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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 recent geometric inequalities', Amer. Math. Monthly 96 (1989), 576-589. [3] CASEY, J.: Gdometrie elementaire rdcente, Gauthier-Villars, 1890. [4] CRELLE, A.L.: Uber einige Eigenschaften des ebenen geradlinigen Dreiecks rilcksichtlich dreier dutch die Winkelspitzen gezogenen geraden Linien, Berlin, 1816. [5] DAVIS, p m j . : 'The rise, fall, and possible transfiguration of triangle geometry: A mini-history', Amer. Math. Monthly 102 (1995), 204-214. [6] EMMERICH,A.: Die Broeardschen Gebilde und ihre Beziehungen zu den verwandten merkwiirdigen Punkten und Kreisen des Dreiecks, G. Reimer, 1891. [7] GUGGENBUHL, L.: 'Henri Brocard and the geometry of the triangle', Math. Gazette 80 (1996), 492-500. [8] HONSBERGER, R.: 'The Brocard angle': Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Math. Assoc. America, 1995, pp. 101-106. [9] JOHNSON, R.A.: Modern geometry: an elementary treatise on the geometry of the triangle and the circle, Houghton Mifflin, 1929, Reprinted as: Advanced Euclidean Geometry, Dover,1960. [10] KIMBERLING,C.: 'Central points and central lines in the plane of a triangle', Math. Mat. 67 (1994), 163-187. [11] MITRINOVIC, D., PECARIC, J.E., AND VOLENEC, V.: Recent advances in geometric inequalities, Kluwer Acad. Publ., 1989. [12] STROEKER, R.J.: 'Brocard points, circulant matrices, and Descartes' folium', Math. Mat. 61 (1988), 172-187. [13] STROEKER,R.J., AND HOOGLAND,H.J.T.: 'Brocardian geometry revisited or some remarkable inequalities', Nieuw Arch. Wisk. ~th Set. 2 (1984), 281-310. [14] YFF, P.: 'An analogue of the Brocard points', Amer. Math. Monthly 70 (1963), 495-501.

R.J. Stroeker MSC 1991:51M04 BROUWER DEGREE, topological degree - A fundamental concept in a l g e b r a i c t o p o l o g y , d i f f e r e n t i a l t o p o l o g y and m a t h e m a t i c a l analysis. It is rooted in the fundamental work of L. Kroneeker [7] for systems of smooth real-valued functions fo,...,fi~ of n real variables such that 0 is a regular value for f0, K := f o l ( ] - e c , 0 ] ) is bounded and the fj do not vanish

fo K f ' w ,

~._l(-1)J-liIxli-nxj dxl A ... A dxj_l A dxj+lA...Adx,~, is equal to ~ x e / _ l ( 0 ) n 0 z sign det f~(x),

where w =

when this sum makes sense, i.e. when the Jacobian of f does not vanish on f - 1 (0) (eft also J a c o b i a n ) . The special case when n = 2 and OK is a closed simple curve was already considered by A. Cauchy in 1837 (the w i n d i n g n u m b e r ) . After several interesting applications to differential equations and function theory by H. Poincar6 in 1882-1886 and P.G. Bohl in 1904, in 1910-1912, L.E.J. Brouwer [2] and J. S a d a m a r d [5] made this gronecker integral a topological tool by extending it to continuous mappings f and more general sets K. Hadamard refined Kronecker's analytical approach, but Brouwer created and used new simplicial techniques to define a (global) degree d[f, M, N] for continuous mappings f : M ~ N between two oriented compact boundaryless connected manifolds of the same finite dimension. He used it to prove the theorems on invariance of dimension and invariance of domain (cf. also B r o u w e r t h e o r e m ) . Kronecker's integral can be seen as a special case of the Brouwer degree d[f/HfH, OK, Sn--1], or of the (local) Brouwer degree deg e [ f , int K, 0], defined as follows (cf. also D e g r e e o f a m a p p i n g ) . If f~ C R n is open and bounded, the Brouwer degree degB[f, f~,y] of a continuous mapping f : ~ C R ~ --+ R ~ can be defined for each y ff f(Oft) using an approximation scheme introduced by M. Nagumo [10] in 1950. The idea consists in defining it first for f smooth and y a regular value of f , through the formula sign det

f' (x),

x~/-l(y) and then to approximate the continuous function f and the point y above by a sequence of such functions and points for which this definition holds. This is possible by the Weierstrass approximation theorem (cf. W e i e r s t r a s s t h e o r e m ) and the S a r d t h e o r e m . The degrees of the approximations stabilize to a common value, denoted by degB[f,f~,y ] and being an algebraic count of the number of counter-images of y under f in f~, which is stable for small perturbations of f and y. A similar approach can be used to define d[f, M, N] when M and N are oriented boundaryless differentiable manifolds. P r o p e r t i e s a n d a x i o m a t i c c h a r a c t e r i z a t i o n . The first basic property of the Brouwer degree is its additivity-excision: if f~l C f~ and ft2 C f~ are disjoint open subsets such that y ¢ f ( ~ \ ( a l U a2)), then one has deg B [f, f~, y] = deg B If, Ftl, y] + deg B If, ft2, y]. 83

BROUWER DEGREE The second property is its homotopy invariance: let U C R ~ x [0,1] be a bounded open set, Ux = {x E R n : (x,A) E U}, let F : U --+ R ~ be continuous, and let y ¢ F(OU); then degB[F(.,,~), U~,y] is independent of A. It has been shown in the 1970s (see [11] for references) that the Brouwer degree can be uniquely characterized as the integer-valued function deg B on the set

[1]: let f~ be a bounded open symmetric neighbourhood of the origin in R n and let f : ~ -~ R ~ be a continuous odd function such that 0 ¢ f(cOf~); then degB[f, ft, 0] is odd. This result and its more recent El-version are basic in critical point theory [9].

P r o d u c t t h e o r e m . In 1934, J. Leray [8] proved a useful

O t h e r a p p r o a c h e s a n d e x t e n s i o n s . The Brouwer degree is a very versatile concept which can be defined through techniques of algebraic topology, differential topology or algebraic geometry. For example, if f : S '~ -+ S ~ is continuous and f*: H*(S ~) -+ H*(S ~) is the induced h o m o m o r p h i s m on the homology groups of S ~ over Z (cf. also H o m o l o g y g r o u p ) , then H~(S ~) is isomorphic to Z and hence f~ becomes multiplication by an integer, which is d[f, S ~, Sn]. If f is any continuous extension of f to the closed unit ball B(1), then d[f, S n, S ~] = degs If, B(1), 0]. In 1995, H. Br@zis and L. Nirenberg [31, [4] defined a Brouwer degree for certain not necessarily continuous mappings f belonging to a Sobolev or other function space. Extensions of the Brouwer degree to various classes of mappings between infinite-dimensional spaces are also known. The most fundamental one is the LeraySchauder degree, defined in 1934 for compact perturbations of the identity defined on the closure of a bounded open subset of a normed vector space (cf. also Degree

product theorem for the Brouwer degree: let f : ft -+ R ~

of a mapping).

and g: A -+ R ~, with A D f ( f t ) , be continuous functions such that y f~ g o f(Oft). Denoting by Ci the bounded components of A \ f(Oft), one has degB[g o f, ft, Y] = ~ i degB [f, ft, Ci] degB [g, Ci, y], where only finitely many terms are different from zero. This result has deep applications in topology, for example the Jordan separation theorem: for homeomorphic compact subsets K1 and K2 of R ~, the sets R ~ \ K1 and R ~ \ K2 have the same number of connected components.

References

ft C R '~ open and bounded, (f, ft, y) : f : ~ ~ R ~ continuous, / '

y C R ~ \ f(Of~) by the additivity-excision and the homotopy invariance properties, together with the following direct consequence of the definition (the normalization property): i f y C ft, then degB[I, ft, y ] = 1. The additivity-excision property implies the existence property: if degB[f,f~,y ] ~ O, then y C f(ft). Easy consequences of the homotopy invariance are the equalities deg B [f, ft, y] = degB [g, ft, y] when f = g on Oft, and deg B[f, ft, y] = deg B[f, ft, z] when y and z belong to the same component Ci of R n \ f ( 0 f t ) (with the common value written degs[f, ft, Ci]). The existence and homotopy properties have many important applications in studying the existence and bifurcation of solutions of various types of equations.

F i x e d - p o i n t t h e o r e m s . An easy consequence of the Brouwer degree is the following Knaster-KuratowskiMazurkiewicz fized-point theorem, first stated and proved in 1929 [6]: let B[R] C R ~ be the closed ball of centre 0 and radius R and let g: B[R] -+ R ~ be a continuous function such that g(OB[R]) C B[R]. Then there is at least one x ~ B[R] such that g(x) = x. The special case where 9: B[R] --+ B[R] is the Brouwer fixed-point theorem [2], which has many different and useful equivalent forms.

Degree of symmetric mappings. Useful computational results hold under symmetry assumptions. The oldest one, which corresponds to Z2-symmetry, was conjectured by S.M. Ulam and proved by K. Borsuk in 1933 84

[1] BORSUI~, K.: 'Drei S~tze fiber die n-dimensionale euklidische Sph/ire', Fundam. Math. 21 (1933), 177-190. [2] BROUWER, L.E.J.: 'Ueber Abbildungen von Mannigfaltigkeiten', Math. Ann. 71 (1912), 97 115. [3] BRI~ZIS, H., AND NIRENBERG, L.: 'Degree theory and BMO', Selecta Math. 1 (1995), 197-263. [4] BRI~ZIS,H., AND NIRENBERG, L.: 'Degree theory and BMO', Selecta Math. 2 (1996), 1 60. [5] HADAMARD,J.: 'Sur quelques applications de l'indice de Kronecker', in J. TANNERY (ed.): Introduction ~t la thdorie des fonctions d'une variable, Vol. 2, Hermann, 1910, pp. 875915. [6] KNASTER, B., KURATOWSKI, C., AND MAZURKIEWICZ, S.: 'Ein Beweis des Fixpunktsatzes fiir n-dimensionale Simplexe', Fundam. Math. 14 (1929), 132-137. [7] KRONECKER, L.: 'Ueber Systeme von Funktionen mehrerer Variabeln', Monatsber. Berlin Akad. (1869), 159-193; 688 698. [8] LERAY, J.: 'Topologie des espaces abstraits de M. Banach', C.R. Acad. Sci. Paris 200 (1935), 1082-1084. [9] M:AWHIN, J., AND WILLEM, M.: Critical point theory and Hamiltonian systems, Springer, 1989. [10] NAGUMO, M.: 'A theory of degree of mapping based on infinitesimal analysis', Amer. J. Math. 73 (1951), 485-496. [11] ZEIDLER, E.: Nonlinear functional analysis and its applications, Vol. I, Springer, 1986. Jean Mawhin

MSC1991:55M25

B R O W N - D O U G L A S - F I L L M O R E THEORY B R O W N - D O U G L A S - F I L L M O R E THEORY, B D F theory - The story of Brown-Douglas-Fillmore theory begins with the Weyl-von N e u m a n n theorem, which, in one of its formulations, says that a bounded selfa d j o i n t o p e r a t o r T = T* on an infinite-dimensional

separable H i l b e r t space ~ is determined up to compact perturbations, modulo unitary equivalence, by its essential spectrum. (The essential spectrum is the spectrum a(re(T)) of the image 7r(T) of T in the Calkin algebra Q(~) = /3(7-/)/K:(7t); it is also the spectrum of the restriction of T to the orthogonal complement of the eigenspaces of T for the eigenvalues of finite multiplicity; cf. also S p e c t r u m o f a n o p e r a t o r . ) In other words, unitary equivalence modulo the compacts K(7/) washes out all information about the s p e c t r a l m e a s u r e of T, and only the essential spectrum remains. This result was extended to normal operators (cf. also N o r m a l o p e r a t o r ) by I.D. Berg [2] and W. Sikonia [12], working independently. However, the theorem is not true for operators that are only essentially normal, in other words, for operators T such that TT* - T * T E h2(7-l). Indeed, the 'unilateral shift' S satisfies S * S = 1 and SS* = 1 - P, where P is a rank-one projection, yet S cannot be a compact perturbation of a normal operator since its Fredholm index (cf. also F r e d h o l m o p e r a t o r ; I n d e x of a n o p e r a t o r ) is non-zero. In [4], L.G. Brown, R.G. Douglas and P.A. Fillmore (known to operator theorists as 'BDF') showed that this is the only obstruction: an operator T in /3(7-/) is a compact perturbation of a normal operator if and only if T is essentially normal and ind(T - A) = 0 for every A ¢ or(re(T)). However, they went considerably further, by putting this theorem in a C*-algebraic context in [4] and [5]. An operator T 'up to compact perturbations' defines an injective *-homomorphism from a C*-algebra A (the closed subalgebra of Q(7-/) generated by re(T) and re(T*)) to Q(7/), and the C*-algebra A is Abelian if and only if T is essentially normal. More generally, an extension of a separable C*-algebra A is an injective .homomorphism A ~-~ Q(7/), since this is equivalent to a commutative diagram with exact rows: 0 0

-+

K(7/)

+

E

-+ -%

A

+

0

-+

0.

BDF defined a natural equivalence relation (basically unitary equivalence) and an addition operation on such extensions, giving a commutative m o n o i d Ext(A), whose 0-element is represented by split extensions (those for which there is a lifting A ~ /3(7-t)). (The essential uniqueness of the split extensions was shown in [14].) It was shown by M.D. Choi and E.G. Effros [6] (see

also [1]) that this monoid is a g r o u p whenever A is nuclear (cf. also N u c l e a r space). (BDF originally worked only with Abelian C*-algebras A = C ( X ) , for which this is automatic, and they used the notation Ext(X) for Ext(A).) BDF showed that X ~ Ext(X) behaves like a generalized homology theory in X (cf. also G e n e r a l i z e d c o h o m o l o g y t h e o r i e s ) , and in fact for finite CW-complexes (cf. also C W - c o m p l e x ) coincides with K I ( X ) , where K . is the homology theory dual to complex K - t h e o r y . This was extended in [7], where it was shown that Ext(X) is canonically isomorphic to K~(X), Steenrod K-homology (cf. also S t e e n r o d S i t n i k o v h o m o l o g y ) , for all compact metric spaces X, and in [3], where it was shown that on a suitable category of C*-algebras, Ext(A) fits into a short e x a c t sequence 0 -+ Ext~(Ko(A), Z) -~ Ext(A) --~ Homz

(K1 (A), Z) --+ 0.

It is now (as of 2000) known that BDF theory is just a special case of a more general theory of C*-algebra extensions. One type of generalization (see [13]) involves replacing K](~) by the algebra of 'compact' operators of a II~ factor (el. also yon Neumann algebra). Another sort of generalization involves replacing ](](7/) by an algebra of the form B ® K](~), where B is another separable (or a-unital) C*-algebra. Theories of this sort were worked out in [9], [10] and in [8], though the theory of [9], [I0] turns out to be basically a special case of Kasparov's theory (see [II]). Kasparov's Ext-theory gives rise to a bivariant functor Ext(A, B), and when A is nuclear, this coincides [8] with Kasparov's bivariant K-functor K K 1 (A, B). References [1] ARVESON, W.: 'Notes on extensions of C*-algebras', Duke Math. 3. 44, no. 2 (1977), 329-355. [2] BERG, I.D.: 'An extension of the Weyl-von Neumann theorem to normal operators', Trans. Amer. Math. Soc. 160 (1971), 365-371. [3] BROWN, L.G.: 'The universal coefficient theorem for Ext and quasidiagonality': Operator Algebras and Group Representations I (Neptun, 1980), Vol. 17 of Monographs Stud. Math., Pitman, 1984, pp. 60-64. [4] BROWN, L.G., DOUGLAS, R.G., AND FILLMORE, P.A.: Unitary equivalence modulo the compact operators and extensions of C*-algebras, Vol. 345 of Lecture Notes in Mathematics, Springer, 1973, pp. 58-128. [5] BROWN, L.G., DOUGLAS, R.G., AND FILLMORE, P.A.: 'Extensions of C*-algebras and K-homology', Ann. of Math. (2) 105, no. 2 (1977), 265-324. [6] CHOI, M.D., AND EFFROS, E.G.: 'The completely positive lifting problem for C*-algebras', Ann. of Math. (2) 104, no. 3 (1976), 585-609. [7] KAMINKER, J., AND SCHOCHET, C.: 'K-theory and Steenrod homology: applications to the Brown-Douglas Fillmore

85

B R O W N - D O U G L A S FILLMORE T H E O R Y theory of operator algebras', Trans. Amer. Math. Soc. 227 (1977), 63-107. [81 KASPAROV, G.G.: 'The operator K-functor and extensions of C*-algebras', Math. USSR Izv. 16 (1981), 513-572. (Izv. Akad. Nauk. SSSR Ser. Mat. 44, no. 3 (1980), 571-636; 719.) [9] PIMSNER, M., POPA, S., AND VOICULESCU,D.: 'Homogeneous C*-extensions of C ( X ) ® 1C(7{). I', J. Opcr. Th. 1, no. 1

(1979), 55-108. [10] PIMSNER, M., POPA, S., AND VOICULESCU, D.: 'Homogeneous C*-extensions of C ( X ) ® 1C(7{). II', J. Oper. Th. 4, no. 2

(1980), 211-249. [Ii] ROSENBERG, J., AND

SCHOCHET,

C.: 'Comparing functors classifying extensions of C*-algebras', d. Oper. Th. 5, no. 2

(1981), 267-282. [12] SIKONIA,W.: 'The yon Neumann converse of Weyl's theorem', Indiana Univ. Math. J. 21 (1971/72), 121-124. [13] SKANDALIS, G.: 'On the group of extensions relative to a semifinite factor', J. Oper. Th. 13, no. 2 (1985), 255-263. [14] VOICULESCU, D.: 'A non-commutative Weyl-von Neumann theorem', Rev. Roum. Math. Pures Appl. 21, no. 1 (1976), 97-113. Jonathan Rosenberg

MSC 1991: 49L80, 19K33, 19K35 B R O W N - G I T L E R SPECTRA - Spectra introduced by E.H. Brown Jr. and S. Gitler [1] to study higherorder obstructions to immersions of manifolds (cf. also I m m e r s i o n ; S p e c t r u m o f spaces). They immediately found wide applicability in a variety of areas of hom o t o p y theory, most notably in the stable homotopy groups of spheres ([9] and [4]), in studying homotopy classes of mappings out of various classifying spaces ([3], [10] and [8]), and, as might be expected, in studying the immersion conjecture for manifolds ([2] and [5]). The modulo-p h o m o l o g y H , X = H , ( X , Z / p Z ) comes equipped with a natural right action of the S t e e n r o d a l g e b r a A which is unstable: at the prime 2, for example, this means that 0=Sqi:H~X-+H~_iX,

2i > n.

Write U, for the c a t e g o r y of all unstable right modules over .4. This category has enough projective objects; indeed, there is an object G(n), n >_ 0, of U, and a natural isomorphism Homu, (G(n), M ) ~- M,~, where Mn is the vector spaces of elements of degree n in M. The module G(n) can be explicitly calculated. For example, i f p = 2 and xn E G(n)n is the universal class, then the evaluation mapping A --+ G(n) sending 0 to x~O defines an isomorphism E~.A/{Sqi: 2i > n } , 4 = G(n). These are the dual Brown Gitler modules. This pleasant bit of algebra can be only partly reproduced in a l g e b r a i c t o p o l o g y . For example, for general n there is no space whose (reduced) homology is G(n); 86

specifically, if p = 2, the module G(8) cannot support the structure of an unstable c o - a l g e b r a over the Steenrod algebra. However, after stabilizing, this objection does not apply and the following result from [1], [4], [7] holds: There is a unique p-complete spectrum T(n) so that H , T ( n ) ~- G(n) and for all pointed CW-complexes Z, the mapping

[T(n), E°°Z] -+ H~Z sending f to f.(Xn) is surjective. Here, E°°Z is the suspension spectrum of Z, the symbol [-, .] denotes stable homotopy classes of mappings, and H is reduced homology. The spectra T ( n ) are the dual Brown-Gitler spectra. The Brown-Gitler spectra themselves can be obtained by the formula B(n) = E n D r ( n ) , where D denotes the Spanier-Whitehead duality functor. The suspension factor is a normalization introduced to put the bottom cohomology class of B ( n ) in degree 0. An easy calculation shows that B(2n) _~ B ( 2 n + 1) for all prime numbers and all n > 0. For a general spectrum X and n ~ =t=1 modulo 2p, the group [T(n), X] is naturally isomorphic to the group D ~ H , f t ° ° X of homogeneous elements of degree n in the Cartier-Dieudonn6 module D , H , f ~ X of the Abelian H o p f a l g e b r a H , Ft°°X. In fact, one way to construct the Brown-Gitler spectra is to note that the functor X ~ D2~H,f~X is the degree-2n group of an extraordinary homology theory; then B(2n) is the p-completion of the representing spectrum. See [7]. This can be greatly, but not completely, destabilized. See [6]. References [1] BROWN JR., t~.H., AND GITLER, S.: 'A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra', Topology 12 (1973), 283-295. [2] BROWN JR., E.H., AND PETERSON, F.P.: 'A universal space for normal bundles of n-manifolds', Comment. Math. Helv. 54, no. 3 (1979), 405-430. [3] CARLSSON, G.: 'G.B. Segal's Burnside ring conjecture for (Z/2) k', Topology 22 (1983), 83-103. [4] COHEN, R.L.: 'Odd primary infinite families in stable homotopy theory', Memoirs Amer. Math. Soc. 30, no. 242 (1981). [5] COHEN, R.L.: 'The immersion conjecture for differentiable manifolds', Ann. of Math. (2) 122, no. 2 (1985), 237-328. [6] GOERSS, P.~ LANNES, J., AND MOREL, F.: 'Vecteurs de Witt non-commutatifs et repr6sentabilit6 de l'homologie modulo p', Invent. Math. 108, no. 1 (1992), 163-227. [7] GOERSS, P., LANNES, J., AND MOREL, F.: 'Hopf algebras, Witt vectors, and Brown-Gitler spectra': Algebraic Topology (Oaxtepec, 1991), Vol. 146 of Contemp. Math., Amer. Math. Soc., 1993, pp. 111-128. [8] LANNES, J.: 'Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe ab61ien 616mentaire', IHES Publ. Math. 75 (1992), 135-244.

BUCHSBAUM RING [9] MAHOWALD, M.: 'A new infinite family in 27r,s', Topology 16, no. 3 (1977), 249-256. [10] MILLER, H.: ' T h e Sullivan conjecture on maps from classifying spaces', Ann. of Math. (2) 120, no. 1 (1984), 39-87.

Paul Goerss

systems of parameters: A d-dimensional Noetherian local ring A with maximal ideal m is Buchsbaum if and only if every system al, • . . , ad of p a r a m e t e r s for A forms a weak A-sequence, t h a t is, the equality

M S C 1991:55P42

( a l , . . . , a i - 1 ) : ai = ( a l , . . . , a i - 1 ) : m

BUCHSBAUM RING The notion of a Buchsb a u m ring (and module) is a generalization of that of a C o h e n - M a c a u l a y r i n g (respectively, module). Let A denote a Noetherian l o c a l r i n g (cf. also N o e t h e r i a n r i n g ) with m a x i m a l i d e a l m and d = dim A. Let M be a finitely-generated A-module with dimd M = s. Then M is called a Buchsbaum module if the difference

gA (M/qM) - e~ (M) is independent of the choice of a parameter ideal q = ( a l , . . . , a s ) of M , where a l , . . . , a s is a system of parameters of M and gA (M/qM) (respectively, e ° (M)) denotes the length of the A-module M / q M (respectively, the multiplicity of M with respect to q). When this is the case, the difference

holds for all 1 _< i <_ d. Therefore, systems a l , . . . , a d of parameters in a Buchsbaum local ring need not be regular sequences, but the differences [(al,...,ai-1) :ai]/(al,...,ai-1),

are very small and only finite-dimensional vector spaces over the residue class field A/re of A. Weak sequences are closely related to d-sequences introduced by C. Huneke [22]. Actually, A is a Buchsbaum ring if and only if every system a l , . . . , a d of p a r a m e t e r s for A forms a dsequence, that is, the equality (al,...,ai-1)

A = t~/(X1,..., Xd) (~ (Y1,-.., Yd), where B = k[[X~,...,Xd, Y1,...,Yd]], with d _> 1, denotes the f o r m a l p o w e r s e r i e s ring in 2d variables over a field k. Then A is a Buchsbaum ring with dim A = d and I(A) = d - 1. A, not necessarily local, N o e t h e r i a n r i n g R is said to be a Buchsbaum ring if the local rings _Re are Buchsb a u m for all f9 E Spec R. The theory of Buchsbaum rings and modules dates back to a question raised in 1965 by D.A. Buchsbaum [3]. He asked whether the difference eA(A/q) - e°(A), with q a p a r a m e t e r ideal, is an invariant for any Noetherian local ring A. This is, however, not the case and a counterexample was given in [33]. Thereafter, in 1973 J. Stiickrad and W. Vogel published the classic paper [34], from which the history of Buchsbaum rings and modules started. In [34] they gave a characterization of Buchsbaum rings in terms of the following property of

: aiaj = (al,...,ai-a)

: aj

holds for a l l l < i < j
H i ( M ) = lim E x t ~ ( d / m n, M )

I ( M ) = eA(M/qM) - e~(V) is called the Buchsbaum invariant of M. The Amodule M is a Cohen-Macaulay module if and only if gA(M/qM) = e~(M) for some (and hence for any) parameter ideal q of M, so that M is a Cohen-Macaulay Amodule if and only if M is a Buchsbaum A-module with I ( M ) = 0. The ring A is said to be a Buchsbaum ring if A is a Buchsbaum module over itself. If A is a Buchsb a n m ring, then A e is a Cohen-Macaulay ring with dim Ap = dim A - dim A / p for every p E Spee A \ {re}. A typical example of Buchsbaum rings is as follows. Let

l
(i e Z)

n--~ oo

denote the ith l o c a l c o h o m o l o g y of M with respect to the maximal ideal re. If M is a Buchsbaurn A-module, then m - H i ( M ) = (0) for all i ¢ s and the equality S--1

i=0

holds, where s = dimA M. Unfortunately, the vanishing does not characterize Buchsbauln modules. Modules M with m . H ~ ( M ) = (0) for all i ¢ dimA M are called quasi-Buehsbaum and constitute a class which is strictly larger t h a n t h a t of Buchsbaum modules. However, if the canonical homomorphism

i q~M:

EXtiA(A/re, M) --+ Hi~(M)

=

lim EXtiA(A/mn, M)

is surjective for all i ¢ dimA M, then M is a Buchsbaum A-module. The converse is also true if the base ring A is regular (cf. also R e g u l a r r i n g (in c o m m u t a t i v e algebra)). After the establishment of the surjectivity criterion, by Stiickrad and Vogel [35] in 1978, the development of the theory became rather rapid. The ubiquity of Buchsb a u m normal local rings was established by S. Goto [6] as an application of the Evans-Gritfith construction [5]. Namely, let d _> 1 and {hi}o 2 and h0 = hi = 0), one may choose the ring A so that A is an i n t e g r a l d o m a i n (respectively, a n o r m a l ring). See [1] for progress in the research 87

BUCHSBAUM RING about the ubiquity of Buchsbaum homogeneous integral domains. Besides, Buchsbaum local rings of multiplicity 2 have been classified [9]. Also, certain famous isolated singularities are Buehsbaum (cf. [23]). The theory of Buchsbaum rings and modules is closely related to that of Cohen-Macaulayness in blowing-ups. Let I be an ideal of positive height in a Noetherian local ring A. Let R(I) = ®n>0I n and call it the Rees algebra of I. Then the canonical morphism Proj R(I) ~ Spec A is the blowing-up of A with centre I (cf. also B l o w - u p algebra). If the ring R(I) is CohenMacaulay, then the scheme Proj R(I) naturally is locally Cohen-Macaulay. The problem when the Rees algebra R(I) is Cohen Macaulay has been intensively studied from the 1980s onwards i[19], [38], [17], [39], [15]). The ring R(I) is Cohen-Macaulay if the ideal I is generated by a regular sequence and if the base ring A is Cohen-Macaulay [2]. However, the converse is not true even for parameter ideals I. Actually, A is a Buchsbaum ring if and only if the Rees algebra R(q) is a CohenMacaulay ring for every parameter ideal q in A, provided that A is an integral domain with dimA = 2. This insightful result of Y. Shimoda [31] in 1979 opened the door towards a further development of the theory. Firstly, Goto and Shimoda [18] showed that a Noetherian local ring A is a Buchsbaum ring with H~(A) = (0) (i ~ 1, dimA) if and only if the Rees algebra R(q) is a Cohen-Macaulay ring for every parameter ideal q in A. When this is the case, the Rees algebras R(q ~) are also Cohen-Macaulay for all n >_ 1. In 1981, Buchsbaum rings were characterized in terms of the blowing-ups of parameter ideals. Let A be a Noetherian local ring with maximal ideal m and d = dim A > 1. Then A~ H ° (A) is a Buchsbaum ring if and only if the scheme Proj R(q) is locally Cohen-Macaulay for every parameter ideal q in A [7]. Subsequently, Goto [10] proved that the associated graded rings G(q) = ®~_>0q~/qn+l of parameter ideals q in a Buchsbaum local ring are always Buchsbaum. In addition, Stiickrad showed that R(q) is a Buchsbaum ring for every parameter ideal q in a Buchsbaum local ring [32]. The systems of parameters in Buchsbaum local rings behave very well and enjoy the monomial property [10]. Buchsbaum rings are yet (2000) the only non-trivial case for which the monomial conjecture, raised by M. Hochster, has been solved affirmatively (except for the equi-characteristic case). See [36] for these results, together with geometric applications and concrete examples. See [36] for researches on the Buchsbaum property in affine semi-group rings and Stanley-Reisner rings of simplicial complexes. Let M be a Buchsbaum module over a Noetherian local ring A. Then M is said to be maximal if 88

dimA M = dimA. Noetherian local rings possessing only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules are said to have finite Buchsbaum-representation type. Buchsbaum representation theory was studied by Goto and K. Nishida [16], [12], [13], and the Cohen-Macaulay local rings A of finite Buchsbaum-representation type have been classified under certain mild conditions. If dim A _> 2, then A must be regular [16]. The situation is a little more complicated if d i m A = 1 [13]. In [12] (not necessarily Cohen-Macaulay) surface singularities of finite Buchsbaum-representation type are classified. Suppose that A is a regular local ring with dim A = d and let M be a maximal Buchsbaum A-module. Then M e is a free A~-module for all p E SpeeA \ {m}, so that the A-module M defines a v e c t o r b u n d l e on the punctured spectrum Spec A \ {m} of A. Thanks to the surjectivity criterion, one can prove the structure theorem of maximal Buchsbaum modules over regular local rings: Every maximal Buchsbaum A-module M has the form d

Oz? i=O

where Ei denotes the ith syzygy module of the residue class field A / m of A, hi = CA(Him(M)) (0 < i < d - 1), and hd = r a n k d M - ~ i =d-1 1 {d-lib" ~i-lJ *, if A is a regular local ring ([4], [11]). This result has been generalized by Y. Yoshino [42] and T. Kawasaki [24]. They showed a similar decomposition theorem of a special kind of maximal Buchsbaum modules over Gorenstein local rings; see [28] for the characterization of Buchsbaum rings and modules in terms of dualizing complexes. (It should be noted here that the main result in [28] contains a serious mistake, which has been repaired in [42].) A local ring A satisfying the condition that all the local cohomology modules H i ( A ) (i 7£ dim A) are finitely generated is said to be an FLC ring (or a generalized Cohen-Macaulay ring). The class of FLC rings includes Buchsbaum rings as typical examples. In fact, a Noetherian local ring A is FLC if and only if it contains at least one system a l , . . . , ad ( d = dimA) of parameters such that the sequence a~l,..., a dnd forms a dsequence in any order for all integers ni >_ 1. Such a sequence is called an unconditioned strong d-sequence (for short, USD-sequence or d+-sequence); they have been intensively studied [29], [37], [20]. Recently (1999), Kawasaki [25] used the results in [20] to establish the arithmetic Cohen-Macaulayfications of Noetherian local rings. Namely, every unmixed local ring A contains an ideal I of positive height with the Cohen-Macaulay Rees

BUCHSBAUM RING algebra R(I), provided dim A _> 1 and all the formal fibres of A are Cohen-Macaulay. Hence, the Sharp conjecture [30] concerning the existence of dualizing complexes is solved affirmatively. Let R = ®~>0Rn be a Noetherian graded ring with k = /:to a field and let if2 = R+. Then R is a Buchsbaum ring if and only if the local ring R ~ is Buchsbaum. When this is the case, the local cohomology modules H~(R) (i ~ direR) are finite-dimensional vector spaces over the field k. The vanishing of certain homogeneous components [H~(R)]n of H~(R) may affect the Buchsbaumness in graded algebras R. For example, if there exist integers {ti}0
for all n • ti and 0 < i < d - 1, then R i s a Buchsbaum ring [8]. Therefore R is a Buchsbaum ring if Hh(R) = for all i # d [27]. Hence the scheme X = Proj R is arithmetically Buchsbaum if X is locally Cohen-Macaulay, provided that R = k[R1] and R is equi-dimensional. See [21] for the bounds of Castelnuovo-Mumford regularities of Buchsbaum schemes X = Proj R. Researches of the Buchsbaumness in Rees algebras recently (1999) started again, although the progress remains tardy (possibly because of the lack of characterizations of Trung-Ikeda type [38] for Buchsbaumness). In [14] the Buchsbaumness in Rees algebras R(I) of certain m-primary ideals I in Cohen-Macaulay local rings is closely studied in connection with the Buchsbaumness in the associated graded rings G(I) = ®n>oIn/I ~+1 and that of the extended Rees algebras RI(I) = ®,~czI n. In [26], [41], [40], Buchsbaumness in graded rings associated to certain m-primary ideals in Buchsbaum local rings is explored. Especially, the Rees algebra R(m) of the maximal ideal m in a Buchsbaum local ring A of maximal embedding dimension (that is, a Buchsbaum local ring A for which the equality v(A) = e°(A) + dim A + I(A) - 1 holds) is again a Buchsbaum ring [40]. References [1] AMASAKI,M.: 'Existence of homogeneous prime ideals fitting into long Bourbakl sequences': Proe. 21st Syrup. Commutative Algebra in Tokyo, Japan, November 23-26, 1999, 1999, pp. 104-111. [2] BARSHAY,J.: 'Graded algebras of powers of ideals generated by A-sequences', J. Algebra 25 (1973), 90 99. [3] BUCHSBAUM, D.A.: 'Complexes in local ring theory': Some Aspects of Ring Theory, C.I.M.E. Roma, 1965, pp. 223-228. [4] EISENBUD, G., AND GOTO, S.: 'Linear free resolutions and minimal multiplicity', Y. Algebra 88 (I984), 89-133. [5] EVANS JR., E.G., AND GRIFFITH, P.A.: 'Local cohomology modules for normal domains', J. London Math. Soc. 19 (1979), 277-284.

[6] GOTO, S.: 'On Buchsbaum rings', J. Algebra 6 7 (1980), 272279. [7] GOTO, S.: 'Blowing-up of Buchsbaum rings': Commutative Algebra, Vol. 72 of Lecture Notes, London Math. Soc., 1981, pp. 140-162. [8] GOTO, S.: 'Buchsbaum rings of maximal embedding dimension', J. Algebra 76 (1982), 383-399. [9] GOTO, S.: 'Buchsbaum rings with multiplicity 2', J. Algebra 74 (1982), 494-508. [10] GOTO, S.: 'On the associated graded rings of parameter ideals in Buchsbaum rings', J. Algebra 85 (1983), 490-534. [11] GOTO, S.: 'Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized CohenMacaulay modules', in M. NAGATA AND H. MATSUMURA (eds.): Commutative Algebra and Combinatories, Vol. 11 of Adv. Stud. Pure Math., Kinokuniya, 1987, pp. 39-46. [12] GOTO, S.: 'Surface singularities of finite Buchsbaumrepresentation type': Commutative Algebra: Proc. Microprogram June 15-July 2, Springer, 1987, pp. 247-263. [13] GOTO, S.: 'Curve singularities of finite Buchsbaumrepresentation type', J. Algebra 163 (1994), 447-480. [14] GoTo, S.: 'Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity', J. Algebra 213 (1999), 604661. [15] GOTO, S., NAKAMURA, Y., AND NISHIDA, K.: 'CohenMacaulay graded rings associated ideals', Amer. J. Math. 118 (1996), 1197-1213. [16] GoTo, S., AND NISHIDA, K.: 'Rings with only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules', J. Math. Soc. Japan 40 (1988), 501-518. [17] GOTO, S., AND NISHIDA, K.: The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, Vol. 526 of Memoirs, Amer. Math. Soc., 1994. [18] GOTO, S., AND SHIMODA, Y.: 'On Rees algebras over Buchsbantu rings', J. Math. Kyoto Univ. 20 (1980), 691-708. [19] GOTO, S., AND SHIMODA,Y.: 'On the Rees algebras of CohenMacaulay local rings', in R.N. DRAPER (ed.): Commutative Algebra, Analytic Methods, Vol. 68 of Lecture Notes in Pure Applied Math., M. Dekker, 1982, pp. 201-231. [20] GOTO, S., AND YAMAGISHI,K.: 'The theory of unconditioned strong d-sequences and modules of finite local cohomology', Preprint (1978). [21] HOA, L.T., AND MIYAZAKI, C.: 'Bounds on CastelnuovoMumford regularity for generalized Cohen-Macaulay graded rings', Math. Ann. 301 (1995), 587-598. [22] HUNEKE, C.: 'The theory of d-sequences and powers of ideals', Adv. Math. 46 (1982), 249-279. [23] ISHIDA, M.-N.: 'Tsuchihashi's cusp singularities are Buchsbaum singularities', Tdhoku Math. J. 36 (1984), 191-201. [24] KAWASAKI, T.: 'Local cohomology modules of indecomposable surjective Buchsbaum modules over Gorenstein local rings', J. Math. Soc. Japan 48 (1996), 551-566. [25] KAWASAKI, T.: 'Arithmetic Cohen-Macaulayfications of local rings': Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999, 1999, pp. 88-92. [26] NAKAMURA,Y.: 'On the Buchsbaum property of associated graded rings', J. Algebra 209 (1998), 345-366. [27] SCHENZEL,P.: 'On Veronesean embeddings and projections of Veronesean varieties', Archly Math. 30 (1978), 391-397. [28] SCHENZEL, P.: Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Vol. 907 of Lecture Notes in Mathematics, Springer, 1982.

89

BUCHSBAUM RING [29] SCHENZEL, P., TRUNG, N.V., AND CUONG, N.T.: 'Verallgemeinerte Cohen-Macaulay-Moduln', Math. Nachr. 85

(1978), 57-73. [30] SHARP, R.Y.: Necessary conditions for the existence of dualizin 9 complexes in commutative algebra, Vol. 740 of Lecture Notes in Mathematics, Springer, 1979, pp. 213-229. [31] SHIMODA, Y.: 'A note on Rees algebras of two-dimensional local domains', J. Math. Kyoto Univ. 19 (1979), 327-333. [32] STi)CKRAD, J.: 'On the Buchsbaum property of Rees and form modules', Beitr. Algebra Geom. 19 (1985), 83-103. [33] STUCKRAD, J., AND VOGEL, W.: 'Ein Korrekturglied in der Multiplizitgtstheorie von D.G. Northcott und Anwendungen', Monatsh. Math. 76 (1972), 264 271. [34] STiJCKRAD, J., AND VOGEL, W.: 'Eine Verallgemeinerung der Cohen-Macaulay-Ringe und Anwendungen auf ein Problem der Multiplitgtstheorie', J. Math. Kyoto Univ. 13 (1973), 513-528. [35] STUCKRAD, J., AND VOGEL, W.: 'Toward a theory of Buchsbaum singularities', Amer. J. Math. 100 (1978), 727 746. [36] STiJCKRAD, J., AND VOGEL, W.: Buchsbaum rings and applications, Springer, 1986. [37] TRUNG, N.V.: 'Toward a theory of generalized Cohen Macaulay modules', Nagoya Math. J. 102 (1986), 1-49. [38] TRUNG, N.V., AND IKEDA, S.: 'When is the Rees algebra Cohen-Macaulay?', Commun. Algebra 17 (1989), 2893-2922. [39] VASCONCELOS, W.: Arithmetic of blowup algebras, Vol. 195 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1994. [40] YAMAGISHI, K.: 'Buchsbaumness in Rees algebras associated to m-primary ideals of minimal multiplicity in Buchsbaum local rings': Proe. 21st Syrup. Commutative Algebra in Tokyo, Japan~ November 23-26, 1999, 1999, pp. 39-45. [41] YAMAGISHI, K.: 'The associated graded modules of Buchsbaum modules with respect to m-primary ideals in equi-Iinvariant case', J. Algebra 225 (2000), 1-27. [42] YOSHINO, Y.: 'Maximal Buchsbaum modules of finite projective dimension', J. Algebra 159 (1993), 240 264.

Shiro Goto MSC1991: 13A30, 13H10, 13H30 BURNSIDE

rank m. The

GROUP - Let Fm be a free g r o u p of free m-generator Burnside group B(m, n)

of exponent n is defined to be the quotient group of Fm by the subgroup F ~ of Fm generated by all nth powers of elements of Fro. Clearly, B(m, n) is the 'largest' mgenerator group of exponent n (that is, a group whose elements satisfy the identity x n -- 1) in the sense that if G is an m-generator group of exponent n then there exists an epimorphism ¢: B(m, n) -+ G. In 1902, W. Burnside [3] posed a problem (which later became known as the Burnside problem for periodic groups) that asks whether every f i n i t e l y - g e n e r a t e d g r o u p of exponent n is finite, or, equivalently, whether the free Burnside groups B(m, n) are finite (cf. also B u r n s i d e p r o b l e m ) . It is easy to show that the free m-generator Burnside group B(m, 2) of exponent 2 is an elementary Abelian 2group and the order IB(m, 2)1 of B(m, 2) is T n. Burnside showed that the groups B(m, 3) are finite for all m. In 1933, F. Levi and B.L. van der Waerden (see [5]) proved 90

that the Burnside group B(m, 3) has the class of nilpotency equal to 3, when m >_ 3, and the order IB(m,3)l 1 2 3 equals 3Cm+cm+C~, where C ~ , . . . are binomial coefficients. In 1940, I.N. Sanov [18] proved that the free Burnside groups B(m, 4) of exponent 4 are also finite. In 1954, S.J. Tobin proved that IB(2, 4)[ = 212 (see [5]). By making use of computers, A.J. Bayes, J. Kautsky, and J.W. Wamsley showed in 1974 that IB(3,4)I = 289 and W.A. Alford, G. Havas and M.F. Newman established in 1975 that IB(4,4)1 = 2422 (see [5]). It is also known (see [5]) that the class of nilpotency of B(m, 4) equals 3 m - 2 when m > 3. On the other hand, in 1978, Yu.P. Razmyslov constructed an example of a non-solvable countable group of exponent 4 (see [5]). In 1958, M. Hall [7] proved that the Burnside groups B ( m , 6 ) of exponent 6 are finite and have the order 1 2 3 given by the formula I B ( m , 6)1 = 2~3c,+cÈ+c,, where oz = l + ( m - 1)3 cm+c~+c~ 1 2 a and ~ = 1 + ( m - l ) 2 m. The attempts to approach the Burnside problem via finite groups gave rise to a restricted version of the Burnside problem (called the restricted Burnside problem) which was stated by W. Magnus [14] in 1950 and asks whether there exists a number f(m,n) so that the order of any finite m-generator group of exponent n is less than f(m, n). The existence of such a bound f(m, n) was proven for prime n by A.I. Kostrikin [11] in 1959 (see also [12]) and for n = pe with a prime number p by E.I. Zel'manov [19], [20] in 1991-1992. It then follows from the Hall-Higman reduction results [6] and the classification of finite simple groups that a bound f(m,n) does exist for all m and n. In 1968, P.S. Novikov and S.I. Adyan [15] gave a negative solution to the Burnside problem for sufficiently large odd exponents by an explicit construction of infinite free Burnside groups B(rn, n), where m >_ 2 and n is odd, n > 4381, by means of generators and defining relators. See [15] for a powerful calculus of periodic words and a large number of lemmas, proved by simultaneous induction. Later, Adyan [1] improved on the estimate for the exponent n and brought it down to odd n _> 665. Using their machinery, Novikov and Adyan obtained other results on the free Burnside groups B(m, n). In particular, the word and conjugacy problems were proved to be solvable for the presentations of B(m, n) constructed in [15], any Abelian or finite subgroup of B(m, n) was shown to be cyclic (for these and other results, see [1]; cf. also I d e n t i t y p r o b l e m ; C o n j u g a t e e l e m e n t s ) . A much simpler construction of free Burnside groups B(m, n) for m > 1 and odd n > 10 l° was given by A.Yu. Ol'shanskii [16] in 1982 (see also [17]). In 1994, further developing Ol'shanskii's geometric method, S.V. Ivanov [9] constructed infinite free Burnside groups B(rn, n), where m > 1, n _> 248 and n is divisible by 29 if n is

BURNSIDE G R O U P even, thus providing a negative solution to the Burnside problem for almost all exponents. The construction of free Burnside groups B ( m , n ) given in [16], [9] is based on the following inductive definitions. Let Fm be a free group over an alphabet A = {a~l,...,a~ml}, m > 1, let n _> 24s and let n be divisible by 29 (from now on these restrictions on m and n are assumed, unless otherwise stated; note that this estimate n >_ 24s was improved on by I.G. Lysenok [13] to n > 213 in 1996). By induction on i, let B(m, n, O) = Fm and, assuming that the group B(m,n,i - 1) with i > 1 is already constructed as a quotient group of Fm, define Ai to be a shortest element of F,~ (if any) the order of whose image (under the natural epimorphism ¢ i - 1 : F,~ --+ B(m, n,i - 1)) is infinite. Then B(m,n,i) is constructed as a quotient group of B(m,n,i - 1) by the normal closure • (Ai). n of ¢~-i Clearly, B(m, n,i) has a presentation of the form B(m,n,i) = ( a l , . . . , a m I A [ ' , . . . , A ~ ) , where A [ ' , . . . , A~~ are the defining relators of B(m,n,i). It is proven in [9] (and in [16] for odd n > 101°) that for every i the word Ai does exist. Furthermore, it is shown in [9] (and in [16] for odd n > 101°) that the direct limit B(m, n, ec) of the groups B(m, n, i) as i --+ ec (obtained by imposing on Fm of relators A~ for all i = 1, 2 , . . . ) is exactly the free m-generator Burnside group B(m, n) of exponent n. The infiniteness of the group B(m,n) already follows from the existence of the word Ai for every i > 1, since, otherwise, B(m,n) could be given by finitely many relators and so Ai would fail to exist for sufficiently large i. It is also shown in [9] that the word and conjugacy problems for the constructed presentation of B(m, n) are solvable. In fact, these decision problems are effectively reduced to the word problem for groups B(m,n,i) and it is shown that each B(m, n, i) satisfies a linear isoperimetric inequality and hence B(m, n, i) is a Gromov hyperbolic group [4] (cf. Hyperbolic group). It should be noted that the structure of finite subgroups of the groups B(m,n,i), B(m,n) is very complex when the exponent n is even and, in fact, finite subgroups of B(m, n, i), B(m, n) play a key role in proofs in [9] (which, like [15], also contains a large number of lemmas, proved by simultaneous induction). The central result related to finite subgroups of the groups B(m, n, i), B(m, n) is the following: Let n = nln2, where nl is the maximal odd divisor of n. Then any finite subgroup G of B(m,n,i), B(m,n) is isomorphic to a subgroup of the direct product D(2nl) x D(2n2) e for some ~, where D(2k) denotes a dihedral group of order 2k. The principal difference between odd and even exponents in the Burnside problem can be illustrated by pointing out that, on the one hand, for every odd n >> 1 there are

infinite 2-generator groups of exponent n all of whose proper subgroups are cyclic (as was proved in [2], see also [17]) and, on the other hand, any 2-group the orders of whose Abelian (or finite) subgroups are bounded is itself finite (see [8]). In 1997, Ivanov and Ol'shanskiY [10] showed that the above description of finite subgroups in B(m, n) is complete (that is, every subgroup of D ( 2 n l ) x D(2n2) ~ can actually be found in B(m, n)) and obtained the following result: Let G be a finite 2-subgroup of B(m, n). Then the centralizer CB(m,~)(G) of G in B(m, n) contains a subgroup B isomorphic to a free Burnside group B(oo, n) of infinite countable rank such that G C) B = {1}, whence (G, B) = G x B. (Since B(oo, n) obviously contains subgroups isomorphic to both D(2na) and D(2n2), an embedding of D ( 2 n l ) x D(2n2) ~ in B(m,n) becomes trivial.) Among other results on subgroups of B(m, n) proven in [10] are the following: The centralizer CB(,~,,~)(S) of a subgroup S is infinite if and only if S is a locally finite 2-group. Any infinite locally finite subgroup L is contained in a unique maximal locally finite subgroup while any finite 2-subgroup is contained in continuously many pairwise non-isomorphic maximal locally finite subgroups. A complete description of infinite (maximal) locally finite subgroups of B(m, n) has also been obtained, in [10]. References [1] ADIAN, S.I.: The Burnside problems and identities in groups, Springer, 1979. (Translated from the Russian.) [2] ATABEKIAN, V . S . , AND IVANOV, S.V.: T w o remarks on groups

[3]

[4] [5] [6]

[7]

[8] [9] [lO]

[11] [12] [13]

of bounded exponent, Vol. 2243-B87, VINITI, Moscow, 1987, (This is kept in the Depot of VINITI, Moscow, and is available upon request). BURNSIDE, W.: 'An unsettled question in the theory of discontinuous groups', Quart. J. Pure Appl. Math. 33 (1902), 230-238. GROMOV, M.: 'Hyperbolic groups', in S.M. GERSTEN (ed.): Essays in Group Theory, Springer, 1987, pp. 75-263. GUPTA, N.: 'On groups in which every element has finite order', Amer. Math. Monthly 96 (1989), 297-308. HALL, PH., AND HIGMAN, G.: 'On the p-length of p-soluble groups and reduction theorems for Burnside's problem', Proc. London Math. Soc. 6 (1956), 1-42. HALL JR., M.: 'Solution of the Burnside problem for exponent 6', Proe. Nat. Acad. Sci. USA 43 (1957), 751-753. HELD, D.: 'On abelian subgroups of an infinite 2-group', Acta Sci. Math. (Szeged) 27 (1966), 97-98. IVANOV, S.V.: 'The free Burnside groups of sufficiently large exponents', Internat. J. Algebra Comput. 4 (1994), 1-308. IVANOV, S.V., AND OL'SHANSKII, A.Yu.: 'On finite and locally finite subgroups of free Burnside groups of large even exponents', J. Algebra 195 (1997), 241-284. KOSTRIKIN, A.I.: 'On the Burnside problem', Math. USSR Izv. 23 (1959), 3-34. (Translated from the Russian.) KOSTRIKIN, A.I.: Around Burnside, Nauka, 1986. LYSENOK, I.G.: 'Infinite Burnside groups of even period', Math. Ross. Izv. 60 (1996), 3-224.

91

BURNSIDE GROUP [14] MAGNUS, W.: 'A connection between the Baker-Hausdorff formula and a problem of Burnside', Ann. Math. 52 (1950), 11-26, Also: 57 (1953), 606. [15] 1NOVIKOV,P.S., AND ADIAN, S.I.: 'On infinite periodic groups I-III', Math. USSR Izv. 32 (1968), 212-244; 251-524; 709 731. [16] OL'SHANSKII,A.YU.: 'On the Novikov-Adian theorem', Math. USSR Sb. 118 (1982), 203-235. (Translated from the Russian.) [17] OL'SHANSKII, A.YU.: Geometry of defining relations in groups, Kluwer Acad. Publ., 1991. (Translated from the Russian.)

92

[18] SANOV, I.N.: 'Solution of the Burnside problem for exponent 4', Uch. Zapiski Leningrad State Univ. Set. Mat. 10 (1940)~ 166-170. [19] ZEL'MANOV,E.I.: 'Solution of the restricted Burnside problem for groups of odd exponent', Math. USSR Izv. 36 (1991), 41-60. (Translated from the Russian.) [20] ZEL'MANOV,E.I.: 'A solution of the restricted Burnside problem for 2-groups', Math. USSR Sb. 72 (1992), 543-565. (Translated from the Russian.) Sergei V. Ivanov

MSC 1991: 20F05, 20F06, 20F32, 20F50

C C A H N - H I L L I A R D EQUATION - An equation modelling the evolution of the concentration field in a binary alloy. When a homogeneous molten binary alloy is rapidly cooled, the resulting solid is usually found to be not homogeneous but instead has a fine-grained structure consisting of just two materials, differing only in the mass fractions of the components of the alloy. Over time, the fine-grained structure coarsens as larger particles grow at the expense of smaller particles, which dissolve. The development of a fine-grained structure from a homogeneous state is referred to as spinodal decomposition, while the coarsening is called Ostwald ripening (cf. also

with minima located at the two coexistent concentration states, labeIled ca and c~ > ca. A similar expression for free energy was introduced much earlier by J.D. van der Waals in [18].

Spinodal decomposition).

Here, A is the Laplacian (cf. Laplace operator), A is a Lagrange multiplier associated with the constraint (cf. also L a g r a n g e m u l t i p l i e r s ) , and n is the normal to 0V. In [8] equations (2)-(3) together with the constraint are used to predict the profile and thickness of onedimensional transitions between concentration phases ca and ca.

If the average concentration, ~, of one of the species and the temperature, T, lie in a particular region of parameter space, spinodal decomposition does not occur and instead, separation into the two preferred concentrations takes place through nucleation. In this scenario, small randomly spaced regions of a preferred state appear due to localized perturbations and then these regions grow. This is similar to the condensation of water droplets in mist, wherein a growing droplet depletes the water in the mist in its immediate vicinity, the depletion being replenished through diffusion-like processes. In 1958, J. Cahn and J. Hilliard [8] derived an expression for the free energy of a sample V of binary alloy with concentration field c(x) of one of the two species. They assumed that the free energy density depends not only upon e(z) but also derivatives of c, to account for interfacial energy or surface tension. To first order in an expansion, the expression for the total free energy takes the form

F = Nv L(fo(c) + ~ IVel 2) dV,

(1)

where Nv is the number of molecules per unit volume, fo is the free energy per molecule of an alloy of uniform composition, and a is a material constant which is typically very small. The function fo has two wells

With the average concentration ~ specified, the equilibrium configurations satisfy the stationary Cahn-

Hilliard equation 2~Ae - f~ (c) = ), 0c

On

0

in V,

on the boundary 0V of V.

(2) (3)

By considering the second variation of the free energy at the homogeneous state c(x) = -d, one can determine the stability of this state. If ~ is such that fg'(~) > 0 (the metastable concentrations), which includes those values near c~ and ca, then the homogeneous state is stable to small perturbations. If f~'(~) < 0, then if ~ is sufficiently small or equivalently, if V is sufficiently large, is unstable with respect to some periodic perturbations. This analysis was performed in [6], where it was also shown that perturbations of a certain characteristic wavelength of order v ~ grow most rapidly. Thus, spinodal decomposition is described mathematically. Likewise, when f~'(~) > 0 and ~ lies strictly between c~ and ca, the homogeneous state is stable but does not minimize the free energy if a is sufficiently small (see [10], [16]). In [9] the existence and properties of a critical nucleus are discussed. This nucleus is a spatially localized perturbation of the homogeneous state which lies on the boundary of the basins of attraction of the stable state

CAHN-HILLIARD EQUATION and the energy minimizing state, and is therefore unstable. Thus, nucleation is accounted for by the free energy proposed by Cahn and Hilliard. The general equation governing the evolution of a non-equilibrium state c(x, t) is put forth in [6] and this is what is now referred to as the Cahn-Hilliard equation:

Oc

0~ = div{M grad[f;(c) - 2~Ac]}

in V,

(4)

with the natural boundary conditions

Oc On

OAc On

- - -

--0

on0V.

(5)

The positive quantity M is related to the mobility of the two atomic species which comprise the alloy. Other derivations for the free energy, the equilibrium equations and the Cahn-Hilliard equation may be found in, e.g., [13], [14], [17], [11]. Further studies of spinodal decomposition as predicted by (4) in one and higher space dimensions and to various degrees of rigour may be found in [7], [14], [12], and [15]. Nucleation, beyond the existence of the canonical stationary nucleus for (4), is discussed in [3], [4] and [19]. The coarsening process is formally described for the one-dimensional version of (4) in [14] and is rigorously shown to be exponentially slow in [1] and [5]. In higher space dimensions, N. Alikakos and G. Fusco show in [2] that (4) predicts Ostwald ripening. It is thus well-established that the Cahn-Hilliard equation is a qualitatively reliable model for phase transition in binary alloys. References [1] ALIKAKOS, N.D., BATES, P.W., AND FUSCO, G.: 'Slow motion for the Cahn-Hilliard equation in one space dimension', g. Diff. Eqs. 90 (1990), 81-135. [2] ALIKAKOS,N.D., AND FUSCO, G.: 'The equations of Ostwald ripening for dilute systems', J. Statist. Phys. 95 (1999), 851866. [3] BATES, P.W., AND FIFE, P.C.: 'The dynamics of nucleation for the Cahn Hilliard equation', S I A M J. Appl. Math. 53 (1993), 990-1008. [4] BATES, P.W., AND FUSCO, G.: 'Equilibria with many nuclei for the Cahn-Hilliard equation', J. Diff. Eqs. 160 (2000), 283-356. [5] BATES, P.W., AND NUN, P.J.: 'Metastable patterns for the Cahn-Hilliard equation. Part I-IF, J. Diff. Eqs. 1 1 1 / 1 1 6

(1994/95), 421-45z/165 216. [6] CAHN, J.W.: 'On spinodal decomposition', Acta Metall. 9 (1961), 795-801. [7] CAHN, J.W.: 'Phase separation by spinodal decomposition in isotropic systems', Y. Chem. Phys. 42 (1965), 93-99.' [8] CAHN, J.W., AND HILLIARD, J.E.: 'Free energy of a nonuniform system I: Interracial energy', Y. Chem. Phys. 28 (1958), 258-266. [9] CAHN, J.W., AND HILLIARD, J.E.: 'Free energy of a nonuniform system III: Nucleation in a two-component incompressible fluid', Y. Chem. Phys. 31 (1959), 688-699.

94

[i0] CARR, J., GURTIN, M., AND SLEMROD,

M.: 'Structured phase

transitions on a finite interval', Arch. Rational Mech. Anal. 86 (1984), 317-357. [11] FIFE, P.C.: 'Models for phase separation and their mathematics', in M. MIMURA AND T. NISHIDA (eds.): Nonlinear Partial Differential Equations with Applications to Patterns, Waves, and Interfaces. Proc. Conf. Nonlinear Partial Differential Equations, Kyoto, 1992, pp. 183-212. [12] GRANT, C.P.: 'Spinodal decomposition for the Cahn-Hilliard equation', Commun. Partial Diff. Eqs. 18, no. 3-4 (1993), 453-490. [13] HILLERT, M.: 'A solid-solution model for inhomogeneous systems', Acta Metall. 9 (1961), 525-535. [14] LANGER, J.S.: 'Theory of spinodal decomposition in alloys', Ann. Phys. 65 (1971), 53 86.

[15] MAIER--PAAPE, S., AND WANNER, T.: 'Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. I. Probability and wavelength estimate', Comm. Math. Phys. 195 (1998), 435 464. [16] MODICA, L.: 'The gradient theory of phase transitions and the minimal interface criterion', Arch. Rational Mech. Anal.

9s (19sD, 123-142. [17] NOVICK-COHEN, A., AND SEGEL, L.A.: 'Nonlinear aspects of the Cahn-Hilliard equation', Phys. D. 10 (1985), 277-298. [18] WAALS, J.D. VAN DER: 'The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation in density', Verh. K. Nederland. Akad. Wetenschappen Amsterdam 1 (1893), 1-56. [19] WEI, J., AND WINTER, M.: 'Stationary solutions for the Cabn-Hilliard equation', Ann. Inst. H. Poincard 15 (1998), 459-492.

P. W. Bates MSC1991: 82B26, 82D35 CAKE-CUTTING PROBLEM, fair division problem - A circular or rectangular cake is to be cut and divided (by radial, respectively vertical, cuts) among n persons. Setting the total size (volume) of the cake to 1, each division among n persons is given by n real numbers xi >_ 0 such that Xl -}- " " + x n = l ,

i.e. by a point x of the standard n-simplex in R n+l. Each of the persons involved can have his/her own preferences: a choice of a segment for each x. Different parts of the cake may have different values for each of the n different persons. The question is whether there is a fair division (or envy-free division), i.e. one for which each of the n persons gets a piece that for him/her is optimal. The answer is yes. A unifying approach to this and similar problems (such as rent partitioning and dispute resolution) can be based on the S p e r n e r l e m m a , giving better and better approximations by means of Sperner labelings of finer and finer subdivisions, [4]. Recently (2000), there has been quite a bit of interest in fair division and cake cutting; see, e.g., [1], [3]. The

CATALAN CONSTANT problem has found its way into recreational mathematics under the name chore-division problem, [2]. References [1] BRAMS, S.J., AND TAYLOa, A.D.: Fair division: from cakecutting to dispute resolution, Cambridge Univ. Press, 1996. [2] GARDNER, M.: Aha! Insight, Freeman, 1978. [3] ROBERTSON, J.M., AND WEBS, W.A.: Cake-cutting algorithms: be fair if you can, A.K. Peters, 1998. [4] Su, F.E.: 'Rental harmony: Sperner's lemma in fair division', Amer. Math. Monthly 106 (1999), 930-942. M. Hazewinkel

MSC 1991: 90Axx, 00A08

[2] LAMBEK, J., AND SCOTT, P.J.: Introduction to higher order categorical logic, Cambridge Univ. Press, 1986. [3] MACLANE, S.: Categories for the working mathematician, Springer, 1971. [4] MACLANE, S., AND MOERmJK, I.: Sheaves in geometry and logic, Springer, 1992.

Named after its inventor, E.Ch. Catalan (1814-1894), the Catalan constant G (which is denoted also by &) is defined by CATALAN

category C such t h a t the following axioms are satisfied: CARTESIAN-CLOSED

CATEGORY

a ::

C-+CxC, U-+C,

( - 1 ) k ..~ (2k + 1)2 =

k:0

(1)

If, in terms of the Digamma (or Psi) function ¢(z), defined by d F'(z) ¢( z) := { l o g r ( z ) } - r(z) (2) or

log F(z) =

¢(t) dt,

one puts

Z(z) := l [ ¢ ( ~ z + 2 ) - ¢ ( ~ z ) ]

=

(3)

(_l)k

=EzT ,

c ~-+ O;

k=0

c~(c,c); a ~-+ a x b

-

= 0.91596 55941 77219 015.. • .

These conditions are equivalent to the following: C is a category with given products such that the functors C --4 1,

CONSTANT

- A

A1) there exists a terminal object 1; A2) for any pair A, B of objects of C there exist a product A x B and given projections Pl : A x B --+ A, p2: A x B - + B; A3) for any pair A, B of objects of C there exist an object A B and an evaluation arrow ev: A B × A -+ B such that for any arrow f : C × A -+ B there is a unique arrow I f ] : C -+ A B with e v o [ f ] × A = f.

M. Eytan

MSC 1991:18D15

where z E C \ Zo,

have each a specified right-adjoint, written respectively

Z o := { 0 , - 1 , - 2 , . . . } ,

then

as:

1 ,(1) G = -~13

,

(4)

0 ~-~ t,

(a,b) ~ a x b , C ~-} Cb .

Some examples of Cartesian-closed categories are: El) any Heyting algebra ?-/; E2) the category $ d s c for any s m a l l c a t e g o r y C with Sets the category of (small) sets - - in particular Sets itself; E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos; E4) any elementary t o p o s £; E5) the category Cat of all (small) categories; E6) the category ~rapO of graphs and their homomorphisms; ET) the category a>CT)O of w-CPOs. These definitions can all be put into a purely equational form. References [1] BARR, M., AND WELLS, C.: Category theory for computing science, CRM, 1990.

which provides a relationship between the Catalan constant G and the Digamma function ¢(z). The Catalan constant G is related also to other functions, such as the CIausen function C12(z), defined by C12(z) := -

log

sin

t

dt=

(5)

sin(kz) = E k=l

k2

'

and the Hurwitz zeta-function ~(s, a), which is defined, when Re s > 1, by 1

~(s,a) := E

(6)

(k + a) s'

k=0

Res>l,

aEC\Z

o.

Thus,

G = c12 (½

)=-el2

=

(7)

16

95

CATALAN CONSTANT Since ¢(~) (z) = ( - 1 ) n + l n ! ~(n + 1, z), heN:={1,2,...},

zeC\Z

(8)

o,

the last expression in (7) would follow also from (4) in light of the definition in (3). A fairly large number of integrals and series can be evaluated in terms of the Catalan constant G. For example, 1 tlog(t -1 4-t) 1 + t4 dt =

f0 f

=

(9)

~ t l ° g ( t : t : t - 1 ) dt 7r G l+t 4 = ~log24-~-,

k=z \ k ! ( h + 1)!//

= 41og2 + 2 -

-

k

4(2G

+

(10)

I),

77

and ¢(2k)

- log

- 1 + --

k=l k(2k + 1)24k

(11)

7r '

where ~(s) = ((s, 1) denotes the familiar R i e m a n n zeta-function. Euler-Mascheroni c o n s t a n t . Another important mathematical constant is the Euler-Mascheroni constant 7 (which is denoted also by C), defined by 7 : = ~lim -~ (1+1 2 +""

1 - loan ) = + -n

(12)

- 0.57721 56649 01532860606512.. • . It is named after L. Euler (1707-1783) and Mascheroni (1750-1800). Indeed, one also has 7 = -g)(1) = - F ' ( 1 ) = =

-log

1+

=-

L. (13)

e-tlogtdt

k=l

and Z

7 =

k(k-+ z)

¢ ( z + 1) =

(14)

k=l ~

2

k=l

1 2k- 1

21092-~

zeC\Z-;

(n+ ~)

Z - := Zo \ {0};

(15)

n C N0 : = N U { 0 } , where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function ~(s), Euler's classical resuRs state: (XD

7 = ] ~ ( - 1 ) k ~(k) _ k k:2

~ ~(2k + 1)2_2k ' 1%~2-z__, 2 k + 1 k=l

96

(16)

References [1] ERDELYI, A., MAGNUS, W., OBERHETTINGER, F., AND TRICOMI, F.G.: Higher transcendental functions, Vol. I, McGraw-Hill, 1953. [2] LEWiN, L.: Polylogarithms and associated functions, Elsevier, 1981. [3] SRIVASTAVA, H.M., AND CHOI, J.: Series associated with the zeta and related functions, Kluwer Acad. Publ., 2001.

Hari M. Srivastava

MSC 1991: 33B15, 11M06, 11M35 CAYLEY GRAPH - Cayley graphs stem from a type of diagram now called a Cayley colour diagram, which was introduced by A. Cayley in 1878 as a graphic representation of abstract groups. Cayley colour diagrams were used in [7] to investigate groups given by generators and relations. A Cayley colour diagram is a directed graph with coloured edges (cf. also G r a p h , o r i e n t e d ) , and gives rise to a Cayley graph if the colours on the edges are ignored. Cayley colour diagrams were generalized to Schreier coset diagrams by O. Schreier in 1927, and both were investigated as 'graphs' in [20]. Cayley graphs and their generalizations - - vertex-transitive graphs - - are systematically" studied in [3], [6], [18]. Cayley graphs provide graphic representations for abstract groups. They are a bridge between groups and surfaces, and they give rise to examples for various extremal graph problems, and good models for interconnection networks. Given a g r o u p G and a subset S C G which does not contain the identity of G, the associated Cayley graph Cay(G,S) is the directed graph F with vertex set G and with x adjacent to y if and only i f y x -1 E S. If S : S - 1 : : {8 - 1 : 8 E S } , then the adjacency relation is symmetric and thus the Cayley graph Cay(G, S) may be viewed as an undirected g r a p h . Some examples of Cayley graphs are

• the well-studied circulant graphs (loop networks) are precisely the Cayley graphs of cyclic groups; • hypercube graphs are Cayley graphs of elementary Abelian 2-groups; more generally, • Hamming graphs are Cayley graphs of elementary Abelian groups. By definition, F = Cay(G, S) has out-valency IS[, and F is connected if and only if (S} = G. Further, the group G acting by right multiplication (that is, g: x --+ xg) is a subgroup of Aut F and acts regularly on the vertex set VF = G (cf. also R e g u l a r g r o u p ) . Thus Aut F contains a subgroup which is regular on VF and isomorphic to G. In particular, Aut F is transitive on VF, and so F is vertex-transitive. It was shown in [20] that an arbitrary graph F is a Cayley graph of a group G if and only

CELLULAR ALGEBRA if Aut F contains a regular subgroup isomorphic to G. Identifying the regular subgroup with G, one has Aut

F =

GNH

GH, = I,

where H = {a ¢ A u t F : v ~ = v} for some v E VF, i.e., Aut F is factorizable. Some part of Aut F can be described in terms of Aut(G): NAut r(G) = G . Aut(G, S), where Aut(G,S) = {a ¢ Aut(G): S ~ = S}. So, part of the information about the graph F (which may be available from Aut F) may be directly read from information about the group G. Some work has been devoted to characterizing Cayley graphs P = Cay(G, S) in terms of Aut(G). See [19], [21] for the study of edge-transitive Cayley graphs, and [2], [14] for determining isomorphism relations between Cayley graphs of G. The extreme case where Aut P = G has received considerable attention, see [3], [9], [13]. Cayley graphs contain long paths (see [3]), and have many other nice combinatorial properties (see [3]). Cayley graphs have been used to construct extremal graphs: see [15], [16] for the constructions of Ramanujan graphs and expanders; see [1], [17] for the constructions of graphs without short cycles. They have also been used to construct other combinatorial structures: see [12], [8] for the constructions of various communication networks; see [4] for difference sets in design theory. Cayley graphs have been used to analyse algorithms for computing with groups, see [3]. For infinite groups, Cayley graphs provide convenient metric diagrams for words in the corresponding groups, and underlie the study of growth of groups, see [3], [10]. Cayley maps are Cayley graphs embedded into certain surfaces, and provide pictorial representations of groups and group actions on surfaces. They have been extensively studied, see [5], [11]. Cayley graphs form a proper subclass of the vertextransitive graphs. The P e t e r s e n g r a p h is the smallest vertex-transitive graph which is not a Cayley graph. B. McKay, C.E. Praeger and G.F. Royle observed that most vertex-transitive graphs of order at most 24 are Cayley graphs, and this led McKay and Praeger to conjecture (1994) that most vertex-transitive graphs are Cayley graphs, see [18]. References

[1] ALON,N.: 'Tools from higher algebra': Handbook of Combinatorics, Elsevier, 1995, pp. 11751-1783. [2] BABAI, L.: 'Isomorphism problem for a class of pointsymmetric structures', Acta Math. Acad. Sci. Hungar. 29 (1977), 329-336.

[3] BABAI, L.: 'Automorphism groups, isomorphism, reconstruction': Handbook of Combinatorics, Elsevier, 1995, pp. 14491540. [4] BETH, T., JUNGNICKEL, D., AND LENZ, H.: Design theory, Vol. I, Cambridge Univ. Press, 1999. [5] BIGGS, L., AND WHITE, A.T.: Permutation groups and combinatorial structures, Vol. 33 of Math. Soc. Lecture Notes, Cambridge Univ. Press, 1979. [6] BIGGS, N.: Algebraic graph theory, second ed., Cambridge Univ. Press, 1992. [7] COXETER, H.S.M., AND MOSER, W.O.J.: Generators and relations for discrete groups, Springer, 1957. [8] FANC, X.G., LI, C.H., AND PRAECEa, C.E.: 'On orbital regular graphs and Frobenius graphs', Discr. Math. 182 (1998), 85 99. [9] GODSIL, C.D.: 'On the full automorphism group of a graph', Combinatorica 1 (1981), 243-256. [10] GROMOV, M.: 'Groups of polynomial growth and expanding maps', Publ. Math. I H E S 53 (1981), 53-73. [11] GROSS, J.L., AND TUCHER, T.W.: Topological graph theory, Wiley, 1987. [12] HEYDEMANN, R., AND DUCOURTHIAL, B.: 'Cayley graphs and interconnection networks': Graph Symmetry: Algebraic Methods and Applications, Vol. 497 of N A T O Ser. C, Kluwer Acad. Publ., 1997, pp. 167-224. [13] LI, C.H.: 'The solution of a problem of Godsil regarding cubic Cayley graphs', Y. Combin. Th. B 72 (1998), 140-142. [14] LI, C.H.: 'Finite CI-gronps are soluble', Bull. London Math. Soc. 31 (1999), 419-423. [15] LUBOTZKY, A.: Discrete groups, expanding graphs and invariant measures, Vol. 125 of Progress in Math., Birkhguser, 1994. [16] LUBOTZKY, A., PHILLIPS, R., AND SARNAK, P.: 'Ramanujan graphs', Combinatorica 8 (1988), 261-277. [17] MARCULIS, G.A.: 'Explicit constructions of graphs without short cycles and low density codes', Combinatorica 2 (1982), 71-78. [18] PRAEGER, C.E.: 'Finite transitive permutation groups and finite vertex-transitive graphs': Graph Symmetry: Algebraic Methods and Applications, Vol. 497 of N A T O Ser. C, Kluwer Acad. PubL, 1997, pp. 277-318. [19] PRAEGER, C.E.: 'Finite normal edge-transitive Cayley graphs', Bull. Austral. Math. Soc. 60 (1999), 207-220. [20] SABIDUSSI,G.O.: 'Vertex-transitive graphs', Monatsh. Math. 68 (1964), 426-438. [21] XU, M.Y.: 'Automorphism groups and isomorphisms of Cayley digraphs', Discr. Math. 182 (1998), 309-320. Cai Heng L i

MSC 1991:05C25 CELLULAR ALGEBRA (in algebraic combinatorics) - Algebras introduced by B.Yu. Weisfeiler and A.A. Leman [9] and initially studied by representatives of the Soviet school of algebraic combinatorics. The first stage of this development was summarized in [8]. Important particular examples of cellular algebras are the coherent algebras (cf. also C o h e r e n t algebra). A cellular algebra W of order n and rank r is a matrix subalgebra of the full matrix algebra C n×~ of (n × n)matrices over C such that: • W is closed with respect to the Hermitian adjoint; 97

CELLULAR ALGEBRA • J E W, where 2 is the all-one matrix; • W is closed with respect to Schur-Hadamard multiplication (cf. also C o h e r e n t algebra).

For each cellular algebra W = ( A 1 , . . . , At) one can introduce its automorphism group Aut(W) = N Aut(Ai). i=1

A coherent algebra is a cellular algebra that contains the unit matrix I. Like coherent algebras, a cellular algebra W has a unique standard basis of zero-one matrices {A1, • •., Ar }, consisting of mutually orthogonal Schur-Hadamard idempotents. The notation W = (A1,... ,At} indicates that W has the standard basis { A I , . . . , Ar }. A cellular algebra W is called a cell if the all unit matrix J belongs to its centre (cf. also C e n t r e of a ring). Cells containing the unit matrix [ are equivalent to Bose-Mesner algebras. If the entries of the matrices in W are restricted to the ring Z, then the corresponding ring of matrices is called a cellular ring. The relational analogue of cellular algebras with the unit matrix I was introduced by D.G. Higman in [3] under the name coherent configuration. For a long time the theories of cellular algebras and coherent configurations were developed relatively independently. After the appearance of Higman's paper [4], where the terminology of coherent algebras was coined, most researchers switched to the terminology of coherent algebras. As a rule, only cellular algebras containing I (that is, coherent algebras) were investigated. Situations where cellular algebras are required properly appear rarely, see for example [7], where a particular kind of such algebras are treated as pseudo-Schur rings. The initial motivation for the introduction of cellular algebras was the graph isomorphism problem (cf. also Graph isomorphism). The intersection of cellular algebras is again a cellular algebra. For each set of matrices of the same order n it is possible to determine a minimal cellular algebra containing this set. In particular, if F is an n-vertex graph and A = A(F) is its adjacency matrix, then ((A)) denotes the smallest cellular algebra containing A. It is called the cellular closure (or Weisfeiler-Leman closure) of W. In [9] and [8], Weisfeiler and Leman described an algorithm of stabilization which has an input A and returns <(A)) in polynomial time, depending on n. Isomorphic graphs have isomorphic cellular closures, and this fact has important applications, see [6]. In general, ((A}} does not coincide with the centralizer algebra of the automorphism group of the graph F; however, they do coincide for many classes of graphs, for example for the algebraic forests introduced in [1]. 98

Here, Aut(Ai) is the automorphism group of the graph Fi with adjacency matrix Ai = A(Fi). For each p e r m u t a t i o n g r o u p (G, fl), its centralizer algebra !U(G, fi) is a cellular algebra with matrix I. Thus, the functors Aut and !U introduce a G a l o i s c o r r e s p o n d e n c e between cellular (coherent) algebras and permutation groups. Some properties and applications of this correspondence are considered in [2], [5]. References [1] EVDOKIMOV, S., PONOMARENKO, I., AND TINHOFER, G.: 'Forestal algebras and algebraic forests (on a new class of weakly compact graphs)', Discr. Math. 225 (2000), 149-172. I2] FARADZEV, I.A., KLIN, M.H., AND ~/iUZICHUK, M.E.: 'Cellular rings and groups of automorphisms of graphs.', in I.A. FARADZEV ET AL. (eds.): Investigations in Algebraic Theory of Combinatorial Objects, Kluwer Acad. Publ., 1994, pp. 1152. [3] HIGMAN, D.G.: 'Coherent configurations I', Rend. Sem. Mat. Univ. Padova 44 (1970), 1-25. [4] HIGMAN, D.G.: 'Coherent algebras.', Linear Alg. ~4 Its Appl.

93 (19s7), 209-239 [51 IVANOV,A.A., FARADZEV,I.A., AND KLIN, M.H.: 'Galois correspondence between permutation groups and cellular rings (association schemes)', Graphs and Combinatorics 6 (1990), 303 332. [6] KLIN, 1Vii.,RUCKER, C., R/.)CKER, G., AND TINHOFER, G.: 'Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras', M A T C H 40 (1999), 7-138. [7] MUZYCHUK,M.E.: 'The structure of rational Schur rings over cyclic groups', Europ. J. Combin. 14 (1993), 479-490. [8] WEISFEILER, B. (ed.): On construction and identification of graphs, Vol. 558 of Lecture Notes in Math., Springer, 1976. [9] WEISEEILER, B.YU., AND LEMAN, A.A.: 'A reduction of a graph to a canonical form and an algebra arising during this reduction', Nauchno-Techn. Inform. Ser. 2 9 (1968), 12-16. (In Russian.) Mikhail K l i n

MSC 1991: 05Exx CHASLES-CAYLEY-BRILL

FORMULA

- Let C

be an irreducible algebraic plane curve of degree n, given by an equation f ( X , Y) = 0 where f is an irreducible bivariate polynomial of degree n over a ground field k (cf. also A l g e b r a i c curve). For simplicity k is assumed to be algebraically closed (cf. also A l g e b r a i c a l l y closed field), although most of what is said below can be suitably generalized without that assumption. For the basic field theory involved, see [6] (or the modernized version [4]) and [3]. For much of the geometry to be discussed, see [5] and [8]; in particular, for the idea of points at infinity of C, see [1]. For an interplay between the geometry and the algebra, see [2].

CHASLES-CAYLEY-BRILL FORMULA One starts by analyzing when the curve C can be rationally parametrized. For example, the unit circle X2 + y 2 = 1 has the rational p a r a m e t r i z a t i o n

Provisionally defining the genus g of C (cf. also G e n u s o f a c u r v e ) by -

g = 1 - t2 X - - -

l + t 2'

Y=

2t l + t 2"

Likewise, the cuspidal cubic y 2 = X 3 has the rational p a r a m e t r i z a t i o n X = t 2 and Y = t 3. However, the nonsingular cubic y 2 = X 3 _ 1 does not have any rational parametrization. To o b t a i n the p a r a m e t r i z a t i o n of the circle, one cuts it by a line of slope t t h r o u g h the point ( - 1 , 0) and notes t h a t it meets the circle in the variable point

-~ t2 , 1 + t 2

.

For the cuspidal cubic one takes a line t h r o u g h the cusp (0, 0) and notes t h a t it meets the cubic in the variable point (t2,t3). This works because a line meets a circle in 2 points, and it meets a cubic in 3 points. In case of a cuspidal cubic two intersections are absorbed in the cusp. In case of a non-singular cubic there is no such point for the absorption. Generalizing this one can show t h a t the curve C c a n n o t have more t h a n ( n - 1 ) ( n - 2 ) / 2 double points and if it does have t h a t many, then it can be parametrized rationally. To this end one first notes t h a t a bivariate polynomial of degree m has

coefficients and hence the dimension of the system S of curves of degree n - 2 passing t h r o u g h (n - 1)(n - 2)/2 double points of C is n(n - 1)/2 - 1 - (n - 1)(n - 2)/2 = n - 2. Next, by the B e z o u t t h e o r e m (which is the oldest t h e o r e m in a l g e b r a i c g e o m e t r y ) , a curve of degree n and a curve of degree m, having no c o m m o n component, meet in nm points, counted properly. In the proper counting a double point of C should be counted twice. Thus, the n u m b e r of free points in which C meets a curve D of degree n - 2 passing t h r o u g h the (n - 1)(n - 2)/2 double points of C is n(n - 2) - (n - 1)(n - 2) = n - 2. Since n - 2 is also the dimension of the system S, the members of S which pass t h r o u g h n - 3 fixed simple points of C form a pencil, i.e., a o n e - p a r a m e t e r family Dr, a variable m e m b e r of which meets C in one variable point whose coordinates are single-valued, and hence rational, functions of t. If C had an e x t r a double point, then one can take a value to of t so t h a t Dto goes t h r o u g h it and this would make the properly counted intersections of C and Dto to be > n(n - 2), contradicting the Bezout theorem because C is irreducible and Dto has smaller degree.

-

2

2)

- ~ d o u b l e points,

one always has g > 0, and g = 0 ~ C is rational, i.e., has a rational parametrization. To make the reverse implication ~ also true, one m u s t learn to count the double points properly. To begin with, one m u s t include singularities of C at infinity. Next, by looking at the curve Y~ = X d, where e > d with gcd(e, d) = 1, which is obviously rational and has a d-fold point at the origin and an (e - d)-fold point at infinity as its only singularities, one decides to count a d-fold point as d ( d - 1)/2 double points. Before discussing infinitely near singularities, one notes t h a t the degree n of C can be geometrically characterized as the n u m b e r of points in which a general line meets it. Likewise, the multiplicity d of a point P of C can be characterized geometrically by saying t h a t n - d is equal to the n u m b e r of points in which a generic line t h r o u g h P meets C outside P ; P is a simple or singular point of C according as d = 1 or d > 1. Algebraically, by translating the coordinates one m a y assume P to be the origin (0, 0), and then d is the order of f , i.e., f has terms of degree d but none of degree < d. By making the quadratic t r a n s f o r m a t i o n X = X ' and Y = X ' Y ' one gets f ( X ' , X ' Y ' ) = X ' g f ' ( X ', Y'), where C~: f f ( X ' , Y~) = 0 is the proper transform of C. T h e exceptional line X ~ = 0 meets C ~ in points P 1 , . . . , Ph whose multiplicities dl, • •., dh add up to _< d. These are the points of C in the first neighbourhood of P. Points in the first n e i g h b o u r h o o d s of these points are the points of C in the second neighbourhood of P , and so on. It is easily seen t h a t all points in a high enough neighbourh o o d of P are simple. Now P is counted as ~(P) double points, where

5(P) = E

d(Q)(d(Q) - 1) 2

with the s u m m a t i o n extended over all points Q in the various n e i g h b o u r h o o d s of P , including P ; here d(Q) is the multiplicity of Q; clearly: 5(P) = 0 ~=~ P is a simple point of C. One arrives at the exact genus formula (n g=

-

1)(n- 2) 2

with s u m m a t i o n over all points P of C. One always has g > 0; and g = 0 ¢~, C is rational. It turns out t h a t g is a birational invariant of C, i.e., it remains u n c h a n g e d when C undergoes a birational t r a n s f o r m a t i o n (of. also B i r a t i o n a l m o r p h i s m ) . The residue class ring of the polynomial ring k[X, Y] modulo the ideal generated by f ( X , Y ) is the af-fine coordinate ring of C and is denoted by k[C]. Note t h a t 99

C H A S L E S - C A Y L E Y - B R I L L FORMULA

k[C] = k[x,y] where x, y are the images of X, Y in k[C]. The quotient field k(C) = k(x, y) of k[C] is the function field of C. A birational correspondence between curves

For i = 1, 2, let Ci be an irreducible algebraic plane curve such that k(C) is a finite separable algebraic field extension of k(Ci) of field degree ui (cf. also E x t e n sion o f a field; S e p a r a b l e e x t e n s i o n ) . This defines a (Yl,u2) correspondence between 9~(C1) and 9l(C2), and hence between C1 and C2; namely, T1 E 9~(C1) and T2 E 9l(C2) correspond if and only if for some T E ~R(C) one has T N k(C1) = T1 and T N k(C2) = T2. Let gi be the genus of Ci, let the different ~ ( C , Ci) be the integer-valued function on 9~(C) whose value at T in iR(C) is given by OrdT(dvi/dQ, where 7-i is a uniformizing parameter of TNk(Ci), and let ~ i = ~ ~(C, Ci)(T) with summation over all T C iR(C). Then the RiemannHurwitz formula says that

C and C* is an almost one-to-one correspondence; it is given by a k-isomorphism between k(C) and k(C*). So one should be able to define g directly in terms of k(C). Following C.G.J. Jacobi one takes any differential of k(C) (cf. also D i f f e r e n t i a l field), i.e., an expression of type u dv with u,v E k(C), and shows that if the differential is not zero, then the number of its zeros minus the number of its poles equals 2g - 2. Having brought the point P of C to the origin, its local ring R(P) is defined to be the subring of k(C) consisting of all quotients r(x, y)/s(x, y) where r(X, Y), s(X, Y) are polynomials with s(0, 0) 7~ 0 (cf. also L o c a l ring); its unique m a x i m a l i d e a l M(R(P)) consists of the above quotients with r(0, 0) = 0. Let C(P) be the conductor of R(P), i.e., the largest ideal in R(P) which remains an ideal in the integral closure R'(P) of R(P) in k(C). It can be shown that iS(P) is the length of g ( P ) in R(P), i.e., the maximal length of strictly increasing chains of ideals g ( P ) = I0 C ... C Ia = R(P) in R(P); moreover, 2~(P) is the length of g ( P ) in R ' ( P ) , which is a ubiquitous result having two dozen proofs in the literature. The ring R'(P) has a finite number of maximal ideals and localizing R'(P) at them gives discrete valuation rings; as P varies over all points of C, including those at infinity, these discrete valuation rings vary over the R i e m a n n s u r f a c e 9~(C) of C, i.e., the set of all discrete valuation rings whose quotient field is k(C) and which contain k. Let !}l(C, P ) denote the localizations of R'(P) at the various maximal ideals in R'(P) (cf. also L o c a l i z a t i o n in a c o m m u t a t i v e a l g e b r a ) ; one calls P the centre on C of the members of 9l(C, P); note that R'(P) = R(P) ¢* P is a simple point of C, and hence for all except a finite number of points of C, the set 9l(C, P ) has exactly one member. For any T E 9l(C) and non-zero r, s C k(C) one puts

tions, the number of these, counted properly, equals ul + v2 + 27g, where the integer 9' is called the valence of the correspondence. For details see [7, pp. 189-194]. In case k is the field of complex numbers, to describe Riemann's approach one topologizes 9I(C) to make it into a compact orientable two-dimensional real manifold, and hence into a sphere with g handles (cf. also R i e m a n n s u r f a c e ) . Likewise, ffl(C1) is made into a sphere with gl handles. Triangulate ffl(C1) by including all the branch points as vertices, and lift this triangulation to a triangulation of 9I(C). Let (V1, El, F1) and (V, E, F ) be the vertices, edges, faces of the bottom and top triangulations respectively. Then V = vlV1 - ~ 1 , E = ulE1, F = ulF1, and hence by the Euler-Poinca% theorem one obtains

OrdT(r/s) = A -- #,

2 g - 2 = Yl(2gl - 2) + ~1.

with rT = M(T) ~ and sT = M ( T ) ' ; take r C T with r T = M(T) and define

This proves the birational invariance of g and the Riemann-Hurwitz formula. For details, see [2] and [4].

o r d r (u dv) = OrdT (u dv/d~); one calls ~- a uniformizing parameter of T. Now the number of zeros minus number of poles of u dv equals ~ o r d T ( u dr) taken over all T in ffl(C). For any point P of C, not at infinity, one has Dedekind's formula

fy(x, y)R'(P) =

x),

where ~ ( P , x) is the different ideal in R'(P) defined by saying that ~ ( P , x ) T = M(T) ~ with e = ordT(dx/dT) for every T C 9I(C, P). I00

2 g - 2 = y~(2gi - 2) + ~ i , and this gives rise to the Zeuthen formula ,1 (291 -- 2) + ~1 = v2(2g2 -- 2) + ~2. Now suppose there is a k-isomorphism ¢: k(C1) -+ k(C2). Then T E ffl(C) is called a fixed place of the correspondence if T n k(C2) = ¢ ( T C/k(C1)). The ChaslesCayley-Brill formula says that under suitable condi-

References [1] ABHYANKAR, S.S.: ' W h a t is the difference between a p a r a b o l a and a hyperbola', Math. Intelligencer 10 (1988), 36-43. [2] A13HYANKAI=t,S.S.: Algebraic geometry for scientists and engineers, Amer. M a t h . Soc., 1990. [3] ABHYANKAR, S.S.: 'Field extensions', in G.A. PILZ AND A.V. MIKHALEV (eds.): Handbook of the Heart of Algebra, Kluwer Acad. Publ., to appear. [4] CHEVALLEY,C.: Introduction to the theory of algebraic functions of one variable, Vol. 6 of Math. Surveys, Amer. M a t h . Soc., 1951. [5] COOLIDGE, J.L.: A treatise on algebraic plane curves, Clarendon Press, 1931.

CHEBYSHEV P S E U D O - S P E C T R A L M E T H O D [6] DEDEKIND, R., AND WEBER, H.: 'Theorie der algebraischen Fkmctionen einer Veriinderlichen', Crelle d. 92 (1882), 181290. [7] LEFSCHETZ, S.: Algebraic geometry, Princeton Univ. Press, 1953. [8] SEVERI, F.: Vorlesungen iiber algebraische Geometric, Teubner, 1921. Shreeram S. A b h y a n k a r

MSC1991: 12F10, 14H30, 20D06, 20E22 CHEBOTAREV

DENSITY

T H E O R E M - Let

L/K

be a normal (finite-degree) extension of algebraic number fields with Galois group Gal(L/K). Pick a prime ideal gl of L and let go be the prime ideal of K under it, i.e. p = AK V19t3, where AK is the ring of integers of K. There is a unique element

of G a l ( L / K ) such that cry = x N(e) m o d ~ f o r x E L integral. Here, N ( p ) , the norm of ~o, is the number of elements of the residue field AK/p. This is the F r o b e n i u s a u t o m o r p h i s m (or Frobcnius symbol) associated to

~p.

If p is unramified in L / K , define FL/K(P) as the conjugacy class of cry in Gal(L/K), where g3 is any prime ideal above ga. This conjugacy class depends only on go. The weak form of the Chebotarev density theorem says that if A is an arbitrary conjugacy class in Gal(L/K), then the set

PA = {fg: FL/K(p) = A} is infinite and has D i r i c h l e t d e n s i t y # A / n , where n = [L : K]. The stonger form specifies in addition that PA is regular (see D i r i c h l e t d e n s i t y ) and that

with NA(X) the number of prime ideals in PA with norm <X.

References [1] CHEBOTAREV, N.G.: 'Determination of the density of the set of primes corresponding to a given class of permutations', Izv. Akad. Nauk. 17 (1923), 205-230; 231 250. [2] CHEBOTAaEV,N.G.: 'Die Bestimmung der Dichtigkeit einer Menge yon Primzahlen welche zu einer gegebenen Substitutionsklasse gehSren', Math. Ann. 95 (1926), 191-228. [3] NARKIEWICZ, W.: Elementary and analytic theory of algebraic numbers, second ed., PWN/Springer, 1990, p. Sect. 7.3.

M. Hazewinkel MSC1991: 11R32, 11R45 CHEBYSHEV

PSEUDO-SPECTRAL

METHOD-

A type of trigonometric pseudo-spectral method (cf. T r i g o n o m e t r i c p s e u d o - s p e c t r a l m e t h o d s ) . See also Fourier pseudo-spectral method.

A Chcbyshev polynomial is defined as T~(x) = cos(n cos -1 x) (cf. also C h e b y s h e v p o l y n o m i a l s ) . If x = cos0, the resulting Chebyshev function is truly an nth order p o l y n o m i a l in x, but it is also a cosine function with a change of variable. Thus, a finite Chebyshev series expansion is related to a discrete cosine transform. The Chebyshev pseudo-spectral method is the most logical choice of pseudo-spectral methods for problems with non-periodic boundary conditions. This comes from the particularly nice characteristics of the Chebyshev interpolating polynomials (cf. also C h e b y s h e v polynomials). Of all (N + 1)st degree polynomials, with leading coefficient 1, TN+I/2 N has the smallest maximum on the interval [-1, 1]. Thus, in Lagrangian interpolation (see also 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 ) , if the interpolation points are taken to be the zeros of this polynomial, the error is minimized. A related and possibly more useful set of interpolation points are the extrema of TN(X): x i = cosOrj/N), j = 0 , . . . , N , called the Gauss-Lobatto points. The trigonometric interpolation PNU of the function u at the Gauss-Lobatto points is PNU = E ; - o u(xj)Cj(x), where Cj is the Cardinal function T~v(x)(-1)J+I

Cj = (1 - x2) ~ - - x T ) ]

,

with T0 = CN = 2, ~j = 1 otherwise. Rearranging, the interpolation polynomial becomes a finite Chebyshev series PNu(x) = ~n=O g anTn(x), where the Chebyshev coefficients are N

an - N -d,~ .= u(xj Suppose the equation Lu = f is to be solved, where L is a differential operator, f is a given function and u is an unknown function. In the Chebyshev pseudo-spectral method, the solution u is approximated by a Chebyshev interpolating polynomial. In the Lagrangian polynomial or 'grid-point representation', the problem can be written as Li,juj -~ fi, where Li,j m nCj(x)lx=x~. The form of Li,j can be found through differentiation of the Cardinal function: Cj (xi) = 5i,j, [ 1 ( 1 + 2N2)

dCj ~xJ-~(1 + 2N 2) dx (xk) = | - - ~ -d~ . ( ( - 1 ) j + k ~j(xk-~j)

forj =k =0, for j = k = N, f o r j = k, 0 < j < N, f o r j # k,

and (dkCj/dxk)(xi) = [(dCj/dx)(xi)] k. Note that derivatives require O(N 2) operations. Another way to find an expression for the derivative is to differentiate the Chebyshev series, which means differentiating the Chebyshev polynomials and 101

CHEBYSHEV P S E U D O - S P E C T R A L M E T H O D using the recurrence relation for derivatives of Chebyshev polynomials, d(PNu)/dx = ~ nN = 0 b~TN(x), where bN = O, bN-1 = 2NAN, -dnbn = bn+2 + 2(n + 1 ) a n + l for 0 <_ n < N - 1. Thus, in coefficient representation, the derivative can be evaluated in O(N) operations while non-linear terms or multiplication by non-constant coefficients require O(N2). The fast discrete cosine transform can be used to switch between spectral and grid point representations. References [1] BOYD, J.P.: Chebyshev and Fourier spectral methods, second ed., Dover, 2000, pdf version: http://wwwpersonal.engin.umich.edu/~j pboyd/book_spectral2000 .html. [2] CANUTO, C., HUSSAINI, M.Y., QUARTERONI, A., AND gANG, T.A.: Spectral methods in fluid dynamics, Springer, 1987. [3] FORNBERG, B.: A practical guide to pseudospeetral methods, Vol. 1 of Cambridge Monographs Appl. Comput. Math., Cambridge Univ. Press, 1996. [4] GOTTLIEB, D., HUSSAINI, M.Y., AND OHSZAG, S.A.: 'Theory and application of spectral methods', in R.G. VOIGT, D. GOTTLIEB, AND M.Y. HUSSAINI (eds.): Spectral Methods for Partial Differential Equations, SIAM, 1984. [5] C-OTTLIEB, D., AND ORSZAG, S.A.:

Numerical

analysis of

spectral methods: Theory and applications, SIAM, 1977.

Richard B. Pelz MSC 1991: 42A10, 41A10, 41A50 CHOQUET

- Let (X,,4) be a m e a s u r -

INTEGRAL

a b l e space. Let m: .d --+ [0, oc] be a monotone set function (cf. also Set f u n c t i o n ) on A, vanishing at the empty set, m(0) = 0. Let f be a non-negative m e a s u r a b l e f u n c t i o n and A • A. The Choquet integral of f on A with respect to m is defined by

(C)

f dm=

/0

m(A n F,) da,

where the right-hand side is an improper integral and F~ = {x: f(x) >_ a} is the a-cut of f , [1], [2], [6]. Specially, let f be a simple measurable non-negative funcn tion on (X,A), f = ~i=1 aiXA~, 0 < al < "" < an, {Ai}~'= 1 C A and Ai N Aj = 0 whenever i ¢ j. One can rewrite f in the following form:

• If fl _< f2 on A, then (C) f A f l dm <_ (C) fA f2 dm. • For co-monotone functions fl and f2, i.e., (fl (x) fl(y))" (f2(x) - f2(y)) _> 0 for all x,y E X , one has (C)/(fl-t-f2)dm=

(C)/fldm~(C)/f2dIn.

For other properties of the Choquet integral, see [2], [6], [7]. R e l a t e d i n t e g r a l s a n d g e n e r a l i z a t i o n s . Let f be a non-negative extended real-valued measurable function on (X, A, m) and A E A. The Sugeno integral [8] of f on A with respect to m is defined by (S) f f d m = JA

sup [ a A m ( A n f , ) ] , aC[o,+oo]

where F~ = {x: f(x) _> a}, a • [0, +oc]. The restrictions of Choquet-like integrals to the unit interval [0, 1] (both for functions and for fuzzy measures) are a special case of the more general t-conorm integrals defined in [31, [41, [5]. References [1] CHOQUET, G.: 'Theory of capacities', Ann. Inst. Fourier

(Grenoble) 5 (1953), 131-295. [2] DENNEBERG,D.: Non-additive measure and integral, Kluwer Acad. Publ., 1994. [3] GRABISCH, M., NGUYEN, H.T., AND WALKER, E.A.: Fundamentals of uncertainity calculi with application to fuzzy inference, Kluwer Acad. Publ., 1995. [4] MESIAR, R.: 'Choquet-like integrals', J. Math. Anal. Appl. 194 (1995), 477-488. [5] MUROFUSHI, T., AND SUGENO, M.: ~A theory of fuzzy measures. Representation, the Choquet integral and null sets', J. Math. Anal. Appl. 159 (1991), 532-549. [6] PAP, E.: Null-additive set functions, Kluwer Acad. Publ./Ister, 1995. [7] SCHMEIDLER,D.: 'Integral representation without additivity', Proc. Amer. Math. Soe. 97 (1986), 253-261. [8] SUGENO, M.: 'Theory of fuzzy integrals and its applications', PhD Thesis Tokyo Inst. Technol. (1974).

E. Pap MSC 1991: 28-XX CLARKE

GENERALIZED

DERIVATIVE

-

G e n e r a l i z e d derivatives, normals and tangent cones are

f = VaixB~,

Bi=~JAi. j=i

i=1

Then n

(C) /.. f d m = J 2~

E(ai

-- ai_l)II~(J~i),

i=1

where a0 = 0. Note that for a m e a s u r e m (i.e., for a a-additive measure) the L e b e s g u e i n t e g r a l and the Choquet integral coincide. The Choquet integral has the following properties:

• (C) f A f d m = ( C ) f f .

XAdm.

• For any constant a • [0, t e e [ , (C) f a .

a. (c) f f 102

f dm=

used in non-smooth analysis, a body of theory concerned with the calculus of functions and sets that do not admit linear approximations in the sense of the customary derivative or the usual tangent space. The growth of the subject from the 1970s onwards reflects the essential need to grapple with non-smoothness (together with absence of convexity) in such topics as optimization, control, viscosity solutions, optimal design, variational methods, and in fact, in non-linear analysis generally. Certain characteristics have become standard in all types of non-smooth analysis: the presence of a calculus of subgradients and normal vectors, in tandem with constructs dual to these (directional derivatives, tangents),

CLOSED M O N O I D A L C A T E G O R Y and especially the placement on an equal footing of sets and of functions, which lends to the subject a distinctly geometric flavour. This p a t t e r n was established in the early 1970s with Clarke's calculus of proximal normals and generalized gradients, and implies that either geometric or functional constructs can be chosen as the basic notions from which the others are derived. The basic definition in a H i l b e r t s p a c e setting can be taken to be t h a t of OFf(x), the set of proximal subgradients of a function f : H --+ R U {oo} at a point x: C belongs to Opf(x) if there exists a ~r > 0 such that

f(y)-f(x)+~lly-xll

2 > (~,Y-X),

Vynearx.

[6] ROCKAFELLAR, R.T., AND WETS, R.: Variational analysis, Springer, 1998.

F. Clarke M S C 1991:90C30 CLOSED MONOIDAL CATEGORY - A category C

is monoidal if it consists of the following data: 1) 2) 3) 4) A1) all a,

a category C; a b i f u n c t o r @: d x C -+ d; an object e E C; and three natural isomorphisms a, ),, p such t h a t a = aa,b,c: a ® (b @ c) = (a ® b) @ c is natural for b, c E C and the diagram

a ® (b ® (c ® d)))

When f is lower semi-continuous (cf. also S e m i c o n t i n u o u s f u n c t i o n ) , Opf(x) is non-empty for a dense set of x, and there is a useful calculus associated with it. In the setting of a B a n a c h s p a c e X, in the case of a locally Lipschitz function (cf. also L i p s c h i t z c o n d i t i o n ) , the consideration of the generalized directional derivative given by

((a ® b) ® c) ® d

a®((b®c)®d)

~+

(a~(b(~c))®d

commutes for all a, b, c, d E d; A2) A and p are natural and A~: e®a -~ a, p~: a g e ~- a for all objects a E C and the diagram

ae(e®c)

-5 (a®e)®e

4id @a

.~p®id

t

a@c

a set which is non-empty, convex and compact. The calculus rules for various generalized derivatives become more precise if the d a t a are regular (for example, smooth or convex). Regarding sets, a basic tool is the (Clarke) tangent cone to a set S at x; it consists of those v which satisfy: For all sequences {xi} in S converging to x, for all sequences {ti} decreasing to 0, there exists a sequence {vi} converging to v such that xi + tivi E S. Other constructs of interest include Dini subderivates, generalized Jacobians, viscosity subdifferentials, and various normal cones. The relationships between the different theories are well understood; together with a variety of applications and other references, they can be found in, e.g., [5]. References [Z] AUBIN, J-P., AND FRANKOWSKA, H.: Set-valued analysis, Birkhguser, 1990. [2] CLARKE, F.: 'Generalized gradients and applications', Trans. Amer. Math. Soc. 205 (1975), 247-262. [3] CLARKE, F.: Optimization and nonsmooth analysis, Wiley/Interscienee, 1983. [4] CLARKE, F.: Methods of dynamic and nonsmooth optimization, Vol. 57 of C B M S / N S F Regional Conf. Ser. Appl. Math., SIAM, 1989.

YU., STERN, R., AND WOLENSKI,

=

a®c

commutes for all a, b, c, d C C; A3) ,~e = Pc: e @ e --+ e.

Of(x) := {~: f ° ( x ; v ) _> ( ~ , v ) , W C X } ,

F., LEDYAEV,

•

~.cz®id

leads to the set Of(x) of generalized gradients:

[5] CLARKE,

(a ® b) ® (c ® d) =

f°(x;v) := liminf f ( y + tv) - f(y) y-+x,t$O

~

.~id ®a

P.:

Nonsmooth analysis and control theory, Vol. 178 of Graduate Texts in Math., Springer, 1998.

These axioms imply that all such diagrams commute. Some examples of monoidal categories are: E l ) any category with finite products is monoidal if one takes a ® b to be the (chosen) product of the objects a and b, with e the terminal object; a, ~ and p are the unique isomorphisms t h a t commute with the appropriate projections; E2) the usual 'tensor products' give monoidal categories - - whence the notation. Note that one cannot identify all isomorphic objects in d. C l o s e d c a t e g o r i e s . A monoidal category C is said to be symmetric if it comes with isomorphisms %,b: a @ b ~ b ® a natural on a, b E C such that the following diagrams all commute: ~/a,b ° ~b,a = 1, Pb = ~b ° ~b,e:

a@(b@c)

b ® e ~- b:

~+ ((a@b)@e)

-~

~id ®~

a®(c®b)

(c@(a®b)) ~a

Z~

((a@c)®b)

2+

(c®a)®b

A closed category Y is a symmetric monoidal category in which each functor - ® b : ]2 -~ y has a specified right-adjoint (.)b: P -+ )2. Some examples of closed monoidal categories are: 103

CLOSED M O N O I D A L

CATEGORY

E3) the category 74cg of relations, whose objects are sets a, b, c,... and in which an arrow a -+ b is a subset cr C_ a ® b; the object a ® b is the Cartesian product of the two sets, which is not the product in this category; E4) the subsets of a monoid M (a poset, hence a category); if A, 13 are two subsets of M, then A ® 13 is {ab: a E A, b E B} while C A is {b C M : ab C

C for all a C A}. References [1] BARR, M., AND WELLS, C.: Category theory for computing science, CRM, 1990. [2] MACLANE, S.: Categories for the working mathematician, Springer, 1971.

