', J. Diff. Eqs. 73 (1988), 215224. [9] KRASNY, R.: 'Desingularization of periodic vortex sheet rollup', Y. Comput. Phys. 65 (1986), 292-313. [10] KRASNY, R.: 'On singularity formation in a vortex sheet and the point vortex approximation', J. Fluid Mech. 167 (1986), 65-93. [11] MEIRON, D.I., BAKER, G.R., AND ORSZAG, S.A.: 'Analytic structure of vortex sheet dynamics, Part 1, Kelvin-Helmholtz instability', Y. Fluid Mech. 114 (1982), 283-298.

71

B I R K H O F F - R O T T EQUATION [12] MOORE, D.W.: 'The s p o n t a n e o u s a p p e a r a n c e of a singularity in the shape of an evolving vortex sheet', Proc. Royal Soc. London A 3 6 5 (1979), 105-119. [13] MOORE, D.W.: 'Numerical a n d analytical aspects of Helmholtz instability', in F.I. NIORDSON AND N. OLHOFF (eds.): Theoretical and Applied Mechanics (Proc. X V I ICTAM), North-Holland, 1984, pp. 629-633. [14] PULLIN, D.I., AND PHILLIPS, W . R . C . : ' O n a generalization of Kaden's problem', ./. Fluid Mech. 104 (1981), 45-53. [15] ROTT, N.: 'Diffraction of a weak shock with vortex generation', J F M 1 (1956), 111. [16] SHELLEY, M.: 'A study of singularity formation in vortexsheet motion by a spectrally accurate vortex m e t h o d ' , J. Fluid Mech. 2 4 4 (1992), 493-526. [17] SULEM, P., SULEM, C., BARDOS, C., AND FRISCH, U.: 'Finite time analyticity for the two and three dimensional K e l v i n Helmoltz instability', Comm. Math. Phys. 80 (1981), 485516.

Russel E. Caflisch MSC 1991:76C05 BISHOP THEOREM - One of the early generalizations of the famous S t o n e - W e i e r s t r a s s t h e o r e m , which itself is a generalization of the celebrated W e i e r s t r a s s t h e o r e m stating that every real-valued continuous function on a closed and bounded interval is a uniform limit (cf. also U n i f o r m c o n v e r g e n c e ) of a sequence of polynomials. Let X be a compact H a u s d o r f f space, and let C(X) denote the set of all complex-valued continuous functions on X equipped with the supremum norm, given by

Ilfll := {llf(x)ll : x E X } . Let A be a non-empty subset of C(X) and let K be a non-empty subset of X. One says that K is a partially Aanti-symmetric set if f E A and f]K (the restriction of f to K ) real imply that flK is a constant. A partially Aanti-symmetric set is called A-anti-symmetric if f E A and flK purely imaginary (that is, Re(flK ) = 0) imply that fiK is a constant. It is easy to see that if A is closed under multiplication by i (that is, f E A implies if E A), then every partially A-anti-symmetric set is A-anti-symmetric. This is not true for arbitrary A. (See [5] for an example.) Every partially A-anti-symmetric set is contained in a maximal partially A-anti-symmetric set. Every maximal partially A-anti-symmetric set is closed. Distinct maximal partially A-anti-symmetric sets are disjoint. Each singleton set is a partially A-anti-symmetric set. Thus, the family of all maximal partially A-anti-symmetric sets forms a partition of X. Proofs of these and many other interesting properties of partially A-anti-symmetric sets can be found in [5]. All these statements are also true for A-anti-symmetric sets. (See [5], [3].) 72

Bishop's theorem. 1) Let A be a uniformly closed real subalgebra of

C(X) containing the constant function 1 and let f E C(X). If fIK E AIK := {fIK: f E A} for every maximal partially A-anti-symmetric set K , then f E A. 2) Let ~- be a h o m e o m o r p h i s m on X such that ToT is the identity mapping on X. Let

C(X,T) := { f E C ( X ) : f(w(x)) = f(x), Vx E X } . Let A be a uniformly closed real subalgebra of C(X, T) containing the constant function 1 and let f E C(X, T). If flK E AIK for every maximal A-anti-symmetric set K, thenfEA. In fact, 1) means that if a c o n t i n u o u s f u n c t i o n f coincides with some function in A for every maximal partially A-anti-symmetric set, then f E A. In view of the comments preceding the statement of the theorem, if A is a complex subalgebra of C(X), then in 1) above one can replace 'partially A-anti-symmetric' by 'A-anti-symmetric'. This was the classical statement by E. Bishop in 1961 [1]. A is said to separate the points of X, if for all x, y E X, x ¢ y, there is an f E A such that f(x) ¢ f(y). A uniformly closed complex subalgebra A of C(X) that contains 1 and separates the points of X is called a complex function algebra. Similarly, a uniformly closed real subalgebra A of C(X, 7) that contains 1 and separates the points of X is called a real function algebra. In view of this, 2) is called an analogue of Bishop's theorem for real function algebras. This was proved in [6]. (See also [5].) If a complex function algebra A is closed under conjugation (that is, f E A implies ] E A), then every maximal A-anti-symmetric set reduces to a singleton. (See [5] for a proof.) Thus, the hypotheses of Bishop's theorem are trivially satisfied by every f E C(X). Hence A = C(X). This is the classical S t o n e - W e i e r s t r a s s t h e o r e m , which has permeated most of modern analysis and has many generalizations. Bishop's theorem is an essential tool in proving many of these generalizations. (See [5], [3].) Similarly, if a real function algebra A is closed under conjugation, then A = C(X, w). This is an analogue of the Stone-Weierstrass theorem for real hmction algebras. (See [5] for a proof.) The proof of Bishop's theorem in [1] uses many nontrivial tools from functional analysis, such as the H a h n B a n a c h t h e o r e m , the Kre~n Mil'man theorem (cf. also L o c a l l y c o n v e x s p a c e ) and the Riesz representation theorem (cf. R i e s z t h e o r e m ) . This proof can also be found in [3] and [10]. J.B. Prolla extended this technique to the case of vector-valued functions [8]. S. Machado formulated a quantitative version of Prolla's theorem and gave an elementary proof of it [7]. A self-contained exposition of Machado's proof can be found in [4]. In

BLACK-SCHOLES FORMULA 1984, T.J. Ransford gave a very short, simple and elementary proof of Machado's version of Bishop's theorem [9] (see also [5]). This proof uses a technique also used in [2].

where

l o g ( S ( t ) / K ) + (r + dl =-

ov~

d2 = l o g ( S ( t ) / K )

+ (r -

ave-

References [1] BISHOP, E.: 'A generalization of the Stone Weierstrass theorem', Pacific J. Math. 11 (1961), 777-783. [2] BROSOWSKI, B., AND DEUTSCH, F.: 'An elementary proof of the Stone-Weierstrass theorem', Proc. Amer. Math. Soc. 81 (1981), 89 92. [3] BURCKEL, R.B.: Characterizations of C ( X ) among its subalgebras, M. Dekker, 1972. [4] BURCKEL, R.B.: 'Bishop's Stone Weierstrass theorem', Amer. Math. Monthly 91 (1984), 22 32. [5] KULKARNI, S.H., AND LIMAYE, B.V.: Real function algebras, M. Dekker, 1992. [6] KULKARNI, S.H., AND SRINWASAN, N.: 'An analogue of Bishop's theorem for real function algebras', Indian J. Pure Appl. Math. 18 (1987), 136-145. [7] MACHADO, S.: 'On Bishop's generalization of the StoneWeierstrass theorem', Indag. Math. 39 (1977), 218 224. (Nederl. Akad. Wetensch. Proc. Ser. A 80 (1977).) [8] PROLLA, J.B.: 'Bishop's generalized Stone-Weierstrass theorem for weighted spaces', Math. Ann. 191 (1971), 283-289. [9] I~ANSFORD, T.J.: 'A short elementary proof of the Bishop Stone-Weierstrass theorem', Math. Proc. Cambridge Philos. Soc. 96 (1984), 309-311. [10] RUDIN, W.: Real and complex analysis, McGraw-Hill, 1966.