[10] and [12]; these also contain numerical analyses concerning the production, costs and work productivity. Mathematical methods and models of long-run growth (in particular, the Solow model) using an enlarged production function are given in [15]. Macro-economical models regarding investment spending and the rental cost capital, as well as the real rate of interest, using a 'production function' are given in [5]. Problems regarding comparative relations of a few national economies have been studied with production functions of the form Q = f(L, N, K, P),

Michel Eytan

(4)

MSC 1991:18D10 C O B B - D O U G L A S FUNCTION - In a mathematical setting, the Cobb-Douglas function is defined as (see [2]): Q = AK~L 1-", (1) where A is a positive constant and c~ is a positive fraction. The primary application of the Cobb-Douglas function has been in agriculture and industrial production. This is the reason that Q, K and L are usually named 'output', 'capital' and 'used labour power to get the output Q'. Setting k = K / L , one may express A as A = cOQ. _1. k l _ a , OK a or

(2)

A=0Q __1

OL 1 - a For given values for K and L, the magnitude of A will proportionately affect the level of Q. Hence A may be considered as an efficient parameter, i.e. as an indicator of the state of technology. The function defined by (1) is homogeneous concerning the factor variables and linear by some logarithm application. Ch.W. Cobb and P.H. Douglas published their article [3] in 1928, and one of its applications has been described in [8] (see also [7]). In [9], a detailed description is given of a mathematical model of the 'production and consumption variation under uncertainty in a one-sector economy', resulting in the equation dK(t) at - F (~;(t), L(t)) - ~ K ( t ) - C(t),

(3)

where ~ is the rate of capital depreciation and C(t) is the aggregate rate of consumption. In (3), F is the production function, and d K / d t is instantaneously determined when K, L and C are known. For applications of statistical and mathematical models using the 'production function', for constructing optimal trajectories in a medium- and long-term run, see 104

where Q, L, K have been defined previously, N represents the natural resources of the nation, and P is the technical progress; see, e.g., [4]. A computer programruing solution tool for statistical and production functions can be found in [6]. The Cobb-Douglas function, regarded as a utility function, and preference functions with applications in micro-economics are given in [14]. The theory and applications of production functions with multiple inputs and of different shapes, not only as a factor products, are well explained in [11]. Mathematical models in insurance and risk theory, constructed using a 'production function' with imperfect and asymmetric information, are very interesting; see [1], [13]. References [1] BOYER, M., DIONNE, G., AND KHILSTROM, l~.: 'Insurance and the value of publicly available information', in T. BAND FOMBY AND TAEKUN SEO (eds.): Studies in the Economics of Uncertainty, 1989. [2] CHANG, A.C.: Fundamental methods of mathematical economics, McGraw-Hill, 1984. [3] COBB, CH.W., AND DOUGLAS, P.H.: 'Theory of production', Amer. Economic Review M a r c h (1928). [4] DIOURY, M.: l~conomie internationale, D~carie t~diteur Montr4al, 1998. [5] DORNBUSH,R., FISHER, S., AND SPARKS, G.: Macro economics, McGraw-Hill Ryerson, 1982. [6] FOGIEL, M.: The statistics problem solver, REA NYNY, 1983. [7] HAYNES, W.W.: Managerial economics: Analysis and cases, The Dorsey Press, 1963. [8] HEADY, E., AND DILLON, J.: Agricultural production functions, Iowa State Univ. Press, 1961. [9] I£ARLIN, S., AND TAYLOR, H.M.: Some stochastic differential equation models, Vol. 2, Acad. Press, p. Chapt 15: Diffusion Processes. [lO] LIPSEY, R.G., AND STEINER, P.O.: Economics, Harper and Row, 1969. [11] SEO, K.K., AND WINGER, B.J.: Managerial economics, Richard D. Irwin, 1979. [12] TODOR, F.: 'Trajectoires optimales dans l'4conomie du Quebec et du Canada', Gazette Sci. Math. Qudbec 13, no. 3 (1990).

C O H E R E N T ALGEBRA [13] TODOR, F.: ' S u r u n mod61e mathematique appliqu6 ~ l'6tude du risque avec l'information imparfaite et asymetrique', Anal. Univ. Ovidiu Constantza Romania Ser. Mat. (1993). [14] VARIAN, H.R.: Intermediate microeconomics: a modern approach, Norton, 1987. [15] WILLIAMS,H.R., AND HUFFNAGLE,J.D.: Macroeconomie theory: Selected readings, Meredith Corp., 1969.

Fabian Todor MSC 1991: 9 0 A l l C O H E R E N T ALGEBRA - Algebras introduced by D.G. Higman, first in relational language under the name coherent configuration [4] and later in terms of matrices [6]. The slightly different axiomatics of cellular algebras were independently suggested by B.Yu. Weisfeiler and A.A. Leman (cf. also C e l l u l a r a l g e b r a ) . Like association schemes (cf. also A s s o c i a t i o n s c h e m e ) and Bose-Mesner algebras, coherent algebras provide a wide and solid foundation for investigations in various areas of algebraic combinatorics. A coherent algebra W of order n and rank r is a matrix subalgebra of the full matrix algebra C ~×n of (n x n)-matrices over C such that:

• W is closed with respect to the Hermitian adjoint, which is defined by A* = (ai,j)* = ( ~ ) for A = (ai,j) • W; • I • W, where I is the unit matrix; • J • W, where J is the all-one matrix; • W is closed with respect to Sehur-Hadamard multiplication o, where A o B = (ai,jbi,j) for A = (ai,j), B = (bi,j), A, B • W. Each coherent algebra W has a unique basis of zero-one matrices { A 1 , . . . , At} such that: 1) ~ i ~ 1 A~ = J; 2) V1 < i < r 31 _< j _< r : AiT = Aj, where AT is the matrix transposed to Ai; Property 1) implies that the basis { A I , . . . , A~} consists of mutually orthogonal idempotents with respect to the Schur-Hadamard product. This basis is called the standard basis of W. The non-negative integer structure constants pk,j are important numerical invariants of W. The notation W = ( A 1 , . . . , A~) indicates that W is a coherent algebra with standard basis { A 1 , . . . , A~}. Let X = { 1 , . . . , n } and denote b y R l = {(i,j): ai,j = 1} a binary relation over X. R1 is called the support of the zero-one matrix At = (ai,j) (or, in other words, At is the adjacency matrix of the graph Pt = (X, Rl) with vertex set X and set Rz of directed edges). The system of relations 9)I = (X, {Ri}I
following combinatorial interpretation: V(x, y) • Rk : k p ,j = t{z • x :

• Ri

(z,y) • R }I.

A coherent configuration 97~ is called homogeneous if one of its basic relations, say R1, coincides with the diagonal relation A = {(x, x): x • X}. In terms of matrices, a coherent algebra W = ( A 1 , . . . , At} is called a Bose-Mesner algebra (briefly BM-algebra) if A1 = I. Note that according to E. Bannai and T. Ito [1], a homogeneous coherent configuration is also called an association scheme (not necessarily commutative; cf. also Association scheme). Let 9)I = (X, {Ri}l_
RiCM 2VRi•M

2=0

(1)

and M is a minimal (with respect to inclusion) subset satisfying condition (1). The coherent algebras with one fibre are exactly the BM-algebras. Coherent algebras with few fibres may be used for a unified presentation and investigation of various combinatorial objects, see, for example, [3], [7], [9]. An important class of coherent algebras consists of the centralizer algebras of permutation groups (not necessarily transitive)[2], [10] (cf. also P e r m u t a t i o n g r o u p ; C e n t r a l i z e r ) . This leads to many important applications of coherent algebras. It was Higman [5], [8] who developed the foundations of the representation theory of coherent algebras as a generalization of the representation theory of finite permutation groups (cf. also F i n i t e g r o u p , r e p r e s e n t a t i o n o f a). References [1] BANNAI, E., AND ITO, T.: Algebraic combinatorics, Vol. I, Benjamin/Cummings, 1984. [2] FARAD~.EV, I.A., KLIN, M.H., AND MUZICHUK, M.E.: 'Cellular rings and groups of automorphisms of graphs', in I.A. tPARADZEVET AL. (eds.): Investigations in Algebraic Theory of Combinatorial Objects, Kluwer Acad. Publ., 1994, pp. 1152. [3] HAEMERS, W.H., AND HIGMAN, D.G.: 'Strongly regular graphs with strongly regular decomposition', Linear Alg. Its Appl. 1 1 4 / 1 1 5 (1989), 379-398. [4] HIGMAN,D.G.: 'Coherent configurations I', Rend. Sere. Mat. Univ. Padova 44 (1970), 1-25. [5] HIGMAN, D.G.: 'Coherent configurations, Part I: Ordinary representation theory', Geom. Dedicata 4 (1975), 1-32. [6] HIGMAN, D.G.: 'Coherent algebras', Linear Alg. ~4 Its Appl. 93 (1987), 209-239. [7] HIGMAN,D.G.: 'Strongly regular designs and coherent configurations of type (a ~),, Europ. d. Combin. 9 (1988), 411-422. [8] HIOMAN, D.G.: 'Computations related to coherent configurations', Congr. Numer. 75 (1990), 9-20. [9] MUZYCHUK, M.E., AND KLIN, M.: 'On graphs with three eigenvalues', Diser. Math. 189 (1998), 191-207.

105

COHERENT ALGEBRA [10] VV-IELANDT,H.: Finite permutation groups, Acad. Press, 1964.

Mikhail Klin MSC 1991: 03E05, 03Exx C O L O M B E A U G E N E R A L I Z E D F U N C T I O N ALGEBRAS - Let g} be an open subset of R n, and Iet 2)(~)

be the algebra of compactly supported smooth functions. In the original definition, J.F. Colombeau [2] started from the space doo(~D([~)) of infinitely Silvadifferentiable mappings from 2~(f~) into C. The space of distributions 7)'(ft) is just the subspace of linear mappings g)(~) -+ C. Let

J Aq(R ~ ) =

~ED(Rn):

dx = 1, }

fx%z(x) dx=O, forl_< Ic~l _
and let

The subalgebra EM(D(f~)) is defined by those members R such that for all compact subsets K C f~ and for all multi-indices c~ E N~ there is an N E N such that for all %oE AN(R'~), the supremum of IO~*R(pc,~)I over x E K is of order O(e -N) as e -~ 0. The ideal JV(2)(f~)) is defined by those members R such that for all compact subsets K C f~ and all a E N~ there is an N E N such that for all q _> N and %o E J [ q ( R n ) , the supremum of [O~R(p~,x)l over x E K is of order O(g q-N) as e -+ 0. The Colombeau generalized function algebra is the factor algebra /M(2)(ft))/A/'(D(f~)). It contains the space of distributions 7?'(f~) with derivatives faithfully extended (cf. also G e n e r a l i z e d f u n c t i o n , d e r i v a t i v e o f a). The asymptotic decay property expressed in A/(~D(f~)) together with an argument using Taylor expansion shows that Coo(ft) is a faithful subalgebra. Later, Colombeau [3], [4] replaced the construction by a reduced power of Coo(f~) with index set h = (0, oc): Let CM(f~) be the algebra of all nets (us)e>0 C Coo(f~) such that for all compact subsets K C f~ and all multiindices ct E N~ there is an N _> 0 such that the supremum of IO%~(x)l over x E K is of order O(e -N) as e -+ 0 (cf. also N e t ( d i r e c t e d set)). Let H(f~) be the ideal therein given by those (u~)~>0 such that for all compact subsets K C f~, all c~ E N~ and all q > 0, the supremum of Ic9~uE(z)l over x E K is of order O(e v) as ¢ + 0. Then set g ( 9 ) = gv(f~)/]~f(f}). There exist versions with the infinite-order S o b o l e v s p a c e W~,P(f~) in the place of Coo(f~), 1 <_ p _< oc, or with other topological algebras. It is possible to enlarge the class of mollifiers (hence the index set A in the reduced power construction) to produce a version for which smooth coordinate changes commute with the imbedding of distributions. This way 106

Colombeau generalized functions can be defined intrinsically on manifolds. Generalized stochastic processes with paths in ~(f~) have been introduced as well. The subalgebra G°°(ft) is defined by interchanging quantifiers: For all compact sets K C ft there is an N _> 0 such that for all c~ E N~, the supremuln of IOc~uE(z)] on K is of order O(g - N ) as e --+ 0. Ogle has that ~oo(f~) n 2)'(ft) = Coo(f~), and ~oo(f~) plays the same role in regularity theory here as C °°(ft) does in distribution theory (for example, u E ~(ft) and Au E Goo(ft) implies u E Goo(f~), where A denotes the Laplace operator).

For applications in a variety of fields of non-linear analysis and physics, see [1], [4], [5], [6], [7]. See also G e n e r a l i z e d f u n c t i o n a l g e b r a s . References [1] BIAOIONI,H.A.: A nonlinear theory of generalized functions, Springer, 1990. [2] COLOMBEAU,J.F.: New generalized functions and multiplication of distributions, North-Holland, 1984. [3] COLOMBEAU,J.F.: Elementary introduction to new generalized functions, North-Holland, 1985. [4] COLOMBEAU,J.F.: Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics, Springer, 1992. [5] GuossEa, M., HORMANN, G., KUNZINGER, M., AND OBERGUGOENBEaGEa, M. (eds.): Nonlinear theory of generalized functions, Chapman and Hall/CRC, 1999. [6] NEDELJKOV, M., PILIPOVIC, S., AND SCAaPAL~ZOS, D.: The linear" theory of Colombeau generalized functions, Longman, 1998. [7] OBERGUGGENBERGER,M.: Multiplication of distributions and applications to partial differential equations, Longman, 1992.

Michael Oberguggenberger MSC 1991:46F30 COMPUTATIONAL

COMPLEXITY

CLASSES

-

Computational complexity measures the amount of computational resources, such as time and space, that are needed to compute a function. In the 1930s many models of computation were invented, including Church's A-calculus (cf. A-calculus), Ghdel's recursive functions, Markov algorithms (cf. also A l g o r i t h m ) and Turing machines (cf. also T u r i n g m a chine). All of these independent efforts to precisely define the intuitive notion of 'mechanical procedure' were proved equivalent. This led to the universally accepted Church thesis, which states that the intuitive concept of what can be 'automatically computed' is appropriately captured by the Turing machine (and all its variants); cf. also C h u r c h thesis; C o m p u t a b l e f u n c t i o n . Computational complexity studies the inherent difficulty of computational problems. Attention is confined to decision problems, i.e., sets of binary strings, S C_ E*,

COMPUTATIONAL C O M P L E X I T Y CLASSES where E = {0, 1}. In a decision problem S the input binary string w is accepted if w E S and rejected if w ¢f S (cf. also D e c i s i o n p r o b l e m ) . For any function t(n) from the positive integers to itself, the complexity class DTIME[t(n)] is the set of decision problems that are computable by a deterministic, multi-tape Turing machine in O(t(n)) steps for inputs of length n.

Polynomial time (P) is the set of decision problems accepted by Turing machines using at most some oo polynomial number of steps, P = [.Jk=l DTIME[ nk] = DTIME[n°(1)]. Intuitively, a decision problem is 'feasible' if it can be computed with an 'affordable' amount of time and hardware, on all 'reasonably sized' instances. P is a mathematically elegant and useful approximation to the set of feasible problems. Most natural problems that are in P have small exponents and multiplicative constants in their running time and thus are also feasible. This includes sorting, inverting matrices, pattern matching, linear programming, network flow, graph connectivity, shortest paths, minimal spanning trees, strongly connected components, testing planarity, and convex hull. The complexity class NTIME[t(n)] is the set of decision problems that are accepted by a non-deterministic multi-tape ~l-hring machine in O(t(n)) steps for inputs of length n. A non-deterministic Turning machine may make one of several possible moves at each step. One says that it 'accepts an input' if at least one of its possible computations on that input is accepting. The time is the maximum number of steps that any computation may take on this input, not the exponentially greater number of steps required to simulate all possible computations on that input.

The complexity class DSPACE[s(n)] is the set of decision problems that are accepted by a deterministic Turing machine that uses at most O(t(n)) tape cells for inputs of length n. It is assumed that the input tape is read-only and space is only charged for the other tapes. Thus it makes sense to talk about space less than n. Similarly one defines the non-deterministic space classes, NSPACE[s(n)]. On inputs of length n, an NSPACE[s(n)] Turing machine uses at most O(s(n)) tape cells on each of its computations. One says it 'accepts an input' if at least one of its computations on that input is accepting. The main space complexity classes are polynomial space, PSPACE = DSPACE[n°(Z)]; Logspace, L = DSPACE[logn]; and non-deterministic logspace, NL = NSPACE[log n]. For each of the above resources (deterministic and non-deterministic time and space) there is a hierarchy theorem saying that more of the given resource enables one to compute more decision problems. These theorems are proved by diagonalization arguments: use the greater amount of the resource to simulate all machines using the smaller amount, and do something different from the simulated machine in each case. H i e r a r c h y t h e o r e m s . See also [12], [11], [4], [22]. In the statement of these theorems the notion of a space- or time-constructible function is used. Recall that a function s from the positive integers to itself is space constructible (respectively, time constructible) if and only if there is a deterministic Turing machine running in space O(s(n)) (respectively, time O(s(n))) that on every input of length n computes the number s(n) in binary. Every reasonable function is constructible. The following are hierarchy theorems: 1) For all space constructible s(n) > logn, if

t(n)

limo~ ~

= 0,

Non-deterministic polynomial time (NP) is the set of decision problems accepted by a non-deterministic Turing machine in some polynomial time, NP = NTIME[n°(Z)]. For example, let SAT be the set of Boolean formulas that have some assignment of their Boolean variables making them true (cf. also B o o l e a n a l g e b r a ) . The formula ~b - (xl Vx2) A (~-V~-ff)A (~i-Vz3) is in SAT because the assignment that makes zl and x3 true and x2 false satisfies q~. Given a formula with k Boolean variables, a non-deterministic machine can nondeterministically write a binary string of length k and then check if the assignment it has written down satisfies the formula, accepting if it does. Thus SAT 6 NP. The decision versions of many important optimization problems, including travelling salesperson, graph colouring, clique, knapsack, bin packing and processor scheduling, are in NP.

then DSPACE[t(n)] is strictly contained in DSPACE[s(n)], and NSPACE[t(n)] is strictly contained in NSPACE[s(n)]. 2) For all time constructible t(n) > n, if

t(n)

li~Inoo s - ~ = 0, then NTIME[t(n)] is strictly contained in NTIME[s(n)]. 3) For all time constructible t(n) >_n, if lim t(n)(log t(n))/s(n) = O, n-+oo

then DTIME[t(n)] is strictly contained in DTIME[s(n)] The slightly stronger requirement in the last of the above theorems has to do with the extra factor of time log(t(n)) that is required for a two-tape Turing machine to simulate a machine with more than two tapes. If one 107

C O M P U T A T I O N A L C O M P L E X I T Y CLASSES restricts attention to multi-tape Turing machines with a fixed number of tapes, then a strict hierarchy theorem for deterministic time results [7]. Much less is know when comparing different resources: One can simulate NTIME[t(n)] in DSPACE[t(n)] by simulating all the non-deterministic computations in turn. One can simulate DSPACE[s(n)] in DTIME[2 °(s(~))] because a DSPACE[s(n)] computation can be in one of at most 2 °('(n)) possible configurations. Savitch's theorem provides a non-trivial relationship between deterministic and non-deterministic space [21]: For s(n) > logn, DSPACE[s(n)] C_ NSPACE[s(n)] c DSPACE[(s(n))2]. For each decision problem S g E*, its complement = E* - S is also a decision problem. For each complexity class C, define its complementary class by

coe = {s:

e c}

Most natural complexity classes include a large number of interesting complete problems. See [9], [10], [18] for substantial collections of problems that are complete for NP, P, and NL, respectively. A non-deterministic machine can be thought of as a parallel machine with weak communication. At each step, each processor p may create copies of itself and set them to work on slightly different problems. If any of these offspring ever reports acceptance to p, then p in turn reports acceptance to its parent. Each processor thus reports the 'or' of its children. A. Chandra, D.C. Kozen and L. Stockmeyer generalized non-deterministic Turing machines to alternating Turing machines, in which there are 'or' states, reporting the 'or' of their children, and there are 'and' states, reporting the ~and' of their children. They proved the following alternation theorem [3]: For s(n) >_logn and

t(n) > n, ATIME[(t(n)) °(1)] = DSPACE[(t(n))°(1)]; ASPACE[s(n)] = DTIME[2°(s(n))].

For deterministic classes C, such as P, L and PSPACE, it is obvious that C = co C. However, this is much less clear for non-deterministic classes. Since the definitions of NTIME and NSPACE were made, it was widely believed that NP # c o N e and NSPACE[n] 7~ co NSPACE[n]. However, in 1987 the latter was shown to be false [16], [2@ In fact, the following Immerman-Szelepcsgnyi theorem holds: For s(n) > logn, NSPACE[s(n)] = co NSPACE[s (n)]. Even though complexity classes are all defined as sets of decision problems, one can define functions computable in a complexity class as follows. For a complexity class C, define F(C) to be the set of all polynomiallybounded functions f : E* --+ E* such that each bit of f (thought of as a decision problem) is in C and co C. One compares the complexity of decision problems by reducing one to the other as follows. Let A, B C_ E*. A is reducible to /3 (A _< B) if and only if there exists a function f E F(L) such that for all w E E*, w E A if and only if f(w) E B. The question whether w is a member of A is thus reduced to the question of whether f(w) is a member of B. If f(w) E B, then w E A. Conversely, if f(w) f[ B, then w ¢ A. The complexity classes, i.e., L, NL, P, NP, co NP, and PSPACE, are all closed under reductions in the following sense: If C is one of the above classes, A _< B, and B E C, then A E C. A problem A is complete for complexity class d if and only if A E C and for all B E C, B _< A. If C is closed under reductions, then any complete problem, say A, characterizes the complexity class, because for all /3, B E C implies B < A. 108

In particular, ASPACE[logn] = P and ATIME[n °(1)] = PSPACE. Let the class ASPACETIME[s(n), t(n)] (respectively, ATIMEALT[t(n), a(n)]) be the set of decision problems accepted by alternating machines simultaneously using space s(n) and time t(n) (respectively, time t(n) and making at most a(n) alternations between existential and universal states, and starting with existential). Thus ATIMEALT[n ° 0 ) , 1] = NP. The polynomial-time hierarchy (PH) is the set of decision problems accepted in polynomial time by alternating Turing machines making a bounded number of alternations between existential and universal states, PH = ATIMEALT[n °(i), O(1)]. NC is the set of decision problems recognizable by a p a r a l l e l r a n d o m a c c e s s m a c h i n e (a PRAM) using polynomially much hardware and parallel time (log n) °(i). NC is often studied using uniform classes of acyclic circuits of polynomial size and depth (log n) °(1). NC is characterized via alternating complexity in the following way, NC = ASPACETIME[log n, (log n)o(i)]. A decision problem is complez if and only if it has a complex description. This leads to a characterization of complexity via logic. The input object, e.g., a string or a graph, is thought of as a (finite) logical structure. R. Fagin characterized NP as the set of second-order expressible properties (NP = SO(3)), N. Immerman and M.Y. Vardi characterized P as the set of properties expressible in first-order logic plus a least-fixed-point operator (P = FO(LFP)). In fact, all natural complexity classes

C O N D O R C E T JURY T H E O R E M have natural descriptive characterizations [6], [15], [25],

[5], [17]. A probabilistic Turing machine is one that may flip a series of fair coins as part of its computation. A decision problem S is in bounded probabilistic polynomial time (BPP) if and only if there is a probabilistic polynomialtime Turing machine M such that for all inputs w, if (w C S), then P ( M accepts w) > 2/3, and if (w ¢ S), then P(M accepts w) _< 1/3. It follows by running M(w) polynomially many times that the probability of error can be made at most 2 -~k for any fixed k and inputs of length n. It is believed that P = BPP. An important problem in B P P that is not known to be in P is primality, [23]. Recent work (1998) on randomness and cryptography has led to a new and surprising characterization of NP via interactive proofs [2]. This in turn has led to tight bounds on how closely many NP optimization problems can be approximated in polynomial time, assuming that P ¢ NP [14]. It is known that NC C DSPACE[(logn)°(1)], so by the space hierarchy theorem NC (and thus of course NL and L) is strictly contained in PSPACE. Strong lower bounds have been proved on some circuit complexity classes below L [1], [8], [13], [20]; but even now (2001), thirty years after the introduction of the classes P and NP, no other inequality concerning the following containments, including that L is not equal to PH, is known: L c_ NL C_ NC _c P c_ NP c PH _c PSPACE. For further reading, an excellent textbook on computational complexity is [19].

References [1] AJTAI, M.: 'E 1 formulae on finite structures', Ann. Pure Appl. Logic 24 (1983), 1-48. [2] ARORA, S., LUND, C., MOTWANI, R., SUDAN, M., AND SZEGEDY, M.: 'Proof verification and the hardness of approximation problems', J. Assoc. Comput. Mach. 45, no. 3 (1998), 501-555. [3] CHANDRA, A., AND STOCKMEYER, L.: 'Alternation': Proc. 17th IEEE Symp. Found. Computer Sci., 1976, pp. 151-158. [4] COOK, S.A.: 'A hierarchy for nondeterministic time complexity', J. Comput. Syst. Sci. 7, no. 4 (1973), 343-353. [5] EBBINGHAUS, H.-D., AND FLUM, J.: Finite model theory, Springer, 1995. [6] FAGIN, R.: 'Generalized first-order spectra and polynomialtime recognizable sets', in R. KAaP (ed.): Complexity of Computation SIAM-AMS Proc., Vol. 7, 1974, pp. 27-41. [7] F/JRER, M.: 'The tight deterministic time hierarchy': l~th ACM STOC Syrup., 1982, pp. 8-16. [8] FURST, M., SAXE, J.B., AND SIPSER, M.: 'Parity, circuits, and the polynomial-time hierarchy', Math. Systems Th. 17 (1984), 13-27. [9] GAREY, M.R., AND JOHNSON, D.S.: Computers and intractability, Freeman, 1979.

[10] GREENLAW, R., HOOVER, H.J., AND RUZZO, L.: Limits to parallel computation: P-completeness theory, Oxford Univ. Press, 1995. [11] HARTMANIS, J., LEWIS, P.M., AND STEARNS, R.E.: 'Hierarchies of memory limited computations': Sixth Ann. IEEE Symp. Switching Circuit Theory and Logical Design, 1965, pp. 179-190. [12] HARTMANIS, J., AND STEARNS, R.: 'On the computational complexity of algorithms', Trans. Amer. Math. Soc. 117, no. 5 (1965), 285-306. [13] HASTAD, J.: 'Almost optimal lower bounds for small depth circuits': 18th ACM STOC Syrup., 1986, pp. 6-20. [14] HOCHBAUM,D. (ed.): Approximation algorithms for NP hard problems, PWS, 1997. [15] IMMERMAN,N.: 'Relational queries computable in polynomial time': 14th ACM STOC Symp., 1982, pp. 147-152, Revised version: Inform. &: Control 68 (1986), 86-104. [16] IMMERMAN,N.: ~Nondeterministic space is closed under complementation', SIAM J. Comput. 17, no. 5 (1988), 935-938. [17] IMMERMAN, N.: Descriptive complexity, Graduate Texts in Computer Sci. Springer, 1999. [18] JONES, N., LIEN, E., AND LAASER, W.: 'New problems complete for nondeterministic logspace', Math. Systems Th. 10 (1976), 1-17. [19] PAPADIMITRIOU,C.H.: Complexity, Addison-Wesley, 1994. [20] RAZBOROV, A.A.: 'Lower bounds on the size of bounded depth networks over a complete basis with logical addition', Math. Notes 41 (1987), 333-338. (Mat. Zametki 41 (1987), 598-607.) [21] SAVITCH, W.: 'Relationships between nondeterministic and deterministic tape complexities', J. Comput. Syst. Sci. 4 (1970), 177-192. [22] SEIFERAS, J.I., FISCHER, M.J., AND MEYER, A.R.: 'Refinements of nondeterministic time an space hierarchies': Proc. Fourteenth Ann. IEEE Syrup. Switching and Automata Theory, 1973, pp. 130-137. [23] SOLOVAY, R., AND STRASSEN, V.: 'A fast Monte-Carlo test for primality', SIAM J. Comput. 6 (1977), 84-86. [24] SZELEPCSI%,NYI,R.: 'The method of forced enumeration for nondeterministic automata', Acta Inform. 26 (1988), 279284. [25] VARDI, M.Y.: 'Complexity of relational query languages': 14th Symp. Theory of Computation, 1982, pp. 137-146. Nell Immerman

MSC1991: 68Q15, 03D15 CONDORCET JURY T H E O R E M - M.J.A.N. de Caritat, Marquis de Condorcet, studied the mathematical problem of how best to combine the opinions of several individuals so as to form a group decision. Under Rousseau's theory of the general will, it is assumed that all group members wish to obtain what is best for the group; the problem is that they differ as to their opinion of what the best decision should be. The situation is best exemplified nowadays by a trial jury, in which all members have the same desire - - to convict a guilty party, and acquit an innocent one - - but have different opinions as to the accused party's innocence or guilt (cf. also S o c i a l choice). 109

CONDORCET

JURY

THEOREM

Condorcet makes the simplifying assumption that all the individuals have equal competence (probability of making the correct choice), that this competence is greater than 0.5, and that these probabilities are independent (cf. also Independence). He also assumes that there are only two alternatives available. Moreover, he implicitly assumes that, for each individual, the probability of a type-I error (convicting an innocent man) is the same as that of a type-If error (freeing a guilty man); see also Statistical test. Under these circumstances, it is not difficult to prove that the decision of a majority of the voters is more likely to be correct than that of the minority. Moreover, as the number of jury members increases, the probability that the group majority will make the correct decision approaches I. It is reasonable to look for modifications of the assumptions. The easiest modification assumes that different individuals have different levels of competence: each individual, i, has probability Pi of making the correct choice. In this case, the probability of a correct group decision is maximized by weighted voting, in which individual i is give a weight wi proportional to the logarithm of p i / ( 1 - Pi).

Another modification assumes different probabilities for type-I and type-II errors. This can be handled by, essentially, giving an artificial advantage to one of the two sides: alternative A will be chosen if its vote tally surpasses t h a t of B by a sufficiently large margin. A more complicated modification assumes that the different individuals' competences are not independent. In this case it is still possible, on the basis of voting, to decide which alternative is more likely to be correct, but the formulas for this can be quite complex. Finally, Condorcet tried to generalize the method to the case of three or more alternatives. In this case, he found that his method can easily lead to contradictions: this is known as the C o n d o r c e t p a r a d o x . References

[1] CONDOReET,N.C. DE: Essai sur l'application de l'analyse gL la probabilitd des ddcisions rendues d la pluralitd des voix,

Paris, 1785. [2] GROFMAN,B.: 'Judgmental competence of individuals and groups in a dichotomous choice situation', J. Math. Sociology 6 (1978), 4~60. [3] NITZAN,S., AND PAROUSH,J.: 'Optimal decision rules in uncertain dichotomous choice situations', Internat. Economic Review 23 (1982), 289-297. [4] SHAPLEY,L.S., AND GROFMAN,B.: 'Optimizing group judgmental accuracy in the presence of interdependencies', Public Choice 43 (1984), 329-343. Guillermo Owen

MSC 1991:90A28

110

CONDORCET PARADOX M.J.A.N. de Caritat, Marquis de Condorcet, studied the problem of determining the most likely correct choice, under voting by a group of decision-makers. In this, his work is closely related to that of J.-Ch. Borda. In the case of dichotomous choice (two alternatives), Condorcet obtained valuable results (see also Condorcet jury theorem), which have been extended in recent times (as of 2000). For the case of three or more alternatives (candidates), however, serious difficulties occur (see also Social choice). Briefly, one can say that candidate A defeats candidate B if a majority of the voters prefer A to B. With only two candidates, there is little more to say: barring ties (which are assumed to have extremely low probability), one of the two candidates will defeat the other. Where there are three or more candidates, however, cyclic situations might occur, wherein A defeats B, who defeats C, who in turn defeats A. It is of course possible that one candidate defeats all the others; if so, this candidate is said to be the Condorcet winner. However, even if there is a Condorcet winner, standard methods of voting need not produce this winner. This is Condorcet's paradox (cf. also V o t i n g p a r a d o x e s ) . The easiest method of voting is, of course, straight plurality voting. Another might be t w o - r o u n d voting,

with a second round only if there is no majority on the first round. A third, more sophisticated method, is the following: voters state their first choice. If no candidate has a majority of the votes, then that candidate with the least votes is eliminated, and voters are asked to choose among the remaining candidates. These steps are repeated until one candidate is left with a majority. This method is frequently used in elections. With any of these three methods, suppose there are three candidates, A, B and C, and nine voters. Suppose two voters rank the candidates A, B, C; three rank them B, A, C; and four rank them C, A, B. In this case, C would be the winner under straight plurality voting. For either of the other two methods, A (with only two votes) would be eliminated in the first round; B would then defeat C in the second round. Note, however, that A defeats B by 6 votes to 3, and also defeats C by 5 votes to 4. Thus A, the first eliminated, is the Condorcet winner. An alternative idea is to count the number of votes that a candidate would have in one-on-one contests against each of the other candidates. This is known as the Borda c o u n t (cf. also V o t i n g p a r a d o x e s ) , and Borda suggested t h a t the candidate with highest count should be the winner. Again, the Condorcet winner (if

CONLEYINDEX one exists) need not be the Borda winner. As an example, suppose there are three candidates, and 11 voters, with rankings as follows: • • • •

5 1 2 3

voters rank voter ranks voters rank voters rank

A, B, C, C,

C, A, A, B,

B; C; B; A.

In this case, A defeats both B (by 7 votes to 4) and C (by 6 to 5). Thus A is the Condorcet winner, and has a Borda count of 13. However, C defeats B by 10 votes to 1, and so C's Borda count is 15. Thus, C is the Borda winner. In general, the problem of group decision when there are three or more alternatives leads to contradictions. These are best summarized in the A r r o w i m p o s s i b i l i t y t h e o r e m , which states (briefly) that there is no method (for such decisions) satisfying certain eminently reasonable axioms. References

[1] ARROW,K.J.: Social choice and individual values, Vol. 12 of Cowles Commission Monograph, Wiley, 1951. [2] BORDA,J.-Cm: 'Sur la forme des ~lections au scrutin', Mdm. Acad. [loyal Sci. Paris (1781/4), 657-665. [3] CONDORCET, N.C. DE: Essai sur l'application de l'analyse it la probabilitd des ddcisions rendues d la pluralitd des voix, Paris, 1785. Guillermo Owen MSC 1991:90A28 A tool to analyze the dynamics of continuous or discrete dynamical systems. It can be used, for instance, to find special orbits like stationary, periodic or heteroclinic orbits, or to prove chaotic behaviour of the system. It has been applied to a wide range of problems, e.g. to find travelling-wave solutions of partial differential equations; to investigate the structure of global attractors of reaction-diffusion equations or delay equations; to find periodic solutions of Hamiltonian systems; to give a rigorous computer-assisted proof of chaos in Lorenz equations; to prove bifurcation and to analyze the set of bifurcating solutions in various settings. The original work of C. Conley and his school took place in the 1970s and early 1980s. Standard references for this work are [3], [11] and [12]. A recent overview on the Conley index and its applications is [6]. In order to describe the basic version of the Conley index, consider a flow ~: R x X -+ X on a locally compact m e t r i c s p a c e X (cf. also F l o w ( c o n t i n u o u s - t i m e d y n a m i c a l s y s t e m ) ) . A compact subset S C X is called isolated invariant if there exists a compact neighbourhood N of S in X such that S is the invariant part of N: CONLEY

INDEX

-

S = i n v ( N ) : = {x E N : ~(t,x) E N f o r allt e R } .

In that case N is said to be an isolating neighbourhood of S. The Conley index associates to an isolated invariant set S the homotopy type of a pointed topological space in the following way. An index pair (N, L) for an isolated invariant set S consists of compact subsets L C N of X such that: i) clos(N \ L) is an isolating neighbourhood of S; ii) L is positively invariant in N: Given x E L and t > 0 with ~([0, t], x) C N, then ~(t, x) E L; iii) L is an exit set for N: Given x E N and t > 0 with ~(t, x) ¢ N, there exists a to C [0, t] with ~(t0, x) E L. Given an isolated invariant set S, it can be proved that index pairs exist. Moreover, if ( N , L ) and ( N ' , L ' ) are two index pairs for S, then the quotient spaces N I L and N ' / L ' are homotopy equivalent with base points [L] and [L'] fixed (cf. also H o m o t o p y ) . The Conley index h(S) = h(S, ~) of S is by definition the h o m o t o p y t y p e of the pointed space ( N / L , ILl), where (N, L) is an index pair for S. As an example, consider the flow T(t,x) = etAx on X = R ~, where A E £ ( R ~) has no eigenvalues on the imaginary axis; e.g. A = d i a g ( ) ` l , . . . , )`~) with ),1 _> " " _> ),k > 0 > )`k+l _> "

_> )`~.

The origin is a hyperbolic stationary point of ~ and S = {0} is an isolated invariant set. Any compact neighbourhood of S is an isolating neighbourhood of S. Suppose that the generalized eigenspace of A corresponding to the eigenvalues with positive real part is spanned by e l , . . . ,ek, and the complementary generalized eigenspace is spanned by e k + l , . . . , en. Then (B k x B n-k, S k - 1 X B n-k) is an index pair for S; here B l is the unit ball in R I with boundary S l-1. Since B ~-k is contractible (cf. also C o n t r a c t i b l e space), the Conley index of S is equal to the homotopy type of ( B k / S k - l , [Sk-1]), which is the same as the homotopy type of (S k, .). In this example one recovers the M o r s e i n d e x of the hyperbolic fixed point. Therefore the Conley index can be interpreted as a generalized Morse index. In applications one usually first has a set N which is an isolating neighbourhood of some a priori unknown isolated invariant set S := inv(N). Then one tries to compute h(S) or to obtain some information, like its homology groups. For this computation the invariance of the Conley index under certain deformations of the flow, the continuation invariance, is very useful - - in analogy to the homotopy invariance of the B r o u w e r d e g r e e . Finally one can use the knowledge about h(S) in order to investigate the invariant set S itself. Whereas one can immediately deduce that S is not empty if h(S) is not trivial, additional information on the flow inside N is needed in order to 111

CONLEY INDEX o b t a i n m o r e d e t a i l e d r e s u l t s a b o u t S, for e x a m p l e t h a t S contains a periodic orbit.

CONSECUTIVE

k-OUT-OF-n:

F-SYSTEM,

con-

T h e o r i g i n a l version of t h e Conley i n d e x has b e e n

secutive k - o u t - o f - n s t r u c t u r e , consecutive s y s t e m - A n o r d e r e d sequence of n c o m p o n e n t s such t h a t t h e s y s t e m

refined a n d e x t e n d e d in several directions. For an equiv a r i a n t version t o g e t h e r w i t h a p r o d u c t s t r u c t u r e on t h e c o h o m o l o g y level, see [4]. T h e C o n l e y i n d e x a n d this a d d i t i o n a l s t r u c t u r e p l a y e d an i m p o r t a n t role in F l o e r ' s w o r k on t h e A r n o l ' d c o n j e c t u r e a n d in t h e d e v e l o p m e n t

fails if a n d o n l y if at l e a s t k c o n s e c u t i v e c o m p o n e n t s fail. It is a consecutive k - o u t - o f - n : G - s y s t e m if it works if a t least k consecutive c o m p o n e n t s work. T h e s e s y s t e m s are called circular, r e s p e c t i v e l y linear, if t h e c o m p o n e n t s are a r r a n g e d in a circle, r e s p e c t i v e l y on a line.

of Floer homology. In [I0] and [2] the Conley index has been generalized to semi-flows on metric spaces which need not be locally compact. This has been applied to parabolic differential equations and delay differential equations. Discrete dynamical systems are being considered in [9] and [7], multi-valued discrete dynamical systems in [5]. The multi-valued version is the basis for rigorous numerical computations of the Conley index for concrete dynamical systems, since it allows one to incorporate interval arithmetic. Parametrized versions of the Conley index have been defined in [i] and [8]; an abstract categorical approach is given in [13].

T h e reliability of such s y s t e m s , which in simple cases

References [1] BARTSCH,

T.: 'The Conley index over a space', Math.

Z.

209

(1992), 167-177. [2] BENCI, V.: 'A new approach to the Morse Conley theory and some applications', Ann. Mat. Pura Appl. (~{)4 (1991), 231 305. [3] CONLEY, C.: Isolated invariant sets and the Morse index, Vol. 38 of CBMS Regional Conf. Ser., Amer. Math. Soc., 1978. [4] FLOER, A.: 'A refinement of the Conley index and an application to the stability of hyperbolic invariant sets', Ergod. Th. Dynam. Syst. 7 (1987), 93-103. [5] KACZYI~!SKI,T., AND MROZEK, M.: 'Conley index for discrete multivalued dynamical systems', Topoi. Appl. 65 (1995), 8396. [6] MISCHAIKOW,K., AND MROZEK, M.: 'Conley index theory', in B. FIEDLER, G. IOOSS, AND N. I£OPELL (eds.): Handbook of Dynamical Systems III: Towards Applications, Elsevier, to appear. [7] MROZEK, M.: 'Leray functor and the cohomological Conley index for discrete dynamical systems', Trans. Amer. Math. Soc. 318 (1990), 149-178. [8] MROZEK, M., REINECK, J., AND SRZEDNICKI, R.: 'The Conley index over a base', Trans. Amer. Math. Soe. 352 (2000), 4171-4194. [9] ROBBIN, J., AND SALAMON,D.: 'Dynamical systems, shape theory and the Conley index', Ergod. Th. Dynam. Syst. 8 (1988), 375-393. [10] RYBAKOWSKI, I~.: The homotopy index and partial differential equations, Springer, 1987. [11] SALAMON,D.: 'Connected simple systems and the Conley index of isolated invariant sets', Trans. Amer. Math. Soc. 291 (1985), 1-41. [12] SMOLLER, J.: Shock waves and reaction-diffusion equations, Springer, 1983. [13] SZYMCZAK,A.: 'The Conley index for discrete dynamical systems', Topoi. Appl. 66 (1995), 215-240. Thomas Bartsch M S C 1991: 5 8 F x x

112

a m o u n t s to p r o b a b i l i t i e s of r u n s of consecutive successes or failures of B e r n o u l l i t r i a l s , has c o n n e c t i o n s w i t h Fibonacci polynomials and Lucas-type polynomials (see L u c a s p o l y n o m i a l s ) . References

[1] CHARALAMBIDES, CH.A.: 'Lucas numbers and polynomials of order k and the length of the longest circular success run', Fibonacci Quart. 29 (1991), 290-297. [2] CHARALAMBIDES,CH.A.: ~Suecess runs in a circular sequence of independent Bernoulli trials', in A.P. GODBOLEAND ST.G. PAPASTAVRIDES (eds.): Runs and Patterns in Probability, Kluwer Acad. Publ., 1994, pp. 15-30. [3] PEKOEZ, E.A., AND ROSS, S.M.: 'A simple derivation of extended reliability formulas for linear and circular consecutive k-out-of-n: F-systems', J. Appl. Probab. 32 (1995), 554-557. [4] PHILIPPOU, A.N., AND MAKRI, F.S.: 'Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k', in G.E. BERGUM ET AL. (eds.): Applications of Fibonacci Numbers, Vol. 3, Kluwer Acad. Publ., 1990, pp. 281 286. [5] PREUSS, W.: 'On the reliability of generalized consecutive systems', Nonlin. Anal. Th. Meth. Appl. 30, no. 8 (1997), 5425-5429. M. Hazewinkel M S C 1991: 60C05, 60K10 CONWAY

ALGEBRA

- A n a b s t r a c t a l g e b r a which

yields an i n v a r i a n t of links in R 3 (cf. also L i n k ) . T h e c o n c e p t is r e l a t e d t o t h e e n t r o p i c right quasig r o u p (cf. also Q u a s i - g r o u p ) . A C o n w a y a l g e b r a consists of a sequence of 0 - a r g u m e n t o p e r a t i o n s (constants) al, a2,.. • and two 2-argument operations I and *, which satisfy the following conditions: Initial conditions:

C1) a n l a n + l = an; C2) an * a n + l = an. Transposition properties: C3) (a]b)l(c[d) = (ale)](bid); C4) ( a l b ) * (cfl) = ( a * c ) i ( b * d ) ; C5) ( a * b ) * ( c * d ) = (a*c)*(b*d). Inverse o p e r a t i o n p r o p e r t i e s : C6) (alb) * b = a; C7) (a * b)lb = a. T h e m a i n link i n v a r i a n t y i e l d e d b y a C o n w a y a l g e b r a is the Jones-Conway p o l y n o m i a l , [3], [5], [4].

COX REGRESSION MODEL

A nice example of a four-element Conway algebra, which leads to the link invariant distinguishing the lefthanded and right-handed trefoil knots (cf. also T o r u s k n o t ) is described below: al = 1, a 3 = 4,

a2 = 2, a i + 3 = a i.

The operations [ and * are given by the following tables: I

1

2

3

1 2 3 4

2 3 1 4

1 4 2 3

4 1 3 2

, 1 2 3 4

1 3 1 2 4

2 1 3 4 2

3 2 4 3 1

4 3 2 4 1

smallest equivalence relation on ambient isotopy classes of oriented links, denoted by He, that satisfies the followrL/+, L I_, LI~ ing condition: If (L+, L_ , Lo) and ~ 0J are Conway skein triples (cf. also C o n w a y s k e i n t r i p l e ) such that if L_ ~c L~ and L0 "-.c L~ then L+ ~ L~_, and, furthermore, if L+ ~c L~_ and L0 Nc L~ then L_ ~¢ L~_. Skein equivalent links have the same Jones-Conway polynomials (cf. also J o n e s - C o n w a y p o l y n o m i a l ) and the same Murasugi signatures (for links with nonzero determinant, cf. also S i g n a t u r e ) . The last property generalizes to Tristram-Levine signatures. References

4

4 2 1 3 If one allows partial Conway algebras, one also gets the Murasugi signature and Tristram-Levine signature of links [2]. The skein calculus (cf. also Skein module), developed by J.H. Conway, leads to the universal partial Conway algebra. Invariants of links, wL, yielded by (partial) Conway algebras have the properties that for the Conway skein triple L+, L_ and Lo:

[1] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): C o m p u t a t i o n a l Problems in Abstract Algebra, Pergamon, 1969, pp. 329-358. [2] GILLER, C.A.: 'A family of links and the Conway calculus', Trans. A m e r . Math. Soc. 270, no. 1 (1982), 75-109.

Jozef Przytycki MSC 1991:57P25 CONWAY S K E I N T R I P L E - Three oriented link diagrams, or tangle diagrams, L+, L _ , L0 in R 3, or more generally, in any t h r e e - d i m e n s i o n a l m a n i f o l d , that are the same outside a small ball and in the ball look like

L+

L

L0

WL+ = W L _ [WL o, eL_

=

Similarly one can define the Kauffman bracket skein triple of non-oriented diagrams L+, L0 and L ~ , and the Kauffman skein quadruple, L+, L_, Lo and L ~ , used

W L + * W L o.

References [1] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): C o m p u t a t i o n a l Problems in Abstract Algebra, Pergamon, 1969, pp. 329-358. [2] PRZYTYCKI, J.H., AND TRACZYK, P.: 'Conway algebras and skein equivalence of links', Proc. A m e r . Math. Soc. 100, no. 4 (1987), 744-748. [3] PRZYTYCKI, J.H., AND TI~ACZYK, P.: 'Invariants of links of Conway type', Kobe J. Math. 4 (1987), 115-139. [4] SIKORA, A.S.: ~On Conway algebras and the Homflypt polynomial', J. K n o t Th. Ramifications 6, no. 6 (1997), 879-893. [5] SMITH, J.D.: 'Skein polynomials and entropic right quasigroups Universal algebra, quasigroups and related systems (Jadwisin 1989)', D e m o n s t r a t i o Math. 24, no. 1-2 (1991), 241-246.

Jozef Przytycki MSC 1991:57P25 CONWAY POLYNOMIAL Conway polynomial.

See A l e x a n d e r -

MSC 1991:57P25 CONWAY SKEIN EQUIVALENCE - An equivalence relation on the set of links in R 3 (cf. also Link). It is the

to define the B r a n d t - L i c k o r l s h - M i l l e t t - H o n o m i a l and the K a u f f m a n p o l y n o m i a l :

/, L+

poly-

)(

L.

L0

L~

Generally, a skein set is composed of a finite number of k-tangles and can be used to build link invariants and skein modules (cf. also S k e i n m o d u l e ) . References [1] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): C o m p u t a t i o n a l Problems in Abstract Algebra, Pergamon, 1969, pp. 329-358.

Jozef Przytycki MSC 1991:57P25 A regression model introduced by D.R. Cox [4] and subsequently proved to be one of the most useful and versatile statistical models, in particular with regards to applications in survival analysis (cf. also R e g r e s s i o n analysis). Let X 1 , . . . , X n be stochastically independent, strictly positive random variables (cf. also R a n d o m COX

REGRESSION

MODEL

-

113

COX REGRESSION MODEL v a r i a b l e ) , to be thought of as the failure times of n different items, such that X~ has hazard function uk

(i.e. P(Xk > t ) = e x p

(/o --

Uk(S) ds

)

for t _> 0) of the form k(t) =

Here, c~ is an unknown hazard function, the baseline hazard obtained if 27 = 0, and ~ r = (/31,...,~p) is a vector of p unknown regression parameters. The z[ (t ) = ( zk,l (t ), . . . , zk,p( t ) ) denote known non-random vectors of possibly time-dependent covariates, e.g. individual characteristics of a patient referring to age, sex, method of treatment as well as physiological and other measurements. The parameter vector f is estimated by maximizing the partial likelihood [5]

c(f)

exp(zTj (Tj)9) :

j=l EkERi

exp(z[(Tj)f)'

(1)

where T1 < " " < Tn are the Xk ordered according to size, Yj = i if it is item i that fails at time Tj, and Rj = {k: Xk > Tj} denotes the set of items k still at risk, i.e. not yet failed, immediately before Tj. With this setup, the j t h factor in C(~) describes the c o n d i t i o n a l d i s t r i b u t i o n of Yj given T 1 , . . . ,Tj and Y1,..-, Yj-1. For many applications it is natural to allow for, e.g., censorings (cf. also E r r o r s , t h e o r y of) or truncations (the removal of an item from observation through other causes than failure) as well as random covariate processes Zk(t). Formally this may be done by introducing the counting processes Nk(t) = l(xk<_tJ~(X~)=l) registering the failures if they are observed, where Ik (t) is a 0 - l-valued s t o c h a s t i c p r o c e s s with Ik(t) = 1 if item k is at risk (under observation) just before time t. If St denotes the a-algebra for everything observed (failures, censorings, covariate values, etc.) on the time interval [0, t], it is then required that Nk have .Pt-intensity proce88

=

(2)

i.e. Nk(t) - f~ Ik(s) ds defines a ¢'t-martingale (cf. also M a r t i n g a l e ) , while intuitively, for small h > 0, the conditional probability given the past that item k will fail during the interval ]t, t + h] is approximately ha(t)eZ[( t)~, provided k is at risk at time t. For known, (2) is then an example of Aalen's multiplicative intensity model [1] with the integrated baseline hazard A(t) = f t c~(s) ds estimated by, for any t, A(tl~) = 114

0,t] Ek=~ 1 ~ Ik(s-)eZ~( s-)~ dN(s),

(3)

writing N = ~ k Nk and where s - signifies that it is the values of Ik and Zk just before the observed failure times that should be used. Since in practice f is unknown, in (3) one of course has to replace/3 by the estimator ~, still obtained maximizing the partial likelihood (1), replacing n by the random number of observed failures, replacing zk by Zk, and using Rj = {k: I k ( T j - ) = 1} with Tj now the j t h observed failure. (Note that in contrast to the situation with non-random covariates described above, there is no longer an interpretation of the factors in C(27) as conditional distributions.) Using central limit theorems for martingales (cf. also C e n t r a l l i m i t t h e o r e m : M a r t i n g a l e ) , conditions may be given for consistency and asymptotic normality of the estimators ~ and A(tl/3), see [3]. It is of particular interest to be able to test for the effect of one or more covariates, i.e. to test hypothesis of the form fie = 0 for one or more given values of g, 1 < g < p. Such tests include likelihood-ratio tests derived from the partial likelihood (cf. also L i k e l i h o o d r a t i o t e s t ) , or Wald test statistics based on the asymptotic normality of ~. A thorough discussion of the tests in particular and of the Cox regression model in general is contained in [2, Sect. VII.2]; [2, Sect. VII.3] presents methods for checking the proportional hazards structure assumed in (2). Refinements of the model (2) include models for hartdling e.g. stratified data, Markov chains with regression structures for the transition intensities, etc. It should be emphasized that these models, including (2), are only partially specified in the sense that with (2) alone nothing much is said about the distributions of the Zk or Ik. This, in particular, makes it extremely difficult to use the models for, e.g., the prediction of survival times. References [1] AALEN, O.O.: 'Nonparametric inference for a family of counting processes', A n n . Statist. 6 (1978), 701-726. [2] ANDERSEN, P.K.A., BORGAN, ~., GILL, R.D., AND KEIDING, N.: Statistical models based on counting processes, Springer, 1993. [3] ANDERSEN, P.K.A., AND GILL, R.D.: 'Cox's regression model for counting processes: A large sample study', A n n . Statist. 10 (1982), 1100-1120. [4] Cox, D.R.: 'Regression models and life-tables (with discussion)', J. Royal Statist. Soc. B 34 (1972), 187 220. [5] Cox, D.R.: 'Partial likelihood', B i o m e t r i k a 62 (1975), 269276. Martin Jacobsen

MSC 1991: 62Jxx, 62Mxx C U R R E N T - Let V be an n-dimensional C °O_ manifold with countable basis (cf. also D i f f e r e n t i a b l e n j , where ~)j denotes the m a n i f o l d ) and let 19 = ®j=019 v e c t o r s p a c e of compactly supported differential forms of degree j on V (cf. also D i f f e r e n t i a l f o r m ) . Endow i9

CURRENT with the usual structure of a F r d c h e t s p a c e by declaring t h a t {¢j E D} tends to ¢ if there exists a compact set K C V such t h a t supp Cj C K for all j and the coefficients of Cj and all their derivatives tend uniformly to those of ¢. A current on V is an element of the dual space D'. The idea of currents was introduced by G. de R h a m in [6], to obtain a homology theory including both forms and chains, but a precise definition, see [7], [8], became only possible after distributions (cf. also G e n e r a l i z e d f u n c t i o n ) had been introduced by L. Schwartz. See also (the editorial comments to) D i f f e r e n t i a l f o r m , whose notation is used here too. While exterior products of currents are in general undefined, exterior differentiation can be defined by duality. If the action of a current T of degree p on a form ¢ is denoted by {T, ¢), then one defines the ezterior differential d T by (dT, ¢} = ( - 1 ) p + I ( T , de). In particular, the notions of closed and exact currents are defined. Now, let V be a c o m p l e x m a n i f o l d . One has the splitting d = 0 + 0 for currents just as for forms. A theorem of P. Lelong [4] states that any pure pdimensional analytic subset A of a Hermitian complex manifold has locally finite 2p-volume. As a consequence one can define the current of integration over A by

([A],¢) = /

Jre gA

¢.

Here, the integration is over the regular points of A (cf. also A n a l y t i c set). [A] is a d-closed current of bi-dimension (p,p). Moreover, [A] is positive, t h a t is, ([A], ¢} is positive for forms ¢ = )~dVA, with ,~ > 0 and alva being the volume form on the regular points of A. See also [2], [51. Thus, currents can be viewed as an extension of the notion of a n a l y t i c m a n i f o l d . This idea has been very fruitful in complex analysis. See e.g. [1], [3] and their references. See also G e o m e t r i c m e a s u r e t h e o r y . References [1] BEN MESSAOUD,H., AND EL MIR, H.: 'Tranchage et prolongement des courants positifs ferm~s', Math. Ann. 307 (1997), 473-487.

[2] CHmKA, E.M.: Complex analytic sets, Vol. 46 of MAIA, Kluwer Acad. Publ., 1989. (Translated from the Russian.) [3] DUVAL, J., AND SmONY, N.: 'Hulls and positive closed currents', Duke Math. Y. 95 (1998), 621-633. [4] LELONO, P.: 'Integration sur un ensemble analytique complexe', Bull. Soe. Math. France 85 (1957), 239-262. [5] LELONO, P.: Fonctions plurisousharmoniques et formes diffdrentielles positives, Gordon & Breach, 1968. [6] RHAM, G. DE: 'Sur l'analyse situs des varietds a n dimensions (Th~se)', J. Math. Pures Appl. 10 (1931), 115-200. [7] RHAM, G. DE: Differentiable manifolds, third ed., Springer, 1984. (Translated from the French.) [8] SCHWARTZ,L.: Thdorie des distributions, Hermann, 1966. J. Wiegerinck M S C 1991: 58A25, 53C65, 32C30

115

D D'ALEMBERT EQUATION FOR FINITE SUM DECOMPOSITIONS - Consider the decomposition of a function h(x, y) into a finite sum of the form

k----1

For sufficiently smooth h, a necessary condition for such a decomposition involves determinants of the form h

hy

---

hy~ /

hx

hxy

""

hxy,~

.

'.

hxny

...

det h n

. h x y.

These determinants were introduced in [8] and [9], and a correct formulation of the sufficient condition was given in [4]; see also [3]. A sufficient and necessary condition for not sufficiently smooth functions h(x,y) defined on arbitrary (even discrete) sets without any regularity conditions was formulated in [4], [3] by introducing a new, special matrix

h(xl,yl) h(x2,yl)

I

•

\h(xn,

Yl)

h(xl,y2) h(x2,y2)

"'" ...

h(xl,y~)~ h(x2,yn)|

. h ( x n , Y2)

•

""

• J h(Xn, Yn)]

'

see also [6] and [7]. Several authors have dealt with problems concerning decompositions of functions of several variables and similar questions, see, e.g., [1], [2], [6]. However, several open problems in this area remain (as of 2000), e.g.: find a characterization of functions h(x, y) of the form

h(x,y)=F(fifk(x).gk(y)),k=l see [5].

References [1] CADEK, M., AND SIMSA, J.: 'Decomposable functions of several variables', Aequat. Math. 40 (1990), 8-25.

[2] GAUCHMAN,H., AND I=~UBEL,L.A.: 'Sums of products of functions of x times functions of y', Linear Alg. ~4 Its Appl. 125 (1989), 19-63. [3] NEUMAN, F.: 'Functions of two variables and matrices involving factorizations', C.R. Math. Rept. Acad. Sci. Canada 3 (1981), 7-11. [4] NEUMAN,F.: 'Factorizations of matrices and functions of two variables', Czech. Math. J. 32, no. 107 (1982), 582-588. [5] NEUMAN, F., AND RASSIAS, TH.: 'Functions decomposable into finite sums of products". Constantin Catathdodory-An lnternat. Tribute, Vol. II, World Sci., 1991, pp. 956-963. [6] RASSIAS, TH.M., A N D SIMSA, J.: Finite sum decompositions in mathematical analysis, Wiley, 1995. [7] RASSIAS, TH.M., A N D SIMSA, J.: '19 Remark', Aequat. Math. 56 (1998), 310. [8] STEPHANOS, C.M.: 'Sur une categorie d'quations fonctionalles': Math. Kongr. Heidelberg, Vol. 1905, 1904, pp. 200-201. [9] STEPHANOS, C.M.: 'Sur une categorie d'quations fonctionalles', Rend. Circ. Mat. Palermo 18 (1904), 360-362. F. N e u m a n

M S C 1991:26B40

DARBO FIXED-POINT THEOREM - The notion of 'measure of non-compactness' was first introduced by C. Kuratowski [4]. For any bounded set B in a m e t ric s p a c e its measure of non-compactness, denoted by a(B), is defined to be the infimum of the positive numbers d such that B can be covered by a finite number of sets of diameter less than or equal to d. Another measure of non-compactness is the ball measure #(B), or Hausdorff measure, which is defined as the infimum of the positive numbers r such that B can be covered by a finite number of balls of radii smaller than r. See also H a u s d o r f f m e a s u r e . Roughly speaking, a measure of non-compactness is some function defined on the family of all non-empty bounded subsets of a given metric space such that it is equal to zero on the whole family of relatively compact sets. G. Darbo used a measure of non-compactness to investigate operators whose properties can be characterized as being intermediate between those of contraction

DAUBECHIES WAVELETS and compact mappings (cf. also C o m p a c t m a p p i n g ; C o m p a c t o p e r a t o r ; C o n t r a c t i o n ) . He was the first to use the index a in the theory of fixed points [3]. Darbo's fixed-point theorem is a generalization of the well-known Schauder fixed-point theorem (cf. also S c h a u d e r theorem). It states that if S is a non-empty bounded closed convex subset of a B a n a c h space X and T: S --+ S is a c o n t i n u o u s m a p p i n g such that for any set E C S,

a ( T E ) < ka(E),

N \ {0}, that satisfy some special properties. First of all, the collection ON(X--k), k E Z, is an o r t h o n o r m a l syst e m for fixed N E N \ {0}. Furthermore, each wavelet 0N is compactly supported (cf. also F u n c t i o n of comp a c t s u p p o r t ) . Moreover, supp(0N) = [0, 2 N - 1]. The index number N is also related to the number of vanishing moments, i.e.,

F xkON(x) dx

(1)

where k is a constant, 0 < k < 1, then T has a fixed point. This theorem is true for the measure # also. Note that every completely-continuous mapping (or c o m p a c t m a p p i n g ; cf. also C o m p l e t e l y - c o n t i n u o u s o p e r a t o r ) satisfies (1) with k = 0, while all Lipschitz mappings with constant k (cf. L i p s c h i t z c o n d i t i o n ) also satisfy (1). Further, mappings that are not completely continuous but satisfy the condition (1) are of the form T = 971+T2, where 971 is completely continuous and T2 satisfies the Lipschitz condition with constant k. The significance of this type of mapping is due to the fact that compactness of either the domain or the range is not required. Methods for determining the value of It(B) for a given set B in a Banach space are given in [2]. Darbo's fixed-point theorem is useful in establishing the existence of solutions of various classes of differential equations, especially for implicit differential equations, integral equations and integro-differential equations, see [2]. It is also used to study the controllability problem for dynamical systems represented by implicit differential equations [1].

O,

0 < k < N.

O0

A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact, 3A > 0 VN E N, N > 2: ON E C ;~N. For large N one has )~ ~ 0.2. The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for 01, which is in fact the Haar wavelet [3]. Satisfying symmetry conditions cannot go together with all other properties of the Daubechies wavelets. The Daubechies wavelets can also be used for the continuous wavelet transform, i.e.

W¢[fl(a,b) = - ~

oo f(x)O

dx,

for f E L 2(R), a E R + and b E R. The parameters a and b denote scale and translation/position of the transform. A stable reconstruction formula exists for the continuous wavelet transform if and only if the following admissibility condition holds:

References

[1] BALACHANDRAN, K., AND DAUER, J.P.: 'Controllability of nonlinear systems via fixed point theorems', J. Optim. Th. Appl. 53 (1987), 345-352. [2] BANAS, J., AND GOEBEL, K.: Measure of noncompactness in Banach spaces, M. Dekker, 1980. [3] DARBO, G.: 'Punti uniti in transformazioni a condominio non compacto', Rend. Sere. Mat. Univ. Padova 24 (1955), 84-92. [4] KURATOWSK~,C.: 'Sur les espaces complets', Fundam. Math.

0 < C ¢ = 27r

-

da < o%

where ~ denotes the F o u r i e r t r a n s f o r m of 9. The reconstruction formula reads:

f ( x ) = ~1 L ~ S F

oo l/V~[f](a,b)O ( ~ _ ~ )

da db a----~"

15 (1930), 301-309.

Krishnan Balachandran MSC 1991:47H10 A wavelet is a function ¢ E L2(R) that yields a basis in L2(R) by means of translations and dyadic dilations of itself, i.e., DAUBECHIES

WAVELETS -

i(x) =

aj,kC(2Jx- k), j = - o o k=--oo

for all f E L2(R) (cf. also W a v e l e t analysis). Such a decomposition is called the discrete wavelet transform. In 1988, the Belgian mathematician I. Daubechies constructed [1] a class of wavelet functions ON, N E

This result holds weakly in L2(R). For f E L I(R) A L 2 (R) and f C L 1(R), this results also holds pointwise. All Daubechies wavelets satisfy the admissibility condition and thus guarantee a stable reconstruction. References

[1] DAUBECHIES,[.: 'OrthonormaI bases of compactly supported wavelets', C o m m u n . Pure Appl. Math. 41 (1988), 909-996. [2] DAUBEOHIES,I.: Ten lectures on wavelets, SIAM, 1992. [3] HAAR, A.: 'Zur theorie der orthogonalen Funktionensysteme', Math. A n n . 69 (1910), 331-371.

P.J. Oonincz MSC 1991: 42Cxx

117

DEDEKIND DOMAIN D E D E K I N D DOMAIN - See D e d e k i n d ring.

[5] MACWILLIAMS, F.J., AND SLOANE, N.J.A.: The theory of

error-correcting codes, North-Holland, 1977. G. McGuire

M S C 1991:13F05 MSC1991: 94Bxx D E L S A R T E - G O E T H A L S CODE - A code belonging to a family of non-linear binary error-correcting codes (cf. also E r r o r - c o r r e c t i n g code). Delsarte-Goethals codes were first presented in a joint paper [2] by Ph. Delsarte and J.-M. Goethals. Let m > 4 be an even integer. Let r be an integer satisfying 0 < r < m / 2 - 1. For each m and r there is a Delsarte Goethals code, denoted D G ( m , r ) . This code has length 2 "~, and is sandwiched between the Kerdock code K(m) and the second-order Reed-Muller code R M ( 2 , m ) of the same length (cf. also K e r d o c k a n d P r e p a r a t a c o d e s ; E r r o r - c o r r e c t i n g code): K(m) C_ D G ( m , r) C RM(2, m). The number of codewords in D G ( m , r) is 2 "('~-1)+2"~ and the minimum distance is 2 "~-1 - 2 "~/2-1+r. As r increases, the number of codewords increases and the minimum distance decreases. When r = 0, the DelsarteGoethals code coincides with the Kerdock code K(m), and when r = m / 2 - 1 the Delsarte Goethals code coincides with RM(2, m). The construction of D G ( r , m ) involves taking the union of certain cosets of RM(1, m) in RM(2, m). These cosets are determined by certain bilinear forms. The rank of these forms, and the rank of the sum of any two of them, is at least m - 2r, and this property determines the minimum distance. The fact that it is possible to find 2 r('~-l)+ra-1 such forms is proved in [2] (see also

[5]). The Delsarte-Goethals codes have been shown to have another construction. It was shown in [3] that they are the Gray image of a Z4-1inear code. A direct proof of the minimum distance from the Z4 construction was given in [1]. There exist non-linear binary codes whose distance distribution is the MacWilliams transform of the distribution of the Delsarte-Goethals codes, see [4]. These codes act like dual codes, and the Z4 construction gives an explanation for their existence, see [3]. References [1] CALDERBANK, A.R., AND McGumE, G.: 'Z4-1inear codes

obtained as projections of Kerdock and Delsarte-Goethals codes', Linear Alg. £3 Its Appl. 226-228 (1995), 647 665. [2] DELSARTE, P., AND GOETHALS,J.M.: 'Alternating bilinear forms over GF(q)', J. Combin. Th. A 19 (1975), 26-50. [3] HAMMONS, A.R., KUMAR, P.V., CALDERBANK, A.R., SLOANE, N.J.A., AND SOLE, P.: 'The Z4-1inearity of Kerdock, Preparata, Goethals, and related codes', IEEE Trans. Inform. Th. 40 (1994), 301-319. [4] HErmERT, F.B.: 'On the Delsarte-Goethals codes and their formal duals', Discr. Math. 83 (1990), 249-263. 118

D E M P S T E R - S H A F E R THEORY, mathematical theory of evidence, belief function theory - A theory initiated by A.P. Dempster [2] and later developed by G. Sharer [6]. It deals with the representation of nonprobabilistic uncertainty a b o u t sets of facts (belief function) and the accumulation of 'evidence' stemming from independent sources (Dempster's rule of evidence combination) and with reasoning under incomplete information (Dempster's rule of conditioning); see below. Extensions to infinite countable sets and continuous sets have been studied. However, below finite sets of facts (elementary events) are considered. As with p r o b a b i l i t y t h e o r y , four different approaches to handling D e m p s t e r - S h a f e r theory may be distinguished: the axiomatic approach (formal properties of belief functions are analyzed); the naive casebased approach (a direct case-based interpretation of properties of belief functions is sought); the in-the-limit approach (properties of the belief function are considered as in-the-limit properties of sets of cases); and the subjectivist approach (predominantly, the qualitative behaviour of subjectively assigned beliefs is studied, no case-based interpretation is sought and belief is considered as a subjective basis for decision-making). A x i o m a t i c a p p r o a c h . Let -~ be a finite set of elements, called elementary events. Any subset of ~ with cardinality 1 is also called a frame of discernment and any other subset of £ is called a composite event. The central concept of a belief function is understood as any function Bel: 2 = ~ [0, 1] fulfilling the axioms: • B e l ( 0 ) = 0;

• Bel(E) = 1; • Bel(A1 U . . . U Ak) >_ ->

E ( - 1 ) 15+1Bel(miuAi). ig{1 ..... k},i#¢

Due to the last axiom, a belief function Bel is actually a Choquet capacity, monotone of infinite order (cf. also C a p a c i t y ) . By introducing the so-called basic probability assignment function (bpa function), or mass function, m : 2"- --+ [0,1] such t h a t ~Ae2~ m(A) = 1 and m(0) = 0, then Bel can be expressed as Bel(A) = ~BCA re(B). Other uncertainty measures can also be defined, like the plausibility function Pl(A) = 1 Bel(E - A) and the commonality function Q(A) = EB;ACB re(B). Given one of these functions Bel, P1, m, Q, any other of t h e m m a y be deterministically derived.

D E M P S T E R - S H A F E R THEORY Hence the function m is frequently used in the definition of further concepts, e.g., any set A with re(A) > 0 is called a focal point of the belief function. If m(E) = 1, then the belief function is called vacuous. Another central concept, the rule of combination of two independent belief functions BelE1, BeiE2 over the same frame of discernment (the so-called Dempster rule of evidence combination), denoted by BelEl,E2 ---BelE1 D BelE2, is defined as follows: "

E1,E2(A) = c B,C;A=BnC

(with c a constant normalizing the sum of reEl,E2 to 1). Suppose t h a t a frame of discernment ~ is equal to the cross product of domains E1,...,-E~, with n discrete variables X 1 , . . . , X ~ spanning the space F~. Let ( x l , , . . . , x ~ ) be a vector in the space spanned by the variables X ~ , , . . . , X ~ . Its projection onto the subspace spanned by the variables X j l , . . . , X j k (with j l , . . . , jk distinct indices from 1 , . . . , n) is then the vector ( x j ~ , . . . , xj~). The vector ( X l , . . . , x~) is also called an extension of (xj~,..., xj~). The projection of a set A of such vectors is the set A 4zj~ .....xj~ of projections of all individual vectors from A onto X j l , . . . ,Xjk. A is also called an extension of A ~xjl .....zj~. A is called a vacuous extension of A*XJ~ .....xJk if (and only if) A contains all possible extensions of each individual vector in A,XJ~ .....xj~. The fact t h a t A is a vacuous extension of B onto X 1 , . . . , X n is denoted by A = B tx~ ..... x~. Let m be a basic probability assignment function on the space of discernment spanned by the set of variables X = { X 1 , . . . , X ~ } , and let Bel be the corresponding belief function. Let Y be a subset of X. The projection operator (or marginalization operator) $ of Bel (or m) onto the subspace spanned by Y is defined as

A;B:ASY

The vacuous extension operator $ of Bel (or m) from Y onto the superspace spanned by X is defined as follows: for any A in X and any B in Y such that A = B $x one has m$X(A) = re(B), and for any other A from X , m t x (A) = 0. To denote that a belief function Bel is defined over the space spanned by the set of variables X one frequently writes Belx. By convention, if one wants to combine, using Dempster's rule, two belief functions not sharing the frame of discernment, then one looks for the closest common vacuous extension of their frames of discernment without explicitly mentioning this. The last important concept of Dempster-Shafer theory is the Dempster rule of conditioning: Let B be a subset of E, called evidence, and let mB be a basic probability assignment such that mB (B) = 1 and mB (A) = 0 for all A different from B. Then the conditional belief

function Bel(.l[B), representing the belief function Bel conditioned on evidence B, is defined as: Bel(.llB ) -- Bel G B e l s . The conditioning as defined by the above rule is the foundation of reasoning in D e m p s t e r - S h a f e r theory: One starts with a belief function Belz,know defined in a multivariable space X (being one's knowledge), makes certain observations on the values taken by some observational variables Y C X , e.g. Y1 C {Yl,1, Yl,3, Yl,s}, denotes this knowledge by myi,obs({Yl,l,yl,3,yi,s}) = 1, and then one wishes to know what value will be taken by a predicted variable Z C X. To t h a t end one calculates the belief for the predicted variable as (Belx,know G Belyl,obs ® Bely2,ob~ @ . . . ) $ z . Due to the large space, the calculation of such a margin is prohibitive unless one can decompose Belz,know into a set of 'smaller' belief functions Belh~,know over a set H of subsets of X such t h a t Belx,know ~- O

Delhi,know.

hicH

The set H is hence a h y p e r g r a p h . If H is a hypertree (a special type of hypergraph), then one can efficiently reason using the Shenoy-Shafer algorithms [8]. Any hypergraph can be transformed into a hypertree, but the task aiming to obtain the best hypertree for reasoning (with smallest subsets in H ) is prohibitive (AlP hard, cf. also AFT)), hence suboptimal solutions are elaborated. In the Shenoy-Shafer framework, both forward, backward and mixed reasoning is possible. Note t h a t in the above decomposition it is not assumed that the Belh~,know can be calculated in any way from Belx,know. As Bel is known to have socalled graphoidal properties [7], a decomposition similar to Bayesian networks for probability distributions has also been studied. An a priori-condition belief function BelzIY of variables Z given Y (defined over Z O Y), both sets with e m p t y intersection and both subsets of X, is introduced as: B el,~zuY = Belzly ® BelSxY X In general, m a n y such functions may exist. In these settings one says t h a t for a belief function Belx two nonintersecting sets of variables T C_ X and R C_ X are independent given X - T - R if B e l X _~

BelSX-n

R_ISX--T T I X _ T _ R ~) ~ I R I X _ T _

f., D^I.~X--T--R R ~:~ l ~ l X

The a priori-conditional belief function is usually not a belief function, as it usually does not match the third axiom for belief functions, and even may take negative values (and so do the corresponding plausibility and mass functions). Only conditional commonality functions are 119

D E M P S T E R - S H A F E R THEORY always non-negative everywhere. As a partial remedy, the so-called K function has been proposed:

Kzly(A) =

" zlY(B) B ; A 4Y C B SY , A t z = B S Z

It may be viewed as an analogue of the true mass function for 'a priori conditionals', as it is non-negative and for any fixed value of lz the sum over Z equals 1. Contrary to intuitions with probability distributions, the combination of an a priori conditional belief function with a (true) belief function by Dempster's rule need not lead to a belief function. Hence such a priori functions are poorly investigated so far (2000).

N a i v e c a s e - b a s e d a p p r o a c h . Currently (as of 2000), at least three naive case-based models compatible with the definition of belief function, Dempster's rule of evidence combination and D e m p s t e r ' s rule of conditioning exist: the marginally correct approximation, the qualitative model and the quantitative model. Marginally correct approximation. This approach [4] assumes that the belief function shall constitute lower bounds for frequencies; however, only for the marginals and not for the joint distribution. Then the reasoning process is expressed in terms of so-called Cano conditionals [1] - - a special class of a priori conditional belief functions that are everywhere non-negative. As for a general belief function, the Cano conditionals usually do not exist, they have to be calculated as an approximation to the actual a priori conditional belief function. This approach involves a modification of the reasoning mechanism, because the correctness is maintained only by reasoning forward. Depending on the reasoning direction, one needs different 'Markov trees' for the reasoning engine. Qualitative approach. This approach [5] is based on earlier rough set interpretations in Dempster-Shafer theory [9], but makes a small and still significant distinction. All computations are carried out in a strictly 'relational' way, i.e. indistinguishable objects in a database are merged (no object identities). The behaviour under reasoning fits strictly into the reasoning model of Dempster Shafer theory. Factors of the hypergraph representation can be expressed by relational tables. Conditional independence is well defined. However, there is no interpretation for conditional belief functions in this model. Quantitative approach. The quantitative model [3], [11] assumes that during the reasoning process one attaches labels to objects, hiding some of their properties. There is a full agreement with the reasoning mechanism of Dempster Sharer theory (in particular, Dempster's rule

120

of conditioning). When combining two independent belief functions, only in the limit agreement with Dempster's rule of evidence combination can be achieved. Conditional independence and conditional belief functions are well defined. Processes have also been elaborated that, in the limit, can give rise to well-controlled graphoidally structured belief functions, and learning procedures for the discovery of graphoidal structures from data have been elaborated. The quantitative model seems to be the model best fitting for belief functions. S u b j e c t i v i s t a p p r o a c h . One assumes that among the elements of the set f~, called 'worlds', one world corresponds to the 'actual world'. There is an agent who does not know which world is the actual world and who can only express the strength of his/her opinion (called the degree of belief) that the actual world belongs to a certain subset of f~. One such approach is the so-called transferable belief model [10]. Besides the two already mentioned rules of Dempster (combination and conditioning), many more rules handling various sources of evidence have been added, including disjunctive rules of combination, alpha-junctions rules, cautious rules, pignistic transformation, a specialization concept, a measure of information content, canonical decomposition, concepts of confidence and diffidence, and a generalized Bayesian theorem. Predominantly, the qualitative behaviour of subjectively assigned beliefs is studied. So far (as of 2000), no attempt paralleling the subjective probability approach of B. de Finetti has been made to bridge the gap between subjective belief assignment and observed frequencies. References

[1] CANO, J., DELGADO, M., AND MORAL~ S.: 'An axiomatic framework for propagating uncertainty in directed acyclic networks', Internat. J. Approximate Reasoning 8 (1993), 253-280. [2] DEMPSTER, A.P.: 'Upper and lower probabilities induced by a multi-valued mapping', Ann. Math. Stat. 38 (1967), 325339. [3] KLOPOTEK, M.A.: 'On (anti)conditional independence in Dempster-Shafer theory', J. Mathware and Softcomputing 5, no. 1 (1998), 69-89. [4] KLOPOTEK, M.A., AND WIERZCHO~, S.T.: 'On marginally correct approximations of Dempster-Shafer belief functions from data': Proc. IPMU'96 (Information Processing and Management of Uncertainty), Grenada (Spain), 1-5 July,

Vol. II, Univ. Granada, 1996, pp. 769-774. [5] KLOPOTEK,M.A., ANDWIERZeHOr~,S.T.: 'Qualitative versus quantitative interpretation of the mathematical theory of evidence', in Z.W.RAg ANDA. SKOWRON(eds.): Foundations of Intelligent Systems 7. Proc. ISMIS'97 (Charlotte NC, 15-17 Oct., 1997), Vol. 1325 of Lecture Notes in Artificial Intelligence, Springer, 1997, pp. 391-400. [6] SHAFER, G.: A mathematical theory of evidence, Prince-

ton Univ. Press, 1976.

DEN J O Y - W O L F F T H E O R E M [7] SHENOY, P.P.: 'Conditional independence in valuation based [8]

[9]

[10]

[11]

systems', Internat. J. Approximate Reasoning 109 (1994). SHENOY, P., AND SHAFER, (].: 'Axioms for probability and belief-function propagation', in R.D. SHACHTER, T.S. LEVITT, L.N. KANAL, AND J.F. LEMMER(ads.): Uncertainty in Artificial Intelligence, Vol. 4, Elsevier, 1990. SKOWRON,A., AND GRZYMALA-BUSSE,J.W.: 'Prom rough set theory to evidence theory', in R.R. YAGER, J. KASPRZYK, AND M. FEDRIZZI(ads.): Advances in the Dempster-Shafer Theory of Evidence, Wiley, 1994, pp. 193-236. SMETS, Pro: 'Numerical representation of uncertainty', in D.M. GABBAYAND PH. SMETS (ads.): Handbook of Defeasible Reasoning and Uncertainty Management Systems, Vol. 3, Kluwer Acad. Publ., 1998, pp. 265-309. WmRZCHOr~,S.T., AND KLOPOTEK, M.A.: 'Modified component valuations in valuation based systems as a way to optimize query processing', J. Intelligent Information Syst. 9 (1997), 157-180.

M.A. Klopotek M S C 1991: 92Jxx, 92K10, 68T30, 68T99 DENJOY-PERRON I N T E G R A L - A generalization of the L e b e s g u e i n t e g r a l . The narrow Denjoy integral (see D e n j o y i n t e g r a l ) is equivalent to the P e r r o n i n t e g r a l . Denjoy-Perron integrability is equivalent to Henstock integrability or Kurzweil-Henstock integrability (cf. also K u r z w e i l - H e n s t o c k i n t e g r a l ) .

M. Hazewinkel MSC1991:28A25 D E N J o Y - W O L F F THEOREM, Wolff-Denjoy theorem - For a domain 7? in a complex B a n a c h s p a c e X one denotes by Hol(7?) the set of all holomorphic selfmappings of 7? (cf. also A n a l y t i c f u n c t i o n ) . The classical Denjoy-Wolff theorem is the following one-dimensional result: Let A be the open unit disc in the complex plane C. If F C Hol(A) is not the identity and is not an automorphism of A with exactly one fixed point in A, then there is a unique point a in the closed unit disc A such t h a t the iterates {Fn}~_l of F converge to a, uniformly on compact subsets of A. This result is, in fact, a s u m m a r y of the following three assertions A ) - C ) due to A. Denjoy and J. Wolff [9], [35], [34], [36], [37]. For ~ E cgA and R > 0, the set

D~=D(~,R):={zCA:

] ~ <- zR~}' Z 2 iz,2

(1)

is called a horocycle at ~ with radius R. This set is a disc in A which is internally tangent to cOA at ~ (cf. also Horocycle). A) The Wolff-Schwarz lemma: If F C Hol(A) has no fixed point in A, then there is a unique unimodular point a E cOA such that every horocycle Da in A, internally tangent to OA at a, is F-invariant, i.e.,

f ( D a ) C Da.

(2)

This assertion is a natural complement of the J u l i a W o l f f - C a r a t h 6 o d o r y t h e o r e m [8]. B) If F E Hol(A) has no fixed point in A, then there is a unique unimodular point b C cgA such that the sequence {Fn}n~_l converges to b, uniformly on compact subsets of A. C) If F E Hol(A) is not an automorphism of A but has a fixed point c in A, then this point is unique in A, and the sequence {F'~}~_I converges to c uniformly on compact subsets of A. The limit point in B) is sometimes called the DenjoyWolff point of F. The point a in A) and the point b in B) are one and the same. However, this is not always the case in higherdimensional situations. Therefore, in the general case, the point a in A) is usually called the sink point of F. So, the sink point is the Denjoy-Wolff point if it is also attractive. By using the S c h w a r z l e m m a , assertion C) can be rephrased as follows: D) Let F E Hol(A) have a fixed point c E A. K F is not the identity, then F is power convergent if and only if ]F'(c)I < 1. Since 1926, these results have been developed in several directions. For a nice exposition of the onedimensional case see [5]. When X = 7-/is a complex H i l b e r t s p a c e with the i n n e r p r o d u c t (., .), and B is its open unit ball, the following generalization of the Wolff-Schwarz l e m m a is due to K. Goebel [12]: If F E Hol(B) has no fixed point, then there exists a unique point a C 0B such t h a t for each 0 < R < cc the set

E(a,R) = { x E B: II - (x'a)12 1 -[Ixll 2

} < R

(3)

is F-invariant. Geometrically, the set E(a, R) is an ellipsoid the closure of which intersects the unit sphere 0B at the point a. It is a natural analogue of the horocycle D(a, R). In the finite-dimensional case, 7-I = C n, the sink point a is also the Denjoy-Wolff point of F; see [15], [24], [20],

[6], For infinite-dimensional Hilbert balls, A. Stachura [31] has given a counterexample to show that the convergence result fails even for biholomorphic self-mappings. Nevertheless, some restrictions on a mapping from Hol(B) lead to a generalization of assertion B). In particular, C.-H. Chu and P. Mellon [7] showed that if F E Ilol(B) is a c o m p a c t m a p p i n g with no fixed point in B, then the sink point a in (3) is attractive in the topology of locally u n i f o r m c o n v e r g e n c e . For weak convergence results see, for example, [13]. 121

DEN J O Y - W O L F F T H E O R E M In 1941, M.H. Heins [14] extended the Denjoy-Wolff theorem to a finitely connected domain bounded by Jordan curves in C (cf. also J o r d a n c u r v e ) . His approach is specific to the one-dimensional case. Another look at the Denjoy-Wolff theorem is provided by a useful result of P. Yang [38] concerning a characterization of the horocycle in terms of the Poincarfi hyperbolic metric in A (cf. also P o i n c a r f i m o d e l ) . More precisely, he established the following formula: 1

[1 -

,-*alim[p(A, #) - p(0, p)] = [ log 1

2

IA[2 .

(4)

So, in these terms the horocycle Da in A can be described by the formula

D(a, R) = =

z E A: ~alim[p(z,w) - p(O,w)] < [ l o g R

(5) .

Since a hyperbolic metric can be defined in each bounded domain in C a, one can try to extend this formula and use it as a definition of the horosphere in a domain in C ~. Unfortunately, in general the limit in (4) does not exist. To overcome this difficulty, M. Abate [2] introduced two kinds of horospheres. More precisely, he defined the small horosphere E~ o (x, R) of centre x, pole z0 and radius R by the formula Ezo

=

n) =

z E 79: limsup[Kz~(z,w) -K~)(Zo,W)] < ~ l o g R

,

w .-+ x

and the big horosphere Fzo (x, R) of centre x, pole z0 and radius R by the formula n) =

=

z E 79: liminf[K~(z,w)~

- Kz~(zo,w)] < ~ l o g R

,

where D is a bounded domain in C a and Kz~ is its Kobayashi metric (cf. H y p e r b o l i c m e t r i c ) . For the Euclidean ball in C '~, Ezo (x, R) = F~o (x, R). Thus, each assertion which states for a domain 7) in C ~ the existence of a point a C 079 such that F~(Ez(a, n)) C F~(a, R)

(6)

for all z C D, R > 0, F C Ho1(79) and n = 1 , 2 , . . . is a generalization of the Wolff-Schwarz lemma. This is true, for example, for a bounded convex domain in C ~ [2]. However, in this case B) does not hold in general. Nevertheless, the convergence result does hold for bounded strongly convex C 2 domains, and for strongly pseudo-convex hyperbolic domains with a C 2 boundary [2], [1]. 122

Assertion A) can be generalized to the operator ball over a Hilbert space 7 / a n d , more generally, to the open unit ball Y of a so-caned J*-algebra (see [33], [25] and the references there), while B) fails in general even if the compactness of F is assumed [7]. For the particular case when F is defined by the Riesz-Dunford integral in the sense of the f u n c t i o n a l c a l c u l u s (cf. also D u n f o r d i n t e g r a l ) , a full analogue of the Denjoy Wolff theorem is due to K. Fan [10], [11]. For general Banach spaces there are examples showing that there are situations where even a sink point does not exist [19]. However, the question is still open (as of 2000) for reflexive Banach spaces. Moreover, when D is the open unit ball of a strictly convex Banach space X and F is compact or, more generally, condensing (cf. also C o n t r a c t i o n o p e r a t o r ) , then the analogue of B) is valid [17], [16], [21]. The situation is more fully understood when F has a fixed point c inside a bounded domain 79 C X. Simple examples show that one cannot always expect c to be an attractive fixed point, even if F is not an automorphism. Nevertheless, rephrasing C) in the form D), one can show (see, for example, [19]) that c is an attractive fixed point of F if and only if the spectral radius of the F r 6 c h e t d e r i v a t i v e F ~(c) is strictly less than 1. In the one-dimensional case, if F C Hol(:D) is not the identity, has an interior fixed point and is power convergent, then c is unique. However, this is no longer true in higher dimensions. A full description of such a situation was obtained by E. Vesentini [32]: Suppose that F has a fixed point c C 79, and denote the spectrum of the linear operator F'(c) by a(F'(c)) (cf. also S p e c t r u m o f a n o p e r a t o r ) . Then F is power convergent if and only if the following two conditions hold: i) c~(F'(c)) C A t2 {1}; and ii) 1 is a pole of the resolvent of F I(e) of order at most one.

Condition ii) is actually equivalent to the condition K e r ( I - Fl(c)) ® I m ( I - F'(c)) = X (see, for example, [23]). It is also known that conditions i) and ii) are equivalent to F ~(c) being power-convergent to a projection P onto K e r ( I - U(c)). So, if the retraction R E Ho1(79) is the limit point of { F a} under these conditions, then R = c is constant if and only if P = 0. The family {Fa}~=l of iterates of F E Hol(D) can be considered a one-parameter discrete sub-semi-group of Ho1(79) (cf. also S e m i - g r o u p o f h o l o m o r p h i c m a p pings). Therefore, another direction is concerned with analogues of the Denjoy-Wolff theorem for continuous semi-groups of holomorphic self-mappings of D. This

DICKMAN FUNCTION approach has been used to study the asymptotic behaviour of solutions to Cauchy problems (see, for example, [4], [1], [3], [18], [29], [28], and [21]). E. Berkson and H. Porta [4] have applied their continuous analogue of the Denjoy-Wolff theorem to the study of the eigenvalue problem for composition operators on Hardy spaces. For results on the asymptotic behaviour (in the spirit of the Denjoy-Wolff theorem) of (not necessarily holomorphic) mappings and semi-groups that are nonexpansive with respect to hyperbolic metrics, see, for

example, [22], [26], [27], [30]. References [1] ABATE, M.: 'Converging semigroups of holomorphic maps', Atti Accad. Naz. Lincei 82 (1988), 223-227. [2] ABATE, M.: 'Horospheres and iterates of holomorphic maps', Math. Z. 198 (1988), 225-238. [3] ABATE, M.: 'The infinitesimal generators of semigroups of holomorphic maps', Ann. Mat. Purl Appl. 161 (1992), 167180. [4] BERKSON, E., AND PORTA, H.: 'Semigroups of analytic functions and composition operators', Michigan Math. J. 25 (1978), 101-115. [5] BURCKEL, R.B.: 'Iterating analytic self-maps of discs', Amer. Math. Monthly 88 (1981), 396-407. [6] CHEN, G.N.: 'Iteration for holomorphic maps of the open unit ball and the generalized upper half-plane of C ~', J. Math. Anal. Appl. 98 (1984), 305-313. [7] CHU, C.-H., AND MELLON, P.: 'Iteration of compact holomorphic maps on a Hilbert ball', Proc. Amer. Math. Soc. 125 (1997), 1771-1777. [8] COWEN, C.C., AND MACCLUER, B.D.: Composition operators on spaces of analytic functions, CRC, 1995. [9] DENJOY, A.: 'Sur Fit@ration des fonctions analytiques', C.R. Acad. Sci. Paris 182 (1926), 255-257. [10] FAN, K.: 'Iteration of analytic functions of operators I', Math. Z. 179 (1982), 293-298. [11] FAN, K.: qteration of analytic functions of operators II', Linear and Multilinear Algebra 12 (1983), 295-304. [12] GOEBEL, K.: 'Fixed points and invariant domains ofholomorphic mappings of the Hilbert ball', Nonlin. Anal. 6 (1982), 1327-1334. [13] GOEBEL, K., AND REICH, S.: Uniform convexity, hyperbolic geometry and nonexpansive mappings, M. Dekker, 1984. [14] HEINS, M.H.: 'On the iteration of functions which are analytic and single-valued in a given multiply-connected region', Amer. J. Math. 63 (1941), 461-480. [15] HERVI~,M.: 'Quelques propri6t6s des applications analytiques d'une boule & m dimensions dans elle-m@me', J. Math. Pures Appl. 42 (1963), 117-147. [16] KAPELUSZNY, J., KUCZUMOW: W., AND REICH, S.: 'The Denjoy-Wolff theorem for condensing holomorphic mappings', J. Funct. Anal. 167 (1999), 79-93. [17] KAPELUSZNY, J., KUCZUMOW, T., AND REICH: S.: 'The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space', Adv. Math. 143 (1999), 111-123. [18] KHATSKEVICH, V., REICH, S., AND SHOIKHET, D.: 'Asymptotic behavior of solutions of evolution equations and the construction of holomorphic retractions', Math. Nachr. 189 (1998), 171-178. [19] KHATSKEVICH,V., AND SHOIKHET, D.: Differentiable operators and nonlinear equations, Birkh~user, 1994.

[20] KUBOTA, Y.: 'Iteration of holomorphic maps of the unit ball into itself', Proc. Amer. Math. Soc. 88 (1983), 476-480. [21] KuCZUMOW, T., REICH, S., AND SHOIEHET, D.: 'The existence and non-existence of common fixed points for commuting families of holomorphic mappings', Nonlin. Anal. (in press). [22] KUCZUMOW,T., AND STACHURA,A.: 'Iterates of holomorphic and kD-nonexpansive mappings in convex domains in C n', Adv. Math. 81 (1990), 90-98. [23] LYUBICH, YU., AND ZEMANEK, J.: 'Precompactness in the uniform ergodic theory', Studia Math. 112 (1994), 89-97. [24] ]VIAcCLUER, B.D.: 'Iterates of holomorphic self-maps of the unit ball in C n', Michigan Math. J. 30 (1983), 97-106. [25] MELLON, P.: 'Another look at results of Wolff and Julia type for J*-algebras', J. Math. Anal. Appl. 198 (1996), 444-457. [26] REICH, S.: 'Averaged mappings in the Hilbert ball', J. Math. Anal. Appl. 109 (1985), 199-206. [27] REICH, S.: 'The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball', J. Math. Anal. Appl. 157 (1991), 237-242. [28] REICH, S., AND SHOIKHET, D.: 'The Denjoy-Wolff theorem', Ann. Univ. Marine Curie-Sk~odowska 51 (1997), 219-240. [29] REICH, S., AND SHOIKHET, D.: 'Semigroups and generators on convex domains with the hyperbolic metric', Atti Accad. Naz. Lincei 8 (1997), 231-250. [30] SINE, R.: 'Behavior of iterates in the Poincar6 metric', Houston J. Math. 15 (1989), 273-289. [31] STACHURA,A.: 'Iterates of holomorphic self-maps of the unit ball in Hilbert space', Proc. Amer. Math. Soc. 93 (1985), 88-90. [32] VESENTINI,E.: 'Su un teorema di Wolffe Denjoy', Rend. Sere. Mat. Fis. Milano 53 (1983), 17-26. [33] WLODARCZYK, K.: 'Julia's lemma and Wolff's theorem for J*-algebras', Proc. Amer. Math. Soc. 99 (1987), 472-476. [34] WOLFF, J.: 'Sur l'it6ration des fonctions born6es', C.R. Acad. Sci. Paris 182 (1926), 200-201. [35] WOLFF, J.: 'Sur l'it6ration des fonctions holomorphes darts une r6gion, et dont les valeurs appartiennent & cette r6gion', C.R. Acad. Sci. Paris 182 (1926), 42-43. [36] WOLFF, 3.: 'Sur une g6n6ralisation d'un th6or~me de Schwarz', C.R. Acad. Sci. Paris 182 (1926), 918-920. [37] WOLFF, J.: 'Sur une g6n6ralisation d'un th6or~me de Schwarz', C.R. Acad. Sci. Paris 183 (1926), 500-502. [38] YANG, P.: 'Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains', preprint (1978). Simeon Reich David Shoikhet

MSC1991: 30D05, 32H15, 46G20, 47H17 DICKMAN FUNCTION - The function p(u) defined on (0, co) by the initial condition p(u) = 1 for 0 1. Interest attaches to this function because of its connection to 'smooth' numbers, i.e. numbers that are the product of many small prime numbers. Let ~(x, y) denote the number of positive integers less than or equal to x and free of prime divisors greater than y. When x is much larger than y, it is a simple matter of inclusion-and-exclusion counting (cf. also I n c l u s i o n - e x c l u s i o n f o r m u l a ) to show that ~(x, y)

123

DICKMAN F U N C T I O N

x l-Ip y, the resulting • (x, y) is approximated by x w ( u ) / l o g y, where w(u) is the Bukhstab function, defined by w(u) = 1/u, 1 2, where for u = 2 the right-hand derivative has to be taken, [2]. Unlike p, w oscillates and tends to a positive limit, equal to e -~. There are two combinatorial identities linking the Dickman function to ~(x, y). Early work is based on the Bukhstab identity: With p denoting a prime number, for y
z
The usual heuristic device of replacing a sum over prime numbers by an integral with 'prime density' 1/log t and replacing ~(r, s) with r p ( l o g r / l o g s ) leads to an identity which, when y = x 1/~ and z = x 1/("-1), simplifies to an integral equivalent to the definition of p(u). N.G. de Bruijn carried this idea to its limits in [1], but accuracy suffers when large and comparable estimated quantities must be subtracted. The more recent Hildebrand identity involves only additions and has the further advantage that the second input is the same y throughout:

extensive analysis is due to De Bruijn, and the function p(u) is sometimes termed the Dickman-De Bruijn function. One has [12]: a) up(u) = f:~-i p(v) dv for u > itive for all positive u, and hence, decreasing); b) p(u) is log-concave, that is, p t ( u ) p l - t ( v ) for u , v > 0 and 0 < t c) The L a p l a c e t r a n s f o r m c p ( s ) :=

=

-

exp

-

/0

t-le

1 (so that p is posfrom the definition,

p(tu + (1 - t)v) > < 1 (K. Alladi);

e-SUp(u) du =

dt

s

d) p(u) = exp(-u(logu O ( l o g l o g u / l o g u ) ) ) as u --~ ec.

= exp

-

in(s)] ;

+

loglogu

-

1 +

The Dickman function is one of a parameterized family of related functions p,(u), [14], and a wider class of similar delay-differential equations has been studied in [9]. A quick and simple bit of Mathematiea code suffices to calculate p to reasonable accuracy in the interval 0 < u < 8 using step-size e = 10 -4. This code calculates a table of values of p at intervals of length e, working back recursively into (0, 2]: p[u_,¢ ] : = p [ ~ , 4 =

Module[{}, If[u < 1, Return[Ill; If[~ ~ 2,, Return[N[1 -- Log[u]]]]; p[u -- c,c] - (e/2)(p[u -- 1 -- e,e]/(u -- e) + p[u -- 1,~]/u) ]

• (x,y) l o g x =

~(t,y)-[-+

~

•

,y

logp.

p~<_x p<_y

Applications require estimates uniform in u; the best known estimate along these lines, due to A. Hildebrand and based on the identity above, is

ff~(x,y)=xp(u) (1+O~\

( l o g ( u + 1)

]-o--og-y ) )

uniformly in y > 2 and 1 < u < exp(log (3/5)-¢ y). There are similar results for algebraic integers, [4]. There are also results concerning the number of smooth integers in an interval, and concerning the distribution of smooth integers into congruence classes [5], [6]. The Riemann hypothesis (cf. R i e m a n n h y p o t h e ses) implies ~(y~, y) = y~p(u) exp(O~ ( l o g ( u + l ) / l o g y ) )

[7]. The analytical properties of p are reasonably well understood; calculus, analysis of the Laplace transform, and the saddle-point method are the key tools. The first 124

References [1] BRUIJN, N.G. DE: 'On the number of uncancelled elements in the sieve of Eratosthenes', Indag. Math. 12 (1950), 247-256. (Nederl. Akad. Wetensch. Proc. 53 (1950), 803-812.) [2] BRUIJN, N.G. DE: 'On the number of positive integers _ x and free prime factors > y. II', Indag. Math. 28 (1966), 239-247. (Nederl. Akad. Wetensch. Proc. Set. A 69 (1966).) [3] DICKMAN,K.: 'On the frequency of numbers containing prime factors of a certain relative magnitude', Ark. Mat., Astron. oeh Fysik 22A, no. 10 (1930), 1-14. [4] FRIEDLANDER, J.: 'On the number of ideals free from large prime divisors', J. Reine Angew. Math. 255 (1972), 1-7. [5] FRIEDLANDER, J., AND GRANVILLE, A.: 'Integers without large prime factors, in short intervals', Philos. Trans. Royal Soc. 345 (1993), 339 348. [6] GRANVILLE, A.: 'On integers, without large prime factors, in arithmetic progressions II', Philos. Trans. Royal Soc. 345 (1993), 349-362. [7] HILDEBRAND,A.: 'Integers free of large prime factors and the Riemann hypothesis', Mathematika 31, no. 2 (1985), 258271. [8] HILDEBRAND, A., AND TENENBAUM, G.: 'Integers without large prime factors', J. Th4or. Nombres Bordeaux 5, no. 2 (1993), 411-484.

DIRAC ALGEBRA [9] HILDEBRAND, A., AND TENENBAUM, G.: 'On a class of differential-difference equations arising in number theory', J. Anal. Math. 61 (1993), 145 179. [1O] HUNTER, S., AND SORENSON, J.: 'Approximating the number

L. Lovisz (eds.): Handbook of Combinatorics, Elsevier, 1995, pp. 2003-2038. M. Hazewinkel M S C 1 9 9 1 : 05C12, 90C27

of integers free of large prime factors', Math. Comput. 66, no. 220 (1997), 1729-1741. [11] MITmNOVId, D.S., SANDOR, J., AND CRSTICI, B.:

Handbook

of number theory, Kluwer Acad. Publ., 1996, p. Sect. IV.21. [12] MOREE, P.: 'Psixyology and Diophantine equations', Diss. Univ. Leiden (1993). [13] SAIAS, E.: 'Sur le hombre des entiers sans grand facteur premier', J. Number Theory 32, no. 1 (1989), 78-99. [14] WHEELER, F.S.: 'Two differential-difference equations arising in number theory', Trans. Amer. Math. Soc. 318, no. 2

DIRAC ALGEBRA - T h e Dirac algebra arises from Dirac's solution [3] to the relativistic electron equation:

H 2 =

References [1] FRANK, A.: 'Connectivity and network flows', in R.L. GRAHAM, M. GR6TSCHEL, AND L. LOV£SZ (eds.): Handbook of Combinatorics, Elsevier, 1995, pp. 111-178. [2] GASS, S.I., AND HARMS, C.M. (eds.): Encyclopedia of Operations Research and Management Science, Kluwer Acad. Publ., 1996, pp. 166-167. [3] LovJ~sz, L., SHMOYS,D.B., AND TARDOS,E.: 'Combinatorics in computer science', in R.L. GRAHAM, M. GROTSCHEL,AND

4

TY~2 C2

=

= (a:~px + O@py -]- OZzpz +/3m0e) 2.

M S C 1991: l l A x x

The algorithm stops as soon as T spans the connected c o m p o n e n t of s in F. T h e shortest p a t h from s to any v in the connected c o m p o n e n t of s in F is given by the unique p a t h in T from s to v; the length is d(s, v). A very rough implementation of Dijkstra's algorithm takes O ( m n ) steps, where n = # V ( F ) , the cardinality of the vertex set of F, and m = ~ E ( F ) , the cardinality of the set of edges of F. It can be done much more efficiently, [3]. The algorithm can be easily a d a p t e d to directed edgeweighted graphs and networks. As of 2001, there is no efficient algorithm for finding longest paths in loop-free (directed) graphs.

)c 2

2 (P~+Py+ P2z)+

D. Hensley

Now, look through V(r) \ V(T) for a vertex v e V(F) \ V ( T ) and a u E V ( T ) for which d ( s , v ) + c ( u v ) is minimal, and add v and the edge uv to T.

+ p

Dirac found h y p e r c o m p l e x elements ax, ay, a~ and /3 (cf. also H y p e r c o m p l e x n u m b e r ) such t h a t

(1990), 491-523.

D U K S T R A A L G O R I T H M , Dijkstra shortest-path algorithm - Let r = ( V ( F ) , E ( F ) ) be a g r a p h with a specified vertex s and for every edge e a non-negative cost (length) c(e). T h e shortest-path problem is to find a shortest p a t h from s to every other vertex (node) i. Basically, the Dijkstra algorithm, which solves this problem, is a node-labeling g r e e d y a l g o r i t h m . It proceeds by constructing, one node at a time, a subtree T rooted at s (an s-arborescenee). If F is connected, T will be a spanning subtree (at the end). Initially, T = {s}. At any further step one knows the shortest distance d(s, v) and corresponding p a t h from s to any vertex v E T. This is the label of v. Of course, d(s, s) = O.

+

T h e Hamiltonian is given, after the usual substitutions for the linear m o m e n t u m c o m p o n e n t s , by

H=c-a

h_+

+

•

T h e time-dependent e q u a t i o n is known as Dirac's equation:

(c

V+

.

oc

)

e=i

°¢

N"

The elements ax, ay, az and satisfy a 2x = ay2 = a z2 = /32 = 1 and the a n t i - c o m m u t a t i v i t y relations:

aiaj+ajai

=O

fori,j E {z,y,z},

i•

j,

for i, j E {x, y, z}.

hi/3 + / 3 h i = 0

T h e Dirac representation of the matrices a and fl is

0 1 0

C~z

~=

1 0 0

0

0 Crx

(!01i) 0 0 -1

0 0

0 0 0

1

=

0 Crz

0

'

'

0

1

where rrx, Cry and Crz are the Pauli spin matrices (cf. also P a u l i m a t r i c e s ; D i r a c m a t r i c e s ) . This choice is not unique; pre-multiplying by any unitary matrix S and post-multiplying by S -1 will produce a new set of matrices satisfying the conditions. T h e defining relations 125

DIRAC ALGEBRA are often expressed more abstractly by the Dirac gamma matrices 7~=1, 7 i T j @ 7 j 7 i ~- O,

i=1,2,3,4,

MSC1991: 46Fxx

i C j, i , j = 1 , 2 , 3 , 4 .

The Dirac algebra is the 24-dimensional complex C l i f f o r d a l g e b r a generated by the gamma matrices under the usual matrix operations and is isomorphic to C(4), the ring of four-by-four matrices over the complex numbers C. The use of the complex numbers as scalars apparently is motivated by the fact that complex numbers are use to express solutions to the Schrhdinger wave equations (cf. also S c h r 5 d i n g e r e q u a t i o n ) . Two other 'Dirac algebras' commonly appear in the literature [4]; in each the relation among the squares of the generating elements, the metric, has been modified and the scalar field is the real numbers. Upon eomplexification, both become the algebra C(4). The modified metrics are the metrics of Minkowski space-time (cf. also M i n k o w s k i s p a c e ) and more easily illustrate the physics or the geometry. In one case (see, for example [2]) the metric is given by

"72=1,1 722=72 =72 =-1, the 7i generate a 24-dimensional real Clifford algebra that is isomorphic to H (2), the ring of two-by-two matrices over the real quaternion division ring. Every Clifford algebra C admits a Z2-grading, C = Co ® C1,

such that for all ri C Ci and sj C Cj, risj E C(i+j ) mod2" The subspace Co, spanned by the identity element and all products of an even number, is, in the present case, a subalgebra isomorphic to to the 23-dimensional Paul± algebra. The Majorana representation is the 24-dimensional real Clifford algebra with metric (as in [1]) =

-I,

=

=

=

1,

that is isomorphic to Re(4). Here, the elements q'2, % and 74 generate a subalgebra isomorphic to to the 23dimensional Paul± algebra. References [1] CARTAN,E.: The theory of spinors, Dover, 1966. [2] CORSON, E.M.: Introduction to tensors, spinors, and relativistic wave-equations, Chelsea, 1953. [3] DmAC, P.A.M.: 'The quantum theory of the electron', Proc. Royal Soc. London A l l 7 (1928), 610-624. [41 SALINGAaOS, N.A., AND WENE, G.P.: 'The Clifford algebra of differential forms', Acta Applic. Math. 4 (I985), 271 191.

G.P. Wene MSC 1991: 15A66, 81R25, 83C22, 81Q05

126

DIRAC DISTRIBUTION - Another term for d e l t a f u n c t i o n or D i r a c d e l t a - f u n c t i o n .

DIRAC MONOPOLE - A solution to the M a x w e l l e q u a t i o n s describing a point source of a m a g n e t i c field. In 1931, P.A.M. Dirac [1] considered the quantum mechanics of the electron in a magnetic field (due to a point source),

r B = g~5-,

(1)

where r = V/X2 + y2 + z ~ is the length of the position vector r = (x, y, z) in the Cartesian coordinates and g is a constant determining the strength of the field, known as a magnetic charge of the monopole. Since the induction vector B in (1) is central, it can be conveniently written in the s p h e r i c a l c o o r d i n a t e s r, 0, ¢ defined by x = r sin0cos¢, y = r s i n 0 s i n ¢ , z = r cos0, 0 < 0 < 7r, 0 _< ¢ < 27r. In these coordinates, only the radial component of B is non-zero and equals Br = g/r 2. Maxwell's equations imply that there is no single vector potential corresponding to B defined on the whole of R 3. However, Dirac found that B = V x A +, with vector potentials A ± whose only non-zero components are in the azimuthal direction and read

A~: - r sing 0 ( + 1 - cos0).

(2)

The potentials A +, A - are singular at 0 = rc (the negative z-axis) and 0 = 0 (the positive z-axis), respectively. These singularities are known as Dirac's string singularities. The union of the regions in which A ± are welldefined covers the whole of R ~. In the intersection of these regions (0 < 0 < ~r) the vector potentials A ± are related by the g a u g e t r a n s f o r m a t i o n , A + = A - + V x , with X = 2g¢. If there is an electron in the magnetic field B, then in the region where both A + and A - are welldefined, the wave functions of the electron corresponding to different vector potentials should be related by the gauge transformation X, i.e., ~ + = eieX/h Tuuw_= e 2 i e g ¢ / h ~ _ ,

where e is the electric charge of the electron and h is the P l a n c k c o n s t a n t divided by 2~r. The wave function ~ + is single valued if and only if 2eg/h = n for an integer n, i.e. if and only if the magnetic charge attains discrete values h

g = n~,

n = o, +1, + 2 , . . . .

(3)

Thus, the consistency of the monopole field (1) with quantum mechanics can be achieved, provided the magnetic charge g be quantized. Equation (3) expresses also 'duality' (reciprocity) between magnetic and electric charges: If g and e are interchanged, (3) remains the

DIRAC QUANTIZATION same. Dirac used this fact to explain the observed quantization of the electric charge: Should a magnetic monopole of charge, say, g exist, then by the above argument the electric charge would be allowed to have only discrete values e = nh/29. This argument, however, would leave the quantization of magnetic charge unexplained, a fact that Dirac found disappointing [1]. In 1975, T.T. Wu and C.N. Yang [9] observed that Dirac's monopole of magnetic charge g = nh/2e has a natural topological interpretation as a c o n n e c t i o n in the U(1) principal bundle over the two-sphere S 2 with the first Chern number (the winding number) - n (cf. Connections on a manifold; Principal fibre bundle; or [3] for a review). In natural units h = e = 1, the potentials A i can be written as one-forms A 4- =

2(=I=1 -

cos0) de,

and they are a connection one-form written in two charts covering S 2. More precisely, ¢, 0 above are coordinates of the two-sphere. Then 0 = 0 is the north pole and A - is well-defined everywhere outside the north pole, for example on a chart H_ covering the southern hemisphere including the equator (0 > ~ c / 2 - e). On the other hand, 0 = 7r is the south pole, and thus A + is well-defined everywhere except the south pole, for example on a chart H+ covering the northern hemisphere including the equator (0 < 7r/2 + e). The intersection H+ N H _ is parametrized by the azimuthal angle ¢. In order to combine this local system into a U(1)-principal bundle, on H+ n H _ the U(1)-coordinate ¢+ over H+ must be related to the U(1)-coordinate ¢ _ over H_ by ¢+ = ¢ _ - n¢, with integer n. This explains the appearance of Dirac's string singularity when the A T are extended to H+, and the fact that it can be removed by a gauge transformation which requires Dirac's quantization condition. Thus, the trivial bundle S ~ x U(1) admits no monopole (charge 0-monopole). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge h/2e is the connection in the H o p f f i b r a t i o n S 3 --+ S 2, while the monopole of charge with n > 1 corresponds to the U(1)bundle over S 2 with the lens s p a c e Ln = SU(2)/Z~ as a total space (Zn is viewed inside SU(2) as a subgroup of nth roots of the unit matrix) [7]. The Dirac monopole is an example of an Abelian monopole, i.e., a solution of field equations of gauge theory with Abelian gauge group U(1). Since the mid1970s there has been a considerable interest in nonAbelian monopoles, in particular those related to the SU(2) gauge theories. In pure mathematics this was triggered in particular by the appearance of SU(2) gauge theory in the classification of four-manifolds by S.K. Donaldson [2]. However, in 1994, E. Witten [8] showed

that certain Abelian monopole equations motivated by the supersymmetric quantum field theory [5], [6] and known as the Seiberg-Witten equations, can be used to derive both the Donaldson invariants of four-manifolds as well as new ones (the Sciberg-Witten invariants; cf. also F o u r - d i m e n s i o n a l m a n i f o l d ) . It was soon noted [4] that the Dirac gauge potential A - with n = - 1 provides a bosonic part of the simplest (not L 2) solution to Seiberg-Witten equations. Witten's observation, as well as the appearance of magnetic monopoles in string theory, revived the interest in both monopoles and the reciprocity between electric and magnetic charges (electricmagnetic duality). References [1] DmAC, P.A.M.: 'Quantized singularities in the electromagnetic field', Proc. Royal Soc. London A133 (1931), 60-72. [2] DONALDSON,S.K., AND KRONHEIMER, P.B.: The geometry of four-manifolds, Clarendon Press/Oxford Univ. Press, 1990. [3] EGUCHI, T., GILKEY, P.B., AND HANSON, A.J.: 'Gravitation, gauge theories and differential geometry', Phys. Rept. 66, no. 6 (1980), 213-393. [4] FREUND, P.G.O.: 'Dirac monopoles and the Seiberg-Witten monopole equations', J. Math. Phys. 36 (1995), 2673-2674. [5] SEIBERG, N., AND WITTEN, E.: 'Electric-magnetic duality: monopole condensation, and confinement in N -- 2 supersymmetric Yang-Mills theory', Nucl. Phys. B426 (1994), 19-52. [6] SEIBERG, N., AND WITTEN, E.: 'Monopoles, duality and chiral symmetry breaking in N ----2 supersymmetric QCD', Nucl. Phys. B431 (1994), 484-550. [7] TRAUTMAN, A.: 'Solutions of Maxwell and Yang-Mills equations associated with Hopf fiberings', Internat. J. Theoret. Phys. 16 (1977), 561-565. [8] WITTEN, E.: 'Monopoles and four-manifolds', Math. Res. Lett. 1 (1994), 769-796. [9] Wu, T.T., AND YANG, C.N.: 'Concept of nonintegrable phase factors and global formulation of gauge fields', Phys. Rev. DI2 (1975), 3845-3857.

T. Brzezinski MSC1991:81V10 DIRAC QUANTIZATION, canonical quantizationA term referring to a proceeding that associates to a c o m m u t a t i v e a l g e b r a of physical observables, of a classical mechanical system, a non-commutative algebra of linear operators on a suitable H i l b e r t s p a c e (or, more generally, on a locally convex t o p o l o g i c a l v e c t o r space; cf. also L i n e a r o p e r a t o r ) . Such a proceeding, called canonical quantization, has been first mathematically axiomatized by P.A.M. Dirac [7] (which justifies the name). Subsequently, many other contributions have been given to generalize this concept in a geometrical way, by obtaining constructive representations of commutative algebras characterizing differential manifolds in non-commutative algebras. The most remarkable examples are geometric quantization (B. Kostant and J.M. Souriau [18], [34], [38]) and deformation quantization (F. Bayen, M.V. Karasev, M. Flato, C. Fronsdal, A. 127

DIRAC QUANTIZATION Lichnerowicz, D. Sternheimer, and V.P. Maslov [4], [3], [12], [17]). These coincide for non-relativistic systems of a finite number of particles with the (Dirac) canonical quantization. So, 'Dirac quantization' can be used also as synonymous of 'canonical quantization'. However, nowadays (2000) the term 'Dirac quantizations' means quantizations of partial differential equations that not necessarily coincide with canonical quantizations. For an example, see the Crumeyrolle-Pr~staro quantizations of partial differential equations [24], [25], [28], [27]. Furthermore, for Lagrangian field theories, an approach of functional type, called the Feynman path method, has had a big success. In fact, this allows one to obtain approximated descriptions of electroweak nuclear phenomena, where the perturbative methods can be of practical convenience. However, the Feynman path method is, in general, not well mathematically founded, as it requires integration on infinite-dimensional manifolds. In some sense, this aspect has been improved in the framework of gauge theory, as the quotient with respect to gauge groups produces finite-dimensional manifolds [2], [8], [9], [10], [11], [14], [15], [1@ (A lot of recent mathematical studies are in some sense related to such a point of view and have given new interesting prospects in pure mathematics. See e.g. [13].) Moreover, the Feynman path method is related to the so-called covari-

ant quantization, which prescribes the quantum bracket [¢J (x), ~i(x,)] for the operators ~i(x) corresponding to the local components ¢i of a field ¢, 'localized' at the point x of the space-time M: [¢~ (x), ~i (x')] = ihG id (x, x')ln, N . .

where G ~ (x, x') is the propagator of the theory [19]. This approach is essentially related to the Peierls bracket [22], but has many limitations and inconsistencies from the mathematical point of view. In fact, first of all it refers to linear dynamic equations of variational type; furthermore, it does not work well for chiral fields, i.e., fields that are sections of non-vector bundles (see Q u a n t u m field t h e o r y ) . Any attempt to extend such proceedings to theories described by means of nonlinear and non-Lagrangian partial differential equations did fail, until some recent geometric studies on the quantization of partial differential equations [24], [25], [28], [27]. More precisely, in [24], [25], [28], [27] the concept of formal Dirac quantization of partial differential equations is introduced, that is, roughly speaking, a procedure that associates a m e a s u r e space (quantum situs) to a partial differential equation. This quantization becomes effective if on (the classic limit of) the quantum situs one recognizes (pre-)spectral measures (quantum

spectral measures of partial differential equations). 128

The axiomatization of the concept of (Dirac) quantization of a classical system, represented by a partial differential equation Ek C J~k(W), can be given on the ground of mathematical logic by means of algebra homomorphisms P(f~(Ek)c) ~ N, where 7)(ft(Ek)c) is the logic of Ek, that is the B o o l e a n a l g e b r a of subsets of the classic limit f~(Ek)~ of the quantum situs f~(Ek) of Ek (in other words, f~(Ek)~ is the set of solutions of Ek), and .4 is a quantum logic, that is, an algebra of (self-adjoint) operators on a locally convex topological vector (Hilbert) space 7/ (el. also H i l b e r t space; L o c a l l y c o n v e x space; S e l f - a d j o i n t o p e r a tor): .4 C L(7/). This is equivalent to the assignment of pre-spectral measures on ft(Ek)c: ft(Ek)~ o-+ L(7/) [24], [25], [28], [27], [33]. In this way it is possible to give a generalization of the concept of covariant quantization in the general framework of the geometric theory of partial differential equations. (Of course, there are many effective quantizations, but the most interesting from the physical point of view is the covariant quantization or the canonical quantization, that is, the covariant quantization observed by a physical frame.) In fact, in that geometric context, it is proved that any physical observable deforms the original partial differential equation around a classical solution. In this way one can associate to the Lie a l g e b r a of classical observables a non-commutative algebra, i.e., the quantum algebra of the system, defined by means of the bracket [~ (s), ~(s)] = ihG(fl, f2; s)lT/(s), for any two observables fi, i = 1,2, at the solutionsection s of Ek. Here, ~(s) are operator-valued distributions, at the section s, on a locally convex topological vector space 7/(s), depending on s, and G is a distributive kernel, which generalizes the usual concept of propagator made for linear differential operators [6], [19], and which is canonically associated to the non-linear dynamic equation of the theory at the section s [24], [25], [28], [27]. In [24], [25], [28], [27], a geometric interpretation of the concept of propagator for non-linear partial differential equations is given. This is related to the concept of (integral) bordism [29], [31], [30]. In this way the quantization of partial differential equations is connected to this important sector of a l g e b r a i c t o p o l o g y , introduced by R. Thorn and L.S. Pontryagin [1], [23], [35], [36]. This geometric approach justifies in some sense the belief that 'quantization' is synonymous of 'deformation' (see e.g., [4], [3], [12], [17] and also the modern concept of quantum geometry in [5], [21], [37]). More recently (1990s), A. Pr£staro has generalized the concept of Dirac quantizations for partial differential equations

DIRICHLET CONVOLUTION also to n o n - c o m m u t a t i v e ( q u a n t u m ) p a r t i a l differential e q u a t i o n s , i.e., p a r t i a l differential e q u a t i o n s b u i l t in t h e c a t e g o r y of q u a n t u m m a n i f o l d s (see [27], [26], [32]). In

this way one gets a mathematically well-founded geometric t h e o r y of q u a n t u m p a r t i a l differential e q u a t i o n s t h a t is useful t o f o r m u l a t e a q u a n t u m field t h e o r y unifying g r a v i t y a n d e l e c t r o m a g n e t i c forces with nuclear forces. See also t h e a l g e b r a i c c a t e g o r i a l f o r m u l a t i o n of q u a n t i z a t i o n s on H o p f a l g e b r a s given b y V. L y c h a g i n [20] (cf. also H o p f a l g e b r a ) . Since t h e q u a n t u m g r o u p is f o r m u l a t e d in t h e l a n g u a g e of H o p f a l g e b r a s (cf. also Q u a n t u m g r o u p s ) , m a n y f o r m a l q u a n t u m theories are given in t h e f r a m e w o r k of such an algebra. However, t h e r e is also a m o r e s t r u c t u r a l g e o m e t r i c reason t h a t emphasizes this a l g e b r a . In fact, in [29], [31], [30], [26], [32] it is p r o v e d t h a t on t h e space of all c o n s e r v a t i o n laws of a ( q u a n t u m ) p a r t i a l differential e q u a t i o n the s t r u c t u r e of ( q u a n t u m ) H o p f a l g e b r a can be recognized. References

[15] HAAG, R.: Local quantum physics, fields, particles, algebras, Springer, 1992. [16] HORZ~Y, S.S.: Introduction to algebraic quantum field theory, Kluwer Acad. Publ., 1990. [17] KARASEV, M.V., AND MASLOV, V.P.: 'Asymptotic and geometric quantization', Russian Math. Surveys 39, no. 6 (1984), 133-205. [18] KOSTANT, B.: Graded manifolds, graded Lie theory and prequantization, Vol. 570 of Lecture Notes in Mathematics, Springer, 1991, pp. 229-232. [19] LICHNEROWICZ, A.: 'Champs spinoriels et propagateurs on en relativit~ g~n~rale', Bull. Soc. Math. France 92 (1964), 11-100. [20] LYCHAGIN,V.: 'Calculus and quantizations over Hopf algebras', Acta Applic. Math. 51 (1998), 303-352. [21] MANIN, YU.I.: 'Quantum groups and non-commutative geometry', Montreal Univ. Preprint CRM-1561 (1988). [22] PEmRLS, R.: 'The commutation laws of relativistic field theory', Proc. Royal Soc. London A214 (1952), 143-157. [23] PONTRJAGIN,L.S.: 'Smooth manifolds and their applications in homotopy theory', Amer. Math. Soc. Transl. 11 (1959), 1-114. [24] PR£STARO, A.: 'Quantum geometry of PDE's', Rept. Math. Phys. 30, no. 3 (1991), 273. [25] PR£STARO, A.: 'Geometry of quantized super PDE's', Amer. Math. Soc. Transl. 167 (1995), 165. [26] PRJ~STARO,A.: '(Co)bordisms in PDEs and quantum PDEs', Rept. Math. Phys. 38, no. 3 (1996), 443-455. [27] PRJ~STARO, A.: Geometry of PDEs and mechanics, World Sci., 1996. [28] PRJ~STARO, A.: 'Quantum geometry of super PDEs', Rept. Math. Phys. 37, no. 1 (1996), 23-140. [29] PRJ~STARO, A.: 'Quantum and integral (co)bordisms in partial differential equations', Acta Applic. Math. 51 (1998), 243-302. [30] PRJ~STARO, A.: '(Co)bordism groups in PDEs', Acta Applic. Math. 59, no. 2 (1999), 111-201. [31] PR~.STARO,A.: 'Quantum and integral bordism groups in the Navier-Stokes equation', in J. SZENTHE (ed.): New Developments in Differential Geometry (Budapest, 1996), Kluwer Acad. Publ., 1999, pp. 344-360. [32] PRA.STARO,A.: '(Co)bordism groups in quantum PDEs', Acta Applic. Math. 64 (2000), 111-127. [33] PRJ~STARO,A.: 'Quantum manifolds and integral (co)bordism groups in quantum partial differential equations', Nonlin. Anal. to a p p e a r (2001). [34] SOURIAU,J.M.: Structure des syst~mes dynamiques, Dunod, 1970. [35] THOM, R.: 'Quelques propri~tds globales des vari~t~s diffdrentiables', Comment. Math. Helv. 28 (1954), 17-86. [36] THOM, R.: 'Remarques sur les probl~mes comportant des in~qualities diffdrentielles globales', Bull. Soc. Math. France 8T (1959), 455-461. [37] VILENKIN,N.JA., AND KLIMYK, A.V.: Representations of Lie groups and special functions, Vol. I-III, Kluwer Acad. Publ.,

[1] ATIYAH, M.: The geometry and physics of knots, Cambridge Univ. Press, 1990. [2] BAEZ, J., SEGAL, I.E., AND ZHOU, Z.: Introduction to algebraic and constructive quantum field theory, Princeton Univ. Press, 1992. [3] BAYEN, F., FLATO, M., FRONSDAL, C., AND LICHNEROWICZ, A.: 'Quantum mechanics as a deformation of classical mechanics', Left. Math. Phys. 1 (1975/77), 521-570. [4] BAYEN, F., FLATO, IV!., FRONSDAL, C., LICHNEROWICZ, A., AND STERNHEIMER, D.: 'Deformation theory and quantization I-IF, Ann. Phys. Iii (1978), 61-110. [5] CONNES, A.: Noncommutative geometry, Acad. Press, 1994. [6] DIMOCK, J.: 'Algebras of local observables on manifold', Comm. Math. Phys. 77 (1980), 219-228. [7] DIRAC, P.M.A.: The principles of quantum mechanics, Oxford Univ. Press, 1958. [8] DOPLICHER, S., HAAG, R., AND ROBERTS, J.E.: 'Fields, observables and gauge transformations I', Comm. Math. Phys. 13 (1969), 1. [9] DOPLICHER, S., HAAG, R., AND ROBERTS, J.E.: 'Fields, observables and gauge transformations, II', Comm. Math. Phys. 1 5 (1969), 173. [10] DOPLICHER, S., HAAG, R., AND ROBERTS, J.E.: 'Local observables and particle statistics, I', Comm. Math. Phys. 23 (1971), 199. [Ii] DOPLICHER, S., HAAG, R., AND ROBERTS, J.E.: 'Local observables and particle statistics, II', Comm. Math. Phys. 35 (1974), 49. [12] FLATO, M., AND STERNEEIMER, D.: 'Quantum groups, star products and cyclic cohomology', in H. ARAKI, K.R. ITO, A. KISHIMOTO, AND I. OJIMA (eds.): Quantum and NonCommutative Analysis, Math. Phys. Stud., Kluwer Acad. Publ., 1993, pp. 239-251. [13] FUKAYA, K.: 'Geometry of gauge field', in T. KOTAKE, S. NISHIKAWA,AND R. SCHOEN (eds.): Geometry and Global

M S C 1991: 81Qxx

Analysis (Rept. First MSJ Internat. Res. Inst. (July I2-23, 1993), TShoku Univ., Sendal, 1993. [14] GLIMM, J., AND JAFEE, A.: Quantum physics. A functional integral point of view, Springer, 1981.

DIRICHLET CONVOLUTION The Dirichlet convolution of two arithmetical functions f and g is defined

1991/9a. [38] WOODHOUSE, M.: Geometric quantization, Press, 1980.

Oford Univ. A. Prdstaro

129

DIRICHLET

CONVOLUTION

as

(f . g)(n) = E f(d)g din

where the sum is over the positive divisors d of n (cf. also A r i t h m e t i c f u n c t i o n ) . General background material on the Dirichlet convolution can be found in, e.g., [1], [6], [8]. Sums of the form ~dln f(d)g(n/d) played an important role from the very beginning of the theory of arithmetical functions. Many results from early times involved these sums. For example, in 1857 J. Liouville published a long list of arithmetical identities of this type (see [5]). It is fruitful to treat the sums ~ d l ~ f(d)g(n/d) as giving a binary operation on the set of arithmetical functions (cf. also B i n a r y r e l a t i o n ) . This aspect was introduced by E.T. Bell [2] and M. Cipolla [3] in 1915. The set of arithmetical functions forms a c o m m u t a t i v e r i n g with unity under the usual addition and the Dirichlet convolution. An arithmetical function f possesses a Diriehlet inverse if and only if f(1) ~ 0. For example, the Dirichlet inverse of the constant function 1 is the M S b i u s f u n c t i o n #. The Mb'bius inversion formula states that

f(n) = E g ( d ) ~=~g(n) = E f(d)p din

@

din

The relation of the Dirichlet convolution with D i r i c h l e t s e r i e s is also important. There are many analogues and generalizations of the Dirichlet convolution; for example, E. Cohen [4] defined the unitary convolution as

(feg)(n)

3--:.f( ) g

-j

,

dH~ where the sum is over the positive divisors d of n such that GCD(d, n/d) = 1, see also [10]. W. Narkiewicz [7] developed a more general convolution:

(f *Ag)(n) = E f(d)g deA(n) where, for each n, A(n) is a subset of the set of the positive divisors of n. See [9] for a survey of various binary operations on the set of arithmetical functions. References [1] APOSTOL, T.M.: Introduction to analytic number theory, Springer, 1976. [2] BELL, E.T.: 'An arithmetical theory of certain numerical functions', Univ. Wash. Publ. Math. Phys. Sci. I, no. 1

[6] MCCARTHY, P.J.: Introduction to arithmetical functions, Springer, 1986. [7] NARKIEWICZ, W.: ' O n a class of a r i t h m e t i c a l convolutions', Colloq. Math. 10 (1963), 81-94. [3] SIVARAMAKRISHNAN, R.: Classical theory of arithmetic functions, Vol. 126 of Monographs and Textbooks in Pure and Applied Math., M. Dekker, 1989. [9] SUBBARAO, M.V.: 'On some a r i t h m e t i c convolutions': The Theory of Arithmetic Functions, Vol. 251 of Lecture Notes in Mathematics, Springer, 1972, pp. 247-271. [10] VAIDYANATHASWAMY,R.: ' T h e theory of multiplicative arithmetic functions', Trans. Amer. Math. Soc. 33 (1931), 579 662. Pentti Haukkanen

MSC1991:11A25 D I R I C H L E T DENSITY Let / ( be an algebraic number field (cf. also A l g e b r a i c n u m b e r ) and let A be a set of prime ideals (of the ring of integers A~:) of K . If an equality of the form Z

- s = a log

1

+ g(s)

pEA

holds, where g(s) is regular in the closed half-plane Re(s) _> 1, then A is a regular set of prime ideals and a is called its Dirichlet density. Here, N ( p ) is the norm of p, i.e. the number of elements of the residue field AK/p.

Examples. i) The set of all prime ideals of K is regular with Dirichlet density 1. ii) Let L / K be a finite extension and A the set of all prime ideals ~ in L t h a t are of degree 1 over K (i.e. [AL/f~: AK/p] = 1, where ~ is the prime ideal ~ n AK under ~ ) . Then A is regular with Dirichlet density 1. iii) Let L / K be a finite normal extension and A the set of all prime ideals ~ in K that split in L (i.e. pAL is a product of [L : K] prime ideals in L of degree 1). Then A is regular with Dirichlet density [L : K] -1. The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set A has Dirichlet density a if N ( P ) -~ = 1. a log 1

lira ~eA

s~l

1--s

References [1] NARKIEWICZ, W.: Elementary and analytic theory of algebraic numbers, second ed., P W N / S p r i n g e r , 1990, p. Sect. 7.2. M. Hazewinkel

(1915). [3] CIPOLLA,M.: 'Sol principi del calculo arithmetico integrale',

MSC1991: 11R44, 11R45

Atti Accad. Gioenia Cantonia 5, no. 8 (1915). [4] COHEN, E.: ' A r i t h m e t i c a l functions associated with the unit a r y divisors of an integer', Math. Z. 74 (1960), 66-80. [5] DICKSON, L.E.: History of the theory of numbers, Vol. I, Chelsea, reprint, 1952.

DIRICHLET E I G E N V A L U E - Consider a bounded domain ~ C R ~ with a piecewise smooth boundary cgfl. A is a Dirichlet eigenvalue of ~ if there exists a function

130

DIRICHLET EIGENVALUE u C C2(f/) r~ C°(~) (a Dirichlet eigenfunction) satisfying the following Dirichlet boundary value problem (cf. also D i r i c h l e t b o u n d a r y c o n d i t i o n s ) : -Au=Au u = 0 where A

is the

Laplace

in~,

(2)

operator

(i.e., A

=

~"=1 02/Ox~) • Dirichlet eigenvalues (with n = 2) were introduced in the study of the vibrations of the clamped membrane in the nineteenth century. In fact, they are proportional to the square of the eigenfrequencies of the membrane with fixed boundary. See [9] for a review and historical remarks. Provided f / i s bounded and the boundary Of/is sumciently regular, the Dirichlet Laplacian has a discrete spectrum of infinitely many positive eigenvalues with no finite accumulation point [13]: 0 < ~ l ( a ) < ~2(~) _<""

(3)

oo as k -+

The Dirichlet eigenvalues are characterized by the

max-rain princ~gle [5]: Ak = sup inf f a (Vu)2 dx '

(4)

where the inf is taken over all u C H l ( ~ ) orthogonal to ~ 1 , - . - , ~ k - 1 E Hol(f/), and the sup is taken over all choices of {~i}i=1. k-1 For simply-connected domains it follows from the max-min principle (4) that ;kl (~) is non-degenerate and the corresponding eigenfunction ul is positive in the interior of ~. For higher values of k the nodal lines of the kth eigenfunction divide f/ into no more than k - 1 subregions (nodal domains; this is Courant's nodal line theorem [5]). Along this subject, notice the proof of A.D. Melas [11] of the nodal line conjecture for plane domains (if ~ is a bounded, smooth, convex domain, the nodal line of u2 always meets 0~). W e y l a s y m p t o t i c s . For large values of k, if ~ C R ~, H. Weyl [18], [17] proved

4rc2k2/,~

(5)

where ]~] and Cn = 7r~/2/F(n/2 + 1) are, respectively, the volumes of ft and of the unit ball in R n. P61ya c o n j e c t u r e . For any plane-covering domain (i.e., a domain that can be used to tile the plane without gaps, nor overlaps, allowing rotations, translations and reflections of itself), G. Pdlya [14] proved that 4~k )~k > ~

fork=l,2,...,

47r2k2/n Ak >_ (Cnlf/])2/~

(1)

incOf/,

fa u2 dx

the Weyl asymptotics of )~k, (5), is a lower bound for Ak, i.e.,

(6)

and conjectured the same bound for any bounded domain in R 2 (here A is the area of the domain). P61ya's conjecture in n dimensions is equivalent to saying that

fork = 1,2, . . . .

(7)

A result analogous to (6) for the Neumann eigenvalues of tiling domains, with the sign of the equalities reversed, also holds (cf. also N e u m a n n eigenvalue). The best result to date (2000) towards the proof of the P61ya conjecture is the bound [10] k

E Ai > n 4~r2k1+2/n i=1 _ n+2(C~lf/l)2/n

k=l,2,...,

(8)

proven using the asymptotic behaviour of the heat kernel of ~2 (cf. also H e a t e q u a t i o n ) and the connection between the heat kernel and the Dirichlet eigenvalues of a domain (see, e.g., [6] for a review and related results). K a c p r o b l e m . Dirichlet eigenvalues are completely characterized by the geometry of the domain ft. The inverse problem, i.e., up to what extent the geometry of oo f / c a n be recovered from the knowledge of {~}~=1, was posed by M. Kac in [8]. If n = 2, for example, and Oft is smooth (in particular Oft does not have corners), then the distribution function behaves as A

E e-Xkt ~ 4~----t+ @

L

1 + 6 (1 - r) + O(t),

(9)

k=l

as t -+ 0, where A is the area, L the perimeter and r the number of holes of ~, so at least these features of the domain can be recovered from knowledge of all the eigenvalues (the first term in (9) is just a consequence of Weyl's asymptotics). However, complete recovery of the geometry is impossible, as was later shown by C. Gordon, D. Web and S. Wolpert, who constructed two isospectral domains in R 2 with different geometries [7]. E i g e n v a l u e s a n d g e o m e t r y . The inverse of the square root of a Dirichlet eigenvalue is a length that may be compared with other characteristic lengths of the domain ~. A typical such comparison is the R a y l e i g h F a b e r - K r a h n i n e q u a l i t y . Another inequality along these lines is the following: If ~ is a simply connected domain in R 2 and ra is the radius of the largest disc contained in ~, then there is a universal constant a such that a ~l(f/) _> r 7 (10) (as of 2000, the best, not yet optimal, constant in (10) is a = 0.6197; see [2] for details and historical facts). For other isoperimetric inequalities, see, e.g., [1], [12], [15]. In the same vein, one can also compare Dirichlet and Neumann eigenvalues (see N e u m a n n eigenvalue). 131

DIRICHLET

EIGENVALUE

Because of the connection between p o t e n t i a l t h e o r y and B r o w n i a n m o t i o n , it is possible to use probabilistic methods to find properties of Dirichlet eigenvalues. One such property was found by H. Brascamp and E.H. Lieb [3] for At: If ~11 and ~2 are domains in R n, and one sets ~t -- t~1 + (I - t)~2, then A1(~t) _~ tA1(~1) + (i - t)A2(f~2) for all t E (0, I). Another example of the use of probabilistic methods is the proof of (i0) by R. Bafiuelos and T. Carroll [2]. To conclude, note that it is possible to define Dirichlet eigenvalues for much more general domains in R ~ (see, e.g., [16, p. 263]), and also for the Laplace-Beltrami operator defined on domains in Riemannian manifolds (see, e.g., [4]). References

[1] ASHBAUGH, M.S., AND BENGURIA, R.D.: 'Isoperimetrie inequalities for eigenvalue ratios': Syrup. Math., Vol. 35, Cambridge Univ. Press, 1994, pp. 1-36. [2] BAI~IUELOS,R., AND CARROLL, T.: 'Brownian motion and the fundamental frequency of a drum', Duke Math. J. 75 (1994), 575-602. [3] BRASCAMP, H., AND LIEB, E.H.: 'On extensions of the B r u n n Minkowski and Pr~kopa-Leindler theorem, including inequalities for log-concave functions, and with an application to the diffusion equation', Y. Funct. Anal. 22 (1976), 366 389. [4] CHAVEL, I.: Eigenvalues in Riemannian geometry, Vol. 115 of Pure Appl. Math., Acad. Press, 1984. [5] COURANT, R., AND HILBERT, D.: Methoden der mathematischen Physik, Vol. I, Springer, 1931, English transl.: Methods of mathematical physics, vol. I., Interscience, 1953. [6] DAVIES, E.B.: Heat kernels and spectral theory, VoI. 92 of Tracts in Math., Cambridge Univ. Press, 1989. [7] GORDON, C., WEBB, D., AND WOLPERT, S.: 'Isospectral plane domains and surfaces via Riemannian orbifolds', 1nvent. Math. 110 (1992), 1-22. [8] KAC, M.: 'Can one hear the shape of a drum?', Amer. Math. Monthly 73, no. 4 (1966), 1-23. [9] KUTTLER, J.R., AND SIGILLITO, V.G.: 'Eigenvalues of the Laplacian in two dimensions', S I A M Review 26 (1984), 163193. [10] LI, P., AND YAU, S.W.: 'On the Schr5dinger equation and the eigenvalue problem', Commun. Math. Phys. 88 (1983), 309-318. [11] MELAS, A.D.: 'On the nodal line of the second eigenfunction of the Laplacian in R 2', J. Diff. Geom. 35 (1992), 255-263. [12] OSSERMAN, R.: 'Isoperimetric inequalities and eigenvalues of the Laplacian': Proc. Internat. Congress of Math. Helsinki, Acad. Sci. Fennica, 1978, pp. 435-441. [13] POCKELS, F.: 'i)ber die partielle Differentialgleichung Au ÷ k2u z 0 und deren Auftreten in die mathematischen Physik', Z. Math. Physik 37 (1892), 100-105. [14] POLYA, G.: 'On the eigenvalues of vibrating membranes', Proc. London Math. Soc. 11, no. 3 (1961), 419-433. [15] POLYA, G., AND SZEGO, G.: Isoperimetric inequalities in mathematical physics, Vol. 27 of Ann. of Math. Stud., Princeton Univ. Press, 1951. [16] REED, IV[., AND SIMON, B.: Methods of modern mathematical physics IV: Analysis of operators, Acad. Press, 1978.

132

[17] WEYL, H.: 'Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen', Math. Ann. 71 (1911), 441-479. []8] WEYL, H.: 'Ramifications, old and new, of the eigenvalue problem', Bull. Amer. Math. Soc. 56 (1950), 115-139. R a f a e l D. B e n g u r i a

MSC 1991: 35J05, 35J25 DITKIN S E T - A closed subset E of a l o c a l l y c o m p a c t s p a c e X is called a Ditkin set (with respect to a regular function algebra .A(X) defined on X; cf. Alg e b r a o f f u n c t i o n s ) if each f E A ( X ) vanishing on E can be approximated, arbitrarily closely, by functions fg with g E A(X) and g vanishing 'near' E (i.e. on a neighbourhood of E). The notion of a Ditkin set is closely related to, but more restrictive than, that of a set of spectral synthesis (cf. S p e c t r a l s y n t h e s i s ) : for such a set the requirement is that each f C A(X) vanishing on E can be approximated by functions g E A(X) vanishing near E. The closed ideal of all f C .A(X) vanishing on E is usually denoted by IE. Denoting the ideal of all f C A(X) vanishing near E by J~ and its closure by JE, one has JE C IE. Now E is a set of spectral synthesis if JE = IE, whereas E is a Ditkin set if each f C IE belongs to the closure of f J ~ (or, equivalently, to the closure of fJE). It is a famous open problem (as of 2000) whether (in specific cases) each set of spectral synthesis is actually a Ditkin set (this problem may be called the synthesis-Ditkin problem; in [1] it is called the

C-set-S-set problem). Ditkin sets were first studied for the F o u r i e r a l g e b r a A(G) -~ LI(G), with the norm defined by II~l = IIflll; here, G is any locally compact Abelian group, G is its dual group, and f" is the Fourier transform of f (cf. also H a r m o n i c analysis; F o u r i e r t r a n s f o r m ) . A.P. Calder6n (1956) studied this kind of set in an effort to obtain results about sets of spectral synthesis. Therefore, Ditkin sets are sometimes called Calderdn sets or C-sets; cf. [5] and [10], respectively. The name 'Ditkin set', attributed in [6, p. 183] to C.S. Herz, refers to work of V.A. Ditkin (1910-1987) in his seminal paper [2]; results from this paper were later studied and generalized in [11]. In [8] the term Wiener-Ditkin set is used. The union of two Ditkin sets is again a Ditkin set; this follows easily from a triangle inequality like I l l - f ghll <- IIf - f gll + Ilf g - f ghll. More generally, if a closed set is the union of countably many Ditkin sets, then it is again a Ditkin set. In contrast, for most function algebras it is unknown (as of 2000) whether the union of two sets of spectral synthesis is again of spectral synthesis: this is the famous union problem for this class of sets. Of course, the union problem becomes trivial if the synthesis-Ditkin problem gets a positive

D R I N F E L ' D - T U R A E V QUANTIZATION answer. Also, if E1 and E2 are sets of spectral synthesis such that E1 N E2 is a Ditkin set, then E1 U E2 is a set of spectral synthesis. It is also easy to prove that if the boundary of a closed set E is a Ditkin set, then so is E itself; cf. [10] (for the case A(G)), [8] or [9]. Ditkin sets are of particular interest if A ( X ) satisfies Ditkin's condition, i.e. if single points are Ditkin sets for A ( X ) . This notion is older than that of a Ditkin set; of., e.g., [7, p. 86]. If A ( X ) has approximate units (i.e. if each f E A ( X ) can be approximated by functions fg with 9 E A ( X ) ) , then A ( X ) satisfies Ditkin's condition if and only if for each x E X and each f E A(X) such that f ( x ) = 0 (i.e. such that f belongs to the maximal ideal Ix), the zero function can be approximated, arbitrarily closely, by functions f~- with ~- E A ( X ) and ~- equals 1 near x. It follows that if A ( X ) satisfies Ditkin's condition, then closed scattered sets (cf. S c a t t e r e d s p a c e ) are Ditkin sets. The following results can be found in [9, Sec. 7.4]. First, closed subgroups of G are Ditkin sets for the Fourier algebra A(G), and the same result still holds for certain Beurling algebras. Secondly, the following injection theorem for Ditkin sets holds: If r is a closed subgroup of G, and E is a closed subset of F, then E is a Ditkin set for A(G) if and only if it is one for A(F) ~- L I(G/H), where H = F -L, the subgroup of G orthogonal to P (cf. also O r t h o g o n a l i t y ) . In the literature a more restrictive class of sets is also considered, especially in the case of the Fourier algebra A(G). A closed set E is called a strong Ditkin set if there exists a net (g~) (cf. also Net (directed set)), bounded in the operator norm (i.e. the mapping a ~-~ sup{llfg~ll/]lflt: f E IE} is bounded), such that lima fgc~ = f. If X is metrizable (cf. Metrizable space), then one can require, equivalently, the existence of a s e q u e n c e (gn)n>_l in JR such that limn-.~ fgn = f for all f E IE, the boundedness in operator norm then being automatically satisfied, by the uniform boundedness theorem (of. Uniform boundedness) Strong Ditkin sets were first considered by I. Wik [12]. Subsequently it was proved that a closed subset E of G without interior is a strong Ditkin set for A(G) if and only if E belongs to the coset ring of G (of., e.g., [5], [3], [9] for details). A closed interval in the circle group T is a strong Ditkin set; cf. [12]. Therefore, it is essential, for the criterion above, to consider closed sets with empty interior. Also, a line segment in T 2 is not a strong Ditkin set for A(T2), because it has empty interior but does not belong to the coset ring. Consequently, the abovementioned injection theorem does not hold for strong Ditkin sets.

If E is not a set of spectral synthesis, then only functions f E JE have a chance of being approximable in the Ditkin sense. This motivates the following definition, given in [9]. A closed set E is called a Ditkin set in the wide sense if each f E JE can be approximated by functions f 9 with g E JR' This notion is, in a way, more natural than that of a Ditkin set; but in 1956 it was not yet known that sets not of spectral synthesis abound in the case of the Fourier algebra: Malliavin's result (cf. S p e c t r a l s y n t h e s i s ) dates from 1959. It is not known in general (for instance in the case of the Fourier algebra) whether all closed subsets are Ditkin sets in the wide sense. This problem is a natural generalization of the synthesis-Ditkin problem.

References [1] BENEDETTO, J.J.: Spectral synthesis, Teubner, 1975. [2] DITKIN, V.A.: ~On the structure of ideals in certain normed rings', Uchen. Zap. Mosk. Gos. Univ. Mat. 30 (1939), 83120. [3] GRAHAM, C.C., AND MCGEHEE, O.C.: Essays in commutative harmonic analysis, Springer, 1979. [4] HERZ, C.S.: 'The sprectral theory of bounded functions', Trans. Amer. Math. Soc. 94 (1960), 181-232. [5] HEWITT, E., AND ROSS, K.A.: Abstract harmonic analysis, Vol. 2, Springer, 1970. [6] KAHANE, J.-P., AND SALEM, ]:~.: Ensembles parfaits et sdries trigonomdtriques, Hermann, 1963. [7] LOOMIS, L.H.: An introduction to abstract harmonic analysis, Van Nostrand, 1953. [8] REITER, H.: Classical harmonic analysis and locally compact groups, Oxford Univ. Press, 1968. [9] REITER, H., AND STEGEMAN, J.D.: Classical harmonic analysis and locally compact groups, Oxford Univ. Press, 2000. [10] RUmN, W.: Fourier analysis on groups, Interscience, 1962. [1.1] SmLov, G.E.: 'On regular normed rings', Tray. Inst. Math. Steklov 21 (1947), English summary. (In Russian.) [12] WIx, I.: 'A strong form of spectral synthesis', Ark. Mat. 6 (1965), 55-64.

Jan D. Stegeman MSC 1991: 43A45, 43A46

DOMAIN (IN RING THEORY) - An (associativecommutative) ring in which the product of two non-zero elements is again non-zero. See also Associative rings

and algebras; Commutative ring. M. Hazewinkel MSC1991: 13-XX, 16-XX D R I N F E L ~ D - T U R A E V QUANTIZATION - A type of quantization typically encountered in k n o t t h e o r y , for example in Jones-Conway, homotopy or Kauffman bracket skein modules of three-dimensional manifolds ([3], [1], [2], cf. also S k e i n m o d u l e ) . Fix a commutative ring with identity, R. Let P be a Poisson algebra over R and let A be an algebra over R[q ±i] which is free as an R[q±l]-module (cf. also F r e e 133

D R I N F E L ' D - T U R A E V QUANTIZATION m o d u l e ) . An R-module e p i m o r p h i s m ¢ : A --+ P is called a Drinfel'd-Turaev quantization of P if i) ¢(p(q)a) = p(1)O(a) for all a e A and all p(q) C R[q±l]; and ii) ab - ba 6 (q - 1)¢-l([¢(a),¢(b)]) for all a,b • P. If A is not required to be free as an R[z]-module, one obtains a so-called weak Drinfel'd Turner quantization. References [1] HOSTE, J., AND PRZYTYCKI, J.H.: 'Homotopy skein modules of oriented 3-manifolds', Math. Proc. Cambridge Philos. Soc. 108 (1990), 475-488. [2] PRZYTYCKI, J.H.: 'Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs': Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, March, 14-15 , 1998), Internat. Press, 2000. [3] TURAEV, V.G.: 'Skein quantization of Poissou algebras of loops on surfaces', Ann. Sci. t~cole Norm. Sup. 4, no. 24 (1991), 635-704.

Jozef Przytycki

The second bifurcation is the H o p f b i f u r c a t i o n , where Gz has a conjugate pair of pure imaginary eigenvalues, i.e. with real part zero. Generically, a curve of periodic orbits is born in a Hopf point. If the equilibrium was initially stable, then generically it loses stability.

A periodic orbit is a solution of (1) for which there exists a period T > 0 such that x(t + T) = x(t) for one and hence all values of t. The linearized return mapping of a periodic orbit (cf. also P o i n c a r ~ r e t u r n m a p ) is called the monodromy matrix. The eigenvalues of the monodromy matrix are the Floquet multipliers (cf. also F l o q u e t e x p o n e n t s ; F l o q u e t t h e o r y ) . There is always one multiplier equal to 1. If all other multipliers have moduli strictly less than 1, then the periodic orbit is asymptotically stable. If at least one multiplier has modulus strictly larger than 1, then the periodic orbit is unstable. In the remaining cases the stability depends on the non-linear terms in the Taylor expansion of the return mapping.

MSC1991: 57P25, 16Wxx

Again, if a component of a is freed, then curves of periodic orbits can be computed.

DYNAMICAL SYSTEMS SOFTWARE PACKAGES, software for dynamical systems - Mathematical background on dynamical systems can be found in [2], [6] or [7] (cf. also D y n a m i c a l s y s t e m ) . Numerical methods are described in [2], [5] and [7]. In its basic form a dynamical system is a system of ordinary differential equations of the form

The periodic orbits that originate at a Hopf point can be either stable or unstable. The stability is guaranteed if the equilibrium preceding to the Hopf point is stable and a quantity, called the first Lyapunov coefficient ~1, is negative (cf. also L y a p u n o v c h a r a c t e r i s t i c exp o n e n t ) . The bifurcation is then supereritieal, i.e. the stable periodic orbits are found at the side where the equilibria are unstable. If the equilibrium preceding to the Hopf point is stable and g~ is positive, then the periodic orbits are unstable and the bifurcation is suberitieal, i.e., the periodic orbits are found at the side of the stable equilibria. The intermediate case where ~1 = 0 is called a generalized Hopf or Bautin point.

= G(x, 5),

(1)

where x E R n is the state variable, a E R TM is a parameter vector and G(x, 5) is a non-linear function of x and a. The independent variable t is usually identified with time. The equilibria of (1) are its constant solutions, i.e. the solutions of the non-linear system

a ( x , 5) = 0,

(2)

for a given parameter vector 5. Equilibria are asymptotically stable if all eigenvalues of the Jacobian matrix Gx have a strictly negative real part (cf. also J a c o b i m a t r i x ) . They are unstable if at least one eigenvalue has a strictly positive part. In the remaining cases the stability depends on the non-linear terms in the Taylor expansion of G (cf. also S t a b i l i t y t h e o r y ) . If a component of (~ is freed, then curves of equilibria can be computed. Generically, curves of equilibria can bifurcate in two ways (cf. also B i f u r c a t i o n ) . The first is the limit point bifurcation, where Gx becomes singular, i.e. has an eigenvalue zero (cf. also L i m i t p o i n t o f a t r a j e c t o r y ) . Generically, this indicates a turning point of equilibria. If the equilibrium was initially stable, then it generically loses the stability. 134

The notion of a dynamical system can be extended in several ways. A discrete dynamical system is an iterated mapping

x -+ a ( x , 5).

(3)

A delay differential equation is an equation of the form (1) where G is also explicitly dependent on the values x(t - 7-i) for one or several delays Ti (cf. also D i f f e r e n t i a l e q u a t i o n s , o r d i n a r y , r e t a r d e d ) . It is a neutral differential equation if G is also explicitly dependent on the values 2(t - 7-i) for one or several delays ~-i (cf. also N e u t r a l d i f f e r e n t i a l e q u a t i o n ) . A partial differential equation of evolution type is also considered as a dynamical system (cf. also E v o l u t i o n e q u a t i o n ) . S o f t w a r e . A website on dynamical systems software is [9].

DYNAMICAL SYSTEMS S O F T W A R E PACKAGES

AUTO. The most widely used software package for dynamical systems computations is AUTO97 [3]. This software is distributed freely; see [10]. A manual is also available from this site. AUTO has many interesting features: • It can compute solution branches of (2), detect and compute branch points and compute the bifurcating branches. It can also detect and compute limit points and Hopf points and continue these in two parameters. Also, it can find extrema of an objective function along solution branches and continue such extrema in more parameters. • It can compute fixed points for the discrete dynamical system (3). It can compute branches of such fixed points, detect, compute and continue fold points, period-doubling (flip) and NeYmark-Sacker bifurcations of fixed points. • It can perform a bifurcation analysis of (1). It can compute branches of stable and unstable periodic orbits and compute the Floquet multipliers. Periodic orbits can be started from Hopf bifurcation points. Along branches of periodic orbits branch points, fold points, perioddoubling, and torus bifurcations can be computed. In branch and period doubling bifurcations branch switching is possible. Period-doubling bifurcations, folds, torus bifurcation points, and orbits with fixed period can be continued in two parameters. • It can follow curves of homoclinic orbits and detect and continue various codimension-2 homoclinic orbits. • It can locate extrema of an integral objective function along a branch of periodic solutions and continue such extrema in more parameters. • It can also compute curves of solutions to (i) on a fixed interval [0, I] subject to general non-linear integral and boundary conditions. Folds and branch points can be computed along such curves. Curves of folds can be computed and branch switching at branch points is

provided. • It can further do some stationary and wave calculations for partial differential equations of the form 2 = Dx~s + a ( z , a),

(4)

where D is a diagonal matrix of diffusion constants and x depends on time t and a one-dimensional space variable s. In AUTO, the numerical quality of the algorithms is strongly emphasised and the graphical user interface got less attention. In fact, AUTO can be used in command mode, i.e. without any graphical interface.

CONTENT. Another important package is C O N T E N T [8], whose main developer is Yu.A. Kuznetsov.

C O N T E N T is a CONTinuation EnvironmeNT and the user interaction is via a windowing system. For algebraic equations (2) as equilibrium solutions of (1), C O N T E N T provides more routines than does AUTO. In fact, it allows one to detect all e o d i m e n s i o n - t w o bif u r c a t i o n s and to continue them numerically if a third parameter is freed. These codimension-two bifurcations are: Bogdanov-Takens, zero-Hopf, double Hopf, cusp, and generalized Hopf. The behaviour of dynamical systems near codimension-two equilibrium bifurcations is described in [6] and [7]. Generically, periodic orbits, homoclinic orbits, invariant tori, and chaotic behaviour can all be detected. C O N T E N T even allows one to detect and compute certain codimension-3 bifurcations, such as triple zero, swallowtail, resonant double Hopf, and a few others. Also, in most cases C O N T E N T offers several computational routines to compute and continue bifurcation points. For discrete dynamical systems (3), C O N T E N T offers the same possibilities as AUTO but leaves the user options to use several methods. For dynamical systems (1), C O N T E N T offers less routines than AUTO. However, it allows one to compute curves of periodic orbits and to detect the fold, flip and Ne~mark-Sacker bifurcations. For partial differential equations, C O N T E N T allows a wider class of one-dimensional problems than does AUTO; actually, in (4) the right-hand side can be replaced by practically any 'reasonable' function and the boundary conditions can be quite general. On the other hand, only the time evolution computation of such systems is at present (2000) supported and only by the implicit Euler and Crank-Nicolson methods.

Other packages. A third roughly comparable package is CANDYS/QA (see [9] for more information). DsTool [9] can compute equilibria of ordinary differential equations and diffeomorphisms and compute their stable and unstable manifolds. Several packages, notably DsTool, Dynamics Solver and X P P simulate and numerically solve dynamical systems equations. Several other packages, notably Global Manifolds 1D, Global Manifolds 2D, GAIO and BOV-method compute invariant manifolds. See [9] for details. For partial differential equations, the choice of software is limited. In addition to the capabilities of AUTO and C O N T E N T there is P D E C O N T [9] for the continuation of periodic solutions of partial differential equations. Next, there exists the software package P L T M G [1] that allows one to solve a whole class of boundary value problems on regions in the plane, to continue the solution with respect to a parameter, and even to 135

DYNAMICAL SYSTEMS SOFTWARE PACKAGES c o m p u t e b r a n c h i n g p o i n t s a n d l i m i t points. T h i s software combines a s o p h i s t i c a t e d finite-element discretization w i t h a d v a n c e d linear a l g e b r a techniques. For d e l a y differential e q u a t i o n s t h e r e is t h e bifurcation package DDE-BIFTOOL

[4].

References [1] BANK, R.E.: pltmg: A software package for solving elliptic partial differential equations, Users' Guide 8.0, SIAM, 1998. [2] BROER, H., AND TAKENS, F. (eds.): Handbook of dynamical systems, Elsevier, to appear, Vol. I: Ergodic Theory (eds. B. Hasselblatt, A. Katok); Vol II: Bifurcation Theory (eds. H. Broer, F. Takens); VoI III: Towards Applications (eds. B. Fiedler, G. Iooss and N. Kopeii). [3] DOEDEL, E.J., CHAMPNEYS, A.R., FAIRGRIEVE, T.F., KUZNETSOV~ Yu.A., SANDSTEDE, B., AND WANG, X.J.: auto97: Continuation and bifurcation software for ordinary differential equations (with HomCont), User's Guide, Concordia Univ., 1997.

136

[4] ENGELBORHGS, K.: 'dde-biftooh A Matlab package for bifurcation analysis of delay differential equations', www. cs. kuleuven, ac. be/~ koen/delay/ddebiftool, shtml

(2ooo). [5] GOVAERTS, W.: Numerical methods for bifurcations of dynamical equilibria, SIAM, 2000. [6] GUCKENHEIMER, J., AND HOLMES, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Vol. 42 of Applied Math. Sci., Springer, 1983. [7] KUZNETSOV, Yu.A.: Elements of applied bifurcation theory, Vol. 112 of Applied Math. Sci., Springer, 1995/98. [8] KUZNETSOV, Yu.A., AND LEVITIN, V.V.: 'content: A multiplatform environment for analyzing dynamical systems', Dynamical Systems Lab. CWI, Amsterdam (1995/97), ftp.cwi.nl/pub/cont ent. [9] OSINGA, H.: 'Website on dynamical systems software',

www.maths.ex.ac.uk/~hinke/dss/ (2000). [10] WEB,

http://indy.cs.concordia.ca/auto/doc/index.html

(2ooo). W. Govaerts

M S C 1 9 9 1 : 58F14, 58-04, 34-04, 35-04

E EFFECTIVE NULLSTELLENSATZ Let f, f l , . . . , fm E R := k [ x l , . . . , Xn], where k is a field. Hilbert's Nullstellensatz [13] says that if f vanishes on all the common zeros of the fi with coordinates in an algebraic closure of k, then, for some integer p, fP C I := ( f l , . . . , f m ) , i.e. there exist a l , . . . , a m E R such that

fP = a l f l + " " + am fro. An effective Nullstellensatz gives information on some aspect of the complexity of such a representation. D e g r e e b o u n d s . If deg fi <_ d, how large might one have to take p and max deg ai? The projective version of the MasserPhilippon/Lazard-Mora e x a m p l e shows that this bound must be at least d n. Although classical work of G. Hermann [12] produces an explicit bound [17], it is known that d n is the correct one [5], [4], [14]. H e i g h t b o u n d s . How large might a common denominator or the numerators of the ai be? Assume, in addition, that k = Q and that when the coefficients of all the fi are put over a common denominator, the absolute values of that denominator and the numerators of the coefficients are at most H > 0. One can, of course, first bound the degrees, and then apply estimates from linear algebra to obtain a bound of n2

the unfortunate shape (dH) c~d . In fact, by [19], a denominator of absolute value at most expeJ'~(d + h)q suffices, with explicit ca. The same is true of the numerators if one allows slightly larger than optimal bounds on the degrees of the fi: deg fi <_ end n [1], [2], [16]. C o m p l e x i t y b o u n d s . In the various models of computation, how many steps might be involved in finding such a representation? In these models, as in the above classical case for heights, the complexity bounds turn out to be better than that obtained by simply applying linear algebra and using the degree bounds. Many interesting aspects,

such as sparsity of the coefficients, arise. See [11] for an introduction to a sizable and growing literature. G e n e r a l i z a t i o n s . Various effective generalizations of Hilbert's Nullstellensatz exist, cf. [7], [9], [15], as well as insight into important special situations [18]. See [3] for the link between the complexity of the Nullstellensatz and the problem P ¢ AlP (cf. ALP). References

[1] BERENSTEIN, C.A., AND YGER, A.: 'Effective Bezout identities in Q [ z l , . . . , z n ] ' , Acta Math. 166 (1991), 69-120. [2] BERENSTEIN, C.A., AND YGER, A.: 'Residue calculus and effective Nullstellensatz', Amer. J. Math. 121 (1999), 723-796. [3] BLUM, L., CUKER, F., SCHUB, M., AND SMALE, S.: Complexity and real computation, Springer, 1998. [4] BROWNAWELL, W.D.: 'Borne effective pour l'exposant dans le th~or~me des z~ros', C.R. Acad. Sci. Paris 305 (1987), 287-290. [5] BROWNAWELL,W.D.: 'Bounds for the degrees in the Nullstellensatz', Ann. of Math. 126 (1987), 577-591. [6] BROWNAWELL, W.D.: 'Local diophantine Nullstellen equalities', J. Amer. Math. Soe. 1 (1988), 311-322. [7] BROWNAWELL,W.D.: 'A pure power product version of the Hilbert Nullstellensatz', Michigan Math. J. 45 (1998), 581597. [8] CANIGLIA, L., GALLIGO, A., AND HEINTZ, J.: 'Borne simplement exponentielle pour les degr~s dans le thfior~me des z~ros sur un corps de characteristique quelconque', C.R. Aead. Sci. Paris 307 (1988), 255-258. [9] EIN, L., AND LAZARSFELD, R.: 'A geometric effective Nullstellensatz', Invent. Math. 137 (1999), 427-448. [lO] FITCHAS, N., AND GALLIGO, A.: 'Nullstellensatz effectif et conjecture de Serre (th~or~me de Quillen-Suslin) pour le calcul formel', Math. Nachr. 149 (1990), 231-253. [11] GIUSTI, M., HEINTZ, J., MORAIS, J.E., MORGENSTERN, J., AND PARDO, L.M.: 'Straight-line programs in geometric elimination theory', J. Pure Appl. Algebra 124 (1998), 101-146. [12] HERMANN, G.: 'Die Frage der endlich vielen Schritte in der Theorie der Polynomideale', Math. Ann. 95 (1926), 736-788. [13] HILBERT, D.: ' 0 b e r die vollen Invariantesysteme', Math. Ann. 42 (1893), 313-373. [14] KOLL;~R, J.: 'Sharp effective Nullstellensatz', J. Amer. Math. Soc. 1 (1988), 963-975. [15] KOLL~.R, 3.: 'Effective Nullstellensatz for arbitrary ideals', J. Europ. Math. Soc. (JEMS) 1 (1999), 313-337.

E F F E C T I V E NULLSTELLENSATZ [16] KRICK, T., PARDO, L.M., AND SOMBRA, M.: 'Sharp estimates for the arithmetic Nullstellensatz', http://front.math.ucdavis.edu math.AG/9911094 (1999). [17] MASSER, D.W., AND W{}SHOLZ, G.: 'Fields of large transcendence degree generated by values of elliptic functions', Invent. Math. 72 (1983), 407-464. [18] OJEDA, I., AND PIEDRA, R.: 'Effective Nullstellensatz for binomial ideals', M a n u s c r i p t (2000). [19] PHILLIPON, P.: 'D6nominateurs darts le th6or~me des z6ros de Hilbert', Acta Arith. 58 (1990), I 25.

W. Dale Brownawell MSC1991: 14A10, 14Q20 E G O R O V GENERALIZED F U N C T I O N A L G E B R A

- Given an open subset f~ of R n, Yu.V. Egorov [1] defined the generalized function algebra A(~) as the factor algebra of (g°°(~)) N modulo the ideal of sequences (Uj)je N which vanish eventually on every compact subset of ~. The family {A(Ft) : f~ open} provides a s h e a f of differential algebras on R n. Convolution with a sequence of mollifiers ( ~ j ) j E N , w h e r e ~oj converges to the Dirac measure, gives an imbedding of the space E'(~) of compactly supported distributions into A(~) which respects derivatives as well as supports. It can be extended as a sheaf morphism to an imbedding of the space of distributions 7)' (~). As a generalized function algebra, A(~) can be employed to study non-linear partial differential equations. In particular, Egorov has used the algebra to construct generalized solutions to boundary value problems as well as evolution equations. In the latter case the spatial derivative (O/Oxk)u(x) may be replaced by the difference operator

of degree n,

Mn =

a , X " Y ~-" : au e q

P(X,Y) =

,

vzO

where7 :

( ca

:)E

G L 2 ( Q ) a c t s by 7 P ( X , Y )

:

e ( a X + cY, bX + dY) det(-y) d. Starting from these data one can construct a s y m m e t r i c s p a c e X = G ( R ) / K o o , the locally symmetric space P \ X and a sheaf M on F \ X. In the example one can take K ~ = SO(2) x Z ( R ) °, where Z ( R ) ° is the connected component of the identity of the real points of the centre. One can consider the cohomology groups H ° ( F \ X , 54). It is a consequence of reduction theory that they form a finite-dimensional (graded) vector space. It carries some further structure: It can be endowed with an action of the so-called Hecke algebra on it. It is a fundamental problem to understand these cohomology groups as a module under this action of the Hecke algebra. If F is torsion free, then P \ X is a R i e m a n n i a n m a n i f o l d with finite volume and sometimes it even carries the structure of a quasi-projective variety. If, for instance, P C SL2(Z), then P \ X is a compact Riemann surface with a finite non-empty set of points removed. Form the tensor product M ® C = M c . A method for investigating these cohomology groups is provided by the de Rham isomorphism H ' ( F \ X, 54 @ C) -% H ' ( F \ X , a ' ( M c ) ) , where f ~ ° ( M c ) is the de Rham complex of fl4c, and this de Rham complex is isomorphic to gt°(54c) --% HomK~(A'(g/e),C~o(P \ G(R) ® 5 4 c ) ) ,

where ek denotes the kth unit vector in R n. This way partial differential equations are approximated by ordinary difference-differential equations in the algebra See also G e n e r a l i z e d f u n c t i o n a l g e b r a s . References

[1] EGOROV,Yu.V.: 'A contribution to the theory of generalized functions', Russian Math. Surveys 45, no. 5 (1990), 1-49. Michael Obergugyenberger MSC 1991:46F30 E I S E N S T E I N C O H O M O L O G Y - The basic ingredients for Eisenstein cohomology are a r e d u c t i v e g r o u p G / Q , an arithmetic subgroup (cf. also A r i t h m e t i c g r o u p ) F C G(Q) and a rational representation p: G / Q ~ GL(54) (cf. also R e p r e s e n t a t i o n o f a g r o u p ) . The simplest example is given by the group G = G L 2 / Q , a congruence subgroup F C GL2(Z) with as representation the space of homogeneous polynomials

138

where 9 (respectively, ~) is the Lie a l g e b r a of G(R) (respectively, Koo ). This opens the door for the application of representation-theoretical methods, because Coo(F \ G(R) ® 5 4 c ) is a module under the group G(R). However, since the quotient F \ G(R) is not compact in general, one has to be careful: the space Coo(F \ G(R) ® 5 4 c ) ~ L2(F \ G(R) ® J~4c) and the Hilbert space technique of decomposing a module into irreducibles cannot be applied so easily. It is possible to define various subspaces, namely the space of cusp forms C0 (F \ G(R)), or the space of compactly supported functions gc(F \ G(R)). These define subspaces in the coh o m o l o g y , namely the euspidal cohomology and the so-called 'inner cohomology' H.,'(F \ X, 54). Classes in one of theses subspaces vanish 'at infinity'. The cuspidal cohomology can be investigated using classical Hilbert space techniques; the space L20(F \ G(R)) is a countable sum of irreducible subspaces under the action of G(R).

EISENSTEIN COHOMOLOGY One wants to understand the rest of the cohomology. To that end one can embed the locally symmetric space F \ X into the Borel-Serre compaetification F \ X; the sheaf f14 extends to a local system on this compactification. The Borel-Serre compactification is a manifold with corners; let a(F \ X) be its boundary. Then there is an exact sequence •..-~ H'-I(O(F\X),~)

--+ H ' ( F \ X , ~ )

~ H:(r\X,~)

A H °(cg(F \ X ) , ~ )

-~

-+....

This raises the question to understand such sequences: What is the cohomology of the boundary H ' ( a ( F \ X), Ad) and what is the image of the 'global' cohomology H ' ( F \ X, A/t) in the cohomology of the boundary? Understanding the cohomology of the boundary requires understanding the cohomology of groups of lower rank, which is sometimes not so difficult. For instance, in the special example above the boundary consists of a finite number h of circles and

/-/'(c~(F \ X), ~ ) = H ° e H 1 -% Qh eQh. Eisenstein cohomology is designed to provide some understanding of the restriction mapping r. One starts from a class w (or a certain space of classes) in the cohomology of the boundary. Viewing it as a class in the de R h a m c o h o m o l o g y , one can represent it by a form which is not invariant under F, but which is invariant under the smaller group Fp given by the intersection of F with a p a r a b o l i c s u b g r o u p P which enters in the datum w. Then one forms the sum

Eis( )= giving a global class. The only difficulty is that this sum need not converge. Hence the form w has to be 'twisted' by a parameter s varying in a space C r and whose real part should be large. Then the sum

Z

This program has been carried out successfully in some low-dimensional cases, see, for instance, [2], [3], [7]. In the example, the restriction mapping r is surjectire in degree one and zero in degree zero if n > 0. If n = 0, then the image of r in degree one has codimension 1 and dimension 1 in degree zero. Actually, in this case the theory of Eisenstein series is not needed, since purely topological arguments are sufficient. It has been demonstrated in [2], [3], [7] that a detailed understanding of the Eisenstein cohomology may have certain arithmetic implications; for instance, one obtains rationality results for special values of L-functions. One may also hope that via the influence of the values of the L-functions on the structure of the cohomology as a module under the Hecke algebra, some interesting arithmetic objects (mixed motives, unramified field extensions) can be constructed that owe their existence to the (arithmetic) properties of certain L-values; see [4],

[5], [8]. Finally, there is the following fundamental and very general theorem of J. Franke [1]. Using the Eisenstein series and their residues and derivatives one can define the subspace Jt(F \ G ( R ) ) C C0(F \ G(R)). This space can also be characterized as a space of functions that satisfy certain growth conditions and differential equations. This subspace is 'very small' and Franke's theorem says that the mapping

HOmKo(A°(9/t), A(F \ G(R)) ® Mc) --%

~EF/Fp

Eis(w,s)=

above questions depends on the behaviour of certain of these L-functions at s = 0.

7w~

7cr/rp

will be convergent and represent a h o l o m o r p h i c funct i o n in s. Langlands' general theory of Eisenstein series implies that this function has a meromorphic continuation and hence one can 'evaluate' at s = 0. However, various things may happen. One may encounter a pole or the class Eis(~, 0) need not be closed. If it is closed, one has to compute its restriction to the boundary. What happens exactly depends, of course, on the original data. The original form w should be specified more precisely; for instance, one may assume that it is an eigenform for a certain subalgebra of the Hecke algebra. Then as such it produces certain L-functions L(w, r, s) (cf. also L - f u n c t i o n ) and the answer to the

--% HomKoo(A°

e (r \ a(R) ® Mc))

induces an isomorphism in cohomology. References [1] FRANKE, J.: 'Harmonic analysis in weighted L2-spaces', Ann. Sci. l~cole Norm. Sup. (4) 31 (1998), 181-279. [2] HARDER, G.: 'Eisenstein cohomology of arithmetic groups: The case GL2', Invent. Math. 89 (1987), 37-118. [3] HARDER, G.: 'Some results on the Eisenstein cohomology of arithmetic subgroups of GLn', in J.P. LABESSE AND J. SCHWERMER (eds.): Cohomology of Arithmetic Groups. Proc. Conf. CIRM, Vol. 1447 of Lecture Notes in Mathematics, Springer, 1990. [4] HARDER, G.: 'Eisenstein cohomology of arithmetic groups and its applications to number theory': Proc. ICM (Kyoto, 1990), Math. Soc. Japan, 1991, pp. 779-790. [5] HARDER, G.: Eisensteinkohomologie und die Konstruktion gemischter Motive, Vol. 1562 of Lecture Notes in Mathematics, Springer, 1993. [6] HARDER, G., AND PINK, R.: 'Modular konstruierte unverzweigte abelsche p-Erweiterungen yon Q(4(p)) und die Struktur ihrer Galoisgruppen', Math. Nachr. 159 (1992), 8399.

139

EISENSTEIN COHOMOLOGY [7] SCHWERMER, J.: 'On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties', Forum Math. 7 (1995), 1-28.

G. Harder M S C 1991:11F67 ENSS METHOD - In 1977, V. Enss [1] introduced a new approach to the study of spectral and scattering properties of Schr6dinger operators (cf. also S c h r 5 d i n g e r e q u a t i o n ) . It was based on a combination of time-dependent scattering theory and phasespace analysis. In his first work, Enss solved the twobody scattering problem (with short-range potentials) and a few years later extended the method to the threebody problem (both short- and long-range potentials). Previous results on this problem were based on timeindependent methods, primarily due to L.D. Faddeev, who worked out the three-body case in 1963. Faddeev's work was later clarified and some generalizations were made, but it remained limited to the three-body case and required further assumptions on the spectral properties of the Hamiltonian and other restrictions on the potentials. Enss' method, on the other hand, removed all the artificial assumptions. It also initiated the fruitful approach of phase-space analysis, later further developed by E. Mourre [3] (1979-1981) and finally led to a general phase-space theory of N - b o d y Hanfiltonians by I.M. Sigal and A. Softer [5] (1987). The Enss method is based on using classical intuition to the study of the large-time behaviour of a quantum system. Consider the case of two-particle scattering; this system can be reduced to the study of the large-time behaviour of a quantum particle interacting with a force field, which decays to zero at large distances. The states of such a particle, being quantal, can only be localized in some energy interval, in general. If the energy is localized near a positive number, one expects the particle to escape to infinity. The problem of scatterin 9 theory is to show that every state t h a t escapes to infinity, moves like a free particle system, for large enough times. This idea is captured using the notion of wave operator. Say the state of the system at time zero is given by a wave function ¢(0). One introduces the dynamics U(t) to be an operator that moves the state of the system by a time t. Hence V(t)~(0) = ¢(t), the state of the system at time t. One can also use a different, free dynamics U0(t); U0(t) is the dynamics of a particle moving without any force acting on it. Suppose now one constructs the following state: 2 ( t ) ¢ ( 0 ) = Uo(-t)U(t)¢(O). 140

T h a t is, f~(t)~(0) is the state of a system moved forward in time under the true (or full) dynamics, for a time t, and then backward under the free dynamics. In the limit, as t goes to infinity, f~(t)¢(0) should approach a new state, ~+ if t -+ +e~ and ~ _ if t -+ - e c . The main problem of scattering theory is to show that for any ~(0) for which U(t)~(O) disperses to infinity as t approaches infinity, the limiting states ~+ exist. To prove such results, Enss begins with proving the following basic property of states which disperse to infinity: Assuming t h a t the force field is regular enough (that is, its value does not j u m p from one point to another), the wave function decays to zero inside any finite ball in space. This decay to zero is, furthermore, uniform in the choice of states, provided they all have their energy support in a same fixed finite interval (a, b) with a > 0. The proof of this results essentially follows from a similar theorem of D. Ruelle [4] (1969). Now, note that the wave operators ft± which map ~p+ to ~(0) measure the 'difference' between the free and full dynamics (when U(t) = Uo(t), one sees t h a t ft+ = 1). Hence one expects t h a t the wave operators applied to a state which is very far from the force field act like 1. This is a key observation in the Enss method. It reduces the problem of scattering and asymptotic completeness to showing that (ft+ - 1)~(t) goes to zero as t goes to infinity. To prove that, one now decomposes the state in the phase space, that is, in the bigger space of position and velocities of the system/particle: P+ will denote the projection on the part of the state where the position vector and velocity vector are related by a sharp angle between them: £.g>

0,

and P_ will be the complement. Then, ( a + - 1)O(t) : (ft+ - 1)g¢(t) = = (a+ - 1)(g - g0)¢(t)

+ (a+ - 1)g0¢(t),

where g stands for the projection on states with total energy in the (fixed) interval (a, b) and g0 stands for the projection on states with kinetic energy in the interval (a, b).

As t approaches infinity, Ruelle's theorem implies that the state moves away to infinity. Hence it does not interact with the force any more; this means that all the energy of the state is kinetic. Hence one concludes that (g - go)

(t)

vanishes as t -+ ec, and so is the term (a+ - 1)(g - 90)¢(t).

E U L E R - P O I S S O N - D A R B OUX E Q U A T I O N There remains the t e r m

(~+

- 1)g0¢(t) =

= (~+ - 1 ) g o P + ¢ ( t ) + (~+ - 1 ) g o P _ ¢ ( t ) .

It is easy to see t h a t when a free particle moves, its velocity becomes more and more parallel to its position vector. Hence the derivative with respect to time of ~7-gis positive under the free flow. This same derivative under the full flow will then be a sum of a positive t e r m (coming from the free part of the motion) and another term, depending on the force. Since for t large the force can be neglected, by Ruelle's theorem, one sees that also under the full flow, ~ - g will have positive growth. Hence, for large enough times, the support of the state will move to the region of phase space where ~. ~7> 0. Hence P _ ¢ ( t ) , the projection on the part of the state where ~ . g < 0, will tend to zero as t approaches infinity. To complete the proof it is then left to show that

References [1] ENSS, V.: 'Asymptotic completeness for quantum-mechanicM potential scattering I. Short range potentials', Comm. Math. Phys. 61 (1978), 285-291. [2] ENSS, V.: 'Quantum scattering theory of two- and three-body systems with potentials of short and long range', in S. GRAFFI (ed.): Schr6dinger Operators, Vol. 1159 of Lecture Notes in Mathematics, Springer, 1985. [3] MOURRE, E.: 'Link between the geometrical and the spectral transformation approaches in scattering theory', Comm. Math. Phys. 68 (1979), 91-94. [4] RUELLE, D.: 'A remark on bound states in potential scattering theory', Nuovo Cimento 61A (1969), 655 662. [5] SIGAL, I.M., AND SOFFER, A.: 'The N-particle scattering problem: Asymptotic completeness for short range systems', Ann. of Math. 126 (1987), 35-108.

Avy Softer M S C 1991: 81Uxx E U L E R - P O I S S O N - D A R B O U X EQUATION - The second-order 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 tion x --v9 0 ~

+ ~-~F

= E

nCZ

=

[A + c~; n][# - n + 1; n] x;~+no ~_~ [#_n+9;n][A+l;n] Y '

where [A; n] = F(A + n)/r(~) and F(A) is the g a m m a function. By conjugate transformation of the differential operator L ( a , / 3 ) with (x - y ) - ~ one obtains the operator x-y

also vanishes as t approaches infinity. The proof t h a t this last t e r m vanishes as t -+ c~ is the most technical p a r t of the Enss method. It is based on Cook's original proof of the existence of the limit defining ~ + , combined with the ideas of Ruelle's argument. It should be remarked t h a t the above description is improved over the original argument of Enss, using P~= motivated by Mourre's work.

OxO~

¢(~, ~; a,#;x,y)

~(~, #) = 0xa~ - ~--2--0x +

(~+ - 1)g0P+¢(t)

o = r(~,9)~ =

This equation appears in various areas of mathematics and physics, such as the theory of surfaces [4], the propagation of sound [3], the colliding of gravitational waves [6], etc.. The Euler-Poisson-Darboux equation has rather interesting properties, e.g. in relation to Miller symmetry and the Laplace sequence, and has a relation to, e.g., the Toda molecule equation (see [4]). A formal solution to the Euler-Poisson-Darboux equation has the form [8]

u = O,

where a and fl are real positive parameters such that a + fl < 1 (see [8]) and a~u denotes the partial derivative of the function u with respect to x.

~ 0~.

(1)

x-y

Many papers deal with the equation

E(~, ~) = 0

(2)

(see, e.g., [11], [8], [7], [10], [12]). In the characteristic triangle ~ = {(x,y) E R 2 : 0 < x < y < 1} and under the conditions uIx=y -- ~-(x),

( v - x ) ~+~

~

(3)

~

=~(x),

the solution of (2) can be expressed as (see [12]): ~(~,y) =

r(~ +/3) [ ~ - r ( ~ ) r ( ~ ) ~0 ~ (x + (v - ~)t) t ~-~(1 - t) ~-~ ~t+ r(1 - a

2F(1 •

/o 1u ( x

-

-

~)r(1 + (y

-

8) -

(y

_

x)t_~_~.

9)

x)t)

t-~(1

-

t) -~ dr.

Formulas for the general solution of (2) are known for I~1 < 1, 191 < 1; ~ = 9; a n d ~ + 9 = 1. For o t h e r

values of the parameters, an explicit representation of the solution can be given using a regularization method for the divergent integral (see [7]). The unique solvability of a boundary value problem for (2) with a non-local boundary condition, containing the Szeg5 fractional integration and differentiation (ef. F r a c t i o n a l i n t e g r a t i o n a n d d i f f e r e n t i a t i o n ) operators, is proved in [11]. For (2) local solutions, propagation of singularities, and holonomic solutions of hypergeometric type are studied in [14]. For hypergeometric functions of several variables occurring as solutions of boundary value problems for (2), see also [14]. 141

EULER-POISSON-DARBOUX

EQUATION

A q-difference analogue of the operator E ( a , ~ ) = (x - y ) E ( a , / ~ ) is considered in [8]; it has been proved t h a t the q-deformation of E ( a , / 3 ) is the q-difference operator Eq(a,/3) = [Ox + a]q[Ov]q - [ O r + £][Ox]q. The existence and uniqueness of global generalized solutions of mixed problems for the generalized EulerPoisson-Darboux equation utt -

~ (aijUxj)Xi+~ u t

-=

[13] SMmNOV,M.M.: Degenerate hyperbolic equations, Izd. Vysh. Shkola, Minsk, 1977. (In Russian.) [14] TAKAYAMA,N.: 'Propagation of singularities of solutions of the Euler-Poisson-Darboux equation and a global structure of the space of holonomic solutions I', Funkc. Ekvacioj, Ser. Internat. 35 (1992), 343 403. [15] WANG, J.: 'Mixed problems for nonlinear hyperbolic equations with singular dissipative terms', Acta Math. Appl. Sin. 16 (1993), 23-30, 1. (In Chinese.) C. Moro~anu

(4)

i,j=l

MSC 1991:35L15

= f ( t , x, u, ut, V u ) are studied in [15], using Galerkin approximation. Moreover, the classical solution of (4) has been obtained by using properties of Sobolev spaces and imbedding theorems (cf. also I m b e d d i n g t h e o r e m s ) . See [2], [11], [1], [9] for various aspects of (4). See [5] for necessary and sufficient conditions for stabilization of the solution of the Cauchy problem for the E u l e r - P o i s s o n - D a r b o u x equation in a homogeneous symmetric space.

References [1] CHAN, C.Y., AND NIP, K.K.: 'Quenching for semilinear

Euler-Poisson-Darboux equations', in J. WIENER(ed.): Partial Differential Equations. Proc. Internat. Conf. Theory Appl. Differential Equations (Univ. Texas-Pan American, Edinbur9, Texas, May 15-18, 1991), Vol. 273 of Pitman Res. Notes, Longman, 1992, pp. 39-43. [2] CHAN, C.Y., AND NIP, K.K.: 'On the blow-up of luttl at quenching for semilinear Euler-Poisson-Darboux equations', Comput. Appl. Math. 14, no. 2 (1995), 185 190. [3] COPSON, E.T.: Partial differential equations, Cambridge Univ. Press, 1975. [4] DARBOUX, G.: Sur la thdorie gdndrale de surfaces, Vol. II, Chelsea, reprint, 1972. [5] DENISOV,V.N.: 'On the stabilization of means of the solution of the Cauchy problem for hyperbolic equations in symmetric spaces', Soviet Math. Dokl. 42, no. 3 (1991), 738 742. (Dokl. Akad. Nauk. SSSR 315, no. 2 (1990), 266-271.) [6] HAUSER,I., AND ERNST, F.J.: 'Initial value problem for colliding gravitational plane wave', J. Math. Phys. 30, no. 4 (1989), 872-887. [7] KHAIRULHN, R.S.: 'On the theory of the Euler-PoissonDarboux equation', Russian Math. 37, no. 11 (1993), 67-74. (Izv. Vyssh. Uchebn. Zaved. Mat., no. 11 (1993), 69-76.) [8] NAGAMOTO,K., AND KOGA, Y.: 'q-difference analogue of the Euler-Poisson-Darboux equation and its Laplace sequence', Osaka J. Math. 32, no. 2 (1995), 451-465. [91 PAN'KO,S.V.: 'On a representation of the solution of a generalized Euler-Poisson-Darboux equation', Diff. Uravnen. 28, no. 2 (1992), 278-281. (In Russian.) [10] REPIN, O.A.: Boundary value problems with shift for equations of hyperbolic and mixed type, Samara: Izd. Sartovsk. Univ., 1992. (In Russian.) [11] REPIN, O.A.: 'A nonlocal boundary value problem for the Euler-Poisson-Darboux equation', Diff. Eqs. 31, no. 1 (1995), 160-162. (Diff. Uravn. 31, no. 1 (1995), 171-172.) [12] SAICO,M.: 'A certain boundary value problem for the Euler Poisson Darboux equation', Math. Japon. 24, no. 4 (1979), 377-385. 142

EXPONENTIAL LAW (IN TOPOLOGY) - T h e idea for a t o p o l o g y on spaces of functions goes back to the metric d ( f , 9) = s u p { d ( f c , gc): c C C } on functions from a c o m p a c t s p a c e C to a m e t r i c s p a c e X . It was found desirable to extend this to the case when C is only locally c o m p a c t (cf. also L o c a l l y c o m p a c t s p a c e ) . To this end, R.H. Fox introduced the compact-open topology on the set of continuous functions Y -+ X , where Y and X are topological spaces (cf. also C o m p a c t - o p e n t o p o l o g y ; T o p o l o g i c a l s p a c e ) . This has a sub-base of sets W ( C , U) for C c o m p a c t in Y and U open in X , where W ( C , U) is the set of continuous functions Y -+ X such t h a t f ( C ) C U. Fox also began the investigation of the relation of this to the 'exponential law'. T h e exponential law for sets uses the set X Y of functions X ~ Y and states t h a t for any sets X , Y, Z there is a n a t u r a l bijection e: X z x Y --+ ( x Y ) z , given by e ( f ) ( z ) ( y ) = f ( z , y ) , z E Z, y C Y . This law is an expression of the s t a n d a r d idea t h a t a function of two variables can be t h o u g h t of as a variable function of one variable. Fox sought a similar result when X , Y, Z are topological spaces and X Y is replaced by C ( Y , X ) , the set of continuous functions Y -+ X . This required finding an appropriate t o p o l o g y on C(Y, X ) . Unfortunately, it was found t h a t this worked well only for Y locally compact, in the sense of having a n e i g h b o u r h o o d base of comp a c t sets, and t h a t the a p p r o p r i a t e topology was the c o m p a c t - o p e n t o p o l o g y . A careful analysis of topologies on 6(II, X ) in relation to the exponential law was given by R. Arens and J. Dugundji. T h e restriction to locally c o m p a c t spaces for the validity of the exponential law was awkward for topology. It was suggested by E. Spanier in [14] that the situation could be remedied by using 'quasi-topological spaces', which specify for X a set of mappings C ~ X for all c o m p a c t Hausdorff spaces, satisfying appropriate axioms (cf. also H a u s d o r f f s p a c e ) . This suggestion was subsequently felt to be vitiated by the fact t h a t a twopoint set had a class of quasi-topological structures (see the discussion and references in [13]).

E X P O N E N T I A L SUM ESTIMATES R. Brown in [4] found that the exponential law was satisfied in the category of Hausdorff k-spaces (cf. S p a c e o f m a p p i n g s ~ t o p o l o g i c a l ) and continuous mappings. In [5] it was suggested that this category 'may be adequate and convenient for all purposes of topology'. The exposition in [6] suggested the equivalent category of Hausdorff spaces and mappings continuous on compact subsets. It also explained the failure of the exponential law for all spaces, by giving a law of the form C(Z xs Y,X) ~- C(Z,C(Y,X)) for a new product topology Z x s Y. The theme of 'convenient categories' was also taken up in the expository paper [15], again using Hausdorff k-spaces, but called 'compactly-generated spaces'. It was known about that time (1967) that the Hausdorff condition could be removed by taking compactly generated to mean 'having the final topology with respect to all mappings of compact Hausdorff spaces into the space'. In [2] this is generalized to the case of certain classes M of compact Hausdorff spaces, considering the set A(X, Y) of M-continuous mappings between spaces, and giving this set a topology with a sub-base of sets W(t, U) = {f e A(X, Y ) : / t ( A ) C U} for all open sets in Y and all 'test' mappings t: A --+ X for A C M. An important extension of results on the exponential law involves spaces of partial maps on closed subsets. A useful trick here is the representability of such partial mappings, an idea which comes from t o p o s theory: The set C(Y, X) of partial mappings with closed domain is bijective with C(Y, 2 ) where )( = X U {co}, where co ¢ X, and C is closed in J( if and only if C is closed in X or co E C - - thus {co} is open but not closed in 0(. Using this device one can obtain an exponential law in the slice category C T o p / B of compactly generated spaces over the compactly generated space B provided B is a T0-space (cf. S e p a r a t i o n a x i o m ) . If q: Q + B, r: R + B are spaces over B, then the space of functions (q, r): (Q, R) --+ B has as fibre over b C B the space of mappings q-lb --+ r-lb. The topology is the join (in the given convenient category) of the topology on partial mappings with closed domain and that which makes (q, r) continuous. A consequence of the fibred exponential law is that a mapping Q --+ R over B corresponds exactly to a section of the mapping (q, r). This law has been extended to more general situations in [1]. These laws are very useful tools in a l g e b r a i c t o p o l o g y . A dual device yields a topology on spaces of mappings with open domain [8], but this has not yet (2000) been much exploited. This is surprising, since the solutions of many standard problems, such as differential equations, are often partial functions with variable open domain. In [11] the category of sequential spaces is embedded into a topos.

Approaches based on other kinds of set-open topologies and on graph topologies, with the aim of such applications, can be found in, for example, [3], [9], which point to a substantial literature in this area. However, [12] uses categorical concepts and constructions to give a fairly comprehensive theory of differentiation in fairly general linear spaces of arbitrary dimension. References [1] BOOTH, P.I., HEATH, P.R., AND PICCININI, R.: 'Fibre preserving maps and functions spaces': Algebraic Topology, Proc. Vancouver, 1977, Vol. 673 of Lecture Notes in Mathematics, Springer, 1978, pp. 158-167. [2] BOOTH, P.I., AND TILLOTSON, A.: 'Monoidal closed, cartesian closed and convenient categories of topological spaces', Pacific J. Math. 88 (1980), 35-53. [3] BRANDI, P., AND CEPPITELLI, R.: 'A new graph topology. Connections with the compact open topology', Applic. Anal. 53 (1994), 185-196. [4] BROWN, R.: 'Some problems of algebraic topology: a study of function spaces, function complexes and FD-complexes', DPhil Thesis, Oxford (1961). [5] BROWN, R.: 'Ten topologies for X x Y', Quart. J. Math. 2, no. 14 (1963), 303-319. [6] BROWN, R.: 'Function spaces and product topologies', Quart. J. Math. 2, no. 15 (1964), 238-250. [7] BROWN, R.: Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, 1988. [8] BROWN, R., AND ABD-ALLAH, A.M.: 'A compact-open topoiogy on partial maps with open domain', d. London Math. Soc. 2, no. 21 (1980), 480-486. [9] CONCILIO, A., AND NAIMPALLY, S.: 'Proximal set-open topologies', Aeta Math. Aead. Sei. Hungar. 88 (2000), 227237. [10] HERRLICH, H.: 'Topological improvements of categories of structured sets', Topol. Appl. 27 (1987), 145-155. [11] JOHNSTONE,P.: 'On a topological topos', Proc. London Math. Soe. 3, no. 38 (1979), 237-271. [12] KRIEGL, A., AND MICHOR, P.W.: The convenient setting of global analysis, Vol. 53 of Math. Surveys and Monographs, Amer. Math. Soc., 1997. [13] MIN, K.C., KIM, Y.S., AND PARK, J.W.: 'Fibrewise exponential laws in a quasitopos', Cah. Topol. Gdom. Diff. Cat. 40 (1999), 242-260. [14] SPANIER,E.H.: 'Quasi-topologies', Duke Math. d. 30 (1963), 1-14. [15] STEENROD,N.: 'A convenient category of topological spaces', Michigan Math. d. 14 (1967), 133-152.

R. Brown MSC1991:54C35 EXPONENTIAL SUM E S T I M A T E S - Exponential

sums have the form S = E

e 2~if(n),

nEA

where A is a finite set of integers and f is a realvalued function (cf. also T r i g o n o m e t r i c s u m ) . The basic problem is to show, under suitable circumstances, 143

E X P O N E N T I A L SUM E S T I M A T E S t h a t S = o(#A) as # A --+ oo. Unless there are obvious reasons to the contrary one actually expects S to have order around ( # A ) 1/2. Exponential sums in more t h a n one variable also occur, and much of what is stated below can be generalized to such sums.

' Methods due to H. Weyl for suitable constants Ck, %. (see [6, Chap. 2]), J. van der Corput (see [1]), I.M. Vinogradov and N.M. Korobov (see [3, Chap. 6]), and E. Bombieri and H. Iwaniec (see [2]) have been used for sums of this type.

A r i t h m e t i c s u m s . There are two common types of exponential sum encountered in a n a l y t i c n u m b e r t h e o r y . In the first type, one starts with polynomials g(X), h(X) E Z[X], a positive integer modulus q, and a finite interval I C R. One then takes A as the set of integers n 6 I for which GCD(h(n),q) = 1, and sets f(n) = g(n)h(n)/q, where h(n) is any integer for which h(n)h(n) - 1 (mod q). When I = (0, q], such a sum is called complete. When I C_ (0, q] one may estimate the incomplete sum in terms of complete ones via the bound

Van der Corput's method. Of the above approaches, the method of van der Corput is perhaps the most versatile. It is based on two processes, which convert the original sum S into other sums. The A-process uses the inequality

E e2~zf(n)"2 ~ M
N

<< H - ~- U 1 e27rig(n)h(n)/q

-

0<m_
-

E

e2~i(mn+g(~)h(~))/q

O
Complete sums have multiplicative properties which enable one to reduce consideration to the case in which q is a prime number or a power of a prime number. Moreover, Well's Riemann hypothesis for curves over finite fields (see [5], for example) shows that, if q is prime, then

E

e2~ig(~)h(~)/q <- (deg(g) + deg(h))x/~,

n6I (h(n),q)=l except when 9(n)h(n) is constant modulo q. For powers of a prime number the situation is more complicated, but broadly similar. It follows that, for fixed g and h and q prime, the incomplete sum will be o(~A) as soon as ~¢A/(v~logq ) --+ co. It is an important outstanding problem (as of 2000) to improve on this in general. A n a l y t i c s u m s . The second type of exponential sum arises when f extends to a suitably smooth function on a real interval I. I m p o r t a n t examples correspond to f(n) = a n k for k E N, or f(n) = (t/2~r)logn, which occur in the theory of Waring's problem and the R i e m a n n z e t a - f u n c t i o n , respectively (cf. also W a r i n g p r o b l e m ; Z e t a - f u n c t i o n ) . Let I = (N, N + M] with positive integers M _< N. One typically imposes the condition that f e C a [N, N + M] with

ekTN-a <_ f(a)(x) <_c~TN -k (kEN, N <x
e2~(f(~+h)_S(n))

H M
where H C N satisfies H < N. This has the effect of replacing f by a function f(x + h) - f(x) which satisfies (1) with a smaller value of T. To describe the B-process it is assumed t h a t f ' is decreasing, and one writes f'(M + N) = A, if(N) = B and f'(x,~) = m for A < m < A + B. The B-process then derives from the the estimate

(h(~),q)=l

< (1 + log q) m a x

~

(1)

E e2Triy(n) : M 2 be an integer, and let f C C k IN, N + M] with M, N integers. Suppose that 0 < Ak _< ]f(k)(x)l _< AAk on [N,N+M]. Then, if K = 2 k - l , one has

E e27rif(n) << M
Exponent pairs. Many bounds of the form S << (T/N)PN q subject to (1), or similar slightly stronger conditions, have been proved. If one has such a bound, one says that (p,q) is an exponent pair. Thus, the

E X P O N E N T I A L SUM ESTIMATES van der Corput third-derivative estimate shows that (1/6, 2/3) is an exponent pair. The case q = p + 1/2 is of particular interest, since it leads to a bound for the Riemann zet a-function, of the form ~ (1/2 + it ) ~ t p log t for t > 2. Although van der Corput's method leads to a rich source of exponent pairs, better results can be derived by more complicated methods. Thus, M.N. Huxley [2, Chap. 17] shows that (p,p+ 1/2) is an exponent pair whenever p > 89/570, using the B o m b i e r i - I w a n i e c m e t h o d . It is conjectured that (p,p + 1/2) is an exponent pair for any positive p. This can be seen as a generalization of the L i n d e l S f h y p o t h e s i s for the Riemann zeta-function.

The Vinogradov-Korobov method. When (1) holds with T larger than about N 2°, the van der Corput method is inferior to that given by Vinogradov and Korobov. This range of values is important in establishing zerofree regions for the Riemann zeta-function, for example. The method reduces the problem to an estimate

for the number of solutions of the simultaneous equations ~ ki : l ?l~ih z E i :kl •i h for 1 < -- h < -- H with positive integer variables mi, ni < P. This is provided by Vinogradov's mean value theorem (cf. also V i n o g r a d o v method). References Ill GRAHAM, S.W., AND KOLESNIK, G.: Van der Corput's method for exponential sums, Vol. 126 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1991. [2] HUXLEY, M.N.: Area, lattice points and exponential sums, Voi. 13 of London Math. Soc. Monographs, Clarendon Press, 1996. [3] IvI~, A.: The Riemann zeta-function, Wiley, ]985. [4] KOROBOV, N.M.: Exponential sums and their applications, Kluwer Acad. Publ., 1992. [5] SCHMIDT, W.M.: Equations over finite fields. A n elementary approach, Vol. 536 of Lecture Notes in Mathematics, Springer, 1976. [6] VAUGHAN, R.C.: The Hardy-Littlewood method, Vol. 80 of Tracts in Math., C a m b r i d g e Univ. Press, 1981. D.R. Heath-Brown

MSC1991:11L07

145

F FACTORIZATION OF POLYNOMIALS, factoring polynomials - Since C.F. Gauss it is known that an arbitrary p o l y n o m i a l over a field or over the integers can be factored into irreducible factors, essentially uniquely (cf. also F a c t o r i a l ring). For an efficient version of Gauss' theorem, one asks to find these factors algorithmically, and to devise such algorithms with low cost.

Based on a precocious uncomputability result in [13], one can construct sufficiently bizarre (but still 'computable') fields over which even square-freeness of polynomials is undecidable in the sense of A.M. Turing (cf. also Undecidability; Turing machine). But for the fields of practical interest, there are algorithms that perform extremely well, both in theory and practice. Of course, factorization of integers and thus also in Z[x] remains difficult; much of cryptography (cf. Cryptofogy) is based on the belief that it will remain so. The base case concerns factoring univariate polynomials over a finite field Fq with q elements, where q is a prime power. A first step is to make the input polynomial f E Fq[x], of degree n, square-free. This is easy to do by computing gcd(f, 0 f / 0 x ) and possibly extracting pth roots, where p = charFq. The main tool of all algorithms is the F r o b e n i u s a u t o m o r p h i s m a: /~ -+ R on the Fq-algebra R = Fq[x]/(f). The pioneering algorithms are due to E.R. Berlekamp [1], [2], who represents cr by its matrix on the basis 1,x, x 2 , . . . , x n-1 m o d f of R. A second approach, due to D.G. Cantor and H. Zassenhaus [3], is to compute cr by repeated squaring. A third method (J. von zur Gathen and V. Shoup [7]) uses the so-called polynomial representation of cr as its basic tool. The last two algorithms are based on Gauss' theorem that x qd - x is the product of all monic irreducible polynomials in Fq[x] whose degree divides d. Thus, fl = gcd(x q - x, f ) is the product of all linear factors of f; next, f2 = gcd(x q~ - x, f / f l ) consists of all quadratic factors, and so on. This yields the distinctdegree faetorization (fl, f2,. . .) of f .

The equal-degree factorization step splits any of the resulting factors fi. This is only necessary if deg fi > i. Since all irreducible factors of fi have degree i, the algebra Ri = Fq[x]/(fi) is a direct product of (at least two) copies of Fq,. A r a n d o m element a of Ri is likely to have L e g e n d r e s y m b o l +1 in some and - 1 in other copies; then gcd(a (q;-1)/2 - 1, fi) is a non-trivial factor of fi. To describe the cost of these methods, one uses fast arithmetic, so t h a t polynomials over Fq of degree up to n can be nmltiplied with O(n log n log log n) operations in Fq, or O~(n) for short, where the so-called 'soft O' hides factors that are logarithmic in n. Furthermore, w is an exponent for matrix multiplication, with the current (in 2000) world record a~ < 2.376, from [4]. All algorithms first compute x q modulo f , with O - ( n l o g q ) operations in Fq. One can show that in an appropriate model, f~(log q) operations are necessary, even for n = 2. The further cost is as follows:

Fq[x]/(f)

as

C o s t in O ~

Berlekamp

Fq-vector space

nw

Cantor

multiplicative semi-group

n 2 log q

Fq-algebra

n2

Zassenhaus

von zur Gathen-Shoup

Table 1. For small fields, even better algorithms exist. The next problem is faetorization of some f C Q[x]. The central tool is Hensel lifting, which lifts a factorization of f modulo an appropriate prime number p to one modulo a large power pk of p. Irreducible factors of f will usually factor modulo p, according to the C h e b o t a r e v d e n s i t y t h e o r e m . One can then try various factor combinations of the irreducible factors modulo pk to recover a true factor in Q[x]. This works quite well in practice, at least for polynomials of moderate degree, but uses exponential time on some inputs (for example, on the Swinnerton-Dyer polynomials). In a celebrated paper, A.K. Lenstra, H.W. Lenstra, Jr. and L. Lovdsz [11] introduced basis reduction of integer lattices (aft L L L b a s i s r e d u c t i o n m e t h o d ) , and applied this to obtain a polynomial-time algorithm. Their reduction

FADDEEV-POPOV GHOST method has since found many applications, notably in cryptanalysis (cf. also C r y p t o l o g y ) . A method in [9] promises an even faster factorizing method. The next tasks are bivariate polynomials. It can be solved in a similar fashion, with Hensel lifting, say, modulo one variable, and an appropriate version of basis reduction, which is easy in this case. Algebraic extensions of the ground field are handled similarly. Multivariate polynomials pose a new type of problem: how to represent them? The dense representation, where each term up to the degree is written out, is often too long. One would like to work with the sparse representation, using only the non-zero coefficients. The methods discussed above can be adapted and work reasonably well on many examples, but no guarantees of polynomial time are given. Two new ingredients are required. The first are efficient versions (due to E. Kaltofen and yon zur Gathen) of Hilbert's irreducibility theorem (cf. also H i l b e r t t h e o r e m ) . These versions say that if one reduces many to two variables with a certain type of random linear substitution, then each irreducible factor is very likely to remain irreducible. The second ingredient is an even more concise representation, namely by a black box which returns the polynomial's value at any requested point. A highlight of this theory is the random polynomial-time factorization method in [10]. Each major computer algebra system has some variant of these methods implemented. Special-purpose software can factor huge polynomials, for example of degree more than one million over F2. Several textbooks describe the details of some of these methods, e.g. [8], [12], [5], [1@ Factorization of polynomials modulo a composite number presents some surprises, such as the possibility of exponentially many irreducible factors, which can nevertheless be determined in polynomial time, in an appropriate data structure; see [6]. For a historical perspective, note that the basic idea of equal-degree factorization was known to A.M. Legendre, while Gauss had found, around 1798, the distinct-degree factorization algorithm and Hensel lifting. They were to form part of the eighth chapter of his 'Disquisitiones Arithmeticae', but only seven got published, due to lack of funding. References [1] BERLEKAMP, E.R.: 'Factoring polynomials over finite fields', Bell @st. Techn. J. 46 (1967), 1853-1859. [2] BERLEKAMP, E.R.: 'Factoring polynomials over large finite fields', Math. Comput. 24, no. 11 (1970), 713-735. [3] CANTOR, D.G., A~D ZASSENHAUS, H.: 'A new algorithm for factoring polynomials over finite fields', Math. Comput. 36, no. 154 (1981), 587-592.

[4] COPPERSMITH, D., AND WINOGRAD, S.: 'Matrix multiplica-

[6] [6] [7]

[8] [9]

[10]

tion via arithmetic progressions', J. Symbolic Comput. 9 (1990), 251-280. GATHEN, J. VON ZUR, AND GERHARD, J.: Modern computer algebra, Cambridge Univ. Press, 1999. GATHEN, J. VON ZUR, AND HARTLIEB, S.: 'Factoring modular polynomials', J. Symbolic Comput. 26, no. 5 (1998), 583-606. GATHEN, J. VON ZUR, AND SHOUP, V.: 'Computing Frobenius maps and factoring polynomials', Comput. Complexity 2 (1992), 187-224. GEDDES, K.O., CZAPOR, s . a . , AND LABAHN, G.: Algorithms for computer algebra, Kluwer Acad. Publ., 1992. HOEIJ, M. VAN: 'Factoring polynomials and the knapsack problem', www.math.fsu.edu/~hoeij/knapsack/ paper/knapsack.ps (2000). KALTOFEN, E., AND TRAGER, B.M.: 'Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators', J. Symbolic Comput. 9 (1990), 301-320.

[Ii] LENSTRA, A.K., H.W. LENSTRA, JR., AND LOVASZ, L.: 'Factoring polynomials with rational coefficients', Math. Ann. 261 (1982), 515-534. [12] SHPARLINSKI, I.E.: Finite fields: theory and computation, Kluwer Acad. Publ., 1999. [13] WAERDEN, B.L. VAN DER: 'Eine Bemerkung fiber die Unzerlegbarkeit von Polynomen', Math. Ann. 102 (1930), 738-739. [14] YAP, CHEE KENG: Fundamental problems of algorithmic algebra, Oxford Univ. Press, 2000.