S.H. Kulkarni

2/2)(T - t)

- t

' / 2 ) ( T - t)

t

The meaning of the symbols is as follows: C(t) is the price at time t of a call option expiring at time T > t with strike price K ; S(t) is the price of the underlying asset at time t; N(-) is the cumulative n o r m a l d i s t r i b u t i o n function; r is the interest rate; and a is a par a m e t e r known as the volatility of the underlying asset. The theorem behind the Black-Scholes formula can be stated as follows. Let the real-valued stochastic process St and the deterministic function Bt satisfy the It5 stochastic differential equations dSt = list

dt + aSt dwt,

(2)

dBt = rBt dt, where #, a, and r are constants, and wt denotes a standard B r o w n i a n m o t i o n (cf. also S t o c h a s t i c p r o cess; S t o c h a s t i c d i f f e r e n t i a l e q u a t i o n ) . Define a selffinancing portfolio strategy as a pair of processes (¢t, ~bt) adapted to the filtration associated to the Brownian motion in (2), such t h a t the process defined by lit = ¢tSt + ~tBt (the portfolio value process) satisfies

dVt = ~t dSt + ~bt dBt. MSC 1991: 54C35, 46E25 B L A C K - S C H O L E S FORMULA In 1973, F. Black and M. Scholes published a formula for the price of a financial contract whose pay-off at a future time depends in a non-linear way on the value of a given asset at that time. The Black-Scholes formula has been remarkably successful both in terms of use within the financial industry and as a starting point for further mathematical research. Nowadays (2000), the theory of pricing of so-called derivative contracts and related subjects has grown into a well-developed mathematical discipline. A survey of the area is available, for instance in [8]. The best-known example of a derivative contract is the European call option, which gives the holder the right, but not the obligation, to acquire a certain asset (the underlying) at a specified future time (the expiration time) for a specified price (the strike price). If the value of the underlying asset at expiration time T is denoted by ST and the strike price is denoted by K , then the value of the call option at expiration is m a x ( S T - - K, 0). The formula given by Black and Scholes for the value of a European call option at a time t < T is c(t)

= S(t)iV(d

) -

(1)

(The economic interpretation of this formula is t h a t no funds are added to or withdrawn from the portfolio, whence the terminology 'self-financing'.) Given fixed times t and T with t < T and a current asset value S(t) = St, there exists a seff-financing strategy (¢, ~b) defined on It, T] such t h a t VT = max(ST -- K, 0) almost surely if and only if Vt = C(t), where C(t) is given by

(I). In the proof of the above theorem, it is shown that the portfolio 'weight processes ¢t and ~ t c a n actually be constructed by ¢t = ¢(t, St) and ~bt = ~(t, St) where ¢(., .) and ~(., .) are smooth functions. The pair of functions ( ¢ , ~ ) is called a hedge strategy. To practitioners, the computation of hedge strategies is at least as important as the computation of prices. The Black-Scholes formula has been extended in m a n y directions. One can consider options that depend on the final value ST in other ways, options that depend not just on the final value but also on the p a t h taken by the variable St, and options on several underlying variables; moreover, one may consider different specifications of the processes followed by the underlying variables. Options whose time of expiry is fixed are called European options; there are also American options, which expire at a point in time to be selected by the holder. In a number of cases analytical solutions 73

BLACK SCHOLES F O R M U L A analogous to the Black Scholes formula can be given, but in many other instances one has to resort to numerical methods. Analytical as well as numerical techniques can be distinguished in two types. One approach is based on the characterization of the option value in terms of the expectation of the value of the option at expiration under a so-called equivalent martingale measure, which is related to the originally given measure through a Girsanov transformation (cf. also M a r t i n gale; C o n t r o l l e d s t o c h a s t i c p r o c e s s ) . The second approach uses a d i f f u s i o n e q u a t i o n t h a t can be written down for the evolution of the option price as a function of the underlying variables. The two approaches are related via a F e y n m a n - K a c formula. The idea of using Brownian motion to describe the behaviour of asset prices dates back to the doctoral dissertation of L. Bachelier, which appeared in 1900 [1]. The model (2), known as geometric Brownian motion, was proposed by P.A. Samuelson in 1964 [7]. The original formulation of the Black-Scholes formula can be found in [2]. Attempts at rigorizing the arguments given in this paper started with [6]. The interpretation given above in terms of self-financing portfolio strategies is due to J.M. Harrison and D. Kreps [3] and Harrison and S. Pliska [4]. Systematic accounts of the theory of option pricing and related issues of financial risk management can be found in several textbooks and at various levels of sophistication; see [5] for a mathematical treatment. For additional references, see O p t i o n p r i c i n g . References [1] BACHELIER, L.: 'Th6orie de la sp6culation', Ann. Sci. t~eole Norm. Sup. III 1"/" (1900), 21-86. [2] BLACK, F., AND SCHOLES, M.: 'The pricing of options and corporate liabilities', J. Political Economy 81 (1973), 637 659. [3] HARRISON, J.M., AND KREPS, D.: 'Martingales and arbitrage in multi-period security markets', J. Economic Th. 20 (1979), 381-408. [41 HARRISON, J.M., AND PLISKA, S.: 'Martingales and stochastic integrals in the theory of continuous trading', Stochastic Processes Appl. 11 (1981), 215 260.

[5] KARATZAS,I.,

AND SHREVE,

S.E.: Methods of mathematical

finance, Springer, 1998. [6] MERTON, R.C.: 'Theory of rational option pricing', Bell J. Economies and Management Sci. 4 (1973), 141-183. [7] SAMUELSON, P.A.: 'Rational theory of warrant pricing', in P.H. COOTNER (ed.): The Random Character of Stock Market Prices, MIT, 1964, pp. 506 525. [8] WILMOTT, P.: Derivatives. The theory and practice of financial engineering, Wiley, 1998.

J.M. Schumacher

M S C 1991: 90A09, 93Exx, 60Hxx B M O A - S P A C E , space of analytic functions of bounded mean oscillation - In 1961, F. John and L. Nirenberg [4] introduced the space of functions of 74

bounded mean oscillation, BMO, in their study of differential equations (cf. also B M O - s p a c e ) . About a decade later, C. Fefferman proved his famous duality theorem [1] [2], which states t h a t the dual of the Hardy space H 1 is BMOA (cf. also H a r d y s p a c e s ) . In these early works, BMO was studied primarily as a space of real-valued functions, but Fefferman's result raised questions a b o u t the nature of analytic functions in the Hardy spaces of the unit disc whose boundary values are in BMO. This is the definition of B M O A and the duality theorem provides the alternative t h a t BMOA consists of those analytic functions t h a t can be represented as a sum of two analytic functions, one with a bounded real p a r t and the other with a bounded imaginary part (cf. also A n a l y t i c f u n c t i o n ; H a r d y s p a c e s ) . Ch. P o m m e r e n k e [5] proved t h a t a u n i v a l e n t f u n c t i o n is in B M O A if and only if there is a bound on the radius of the discs contained in the image. Subsequently, W. H a y m a n and P o m m e r e n k e [3] and D. Stegenga [6] proved t h a t any analytic function is in BMOA provided the complement of its image is sufficiently thick in a technical sense that uses the notion of l o g a r i t h m i c cap a c i t y . As an example, any function whose image does not contain a disc of a fixed radius e > 0 centred at a + ib, where a, b range over all integers, satisfies this criterion and hence is in BMOA. In a similar vein, K. Stephenson and Stegenga [7] proved that an analytic function is in BMOA provided its image R i e m a n n s u r f a c e (viewed as spread out over the complex plane) has the following property: There are 0 < R < ec, 0 < e < 1 so that a Brownian traveller will, with probability at least e, fall off the edge of the surface before travelling outward R units (cf. also B r o w n i a n m o t i o n ) . As an example, an overlapping infinite saussage-shaped region can be constructed so that the Riemann mapping function maps onto to the entire complex plane but is nevertheless in BMOA. BMOA comes up naturally in m a n y problems in analysis, such as on the composition operator, the corona problem (cf. also H a r d y classes), and on functions in one and several complex variables. BMO and its variants has become an indispensable tool in real and complex analysis. References [1] FEFFERMAN, C.: 'Characterization of bounded mean oscillation', Bull. Amer. Math. Soc. g7 (1971), 587 588. [2] FEFFERMAN, C., AND STEIN, E.: ~Hp spaces of several variabIes', Acta Math. 129 (1974), 137-193. [3] HAYMAN, W., AND POMMERENKE, CH.: 'On analytic functions of bounded mean oscillation', Bull. London Math. Soc. 10 (1978), 219 224. [4] JOHN, F., AND NmENBERC, L.: 'On functions of bounded mean oscillation', Commun. Pure Appl. Math. 14 (1961), 415 426.

BOMBIERI-IWANIEC M E T H O D [5] POMMERENKE, CH.: 'Schlichte Funktionen und analytische Funktionen von beschr~inkter mittlerer Oszillation', Comment. Math. Helv. 152 (1977), 591-602. [6] STEGENGA,D.: 'A geometric condition t h a t implies BMOA': Proc. Symp. Pure Math., Vol. XXXV:I, Amer. Math. Soc., 1979, pp. 427-430. [7] STEPHENSON, K., AND STEGENGA, D.: 'A geometric characterization of analytic functions of bounded mean oscillation', J. London Math. Soc. (2) 24 (1981), 243 254. D. Stegenga

MSC1991: 30Axx, 46Exx B O M B I E R I - I W A N I E C METHOD - Invented in [1] and [2], where it was used to bound the R i e m a n n z e t a f u n c t i o n ~(s) along the mid-line of its critical strip, this is currently (2000) the strongest method around for establishing upper bounds on several wide classes of Weyl sums that have many applications within a n a l y t i c n u m b e r t h e o r y (cf. also W e y l s u m ) . General application to the first such class, exponential sums within the scope of Van der Corput's exponent pairs theory (see [3]), was done by M.N. Huxley and N. Watt in [10]. Improvements to one aspect of the method (the first spacing problem below) were found by Watt [16] and Huxley and G. Kolesnik [8], while in [11] H. Iwaniec and C.J. Mozzochi made an important adaptation, forming a second branch of the method that yielded new results for the circle and divisor problems. In a series of papers beginning with [5] Huxley has generalized and refined this adaptation, applying it to bound lattice point discrepancy (the difference between an area and the number of integer lattice points within the area; cf. also L a t tice o f p o i n t s ; G e o m e t r y o f n u m b e r s ) . The original insight of E. Bombieri and Iwaniec into the second spacing problem (see below), which is of crucial importance for both branches of the method, has been significantly augmented by the theory of resonance curves invented and refined by Huxley in [6] and [4]. See [3] for a brief introduction to the method; [7] is the most recent book on the method, and covers almost all of its aspects, including applications to, e.g.: exponential sums with a Dirichlet character factor; the mean-square of 1~(1/2 + it)l over a short interval; and counting integer lattice points near a curve. It also covers Jutila's independent (but related) method of [12] and [13], used for the estimation of exponential sums with a Fourier coefficient of a cusp form, or a divisor function, as a factor. The original branch of the Bombieri-Iwaniec method deals with a sum

S(f; M1, M2) =

~

e(f(m)),

M1
where e(x) = exp(27rix), M1,M2 E [M, 2M], and the derivatives (from first to fourth, say) of F(x) =

f(Mx) have absolute values ranging within a uniformly bounded factor of a single parameter T (sufficiently large) with a = logM/logT C (0,1). The method, which only applies directly when a C (1/3,2/3), has five main steps: i) Division of S(f; M1, M2) into consecutive shorter sums within each of which f ( m ) is approximated, via T a y l o r series, by a cubic polynomial with rational quadratic coefficient a/q. ii) Use of Poisson summation (cf. also P o i s s o n s u m m a t i o n m e t h o d ) , evaluation of Gauss sums (cf. also G a u s s s u m ) and the stationary phase method (cf. also S t a t i o n a r y p h a s e , m e t h o d o f t h e ) to transform the short sum S(a/q) belonging to a/q into an even shorter one:

= (h 2,h,h 3/2,h 1/2,h -1/2) and yl(a/q) = -g/q, where ag = 1 (rood q). iii) Estimation of the sum over a/q of [S*(a/q)l, in

with x(h)

terms of the pth power moment, using the H S l d e r ine q u a l i t y and the double large sieve introduced in [1] (cf. also L a r g e sieve). This leaves two steps to provide data for the sieve. iv) The first spacing problem: Count pairs of 'neighbours' amongst vectors of the form x ( h l ) + . . . + x(hp). v) Second spacing problem: Count pairs of 'neighbours' y(a/q) and y ( a l / q a ) (coincidences). Steps iv) and v) remain open problems (as of 2000), as does the question of more radical improvement of the whole. E. Fouvry and Iwaniec, using only Van der Corput's method with the double large sieve from Step iii), found useful new bounds for their monomial exponential sums (see [7]). The method's second branch treats

U(f; M1, M2"H1, H2) = E S(hf'; Mx, M2) h

h

where the summation is over h C [H1,//2] C [H, 2HI and f , M1 and M2 are as above. Since f has been replaced by F , so must the cubic approximation of Step i) be replaced by its derivative. Steps ii)-iv) also change, but the second spacing problem does not. These sums U(f; MI, 21//2;H1,//2) can arise when a truncated form of the F o u r i e r s e r i e s for the sawtooth function

p(f') = [f'] - f ' +

1

is used in the estimation of the lattice point discrepancy of an area of size f(M2) - f(M1) << T bounded by x = M1, y = 0, x = M2 and y = if(x) in the (x, y)-plane. If M gets too large, then one can switch to summing the discrepancies for columns of width 75

B O M B I E R I IWANIEC M E T H O D 1 to summing those for rows of depth 1, the number of which required will be O(T/M). Poisson summation makes a similar switch possible in estimating S(f; M1, M2) alone. In both contexts the critical cases are those around a = 1/2 ( M = v ~ ) , so the a/q, being values of f"(x)/2, will satisfy 1 << la/ql << 1; something like the situation in applications of the H a r d y Littlewood circle m e t h o d . Jutila's method [13] utilizes a 'twisted' Wilton summation formula, corresponding to a cusp form of even weight k for the full modular group:

invite one to apply the Bombieri-Iwaniec method iteratively. P. Sargos m a n a g e d to do this in [15], using his own simpler theory of (quite different) resonance curves. He obtained results of greatest interest for c~ near 2/5 = 0.4. A third construction of resonance curves (different again) has been given by Huxley and Kolesnik [9], and is of interest for c~ near 7/17 = 0.4118. • .. They were able to apply this iteratively, but got better results (in most cases) with a single step of an elementary method once employed by H. Swinnerton-Dyer to count lattice points exactly on a curve. References