Joachim yon zur Gathen

MSC1991:12D05 F A D D E E V - P O P O V GHOST - An auxiliary field, not physical and known as a ghost field, which was introduced in the quantization procedure of non-Abelian gauge theories. The quantization procedure of non-Abelian gauge theories demands the introduction of certain auxiliary fields, known as ghosts fields, which are not physical. The need for such fields was first observed by R. Feynman [6], based on unitary arguments. Later, the quantization procedure of Yang-Mills theory (cf. also Y a n g - M i l l s field) based on a path integral (functional integral) was developed by L.D. Faddeev and V.N. Popov [5], and this procedure as a whole is known as the Faddeev-Popov method. Faddeev-Popov ghosts (anti-ghosts) are fictitious anti-commuting complex scalar fields ca(x) (respectively, "~a(x)), where a = 1, 2, 3 and x is a point of the space-time, which are used in the Faddeev-Popov method to represent the Faddeev-Popov determinant appearing in the generating functional of the S-matrix in the form of a fermionic Gaussian integral (Berezin integral). If Ta, a = 1, 2, 3, are the generators of the Lie a l g e b r a su(2), then the Faddeev-Popov ghosts (anti-ghosts) are usually combined into the Lie algebra su(2)-valued function e(x) = ca(x)Ta (respectively, ~(x) = ~a (x)Ta). Although the Faddeev-Popov ghosts anti-commute, they are not physical fermionic fields. 147

FADDEEV-POPOV GHOST From a mathematical point of view, the FaddeevPopov ghosts are the generators of the infinitedimensional Grassmann algebra, whose description can be found in [3]. It follows from the structure of an infinite-dimensional Grassmann algebra that the Faddeev-Popov ghosts satisfy the following commutation relations:

ca(x)cb(v) e (x)cb(y)

(1) (2)

(3) The introduction of Faddeev-Popov ghosts leads to the appearance of additional terms in the exponent of the generating functional; these terms are combined with the classical Yang-Mills Lagrangian into the quantum Lagrangian. It turns out that the quantum Lagrangian is invariant under the Becchi-Rouet-StoraTyutin transformations 5BRST (BRST transformations; 2 cf. [2], [10]), which are nilpotent: 5BRST = 0, i.e. applied twice to any field they give zero. Later, the BRST transformations 5BRST were complemented by the anti-BRST transformations ~BRST [9], which are also nilpotent. It is well known that an appropriate geometric framework for the Yang-Mills theory is the theory of connections on fibre bundles (cf. also C o n n e c t i o n ; P r i n c i pal fibre b u n d l e ) . This fact and the nilpotency of the BRST transformations suggested an idea to construct a geometric interpretation for the Faddeev-Popov ghosts and BRST transformations in terms of exterior differentiation and differential forms on a p r i n c i p a l f i b r e b u n d l e (cf. also E x t e r i o r a l g e b r a ) . A first geometric interpretation of this kind, identifying the FaddeevPopov ghost c with a Lie algebra-valued 1-form on a principal bundle, was proposed by Y. Ne'eman and J. Thierry-Mieg [8]. In order to incorporate the anti-ghost and anti-BRST transformations into this geometric Ne'eman-Thierry-Mieg interpretation, the formalism of q-vector fields, considered as (-q)-forms, and a corresponding analogue of exterior differentiation was developed by id. Lumiste [7]. A geometric interpretation of the Faddeev Popov ghost c and anti-ghost ~, identifying them with the components of a connection form on a super fibre bundle, was proposed by L. Bonora and M. Tonin [4] and was subsequently specified by Lumiste in [7]. It should be noted that all geometric interpretations of the Faddeev-Popov ghosts mentioned above lay aside the structure of infinite-dimensional Grassmann algebra (1)-(3) generated by the ghosts and anti-ghosts. In other words, identification of the Fadeev-Popov ghost c(x) with a differential form leads to anti-commutative behaviour of ghosts only with respect to superscripts: 148

ca(x)cb(x) = -cb(x)ca(x), but not in different points of the space-time. The infinite-dimensional structure of the Grassmann algebra generated by the FaddeevPopov ghosts was used in [1] to construct an infinitedimensional s u p e r - m a n i f o l d with underlying infinitedimensional manifold of all connections of a principal fibre bundle. In this approach, the Faddeev-Popov ghosts play the role of odd coordinates of a super-manifold. It was shown that the quantum Lagrangian, considered as a function on an infinite-dimensional super-manifold, can be obtained by a procedure of continuation of the classical Yang-Mills Lagrangian from the underlying manifold of all connections to the super-manifold. The Faddeev-Popov ghosts allow one to develop a BRST method of quantization based on the BRST transformations. This method was later used in quantization of several field theories. It also plays an essential role in developing topological field theories in dimension four [11]. References [1] ABaAMOV,V., AND LUMISTE, 0.: 'Superspace with underlying Banaeh fiber bundle of connections and the supersymmetries of effective action', Soviet Math. (Iz. VUZ) 30, no. 1 (1986), 1-13. [2] BECCHI, C., ROUET, A., AND STORA, R., Commun. Math. Phys. 42 (1975), 127. [3] BEREZIN, F.A.: The method of second quantization, Acad. Press, 1966. [4] BONORA, L., AND TONIN, M.: 'Superfield formulation of extended BRS symmetry', Phys. Lett. 98B (1981), 48-50. [5] FADDEEV, L.D., AND POPOV, V.N., Phys. Lett. B25 (1967), 29. [6] FEYNMAN, P~.P., Acta Physiea Polonica 24 (1963), 697. [7] LUMISTE, I.J.: 'Connections in geometric interpretation of Yang Mills and Faddeev-Popov fields', Soviet Math. (Iz. VUZ) 27, no. 1 (1983), 51-62. [8] NE'EMAN, Y., AND THIERRY-MIEG, J.: 'Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries', Proc. Nat. Acad. Sci. USA 77, no. 2 (1980), 720-723. [9] OJIMA, I.: 'Another BRS transformation', Progr. Theoret. Phys. 64, no. 2 (1980), 625 638. [10] TYUTIN, I.V., Preprint P h I A N 39 (1975). [11] WITTEN, E.: 'Topological quantum field theory', Comm. Math. Phys. 117 (1988), 353-386.

V. Abramov U. Lumiste MSC 1991: 81Qxx, 81Sxx, 81T13 F C - G R O U P , finite conjugate group - A g r o u p G such that each x E G has only finitely many conjugates. This is one of several important possible finiteness conditions on an (infinite) group (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 ) . FC-groups are similar to finite groups in several respects. Let G be an arbitrary group. An element x C G is an FC-element if it has only finitely many conjugates. The FC-elements form a characteristic subgroup F, and

F E R M A T - T O R R I C E L L I PROBLEM

G / C a ( F ) is residually finite (here, C a ( F ) is the centralizer of F in G). An FC-group is thus a group in which all elements are FC-elements. The commutator subgroup of an FC-group is periodic (torsion). A group G is a finitely-generated FC-group if and only if it has a free Abelian subgroup A of finite rank in its centre such that A is of finite index in G. For further results, see [1, Part 1, Sect. 4.3; Part 2, pp. 102-104], and [2, Sect. 15.1]. See also C C - g r o u p . References [1] ROBINSON, D.J.S.: Finiteness conditions and generalized soluble groups, Parts 1-2, Springer, 1972. [2] SCOTT, W.R.: Group theory, Dover, reprint, 1987.

M. Hazewinkel MSC 1991:20F24 F E D O S O V T R A C E F O R M U L A - An asymptotic formula as h + 0 for the 'localized' trace of the exponential of a Hamiltonian H(t). The leading terms of this expansion can be calculated in terms of the fixed points of the classical Hamiltonian flow associated to H (provided that it has only isolated fixed points, see below). Explicitly, n

Tr [Aexp ( - i h - l H ( t ) ) ]

= E

ao(xk)dk e bk + O(h).

k=l

Here, the meaning of xk, dk and bk is the following. First, A is a p s e u d o - d l f f e r e n t i a l o p e r a t o r with compactly supported Weyl symbol (cf. also S y m b o l of a n operator). Let H0 and HI be the homogeneous components of H, and denote by ft the Hamiltonian flow associated to H0 (cf. also H a m i l t o n i a n s y s t e m ) . The formula above is proved under the assumption that, on the support of A, the flow ft has only isolated fixed points, denoted by X l , . . . , z ~ . Then dk = d e t ( 1 - f{(xk)) 1/2 and bk = - i h - l H o ( x k ) t - iHl(xk)t. See [1]. References [1] FEDOSOV, B.: 'Trace formula for SchrSdinger operator', Russian J. Math. Phys. 1 (1993), 447 463.

Victor Nistor MSC1991:81Q05 FERMAT-TORRICELLI PROBLEM, TorricelliFermat problem The (generalized) Fermat-Torricelli problem, incorrectly also called the Steiner-Weber problem, refers to finding the unique point x0 C R n, n > 2, minimizing the function m

f(x)

~

IIP~ - xll = c

z e R

(1)

(a constant),

(2)

i=1

which led Fermat to ask in 1643 for the minimum point with respect to the unweighted subcase rn = 3 of (1) (i.e., Wl = we = wa), cf. [14, p. 153]. The first solution to this subcase was obtained by E. Torricelli (see [29], [30]) using the focal property of the ellipse, and further ruler-and-compass constructions were given by B. Cavalieri, V. Viviani, Th. Simpson, and H. Lebesgue; the unweighted case m = 4 was solved by G. Fagnano, cf. [34], [20], and [2, Chap. II] for extensive historical discussions and corrections. With the help of Galois t h e o r y it was proved in [6] and [1] that for m > 5 points in general p o s i t i o n the Fermat-Torricelli problem does not allow exact algorithms under computation models with arithmetic operations and extraction of kth roots. On the other hand, c-approximative solutions for n = 2 can be constructed in polynomial time, see [4]. The weighted case m = 3 was completely solved by W. Launhardt [21]; its relations to generalizations of the Napoleon theorem (also for higher dimensions) were investigated in [24]. In 1846, E. Fasbender [13] proved that the unweighted case m = 3 of minimizing (1) is dual to the construction of the largest equilateral triangle circumscribed to the triangle Plp2Pa, and H.W. Kuhn [19] pointed out that this Vecten-Fasbender duality is historically the first example of dualizing a problem in the spirit of n o n - l i n e a r programming. For n = 2, the level curves (2) of the function (1) (with equal weights) are called poly-ellipses or multifocal ellipses. They were first studied by E.W. yon Tschirnhaus [31], who extended the classical gardener's string construction from m = 2 in (2) to m > 3. Many further properties and applications of these curves (and of their analogues in the weighted and higher-dimensional situation) are collected in [25], [12] and [16]. Analytic approaches to the minimum point x0 of (1) were presented by J. Bertrand, R. Sturm, L.L. LindelSf, Kuhn, C. Witzgall, and many others; in modern terms they can be summarized by the following theorem (see, e.g., [18]):

I) The minimum point Xo of (1) exists and is unique. II) If for each pi E { p l , . . . , p ~ } , j =~-~

lip, -
=

for a set P := {Pl,... ,P,~} of m > 3 given non-collinear points with corresponding positive weights W l , . . . , win, where II'll denotes the Euclidean norm. In 1638, R. Descartes invited P. de Fermat to investigate (for m = 4) curves of the form

P j -- P i

wj . IlPj - Pi[I

> wi,

i C j,

i=1

149

FERMAT-TORRICELLI PROBLEM holds, then x0 ~ { P l , . . - , P m } and fn

Pi -- xo

i=1

III) If there is a point Pi E { P l , . . . ,Pro} satisfying j=~

Pj-Pi w j . ][pj _ pill

<wi,

it

j,

--

then Pi = xo. These characterizations of Xo have a realistic appeal, since (for n = 2) there is even a mechanical device based on the so-called Varignon f r a m e : A board is drilled with m holes at the points P l , . . . , P m ; m strings are tied together in a knot at one end, the loose ends are passed through the m holes and are attached to physical weights below the board. The equilibrium position of the knot yields the solution. This was presented by G. Pick in [32, Math. Appendix], but the early history of this approach is discussed in [15]. The first algorithmic approach to x0 was provided by E. Weiszfeld [33], see also [18]. It is gradient descent but not convergent at the foci pi (cf. also G r a d i e n t m e t h o d ) . L.M. Ostresh [27] proposed the first completely convergent iteration procedure, which can even be applied in Banach spaces, see [11]. The Fermat Torricelli problem has been one of the starting points of location science from o p e r a t i o n s res e a r c h , in particular belonging to the field of continuous location theory, where it is usually called the l - m e d i a n problem or single facility location p r o b l e m , cf. [8], [22], and from the historical point of view also [21], [32] (see also W e b e r p r o b l e m ) . Since local conditions for Steiner points in Steiner minimal trees (cf. also S t e i n e r t r e e p r o b l e m ; S t e i n e r p o i n t ) are based on properties of Fermat-Torricelli points, a related passage in [7, pp. 354-361] can be considered as starting point of investigations in this direction (although the respective question goes back even to C.F. Gauss, 1836: To connect four towns by a network of minimal total length). For historical corrections with respect to Steiner minimal trees (including the fact that even the name is not justified, analogous to the wrong phrase 'Steiner-Weber problem' instead of 'Fermat-Torricelli problem') see [20] and [2, Sect. 23.9]; a modern treatment of Steiner minimal trees is [5]. G e n e r a l i z a t i o n s . Generalizations of the problem to minimize (1) were mainly studied in two directions. • Extensions to finite-dimensional normed spaces M n (i.e., Minkowski spaces) or other non-Euclidean spaces: For example, denoting by S the (in Minkowski spaces not necessarily point-shaped) solution set, the property S N a f f P ~ 0 holds for all finite point sets 150

P C M ~, n _> 3, if and only if M n is an i n n e r p r o d u c t space [9]. Various further properties of S C M n were investigated in [10], see also [3]. Extensions to other nonEuclidean spaces are considered in [11], [26]. • Modification of the given geometric configuration: For example, replacing the searched point in (1) by a hyperplane H , one gets the m e d i a n h y p e r p l a n e p r o b l e m (also called linear fit p r o b l e m , or L1 regression problem), which can be formulated with respect to vertical and orthogonal distances. The importance of this problem (e.g., compared with the known least-squares regression) for r o b u s t s t a t i s t i c s is based on the fact t h a t L1 estimates are technically robust against arbitrary outliers, cf. [28]. Also, such problems are studied in linear a p p r o x i m a t i o n t h e o r y and c o m p u t a t i o n a l g e o m e t r y (also in relation to the k-set problem). Position criteria for median hyperplanes of weighted point sets and algorithmical approaches are presented in [17] (for Euclidean n-spaces) and in [23] (for other Minkowski n-spaces). References

[1] BAJAJ, C.: 'The algebraic degree of geometric optimization problems.', Discr. Comput. Geom. 3 (1988), 177-191. [2] BOLTYANSKI, V., MARTINI, H., AND SOLTAN, V.: Geometric methods and optimization problems, Kluwer Acad. Publ.,

1999. [3] CHAKEmAN, G.D., AND GHANDEHARI,M.A.: 'The Fermat problem in Minkowski spaces.', Geom. Dedicata 17 (1985), 227 238. [4] CHANDRASEKARAN,R., AND TAMIa, A.: 'Algebraic optimization: The Fermat-Weber problem', Math. Programming 46

(1990), 219 224. [5] CmSLIK, D.: Steiner minimal trees, Kluwer Acad. Pub1., 1999. [6] COCKAYNE, E.J., AND MELZAK, Z.A.: 'Euclidean constructibility in graph-minimization problems', Math. Mag. 42 (1969), 206-208. [7"] COURANT, R., AND ROBBINS, H.: What is mathematics?, Oxford Univ. Press, 1941. [8] DREZNER,Z. (ed.): Facility location: A survey on applications and methods, Ser. in Operations Research. Springer, 1995.

[9] DURIER, l~.: 'The Fermat-Weber problem and inner product spaces', J. Approx. Th. 78 (1994), 161-17'3. [10] DURIER, R., AND MICHELOT, C.: 'Geometrical properties of the Fermat-Weber problem', European J. Oper. Res. 20

(1985), 332-343. [11] ECKHARDT, U.: 'Weber's problem and Weiszfeld's algorithm in general spaces', Math. Programming 18 (1980), 186-196. [12] ERDSS, P., ANDVINCZE,I.: 'On the approximation of convex, closed plane curves by multifocal ellipses', J. Appl. Probab. 19A (1982), 89-96. [13] FASBENDER, E.: @her die gleichseitigen Dreiecke, welche um ein gegebenes Dreieck gelegt werden kSnnen', J. Reine Angew. Math. 30 (1846), 230-231. [14] FERMAT,P. DE: (Evres, Vol. I, H. Tannery (ed.), Paris, 1891, Supplement: Paris 1922. [15] FRANKSEN, O.I., AND GRATAN--GUINNESS,I.: 'The earliest contribution to location theory? Spatio-economic equilibrium with Lam~ and Clapeyron (1829)', Math. and Computers in Simulation 31 (1989), 195-220.

FIBONACCIGROUP

[16] GROSS, C., AND STREMPEL, T.-t~.: 'On generalizations of conics and on a generalization of the Fermat-Torricelli point', Amer. Math. Monthly 105 (1998), 732 743. [17] KORNEENKO, N.M., AND MARTINI, H.: 'Hyperplane approximation and related topics', in J. PACH (ed.): New Trends in Discrete and Computational Geometry, Springer, 1993,

pp. 135-162. [18] KUHN,H.W.: 'Steiner's problem revisited', in G.B. DANTZIG AND B.C. EAVES (eds.): Studies in Optimization, Vol. 10 of Studies in Math., Math. Assoc. Amer., 1974, pp. 52 70. [19] KUHN,H.W.: 'Nonlinear programming: A historical view', in R.W. COTTLEAND C.W. LEMKE(eds.): S I A M A M S Proc., Vol. 9, Amer. Math. Soc., 1976, pp. 1-26. [20] KUPITZ, Y.S., AND MARTINI, H.: 'Geometric aspects of the generalized Fermat-Torricelli problem', in I. BiSR~.SNY AND K. BSRSCZKY (eds.): Intuitive Geometry (Budapest, 1995),

Vol. 6, Bolyai Soc. Math. Studies, 1997, pp. 55-127. [21] LAUNHARDT,W.: Kommereiellc Tracirung der Verkehrswege ,

Hannover, 1872. [22] LOVE,R.F., MORRIS, J.G., AND WESOLOWSKY, G.O.: Facilities location: models and methods, North-Holland, 1988. [23] MARTINI, H., AND SCHOBEL, A.: 'Median hyperplanes in normed spaces - - a survey', Discr. Appl. Math. 89 (1998), 181-195. [24] MARTINI, H., AND WEISSBACH, B.: 'Napoleon's theorem with weights in n-space', Geom. Dedicata 74 (1999), 213-223. [25] MELZAK, Z.A., AND FORSYTH, J.S.: 'Polyconics 1: Polyellipses and optimization', Quart. Appl. Math. 35 (1977), 239255. [26] NODA, R., SAKAI, W., AND MORIMOTO, M.: 'Generalized Fermat's problem.', Canad. Math. Bull. 34 (1991), 96 104. [27] OSTRESH, L.M.: 'On the convergence of a class of iterative methods for solving the Weber location problem', Operat. Res. 26 (1978), 597-609. [28] ROUSSEEUW, P.J., AND LEROY, A.M.: Robust regression and outlier detection, Wiley, 1987. [29] TORRICELLI, E.: Opere, Vol. I/2, Fa~nza, 1919, pp. 90-97. [30] TORRICELLI, E.: Opere, Vol. III, Fa~nza, 1919, pp. 426-431. [31] TSCHIRNHAUS, E . g . VON: Medicina mentis, Lipsiae, 1695, German ed. by R. Zaunick, Acta Historica Leopoldina, J.A. Barth, Leipzig 1963. [32] WEBER, A.: Ober den Standort der [ndustrien, Tell I: Reine Theorie des Standorts, J.C.B. Mohr, Tiibingen, 1909, English ed. by C.J. Priedrichs, Univ. Chicago Press, 1929. [33] WEISZFELD, E.: 'Sur le point pour lequel la somme des distances de n points dorm,s est minimu', Tdhoku Math. d. 43

(1937), 355-386. [34] WESOLOWSKY,G.O.: 'The Weber problem - - history and perspectives', J. Location Sci. 1 (1993), 5-23.

where indices are taken modulo m. Fibonacci groups were introduced by J.H. Conway [2] and are related to the F i b o n a c c i n u m b e r s with inductive definition ai + ai+l = ai+2 (with al = a2 = 1 as initial ones). Several combinatorial studies (see [1] for references) answered some questions on F(2, m), including their non-triviality and finiteness: F(2, m) is finite only for m = 1, 2, 3, 4, 5, 7. H. Helling, A.C. Kim and J. Mennicke [3] provided a geometrization of F(2, m), by showing that the groups F ( 2 , 2 n ) , n _> 2, are the fundamental groups of certain closed orientable threemanifolds (so-called Fibonacci manifolds, denoted by M~). See also F i b o n a c c i m a n i f o l d . In fact, for n > 4, F ( 2 , 2 n ) = 7h(Mn), where M,~ is a closed hyperbolic three-manifold; F ( 2 , 6 ) = 771(M3), where M3 is the Euclidean Hantzche-Wendt manifold; F ( 2 , 4 ) = 771(L(5, 2)), with L(5, 2) a lens s p a c e . This and properties of the fundamental groups of these three-manifolds imply that F ( 2 , 2 n ) are Noetherian groups, i.e. every finitely-generated subgroup of F ( 2 , 2 n ) is finitely presented (cf. also N o e t h e r i a n g r o u p ) . Since M3 is an affine R i e m a n n i a n m a n i f o l d , F(2, 6) is a torsion-free finite extension of Z a. Due to hyperbolicity for n _> 4 (cf. also H y p e r b o l i c g r o u p ) , the F ( 2 , 2 n ) are torsion-free, their Abelian subgroups are cyclic (cf. also C y c l i c g r o u p ) , there are explicit imbeddings F(2, 2n) C PSL2 (C), and the word and conjugacy problems are solvable for them (cf. also G r o u p calculus; I d e n t i t y p r o b l e m ) . Also, the groups F(2, 2n), n _> 4, are arithmetic if and only if n = 4, 5, 6, 8, 12; see [3], [4] and A r i t h m e t i c g r o u p . There are several generalizations of Fibonacci groups, related to generalizations of Fibonacci numbers. D.L. Johnson [5] has introduced the generalized Fibonacci groups (see [9] for a survey) F(r,m)

~- ( X l , . . . , Z m

[ Zi'''Xi+r--1

= Zi--r),

where indices are taken modulo m. Another generalization of Fibonacci groups is due to C. Maclachlan [7] (see [8] for their geometrization):

Horst Martini

F k (2, m ) =

MSC 1991:90B85 = (Xl,...

FERMAT-WEBER Weber MSC

PROBLEM

-

The

same

as the

problem. 1991:90B85

, Z m I Xi2gki+l = Xi+2;

indices

(rood m ) ) .

Fractional Fibonacci groups were introduced by A.C. Kim and A. Vesnin in [6] (which contains their geometrization as well):

F k/1 (2, m) =

FIBONACCI GROUP - The Fibonacci group F ( 2 , m ) has the presentation (cf. also F i n i t e l y presented group; Presentation): F(2, m) = < x l , . . . , x ~ I ~ix~+l = xi÷2>,

z

(xl, . . . , x,~ I X itX ik+ l ~ Xi+2 t .~ indices

(mod m))

References [1] CAMPBELL, C.M.: Topics in the theory of groups, Vol. I of Notes on Pure Math., Pusan Nat. Univ., 1985.

151

FIBONACCIGROUP

[2] CONWAY, J.H.: 'Advanced problem 5327', Amer. Math. Monthly 72 (1965), 915.

[3] HELLING, H., KIM, A.C., AND MENNICKE, J.: 'A geometric study of Fibonacci groups', Y. Lie Theory 8 (1998), 1 23. [4] H~LDEN, H.M., LOZANO, M.T., aND MONTESINOS, J.M.: 'The arithmeticity of the figure-eight knot orbifolds', in B. APANASOV,W. NEUMANN,t . REID, AND L. SIEBENMANN (eds.): Topology'90, de Gruyter, 1992, pp. 169-183. [5] JOHNSON,D.L.: 'Extensions of Fibonacci groups', Bull. London Math. Soc. 7 (1974), 101-104. [6] KIM, A.C., ANDVESNIN,A.: 'The fractional Fibonacci groups and manifolds', Sib. Math. J. 38 (1997), 655-664. [7] MACLACHLAN,C.: 'Generalizations of Fibonacci numbers, groups and manifolds': Combinatorial and Geometric Group Theory (1993), Vol. 204 of Lecture Notes, London Math. Soc., 1995, pp. 233-238. [8] MACLACHLAN,C., AND REID, A.W.: 'Generalized Fibonacci manifolds', Transformation Groups 2 (1997), 165-182. [9] THOMAS, R.M.: 'The Fibonacci groups revisited', in C.M. CAMPBELLAND E.F. ROBERTSON(eds.): Groups II (St. Andrews, 1989), Vol. 160 of Lecture Notes, London Math. Soc., 1991, pp. 445-456. Boris N. Apanasov

MSC 1991:20F38 F I B O N A C C I M A N I F O L D - T h e Fibonacci manifold M~, n >_ 2, is a closed orientable t h r e e - d i m e n s i o n a l m a n i f o l d whose f u n d a m e n t a l group is the F i b o n a c c i g r o u p F(2, 2n) (cf. also O r i e n t a t i o n ) . Such manifolds were discovered by H. Helling, A.C. Kim and J. Mennicke [3] as geometrizations of Fibonacci groups. For n > 4, the manifolds M~ are closed hyperbolic threemanifolds (cf. also H y p e r b o l i c m e t r i c ) , 2/43 is the Euclidean H a n t z c h e - W e n d t manifold, and M2 is the l e n s s p a c e L(5, 2) (see [3]). M a n y interesting properties of Fibonacci manifolds can be obtained from their relation to links and branched coverings over the three-dimensional sphere S 3, discovered by H.M. Hilden, M.T. Lozano and J.M. Montesinos [4]. In fact,

1) Mn is the n-fold cyclic covering of the threedimensional sphere S a, branched over the figure-eight knot (cf. L i s t i n g k n o t ) , see [4]; 2) MR can be obtained by D e h n s u r g e r y with parameters 1 and - 1 on the components of the chain of 2n linked circles in S 3, see [2]; 3) M n is the two-fold covering of S 3, branched over the link T~ corresponding to the closed 3-string braid

see [10]. The above well-known family Tn of links in S 3 includes the figure-eight knot as T2, the B o r r o m e a n rings as T3, the Turk's head knot 81s as T4, and the knot 10123 as T5 (in the notation of [8]). T h e last description of Mn also shows t h a t the hyperbolic volumes of the c o m p a c t Fibonacci manifolds M2~, n > 2, coincide with those ones of the (non-compact) link complements S 3 \ T,~, see [9], 152

[10]. Also, since the M~ are arithmetic if and only if

n = 4, 5, 6, 8, 12 (see [31, [4] and A r i t h m e t i c

group),

this shows t h a t hyperbolic manifolds with the same volu m e can be b o t h arithmetic and non-arithmetic, see [9]. There are several generalizations of Fibonacci manifolds, related to generalizations of the Fibonacci groups, see [1], [5], [6], [7] and F i b o n a c e i g r o u p . References [1] APANASOV,B.N.: Conformal geometry of discrete groups and manifolds, de Gruyter, 2000. [2] CAVICCHIOLI,A., AND SPACGIARI,F.: 'The classification of 3manifolds with spines related to Fibonacci groups': Algebraic Topology, Homotopy and Group Cohomology, Vol. 1509 of Lecture Notes in Mathematics, Springer, 1992, pp. 50-78. [3] HELLING,H., t~IM, A.C., AND MENNICKE, J.: 'A geometric study of Fibonacci groups', J. Lie Theory 8 (1998), 1-23. [4] HILDEN, H.M., LOZANO, M.T., AND MONTESINOS, J.M.: 'The arithmeticity of the figure-eight knot orbifolds', in B. APANASOV,W. NEUMANN,n. REID, AND L. SIEBENMANN (eds.): Topology'90, de Gruyter, 1992, pp. 169-183. [5] KIM, A.C., ANDVESNIN, n.: 'The fractional Fibonacci groups and manifolds', Sib. Math. J. 38 (1997), 655-664. [6] MACLACHLAN, C.: 'Generalizations of Fibonacci numbers, groups and manifolds', in A.J. DUNCAN,N.D. GILBERT,AND J. HOWIE (eds.): Combinatorial and Geometric Group Theory (Edinburgh, 1993), Vol. 204 of Lecture Notes, London Math. Soe., 1995, pp. 233-238. [7] MACLACHLAN,C., AND REID, A.W.: :Generalized Fibonaeci manifolds', Transformation Groups 2 (1997), 165-182. [8] ROLFSON,D.: Knots and links, Publish or Perish, 1976. [9] VESNIN, A.Yu., AND MEDNYKH, A.D.: 'Hyperbolic volumes of Fibonacci manifolds', Sib. Math. J. 36, no. 2 (1995), 235245. [10] VESNIN, A.YU., AND MEDNYKH, A.D.: 'Fibonacci manifolds as two-fold coverings over the three-dimensional sphere and the Meyerhoff-Neumann conjecture', Sib. Math. J. 37, no. 3 (1996), 461-467. Boris N. Apanasov

M S C 1991: 57Mxx FIBONACCI POLYNOMIALS Un(x) (cf. [1] and [4]) given by

{

The polynomials

u0(x) = 0 , Ui(x) =

1,

Un(x)

xU,,-l(x)+gn-2(x),

(1) n = 2,3,....

T h e y reduce to the F i b o n a c c i n u m b e r s Fn for x = 1 and they satisfy several identities, which m a y be easily proved by induction, e.g.:

U-n(x) = ( - 1 ) n - l U n ( x ) ; um+n(x) =

(2)

+ um(x)u

_l(x);

Un-}-l(X)~r~_l(X) - U2(z) = ( - 1 ) n ;

- 9(x)

'

(3) (4)

(5)

FIBONACCI POLYNOMIALS

(cf. M u l t i n o m i a l

where x -t- (5 2 + 4) 1/2 2

x -- (x 2 + 4) 1/2 ,

2

'

gn(k+1(x) )

coefficient):

v " (nl + ' "

[~/21 ( n - j ) !

(10)

n=0,1,...,

so that a ( x ) ~ ( x ) = - 1 ; and

Un+l(X) = J::0

+ nk)!Zk(nl+...+~k)_n '

~-2j

,

n=0,1,...,

(6)

where [y] denotes the greatest integer in y. W.A. Webb and E.A. Parberry [14] showed that the U,~(x) are irreducible polynomials over the ring of integers if and only if n is a prime number (cf. also Irr e d u c i b l e p o l y n o m i a l ) . They also found that xj = 2i cos(jTr/n), j = 1 , . . . , n - l , are the n - 1 roots of U~(x) (see also [2]). M. Bicknell [1] proved that Urn(x) divides U,~(x) if and only if m divides n. V.E. Hoggatt Jr., and C.T. Long [3] introduced the bivariate Fibonacei polynomials Un(x, y) by the recursion

where the sum is taken over all non-negative integers n l , . • •, nk such that nl + 2n2 +. • • + knk = n. They also obtained a simpler formula in terms of binomial coefficients. As a byproduct of (10), they were able to relate these polynomials to the number of trials Ark until the occurrence of the kth consecutive success in independent trials with success probability p. For p = 1/2 this formula reduces to

TT(k)

P(Nk = n + k) - V n+l 2n+k,

n = 0,1, . . . .

(11)

The Fibonacci-type polynomials of order k, F (k)(x), defined by v0 (k)(5) = 0,

I

F (k)(x) = 1,

Uo(x,y) = O, Ul(x, y) = 1, U~(x,y) = x U n _ l ( x , y ) + yU~-2(x,y),

(7)

n = 2,3,..., and they showed that the U~(x, y) are irreducible over the rational numbers if and only if n is a prime number. They also generalized (5) and proved that

n=2,..,k,

(12)

(k) (5), / F(k)~ (5) = x E j =k I F~_j (

n=k+l,k+2,...,

have simpler multinomial and binomial expansions than U(k) (x). The two families of polynomials are related by

U(nk)(x) = xl-nF(~k)(Xk),

n=1,2,....

(13)

Furthermore, with q = 1 - p ,

[~/2]

(n-j)!

. .

=

(s)

j=0 n = 0, 1, . . . .

/

(14)

n = k,k + l,....

In a series of papers, A.N. Philippou and his associates (cf. [5], [6], [7], [8], [9], [12], [131, [10], [11])introduced and studied Fibonacci, Fibonacci-type and multivariate Fibonacci polynomials of order k, and related them to probability and reliability. Let k be a fixed positive integer greater than or equal to 2. The Fibonacci polynomials of order k, U(~k) (x), are defined by

Assuming that the components of a c o n s e c u t i v e kout-of-n: F - s y s t e m are ordered linearly and function independently with probability p, Philippou [6] found that the reliability of the system, Rl (p; k, n), is given by

Rl(p;k,n) = p-lqn+lF~+2 ( P ) ,

(15)

n = k,k + l,.... If the components of the system are ordered circularly, then its reliability, Rc(p; k, n), is given by (cf. [12])

u0 = 0, U} k) (x) = 1,

U(k)(x v'n x k - J u n(k) n ~, /~- -- Z--~j=I - - j k(x~/,

P(Nk = n) = - n F n+l-k ( q ) ,

k =2,...,k,

(9)

TT(k)[a.~ -- X-,k 5 k - j T [ ( k ) (X ~ v n ~ / -- A.~j~-I ~ n - - j k I,

n = k + 1, k + 2 , . . . .

For k = 2 these reduce to Un(x), and for x = 1 these reduce to U(k), the Fibonacci numbers of order k (cf. [13]). Deriving and expanding the g e n e r a t i n g funct i o n of U(k)(5), they [10] obtained the following generalization of (6) in terms of the multinomial coefficients

Rc(p;k,n)=pq

n-l~-~'~(k)2.~)/~n_j+l ( P )

,

(16)

j=l n=k,k+l,.... Next, denote by Nk,~ the number of independent trials with success probability p until the occurrence of the rth kth consecutive success. It is well-known [5] that Nk,~ has the negative b i n o m i a l d i s t r i b u t i o n of order k with parameters r and p. Philippou and C. Georghiou [9] have related this probability distribution to the (r - 1)-fold 153

FIBONACCI POLYNOMIALS (k) convolution of F (k)(x) with itself, say Fa,~ (x), as follows: P(Nk# = n) -- - ~ Fn v*( kn 4)- 1 - - k r , r

(q)

(17)

n = kr, kr + 1 , . . . , which reduces to (14) for r = 1, and they utilized effectively relation (17) for deriving two useful expressions, a binomial and a recurrence one, for calculating the above probabilities. Let x = ( X l , . . . , x k ) . The multivariate Fibonacci polynomials of order k (cf. [8]), H(k)(x), are defined by the recurrence :0,

g~ k)(x) = 1, H(~k)(x)=E]_~xjH(~j(x),

n = 2,...,k,

(18)

n = k + l,k + 2,....

[11] PHILIPPOU, A.N., GEORGHIOU, C., AND PHILIPPOU, G.N.:

For x = ( X k - l , x k - 2 , . . . , 1 ) , Hn(k)(x) = U,(f)(x), n = 0 , 1 , . . . , and for x = ( x , . . . , x ) , H (k)(x) ---- F (k)(x). These polynomials have the following multinomial expansion:

(%+ '

(19)

n=0,1,..., where the sum is taken over all non-negative integers n, , . . . , nk such t h a t n z + 2 n 2 + . . . + k n k = n. Let the r a n d o m variable X be distributed as a multi-parameter negative binomial distribution of order k (cf. [7]) with parameters r, q l , . . . , q k (r > 0, 0 < qj < 1 for j = 1 , . . . , k and 0 < ql + " " + qk < 1). Philippou and D.L. Antzoulakos [8] showed t h a t the (r - 1)-fold convolution, H~(,k~ ) (x), of H(~k) (x) with itself is related to this distribution by P(X

----

rt) = p r H ( k ) + l , r ( q l , . . . , qk),

(20)

n : 0, 1, . . . . Furthermore, they have effectively utilized relation (20) in deriving a recurrence for calculating the above probabilities. References [1] BICKNELL, M.: 'A primer for the Fibonacci numbers VII', Fibonacci Quart. 8 (1970), 407-420. [2] HOGGATT JR., V.E., AND BICKNELL, M.: ' R o o t s of Fibonacci

polynomials', Fibonacci Quart. 11 (1973), 271-274. [3] HOGGATT JR., V . E . , AND LONG, C.T.: 'Divisibility properties of generalized Fibonacci polynomials', Fibonacci Quart. 12 (1974), 113-120. [4] LUCAS, E.: 'Theorie de fonctions numeriques simplement periodiques', Amer. J. Math. 1 (1878), 184-240; 289-321.

154

[10] PHILIPPOU, A.N., GEORGHIOU, C., AND PHILIPPOU, G.N.: 'Fi-

bonacci polynomials of order k, multinomial expansions and probability', Internat. J. Math. Math. Sci. 6 (1983), 545-550.

H~(k)(x) = E5=1 x j H ( ~ j ( x ) ,

=

[5] PHILIPPOU, A.N.: 'The negative binomial distribution of order k and some of its properties', Biota. J. 26 (1984), 789 794. [6] PHILIPPOU, A.N.: 'Distributions and Fibonacci polynomials of order k, longest runs, and reliability of concecutive-k-outof-n : F systems', in A.N. PHILIPPOU, G.E. BERGUM, AND A.F. HORADAM (ed8.): Fibonacci Numbers and Their Applications, Reidel, 1986, pp. 203-227. [71 PHILIPPOU, A.N.: 'On multiparameter distributions of order k', Ann. Inst. Statist. Math. 40 (1988), 467-475. [81 PHILIPPOU, A.N., AND ANTZOULAKOS, D.L.: ' M u l t i v a r i a t e Fibonacci polynomials of order k and the multiparameter negative binomial distribution of the same order', in G.E. BERGUM, A.N. PHILIPPOU, AND A.F. HORADAM (eds.): Applications of Fibonacci Numbers, Vol. 3, Kluwer Acad. Publ., 1990, pp. 273-279. [9] PmLmeOU, A.N., AND GEOaGmOU, C.: 'Convolutions of Fibonacci-type polynomials of order k and the negative binomial distributions of the same order', Fibonaeci Quart. 27 (1989), 209-216.

'Fibonacci-type polynomials of order k with probability applications', Fibonacci Quart. 23 (1985), 100-105. [12] PHILIPPOU, A.N., AND MAKRI, F.S.: 'Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k', in G.E. BERGUM, A.N. PHILIPPOU, AND A.F. HORADAM (eds.): Applications of Fibonaeei Numbers, Vol. 3, Kluwer Acad. Publ., 1990, pp. 281-286. [13] PHILIPPOU, A.N., AND MUWAFI, A.A.: 'Waiting for the kth consecutive success and the Fibonacci sequence of order k', Fibonacci Quart. 20 (1982), 28-32. [14] WEBB, W.A., AND PARBERRY, E.A.: 'Divisibility properties of Fibonacci polynomials', Fibonacci Quart. 7 (1969), 457463.

Andreas N. Philippou MSC1991: 33Bxx F I G ) t - T A L A M A N C A A L G E B R A - Let G be a locally compact group, 1 < p < oo and p' = p / p - 1. Conoo oo sider the set Ap(G) of all pairs (( k ~)~=1, (ln)~=l), with (kn)~__l a sequence in £ ~ ( G ) and (l~)~°°__1 a sequence

in P c ( G ) such that E _lNp(k )Np,(l ) < oo. Here, p(f) is defined by Np(f) = (fG If(x)J dm(x))l/< where m is some left-invariant H a a r m e a s u r e on G. Let Ap(G) denote the set of all u E C G for which oo oo there is a pair ((k~)~=l, (I~)~=1) E Ap(G) such t h a t u(x) = ~°°__1 k~*l~(x), where ~5(x) = qo(x-1). The set Ap(G) is a linear subspace of the C-vector space of all continuous complex-valued functions on G vanishing at infinity. For u E Ap(G) one sets [l ltA (c) = inf (()n=l,()n=l)

Np(k~)Np,(l~):

• u = ~ =oo with l k n-.-l n

P( ~

)

•

1) For the pointwise product on G, Ap(G) is a B a nach algebra.

FIGi-TALAMANCA ALGEBRA This algebra is called the Fig&Talamanca algebra of G. If G is Abelian, A2(G) is isometrically isomorphic to L~ (G), where G is the dual group of G. For G not necessarily Abelian, A2(G) is precisely the Fourier algebra of G. 2) If G is amenable, then A~(G) C A;(G). The algebra Ap(G) is a useful tool for studying the pconvolution operators of G (see [2], [7], [8]). For a function ~ on G and a , x E G one sets ~ ( x ) = ~(ax). A continuous linear operator T on L~(G) is said to be a p-convolution operator of G if T ( ~ ) = ~(T(~)) for every a E G and every ~ E LPc(G). Let CV;(G) be the set of all p-convolution operators of G. It is a closed subalgebra of the Banach algebra £(LPc(G)) of all continuous linear operators on L~(G). For a complex bounded m e a s u r e # on G (i.e. # E M~(G)) and a continuous complex-valued function p with compact support on G (~ C C00(G; C)), the rule AP(#)[~] = r[~ , ~ala/ p ~/2] defines a p-convolution operator AP(#). Of course, for f C C a, If] denotes the set of all g C C a with g(x) = f(x) malmost everywhere. Even for G = R one has CVp(G) ¢ AP(M*(G)). Let PMAG) be the closure in CVp(G) of Ap(MI(G)) with respect to the ultraweak operator topology on

C(LPc(G)). 3) The dual Ap(G)' of the Banach space Ap(G) is canonically isometrically isomorphic to PMp(G). Also, A;(G)' with the topology a(Ap(G)', Ap(G)) is homeomorphic to PMp(G) with the ultraweak operator topoiogy on I:(LPc(G)). As a consequence, for G amenable or for G arbitrary but with p = 2, PMp(G) = CVp(G). This duality between Ap(G) and P~/~;(G) also permits one to develop (see [1]) a kind of 'non-commutative harmonic analysis on G', where (for G Abelian) Ap(G) replaces L~(O) and CVp(G) replaces L ~ ( O ) . (Cf. also

Harmonic analysis, abstract.) Let T E CVp(G). Then the support of T, denoted by suppT, is the set of all x E G for which for all open subsets U, V, of G with e C U and x C V there are ¢,~b C C00(G; C) with s u p p ¢ C U, supp~b C V and

¢ 0.

If > E M~(G), then s u p p k ~ ( > ) = (supp#) -1. For G Abelian, let e be the canonical mapping from G onto G. Then f ~-+ (f'o ¢)-, where qo(X) = qo(X-1), is an isometric isomorphism of the Banach algebra L I(G) onto Au(G). Let u E L ~ ( 0 ) and x C G. Then x 'belongs to the spectrum of u' (written as x C spu) if [~(x)] lies in the closure of the linear span of {xu: X C G} in L ~ ( G ) , for the w e a k t o p o l o g y cr(L~(G), L~(G)). Let

T E CVp(G); then s p T = (suppT) -1. For G not necessarily amenable and T E CVp (G), T = 0 if and only s u p p T is empty. This assertion is a non-commutative version of the Wiener theorem! Similarly, there is also a version of the Carleman-Kaplansky theorem: for T E CVp(G), s u p p T = {Xl,... ,Xn} if and only there exist Cl,... ,ca C C such that T = ClA;(bx,)+...+c~AP(5~,), where 5x denotes the Dirac measure in x (cf. also DirGe distribution). In fact, even for G = T or for G = R (but p ¢ 2) the situation is not classical! The Banach space Ap(G) has been first introduced by A. Figh-Talamanca in [3] for G Abelian or G nonAbelian but compact. For these classes of groups he obtained assertion 3) above. The statement for a general locally compact group is due to C.S. Herz [5]. Assertion 1) is also due to Herz [4]. The Banach algebra Ap also satisfies the following properties: a) Let H be a closed subgroup of G. Then R e s H A ; ( G ) = Ap(H). More precisely, for every u E Av(H) and for every e > 0 there is a v C Ap(G) with

Res/

=

and IlvlJAp(< < II llAp(, )+e (see [5]).

b) The Banach algebra Ap(G) has bounded approximate units (i.e. there is a C > 0 such that for every u E A;(G) and for every e > 0 there is a v C Ap(G) with llVlIA~(a) <_c and Ilu - uv[]A~(a) < e) if and only if the locally compact group G is amenable (see [5] and [6] for p = 2). This algebra is often called the Figd-Talamanca-Herz

algebra. See also F o u r i e r a l g e b r a .

References [1] DERIOHETrrI, A.: 'Quelques observations concernant les ensembles de Ditkin d'un groupe loealement compact', Monatsh. Math. 101 (1986), 95 113. [2] EYMARD, P.: 'Alg~bres Ap et convoluteurs de LP(G)': Sere. Bourbaki 1969/70, Exp. 367, Vol. 180 of Lecture Notes in Math., Springer, 1971, pp. 364-381. [3] Fm3,-TALAMANCA, A.: 'Translation invariant operators in L p', Duke Math. J. 32 (1965), 495-501. [4] HERZ, C.: 'The theory of p-spaces', Trans. Amer. Math. Soc. 154 (1971), 69-82. [5] HERZ, C.: 'Harmonic synthesis for subgroups', Ann. Inst. Fourier (Grenoble) 23, no. 3 (1973), 91-123. [6] LEPTIN, H.: 'Sur l'alg~bre de Fourier d'un groupe localement compact', C.R. Acad. Sci. Paris Sdr. A 266 (1968), 11801182. [7] LOHOU~, N.: 'Alg~bres Ap et convolnteurs de LP(G)', Th~se, Univ. Paris-Sud (1971). [8] LOHOU~, N.: 'Estimations L p des coefficients de representations et op~rateurs de convolution', Adv. Math. 38 (1980), 178 221. [9] MCMULLEN, J.R.: 'Extensions of positive-definite functions', Memoirs Amer. Math. Soc. 117 (1972).

155

FIGh-TALAMANCA ALGEBRA

[10] PIER, J.-P.: Amenable locally compact groups, Wiley, 1984. Antoine Derighetti MSC1991: 43A15, 43A07, 43A45, 43A46, 46J10 F I T T I N G CHAIN, Fitting series, nilpotcnt series A standard way to decompose a finite s o l v a b l e g r o u p into nilpotent sections (cf. also N i l p o t e n t g r o u p ) . The Fitting chains achieve this with shortest possible length. This shortest possible length, the length of a Fitting series, is called the F i t t i n g l e n g t h of the group. The ascending Fitting chain (or upper nilpotent series) starts at the identity subgroup and builds up, each time with the largest subgroup of the group which contains the previous one and whose quotient by the previous one is nilpotent. The descending Fitting chain (or lower nilpotent series) starts with the group itself and builds successive smallest normal subgroups (cf. N o r m a l subg r o u p ) such that the quotient of the previous subgroup by the new subgroup is nilpotent. In the case of solvable groups, both series terminate, one with the whole group and the other with the identity. Let G be a f i n i t e g r o u p . Let ~-(G) denote the F i t t i n g s u b g r o u p of G, and let G • denote the smallest normal subgroup of G such that G / G H is nilpotent. The upper Fitting series is 2W0 z .ri0(G ) ~ F1 ~_~ ~Ul(G) ~ . . . ~ Fn : .fin(G) ~ ' ' " ,

where F0 = 1 and Fi = F ( G / F i - 1 ) for i = 1, 2 , . . . . The lower Fitting series is

Ho >_HI > H2 > ' "

>_Hn > _ ' " ,

where H0 = G and Hi = ( H i - l ) H for i = 1, 2 , . . . . Each term in each of these series is a c h a r a c t e r i s t i c s u b g r o u p of G. These characteristic subgroups are basic in describing the solvable group G. The ascending Fitting factors F i ( G ) / F i I(G) are nilpotent subgroups, and they give important information about the structure of G. Analogues of these concepts can also be obtained by replacing throughout the term 'nilpotent' by some suitable other collection of groups, such as a Fitting class. Fhrther details can be found in [1], [2], [3]. References [1] DOERK, K., AND HAWKES, T.: Finite soluble groups, de Gruyter, 1992. [2] HUPPERT, B.: Endliche Gruppen I, Springer, 1967. [3] HUPPERT, B., AND BLACKBURN, N.: Finite groups H, Springer, 1982.

Alexandre Turull MSC1991: 20F17, 20F18 F I T T I N G LENGTH, nilpotent length - The Fitting length, also known as nilpotent length, of a finite solvable g r o u p (cf. also F i n i t e g r o u p ) provides a measure of how far the group is from being nilpotent. For any 156

finite solvable group G, the ascending and the descending Fitting chains of G both have the same number of distinct elements (cf. also F i t t i n g chain). The length of the chain (that is, the number of distinct elements in either chain minus one) is called the Fitting length of G. Thus, the trivial group has Fitting length 0, and any non-trivial n i l p o t e n t g r o u p has Fitting length 1, whereas any finite solvable non-nilpotent group will have Fitting length at least 2; see any standard reference such as [2], [3], [4] for details. As one measure of the complexity of a solvable group, the Fitting height is related to many other such measures. Thus, it is related to the number of distinct irreducible character degrees, to the derived length, to the number of elements needed to generate the Sylow subgroups of the group, to the derived length of the Sylow subgroups or to their nilpotent class. It is, however, its relationship with fixed points of automorphism groups that is the most striking.

Frobenius's conjecture, proved by J.G. Thompson [6], states that if G is any finite group admitting an a u t o m o r p h i s m ¢ such that ¢ has prime order and no element of G except the identity is fixed by ¢, then G is nilpotent. This can be extended to the following conjecture. Let G be a finite group and let A be a group of automorphisms of G such that ([A[, IGI) = I and no element of G except the identity is fixed by all the elements of A. Then G is solvable and the Fitting length of G is bounded above by the length of the longest chain of subgroups in A. Denoting by h(G) the Fitting length of G and by ~(A) the length of the longest chain of subgroups of A, the conjecture states that h(G) _< g(A). This is known to be true in many cases [9] and to be the best possible in all cases [8]. Similar bounds can be obtained when the group of automorphism does have some fixed point. For example, [7], if G is a finite solvable group and A is a solvable group of automorphisms of G such that (IAI, I a l ) = 1, then h(G) _< h(Ca(A)) + 2e(A), where Ca(A) denotes the subgroup of the elements of G that are fixed under every automorphism in A. In some cases one can also give bounds when A is not solvable, see [5], but these bounds are bigger. Fixed-point-free automorphisms are closely related to Carter subgroups of solvable groups (cf. also C a r t e r s u b g r o u p ) . In [1], E.C. Dade proved that there is an exponential function f such that if G is a finite solvable group and C is its Carter subgroup, then h(G) _< f(g(C)), and conjectured that there actually exists a linear function f with the same properties. In [8], it is proved that there exits a quadratic function f which satisfies the condition whenever C is a direct product of elementary Abelian groups.

FLAT C O V E R References [1] DADE, E.C.: 'Carter subgroups and Fitting heights of finite solvable groups', Illinois Y. Math. 13 (1969), 449-514. [2] DOERK, K., AND HAWKES, T.: Finite soluble groups, de Gruyter, 1992. [3] HUPPERT, B.: Endliche Gruppen I, Springer, 1967. [4] HUPPERT, B., AND BLACKBURN, N.: Finite Groups II, Springer, 1982. [5] KURZWEIL, H.: 'AuflNsbare Gruppen auf denen nicht auflNsbare Gruppen operieren', Manuscripta Math. 41 (1983), 233-305. [6] THOMPSON, J.G.: 'Finite groups with fixed-point-free automorphisms of prime order', Proc. Nat. Acad. Sci. USA 45 (1959), 578-581. [7] TURULL, A.: 'Fitting height of groups and of fixed points', J. Algebra 86 (1984), 555 566. [8] TURULL, A.: 'Character theory and length problems': Finite and Locally Finite Groups (Istanbul, 1994) , Kluwer Acad. Publ., 1995, pp. 377-400. [9] TURULL, A.: 'Fixed point free action with some regular orbits', J. Algebra 194 (1997), 362-377.

Alezandre TurulI M S C 1991:20F18

Bass' result says that if every module has a projective cover, then these projective covers are fiat covers. In [7], E. Enochs proved that if a module has a fiat pre-cover, then it has a fiat cover. He also conjectured that every module has a flat cover. J. Xu [12] proved that the conjecture holds for all commutative Noetherian rings of finite Krull dimension (cf. also D i m e n sion; N o e t h e r i a n r i n g ) and L. Bican, R. E1 Bashir and Enochs [4] gave two different solutions of the conjecture for any ring. One proof uses a result of E1 Bashir which shows t h a t any m o r p h i s m F --+ M of a fiat roodule into M with F sufficiently large has a non-zero pure submodule of F in its kernel. The other proof is an ap-

plication of a theorem of P.C. Eklof and J. Trlifaj [6, Thm. 2] guaranteeing 'enough injectives and projectives' for certain cotorsion theories (as defined in [ii]). Eklof and Trlifaj attribute the inspiration for their theorem to a construction of [9]. D. Quillen [I0, Lemma II 3.3] gives essentially the same argument in the setting of homotopical algebra. In [3], Bass defined what were subsequently

FLAT COVER, fiat covering - In [1], R. Baer proved that every m o d u l e M can be embedded in an i n j e c t i v e m o d u l e E. In [5], B. Eckmann and A. SchNpf defined an injective envelope of a module to be an embedding M C E with E injective and with M essential in E, i.e. such that S N M # 0 for every submodule S C E, S # 0. They proved t h a t every module has an injective envelope and that if M C E1 and M C E2 are two injective envelopes of M, then any m o r p h i s m E1 -+ E2 which is the identity on M (and such exists) is an i s o m o r p h i s m . So, injective envelopes are unique up to isomorphism. In [2], H. Bass considered projective covers of roodules. The notion of a projective cover is categorically dual to that of an injective envelope. If R is a r i n g , Bass proved that every left R-module has a projective cover if and only if every flat left R-module is projective (cf. also P r o j e c t i v e m o d u l e ) . Projective covers (when they exist) are also unique up to isomorphism. In [7] the following definition can be found: If j r is a class of objects in a c a t e g o r y C, then a morphism ¢: F ~ X in C with F C j r is called an jr-pc if H o m ( G , F ) ~ Horn(G, X) is surjective for all G E j r and is called an jr-cover if, moreover, every f : F ~ F such that ¢ o f = ¢ is an a u t o m o r p h i s m of 5 . When F-covers exist, they are unique up to isomorphism. Both pre-covers and covers are frequently named after the class jr. So, fiat covers are jr-covers with jc the class of flat modules in the category of modules. Preenvelopes and envelopes are defined dually. Then this terminology agrees with the earlier terminology of injective envelopes and projective covers. In this language,

called

the Bass numbers of a module M over a commutative Noetherian ring. These are computed using the minimal injective resolution of the ring. Xu [8] showed that flat covers can be used to define dual Bass numbers of modules. These numbers have properties in some sense dual to the properties of the original Bass numbers. References [1] BAER, R.: 'Abelian groups which are direct summands of every containing group', Bull. Amer. Math. Soc. 46 (1940), 800-806. [2] BASS, H.: 'Finitistic dimension and a homological generalization of semiprimary rings', Trans. Amer. Math. Soc. 95 (1960), 466 488. [3] BASS, H.: 'On the ubiquity of Gorenstein rings', Math. Z. 82 (1963), 8-28. [4] BICAN, L., EL BASHIR, R., AND ENOCHS, E.: 'All modules have flat covers', Bull. London Math. Soc. (to appear). [5] ECKMANN, B., AND SCH()PF, A.: @ber injektive Moduln', Archly Math. 4 (1953), 75-78. [6] EKLOF, P., AND TRLIFAJ, J.: 'How to make Ext vanish', Bull. London Math. Soc. (to appear). [7] ENOCHS, E.: 'Injective and fiat covers, envelopes and resolvents', Israel J. Math. 39 (1981), 33-38. [8] ENOCHS, E., AND XU, J.: 'On invariants dual to the Bass numbers', Proc. Amer. Math. Soc. 125 (1997), 951-960. [9] G()BEL, R., AND SHELAH, S.: 'Cotorsion theories and splitters', Trans. Amer. Math. Soc. (to appear). [10] QUILLEN,D.: Homotopical algebra, Vol. 43 of Lecture Notes in Mathematics, Springer, 1967. [!1] SALCE, L.: 'Cotorsion theories for Abelian groups': Syrup. Math., Vol. 23, Amer. Math. Sou., 1979, pp. 11-32. [12] Xu, J.: 'The existence of flat covers over noetherian rings of finite Krull dimension', Proc. Amer. Math. Soc. 123 (1995), 27-32.

157

FLAT C O V E R [13] Xu, J.: Flat covers of modules, Vol. 1634 of Lecture Notes in Mathematics, Springer, 1996.

E. Enochs M S C 1991:16D40 FLECNODE on a planar curve A point at which a node (or double point; cf. also N o d e ) and an inflection (cf. also P o i n t o f i n f l e c t i o n ) coincide. Thus, one of the tangents at the node has intersection multiplicity at least 4 with the curve at that point.

#•p (MR) = #Q(R/P)(M ®n/P Q ( R / P ) ) .

References [1] BRUCE, J.W.: 'Lines, surfaces and duality', Math. Proc. Cambridge Philos. Soc. 112 (1992), 53-61. [2] TSUBOKO, M.: 'On the line complex determinant of flecnode tangents of a ruled surface and its flecnodal surfaces', Memoirs Ryojun Coll. Engin. 11 (1938), 233-238.

M. Hazewinkel M S C 1991: 14Hxx FLOOR FUNCTION, entier function, greatest integer function, integral part function - The function of a real variable that assigns to a real number x the largest integer < x. The modern notation is Ix J; the classical notation is Ix]. In computer science and computer languages it is often denoted by int(x). The related ceiling function Ix 1 gives the smallest integer _> x. The fractional part function is defined as frac(x) = {~ - [xJ [x J - 1

f o r x > 0, forx
The nearest integer function is nint(x) = round(x) = x - frac(x).

References [1] GRAHAM, R.L., KNUTH, D.E., AND PATASHNIK, O.: Concrete mathematics: a foundation for computer science, AddisonWesley, 1990. [2] WOLFRAM, S.: Mathematica: Version 3, Addison-Wesley, 1996, pp. 718-719.

M. Hazewinkel MSC 1991: 26Axx F O R S T E R - S W A N THEOREM, Swan-Forster theorem - An example of a local-global principle in commutative algebra (cf. also L o c a l - g l o b a l p r i n c i p l e s for large r i n g s o f a l g e b r a i c i n t e g e r s ; L o c a l - g l o b a l p r i n c i p l e s for t h e r i n g o f a l g e b r a i c i n t e g e r s ) . That is, it provides a method by which local information can be lifted to the whole ring or module. In the case of the Forster-Swan theorem the information consists of the minimal number of elements required to generate a given finitely-generated module. The theorem itself, and also the methods developed in proving it, have found applications in commutative algebra, algebraic geometry and algebraic K-theory; see [5] and [6]. 158

Let R be a c o m m u t a t i v e r i n g , and let M be a finitely-generated R-module (cf. also M o d u l e ) . Suppose that one wants to compute # n ( M ) , the minimal number of generators of M. This can be a difficult task in general, but one can easily compute the number of generators of M locally. Thus, let P be a p r i m e i d e a l of R, and denote by Q ( R / P ) the quotient ring of R / P . By the Nakayama lemina (cf. also J a c o b s o n r a d i c a l ) , (1)

This number will be denoted by #(M, P), and it will be called the local number of generators of M at P. Note that the right-hand side of (1) is equal to the d i m e n sion of a v e c t o r s p a c e , and as such is easily computed. Hence, an upper bound for # n ( M ) in terms of the numbers #(M, P), for the various prime ideals P of R, would be a very desirable result. In order to have a means of guessing what upper bound one might expect, one turns to the equivalence between projective modules and fibre bundles. Let X be an n-dimensional C W - c o m p l e x and denote by C(X) its ring of continuous functions. If ~ is a real vector bundle of dimension k over X, then there exists a v e c t o r b u n d l e r1 such that ~ ® ~ is a trivial bundle. Moreover, one can assume that r/ has dimension smaller than or equal to n; see [4, Chapt. 8, Thins. 1.2, 1.5]. Thus, the C(X)-Inodule of sections F(~ ® r/) is isomorphic to a free module of rank at most k + n. Since F(~) is a homomorphic image of this free module, k + n elements must be sufficient to generate it. Using the above as a guide, one may now return to the general algebraic setting. First, the rank of a projective module at a prime is equal to its minimal number of generators. Secondly, the role of the dimension of the topological space will be played, in the algebraic setting, by the Krull dimension of the base ring (cf. also D i m e n sion). Finally, taking into account the fact that the local number of generators of a module need not be the same at every prime, the topological analogy suggests that

#R(M) < max {p(M, P ) : P E Spec(R)} + Kdim(R). Assuming that R is a commutative N o e t h e r i a n ring, this is just what R.O. Forster proved in [3]. Following a suggestion of J.-P. Serre, Forster's result was later generalized by R.G. Swan to rings whose maximal spectrum is Noetherian, [9] (cf. also S p e c t r u m o f a ring). Denoting by j - Spec(R) the set of prime ideals that are intersections of maximal ideals, the Forster-Swan theorein can be stated as follows. Let R be a commutative ring and assume that j - Spec(R) is a Noetherian space.

F O U R I E R - H A A R SERIES If M is a finitely-generated R-module, then

Fourier algebra of G. In fact, A2(G) = {k*/: k,l e £ ~ ( G ) }

R(M) < _< max {p(M, P) + Zdim(R/P): P E j - Spec(R)}. A number of further improvements are possible. For instance, one can discard all j-primes outside the support of M, and the Krull dimension can be replaced by the j-dimension. For details on these and other improvemeats, see [2] or [5], where a proof of the theorem is given based on the notion of a 'basic element'. These methods also produce a similar bound for the stable rank of a module; see [2, Corollary 6]. Given the importance of the Forster-Swan theorem, it was natural to ask whether it could be generalized to non-commutative rings. T h e obstacle was the usual problem of localizing at a prime in a non-commutative algebra. However, if P is a prime ideal of a Noetherian ring R, then R / P is a prime Noetherian ring. Hence it must have a quotient ring Q(R/P) by Goldie's theorem. Therefore, the right-hand side of (1) makes sense, and there is still a good definition of #(M, P) in this case. This led J.T. Stafford, building on earlier work of R.B. Warfield, to a proof of the Forster-Swan theorem for right and left Noetherian rings [7]. A simpler proof, that also works for right Noetherian rings, can be found in [1]. For applications of the Forster-Swan theorem to non-commutative algebra, see [8]. References [1] COUTINHO, S.C.: 'Generating modules efficiently over noncommutative noetherian rings', Trans. Amer. Math. Soc. 323 (1991), 843-856. [2] EISENBUD, D., AND EVANS, JR., E.G.: 'Generating modules efficiently: Theorems from algebraic K-theory', J. Algebra 27 (1973), 278-305. [3] FORSTER, O.: '0ber die Anzahl der Erzeugenden eines Ideals in einem Noetherschen l~ing', Math. Z. 84 (1964), 80-87. [4] HUSEMOLLER, D.: Fibre bundles, second ed., Springer, 1966. [5] KUNZ, E,: Introduction to commutative algebra and algebraic geometry, Birkb~iuser, 1985. [6] McCONNELL, J.C., AND ROBSON, J.C.: Noncommutative noetherian rings, Ser. in Pure and Applied Math. Wiley, 1987. [7] STAFFORD, J.T.: 'Generating modules efficiently: algebraic K-theory for noncommutative noetherian rings', J. Algebra 69 (1981), 312-346. [8] STAFFORD, J.T.: 'The Goldie rank of a module': NoetherJan rings and their applications (Oberwolfach, 1983), Vol. 24 of Math. Surveys and Monographs, Amer. Math. Soc., 1987, pp. 1-20. [9] SWAN, R.G.: 'The number of generators of a module', Math. Z. 102 (1967), 318-322.

S. C. Coutinho MSC1991: 13C15, 13B30, 16Lxx, 16P60 FOURIER ALGEBRA, Eymard algebra Let G be a locally c o m p a c t group. The algebra As (G) (see also F i g h - T a l a m a n c a a l g e b r a for notations) is called the

and for u E A2 (G),

[lU][A2(a) = inf {Ns(k)N2(1): k, 1 e £~(G), u = k • [}. Let G be Abelian and let ¢ be the canonical mapping from G onto G. Then f ~ (f'oe)', where %(X) = ~o(X-1), is an isometric isomorphism of the B a n a c h a l g e b r a L 1(G) onto A2 (G). Therefore As (G) can be considered as a substitute of L~ (G) if G is non-Abelian. One always has PM2(G) : CV2(G): As(G) is precisely the pre-dual of the y o n N e u m a n n alg e b r a CVs(G). For the definition of CV2(G) and PM2 (G), see F i g ~ - T a l a m a n c a a l g e b r a . Consequently, in analogy with the Abelian case, As(G) is weakly a(As (G), CV2 (G) ) sequentially complete. The existence, for G not amenable, of approximate units in A2(G) is still in doubt (as of 2000). However, such exist for all closed subgroups of SO(n, 1) and SU(n, 1). Such approximate units in A2 can be used in the study of lattices of non-compact simple Lie groups of real rank one ([1]). For a study of certain ideals of A2(G), see [2], [4]. The unimodular case was first investigated by W.F. Stinespring, using a very interesting non-commutative integration theory [6]. The case of a general locally compact group G was initiated by P. Eymard [3] on the basis of an extensive use of the theories of C*-algebras and von Neumann algebras. If G is non-Abelian, As (G) is also called the Eyrnard algebra of G and is denoted by A(G). See also F o u r i e r - S t i e l t j e s algebra. References [1] COWLINa, M., AND HAAGERUP, U.: 'Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one.', Invent. Math. 96 (1989), 507-549. [2t DELAPORTE, J., AND DERIOHETTI, A.: 'Best bounds for the approximate units of certain ideals of L 1 (G) and of Ap(G)', Proc. Amer. Math. Soc. 124 (1996), 1159-1169. [3] EYMARD, P.: 'L'alg6bre de Fourier d'un groupe localement compact', Bull. Soc. Math. France 92 (1964), 181-236. [4] KANIUTH, E., AND LAU, A.T.: 'A separation property of positive definite functions on locally compact groups and applications to Fourier algebras', J. Funet. Anal. 175 (2000), 89-110. [5] PIER, J.-P.: Amenable locally compact groups, Wiley, 1984. [6] STINESPRING, W.P.: 'Integration theorems for gages and duality for unimodular groups', Trans. Amer. Math. Soc. 90 (1959), 15-56.

Antoine Derighetti MSC 1991: 22D10, 43A46, 46J10

43A07,

43A30, 43A35, 43A45,

FOURIER-HAAR SERIES, Haar-Fourier series Consider an interval (a,b), a m e a s u r e # on it and a -

159

F O U R I E R HAAR SERIES corresponding complete o r t h o n o r m a l s y s t e m of functions ¢0, ¢ 1 , . . . (so that f : Ck(X)¢l(x)d#(x) = 5kl). The Fourier series of a function f with respect to such an orthonormal system of functions is: C~

(1) k=O

with coefficients c~ =

f(~)¢~(x)

a~(x).

See Fourier series; Fourier series in orthogonal polynomials. Depending on the orthonormal system used, one thus obtains

• F o u r i e r - B e s s e l series (see also Bessel functions); • Fourier-Chebyshev series (see also C h e b y s h e v polynomials); • Fourier-Franklin series (see also F r a n k l i n system); • Fourier Haar series (see also H a a r s y s t e m ) ; • Fourier-Jacobi series (see also J a c o b i p o l y n o mials); • Fourier-Laguerre series (see also L a g u e r r e p o l y nomials); • Fourier-Legendre series (see also Legendre polynomials); • Fourier-Walsh series (see also W a l s h s y s t e m ) . There are corresponding notions of coefficients, expansions and transforms (i.e., Fourier-Bessel coefficients,

Fourier-Chebyshev coefficients, Fourier-Franklin eoefficients, Fourier-Haar coefficients, Fourier-Jacobi coefficients, Fourier-Laguerre coefficients, Fourier-Legendre coefficients, Fourier- Walsh coefficients, etc.). The properties of these coefficients and the convergence properties of the series (1) often differ sharply from those in the trigonometric case; see, e.g., Fourier-

Bessel integral; Laguerre transform. There is a fair amount of variation in the terminology used: names can be switched and sometimes the word 'Fourier' is left out altogether. M. Hazewinkel MSC 1991: 42C15, 42C10

directions with periodic boundary conditions. In nonperiodic directions, Chebyshev methods, finite element or difference methods of high order should be used. Suppose the equation Lu = f is to be solved where L is a differential operator, f is a given periodic function and u is an unknown periodic function (period 27r). In the Fourier pseudo-spectral method, the solution u is approximated by a t r l g o n o m e t r i e p o l y n o m i a l PNU that interpolates u at equally spaced points xj = 7cj/N, j =0,...,2N1. The Lagrange interpolation polynomial (cf. also Lagrange interpolation formula) has the form PNn = y~2N--1 u(xj)Cj(x), where Cj is the Cardinal function, j=0 having the form 1 s i n N ( x - xj) cot (x - xj_______~) 2N 2 An equivalent way of defining the interpolating polynomial is by the discrete Fourier transform: PNU -= ~ N _ N ake ~kx, where the Fourier coefficients are 2N--1

1

j=0 CN

=

C--N

=

1,

cj = 2 otherwise. In the Lagrange polynomial or 'grid-point representation', the problem can be written as Li,juj = fi, where Lid = LCj(x)lx=x~ The form of Lid can be found through differentiation of the Cardinal function: C j ( x d = 5i,j,

dCj

f 0

a x ( X d = l l (~ d2Cj

dx 2 ( x i ) =

for i = j, - l ? + J c o t -~,-~J w-½ 2N2+1

6

( - 1 ) i-}-j+l

sin2 ~ j

foriCj,

fori = j, foriCj,

to give a few. The derivative matrices are full and O(N 2) operations are required for evaluation. The problem can also be formulated using the spectral coefficient representation. Derivatives are found through differentiation of the discrete Fourier series, for example,

(d);PNu(x):E(ik)2aaeikx, k

FOURIER PSEUDO-SPECTRAL M E T H O D - A type of trigonometric pseudo-spectral method (cf. Trigonometric p s e u d o - s p e c t r a l m e t h o d s ) used to solve differential and integral equations. See also Chebyshev pseudo-spectral method. The Fourier pseudo-spectral method is used for problems in which there is a natural periodicity. In multidimensional problems it should be used only in those 160

and have an O(N) operation count. If the operator L is non-linear or has non-constant coefficients multiplying the derivatives, evaluation of L is much more complicated. The fast Fourier transform provides a way of switching between grid-point and spectral representations in O ( N l o g N) operations, instead of the O ( N 2) using the definitions above. Derivatives can be accomplished i n

FOURIER-STIELTJES ALGEBRA spectral space and multiplication of non-constant coefficients or evaluation of non-linear terms can be accomplished in grid point space - - each with O(N) operations. Particularly for time-dependent problems in which boundary value problems must be solved at each time step, it is very important to minimize operation counts by taking full advantage of each representation and the fast transformation between representations. As an example, consider the operator

let llfl] = sup{ll (f)l[: e z } , where E is the set of all equivalence classes of unitary continuous representations of G (cf. also U n i t a r y r e p r e s e n t a t i o n ) . The completion of Lie(G) with respect to this norm is a B a n a e h a l g e b r a , denoted by C* (G) and called the full C*-algebra of G. If G is Abelian and G its dual group, then C*(G) is isometrically isomorphic to the Banach algebra Co (G; C) of all complex-valued continuous functions on G vanishing at infinity. Let B(G) be the complex linear span of the set of all continuous positive-definite functions on G.

•

Lu = sm(X)~x2 -

\dx/

for the time-dependent problem du/dt = Lu. Suppose at a particular time the spectral coefficients aj, j = 0 , . . . , N - 1, are known. To compute Lu, first the spectral coefficients of the first- and second-order derivatives are found: ijaj and -j2aj. Then both of these arrays are transformed into their grid point representation using a fast Fourier transformation. Then Lu can be evaluated in O(N) operations. The complete evaluation of the operator is done in O(N log N) operations rather than O(N 2) if done completely in one space. This property becomes even more critical for multi-dimensional problems; the operation counts are O(N d log N) versus O(N 2d) for a d-dimensional problem. References

[1] BOYD, J.P.: Chebyshev and Fourier spectral methods, second ed., Dover, 2000, pdf version: http://www-personal.engin.umich.edu/ Njpboyd/book_spectral2000.html. [2] CANUTO,C., HUSSAINI,M.Y., QUARTEFtONI,A., AND ZANG, T.A.: Spectral methods in fluid dynamics, Springer, 1987. [3] FORNBERG,B.: A practical guide to pseudospectral methods, Vol. 1 of Cambridge Monographs Appl. Comput. Math., Cambridge Univ. Press, 1996. [4] GOTTLIEB,

D., HUSSAINI, M.Y., AND ORSZAG, S.A.: 'Theory and application of spectral methods', in R.G. VOIGT, D. GOTTLIEB, AND M.Y. HUSSAINI(eds.): Spectral Methods

for Partial Differential Equations, SIAM, 1984. [5] GOTTLIEB, D., AND ORSZAG, S.A.: N u m e r i c a l a n a l y s i s o f spectral m e t h o d s : T h e o r y a n d applications, SIAM, 1977.

Richard B. Pelz

1) The C-vector space B(G) is isomorphic to the dual space of C* (G). With the dual norm and the pointwise product on G, B(G) is a commutative Banach algebra [4]. This Banach algebra is called the Fourier-Stieltjes algebra of G. If G is Abelian, then B(G) is isometrically isomorphic to the Banach algebra of all bounded Radon measures

on

G.

2) On the boundary of the unit ball of B(G) (i.e. e B(G): [l [Imc) = 1 } ) t h e weak topology ~(12~(G), £~ (G)) coincides with the compact-open topology on G ([3]; see also [12], [6]). 3) The following properties are satisfied ([4]): a) The F o u r i e r a l g e b r a A(G) is a closed ideal of

on {u

B(G); b) B(G) N C00(G; C) C A(G); c) A(G) coincides with the closure in B(G) of

B(C) n C00(C; C); d) B(G) = B(C ) n C(C; C), with equality of the corresponding norms. Here, C00(G; C) is the algebra of functions of compact support on G. In [14], M.E. Walter showed that B(G) (and also A(G)) completely characterizes G. More precisely, assume that G1 and G2 are locally compact groups; then the following assertions are equivalent:

MSC 1991: 65Txx F O U R I E R R E P R E S E N T A T I O N , Fourier series rep-

resentation -

The representation of a f u n c t i o n (or other suitable object) f by means of a F o u r i e r series, especially a t r i g o n o m e t r i c series: cx~

f(x)=

Z

ck e ikx

He also gave a description of the dual of B(G).

k = -- oc)

MSC 1991: 42Axx FOURIER-STIELTJES

arbitrary locally c o m p a c t

A L G E B R A - Let G be an

g r o u p . For f

• the locally compact groups G1 and G2 are topologically isomorphic; • the Banach algebras B(G1) and B(G2) are isometrically isomorphic; • the Banach algebras A(Ga) and A(G2) are isometrically isomorphic.

E Lie(G),

For a connected semi-simple Lie group G, M. Cowling [1] has given a description of the spectrum of B(G); surprisingly, if G is Abelian, then the spectrum of B(G) seems to be much more complicated than in the nonAbelian case! 161

FOURIER-STIELTJES

ALGEBRA is, probably, the simplest m e t h o d of showing t h a t the trefoil knot is non-trivial (see Fig. 1).

If G is amenable, then ([3]) B(G)

= {u E C a : u v C A(G) for every v E A ( G ) } .

(1) V. Losert [11] proved the converse assertion: if (1) holds, then G must be amenable! In a difficult p a p e r [7], C.S. Herz tried to extend the preceding results, replacing u n i t a r y representations by representations in Banach spaces. He partially succeeded in the a m e n a b l e case. See also [2], [5]. M. Lefranc generalized Paul Cohen's idempotent theorem to B ( G ) for a r b i t r a r y locally c o m p a c t groups G ([9], [8]; see also [10] for detailed proofs). See also F i g h - T a l a m a n c a a l g e b r a . References [1] COWLINC,M.: 'The Fourier StieRjes algebra of a semisimple Lie group', Colloq. Math. 41 (1979), 89-94. [2] COWLING,M., AND FENDLER,CI.: ~On representations in Banach spaces', Math. Ann. 266 (1984), 307-315. [3] DERIGHETTI, A.: 'Some results on the Fourier-Stieltjes algebra of a locally compact group', Comment. Math. Helv. 45 (1970), 219-228. [4] EYMARD, P.: 'L'alg~bre de Fourier d'un groupe localement compact', Bull. Soc. Math. France 92 (1964), 181-236. [5] FENDLER, G.: 'An LP-version of a theorem of D.A. Raikov', Ann. Inst. Fourier (Grenoble) 35, no. 1 (1985), 125-135. [6] GRANIRER, E.E., AND LEINERT, M.: 'On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G)', Rocky Mount. J. Math. 11 (1981), 459 472. [7] HERZ, C.: 'Une g6n6ralisation de la notion de transform6e de Fourier-Stieltjes', Ann. Inst. Fourier (Grenoble) 24, no. 3 (1974), 145-157. [8] HOST, B.: 'Le th6or~me des idempotents dans B(G)', Bull. Soc. Math. France 114 (1986), 215-223. [9] LEFRANC, M.: 'Sur certaines algebras sur un groupe', C.R. Acad. Sci. Paris Sdr. A 274 (1972), 1882-1883. [10] LEFRANC, M.: ~Sur certaines alghbres sur un groupe', Th~se de Doctorat d@tat,

Univ. Sci. ct Techn. du Languedoc

(1972). [11] LOSERT, V.: 'Properties of the Fourier algebra that are equivalent to amenability', Pwc. Amer. Math. Soc, 92 (1984), 347-354. [12] McKENNON, K.: 'Multipliers, positive functionals, positivedefinite functions, and Fourier-Stieltjes transforms', Memoirs Amer. Math. Soc. 111 (1971). [13] PIER, J.-P.: Amenable locally compact groups, Wiley, 1984. [14] WALTER,M.E.: 'W*-algebras and nonabelian harmonic analysis', J. Funct. Anal. 11 (1972), 17-38. A n t o i n e Derighetti

M S C 1991: 22D10, 22D25, 43A30, 43A35, 46J10

43A15,

43A07,

43A25,

F O X n - C O L O U R I N G - A colouring of a non-oriented link diagram (cf. also K n o t a n d l i n k d i a g r a m s ) , leading to an Abelian group invariant of links in R 3 (cf. also L i n k ) . It was introduced by R.H. Fox a r o u n d 1956 to visualize dihedral representations of the knot group [1] (cf. also K n o t a n d l i n k g r o u p s ) . Using 3-colourings 162

One says t h a t a link (or tangle) diagram, D, is nc o l o u r e d if every arc is coloured by one of the numbars 0 , . . . , n - 1 in such a way t h a t at each crossing the sum of the colours of the undercrossings is equal to twice the colour of the overcrossing m o d u l o n. T h e set of n-colourings forms an A b e l i a n g r o u p , denoted by Col~(D). This g r o u p can be interpreted using the first h o m o l o g y g r o u p ( m o d u l o n) of the double branched cover of S 3 with the link as the b r a n c h e d point set. T h e group of 3-colourings is d e t e r m i n e d by the Jones polynomial (at t = e27rff6), and the g r o u p of 5-colourings by the K a u f f m a n p o l y n o m i a l (at a = 1, z = 2cos(27r/5)), [2]. n-moves preserve the g r o u p of n-colourings and 3moves lead to the M o n t e s i n o s - N a k a n i s h i conjecture.

f

\ Fig. 1. T h e linear space of p-colourings of the b o u n d a r y points of an n-tangle has a symplectic form (cf. also S y m p l e c t i c s t r u c t u r e ) , so t h a t tangles correspond to L a g r a n g i a n s u b s p a c e s (i.e. m a x i m a l totally degenerate subspaces) of the symplectic form. T h e Alexander module is a generalization of the group of n-colourings. References [1] CROWELL, R.H., AND FOX, R.H.: An introduction to knot theory, Ginn, 1963. [2] PRZYTYCKI, J.: '3-coloring and other elementary invariants of knots': Knot Theory, Vol. 42 of Banach Center Publications, 1998, pp. 275 295. Jozef Przytyeki

MSC 1991:57M25

FREDHOLM

SOLVABILITY

n ) - m a t r i x and b C R ~ a vector.

- Let A be a real (n ×

F R E U D E N T H A L - K A N T O R T R I P L E SYSTEM The Fredholm alternative in R n states that the equation Ax = b has a solution if and only if bTv = 0 for every vector v C R n satisfying ATv = O. This alternative has many applications, e.g. in bifurcation theory. It can be generalized to abstract spaces. So, let E and F be Banach spaces (cf. B a n a c h s p a c e ) and let T : E -+ F be a continuous l i n e a r o p e r a t o r . Let E*, respectively F*, denote the topological dual of E, respectively F, and let T* denote the adjoint of T (cf. also D u a l i t y ; A d j o i n t o p e r a t o r ) . Define (KerT*) ± = {y E F : (y,y*} = O f o r ally* e K e r T * } . An equation T x = y is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever y E (KerT*) ± (cf. also N o r m a l s o l v a b i l i t y ) . A classical result states that T x = y is normally solvable if and only if T(E) is closed in F. In non-linear analysis, this latter result is used as definition of normal solvability for non-linear operators. The phrase 'Fredholm solvability' refers to results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, Fredholm-type properties of the operator involved. References [1] HAUSDORFF, F.: 'Zur Theorie der linearen metrischen Rgume', J. Reine Angew. Math. 167 (1932), 265. [2] KOZLOV, V.A., MAZ'YA, V.G., AND ROSSMANN, J.: Elliptic boundary value problems in domains with point singularities, Amer. Math. Soc., 1997. [3] ORLOVSKIJ, D.G.: 'The Fredholm solvability of inverse problems for abstract differential equations', in A.N. TmHONOV ET AL. (eds.): Ill-Posed Problems in the Natural Sciences, VSP, 1992. [4] PRILEPKO, A.T., ORLOVSKY, D.G., AND VASIN, I.A.: Methods for solving inverse problems in mathematical physics, M. Dekker, 2000.

G. Isac Themistocles M. Rassias MSC 1991:47A53 FREUDENTHAL-KANTOR

TRIPLE SYSTEM - A

triple system considered for constructing all simple Lie algebras (cf. Lie a l g e b r a ) , and introduced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of a generalized J o r d a n t r i p l e s y s t e m of second order developed by I.L. Kantor. Recall that a triple system is a v e c t o r s p a c e V over a field ~ together with a ¢-trilinear mapping

For e = +1, a vector space U(e) over a field with the trilinear product (xyz) is called a FreudenthalKantor triple system if

(ab(cde)} = ((abe)) +¢(c(bad)e) + (cd(abe)} ,

(1)

K ( L ( a , b)c, d) + K(c, L(a, b)d) + K(a, K(c, d)b) = O,

(2) where L(a, b)e = (abe> and K(a, b)e = -(bca>. In particular, a Freudenthal-Kantor triple system U(c) is said to be balanced if there exists a bilinear form (., .) such that K(a, b) = {a, b) Id, for all a, b ff U(¢). This balancing property is closely related to metasymplectic geometry. Note that if c = - 1 and K(a, b) = 0 (identically), then the Freudenthal-Kantor triple system reduces to a

Jordan triple system. As the notion of a Freudenthal-Kantor triple system includes the notions of a generalized Jordan triple system of second order, a structurable algebra, and an A l l i s o n - H e i n t r i p l e s y s t e m , it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices. Let V be a vector space with a bilinear form {x, y) = - c ( y , x/. Then V is a Freudenthal-Kantor triple system with respect to the triple product (xyz} := (y, z)x. In particular, it is important that the linear span k := {K(a, b)}span of the set K(a, b) makes a Jordan triple system of (End U(c)) + with respect to the triple product { A B C } := 1 / 2 ( A B C + CBA). Let U(c) be a Freudenthal-Kantor triple system. The vector space U(c) ® U(c) becomes a Lie t r i p l e s y s t e m with respect to the triple product defined by

[(0(:)0)]: := (L(a, d) - L(c, b)

\

-cK(b,d)

K ( a , e)

c(L(d,a)-L(b,c)))(~)"

Using this, one can obtain the Lie triple system U(¢) ® U(¢) associated with U(¢); it is denoted be T(¢). Using the concept of the standard embedding Lie algebra L(¢) = Inn Der T(e) • T(c) associated with a Lie triple system T(c), one can obtain the construction of L@) associated with a Freudenthal-Kantor triple system U(c). In fact, put • L2 equal to the linear span of the endomorphisms

• L1 := U ( ¢ ) • (0); • L _ I := (0) e

• L0 equal to the linear span of the endomorphisms

VxVxV~V. 163

F R E U D E N T H A L K A N T O R T R I P L E SYSTEM • L - 2 equal to the linear span of the endomorphisms

Then one obtains the decomposition

L(e) = L-2 ® L-1 ® Lo ® L, ® L2, and, more precisely, [(Id

_Oid),Li] = i L i

(-2
These results imply the dimensional formula d i m . n(e) = 2 (dime U(e) + dim.{K(x, Y)}span) + + dime {n(x, Y)}span = = dime T(e) + d i m . Inn Der T(c). This algebra L(e) is called the Lie algebra associated

with U(e). The concepts of a triple system and a supertriple system are important in the theory of quarks and YangBaxter equations. Note that a 'triple system' in the sense discussed above is totally different from 'triple system' in combinatorics (see, e.g., S t e i n e r s y s t e m ) . References [1] FREUDENTHAL, H.: 'Beziehungen der E7 und Es zur Oktavenebene I-IF, Indag. Math. 16 (1954), 218-230; 363-386. [2] KAMIYA,N.: 'The construction of all simple Lie algebras over C from balanced Freudenthal-Kantor triple systems': Contributions to General Algebra, Vol. 7, H51der-Pichler-Tempsky, Wien, 1991, pp. 205-213. [3] I(AMIYA,N.: 'On Freudenthal-Kantor triple systems and generalized structurable algebras': Non-Associative Algebra and Its Applications, Kluwer Acad. Publ., 1994, pp. 198-203. [4] KAMIYA, N., AND OKUBO, S.: 'On a-Lie supertriple systems associated with (e,a)-Freudenthal-Kantor supertriple systems', Proc. Edinburgh Math. Soc. 43 (2000), 243-260. [5] KANTOR, I.L.: 'Models of exceptional Lie algebras', Soviet Math. Dokl. 14 (1973), 254-258. [6J OKUBO, S.: Introduction to octonion and other nonassociative algebras in physics, Cambridge Univ. Press, 1995. [7] YAMAGUTI,K.: 'On the metasymplectic geometry and triple systems', Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. 306 (1977), 55-92. (In Japanese.) Noriaki K a m i y a

MSC1991:17A40 F R I T Z J O H N C O N D I T I O N - A necessary condition for local optimality in problems with inequality constraints. It is closely related to the classical problem of minimizing a function f on a constraint set F = {x • R n : hi(x) = 0, i • P}. The functions are defined on R n, the objective f is assumed to be differentiable (cf. also D i f f e r e n t i a b l e f u n c t i o n ) and the constraints h i, i • P, are continuously differentiable at a feasible point x* • F tested for optimality; P is a finite index set. If f(x*) <_ f(x) for every

164

x C F n N(x*), where N(x*) is a neighbourhood of x*, then x* is said to be a constrained local minimum. At such a point the gradients of the objective function and the constraints are linearly dependent, i.e., there exist multipliers hi E R, i E {0} U P , not all zero, such that AoVf(x*) -t-EiEP AiVhi(x*) = 0. If the gradients of the constraints are linearly independent, then one can specify ),0 = 1. This result, usually proved by the implicit function theorem (cf. I m p l i c i t f u n c t i o n ) , is used to formulate the L a g r a n g e m e t h o d in calculus. Now assume that the feasible set is determined by inequality constraints, i.e., consider

(NP) min

f(x),

where

xcF=

{xERn:gi(x)

<0, i C P } .

The Fritz John condition describes local optimality of a feasible point x* using the gradients of the objective function and the 'active' constraints P(x*) = {i C P: gi(x*) = 0}, [10]. The basic Fritz John condition is as follows. Consider the problem (NP) where all functions are differentiable at some feasible x*. If x* is a constrained local minimum, then there exist multipliers ui > 0, i E {0} U P(x*), not all zero, such that

uoVf(x*) +

~ uiVgi(x *) = O. i~p(x*)

By Gordan's theorem of the alternative (e.g., [14]), or the DubovitskiY-Milyutin theorem (e.g., [7]), the Fritz John condition is equivalent to the inconsistency of the system

V f(x*)d < O,

Vgi(x*)d < O, i • P(x*),

which is a 'primal' optimality condition. For problems with both equality and inequality constraints, the Fritz John condition requires that the equality constraints be continuously differentiable while the active inequality constraints need only be differentiable. The multipliers that correspond to the equality constraints are unrestricted in sign. This result, referred to as the Mangasarian-Fromovitz condition, does not follow from the Fritz John condition; e.g., [14], [15]. Conditions that guarantee that the leading multiplier in the Fritz John condition is positive, in which case it can be set Uo = 1, are called regularization conditions or constraint qualifications. One of these is that the gradients of active constraints are linearly independent. This condition is typically violated in multi-level and multi-objective optimization problems; e.g., [4], [23]. C o n v e x p r o g r a m s . The Fritz John condition is not sufficient for optimality even for linear programs; e.g., [3, p. 150], [4], [14]. However, a reformulation of it is

FRITZ JOHN CONDITION both necessary and sufficient for optimality of a feasible point x* for a convex program, i.e., for the problem (NP) when all functions are convex (cf. C o n v e x f u n c t i o n ( o f a r e a l v a r i a b l e ) ) . First, using properties of feasible directions, optimality of x* is characterized by the inconsistency of the system

V f(x*)d < O, i E P(x*) \ P=,

Vgi(x*)d < O, d E D(x*),

where P = is the set of the constraints that are equal to zero on the entire feasible set and D(x*) is the intersection of the cones of directions of constancy at x* of all such constraints. This is equivalent to the Fritz John formulation uoV f(x*) + EiEP(x.)\e= uiVgi(x *) E {D(x*)} +, where the multipliers are non-negative and not all equal to zero, and {D(x*)} + is the p o l a r set of D(x*), [4]. Moreover, feasibility of x* guarantees here u0 = 1. If the gradients of active constraints are linearly independent, or, more generally, if P = is an empty set (this is known as Slater's condition, cf. M a t h e m a t i c a l p r o g r a m m i n g ) , then the Fritz John condition becomes Vf(x*)+

uiVgi(x*) = O'

E

ui >_ O,

i E P(x*).

iEp(x*) Consistency of this system is known as the Karushcondition (cf. also K a r u s h - K u h n Tucker conditions). Alternatively, in c o n v e x p r o g r a m m i n g the Fritz John condition is equivalent to the existence of a s a d d l e p o i n t of the Lagrangian L(x,u) = f(x) + X EiEP\p= uig i (). First, define F = = {x: gi(x) = O, i E P = } and let c be the cardinality of P \ P = . Consider the convex program (NP). Then a point x* E F = is optimal if and only if there exists a u* >_ 0 in R c such that L(x*,u) <_ L(x*,u*) <_ L(x,u*) for every u > 0 and every x E F = (saddle-point characterization of op-

Kuhn-Tucker

timality). This formulation is useful when some functions are not differentiable. Using non-smooth analysis, one can replace derivatives by other objects such as subgradients

(see, e.g., [3]). O p t i m a l i t y a n d s t a b i l i t y . Descriptions of optimality often require stability in the convex model (NP,0)

{

min(~)

f(x,O),

where

x

F(0) = {x: ¢ ( x , 0) __ 0, i e P);

all functions are continuous, 0 E R p is considered as a 'parameter', and f(.,O),gi(.,O): R n --+ R, i E P are convex for every 0. (The functions need not be convex in 0.) Assume that the set of optimal solutions x ° = x ° (0) of the corresponding convex program is nonempty and bounded at some 0 = 0". Perturbations of

0 from 0* that locally preserve lower semi-continuity of the feasible set mapping F : 0 --+ F(O) form a 'region of stability' at 0", denoted by S(O*). Denote the constraints that are equal to zero on F(O) by P=(0). Let F,(O) = {x: gi(x,O) <_ O, i E P=(0*)}, let c be the cardinality of the set P \ P=(0*), and consider the Lagrangian L. (x, u; 0) = f ( x , O) + ~icp\p=(o.) uigi(x, 0). Then the following condition characterizes local optimality of 0", relative to stable perturbations, for the optimal value function f°(O) = f(x°(O), O) (cf., e.g., [23]). Consider the convex model (NP,0) around some 0". Then 0* locally minimizes f°(O) relative to perturbations in S(O*) if and only if there exists a non-negative function U: S(O*) n N(O*) ~ R c such that, whenever

0 e S(O*)

n

iv(o*),

L,(x°(0*),

0 *) <

C,(x°(O*),U(O*);O *) <_

_< L. (x, U(0); 0), for every non-negative u E R c and every x E FZ(O* ) (a

characterization of local optimality on a region of stability). The above claim does not generally hold if the stability requirement is omitted. As an example, consider the convex model m i n ( x ) { x l 0 : x 2 <_ O, (x~ - x2)O 2 <_ 0, m a x { x 2 + x22 - 1,0} _< 0}. Here f°(O) = 0 for any 0, hence 0* = 0 is a global minimum. The saddle-point condition requires U1 = UI (O) >_ 0 such t h a t XlO+Ulx2 >_ 0 for all x~ < x2. But such a multiplier function does not exist outside the region of stability, e.g., for 0 < 0, as a sequence on x~ = x2 with xl -+ + 0 shows. The necessary conditions for local optimality of parameters, subject to stable perturbations, are simplified under 'input constraint qualifications' [18]. Global optimality of a p a r a m e t e r 0* for the convex model can be characterized by a saddle-point condition on the region of cooperation of 0". This is the set of all 0 for which P = ( 0 ) C P=(0*). No stability is required in this case [22]. Programs t h a t assume the form of a convex model, after 'freezing' some variables, are called partly convex. Every p r o g r a m with twice continuously differentiable functions can be transformed into an equivalent partly convex p r o g r a m (cf., e.g., [8], [12]). A b s t r a c t p r o g r a m s . Various extensions of the basic Fritz John condition have been studied in abstract spaces with finite or infinite number of constraints; e.g., [2], [7], [11], [13], [24]. The weakest derivative for which this condition holds appears to be the Gil de Lamadrid and Sova compact derivative [16] in non-sequential locally convex Hausdorff spaces. A saddle-point condition that characterizes locally optimal parameters relative to their stable perturbations for the abstract convex model min(x) f(x,O) on the set x E F(O) = {x: gt(x,O) <_ 0, t E T}, where x and the parameter 0 belong to 165

F R I T Z JOHN

CONDITION

some s u i t a b l e a b s t r a c t spaces a n d T is a c o m p a c t subset of R , can b e given in t e r m s of t h e existence of a finite B o r e l m e a s u r e ~ for t h e L a g r a n g i a n L ( x , ~; O) = f (x, O) + f T gt( x, O) ~(dt) (cf. e.g. [1]). T h e a b s t r a c t ver-

sions of t h e F r i t z J o h n a n d K a r u s h - K u h n - T u c k e r conditions a r e a p p l i c a b l e to a wide r a n g e of p r o b l e m s , from o p t i m a l c o n t r o l p r o b l e m s in engineering [5], [6], [17] a n d m a n a g e m e n t [19], [21] to t h e prior selection p r o b l e m in B a y e s i a n s t a t i s t i c a l inference [1]. T h e s e c o n d i t i o n s are used for i d e n t i f i c a t i o n of o p t i m a l solutions, p a r a m e t e r s , controls, a n d s t a t e s , a n d for f o r m u l a t i o n s of d u a l i t y theories a n d n u m e r i c a l m e t h o d s . T h e m u l t i p l i e r s describe s e n s i t i v i t y of t h e o p t i m a l value function s u b j e c t to t h e r i g h t - h a n d side p e r t u r b a t i o n s a n d can b e i n t e r p r e t e d as values of t h e c o n s t r a i n t s ( ' s h a d o w prices' in linear prog r a m m i n g ) . Some o p t i m i z a t i o n p r o b l e m s can a l t e r n a tively b e s t u d i e d by calculus of v a r i a t i o n s which, fundamentally, is t h e s a m e m e t h o d ; e.g., [6], [9], [19], [20], [21, p. 17]. References [1] ASGHARIAN, M., AND ZLOBEC, S.: 'Abstract parametric programming', Preprint McGiU Univ. M a r c h (2000). [2] BARBU, V., AND PRECUPANU, TH.: Convexity and optimization in Banach spaces, Sijthoff & Noordhoff, 1978. [3] BAZARAA,M.S., SHERALI,H.D., AND SHETTY, C.M.: Nonlinear programming: Theory and algorithms, second ed., Wiley, 1993. [4] BEN-ISRAEL, A., BEN-TAL, A., AND ZLOBEC, S.: Optimality in nonlinear programming: A feasible directions approach, Wiley/Interscience, 1981. [5] BRYSON, JR., E., AND HO, Yu-CHI: Applied optimal control, Blaisdell, 1969. [6] CANON, M.~ CULLUM, C., AND POLAK, E.: Theory of optimal control and mathematical programming, McGraw-Hill, 1970. [7] GIRSANOV, I.V.: Lectures on mathematical theory of extremum problems, Vol. 67 of Lecture Notes in economics and math. systems, Springer, 1972. [8] GUDDAT, J., AND JONGEN, H.TH.: 'On global optimization based on parametric optimization', in J. GUDDAT ET AL. (eds.): Advances in Mathematical Optimization, Akad. Berlin, 1988, pp. 63 79. [9] HALKIN, H.: 'Maximum principle of the Pontryagin type for systems described by nonlinear difference equations', S I A M J. Control 4 (1966), 90 111. [10] JOHN, F.: 'Extremum problems with inequalities as subsidiary conditions', in K.O. FRIEDRICHS ET AL. (eds.): Studies and Essays, Courant Anniversary Volume, Wiley/Interscience, 1948, Reprinted in: J. Moser (ed.): Fritz John Collected Papers 2, Birkhguser, 1985, pp. 543-560. [11] LIGNOLA, M.B., AND MORGAN, J.: 'Existence of solutions to generalized bilevel programming problem', in A. MIGDALAS ET AL. (eds.): Multilevel Optimization: Algorithms and Applications, Kluwer Acad. Publ., 1998, pp. 315 332. [12] LIU, W.B., AND FLOUDAS, C.A.: 'A remark on the GOP algorithm for global optimization', J. Global Optim. 3 (1993), 519-521. [13] LUENBERGER, D.G.: Optimization by vector space methods, Wiley, 1969.

166

[14] MANGASARIAN, O.L.: Nonlinear programming, McGraw-Hill, 1969. [15] MANGASARIAN,O.L., AND FROMOVITZ, S.: 'The Fritz John optimality conditions in the presence of equality and inequality constraints', J. Math. Anal. Appl. 17 (1967), 37-47. [16] MASSAM, H., AND ZLOBEC, S.: 'Various definitions of the derivative in mathematical programming', Math. Programming 7 (1974), 144 161, Addendum: ibid 14 (1978), 108-111. [17] PONTRYAGIN,L.S., BOLTYANSKI,V.G., GAMKRELIDZE,R.V., AND MISHCHENKO, E.F.: The mathematical theory of optimal processes, Wiley, 1962. [18] ROOYEN, M. VAN, SEARS, M., AND ZLOBEC, S.: 'Constraint qualifications in input optimization', J. Austral. Math. Soc. Ser. B 30 (1989), 326-342. [19] SETHI, S.P.: A survey of management science applications of the deterministic maximum principle, Vol. 9 of TIMS Studies in the Management Sci., North-Holland, 1978, pp. 33-67. [20] SMITH, D.R.: Variational methods in optimization, PrenticeHall, 1974. [21] TAPIERO, C.S.: Time, dynamics and the process of management modeling, Voh 9 of TIMS Studies in the Management Sei., North-Holland, 1978, pp. 7-31. [22] ZLOBEC, S.: 'Partly convex programming and Zermelo's navigation problems', Y. Global Optim. 7 (1995), 229-259. [23] ZLOBEC, S.: 'Stable parametric programming', Optimization 45 (1999), 387-416, (Augmented version forthcoming as research monograph, Kluwer Acad. Publ., Applied Optim. Series.). [24] ZLOBEC, S., AND CRAVEN, B.D.: 'Stabilization and determination of the set of minimal binding constraints in convex programming', Math. Operationsforschung und Statistik, Ser. Optim. 12 (1981), 203-220. S. Zlobec

M S C 1991: 9 0 C x x FROBENIUS

M A T R I X N O R M , Frobenius n o r m -

Let A be an (n x m ) - m a t r i x , a n d let I1"[I be t h e n o r m in t h e u n d e r l y i n g field ( u s u a l l y R or C w i t h t h e s t a n d a r d n o r m ( a b s o l u t e value)). T h e P r o b e n i u s n o r m of A is defined as

IIAII~ = ~

lagjl 2 •

i,j

N o t e t h a t this n o r m differs from t h e o p e r a t o r n o r m of A (for i n s t a n c e b e c a u s e I I ~ l l s = n ; cf. N o r m ) . If U a n d V are u n i t a r y m a t r i c e s of a p p r o p r i a t e size,

IIUAVI[~ :

IIAIIF.

References

[1] NOBLE, B., AND DANIELS, J.W.: Applied linear algebra, second ed., Prentice-Hall, 1969, p. 328ff. M. Hazewinkel

MSC 1991:15A60 FUNCTION V A N I S H I N G A T I N F I N I T Y - Let X be a t o p o l o g i c a l space. A real- or c o m p l e x - v a l u e d func-

t i o n on X is said to vanish at i n f i n i t y if for each e > 0 t h e r e is a c o m p a c t set K~ such t h a t ] f ( x ) l < e for all x C X \ / ( ~ . For n o n - c o m p a c t X , such a f u n c t i o n can be

FUZZY P R O G R A M M I N G extended to a c o n t i n u o u s f u n c t i o n on the one-point compactification X t0 {,} of X (with value 0 at *). The algebra of functions on X vanishing at infinity is denoted by Co(X). In many cases Co(X) determines X, see e.g. B a n a c h - S t o n e t h e o r e m . If X is compact, Co(X) = C(X). The space Co(X) identifies with { f E C(X 0 {*}): f(*) = 0}. References

[1] BEHRENDS, E.: M-structure and the Banach-Stone theorem, Springer, 1979. [2] JAROSZ, K.: Perturbations of Banach spaces, Springer, 1985.

M. HazewinheI MSC 1991:54C35

approximately smaller than b', one treats the soft constraints as Ax C/3, by controlling the membership value P15 (Ax). Such a fuzzy set/~ is called a fuzzy constraint. The coefficients e and A in the above linear programming problem are supposed to be given as real numbers. However, in real world problems one's knowledge is often insufficient to determine the coefficients as real numbers, even when the possible ranges of coefficients is known, such as 'ci is about 3'. Such possible ranges can be modelled by fuzzy sets. Let ~ and A~be fuzzy sets representing possible ranges of c and A. Then the possible range of Ax is obtained as the fuzzy set Ax defined by the membership function: p,~x(Z) = sup #x(A).

Fuzzy programming deals with m a t h e m a t i c a l p r o g r a m m i n g problems under non-probabilistic uncertainty. The idea of fuzzy programming was first given by R.E. Bellman and L.A. Zadeh [1] and then developed by H. Tanaka [7] and H.3. Zimmermann [8]. Those approaches treat soft constraints and vagueness of aspiration levels of objective function values and are called flexible programming. Later, C.V. Negoita, S. Minoiu and E. Stan [4] dealt with the ambiguity of coefficients in mathematical programming problems by fuzzy set theory (cf. also N o n p r e c i s e d a t a ) . This fuzzy programming is called robust programming. D. Dubois [2] proposed the treatment of fuzzy coefficients in fuzzy programming problems based on possibility theory. Fuzzy programming based on possibility theory is called possibilistic programming. Until now (2000), a lot of fuzzy programming approaches have been proposed, as shown in [6], [5], [3]. The traditional linear programming problem can be written as (of. also L i n e a r p r o g r a m m i n g ) FUZZY

PROGRAMMING

z~gx

-

min

cTx

s.t.

Ax < b.

In this problem, the decision variable vector x is imposed to satisfy the constraints Ax 0 is extremely small. However, in the real world it frequently occurs that the right-hand values are not determined as rigorous restrictions and sometimes the right-hand values show the aspiration levels of the left-hand function values elicited from the decision maker. In such cases, the constraints should be treated 'softly'. Fuzzy programming can treat such soft constraints. Let/3 = {r: r < b}; then the constraints of the above linear programming problem can be represented as Ax E B. In fuzzy programming, replacing B by a fuzzy set/~ representing a 'set of values

The possibility and necessity degrees of Ax E B are defined by: n g × ( / ? ) = sup min(#~×(z), plX(z)), g

N~× (/?) = inf max(1 - #X× (z), P*5 (z)). The possible range of cTx and the possibility and necessity degrees of cTx E are defined similarly, where is a fuzzy set. Soft constraints with ambiguous coefficients are treated as IIXx(B ) >_ h n and NXx(/~) > h N, where h n E [0, 1] and h N E [0, 1] are constants given by the decision maker. In order to treat the objective function N with ambiguous coefficients, a fuzzy set G of satisfactory values for the decision maker is established. Such a fuzzy set is called a fuzzy goal. Then the objective function is treated as max II~%¢(G) or max N~Tx(G). Namely, in the former case, the problem is formulated as max

H~Tx(0 )

s.t.

H x(h ) > hn, Nxx( ) > hN.

It is known that such problems can be reduced to linear programming problems under certain reasonable assumptions that are applicable to real world problems [3]. References [1] BELLMAN, R.E., AND ZADEH, L.A.: 'Decision-making in a

fuzzy environment', Management Sci. 17B (1970), 141-164. [2] DUSOIS, D.: 'Linear programming with fuzzy data', in J.C. BEZDEK (ed.): Analysis of Fuzzy Information, Vol. 3: Applications in Engineering and Sci., CRC, 1987, pp. 241-263. [3] INUIGUCHI, M., AND RAMIK~ J . : ' F u z z y linear p r o g r a m m i n g : A

brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem', Fuzzy Sets and Syst. 111 (2000), 3-28. [4] NEGOITA, C.V., MINOIU, S., AND SWAN,E.: 'On considering imprecision in dynamic linear programming', E C E C S R J. 3 (1976), 83-95.

167

FUZZY PROGRAMMING [5] SAKAWA,Z.: Fuzzy sets and interactive multiobjective optimization, Plenum, 1993. [6] SLOWINSKI,l~., AND TEGHEM, J.: Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty, Kluwer Acad. Publ., 1990. [7] TANAKA, H., OKUDA, T., AND ASAI, K.: 'On fuzzy mathematical programming', Y. Cybernet. 3 (1974), 37-46. [8] ZIMMERMANN, H.-J.: 'Description and optimization of fuzzy systems', Int. J. General Syst. 2 (1976), 209-215.