~

n

where b(m) is the ruth Fourier coefficient of the cusp form, 9(x) is a smooth function supported in [M, 2M], and the linear integral transform 12 = £k,q has integrand containing the Bessel J-function of order k - 1 as a factor (cf. also B e s s e l f u n c t i o n s ) . Step ii) above is essentially the extension of Jutila's formula to the case where the cusp form is replaced by the t h e t a - s e r i e s of weight k = 1/2 with Fourier coefficients b(m) = ~ { n C Z: n 2 = m}. M. Jutila has an analogous 'twisted' VoronoY summation formula for the more delicate case where b(m) is the divisor function (cf. also N u m b e r o f d i v i s o r s ) , for which the corresponding modular form is non-holomorphic; this enabled a more elementary proof of a famous theorem of Iwaniec (see [14]). The original t r e a t m e n t by Bombieri and Iwaniec of the second spacing problem rested on the observation that proximity of g/q and a T / q l (modulo Z) implies that the row vectors vl = [al, ql] and v = [a, q] will be linked by a relation v~ = B v t, with B being an integer (2 x 2)matrix of determinant 1 having a 'small' lower left entry. Huxley's theory of resonance curves showed t h a t the coincident pairs y(a/q), y(al/ql) for a given choice of B correspond to integer points lying near a certain plane resonance curve, this curve being determined (up to a minor transformation) by the choice of f(x) and B. This led, in [6], to better bounds in the second spacing problem, to improved exponential sum estimates, and (for example) to the result t h a t

when t --+ + e c , with fixed/3 > 89/570 = 0.1561... (an unpublished improvement of this and of resonance curve theory is given in [4]). For comparison, Bombieri and Iwaniec's original paper [1] had fl > 9/56 = 0.1607..., while even a complete resolution of both the first and second spacing problems (alone) could not get/~ below 3/20 = 0.15. Integer points near a suitable curve may be counted using exponential sum estimates, so resonance curves 76

[1] BOMBIERI, E., AND IWANIEC, H.: COn the order of ¢(1/2 + i t ) ' , Ann. Seuola Norm. Sup. Pisa CI. Sci. 13 (1986), 449-472. [2] BOMBIERI, E., AND IWANIEC, H.: 'Some mean value theorems for exponential sums', Ann. Scuola Norm. Sup. Pisa Cl. Sci.

13 (1986), 473-486. [3] GRAHAM, S.W., AND KOLESNIK, C-.: Van der Corput's method for exponential sums, Vol. 126 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1991. [4] HUXLEY, M.N.: Exponential sums and the Riemann zetafunction !7, unpublished notes. [5] HUXLEY, M.N.: 'Exponential sums and lattice points', Proc. London Math. Soc. 60 (1990), 471-502, Corrigenda,

66 (1993), 70. [6] HUXLEY, M.N.: 'Exponential sums and the Riemann zetafunction IV', Proc. London Math. Soc. 66 (1993), 1-40. [7] HUXLEY, M.N.: Area, lattice points and exponential sums, Vol. 13 of London Math. Soc. Monographs, Oxford Univ. Press, 1996. [81 HUXLEY, M.N., AND KOLESNIK, G.: 'Exponential sums and the Riemann zeta-function III', Proc. London Math. Soc. 62 (1991), 449-468, Corrigenda 66 (1993), 302. [9] HUXLEY, M.N., AND KOLESNIK, G.: 'Exponential sums with a large second derivative', in M. JUTILA AND T. METSANKYLA (eds.): Number Theory (Proc. Turku Conf. Number Theory in Memory of Kustaa Inkeri), de Gruyter, 2000, pp. 131-143. [10] HUXLEY, M.N., AND WATT, N.: 'Exponential sums and the Riemann zeta-function', Proc. London Math. Soc. 57 (1988), 1-24. [11] IWANIEC, H., AND MOZZOCH, C.J.: 'On the divisor and circle problems', J. Number Theory 29 (1988), 60-93. [12] JUTILA, M.: 'On exponential sums involving the divisor function', J. Reine Angew. Math. 355 (1985), 173-190. [13] JUTILA, M.: Lectures on a method in the theory of exponential sums, Vol. 80 of Tata Inst. Fundam. Res. Lect. Math. and Physics, Springer, 1987. [14] JUTILA, M.: 'The fourth power moment of the Riemann zetafunction over a short interval': Number Theory I (Budapest, 1987), Vol. 51 of Colloq. Math. Soc. J . Bolyai, NorthHolland, 1990, pp. 221-244. [15] SARCOS, P.: 'Points entiers au voisinage d'une courbe, sommes trigonom6triques courtes et pairs d'exposants', Proe. London Math. Soc. 70 (1995), 285-312. [16] WATT, N.: 'Exponential sums and the Riemann zeta-function II', Y. London Math. Soc. 39 (1989), 385-404.

N. Watt MSC1991: 11L03, 11L05, 11L15, 11Lxx

BORCHERDS LIE A L G E B R A BORCHERDS LIE ALGEBRA, Borcherds algebra While a K a c - M o o d y a l g e b r a is generated in a fairly simple way from copies of ~2, a Borcherds or generalized Kac-Moody algebra [1], [7], [9], [11] can also involve copies of the 3-dimensional Heisenberg algebra. Nevertheless, it inherits many of the Kac Moody properties. Borcherds algebras played a key role in the proof of the Monstrous Moonshine conjectures [4], and also led to the development of a theory of automorphie products

H, First recall the definition of a Kac Moody algebra. By a (symmetrizable) Caftan matrix A = (aij) one means an integral (~ x ~)-matrix obeying C1) a i i = 2 and aij < 0 for all i ~ j; and C2) there is a diagonal matrix D with each dii > 0 such that D A is symmetric. A (symmetrizable) Kac-Moody algebra g = g(A) [10], [12] is the Lie a l g e b r a on 3~ generators ei, fi, hi, obeying the relations: R1) [eifj] = 5ijhi, [hiej] = aijej, [hifj] = - a i j f j , and [hihj] = 0, for all i,j; and R2) ( a d e i ) l - ~ J e j = ( a d f i ) ~ - ~ f y = 0 for a l l / # j. A Borcherds algebra is defined similarly. By a generalized Caftan matrix A one means a (possibly infinite) matrix A = (a/j), aij 6 R, obeying GC1) either aii = 2 o r aii <_ 0 ; GC2) aij < 0 for i # j, and aij C Z when a i i = 2; and GC3) there is a diagonal matrix D with each dii > 0 such that D A is symmetric. By the (symmetrizable) universal Borcherds algebra ~j = ~(A) one means the Lie a l g e b r a (over R say) with generators ei, fi, hij, subject to the relations [3]: GR1) [eifj] = hij, [hijek] = 5ijaike~ and [hijA ] = - h i j a i k A , for all i,j; GR2) ( a d e i ) ~ - ~ e j = ( a d f i ) ~ - ~ f j = 0, whenever both a i i = 2 and i # j; and GR3) [eiey] = [Afj] = 0 whenever aiy = 0. Note that for each i, span{ei, fi, hii} is isomorphic to ~ge(R) when aii ¢ O, and to the 3-dimensional He±senberg algebra when a i i = 0. Immediate consequences of the definition are that: i) [hi3, h,~,~] = 0; ii) hij = 0 unless the ith and j t h column of A are identical; i/i) the hij for i ¢ j lie in the centre of ~. Setting all hij = 0 for i # j gives the definition of the (symmetrizable) Borcherds algebra g = I~(A) [1]. This central extension ~ of ~ is introduced for its role in the characterization of Borcherds algebras below. If A has no zero columns, then ~ equals its own universal central