Masahiro Inuiguchi

MSC 1991:90C70

f ~ ( a ) ( y ) = V {a(x) : f ( x ) = y},

FUZZY TOPOLOGY, lattice-valued topology, pointset lattice-theoretic topology, poslat topology - A branch of mathematics encompassing any sort of topology using lattice-valued subsets. The following description is based on the standardization of this discipline undertaken in [3], especially [2], [8]; much additional information is given in the references below. Let X be a set and (L,_<,®) any complete quasimonoidal lattice (a cqml; i.e., (L, <) is a c o m p l e t e latrice with bottom element _1_ and top element T, and the t e n s o r p r o d u c t ®: L x L --+ L is isotone in both arguments with T ® T = T). Examples of complete quasi-monoidal lattices are:

• complete lattices with ® = A (binary meet); • L = [0, 1] with ® any of the t-norms Tmin(a, b) = a A b, Tprod(a,b) = a. b, or Tm(a,b) = (a + b - 1) V0; or • L = [0,1] x [0,1] with ® any of Train x Tprod, Tprod X Tin, etc. L-subsets of X comprise the L-powerset L x = {a: X --+ L, a a function}, a complete quasi-monoidal lattice via the lifting of the structure of L. A subfamily 7- C L x is an L-topology on X, and (X,T) is an Ltopological space, if v is closed under ® and arbitrary V and contains the constant mapping T. A function T : L X --+ L is an L-fuzzy topology on X, and (X, T) is an L-fuzzy topological space, if T satisfies (reading V as 'for each'):

1) V set J,

V{Uj: j i j6J

E J} C L X,

T(uj)<-T(Vuj \j6g

I • f

2) V two-element set J, V{W: j E J} C L X,

3) 7 - ( 2 ) ---= T . The member T ( u ) of L is interpreted as the 'degree of openness' of u. 168

Important examples of L-topological and L-fuzzy topological spaces can be found in [9, Chap. 11]; [4, Kubiak's paper]; [11]; [3, Chaps. 6, 8, 10]; [3, Chap. 7, Sect. 2.15-2.16]; [2, Sect. 7]; [8, Sect. 7]. For a complete quasi-monoidal lattice L and function f : X --+ Y, one defines the powerset operators f~+: L x --+ L Y (the image operator) and f~-: L z -+ L x (the pre-image operator) by

f g (b) = b o f.

It is well-known that f~ 4 f~- and that these operators generalize the traditional operators f-+ and f~-. Given L-topological spaces (X,T) and (]1, a), a mapping f : X -+ Y is L-continuous from (X,T) to (Y,a) if f l ~ : 7 +-- a; and given L-fuzzy topological spaces (X, 7-) and (Y, $), a mapping f : X --+ Y is L-fuzzy continuous from (X, 7-) to (]I,S) if 7-o f~- > $ on L Y. The c a t e g o r y L - T O P comprises L-topological spaces, L-continuous mappings, and the composition and identities from the category S E T (cf. also Sets, c a t e g o r y of); and the category L - F T O P comprises L-fuzzy topologicM spaces, L-fuzzy continuous mappings, and the composition and identities from the category SET. It is a theorem that for all complete quasi-monoidal lattices L, the categories L - T O P and L - F T O P are topological categories over S E T , in the sense of [1] and [8, Sect. 1], and hence topological constructs. The above briefly describes 'fixed-basis topology' - topology where the complete quasi-monoidal lattice L, viewed as the lattice-theoretic base of powersets L x and spaces (X,T) or (X, 7-), is fixed relative to the spaces and mappings of the category L - T O P or L - F T O P . 'Variable-basis topology' permits the base to change within a category, so that each space has its own latticetheoretic base. To outline variable-basis topology, note that all complete quasi-monoidal lattices form a category, C Q M L , in which morphisms are mappings preserving ®, arbitrary V, and T; and also note that F R M and S F R M embed into C Q M L . One then considers L O Q M L = C Q M L °p, with objects the same as those of C Q M L , now called localic quasi-monoidal lattices, but morphisms reversed from those of C Q M L ; and one notes that L O C and S L O C embed into L O Q M L . Now, let C C L O Q M L . The category C - T O P for variable-basis topology and the category C - F T O P for variable-basis fuzzy topology are both 'concrete' categories over S E T x C as a 'ground' or 'base' category. For a S E T x C morphism (f, ¢): (X, L) --+ (Y, M), the pre-image operator (f, ¢ ) ~ - : L X +- M z is defined by (f, ¢)*-- (b) = Cop o b o f. An image operator (f, ¢)--~ is

FUZZY T O P O L O G Y also available which, if ¢ °p preserves arbitrary A, satisfies (f, ¢)-+ -q (f, ¢)+-; and if L = M, ¢ °p = idL, these operators reduce to their fixed-basis counterparts. Data for the category C - T O P include:

as singleton spaces into L O C - T O P . Thus, the variablebasis approach categorically unifies topology and fuzzy topology as a discipline.

• objects are topological spaces ( X , L , r ) (cf. also T o p o l o g i c a l space), where (X,L) E ISET x C I and (X,T) e I n - w o P l ; • morphisms are continuous mappings

[1] ADAMEK, J., HERHLICH, H., AND STHECKER, G.E.: Abstract and concrete categories, Wiley, 1990. [2] H(hHLE, U., AND SOSTAK, A.: 'Axiomatic foundations of fixedbasis fuzzy topology', in U. H()HLE AND S.E. I:{ODABAUGH (eds.): Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3 of The Handbooks of Fuzzy Sets, Kluwer Acad. Publ., 1999, pp. 123-272. [3] H{hHLE, V., AND RODABAUGH, S.E. (eds.): Mathematics of fuzzy sets: Logic, topology, and measure theory, Vol. 3 of The Handbooks of Fuzzy Sets, Kluwer Acad. PubI., 1999. [4] H()HLE, U., RODABAUGH,S.E., AND SOSTAK, A. (EDS.): 'Special issue on fuzzy topology', Fuzzy Sets and Syst. 73, no. 1 (1995). [5] H()HLE, U. (ED.): 'Mathematical aspects of fuzzy set theory', Fuzzy Sets and Syst. 40, no. 2 (1991), Special Memorial Volume-Second Issue. [6] JOHNSTONE, P.T.: Stone spaces, Cambridge Univ. Press, 1982. [7] KOTZE, W. (ED.): 'Special issue', Quaestiones Math. 20, no. 3 (1997). [8] RODABAUGH,S.E.: 'Categorical foundations of variable-basis fuzzy topology', in U. HOHLE AND S.E. RODABAUGH(eds.): Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3 of The Handbooks of Fuzzy Sets, Kluwer Acad. PubI., 1999, pp. 273 388. [9] RODABAUGH, S.E., KLEMENT, E.P., AND H()HLE, U. (eds.): Applications of category theory to fuzzy subsets, Kluwer Acad. Publ., 1992. [10] WANG, G.-J.: Theory of L-fuzzy topological spaces, Shanxi Normal Univ. Publ. House, 1988. (In Chinese.) [11] YING-MING, LIU, AND MAO-KANG, LUG: Fuzzy topology, World Sci., 1997.

(f, ¢): (X, L, r) + (Y, M, ~) (cf. also C o n t i n u o u s f u n c t i o n ) , where (f, ¢): (X, L) --+ (Y, M) is in S E T × C and (f, ¢)+-I~: r +-- ~. Data for the category C - F T O P include: • objects are fuzzy topological spaces (X,L,T), where (X,L) E ]SET x C I and (X,T) E IL-FTOPI; • morphisms are fuzzy continuous mappings (f, ¢): (X, L, 7") --+ (Y, M, S), where (f, ¢): (X, L) --+ (Y, M) is in S E T × C and T o (f, ¢)~- _> Cop oS on M Y. In both categories, compositions and identities are those of S E T x C. It is a theorem that for all C C L O Q M L , C - T O P and C - F T O P are topological over the ground S E T × C in the sense of [1] and [8, Sect. 1]. Further, these frameworks unify all the fixed-basis categories for topology given above and hence unify all important examples (referenced above) over different lattice-theoretic bases (e.g. two fuzzy real lines R(L) and R(M)). Moreover, all purely lattice-theoretic or point-free approaches to topology - - locales, topological molecular lattices, uniform lattices, etc. (see [6], [10], [11]) - - categorically embed into C - T O P or C - F T O P (for appropriate C) as subcategories of singleton spaces; e.g. L O C embeds

References

Stephen E. Rodabaugh MSC 1991: 54A40, 03G10, 06Bxx

169

G G A L O I S F I E L D S T R U C T U R E , Galois field (update) - This article contains some additional information concerning the structural properties of a G a l o i s f i e l d extension E / F , where E = GF(q n) and F = GF(q); this is also of interest for c o m p u t a t i o n a l applications. Usually E is represented as an n-dimensional v e c t o r s p a c e over F, so t h a t addition of elements of E becomes trivial, given the arithmetics in F (which, in applications, usually is a prime field GF(p) represented as the residues modulo p). However, the choice of a basis is crucial for performing multiplication, inversion and exponentiation. Various types of bases have been studied extensively. The most obvious choice is t h a t of a polynomial basis {1, c ~ , a 2 , . . . , a ~ - l } , where a is a root of an irr e d u c i b l e p o l y n o m i a l of degree n over F (so t h a t c~ generates E over F , cf. G a l o i s field). In this context, one often prefers a to be a generator of the c y c l i c g r o u p E* (cf. G a l o i s field); then a is usually called a primitive element or a primitive root for E, and the polynomial f is called a primitive polynomial. Note t h a t these terms carry a different m e a n i n g in the context of Galois fields than in algebra in general, see G a l o i s t h e o r y and Primitive

polynomial.

The s t a n d a r d alternative to using a polynomial basis is a normal basis, t h a t is, a basis of the form qn--1 {a, a q , . . . , a }, cf. N o r m a l b a s i s t h e o r e m . Hence such a basis consists of an orbit of maximal length n under the F r o b e n i u s a u t o m o r p h i s m x ~-~ x q. T h e element a is called a free element (or a normal element) in E / F . A stronger result is the existence of an element cJ E E t h a t is simultaneously free in E / K for every intermediate field K ; such an element is called completely free (or completely normal). A constructive t r e a t m e n t of normal bases and completely free elements in Galois fields can be found in [8]. Much current research (as of 2001) concerns the construction of primitive a n d / o r free elements with additional properties. The seminal result in this direction is

the primitive normal basis theorem: There always exists a primitive element w E E t h a t is simultaneously free over F . This result is due to A.K. Lenstra and R.J. Schoof [15], see also [9]. In this context the concepts of trace and n o r m play an i m p o r t a n t role. For any G a l o l s e x t e n s i o n E / F with Galois group G, one defines the trace and the n o r m (over F ) of an element z E E as the sum and the p r o d u c t of all conjugates z ~, a E G, respectively (each taken with the a p p r o p r i a t e multiplicity). In the special case u n d e r consideration, there are explicit formulas:

T r E / F ( Z ) = z + z q + . . . + z qn-x

(1)

N E / F ( Z ) = z . z q . . . . . Z qn-I.

(2)

and Now, let f = z n + a ~ _ l x n-1 + . . . + a l x + ao be an irreducible polynomial over F , and let a be a root of f (generating E). T h e n a~-i = -Tr(a)

and

ao = ( - 1 ) n N ( a ) .

T h e r e are m a n y results on the existence of primitive a n d / o r (completely) free elements a with prescribed trace a n d / o r norm, or with other prescribed coefficients. T h e first of these is due to S.D. Cohen [4]: Given a E F , where a ¢ 0 if either n = 2 or (n, q) = (3, 4), there exists a primitive element c~ of E with TrE/F (aJ) = a. For more results of this type, see [9]. Given any ordered basis B = (/30,...,/3~-1) of E , there exists a unique dual basis 13" = ( 7 0 , - . . , 7 n - 1 ) , defined by the p r o p e r t y

TrE/F(/3iTj) = (~ij

f o r i , j = 0 , . . . , n - 1.

One calls B self-dual if B = B*. A self-dual basis for E / F exists if either q is even or b o t h q and n are odd. It is easily checked t h a t the dual basis of a normal basis is likewise a n o r m a l basis; a self-dual normal basis for E / F exists if either q is even and n is not a multiple of 4, or b o t h q and n are odd. T h e n u m b e r of bases of these types has also been determined. For c o m p u t a t i o n a l purposes (in particular, for hardware implementations), it

GALOIS FIELD STRUCTURE would be desirable to have a self-dual polynomial basis; unfortunately, such bases do not exist. If one slightly weakens the requirements, a suitable substitute can be found in the so-called weakly self-dual polynomial bases; these belong to irreducible binomials and irreducible trinomials with constant term -1. Therefore the existence of such trinomials is an important (as of 2001 still open) question. These topics are discussed in detail in [9] and [13]. There is an alternative to using basis representations for finite fields: If one represents the non-zero elements of a Galois field F = GF(q) as the powers of a primitive element co, multiplication is trivial, but addition then becomes difficult. For any element 2/C F*, the discrete logarithm of 7 (to the base w) is the unique integer c with 0 < c < q - 2 satisfying cJ~ = 7; one writes c = log~ 7 and also puts log~ 0 = oe. Identifying the elements of F with their discrete logarithms, multiplying two elements reduces to adding the corresponding discrete logarithms: log~ (75) = log~ 3' + log~ 5, where the addition is done modulo q - 1. In order to perform additions in this representation, one needs to determine the discrete logarithm of 7 + d for 7, 5 E F*. Since o~c + w d = coo(1 + ajd-c), it suffices to determine the discrete logarithms for sums involving 1. This motivates the definition of the so-called Zech logarithm Z(e) = log~(1 + coe) (which is actually due to C.G.J. Jacobi); thus, Z(e) is determined from the equation 1 +cu~ = w z(*). Using discrete logarithms in conjunction with Zech logarithms is a useful representation in practical applications where repeated computations over a comparatively small finite field are required (with applications in coding theory being typical examples), since then the Zech logarithms can be pre-computed and, when needed for addition, retrieved by a simple table lookup. It is clear that a table lookup of Zech logarithms becomes impractical for large Galois fields. Thus the possibility of using this type of representation for large fields depends on the practicality of actually computing discrete logarithms, which is generally believed to be a very difficult problem. In fact, some systems in public-key cryptography (see C r y p t o g r a p h y ) are based on the intractability of computing discrete logarithms in sufficiently large Galois fields or, for state-ofthe-art systems, in elliptic curves over Galois fields (cf. also Elliptic curve); see, e.g., [5], [14], [19], [20]. As of 2001, the standard reference on Galois fields is [16]. In recent years there has been a resurgence of research in finite fields due to the wide variety of applications of various theoretical aspects of finite fields, e.g. in Galois g e o m e t r y , coding theory (cf. C o d i n g

a n d d e c o d i n g ) , design theory (cf. B l o c k design; Diff e r e n c e set; S y m m e t r i c design), cryptography (cf. C r y p t o g r a p h y ; C r y p t o l o g y ) , and signal processing. These applications usually require the use of efficient arithmetics, often in very large Galois fields; e.g., both GF(2593) and GF(2155) have been used in commercial cryptographical devices. This has been one of the major motivations for studying the structural properties of proper Galois fields as sketched above in more detail. The interplay of structural and arithmetical properties is discussed in detail in [9] and [13]; computational and algorithmic aspects are treated in [22]. Some good references for actual applications of Galois fields in the areas mentioned above are [1], [2], [3], [5], [6], [9], [10], [11], [12], [14], [17], [18], [19], [20], [21]. A good reference for computational aspects is [7]. References [1] ASSMUS,E.F., AND KEY, J.D.: Designs and their codes, Cambridge Univ. Press, 1992. [2] BERLEKAMP, E.R.: Algebraic coding theory, McGraw-Hill, 1968. [3] BETH, T., JUNGNICKEL, D., AND LENZ, H.: Design theory, second ed., Cambridge Univ. Press, 1999. [4] COHEN, S.D.: 'Primitive elements and polynomials with arbitrary trace', Diser. Math. 83 (1990), 1-7. [5] ENGE, A.: Elliptic curves and their applications to cryptography, Kluwer Acad. Publ., 1999. [6] FAN, P., AND DARNELL, M.: Sequence design for communication applications, Wiley, 1996. [7] GATHEN, J. VON ZUH, AND GERHARD, J.: Modern computer algebra, Cambridge Univ. Press, 1999. [8] HACHENBERGER, D.: Finite fields: Normal bases and completely free elements, Kluwer Acad. Publ., 1997. [9] HACHENBERGER,D., AND JUNGNICKEL, D.: Topics in Galois fields, Springer, to appear. [10] HIRSCHFELD,J.W.P.: Finite projective spaces of three dimensions, Oxford Univ. Press, 1985. [11] HIRSCHFELD,J.W.P.: Projective geometries over finite fields, second ed., Oxford Univ. Press, 1998. [12] HIRSCHFELD, J.W.P., AND THAS, J.A.: General Galois geometries, Oxford Univ. Press, 1991. [13] JUNGNICEEL, D.: Finite fields: Structure and arithmetics, Bibliographisches Inst. Mannheim, 1993. [14] KOBLITZ, N.: Algebraic aspects of cryptography, Springer, 1998. [15] LENSTRA, A.K., AND SCHOOF, R.J.: 'Primitive normal bases for finite fields', Math. Comput. 48 (1987), 217-231. [16] LIDL, R., AND NIEDERREITER, H.: Finite fields, AddisonWesley, 1983. [17] LINT, J.H. VAN: Introduction to coding theory, third ed., Springer, 1999. [18] MACWILLIAMS, F.J., AND SLOANE, N.J.A.: The theory of error-correcting codes, North-Holland, 1977. [19] MENEZES, A.J. (ed.): Applications of finite fields, Kluwer Acad. Publ., 1993. [20] MENEZES, A.J.: Elliptic curve public key cryptosystems, Kluwer Acad. Publ., 1993. [21] POTT, A., KUMAR, P.V., HELLESETH, W., AND JUNGNICKEL, D. (eds.): Difference sets, sequences and their correlation properties, Kluwer Acad. Publ., 1999.

171

GALOIS FIELD S T R U C T U R E [22] SHPARLINSKI,I.E.: Computational and algorithmic problems in finite fields, Kluwer Acad. Publ., 1992. Dieter Jungnickel

MSC1991:12E20 GEL~FOND-SCHNEIDER

METHOD -

In

1934

Hilbert's seventh problem (cf. also Hilbert problems)

was solved independently by A.O. Gel'fond [4] and Th. Schneider [9]: If a is a non-zero a l g e b r a i c n u m b e r , log a a non-zero logarithm of a and/~ an irrational algebraic number, then the number a z = exp{fl log c~} is transcendental (cf. T r a n s c e n d e n t a l n u m b e r ) . The transcendence of e ~ (corresponding to a = - 1 , l o g a = iTc, fl = - i ) had already been proved by Gel'fond in 1929 [3] using interpolation formulas for the function e ~z, like in Pdlya's work [8] on integral-valued entire functions. One main common feature of both the Gel'fond and the Schneider method is to start with the construction of an auxiliary function by means of Dirichlet's box principle (the Thue-Siegel lemma; cf. also D i r i c h l e t p r i n ciple). While Schneider's proof (cf. Schneider m e t h o d ) is based on the addition theorem for the exponential function e zl+z2 = eZle z2, the main ingredient in Gel'fond's proof is the differential equation ( d / d z ) e z = e z. Gel'fond considers the two functions e ~ and e ~ ; his auxiliary function has the form F ( z ) = P(e~,eZ~), where P is a polynomial with algebraic coefficients. He investigates the values of F as well as its derivatives at the points s loga, s C Z. An extrapolation is an essential feature of his proof. This method has been developed by Gel'fond himself for proving quantitative Diophantine approximation estimates (see [5]; see also G e l ' f o n d - B a k e r m e t h o d ; D i o p h a n t i n e a p p r o x i m a t i o n s ) , and by Schneider, who obtained an extension of the Gel'fond-Schneider theorem to elliptic and Abelian functions: he proved the transcendence of elliptic integrals of the first or second kind [10] and of Abelian integrals [11], including the transcendence of the values B ( a , b) of the beta-function at rational points (a,b) C (Q \ Z) 2. Next, Schneider [12], [13] provided general statements on the algebraic values of analytic functions satisfying differential equations; these results have been simplified and improved in the 1960s by S. Lang [6], who extended Schneider's results to commutative algebraic groups. The following far-reaching statement is called the Schneider-Lang criterion: Let K be a n u m b e r field and let f l , . . . , fd be meromorphic functions in C of finite order of growth (cf. also M e r o m o r p h i c f u n c t i o n ) . Assume fl, f2 are algebraically independent (cf. also A l g e b r a i c i n d e p e n dence). Assume also that for i = 1 , . . . , d, the derivative 172

( d / d z ) f i of fi belongs to the ring K [ f l , . . . , fd]. Then the set of w E C that are not poles of any f l , . . •, fd and

such that fi(w) C K , for 1 < i < d, is finite. Schneider and Lang extended their criterion to several variables by considering Cartesian products; a deeper result, involving algebraic hypersurfaces and suggested by M. Nagata [6], has been obtained by E. Bombieri [2]. A clever modification of the Gel'fond-Schneider method has been applied to modular functions in [1], solving Mahler's conjecture: For any algebraic number a with 0 < la] < 1 the value J ( a ) of the m o d u l a r f u n c t i o n is transcendental. Gel'fond proved in 1949 the algebraic independence of 2"~5 and 2 ~ . More generally, he proved that for algebraic (~ and fl with a ~ {0,1} and fl of degree d _> 3, the transcendence degree over Q of the field Q ( a ~ , . . . , a ~d-1) is _> 2 (cf. also T r a n s c e n d e n t a l e x t e n s i o n ) . After the work of G.V. Chudnovskii, P. Philippon and G. Diaz, it is known that this transcendence degree is >_ [(d + 1)/2]. This method not only provides a new proof of the L i n d e m a n n - W e i e r s t r a s s theorem on the algebraic independence of numbers e ~1 , . . . , e ~" when/~1,. • •, fin are Q-linearly independent algebraic numbers, but also yields a similar result for elliptic functions (and, more generally, Abelian functions), as shown by Philippon and G. W/istholz. Also, Chudnovskii proved the algebraic independence of the two numbers 7r, F(1/4) (showing therefore that F(1/4) is transcendental), and later Yu.V. Nesterenko adapted the method of [1] and obtained remarkable results of algebraic independence on values of modular functions, including the algebraic independence of the three numbers % F(1/4) and e ~ [7]. In another direction, both the Gel'fond and the Schneider method have been extended in order to prove results of linear independence over the field of algebraic numbers of logarithms of algebraic numbers (see Schneider m e t h o d and G e l ' f o n d - B a k e r m e t h o d ) . References

[1] BARRE-SIRIEIX,K., DIAZ, G., GRAMAIN,F., AND PHILIBERT, G.: 'Une preuve de la conjecture de Mahler-Manin', Invent. Math. 124, no. 1-3 (1996), 1-9. [2] BOMmERI, E.: 'Algebraic values of meromorphic maps', Invent. Math. 10 (1970), 267-287, Addendum, 11 (1970), 163166. [3] GEL'FOND,A.O.: 'Sur ins propri6t~s arithm6tiques des fonctions enti~res', Tdhoku Math. J. 30 (1929), 280-285. [4] GEL'FOND,A.O.: 'Sur le septi~me probl~me de Hilbert', Izv. Akad. Nauk. SSSR 7 (1934), 623-630. (Dokl. Akad. Nauk. SSSR 2 (1934), 1-6.) [5] GEL'FOND, A.O.: Transcendental and algebraic numbers, Dover, 1960. (Translated from the Russian.)

GENERALIZED FUNCTION ALGEBRAS [6] LANG, S.: Introduction to transcendental numbers, AddisonWesley and Don Mills, 1966, reprinted in: Collected Papers, Vol. I, Springer, 2000, pp. 396-506. [7] NESTERENKO, Y.V., AND PHILIPPON, P. (eds.): Introduction to algebraic independence theory. Instructional Conference ( C I R M Luminy, 1997), Vol. 1752 of Lecture Notes in Mathematics, Springer, 2001. [8] PdLYA, G.: 'Uber ganzwertige ganze Funktionen', Rend. Circ. Mat. Palermo 40 (1915), 1-16, See also: Collected papers

I Singularities of analytic functions, (ed. R.P. Boas), MIT

(1974), 1-16. [9] SCHNEIDER, TH.: 'Transzendenzuntersuchungen periodischer ~mktionen I', J. Reine Angew. Math. 172 (1934), 65-69. [10] SCHNEIDER, TH.: 'Transzendenzuntersuchungen periodischer Funktionen II', J. Reine Angew. Math. 172 (1934), 70-74. [11] SCHNEIDER, TH.: 'Zur Theorie der Abelschen Fanktionen und Integrale', J. Reine Angew. Math. 183 (1941), 110-128. [12] SCHNEIDER, TH.: 'Ein Satz fiber ganzwertige Funktionen als Prinzip fiir Transzendenzbeweise', Math. Ann. 121 (1949), 131-140. [13] SCHNEIDER, TH.: Einfiihrung in die transzendenten Zahlen, Springer, 1957.

Michel Waldschmidt MSC 1991:11J85 GENERALIZED FUNCTION ALGEBRAS - L e t ft be an open subset of R ~. A generalized function algebra is an associative, commutative d i f f e r e n t i a l algeb r a A(ft) containing the space of distributions 7P'(f~) or other distribution spaces as a linear subspace (cf. also G e n e r a l i z e d f u n c t i o n s , space of). An early construction of a non-associative, non-commutative algebra was given by H. K5nig [6]. The main current (2000) direction has been to construct associative, commutative algebras as reduced powers FA/2; of classical function spaces Y. A further approach uses analytic continuation and asymptotic series of distributions. To describe the principles, consider the space 12 = Coo(fl) of infinitely differentiable functions on ft (cf. also D i f f e r e n t i a b l e f u n c t i o n ) . Let A be an infinite index set,/3 a differential subalgebra of ];A and Z a differential ideal in/3. The generalized function algebra A(~) is defined as the factor algebra A(12) =/3/2;. Assuming that A is a directed set, let (~x)XcA be a net in Coo(R n) (cf. also N e t ( d i r e c t e d set)) converging to the Dirac measure in 79~(R n) (cf. also G e n e r a l i z e d f u n c t i o n s , space of). Any compactly supported distribution w E g'(f~) can be imbedded in FA by convolution (cf. also G e n e r a l i z e d function): w ~-~ (w * ~X)XEA. Appropriate conditions on/3 and 5[ will guarantee that this extends to an imbedding of g'(~t) into A(f~). An imbedding of 79' (f~) is obtained, provided the family {A(f~) : f~ open} forms a s h e a f of differential algebras on R '~ (the restriction mappings are defined componentwise on representatives). This imbedding preserves the derivatives of distributions. It follows from the impossibility result of L. Schwartz (see M u l t i p l i c a t i o n of d i s t r i b u t i o n s )

that it cannot retain the pointwise product of continuous functions at the same time. If 2; is contained in the subspace Z of ~)A comprised by those nets which converge weakly to zero, then an equivalence relation u ~ v can be defined on A(ft) by requiring that (u~, - VX)XEA E Z for representatives (UX)XEA and (Vx)XEA of u and v. The pointwise product of continuous functions (as well as all products obtained by multiplication of distributions) are retained up to this equivalence relation. A

list of typical examples of generalized function algebras follows: 1) 13 = (Coo(ft)) N, 2;o = {(uj)jCN: there is j0 such that uj -= 0 for j _> j0}. The algebra A(fl) =/3/Zo was introduced by C. Schmieden and D. Laugwitz [10] in their foundations of infinitesimal analysis. 2) Let L/ be a free u l t r a f i l t e r on the infinite set A and define 2;u = {(u~)xea: the set of indices {A: ux = 0} belongs to L/}, let/3 = (Coo(~t)) a. Then *C°°(a) = / 3 / Z u is an instance of the ultrapower construction of the algebra of internal smooth functions of n o n - s t a n d a r d a n a l y s i s (A. Robinson [8]). Neither 1) nor 2) provide sheaves on R n. To get a sheaf, localization must be introduced: 3) Let ]3 :

(Cc<~(~-~))N, 2;O,loc ~- {(?~j)jEN: for each

compact subset I4/C ~ there is a jo such that ujiK =_0 for j _> jo}. The algebra M(f~) = /3/2;0,~oc was introduced by Yu.V. Egorov [3] (cf. also E g o r o v generalized f u n c t i o n algebra). 4) Let SM = {(u~)~>o E Coo(f~)(°'oo): for each compact subset K C f~ and each multi-index a E N~ there is an N > 0 such that the supremum of IO~u~(x)l over x E K is of order O(e -N) as e ~ 0}. Let iV = {(u~)~>0 E gM: for each compact subset K C f~, each multi-index a E N~ and each q _> 0, the supremum of [O~u~(x)l over x E K is of order O@q) as ~"~ 0}. Then G(f~) = gM/iV is one of the versions of the algebras of J.F. Colombeau [1] (cf. also C o l o m b e a u g e n e r a l i z e d f u n c t i o n algebras). It is distinguished by the fact that the imbedding of 79'(f~) gives Coo(ft) as a faithful subalgebra. 5) Let /3 = (Coo(f~)) N, 2;nd : {(Uj)jEN: there is a closed, nowhere-dense subset r C f~ such that for all x E f t \ F there are a J0 and a neighbourhood V C ~ \ F o f x such that ujlv = 0 for j _> J0}. This is the nowhere dense ideal introduced by E.E. Rosinger [9] (cf. also R o s i n g e r n o w h e r e - d e n s e g e n e r a l i z e d f u n c t i o n algebra). The algebra 7~nd(a) -~- /3/Znd contains the algebra C ~ ( ~ ) of smooth functions defined off some nowhere-dense set as a subalgebra. Since 2;nd ~ Z, the imbedding of D'(ft) cannot be done by convolution, but uses an algebraic basis. 173

GENERALIZED F U N C T I O N ALGEBRAS There are many variations on this theme, different sets A, different spaces Y. The algebras can be defined on smooth manifolds as well. Usually, further operations can be applied to the elements of these algebras: superposition with non-linear mappings, restriction to submanifolds, pointwise evaluation (with values in the corresponding ring of constants). The algebras offer a general framework for studying all problems involving non-linear operations, differentiation, and distributional or otherwise non-smooth data and coefficients. Applications include non-linear partial differential equations, stochastic partial differential equations, Lie symmetry transformations, distributional metrics in general relativity, quantum field theory. For a survey of current applications, see [4]. A second approach is based on the algebras constructed by V.K. Ivanov [5] by means of analytic or harmonic regularization of homogeneous distributions and on the weak asymptotic expansions of V.P. Maslov (see e. g. [7]). a simple, specific example is given by the space h of distributions spanned by {xi,vpl/xJ,6(k)(x): i , j , k E No} in one dimension, where vp(.) denotes the principal value distribution and c~(k) (-) the kth derivative of the Dirac measure (cf. also G e n e r a l i z e d f u n c t i o n ) . Their harmonic regularizations generate a function algebra h* of smooth functions f * ( x , c ) defined on (x, e) E R x (0, oc). Each f * ( x , e ) has a unique weak asymptotic expansion of the form OO ~ j = r n f J ( x ) ej as e ~ 0 w i t h coefficients fj(x) in the original space h; the summation starts at some, possibly negative, rn E Z. The approach was extended [2] to the class of associated homogeneous distributions. This way the structure of an algebra may be introduced on certain subspaces of the space of asymptotic series with distribution coefficients. As an application, asymptotic solutions to non-linear partial differential equations can be constructed by direct computation with the asymptotic series. A relation with the previous construction of generalized function algebras is obtained by observing that harmonic regularization amounts to convolution with the kernel

= c_(x2 + c2)_1 7~ References

[1] COLOMBEAU,J.F.: New generalized functions and multiplication of distributions, North-Holland, 1984. [2] DANILOV, V.G., MASLOV, V.P., AND SHELKOVICH, V.M.: 'Algebras of singularities of singular solutions to first-order quasilinear strictly hyperbolic systems', Theoret. Math. Phys. 114, no. 1 (1998), 3-55.

[3] EGoaov, Yu.V.: 'A contribution to the theory of generalized functions', Russian Math. Surveys 45, no. 5 (1990), 1-49. 174

[4] GROSSER, M., HORMANN, G., KUNZINGER, M., AND OBERGUGGENBERGER, 1V[. (eds.): Nonlinear theory of generalized functions, Chapman and Hall/CRC, 1999. [5] IVANOV,V.K.: 'An associative algebra of the simplest generalized functions', Sib. Math. Y. 20 (1980), 509-516. [6] KONIG, H.: 'Multiplikation von Distributionen I', Math. Ann. 128 (1955), 420-452. [7] 1V[ASLOV,V.P., AND OMEL'YANOV,G.A.: 'Asymptotic solitonform solutions of equations with small dispersion', Russian Math. Surveys 36, no. 3 (1981), 73-149. [8] ROBINSON, A.: Non-standard analysis, North-HolIand, 1966. [9] ROSINGER, E.E.: Nonlinear partial differential equations. Sequential and weak solutions, North-Holland, 1980. [10] SCHMIEDEN, C., AND LAUGWITZ, D.: 'Eine Erweiterung der Infinitesimalrechnung', Math. Z. 69 (1958), 1 39.

Michael Oberguggenberger MSC 1991:46F30 An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century [3]: Given a boundary wire, how does one find the (minimal) soap film which spans it? Slightly more mathematically: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also P l a t e a u p r o b l e m . ) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, harmonic analysis, and fractals. Successes in the field include: classifying the structure of singularities in soap fihns (see [18], together with the fine descriptive article [4]); showing that the standard 'double bubble' is the optimal shape for enclosing two prescribed volumes in space [13], and developing powerful computer software for modelling the evolution of surfaces under the action of physical forces [7]. The main reference text for the subject is [10]. It is very densely written and [15] serves as a useful guide through it; [11] provides a comprehensive overview of the subject and contains a summary of its main results. For suitable introductions, see also [17], which contains an introduction to the theory of varifolds and Allard's regularity theorem, and [14], which includes information about tangent measures and their uses. For a slightly different slant, [9] discusses applications of some of the ideas of geometric measure theory in the theory of Sobolev spaces and functions of bounded variation. Many variational problems (cf. also V a r i a t i o n a l calculus) are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged GEOMETRIC

MEASURE

THEORY

-

GEOMETRIC MEASURE THEORY class. Much of the work in geometric measure theory has been directed towards placing this informal description on a formal footing appropriate for the study of surfaces. R e c t i f i a b i l i t y for s e t s . The key concept underlying the whole theory is t h a t of rectifiability, a measuretheoretic notion of smoothness (cf. also R e c t i f i a b l e c u r v e ) . A set E in Euclidean n-space R n is (countably) m-rectifiable if there is a sequence of C 1 mappings, fi: R m -9 R n, such t h a t

~m (E \ Ui=l/i ( R

m

)) -- O.

It is purely m-unrectifiable if for all C 1 mappings f : R TM -9 R ~, ~m(E n/(Rm))

: 0.

(Here, 7/m denotes the m-dimensional Hausdorff (outer) measure, defined by ~(E)

= supinf 5>0

c~

IEit~ = IEgl < ~for alli

'

where I'1 denotes the diameter and the constant c~ is chosen so that, when m = n, H a u s d o r f f m e a s u r e is just the usual L e b e s g u e m e a s u r e . ) A basic decomposition theorem states that any set E C R ~ of finite m-dimensional Hausdorff measure m a y be written as the union of an m-rectifiable set and a purely m-unrectifiable set, with the intersection necessarily having ~ m - m e a s u r e zero. In practice, the definition of rectifiability is commonly used with Lipschitz mappings replacing C 1 mappings: it may be shown that this does not change anything, see [14, Thm. 15.21]. A standard example of a l-rectifiable set in the plane is a countable union of circles whose centres are dense in the unit square and with radii having a finite sum; the closure of the resulting set contains the unit square, and yet, as indicated below, the set itself still has 'tangents' at ~/l-aImost every point. An example of a purely l-unrectifiable set is given by taking the cross-product of the 1/4-Cantor set with itself. (The 1/4-Cantor set is formed by removing 2 k intervals of diameter 4 -~, rather than 3 -k as for the plain C a n t o r set, at each stage of its construction.) A p p r o x i m a t e t a n g e n t s . The main importance of the class of rectifiable sets is that it possesses m a n y of the nice properties of the smooth surfaces which one is seeking to generalize. For example, although, in general, classical tangents may not exist (consider the circle example above), an m-rectifiable set will possess a unique approximate tangent at 7/'~-almost every point: An mdimensional linear subspace V of R ~ is an approximate

m-tangent plane for lira sup

E at x if 7 / ~ ( E n / 3 ( x , r))

r--+O

and for a l l 0 < s < ~m lira

r-+O

(

> 0

rm

1,

{y ~ E n B ( x , r ) : r rn

dist(y-x,V)> > s l Y - x]

}

) =0.

Conversely, if E C R ~ has finite 7/'~-measure and has an approximate m - t a n g e n t plane for ~ m - a l m o s t every x C E, then E is m-rectifiable. B e s i c o v i t c h - F e d e r e r p r o j e c t i o n t h e o r e m . Often, one is faced with the task of showing that some set, which is a solution to the problem under investigation, is in fact rectifiable, and hence possesses some smoothness. A m a j o r concern in geometric measure theory is finding criteria which guarantee rectifiability. One of the most striking results in this direction is the Besicovitch-Federer projection theorem, which illustrates the stark difference between rectifiable and unrectifiable sets. A basic version of it states that if E C R n is a purely m-unrectifiable set of finite m-dimensional Hausdorff measure, then for almost every orthogonal projection P of R n onto an m-dimensional linear subspace, ~'~(P(E)) = O. (It is not particularly difficult to show that in contrast, m-rectifiable sets have projections of positive measure for almost every projection.) This deep result was first proved for 1-unrectifiable sets in the plane by A.S. Besicovitch, and later extended to higher dimensions by H. Federer. Recently (1998), B. White [19] has shown how the higher-dimensionM version of this theorem follows via an inductive argument from the planar version. R e c t i f i a b i l i t y for m e a s u r e s . It is also possible (and useful) to define a notion of rectifiability for Radon (outer) measures: A R a d o n m e a s u r e # is said to be m-rectifiable if it is absolutely continuous (cf. also A b s o l u t e c o n t i n u i t y ) with respect to m-dimensional Hausdorff measure and there is an m-rectifiable set E for which # ( R n \ E) = 0. The complementary notion of a measure # being purely m-unreetifiable is defined by requiring t h a t # is singular with respect to all mrectifiable measures (cf. also M u t u a l l y - s i n g u l a r m e a s u r e s ) . Thus, in particular, a set E is m-rectifiable if and only if "]-/mlE (the restriction of ~a~m t o E) is m-rectifiable; this allows one to study rectifiable sets through m-rectifiable measures. It is common in analysis to construct measures as solutions to equations, and one would like to be able to deduce something about the structure of these measures (for example, that they are rectifiable). Often, the only a priori information available is some limited metric information about the measure, perhaps how the mass of 175

GEOMETRIC MEASURE THEORY small balls grows with radius. Probably the strongest known result in this direction is Preiss' density theorem [16] (see also [14] for a lucid sketch of the proof). This states t h a t if # is a Radon measure on R ~ for which limr-~0 #(B(x,r))/r "~ exists and is positive and finite for #-almost every x, then # is m-rectifiable. Preiss' main tool in proving this result was the notion of tangent measures. A non-zero Radon measure ~ is a tangent measure of # at x if there are sequences ri "N 0 and ci > 0 such t h a t for all continuous real-valued functions with compact support, i~o~limci f

¢ (--~-i y - x ) d#(y) = f ¢(y) du.

Thus, an m-rectifiable measure will, for almost-every point, have tangent measures which are multiples of mdimensional Hausdorff measure restricted to the approximate tangent plane at that point; for unrectifiable measures, the set of tangent measures will usually be much richer. The utility of the notion lies in the fact that tangent measures often possess more regularity than the original measure, thus allowing a wider range of analytical techniques to be used upon them. C u r r e n t s . A natural approach to solving a minimal surface problem would be to take a sequence of approximating sets whose areas are decreasing and finally extract a convergent subsequence with the hope that the limit would possess the required properties. Unfortunately, the usual notions of convergence for sets in Euclidean spaces are not suited to this. The theory of currents, introduced by G. de R h a m and extensively developed by Federer and W.H. Fleming in [12] (see [11] for a comprehensive outline of the theory and [10] for details), was developed as a way around this obstacle for oriented surfaces. In essence, currents are generalized surfaces, obtained by viewing an m-dimensional (oriented) surface as defining a continuous linear functional on the space of differential forms with compact support of degree m (cf. also C u r r e n t ) . Using the duality with differential forms, it is then possible to define m a n y natural operations on currents. For example, the boundary of an m-current can be defined to be the (m - 1)-current, OS, which is given via the exterior derivative for differential forms (cf. also E x t e r i o r a l g e b r a ) by setting 0S(¢) = S(d¢) for a d i f f e r e n t i a l f o r m ¢ of degree (m - 1). Of particular importance is the class of m-rectifiable currents: this class consists of the currents that can be written as S(¢)

176

=[

3

¢(x)) O(x)

where R is an m-rectifiable set with ?-/'~(R) < 0% O(x) is a positive integer-valued function with f 0 dT/"~ IR < oc and ~(x) can be written as vl A " - A V m with V l , . . . , V m forming an orthonormal basis for the approximate tangent space of R at x for 7-tin-almost every x E R. (That is, ~(x) is a unit simple m-vector whose associated m-dimensional vector space is the approximate tangent space of R at x for 7-/'~-almost every x E R.) The mass of a current given in this way is defined by M ( S ) = fO(x) d~'~lR(x). If the b o u n d a r y of an mrectifiable current is itself an (m - 1)-rectifiable current, then the m-current is said to be an integral current. These are the class of currents suitable for investigating Plateau's problem. The celebrated Federer-Fleming closure theorem says t h a t on a not too wild compact domain (it should be a Lipschitz retract of some open neighbourhood of itself), those integral currents S on the domain which all have the same b o u n d a r y T, an (m - 1)-current with finite mass, and for which M ( S ) is bounded above by some constant c, form a compact set. (The topology is t h a t generated by the integral fiat distance, defined for m-integral currents $1, $2 by

SK(SI,S2) = inf { M ( U ) + M ( V ) : U + 0V = $1 - $2}, where the infimum is over U and V such t h a t U is an mrectifiable current on K and V is an (m + 1)-rectifiable current on K.) In particular, if the constant c is chosen large enough so that this set is non-empty, then one can deduce the existence of a mass-minimizing current with the given boundary T. V a r i f o l d s . The theory of currents is ideally suited for investigating oriented surfaces, but for unoriented surfaces problems arise. The theory of varifolds was initiated by F.J. Almgren and extensively developed by W.K. A1lard [1] (see also [2] for a nice survey) as an alternative notion of surface which did not require an orientation. An m-varifold on an open subset ~ of R n is a Radon measure on f~ × G(n,m). (Here, G(n,m) denotes the G r a s s m a n n m a n i f o l d of m-dimensional linear subspaces of R n.) The space of m-varifolds is equipped with the w e a k t o p o l o g y given by saying that ~i --+ ~ if and only if f f &'i -4 f f d~ for all compactly supported, continuous real-valued functions on f t x G(n, m). Given an m-varifold u, one associates a Radon measure on ft, II'll, by setting II~ll (A) -- ~,(A x a(n, m)) for A C ft. As a partial converse, to an m-rectifiable measure I1~11 one can associate an m-rectifiable varifold # by defining for

t3 C f~ x G(n, m),

#(B) = II ll {x: (x, Tx) e B}, where T~ is the approximate tangent plane at x. The

first variation of an m-varifold ~ is a mapping from the space of smooth compactly supported vector fields on f~

GEOMETRIC TRANSVERSAL THEORY

to R, defined by

= f (X(x), v) d (x, V). If 5u = 0, then the varifold is said to be stationary. The idea is that the variation measures the rate of change in the 'size' of the varifold if it is perturbed slightly. A key result in the theory of varifolds is Allard's regularity theorem, which states t h a t stationary varifolds which satisfy a growth condition (detailed below) are supported on a smooth manifold. More precisely: For all e E (0, 1) there are constants 5 > 0, C > 0 such that whenever a E R ~, 0 < R < ~ , and v is an m-dimensional stationary varifold on the open ball U(a, R) with 1) a E spt v; 2) limr~0 ]]v[](B(a, r ) ) / ( c m r TM) existing and equal to at least one for Nvl]-almost every x; and

3) II-II(B(a,R)) < m(1

m,

then s p t ( l l v H ) N B ( a , ( 1 e)R) is a continuously differentiable embedded m-submanifold of R n, and dist(Tx, Ty) <_ C(r[x - yl) 1-~ for points in this submanifold. (The distance between the tangent spaces is given by the distance between their corresponding orthogonal projections.) This is a theorem which gives much more than just rectifiability; it gives information about the degree of smoothness as well. See [17] for some variants and a proof of this result. G e n e r a l i z a t i o n . Given the success of the theory in Euclidean spaces, it is natural to ask whether a similar theory holds in more general spaces [8]. There are m a n y difficulties to be overcome, but [6], [5] suggest that it may be possible. References [1] ALLARD, W.K.: 'On the first variation of a varifold', Ann. of Math. 95 (1972), 417-491.

[2] ALLARD,W.K.: 'Notes on the theory of varifolds. Th~orie des vari~t~s minimales et applications', Astdrisque 154/5 (1987), 73-93. [3] ALMGREN,JR., F.J.: Plateau's problem: An invitation to varifold geometry, W.A. Benjamin, 1966. [4] ALMGREN,JR., F.J., AND TAYLOR, J.E.: 'The geometry of soap bubbles and soap films', Scientific Amer. July (1976), 82-93. [5] AMBROSIO, L., AND KIRCHHEIM, B.: 'Currents in metric spaces', Acta Math. (to appear). [6] AMBROSIO, L., AND KIRCHHEIM, B.: 'Rectifiable sets in metric and Banach spaces', Math. Ann. (to appear). [7] BRAKKE, K.: 'The surface evolver V2.14', www.susqu.edu/facstaff/b/brakke/evolver/evolver.html

(2000). [8] DAVID, G., AND SEMMES, S.: Fractured fractals and broken dreams. Self-similar geometry through metric and measure, Vol. 7 of Oxford Lecture Ser. in Math. Appl., Clarendon Press, 1997. [9] EVANS, L.C., AND GARmPY, R.F.: Measure theory and fine properties of functions, Stud. Adv. Math. CRC, 1992.

[10] FEDERER, H.: Geometric measure theory, Vol. 153 of Grundl. Math. Wissenschaft., Springer, 1969. [11] FEDERER, H.: 'Colloquium lectures on geometric measure theory', Bull. Amer. Math. Soc. 84, no. 3 (1978), 291-338. [12] FEDERER, H., AND FLEMING, W.H.: 'Normal and integral currents', Ann. of Math. 72, no. 2 (1960), 458-520.

[13] HUTCHINGS, M., MORGAN, F., RITORI~, M., AND ROS, A.: 'Proof of the double bubble conjecture', Preprint (2000). [14] MATTILA, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Vol. 44 of Stud. Adv. Math., Cambridge Univ. Press, 1995. [15] MORGAN~ F.: Geometric measure theory. A beginner's guide, second ed., Acad. Press, 1995. [16] PREISS, D.: 'Geometry of measures in R n : distribution, rectifiability, and densities', Ann. of Math. (2) 125, no. 3 (1987), 537-643. [17] SIMON, L.: 'Lectures on geometric measure theory': Proc. Centre Math. Anal. Austral. National Univ., Centre Math. Anal. 3 Austral. National Univ., Canberra, 1983. [18] TAYLOR, J.E.: 'The structure of singularities in soap-bubblelike and soap-film-like minimal surfaces', Ann. of Math. (2) 103, no. 3 (1976), 489-539. [19] WHITE, B.: 'A new proof of Federer's structure theorem for k-dimensional subsets of R N', J. Amer. Math. Soc. 11, no. 3 (1998), 693-701.

T.C. O'Neil M S C 1991: 49Q15, 28A78, 49Qxx, 53C65, 58A25 GEOMETRIC

TRANSVERSAL

THEORY

- A the-

ory having its origin in a beautiful theorem of E. Helly in combinatorial geometry and a provocative question by P. Vincensini whose answer turned out to be negative. Helly's theorem states [14]: If .4 is a collection of compact convex sets in R d, every d + 1 or fewer members of which have a point in common, then there is a point common to all the members of A. A k-transversal to a collection of sets is a k-flat (cf. H i g h e r - d i m e n s i o n a l g e o m e t r y ) that meets every set in the collection. With this notion Helly's theorem can be reformulated to assert that if every d + 1 or fewer members of A have a 0-transversal, then ,4 has a 0-transversal. In 1935, Vincensini [17] asked whether there was a corresponding theorem for k-transversals to a collection of convex sets in Rd: Does there exist an r = r(k, d) such that for every collection M of compact convex sets in R d, if every subcollection of at most r members of M has a k-transversal, then .4 has a ktransversal? Vincensini gave an incorrect proof that r(1, 2) = 6. Subsequently, several examples showed that no finite r(1, 2) exists, even if it is assumed that the sets of A are pairwise disjoint. It is perhaps counterintuitive that the existence of such a number should depend on whether the sets are pairwise disjoint; however, L. Danzer [3] pointed out that although families of disjoint unit discs in the plane admit such a number (see below), there are examples, with m arbitrarily large, of m + 1 unit 177

GEOMETRIC TRANSVERSAL THEORY discs in the plane (not disjoint) that do not have a line transversal even though any m of them do. He also gave examples of m + 1 disjoint unit balls in R 3 that have no plane transversal even though any m of them do, for m arbitrarily large. With hopes dashed for a generalization of Helly's theorem in the form proposed by Vincensini, other Hellytype transversal theorems were found, where the elements of the collection A were subject to significant restrictions on either their shape or their relative position. Some noteworthy examples of such theorems are as follows.

Santald's theorem. ([15]) Let `4 be a collection of parallel rectangles in R 2. If every 6 or fewer elements of `4 admit a line transversal, then ,4 admits a line transversal. The number 6 cannot be reduced in general. This generalizes in two directions to: a) Let ,4 be a collection of parallelotopes in R d with edges parallel to the coordinate axes. If every 2 d - 1 ( 2 d - 1) or fewer elements of ,4 admit a line transversal, then ,4 admits a line transversal. b) Let ,4 be a collection of parallelotopes in R d with edges parallel to the coordinate axes. If every 2 d-1 ( d + 1) or fewer elements of ,4 admit a hyperplane transversal, then A admits a hyperplane transversal.

Hadwiger's theorem. ([11], [4]) Suppose `4 is a collection of compact convex sets in R a whose union is unbounded, each having diameter bounded by a constant. If every d + 1 or fewer sets of `4 have a line transversal, then all the sets do. Griinbaum's theorem. ([9], [10]) a) Let ,4 be a collection of compact convex sets in R d, each lying in one of a collection of parallel hyperplanes. If every 2d - 1 or fewer sets of ,4 have a line transversal, then ,4 has a line transversal. b) Let ,4 be a collection of pairwise disjoint translates of a parallelogram. If every 5 or fewer elements of ,4 admit a line transversal, then ,4 has a line transversal.

Danzer's theorem. ([3]) Let ,4 be a collection of pairwise disjoint unit discs in the plane. If every 5 or fewer elements of ,4 admit a line transversal, then A has a line transversal. G r i i n b a u m c o n j e c t u r e . In view of these last two theorems, it was natural for B. Gr/inbaum to conjecture, in 1958 [9], that for a collection of disjoint translates of a convex set, if every 5 or fewer sets of the collection admit a line transversal, then the entire collection has a line transversal. This conjecture remained open for over 30 years and its eventual solution involved an idea t h a t had first appeared in a theorem of H. Hadwiger. The key observation is that while, without additional assumptions, a 178

Helly-type theorem does not hold for arbitrary convex sets in the plane, by imposing a consistency condition on the order in which triples of a collection of convex sets are met by a line, one can recover the Helly property:

Hadwiger's transversal theorem. ([12]) Let ,4 be a finite collection of pairwise disjoint convex sets in R 2. If there exists a linear ordering of ,4 such that every 3 members of ,4 are met by a directed line in the corresponding order, then A has a line transversal. A line transversal to any collection of disjoint convex sets meets the sets in a given order or its reverse, depending on the direction of the transversal. This pair of orders (permutations) is called a geometric permutation. Building on the fact t h a t disjoint translates admit only few geometric permutations, H. Tverberg was able to prove Griinbaum's conjecture.

Tverberg's theorem. ([16]) Let A be a collection of disjoint translates of a convex set in R 2. If every 5 or fewer members of `4 admit a line transversal, then `4 has a line transversal. Hyperplane t r a n s v e r s a l s . Hadwiger's transversal theorem restored the generality and simplicity of Helly's theorem to the situation of Vincensini's problem. While it has been shown that no theorem of Hadwiger type holds for line transversals in a d for d > 3, it turns out nevertheless that Hadwiger's theorem does generalize to hyperplane transversals in R d. This requires a notion of order for finite point sets in higher dimensions. By the k-order type of a numbered set of points P = { P l , - - - , P n } C R k is meant the family of orientations of its (k + 1)-tuples, i.e., the collection

sign det

'

".. -

.

-

" k l <_io<...
This order type is simple if the set P lies in g e n e r a l p o s i t i o n in R k, i.e., if every determinant is non-zero. A generalization of disjointness is also needed. Just as every directed line transversal to a collection of convex sets meets the sets in a unique, well-defined order if and only if the sets are disjoint, every k-transversal will meet them in a unique, simple k-ordering if and only if no k + 1 sets of the collection admit a (k-1)-transversal; such a collection is said to be (k-1)-separated. Thus, a collection is pairwise disjoint if and only if it is 0-separated. With these definitions, J.E. G o o d m a n and R. Pollack proved the following generalization of Hadwiger's transversal theorem in arbitrary dimension.

Goodman Pollack theorem. ([6]) A ( d - 2 ) - s e p a r a t e d collection ,4 of compact convex sets in R d has a hyperplane transversal if and only if there is a one-to-one correspondence between A and a set S of points in R d-1 such

G E O M E T R I C TRANSVERSAL T H E O R Y that every d + 1 sets of A are met by an oriented hyperplane consistently with the (d-1)-ordering induced on the corresponding points of S. For this to hold, moreover, it is sufficient to find an acyclic rank-d oriented m a t r o i d structure on `4 whose (d+ 1)-tuples satisfy the consistency condition of the theorem. (For the notion of oriented matroid, which can be thought of as a 'locally realizable' generalization of the order type of a set of points, see also [2].) This theorem was subsequently generalized in several directions, the most comprehensive statement (which subsumes intermediate results of M. Katchalski and of Pollack and R. Wenger) being:

Anderson-Wenger theorem. ([1]) Let `4 be a finite collection of connected sets in R d. `4 has a hyperplane transversal if and only if for some k, 0 _< k < d, there exists a rank-(k + 1) acyclic oriented matroid structure on .4 such that every k + 2 members of `4 are met by an oriented k-flat consistently with that oriented matroid structure. Other directions. An effort to understand intermediate-dimensional transversals leads to considering the set of all k-transversals to a collection of convex sets. While there are as yet (2000) very few non-trivial results about transversals of dimensions between 1 and d - 1, there is a good deal known about the structure of these complete sets of transversals to collections of convex sets. It turns out that these subsets of the 'affine Grassmannian' themselves behave very much like convex point sets. Although they need not be connected, they nevertheless share many properties with convex sets, such as being defined by a convex hull operator satisfying the anti-exchange property that commutes with non-singular affine transformations, and satisfying the KreYn-Mil'man theorem [7] (cf. also L o c a l l y c o n v e x space). Other streams in geometric transversal theory include: • Gallai-type theorems, in which a Helly-type hypothesis leads to the conclusion that several transversals cover the entire collection (see [5], [18]); • the more general theorems of (p, q) type, in which a hypothesis of the form 'for every choice of p sets from the collection, some q have a common transversal' leads to a Gallai-type conclusion (see [5], [18]); • generalizations from transversal flats to transversal curves and surfaces (see [8]); • ongoing work of L. Montejano on a topological generalization of Hadwiger's transversal theorem and on the related notion of a 'separoid'; • the problem of bounding the number of geometric permutations of a collection of n convex bodies in R d,

and its generalization (via order types) to k-transversals (see [19]); • algorithmic geometric transversal theory, a branch of c o m p u t a t i o n a l g e o m e t r y (see [19]).

Surveys include [4], [5], [8], [13], [18], [19], where many other references can be found. References [1] ANDERSON, L., AND WENGER, R.: 'Oriented matroids and hyperplane transversals', Adv. Math. 119 (1996), 117-125. [2] BJORNER, A., LAS VERGNAS, M., WHITE, N., STURMFELS, B., AND ZIEGLER, G.M.: Oriented matroids, Cambridge Univ. Press, 1993. [3] DANZER, L.: ' 0 b e r ein Problem aus der kombinatorischen Geometric', Archly Math. 8 (1957), 347-351. [4] DANZER, L., GR/JNBAUM,B., AND KLEE, V.: 'Helly's theorem and its relatives', in V. KLEE (ed.): Convexity, Vol. 7 of Proc. Syrup. Pure Math., Amer. Math. Soc., 1963, pp. 101-180. [5] ECKHOFF, J.: 'Helly, Radon, and Carath6odory type theorems', in P.M. GRUBER AND J.M. WILLS (eds.): Handbook of Convex Geometry, North-Holland, 1993, pp. 389-448. [6] GOODMAN, J.E., AND POLLACK, R.: 'Hadwiger's transversal theorem in higher dimensions', J. Amer. Math. Soc. 1 (1988), 301-309. [7] GOODMAN,J.E., AND POLLACK, R.: 'Foundations of a theory of convexity on affine Grassmann manifolds', Mathematika 42 (1995), 305-328. [8] GOODMAN, J.E., POLLACK, R., AND WENGER, R.: 'Geometric transversal theory', in J. PACH (ed.): New Trends in Discrete and Computational Geometry, Vol. 10: Algorithms and Combinatorics, Springer, 1993, pp. 163-198. [9] GRUNBAUM, B.: 'Oil common transversals.', Archly Math. 9 (1958), 465 469. [10] GRONBAUM, B.: 'Common transversals for families of sets.', J. London Math. Soc. 35 (1960), 408-416. [11] HADWIGER, H.: 'fiber einen Satz Hellyscher Art.', Archly Math. 7 (1956), 377-379. [12] HADWIOER, H.: '~Jber Eibereiche mit gemeinsamer Treffgeraden', Portugal. Math. 16 (1957), 23-29. [13] HADWIGER, H., DEBRUNNER, H., AND KLEE, V.: Combinatorial geometry in the plane, Holt, Rinehart & Winston, 1964. [14] HELL'/, E.: ' 0 b e r Mengen konvexen KSrper mit gemeinschaftlichen Punkten', Jahresber. Deutsch. Math. Ferein. 32

(1923), 175-176. [15] SANTAL6, L.: 'Un teorema sobre conjuntos de paralelepipedos de aristas paraIelas', Publ. Inst. Mat. Univ. Nae. Litoral 2 (1940), 49-60. [16] TVERBERG, H.: 'Proof of Grfinbaum's conjecture on common transversals for translates', Discr. Comput. Geom. 4 (1989), 191-203. [17] VINCENSIm, P.: 'Figures convexes et vari6t6s lin6aires de l'espace euelidien &n dimensions', Bull. Sci. Math. 59 (1935), 163-174. [18] WENGER, R.: 'Helly-type theorems and geometric transversals', in J.E. GOODMANAND J.O'RoURKE (eds.): Handbook of Discrete and Computational Geometry, CRC, 1997, pp. 6382 (Chap. 4). [19] WENGER, R.: 'Progress in geometric transversal theory', in ]3. CHAZELLE, J.E. GOODMAN, AND R. POLLACK (eds.): Advances in Discrete and Computational Geometry, Vol. 223

179

GEOMETRIC TRANSVERSAL THEORY of Contemp. Math., Amer. Math. Soc., 1999, pp. 375-393.

Jacob E. Goodman Richard Pollack M S C 1991:52A35

GERSHGORIN THEOREM, Gerschgorin theorem, Ger@orin theorem - Given a complex (n x n)-matrix, A = [ a i d ] , with n > 2, then finding the eigenvalues of A is equivalent to finding the n zeros of its associated

characteristic polynomial p~(z) := d e t { z I - A}, where I is the identity (n x n)-matrix (cf. also M a t r i x ; E i g e n v a l u e ) . But for n large, finding these zeros can be a daunting problem. Is there an 'easy' procedure which estimates these eigenvalues, without having to explicitly form the characteristic polynomial p~(z) above and then to find its zeros? This was first considered in 1931 by the Russian mathematician S. Gershgorin, who established the following result [2]. If Aa(c~) := {z E C: Iz - c~I _< 5} denotes the closed complex disc having centre c~ and radius 5, then Gershgorin showed that for each eigenvalue A of the given complex (n x n)-matrix A = [aid] there is a positive integer i, with 1 0 in (3) gives t h a t A E Gi(A). In the same paper, Gershgorin also established the following interesting result: If the n discs Gi(A) of (2) consist of two non-empty disjoint sets S and T, where S consists of the union of, say, k discs and T consists of the union of the remaining n - k discs, then S contains exactly k eigenvalues (counting multiplicities) of A, while T contains exactly n - k eigenvalues of T. (The proof of this depends on the fact t h a t the zeros of the characteristic polynomial p~(z) vary continuously with the entries ai,j of A.) One of the most beautiful results in this area, having to do with the sharpness of the inclusion of (2), is a result of O. Taussky [4], which depends on the following use of directed graphs (cf. also G r a p h , o r i e n t e d ) . Given a complex (n x n ) - m a t r i x A = [ai,j], with n _> 2, let { p i}i=l be n distinct points, called vertices, in the plane. Then, for each a<j ¢ O, let PiP~ denote an arc from vertex i to vertex j. The collection of all these arcs defines the directed graph of A. Then the matrix A = [aid], with n >_ 2, is said to be irreducible if, given any distinct vertices i and j, there is a sequence of abutting arcs from i to j, i.e.,

(1)

Pi P& , Pe l Pg 2 , . . .

, Pe.~Pgm+l ,

where g,~+, = j.

with

Taussky's theorem is this. Let A = [ai,j] be any irre-

r~(A) := ~ la<jl. j=l

(G~ (A) is called the ith Gershgorin disc for A.) As this is true for each eigenvalue A of A, it is evident that if a(A) denotes the set of all eigenvalues of A, then

o-(A) C_ 0 Gi(A).

(2)

i=1

Indeed, let A be any eigenvalue of A [aid], so that there is a complex vector x = Ix1 ... x,~]T, with x ¢ O, such that Ax = Ax. As x ¢ O, then maxl_<j_<~ lxjl > O, and there is an i, with 1 <_ i _< n, such that lxil = maxl_<j_<~ Ixjt. Taking the ith component of Ax = Ax =

gives E j = I

ai,jzj

= /~xi, o r

equivalently

ducible complex (n x n)-matrix, with n _> 2. If k is an eigenvalue of A which lies on the boundary of the union of the Gershgorin discs of (2), then k lies on the boundary of each Gershgorin circle, i.e., from (1) it follows

that

IA-ai,il =ri(A)

for eachl
Next, there is related work of A. Brauer [I] on estimating the eigenvalues of a complex (n x n)-matrix (n > 2), which uses Cassini ovals instead of discs. For any integers i and j (i _< i,j <_ n) with i ~ j, the (i,j)th Cassini oval is defined by (cf. also Cassini oval)

Kid(A) :=

(4)

:= {z E C: I z - a i , i l . Iz -aj,jl < ri(A).rj(A)}. Then Brauer's theorem is that, for any eigenvalue A of A, there exist i and j, with i ¢ j, such that A E Kid(A), and this now gives the associated eigenvalue inclusion

7~ j=l

j¢i

On taking absolute values in the above expression and using the triangle inequality, this gives

la-

Ix l _51 -< j=l

180

Ix t,

(a)

a(A) C_ 0

K<j(A).

(5)

i,j=l

Note that there are now n(n - 1)/2 such Cassini ovals in (5), as opposed to the n Gershgorin discs in (2). But it is equally important to note that the eigenvalue inclusions

GLEAS O N - K A H A N E - Z E L A Z K O T H E O R E M of (2) and (5) use the exact same data from the matrix A = [ai,j], i.e., {ai,i}i~=l and {ri(A)}~ 1. So, which of the eigenvalue inclusions of (2) and (5) is smaller and hence better? It turns out that

[7] VARGA, R.S., AND KRAUTSTENGL, A.: 'On Ger~gorin-type problems and ovals of Cassini', Electron. Trans. N u m e r . Anal. 8 (1999), 15-20.

Richard S. Varga MSC1991:15A18

0

0

i,j=l

(6)

i=1

~¢j

GLEASON-KAHANE-ZELAZKO

for any complex (n x n)-matrix A, so that the Cassini ovals are always at least as good as the Gershgorin discs. (The result (6) was known to Brauer, but was somehow neglected in the literature.) Finally, as both eigenvalue inclusions (2) and (5) depend only on the row sums ri(A), it is evident that these inclusions apply not to just the single matrix A, but to a whole class of (n x n)-matrices, namely, a(A) := := {B = [hi,d]: bi,i = ai,i, and ri(B) = ri(A), 1 < i < n } . Thus,

i=1

i,j= l

iCj

for each B in Ft(A). Then, if cr(a(A)) denotes the set of all eigenvalues of all B in ft(A), it follows that

c £J K,j(A)c 0hi(A).

(7)

i=1

i,j=l

iCj

How sharp is the first inclusion of (7)? It was shown in 1999 by R.S. Varga and A. Krautstengl [7] that boundary of K1,2(A) o-(a(A))

=

Uit~j= 1

Kid(A)

n = 2; n

_> 3.

(8)

Thus, for n > 3, it can be said that the Cassini ovals give 'perfect' results. Gershgorin's discs and Brauer's Cassini ovals are mentioned in [5], [3]. A more detailed treatment of these topics can be found in [6]. References [1] BRAUER, A.: 'Limits for the characteristic roots of a matrix', Duke Math. d. 13 (1946), 387-395. [2] GERSCHGORIN, S.: 'Ueber die Abgrenzung der Eigenwerte einer Matrix', Izv. Akad. Nauk. S S S R Ser. Mat. 1 (1931), 749-754. [3] HORN, R.A., AND JOHNSON, C.R.: Matrix analysis, Cambridge Univ. Press, 1985. [4] TAUSSKY, O.: 'Bounds for the characteristic roots of matrices', Duke Math. J. 15 (1948), 1043-1044. [5] VAROA, R.S.: M a t r i x iterative analysis, second ed., Springer, 2000. [6] VARGA, R.S.: Gerggorin and his circles, Springer, to appear.

THEOREM

-

Let F be a non-zero linear and multiplicative f u n c -

t i o n a l on a complex B a n a c h a l g e b r a M with a unit e, and let A -1 denote the set of all invertible elements of ~4. Then F(e) = 1, and for any a E A -1 one has F(a) ~ O. A.M. Gleason [1] and, independently, J.P. Kahane and W. Zelazko [5], [6] proved that the property characterizes multiplicative functionals: If F is a linear functional on a complex unital Banach algebra A such that F(e) = 1 and F(a) ~ 0 for a C A -1, then F is multiplicative. Equivalently: a l i n e a r f u n c t i o n a l F on a commutative complex unital Banach algebra A is multiplicative if and only if F(a) C a(a) for all a C A, where a(a) stands for the spectrum of a (cf. also S p e c t r u m o f a n e l e m e n t ) . As there is a one-to-one correspondence between linear multiplicative functionals and maximal ideals, the theorem can also be phrased in the following way: A codimension-one subspace M of a commutative complex unital Banach algebra A is an ideal if and only if each element of M is contained in a non-trivial ideal. The theorem is not valid for real Banach algebras. The Gleason-Kahane-Zelazko theorem has been extended into several directions: 1) If p is non-constant e n t i r e f u n c t i o n and F is a linear functional on a complex unital Banach algebra M, such that F(e) = 1 and f(a) ¢ 0 for a C ~(A), then F is multiplicative [3]. 2) Let M be a finite-codimensional subspace of the algebra C(X) of all continuous complex-valued functions on a compact space X. If each element of M is equal to zero at some point of X, then the functions from M have a common zero in X [2]. It is not known if the analogous result is valid for all commutative unital Banach algebras. 3) The assumption of linearity of the functional F has been weakened, and the result has been extended to mappings between Banach and topological algebras. See [4] for more information about the history, related problems, and further references. References [1] GLEASON, A.M.: 'A characterization of maximal ideals', y. d'Anal. Math. 19 (1967), 171-172. [2] JAROSZ, K.: 'Finite codimensional ideals in function algebras', Trans. A m e r . Math. Soc. 287, no. 2 (1985), 779-785. [3] JAROSZ, K.: 'Multiplicative functionals and entire functions II', Studia Math. 124, no. 2 (1997), 193-198.

181

GLEAS ON-KAHANE-ZELAZKO

THEOREM

[4] JAROSZ, K.: 'When is a linear functional multiplicative?': Function Spaces: Proc. 3rd Conf. Function Spaces, Vol. 232 of Contemp. Math., Amer. Math. Soc., 1999. [5] KAHANE, J.-P., AND ZELAZKO, W.: 'A characterization of maximal ideals in commutative Banach algebras', Studia Math. 29 (1968), 339-343.

[6] ZELAZKO, W.: 'A characterization of multiplicative linear functional's in complex Banach algebras', Studia Math. 30 (1968), 83-85. K. Jarosz M S C 1991: 4 6 H x x

GMANOVA,

182

generalized

variance -

See A N O V A .

MSC1991:

62Jxx

multivariate

analysis of

H HAKEN MANIFOLD, sufficiently-large 3-manifold, sufficiently-large three-dimensional manifold - A compact, P2-irreducible t h r e e - d i m e n s i o n a l manifold which contains a properly embedded, incompressible, two-sided surface. All objects and mappings are in the piecewise-linear category (cf. also P i e c e w i s e - l i n e a r t o p o l o g y ) . The surface S 2 denotes the two-dimensional sphere, while p2 denotes the p r o j e c t i v e p l a n e . A surface F properly embedded in a three-dimensional manifold M is two-sided in M if it separates its regular neighbourhood in M. A three-dimensional manifold M is reducible (reducible with respect to c o n n e c t e d s u m decomposition) if it contains a properly embedded two-dimensional sphere t h a t does not bound a three-dimensional cell in M. Otherwise, the three-dimensional manifold M is irreducible. If the three-dimensional manifold M is irreducible and does not contain an embedded, twosided p 2 it is said to be p2-irreducible. An orientable three-dimensional manifold is P2-irreducible if it is irreducible. A surface F ¢ S 2 which is properly embedded in a three-dimensional manifold M is compressible in M if there is a disc D embedded in M such t h a t D N F = OD and the simple closed curve 0 D does not bound a disc in F. Otherwise, such a surface F is said to be incompressible in M. For two-sided surfaces it follows from the D e h n l e m m a that this geometric condition is equivalent to the inclusion mapping of fundamental groups, 7rl(F) ~ 7rl(M), being injective. The three-dimensional cell is a Haken manifold, as is any compact, P2-irreducible three-dimensional manifold with non-empty boundary. In fact, a sufficient condition for a compact, p2-irreducible threedimensional manifold M to be a Haken manifold is that its first h o m o l o g y g r o u p with rational coefficients, H I ( M , Q ) , be non-zero. This condition is not necessary. Any closed three-dimensional manifold which is not a Haken manifold is said to be a non-Haken manifold. It is conjectured that every closed, P%irreducible

three-dimensional manifold with infinite f u n d a m e n t a l g r o u p has a finite sheeted covering space (cf. also C o v e r i n g s u r f a c e ) t h a t is a Haken manifold. An embedded, incompressible surface in a threedimensional manifold is a very useful tool. However, in the case of three-dimensional manifolds with none m p t y boundary, it is necessary to add an additional condition related to the b o u n d a r y to achieve the same geometric character of the incompressible surface in a closed three-dimensional manifold. A properly embedded surface F in a three-dimensional manifold M with non-empty boundary is boundary compressible, written O-compressible, if there is a disc D embedded in M such that OD is the union of two arcs c~ and fl, o~nfl = Oc~ = Off, D A F = (~, D A O M = fl, and c~ does not cobound a disc in F with an arc in OF. If a properly embedded surface F in a three-dimensional manifold M is not &compressible, it is said to be boundary incompressible (O-incompressible). There is an algebraic interpretation of this geometric condition in terms of relative fundamental groups analogous to that given above in the case of an incompressible surface. Any compact, P%irreducible three-dimensional manifold M with non-empty boundary, other t h a n the three-dimensional cell, contains a properly embedded, incompressible and 0-incompressible surface t h a t is not a disc parallel into

OM. Just as two-dimensional manifolds have families of embedded simple closed curves t h a t split them into more simple pieces, the existence of incompressible and 0incompressible surfaces in Haken manifolds allow them to be split into more simple pieces. If F is a properly embedded, two-sided surface in a three-dimensional manifold M and U(F) is the interior of some regular neighbourhood of F in M, then M' = M \ U(F) is the threedimensional manifold obtained by splitting M at F. A partial hierarchy for M is a finite or infinite sequence of manifold pairs (M1,F1),.

. . , (Mi,Fi),

. . .,

. . ,

HAKEN MANIFOLD where Fi is a properly embedded, two-sided, incompressible surface in Mi which is not parallel into the boundary of Mi, and Mi+l is obtained from Mi by splitting M.i at Fi. A partial hierarchy is said to be a hierarchy for M if for some n, each component of M~ is a a threedimensional cell. Necessarily, a hierarchy for M is a finite partial hierarchy, ( M1, F1), . . . , ( Mi, Fi), . . . , ( M~, F~), and n is called the length of the hierarchy. The fundamental theorem for Haken manifolds is that every Haken manifold has a hierarchy, (MI,F1),...,(Mi,Fi),...,(M~,F~), where each Fi is incompressible and c0-incompressible in M~. The existence of a hierarchy admits inductive methods of analysis and proof. This is exhibited in many of the m a j o r results regarding three-dimensional manifolds. For example, the problem of determining if two given three-dimensional manifolds are homeomorphic (the homeomorphism problem), the problem of determining if a given loop in a three-dimensional manifold is contractible (the word problem), and the problem of determining if a closed, orientable, irreducible, atoTvidal (having no embedded, incompressible tori) three-dimensional manifold admits a metric with constant negative curvature, (uniformization), have all been solved for Haken manifolds but remain open (as of 2000) for closed, orientable, irreducible threedimensional manifolds in general. The study of Haken manifolds is quite rich and includes a large part of the knowledge and understanding of three-dimensional manifolds. References [1] FOMENKO, A.T., A~D MATVEEV, S.V.: Algorithmic and computer methods for three-manifolds, Kluwer Acad. PUN., 1997. [2] HAKEN, W.: 'Theorie der Normal Flgochen I', Acta Math. 105

(1961), 245

375.

[3] THURSTON, W.: 'Three-dimensional manifolds, Kleinian groups and hyperbolic geometry', Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357-381. [4] WALDHAUSEN,F.: :On irreducible 3-manifolds which are sufficiently large', Ann. of Math. 87 (1968), 56-88. [5] WALDHAUSEN,F.: 'The word problem in flmdamental groups of sufficiently large irreducible 3-manifolds', Ann. of Math. 88 (1968), 272-280.

William Jaco MSC 1991:57N10 H A L E S - J E W E T T THEOREM, ffewett-Hales theorem - One of the fundamental results in Ramsey theory. Let A = { a l , . . . , a q } be a finite a l p h a b e t and let A n denote the set of n-tuples with entries from A. A set { w l , . . . , w q} C A N consisting of q n-tuples w t = (w~,...,wln), 1 = 1 , . . . , q , is a combinatorial line if there exists a non-empty subset I C { 1 , . . . , n} such that f o r i E I a n d l = 1 , . . . , q o n e h a s w ~ = a t and for q An alternative i E { 1 , . . . , n } \ I one has w it . . . . . w i. 184

way of describing a combinatorial line is the following. Let t 9( A be a 'variable'. A variable word w(t) is a word over the alphabet A U {t} in which t occurs. For a E A, denote by w(a) the word which results from replacing each occurrence of t in w(t) by a. T h e n the set {w(a)}~ca is a combinatorial line. For example, if A = {0, 1, 2, 3, 4} and w(t) = 2tlt213 then {2tlt213}t~A = {2010213, 2111213, 2212213, 2313213, 2414213} is a combinatorial line in A 7. The Hales-Jewett theorem, proved in [12], states t h a t for any q,r E N there exists an N = N ( q , r ) E N such that if A N is partitioned into r classes, then at least one of these classes contains a combinatorial line. The following is an equivalent formulation of the Hales-Jewett theorem. Given a set S, let 5c(S) denote the set of all finite subsets of S. For any q, r E N there exists an N = N ( q , r ) so t h a t if S is a set with >_ N elements, then for any r-colouring of 5c(S) q there exist a non-empty set 7 E 5c(S) and sets a l , . . . , a q E 5 ( S ) such that 7 n a l . . . . . 7 n aq = (3 and such that (OQ,...,O~q), (oL1U~,oz2,... ,Ozq), (O~l,O~2U'~,... ,OLq),...,

(al, a 2 , . . . , aq U ~/) E ~T(s)q all have the same colour. Introduced originally in [12] as a tool for analyzing certain types of games (cf. also G a m e s , t h e o r y of), the Hales-Jewett theorem was soon recognized as a cornerstone of R a m s e y theory. Among the numerous corollaries of the H a l e s - J e w e t t theorem one has the v a n d e r W a e r d e n t h e o r e m on arithmetic progressions and its multi-dimensional version (the Gallai theorem). For example, to see that the Hales Jewett theorem implies the van der Waerden theorem, take A = { 0 , . . . , q - 1} and interpret the words over A as base-q expansions. If w(t) is a variable word over A U {t}, then {w(a)}aeA is a length-q arithmetic progression in N. Also, in [17] the Hales-Jewett theorem has been used to provide a significantly simplified proof of the geometric Ramsey theorem due to R. Graham, K. Leeb and B. Rothschild ([10]). See [11] for more details and discussion. The Furstenberg Katznelson density Hales-Jewett theorem ([9]), which confirmed a conjecture made by G r a h a m and which bears the same relation to the HalesJewett theorem as the Szemer~di theorem on arithmetic progressions bears to van der Waerden's theorem, states that for any q E N and e > 0 there exists an M = M ( q , ~) so that if n > M , A is a q-element set and R C_ A n satisfies IRI > cq ~, then R contains a combinatorial line. A polynomial extension of the Hales-Jewett theorem was obtained in [3], where it was shown that for any q , r , d E N there exists an N = N ( q , r , d ) so that if S is a set with _> N elements, then for any r-colouring of Jz(sd) q there exist a non-empty set ~, E 5c(S) and sets a l , • . . , aq E :P(S d) such that ~,d (-/al . . . . . ~,d n

HANKEL M A T R I X

aq = 0 and such that (O:1,... , O~q), ((21 ~7 d, OL2,... , OLq), (OLl,(2 2 u T d , . . . , O ~ q ) ,

. . . , (O~1,O~2,... , O~q [.]7 d) e ~2"(s d ) q

all have the same colour. The method utilized in [3] is that of t o p o l o g i c a l d y n a m i c s ; see [14] or [18] for a purely combinatorial proof. See [4], [13], [5] for applications of the polynomial Hales-Jewett theorem to density Ramsey theory.

Infinitary generalizations of the Hales-Jewett theorem were obtained in [7], [6], [8] and [2]. For an infinitary version of the polynomial Hales-Jewett theorem, see [14, Sect. 2.6].

[15] NILLI, A.: 'Shelah's proof of the Hales-Jewett theorem': Mathematics of Ramsey theory (Algorithms Combin.), Vol. 5, Springer, 1990, pp. 150-151. [16] SHELAH,S.: 'Primitive recursive bounds for van der Waerden numbers', J. Amer. Math. Soc. 1, no. 3 (1988), 683-697. [17] SPENCER, J.: 'Ramsey's theorem for spaces', Trans. Arner. Math. Soc. 249 (1979), 363-371. [18] WALTEaS, M.: 'Combinatorial proofs of the polynomial van der Waerden theorem and the polynomial Hales-Jewett theorem', J. London Math. Soc. (2) 61, no. 1 (2000), 1-12.

V. Bergelson MSC 1991:05D10

HANKEL M A T R I X - A m a t r i x whose entries along a parallel to the main anti-diagonal are equal, for each parallel. Equivalently, H = (hi,j) is a Hankel matrix if and only if there exists a sequence Sl, 82,..., such that hi,j = si+j-1, i , j = 1 , 2 , . . . . If sk are square matrices, It is conjectured that the polynomial Hales-Jewett then H is referred to as a block Hankel matrix. Infinite theorem should admit a density generalization. See [1, Hankel matrices are associated with the representation p. 57] for a formulation. of Hankel operators acting on the H i l b e r t s p a c e of References square summable complex sequences. Hankel matrices are frequently encountered in appli[1] BERGELSON, V.: 'Ergodic Ramsey theory - - An update', in cations where the close interplay between polynomial M. POLLICOTT AND K. SCHMIDT (eds.): Ergodic Theory of Zd-actions, Vol. 228 of Lecture Notes, London Math. Soc., and matrix computations is exploited in order to de1996, pp. 1-61. vise very effective numerical solution algorithms. Fa[2] BERGELSON, V., BLASS, A., AND HINDMAN, N.: 'Partition themous special cases are the Hilbert matrix and Hilbertorems for spaces of variable words', Proc. London Math. Soc. Hankel operator, defined by si+j-1 = ( i + j - 1 ) -1, which 68 (1994), 449-476. play an important role in the study of spectral properties [3] BERGELSON, V., AND LEIBMAN, n.: 'Set-polynomials and polynomial extensions of the Hales Jewett theorem', Ann. of integral operators of Carleman type (cf. also C a r l e of Math. (2) 150, no. 1 (1999), 33-75. m a n o p e r a t o r ) . For theoretical properties of Hankel [4] BERGELSON, V., AND MCCUTCHEON, R.: 'Uniformity in matrices, see [18]. For an overview of Hankel operators, polynomial Szeme%di theorem', in M. POLLICOTT AND see [25]. For references to the varied and numerous reK. SCHMIDT (eds.): Ergodic Theory of Zd-actions, Vol. 228 lations among Hankel matrices and polynomial compuof Lecture Notes, London Math. Soc., 1996, pp. 273-296. [5] BERGELSON, V., AND MCCUTCHEON, R.: A polynomial IP tations, see [5], [12], [7]. Szemerddi theorem for finite .families of commuting transTo a Hankel operator H = (si+j-1) one naturally asformations, Memoirs. Amer. Math. Soc., to appear. sociates the function r(z) = ~k~=l SkZ -k, which is called [6] CARLSON, T.: 'Some unifying principles in Ramsey theory', i t s symbol. Kronecker's theorem [12] says that H has fiDiscr. Math. 68 (1988), 117 169. nite rank n if and only if its symbol is a rational function, [7] CARLSON, T., AND SIMPSON, S.: 'A dual form of Ramsey's theorem', Adv. Math. 53 (1984), 265-290. that is, r(z) = p(z)/q(z), where p(z) and q(z) are mu[8] FURSTENBERG, H., AND JS[ATZNELSON, Y.: 'Idempotents in tually prime polynomials. In this case n is the number compact semigroups and Ramsey theory', Israel J. Math. 68 of poles of r(z) and, therefore, the degree of q(z). In ad(1990), 257-270. dition, assuming that Hk is the k × k leading principal [9] FURSTENBERG, H., AND KATZNELSON, Y.: 'A density version submatrix of H , i.e., the submatrix made up by the enof the Hales-Jewett theorem', Y. d'Anal. Math. 57 (1991), 64-119. tries in the first k rows and columns of H, then Hn is [10] GRAHAM, R., LEER, K., AND ROTHSCHILD, B.: 'Ramsey's thenon-singular, whereas H i is singular for any j > n. orem for a class of categories', Adv. Math. 8 (1972), 417-433. Given a Hankel operator H with symbol r(z), then [11] GRAHAM, R., ROTHSCHILD, B., AND SPENCER, J.: Ramsey the problem of finding two polynomials p(z) and q(z) of theory, Wiley, 1980. [12] HALES, A.W., AND JEWETT, R.I.: 'Regularity and positional degree at most n - 1 and n, respectively, such that

For a proof of the Hales-Jewett theorem which yields a primitive recursive upper bound for N(q,r), see [16] or [15].

games', Trans. Amer. Math. Soc. 106 (1963), 222-229. [13] LEIBMAN, A.: 'Multiple recurrence theorem for measure preserving actions of a nilpotent group', Geom. Funct. Anal. 8 (1998), 853-931. [14] McCUTCHEON, R.: Elemental methods in ergodic Ramsey theory, Vol. 1722 of Lecture Notes in Mathematics, Springer, 1999.

T(Z - 1 )

z

p ( z ) _ WoZ2 n -~- WlZ2n+ 1 ~ - ' ' " ,

(1)

q(z)

is a particular instance of the P a d ~ a p p r o x i m a t i o n problem. If H~ is non-singular, these polynomials are uniquely determined up to a suitable normalization, say 185

HANKEL MATRIX q(0) = 1, and their computation essentially amounts to solving a linear system with coefficient matrix H~. See [1], [6] and [13] f o r a survey of both the theory and applications of general Pad5 approximation problems. In [22] this theory is first generalized and then applied to the inversion of (block) Hankel matrices. Other variants of (1) can also be considered, generally leading to different computational problems. From a system-theoretic point of view, the possibility of recovering a rational function p(z)/q(z), where q(z) is monic, by its MacLaurin expansion at infinity has been extensively studied as the partial realization problem of system theory (see, for instance, [14]). It is intimately connected to such topics as the B e r l e k a m p - M a s s e y a l g o r i t h m in the context of coding theory and Kalman filtering. For applications of the theory of Hankel matrices to engineering problems of system and control theory, see [19] and [10]. The connection between Hankel matrices and ort h o g o n a l p o l y n o m i a l s arises in the solution of moment problems (el. also M o m e n t p r o b l e m ) . Given a positive B o r e l m e a s u r e r/ on ( - 1 , 1 ) , then the Hankel operator H = (si+j-1) defined by si+j-1 =

f J l zi+J-2 dr/(z), i, j = 1, 2 , . . . , is positive definite and, moreover, the last columns of H k-1 , k = 1, 2 , . . . , gencrate a sequence of orthogonal polynomials linked by a three-term recurrence relation. The converse is known as the Hamburger moment problem (cf. also M o m e n t p r o b l e m ) . The underlying theory is very rich and can be suitably extended to both finite Hankel matrices, by considering discrete measures, and to indefinite Hankel matrices, by means of formal orthogonal polynomials. A survey of results on Hankel matrices generated by positive measures can be found in [26]. See [11] and [15] for an introduction to the theory of formal orthogonal polynomials in the context of the algorithms of numerical analysis, including Lanczos' tridiagonalization process, rational interpolation schemes, the E u c l i d e a n alg o r i t h m , and inverse spectral methods for Jacobi matrices. Since orthogonal polynomials on the real axis gencrate Sturm sequences (cf. also S t u r m t h e o r e m ) , it follows that the use of quadratic forms associated with Hankel matrices provides a means for solving real root counting problems and real root localization problems; see [24] and [3]. Moreover, certain properties of sequences of Hankel determinants give the theoretical bases on which both Koenig's method and the Rutishauser qd algorithm, for the approximation of zeros and poles of meromorphic functions, rely; see [17]. The problem of inverting a finite non-singular Hankel matrix H~ has been extensively studied in the literature 186

on numerical methods and the connections shown earlier between the theory of Hankel matrices and m a n y other fields have been exploited in order to derive m a n y different Hankel system solvers. As mentioned above (the Kronecker's theorem), if the Hankel operator H has a rational symbol r(z) = p(z)/q(z) with p(z) and q(z) mutually prime and q(z) of degree n, then H~ is invertible. On the other hand, if H,~ is an invertible finite Hankel matrix of order n determined by its entries 8i+j_1, 1 <_ i,j <_ n, then the rational function r(z) = Z--~i=l~2n--1siz_; can uniquely be extended to a power series g(z) = r(z) + CX3 ~i=1 s2,~+iz-(2"~+*) in such a way t h a t g(z) is the expansion at infinity of p(z)/q(z), where p(z) and q(z) are mutually prime with q(z) monic of degree n. In this case the inverse of Hn is given by H~-1 = B(q,t), where t(z) is a polynomial of degree less than n that satisfies the Bezout equation t(z)p(z) + q(z)v(z) = 1, and B(q,t) = (bi,j) denotes the Bezout matrix, whose entries are defined by

q(z)t(w) - q(w)t(z) = i Z -- ~l)

bi'JZi-lwJ-l"

i,j=l

Since the Bezout equation can be solved by means of the Euclidean algorithm, this brings up the possibility of a recursive approach to the inversion of Hankel matrices. Analogously, in a matrix setting this recursive structure is fully exploited by the observation that Hankel matrices have low displacement rank in a sense first defined in [20]. Based on these properties, many fast algorithms for solving Hankel linear systems with cost O(n 2) have been developed. Superfast O(nlog 2 n) algorithms have also appeared; see [16], [21], [7], [5], [8] and [2] for extensive bibliographies on these methods. Generalizations to the case where Hn has entries over an integral domain are discussed in [4], where the subresultant theory [9] is described in terms of factorization properties of Hankel matrices. This becomes significant for the efficient solution of many problems in real algebraic geometry involving polynomials and rational functions [23]. References [I] BAKER,

G.A., AND

GRAVES-MORRIS,

P.R.: Padg approxi-

m a n t s , Addison-Wesley, 1981.

[2] BAREL, M. VAN, AND KRAVANJA, P.: 'A stabilized superfast solver for indefinite Hankel systems', L i n e a r Alg. ~ Its A p p l .

284 (1998), 335 355. [3] BARNETT, S.: Polynomials and linear control systems, M. Dekker, 1983. [4] BIN1, D.A., AND GEMIGNANgL.: 'Fast fraction free triangularization of Bezoutians with applications to sub-resultant chain computation', Linear Alg. £4 Its Appl. 284 (1998), 1939. [5] BIN% D.A., AND PAN, V.: Matrix and polynomial computations 1: Fundamental algorithms, Birkhguser, 1994.

H A R D Y - W E I N B E R G LAW [6] BREZINSKI, C.: Padd-type approximation and general orthogonal polynomials, Birkhguser, 1980. [7] BULTHEEL, A., AND BAREL, M. VAN: Linear algebra: Rational approximation and orthogonal polynomials, Studies in Computational Math. North-Holland, 1997. [8] CABAY, S., AND MELESHKO, R.: 'A weakly stable algorithm for Pad6 approximation and the inversion of Hankel matrices', S I A M J. Matrix Anal. Appl. 14 (1993), 735-765. [9] COLLINS, G.E.: 'Sub-resultants and reduced polynomial remainder sequences', J. Assoc. Comput. Maeh. 14 (1967), 128-142. [10] DATTA, B.N., JOHNSON, C.R., KAASHOEK,M.A., PLEMMONS, R., AND SONTAG, E.D. (eds.): Linear algebra in signals, systems and control, SIAM, 1988. [11] DRAUX, A.: P o l y n f m e s orthogonaux: Formels-applieations, Vol. 974 of Lecture Notes in Mathematics, Springer, 1983. [12] FUHRMANN, P.A.: A polynomial approach to linear algebra, Springer, 1996. [13] GRAOO., W.B.: 'The Pad~ table and its relation to certain algorithms of numerical analysis', SIAM Review 14 (1972), 1-61. [14] GRAGG, W.B., AND LINDQUIST, A.: 'On the partial realization problem.', Linear Alg. ~¢ Its Appl. 50 (1983), 277-319. [15] GUTKNECHT, M.H.: 'A completed theory of unsymmetric Lanczos process and related algorithms', S I A M J. Matrix Anal. Appl. 13 (1992), 594-639. [16] HEINIG, G., AND ROST, K.: Algebraic methods for Toeplitzlike matrices and operators, Akad. Berlin & Birkh~user, 1984. [17] HOUSEHOLDER, A.S.: The numerical treatment of a single nonlinear equation, McGraw-Hill, 1970. [18] IOHVIDOV., I.S.: Hankel and Toeplitz matrices and forms, Birkh~user, 1982. [19] KAILATH, T.: Linear systems, Prentice-Hall, 1980. [20] KAILATH, T., KUNG, S.Y., AND MORF, M.: 'Displacement ranks of matrices and linear equations', J. Math. Anal. Appl. 68 (1979), 395-407. [21] KAILATH, T., AND SAYED, A.H. (eds.): Fast reliable algorithms for matrices with structure, SIAM, 1999. [22] LABAHN, G., CHOI, D.K., AND CABAY, S.: 'The inverses of block Hankel and block Toeplitz matrices', S I A M J. Cornput. 19 (1990), 98-123. [23] LICKTEIG, W., AND ROYAL, M.F.: 'Cauchy index computation', Calcolo 33 (1996), 337-352. [24] NAIMARE, M.A., AND KREIN., M.G.: 'The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations', Linear and Multilinear Algebra 10 (1981), 265-308. [25] POWER, S.C.: Hankel operators in Hilbert spaces, Research Notes in Math. Pitman, 1982. [26] WIDOM, H.: 'Hankel matrices', Trans. Amer. Math. Soc. 127

(1966), 179-203.

pool. The biological sciences now generally define evolution as being 'the sum total of the genetically inherited changes in the individuals who are the members of a population's gene pool'. It is clear that the effects of evolution are felt by individuals, but it is the population as a whole that actually evolves. The gene frequency of an allele is the number of times an allele for a particular trait occurs compared to the total number of alleles for that trait. For evolution to occur in real populations, some of the gene frequencies must change with time. The Hardy-Weinberg law provides a baseline to determine whether or not gene frequencies have changed in a population, and thus whether evolution has occurred. As a result of independent work in the early 20th century, G.H. Hardy, an English mathematician [1], and W. Weinberg, a German physician [3], concluded that gene pool frequencies are inherently stable but that evolution should be expected in all populations virtually all of the time. They resolved this apparent paradox by analyzing the probable net effects of evolutionary mechanisms using mathematical models. An important way of discovering why real populations change with time is to construct a model of a popnlation that does not change. This is just what Hardy and Weinberg did. Their principle describes a hypothetical situation in which there is no change in the gene pool (frequencies of alleles), hence no evolution. Consider a population whose gene pool contains the alleles A and a. Hardy and Weinberg assigned the letter p to the frequency of the dominant allele A and the letter q to the frequency of the recessive allele a. In other words, p equals all of the alleles in individuals who are homozygous dominant (AA) and half of the alleles in people who are heterozygous (Aa) for this trait. Likewise, q equals all of the alleles in individuals who are homozygous recessive (aa) and the other half of the alleles in people who are heterozygous (Aa). In mathematical terms, these are 1

p = A A + -~Aa,

1

q = aa + -~Aa.

Since the sum of all the alleles must equal I00~, then

p+q -- i. They then reasoned that all the random possiLuca Gemignani

MSC1991: 15A57, 47B35, 65F05, 93B15 H A R D Y - W E I N B E R ( ] LAW - An understanding of evolution depends upon knowledge of population genetics. A population is a group of individuals of the same species in a given area whose members can interbreed. Because the individuals of a population can interbreed, they share a common group of genes known as the gene

ble combinations of the members of a population would equal (p + q)2 = p2 + 2pq + q2 = I. This has become known as the Hardy-Weinberg equilibrium equation. In this equation, p2 is the frequency of homozygous dominant (AA) people in a population, 2pq is the frequency of heterozygous (Aa) people, and q2 is the frequency of homozygous recessive (aa) ones. The Hardy-Weinberg equation can be used to discover the genotype frequencies in a population and to track their changes from one generation to another. 187

H A R D Y - W E I N B E R G LAW From observations of phenotypes, it is usually only possible to know the frequency of homozygous recessive people, or q2 in the equation, since they will not have the trait. Those who express the trait in their phenotype could be either homozygous dominant p2 or heterozygous 2pq. The H a r d y - W e i n b e r g equation allows one to determine which ones they are. Since p = 1 - q and q is known, it is possible to calculate p as well. Knowing p and q, it is a simple m a t t e r to plug these values into the Hardy-Weinberg equilibrium equation. This then provides the frequencies of all three genotypes for the seleered trait within the population. Using phenotype frequencies from the next generation in a population, one can also learn whether or not evolution has occurred and in what direction and rate for the selected trait. However, the H a r d y - W e i n b e r g equation cannot determine which of the various possible causes of evolution were responsible for the changes in gene pool frequencies. Hardy, Weinberg, and the population geneticists who followed them came to understand that evolution will not occur in a population if seven conditions are met: 1) 2) 3) 4) 5) 6) 7)

the population is infinitely large; all mating is totally random; mutation is not occurring; natural selection is not occurring; there is no migration in or out of the population; all members of the population breed; and everyone produces the same number of offspring.

So long as the above conditions are met, gene frequencies and genotype ratios in a randomly-breeding population remain constant from generation to generation. In other words, if no mechanisms that can cause evolution to occur are acting on a population, evolution will not occur - - the gene pool frequencies will remain unchanged. However, since it is highly unlikely that any of these seven conditions, let alone all of them, will occur in the real world, evolution is the inevitable result. W h a t the law expresses is that populations are able to maintain a reservoir of variability so that if future conditions require it, the gene pool can change. If recessive alleles were continually tending to disappear, the population would soon become homozygous. Under Hardy-Weinberg conditions, genes t h a t have no present selective value will nonetheless be retained.

References [1] HARDY, G.H.: 'Mendelian proportions in a mixed population', Sci. 28 (1908), 49 50. [2] STREN, C.: 'The Hardy-Weinberg law', Sci. 97 (1943), 137 138. [3] WEINBERG,W.: 'On the demonstration of heredity in man': Papers on Human Genetics, Prentice-Hall, 1963, Original: 1980; Translation by S.H. Boyer. Sujit K. Ghosh M S C 1991:92D10 188

HARRY DYM EQUATION - The non-linear partial differential equation at - ~Ox

(1)

for a real-valued function u ( x , t ) of one space variable x and time t. It belongs to a privileged class of non-linear partial differential equations known as completely-integrable systems, whose members share a number of remarkable properties including soliton solutions, B/icklund transformations, the Painlev~ property, and infinitely m a n y conservation laws [16], [18] (cf. also

B~icklund transformation; Completely-integrable differential equation; Non-linear partial differential equation; Painlev~-type equations; Painlev5 test; Soliton). (However, the H a r r y D y m equation is somewhat exceptional amongst known completelyintegrable systems. One reason is that, although it possesses the other listed properties, it does not possess the Painlev~ property.) These and other properties are the concomitants of the central technique used to solve the initial-value problem for completely-integrable systems, namely, the inverse scattering transform (or inverse spectral transform; cf. also Inverse scattering, fullline case; Korteweg-de Vries equation; HamiltonJan system; Painlev6-type equations). Non-linear partial differential equations such as (1), where a single time derivative is expressed in terms of space derivatives, are often referred to as non-linear evolution equations (cf. also Evolution equation). For this circle of ideas, see e.g. [11, [4], [16], [18]. The Harry D y m equation (1) was discovered by H. D y m in 1973-1974; however, its first appearance in the literature occurred in a 1975 paper of M.D. Kruskal [13], where it was named after its discoverer. It arises, e.g., in the analysis of the Saffman-Taylor problem with surface tension [9]. Relationship to the K o r t e w e g - d e Vries equation. The inverse scattering transform, which may be used to solve the characteristic initial value problem for the Harry D y m equation, Ot

Ox 3

,

-e~<x<ec,

t>0,

(2)

u(x, o) =

was discovered by C.S. Gardner, J.M. Greene, Kruskal and R.M. Miura [6], who applied it to the solution of the corresponding problem for the Korteweg-de Vries

equation Ou 9-[ +

au

~3u +

= 0.

(3)

This equation (3) was first derived by D.J. Korteweg and G. de Vries in a study of long waves in a shallow rectangular canal [12], where u(x, t) is the height of

HARRY DYM EQUATION the fluid above the undisturbed level, at position x and time t. The equation was rediscovered by Kruskal and N.J. ZabuskiY in their study of the Fermi-Pasta-Ulam problem, a proposed model of thermalization in metals. Since these works, (3) has found numerous applications in physics and is widely regarded as the prototypical example of a completely-integrable system. To see how problem (2) is solved by the inverse scattering transform and to describe its relationship to the Korteweg-de Vries equation, the approach taken in [6] to solve the corresponding problem for (3) is briefly sketched below. In [6], the one-parameter family of eigenvalue problems based on the second-order ordinary differential equation

d2¢ dx 2 + [ ~

(4) given by I.M. Gel'fand and B.M. Levitan [7], V.A. Marchenko [15] and I. Kay and H.E. Moses [10] to the scattering data S(t) and solve a linear Fredholm integral equation (cf. F r e d h o l m e q u a t i o n ) to recover the potential u(x, t) for any time t > 0. It follows that this potential is the solution of the initialvalue problem in question. Fundamental to the above solution scheme for the Korteweg-de Vries equation is its association with the eigenvalue problem (4). The discovery of the Harry Dym equation arose precisely by positing a slight variation of the eigenvalue problem (4), namely one where the eigenvalue A multiplies the potential instead of adding to it. That is, one considers the eigenvalue problem d2¢ + 5p(z, t ) ¢ = 0,

u(x,t)]¢

0,

-00<x<00,

(4)

is studied, where for parameter value t = 0, the potential u(x, 0) decays to zero sufficiently rapidly as x --+ 4-00. A sufficient decay rate is given by the Faddeev condition f _ ~ ( 1 + [xI)lu(x,O)l dx < 00. For such an eigenvalue problem it is known that the spectrum splits into a discrete and a continuous part, corresponding to ), < 0 and > 0, respectively (cf. also S p e c t r a l t h e o r y of diff e r e n t i a l o p e r a t o r s ) . In the discrete case, there are a finite number of eigenvalues {A~ = - - / {2n }N n = l , /~n > 0, and corresponding eigenfunctions %bn C L 2 (-00, 00), the bound-state eigenfunctions. The continuous spectrum A = k 2, k > 0, leads to the transmission and reflection coefficients, a(k) and b(k), respectively, via the asymptotic behaviour of the corresponding eigenfunctions,

~ e -ikx

+

b(k )e ikx as x -+ 00,

~(x, k) ~ La(k)e_,kx

as x -~ - 0 0 ,

If the potential i n (4) evolves from an initial condition u(x, 0) according to the Korteweg-de Vries equation, then the corresponding discrete eigenvalues are constants of the motion while the transmission and reflection coefficients together with the L2(-00, c~)-norm of the bound-state eigenfunctions have a very simple evolution. This suggested the following procedure for solving the characteristic initial-value problem for the Korteweg-de Vries equation: i) compute the bound-state eigenvalues and eigenfunctions, and the transmission and reflection coefficients for an initial potential u(x, 0), obtaining scattering data S(0) at time t = 0; ii) time evolve the initial potential u(x, 0) by the Korteweg-de Vries equation, obtaining scattering data S(t) for any time t > 0; iii) apply the solution of the inverse scattering problem for the time-independent S c h r S d i n g e r e q u a t i o n

dx 2

- 0 0 < x < 00,

(5)

and seeks the lowest-order non-linear evolution equation for p(x,t) so that the bound-state eigenvalues of problem (5) are constant in time. In the language of the inverse scattering transform, the linear eigenvalue problem (5) is said to be isospectral for the Harry Dym equation (1), just as problem (4) is isospectral for the Korteweg-de Vries equation. See [2] for a textbook account. Though the isospectral problem for the Harry Dym equation described above is fundamental, to date (2000) it has proved difficult to obtain solutions of the Harry Dym equation as explicitly as those available for the Korteweg-de Vries equation and other completelyintegrable systems. This, despite the existence of a reciprocal B~icklund t r a n s f o r m a t i o n [21] linking solutions of the Harry Dym and Korteweg-de Vries equations; see [8]. A class of eigenvalue problems that includes (4) and (5) as special cases was studied by P.C. Sabatier [23] and Li Yi-Shen [28]. They study the one-parameter family (t) of eigenvalue problems d2¢

dx--Y + [~p(x, t) - u(x, t)] ¢ = 0,

- 0 0 < x < 00, (6)

and compute the lowest-order non-linear evolution equation for which (6) is the isospectral problem. The Korteweg-de Vries and Harry Dym equations arise from the appropriate specializations.