extension [3]. An important technical point is that both g(A) and ~(A) have trivial radical. The basic structure theorem [1] is that of K a c - M o o d y algebras. Let g = g(A) be a symmetrizable Borcherds algebra. Then: a) g has triangular decomposition g = g+ ® b ® g_, where g+ is the subalgebra generated by the ei, g - is generated by the fi, and b = span{hi} is the C a r t a n s u b a l g e b r a . Also, [g+, g_] C 0 and [~),g±] C 9±. b) g has a root space decomposition: formally calling ei degreec~i and fi degree-c~i, and defining g~ to be the subspace of degree c~ E Z a l + Zc~2 + ..., one gets f) = g0 and g± = e ~ c A ~ ~, where [9~,g 9] C g~+~ and A_ = - A + ; c) there is an involution w on g for which wei = fi, cohi = - h i , and wl~~ = 9-~; d) dim~ ~ < oc and dimg ±~; = 1; e) there is an invariant symmetric b/linear form ('l') on g such that for each root a # 0, the restriction of ('1") to g~ x g-~ is non-degenerate, and (g~lg 9) = 0 whenever fl # - a ; f) there is a linear assignment c~ ~ x~ 6 b such that for all a 6 g~, b 6 g-~, one has [a, b] = (alb) x~. The condition that 9 be symmetrizable (i.e. condition GC3)) is necessary for the existence of the b/linear form in e). For representation theory it is common to add derivations, so that the roots c~ will lie in a dual space t)e*. In particular, define Di(a) = nia for any a 6 g~1~1+-..; then each linear mapping Di is a derivation, and adjoining these to 0 defines an Abel±an algebra ~)e. The simple root c~i can be interpreted as the element of ~* obeying c~j(hi) = aij and c~j(Di) = 5ij. Construct the induced b/linear form ('1") on b ~*, obeying (O~ilO~j) : diaij. The properties a)-f) characterize Borcherds algebras. Let G be a Lie algebra (over R) satisfying the following conditions: 1) G has a Z-grading ®iGi (cf. also Lie a l g e b r a , g r a d e d ) , and dim Gi < oo for all i ¢ 0; 2) G has an involution w sending Gi to G-i and acting as - 1 on Go; 3) G has an invariant b/linear form ('l') invariant under cJ such that (GiIGy) = 0 if i ~ - j , and such that -(alcJ(a)) > 0 if 0 7£ a E Gi for i ¢ 0. Then there is a homomorphism 7r from some ~(A) to G whose kernel is contained in the centre of ~, and G is the semi-direct product of the image of lr with a subalgebra of the Abel±an subalgebra Go. T h a t is, G is obtained from ~ by modding out some of the centre and adding some commuting derivations. See e.g. [4] for details. Define II ~ to be the set of all real simple roots, i.e. all ~i with aii = 2 ; the remaining simple roots are

77

B O R C H E R D S LIE A L G E B R A the imaginary simple roots a E Him. The Weyl group (cf. also W e y l g r o u p ) W of g is the group generated by the reflections ri: b e* -+ b e* for each ai G IIre: r~(A) = ~-- A(hi)ai. It will be a (crystallographic) C o x e t e r g r o u p . The real roots of g are defined to be those in W(IIr~); all other roots are called imaginary. For all real roots, dimg ~ = 1 and (ala) > O. V is called an integrable module if V = ®x~o~.V x, where the weight space V x := {v C V: h- v = ~(h)v}, with dim V x < ec, and for each i with aii = 2 both ei and fi are locally nilpotent: i.e. for all v C V and all sufficiently large k, (ei) k .v = 0 = (fi) k .v. By the character one means the formal sum chv:=

~

(dimVX)e x.

Let P+ be the set of all weights A E 0* obeying A(hi) E Z>0 whenever aii = 2, and A(hi) > 0 for all i. Define the highest-weight g-module L(A) in the usual way as the quotient of the Verma module (cf. also R e p r e s e n t a t i o n o f a L i e a l g e b r a ) by the unique proper graded submodule. Then one obtains the W e y l - K a e Borcherds character formula: Choose p E 0* to satisfy (pl

i) =

1

for all i, and define SA = e A+z ~

e(s) e ~, where s runs ( - l ) m if s is the sum of m distinct mutually orthogonal imaginary sireple roots, each of which is orthogonal to A, otherwise e(s) = 0. Then o v e r all s u m s of ~i E !q im a n d c(8) :

(SA) ChL(A) = eP YIc~cA+ (1 - e - a ) muIr a '

where mult c~ = dim g~. SA is the correction factor due to imaginary simple roots, much as the 'extra' terms in the Macdonald identities are due to the imaginary affine roots. Putting A = 0 gives the denominator identity, as usual. Thus, Borcherds algebras strongly resemble K a c Moody algebras and constitute a natural and non-trivial generalization. The main differences are that they can be generated by copies of the Heisenberg algebra as well as ~ 2 (R), and that there can be imaginary simple roots. Interesting examples of Borcherds algebras are the Monster Lie algebra [4], whose (twisted) denominator identity supplied the relations needed to complete the proof of the Monstrous Moonshine conjectures, and the fake Monster [2]. A Borcherds algebra can be associated to any even Lorentzian lattice. The denominator identities of Borcherds algebras are often automorphic forms on the automorphism group O~+2,2 (R) of the even selfdual lattice II~+2,2 [5]. T h e y can serve as 'automorphic 78

corrections' to Lorentzian K a c - M o o d y algebras (see, for instance, [6]). T h e space of BPS states in string theory carries a natural structure of a Borcherds-like algebra [8]. References

[1] BORCHERDS,R.E.: 'Generalized Kac-Moody algebras', J. Algebra 115 (1988), 501 512. [2] BORCHERDS,R.E.: 'The monster Lie algebra', Adv. Math. 83 (1990), 30-47. [3] BORCHERDS, R.E.: 'Central extensions of generalized KacMoody algebras', ft. Algebra 140 (1991), 330-335. [4] BORCHERDS,R.E.: 'Monstrous moonshine and monstrous Lie superalgebras', Invent. Math. 109 (1992), 405-444. [5] BORCHERDS, R.E.: 'Automorphic forms on O~+2,2(R) and infinite products', Invent. Math. 120 (1995), 161-213. [6] GRITSENKO, V.A., AND NIKULIN, V.V.: 'Siegel automorphic form corrections of some Lorentzian Kac Moody Lie algebras', Amer. J. Math. 119 (1997), 181-224. [7] HARADA, K., MIYAMOTO, M., AND YAMADA, H.: 'A generalization of Kac-Moody algebras': Groups, Difference Sets, and the Monster, de Gruyter, 1996. [8] HARVEY, 3.A., AND MOORE, C-.: 'On the algebras of BPS states', Commun. Math. Phys. 197 (1998), 489 519. [9] JURISICH, E.: 'An exposition of generalized Kac-Moody algebras', Contemp. Math. 194 (1996), 121-159. [10] KAC, V.G.: 'Simple irreducible graded Lie algebras of finite growth', Math. USSR Izv. 2 (1968), 1271-1311. [11] KAC, V.G.: Infinite dimensional Lie algebras, third ed., Cambridge Univ. Press, 1990. [12] MOODY, R.V.: 'A new class of Lie algebras', J. Algebra 10 (1968), 211-230.

Terry Gannon MSC 1991: 17B67, 11Fxx, 20D08 The words 'Braess's paradox' refer to a surprising decrease in performance for some network, which is a result of it being improved locally. The occurrence of this decrease has been studied in mathematical models of equilibrium flow in road/rail traffic, computer networks, telephone networks, water supply systems, electrical circuits, spring systems, and so on. This has been done under various assumptions on routing schemes, such as being state-dependent or fixed. Below, the occurrence of Braess's paradox in a classical model of user equilibrium traffic flow is given. Let (N, B) be a finite directed g r a p h , with node set N and arc, branch or link set B (cf. also G r a p h , orie n t e d ) . A path in which all links are similarly directed is called a route, with the initial and final nodes forming an origin/destination pair (or O / D pair). One considers a set W of O / D pairs, and for each w C W, supposes a f l o w demand d~ > 0 to be given. Let R~ be a set of routes joining w. For each w E W, and r C R~, one considers Fr _> 0 such that ~ r c R ~ Fr = d~, giving a route flow vector F = (Fr)~cRw,~ew. This route flow induces a link flow f = (fb)bcB, by fb = ~ b F~ for each b, where one identifies a route with the set of its links. For each link BRAESS

PARADOX

-

BRAMBLE-HILBERT LEMMA a, one supposes a link cost ca = ~ V c B gabfb + ha, where gab and ha are given. For r C Rw, w E W , one defines a route cost by C~ = ~ a e r ca. A route flow H is a user equilibrium if it satisfies the condition t h a t for all r, s E Rw, w E W, if C~ < Cs then H~ = 0. In other words, there is, for each w, a common route cost ~,w for all routes r E R~ with non-zero H,,. A user equilibrium flow exists always [5], and if the matrix g = gab is such t h a t g + gT is positive-definite, then the equilibrium link flows, and hence the route costs 7~, are unique. In [3], Braess's paradox is said to occur if adding a new route r to some R~ results in 7~ being increased. See also [3] for necessary and sufficient conditions for this to happen under the assumption that there is a strictly positive equilibrium flow in all routes. There is no fixed definition of Braess's paradox in all systems, but there is a common theme. One assumes some measure of performance, t, such as 7~- On the network given by any one link b, t depends on the flow f and a parameter kb. For example, t = 1/(kb -- f ) if the link is given by an M / M / 1 queue (cf. also Q u e u e ) . Note that t decreases if kb increases, for f fixed. Suppose some flow demand is fixed and link flows are given by some requirement about equilibria, or by a given dynamical process not in equilibrium. One says that Braess's paradox occurs if, for the network as a whole, t increases when some kb increases. Adding links may be thought of as changing a p a r a m e t e r kb from zero or infinity. A different language would be used to describe certain other types of networks, such as electrical circuits. Under this description, the D o w n s - T h o m s o n paradox [4] is a particular type of Braess's paradox. If one distinguishes these paradoxes mathematically, it is by requiring the link costs t in the Braess paradox to be increasing functions of the link flow f , while in the DownsThomson paradox there is a link with t a decreasing function of f . Independent discoveries of Braess's paradox can be attributed to D. Braess [1], A. Downs [4], J.M. Thomson [6] and C.A. Zukowski and J.L. W y a t t [7]. [2] contains an ample list of references.

References [1] BRAESS, D.: @ber ein Paradoxon aus der Verkehrsplannung', Unternehmcnsforschung 12 (1968), 258-268. [2] BRAESS, D., http://homepage.ruhr-uni-bochum.de/Dietrich.Braess (2000). [3] DAFERMOS, S., AND NAOURNEY, A.: 'On some traffic theory equilibrium paradoxes', Transportation l~es. B 18 (1984), i01 II0. [4] DOWNS, A.: 'The law of peak-hour expressway congestion', Traffic Quart. 16 (1962), 393-409. [5] SMITH, M.J.: 'The existence, uniqueness, and stability of traffic equilibria', Transportation F&s. B 13 (1979), 295-304. [6] THOMSON, J.M.: Great cities and their traffic, Gollanez, London, 1977.

[7] ZUKOWSKI, C.A., AND WYATT, J.L.: 'Sensitivity of nonlinear one-port resistor networks', IEEE Trans. Circuits Syst. C A S - 3 1 (1984), 1048-1051.

B.D. Calvert M S C 1991: 90B10, 90B15, 68M10, 68M20, 94C99

90B18,

90B20,

60K30,

BRAMBLE-HILBERT LEMMA, Hilbert-Bramble lemma - An abstract theoretical tool for studying the approximation error of functions in Sobolev spaces (cf. also A p p r o x i m a t i o n o f f u n c t i o n s ; S o b o l e v s p a c e ) by algebraic polynomials. The general formulation of the l e m m a was given and proven first by J.H. B r a m ble and S.R. Hilbert [3] in terms of a class of linear functionals on Sobolev spaces t h a t annihilate the set of polynomials PK (see the definition below), an intermediate to Pk-1 (polynomials of degree k - 1) and Pk, i.e. Pa-1 C PK C Pk. This formulation was motivated by the work of G. Birkhoff, M. Schultz and R.S. Varga [1] and applied to estimate the error of the H e r m i t e int e r p o l a t i o n f o r m u l a [3], s p l i n e i n t e r p o l a t i o n and Fourier transformation (cf. F o u r i e r t r a n s f o r m ) [2]. The Bramble-Hilbert l e m m a has a wide range of applications. It is used in the analysis of projection operators in the function spaces L2 and W21 for designing optimal domain decomposition and multi-level methods. Its application to the L2-error of the finite element approximations of elliptic problems of second order leads to sharp estimates with respect to both the regularity of the solution and rate of convergence. It is an indispensible tool in the error analysis of finite element [4], [5], finite volume [8], finite difference [9], collocation, and boundary element methods for solving partial differentim equations. The complete error analysis of the finite difference and finite volume schemes in [9] is based on the Bramble Hilbert lemma. To formulate the lemma, some notation regarding Sobolev spaces of real-valued functions on a bounded domain T in N-dimensional Euclidean space R N is needed. It is assumed that T satisfies the strong cone condition (cf. C o n e c o n d i t i o n ) and has diameter p = p(T) = diam(T). The boundary of T is denoted by 0T. The notations a, /3 and ~ will be used for multiindices, with lal = ~ j =Nl a J and D ~ . . D~I aN , . . DN where D i = O/Ox i. Let Lp(T) be the set of all functions u such that fT lu(x)l • dx exists and is finite. The n o r m in Lp is given by IlUllp,T = (fT ]u(x)] p dx) 1/p" Let Wp~(T) be the set of all functions in Lp(T) whose distributional derivatives of order less than equal to m (a non-negative integer) are in Lp(T). It will be assumed that 1 _< p < ~ . The norm and the s e m i - n o r m on 79

B R A M B L E - H I L B E R T LEMMA

Wpr~(T) are, respectively,

IlUllp,m,T = ~

IlD~Ullp,T,

in the whole domain ~ follows by summing all local estimates, taking the m a x i m a l p, and using the additivity of the integrals:

I~l<_m

lulp,m,T = ~

IlD~ullp,T •

151=m Let K be any subset of the set of multi-indices 7 of length m (i.e. [71 = m) which contains the indices of the form 7l = m, 7j = 0 for j ¢ l, I = 1 , . . . , N . The set of polynomials q such that D~q = 0 for all 7 C K will be denoted by PK. Obviously, Pk-1 C P~c C Pk. The Bramble-Hilbert lemma ([3]) is as follows: Let F ( u ) be a bounded l i n e a r f u n c t i o n a l on W ~ ( T ) that vanishes for polynomials in PK, i.e.