Generalized and extended Harry Dym equations. Since its discovery, the Harry Dym equation has attracted a good deal of attention from researchers. See, for example, the brief list [3], [5], [14], which is by no means exhaustive. A brief description of a number of results related to extensions and generalizations of the equation follows. In 1984, B.G. Konopelchenko and V.G. DubrovskiY [11] discovered a linear isospectral problem (which forms one half of a Lax pair, cf. M o u t a r d t r a n s f o r m a t i o n ; 189

HARRY DYM EQUATION

Darboux

transformation)

for the n o n - l i n e a r evolu-

tion e q u a t i o n

-Ot -

~

-2-~-S

+ 6ue

u _ 1 0 ~ 1 ~0y

1

'

(7) where O~-1 is the o p e r a t o r f . ~ ds. E q u a t i o n (7), which is s o m e t i m e s called the 2 + 1 - d i m e n s i o n a l H a r r y D y m equation, generalizes the H a r r y D y m e q u a t i o n (1) to two

space d i m e n s i o n s x a n d y. In [19], C. Rogers showed t h a t the 2 + 1 - d i m e n s i o n a l H a r r y D y m e q u a t i o n a d m i t s a reciprocal B ~ c k l u n d t r a n s f o r m a t i o n l i n k i n g its solutions with those of the s i n g u l a r i t y m a n i f o l d equation, first int r o d u c e d by J. Weiss [26] (see also [25], [27]), o b t a i n e d by a p p l i c a t i o n of the P a i n l e v & t e s t to the K a d o m t s e v P e t v i a s h v i l i e q u a t i o n (see K P - e q u a t i o n ) . This m a y be c o m p a r e d with the invariance of the H a r r y D y m equation (1) u n d e r a reciprocal t r a n s f o r m a t i o n as n o t e d in [22]. This i n v a r i a n c e e x t e n d s to hierarchies, and, c o n j u g a t e d by a G a l i l e a n t r a n s f o r m a t i o n , induces the u s u a l a u t o - B ~ c k l u n d t r a n s f o r m a t i o n for the K o r t e w e g de Vries hierarchy [20]. These results have been usefully revisited, using the t h e o r y of generalized Lax equations, in [17], where 2 + 1-dimensional c o m p l e t e l y - i n t e g r a b l e systems are studied, i n c l u d i n g the 2 + 1-dimensional H a r r y D y m equation. More recently (1999), W . K . Schief a n d Rogers [24] have a derived a n e x t e n d e d H a r r y D y m equation, shown to be completely integrable, as a flow on a special family of curves in t h r e e - d i m e n s i o n a l Euclidean space, where each m e m b e r curve has c o n s t a n t c u r v a t u r e or c o n s t a n t torsion a n d where the t i m e derivative of its p o s i t i o n vector p o i n t s in the direction of the u n i t b i n o r m a l vector. References [1] ABLOWITZ, M.J., AND CLARKSON, P.A.: Solitons, nonlinear evolution equations and inverse scattering, Vol. 149 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1991. [2] CALOGERO, F., AND DEGASPERIS,A.: Spectral transform and solitons i, Vol. 13 of Studies Math. Appl., North-Holland, 1982. [3] DMITRIEVA, L.A.: 'Finite-gap solutions of the Harry Dym equation', Phys. Lett. A 182, no. 1 (1993), 65-70. [4] DODD, R.K., EILBECK, J.C., GIBBON, J.D., AND MORRIS, H.C.: Solitons and nonlinear waves, Acad. Press, 1982. [5] FUCHSSTEINER, B., SCHULZE, T., AND CARILLO, S.: 'Explicit solutions for the Harry Dym equation', J. Phys. A 25, no. 1 (1992), 223 230. [6] GARDNER, C.S., GREENE, J.M., t~RUSKAL, M.D., AND MIURA, R.M.: 'Method for solving the Korteweg de Vries equation', Phys. Rev. Lett. 19 (1967), 1095-1097. [7] GEL'FAND, I.M., AND LEVITAN, B.M.: 'On the determination of a differential equation from its spectral function', Izv. Akad. Nauk. SSSR Ser. Mat. 15 (1951), 309-366. [8] HEREMAN, W., BANERJEE, P.P., AND CHATERJEE, M.R.: 'Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation', J. Phys. A 22, no. 3 (1989), 241-255. 190

[9] KADANOFF, L.P.: 'Exact solutions for the Saffman-Taylor problem with surface tension', Phys. Rev. Lett. 65, no. 24 (1990), 2986-2988. [10] KAY, I., AND MOSES, H.E.: 'The determination of the scattering potential from the spectral measure function, III. Calculation of the scattering potential from the scattering operator for the one-dimensional Schrhdinger equation', Nuovo Cimento 3, no. 10 (1956), 276 304. [11] KONOPELCHENKO, B.G., AND DUBaOVSKY, V.G.: 'Some integrable nonlinear evolution equations in 2 + 1 dimensions', Phys. Lett. A 102 (1984), 15-17. [12] KORTEWEG,D.J., AND VRIES, G. DE: 'On the change in form of long waves advancing in a rectangular canal and on a new type of long stationary waves', Philos. Mag. 39, no. 5 (1895), 422 443. [13] KRUSKAL, M.D.: 'Nonlinear wave equations', in J. MOSER (ed.): Dynamical Systems, Theory and Applications, Vol. 38 of Lecture Notes in Physics, Springer, 1975. [14] LEO, M., LEO, R.A., SOLIANI, G., SOLOMBRINO, L., AND MARTINA, L.: 'Lie-B~cklund symmetries for the Harry Dym equation', Phys. Rev. D 27, no. 6 (1983), 1406-1408. [15] MARCHENKO,V.A.: 'On the reconstruction of the potential energy from phases of the scattered waves', Dokl. Akad. Nauk SSSR 104 (1955), 695-698. [16] NEWELL, A.C.: Solitons in mathematics and physics, Vol. 48 of CBMS-NSF, SIAM, 1985. [17] OEVEL, W., AND ROGERS, C.: 'Gauge transformations and reciprocal links in 2+1 dimensions', Rev. Math. Phys. 5 (1993), 299 330. [18] PALAIS, R.S.: 'Symmetries of solitons', Bull. Amer. Math. Soc. 34, no. 4 (1997), 339-403. [19] ROGERS, C.: 'The Harry Dym equation in 2 + 1 dimensions: a reciprocal link with the Kadomtsev-Petviashvili equation', Phys. Lett. A 120 (1987), 15-15. [20] ROGERS, C., AND NUCCI, M.C.: 'On reciprocal B~cklund transformations and the Korteweg-de Vries hierarchy', Physica Scripta 33 (1988), 289-292. [21] ROGERS, C., AND SHADWICK, W.F.: Biicklund transformations and their applications, Vol. 161 of Math. Sci. and Engin., Acad. Press, 1982. [22] ROGERS, C., AND WONG, P.: 'On reciprocal transformations of inverse schemes', Physica Scripta 30 (1984), 10-14. [23] SABATIER, P.C.: 'On some spectrM problems and isospectral evolutions connected with the classical string problem. I: Constants of the motion; II: Evolution equations', Lett. Nuovo Cimento 26 (1979), 477-482; 483-486. [24] SCHIEF, W.K., AND ROGERS, C.: 'Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces', Proc. Royal Soc. London 455 (1999), 3163-3188. [25] WEISS, J.: 'The Painlev& property for partial differential equations II: B~cklund transformations, Lax pairs, and the Schwarzian derivative', J. Math. Phys. 24, no. 6 (1983), 14051413. [26] WEISS, J.: 'Modified equations, rational solutions and the Painlev~ property for the Kadomtsev-Petviashvili and Hirota-Satsuma equations', Y. Math. Phys. 26, no. 9 (1985), 2174 2180. [27] WEISS, J.: 'B~cklund transformation and the PainlevO property', J. Math. Phys. 27, no. 5 (1986), 1293-1305. [28] YI-SHEN, LI: 'Evolution equations associated with the eigenvalue problem based on the equation ¢xx = [u(x)- k2p(x)]¢ ', Nuovo Cimento 70B, no. N1 (1982), 1-12. P.J. Vassiliou

M S C 1991: 58F07, 35Q53

HERMANN ALGORITHMS I-IECKE O P E R A T O R - Let M ( k ) be the vector space of (entire) modular forms of weight k, see M o d u l a r f o r m or [1]. Then the Hecke operator Tn is defined for f C M ( k ) by

d-1 (T~f)(~-) = n k-1 ~

d-k

Eb=0f

( n T + bd~ \ ~-~ j ,

(1)

where ~- C H , the upper half-plane. One (easily) proves that T ~ f E M ( k ) if f E M ( k ) . If f ( z ) = E ~ = o C ( m ) q m ( z ) , q(z) = e 2~iz, is the Fourier expansion of f , then

(The restriction of the D a D to a C / 9 is needed to keep things, e.g. the sets A, B, finite.) Let X be a subset o f / 9 containing D and multiplicatively closed. Then one defines R o ( X , D) as the submodule of R spanned by the D~D for ~ E X. This gives a subring of R. Finally, one defines R ( X , D), the Heeke algebra of (X, D) as R0(X, D) ® Q. In m a n y situations the double cosets D~D act on forms, functions, etc., which gives Hecke operators. See [7] for an example in the case of double cosets with respect to the principal congruence subgroup r(n) =

T f(z)

=

m=0

with mn

dl(n,-~) Note that

TnTm =

E

dk-lTmn/d2,

dl(n,rn)

so that, in particular, the Tn commute. The discriminant form O0

Zx(z) =

12 Z

e M(12),

rn= l

where ~-(m) is the R a m a n u j a n function, is a simultaneous eigenfunction of all Tn. Formula (1) can be regarded as coming from an operation on lattices in the complex plane, T~(L) = ~ L I, where the sum is over all sublattices of L of index n. This geometric definition, [6], makes (1) easier to understand. There are Hecke operators in much more general settings, e.g. for suitable subgroups of the m o d u l a r g r o u p F. A quite abstract group setting follows, [8]. Let G be a g r o u p and D a subgroup. Another subgroup D I is commensurable with D if D n D I is of finite index in both D and D ~. Let /9 = {a C G: a D a -1 is commensurable with D}. This is a subgroup of G that contains D. Now, let R be the Z-module of all formal sums ~ c ~ D a D , i.e. the free Abelian group on the double cosets of D i n / 9 . There is an associative multiplication on R, defined as follows. Let u = D a D , v = D/3D. Then the product uv = DaD/3D is clearly a (disjoint) union of double cosets. It gives a product u. v, provided multiplicities are taken into account. More precisely, let D a D = H a ' E A DR', D/3D = L[Z,cA D/3'. Then

(DaD)(D/3D) = DaD/3D = D a (Uz,D/3') = = U~,DaD/3 t = Us,,~, D a /'3 , . Now, let # ( u . v,w) = # { ( a ' , / 3 ' ) E A x B : Da'fl' = D ~ w i t h w = D~D}. Then u - v = ) - - ~ # ( u . v , w ) w .

which gives rise to the (usual) Hecke operators for modular forms. In [8] this setting is used to define Hecke operators for the case of adelic groups. Modular forms turn up all over mathematics and physics and, hence, so do the Hecke operators. See the references for a variety of uses of them. References

[1] APOSTOL, T.M.: Modular functions and Dirichlet series in number theory, Springer, 1976, p. 120ft. D.: Automorphic forms and representations, Cambridge Univ. Press, 1997. [3] HURT,N.E.: 'Exponential sums and coding theory. A review', Acta Applic. Math. 46 (1997), 49-91. [4] HURT, N.: Quantum chaos and mesoscopic systems, Kluwer Acad. Publ., 1997, p. 101; 163ff. [5] KNOPP~ M.I.: Modular functions in analytic number theory, Markham Publ., 1970. [6] OGG, A.: Modular forms and Dirichlet series, Benjamin, 1969, p. Chap. II. [7] RANKIN, R.A.: Modular forms and functions, Cambridge Univ. Press, 1977, p. Chap. 9. [8] SHIMURA, G.: Euler products and Eisenstein series, Amer. Math. Soc., 1997, p. Sect. 11. [9] VENKOV, A.B.: Spectral theory of automorphic functions, Kluwer Acad. Publ., 1990, p. 34; 59.

[2] BUMP,

M. Hazewinkel MSC1991: 11F25, 11F60 HERMANN ALGORITHMS - In her famous 1926 paper [3], G. H e r m a n n set out to show that all standard objects in the theory of polynomial ideals over fields k, including the prime ideals associated to a given ideal, can be determined by means of computations involving finitely m a n y steps, i.e. field operations in k. Any Rmodule for R := k [ X 1 , . . . , Xn] is determined by giving a finite set of generators, called a basis. Hermann states explicitly t h a t one can give an upper bound for the number of operations necessary for each sort of computatiolL Building on previous work [2] by K. Henzelt and E. Noether, Hermann's work set a milestone in effective

191

HERMANN ALGORITHMS algebra. While the structural approach to algebra continued to flourish, Hermann's contribution lay fallow for decades except mainly for the notice of a few gaps: • Condition (F): B.L. van der Waerden pointed out in [11] t h a t it is necessary to assume that one can completely factor an arbitrary polynomial over k. • Condition (P): A. Seidenberg pointed out in [10] that, in characteristic p, it is necessary to assume roughly the decidability of whether [kP(al,... ,as) : k;] = pS. • Condition (F'): M. Reufel pointed out in [9] that, in order to obtain a normal basis for a finitely-generated free module over a polynomial ring, one needs only the factorization of polynomials into prime powers. This is weaker than (F) in positive characteristic. • Numerical corrections: C. Veltzke, cf. [7], [8], (and later Seidenberg [10] and D. Lazard [5]) noted and removed numerical inaccuracies in Hermann's bounds. The conditions are vital in Hermann's manipulations of 'Elementarteilerformen' (Chow forms). Starting in mid-twentieth century, the work of W. Krull [4], A. FrShlich and J.C. Sheperdson [1], Reufel [9], and Lazard [5] made even more explicit that Herm a n n ' s computations give an algorithm. For more detailed historical remarks and a complete bibliography up to 1980, see [7], [8]. Nowadays (as of 2000), the main practical methods for computation in c o m m u t a t i v e algebra are implemented using Gr6bner bases (cf. also

GrSbner basis). However, even from a thoroughly modern point of view, Hermann's algorithm for linear algebra over R retains interest because it gives directly the correct order of magnitude of complexity of the fundamental membership problem over R. T h a t algorithm is embodied in the following basic result. Let aij E R have degree at most D, 1 < i < m, 1 < j < I. An R-basis for the solutions f = ( f x , . . . , fl) C R z of the related homogeneous system of equations

ailfl + ' " + a i l f l

=O,

i= l,...,m,

can be determined in a finite number of steps. T h a t basis will have entries whose degrees are bounded by a function B(m, D, n) satisfying the recursion

B ( m , D , n ) < mD + B ( m D + mD 2,D,n - 1 ) , B ( m , D , 1 ) < roD. To see this, let t be a new indeterminate over k. If f E R(t) 1 is a solution of the system, one can clear out the denominators to assume that a E R[t] 1. Then the coefficients of each fixed power of t give a solution. So a basis of solutions over k(t)[X1,..., Xn] leads to a basis of solutions over R, with the same bounds, and one can assume that k is infinite. 192

One may re-index the equations and unknowns, if necessary, to arrange t h a t the upper left (r x r)submatrix of coefficients has maximal rank r. Set all

•••

alr I

\at1

•••

art/

Now, since k is infinite, one m a y arrange by a change of variables Xi ~ Xi + c~iX~, ai E k, t h a t X~ occurs in A with exponent equal to deg A. Next one may apply Cramer's rule (cf. Cramer rule) to think of the original system of equations as being of the form:

Af~ = Ai,r+lf~+l + ' " + Ai,lfl, Aid c k,

i= l,...,r.

Subtracting as necessary multiples of the obvious solutions

(Ai,~+j,Ai+l,r+j,...,Ar,r+j;Aej),

j= 1,...,l-r,

where ej denotes the j t h standard basis element of k l-~, allows one to restrict the search for possible further solutions to those with degfj+~,...,degfl

< d e g A = rD.

Thus, for these remaining solutions one may bound the degrees of all fj with respect to Xn by rD, and hence one m a y think of the fj as linear combinations of X h, h < rD, with coefficients from R7~-1 := k[Xz,... ,Xn-1]. Setting to zero the coefficients of the resulting D + rD powers of X~ from the original system of equations gives a linear homogeneous system of at most m(D + rD) equations with coefficients from R ~ - I of degree at most D. Tracing through the argument verifies the recurrence. It is easy to verify that when n > 2,

S ( m , D , n ) < (2m(,~ + 1))~°-~D~-~. For consistent systems of inhomogeneous equations, Cramer's rule gives a particular solution, and the above procedure gives a basis for the related homogeneous system: One can determine in a finite number of steps whether a given system of R-linear inhomogeneous equations

aiifl + ' " + a i l f l

=bi,

i=l,...,m,

has a solution f E R I. If it does, one can be found in a finite number steps with m a x deg fj < B(m, D, n). Thus, the H e r m a n n algorithm gives explicit bounds for the ideal membership problem. According to the examples in [6], such bounds are necessarily doubly exponential. This is in contrast with the singly exponential bounds for the Hilbert Nullstellensatz (cf. E f f e c t i v e

Nullstellensatz).

HNN-EXTENSION

References [1] FROHLICH, A., AND SHEPERDSON, J.C.: 'Effective procedures in field theory', Philos. Trans. Royal Soc. A 248 (1956), 407432. [2] HENZELT, K.: 'Zur Theorie der Polynomideale und Resultanten, bearbeitet von Emmy Noether', Math. Ann. 88 (1923), 53 79. [3] HERMANN, G.: 'Die Frage der endlich vielen Schritte in der Theorie der Polynomideale', Math. Ann. 95 (1926), 736-788. [4] KRULL, W.: 'Parameterspezialisierung in Polynomringen', Archly Math. 1 (1948/49), 57-60. [5] LAZARD, D.: 'Alg~bre lin~aire sur K[X1,... ,Xn] et ~limination', Bull. Soc. Math. Prance 105 (1977), 165-190. [6] MAYR, E.W., AND MEYER, A.R.: 'Complexity of the word problems for commutative semigroups and polynomial ideals', Adv. Math. 46 (1982), 305-329. [7] RENSCHUCH, B.: 'Beitr~ge zur konstruktiven Theorie der Polynomideale XVII/1: Zur Henzelt/Noether/Hermannschen Theorie der endlich vielen Schritte', Wiss. Z. Pildagog. Hochsch. Karl Liebknecht, Potsdam 24 (1980), 87-99. [8] RENSCHUCH, B.: 'Beitr~ge zur konstruktiven Theorie der Polynomideale XVII/2: Zur Henzelt/Noether/Hermannschen Theorie der endlich vielen Schritte', Wiss. Z. Piidagog. Hochsch. Karl Liebknecht, Potsdam 25 (1981), 125-136. [9] REUFEL, M.: 'Konstruktionsverfahren bei Moduln fiber Polynomringen', Math. Z. 90 (1965), 231-250. [16] SEIDENBERG, A.: 'Constructions in algebra', Trans. Amer. Math. Soc. 197 (1974), 273-313. [11] WAERDEN, B.L. VAN DER: 'Eine Bemerkung fiber die Unzerlegbarkeit von Polynomen', Math. Ann. 102 (1930), 738 739. W. Dale Brownawell M S C 1991: 14Q20, 1 3 P x x

[3] RADHAKRISHNA, L.: 'History, culture, excitement, and relevance of mathematics', Rept. Dept. Math. Shivaji Univ. (1982). M. Hazewinkel M S C 1991: 04A99, 03E99 I-INN-EXTENSION

T h e easiest w a y to define a n H N N - g r o u p is in t e r m s of p r e s e n t a t i o n s of groups.

P r e s e n t a t i o n o f g r o u p s . A p r e s e n t a t i o n of a g r o u p G is a p a i r ( X : Y) w h e r e Y is a s u b s e t of F ( X ) , t h e free g r o u p on t h e set X , a n d G is i s o m o r p h i c (cf. also Isomorphism) to t h e q u o t i e n t g r o u p F ( X ) / N ( Y ) , w h e r e N ( Y ) is t h e i n t e r s e c t i o n of all n o r m a l s u b g r o u p s of F ( X ) c o n t a i n i n g Y (cf. also N o r m a l s u b g r o u p ) . T h e s u b g r o u p N ( Y ) is called t h e n o r m a l closure of Y in F ( X ) . See also P r e s e n t a t i o n . G i v e n an a r b i t r a r y g r o u p G, t h e r e is an obvious h o momorphism ~-G: F ( G ) --+ G such t h a t 7-a(g) = g for all g E G. Clearly, (G : ker(7-c)) is a p r e s e n t a t i o n for G. HNN-extensions.

H I L B E R T I N F I N I T E H O T E L , Hilbert paradox, infinite hotel paradox, Hilbert hotel - A nice i l l u s t r a t i o n of some of t h e s i m p l e r p r o p e r t i e s of ( c o u n t a b l y ) infinite sets. A n infinite hotel w i t h r o o m s n u m b e r e d 1, 2 , . . . can b e full a n d yet have a r o o m for an a d d i t i o n a l guest. Indeed, s i m p l y shift t h e existing guest in r o o m 1 to r o o m 2, t h e one in r o o m 2 to r o o m 3, etc. (in general, t h e one in r o o m n to r o o m n + 1), to free r o o m 1 for t h e newcomer. T h e r e is also r o o m for an infinity of new guests. Indeed, shift t h e existing guest in r o o m 1 to r o o m 2, the one in r o o m 2 to r o o m 4, etc. (in general, t h e one in r o o m n to r o o m 2n), to free all r o o m s w i t h o d d n u m b e r s for t h e newcomers. T h e s e e x a m p l e s i l l u s t r a t e t h a t an infinite set can be in bijective c o r r e s p o n d e n c e w i t h a p r o p e r subset of itself. T h i s p r o p e r t y is s o m e t i m e s t a k e n as a definition of infinity (the Dedekind definition of infinity; see also

Infinity). References [1] ERICKSON, G.W., AND FOSSA, J.A.: Dictionary of paradox, Univ. Press Amer., 1998, p. 84. [2] HERMES,

H., AND MARKWALD,

W.: 'Foundations

of mathe-

matics', in H. BEHNKE ET AL. (eds.): Fundamentals of Math-

ematics, Vol. 1, MIT, 1986, pp. 3-88.

- In 1949, G. H i g m a n , B.H.

N e u m a n n a n d H. N e u m a n n [4] p r o v e d several f a m o u s e m b e d d i n g t h e o r e m s for g r o u p s u s i n g a c o n s t r u c t i o n l a t e r called t h e H N N - e x t e n s i o n . T h e t h e o r y of H N N g r o u p s is c e n t r a l t o g e o m e t r i c a n d c o m b i n a t o r i a l g r o u p t h e o r y a n d s h o u l d b e viewed in p a r a l l e l w i t h a m a l g a m a t e d p r o d u c t s (cf. also A m a l g a m o f g r o u p s ) .

S u p p o s e # : A --+ B is an i s o m o r -

p h i s m of s u b g r o u p s of a g r o u p G a n d t is n o t in G. T h e H N N - e x t e n s i o n of G w i t h r e s p e c t t o # has p r e s e n t a t i o n

(C U {t}: (ker(7o)) U {t-la-lt#(a):

Va e d } ) .

T h e g e n e r a t o r t is called t h e stable letter, G t h e base group a n d A a n d B t h e associated subgroups of this H N N - e x t e n s i o n . W h e n A = G, t h e H N N - e x t e n s i o n is called ascending. Shorthand n o t a t i o n for ( G , t : t - l A t = B , # ) or G * , .

the

above

group

is

In [4] it was shown t h a t t h e m a p p i n g G --+ G * , t a k ing g + g for all g E G is a m o n o m o r p h i s m . T h e rest of t h e n o r m a l form t h e o r e m for H N N - e x t e n s i o n s was p r o v e d by J.L. B r i t t o n in 1963 [1] (Britton's lemma): Let go,. • •, gn be a sequence of elements of G a n d let t h e l e t t e r e, w i t h or w i t h o u t s u b s c r i p t s , d e n o t e 4-1. A sequence go, t ~1, g l , . •., t c~ , g~ will be called reduced if t h e r e is no consecutive s u b s e q u e n c e t - 1 , Hi, t with gi E A or t, g i , t -1 w i t h gi E B . F o r a r e d u c e d sequence a n d n _> 1, t h e e l e m e n t got¢l g 1 • . . t e ~ g n of G~ is different from t h e unit element. In t h e original reference [4], t h e following t h e o r e m is proved: E v e r y g r o u p G can be e m b e d d e d in a group 193

HNN-EXTENSION G* in which all elements of the same order are conjugate (cf. also C o n j u g a t e e l e m e n t s ) . In particulaL every torsion-free group can be embedded in a group G** with only two conjugacy classes. If G is countable, so is G**. Also, every countable group C can be embedded in a group G generated by two elements of infinite order. The group G has an element of finite order n if and only if C does. If C is finitely presentable, then so is G. For an excellent account of the history of HNNextensions, see [2]. See [7, Chap. IV/ for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the B a u m s l a g - S o l i t a r g r o u p (b, t : t - l b 2 t = b3) is non-Hopfian; cf. also N o n - H o p / g r o u p ) ; decompositions of 1-relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of a m a l g a m a t e d products and HNN-extensions; and Stallings' characterization of bipolar structures on groups. HNN-extensions are of central importance in, e.g., the modern version of the Van K a m p e n theorem (based on topological results in [6], [5]); the Bass-Serre theory of groups acting on trees and the theory of graphs of groups (see [9]); Dunwoody's accessibility theorem [3]; and JSJ decompositions of groups [8]. References [1] BRITTON,J.L.: 'The word problem', Ann. o/Math. 77 (1963), 16-32. [2] CHANDLER, B., AND MAGNUS, W.: The history of combinatorial group theory: A case study in the history of ideas, Vol. 9

of Studies History Math. and Phys. Sci., Springer, 1982. [3] DUNWOODY, M.J.: 'The accessibility of finitely presented groups', Invent. Math. 81 (1985), 449-457. [4] HIGMAN, G., NEUMANN, B.H., AND NEUMANN, H.: 'Embedding theorems for groups', J. London Math. Soc. 24 (1949),

247-254; II.4, 13. [5] KAMPEN, E.R. VAN: 'On some lemmas in the theory of groups', Amer. J. Math. 55 (1933), 268-273. [6] KAMPEN,E.R. VAN: 'On the connection between the fundamental groups of some related spaces', Amer. J. Math. 55 (1933), 261-267. [7] LYNDON,R., AND SCHUPP, P.: Combinatorial group theory, Springer, 1977. [8] 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. [9] SEHHE,J.P.: 'Arbres, amalgams, SL2', Astdrisque 46 (1977). Mike Mihalik MSC1991: 20F05, 20F06, 20F32

HOMOTOPY POLYNOMIAL - An invariant of oriented links (cf. also L i n k ) . It is a polynomial of two variables associated to homotopy classes of links in R 3, depending only on linking numbers between components ([1], cf. also K n o t t h e o r y ) . It satisfies the skein relation (cf. also C o n w a y skein triple) q - I H L + -- qHL_ = zHLo

for a mixed crossing. The h o m o t o p y polynomial of a link with diagram D is closely related to the dichromatic polynomial of the graph associated to D (cf. also G r a p h c o l o u r i n g ) . The h o m o t o p y polynomial can be generalized to homotopy skein modules of three-dimensional manifolds (cf. also S k e i n m o d u l e ) . References [1] PRZYTYCKI,J.H.: 'Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs': Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, March, 14-15, 1998), Internat. Press, 2000. Jozef Przytycki M S C 1991:57M25

HYERS-ULAM-RASSIAS STABILITY, H y e r s Ulam stability - In almost-all areas of mathematical analysis one can ask the following question: 'When is it true that a m a t h e m a t i c a l object satisfying a certain property approximately must be close to an object satisfying the property exactly?' If one applies this question to the case of functional equations, one can particularly ask when the solutions of an equation differing slightly from a given one must be close to a solution of the given equation. The stability problem of functional equations originates from such a fundamental question. In 1940, S.M. Ulam [31] raised a question concerning the stability of homomorphisms: Let G1 be a g r o u p and let G2 be a metric group with a m e t r i c d(.,.). Given s > 0, does there exist a 5 > 0 such that if a function h: G1 -+ G2 satisfies the inequality d ( h ( x y ) , h ( x ) h ( y ) ) < 5 for all x, y C G1, then there is a homomorphism H : G1 --+ G2 with d ( h ( x ) , H ( x ) ) < s for all x E G l ? In the following year 1941, D.H. Hyers [11] gave a partial solution to Ulam's question. H y e r s ' theorem says that if a function f : E1 --~/?72 defined between Banach spaces (cf. also B a n a c h s p a c e ) satisfies the inequality I i f ( x + y) - f ( x ) - f ( y ) l l < c

HOMFLY POLYNOMIALpolynomial.

See J o n e s - C o n w a y

for all x , y C E l , then there exists a unique additive function a: E1 -+ E2 with

Jozef Przytycki

I / / ( x ) - a(x)l] <

M S C 1991:57M25 for all x C E l . 194

HYERS-ULAM-RASSIAS STABILITY For the above case one says that the additive Cauchy equation f ( x + y) = f ( x ) + f ( y ) has the Hyers-Ulam stability on (El, E2). Hyers explicitly constructed the additive function a directly from the given function f by the formula a(x) =

lira 2 - n f ( 2 n x ) .

n---~oo

This method is called a direct method and is a powerful tool for studying the stability of several functional equations. It is often used to construct a certain function which is a solution of a given functional equation. Following its appearance, Hyers' theorem was further extended in various directions (see [3], [5], [7], [10], [14], [16], [22], [23], [27], [28], [29]). In particular, Th.M. Rassias [22] considered a generalized version of it in which the Cauchy difference is allowed to become unbounded. He assumed that a function f : E1 ~ E2 between Banach spaces satisfies the inequality Hf(x + y) - f ( x ) - f(y)ll < e(llxll p + Ilyll p) for some 0 _> O, 0 _< p < 1 and for all x,y E El. Functions such as f are called approximately additive functions. Using a direct method, he proved that in this case too there exists a unique additive function a: E1 -+ E2 such that IIf(x) - a(x)rl < K Irxll p

for all x E El, where K > 0 depends on p as well as on 0. This phenomenon of Hyers-Ulam-Rassias stability was later extended to all p ¢ 1 and generalizations of it were given (see [7], [10], [14], [16], [27], [17]). In general, Hyers-Ulam stability is a special case of HyersUlam-Rassias stability. S u p e r s t a b i l i t y . An equation involving homomorphisms is called superstable if each approximate homomorphism is actually a true homomorphism. For example, in [2] it is proved that if a real-valued function f defined on a vector space V satisfies the inequality [f(x + y) - f ( x ) f ( y ) [ ~ for some c > 0 and for all x , y E V, then either f is a bounded function or f ( x + y ) = f ( x ) f ( y ) for all x, y C V. This result was further generalized in [1] and [30] (cf. also [8], [9]). Superstability phenomena can also be regarded as special cases of Hyers-Ulam-Rassias stability. For results concerning the stability of other equations, see [4], [18], [19], [21], [26], or the references listed in [12], [20]. The survey papers [13], [6], [17], [25], [24] contain general information on stability. See [12], [20] for a comprehensive introduction to the general theory of Hyers-Ulam-Rassias stability of functional equations.

In the same vein there is a theory of almost isomorphisms of Banch algebras, e-perturbations of the multiplication and e-isometries of Banach algebras. See [15] for a selection of results. References [1] BAKER, J.: 'The stability of the cosine equation', Proc. Amer. Math. Soc. 80 (1980), 411-416. [2] BAKER, J., LAWRENCE, J., AND ZORZITTO, F.: 'The stability of the equation f ( x + y ) = f ( x ) f ( y ) ' , Proc. Amer. Math. Soc. 74 (1979), 242-246. [3] BORELLI, C., AND FORTI, G.L.: 'On a general Hyers-Ulam stability result', Internat. Y. Math. Math. Sci. 18 (1995), 229-236. [4] CZERWIK, S.: 'On the stability of the quadratic mapping in normed spaces', Abh. Math. Sere. Univ. Hamburg 62 (1992), 59 64. [5] FORTI, G.L.: 'The stability of homomorphisms and amenability with applications to functional equations', Abh. Math. Sem. Univ. Hamburg 57 (1987), 215-226. [6] FoawI, G.L.: 'Hyers-Ulam stability of functional equations in several variables', Aequat. Math. 50 (1995), 143-190. [7] CAJDA, Z.: 'On stability of additive mappings', Internat. J. Math. Math. Sci. 14 (1991), 431-434. [8] GER, R.: 'Superstability is not natural', Rocznik NaukowoDydaktyczny W S P w Krakowie, Prace Mat. 159 (1993), 109123. [9] GER, R., AND SEMRL, P.: 'The stability of the exponential equation', Proc. Amer. Math. Soc. 124 (1996), 779-787. [10] GAVRUTA, P.: 'A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings', Y. Math. Anal. Appl. 184 (1994), 431-436. [11] HYERS, D.H.: 'On the stability of the linear functional equation', Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. [12] HYERS, D.H., ISAC, G., AND RASSIAS, TH.M.: Stability of functional equations in several variables, Birkh~user, 1998. [13] HYERS, D.H., AND RASSIAS, TH.M.: 'Approximate homomorphisms', Aequat. Math. 44 (1992), 125-153. [14] ISAC, G., AND RASSIAS, TH.M.: 'On the Hyers Ulam stability of e-additive mappings', J. Approx. Th. 72 (1993), 131-137. [15] JAROSZ, K.: Perturbations of Banach algebras, Springer, 1985. [16] JUNG, S.-M.: 'On the Hyers-Ulam-Rassias stability of approximately additive mappings', J. Math. Anal. Appl. 204 (1996), 221-226. [17] JUNG, S.-M.: 'Hyers-Ulam-Rassias stability of functional equations', Dynamic Syst. Appl. 6 (1997), 541-566. [18] JUNG, S.-M.: 'Hyers-Ulam-Rassias stability of Jensen's equation and its application', Proc. Amer. Math. Soc. 126 (1998), 3137-3143. [19] JUNG, S.-M.: 'On the Hyers-Ulam stability of the functional equations that have the quadratic property', J. Math. Anal. Appl. 222 (1998), 126-137. [20] JUNG, S.-M.: Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, 2001. [21] KOMINEK, Z.: 'On a local stability of the Jensen functional equation', Demonstratio Math. 22 (1989), 499-507. [22] RASSIAS, TH.M.: 'On the stability of the linear mapping in Banach spaces', Proc. Amer. Math. Soc. 72 (1978), 297-300. [23] RASSIAS, TH.M.: 'On a modified Hyers-Ulam sequence', J. Math. Anal. Appl. 158 (1991), 106-113. [24] RASSIAS,TH.M.: 'On the stability of functional equations and a problem of Ulam', Acta Applic. Math. 62 (2000), 23-130.

195

H Y E R S - U L A M - R A S S I A S STABILITY [25] RASSIAS,

TH.M.:

'On the stability of functional equations

originated by a problem of Ulam', Studia Univ. Babes-Bolyai (to appear). [26] RASSIAS,TH.M.: 'On the stability of the quadratic functional equation', Mathematica (to appear). [27] RASSIAS,TH.M., AND SEMRL, P.: 'On the Hyers-Ulam stability of linear mappings', J. Math. Anal. Appl. 173 (1993), 325-338. [28] RASSIAS,TH.M., AND TABOR, J.: 'What is left of Hyers-Ulam stability?', J. Natural Geometry 1 (1992), 65-69. [29] SKOF, F.: 'Sull'approssimazione delle applicazioni localmente &additive', Atti Accad. Sci. Torino 117 (1983), 377-389.

[30] SZEKELYHIDI,

L.: 'On a theorem of Baker, Lawrence Zorzitto', Proc. Amer. Math. Soc. 84 (1982), 95-96.

and

[31] ULAM, S.M.: A collection of mathematical problems, Inter-

science, 1960. Soon-Mo Jung

M S C 1991: 39B72, 46B99, 46Hxx HYPERBOLIC CROSS - A s u m m a t i o n domain of multiple F o u r i e r s e r i e s (cf. also P a r t i a l F o u r i e r s u m ) . Let f ( x ) be an integrable periodic function of n variables defined on T n, T = ( - % 7r]. It has an expansion as a Fourier series, E k Ckeik'x, k = (kl, • • • , kn), X = ( X l , . . . , X ~ ) , k • x = k l x l + . . . + k~x~. Unlike in the one-dimensional case, there is no n a t u r a l ordering of the Fourier coefficients, so the choice of the order of s u m m a t i o n is of great importance. Let r = ( r l , . . . , r ~ ) E R " with all coordinates positive, rj > 0. Consider the d i f f e r e n t i a l o p e r a t o r D r = Orl+'"+rn/Orlxl "''O~nx~ with periodic boundary conditions on T n. T h e n the eigenvalues (cf. E i g e n

196

v a l u e ) of D r are )~k = irl+'"+r~k~l . . . . . k~~, while the corresponding eigenfunctions are e ik'x. T h e partial sums of the Fourier series c o r r e s p o n d i n g to the eigenfunctions with all eigenvalues ['~k{ ~ N are called hyperbolic partial Fourier s u m s of order N (or hyperbolic crosses). This approach, in which the m e t h o d of s u m m a t i o n of the Fourier series is defined by the differential operator, is due to K. B a b e n k o [1], who applied it to various problems in a p p r o x i m a t i o n t h e o r y (e.g., K o l m o g o r o v widths, c-entropy, etc.). Subsequently the hyperbolic cross itself b e c a m e the object of s t u d y in connection with Lebesgue constants, the BernshteYn inequality, etc. Also, this a p p r o a c h initiated a detailed s t u d y and applications of spaces of functions with b o u n d e d mixed derivative (in Lp). M a n y of these and related classes, as well as various problems in a p p r o x i m a t i o n theory, are considered in [2]. This m e t h o d of s u m m a t i o n has also been applied to other series expansions, e.g., multiple wavelets systems. References

[1] BABENKO,K.: 'Approximation of periodic functions of many variables by trigonometric polynomials', Soviet Math. 1 (1960), 513-516. (Dokl. Akad. Nauk. SSSR 132 (1960), 247-

250.) [2] TEMLYAKOV,V.: Approximation of periodic functions, Nova Sci., 1993. E.S. Belinsky

M S C 1991: 42B05, 42B08

I In connection with his work on the representation theory of groups (cf. also Schur funct i o n s in a l g e b r a i c c o m b i n a t o r i c s ) , I. Schur introduced the following class of matrix functions. For a character ) / o f a subgroup G of the symmetric group S~ (cf. also S y m m e t r i c g r o u p ; C h a r a c t e r of a group), the associated generalized matrix function dCx: C nxn -+ C acts on the (n x n)-matrix A = [aij] by IMMANANT

-

If one defines the normalized immanant -~ to be

n

dax(A) := ~

following spectacular generalization of the Hadamard Fischer inequality for positive semi-definite Hermitian matrices. Schur's inequality: Let X be a character of the subgroup G of Sn and let id be the identity permutation. For a positive semi-definite H e r m i t l a n m a t r i x A one has d~(A) > x(id) det(A). (1)

X(a) l-I ai~(0"

c~EG

--G

d x (A) := dGx(A)/x(id) = dCx(A)/d~(In),

i=l

When G = S~ and X = Xx is an irreducible character of S~ indexed by the partition A of n, D.E. Littlewood [24, Chap. VII called the matrix function dS~ an immanant. He used the immanants of certain matrices whose entries are symmetric functions to define the Schur functions (cf. also Schur f u n c t i o n s in algebraic combin a t o r i c s ) . The immanant dS~ is also denoted by the less cumbersome notation dx. The familiar determinant and permanent functions of a matrix (cf. also D e t e r m i n a n t ; P e r m a n e n t ) are examples of immanants, and they correspond to the immanants associated with the alternating, respectively the trivial, character of Sn. Indeed, for a matrix A =

n

d(l~)(A) = det(d) = E

sgn(a) I-I ai~(i),

~ESn

where In is the identity (n × n)-matrix, then Schur's inequality (1) may be written as --G

dx (d) >>_det(A) = d(ln)(A).

The permanental analogue of the Hadamard inequality for the determinant of a positive semi-definite Hermitian matrix was obtained by M. Marcus [25] in 1963, when he showed that for any positive semi-definite Hermitian matrix A = [aij], n

per(A) _> I ~ aii. i----1

In 1966, Marcus' inequality was generalized by E.H. Lieb [23], who showed that for a positive semi-definite Hermitian matrix A that is partitioned in the form A=

(B,

C ) , where C* = CT denotes the transpose

i=1

conjugate of C, one has where sgn(a) = 1 if a is an even permutation and - 1 otherwise (cf. also P e r m u t a t i o n ) , and

per(A) _> per(B) per(D) _> f l aii. i=1

d(,~)(A) : p e r ( A ) = E

~I ai~(O.

crES~ i = 1

Given the plethora of inequalities and identities that involve the determinant and permanent functions, it is natural to seek generalizations of these relations to other immanants. For instance, in 1918 Sehur [40] obtained the

In the same paper [23], the following permanental analogue of Schur's inequality was conjectured. The permanental dominance conjecture, or permanent-on-top conjecture (POT conjecture), states that for all positive semi-definite Hermitian matrices A, 3(n)(A) = per(A) > dx(A).

(2)

IMMANANT More generally, given two irreducible characters X~ and X~ of Sn, one writes Xx --<-gu if a~(A) < d , ( A ) for all positive semi-definite Hermitian matrices A. This gives a p a r t i a l o r d e r on the characters and also the immanants. In this context, Schur's inequality says that the alternating character X(ln) is the smallest element of the resulting partially ordered set and the permanenton-top conjecture asserts that the trivial character X(,~) is the largest element. A clear exposition on this topic can be found in [30, Chap. 7]. There was little work done in this area before the appearance of a series of papers [12], [18], [20], [21], [32] by several authors in the mid 1980s. These papers provide a partial resolution to the conjectured inequality in (2), by showing that the permanent dominates the immanants associated with various characters. Since then there has been much progress; see, e.g., [13], [14], [22], [28], [41]; and especially [16], [17], [19], [aa], [34], [35], [36], [37], [38], for the nature of the partial order on the characters. For instance, P. Heyfron [16] proved that the hook immanants d(k,ln-k) are ordered as d(1 n ) -~ d(2,1n--2) ~- "' ' "~ 3(k,ln--k ) -~ " "" -~ d(n ) .

(3)

The ordering in (3) is part of a more general result of Th.H. Pate [33] saying that if A = ( A , , . . . , A~,..., At) is a non-increasing partition of n with ),, > max{A,+l, 1} and A' = ( A i , . . . , A , - 1, A , + I , . . . , A t , 1), then Xx' -~ Xx. The P O T conjecture has been established for all n < 15 with the exception of the three partitions (42, 32), (35) and (5, 4, 32) by Pate in [38]. The article [38] gives a good survey of the results related to the P O T conjecture and also discusses some topics related to immanants. In [41], G.W. Soules considered an alternative approach to the P O T conjecture through the eigenvalues of the Schur power matrix. In a similar vein, R. Merris [28] investigated the extent to which the P O T conjecture depends on the theory of group representations. Immanants of other families of matrices besides the positive semi-definite Hermitian matrices have been studied. In [10] I.P. Goulden and D.M. Jackson studied immanants of Jacobi-Trudi matrices and incidence matrices of directed trees. Their work inspired further results and conjectures about immanants of Jacobi-Trudi matrices and totally positive matrices, see, e.g., [11], [15], [42], [44]. Immanants of combinatorial matrices, such as the Laplacian matrix of graphs, have been considered in [3], [5], [6], [29]. Given a g r a p h G on n vertices {Vl,..., v,~}, the Laplacian matrix L(G) = [lij] of G is the (n x n)matrix whose diagonal element lii is the degree of the vertex vi and for i ~ j, l~j = - 1 or 0 according to whether vi and vj are adjacent or not. Given two graphs G1 and G2 on n vertices, it is of interest to determine the 198

set of immanants dx for which d~(L(G1)) < d),(L(G2)). See [3], [6], for a characterization of the trees in various families that attain the smallest immanant values. See [7] for a linear relation between the third hook immanant d(a,l~-a)(L(T)) of the tree T and the chemical index, called the Wiener number of T. The relation between d(a,>-a)(L(T)) and other chemical indices is explored in [4]. The results of Ph. Botti and Merris in [1] show that immanants of Laplacian matrices of trees do not distinguish non-isomorphic trees in general. More specifically, almost every tree shares the same set of immanant values with another non-isomorphic tree on the same number of vertices. Linear transformations

of a matrix

that preserve its

immanant values have been investigated in [8], [9], [39]. When an indeterminate x is introduced, the immanant dx (x[~ - A) may be expanded as a polynomial in x of degree n. This polynomial is known as the immanantal polynomial of A. The immanantal polynomials of graph Laplacians have been studied in [1], [26], [27], [311, [45] Computational issues related to the evaluation of immanants are considered in [2]. References [1] BOTTI, PH., AND MERRIS, R.: 'Almost all trees share a complete set of immanantal polynomials', J. Graph Th. 17, no. 4 (1993), 467-476. [2] BORGISSER, P.: 'The computational complexity of immanants', S I A M J. Comput. (to appear). [3] BaUALDI, R.A., AND GOLDWASSER, J.L.: 'Permanent of Laplacian matrix of trees and bipartite graphs', Discr. Math. 48 (1984), 1-21. [4] CHAN, O., GUTMAN, I., LAM, TAO-KAI, AND MERRIS, R.: 'Algebraic connections between topological indices', J. Chemical Inform. ~ Computer Sci. 38, no. 1 (1998), 62-65. [5] CHAN, O., AND LAM, T.K.: 'Hook immanantal inequalities for trees explained', Linear All. ~ Its Appl. 273 (1998), 119-131. [6] CHAN, O., AND LAM, T.K.: ' I m m a n a n t inequalities for Laplacians of trees', S I A M J. Matrix Anal. Appl. 21, no. 1 (1999), 129-144. [7] CHAN, O., LAM, T.K., AND MERRIS, m.: 'Wiener number as an i m m a n a n t of the Laplacian of molecular graphs', J. Chemical Inform. ~ Computer Sci. 87, no. 4 (1997), 762-765. [8] COELHO, M.P., AND DUFFNER, M.A.: 'Linear preservers of immanants on symmetric matrices', Linear All. ~ Its Appl. 255 (1997), 315-334. [9] DUFFNER, M.A.: 'Linear transformations that preserve immanants', Linear All. ~ Its Appl. 1 9 7 / 1 9 8 (1994), 567-588. [10] GOULDEN, I.P., AND JACKSON, D.M.: 'Immanants of combinatorial matrices', J. Algebra 148 (1992), 305-324. [11] GREENE, C.: 'Proof of a conjecture on immanants of the Jacobi Trudi matrix', Linear Alg. ~ Its Appl. 171 (1992), 65-79. [12] GRO~E, R.: 'An inequality for the second immanant', Linear and Multilinear Algebra 18 (1985), 147-152. [13] CRONE, R., AND MERRIS, R.: 'A Hadamard inequality for the third and fourth i m m a n a n t s ' , Linear and Multilinear Algebra 21

(1987), 201-209.

INCLUSION-EXCLUSION [14] GRONE, R., MERRIS, R., AND WATKINS, W.: 'A Hadamard dominance theorem for a class of immanants', Linear and Multilinear Algebra 19 (1986), 167-171.

[15] HAIMAN, M.: 'Hecke algebra characters and immanant conjectures', J. Amer. Math. Soc. 6 (1993), 569-595. [16] HEYFRON, P.: 'Immanant dominance orderings for hook partitions', Linear and Multilinear Algebra 24, no. 1 (1988), 6578. [17] HEYFRON, P.: 'Some inequalities concerning immanants', Math. Proc. Cambridge Philos. Soc. 109 (1991), 15 30. [18] JAMES, G.D.: 'Permanents, immanants, and determinants': Proc. Sympos. Pure Math., Vol. 47, Amer. Math. Soc., 1987, pp. 431-436. [19] JAMES, G.D.: 'Hecke algebras and immanants', Linear Alg. Its Appl. 1 9 7 / 1 9 8 (1994), 659-670. [20] JAMES, G.D., AND LIEBECK, M.W.: 'Permanents and iramananas of Hermitian matrices', Proc. London Math. Soc. 55, no. 3 (1987), 243-265. [21] JOHNSON, CH.R.: 'The permanent-on-top conjecture: a status report', in F. UHLIG AND R. CRONE (eds.): Current Trends in Matrix Theory, Elsevier, 1987.

[22] JOHNSON, CH.R., AND PIERCE, STEPHEN: 'Inequalities for

[23] [24] [25]

[26] [27]

[28]

[29] [30] [31]

[32]

single-hook immanants', Linear Alg. ~ Its Appl. 102 (1988), 55-79. LmB, E.H.: 'Proofs of some conjectures on permanents', Y. Math. Mech. 16, no. 2 (1966), 127-134. LITTLEWOOD, D.E.: The theory of group characters, second ed., Oxford Univ. Press, 1950. MARCUS, M.: 'The permanent analogue of the Hadamard determinant theorem', Bull. Amer. Math. Soc. 69 (1963), 494496. MERRIS, R.: 'The Laplacian permanental polynomial for trees', Czech. Math. J. 32 (1982), 397-403. MERRIS, R.: 'The second immanantal polynomial and the centroid of a graph', S I A M J. Algebraic Discr. Meth. 7, no. 3 (1986), 484-497. MERRIS, R.: 'The permanental dominance conjecture', in F. UHLIG AND R. GRONE (eds.): Current Trends in Matrix Theory, Elsevier, 1987. MERRIS, R.: 'Laplacian matrices of graphs: A survey', Linear AIM. ~ Its Appl. 1 9 7 / 1 9 8 (1994), 143-176. MERRm, R.: Multilinear algebra, Gordon & Breach, 1997. MERRIS, R., REBMAN, K.R., AND WATKINS,W.: 'Permanental polynomials of graphs', Linear AIM. ~ Its Appl. 38 (1981), 273-288. MERRIS, R., AND WATKINS, W.: 'Inequalities and identities for generalized matrix functions', Linear Alg. ~ Its Appl. 64

(1985), 223-242. [33] PATE, TH.H.: 'Immanant inequalities and partition node diagrams', J. London Math. Soc. 46 (1991), 65-80. [34] PATE, TH.H.: 'Partitions, irreducible characters and inequalities for generalized matrix functions', Trans. Amer. Math. Soc. 325 (1991), 875-894. [35] PATE, TH.H.: 'Descending chains of immanants', Linear Alg. Its Appl. 1 6 2 / 1 6 4 (1992), 639-650. [36] PATE, TH.H.: 'Psi-functions, irreducible characters and a conjecture of Merris and Watkins', Linear and Multilinear Algebra 35 (1993), 195-212. [37] PATE, TH.H.: 'Immanant inequalities, induced characters, and rank two partitions', J. London 40-60.

Math.

Soc. 49 (1994),

FORMULA

[38] PATE, TH.H.: 'Row appending maps, ~-functions, and iramananA inequalities for Hermitian positive semi-definite matrices', Proc. London Math. Soc. 76, no. 3 (1998), 307-358. [39] ROSARIOFERNANDES,M. DO: 'Pairs of matrices that have the same immanant', Linear and Multilinear Algebra 40 (1996), 193-201. [40] SCHUR, I.: @ber endlicher Gruppen und Hermiteschen Formen', Math. Z. 1 (1918), 184-207. [41] SOULES, G.W.: 'An approach to the permanental-dominance conjecture', Linear Al 9. ~4 Its Appl. 201 (1994), 211-229.

[42] STANLEY,

R.P., AND STEMBRIDGE, J.R.: 'On immanants of Jacobi Trudi matrices and permutations with restricted position', g. Combin. Th. A 62 (1993), 263-279. [43] STEMBRIDGE, J.R.: 'Immanants of totally positive matrices are non-negative', Bull. London Math. Soc. 23 (1991), 422428. [44] STEMBRIDGE, J.R.: 'Some conjectures for immanants', Canad. J. Math. 44 (1992), 1079-1099. [45] TURNER, J.: 'Generalized matrix functions and the graph isomorphism problem', SIAM Y. Appl. Math. 16 (1968), 520526. O. Chan

M S C 1991: 15A15, 2 0 C 3 0

INCLUSION-EXCLUSION exclusion principle,

FORMULA,

inclusion-exclusion

inclusion-

method

- The

i n c l u s i o n - e x c l u s i o n p r i n c i p l e is u s e d in m a n y b r a n c h e s of p u r e a n d a p p l i e d m a t h e m a t i c s . I n p r o b a b i l i t y

the-

o r y it m e a n s t h e f o l l o w i n g t h e o r e m : Let A 1 , . • •,

An b e

e v e n t s in a p r o b a b i l i t y

Sk =

space and

(1)

P(Ai, n . . . n Ai~),

E l <_il <...
k=l,...,n.

Then

one has the relation

P(A1 U . . . U An) = S1 - S2 + " "

+ (--1)n--lSn.

(2)

This theorem can easily be proved by induction on n . A n o t h e r p r o o f c a n b e g i v e n b y t h e use of i n d i c a t o r v a r i a b l e s , as follows. I n c o n n e c t i o n w i t h a n e v e n t A o n e defines I A = 1 if A o c c u r s , a n d I A = 0 o t h e r w i s e . T h e n the equation

]AIU'"UAn=£ k=l

E

IAil'''IAik

(3)

l
holds. T a k i n g e x p e c t a t i o n s o n b o t h sides, t h e t h e o r e m follows. If o n e w a n t s to find t h e p r o b a b i l i t y of A1 N .. • N A n , t h e n first o n e o b s e r v e s t h a t P(A1 N . . . N A n ) = 1 - P(A1 U • '" U A n ) a n d t h e n uses (2) for t h e e v e n t s A 1 , . . . , A n . T h e t h e o r e m is f r e q u e n t l y a t t r i b u t e d to H. P o i n c a r ~ [8]. However, it was a l r e a d y k n o w n to A. De M o i v r e [6] a n d t h e e v e n m o r e g e n e r a l f o r m u l a for t h e p r o b a b i l i t y t h a t at least r e v e n t s o c c u r h a s b e e n f o u n d b y C. J o r d a n [3]. See also [4] for f u r t h e r references. All these f o r m u l a s c o n c e r n B o o l e a n f u n c t i o n s of r a n d o m events.

199

INCLUSION-EXCLUSION FORMULA An example for the application of formula (2) is the solution of the classical problem of coincidences proposed by P.R. Montmort [7]: A box contains n cards marked 1 , . . . , n and all cards are drawn, one by one, without replacement, where each sequence has the same probability; one says that a coincidence occurs at the ith drawing, or event A i occurs, if the card drawn is marked i; the problem is to find the probability that at least one coincidence occurs (cf. also M o n t m o r t m a t c h i n g p r o b l e m ) . One easily sees that for any 1_
(n - k)! n!

n!

and

n P(A1

n...

n An) = E ( - 1 )

1

k=l

+''-+IA~,i.e.,

X is the number of events which occur. One says that Sk is the kth binomial m o m e n t of X. For a proof of the above equation, see, e.g., [11]. There are m a n y practical applications where one needs to compute the probability of a union, or other Boolean function of events. Prominent are those in reliability theory. For example, in a communication network, where the links randomly work or fail, one m a y want to find a two-terminal reliability or the all-terminal reliability. In the former case one has to find the probability that all links in at least one p a t h connecting the two terminal nodes work, and in the latter case the probability that all links in at least one spanning tree work. In both cases the number of random events is too large to apply inclusion-exclusion. In fact, one may be able to compute only a few of the binomial moments involved. In such cases one looks for bounds. The simplest ones, for the probability of the union, are the B o n f e r r o n i bounds: P(A1 U . . . U A~) _> $1

-

$2 + . - . + S,~_~ - S,~

if m is even, and P(A1 U - . . U A~) < S1 - $2 + - " + S,~-2 - Sm_x + S,~ if m is odd, where m < n (cf. also B o n f e r r o n i ine q u a l i t i e s ) . However, in m a n y cases these bounds are not sufficiently tight. A general, efficient method for bounding P(A1 U. • • U A,~) and other Boolean functions of events has been worked out by A. Pr~kopa (see [10] and the references therein). In the case of bounding the probability of the o

union one observes that Sk = E = E i = l (k)Pi, where p~ is the probability that exactly i events occur. 200

E i : i Pi

subject to

E i = l (~)Pi = S~, k = 1 , . . . , r n ,

(4)

Pi _> 0, i = 1 , . . . , n . and Pmax are the o p t i m u m values corresponding to the min and m a x problems, respectively, then these are sharp bounds for the probability of the union, i.e. If Pmin

Pmin ~ P(A1 U . . . U AN) _< Pmax

m

m

k=l

k~l

k-1

The numbers Sk have another probabilistic interpretation:Sk :E[(X)],whereX:IA~

min(max)

and no better bounds can be given based on the knowledge of S 1 , . . . , S m . There are simple and efficient algorithms to solve problems (4) which are variants of the dual algorithm of linear programming. The sharp bounding formulas can also be written as

'

hence Sk =

Then, if one knows S 1 , . . . , S m , where m < n, but the Pi are unknown, one solves the linear programs:

where x = ( X l , . . . , x , ~ ) T and y = ( Y l , . . . , Y , ~ ) T are optimal solutions of the duals of the min and m a x problems, respectively. It has been shown that the components of x and y have alternating signs, starting with +, and Ixll __ .-. _> I x m l , lY I --> "" --> lYml. The optimum values can be expressed by formulas if m _< 4 (see the references in [10]). Formula (2) holds for any finitely additive set function # defined on an algebra S of some subsets of f~ (if

A ¢ S , thenA¢S, andifA, B E S , thenAOBES). In particular, one may take f~ an arbitrary finite set, 8 all subsets of f~ and #(A) = IAI, the cardinality of the set A. The latter case has m a n y applications in combinatorics, especially in enumeration problems. A good sample of combinatorial problems, where inclusion-exclusion is used, is presented in [5]. Inclusion-exclusion plays also an important role in number theory. Here one calls it the sieve f o r m u l a or sieve method. In this respect, V. Brun did pioneering work [1] (cf. also S i e v e m e t h o d ; B r u n sieve). For a s u m m a r y of the sieve methods, see [9] and [2]. A simple but revealing example is the formula for ~(n), the number of those positive integers t h a t are relative prime to, and smaller than, n. If n has prime divisors P l , . . . ,Pk, then the inclusion-exclusion principle gives: =

. . .n . . . . . .

Px +

n

PiP2 n

. . . . .

PlP2P3

n

+

Pk n

+ ... + - - - F Pk-lPk

+ (-I)

n

Pl • • •Pk

References [1] BRUN, V.: 'Le crible d ' E r a t o s t h ~ n e et le th~or~me de Goldbach', S k r i f t e r N o r s k e V i d . - A k a d . K r i s t i a n a I 3 (1920).

INDEX T H E O R Y [2] HALBERSTAM, H., AND P~ICHERT, H.E.: Sieve methods, Acad. Press, 1974. [3] JORDAN, C.: 'De quelques formules de probabilitY', C.R. Acad. Sci. Paris 65 (1867), 993-994. [4] JORDAN, K.: Chapters on the classical calculus of probability, Akad. Kiad6, 1972. [5] Lov~.sz, L.: Combinatorial problems and exercises, Akad. Kiad6, 1993. [6] MOIVRE, A. DE: Doctrine of chances, London, 1718. [7] MONTMORT, P.R.: Essai d'analyse sur les jeux de hazard, Paris, 1708. [8] POINCAR~, H.: Calcul des probabilitds, Ganthier-Villars, 1896. [9] PRACHAR, K.: Primzahlverteilung, Springer, 1957. [10] PR~KOPA, A.: Stochastic programming, Kluwer Acad. Publ., 1995. [11] TAKAeS, L.: 'On the method of inclusion and exclusion', Y. Amer. Statist. Assoc. 62 (1967), 102-113.

Andrds Prdkopa MSC1991: 60A99, 60E15, 05A99, 11N35 The area of mathematics whose main object of study is the index of operators (cf. also Index of an operator; Index formulas). The main question in index theory is to provide ind e x f o r m u l a s for classes of Fredholm operators (cf. also F r e d h o l m o p e r a t o r ) , but this is not the only interesting question. First of all, to be able to provide index formulas, one has to specify what meaning of 'index' is agreed upon, then one has to specify to what classes of operators these formulas will apply, and, finally, one has to explain how to use these formulas in applications. A consequence of this is that index theory also studies various generalizations of the concept of Fredholm index, including K-theoretical and cyclic homology indices, for example. Moreover, the study of the analytic properties necessary for the index to be defined are an important part of index theory. Here one includes the study of conditions for being Fredholm or non-Fredholm for classes of operators that nevertheless have finitedimensional kernels. Soon after (1970s), other invariants of elliptic operators have been defined that are similar in nature to the analytic index. The study of these related invariants is also commonly considered to be part of index theory. The most prominent of these new, related invariants are the Ray-Singer analytic torsion and the e t a - i n v a r i a n t . Fixed-point formulas are also usually considered part of index theory, see [4]. Finally, one of the most important goals of index theory is to study applications of the index theorems to geometry, physics, group representations, analysis, and other fields. There is a very long and fast growing list of papers dealing with these applications. Index theory has become a subject on its own only after M.F. Atiyah and I. Singer published their index theorems in the sequence of papers [10], [11], [12] (of. INDEX

THEORY

-

also I n d e x f o r m u l a s ) . These theorems had become possible only due to progress in the related fields of K t h e o r y [2], [9] and pseudo-differential operators (cf. also 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 ) [34], [36], [47]. Important particular cases of the Atiyah-Singer index theorems were known before. Among them, Hirzebruch's signature theorem (cf. also S i g n a t u r e ) occupies a special place (see [32], especially for topics such as multiplicative genera and the Langlands formula for the dimension of spaces of automorphic forms). Hirzebruch's theorem was generalized by A. Grothendieck (see [22]), who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the AtiyahSinger index theorems (see also I n d e x f o r m u l a s for some index formulas that preceded the Atiyah-Singer index formula). A t i y a h - S i n g e r i n d e x f o r m u l a s . A common characteristic of the first three main index formulas of AtiyahSinger and Atiyah-Segal is that they depend only on the principal symbol of the operator whose index they compute. (For a differential operator, the principal symbol is given by the terms involving only the highest-order differentials and is independent of the choice of a coordinate system; cf. also P r i n c i p a l p a r t o f a d i f f e r e n t i a l o p e r a t o r ; S y m b o l o f a n o p e r a t o r . ) The main theorems mentioned above are: • the index theorem for a single elliptic operator P acting between sections of vector bundles on a smooth, compact manifold M (Atiyah-Singer, [10]); • the equivariant index theorem for a single elliptic operator equivariant with respect to a compact group G (Atiyah Segal, [9]); and • the index theorem for families (Pb)beB of elliptic operators acting on the fibres of a fibre bundle Y -~ B ( Atiyah-Singer, [12]). These results are briefly reviewed below.

A single elliptic operator acting between sections of vector bundles. If P is an elliptic differential, or, more generally, an elliptic 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 acting between sections of two smooth vector bundles (cf. also E l l i p t i c o p e r a t o r ) , then P defines a continuous operator between suitable Sobolev spaces with closed range and finite-dimensional kernel and cokernel, that is, a F r e d h o l m o p e r a t o r . The first of the index theorems gives an explicit formula for the Fredholm, or analytic, index ind(P) of P: ind(P) := dim(ker(P)) - dim(coker(P)). Denote by T ( M ) the T o d d class of the complexification of the tangent bundle T M of M. If P is an elliptic 201

INDEX T H E O R Y operator as above, its principal symbol a = a ( P ) defines a K - t h e o r y class [a] with compact supports on T*M whose C h e r n c h a r a c t e r , denoted Ch([a]), is in the even c o h o m o l o g y of T*M with compact supports. The Atiyah Singer index formula of [11] then states that ind(P) = ( - 1 ) n Ch([a])T(M)[T*M], n being the dimension of the manifold M and [ T ' M ] being the f u n d a m e n t a l class of T*M. (The factor ( - 1 ) n reflects the choice of the o r i e n t a t i o n of T*M in the original articles. Other choices for this orientation will lead to different signs.) In other words, the index is obtained by evaluating the compactly supported cohomology class Ch([a])T(M) on the fundamental class of

T*M. Equivariant indez theorem. The second of the index for-

convolution algebra of G, in the equivariant index theorem. For the C h e r n c h a r a c t e r of the family index of a family of elliptic operators (Pb) as above, there is a formula similar to the fornmla for the index of a single elliptic operator. The principal symbols ab = a(Pb) of the operators Pb define, in this case, a class [a] in the K - t h e o r y with compact supports of T,~ertY := T*Y/Tr*(T*B), the vertical cotangent bundle to the fibres of 7r: Y --+ B, as in the case of a single elliptic operator. Denote by T(MIB ) the Todd class of the complexification of T¢ertY and by 7r. : H*(Tv*ertY) --~ H * - 2 n ( B ) the morphism induced by integration along the fibres, with n being the common dimension of the fibres of 7r. Then C h ( i n d ( r ) ) -- ( - 1 ) ~ r . (ind([a])T(MlB)). This completes the discussion of these three main theorems of Atiyah and Singer.

mulas refines the index when the operator P above is invariant with respect to a compact Lie g r o u p , see [9], [11]. Recall that the representation ring of a compact group G is defined as the ring of formal linear combinations with integer coefficients of equivalence classes of irreducible representations of G (cf. also I r r e d u c i b l e r e p r e s e n t a t i o n ) . For operators P equivariant with respect to a compact group G, the kernel and cokernel are representations of G, so their difference can now be regarded as an element of R(G), called the equivariant indez of P. The Atiyah-Singer index formula in [11] gives the value indg(P) of the (character of the) index of P at g C G in terms of invariants of M g, the set of fixed points of g in M. Denote by a[T*Mo the restriction of a to the cotangent bundle of M~ and by T(M g) the Todd class of the complexification of the cotangent bundle of Mg. In addition to these ingredients, which are similar to the ingredients appearing in the formula for ind(P) above, the formula for indg (P) involves also a Lefschetz-type contribution, denoted below by L(N, g), obtained from the action of g on the normal bundle to the set Mg:

K - t h e o r y in i n d e x t h e o r y . The role of K-theory in the proof and applications of the index theorems can hardly be overstated and certainly does not stop at providing an interpretation of the index as an element of a K-theory group. A far-reaching consequence of the use of K-theory, which depends on Bott periodicity (or more precisely, the T h o r n i s o m o r p h i s m , cf. also B o t t per i o d i c i t y t h e o r e m ) , is that all elliptic operators can be connected, by a homotopy of Fredholm operators, to certain operators of a very particular kind, the so-called generalized Dirac operators (see below). It is thus sufficient to prove the index theorems for generalized Dirac operators. Due to their differential-geometric properties, it is possible to give more concrete proofs of the AtiyahSinger index theorem for generalized Dirac operators, using heat kernels, for example (cf. also H e a t c o n t e n t a s y m p t o t i c s ) . The generalized Dirac operator with coefficients in the spin bundle is called simply the Dirac operator (sometimes called the Atiyah-Singer operator). See below for more about generalized Dirac operators.

indg (P) = ( - 1 ) ~ Ch([aIT.M~])T(Mg)L(N, g)[T*Mq.

A p p l i c a t i o n s o f i n d e x t h e o r e m s . After the publication of the first papers by Atiyah and Singer, index theory has evolved into essentially three directions:

Families of elliptic operators. For families of elliptic operators acting on the fibres of a fibre bundle 7r: Y -+ B (cf. also F i b r a t i o n ) , a first problem is to make sense of the index. The solution proposed by Atiyah and Singer in [12] is to define the index as an element of a K-theory group, namely K °(B) in this case (cf. also K - t h e o r y ) . This fortunate choice has opened the way for many other developments in index theory. Actually, in the two index theorems mentioned above, the index can also be interpreted using a K-theory group, the K-theory of the algebra C of complex numbers in the first index theorem and the K-theory group of C* (G), the norm closure of the 202

• a direction which consists of applications and new proofs of the index theorems (especially 'local' proofs using heat kernels); • a direction which studies invariants other than the index; and • a direction which aims at more general index theorems. There is a very large number of applications of index theorems to topology and other areas of mathematics. A few examples follow. In [8], Atiyah and W. Schmid used Atiyah's L2-index theorem for coverings [3] to construct

INDEX

discrete series representations. In [33], N.J. Hitchin used the families index theorem to prove that there exist metrics whose associated Dirae operators have non-trivial kernels (in suitable dimensions). An index theorem for foliations that is close in spirit to Atiyah's L2-index theorem was obtained by A. Connes [25]. The index of Dirac (or Atiyah-Singer) operators was used to formulate and then prove the Gromov-Lawson conjecture [31], which states that a compact, spin, simply connected manifold of dimension > 5 admits a metric of positive scalar curvature if and only if the index of the spin Dirac operator (in an appropriate K-theory group) is zero. This conjecture was proved by S. Stolz, [49]. Dirac operators have been used to give a concrete construction of K-homology [13]. Some of the applications of the index theorems require new proofs of these theorems, usually relying on the 'heat-kernel method'. The main idea of this method is as follows. H. McKean and Singer [38] stated the problem of investigating the behaviour, as t -~ 0, t > 0, of the (super-trace of the) heat kernel. More precisely, let

kt(x, y) = str(e -tD2) = tr(e -tDJ~D+) - t r ( e -tD+D; ) be the well-known term appearing in the McKeanSinger index formula, where D = D+ + D~_ is a selfadjoint geometric operator (cf. also Self-adjoint operator) with D+ mapping the subspace of even sections to the subspace of odd sections. They considered the case of the de Rham operator D+ + D~, where D+ is then the de Rham differential (cf. also de R h a m cohom o l o g y ) . It was known that the integral over the whole manifold of kt(z, x) gives the analytic index of D+, and they expressed the hope that kt (z, z) will have a definite limit as t -+ 0. This was proved for various particular cases by V. Patodi in [43] and then by P. Gilkey [29], [28] using invariant theory (see [30] for an exposition of this method). This method was finally refined in [5] to give a clear and elegant proof of the local index theorem for all Dirac operators. Inspired by a talk of Atiyah, J.-M. Bismut investigated connections between p r o b a b i l i t y t h e o r y and index theory. He was able to use the stochastic calculus (ef. also M a l l i a v i n calculus) to give a new proof of the local index theorem [15]. His methods then generalized to give proofs of the local index theorem for families of Dirac operators [17] using Quillen's theory of super-connections [44], and of the Atiyah-Bott fixedpoint formulas [16]. An application of his results is the determination of the Quillen metric on the determinant bundle [19]. The local index theorems have many connections to physics, where Dirae operators play a prominent role.

THEORY

Actually, several physicists have come up with arguments for a proof of the local index theorem based on supersymmetry and functional integration, see [i] and [52], for example. Building on these arguments, E. Getzler has obtained a short and elegant proof of the local index theorem [14], [30], which also uses supersymmetry. Moreover, ideas inspired from physics have lead E. Witten to conjecture that certain twisted Dirac operators on St-manifolds have an index that is a trivial representation of S I, see [54]. This was proved by C.H. Taubes [50] (see also [23] and [53]). For the Dirac operator, this had been proved before by Atiyah and F. Hirzebruch [6]. Other invariants. Heat-kernel methods have proved very useful in dealing with non-compact and singular spaces. A common feature of these spaces is that the index formulas for the natural operators on them depend on more than just the principal symbol, which leads to the appearance of non-local invariants in these index formulas. In general, there exists no good understanding, at this time (2000), of what these non-local invariants are, except in particular cases. The most prominent of these particular cases is the Atiyah-Patodi-Singer index theorem for manifolds with boundary. Other results in these directions were obtained in [24], [39], [41], [48]. In all these cases, eta-invariants of certain boundary operators must be included in the formula for the index. Moreover, one has to either work on complete manifolds or to include boundary conditions to make the given problems Fredholm. The Atiyah-Patodi-Singer index theorem [7], e.g., requires such boundary conditions; see below. Let M be a compact manifold with boundary cgM and metric g which is a product metric in a suitable cylindrical neighbourhood of OM. Fix a Clifford module W on M (cf. also C l i f f o r d a l g e b r a ) and an admissible c o n n e c t i o n V. Denote by D := ~ e(ei)V~ the generalized Dirac operator on W, where c: T*M ~- T M End(W) is the Clifford multiplication and ei is a local orthonormal basis (cf. also O r t h o g o n a l basis). Also, let Do be the corresponding generalized Dirac operator on OM, which is (essentially) self-adjoint because OM is compact without boundary. Then the eigenvalues of Do will form a discrete subset of the real numbers; denote by P+ the spectral projection corresponding to the eigenvalues of Do that are >_ 0. Decompose D = D+ + D~_ using the natural Z/2Z-grading on W. The operator D+, the chiral Dirac operator, acts from sections of W+ to sections of W_, and has an infinite-dimensional kernel. Because of that, Atiyah, Patodi and Singer have introduced a non-local boundary condition of the form P+f = 0, for f a smooth section of W+ over OM, which is a compact perturbation of the Calderdn projection boundary condition. The effect of this boundary condition is that the restriction of D to the subspace of

203

INDEX THEORY

sections satisfying this boundary condition is Fredholm. Assume that M is spin e with spinor bundle S, such that W = S ® E , and let h denote the dimension of the kernel of Do. The index of the resulting operator D+ with the above boundary conditions is then inda(D+) = /M .4(M) Ch(E)

rl(D°)2+ h

This formula was generalized by Bismut and J. Cheeger in [18] to families of manifolds with boundary, the result being expressed using the 'eta form' ~. More precisely, using the notation above, they proved that Ch(inda(D+)) = 7r.(A(M) Ch(E))

2' provided that all Dirac operator associated to the boundaries of the fibres are invertible. Presently (2000), cyclic homology (cf. also Cyclic e o h o m o l o g y ) is probably the only general tool to deal with index problems in which the index belongs to an abstract, possibly unknown, K-theory group, or to deal with index theorems involving non-local invariants. See [26], [35], [37], or [51] for the basic results on cyclic homology. The relation between the K-theory of the algebra A and the cyclic homology of A is via Chern characters Ch: Ko(A) -+ HC2~(A), n >_ 0, and is due to Connes and M. Karoubi. G e n e r a l i z e d i n d e x t h e o r e m s . In [27], Connes and H. Moscovici have generalized Atiyah's L2-index theorem, which allowed them to obtain a proof of the Novikov conjecture (cf. also C*-algebra) for certain classes of groups. The index theorem, also called the higher index theorem for coverings, is as follows. Let M ~ M be a covering of a compact manifold M with group of deck transformations F (cf. also M o n o d r o m y t r a n s f o r m a tion). If D is an elliptic differential operator on M invariant with respect to F (such as the signature operator), then it has an index ind(D) C K0(C~(F)), the K0group of the closure of the group algebra of F acting on /2(r). This index was defined by A.T. Fomenko and A. Mishchenko in [40]. This index can be refined to an index in Ko (7~® C[F]), where 7~ is the algebra of infinite matrices with complex entries and with rapid decrease. Using cyclic cohomology and the Chern character in cyclic homology, every cohomology class ¢ E H*(r) = H*(BF) gives rise to a morphism ¢. : K0(g~GC[F]) --~ C, and the problem is to determine ¢.(ind(D)). If ¢ = 1 C H°(F), then this number is exactly the yon Neumann index appearing in Atiyah's L2-index formula. Let f : M --+ BF be the mapping that classifies the covering M --+ M and let T ( M ) be the Todd class of the complexification of T,~ertY. If D is an elliptic invariant differential operator, its principal symbol a = a(P) defines a K-theory class [a] with compact supports on 204

T ' M , whose Chern character Ch([a]) is in the even cohomology of T * M with compact supports, as in the case of the Atiyah-Singer index theorem for a single elliptic operator. Suppose ¢ E H2"~(F); then the ConnesMoseovici higher index theorem for coverings [27] states that ¢, (ind(D)) = (-1)n(2~ri) -m (Ch([a])T(M)f*¢)IT*M]. The Chern character in cyclic cohomology turns out to be a natural mapping, and this can be interpreted as a general index theorem in cyclic cohomology [42]. It is hoped that this general index theorem will help explain the ubiquity of the Todd class in index theorems. For more information on index theory, see, e.g., [14], [21], [46]. To get a balanced point of view, see also [20] for an account of the original approach to the AtiyahSinger index theorems, which also gives all the necessary background a student needs. References [1] ALVAREZ-GAUMi,L.: 'Supersymmetry and the Atiyah-Singer index theorem', Comm. Math. Phys. 90 (1983), 161-173. [2] ATIYAH, M.: K-theory, Benjamin, 1967. [3] ATIYAH, M.: 'Elliptic operators, discrete subgroups, and yon Neumann algebras', Astdrisque 3 2 / 3 3 (1969), 43-72. [4] ATIYAH, M., AND BOTT, R.: 'A Lefschetz fixed-point formula for elliptic complexes II: Applications.', Ann. of Math. 88 (1968), 451-491. [5] ATIYAH, M., BOTT, R., AND PATODI, V.: 'On the heat equation and the index theorem', Invent. Math. 19 (1973), 279330, Erata ibid. 28 (1975), 277-280. [6] ATIYAH,M., AND HmZEBRUCH, F.: 'Spin manifolds and group actions': Essays in Topology and Related subjects, Springer, 1994, pp. 18-28. [7] ATIYAH, M., PATODI, V., AND SINGER, I.: 'Spectral asymmetry and Riemannian geometry I', Math. Proe. Cambridge Philos. Soc. 77 (1975), 43-69. [8] ATIYAH, M., AND SCHMID, W.: 'A geometric construction of the discrete series', Invent. Math. 42 (1977), 1-62. [9] ATIYAH, M., AND SEGAL, G.: 'The index of elliptic operators II', Ann. of Math. 87 (1968), 531-545. [10] ATIYAH, M., AND SINCER, I.: 'The index of elliptic operators I', Ann. of Math. 87 (1968), 484-530. [11] ATIYAH, M., AND SINGER, I.: 'The index of elliptic operators III', Ann. of Math. 93 (1968), 546-604. [12] ATIYAH, M., AND SINGER, I.: 'The index of elliptic operators IV', Ann. of Math. 93 (1971), 119-138. [13] BAUM, P., AND DOUGLAS, R.: 'Index theory, bordism, and K-homology': Operator Algebras and K - T h e o r y (San Francisco, Calif., 1981), Vol. 10 of Contemp. Math., Amer. Math. Soc., 1982, pp. 1-31. [14] BERLINE, N., GETZLER, E., AND Vt~RGNE, M.: Heat kernels and Dirac operator, Vol. 298 of Grundl. Math. Wissenschaft., Springer, 1996. [15] BISMUT, J.-M.: 'The Atiyah-Singer theorems: a probabilistic approach', J. Funct. Anal. 57 (1984), 56-99. [16] BISMUT, J.-M.: 'The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas', J. Funct. Anal. 57, no. 3 (1984), 329-348.

INTEGRABILITY OF TRIGONOMETRIC SERIES [17] BISMUT, J.-M.: 'The index theorem for families of Dirac operators: two heat equation proofs', Invent. Math. 83 (1986), 91-151. [18] BISMUT, J.-M., AND CHEEGER, J.: '~-invariants and their adiabatic limits', J. Amer. Math. Soc. 2 (1989), 33 70. [19] BISMUT, J.-M., AND FREED, D.: 'The analysis of elliptic families: Metrics and connections on determinant bundles', Comm. Math. Phys. 106 (1986), 103-163. [20] BOOSS-BAVNBEK,B., AND BLEECKER, D.: Topology and analysis. The Atiyah Singer index formula and gauge-theoretic physics, Universitext. Springer, 1985. [21] BOOSS-BAVNBEK, B., AND WOJCIECHOWSKI, K.: Elliptic boundary problems for Dirac operators, Math. Th. Appl. Birkhguser, 1993. [22] BOREL, A., AND SERRE, J.-P.: 'Le t~or~me de Riemann-Roch (d'apre~s Grothendieck)', Bull. Soc. Math. France 86 (1958), 97-136. [23] BOTT, R., AND TAUBES, C.: 'On the rigidity theorems of Witten', J. Amer. Math. Soc. 2, no. 1 (1989), 137-186. [24] CHEEGER, J.: 'On the Hodge theory of Riemannian pseudomanifolds': Geometry of the Laplace operator (Univ. Hawaii, 1979), Vol. XXXVI of Proc. Syrup. Pure Math., Amer. Math. Soc., 1980, pp. 91-146. [25] CONNES, A.: 'Sur la th6orie noncommutative de l'intfigration': Alg~bres d'Opdrateurs, Vol. 725 of Lecture Notes in Mathematics, Springer, 1982, pp. 19-143. [26] CONNES, A.: 'Non-commutative differential geometry', Publ. Math. IHES 62 (1985), 41-144. [27] CONNES, A., AND MOSCOVICI, H.: 'Cyclic cohomology, the Novikov conjecture and hyperbolic groups', Topology 29 (1990), 345-388. [28] GILKEY, P.: 'Curvature and the eigenvalues of the Dolbeault complex for Kaehler manifolds', Adv. Math. 11 (1973), 311325. [29] GILKEY, P.: 'Curvature and the eigenvalues of the Laplacian for elliptic complexes', Adv. Math. 10 (1973), 344-382. [30] GILKEY, P.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC, 1994. [31] GROMOV, M., AND LAWSONJR., H.: 'The classification of simply connected manifolds of positive scalar curvature', Ann. of Math. 111 (1980), 423-434. [32] HIRZEBRUCH,F.: Topological methods in algebraic geometry, third ed., Vol. 131 of Grundl. Math. Wissenschaft., Springer, 1966. [33] HITCHIN, N.: 'Harmonic spinors', Adv. Math. 14 (1974), 1-55. [34] HORMANDER, L.: 'Pseudo-differential operators', Commun. Pure Appl. Math. 18 (1965), 501-517. [35] KAROUBI, M.: 'Homology cyclique et K-theorie', Astdrisque 149 (1987), 1-147. [36] KOHN, J., AND NIRENBERG, L.: 'An algebra of pseudodifferential operators', Commun. Pure Appl. Math. 18 (1965), 269305. [37] LODAY, J.-L., AND QUILLEN, D.: 'Cyclic homology and the Lie homology of matrices', Comment. Math. Helv. 59 (1984), 565-591. [38] MCKEAN JR., H., AND SINGER, I.: 'Curvature and the eigenvalues of the Laplacian', Y. Diff. Geom. 1 (1967), 43-69. [39] MELROSE, H.: The Atiyah-Patodi-Singer index theorem, Peters, 1993. [40] MISCENKO,A., AND FOMENKO, A.: 'The index of elliptic operators over C*-algebras', Izv. Akad. Nauk. SSSR Ser. Mat. 43 (1979), 831-859.

[41] MiJLLER, W.: Manifolds with cusps of rank one, spectral theory and an L 2-index theorem, Vol. 1244 of Lecture Notes in Mathematics, Springer, 1987. [42] NISTOR, V.: 'Higher index theorems and the boundary map in cyclic cohomology', Documenta Math. (1997), 263-296, (electronic). [43] PATODI, V.: 'Curvature and the eigenforms of the Laplace operator', J. Diff. Geom. 5 (1971), 233-249. [44] QUILLEN, D.: 'Superconnections and the Chern character', Topology 24 (1985), 89-95. [45] RAY, D., AND SINGER, I.: ~R-torsion and the laplacian on Riemannian manifolds', Adv. Math. 7 (1971), 145-210. [46] ROE, J.: Elliptic operators, topology and asymptotic methods, Vol. 179 of Pitman Res. Notes in Math. Ser., Longman, 1988. [47] SEELEY, R.T.: 'Refinement of the functional calculus of Calderbn and Zygmund', Indag. Math. 27 (1965), 521-531. (Nederl. Akad. Wetensch. Proc. Ser. A 68 (1965).) [48] STERN, MARK: 'L2-index theorems on locally symmetric spaces', Invent. Math. 96 (1989), 231 282. [49] STOLZ, S.: 'Simply connected manifolds of positive scalar curvature', Ann. of Math. 136, no. 2 (1992), 511-540. [50] TAUBES, C.: 'S 1 actions and elliptic genera', Comm. Math. Phys. 122 (1989), 455-526. [51] TSYGAN, B.L.: 'Homology of matrix Lie algebras over rings and Hochschild homology', Uspekhi Mat. Nauk. 38 (1983), 217-218. [52] WITTEN, E.: 'Constraints on supersymmetry breaking', Nucl. Phys. B 202 (1982), 253-316. [53] WITTEN, E.: 'Supersymmetry and Morse theory', J. Diff. Geom. 17 (1982), 661-692. [54] WITTEN, E.: 'Elliptic genera and quantum field theory', Comm. Math. Phys. 109 (1987), 525-536.

Victor Nistor MSC 1991: 58G10, 55N15 INTEGRABILITY

58Gll,

58G12, 46L80, 46L87,

OF T R I G O N O M E T R I C

SERIES

series

- Given a trigonometric O<3 a0

y + E ( a k coskx + bk sinkx),

(1)

k=l

the problem of its integrability asks under which assumptions on its coefficients this series is the Fourier series of an integrable function (i.e., belonging to L1). Frequently, the series a0

O<3

ak cos kx

+

(2)

k=l

and oo

bk sin

(3)

k=l

are investigated separately, since there is a difference in their behaviour, and usually integrability of (3) requires additional assumptions. Of course, one may also consider trigonometric series in complex form. There exists no convenient description of L I in terms A of a given sequence alone. Hence, subspaces of L I are studied. In view of the Riemann-Lebesgue lemma (cf. 205

INTEGRABILITY

OF TRIGONOMETRIC

SERIES

A

F o u r i e r s e r i e s ) L 1 is a subspace of the space of null sequences, while the space of sequences of b o u n d e d variation bv =

LAdkl < ec

d = {dk}: ]ldllbv = k=0

is not a subspace of L 1. Here Adk = dk -- dk+l. Having a null sequence of b o u n d e d variation as its Fourier coefficients, the series (2) converges for every x ¢ 0 (rood 27r), while (3) converges everywhere. In 1913, W.H. Young [20] proved t h a t if {ak} is a convex null sequence, t h a t is, A2ak = A ( A a k ) > 0 for k = 0 , 1 , . . . , then (1) is the Fourier series of an integrable function (cf. also T r i g o n o m e t r i c s e r i e s ) . In 1923, A.N. K o l m o g o r o v [12] extended this result to the class of quasi-convex sequences {ak}, namely, those satisfying

k=0

Such a sequence is the difference of two convex sequences. In 1956, R.P. Boas generalized all previous re2 sults [4]. Subsequently, m o r e general subspaces of L 1 were considered: 1) T h e so-called Boas-Telyakovskif space bt (see, e.g., [16]):

Ildllbt = Ildllbv +

~-~. ~ A d n - k - A d n + k

k

n=2 k=l

2) T h e Fomin space ap [7] for 1 < p < oo, 1 / p + 1/p I = 1:

IIdlla =

IzXdklp

2n/P' n=0

< oo.

~, k=2 ~

3) T h e Sidon-Telyakovskif space [17]: oo

Ak $ O (k -~ oc),

y ~ Ak < OO,

IAdkl
k=0

4) T h e B u n t i n a s - T a n o v i c - M i l l e r

spaces (see, e.g.,

[5]). 5) The amalgam space [1], [6]: 2 } 1/2

]Adk n=0

L

m=l

i]

k=m2~

A classical way to prove such results is by using Sidon-type inequalities (see, e.g., [8]), a typical example of which is the one o b t a i n e d by S. Sidon [15]: (N+I)

-1

k=oCkDk L 1 < 0 <mka<xN ICkl ' -

-

where Dk is the D i r i c h l e t k e r n e l of order k. 206

In [13] a new approach to these problems was suggested. First, a locally absolutely continuous function f on [0, oc) is considered such that limz_~ f(x) = 0 (cf. also Absolute continuity) and f E X, where X is a subspace of the space of functions of bounded variation BV and is a generalization of a known space of sequences; e.g., 1)-3) above. Then the asymptotic behaviour of the Fourier transform of a function from 2( is investigated. Using the following result from [19] (an earlier version for functions with compact support can be found in [3]),

sup

0
(xDe - yx dx -

(k)e -ik

< C

II llgv,

one obtains even stronger results t h a n those known earlier (for early results, see [18], [19]). Results on integrability of t r i g o n o m e t r i c series have numerous applications to a p p r o x i m a t i o n problems. T h e L e b e s g u e c o n s t a n t s of linear m e a n s of F o u r i e r ser i e s can be efficiently e s t i m a t e d in this way (see, e.g., [16]). For applications to multiplier problems, see [11] and [14]. Other integrability conditions (see, e.g., [2] and [16]) were surprisingly applied to the a p p r o x i m a t i o n of infinitely differentiable functions in [10] and [9]. There exist various extensions of integrability conditions for t r i g o n o m e t r i c series to the multi-dimensional case (see, e.g., [13]). References [11 AVBERTIN,B., AND FOURNIER, J.J.F.: 'Integrability theorems for trigonometric series', Studia Math. 107 (1993), 33-59. [2] BAusov, L.: 'On linear methods for the summation of Fourier series', Mat. Sb. 68 (1965), 313-327. (In Russian.) [3] BELINSKII, E.: 'On asymptotic behavior of integral norms of trigonometric polynomials': Metric Questions of the Theory of Functions and Mappings, Vol. 6, Nauk. Dumka, Kiev, 1975, pp. 15-24. (In Russian.) [4] Boas, R.P.: 'Absolute convergence and integrability of trigonometric series', J. Rat. Mech. Anal. 5 (1956), 621-632. [5] BUNTINAS, M., AND TANOVId-MILLEa, N.: 'Integrability classes and summability', Israel Math. Conf. Proc. 4 (1991), 75-88. [6] BUNTINAS, M., AND TANOVIC-MILLER, N.: ~New integrability and Ll-convergence classes for even trigonometric series II', in J. SZABADOS AND K TANDORI (eds.): Approximation Theory, Vol. 58 of Colloq. Math. Soc. Jdnos Bolyai, NorthHolland, 1991, pp. 103-125. [7] FOMIN, G.A.: 'A class of trigonometric series', Math. Notes 23 (1978), 117-123. (Mat. Zametki 23 (1978), 213-222.) [8] FRIDLI, S.: 'Integrability and L 1 convergence of trigonometric and Walsh series', Ann. Univ. Sci. Budapest, Sect. Comput. 16 (1996), 149-172. [9] GANZBURG, M.: 'Best a p p r o x i m a t i o n of functions like Ixl ~ e x p ( - A l x l - a ) ' , J. Approx. Th. 92 (1998), 379-410. [10] GANZBURG, M., AND LIFLYAND, E.: ' E s t i m a t e s of best approximation and Fourier transforms in integral metrics', J. Approx. Th. 83 (1995), 347-370. [11] GIANt, D.V., AND MORICZ, F.: 'Multipliers of Fourier transforms and series on L 1', Archiv Math. 62 (1994), 230-238.

INVERSE SCATTERING, FULL-LINE CASE [12] KOLMOGOROV,A.N.: 'Sur l'ordre de grandeur des coefficients de la s6rie de Fourier-Lebesgue', Bull. Acad. Polon. (1923), 83-86. [13] LWLYAND,E.R.: 'On asymptotics of Fourier transform for functions of certain classes', Anal. Math. 19, no. 2 (1993), 151-168. [14] LWLYAND,E.R.: 'A family of function spaces and multipliers', Israel Math. Conf. Proc. 13 (1999), 141-149. [15] SIDON, S.: 'Hinreichende Bedingungen fiir den FourierCharakter einer trigonometrischen Reihe', d. London Math. Soc. 14 (1939), 158-160. [16] TELYAKOVSKII,S.A.: 'An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients', Proc. Steklov Inst. Math. 109 (1971), 73-109. (In Russian.) [17] TELYAKOVSKII,S.A.: 'Concerning a sufficient condition of Sidon for the integrability of trigonometric series', Math. Notes 14 (1973), 742-748. (Mat. Zametki 14 (1973), 31~

r ± ( - k ) = r~=(k). T h e m a t r i x t(k) r+(k)

is called the S-matrix (cf. S c a t t e r i n g m a t r i x ) . Conservation of energy implies It(k)l 2 + Ir(k)l 2 = 1. Let f ( x , k) and g(x, k) be the solutions to (1) satisfying the conditions:

f(x, k) = e ikx + o(1), g(x, k) = e -ik~ + o(1),

315-333.) [19] TRIGUB, R.M.: 'Multipliers of Fourier series and approximation of functions by polynomials in spaces C and L', Soviet Math. Dokl. 39, no. 3 (1989), 494-498. (Dokl. Akad. Nauk SSSR 306 (1989), 292-296.) [20] YOUNG,W.H.: 'On the Fourier series of bounded functions', Proc. London Math. Soe. 12, no. 2 (1913), 41-70.

x -+ c¢, x --+ - o c .

Then oo

f (x, k) = e ikx +

328.) [18] TRIGUB, R.M.: 'On integral norms of polynomials', Math. USSR Sb. 30 (1976), 279-295. (Mat. Sb. 101 (143) (1976),

r_(k)) = s(k ) t(k) )

L

A+ (x, y)e iky dy,

g(x, k) = e -ikx +

A_ (x, y)e -iky dy, oo

where A ! ( x , y) are the kernels which define the transformation operators. One has

f ( x , k) = b(k)g(x, k) + a ( k ) g ( x , - k ) , g(x, k) = - b ( - k ) f ( x ,

k) + a ( k ) f ( x , - k ) ,

where

E.R. Liflyand

a ( - k ) = a(k),

M S C 1 9 9 1 : 42A20, 42A32, 42A38

b ( - k ) = b(k),

la(k)q 2 = 1 + Ib(k)J 2 ,

INVERSE SCATTERING, FULL-LINE CASE- Let q(x) E / 1 , i : = {q: f _ ~ ( 1 + ]xl)lq(x)7 dx < ee, q = ~},

r_(k)-

where the bar stands for complex conjugation. Consider the (direct) scattering problem:

~u_ - k ~ _

:= ( - u " + q(~) - k2)~_ = 0,

(I)

x 6 R : = ( - o c , co), ~_

=

[ e -ikx + r_(k)e -ikx / t _ ( k ) e ik~,

'

x --+

(2)

x ~ +~.

T h e coefficients r(k) and t(k) are called the reflection and transmission coefficients. One can prove t h a t t_ (k) is analytic in C + : = {k: I m k > 0} except at a finite n u m b e r of points ikj, I <_ j <_ J, kj > O, which are simple poles of t(k). P r o b l e m (1)-(2) describes scattering by a plane wave e ikx falling from - o o and scattered by the potential

q(x). One can also consider the scattering of the plane wave falling from +~c:

gu+-k2u+=O, f t + ( k ) e -ik~, 11,+ ---- [ e - i k x + r + e i k x ,

xCR,

(3)

x -~ + ~ , X ~

--(~.

(4)

One proves that t_(k) = t+(k) : = t(k), t ( - k ) = t(k), k E R , where the bar stands for complex conjugation,

b(k) a(k)'

t(k)T h e function a(k) is and has finitely m a n y the points ikj, 1 < j & : = da/dk. If k = ikj, then f ( x ,

r+(k)-

b(-k) a(k) '

1 a(k)"

analytic in C + := {k: I m k > 0} simple zeros all of which are at < J, a(ikj) = 0, &(ikj) ~ 0,

ikj) C L 2 ( R ) ,

- f " ( x , ikj) + q ( x ) f ( x , ikj) + k~f(x, ikj) = O,

/_

=~ [f(x, ikj)l 2 dx = (m+) -2,

f f f l(x)g ( x ,

ikj)l ~ dx = (,~/)-~.

T h e numbers - k ] are the eigenvalues of the operator - d 2 / d x 2 + q(x) in L 2 ( R ) . T h e y are called the bound

states. T h e scattering d a t a are the values S := { r + ( k ) , i k j , ( m + ) 2 : Vk > O, 1 < j <_ J } . T h e inverse scattering problem (ISP) consists of finding q(x) E L1,1 from S. T h e inverse scattering problem has at most one solution in the class L1,1. This solution can be calculated by the following Marchenko method: 207

INVERSE SCATTERING, FULL-LINE CASE

1

In [4] the above theory is generalized to the case when

Define

J

p+(x) =

1ff

x +

j:1

q(x) tends to a different constants as x --+ +ec and r + ( k ) c ikx dk

(5)

cx)

and solve the f o l l o w i n g Marchenko equation f o r A+ (z, y) :

A+(x,y) + F+(x + y) +

/5 A(x,t)F+(t,y) dt = O,

y>x.

I f t h e data { r + ( k ) , i k j , ( m + ) 2 : 1 < j <_ J} correspond C LI,1, then equation (5) is uniquely solvable in LI(x,oo) for every x >--oo. If A+(x,y) is found, then q ( x ) : - 2 d A + ( z , x ) / d x .

to a q

2

Marchenko method. The main result [7] is the characterization property for the scattering data: In order that $ := {r+(k),ikj,(m+)2:1 < j < J, kj > O, m + > O, k > 0} be the scattering data corresponding to a q(x) E LI,I(R), it is necessary and sufficient that the following conditions hold: i) r ( - k ) = r(k) for k > 0, the function r(k) for k # 0 is continuous,

Ir+(k)l <_ 1 - ck2(1 + k2) -1, where c = const > 0, and r+(k) = O(1/k) as k --+ +ce. ii) The function R+(x) : :

1

?

r+(k)e ikx dk

is absolutely continuous and

dx

iii) Denote

{-~i 1 /?

~ ln(1-1r_+(k)l 2) d k } . oc

The function a(z) is continuous in C+ and lim ka(k)[r+ (k) + 1] = 0.

k--+0

iv) The function

1 /-~

R_(x) . -

a(-k)

-ikx

2~r~_oor+(-k)~(k) e

dk

is absolutely continuous and

(1 + Ixl) IR'(x) I dx < oo for every s > -oo. A similar result holds for the data

{r_(k),ikj,(mi)2:1

{

q: q = q,

?

(1 +

x 2) [q(x)l

OO

d x < oo

}

The approach in [5] is based on a trace formula. If q(x) = 0 for z < x0 < 0% then the reflection coefficient {r+(k) : Vk > 0} alone, without the knowledge of ikj and (m+) 2, determines q(x) uniquely. A simple proof of this and similar statements, based on property C for ordinary differential equations (cf. O r d i n a r y different i a l e q u a t i o n s , p r o p e r t y C for), is given in [10]. An inverse scattering problem for an inhomogeneous S c h r S d l n g e r e q u a t i o n is studied in [5]. The inverse scattering method is a tool for solving many evolution equations (cf. also E v o l u t i o n equation) and is used in, e.g., soliton theory [7], [1], [2], [6] (cf. also K o r t e w e g - d e Vries e q u a t i o n ; H a r r y D y m equation). Methods for adding and removing bound states are described in [5]. They are based on the Darboux-Crum transformations and commutation formulas. A large bibliography can be found in [3].

tering transform, SIAM, 1981. [2] CALOGEHO, F., AND DEGASPERIS, A.: Solutions and the spectral transform, North-Holland, 1982. [3] CHADAN, K., AND SABATIER, P.: Inverse problems in quantum scattering, Springer, 1989. [4] COHEN, A., AND KAPPELER, T.: 'Scattering and inverse scattering for step-like potentials in the Schr5dinger equation', Indiana Math. J. 34 (1985), 127-180. [5] DEIFT~ P., AND TRUBOWITZ, E.: 'Inverse scattering on the line', Commun. Pure Appl. Math. 32 (1979), 121-251. [6] FADDEEV, L., AND TAKHTADJIAN, L.: Hamiltonian methods in the theory of solutions, Springer, 1986. [7] MARCHENKO, V.: Sturm-Liouville operators and applications, BirkhS.user, 1986. [8] RAMM, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992. [9] RAMM, A.G.: 'Inverse problem for an inhomogeneons SchrSdinger equation', J. Math. Phys. 40, no. 8 (1999), 38763880. [10] RAMM, A.G.: 'Property C for ODE and applications to inverse problems', in A.G. RAMM, P.N. SHIVAKUMAR,AND A.V. STRAUSS (eds.): Operator Theory and Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc., 2000, pp. 15-75. A.G. Ramm

< j _< J, Vk > O}

and the potential q(x) can be obtained by the Marchenko method, q(x) = - 2 d A _ (x, x)/dx. 208

q 6 L1,2 : =

[I] ABLOWITZ, M., AND SEGUR, H.: Solutions and inverse scat-

<

for every 8 > --oo.

j=l

In [5] a different approach to solving the inverse scattering problem is described for

References

/ff l<(x)l(1+

a ( z ) : = r i ~3 e xz -pi k j

x -+ - o o .

MSC 1991: 58F07, 81U20, 35P25, 47A40

INVERSE SCATTERING, HALF-AXIS CASE

INVERSE SCATTERING~ HALF-AXIS CASE- The direct scattering problem on the half-axis consists of finding the solution u(x, k) to the problem u" + k2u - q(x)u = 0,

x > 0,

(1) (2)

k) : o,

u(x, k) = e i~ sin(kx + 5) + o(1),

as x -+ oc.

1

Calculate

r(x) := ZJ sJ e-kj 2

Solve the following

{/0 q:

x[q(x)[ dx < oc, q = ~

}

,

where the bar stands for complex conjugation. The solution f ( x , k) to (1) which satisfies the relation f ( x , k) = e ik~ +o(1), as x --+ +oc, is called the Jost solution. The function f(O, k) :-- f ( k ) is called the Yost function. One has

f ( k ) = If(k)l e -~5(k), 5(-k) = -5(k),

k e R,

5(cc) = 0.

If q E L1,1, then f ( x , k) exists and is unique, f ( k ) is analytic in C+ := {k: I m k > 0} and has at most finitely many zeros in C+, all of which are simple and of the form ikj, kj > 0, 1 ~_ j _< J. The numbers - k ~ are the eigenvalues of the s e l f - a d j o i n t o p e r a t o r

d2

I .--

dx ~ + q(x),

which is determined by the Dirichlet boundary condition at x = 0 in the Hilbert space L2(R+), R + := [0, co) (cf. also D i r i c h l e t b o u n d a r y c o n d i t i o n s ) . In physics, -k~ are called the bound states. The positive numbers sj := [If(x, ikj)ll~(a+) are called the norming con-

stants. The function S(k) := f ( - k ) / f ( k ) = e 2i5(k) is called the S-matrix (cf. S c a t t e r i n g m a t r i x ) . The triple $ := {S(k),ikj, sj: 1 < j < J} is called the scattering data. The inverse scattering problem consists of finding q(x) given 3. The point k = 0 can also be a zero of f(k). It is called a resonance at k = 0. If f(0) = 0, then f'(0) ~ 0. The basic results of inverse scattering theory are (see [5], [7]): 1) The uniqueness theorem: S ~ q; that is, the scattering data determine q E L1,1 uniquely. 2) The reconstruction theorem: If $, corresponding to a q E L1,1, is given, then q(x) can be reconstructed by the Marchenko method, as follows:

F

for

A(x,y):

A(x, s)F(s + y) ds = O,

(4)

y>x>0.

This e ~ a t i o n

L1,1:=

equation

A(x, y) + F(x, y) +

(3)

Here, 5 = 6(k) is to be determined. The function 6 = 5(k) is called the phase shift. The coefficient q(x) is called the scattering potential. It is assumed to be a real-valued function in the class

[1 - S(k)]e ikx dk.

+ g1

j=l

is uniquely solvable ~ d

is called

t h e Ma~chenko equation. 3

Calculate

q(x) = - 2 d m ( x , x ) / d x .

Marchenko method. 3) The characterization theorem: For $ to be the scattering data corresponding to a q E L1,1 it is necessary and sufficient that the following conditions hold: i) S(k) = S ( - k ) = S-~(k), k e R+; S(o~) = 1,

kj > O, 8j > O, 1 < j <_ J; ii) indS(k) = -to, t~ = 2J or a = 2 J + 1; iii) I]F(x)l]L~(a+) + ]IF(x)IILI(R+)+

+ IIxF'(x)IILI(m ) < Here, indS(k) := ( 1 / 2 7 r ) f _ ~ d l n S ( k ) . Note that t~ = 2 J if f(0) # 0, and ~ = 2 J + 1 if f(0) = 0. The mapping T : q -+ $ is a h o m e o m o r p h i s m between L1,1 and the space of the scattering data equipped with the norm IlSII := f ~ ( l + x ) l F ' ( x ) l dx (see

[4], [5] [7]). One can prove (see [7], [14]) the diagram

each step of which is invertible. Here, F = F(x) and A = A(x, y) are defined above. This result guarantees, in particular, that the potential recovered by the Marchenko method generates the original scattering data (provided that q E L1,1 or $ satisfies the characterization conditions). Other methods for solving the inverse scattering problem on the half-axis are based on the solution of the inverse problem of recovery of q(x) from the s p e c t r a l f u n c t i o n p := p(A) ($ ~ p ~ q) and the KreYn method ([1], [3], [5] [7], [13]). The scattering data are in one-to-one correspondence with the spectral function [7] [6], [14]. Recovery of q(x) given the spectral function is discussed in [1], [3], [5], [7]. The original work of M.G. KreYn [2] and its review in [1] do not contain proofs. A detailed presentation of KreYn's theory with complete proofs is given in [13] for the first time. Also, a proof of consistency of KreYn's method is given in [13]. In [2] (and in [1]) there is no discussion of the consistency of KreYn's method. By the consistency of an inversion method one means a proof of the implication q ~ $ (the reconstructed potential generates the data from which it was reconstructed). 209

INVERSE SCATTERING, HALF-AXIS CASE Below, Kre{n's method is described under the simplifying assumption n = 0 (no bound states and no resonance at k = 0). The general case is treated in [13].

For q E LI,1 to belong to L 2 ( R + ) it is necessary and sufficient [7] that

k[l-S(k)+~] Given

S(k) = e 2iS(k), i n d S ( k )

=

0, 6(oo)

=

0, one

finds 6(k), then calculates

g(t):= __~2fo~ 5(k)sin(kt)dk, f(k) : e x p and

H(t) = ~ Given

H(t),

oo

(/7

)

If(k)[ -2

--

)

1

e -ikt

dk.

one solves the equation

(I-k H~)Px : :

:= r~(t,~)+

/? H(t-u)r~(u,s)au=H(t-~), O<_t,s<_x,

for

F~(t,s)

and f i n d s

F~x(2x,0),

0 _< x < oc.

A(x) : 2 F 2 x ( 2 x , O), and c a l c u l a t e s q(x) = A 2 ( x ) + A'(x). Alternatively, One d e f i n e s

q(x) 2¢[F2~(2x, 0) - F2~(0, 0)]. ax =

Kreln's method. In Step 1, one can find f(k) by a different method: Solve the Riemann problem

~+ (k) : s ( - k ) ~ _ (k), kcR, p±(~)=l.