CE~,_opJ-N/Pl~lpjT

1) IF(u)l _< where the constant C is independent of p and the function u; 2) F(q) = 0 for q C PK. Then there is a constant C1, independent of p and u, such that

Ie(u)l <_ Cx ~

/).~-N/p IID%lip,r.

c~CK

Note that the l e m m a deals with function spaces of m a n y variables. For functions in a single variable, the inequality derived by the l e m m a follows immediately from the Peano kernel theorem for the remainder of the Taylor expansion (see, e.g., [10]). In many applications, a given domain ft is split into a number of non-overlapping subdomains T, usually triangles, rectangles, tetrahedra, hexahedra, etc., so that p = p(T) is small. Consider the error of the L a g r a n g e i n t e r p o l a t i o n f o r m u l a for a function u C W ~ ( ~ ) with polynomials of degree m - 1 over each T and take = UT. Here, the domains T are non-degenerate simplices in R N, /) = m a x T p ( T ) , K = {7: I~/f = m}, and PK = Pm-1. Assume that p > 2/- and m = 2 and let ql be a piecewise polynomial of degree 1 which interpolates the values of the function u at the verrices of each simplex T. Consider the linear functional F ( u ) = u(x) - qi(x) at a fixed point x • T. By the Sobolev imbedding theorem (cf. also I m b e d d i n g t h e orems), 2

I~(x)l <__ c}-~./) N-N/~ I~lp,j,z. j=0

Therefore F(u) is bounded in W ~ ( T ) . Obviously, F(u) = 0 if u is a polynomial of degree 1, since the linear interpolant recovers u exactly. Therefore, by the Bramble-Hilbert lemma: ]F(u)l < C1/)2-N/P[Ulp,2,T • An estimate of the Lp-error for the interpolant q1 over T follows immediately after taking this inequality to power p and integrating it over T. The estimate for the error 80

Ilu - qillp,~ < c p 2 lUlp,2,a . If p tends to 0, this estimate gives the rate at which the piecewise-linear interpolant converges to u in the L p norm. Similarly, one can obtain bounds for the Lp-norm of the first derivatives. This is a typical application of the B r a m b l e - H i l b e r t l e m m a in constructive function theory and in numerical methods for differential equations. The original proof of the l e m m a was abstract and the size of the constant C1 could not be estimated. This was the reason t h a t T. Dupont and L.R. Scott [7] gave a new constructive proof of the main result. Consider the polynomial Qmu which is an averaged Taylor polynomial of degree m - 1 over T (for the exact construction, see [4, Chap. 4]). This construction of QmU is related to the polynomials used by S.L. Sobolev [11] to study the spaces t h a t bear his name. Here, T is considered to be a star-shaped domain with respect to a ball B, that is, for all x E B, the closed convex hull of {x} U B is a subset of T, and one defines Pmax = sup {p = p ( B ) : T star shaped w.r.t. B } . The revised B r a m b l e - H i l b e r t l e m m a now reads ([4, p. 100]): Let T be a star-shaped domain with respect to the ball B. Then

lu - Q.~ulp,k,v __ < C ~n,N,~Pm-k lu _ Omu]p,m, T k=0,...,m. Here, e is the quotient of p and the diameter/)max, and the constant C,~,N,c does not depend on p while its dependence on m and k is estimated explicitly from above. This variant of the l e m m a was proved by Dupont and Scott [7] for m non-integer and for domains that are finite unions of domains t h a t are star-shaped with respect to a ball, while more general results were established in [6]. References [1] BIRKHOFF, G., SCHULTZ, M., AND VARGA, R.: 'Hermite interpolation in one and two variables with application to partial differential equations', Numer. Math. 11 (1968), 232-256. [2] BRAMBLE, J.H., AND HILBERT, S.R.: ' E s t i m a t i o n of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation', S I A M J. Numer. Anal. 7 (1970), 112-124. [3] BRAMBLE, J.H., AND HILBERT, S.R.: ~Bounds for a class of linear functionals with applications to Hermite interpolation', Numer. Math. 16 (1971), 362 369. [4] BRENNER, S.C., AND SCOTT, L.R.: The mathematical theory of finite element methods, Springer, 1994. [5] CIAaLET, P.G.: The finite element method for elliptic problems, North-Holland, 1978. [6] DECHEVSKI, L.T., AND QUAK, E.: 'On the B r a m b l e - H i l b e r t l e m m a ~, Numer. Funct. Anal. Optim. 11 (1990), 485-495.

BRANCH GROUP [7] DUPONT, W., AND SCOTT, L.R.: 'Polynomial approximation of functionals in Sobolev spaces', Math. Comput. 34 (1979), 441-463.

[8] LI, R., CHEN, Z., AND Wu, W.: Generalized difference methods for differential equations. Numerical analysis of finite volume methods, M. Dekker, 2000. [9] SAMARSKn, A.A., LAZAROV,R.D., AND MAKAROV,V.L.: Fi-

nite difference schemes for differential equations having generalized solutions, Visshaya Shkola Publ., Moscow, 1987. (In Russian.)

[10] SARD, A.: Linear approximation, Vol. 9 of Math. Surveys, Amer. Math. Soc., 1963. [11] SOBOLEV, S.L.: Applications o/ the functional a~alysis in mathematical physics, Amer. Math. Soc., 1963. (Translated from the Russian.)

Raytcho D. Lazarov

MSC1991: 46E35, 65N30 BRANCH GROUP - The class of branch groups is recently (1999) defined, and consists of groups with m a n y remarkable properties. It is one of three classes that naturally partition the class of just infinite groups (i.e. infinite groups with finite proper quotients). There are two (non-equivalent) ways to define the class of branch groups. The first definition is as follows: A branch group is a g r o u p G which has a descending sequence {Hn}.~__l of normal subgroups (cf. also N o r m a l s u b g r o u p ) such that for any n > 1:

i) [ G : H ] < oc; ii) H~ ~ L~ x ... x L~ (with M~ factors) for some group Ln and integer M~; iii) the factorization for H~+I subdivides the factorization for H~; in particular, M~+I/M~ is an integer _> 2; iv) the group G acts transitively by conjugation on the set of factors of H~ (cf. also T r a n s i t i v e g r o u p ) . Another version of the definition uses an action of G on rooted trees. Let ~ = {m~}~=0 be a sequence of integers > 2, called a branch index. Let Tm be a spherically homogeneous rooted tree (cf. also T r e e ) determined by ~ . It has a root vertex O, it has Mn = m o • - • ran-1 vertices on level n, and rn~ is a branch index for level n (i.e. every vertex u of the level ]u] = n has mn successors). Let V be a set of vertices of the tree T. For a group G acting by automorphisms on T (cf. also A u t o m o r p h i s m ) one defines following subgroups: • Sta(u) = {g 6 G: u a = u}, a stabilizer of the vertex u E V (cf. also S t a b i l i z e r ) ; • Stc(n) = nl~l=,~ StG(u), a stabilizer of level n; • ristG(u) = {g E G: g acts trivially on T \ T~}, a rigid stabilizer of vertex u (T~ is a subtree of T with a root vertex u);

• ristG(n) -- , a rigid stabilizer of level n (i.e. the group generated by the rigid stabilizers of the vertices of level n). It is clear that ristG(n) decomposes as a direct product of groups rista(u), lul = n. The subgroups S t a ( u ) and S t a ( n ) have finite index in G, while ristc(u) and rista(n) can be trivial subgroups. An action of G on T is called spherically transitive if it is transitive on each level n = 1, 2 , . . . ; in this case stabilizers and rigid stabilizers of vertices of the same level are conjugate in G. Now, a group G is called a branch group if there is a faithful spherically transitive action of G on some tree Tm such that [G: rista (n)] < ~ for any n ~ 1. A group satisfying the last definition also satisfies the first, with /In = rista(n) and Ln being an isomorphic type of groups rista (u), ]u I = n. The opposite is not correct. For the class of just infinite groups both definitions are equivalent. A profinite branch group is defined in the same manner as above, only all groups involved have to be closed subgroups in G or in Aut T, considered as a p r o f i n i t e group. The importance of the class of branch groups follows from the following theorem [7], [10]: Let G be an abstract just infinite group. Then either G is a branch group or G contains a n o r m a l s u b g r o u p of finite index which is isomorphic to a direct product of a finite number of copies of a group L, where L is either a s i m p l e g r o u p or a hereditarily just infinite group (i.e. a r e s i d u a l l y - f i n i t e g r o u p with just infinite subgroups of finite index). For profinite just infinite groups, this trichotomy becomes a dichotomy, as simple groups cannot occur. The class of just infinite branch groups coincides with the class of just infinite groups with an infinite structural lattice of normal subgroups [9]. The first finitelygenerated just infinite branch groups were constructed in [3], [4], [5], [8], [6]. Since every finitely-generated infinite group can be m a p p e d onto a just infinite group, the above theorem shows that the class of branch groups should contain groups with m a n y specific properties that are stable under the factorization. This has been confirmed by m a n y investigations. Namely in [3], [4], [5], [8], [6] it was shown that for any prime number p there is a finitely-generated branch torsion p-group (cf. also p - g r o u p ) . In [4], [5], [6] the first examples of groups of intermediate growth between polynomial and exponential are constructed (cf. also P o l y n o m i a l a n d e x p o n e n t i a l g r o w t h in g r o u p s a n d a l g e b r a s ) . Examples of branch groups of finite width (i.e with uniformly bounded ranks of quotients of lower central series) are considered in [1]. 81