Note that the data $ allow one to find a unique f(k) by solving the Riemann problem (5) with the additional conditions: w+(k) has J simple zeros at the points ikj if = - 2 J and, if ~ = - 2 J - 1, ~+(k) has, in addition, a simple zero at k = 0. Thus, the data $ is equivalent to the data { f ( k ) , s j : 1 < j _< J}. An inverse problem of recovery of q(x) from incomplete scattering data but with an a priori assumption that q(x) has compact support is investigated in [10] [8]. It is proved that if q E L1,1 is compactly supported and if 6(k) is known for a sequence k = kn > 0 which has a finite limit point inside (0, oc), then q(x) is determined uniquely. An algorithm for finding a compactly supported q(x) from 6(h) (that is, from S(k)) known for all k > 0 is given in [10]. A uniqueness theorem for the problem of finding a compactly supported q(x) from the knowledge of if(0, k), Vk > 0, is proved in [14]. In [6], [12] an algorithm for recovery of q(x) from t h e / - f u n c t i o n is given, where the I-function is identical with the Weyl function.

q(t) dt = - 2 i lira {k[f(k) - 1]}. k--+oc

If q(x) 6 L l a N L 2 ( R + ) , q = 0 for x _> a, is compactly supported, then f(k) is an e n t i r e f u n c t i o n of exponential type _< 2a. Its zeros in C _ := {k: I m k < 0} are called resonances. If q(x) ~ 0, foxnlq(x)[ dx = o(nbn), 0 e, where e > 0 is arbitrary small, which generates infinitely many purely imaginary resonances [7]. If q(x) 6 L1,1, q(x) = 0 for x > a and q(x) does not change sign in an interval (a - 6, a), where 6 > 0 is arbitrarily small, then q(x) generates only finitely many purely imaginary resonances (6). If q E L l a , then the following estimate (see [5]) is useful: F'(2x) -

q(t) dt

+ ~

a(x) :=

/5

<_ca2(x),

Iq(t)l dt.

The Jost solution f(x, k) can be written as f(x, k) = e ikx + f ~ A(x, y)e iky dy, where A(x, y) is the kernel of the transformation operator. If q E L1,1, then IF(2x) + A(x, x)[ <_ca(x),

f(-k).

210

/7

(~)

If indS(k) = 0, this problem has the unique solution { ~ + ( k ) , p _ ( k ) } . One has p+(k) = f(k), p _ ( k ) =

(6)

where Q :=

9(t)e ikt d t ,

E L2(R),

IF(2x)l < ca(x),

]A(x,y), < ca ( ~ )

,

where c > 0 is a constant. The function A(x,y) solves the Volterra-type equation

A(x, y) = -~ +

ds

+y)/2q(t) dt+

q ( s - t ) A ( s - t,s + t) dr.

If q E L1,1 and q(x) = 0 for x > a, then A(x,y) = 0 for y > x > a, F(x) = 0 for x = 2a, and A(y) := A(0, y) = 0 for y _> 2a. Since f(k) = l + f o A(y)eikydy, it follows that f(k) is an e n t i r e f u n c t i o n of order 1 and type _< 2a, and S(k) = f ( - k ) / f ( k ) is meromorphic

INVERSE SCATTERING, MULTI-DIMENSIONAL CASE on the whole complex k-plane (cf. also M e r o m o r p h i c function). Conversely, if the scattering data $ correspond to a q C L1,1 (necessary and sufficient conditions for this were given above) and generate (by solving the Riemann problem mentioned above) the function f(k) which is an entire function of exponential type < 2a, then q(x) = 0 for x > a, (see [7]).

Inverse potential scattering. To formulate the inverse potential scattering problem, consider first the direct scattering problem (see [1], [2], [4], [5], [7, Appendix]):

[-V ~ + q(x) - k2]u = 0 in R 3, u=e ik~'x+v,

k = c o n s t > 0, (1)

a C S 2,

(2)

2

lira ~

OV _ ikv

ds = O,

(3)

References [1] CHADAN, K., AND SABATIER, P.: I n v e r s e problems in q u a n turn scattering theory, Springer, 1989. [2] KREIN, M.: 'Theory of accelerants and S-matrices of canonical differential systems', Dokl. Akad. Nauk. USSR III, no. 6 (1956), 1167-1170. (In Russian.) [3] LEVITAN, B.: Inverse Sturm-Liouville problems, VNU Press, 1987. [4] MARCHENKO,V.: 'Stability in the inverse problem of scattering theory', Mat. Sb. 77 (1968), 139-162. (In Russian.) [5] MARCHENKO, V.: Sturm-Liouville operators and applications, Birkh/iuser, 1986. [6] RAMM, A.G.: 'Recovery of the potential from I-function', Math. Rept. Acad. Sci. Canada 9 (1987), 177-182. [7] RAMM, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992. [8] RAMM, A.G.: 'Compactly supported spherically symmetric potentials are uniquely determined by the phase shift of swave', Phys. Lett. A 242, no. 4-5 (1998), 215-219. [9] RAMM, A.G.: 'Recovery of a quarkonium system from experimental data', J. Phys. A 31, no. 15 (1998), L295-L299. [10] RAMM, A.G.: 'Recovery of compactly supported spherically symmetric potentials from the phase shift of s-wave', in A.G. RAMM (ed.): Spectral and Scattering Theory, Plenum, 1998, pp. 111-130. [11] RAMM, A.G.: 'Inverse scattering problem with part of the fixed-energy phase shifts', Comm. Math. Phys. 207, no. 1 (1999), 231-247. [12] RAMM, A.G.: 'Property C for ODE and applications to inverse scattering', Z. Angew. Anal. 18, no. 2 (1999), 331-348. [13] RAMM, A.G.: 'Krein's method in inverse scattering': Operator Theory and Applications, Amer. Math. Soc., 2000, pp. 441456. [14] RAMM, A.G.: 'Property C for ODE and applications to inverse problems', in A.G. RAMM, P.N. SHIVAKUMAR,AND A.V. STRAUSS (eds.): Operator Theory and Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc., 2000, pp. 15-75. [15] RAMM, A.G., AND SCHEID, W.: 'An approximate method for solving inverse scattering problem with fixed-energy data', J. Inverse Ill-Posed Probl. 7", no. 6 (1999), 561-571.

A. G. I~amm MSC 1991: 35P25, 47A40, 81U20

INVERSE SCATTERING~ MULTI-DIMENSIONAL CASE - There are many multi-dimensional inverse scattering problems. Below, inverse potential scattering and inverse geophysical scattering are briefly discussed; see O b s t a c l e s c a t t e r i n g for inverse obstacle scattering problems.

where a is given, S 2 is the unit sphere, v is the scattered field, u is the scattering solution, condition (3) is called the (outgoing) radiation condition, e ik~x is the incident plane wave, and q(x) is a real-valued function, called a

potential, q(x) C L~oc(R3), Iq(x)l < c(1 + Ixl) -b,

b > 2,

for large Ixl.

The existence and uniqueness of the solution to (1)(3) has been proved under less restrictive assumptions on q(x) [2]. The function v has the form --+o r

r --F 0o~

(!)

x -- --~ OZ, r

where the coefficient A(a', a, k) is called the scattering

amplitude. The inverse potential scattering problem consists of finding q(x) given A(a ~,a, k) on some subsets of S 2 x S 2 x R+. The first result is simple: If A(a ~, a, k) is known for all a~, a E S 2 and all k > 0, then q(x) is uniquely determined. If q C Qm :=

{

q:

Iq(x)l + IVmql < c(1 + fxf) -b, b>3

}

,

then it is known (e.g. [7, p. 233], see also [4]) that

A(a',a,k) -

i fR 3 eik(c~-~')'Xq(x)dx+O(k ) 47r k --+ oo ,

so that ~'(~) := fa3 e-i~'Xq(x) dx can be found: ~(() = -47r

lim A(a', a, k). k-+oo k(~-~')=~

The second result is much more difficult. For decades it was not known if the data A(a ~,a) := A(a', k0), Va', a C S 2 and k0 > 0 fixed, determine q(x) uniquely. In 1987 the uniqueness result has been established by A.G. Ramm (see [8], [6]) under the assumptions q(x) e L2(R3), q(x) = 0 for ]x I > a, where a > 0 211

INVERSE SCATTERING, MULTI-DIMENSIONAL CASE is an arbitrary large fixed number, and in 1988 inversion procedures were published; see [8]. One of them, proposed by Ramm, is based on the formula ~(~) = -47~ lim

f

0--4oo ~ 2 O,O' EM 0-0'=~

A(O',a)v(a,O) da,

where M := { 0 : 0 E C 3 , 0 ' 0 = k~}, O.w := E ~ : l Oj.wj, v(a,O) E L2($2), and ~ E R a is an arbitrary point. Another inversion procedure ([3], [8]) is based on the reconstruction of the Dirichlet-to-Neumann mapping and then finding q(x). Error estimates for Ramm's inversion procedure in the case of noisy data and an algorithm for calculating the function v(a, 0) in the inversion formula are obtained in [9]. The uniqueness problem for inverse potential scattering with the data A(a',ao,k), Va' E S 2, Vk > O, ao E S 2, fixed, is still open (as of 2000). The same is true for the uniqueness problem for inverse potential scattering with the (backscattering) data A ( - a , a, k), Va E S 2, Vk > 0, although for this problem a uniqueness theorem for small q(x) holds.

Inverse geophysical scattering. The inverse geophysical scattering problem consists of finding the unknown coefficient v(x) in the equation (V 2 + k~ + k~v(x))u(x, y, ko) = -6(x - y)

sup a,a' E S 2

Inverse potential scattering: Open problem. An interesting open problem (as of 2000) in inverse potential scattering is the problem of finding discontinuities of q(x) and the number of bound states of the Schrhdinger operator generated by the expression - V 2 + q(x) in

IAh(a',a) - A(a',a)l < 6

(see [8] for a proof).

References [1] CYCON, [2] [3] [4] [5] [6]

in R 3, (4)

where u := u(x, y) := u(x, y, ko) satisfies the outgoing radiation condition (3), k0 = const > 0 is fixed, and v(x) is a real-valued L~oc function with compact support in R a_ := {x: zs < 0}. The scattering data are the values u(x,y), Vx, y E P := {x: x3 = 0}, that is, the values of u on the surface of the Earth. The function v(x) describes an inhomogeneity in the velocity profile (in the refraction coefficient), u can be an acoustic pressure. Uniqueness of the solution to inverse geophysical scattering problem was proved in 1987 [6], [8]. The uniqueness problem for inverse geophysical scattering with data u(x, Y0, k), Vx E P, Vk > 0, and Y0 E P fixed, is open (as of 2000). A reduction of the inverse geophysical scattering problem with the data u(x,y, ko), Vx, y E P, to the inverse potential scattering problem with the data A(a', a, k0), Va, a' E S~_, k0 > 0 fixed, S~_ := { a : a E S 2, a . ea > 0}, with e3 the unit vector along x3-axis, is done in [8].

212

L~(R3) from the knowledge of fixed energy scattering data A(a', a, ko), Va', a E S 2. If q E Lo2(R3), then A(a',a) is an analytic function of a', a E M. Therefore, knowledge of A(a', a) on an open set in S 2 x S 2, however small, allows one to recover A(a', a) on M × M. The assumption concerning compactness of the support of q(x) is natural in inverse potential scattering because the scattering data are always noisy and it is not possible in principle to recover the tail of a q(x) E Q (that is, q(x) for Ixl > R, where R > 0 is sufficiently large) fl'om knowledge of noisy data A5 (a', a),

[7] [8] [9J [1O]

H., FROESE,

R., KIRSCH,

W.,

AND

SIMON,

B.:

Schrb'dinger operators, Springer, 1986. H6RMANDER, L.: Analysis of linear partial differential operators, Vol. IV, Springer, 1985. NACHMAN, A.: 'Reconstruction from boundary measurements', Ann. Math. 128 (1988), 531-578. NEWTON, R.: Inverse Schrhdinger scattering in three dimensions, Springer, 1989. PEARSON, D.: Quantum scattering and spectral theory, Acad. Press, 1988. RAMM, A.G.: 'Recovery of the potential from fixed energy scattering data', Inverse Probl. 4 (1988), 877-886, See also: Ibid. 3 (1987), L77-82. RAMM, A.G.: Random fields estimation theory, Longman/Wiley, 1990. RAMM, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992. RAMM, A.G.: 'Stability estimates in inverse scattering', Acta Applic. Math. 28, no. 1 (1992), 1-42. RAMM, A.G.: 'Stability of solutions to inverse scattering problems with fixed-energy data', Rend. Sere. Mat. e Fisico

(2001), 135-211. A.G. Ramm

MSC1991: 35P25, 47A40, 81U20 ISOGONAL - Literally 'same angle'. There are several concepts in mathematics involving isogonality.

Isogonal trajectory. A trajectory that meets a given ramily of curves at a constant angle. See I s o g o n a l trajectory. Isogonal mapping. A (differentiable) mapping that preserves angles. For instance, the stereographic projection of cartography has this property [6]. See also Conformal mapping; Anti-conformal mapping. Isogonal circles. A circle is said to be isogonal with respect to two other circles if it makes the same angle with these two, [3].

Isogonal line. Given a triangle A1A2As and a line L1 from one of the vertices, say fl'om A1, to the opposite

IWASAWA T H E O R Y side. The corresponding isogonal line L~ is obtained by reflecting L1 with respect to the b i s e c t r i x in A1. If the lines L1 = ALP1, L2 = AjP2 and L3 = A3P3 are concurrent (i.e. pass through a single point X, i.e. are Cevian lines), then so are the isogonal lines L~, L~, L~. This follows fairly directly from the C e v a t h e o r e m . The point X ' = L~ N L~ = L~ N L~ = L~ A L~ is called the isogonal conjugate point. If the b a r y c e n t r i c c o o r d i n a t e s of X (often called trilinear coordinates in this setting) are (c~ : ~ : V), then those of X ' are (ct -1 : ¢?-1 : 3,-1) A1

As

P1

F~

A3

Another notion in rather the same spirit is that of the isotomic line to L1, which is the line L~I = AjP[' such that IAjP~'I = IP1A31. Again it is true that if L1, Lj, L3 are concurrent, then so are L~', L~~, L~3q This follows directly from the C e v a t h e o r e m . A1

[3] BERGER, M.: Geometry, Vol. I, Springer, 1987, p. 327. [4] COXETER, H.S.M.: The real projective plane, third ed., Springer, 1993, pp. 197-199. [5] EDDY, R.H., AND WILKER, J.B.: 'Plane mappings ofisogonalisotomic type', Soochow J. Math. 18, no. 2 (1992), 135-158. [6] HILBERT, D., AND COHN-VOSSEN, S.: Geometry and the imagination, Chelsea, 1952, p. 249. [7] JOHNSON, R.A.: Modern geometry, Houghton-Mifflin, 1929. M. Hazewinkel

MSC 1991:51M04 A theory of Zp-extensions introduced by K. Iwasawa [8]. Its motivation has been a strong analogy between number fields and curves over finite fields. One of the most fruitful results in this theory is the Iwasawa main conjecture, which has been proved for totally real number fields [21]. The conjecture is considered as an analogue of Well's result that the characteristic polynomial of the Probenius automorphism acting on the Jacobian of a curve over a finite field is the essential part of the zeta-function of the curve. A lot of methods and ideas developed in the theory appeared to be widely applicable and have given rise to major advances, for example, results on the Birch-SwinnertonDyer conjecture [4], [7], [16], [18] and on Fermat's last theorem [22] (cf. also F e r m a t last t h e o r e m ) . For details and generalizations of Iwasawa theory, see [3], [9], [12], [20]. IWASAWA

THEORY

-

Z p - e x t e n s i o n o f a n u m b e r field. Let p be a prime number and let k be a finite extension of the rational number field Q. A Zp-extension of k is an extension K / k with Gal(K/k) = Zp, where Zp is the additive group of p-adic integers. Then there is a sequence of fields k = kO C kl C "'" C kn C ' ' ' C K :

As

P1

P~'

Aa

The point X " = L~' A L~ = L~~ f3 L~' = L f N L~' is called the isotornic conjugate point. The barycentric coordinates of X " are (aJo~-1 : b2/3-1 : e2~-1), where a, b, c are the lengths of the sides of the triangle. The G e r g o n n e p o i n t is the isotomic conjugate of the N a g e l point. The involutions X ~-~ X ~ and X ~-~ X " , i.e. isogonal conjugation and isotomic conjugation, are better regarded as involutions of the projective plane P J ( R ) , [5]. References

[1] ALTSHILLER-COUaT, N.: College geometry, Barnes & Noble, 1952. [2] BACHMANN,F.: Aufbau der Geometric aus dem Spiegelungsbegriff, second ed., Springer, 1973.

U kn, n>0

where k~ is a cyclic extension of k of degree p~. Class field t h e o r y shows that there are at least 1 + rj(k) independent Zp-extensions of k (cf. below, the section Leopoldt conjecture). Every k has at least one Zpextension, namely the cyclotomic Zp-extension k~. It is obtained by letting k ~ be an appropriate subfield of Un>O k(pp~), where #m is the group of mth roots of unity. L e o p o l d t c o n j e c t u r e . Let E l ( k ) be the group of units of k which are congruent to 1 modulo every prime ideal f0 of k lying above p. By Dirichlet's unit theorem, rankz E1 (k) = rl (k) + r2 (k) - 1, where rl (k) (resp. 2r2 (k)) is the number of embeddings of k in R (resp. C). Let Ul,e be the group of local units of ke congruent to 1 modulo fo. There is an embedding El(k) --~ I-Irolp Ul,~o (e ~ ( s , . . . ,e)). Let E l ( k ) denote the topological closure of the image. It is Leopoldt's conjecture that the 213

IWASAWA T H E O R Y equality r a n k z E1 (k) = rankz~ E1 (k) holds for every k. A. Brumer [1] proved the conjecture for Abelian extensions k / Q (or an imaginary quadratic field). P u t 5p(k) = r a n k z E l ( k ) - rankz, El(k) > O. Then class field theory shows t h a t there are 1 + r2 (k) + 5p(k) independent Zp-extensions of k. I w a s a w a m o d u l e . Let (9 be the integer ring nite extension of Qp and 7c a uniformizer of F be a compact Abelian group isomorphic to R = O[[r]] = ~ O[F/FP~], where the inverse

of a fi(9. Let Zp and limit is

taken with respect to F/F pm --+ F/F p~ (7 mod F p'~ ~+ 7 rood F p~) for m 2 n. Fix a topological generator 7 of F. Let A = O[[T]] be the ring of formal power series in an indeterminate T with coefficients in (9. P(T) • O[T] is called a distinguished polynomial if P(T) = T n + a n _ l T ~-1 + . . . + C o with ai • (Tr) for 0 < i < n - 1. The prime ideals of A are 0, (~r,T), (~r), (P(T)), where P(T) is distinguished and irreducible. The classification of compact R-modules in [8] was simplified by J.-P. Serre, who pointed out that R is topologically isomorphic to A, hence each compact R-module X admits the unique structure of a compact A-module such that (1 + T)x = 7 • x for every x • X. Finitelygenerated A-modules are called Iwasawa modules. They are classified as follows: for an Iwasawa module X , there is a A-homomorphism

x

A •

• @ i=1

with

Ker~

and

j=l

Cokerqa

finite

A-modules,

where

r,s, li,t, mj • Z>0 and fi(T) is distinguished and irreducible. For a torsion A-module X, i.e., r = 0, one defines

char(X) =

1-I i=1

w: Gal(k(pp)/k) --+ Zpx t

= Z j=l

I w a s a w a i n v a r i a n t . Let K / k be a Zp-extension. Let An(k) denote the p-Sylow subgroup of the ideal class group of kn. Let p*~ be the order of An(k). Iwasawa [8] proved that there exist integers )b(K/k) >_ O, pp(K/k) >_ 0 and up(K/k) such that +

n +

for all sufficiently large n. The invariants AB(K/k) and #p(K/k) can be obtained from the Iwasawa module X = l ~ A n ( k ) , where the inverse limit is taken with respect to the relative norm mappings. P u t P = Gal(K/k). 214

(w(a) - a rood p)

deg(fi(r)h),

i=1

=

I w a s a w a m a i n c o n j e c t u r e . Let p be an odd prime number and k a totally real number field. Fix an embedding of Q into Qp. Let X be a p-adic valued Artin character for k of order prime to p. Let kx be the extension of k attached to X. Assume t h a t k x is also totally real. Fix a topological generator 7 of F = Gal(kx,oo/kx) ~Gal(kx(pp~)/kx(pp)) and let u • Z~ be such t h a t ~ = ¢~ for all ¢ • #p~. Let co be the Teichmiiller character

j=l

s

A(X) = E

en

X is a compact R = Zp[[r]]-module in a natural way. One fixes a topological generator 7 of F. T h e n X is considered as a compact A = Zp[[T]]-module (cf. the section on Iwasawa module above). Since A~(k) is finite, X is a finitely-generated torsion A-module. One has that Ap(K/k) = A(X) and #p(K/k) = #(X). Iwasawa [10] constructed infinitely m a n y noncyclotomic Zp-extensions K / k with #p(K/k) > 0. There are infinitely m a n y Zp-extensions K / k with Ap(K/k) > 0. For k = Q(pp), Ap(ko~/k) > 0 if and only if p is irregular (of. also I r r e g u l a r p r i m e n u m b e r ) . It is Iwasawa's conjecture that #;(ko~/k) = 0 for every k. B. Ferrero and L. Washington [6] proved this conjecture for Abelian extensions k/Q. W. Sinnott [19] gave a new proof of this using the F-transform of a rational function. It is Greenberg's conjecture that Ap(k~/k) = pp(k~/k) = 0 for every totally real k. For small p, it was proved t h a t there are infinitely m a n y real quadratic fields k with Ap(ko~/k) = #p(ko~/k) = ,p(ko~/k) = 0 [14], [15]. There exists a lot of numerical work verifying this conjecture, mainly for real quadratic fields. It is Vandiver's conjecture t h a t p does not divide the class number of the maximal real subfield k of Q(pp) for all p, which implies t h a t Ap(ko~/k) = #p(koo/k) = vp(koo/k) = 0. This conjecture was verified for all p < 12000000 [2].

and let L(s, X) be the classical L-function for k. Following T. K u b o t a and H.W. Leopoldt [11], P. Deligne and K. Ribet [5] proved the existence of a p-adic L-function L p ( s , x ) on s E Z ; (s # 1 if X is trivial) satisfying the following interpolation property:

Lp(1 - n , x ) = L(1 - n, xw -n) I I ( 1 - X w - n ( p ) N p n-l) PIP for n > 1. There exists a unique power series G x(T) 6 Zp[X][[T]] such that Lp(1 - s , x ) = Gx(u s - 1) (if X is trivial, L p ( 1 - s,x) = Gx(u ~ - 1)/(u ~ - 1)), where Zp[X] is the ring generated over Zp by the values of X. By the p-adic Weierstrass preparation theorem (cf. also W e i e r s t r a s s t h e o r e m ) , one can write Gx(T ) =

IWASAWA

THEORY

~gx(T)ux(T), w h e r e #x • Z>0, 9 x ( T ) is a distinguished p o l y n o m i a l , 7r is a u n i f o r m i z e r of Zp[X] , a n d u x ( T ) is a u n i t p o w e r series. Let G ~ ( T ) • Zp[X][[T]] b e such t h a t L p ( s , x ) = G x ( u S - 1) (if X is trivial, L p ( s , X) = G ~ ( u ~ - 1 ) / ( u~ - u ) ) . One can s i m i l a r l y define #~ = Px a n d a d i s t i n g u i s h e d p o l y n o m i a l g;c (T) for

[3] COATES,J., GREENBERG,R., MAZUR, B., AND SATAKE,I.: Algebraic Number Theory - In Honor of K. Iwasawa, Vol. 17 of Adv. Studies in Pure Math., Acad. Press, 1989. [4] COATES, J., AND WILES, A.: 'On the conjecture of Birch and Swinnerton-Dyer', Invent. Math. 39 (1977), 223-251. [5] DELIGNE, P., AND RIBET, K.: 'Values of abelian L-functions at negative integers over totally real fields', Invent. Math. 59

G~(T). Let k' = k x ( # p ) , let L ( k ' ) b e t h e m a x i m a l u n r a m ified A b e l i a n p - e x t e n s i o n of k ~ a n d M ( k ' ) t h e m a x i m a l A b e l i a n p - e x t e n s i o n of k ~ , which are b o t h u n r a m ified o u t s i d e t h e p r i m e s a b o v e p. By class field theory,

(1980), 227-286. [6] FERRERO, B., AND WASHINGTON, pp vanishes for abelian number

Gal(L(k')/k')

Extend g C Gal(k'/k) to

"g C G a l ( L ( k ' ) / k ) . T h e n g acts on x C G a l ( L ( k ' ) / k ~ ) b y g . x = "~x'~-1. P u t X = G a l ( L ( k ' ) / k ~ ) ® Zp[X] a n d Y = G a l ( M ( k ' ) / k ~ ) ® Zp[x]. Let A = G a l ( k ~ / k o o ) -~

Gal(k'/k), X ~'x-~ = { x E X : 5 . x = w x - l ( 6 ) x for 6 E A } , y x = {y e Y : 5. y = X(a)y for a e A } . T h e n one can r e g a r d X ~x-~ a n d y x as A = Zp[X][[T]]modules. Following [13], A. Wiles p r o v e d t h e following equality, i.e., t h e I w a s a w a m a i n c o n j e c t u r e for t o t a l l y real fields: c h a r ( X ~°x-~) = 7r'x 9 ; (T). T h i s e q u a l i t y is equivalent to c h a r ( y x ) = rcU~gx (T). T h e p r o o f uses d e l i c a t e techniques from m o d u l a r forms, especially H i d a ' s t h e o r y of m o d u l a r forms, to c o n s t r u c t u n r a m i f i e d extensions. Following S t i c k e l b e r g e r ' s t h e o r e m , F. T h a i n e a n d V. K o l y v a g i n invented techniques for c o n s t r u c t i n g r e l a t i o n s in ideal class groups. T h e s e m e t h o d s , which use G a u s s s u m s (cyclotomic units or elliptic units) satisfying p r o p erties k n o w n as t h e E u l e r s y s t e m , have given e l e m e n t a r y proofs of t h e I w a s a w a m a i n c o n j e c t u r e for k = Q [12],

[17]. References [1] BRUMER, A.: 'On the units of algebraic number fields', Mathematika 14 (1967), 121-124. [2] BUHLEH, a., CRANDALL, R., ERNVALL, R., METSA.NKYLA, T., AND SHOKROLLAHI, M.A.: 'Irregular primes and cyclotomic

invariants to 12 million', J. Symbolic Comput. 31 (2001), 89-96.

L.: 'The Iwasawa invariant fields', Ann. of Math. 109

(1979), 377-395. [7] GREENBERG,R.: 'On the Birch and Swinnerton-Dyer conjecture', Invent. Math. 72 (1983), 241-265. [8] IWASAWA,Z.: 'On F-extensions of algebraic number fields', Bull. Amer. Math. Soc. 65 (1959), 183-226. [9] IWASAWA,K.: 'On Zl-extensions of algebraic number fields', Ann. of Math. 98 (1973), 246 326. [10] IWASAWA,K.: 'On the winvariants of Zl-extensions': Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, 1973, pp. 1-11. [11] KUBOTA, T., AND LEOPOLDT, H.W.: 'Eine p-adische Theorie der Zetawerte, I. Einffihrung der p-adischen Dirichletschen LFunktionen', J. Reine Angew. Math. 214/215 (1964), 328339. [12] LANG, S.: Cyclotomic fields I-II, Vol. 121 of Graduate Texts in Math., Springer, 1990, with an appendix by K. Rubin. [13] ]~/[AZUR, B., AND WILES, A.: 'Class fields of abelian extensions of Q', Invent. Math. 76 (1984), 179-330. [14] NAKAGAWA, J., AND HOME, K.: 'Elliptic curves with no rational points', Proe. Amer. Math. Soc. 104 (1988), 20-24. [15] ONO, K.: 'Indivisibility of class numbers of real quadratic

fields', Compositio Math. 119 (1999), 1-11. [16] RUmN, K.: 'Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication', Invent. Math. 89 (1987), 527-560. [17] RUmN, K.: 'The "main conjectures" of Iwasawa theory for imaginary quadratic fields', Invent. Math. 103 (1991), 2568. [18] RUBIN, K.: 'Euler systems and modular elliptic curves': Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), Vol. 284 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1998, pp. 351-367. [19] SINNOTT, W.: 'On the #-invariant of the F-transform rational function', Invent. Math. 75 (1984), 273-282.

of a

[20] WASHINGTON, L.: Introduction to cyclotomic fields, second ed., Vol. 83 of Graduate Texts in Math., Springer, 1997. [21] WILES, A.: 'The Iwasawa conjecture for totally real fields', Ann. of Math. 131 (1990), 493-540. [22] WILES, A.: 'Modular elliptic curves and Fermat's last theorem', Ann. of Math. 141 (1995), 443-551. Hiroki Sumida MSC1991:11R23

215

J formula for the J o n e s - C o n w a y p o l y n o m i a l , describing it as a sum of products of the Jones-Conway polynomials of pieces of the diagram. It has its root in the statistical mechanics model of the Jones-Conway polynomial by V.F.R. Jones. It has been applied to periodic links and to the building of a H o p f a l g e b r a structure on the Jones-Conway s k e i n m o d u l e of the product of a surface and an interval [3], [4], [2]. To define the Jaeger composition product it is convenient to work with the following regular isotopy variant of the Jones-Conway polynomial: JAEGER

COMPOSITION

PRODUCT

Q D ( V , Z ) = Zc ° m ( D ) - l v - T a i t ( D ) ( v - 1

- A

-- V ) P D ( V , Z ) ,

where corn(D) is the number of link components and Tait(D) is the algebraic sum of the signs of the crossings of D. It is also convenient to add the empty link, 0, to the set of links and put Qo(v,z) = 1. QD(V,Z) satisfies the skein relation

[QDo QD+ - QD_ = (z2QDo

for

a

self-crossing,

for a mixed crossing,

and QDuo = ( v - 1 -- V)QD. The advantage of working with QD(V, z) is that QD(V, z) E Z[v ±1, z 2] (no negative powers of z) and that the Jaeger composition product has a nice simple form. Indeed ([1]): Let D be a diagram of an oriented link in S 3, then

/\ 1

1

2

2

(possibly i=j)

The set of 2-1abellings of D is denoted by lbl(D). The edges of D with label i form an oriented link diagram, denoted by Df,~. The vertices of D which are neither in D/,1 nor D/,2 are called f-smoothing vertices of D. Let I f l - (respectively, Ifl+) denote the number of negative (respectively, positive) f-smoothing vertices of D. Let If] = I l l - + Ill+ and define (DIf) = (-1)lfl-z Ifl-c°m(DL1)-c°m(D/,2)÷c°m(D).Finally, rot(D) denotes the rotational number of D, i.e. the sum of the signs of the Seifert circles of D, where the sign of such a circle is 1 if it is oriented counterclockwise and - 1 otherwise. References [I] JAEGER, F.: 'Composition products and models for the Homily polynomial', L'Enseign. Math. 35 (1989), 323-361. [2] PRZYTYCKI, J.H.: 'Quantum group of links in a handlebody', in M. GERSTENHABER AND J.D. STASHEFF (eds.): Defor-

mation Theory and Quantum Groups with Applications to Mathematical Physics, Vol. 134 of Contemp. Math., 1992, pp. 235-245. [3] PRZYTYCKI, J.H.: 'A simple proof of the Traczyk-Yokota criteria for periodic knots', Proc. Amer. Math. Soc. 123 (1995), 1607-1611. [4] TURAEV, V.G.: 'Skein quantization of Poisson algebras of loops on surfaces', Ann. Sci. Ecole Norm. Sup. 4, no. 24 (1991), 635-704.

Jozef Przytycki QD(vlv2,z) =

Z

MSC 1991:57M25

fclbl(D)

(Dlf) v~°t(D~'l) QD,.1 (vl, z)v? r°t (D/'2) f~Dy,2 [V Z). The meaning of the used symbols is as follows. To define lbl(D), consider D as a 4-valent graph. Let Edge(D) denote the set of edges of the graph D. A 2-1abelling of D is a function f : Edge(D) -4 {1,2} such that around a vertex the following labellings are allowed:

J A N S O N I N E Q U A L I T Y - There are a couple of inequalities referred to as 'Janson inequality'. They provide exponential upper bounds for the p r o b a b i l i t y that a sum of dependent zero-one random variables (cf. also R a n d o m v a r i a b l e ) equals zero. The underlying p r o b a b i l i t y s p a c e corresponds to selecting a random subset Fp of a finite set F, where p = {Pi: i E F}, in such a

JANSON I N E Q U A L I T Y way that the elements are chosen independently with P(i E Fp) = Pi for each i C F. Let $ be a family of subsets of P and, for every A ff N, let I a be equal to one if A C_ F; and be zero otherwise. Then X = ~ a e s IA counts those elements of $ which are entirely contained in Fp. Set

k < n. Then X is the number of triangles in G ( n , p ) , {4]p5 • Thus, if p = 0(n-1/2), A = (3)p 3 and A = (4) ,2, then l n P ( X = 0) .-- -A, while for p = f~(n-1/2), inequality (2) yields P(X = 0) < e -~O/(np2)). As long as p = o(1), var(X) ,-, A, and the above exponential bounds strengthen the polynomial bound

,~ = E(X),

1

E

E(IAIB),

ACB, A N B ¢ ~

A=A+2A. Then P(X = O) < e x p ( - A + A)

(1)

and, which is better for A > A/2,

P(X : 0)_< exp ( - ~ )

.

(2)

obtained by the method of second moments, i.e. by a corollary of the C h e b y s h e v i n e q u a l i t y . To illustrate the strength of (1), fix p = 105n -2/3 and assume that n is divisible by 100. Then (1) easily implies that with probability tending to one, more than 99% of the vertices of G ( n , p ) are covered by vertex-disjoint triangles. Indeed, otherwise there would be a subset of n/100 vertices spanning no triangle. By (1), the probability that this may happen is smaller than

Research leading to these inequalities was motivated by a ground-breaking proof of B. Bo]lob~s [I],who, in order to estimate the chromatic number of a random graph, used martingales (cf. also G r a p h c o l o u r i n g ; G r a p h , r a n d o m ; M a r t i n g a l e ) to show that the probability of not containing a large clique is very small. Bollob£s presented his proof at the opening day of the 'Third Conference on Random Graphs (Poznafi, July 1987)'. By the end of the meeting S. Janson found a proof of inequality (1) based on Laplace transforms (cf. also L a p l a c e t r a n s f o r m ) , while T. Luczak proved a related, less explicit estimate using martingales. The latter result was restricted to a special, though pivotal, context of small subgraphs of random graphs. Soon after, R. Boppana and J. Spencer [2] gave another proof, resembling the proof of the Lovfisz local l e m m a , of the following version of (1):

P(X=O)_<exp

~

var(X)

P(X = 0) <

H(1-EIA),

(3)

A

where e = max EIA. This version puts emphasis on comparison with the independent case. Under modest symmetry conditions, inequality (2) appeared in [5], while the general version was proved in [3]. See also [7] for another proof of (2), but with an extra factor 1/2 in the exponent. The quantity A is a measure of the pairwise dependence between the IAS. If A = o(A), then the exponents in (1) and (2) are both equal to - E X ( 1 + o(1)), matching asymptotically a lower bound obtained via the F K G i n e q u a l i t y , provided further maxpi --+ 0. Example. Let n be an integer, n > 3, and let F be the set of all two-element subsets of { 1 , . . . , n } . With all Pi = P : p(n), i = 1 , . . . , (~), the random subset Fp is a random graph G(n,p). Let $ be the family of all triples of pairs of the form {ij, i k , j k } , 1 <_ i < j <

2nexp{-(n/~OO)p3+O(n4p5)}

= o(1).

In 1990, Janson [3] generalized inequality (2), obtaining an estimate of the lower tail of the distribution of X. W i t h ¢ ( x ) = ( l + x ) l n ( l + x ) - x , for0
<exp --

¢(-t/A)12 A

<exp

-~-~

--

(4) When $ consists of mutually disjoint sets, X is a sum of independent zero-one random variables and (4) coincides with the (one-sided) Chernoff bound. No corresponding upper tail estimate is true in general. Also in 1990, W.C.S. Suen [8] proved a related inequality, which remains valid for a general probability space (and not only for random subsets). In his setting, X is a sum of arbitrary indicators In, and the bound hinges on the dependency graph induced by them. Suen's inequality has been revived and extended

in [4]. Fore more details on this subject see [6]. References [1] BOLLOBJ~S, B.: 'The chromatic number of random graphs',

Combinatorica 8 (1988), 49-56. [2] BOPPANA, R.~ AND SPENCER,J.: 'A useful elementary corre-

lation inequality', J. Combin. Th. A 50 (1989), 305-307. [3] JANSON, S.: 'Poisson approximation for large deviations', Random Struct. Algor. 1 (1990), 221-230. [4] JANSON, S.: 'New versions of Suen's correlation inequality', Random Struct. Algor. 13 (1998), 467-483. [5] JANSON, S., LUCZAK, T., AND RUCII~SKI, A.: 'An exponential bound for the probability of nonexistence of a specified subgraph in a random graph', in M. KARONSKI, J. JAWORSKI, AND A. RUCINSKI (eds.): Random Graphs '87, Wiley, 1990, pp. 73-87. [6] JANSON, S., LUCZAK, T., AND RUCII~SKI, A.: Random graphs, Wiley, 2000. [7] SPENCER, J.: 'Threshold functions for extension statements', J. Combin. Th. A 53 (1990), 286 305.

217

JANSON I N E Q U A L I T Y [8] SUEN, W.C.S.: 'A correlation inequality and a Poisson limit theorem for nonoverlapping balanced subgraphs of a random graph', Random Struct. A19or. 1 (1990), 231-242. A. Rucigski

MSC 1991: 05C80, 60D05 J B * - T R I P L E - JB*-triples were introduced by W. Kaup [8] in connection with the study of bounded symmetric domains in compIex Banach spaces. A definition of JB*-triples involving holomorphy is as follows: A JB* triple is a complex B a n a c h s p a c e E such that the open unit ball B of E is homogeneous under its full group G of biholomorphic automorphisms (and hence is symmetric, cf. S y m m e t r i c space). The main result in [8] states t h a t to every abstract bounded symmetric domain D in a complex Banach space there exists a unique (up to linear isometry) JB*-triple E whose open unit ball is biholomorphically equivalent to D. The group G is always a real Banach Lie group (of. also Lie g r o u p , B a n a c h ) acting in a natural way on various spaces of holomorphic functions as well as on various submanifolds of the unit sphere in E (in case E has finite dimension, G is semi-simple and also has finite dimension - - the induced u n i t a r y r e p r e s e n t a t i o n on Bergman space, cf. also B e r g m a n spaces, is of special interest in h a r m o n i c analysis). An equivalent, but more algebraic definition for JB*triples is as follows: The complex Banach space E is a JB*-triple if it carries a (necessarily unique) ternary product (called triple product) E x E x E - + E,

( x , y , z ) ~-+ { x y z }

satisfying the following properties for all a, b, x, y, z E E and aV]b* : E --+ E defined by z ~-~ {abz}: i) { x y z } is symmetric complex bilinear in the outer variables x, z and conjugate linear in y; ii) [aDb*, x[:]y*] = {abx}[:]y* - x [ 3 { y a b } * (the Jordan triple identity); iii) aE]a*, as a l i n e a r o p e r a t o r on E, is Hermitian and has spectrum > 0 (cf. also H e r m i t i a n o p e r a t o r ; Spectrum of an operator); iv) Ilarga*li = ilall 2 (the C*-condition). The sesquilinear mapping (a, b) ~ aDb* may be considered as an operator-valued product on E. It satisfies IlaDb*lt < Ilall. Ilbll (not an elementary fact!) and condition iv) is analogous to the characteristic property of C*-algebras (cf. also C * - a l g e b r a ) . On the other hand, by iii), the above mapping may also be considered as a positive Hermitian operator-valued form on E, thus giving a natural orthogonality relation on E. Some examples are: 218

1) Every C*-algebra (more precisely, the underlying complex Banach space). The triple product is given by

{xyz} = (xy*z + zy*x)/2. 2) Every closed (complex) subtriple of a C*-algebra. These are also called JC*-triples. These triples were originally introduced and intensively studied by L.A. Harris [5] under the name J*-algebra (cf. also B a n a c h Jordan algebra). 3) Every JB*-algebra (i.e. Jordan C*-algebra, [15]), with { x y z } = x o (y* o z) + z o (y* o x) - (x o z) o y*. In particular, the famous exceptional 27-dimensional JB*algebra ~3 (O c) (which is not a JC*-triple) of all Hermitian (3 x 3)-matrices over the complex octonian algebra. 4) Every JBW*-triple, i.e. a JB*-triple having a (necessarily unique) pre-dual. Among these are the w*closed subtriples of yon Neumann algebras as well as the Cartan factors, which are the building blocks of the JBW*-triples of type I (in analogy to the von Neumann algebras of type I, cf. also v o n N e u m a n n a l g e b r a ) . The class of all JB*-triples is invariant under taking arbitrary/m-sums, quotients by closed triple ideals, ultrapowers, biduals [1], as well as contractive projections [9]. Notice that the range of a contractive projection on a C*-algebra in general does not have the structure of a C*-algebra, but always is a JC*-triple. The G e l ' f a n d - N a f m a r k theorem of Y. Friedman and B. Russo [3] states that each JB*-triple can be realized as a subtriple of a n / a - s u m A ® B where A is the JBW*-triple of all bounded linear operators on a suitable complex H i l b e r t s p a c e and B is the exceptional JB*-algebra of all 7/3 (OC)-valued continuous functions on a suitable compact t o p o l o g i c a l space. By [6], the classification of JBW*-triples can be achieved modulo the classification of yon Neumann algebras. Furthermore, in [11] all prime JB*-triples have been classified using Zel'manov techniques. The JB*-triples form a large class of complex Banach spaces whose geometry can be described algebraically. Examples of this are: • A bijective linear operator between JB*-triples is an isometry if and only if it respects the Jordan triple product. • The M-ideals in E are precisely the closed triple ideals of E. • The open unit ball of E is the largest convex subset B C E containing the origin such that for every a C B tile Bergman operator z ~+ z - 2{aaz} + { a { a z a } a } is invertible, and a E E is an extreme point of the closed unit ball in E if and only if the Bergman operator associated to a is the zero operator. Real JB*-triples were studied in [7]; these are the real forms of (complex) JB*-triples. In general, a JB*-triple

J O N E S - C O N W A Y POLYNOMIAL may have many non-isomorphic real forms. An important class of real JB*-triples is obtained from the class of JB-algebras, compare [4].

References [1] DINEEN, S.: 'Complete holomorphic vector fields on the second dual of a Banach space', Math. Scand. 59 (1986), 131-42. [2] EDWARDS, C.M., McCRIMMON, K., AND RUTTIMANN, G.T.: 'The range of a structural projection', J. Funct. Anal. 139 (1996), 196-224. [3] FRIEDMAN, Y., AND RUSSO, B.: 'The Gelfand-Naimark theorem for JB*-triples', Duke Math. J. 53 (1986), 139-148. [4] HANCHE-OLSEN, H., AND STORMER, E.: Jordan operator algebras, Vol. 21 of Mon. Stud. Math., Pitman, 1984. [5] HARRIS, L.A.: Bounded symmetric homogeneous domains in infinite dimensional spaces, Vol. 364 of Lecture Notes in Math., Springer, 1973. [6] HORN, G., AND NEHER, E.: 'Classification of continuous J B W * - t r i p l e s ' , Trans. Amer. Math. Soc. 306 (1988), 553578. [7"] ISIDRO, J.M., KAUP, W., AND RODRfGUEZ, A.: 'On real forms of JB*-triples', Manuscripta Math. 86 (1995), 311-335. [8] KAUP, W.: 'A Riemann mapping theorem for bounded symmetric domains in complex Banaeh spaces', Math. Z. 183 (1983), 503-529. [9] KAUP, W.: ~Contractive projections on Jordan C*-algebras and generalizations', Math. Scan& 54 (1984), 95-100. [10] Loos, O.: 'Bounded symmetric domains and Jordan pairs', Math. Lectures. Univ. California at Irvine (1977). [11] MORENO, A., AND RODRIGUEZ, A.: 'On the Zelmanovian classification of prime JB*-triples', J. Algebra 226 (2000), 577613. [12] Russo, B.: 'Stucture of JB*-triples': Proc. Oberwolfach Conf. Jordan Algebras, 1992, de Gruyter, 1994. [13] UPMEIER,H.: Symmetric Banach manifolds and Jordan C*algebras, Vol. 104 of Math. Studies, North-Holland, 1985. [14] UPMEIER, H.; Jordan algebras in analysis, operator theory and quantum mechanics, Vol. 67 of Regional Conf. Set. Math., Amer. Math. Soc., 1987. [15] WRIGHT, J.D.M.: 'Jordan C*-algebras', Michigan Math. J. 24 (1977), 291-302. Wilhelm Kaup

MSC1991: 46-XX, 17Cxx JONES-CONWAY POLYNOMIAL, Homily polynomial, Homflypt polynomial, skein polynomial - An invariant of oriented links. It is a Laurent polynomial of two variables associated to ambient isotopy classes of oriented links in R 3 (or $3), constructed in 1984 by several groups of researchers (thus the acronyms Homily and Homflypt) [3], [4], [14], [22], and denoted by PLO. It generalizes the A l e x a n d e r - C o n w a y p o l y n o m i a l and the Jones polynomial. There are several constructions of the polynomial, using diagrams of links (and Reidemeister moves, cf. also R e i d e m e i s t e r t h e o r e m ) , the braid group (and the Alexander and Markov theorems, cf. also B r a i d theory; A l e x a n d e r t h e o r e m o n b r a i d s ; M a r k o v b r a i d t h e o r e m ) , or statistical mechanics (interpreting

the polynomial as a state sum, cf. also S t a t i s t i c a l m e chanics, m a t h e m a t i c a l p r o b l e m s in). The first approach uses the recursive (skein) relation

v - l pL+ (v, z)

-

vPL_ (V, Z) = ZPLo (V, z),

where L+, L_ and L0 form a C o n w a y s k e i n t r i p l e . The Jones-Conway polynomial is usually normalized to be 1 for the trivial knot. Then for the trivial link of n components, Tn, one gets PTo(v,z)=

v

Setting v = t and z = v/t - 1/v/t yields the Jones polynomial, VL(t), and substituting v = 1 and z = v q - 1 / v ~ yields the A l e x a n d e r - C o n w a y p o l y n o m i a l . In the second approach one considers the Hecke algebra associated to the Artin braid group and constructs on it the Jones-Ocneanu trace, which essentially is invariant under Markov moves. This approach is strongly related to the first approach, as the Hecke algebra quadratic relation is analogous to the skein relation of the first method. The third approach uses the fact that the statistical mechanical systems considered satisfy the Y a n g B a x t e r e q u a t i o n . This approach is immediately related to the first two by the fact that a Yang-Baxter operator satisfies the minimal polynomial (leading to a skein relation) and, on the other hand, yields a linear representation of the braid group. This state sum approach was first developed by V.F.R. Jones and has a very nice reflection in the J a e g e r composition prod-

uct [6]. Properties o/ the Jones-Conway polynomial. 1) PL(V, Z) is an element of the Laurent polynomial ring Z[v ±1, z±l]. Furthermore, (vz)C°m(L)-IpL(V , Z) e Z[v ±2, z2], where corn(L) denotes the number of components of L. In particular,

PL(v,z) -= PL(--v,--z) = (--1)c°m(L)-IPL(--V,Z). For example, for the right-handed Hopf link, 21, one has V -- V3

P21 - - -

z

+ vz.

2) If L denotes the mirror image of a link L, then

PT(v,z) = p n ( - - v - l , z ) . This property often allows one to detect lack of amphicheirality of a link. For example, for the right-handed trefoil knot, 31, one has P31 = 2v 2 - v 4 + v2z 2, so the trefoil knot is not amphicheiral. For the figure eight knot, 41, which is amphicheiral, one gets P41 = v -2 1 + v 2 - z 2. The first non-amphicheiral knot not detected by the Jones-Conway polynomial is the 942 knot. 219

JONES-CONWAY POLYNOMIAL The K a u f f m a n p o l y n o m i a l also does not detect nonamphicheirality of 942. However, one can use the JonesConway polynomial of the 2-cable of 942 to see that it is non-amphicheiral. 3) Next, V-1 --V

PLI#L2 = PL1PL2

and

PLlUL2

-

-

--IPL1PL2, Z

where # denotes the c o n n e c t e d s u m and tO the split sum of links. Because the connected sum of links may depend on the choice of connected components, one can use the connected sum formula to find different links with the same Jones-Conway polynomial, for example the connected stun of three Hopf links can give two different results, both with the polynomial equal to

((v

-

a(n+_n__s(DL)+l),(n_s(DL)+l) 7£ 0

if and only if DL is a positive diagram (i.e. n_ = 0). For example, for a positive diagram of the right-handed trefoil knot with 3 crossings one has s(D31) = 2 and a 2 , 2 = 1.

8) For a knot K, PK(V, z) -- 1 is a multiple of (v -1 V) 2 -- Z 2.

For example, P41(v,z) - 1 = (v -1 - v) 2 - z 2 =

-v-2(P3~ (v, z) - 1) = -v2(ph~ (v, z) - 1). Setting PK(V,Z)=

va)lz + w ) ~.

P~(D1,D=)(V) = P~(D1)(v)P~(D=)(V),

where DI*D2 denotes the planar star (Murasugi) product of the diagrams Dz and D2 [18]. D.L. Vertigan proved that for a fixed i, Pi (v) can be computed in polynomial time on the number of crossings [20]. 6) If PL(v,z) = ~i=~a~(z)v E i (with a¢(z) ~ 0, aE(z) ~ 0), then

(n+--n_)--(s(DL)--l)

PK(V,Z)-- I (v-i

4) If L - denotes the link obtained by reversing the orientation of L, then the Jones-Conway polynomials of L and L - coincide. Thus, each non-reversible knot (817 is the smallest example) gives rise to an example of different knots with the same Jones Conway polynomial. Furthermore, if Kz and/(2 are two non-reversible knots, then/(z ~K2 and Kz ~K 2 are different knots which cannot be distinguished by the Jones-Conway polynomials (nor by the Kauffman polynomials) of their satellites. It is an open problem (as of 2001) whether they can be distinguished by any Vassiliev-Gusarov invariants. If PL(V,Z) = EiM=mPi(v)z i (with P,~(v) # O, PM(V) ¢ 0), then m = 1 - com(L) and M < cr(DL) -s(DL) + 1, where DL is any diagram of L, cr(DL) denotes the number of crossings of DL and S(DL) is the number of Seifert circles of DL. The equality holds, e.g., for homogeneous diagrams (including positive and alternating diagrams). In particular for a non-trivial knot K , M < cr(K) where cr(K) is the crossing number of K , i.e. the minimum over all diagrams of L of the crossing number. 5) If ~(D) = cr(DL) -- S(DL) + 1, then

< e < E <_

(n+--n_)+(S(DL)--l),

where n+ (respectively, n_) is the number of positive (respectively, negative) crossings of the diagram DL. In particular, s(L) > (E - e)/2, where s(L) is the minimal number of Seifert circles of all diagrams DL representing L; it is equal to the braid index of L, see [24]. The knot 942 is the first knot for which the inequality is sharp. 220

The smallest (known) alternating knot for which the inequality is sharp has 18 crossings [19]. 7) If PL(V, z) = ~ ai,jv% j, then

_

_

then PK+ (V, Z) -- PK_ (V, Z) = lk(K0)

n o d (v 2 - 1, z),

where lk(K0) is the linking number of K0. In particular, PK(1,0) = a2 (the second coefficient of the A l e x a n d e r - C o n w a y p o l y n o m i a l and the first nontrivial Vassiliev-Gusarov invariant). More generally, consider PL(V,Z) as an element of the ring R that is the subring of Z [ v + l , z i l ] generated by v +1, z and (v -1 - v ) / z. Then PL ( V, z) -- PToom(L) (% Z) is a multiple of z(((v -1 - v)/z) 2 - 1). Furthermore, PL(V,Z) -- P%om(L)(V,Z)

rood

in R, where lk(L) is the linking number of L. In particular,

P2~ (v, z) z ((~Y-)2

v-~-v z

~--V.

2 1)

The number of components of a link and its linking numher can be recovered from the Jones-Conway polynomial. 9) If PK(v, z) = v 2c ~ ci,j(v 2 - 1 ) i z j for a knot K and some constant c, then ci,j is a Vassiliev-Gusarov invariant of order i + j. Equivalently, PK(V,Z) n o d (((v 2 1), z) k+l) is a Vassiliev-Gusarov invariant of order k. One can obtain a coloured Jones-Conway polynomial by choosing an element of the Jones-Conway s k e i n n o d ule of the solid torus (a linear combination of links in a solid toms) and computing for this the Jones-Conway polynomial. Such an invariant is stratified by VassilievGusarov invariants. Computing the whole Jones-Conway polynomial is A/P-hard [5] (so up to the conjecture that AfT) # 7), the polynomial cannot be computed in polynomial time).

JONES UNKNOTTING

F u r t h e r m o r e , c o m p u t i n g m o s t of t h e s u b s t i t u t i o n s to t h e J o n e s - C o n w a y p o l y n o m i a l is i V ' P - h a r d (even # P - h a r d ) . T h e e x c e p t i o n s h a v e well u n d e r s t o o d i n t e r p r e t a t i o n s [7]: i) v =

4-1. N o w PL reduces to t h e A l e x a n d e r -

Conway polynomial. ii) z = ± ( v - 1 - v). C o m p a r e 8) above. iii) (v, z) = (4-i, 4-ix/2), where

the

+'s

are independent. For example, P ( i , ix/~) = (--v~)C°m(L)--l(--1)Arf(L) if t h e A r f - i n v a r i a n t is defined a n d 0 o t h e r w i s e . iv) (v, z) = (+i, +i). E.g., PL (i, i) = ( i v Y ) dim(ul(M(3),Z:)), where M (k) denotes t h e k-fold cyclic covering of S 3 b r a n c h e d over L. v) ( v , z ) = (±e-4-~/3,4-i). E.g., P L ( e ' i / 3 , i ) ¢ ( L ) i c o m ( i ) - l ( i x / r ~ ) d i m ( H l ( M (2),Z3)), where ¢(L) =

= 4-1

can b e d e r i v e d from t h e m o d u l o 3 linking form in M(2). T h e r e are several c o n s t r u c t i o n s of different links w i t h the same Jones-Conway polynomial: mutation, rotation, cabling, s p e c t r a l p a r a m e t e r tangle, etc. [21]. T. K a n e n o b u [10] has c o n s t r u c t e d an infinite f a m i l y of different links w i t h t h e s a m e J o n e s - C o n w a y p o l y n o m i a l . It is an o p e n p r o b l e m (as of 2001) w h e t h e r t h e r e exists a n o n - t r i v i a l link w i t h J o n e s - C o n w a y p o l y n o m i a l equal to t h a t of a t r i v i a l link (cf. J o n e s u n k n o t t i n g c o n j e c ture). References [1] CROMWELL, P.R.: 'Homogeneous links', Y. London Math. Soc. 39, no. 2 (1989), 535-552. [2] FRANKS,J., AND WILLIAMS,R.F.: 'Braids and the Jones polynomial', Trans. Amer. Math. Soc. 303 (1987), 97-108. [3] FREYD, P., YETTER, D., HOSTE, J., LICKORISH, W.B.R., MILLETT, K., AND OCNEANU, A.: 'A new polynomial invariant of knots and links', Bull. Amer. Math. Soc. 12 (1985), 239-249. [4] HOSTE, J.: 'A polynomial invariant of knots and links', Pacific J. Math. 124 (1986), 295-320. [5] JAEGER, F.: 'On Tutte polynomials and link polynomials', Proc. Amer. Math. Soc. 103, no. 2 (1988), 647-654. [6] JAEGER, F.: 'Composition products and models for the Homily polynomial', L'Enseign. Math. 35 (1989), 323-361. [7] JAEGER, F., VERTIGAN, D.L., AND WELSH, D.J.A.: 'On the computational complexity of the Jones and T~tte polynomials', Math. Proc. Cambridge Philos. Soc. 108 (1990), 35-53. [8] JONES, V.F.R.: 'Hecke algebra representations of braid groups and link polynomials', Ann. of Math. 126, no. 2 (1987), 335-388. [9] JONES, V.F.R.: 'On knot invariants related to some statistical mechanical models', Pacific J. Math. 137, no. 2 (1989), 311-334. [10] KANENOBU, T.: 'Infinitely many knots with the same polynomial invariant', Proc. Amer. Math. Soc. 97, no. 1 (1986), 158-162. [11] KANIA-BARTOSZY~SKA,J., AND PRZYTYCKI, J.H.: 'Knots and links, revisited', Delta, Warsaw J u n e (1985), 10-12. (In Polish.)

CONJECTURE

[12] KOBAYASHI,K., AND KODAMA,K.: 'On the deg z PL(v,z) for plumbing diagrams and oriented arborescent links', Kobe J. Math. 5 (1988), 221-232. [13] LICKORISH, W.B.R.: An introduction to knot theory, Springer, 1997. [14] LICKORISH,W.B.R., AND MILLETT, K.: 'A polynomial invariant of oriented links', Topology 26 (1987), 107-141. [15] MORTON, H.R.: 'Seifert circles and knot polynomials', Math. Proc. Cambridge Philos. Soc. 99 (1986), 107-109. [16] MORTON, H.R., AND SHORT, H.B.: 'The 2-variable polynomial of cable knots', Math. Proc. Cambridge Philos. Soc. 101 (1987), 267-278. [17] MURAKAMI, H.: 'On derivatives of the Jones polynomial', Kobe J. Math. 3 (1986), 61-64. [18] MURASUGI,K., AND PRZYTYCKI, J.H.: 'The Skein polynomial of a planar star product of two links', Math. Proe. Cambridge Philos. Soe. 106 (1989), 273-276. [19] MURASUGI, K., AND PRZYTYCKI, J.H.: An index of a graph with applications to knot theory, Vol. 106 of Memoirs, Amer. Math. Soc., 1993. [20] PRZYTYCKA,T.M., AND PRZYTYCKI, J.H.: 'Subexponentially computable truncations of Jones-type polynomials': Graph Structure Theory, Vol. 147 of Contemp. Math., 1993, pp. 63108. [21] PRZYTYCKI, J.H.: 'Search for different links with the same Jones' type polynomials': Ideas from Graph Theory and Statistical Mechanics, Panoramas of Mathematics, Vol. 34, Banach Center Publ., 1995, pp. 121-148. [22] PRZYTYCKI, J.H., AND TRACZYK, P.: 'Invariants of links of Conway type', Kobe J. Math. 4 (1987), 115-139. [23] TURAEV, V.G.: 'The Yang-Baxter equation and invariants of links', Invent. Math. 92 (1988), 527-553. [24] YAMADA,S.: 'The minimal number of Seifert circles equals to braid index of a link', Invent. Math. 89 (1987), 347-356. Jozef Przytycki MSC 1991:57M25

JONES U N K N O T T I N G C O N J E C T U R E Every n o n - t r i v i a l k n o t h a s a n o n - t r i v i a l Jones p o l y n o m i a l .

Fig. 1. T h e c o n j e c t u r e h a s b e e n confirmed for several families of k n o t s , i n c l u d i n g a l t e r n a t i n g a n d a d e q u a t e k n o t s , k n o t s u p to 18 crossings a n d 2 - a l g e b r a i c k n o t s (cf. K n o t t h e o r y ) u p to 21 crossings [5], [7]. R e c e n t l y (2001), S. Y a m a d a a n n o u n c e d t h a t t h e c o n j e c t u r e holds for k n o t s w i t h up to 20 crossings. T h e a n a l o g o u s conjecture for links does n o t hold, as M.B. T h i s t l e t h w a i t e found a 15crossing link whose J o n e s p o l y n o m i a l coincides w i t h a 221

JONES U N K N O T T I N G C O N J E C T U R E trivial link of two components, cf. Fig. 1. This and similar examples constructed since are 2-satellites on a Hopf link [6], [1]. L.H. Kauffman showed that there are non-trivial virtual knots with Jones polynomial equal to 1, [4]. It is still an open problem (as of 2001) whether a simple (non-satellite) link can have a Jones polynomial of an unlink. References [1] ELIAHOU, S., I(AUFFMAN, L.H., AND THISTLETHWAITE, M.: 'Infinite families of links with trivial Jones polynomial', preprint (2001). [2] JONES, V.F.R.: 'Hecke algebra representations of braid groups and link polynomials', Ann. of Math. 126, no. 2 (1987), 335-388. [3] JONES, V.F.R.: 'Ten problems': Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000, pp. 79-91. [4] KAUFFMAN, L.H.: 'A survey of virtual knot theory': Knots in Hellas '98, Vol. 24 of Set. on Knots and Everything, 2000, pp. 143-202. [5] LICKORISH, W.B.R., AND THISTLETHWAITE, M.B.: 'Some links with non-trivial polynomials and their crossingnumbers', Comment. Math. Helv. 63 (1988), 527-539. [6] THISTLETHWAITE, 7~[.B.: 'Links with trivial Jones polynomial', J. Knot Th. Ramifications 10, no. 4 (2001), 641-643. [7] YAMADA, S.: 'How to find knots with unit Jones polynomials': Knot Theory, Proe. Conf. Dedicated to Professor Kunio Murasugi for his 70th Birthday (Toronto, July 13th-1Yth 1999), 2000, pp. 355-361.

Jozef Przytycki MSC 1991:57M25 J O R D A N TRIPLE SYSTEM - A triple system closely

related to Jordan algebras. 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 mapping V x V x V ~ V, called a triple product and usually denoted by {.,-,.} (sometimes dropping the commas). It is said to be a Jordan triple system if {uvw} = {wvu}, =

(1) (2)

with u , v , x , y , w E V. From the algebraic viewpoint, a Jordan triple system (1/, {xyz}) is a Lie t r i p l e s y s t e m with respect to the new triple product [xyz] := { x y z } - {yxz}. This implies that all simple Lie algebras over an algeb r a i c a l l y c l o s e d field of characteristic zero, except G2, F4 and Es (cf. also Lie a l g e b r a ) , can be constructed using the standard embedding Lie algebra associated with a Lie triple system via a Lie triple system. From the geometrical viewpoint there is, for example, a correspondence between symmetric R-spaces and 222

compact Jordan triple systems [3] as well as a correspondence between bounded symmetric domains and Hermitian Jordan triple systems [2]. For superversions of this triple system, see [5]. E x a m p l e s . Let D be an associative algebra over K (cf. also 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 ) and set V := Mat(p, q; D), the (p x q)-matrices over D. This vector space V is a Jordan triple system with respect to the product { x y z } = x y t z + zytx, where yt denotes the transpose matrix of y. Let V be a vector space over K equipped with a symmetric bilinear form (x,y). Then V is a Jordan triple system with respect to the product {xyz} = <x,y> z + (y,z) x - Y. Let V be a commutative J o r d a n a l g e b r a . Then V is a Jordan triple system with respect to the product { x y z } = x ( W ) + (xy)z - y(xz). Note that a triple system in this sense is completely different from, e.g., the combinatorial notion of a Steiner triple system (cf. also S t e i n e r s y s t e m ) . References [1] JACOBSON, N.: 'Lie and Jordan triple systems', Amer. J. Math. 71 (1949), 149-170. [2] KAUP, W.: 'Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains': Non Associative Algebra and Its Applications (Oviedo, 1993), Kluwer Acad. PUN., 1994, pp. 204-214. [3] Loos, O.: 'Jordan triple systems, R-symmetric spaces, and bounded symmetric domains', Bull. Amer. Math. Soc. 77 (1971), 558-561. [4] NEHR, E.: Jordan triple systems by the graid approach, Vol. 1280 of Lecture Notes in Mathematics, Springer, 1987. [5] OKUBO, S., AND KAMIYA, N.: 'Jordan-Lie super algebra and Jordan-Lie triple system', J. Algebra 198, no. 2 (1997), 388 411.

Noriaki Kamiya MSC 1991:17A40 JULIA-WOLFF-CARATHI~ODORY

THEOREM,

Julia-Carathdodory theorem, Julia-Wolff theorem - A classical statement which combines the celebrated J u lia t h e o r e m fi'om 1920 [18], Carath6odory's contribution from 1929 [7] (see also [8]), and Wolff's boundary version of the S c h w a r z l e m m a from 1926 [31]. Let A be the open unit disc in the complex plane C, and let Hol(A, t2) be the set of all holomorphic functions on A with values in a domain f / i n C (cf. also A n a l y t i c f u n c t i o n ) . For the set Hol(A, A), of holomorphic selfmappings on A, one writes Hol(A); it is a s e m i - g r o u p o f h o l o m o r p h i c m a p p i n g s with respect to composition.

JULIA-WOLFF - C A R A T I ~ O D O R Y THEOREM For w on the unit circle 0A, the boundary of A, and c~ > 1, a non-tangential approach region at w is defined by r(~,~)--{zeZX:

Iz-~l

<~(1-lzt)

}.

(1)

The term 'non-tangential' refers to the fact that at the point w, F(~, a) lies in the sector S (in A) that is the region between two straight lines in A meeting at ~ and symmetric about the radius to w, the boundary curves of S having a corner at ~, with angle less than 7r. A function f E H o l ( A , C ) is said to have a nontangential limit L at w if L = l i m ~ f ( z ) exists in each non-tangential region F(w, a). In this ease one also writes L = Z Zlim f(z). -~Ld For z E A and c~ E 0A, let

Iz - ¥ - iT l ,

(2)

and for k > 0, let E ( k , ~ ) = {z E ZX: ¢~(z) < k}.

(3)

The set E ( k , w ) is a closed disc internally tangent to the circle at w with centre (1/(1 + k))w and radius k/(1 + k). Such a disc is called a horodisc (cf. also H o r o sphere). In 1920, G. Julia [18] identified hypotheses showing how to get the existence of the non-tangential limit at a given boundary point. J u l i a ' s l e m m a . Let F E Hol(A) be not constant. Suppose that there are points w and r/on the boundary 0A, such that for a sequence {z~} C A converging to c~ the sequence {F(z~)} converges to ~ and 1 -If(z~)l

+ d(w) < ec.

(4)

1 - Iz l Then

i) d(w) > 0; ii) ¢ , ( F ( z ) ) < d(cJ)¢~(z), i.e. F ( E ( k , w ) ) E(d(c~)k, ~]) for all k > 0; iii) Z l i m z - ~ F(z) exists and is equal to 7.

C_

Moreover, if the equality in ii) holds for some z E A, then F is an automorphism of the disc. Julia-Carath6odory theorem. In 1929, C. Carath6odory [7] proved that under Julia's hypotheses the derivative also admits a non-tangential limit at the same boundary point. Suppose F E Hol(A). Then the following statements are equivalent: i) liminfz_+~(1-1F(z)I)/(1-1zl) = d(cu) < oc, where the limit is taken as z approaches w unrestrictedly in A; ii) Z l i m z ~ o ( F ( z ) - rl)/(z - w) = ZF'(w) exists for some r1 E 0A;

iii) £ limz-+~ F'(z) exists, and /limz-+w F ( z ) = r1 E 0A. Moreover, a) d(cJ) > 0 in i); b) the boundary points r/in ii) and iii) are the same; c) Zlimz_+~ F'(z) = XF'(w) = w~d(w). After appropriate preliminary rotations, one may assume that w = rI. Thus, these results show that if F has an angular

derivative

at some

boundary

point w such

that

ZlimF(z)

=w,

and

ZF~(w) < 1 ,

z-+co

then F cannot have an interior fixed point in A. Now assume only that F has no interior fixed point in A. The question then is: Does the angular derivative at a certain point on the boundary exist? The affirmative answer was given by J. Wolff [31] in 1926. W o l t F s t h e o r e m . Suppose F E Hol(A) has no fixed point in A. Then there is a unique unimodular point w E 0A such that i) Z l i m z _ ~ F ( z ) = c~; ii) Cw(F(z)) < ¢~(z); iii) ZF~(z) exists and is less than or equal to 1. The latter assertion can be interpreted as a direct analogue of the Schwarz Pick lemma (cf. S c h w a r z l e m m a ) , where the role of the fixed point is taken over by a point on the unit circle. Moreover, this result is the key to all the deeper facts about sequences of iterates. G e n e r a l i z a t i o n s . There are various versions and proofs of the Julia-Carathdodory theorem (sometimes also called the Julia-Wolff-Carathgodory theorem or JuliaWolff theorem). For the one-dimensional case, see, for example, [19], [29], [27], [14], [26], [6], [21] or [8], [22], [28], [9]. Note that D. Sarason [26] gave an interesting proof of the Julia-Carath6odory theorem by using Hilbert space constructions for angular derivatives. A strengthened version of Julia's lemma was established by P.R. Mercer [21], employing techniques for the hyperbolic Poincar6 metric (cf. P o i n c a r 6 m o d e l ) . Different generalizations of the Julia-WolffCarath6odory theorem for bounded domains in C a are known: for the unit ball in C ~ ([16], [25]), for the poly-disc ([17], [3]), for strongly convex and strongly pseudo-convex domains ([1], [2]). Also, M. Abate and R. Tauraso [4] have described a general framework allowing one to generalize the Julia-Wolff-Carath6odory theorem in terms of the Kobayashi metric (cf. also H y p e r b o l i c m e t r i c ; K o b a y a s h i h y p e r b o l i c i t y ) on a bounded domain in C n. 223

JULIA-WOLFF -CARATH~ODORY THEOREM For generalizations of Wolff's theorem in the unit ball of a complex Hilbert space, see [12] and [13]. Earlier, V.P. Potapov [23] extended Julia's lemma to matrix-valued holomorphic mappings of a complex variable. His results, as well as the Julia-WolffCarath6odory theorem, were generalized by K. Fan and T. Ando ([10], [11] and [5]) to operator-valued holomorphic mappings. Also, in these works they extended the Julia-Wolff-Carath6odory theorem to holomorphic mappings of proper contractions on the unit Hilbert bali acting in the sense of functional calculus. K. Wlodarczyk [30] and P. Mellon [20] have presented some more general results in this direction for the holomorphic mappings on the open unit ball of so-called J*algebras, using techniques developed by L.A. Harris [15]. For a survey of work in higher dimensions, see [25], [13], [9], [24], [20], [4]. References [1] ABATE, M.: 'The LindelSf principle and the angular derivarive in strongly convex domains', J. Anal. Math. 54 (i990), 189-228. [2] ABATE, M.: 'Angular derivatives in strongly pseudoconvex domains': Proc. Syrup. Pure Math., Vol. 52/2, Amer. Math. Soc., 1991, pp. 23-40. [3] ABATE, M.: 'The Julia WoIff-Caratheodory theorem in polydisks', J. Anal. Math. 74 (1998). [4] ABATE, M., AND TAURASO, R.: 'The J u l i ~ W o l f f Caratheodory theorem(s)', Contemp. Math. 222 (1999), 161-172. [5] ANDO, T., AND FAN, K.: 'Pick Julia theorems for operators', Math. Z. 168 (1979), 23-34. [6] BURCKEL, R.B.: 'Iterating analytic self-maps of discs', Amer. Math. Monthly 88 (1981), 396-407. [7] CARATHEODORY, C.: 'Uber die Winkelderivierten von beschr/~nkten Analytischen Funktionen', Sitzungsber. Preuss. Akad. Wiss. Berlin, Phys.-Math. Kl. (1929), 39-54. [8] CARATHEODORY,C.: Theory of functions of a complex variable, Chelsea, 1954. [9] COW,N, C.C., AND MACCLUER, B.D.: Composition operators on spaces of analytic functions, CRC, 1995. [10] FAN, K.: 'Julia's lemma for operators', Math. Ann. 239 (1979), 241-245. [11] FAN, K.: 'Iterations of analytic functions of operators', Math. Z. 179 (1982), 293-298.

224

[12] GOEBEL, K.: 'Fixed points and invariant domains of holomorphic mappings of the Hilbert ball', Nonlin. Anal. 6 (1982), 1327--1334. [13] GOEBEL, K., AND REICH, S.: Uniform convexity, hyperbolic geometry and nonexpansive mappings, M. Dekker, 1984. [14] GOLDBERG, J.L.: 'Functions with positive real part in a halfplane', Duke Math. d. 29 (1962), 335-339. [15] HARRIS, L.A.: Bounded symmetric homogeneous domains in infinite-dimensional space, Vol. 364 of Lecture Notes in math., Springer, 1974, pp. 13-40. [16] HERV~, M.: 'Quelques propri6tfs des applications analytiques d'une boule ~ m dimensions dans elle-m~me', J. Math. Pures Appl. 42 (1963), 117-147. [17] JAFARI, F.: 'Angular derivatives in polydisks', Indian J. Math. 35 (1993), 197-212. [18] JULIA, G.: 'Extension nouvelle d'un lemme de Schwarz', Acta Math. 42 (1920), 349 355. [19] LANDAU,E., AND VALIRON, G.: 'A deduction from Schwarz's lemma', d. London Math. Soc. 4 (1929), 162-163. [20] MELLON, P.: 'Another look at results of Wolff and Julia type for d*-algebras', d. Math. Anal. AppL 198 (1996), 444-457. [21] MERCER, P.R.: 'On a strengthened Schwarz-Piek inequality', J. Math. Anal. Appl. 234 (1999), 735-739. [22] NEVANLINNA,R.: Analytic functions, Springer, 1970. [23] POTAPOV, V.P.: 'The multiplieative study of J-contractive matrix functions', Amer. Math. Soc. Transl. (2) 15 (1960), 231 243. [24] REICH, S., AND SHOIKHET, D.: 'The Denjoy-Wolff theorem', Ann. Univ. Mariae Curie-Sktodowska 51 (1997), 219-240. [25] RUDIN, W.: Function theory on the unit ball in C ~, Springer, 1980. [26] SARASON, D.: 'Angular derivatives via Hilbert space', Complex Variables 10 (1988), 1-10. [27] SERRIN, J.: 'A note on harmonic functions defined in a halfplane', Duke Math. J. 23 (1956), 523-526. [28] SHAPIRO, J.H.: Composition operators and classical function theory, Springer, 1993. [29] VALIRON, G.: 'Sur l'iteration des fonctions holomorphes dans un demi-plan', Bull. Sci. Math. 55, no. 2 (1931), 105-128. [30] WLODARCZYK, K.: 'Julia's lemma and Wolff's theorem for Y*-algebras', Proc. Amer. Math. Soc. 99, no. 3 (1987), 472 476. [31] WOLFF, J.: 'Sur une generalisation d'un theoreme de Schwarz', C.R. Acad. Sci. 182 (1926), 918-920. David Shoikhet

MSC 1991: 30C45, 47H10, 47H20

K KAUFFMAN

BRACKET

POLYNOMIAL

-

An in-

variant of unoriented framed links. It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed links in R 3 (or $3), constructed by L.H. Kauffman in the summer of 1985 and denoted by (L>. It is defined recursively as follows: For a trivial link of n components, with zero framing, Tn, one puts (T~> = ( - A 2 - A-2) n-1.

denotes the number of smoothings of type L0 minus the number of smoothings of type L ~ . IsD[ denotes the number of components of the diagram after all ssmoothings on D are performed. The Kauffman bracket polynomial has a straightforward generalization to the solid torus (projected onto the annulus) and to the genus-two handlebody (projected onto the disc with 2 holes). In the first case it has values in Z[A+I,a] and in the second case it has values in Z[A 11 , a, b, c], see Fig. 2.

For the Kauffman bracket skein triple (cf. Fig. 1) one has the Kauffman bracket skein relation:

(L+) = A (Lo> + A -1 (L~> .

X

)(

L+

L 0

L

Fig. 1. I f L (1) is obtained from L by a positive full twist on its framing, then = - A 3 ( L ) . The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves: R~, R3, cf. R e i d e m e i s t e r t h e o r e m ) of diagrams on the plane. (D} is changed by the first Reidemeister move by - A +3. The Kauffman bracket polynomial is related to a substitution of the dichromatic polynomial of signed graphs. This connection also relates the Kauffman bracket polynomial to the Potts model in statistical mechanics. State sum expansions of the dichromatic polynomial have their analogue for the Kauffman bracket polynomial. For example:

(D> = E AT(s)(-AS - A-~)]sDf-l' 8

where the sum is taken over all states of the link diagram D and where a state codes the type of the smoothing performed at each crossing (L0 or L ~ type). T(s)

Fig. 2. For links in a solid torus the bracket polynomial can be used to estimate the wrapping number of the link. The wrapping conjecture says that wrap(D) for a link diagram D in the annulus is equal to the a-degree of (D) [1]. The Kauffman bracket skein module (cf. S k e i n m o d u l e ) is a generalization of the Kauffman bracket polynomial to any 3-dimensional manifold. The Kauffman bracket polynomial is a variant of the Jones polynomial. If one chooses an o r i e n t a t i o n / J on an unoriented link diagram D, then one defines an oriented link invariant f ( D ( A ) ) = (-A3)-Tait(/9)(D), where Tait(/)) is the Tait number (or writhe number) of/~, defined to be the sum of signs over all crossings of £J. Then the Jones polynomial VL(t) = fL(A) for t = A -4. Furthermore, the Kauffman bracket polynomial satisfies: A (D+) - A -1 (D_) = (A 2 - A -2) (Do) and

= (A + A-1)( + ), and it is a specialization of both the J o n e s - C o n w a y p o l y n o m i a l and the K a u f f m a n p o l y n o m i a l .

KAUFFMAN BRACKET POLYNOMIAL The Kauffman bracket polynomial was essential in the proof of the TaR conjectures on alternating links. In particular, for a connected alternating diagram without a nugatory crossing span(D) = 4c(D), where c(D) is the number of crossing points of D. If the diagram is prime and non-alternating, then s p a n / D ) < 4c(D). The Kauffman bracket polynomial has several other applications, for example in the analysis of periodic links. The Reshetikhin-Turaev invariants of 3-manifolds can be constructed using the Kauffman bracket polynomial (via Kirby moves, cf. also K i r b y c a l c u l u s ) [5]. References [1] HOSTE, J., AND PRZYTYCKI, J.H.: 'An invariant of dichromatic links', Proc. Amer. Math. Soc. 105, no. 4 (1989), 10031007. [2] JONES, V.F.R.: 'Hecke algebra representations of braid groups and link polynomials', Ann. of Math. 126, no. 2 (1987), 335 388. [3] KAUFFMAN, L.H.: 'State models and the Jones polynomial', Topology 26 (1987), 395-407. [4] KAUFFMAN, L.H.: ' A n invariant of regular isotopy', Trans. Amer. Math. Soc. 318, no. 2 (1990), 417-471. [5] LIeKOmSH, W . B . R . : An introduction to knot theory, Springer, 1997. [6] MENASCO, W.M., AND THISTLETHWAITE, M.B.: 'The classification of alternating links', Ann. of Math. 1 3 8 (1993), 113171. [7] MURASUGI, K.: 'Jones polynomial and classical conjectures in knot theory', Topology 26, no. 2 (1987), 187-194. [8] THISTLETHWAITE, M.B.: ' K a u f f m a n polynomial and alternating links', Topology 27 (1988), 311-318. [9] TRACZYK, P.: '10101 has no period 7: a criterion for periodic links', Proc. Amer. Math. Soc. 1 8 0 (1990), 845 846.

Jozef Przytycki MSC1991:57M25 KENDALL TAU M E T R I C , Kendall tau - The nonparametric c o r r e l a t i o n c o e f f i c i e n t (or measure of association) known as Kendall's tau was first discussed by G.T. Fechner and others about 1900, and was rediscovered (independently) by M.G. Kendall in 1938 [3], [4]. In modern use, the term 'correlation' refers to a measure of a linear relationship between variates (such as the P e a r s o n p r o d u c t - m o m e n t c o r r e l a t i o n coefficient), while 'measure of association' refers to a measure of a monotone relationship between variates (such as Kendall's tau and the S p e a r m a n r h o m e t r i c ) . For a historical review of Kendall's tau and related coefficients, see [5]. Underlying the definition of Kendall's tau is the notion of concordance. If (xj, yj) and (Xk, Yk) are two elements of a s a m p l e {(xi, yi)}i~ 1 from a bivariate population, one says that ( x j , y j ) and (xk,yk) are concordant if xj < xk and yj < Yk or if xj > xk and yj > Yk (i.e., if (xj -- Xk)(yj -- Yk) > 0); and discordant if xj < xk and yj > Yk or if xj > Xk and yj < Yk 226

(i.e., if (xj - xk)(yj -- Yk) < 0). There are (2) distinct pairs of observations in the sample, and each pair (barring ties) is either concordant or discordant. Denoting by S the number c of concordant pairs minus the number d of discordant pairs, Kendall's tau for the sample is defined as c- d S 2S -

c

+

-

-

.(n

-

1)

When ties exist in the data, the following adjusted formula is used: S ~-n = ~ / n ( n - 1)/2 - T ~ / n ( n - 1)/2 - U ' where T = Y~t t(t - 1)/2 for t the number of X observations t h a t are tied at a given rank, and U = ~ u u(u - 1)/2 for u the n u m b e r of Y observations t h a t are tied at a given rank. For details on the use of Tn in hypotheses testing, and for large-sample theory, see [2]. Note that Tn is equal to the probability of concordance minus the probability of discordance for a pair of observations (xj, yj) and (Xk, Yk) chosen randomly from the sample {(xi,yi)}i~l. The population version T of Kendall's tau is defined similarly for r a n d o m variables X and Y (cf. also R a n d o m v a r i a b l e ) . Let (X1, Y1) and (X2, Y2) be independent r a n d o m vectors with the same distribution as (X, Y ) . Then T

=

P [(X 1

-P [(Xl

--

--

-X-2) (Y1 - Y2) > O] + X2)(Y1 - Y2) < 0] =

= corr [sign(X1 - X2), sign(Y1 - Y2)]. Since ~- is the Pearson p r o d u c t - m o m e n t correlation coefficient of the random variables sign(X1 - X2) and sign(Y1 - Y2), ~- is sometimes called the difference sign correlation coefficient. When X and Y are continuous,

~- = 4

/01/01

C x , y ( u , v) d C x , y ( u , v) - 1,

where C x , y is the c o p u l a of X and Y. Consequently, ~is invariant under strictly increasing transformations of X and Y, a property W shares with Spearman's rho, but not with the Pearson p r o d u c t - m o m e n t correlation coefflcient. For a survey of copulas and their relationship with measures of association, see [6]. Besides Kendalt's tau, there are other measures of association based on the notion of concordance, one of which is Blornqvist's coefficient [1]. Let {(xi, Yi)}L1 denote a sample from a continuous bivariate population, and let 2 and ~ denote sample medians (cf. also M e d i a n (in s t a t i s t i c s ) ) . Divide the (x, y)-plane into four quadrants with the lines x = 2 and y = ~; and let n l be the number of sample points belonging to the first or third quadrants, and n2 the number of points belonging to the second or fourth quadrants. If the sample size n

KNAPSACK PROBLEM is even, the calculation of nl and n2 is evident. If n is odd, then one or two of the sample points fall on the lines x = ~ and y = ~. In the first case one ignores the point; in the second case one assigns one point to the quadrant touched by both points and ignores the other. Then Blomqvist's q is defined as q--

n I -- n 2

n l + n2

For details on the use of q in hypothesis testing, and for large-sample theory, see [1]. The population parameter estimated by q, denoted by /3, is defined analogously to Kendall's tau (cf. K e n d a l l t a u m e t r i c ) . Denoting by .~ and Y the population medians of X and Y , then

/3 =

P

[(x-

2)(y -

> o] + 1

~

p

l(x -

_

< ol

_-

= 4 F x , y (X, Y ) - 1,

two different framed links, L and L I, yield the same 3manifold if and only if one can pass from L to L r by a sequence of these operations. 1) Blow-up: One may add or subtract from L an unknotted circle with framing 1 or - 1 , which is separated from the other circles by an embedded 2-sphere. 2) Handle slide: Given two circles 7i and 7j in L, one may replace 7j with 73 obtained as follows. First, push 7i off itself (missing L) using the framing to get 7~. Then, let 7~ be a band sum of 7~ with 7j. Framing on 7j is changed by taking the sum of framings on 7i and on 7j with 4- algebraic linking number of 7/ with 7j. R.P. Fenn and C.P. Rourke [1] proved that these operations are equivalent to a K-move, where links L and L ~ are identical except in a part where an arbitrary number of parallel strands of L are passing through an unknot 7o with framing - 1 (or +1). In the link L ~ the unknot 70 disappears and the parallel strands of L are given one full right-hand (respectively, left-hand) twist.

where F x , y denotes the joint distribution function of X and Y. Since/3 depends only on the value of F x , y at the point whose coordinates are the population medians of X and Y, it is sometimes called the medial correlation coefficient. When X and Y are continuous,

References

where C x , y again denotes the copula of X and Y. Thus /3, like T, is invariant under strictly increasing transformations of X and Y.

MSC 1991:57M27

[1] FENN, R.P., AND ROURKE, C.P.: 'On Kirby's calculus of links', Topology 18 (1979), 1-15. [2] KIRBY, R.: 'A calculus for framed links in S 3', Invent. Math.

45 (1978), 35-56. [3] LICKORISH, W.B.R.: 'A representation of orientable combinatorial 3-manifolds', Ann. Math. 76 (1962), 531-540. [4] WALLACE, A.H.: 'Modification and cobounding manifolds', Canad. J. Math. 12 (1960), 503-528.

Joanna Kania-BartoszyTiska

References

[1] BLOMQVIST, N.: 'On a measure of dependence between two random variables', Ann. Math. Star. 21 (1950), 503 600. [2] GIBBONS, J.D.: Nonparametric methods for quantitative analysis, Holt, Rinehart & Winston, 1976. [3] KENDALL, M.G.: 'A new measure of rank correlation', Biometrika 30 (1938), 81-93. [4] KENDALL, M.G.: Rank correlation methods, fourth ed., Charles Griffin, 1970. [5] KRUSKAL, W.H.: 'Ordinal measures of association', J. Amer. Statist. Assoc. 53 (1958), 814 861. [6] NELSEN, R.B.: An introduction to copulas, Springer, 1999. R.B. Nelsen

MSC1991:62H20 K I R B Y CALCULUS, Kirby moves - A set of moves between different surgery presentations of a 3-manifold. W.B.R. Lickorish [3] and A.D. Wallace [4] showed that any orientable 3-manifold may be obtained as the result of s u r g e r y on some framed link in the 3-sphere. A framed link is a finite, disjoint collection of smoothly embedded circles, with an integer (framing) assigned to each circle. R. Kirby [2] described two operations (the calculus) on a framed link and proved that

KNAPSACK P R O B L E M - Given a knapsack (container) of total capacity e, and n objects with weights a l , . . . , an and respective values C l , . . . , ca, the problem is to pack as much value in the knapsack as possible. Abstractly the problem can be formulated as follows. Given positive integers c, a l , . . . , an, c l , . . •, ca, the problem is to maximize ~ cixi subject to ~ i aixi < c and xi E {0, 1}. The g r e e d y a l g o r i t h m to 'solve' this proceeds as follows. It is natural to favour objects with the greatest value/weight density. So, relabel, if needed, the objects so that c l / a l >_ ... > c~/a,~. Then select X l , . . . , x ~ recursively according to

ifai <_ c - ~ j = l ajxj, X i ~-

otherwise.

This greedy algorithm has a performance ratio of 2, i.e. it is guaranteed to find a solution which is within a factor 2 of the optimal one. If the xi are allowed to take fractional values (i.e. values in [0, 1] instead of {0, 1}), then the greedy algorithm is optimal. The knapsack problem is A/'7)-hard, cf. N'P. 227

KNAPSACK PROBLEM References

[1] GASS, S.I., AND HARRIS, C.M. (eds.): Encyclopedia o/ Operations Research and Management Science, Kluwer Acad. Publ., 1996, pp. 325 326. [2] GROTSCHEL, M., AND LOVASZ, L.: 'Combinatorial optimization', in R.L. GRAHAM, M. GROTSCHEL, AND L. LOVA.SZ (eds.): Handbook of Combinatorics, Elsevier, 1995, pp. 15411579. M. Hazewinkel

MSC 1991:90C27 In 1755, L. Euler derived from the basic N e w t o n laws o f m e c h a n i c s an equation for the motion of a fluid without taking into account the viscosity (cf. also E u l e r e q u a t i o n ) . To introduce the viscosity, and also compressibility, this equation has to be modified in accordance with the work of L. Navier (1821) and G.C. Stokes (1842) (cf. also N a v i e r - S t o k e s e q u a t i o n s ) . Such an equation can be written in dimensionless form by introducing three dimensionless numbers: the M a c h n u m b e r , the R e y n o l d s n u m b e r and the P r a n d t l n u m b e r . The Mach number is the ratio of the mean velocity u of the fluid to the velocity of sound c, the Reynolds number is the product of the characteristic length of the vessel l with the mean velocity u divided by the viscosity ~, and the Prandlt number is the ratio of the viscosity and the thermometric conductivity ~: KNUDSEN

NUMBER

Ma=-,

u

-

ul Re=--,

Pr=-.

The discovery of atoms and molecules and the understanding of the role they play in the 'human size' world was finalized around the turn into the nineteenth century, starting with A. Avogadro in 1833, giving an order of magnitude of the number N _~ 1()19 of molecules per volume element, continuing with J.R. Clausius, who in 1858 introduced the mean free path, and culminating with the 1905 paper of A. Einstein on B r o w n i a n m o tion. The connection between the macroscopic and the molecular point of view was one of the challenges of the end of nineteenth century and still (as of 2000) attracts the attention of physicists and mathematicians. It is important both for a full understanding of basic laws of physics, like the appearance of irreversibility, and for practical purpose. Computation cannot be done at the level of molecules because the number of degree of freedom of the system, of the order of N ~, is definitely too large. On the other hand, several modern technological problems (the re-entry of a space shuttle in the atmosphere, the computation of the current in a semiconductor which is so small that the gas of electrons cannot reach thermal equilibrium, or the analysis of the 228

ionized air between the reading head and a compact disc) involve gases of particles which are too rarefied to be computed with the macroscopic laws of physics. To bridge such a gap, an intermediate description, the kinetic description, was given by J.C. Maxwell (1860), who introduced a density function f ( x , v , t) which describes the probability of having at time t and at the point x a particle with the velocity v, and by L. Boltzmann who eventually (1872) derived an equation for this density (cf. also B o l t z m a n n e q u a t i o n ) . Therefore, with the number of particles increasing one should deduce from the molecular description the kinetic description and from the kinetic the Navier-Stokes equations. However, in this process it is not the macroscopic density that matters, but the average number of collisions that a molecule of 'radius' cr experiences during a unit time. It is given by the formula: n = 7r~2Ne.

The averaged time for collision is then n -1 and the average distance travelled during this time, i.e. the mean free path, is /~ = n - l e

= (~-~2N)-1 '

Observe that the total volume occupied by the gas (i.e. the packed volume) is given by 4 = ~

2 N,

and therefore is in a regime where N is large, of the order of 1019 and A of the order of unity, the packed volume is of the order 10-s; hence the limit 1 = lim(7~a2N) E ]0, (x~[ corresponds to a rarefied gas, described by the Boltzmann equation. Again, this equation is written in dimensionless form with the introduction of the Knudsen number, the ratio of the mean free path to the characteristic length of the vessel containing the gas: A KH = - . l As M.H.Ch. Knudsen (1871-1949), professor of physic at the university of Copenhagen, discovered, it is rather this number than the mean free path itself that contains the decisive information. As said above, when the Knudsen number is very small, intermolecular collisions dominate and the kinetic approximation becomes valid; if this number is large (in this situation one refers to the Knudsen gas), the molecules evolve almost freely in the vessel and the effect of the collisions with the boundaries dominate. Furthermore, writing Ma Re

u/c ul/v

1v c A'

KRUSKAL-KATONA THEOREM one observes that the expression u/A is independent of the gas motion and homogeneous to a speed; the only such speed can be the speed of sound c (up to some unessential 'pure' constant a) and therefore one obtains the yon Kdrmdn relation Kn=a--

Ma Re '

which is due to Th. yon K£rm£n (1923) and which is of paramount importance for the next step of the discussion: the derivation of the macroscopic equations from the kinetic description. This program was initialized by D. Hilbert (1916), who proved that when the Knudsen number goes to zero, a first-order macroscopic approximation is provided by the compressible Euler equation. The connection with the Navier-Stokes equation turned out to be more subtle. Independently, S. Chapman (1916) and D. Enskog (1917) proved that a second-order correction is provided by the compressible Navier-Stokes equation with Reynolds number and Mach number of the order of the Knudsen number (cf. also C h a p m a n E n s k o g m e t h o d ) . This is in perfect agreement with the von K~rm£n relation. Even more, the ChapmanEnskog derivation may be used to provide a 'mathematical' proof of the von K~rm~,n relation. Eventually, the yon K~rm~n relation implies that it is only in the zero Mach limit, i.e. in the incompressible limit, that the Navier-Stokes equations with finite Reynolds number are relevant. A detailed description of recent mathematical work on this subject can be found in [2], [1]. Passing through the kinetic equation (first having a finite Knudsen number and then letting only this number go to infinity) implies, with relaxation through Maxwellian distributions, that only perfect gases, incompressible or with a pressure law given by the relation

p = pRT, are described at the macroscopic level. For other constitutive relations, in particular with pressure given by the van der Waals law, in spite of some preliminary work [3], [4], the mathematical theory is in the preliminary stage and the role of the Knudsen number is not evident.

[4] MORREY, C.: 'On the derivation of the equations of hydrodynamics from statistical mechanics', Commun. Pure Appl. Math. 8 (1955), 279-326.

C. Bardos MSC 1991: 76Axx KRUSKAL-KATONA

THEOREM-

Let [n]

:=

{ 1 , . . . , ~ } and ([~]) := { X _c [n]: IXl = k}, k -- 0 , . . . , n (cf. also S p e r n e r p r o p e r t y ; S p e r n e r t h e o r e m ) . The elements of ([k]) can be linearly ordered in the following way: X < Y if the largest element in which X and Y differ is in Y (cf. also T o t a l l y o r d e r e d set). If{al+l,...,ak+l},O<_al < - . . < a k _< n-- 1, is the (m + 1)th element in this order, then "~=

+\k-l/+""

+

+

1

'

(1)

since (aN k-element subsets of [n] have maximal element smaller than ak + 1, (kk_l1) have maximal ele-

ak -k-1 but the second largest element smaller than ak-1 + 1, etc.. Equation (1) is called the k-representation

ment

of m, I _< m a2

=

<

l,...,ak_1

representation

(k) (for m = (k)' one takes al = 0, ---- k-2, ak = n). This (unique) can be found directly by choosing ak

maximal such that (akk) _< m,

ak_ 1 maximal

such that

(a;_~) _< . ~ _ ( ~ ) , etc.. For a family f C (I~J), the lower shadow of f is defined by

A(F):={YC (k[n]l):YcXforsomeXCF

}.

If m is given as above, then the lower shadow of the family of the first m elements in ([k]) is the family of all (k - 1)-element subsets of [n] having maximal element smaller than ak + 1, of all (k - 1)-element subsets of [n] having maximal element ak + 1, but the second largest element smaller than ak-1 + 1, etc., i.e. the family of the first Ok(m) elements of (kin]l) where

(ak-lh

Ok(m)=(k~l)+\k_2j+'"+(;1

)"

The Kruskal Katona theorem states that in this way one obtains a lower shadow of minimum size, i.e., if U is any ,~-element family in ([~1) and ,~ is given by (1), then

I~X(~:)I > 0k(m).

References [1] BOUCHUT,F., GOLSE, F., AND PULVIRENTI, M.: Kinetic equations and asymptotic theories, Vol. 4 of Series in Appl. Math., Elsevier/Gauthier-Villars, 2000. [2] CERCIGNANI, C., ILLNER, R., AND PULVIRENTI, M.: The mathematical theory of dilute gases, Applied Math. Sci. Springer, 1994. [3] CERCIGNANI, C., AND LAMPIS, M.: 'On the H-theorem for polyatomie gases', Y. Statist. Phys. 26 (1981), 795-801.

There exist generalizations to similar results for other partially ordered sets, like products of chains, products of stars, the partially ordered set of subwords of 0-1words, and the partially ordered set of submatrices of a matrix. The following result of L. Lov~.sz is weaker but numerically easier to handle: If .P C ([k]) and I)cl = (~) 229

KRUSKAL-KATONA THEOREM with some real x, where k < x < n, then

IA(7)I_>

k-1

'

The original papers by J.B. Kruskal and G.O.H. Katona are [4], [3]. According to [2, p. 1296], the KruskM-Katona theorem is probably the most important one in finite extremal set theory.

This is readily apparent in Fourier space, where one may write (1) with periodic boundary conditions as dA

/uk = (k

-

+

(2) M

where u(x, t) = i ~-~-k~ k ( t ) e x p ( i k x ) , k = nq, q = 27r/L, n E Z, i = v/-21. The zero solution is linearly unstable to modes with Ikl < 1; these modes, whose number is proportional to the bifurcation parameter L, are coupled to each other and to damped modes at Ikl > 1 References [1] ENGEL, K.: Sperner theory, Cambridge Univ. Press, 1997. through the non-linear term. [2] FRANKL, P.: 'Extremal set systems', in R.L. GRAHAM, As L increases beyond 27r, therefore, the zero soluM. GROTSCHEL, AND L. LOVASZ (eds.): Handbook of Comtion destabilizes, initially to a single-humped stationbinatorics, Vol. 2, Elsevier, 1995, pp. 1293-1329. ary 'cellular' state, which then in turn becomes unsta[3] KATONA, G.O.H.: 'A theorem of finite sets': Theory of ble through a complex hierarchy of bifurcations includGraphs. Proc. Colloq. Tihany, Akad. Kiad6, 1966, pp. 187ing multi-modal stationary, oscillatory and chaotic so207. [4] KRUSKAL, J.B.: 'The number of simplices in a complex': lutions, which have been characterized in detail [14], Mathematical Optimization Techniques, Univ. California [15], [18]. Note that as suggested by the presence of Press, 1963, pp. 251-278. chaotic solutions and by a Painlev6 analysis [7] (cf. also K. Engel Painlev~ test), the Kuramoto-Sivashinsky equation is MSC 1991: 05D05, 06A07 non-integrable, and no explicit general analytic solutions exist. A striking feature of the bifurcation behaviour in KURAMOTO-SIVASHINSKY EQUATION, this partial differential equation, especially for relatively S i v a s h i n s k y - K u r a m o t o equation, K S equation - The small L, is the apparent low-dimensionality of the dyKuramoto-Sivashinsky equation in one space dimennamics, and the similarity of the observed bifurcations sion, in 'derivative' form to those found in (low) finite-dimensional systems. Motivated by this observation, extensive analytical study of ut + uxxxx + Uxx + UUx = O, x • [ - L / 2 , L/2], (1) the solutions has shown that the Kuramoto-Sivashinsky or in 'integral' form equation is rigorously equivalent to a finite-dimensional dynamical system (for an overview of analytical results h~ + hxx~x + hx~ + ~h~ = O, in an appropriate functional setting, see [32]). where u = h~, has attracted a great deal of interest as a model for complex spatio-temporal dynamics in spatially extended systems, and as a paradigm for finitedimensional dynamics in a partial differential equation. The Kuramoto-Sivashinsky equation (with various alternative scalings for u, x or t, which can be reduced to the form (1)) has been independently derived in the context of several extended physical systems driven far from equilibrium by intrinsic instabilities, including instabilities of dissipative trapped ion modes in plasmas [20], [3], instabilities in laminar flame fronts [29], phase dynamics in reaction-diffusion systems [19], and fluctuations in fluid films on inclines [30]. Indeed, (1) generically describes the dynamics near long-wave-length primary instabilities in the presence of appropriate (translational, parity and Galilean) symmetries [25]. The u~x term in (1) is responsible for an instability at large scales; the dissipative u ~ term provides damping at small scales; and the non-linear term uu~ (which has the same form as that in the Burgers or one-dimensional Navier-Stokes equations) stabilizes by transferring energy between large and small scales. 230

Analytical results and finite-dimensionality of dynamics. Specifically, a significant feature of the Kuramoto-Sivashinsky dynamics is its dissipativity (cf. also Dissipative system): solutions are attracted to an absorbing ball, with L-dependent radius, in L 2 and higher Sobolev spaces ([26] for odd initial data, [5], [12] for general periodic solutions; cf. also S o b o l e v space). The strong smoothing properties of the linear operator in fact imply boundedness in the Gevrey norm (cf. Gevrey class) and thus space-analyticity of solutions of (i) [4], as well as time-analyticity [16]. The dissipativity of the dynamics has been used to show [26] that the system (I) has a finite number of determining modes, and a compact global attractor with finite fractal and Hausdorff dimension. While the attractor can have very complex structure, a stronger result is the existence of a finite-dimensional inertial manifold, which exponentially absorbs solutions and contains the global attractor [I0], [6]. On restricting the partial differential equation to the inertial manifold, one obtains a system of ordinary differential equations, the inertial form, which completely describes the long-time

KURAMOTO-SIVASHINSKY EQUATION dynamics; thus, the Kuramoto-Sivashinsky equation is rigorously equivalent to a finite-dimensional d y n a m i c a l s y s t e m . The existence of the inertial manifold does not provide an explicit construction, however, so various approximation schemes have been introduced; for instance, one can construct approximate inertial manifolds so that all trajectories of the Kuramoto-Sivashinsky equation approach the approximate inertial manifold at an exponential rate [16]. B i f u r c a t i o n s a n d e l e m e n t a r y solutions. The cellular or 'roll' solutions [11] form the backbone to the spatial structure of solutions of (1) (with periodic boundary conditions) observed as L increases: the N-cell state consists of solutions with periodicity L / N which lie on the branch bifurcating from the trivial solution at L = N.27r, and have rapidly decreasing basin of attraction for increasing N [9]. Other solutions observed numerically for increasing L, and in some cases accounted for analytically, include other families of stationary states, timeperiodic standing and travelling waves, quasi-periodic modulated travelling waves, and heteroclinic cycles [1], [18]. There are also windows in which strange attractors with positive Lyapunov exponents (cf. L y a p u n o v c h a r a c t e r i s t i c e x p o n e n t ) are observed, together with more complex dynamical phenomena associated with chaotic dynamics, including period doubling cascades, Shil'nikov connections and crises of chaos. S p a t i o - t e m p o r a l chaos. As L increases and one passes through an increasingly intricate bifurcation sequence of ordered and chaotic states, eventually one reaches a state of persistent dynamical disorder for (almost) all sufficiently large L [15] (see Fig. 1), and there is strong numerical evidence that the 'simple' solutions destabilize to an (apparently unique) spatio-temporally chaotic attractor (see [8] for a review of spatio-temporal pattern formation).

0

~0

~0

3o

40

so

6o

7o

8o

90

i00

3,"

Fig. 1: A solution u(x, t) of the Kuramoto Sivashinsky equation (1) on the spatio-temporally chaotic attractor, for L = 100, and covering 256 time units separated by At = 1. This regime of 'weak' or 'phase' turbulence [23] is distinct from the 'strong' turbulence exhibited in, for

instance, the N a v i e r - S t o k e s e q u a t i o n s for fluids, in that there are no major excursions from space or time averages. While the individual solutions bifurcating from the zero solution break the translational, parity and Galilean symmetries of (1), the spatio-temporally chaotic state displays 're-emergent order', in that the symmetries are restored in a statistically averaged sense. Numerical evidence indicates that the spatiotemporally chaotic state is characterized by a finite density of positive Lyapunov exponents, that is, the Lyapunov dimension of the attractor is proportional to L [22]. In fact, in general there appears to be 'extensive chaos' for sufficiently large L: that is, due to rapid decay of spatial correlations [33] local dynamics are asymptotically independent of system size L, extensive quantities such as the energy (square of the L 2 norm) scale with L, and one can hope to study the thermodynamic limit, interpreting the large system as being composed of weakly interacting smaller subsystems. However, this picture is as yet by no means wellestablished, and even relatively 'simple' analytical results on intensive properties which might seem rigorously provable, have remained elusive at the time of this writing (2000). For example, the known analytical and numerical solutions all appear to have uniformly bounded lu(x, t) l, that is, the L ~ norm llull~ is bounded independent of L; this would imply the existence of a finite energy density, or that the L 2 norm

Ilul12

L"-s/2~2(~'t)dz

is proportional to L 1/2. While a uniform bound on [lulloo is known for stationary solutions [24] and solutions near these on the attractor, currently (2000) the best known general bound for llul12 is O(L 8/5) [5]. Similarly, based on extensive numerical evidence, it has been conjectured [5], [27], [26] that the attractor and inertial manifold dimensions scale linearly with L, or with the number of linearly unstable Fourier modes, while the radius of the strip of space-analytieity is L-independent [4]; but the best known rigorous bounds for the KuramotoSivashinsky equation do not yet approach this thermodynamic limit. The dynamics on the spatio-temporally complex attractor for large L are best understood in the light of the characteristic shape of the normalized (time-averaged) power spectrum S(k) = L(II~kll21, which appears to be independent of L in the disordered regime, consistent with a finite energy density [27], [28] (see Fig. 2). The power spectrum reveals three distinct regimes of the dynamics, whose dynamical significance is corroborated by other evidence including numerical experiments in which different modes are eliminated or forced [33]: 231

K U R A M O T O - S I V A S H I N S K Y EQUATION The exponential tail in S(k) is due to strong dissipation at small scales (high k), corresponding to the exponential decay of Fourier modes of an analytic function; these modes are strongly damped and essentially irrelevant for the qualitative dynamics. The active scales for k = O(1) have distinctly non-Gaussian distributions and contain most of the energy, with a pronounced peak near k = 1 / v ~ , the most linearly unstable mode; the localized dynamics at these scales, which may be interpreted as cell creation and annihilation events [2], are essential to the spatio-temporal disorder. i(io ............................. --/~"~.. \ 10...,5

\\ \"\

s(~.) H) ' I0

\

',

~0-~ l 0..,.~0; 0,01

j

\ ........ ......