BRANCH GROUP Applications of b r a n c h groups to the theory of the discrete Laplace o p e r a t o r on graphs are given in [2]. For m o r e information on branch groups, see [7]. References 'Lie methods in growth of groups and groups of finite width': Proc. Conf. Group Theory Edinburg 1998, to appear. [2] BARTHOLDI, L., AND GRIGORCHUK,R.I.: On the spectrum of Hecke type operators related to some fractal groups, to appear. [3] GRIGORCHUt%R.I.: 'On the Burnside problem for periodic groups', Funct. Anal. Appl. 14 (1980), 41 43. [4] GRIGORCHUK,l~.I.: 'On Milnor's problem on group growth', Soviet Math. Dokl. 28 (1983), 23-26. [5] GRIGORCItUK,R.I.: 'The growth degrees of finitely generated groups and the theory of invariant means', Izv. Akad. Nauk. SSS[t Set. Mat. 48, no. 5 (1984), 939-985. [6] GRIGORCHUK,R.I.: 'Degrees of growth of p-groups and torsion free groups', Mat. Sb. (N.S.) 126, no. 168:2 (1985), 194214. [7] C,RIGORCHUK, R.I.: 'Just infinite branch groups', in M. DU SANTOY AND D. SEGAL (eds.): Horizons in Profinite Groups, Birkhb.user, to appear. [8] GUPTA, N., AND SIDKI, S: 'On the Burnside problem for periodic groups', Math. Z. 182 (1983), 385-388. [9] W~LSON, J.S.: 'Groups with every proper quotient finite', Proc. Cambridge Philos. Soc. 69 (1971), 373-391. [10] WILSON, J.S.: 'Abstract and profinite just infinite groups', in M. DU SANTOY AND D. SEGAL (eds.): Horizons in Profinite Groups, Birkhguser, to appear. R.I. Grigorchuek M S C 1 9 9 1 : 20E08, 20E18, 20Fxx [1] BARTHOLDI, L., AND GRIGORCHUK, R..I.:

BRANDT-LICKORISH-MILLETT-HO POLYNOMIAL - An invariant of non-oriented links in R 3, invented at the beginning of 1985 [1], [2] and generalized by L.H. Kauffman (the K a u f f m a n p o l y n o m i a l ; cf. also

Link). It satisfies the four t e r m skein relation (cf. also C o n way skein triple)

O,L÷ (z) + O,L_ (z) = z(QLo(z) +

(z) ),

and is normalized to be 1 for the trivial knot. References [1] BRANDT, R.D., LICKORISH, W.B.R., AND MILLETT, K.C.: 'A

polynomial invariant for unoriented knots and links', Invent. Math. 84 (1986), 563-573. [2] Ho, C.F.: 'A new polynomial for knots and links; preliminary report', Abstracts Amer. Math. Soc. 6, no. 4 (1985), 300. Jozef Przytycki MSC 1991:57M25

BROCARD P O I N T - T h e first (or positive) Brocard point of a plane triangle (T) with vertices A, B, C is the interior point ft of (T) for which the three angles Z f ~ A B , Z~2BC, Z f t C A are equal. Their c o m m o n value co is the Broeard angle of (T). 82

The second (or negative) Brocard p o i n t of (T) is the interior point f~' for which A f ~ ' B A = Z f t ' C B = Z f t ' A C . Their c o m m o n value is again w. T h e B r o c a r d angle satisfies 0 < w < 7r/6. T h e two B r o c a r d points are isogonal conjugates (cf. I s o g o n a l ) ; t h e y coincide if (T) is equilateral, in which case w = 7r/6. The Brocard configuration (for an extensive account see [6]), n a m e d after H. B r o c a r d who first published about it a r o u n d 1875, belongs to triangle geometry, a subbranch of Euclidean g e o m e t r y t h a t thrived in the last quarter of the nineteenth c e n t u r y to fade away again in the first quarter of the twentieth century. A brief historical account is given in [5]. A l t h o u g h his n a m e is generally associated with the points ft and ft', B r o c a r d was not the first person to investigate their properties; in 1816, long before B r o c a r d wrote a b o u t them, t h e y were m e n t i o n e d by A.L. Crelle in [4] (see also [8] and [11]). I n f o r m a t i o n on B r o c a r d ' s life can be found in [7]. The B r o c a r d points and B r o c a r d angle have m a n y remarkable properties. Some characteristics of the Brocard configuration are given below. Let (T) be an a r b i t r a r y plane triangle with vertices A, B, C and angles ~ = Z B A C , fl = Z C B A , ~/ = Z A C B . If C B c denotes the circle t h a t is t a n g e n t to the line A C at C and passes t h r o u g h the vertices B and C, then C B c also passes t h r o u g h ft. Similarly for the circles CCA and CAB. So the three circles CBC, CCA, CAB intersect in the first B r o c a r d point f~. Analogously, the circle C ~ c t h a t passes t h r o u g h B and C and is t a n g e n t to the line A B at /3, meets the circles Cbd and CAB in the second B r o c a r d point f~t. Further, the circumcentre O of (T) and the two B r o c a r d points are vertices of a isosceles triangle for which Z f t O f t ~ = 2w. T h e lengths of the sides of this triangle can be expressed in terms of the radius R of the circumcircle of (T), and the B r o c a r d angle w: ftfg - Oft 2 sin w

Of Y

R~/1

4 sin 2 co.

T h e Brocard circle is the circle passing t h r o u g h the two B r o c a r d points and O. T h e L e m o i n e point K of (T), n a m e d after E. Lemoine, is a distinguished point of this circle, and the length of the line segment OK-

Of~ COS co

gives the diameter of the B r o c a r d circle. The B r o c a r d angle co is related to the three angles c~, /~, 3` by the following trigonometric identities: cot co = cot (~ + cot ~ + cot 7, - 1- _ - - 1+ sin 2 co sin 2 a

1 ~

+ - 1 sin 2 3'"

BROUWER DEGREE Due to a remarkable conjecture by P. Yff in 1963 (see [14]), modest interest in the Brocard configuration arose again during the 1960s, 1970s and 1980s. This conjecture, known as Yff's inequality,

simultaneously on OK. Letting f = ( f l , . . - , f~), Kronecker showed in 1869 that the number X[fo,..., f~] defined (in modern notation) by the integral vol S1n - 1

8~ 3 _< a¢~7, is unusual in the sense that it contains the angles proper instead of their trigonometric function values (as could be expected). A proof for this conjecture was found by F. Abi-Khuzam in 1974 (see [1]). In [13] and [2] a few inequalities of similar type were proposed and subsequently proven. References [1] ABI-KHUZAM, F.: 'Proof of Yff's conjecture on the Brocard angle of a triangle', Elem. Math. 29 (1974), 141-142. [2] ABI-KHUZAM, F.F., AND BOGHOSSIAN, A.B.: 'Some re