'~ i ...................................................... L.i

~ 0,i

k

i

i0

Fig. 2: R e s c a l e d p o w e r s p e c t r u m S ( k ) , for L = 100 a n d

L = 800. In the large scale region, there is a shoulder in S(k) which flattens as k -+ 0, reminiscent of a thermodynamic regime with equipartition of energy. These scales exhibit Gaussian statistics and appear to act as a 'heat bath', providing the background excitation needed to maintain the spatio-temporal disorder [33]. There has been considerable effort devoted towards understanding the effective stochastic dynamics at large length and time scales [35], [31], [21], [2]. The (deterministic) chaotic dynamics at active and small scales simulate the effect of random forcing on the largest scales, and act to renormalize the viscosity, so that the scaling of solutions at large scales appears to be well-described by a noisedriven Burgers equation or, equivalently, the K a r d a r Parisi-Zhang equation for kinetic roughening [34], [17] (see [13] for a review). Numerous investigators have extended the abovementioned results on the analysis and dynamics of the Kuramoto-Sivashinsky equation in the small-L and large-L regimes, and have studied generalizations to higher space dimensions and non-periodic boundary conditions (including the unbounded system, x C R), and the effect of additional terms in the partial differential equation. References [i] ARMBRUSTER, D., GUCKENHEIMER, J., AND HOLMES, P.: 'Kuramoto Sivashinsky dynamics on the center-unstable manifold', SIAM J. Appl. Math. 49 (1989), 676-691. [2] CHOW, C.C., AND HWA, T.: 'Defect-mediated stability: an effective hydrodynamic theory of spatiotemporal chaos', Physica D 84 (1995), 494-512.

232

[3] COHEN, B., KROMMES,

J., TANG, W., AND ROSENBLUTH, M.: 'Nonlinear saturation of the dissipative trapped-ion mode by mode coupling', Nucl. Fus. 16 (1976), 971-992. [4] COLLET, P., ECKMANN, J.-P., EPSTEIN, H., AND STUBBE, J.: 'Analyticity for the Kuramoto-Sivashinsky equation', Physica D 67 (1993), 321-326. [5] COLLET, P., ECKMANN, J . - P . , EPSTEIN, H., AND STUBBE, J.:

'A global attracting set for the Kuramoto-Sivashinsky equation', Commun. Math. Phys. 152 (1993), 203-214. [6] CONSTANTIN, P., FOIAS, C., NICOLAENKO, B., AND TEMAM, R.: Integral manifolds and inertial manifolds for dissipative partial differential equations, Vol. 70 of Appl. Math. Sei., Springer, 1989. [7] CONTE, R., AND MUSETTE, iV[.: 'Painlev~ analysis and B~.cklund transformation in the Kuramoto-Sivashinsky equation', J. Phys. A 22 (1989), 169-177. [8] CROSS, M., AND HOHENBERG, P.: 'Pattern formation outside of equilibrium', Rev. Mod. Phys. 65 (1993), 851 1112. [9] ELGIN, J.N., AND WU, X.: 'Stability of cellular states of the Kuramoto-Sivashinsky equation', S l A M J. Appl. Math. 56 (1996), 1621-1638. [10] FOIAS, C., NICOLAENKO, B., SELL, G.R., AND TEMAM, R.: 'Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension', J. Math. Pures Appl. 67 (1988), 197-226. [11] FRISCH, V., SHE, Z.S., , AND THUAL, O.: 'Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model', J. Fluid Mech. 168 (1986), 221-240. [12] GOODMAN,J.: 'Stability of the Kuramoto-Sivashinsky and related systems', Commun. Pure Appl. Math. 47 (1994), 293306. [13] HALPIN HEALY, T., AND ZHANG, Y.-C.: 'Kinetic roughening phenomena, stochastic growth, directed polymers and all that', Phys. Rept. 254 (1995), 215-414. [14] HYMAN, J.M., AND NICOLAENKO, B.: 'The KuramotoSivashinsky equation: A bridge between PDEs and dynamical systems', Physica D 18 (1986), 113-126. [15] HYMAN, J.M., NICOLAENKO,B., AND ZALESKI, S.: 'Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces', Physica D 23 (1986), 265-292. [16] JOLLY, M., KEVREKIDIS, I., AND TITI, E.: 'Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computation', Physica D 44 (1990), 38-60. [17] KARDAR, M., PARISI, G., AND ZHANG, Y . - C . : 'Dynamic scaling of growing interfaces', Phys. Rev. Lett. 56 (1986), 889892. [18] KEVREKIDIS, l.C-., NICOLAENKO, B., AND SCOVEL, J.C.: 'Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation', SIAM J. Appl. Math. 50 (1990), 760-790. [19] KURAMOTO, Y., AND TSUZUKI, T.: 'Persistent propagation of concentration waves in dissipative media far from thermal equilibrium', Progr. Theoret. Phys. 55 (1976), 356-369. [20] LAQUEY, R., MAHAJAN, S., RUTHERFORD, P., AND TANG, W.: 'Nonlinear saturation of the trapped-ion mode', Phys. Rev. Lett. 34 (1975), 391 394. [21] L'VOV, V.S., LEBEDEV, V.V., PATON, M., AND PROCACCIA, I.: 'Proof of scale invariant solutions in the Kardar-ParisiZhang and Kuramoto-Sivashinsky equations in 1 + 1 dimensions: analytical and numerical results', Nonlinearity 6 (1993), 25-47.

KURAMOTO-SIVASHINSKY EQUATION [22] MANNEVILLE, P.: 'Liapounov exponents for the KuramotoSivashinsky model', in U. FRISCH, J. KELLER, G. PAPANICOLAOU, AND O. PmONNEAU (eds.): Macroscopic Modelling of Turbulent Flows, Vol. 230 of Lecture Notes in Physics, Springer, 1985, pp. 319-326. [23] MANNEVILLE,P.: Dissipative structures and weak turbulence, Acad. Press, 1990. [24] MICHELSON, D.: 'Steady solutions of the KnramotoSivashinsky equation', Physica D 19 (1986), 89-111. [25] MISBAH, C., AND VALANCE, A.: 'Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation', Phys. Rev. E 49

(1994), 166-183. [26] NICOLAENKO, B., SCHEURER, B., AND TEMAM, R.: 'Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors', Physica D 16 (1985), 155 183. [27] POMEAU, Y., PUMm, A., AND PELCE, P.: 'Intrinsic stochasticity with many degrees of freedom', J. Statist. Phys. 37 (1984), 39-49. [28] PUMIR, A.: 'Statistical properties of an equation describing fluid interfaces', J. Physique 46 (1985), 511 522.

[29] SIVASHINSKY,G.: 'Nonlinear analysis of hydrodynamic instability in laminar flames I. Derivation of basic equations', Acta Astron. 4 (1977), 1177-1206. [30] SIVASHINSKY, G., AND MICHELSON, D.: 'On irregular wavy flow of a liquid film down a vertical plane', Progr. Theoret. Phys. 63 (1980), 2112-2114.

[31] SNEPPEN, K., KRUG, J., JENSEN, M., JAYAPRAKASH, C., AND BOHR, T.: 'Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation', Phys. Rev. A 46 (1992), R7351-R7354. [32] TEMAM, R.: Infinite-dimensional dynamical systems in mechanics and physics, second ed., Vol. 68 of Applied Math. Sci., Springer, 1997.

[33] WITTENBERG, R.W., AND HOLMES, P.: 'Scale and space localization in the Kuramoto-Sivashinsky equation', Chaos 9 (1999), 452-465. [34] YAKHOT,V.: 'Large-scale properties of unstable systems governed by the Kuramoto-Sivashinski equation', Phys. Rev. A 24 (1981), 642-644. [35] ZALESKI,S.: ' t stochastic model for the large scale dynamics of some fluctuating interfaces', Physica D 34 (1989), 427438. Ralf W. Wittenberg MSC1991: 35Q35, 76Exx, 58F13

233

L LEBESGUE CONSTANTS OF MULTIDIMENSIONAL PARTIAL F O U R I E R S U M S - Let f be an i n t e g r a b l e f u n c t i o n on T n, T = (-%77], n = 2, 3 , . . . , 2It-periodic in each variable. Consider its F o u r i e r s e r i e s ~ k "f(k) eikx, where x = ( x l , . . . , x~) C T n, k = ( k l , . . . , kn) E Z n, the lattice of points in R with integer coordinates, kx = kl xl + . . . + k,~xT~, while =

- n

is the kth Fourier coefficient of f . No natural ordering of Fourier coefficients exists, thus the definition of a multi-dimensional partial Fourier sum presents many problems and points of interest intimately connected to geometry and number theory. To indicate that the partial sum corresponds to a certain summation domain B, one denotes it by

S u ( f ; x ) = ~ f(k)e ik*. kEB Frequently, sums SNB are considered, where N B is the N t h d i l a t a t i o n of a fixed set B; in m a n y cases this is the most natural way of summation. An example of partiai Fourier sums that are not of this kind are the rectangular partial sums. By SN one denotes the p a r t i a l F o u r i e r s u m when the dependence on the p a r a m e t e r N, either scalar or vectorial, is of primary importance. As is well-known, if the F o u r i e r s e r i e s of a c o n t i n u o u s f u n c t i o n fails to converge at each point, then the sequence of norms of the operators SN,

f(x)

sN(I; x),

taking C ( T n) into C ( T n) (or, equivalently, L I ( T ~) into L 1 (T~)) is unbounded and measures the rate of divergence of the Fourier series. This is strongly related to the behaviour of the F o u r i e r t r a n s f o r m of the indicator function of the summation domain B. For known results on this subject, see, e.g., [11], [13], [18].

For the spherical partial Fourier sums SN(f;x) = 7~k~eikx Ik]
CIN (n-1)/2 <_ IISN][ < C2N (n-1)/2.

(1)

The estimate from below has first appeared in [14]; the method used there is the main tool for obtaining lower bounds for L e b e s g u e c o n s t a n t s . Nothing is known about existence or non-existence of the limit of IISNII/N(n-1)/2 as N -+ oc; this is the main open problem in the subject (as of 2000). More general s u m m a t i o n domains B possessing properties of the spherical partial Fourier sums have been considered. E.g., in Yudin's estimate from above [26], the summation domains B are balanced (i.e.: along with each point x the whole set 6x, 161 _< 1, belongs to B), with finite upper Minkowski measure, t h a t is, 1 lim sup - meas {x: p(x, OB) < e} < oc, e--+0

C

where p(x, OB) = infyc0B p(x,y). These natural assumptions provide the bound IISNBII _ C N /~. It turns out that to satisfy the same estimate from below, only local information is needed ([17]): Let the boundary of a domain B contain a simple (non-intersecting) piece of a surface of smoothness [(n + 2)/2] in which there is at least one point with non-vanishing principal curvatures (cf. also P r i n c i p a l c u r v a t u r e ) . Then there exists a positive constant C, depending only on B, such that IISNBII _> CN(n-1)/2 for N large. The estimates in the spherical case and its generalizations are the worst possible if B is compact. Once B has a point with non-vanishing principal curvatures, the Lebesgue constants are t h a t 'bad'. The other side of the scale is called 'polyhedral' and is of 'logarithmic nature'. 0 n l y some natural restrictions have to be put on polyhedra B, for example, the hyperplanes that define the sides of the polyhedron do not contain the origin. In that case there exist two positive constants C1 and C2,

L E B E S G U E CONSTANTS OF MULTI-DIMENSIONAL PARTIAL FOURIER SUMS

C1 < C2, such that for each such polyhedron B:

Cl lnn N < IISNBII < C21nnN.

and C2, Ca < C2, such that C1N n+(n-1)/2 < IISHNI[ <

(2)

Actually, this was proved by E. Belinsky [6]. There are two important problems concerning the polyhedral case. The first is: Can partial Fourier sums have Lebesgue constants with an intermediate rate of growth (i.e. between (1) and (2))? Some trivial solutions were suggested in [27], where an intermediate growth is achieved by the product of the two mentioned situations. Of course, this is possible only for dimension three and greater. Thus, the problem is to find one for dimension two. It is clear that in this case the boundary can possess no point with non-vanishing curvature. On the other hand, any polyhedron matches (2). Thus, the solution can only be a (convex) 'polyhedron' with infinitely many specially located sides. Such a solution was constructed by A. Podkorytov [21]. The next question also seems very natural: Is it possible to have a certain asymptotic relation instead of the order estimate (2)? For rectangular partial sums some special cases were investigated by I. Daugavet [9] and O. Kuznetsova [15]. For the sequence of dilated summation domains, an unexpected result was obtained by Podkorytov [22]. Here n = 2 also causes the main difficulties. There are two main cases. In the first one, the polygons B with sides of rational slopes are dealt with - - then the estimates change insignificantly if only one considers instead of sums the corresponding integrals, that is, the Fourier transform :~xB of the indicator function of the N-dilation of the corresponding set B. This allows one to obtain logarithmic asymptotics; namely, the values IISNBII, l n 2 N and fT2 I•NB(X)I dx are equivalent. When at least one of the slopes is irrational, the situation changes qualitatively: The upper limit and the lower limit of the ratio IISNBI[/In2 N, as N + oo, may differ. In [22] and [20], quantitative estimates of this phenomenon as well as open problems are given. The paper [3] started the interest in various questions of approximation theory and Fourier analysis in R ~ connected with the study of hyperbolic cross partial Fourier sums (see, e.g., [24] and H y p e r b o l i c cross). The exact order of growth of their Lebesgue constants, N (~-1)/2, the same as in the spherical case, was established in the two-dimensional case independently in [5] and in [25], and afterwards was generalized to the case of arbitrary dimension in [1@ Step hyperbolic crosses HN were introduced by B. Mityagin [19] and are defined as follows (cf. also S t e p h y p e r b o l i c cross): H N = U { m E z n : 2 sj < Irnjl < 2 s;+l} for s E Z~_ such that 0 _< sl + ' " + sT~ _< N. These have many important applications too. Belinsky [8] proved that there exist two positive constants C1

C2Nn+(n-1)/2.

When B is unbounded, it may happen that the operator SB is unbounded even for fixed B. It is proved in [4] that the Lebesgue constants HSNBII are either of the usual order of growth or infinite for all values of the parameter N > N0, where t3 is a h y p e r b o l i c cross, depending on whether the hyperbolic cross is turned at a rational or irrational angle, respectively. For n = 2, this was earlier obtained in [7], which also contains similar results for the strip to be a summation domain. For other results on Lebesgue constants and related topics, see [i], [2], [10], [12], [18], [23], [28], [29]. The ideas used to prove many of the results discussed above have also been applied to estimates of the Lebesgue constants of linear means of multiple Fourier series. Some results are known for Lebesgue constants in more abstract settings, e.g., for spherical harmonics expansions or Fourier series on compact Lie groups. References [i] ALIMOV, SH.A., ASHUROV, R.R., AND PULATOV, A.K.: 'Multiple Fourier series and Fourier integrals', in V.P. KHAVIN AND N.K. •IKOLSKII (eds.): Commutative Harmonic Analysis IV, Vol. 42 of Eric. Math. Sci., Springer, 1992, pp. 1-95. [2] ALIMOV, SH.A., ILYIN, V.A., AND NIKISHIN, E.M.: 'Convergence problems of multiple Fourier series and spectral decompositions, I, II', Russian Math. Surveys 3 1 / 3 2 (1976/77), 29-86; 115-139. (Uspekhi Mat. Nauk. 3 1 / 3 2 (1976/77), 2883; 107-130.) [3] BABENKO, K.I.: 'Approximation by trigonometric polynomials in a certain class of periodic functions of several variables', Soviet Math. Dokl. I (1960), 672-675. (Dokl. Akad. Nauk. SSSR 132 (1960), 982-985.) [4] BELINSKII, E.S., AND LIFLYAND, E.R.: 'Behavior of the Lebesgue constants of hyperbolic partial sums', Math. Notes 43 (1988), 107-109. (Mat. Zametki 43 (1988), 192-196.) [5] BELINSKY, E.S.: 'Behavior of the Lebesgue constants of certain methods of summation of multiple Fourier series': Metric Questions of the Theory of Functions and Mappings, Nauk. Dumka, Kiev, 1977, pp. 19-39. (In Russian.) [6] BELINSKY,E.S.: 'Some properties of hyperbolic partial sums': Theory of Functions and Mappings, Nauk. Dumka, Kiev, 1979, pp. 28 36. (In Russian.) [7] BELINSKY,E.S.: 'On the growth of Lebesgue constants of partim sums generated by certain unbounded sets': Theory of Mappings and Approximation of Functions, Nauk. Dumka, Kiev, 1983, pp. 18-20. (In Russian.) [8] BELINSKY, E.S.: 'Lebesgue constants of step hyperbolic partim sums': Theory of Mappings and Approximation of Functions, Nauk. Dumka, Kiev, 1989, pp. 23-27. (In Russian.) [9] DAUGAVET, I.K.: 'On the Lebesgue constants for double Fourier series', Moth. Comput., Lenin9rad Univ. 6 (1970), 8-13. (In Russian.) [10] DYACHENKO, M.: 'Some problems in the theory of multiple trigonometric series', Russian Math. Surveys 47, no. 5 (1992), 103-171. (Uspekhi Mat. Nauk. 47, no. 5 (1992), 97-

162.) 235

LEBESGUE CONSTANTS OF MULTI-DIMENSIONAL PARTIAL FOURIER SUMS [11] GELFAND,I.M., GRAEV, M.I., AND VILENKIN,N.YA.: Generalized functions 5: Integral geometry and problems of representation theory, Acad. Press, 1966. [12] GOLUBOV,B.I.: 'Multiple Fourier series and integrals', J. Soviet Math. 24 (1984), 639-673. (Itogi Nauki i Tekhn. V I N I T I Akad. Nauk. SSSR 19 (1982), 3-54.) [13] HERE, C.S.: 'Fourier transforms related to convex sets', Ann. of Math. 2, no. 75 (1962), 81-92. [14] ILYIN, V.A.: 'Problems of localization and convergence for Fourier series in fundamental systems of the Laplace operator', Russian Math. Surveys 23 (1968), 59-116. ( Uspekhi Mat. Nauk. 23 (1968), 61-120.) [15] KUZNETSOVA,O.I.: 'The asymptotic behavior of the Lebesgue constants for a sequence of triangular partial sums of double fourier series', Sib. Math. J. 18 (1977), 449-454. (Sibirsk. Mat. Zh. X V I I I (1977), 629-636.) [16] LIFLYAND,E.R.: 'Exact order of the Lebesgue constants of hyperbolic partial sums of multiple Fourier series', Math. Notes 39 (1986), 369-374. (Mat. Zametki 39 (1986), 674 683.) [17] LIFLYAND, E.R.: 'Sharp estimates of the Lebesgue constants of partial sums of multiple Fourier series', Proc. Steklov Inst. Math. 180 (1989), 176-177. ( Trudy Mat. Inst. V.A. Steklov. 180 (1987), 151-152.) [18] LIFLYAND, E.R., RAMM, A.G., AND ZASLAVSK¥, A.I.: 'Estimates from below for Lebesgue constants', J. Fourier Anal. Appl. 2 (1996), 287-301. [19] MITYAGIN, B.S.: 'Approximation of functions in LP and C spaces on the torus', Mat. Sb. (N.S.) 58 (100) (1962), 397414. (In Russian.) [20] NAZAROV,F., AND PODKORYTOV, A.: 'On the behavior of the Lebesgue constants for two-dimensional Fourier sums over polygons', St.-Petersburg Math. J. 7 (1995), 663-680. (Algebra i Anal. 7 (1995), 214-238.) [21] PODKORYTOV, A.N.: 'Intermediate rates of growth of Lebesgue constants in the two-dimensional case', J. Soviet Math. 36 (1987), 276-282. (Numerical Methods and Questions on the Organization of Calculations, Part 7 Notes Sci. Sere. Steklov Inst. Math. Leningrad. Branch Acad. Sci. USER, Nauka, Leningrad 139 (1984), 148-155.) [22] PODKORYTOV, A.N.: 'Asymptotic behavior of the Dirichlet kernel of Fourier sums with respect to a polygon', J. Soviet Math. 42 (1988), 1640-1646. (Zap. Nauchn. Sere. L O M I 149 (1986), 142-149.) [23] STEIN, E.M., AND WEISS, G.: Introduction to Fourier on Euclidean spaces, Princeton Univ. Press, 1971. [24] TEMLYAKOV, V.N.: Approximation of periodic functions, Nova Sci., 1993. [25] YUDIN, A.A., AND YUDIN, V.A.: 'Discrete imbedding theoreins and Lebesgue constants', Math. Notes 22 (1977), 702711. (Mat. Zametki 22 (1977), 381-394.) [26] YUDIN, V.A.: 'Behavior of Lebesgue constants', Math. Notes 17 (1975), 369-374. (Mat. Zametki 17" (1975), 401 405.) [271 YUDIN, V.A.: 'A lower bound for Lebesgue constants', Math. Notes 25 (1979), 63-65. (Mat. Zametki 25 (1979), 119-122.) [28] ZHIZHIASHVILI,L.V.: 'Some problems in the theory of simple and multiple trigonometric and orthogonal series', Russian Math. Surveys 28 (1973), 65-127. (Uspekhi Mat. Nauk. 28 (1973), 65-119.) [29] ZHIZHIASHVILI,L.V.: Some problems of multidimensional harmonic analysis, second ed., Tbilisi State Univ., 1996. (In Russian.)

sis

analy-

LEHMER CONJECTURE - A conjecture about the minimal M a h l e r m e a s u r e of a non-zero algebraic integer which is not a root of unity. The Mahler measure M((~) of an a l g e b r a i c n u m b e r c~ is defined by N

M(a) -- a0 H max(l, lai[),

where ao denotes the leading coefficient and N is the degree of the minimal polynomial f (with integral coefficients) of a (cf. also A l g e b r a i c n u m b e r ) and al = OZ,OZ2,... , a N are its conjugates. Since M ( a ) depends only on f, it is also denoted by M ( f ) and called the Mahler measure of f. Jensen's formula (cf. also J e n s e n formula) implies the equality

M(f) = exp (Jilloglf(e27~it)[ dr) and this observation permits one to generalize Mahler's measure to polynomials in several variables (see [11],

[13]). A theorem of L. Kronecker implies that if a is an algebraic integer with M(a) < 1, then a is either zero or a root of unity. D.H. Lehmer [7] asked whether M(c~) could attain values arbitrarily close to 1. This subsequently led to the following formulation of Lehmer's conjecture: There exists a positive constant ~ such that if a ~ 0 is an algebraic integer, not a root of unity, then M(c~) _> 1 + 7/. Lehmer's conjecture is equivalent to the existence of ergodic automorphisms of the infinite-dimensional torus having finite e n t r o p y [8] and its truth would imply the following conjecture stated by A. Schinzel and H. Zassenhaus [14]: There exists a positive constant C with the property that if a is a non-zero algebraic integer of degree N , not a root of unity, then [a], the maximal absolute value of a conjugate of a is at least C It is known ([2], [15]) that Lehmer's conjecture holds for non-reciprocal integers a, i.e. algebraic integers whose minimal polynomials do not have 1 / a as a root. In this case the minimal value for M ( a ) equals 1.32471... and is attained by roots of the polynomial X 3 - X - 1. In 1971, P.E. Blanksby and H.L. Montgomery [1] established, for all algebraic integers a ~ 0 of degree N that are not roots of unity, the inequality

___1 +

236

1

52Nlog(6N)'

and subsequently E. Dobrowolski [4] obtained

~loglogNh 3

E.R. Liflyand

MSC1991: 42B05, 42B08

(1)

M(c~) > l + c \

logN

]

'

LIE TRIPLE SYSTEM

w i t h c = 1/1200, w h e r e a s for N _> N ( c ) he got c = 1 - c . S u b s e q u e n t l y , s e v e r a l a u t h o r s i n c r e a s e d t h e value of c to c = 2 - e ([3], [12]) a n d c = 9 / 4 - e ([9]). Since for nonr e c i p r o c a l i n t e g e r s a one has M ( a )

<_ ~ N / 2 ,

t h e last

result leads t o t h e i n e q u a l i t y 9

(loglogN~

[~>I+~-~\ logN ff

3 '

b u t this h a s b e e n s u p e r s e d e d b y A. Dubickas [5], who p r o v e d for sufficiently l a r g e N t h e i n e q u a l i t y

(04)1

N\

logN /

'

which is t h e s t r o n g e s t k n o w n result t o w a r d t h e S c h i n z e l Z a s s e n h a u s c o n j e c t u r e as of 2000. T h e s m a l l e s t k n o w n value of M ( a ) > 1 is 1.17628.. ,, realized by t h e r o o t of X 1° + X 9 - X 7 - X 6 - X 5 - X 4 X 3 + X + 1 a n d f o u n d in [7]. References

[1] BLANKSBY, P.E., AND MONTGOMERY, H.L.: 'Algebraic integers near the unit circle', Acta Arith. 18 (1971), 355-369. [2] BREUSCH, K.: 'On the distribution of the roots of a polynomial with integral coefficients', Proc. Amer. Math. Soc. 3 (1951), 939-941. [3] CANTOR, D.G., AND STRAUS, E.G.: 'On a conjecture of D.H. Lehmer', Acta Arith. 42 (1982), 97-100; 325. [4] DOBROWOLSKI, E.: 'On a question of Lehmer and the number of irreducible factors of a polynomial', Acta Arith. 34 (1979), 391-401. [5] DUBICKAS,A.: 'On algebraic numbers of small measure', Liet. Mat. Rink. 35 (1995), 421-431. [6] DUBICKAS,A.: 'Algebraic conjugates outside the unit circle': New Trends in Probability and Statistics, Vol. 4, 1997, pp. 1121. [7] LEHMER, D.H.: 'Factorization of certain cyelotomic functions', Ann. Math. 34, no. 2 (1933), 461-479. [8] LIND, D.A., SCHMIDT, K., AND WARD, W.: 'Mahler measure and entropy for commuting automorphisms of compact groups', Invent. Math. 101 (1990), 503-629. [9] LOUBOUTIN, R.: 'Sur la mesure de Mahler d'un nombre alg@brique', C.R. Acad. Sci. Paris 296 (1983), 707-708. [10] MAHLER,K.: 'An application of Jensen's formula to polynomials', Mathematika 7" (1960), 98-100. [11] MAHLER, K.: 'On some inequalities for polynomials in several variables', J. London Math. Soc. 37 (1962), 341-344. [12] RAUSCH,U.: 'On a theorem of Dobrowolski about the product of conjugate numbers', Colloq. Math. 50 (1985), 137-142. [13] SCHINZEL, A.: 'The Mahler measure of polynomials': Number Theory and its Applications (Ankara, 1996), M. Dekker, 1999, pp. 171-183. [14] SCHINZEL,A., AND ZASSENHAUS,H.: 'A refinement of two theorems of Kronecker', Michigan J. Math. 12 (1965), 81-85. [15] SMYTH, C.J.: 'On the product of the conjugates outside the unit circle of an algebraic integer', Bull. London Math. Soc. 3 (1971), 169-175. [16] STEWART, C.L.: 'Algebraic integers whose conjugates lie near the unit circle', Bull. Soc. Math. France 196 (1978), 169-176. Wtadys~aw Narkiewicz M S C 1 9 9 1 : 11C08, 11R04

LEIBNIZ-HOPF

ALGEBRA

AND

QUASI-

SYMMETRIC FUNCTIONS - L e t /~4 b e t h e g r a d e d d u a l of t h e L e i b n i z - H o p f a l g e b r a over t h e integers. T h e strong Ditters conjecture s t a t e s t h a t f14 is a free c o m m u t a t i v e a l g e b r a w i t h as g e n e r a t o r s t h e c o n c a t e n a t i o n p o w e r s of e l e m e n t a r y L y n d o n words. T h i s conject u r e is still o p e n (as of 2001); t h e initial p r o o f c o n t a i n s m i s t a k e s (so t h e a s s e r t i o n of its p r o o f in L e i b n i z - H o p f a l g e b r a is i n c o r r e c t ) , a n d so does a l a t e r v e r s i o n [1] of it. M e a n w h i l e , t h e weak Ditters conjecture, which s t a t e s t h a t A4 is free over t h e integers w i t h o u t giving a c o n c r e t e set of g e n e r a t o r s , has b e e n proved; see Quasi-symmetric f u n c t i o n a n d [2]. References [1] DITTERS, E.J., AND SCHOLTENS, A.C.J.: 'Free polynomial generators for the Hopf algebra Qsym of quasi-symmetric functions', Y. Pure Appl. Algebra 144 (1999), 213-227. [2] HAZEWlNKEL,M.: 'Quasi-symmetric functions', in D. KROB, A.A. MIKHALEV, AND A.V. MIKHALEV (eds.): Formal Power Series and Algebraic Combinatorics (Moscow 2000), Springer, 2000, pp. 30-44. M. Hazewinkel M S C 1 9 9 1 : 05E05, 16W30 LIE TRIPLE SYSTEM - A triple system is a v e c t o r s p a c e V over a field K t o g e t h e r w i t h a K - t r i l i n e a r m a p p i n g V × V × V -+ V. A v e c t o r space U w i t h t r i p l e p r o d u c t [.,., .] is said to be a Lie triple system if [xyz] = - [ y x z ] ,

(1)

[xyz] + [yzx] + [zxy] = 0,

(2)

[xy[~v~]] = [ [ ~ y ~ ] w ] + [~[xyv]~] + [~v[xy~]], for all x , y , z , u , v , w

(3)

E U.

S e t t i n g L ( x , y ) z : = [xyz], t h e n (3) m e a n s t h a t t h e left e n d o m o r p h i s m L ( x , y) is a d e r i v a t i o n of V (cf. also D e r i v a t i o n i n a r i n g ) . T h u s one denotes { L ( x , Y)}span b y I n n Der A. Let A be a Lie t r i p l e s y s t e m a n d let L ( A ) be t h e vect o r space of t h e d i r e c t s u m of I n n Der A a n d A. T h e n L ( A ) is a L i e a l g e b r a w i t h r e s p e c t to t h e p r o d u c t [D + x, E + y] : = [D, E] + D y - E x + L ( x , y), where L ( x , y ) , D , E

C I n n D e r A, x , y E A.

This a l g e b r a is called t h e standard embedding Lie algebra a s s o c i a t e d w i t h t h e Lie t r i p l e s y s t e m A. This implies t h a t L ( A ) / I n n D e r A is a h o m o g e n e o u s s y m m e t ric space (cf. also H o m o g e n e o u s space; Symmetr i c s p a c e ) , t h a t is, it is i m p o r t a n t in t h e correspondence w i t h g e o m e t r i c p h e n o m e n a a n d a l g e b r a i c systems. The relationship between Riemannian globally symmetric spaces and Lie t r i p l e s y s t e m s is given in [4], a n d t h e r e l a t i o n s h i p b e t w e e n t o t a l l y geodesic s u b m a n i f o l d s a n d 237

LIE TRIPLE SYSTEM Lie triple systems is given in [1]. A general consideration of supertriple systems is given in [2] and [5]. Note t h a t this kind of triple system is completely different from the combinatorial one of, e.g., a Steiner triple s y s t e m (cf. also S t e i n e r s y s t e m ) . References

[1] HELGASON, S.: Differential geometry, Lie groups, and symmetric spaces, Acad. Press, 1978. [2] KAMIYA, N., AND OKUBO, S.: 'On d-Lie supertripIe systems associated with (e,d)-Preudenthal-Kantor supertriple systems', Proe. Edinburgh Math. Soc. 43 (2000), 243-260. [31 LISTER, W.G.: 'A structure theory of Lie triple systems', Trans. Amer. Math. Soc. "/2 (1952), 217-242. [4] LOOS, O.: Symmetric spaces, Benjamin, 1969. [5] OKUBO, S., AND KaMWA, N.: 'Jordan-Lie super algebra and Jordan-Lie triple system', J. Algebra 198, no. 2 (1997), 388411. Noriaki Kamiya

T h e linear complexity of a sequence is an i m p o r t a n t aspect in judging its suitability for use in c r y p t o g r a p h y . A high linear c o m p l e x i t y by itself does not guarantee any r a n d o m n e s s p r o p e r t i e s of the sequence considered. T h e linear c o m p l e x i t y profiles of binary r a n d o m sequences are analyzed in [3, C h a p . 4 ], where it is shown t h a t a binary r a n d o m sequence a of length N usually has linear complexity very close to N / 2 with the complexity profile growing in a r o u g h l y (but not exactly!) continuous m a n n e r (so t h a t L k ( a ) is close to k / 2 ) . Moreover, using a to generate a periodic sequence s with period N results in a linear c o m p l e x i t y close to N , provided t h a t N is a power of 2 or a Mersenne prime n u m b e r (cf. M e r s e n n e n u m b e r ) . Consequently, a periodic binary sequence with g o o d r a n d o m n e s s properties should have complexity close to the period length and a profile growing m o r e or less smoothly. References

MSC 1991:17A40

[1] BLAHUT, R.E.: Theory and practice of error control codes, Addison-Wesley, 1983.

L I N E A R C O M P L E X I T Y OF A S E Q U E N C E - For a

s h i f t r e g i s t e r s e q u e n c e a, the linear complexity L(a) is just the degree of its m i n i m a l polynomial m, i.e. the length of a shortest linear feedback shift register (LFSR; cf. S h i f t r e g i s t e r s e q u e n c e ) capable of producing a. T h e linear complexity also equals the m a x i m u m n u m b e r of linearly independent vectors a m o n g the state vectors

a (t) = ( a t , a t + l , . . . , a n + t - 1 )

(t >_0)

[2] ,]UNGNICKEL, D.: Finite fields: Structure and arithmetics,

Bibliographisches Inst. Mannheim, 1993. [31 RUEPPEL, R.: Analysis and design of stream ciphers, Springer, 1986. Dieter Jungnickel

M S C 1991: 94A60, 93B99, 68Q15, 65C10 LINEAR CONGRUENTIAL METHOD - A method

widely used for generating r a n d o m n u m b e r s from the u n i f o r m d i s t r i b u t i o n : A sequence of integers is initialized with a value z0 and continued as zi+l-azi+r

(modrn),

O
(1)

of an a r b i t r a r y L F S R of length n associated with a. Now,

let a denote either a finite sequence (ak)k=O..... N-1 of length N or an infinite sequence (ak)k>_O over a G a l o i s field F = GF(q) (in the latter case, put N = ce). For every positive integer k < N , denote by Lk (a) the length of an L F S R Ak(a) of least length over F capable of producing a shift register sequence s (k) which agrees with a for the first k entries a o , . . . , ak-1. In this way, one obtains a sequence L = (Lk(a)) of the same length N , which is called the linear complexity profile of a. T h e n one defines the linear complexity L(a) of a as the m a x i m u m value of all Lk(a) if these values are bounded, and as ec otherwise. Thus, L(a) = oe if and only if a is an infinite sequence which is not u l t i m a t e l y periodic (cf. U l t i m a t e l y p e r i o d i c s e q u e n c e ) , and L(a) = L N ( a ) if a is a finite sequence of length N . Hence the linear complexity profile constitutes a refinement of the linear complexity of a sequence. T h e linear complexity profile of a and the associated sequence of L F S R s Ak(a) can be c o m p u t e d efficiently by the celebrated B e r l e k a m p - M a s s e y alg o r i t h m , see [1] or [2]. 238

for all i. T h e fractions ui = z i / r n are the derived p s e u d o - r a n d o m n u m b e r s in the interval [0, 1) (cf. also Random and pseudo-random numbers; Pseudor a n d o m n u m b e r s ) . T h e constants m, the modulus, a, the multiplicator, r, the increment, and z0, the starting number, are suitably chosen non-negative integers. T h r e e choices of m, a and r are c o m m o n on most computers: 1) r = 0, m = 2 E, a - 5 ( r o o d S ) , and z0 - 1 (rood 4). All zi - 1 (rood 4) are generated. 2) r = 0, m = p, p prime, a a p r i m i t i v e r o o t roodulo p. All zi = 1 , . . . ,p - 1 are generated. 3) r _-- 1 (mod 2), m = 2 E, a ~ 1 (mod 4). All integers 0 , . . . , 2 E - 1 are generated. For selecting ' g o o d ' r a n d o m n u m b e r generators one has to s t u d y the distribution of the k-tuplets Pa = ( U i + l , . . . , u i + k ) . G e o m e t r i c a l l y these Pk m a y be considered as points of a lattice G in the k-dimensional h y p e r c u b e [0, 1) a. T h e lattice points can also be seen as intersection points of k sets of parallel hyperplanes. Consequently, the following questions m a y be raised:

LINEAR C O N G R U E N T I A L M E T H O D i) Determine the minimal number N~ of parallel hyperplanes on which all points Pk lie. ii) Determine the maximal distance D~ of parallel hyperplanes on which all points Pk lie. Question i) was asked by G. Marsaglia [12], who derived upper bounds for N~ using Minkowski's convex body theorem (cf. also M i n k o w s k i t h e o r e m ) . The 'wave numbers' W~ = 1/D~ were introduced by R.R. Coveyou and R.D. MacPherson [4]. Their algorithm for calculating W~ was simplified by D.E. Knuth [10]. The calculation of both quantities is based on a general procedure to determine non-zero vectors of shortest length in the dual lattice of covering hyperplanes. For the determination of N~ the gl-norm is used, and for D~ the Euclidean norm is appropriate. The algorithm of U. Dieter (1973) gives exact values for both quantities; no exact values for N~ were known before. Knuth included a variant of this algorithm in [10]. A completely different approach was proposed by L. Lovgsz, J.K. Lenstra and H.W. Lenstra, Jr., called the LLL-algorithm (cf. also L L L basis r e d u c t i o n m e t h o d ) . In the case of the Euclidean norm, the final search can be shortened by an idea of U. Flake and M. Pohst [8]. For any sequence {ui} of [0, 1)-uniformly distributed random numbers, the local deviations k

zxk(s, t) = - I I ( t j

- 8j)+

j=l

•{Ui = ( U i + l , . . . , ui-l-k): 8j <~ ui-bj ~ tj, 1 <_ j <_ k} =

and their largest value, the (global) discrepancy Ak = sup{lAk(s,t)l : 0 _< sj <_ tj < 1, 1 <_ j <_ k } ,

(2) are of great importance. For example, for the calculation of k-dimensional integrals by Monte-Carlo methods, the difference of the integral and its approximation by a Riemann sum is bounded by the discrepancy Ak multiplied by the variation of the function V ( f ) (in the sense of Hardy-Krause, cf. also V a r i a t i o n o f a f u n c tion). Since the variation of the function is fixed, the discrepancy has to be as small as possible. See [9]. No methods are known for calculating the discrepancy in dimension greater than two. Dieter derived a lower bound in 1973, and H. Niederreiter found an upper bound in 1978 [13]. Even these bounds are difficult to calculate. In dimension two, i.e. in the case of pairs, all three quantities can be calculated by the E u c l i d e a n algor i t h m for the period length n and a. n is equal to m / 4 if m = 2E and r = 0 and n = m in the two other cases.

Define a sequence {mi} by m0=n,

ml=a,

(3) i=1,2,...,

mi-1 = ai-lmi + mi+j,

where ai-l=

mi-1 , (4) L mi J and LxJ is the integer function (or f l o o r f u n c t i o n ) . Associated is the sequence

p0=0,

(5)

Pl = 1,

Pi+l = a i - l p i + Pi-*,

i = 1, 2, . . . .

Then

(6) and W;-D-~

1 _ min v/~m~ + p2. i V '-~

(7)

Finally, for the discrepancy the following rather sharp bounds hold [6]:

1

~nnmaX{ai:0
1(±

ai+2

)

. (8)

i=0

See [2], [7] for methods for calculating exact values of the discrepancy. The numerical values differ only slightly from the upper bound in (8). If the dimensions become larger, the number of hyperplanes on which the lattice points Pk lie decreases considerably. Therefore a different procedure was proposed by Knuth [i0]: A sequence of integers zi is initialized to (z0,...,zr-1) ~ (0,...,0) and updated by zi - a i z i - I + " " + arZi-r

(mod p)

(9)

for 0 < z~ < p. Here the factors ai are given integers, and for the modulus p only prime numbers are considered. Again, the fractions ui = z i / p are taken as random samples from the interval [0, 1). Since there are only p~ possible r-tuplets ( z k , . . . , z k + ~ - l ) and ( 0 , . . . , 0 ) must not occur, the period length of (9) is at most p~ - 1. It can be shown that this maximum period length of p~ - 1 may in fact be achieved for suitable choices of the factors a l , . . •, at. This means that all r-tuplets (Zk,...,Zk+r-i) ( 0 , . . . ,0) must occur, resulting in perfectly uniform distributions of the ui = z i / p , the pairs (ui,ui+l), the triplets (ui,ui+x,u~+2) (if r >__ 3), etc. In other words, for all dimensions k < r the hypercube [0, 1) k is now evenly filled. For dimensions k > r the quantities N~ and D~ can be calculated exactly, since the points Pk are again points of a lattice. The numerical values show that a sequence of type (9) behaves as good in dimension k x r as a linear congruential generator behaves in dimension k. Furthermore, reduced bases (in the sense of H. Minkowski) can be determined which show how 'good' the specific generator behaves. See [3] and recent

239

LINEAR CONGRUENTIAL METHOD publications of L. Aflterbach and H. Grothe, starting with [1]. T h e case p = 2 is of special interest; it is called the Tausworth generator. Here, bits are p r o d u c e d and a word of, say, 8 bits is taken as a sample from the uniform distribution. See [11] for further information. References [1] AFFLERBAeH, L., AND GROTHE, H.: 'Calculation of Minkowski-reduced lattice bases', Computin 9 35 (1985), 269276. [2] AFFLERBACtt,L., AND WEILBACHEa,R.: 'The exact determination of rectangle discrepancy for linear congruential pseudorandom generators', Math. Comput. 53 (1989), 343-354. [3] BEYER, W.A., ROOF, R.B., AND WlLLIAMSON,D.: 'The lattice structure of multiplicative congruential pseudo-random vectors', Math. Comput. 25 (1971), 345-360. [4] COVEYOU,R.R., AND MAePHERSON, R.D.: 'Fourier analysis of uniform random number generators', J. ACM 14 (1967), 100-119. [5] DIETER,U.: 'Pseudo-random numbers: the exact distribution of pairs', Math. Comput. 25 (1971), 855-883. [6] DIETER, U.: 'How to calculate shortest vectors in a lattice', Math. Comput. 29 (1975), 827-833. [7] DIETER, U.: 'Pseudo-random numbers: The discrepancy in two dimensions', to appear (2002). [8] FrNKE, U., AND POHST, M.: 'Improved methods for calculating vectors of short length in a lattice, including a complexity analysis', Math. Comput. 44 (1985), 463 471. [9] FISHMAN, G.S.: Monte Carlo: Concepts, algorithms, and applications, Springer, 1996. [10] KNUTH, D.E.: The art of computer programming, first ed., Vol. II: Seminumerical algorithms, Addison-Wesley, 1969. [11] LEWrS, T.G., AND PAYNE,W.H.: 'Generalized feedback shift register pseudorandom number algorithms', J. ACM 20 (1973), 456-468. [12] MARSAGLI*,G.: 'Random numbers fall mainly in the planes', Proc. Nat. Acad. Sci. 61 (1968), 25-28. [13] NIEDERREITER,H.: 'Quasi-Monte-Carlo methods and pseudorandom numbers', Bull. Amer. Math. Soc. 84 (1978), 957 1041. U. Dieter MSC1991:65C10 LINEAR

SKEIN-

See S k e i n m o d u l e .

MSC 1991:57M25 L I O U V I L L E - L O J A S I E W I C Z I N E Q U A L I T Y - A Liouville inequality is one e m b o d y i n g the principle in number theory t h a t algebraic n u m b e r s cannot be very well approximated by rational n u m b e r s or, equivalently, t h a t integral polynomials cannot be small and non-zero at algebraic numbers (cf. also L i o u v i l l e t h e o r e m s ) . A Lojasiewicz inequality gives a lower b o u n d for functions in terms of the distance to c o m m o n zeros. These features can be combined [2] in the following L i o u v i l l e - L o j a s i e w i c z inequality. Let each P1, • • •, P ~ E Z [ x l , . . . , Xn] have total degree at most d and coefficients of absolute value at most exp(h). For co e C n, let ]co] > 1

240

be greater than the coordinates to the distance Then there are on n such that

or equal of co and from co explicit

to the largest absolute value of let p < 1 be less than or equal to the common zeros Z of Pi. constants el, c2, c3 depending

log max{lPi(co)l } > - d ' ( c i d

+ e2h) + e3d" log ~-~,

where # : = m i n { m , n - 1 } , , := rain{m, n}. Over a r b i t r a r y fields with an absolute value, the lower b o u n d takes the form - c l + cad ~ log(p/leo]), cf. [1], [4] and [5] (in the last citation, the polynomials Pi are replaced by ideals [i and Ii(co) are taken to be the values of fixed Chow coordinates of Ii). In this setting, M. Hickel [3] obtains the o p t i m a l involvement of Iw] at the right-hand side. Actually, the above arithmetic inequality holds with ca = 1. If, when working over Z, co denotes a zero of an unmixed ideal I and p denotes the distance from w to the zeros of I + ( P 1 , . . . , Pro), t h e n the above upper b o u n d holds with # : = m i n { d i m I , n - 1}, u : = m i n { d i m I , n}, with el replaced by c l d e g I + c 2 1 o g h t I , and e2 by e - 2 deg I. W h e n dim I = 0, the zeros Z ( I ) of I have algebraic coordinates. W h e n m = 1 and P1 does not vanish at any point of Z ( I ) , then one obtains an explicit lower b o u n d on IP1 (w)l , i.e. a Liouville inequality. References [1] BROWNAWELL,W.D.: 'Bounds for the degrees in the Nullstellensatz', Ann. of Math. 126 (1987), 577 591. [2] BROWNAWELL,W.D.: 'Local diophantine Nulistellen equalities', J. Amer. Math. Soe. 1 (1988), 311-322. [3] HICKEL, 1V[.:'Solution d'une conjecture de C. Berenstein-A. Yger et invariants de contact ~ l'infini', Prepubl. Lab. Math. Pures Univ. Bordeaux I 118, no. jan. (2000). [4] JI, S., KOLL~.R,J., AND SHIFFMAN,B.: 'A global Lojasiewicz inequality for algebraic varieties', Trans. Amer. Math. Soc. 329 (1992), 813-818. [5] KOLL~R, J.: 'Effective Nullstellensatz for arbitrary ideals', J. Europ. Math. Soc. (JEMS) 1 (1999), 313-337. W. Dale Brownawell M S C 1991: 14Q20, 11J68 L I S T I N G P O L Y N O M I A L S - Two polynomial invariants of a knot diagram, described by J.B. Listing in 1847 [1], n a m e d by him complexions-symbol (cf. also K n o t a n d l i n k d i a g r a m s ) . Listing associated with every corner of a crossing a symbol 5 (for dexiotropic, i.e. righthanded) or A (for leotropic, i.e. left-handed) according to the convention of Fig. la. He then coloured regions cut out by the d i a g r a m in a checkerboard manner: black and white and such t h a t neighbouring regions have different colours. To each region he associated a 2-variable m o n o m i a l 5Q~j , where i is the n u m b e r of 5 corners and j the n u m b e r of A corners in the region. T h e f i r s t Listing polynomial, or black Listing polynomial, Pb (5, A) is equal to the sum of the above monomials taken over all black

LOCAL T O M O G R A P H Y regions. The second Listing polynomial, or white Listing polynomial, P~((~, A) is defined in a similar manner, summing over white regions. Listing noted that for an alternating knot diagram each region has corners of the same colour, and that his polynomials are in fact one-variable polynomials. He also stated that the left- and right-handed trefoil knots a r e not equivalent (cf. also T o r u s k n o t ) and that there are knots without an alternating diagram. He noticed further that his polynomials can change when the knot diagram is modified without changing the knot type.

~ 2~3 + ~2 (a)

(b)

5

Fig. 1. In his 1849 note, Listing showed that the figure-eight knot, Fig. lb, and its mirror image are equivalent [2]. Because of this, this knot is often called the L i s t i n g knot. In modern knot theory, Listing's ideas are widely used: colouring of regions (e,g. to define the Goeritz matrix), labelling of corners (e.g. to define the K a u f f m a n b r a c k e t p o l y n o m i a l ) , studying labelling of corners of alternating diagrams (e.g. to proof the TaR conjectures). Listing polynomials can be thought as a precursors of the partition function of a state model of a physical system in statistical mechanics (of. also S t a t i s t i c a l m e c h a n i c s , m a t h e m a t i c a l p r o b l e m s in). References

[1] LISTINC, J.B.: 'Vorstudien zur Topologie', G6ttin9er Studien (Abt. 1) 1 (1847), 811-875. [2] LITTLE, C.N.: ' N o n - a R e r n a t e ± knots', Trans. Royal Soc. Edinburgh X X X I X : III, no. 30 (1899), 771-778, Read: July 3, 1899.

Jozef Przytycki MSC 1991:57M25 LOCAL TOMOGRAPHY - Let f(x) be a compactly supported piecewise-smooth function, f(x) = 0 if x ¢ C R 2, D a bounded domain, and let f'(a,p) = f,~, f(x) ds := R f be its R a d o n t r a n s f o r m , where ~c~p :~--- {X: O~-X: p} is the straight line parametrized by the unit vector a and a scalar p. The inversion formula which reconstructs f(x) from the knowledge of f'(a,p) for all a E S 1 and all p E R, where S 1 is the unit circle in R 2, is known to be:

f(x) = ~ 1 f s ~ / / oc ~fP(a'P) ' x ~ P da dp,

o?

(1)

It is non-locah one requires the knowledge of f ( a , p ) for all p in order to calculate f(x). By local tomographic data one means the values of f'(a,p) for those a and p which satisfy the condition [a • x0 - p[ < 5, where Xo is a fixed 'point of interest' and 5 > 0 is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at x0 with radius 5. In many applications only local tomographic data are available, while in medical imaging one wants to minimize the radiation dose of a patient and to use only the local tomographic data for diagnostics. Therefore, the basic question is: What practically useful information can one get from local tomographic data? As mentioned above, one cannot find f(xo) from local tomographic data. What does one mean by 'practically useful information' ? By this one means the location of discontinuity curves (surfaces, if n > 2) of f(x) and the sizes of the jumps of f(x) across these surfaces. Probably the first empirically found method for finding discontinuities of f(x) from local tomographic data was suggested in [11], where the function

(2) which is the standard local tomography function, was proposed. To calculate f(x) one needs to know only the local tomography data corresponding to the point x. It is proved that f(x) and fset(x) have the same discontinuities (but different sizes of the jumps across the discontinuity curves) [10]. For various aspects of local tomography, see the references. See also T o m o g r a p h y . In [6], [7], [8], a large family of local tomography functions was proposed. The basic idea here is to establish a relation between hypo-elliptic pseudo-differential operators and a class of linear operators acting on the functions f ( a , p ) . Let 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 be defined by the formula B / = Y-][b(x,t,a)f], where f := 5~f is the F o u r i e r t r a n s f o r m , f(¢) = fa~ f(x) ei~'x dx, and b(x, t, @ is a smooth function, which is called the symbol of B, a := ~/141, t = I~[. If the symbol is hypo-elliptic, that is, el[~] rnl ~ [b[ _< c2[~[m2, [~[ > R, x E D, cl and c2 are positive constants, then W F ( B f ) = W F ( f ) , where W F ( f ) is the w a v e f r o n t of f . Thus, the singularities of B f and f are the same. One can prove [8] the formula B f = R*(ae ® f) := dr', where R*9 := fsn_l g ( a , a • x) da, where R* is the adjoint to the Radon operator R (cf. also R a d o n t r a n s f o r m ) , and 241

LOCAL T O M O G R A P H Y a ® f := f~_~ a(x, a, p - q) f(q) dq is the convolution operator, with distributional kernel a(x, a , p - q) defined by

1 .-

(2

tn-le-itPb(x, t, a) dt, )n

and with

a(x, a,p) + a ( x , - a , - p ) ae (X, Ct, p) :=

2

the even part of a(x, a,p). An operator A is called a local tomography operator if and only if suppae(X,a,p) C [-(~,~] uniformly with respect to x E D and a E S "-1. A necessary and sufficient condition for A to be a local tomography operator is given in [8]: The kernel b(x, t, a)t~_-1 + b(x, - t , - a ) t n-l_ is an entire f u n c t i o n of t of exponential type _< (~ uniformly with respect to x E D and a E S ~ - l . References [1] FARIDANI, A., RITMAN, E., AND SMITH, K.: 'Local tomography', S I A M J. Appl. Math. 52 (1992), 459-484. [2] KATSEVICH, A.: 'Local tomography for the generalized Radon transform', S I A M J. Appl. Math. 57, no. 4 (1997), 11281162. [3] KATSEVICH, A.: 'Local tomography for the limited-angle problem', J. Math. Anal. Appl. 213 (1997), 160-182. [4] KATSEVICH, A.: 'Local tomography with nonsmooth attenuation II', in A.G. RAMM (ed.): Inverse Problems, Tomography, and Image Processing, Plenum, 1998, pp. 73-86. [5] KATSEVICH, A.: 'Local tomography with nonsmooth attenuation', Trans. Amer. Math. Soc. 351 (1999), 1947-1974. [6] RAMM, A.G.: 'Optimal local tomography formulas', PanAmer. Math. J. 4, no. 4 (1994), 125-127. [7] RAMM, A.G.: 'Finding discontinuities from tomographic data', Proc. Amer. Math. Soc. 123, no. 8 (1995), 2499-2505. [8] RAMM, A.G.: 'Necessary and sufficient conditions for a PDO to be a local tomography operator', C.R. Acad. Sci. Paris 332, no. 7 (1996), 613-618. [9] RAMM, A.G.: 'New methods for finding discontinuities of functions from local tomographic data', J. Inverse Ill-Posed Probl. 5, no. 2 (1997), 165-174. [10] RAMM, A.G., AND KATSEVICH, A.I.: The Radon transform and local tomography, CRC, 1996. [11] VAINBERG, E., I~AZAK, I., AND KURCZAEV, V.: 'Reconstruction of the internal 3D structure of objects based on real-time integral projections', Soviet J. Nondestr. Test. 17 (1981), 415-423. (In Russian.)

A. G. Ramm MSC 1991: 44A12, 92C55, 65R10 LOVASZ LOCAL LEMMA, LLL - A central technique in the probabilistic method. It is used to prove the existence of a 'good' object even when the random object is almost certainly 'bad'. It is applicable in situations in which the bad events are mostly independent. It sieves the bad events to find the rare good one. Let Be, a E I, be a finite family of 'bad' events. A graph G on I is called a dependency graph for the 242

events if each Be is mutually independent of those B~ with a,/3 not adjacent (cf. also I n d e p e n d e n c e ) . S y m m e t r i c ease o f t h e Lovfisz local l e m m a . Let B~, G be as above. Suppose all P[B~] _< p. Suppose all a E I are adjacent to at most d other /3 E I. Suppose 4dp < 1. Then AIB~ ¢ 0. Here, the number of events, III, may be arbitrarily high, giving the Lov~sz local lemma much of its strength. In most applications the underlying probability space is generated by mutually independent choices, each event Be depends on a set X~ of choices, and a,/3 are adjacent when X~, X~ overlap.

Example. Let A~, a E I, be sets of size ten in some universe f~, where every v E f~ lies in at most ten such sets. Then there is a red-blue colouring of ft so that no A~ is monochromatic. The underlying space is a random red-blue colouring of fk The bad event B~ is that A~ has been coloured monochromatically. Each P[B~] = 2 .9 = p. Each A~ overlaps at most 90 other A3, so d = 90. The Lovgsz local lemma gives the existence of a colouring. The lemma was discovered by L. Lov~sz (see [3] for an original application) in 1975. It ushered in a new era for the probabilistic method. G e n e r a l case of t h e Lowlsz local lemma. Let Be, G be as above. If there exist an x~ E (0, 1) with P[B~] _< x~ I-I(1 - x~),

the product over those fl adjacent to a, then ABe ¢ 0. Application of the general case generally requires mild analytic skill in choosing the x~. The proof of the Lovgsz local lemma (in either case) requires only elementary (albeit ingenious) p r o b a b i l i t y t h e o r y and takes less than a page. A breakthrough in algorithmic implementation was given by J. Beck [2] in 1991. He showed that in certain (though not all) situations where the Lovgsz local lemma guarantees the existence of an object, that object can be found by a polynomial-time algorithm. Proofs, applications and algorithmic implementation are explored in [1] and elsewhere. The acronym LLL is also used for the LenstraLenstra-Lovgsz algorithm (see LLL basis r e d u c t i o n method). References [1] ALON, N., AND SPENCER, J.: The probabilistic method, second ed., Wiley, 2000. [2] BECK, J.: 'An algorithmic approach to the Lov~sz local lemma, I': Random Structures and Algorithms, Vol. 2, 1991, pp. 343-365. [3] ERD6S, P., AND LOViSZ, L.: 'Problems and results on

LUCAS 3-chromatic hypergraphs and some related questions', in A. HMNAL ET AL. (eds.): Infinite and Finite Sets, NorthHolland, 1975, pp. 609-628.

POLYNOMIALS

Bergum and Hoggatt Jr. introduced in [1] the bivariate Lueas polynomials V~(x, y) by the recursion

Joel Spencer

Vo(x, y) = 2,

MSC 1991:05C80

V1 (x, y) = x, LUCAS

POLYNOMIALS

V~(x,y) =n 2,3,...,xVn-I(x' = Y) + y V ~ - 2 ( x , y ) ,

The polynomials V~(x)

-

(11)

(cf. [1] and [5]) given by v0(z) = 2,

Vx(x) =

x,

(1)

Vn(X) : xVn_l(X ) -~ Vn_2(x),

n : 2,3,....

generalized (7) for V~(x,y), and showed that the V~(x,y) are irreducible polynomials over the rational numbers if and only if n = 2 k for some positive integer (cf. also I r r e d u c i b l e p o l y n o m i a l ) . The formula

They reduce to the Lucas numbers L~ for x = 1, and they satisfy several identities, which may be easily proved by induction, e.g.:

[n/2]

n

(n - j)[

V~(x,y) = Z n - j J G : ~)~x

n-2j j

y,

(12)

j=0

n = 1,2,..., (2)

V-n(X) = ( - 1 ) n V n ( x ) ; Vm+n(x)

=

Vm(x)Vn(x)

V2n(X)

:

-

V~(x) - 2(-1)~;

(4)

V2n+l(x) = Vn+l(x)V~(x) - (-1)~x; V2~(x)

=

which may be derived by induction on n or by expanding y), generalizes (8). Ch.A. Charalambides [3] introduced and studied the Lucas and Lucas-type polynomials of order k, V (k)(x) and L (k) (x). The Lucas-type polynomials o] order k satisfy the recurrence

(3) the g e n e r a t i n g f u n c t i o n of Vn(x,

(--l)nVm--n(X);

(5) (6)

U~(x)V~(x),

where Urn(x) denote the F i b o n a e e i p o l y n o m i a l s ;

/L~~)(~)

=

~(x)

= ~(x)

+ 9n(x),

(7)

where a(x) = x + ( x 2 2 + 4)1/2'

x,

n = 2,...,k, L(k) (x) = x E j =k I n ~(k) _j(x),

f l ( x ) = x - ( x 22+ 4)1/2'

(13)

n = k + l,k + 2,....

so that a(x)fl(x) = -1; and [~/2] n (n-j)! x~_2j E~(x) = E n - - j j ! ( n - - - - 2 - f ) ! ' j=0

These polynomials have the binomial and multinomiM expansions (8)

L(k)(x) = - 1 +

n = 1,2,...,

[n/(k+x)]

where [y] denotes the greatest integer in y. The Lucas polynomials are related to the C h e b y s h e v p o l y n o m i a l s Tn(x) = cos(n0), cos(0) = x, by V,(x) = 2i-~T, (2)'

i=(-1)

1/2.

J. Riordan [9] considered the polynomials h~(x)

(9) =

i - " V n ( i x ) and the Lucas-type polynomials

n ( n - j)! L~(~)= [~/2] ~j = 0 n-j j~(~--F)~ ~~-J =

+ = E

E j=O

n (n - j k ) ! (-1)J n . 2 j k j ! ( n - - - j - k ~ j )

where the second summation is taken over all nonnegative integers n l , . . . , nk such that nl + 2n2 + ..- + knk = n, and they are related to the Fibonacci-type polynomials of order k (cf. [6] and [8] and F i b o n a c c i p o l y n o m i a l s ) , Fn(k) (x), by min{n,k}

.~ n_j+l(X).

(15)

j=l

= xn/2Vn(xl/2),

in a derivation of Chebyshev-type pairs of inverse relations. V.E. Hoggatt Jr. and M. Bicknell [4] found the roots of V~(x). These are xj = 2icos((2j + 1)Tr/2n), j = 0 , . . . , n - 1. Bicknell [2] showed that V,~(x) divides V~(x) if and only if n is an odd multiple of m. G.E.

x j (1 + x) n - j k - j = !

nl + 2n2 + ' " + knk (nl + " ' + nk)! ,~l+'"+~k nl + " • + nk nl! • • • nk ! x ,

(10)

n= 1,2,...,

(14)

Furthermore, V (k) (x) = x-nn(~ k) (x k) =

(16)

min{n,k}

=

E

jxk--j+l U(nk)-J+1 (X),

j=l

n=1,2,...,

k=2,3,..., 243

LUCAS POLYNOMIALS where the U~(,k) (x) are the Fibonacci polynomials of order k (cf. [7]). Charalambides [3] showed that the reliability of a circular c o n s e c u t i v e k-out-of-n: F - s y s t e m , Rc(p;k,n), whose components function independently with probability p is given by

Rc(p;k,n) = q'~L(k) (P) = --

~

q n

[~/(k+l)] X-" +

z_,

j=o

(17)

(-1); n-

jk j!(n-

jk

- j)! -

u

"

References [1] BERGUM, G.E., AND HOGGATT, JR., V.E.: 'Irreducibility of Lueas and generalized Lucas polynomials', Fibonaeci Quart. 12 (1974), 95 100. [2] BICKNELL, M.: ~A primer for the Fibonacci numbers VII', Fibonacei Quart. 8 (1970), 407-420.

244

[3] CIIARALAMBIDES, CH.A.: 'Lucas numbers and polynomials of order k and the length of the longest circular success run', Fibonacci Quart. 29 (1991), 290-297. [4] HOOGATT JR., V.E., AND BICKNELL, M.: 'Roots of Fibonacci polynomials', Fibonaeci Quart. 11 (1973), 271-274. [5] LUCAS, E.: 'Theorie de fonctions numeriques simplement periodiques', Amer. J. Math. 1 (1878), 184-240; 289-321. [6] PHILIPPOU, A.N.: 'Distributions and Fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-outof-n : F systems', ill A.N. PHILIPPOU, G.E. BERGUM, AND A.F. HORADAM (eds.): Fibonacci Numbers and Their Applications, Reidel, 1986, pp. 203-227. [7] PHILIePOU, A.N., GEORGEIOU, C., AND PHILIPPOU, G.N.: 'Fibonacci polynomials of order k, multinomial expansions and probability', Internat. Y. Math. Math. Sci. 6 (1983), 545-550. [8] PHILIPeOU, A.N., GEOtVGHIOU, C., AND PHILIPPOU, G.N.: 'Fibonacci-type polynomials of order k with probability applications', Fibonacei Quart. 23 (1985), 100 105. [9] RIORDAN, J.: Combinatorial Identities, Wiley, 1968. Andreas N. PhiIippou

MSC 1991:11B39

M MACHINE LEARNING, M L

Machine learning is concerned with modifying the knowledge representation structures (or knowledge base) underlying a computer program such that its problem-solving capability improves (for surveys, cf. [5], [11]). More specifically, a program is said to learn from experience E with respect to some class of tasks T and performance measure 7r, if its performance at tasks t E T, as measured by % improves as training experience E increases. -

Given this definition, any learning system must make choices along the following dimensions: • The type of training experience from which the system will learn, so that it may either directly or, worse, indirectly assess whether its performance on certain tasks has improved. • The type of knowledge which the system will learn, its target function, and how that knowledge will be used after the training phase. Care has to be taken that this function contains an operational description such that it be efficiently computable and properly approximates the 'ideal' target function. • The kind of representation structure the learning system will use in order to describe the target function to be learned. The choice of that representation structure implies a direct correlation between expressiveness and the required size of the training data. • A learning algorithm which evaluates training examples in order to approximate effectively the target function by a learning hypothesis. Each training example is a pair composed of a training instance (a particular state in the problem space) and a corresponding value assignment. The latter is related to the utility of the training instance for properly solving the tasks t C T. The validity ('best fit') of the function approximation (on the training set) might be assessed by computing the minimum of the squared error between the training values and the values predicted by the learning hypothesis, given the particular training instance.

This approach to machine learning can be characterized as a form of inductive inference, where, from a sample of examples of a function f , a hypothesis function f" is guessed that approximates f as closely as possible. Alternatively, one may consider machine learning as a search problem (cf. S e a r c h a l g o r i t h m ) . This typically involves a very large (usually, exponentially growing) search space of possible target functions and requires to determine the one(s) which best fit(s) the data in terms of task accomplishment. A learning system can be viewed as consisting of four main components: • the performance system solves the tasks t E T by using the hypothesis, or learned target function, f'; • a critic, given a trace of each solution, judges the quality of the solution by comparing training instances with their benefits as computed by f; • the generalizer generalizes from this set of training examples in order to better fit the available and, possibly, additional data by hypothesizing a more general function, f", according to the 'best fit' criterion just discussed; and, finally, • the experiment generator takes this new hypothesis ]) and outputs a new task t E T to the performance system, thus closing the update loop. Machine learning is typicMly applied to tasks which involve the classification of new, unseen examples into one of a discrete set of possible categories. The inference of a corresponding target function, a so-called classifier, from training examples is often referred to as concept learning. If the target function involves the representation of 'if-then' rules, this kind of learning is referred to as rule learning. If the learning involves sequences of actions and the task consists of acquiring a control strategy for choosing appropriate actions to achieve a goal, this mode is sometimes referred to as policy learning. A particular learning algorithm is as good or as bad as its performance is on classifying unseen exemplars,

MACHINE LEARNING i.e., whether its decisions are right or wrong. A methodology for checking the prediction quality of a learning algorithm involves the following steps: 1) Collect a large set of examples. 2) Divide it into two disjoint sets: the training set and the test set. 3) Run the learning algorithm with exemplars only taken from the training set and generate an approximation of the target function, the learning hypothesis 4) Evaluate the degree of correct classifications of the current version of f" and update it to reduce the number of misclassifications. 5) Repeat steps 2)-4) for different randomly selected training sets of various sizes, until the learning algorithm delivers a sufficient degree of correct classifications. Inherent to the choice of a large set of examples and the approximation of the target function are several well-known problems. These include, among others, noise in the data (e.g., when two examples have the same description but are assigned different a priori classification categories), and over-fitting of the learned target function in the sense that it does well for the training set but fails for the test set (mostly due to incorporating irrelevant attributes in the target function). A crucial issue for machine learning is the kind of representation structure underlying the target function. The representation formalisms most widely used are attribute-value pairs, first-order predicate logic, neural networks, and probabilistic functions. Attribute-value representations address Boolean functions which deal with different logical combinations of attribute-value pairs (a single attribute may have continuous or discrete values, discrete ones can be Boolean or multi-valued; cf. also B o o l e a n f u n c t i o n ) . More specifically, one assumes given some finite hypothesis space ~ defined over the instance space 32, in which the task is to learn some target function (target concept or classifier) c: X -+ {0, 1}. The learner is given a sequence of training examples {(xl, dl},.. •, (xn, dn) }, where xi E X and di = c(xi). Now, f C ~ , the learning hypothesis, is supposed to approximate c such that f(xi) = c(xi) for the vast majority (ideally, all) of the cases, and f(xi) ¢ c(xi) for only few (ideally, no) cases, certainly not exceeding a given minimum bound. Decision trees [7] are the most commonly used attribute-value representation. Given an exemplar described by a set of attribute values (or a feature vector), a decision tree outputs a Boolean classification decision for that exemplar. The decision is reached by branching through the node structure of the decision tree. Each of its nodes tests the value of one of the attributes and 246

branches to one of its children, depending on the outcome of the test. Hence, a decision tree implements the learned target function. First-order predicate logic (cf. P r e d i c a t e c a l c u l u s ) is certainly the most powerful representation language currently (2000) considered within the machine-learning community, though the one which requires the most sophisticated training environment and most costly computations to determine the target function. For example, inductive logic programming (ILP) [6] combines inductive reasoning and first-order logical representations such that the representation language of target functions is considered as a logic program, i.e., a set of Horn clauses (cf. L o g i c p r o g r a m m i n g ) . Inductive logic programming achieves inductive reasoning either by inverting resolution-based proofs or by performing a generalto-specific, hill-climbing search which starts with the most general preconditions possible, and adds literals, one at a time, to specialize the rule until it avoids all negative examples. Neural networks are continuous, non-linear functions represented by a parametrized network of simple computing elements (cf. N e u r a l n e t w o r k ) . The backpropagation algorithm [1] for learning neural networks begins with an initial neural network with randomized weights and computes the classification error of that network on the training data. Subsequently, small adjustments in the weights of the network are carried out (by propagating classification errors backward through the network) in order to reduce the error. This process is repeated until the error reaches a certain minimum. Probabilistic functions return a probability distribution over a set of propositional (or multi-valued) random variables (possible output values), and are suitable for problems where there may be uncertainty as to the correct answer (cf. P r o b a b i l i t y t h e o r y ) . They calculate the probability distribution for the unknown variables, given observed values for the remaining variables based on the B a y e s f o r m u l a . In machine learning, two basic uses of Bayes' theorem can be distinguished. In the first approach, the nai've Bayesian classifier, a new instance, described by a tuple of attribute values ( a l , . . . , an}, is classified by assigning the most probable target value, taken from some finite set F, the so-called maximum a posteriori hypothesis, VMAP. Using Bayes' theorem one may state VMAP = a r g m a x P ( a l , . . , vj CV

a.lvj). P(vj).

(1)

Incorporating the simplifying assumption that the attribute values are conditionally independent given the target value (cf. also C o n d i t i o n a l d i s t r i b u t i o n ) , one

MACHINE LEARNING may rewrite (1) as

VMAP = a r g m a x I ~ P(ailvy)" P(v~.). vj EI¢

(2)

i

In contrast to the naYve Bayesian classifier, which assumes that all the variables are conditionally independent given the target value, Bayesian networks allow stating conditional independence assumptions that apply to subsets of the variables. In particular, a node in a Bayesian network corresponds to a single random variable Xi, and a link between two nodes represents the causal dependency between the parent and the child node. The strength of the causal relation is represented by the conditional probability P(Xilvi) of each possible value of the variable, given each possible combination of values of the parent nodes (Vi being the set of predecessors of Xd. The problem that is posed by the appropriate choice of the underlying representation formalism is, as with formal reasoning, the fundamental trade-off between expressiveness (is the desired function representable in the chosen representation format?) and efficiency (is the machine learning problem tractable for the given choice of the representation format?). Machine learning as considered from the inductive perspective generates hypotheses by using combinations of existing terms in their vocabularies. These can, however, become rather clumsy and unintelligible. The problem may be overcome by introducing new terms into the vocabulary. In the machine learning community such systems are known as constructive induction or discovery systems [9]. Basically, these systems employ techniques from inductive logic programming (such as constructive induction), or are concept formation systems, which generate definitions for new categories (usually attribute-based descriptions) in order to improve the classification of examples based on clustering algorithms. Inductive learning is typically seen as supervised learning, where the learning problem is formulated as one to predict the output of a function from its input, given a collection of examples with a priori known inputs and outputs. With unsupervised learning such correctly labelled training exemplars are not available. Instead the learning system receives some sort of continuous reward indicating whether it was successful or whether it failed, an approach which underlies a variety of reinforcement learning algorithms [10]. From a technical perspective, in reinforcement learning, environments are usually treated as being in one of a set of discrete states S. Actions cause transitions between states. Hence, a complete model of an environment specifies the probability that the environment will

be in state j E S if action a is executed in state i C S. This probability is denoted by M~j. Furthermore, a reward R(i) is associated with each state i. Together, M and R specify a Markov decision process (cf. M a r k o v p r o c e s s ) . Its ideal behaviour maximizes the expected total reward until a terminal state is reached. One may also draw a distinction between learning based on pure induction and other inference modes for learning that take only few examples, sometimes even only a single example. One of these approaches is based on analogical reasoning. This is an inference process in which the similarity between a source and a target is inferred from the presence of known similarities, thereby providing new information about the target when that information is known about the source. One may provide syntactic measures of the amount of known similarity to assess the suitability of a given source (similarity-based analogy), or use prior knowledge of the relevance of one property to another to generate sound analogical inferences (relevance-based analogy) [8]. The increasing reliance on some form of a priori knowledge is most evident in explanation-based learning (EBL) [2]. It can be viewed as a form of single-instance generalization and uses background knowledge to construct an explanation of the observed learning exemplar, from which a generalization can be constructed. An important aspect of this approach to machine learning is that the general rule follows logically (or at least approximately so) from the background knowledge available to the learning system. Hence, it is based on deduction rather than induction. For explanation-based learning, the learning system does not actually learn anything substantially new from the observation, since the background knowledge must already be rich enough to explain the general rule, which in turn must explain the observation (often this approach, in contrast to inductive learning, is referred to as analytical learning). With the 'new' piece of knowledge compiled-out, the system will, in the future, operate more efficiently rather than more effectively (hence, it can be considered as a form of speedup learning). Unlike inductive logic programming, another knowledge-intensive learning method, explanation-based learning does not extend the deductive closure of the knowledge structures already available to the learning system. This approach can also be viewed at as a third variant of analogybased learning, viz. a kind of derivational analogy which uses knowledge of how the inferred similarities are derived from the known similarities to speed up analogical problem solving. So far, machine learning has been discussed in terms of different approaches which aim at improving the performance of the learning system. One might also raise 247

MACHINE LEARNING more principal questions as to the fundamental notions guiding research in machine learning or its theoretical limits. The notion of simplicity, for instance, is a prim a r y one in induction, since a simple hypothesis that explains a large number of different examples seems to have captured some fundamental regularity in the underlying data. An influential formalization of the notion of simplicity (known as Kolmogorov complexity or minimum description length theory [4]) considers a learning hypothesis as a program of a universal Turing machine (cf. T u r i n g m a c h i n e ) , with observations viewed as output from the execution of the program. The best hypothesis is the shortest program for the universal Turing machine that produces the observations as output (finding the shortest p r o g r a m is, however, an undecidable problem, cf. also U n d e c i d a b i l i t y ) . This approach abstracts away from different configurations of universal T~ring machines, since any universal Turing machine can encode any other with a program of finite length. Hence, although there are m a n y different universal Turing machines, each of which might have a different shortest program, this can make a difference of at most a constant amount in the length of the shortest program. The basic idea behind computational learning theory (or PAC learning) [3] is the assumption t h a t a hypothesis which is fundamentally wrong will almost certainly be identified with high probability after it has been exposed to only a few examples, simply because it makes wrong predictions. Thus, any hypothesis that is consistent with a sufficiently large set of training examples is unlikely to be seriously wrong, i.e., it must be 'probably approximately correct' (PAC). More specifically, assume an unknown distribution or density P over an instance space 2(, and an unknown Boolean target function f over X, chosen from a known class jc of such functions. The finite sample given to the learning algorithm consists of pairs {@1, Yl},..-, (Xm, Ym} }, where xi is distributed according to P and Yi = f(xi). Let ~ be the class of all linearthreshold functions (perceptrons) over n-dimensional real inputs. The question arises whether there is a learning algorithm that, for any input dimension n and any desired error e > O, requires a sample size and execution time bounded by fixed polynomials in n and i/c, and produces, with high probability, a hypothesis function h such that the probability that h(xi) ¢ f(xi) is smaller than e under P. One interesting result in that theory shows that the pure inductive learning problem, where the learning system begins with no prior knowledge about the target function, is computationally infeasible in the worst case. 248

References

[1] CHAUVIN, Y., AND RUMELHART, D.E. (eds.): Backpropagation: Theory, architectures and applications, Lawrence Erlbaum, 1993. [2] ELLMAN, T.: 'Explanation-based learning: A survey of programs and perspectives', A C M Computing Surveys 21, no. 2 (1989), 163-221. [3] I4[EARNS, M., AND VAZIRANI, U.: An introduction to computational learning theory, MIT, 1994. [4] LI, M., AND VITANYI, P.: An introduction to Kolmogorov complexity and its applications, Springer, 1993. [5] MITCHELL, T.: Machine learning, McGraw-Hill, 1997. [6] MUGGLETON, S.: Foundations of inductive logic programming, Prentice-Hall, 1995. [7] QUINLAN, J.R.: C~.5: Programs for machine learning, Kaufmann, 1993. [8] RUSSELL, S.J.: The use of knowledge in analogy and induction, Pitman, 1989. [9] SHRAGER, J., AND LANGLEY, P.: Computational models of scientific discovery and theory formation, Kaufmann, 1990. [10] SUTTON, R.S., AND BARTO, A.G.: Reinforcement learning. An introduction, MIT, 1998. [11] WEISS, S., AND KULIKOWSKI, C.: Computer systems that learn. Classification and prediction methods from statistics, neural nets, machine learning and expert systems, Kaufmann, 1991.

Udo Hahn M S C 1991:68T05 In physics, the phrase 'magnetic monopole' usually denotes a Yang-Mills potential A and Higgs field ¢ whose equations of motion are determined by the Yang-Mills-Higgs action MAGNETIC

MONOPOLE

-

(FA,FA) + (DAO, DA¢) -- :<1 --1f¢112) 2. In mathematics, the phrase customarily refers to a static solution to these equations in the Bogomolny-PrasadSommerfield limit A -+ 0 which realizes, within its topological class, the absolute minimum of the functional

i~3(FA, FA) + (DA¢, DA¢). This means that it is a connection A on a principal Gbundle over R 3 (cf. also C o n n e c t i o n s o n a m a n i f o l d ; P r i n c i p a l G - o b j e c t ) and a section ¢ of the associated adjoint bundle of Lie algebras such that the c u r v a t u r e FA and c o v a r i a n t d e r i v a t i v e DA¢ satisfy the Bogo-

molny equations FA = *DA¢ and the boundary conditions I]¢11 ---- 1 -- m + O(r_2) ' r

IIDAOII = O(r_2) '

Pure mathematical advances in the theory of monopoles from the 1980s onwards have often proceeded on the basis of physically motivated questions. The equations themselves are invariant under gauge transformations and orientation-preserving isometrics.

MAGNETIC MONOPOLE When r is large, ¢/11¢11 defines a mapping from a 2sphere of radius r in R 3 to an adjoint orbit G/K and the homotopy class of this mapping is called the magnetic charge. Most work has been done in the case G = SU(2), where the charge is a positive integer k. The absolute minimum value of the functional is then 87rk and the coefficient m in the asymptotic expansion of II¢II is k/2. The first SU(2) solution was found by E.B. Bogomolny, M.K. Prasad and C.M. Sommerfield in 1975. It is spherically symmetric of charge I and has the form

A

=

eijk -~-~rk dxi,

sin-h r z

¢

- -

tanhr

".

r ~*

In 1980, C.H. Taubes [10] showed by a gluing construction that there exist solutions for all larger k and soon after explicit axially-symmetric solutions were found. The first exact solution in the general case was given in 1981 by R.S. Ward for k = 2 in terms of elliptic functions. There are two ways of solving the Bogomolny equations. The first is by twistor methods. In the formulation of N.J. Hitchin [4], an arbitrary solution corresponds to a holomorphic vector bundle over the complex surface T P 1, the tangent bundle of the projective line. This is naturally isomorphic to the space of oriented straight lines in R 3. The boundary conditions show that the holomorphic bundle is an extension of line bundles determined by a compact a l g e b r a i c c u r v e of genus ( k - 1) 2 (the spectral curve) in T P 1, satisfying certain constraints. The second method, due to W. Nahm [13], involves solving an eigenvalue problem for the coupled Dirac operator and transforming the equations with their boundary conditions into a system of ordinary differential equations, the Nahm equations,

drl d8

dr2 : [T2, T3],

d8

dT3 - [T3' T1],

ds

= [TI' 7'2],

where Ti(s) is a (k x k)-matrix-valued function on (0, 2). Both constructions are based on analogous procedures for instantons, the key observation due to N.S. Manton being that the Bogomolny equations are dimensional reductions of the self-dual Yang-Mills equations (cf. also Y a n g - M i l l s field) in R 4. The equivalence of the two methods for SU(2) and their general applicability was established in [5] (see also [6]). Explicit formulas for A and ¢ are difficult to obtain by either method, despite some exact solutions of Nahm's equations in symmetric situations [7]. The case of a more general Lie g r o u p G, where the stabilizer of ¢ at infinity is a maximal torus, was treated by M.K. Murray [11] from the twistor point of view, where the single spectral curve of an SU(2)-monopole is

replaced by a collection of curves indexed by the vertices of the D y n k l n d i a g r a m of G. The corresponding Nahm construction was described by J. Hurtubise and Murray [9]. The moduli space (cf. also M o d u l i t h e o r y ) of all SU(2) monopoles of charge k up to gauge equivalence was shown by Taubes [16] to be a smooth non-compact manifold of dimension 4 k - 1. Restricting to gauge transformations that preserve the connection at infinity gives a 4k-dimensional manifold Mk, which is a circle bundle over the true moduli space and carries a natural complete hyper-Kghler metric [2] (cf. also K ~ i h l e r E i n s t e i n m a n i f o l d ) . With respect to any of the complex structures of the hyper-K//hler family, this manifold is holomorphically equivalent to the space of based rational mappings of degree k from p1 to itself [3]. The metric is known in twistor terms [2], and its K/~hler potential can be written using the R i e m a n n t h e t a - f u n e t i o n of the spectral curve [6], but only the case k = 2 is known in a more conventional and usable form [2] (as of 2000). This Atiyah-Hitehin manifold, the Euclidean T a u b NUT metric and R 4 are the only 4-dimensional complete hyper-K/ihler manifolds with a non-triholomorphic SU(2) action. Its geodesics have been studied and a programme of Manton concerning monopole dynamics put into effect. Fhrther dynamical features have been elucidated by P.M. Sutcliffe and C.J. Houghton [15] using a mixture of numerical and analytical techniques. A cyclic k-fold covering of Mk splits isometrically as a product Mk x S 1 x R 3, where Mk is the space of strongly centred monopoles. This space features in an application of S - d u a l l t y in theoretical physics, and in [14] G.B. Segal and A. Selby studied its topology and the L 2 harmonic forms defined on it, partially confirming the physical predictions. Magnetic monopoles on hyperbolic three-space were investigated from the twistor point of view by M.F. Atiyah [1] (replacing the complex surface T P 1 by the complement of the anti-diagonal in p1 x p1) and in terms of discrete Nahm equations by Murray and M.A. Singer, [12]. References [1] ATIYAH, M.F.: 'Magnetic monopoles in hyperbolic space': Vector bundles on algebraic varieties, Oxford Univ. Press, 1987, pp. 1-34. [2] ATIYAH, M.F., AND HITCHIN, N.J.: The geometry and dynamics of magnetic monopoles, Princeton Univ. Press, 1988. [3] DONALDSON, S.K.: 'Nahm's equations and the classification of monopoles', Commun. Math. Phys. 96 (1984), 397-407. [4] HITCHIN, N.J.: 'Monopoles and geodesics', Commun. Math. Phys. 83 (1982), 579-602. [5] HITeHIN, N.J.: 'On the construction ofmonopoles', Commun. Math. Phys. 89 (1983), 145-190.

249

MAGNETIC MONOPOLE [6] HITCHIN, N.J.: 'Integrable systems in Riemannian geometry', in C.-L. TERNG AND K. UnLENBEeK (eds.): Surveys in Differential Geometry, Vol. 4, Internat. Press, Cambridge, Mass., 1999, pp. 21-80. [7] HITCHIN, N.J., MANTON, N.S., AND MURRAY, M.K.: 'Symmetric monopoles', Nonlinearity 8 (1995), 661-692. [8] HITCHIN, N.J., AND MURRAY, M.K.: 'Spectral curves and the ADHM method', Commun. Math. Phys. 114 (1988), 463474. [9] HURTUBISE, J., AND MURRAY, M.K.: ' O n the construction of monopoles for the classical groups', Commun. Math. Phys. 122 (1989), 35 89. [10] JAFFE, A., AND TAUBES, C.H.: Vortices and monopoles, Vol. 2 of Progress in Physics, Birkhguser, 1980. Ill] MURRAY, M.K.: 'Monopoles and spectral curves for arbitrary Lie groups', Commun. Math. Phys. 90 (1983), 263-271. [12] MURRAY, M.K.: 'On the complete integrability of the discrete Nahm equations', Commun. Math. Phys. 210 (2000), 497-519. [13] NAHM, W.: 'The construction of all self-dual monopoles by the ADHM method', in N.S. CRAIGIE, P. GODDARD, AND W. NAttM (eds.): Monopoles in Quantum Field Theory, World Sci., 1982. [14] SEGAL, G.B., AND SELBY, A.: 'The cohomology of the space of magnetic monopoles', Commun. Math. Phys. 177 (1996), 775-787. [15] SUTCLI~FE, P.M.: 'BPS monopoles', Internat. J. Modern Phys. A 12 (1997), 4663-4705. [16] TAUBES, C.H.: 'Stability in Yang-Mills theories', Commun. Math. Phys. 91 (1983), 235-263.

N.J. Hitchin MSC1991: 35Qxx, 78A25 In 1929, K. Mahler [1] started the study of transcendence properties of the values of analytic functions f satisfying certain functional equations. A simple example is P(z, f(z), f(zd)) = 0, where P is a p o l y n o m i a l with algebraic coefficients and d > 2 an integer. For instance, the function f(z) = ~ n > 0 za~, w h i c h is analytic in the unit disc of C (cf. also A n a l y t i c f u n c t i o n ) , satisfies f(z s) = f(z) - z, and Mahler proved that f(a) is transcendental (cf. also T r a n s c e n d e n t a l n u m b e r ) whenever a is an a l g e b r a i c n u m b e r satisfying 0 < la[ < 1. The functional equation is used to derive many points from the starting one (in the previous example the points are {a, Re,..., adS,...}), and this iteration yields points close to the origin. Mahler's proof involves the construction of an auxiliary polynomial. This construction is different from Hermite's one, since the polynomial is not explicit, and also different from Siegel's, Gel'fond's or Schneider's ones (cf. also G e l ' f o n d - S c h n e i d e r m e t h o d ; S c h n e i d e r m e t h o d ) , since it rests on an argument of linear algebra rather than on the Thue-Siegel lemma (el. also D i r i e h l e t p r i n c i p l e ) : No bound for the height of the coefficients is required. MAHLER

250

METHOD

-

Mahler also worked with functions of several variables [11, [3], introducing transformations C n -+ C ~ given by n monomials. By this m e t h o d he even obtained results of algebraic independence [2]. The topic was somehow forgotten until 1969 [4]. Thanks to the work of several mathematicians, including J.H. Loxton, A.J. van der Poorten, K.K. Kubota, K. Nishioka, P.G. Becker, M. Amou, and T. Thpfer (see [5]), general results are now available for the transcendence and algebraic independence of values of such functions, in one or several variables. This method turns out to be one of the most efficient ones for proving strong results of algebraic independence. Here is an example. Let n, d be positive integers with d > 2 and a an algebraic number with 0 < [a[ < 1. For 1 _< i < n, let /~(~) = (p~i),/3~i),...) be a sequence of algebraic numbers satisfying a linear recurrence relation. Assume /3(1),..., ~_(n) are linearly independent. Then the n numbers

E / ~ i ) a ak (1 < i < n ) k=0

are algebraically independent. Also, sharp estimates of D i o p h a n t i n e a p p r o x i m a t i o n s (transcendence measures as well as measures of algebraic independence) have been obtained. A farreaching extension of Mahler's vanishing theorem was given by D.W. Masser in 1982. Mahler's early paper [1] contains the transcendence of the Thue-Morse number, whose binary expansion 0.0110100... is given by the fixed point starting with 0 of the substitution 0 ~-~ 01 and 1 ~-~ 10 (the related functional equation is f(z) = ( 1 - z)f(z2)). More generally, Mahler's method has interesting deep connections with automata theory (cf. also F o r m a l l a n g u a g e s a n d a u t o m a t a ) . It is conjectured that a number whose expansion in an integral basis is given by an automaton is either rational or else transcendental. One of Mahler's goals (see [4]) was to derive from his method the transcendence of J(a) for algebraic a with 0 < I~1 < 1. Here, J is the m o d u l a r f u n c t i o n , which satisfies indeed functional equations, namely the modular equations relating J(q) and J(q~) for any n >__ 1. This conjecture was proved only in 1995 (see G e l ' f o n d Schneider method). References [1] MAHLER, K.: 'Arithmetische Eigenschaften der Lhsungen einer Klasse yon Funktlonalgleichung', Math. Ann. 101

(1929), 342-366, Corrigendum: 103 (1930), 532. [2] MAHLER, K.: 'Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen', Math. Z. 32 (1930), 545-585. [31 MAHLER, K.: 'fiber das Verschwinden von Potenzreihen mehrerer Vergnderlichen in speziellen Punktfolgen', Math. Ann. 103 (1930), 573-587.

MASSIVE FIELD [4] MAHLER,K.: 'Remarks on a paper by W. Schwarz', J . Number Theory 1 (1969), 512-521. [5] NISHIOKA, K.: Mahler functions and transcendence, Vol. 1631 of Lecture Notes i n Mathematics, Springer, 1996. Michel Waldschmidt

MSC1991: l l J 9 1 , 11582, l l J 8 5 , l l F l l

MANOVA, multivariate analysis of variance - See ANOVA. MSC 1991: 62Jxx

MARKOVBRAID THEOREM - If two closed braids represent the same ambient isotopy class of oriented links (cf. also B r a i d t h e o r y ) , then One can transform one braid to another by a sequence of Marlcov moves: i) a ii) a

+, +,

bab-' (conjugation). abzl, where a is an element of the nth braid

it is easy to see that deg,-- a1 2 dm-'(d - 1). This lower bound of order dm for the degrees of the coefficients for the Nullstellensatz is much better than the doubly exponential lower bound for the general ideal membership problem given in [3]. Variants of the example, cf. [I], show that the terms in the Liouville-Lojasiewicz inequality are nearly optimal, with the presumed exception depending solely on the degree. Another family of extremal examples for the Nullstellensatz is given in [2]. -

8,.

References [I] BROWNAWELL, W.D.: 'Local diophantine Nullstellen equalities', J. Amer. Math. Soc. 1 (1988), 311-322. [2] KOLLAR,J.: 'Sharp effective Nullstellensatz', J. Amer. Math. Sac. 1 (1988), 963-975. [3] MAYR,E.W., AND MEYER,A.R.: 'The complexity of the word problems in commutative semigroups and polynlomial ideals', Adv. Math. 46 (1982), 305-329. ~

-

W. Dale Brownawell

MSC 1991: 14Axx, 14Q20

+

and bn is the nth generator of the (n 1)th braid group. Markov's braid theorem is an important ingredient in the construction of the Jones polynomial and its generalizations (e.g. the Jones-Conway polynomial). References [I] BIRMAN,J.S.: Braids, links and mapping class groups, Ann. of Math. Stud. Princeton Univ. Press, 1974. [2] MARKOV,A.A.: ' ~ b e rdie freie Aquivalenz der geschlossen Zopfe', Recueil Math. Moscou 1 (1935), 73-78. [3] WEINBERG, N.M.: 'On free equivalence of free braids', C.R. (Dokl.) Acad. Sci. U S S R 23 (1939), 215-216. (In Russian.) Jozef Przytycki

MSC 1991: 20F36, 57M25

MASER-PHILIPPON/LAZARD-MORA EXAMPLE, Lazard-Mora/Masser-Philippon example, LazardMora example, Masser-Philippon example - An extremal family for the degrees in the Hilbert Nullstellensatz (cf. Hilbert t h e o r e m ) is given by the following example, ascribed variously to D.W. Masser and P. Philippon and to D. Lazard and T. Mora: f l := xt, d f2 := XI - x2 , . . . , fm-1 := Xm-2 - Xrn-1,

D

f m := 1 - 2,-1

,.

xd- 1

The f i are readily seen t~ have no common zeros. If a l , . . . , a, are polynomials such that alfi

+...+amfm = 1,

by evaluation on the rational curve tdm-1(d-l)tdm-2(d-l)

,

, ' " , t d - l , l t)

MASSIVEFIELD - A q u a n t u m field t h e o r y contains massive fields if the H i l b e r t space obtained by repeated application of these fields to the vacuum carries a unitary representation of the covering group of the orthochronous proper Poincark group (cf. also Poincar6 group) which fulfills the mass-gap condition: Let P p ( p = 0 , 1 , 2 , 3 with p = 0 referring to the 'time variable') be the generators of the space-time translations and let M = be the mass operator. M is well defined due to the positive-energy condition, i.e. the condition that the joint spectrum of PI lies in the forward lightcone [15]. The mass-gap condition then says that the spectrum of M lies in (0) U [ma, m), where mo > 0 and the multiplicity of the zero-eigenvector of M is one (uniqueness of the vacuum). By the analysis of E. Wigner [16], [lo], all states which describe a single particle form a Hilbert subspace carrying an irreducible representation of the Poincark group which is labelled by a pair [m, s]. Here, m is the eigenvalue of these states with respect to the mass operator M and s E (1/2)Z, called the 'spin' of the particle, labels the finite-dimensional representation of the little group stabilizing a vector p in the Minkowski space-time with p0 > 0, p2 = m2, i.e. the covering group of SO(3). As a consequence of the mass gap assumption, all particles in a theory with massive fields have positive mass m 2 mo > 0. Since one-particle states are usually assumed to be the states of lowest energy (above that of the vacuum), the mass-gap assumption and the assumption that a quantum field theory contains only particle

Jv

states with positive mass, are considered as equivalent assumptions.

MASSIVE FIELD In the case that the one-particle states with the label Ira, s] are separated from the rest of the mass spectrum by a second mass-gap, i.e. the spectrum of M lies in {0} U { ' d U [m + e, oo) for some e > 0, and there is some quantum field A(f) in the theory such that E[m,~]A(f)f~ ~ 0 for some Schwartz test function f (cf. also Generalized functions, space of), with E[,~,~] the projector on the Hilbert subspace on the [m, s]-oneparticle states, one can apply the Haag-Ruelle scattering theory [5], ]11], [14] to A(f): Let fl be Schwartz test functions, such that the Fourier transform S ( f l ) of fl has support in the set {p: p0 > 0, fp2 - m21 < e}. Setting f[ = 2K-l(ei(P°-~)t~P(fl)), one defines asymptotic fields by their action on the vacuum vector fh

lim HA(f?)a,

/=1

t-+:hoc

1--1

where the vectors on the right-hand side converge in the strong Hilbert space topology. The asymptotic fields Ain/°ut(f) are free fields and generate a F o c k s p a c e of multi-particle in- and out-states over the space of oneparticle states with label [m, s]. If these in- and out-Fock spaces span the whole Hilbert space of the theory (the so-called requirement of asymptotic completeness) then, as a corollary to the P C T theorem, the s c a t t e r i n g m a t r i x taking in-states to the related out-states is unitary [7]. The requirements of Haag-Ruelle theory alone suffice to derive the LSZ-reduction formulas [9], which express the scattering matrix elements (scalar product of in- and out-states, which gives the physical transition amplitude) via the time-ordered vacuum expectation values of the field A(f) [6]. This links the general forrealism of quantum fields [15], [7] to the heuristic perturbation expansions for the time-ordered Wightman functions based on the classical Lagrangian and the heuristic path integral (cf. also Quantum field theory). From the 1960s onwards, a systematic construction of rigorous (non-perturbative) models has been started in space-time dimensions d = 2, 3, see [12], [4], [1]; for models with d arbitrary (however with a state space carrying an indefinite inner product), see e.g. [2]. Massive quantum field theory is taken to be an approximation to the real physical situation, where all long range forces, associated with massless fields, can be neglected as 'weak' in comparison with the strong short range forces associated with massive fields. If only massive fields are present in a theory, the mathematical treatment of the theory is simpler, due to the absence of a nmnber of effects connected with massless particles and fields (cf. M a s s l e s s field). However, several features of the contemporary (2000) physical theory of strong interactions, as e.g. 'quarks', 'confinement' and 'asymptotic freeness', are not yet well explained in the 252

given mathematical framework (but see e.g. [3] for an interesting new approach). Massive classical fields are studied in the framework of non-linear hyperbolic partial differential equations

(cf. also Hyperbolic partial differential equation), see e.g. [13], [8].

References [1] ALBEVERIO, S.: 'Mathematical physics and stochastic analysis', Bell. Sci. Math. 117 (1993), 125. [2] ALBEVERIO, S., C-OTTSCHALK, H., AND Wu, J.-L.: 'Scattering behaviour of quantum vector fields obtained from Euclidean covariant SPDEs', Rept. Math. Phys. 44, no. 1 (1999), 21. [3] BUCtIHOLZ, D., AND VRECH, R.: 'Scaling algebras and renormalization group in algebraic quantum field theory', Rev. Math. Phys. 7 (1995). [4] GLIMM, J., AND JAFFE, A.: Quantum physics: A functional integral point of view, second ed., Springer, 1987. [5] HAAG, R.: ' Q u a n t u m field theories with composite particles and asymptotic condition', Phys. Rev. 112 (1958), 669. [6] HEPP, K.: 'Oil the connection between LSZ and Wightman quantum field theory', Cornmun. Math. Phys. 1 (1965), 95. [7] JOST, R.: The general theory of quantized fields, Amer. Math. Sot., 1965. [8J KUKSIN, S.B.: 'On the long-time behaviour of solutions of nonlinear wave equations', in D. IAGOLNITZER (ed.): XIth Int. Cong. Math. Phys., Cambridge Internat. Press, 1995, pp. 273-277. [9] LEHMANN, H., SYMANZIK, a . , AND ZIMMERMANN, W.: 'Zur Formulierung quantisierter Feldtheorien', Il Nuovo Cimento

1 (1954), 205. [10] Ri.~HL, W.: The Lorentz group and harmonic analysis, Benjamin, 1970. [1]] RUELLE, D.: 'On the asymptotic condition in quantum field theory', Helv. Phys. Acta 35 (1962)~ 147. [12] SIMON, B.: The P(~)2 Euclidean (quantum) field theory, Princeton Univ. Press, 1975. [13] STaAUSS, W.: Nonlinear wave equations, Amer. Math. Soc., 1989. [14] STaEATER, R.: 'Uniqueness of the Haag Ruelle scattering states', J. Math. Phys. 8 (1967), 1685-1693. [1.5] STREATER, R.F., AND WIGHTMAN, A.S.: P C T spin ~ statistics and all that..., Benjamin, 1964. [16] WIGNER, E.P.: 'On unitary representations of the inhomogenous Lorentz group', Ann. Math. 40 (1939), 149.

S. Albeverio H. Gottschalk MSC 1991: 81Txx

MASSLESS FIELD A quantum field theory is said to contain massless fields if in the Hilbert space generated by repeated application of the quantum fields to the vacuum state there exist subspaces associated with one-particle states of mass zero. According to the concept of a relativistic particle, introduced by E. Wigner [14], the one-particle states associated with a particle of type [re, s] are given by Hilbert subspaces transforming irreducibly under the unitary representation of the covering group of the orthochronous proper Poinca% group (cf. also Poincar~

MASSLESS FIELD

g r o u p ) ~ + . Here, m is the eigenvalue of the states in these subspaces with respect to the mass operator M = x / P ~ P ~, where the PU (# = 0, 1, 2, 3 with 0 referring to the 'time variable') are the generators of spacetime translations (the four-vector P is also called the e n e r g y - m o m e n t u m operator), and one finds that rn > 0 is well-defined by the spectral condition that the (joint) spectrum of P lies in the forward lightcone [10]. Furthermore, s stands for the spin associated with the representation of the little group, i.e. the subgroup in i5¢+ stabilizing a vector p in the Minkowski space-time with Minkowski inner product p2 = rn 2 and p0 > 0. This representation of the little group is furthermore assumed to be finite dimensional for subspaces associated with oneparticle states. In the massless case m = 0, the little group stabilizing p = (1, 1, 00) is ISO(2) and the finite-dimensional representations of this group are characterized by a number a C (1/2)Z called the helicity, which has the physical interpretation of the amount of the internal angular momentum (respectively, 'spin') directed in the flight direction of the particle. The particle type associated with a massless particle is thus denoted by the pair [0, or]. Denote the field operators of the theory by A(f), with f from a suitable space of Schwartz test functions (cf. also G e n e r a l i z e d f u n c t i o n s , s p a c e of). If these field operators connect the vacuum ~ of the theory with the one-particle states with label [0, a], i.e. E [ o ¢ ] A ( f ) ~ 7~ 0 (E[0,¢] being the projector associated with the oneparticle Hilbert space of [0, a]-states), one can develop a scattering theory for the quantum field A, following [3], [4]: For a test function f , let A / ( x ) = A ( f x ) with fx (y) = f ( y - x). Take ht(s) = h((s - t ) / l o g It[)/log tt[, with h a positive test function of compact support with f h ( s ) ds = 1 and set Atf = - 2

i

ht(s) x

, dco x

out-Fock space define the s c a t t e r i n g m a t r i x for scattering processes, which only involve massless incoming and outgoing particles. A number of mathematical problems and physical effects arise in the presence of massless fields, as, for example (see the literature for further details): massless fields are intimately connected with long-range forces in elementary particle physics such as e.g. electro-magnetism, see e.g. [13]. Among others, this leads to mathematical problems in the theory of superselection sectors associated to the algebra of observables of a quantum field theory involving long-range forces [5]. In the quantum field theory of gauge fields, massless gauge fields are being coupled to Fermionic currents by the Gauss law, which makes it necessary to introduce an indefinite inner product on the state space underlying the quantum field theory and to single out a subspace of 'physical states' with positive norm by a gauge principle [12], [11], [8] (for models partly implementing this, see [1], [2]). If massive particles (cf. also M a s s i v e field) interact with massless particles, the massive particle can be accompanied by a 'cloud' of infinitely many massless particles with finite total energy, which can 'smear out' the mass of the massive particle and give rise to the infraparticle problem [9], [6]. In the perturbation theory of quantum fields this effect is assumed to justify the mass renormalization, cf. [13]. Furthermore, mass-zero particles in perturbation theory cause infrared divergences and the problem of summing up Feynman graphs of all orders with 'soft' (i.e. low-energy) massless particles [13]. The occurrence of so-called Goldstone bosons, which are massless particles, is related to symmetry breaking in quantum field theory; for two different aspects of this phenomenon, see e.g. [13, Vol. II], [7].

(0.),.,q..., ~

I

where S 2 is the unit sphere in R a and dco means integration over all unit vectors co in S 2. Then it was shown in the above-mentioned references (in the axiomatic framework described in [10], [3], [4]), using the Reeh-Schlieder theorem [10] together with a kind of H u y g e n s p r i n c i p l e and locality, that the ('adiabatic') limit Ain/°ut(f) = limt++oo A) can be defined on a suitable dense domain. The quantum fields Ain/°ut(f) are by definition the free asymptotic quantum fields associated to the field A ( f ) . Repeated application of the field operators Ain/°ut(f) to the vacuum f~ generates the incoming and outgoing multiple particle states of particle type [0, or], which define in- and outFock spaces (cf. also F o c k space). The scalar product of states from the in-Fock space with states from the

References

[1] ALBEVERIO,S., GOTTSCHALK,H., ANDWU, J.-L.: 'Models of local, relativistic quantum fields with indefinite metric (in all dimensions)', Commun. Math. Phys. 184 (1997), 509. [2] ALBEVERIO,S., GOTTSCHALK,H., AND WU, J.-L.: 'Nontrivial scattering amplitudes for some local, relativistic quantum field models with indefinite metric', Phys. Lett. B 405 (1997), 243. [3] BUCHHOLZ, D.: 'Collision theory for massless Fermions', Commun. Math. Phys. 42 (1975), 269. [4] BUCHHOLZ,D.: 'Collision theory for massless Bosons', Commun. Math. Phys. 52 (1977), 147. [5] BUCHHOLZ,D.: 'The physical state space of quantum electrodynamics', Commun. Math. Phys. 85 (1982), 49. [6] BUCHHOLZ,D.: 'On the manifestation of particles', in A.N. SEN AND A. GERSTEN(eds.): Proc. Beer Sheva Conf. (1993): Math. Phys. Towards the 21st Century, Ben Gurion of the Negev Press, 1994. 253

MASSLESS FIELD [7] BUCHHOLZ, D., DOPLICHER, S., LONGO, R., AND ROBERTS, J.E.: 'A new look at Goldstone's theorem', Rev. Math. Phys., Special Issue 49 (1992). [8] MORCHIO, G., AND STROCCHI, F.: 'Infrared singularities, vacuum structure and pure phases in local quantum field theory', Ann. Inst. H. Poincard B33 (1980), 251. [9] SCHROER, B.: 'Infrateilchen in tier Quantenfeldtheorie', Fortschr. Phys. 173 (1963), 1527. [10] STREATER, R.F., AND WIGHTMAN, A.S.: P C T spin ~4 statistics and all that..., Benjamin, 1964. [11] STROCCHI, F.: 'Local and covariant gauge quantum field theories, cluster property, superselection rules and the infrared problem', Phys. Rev. D 1 7 (1978), 2010. [12] STROCCm, F., AND WIGHTMAN, A.S.: 'Proof of the charge superselection rule in local, relativistic quantum field theory', J. Math. Phys. 15 (1974), 2198. [13] WEINBERG, S.: The quantum theory of fields, Vol. I-II, Cambridge Univ. Press, 1995. [14] WIGNER, E.P.: 'On unitary representations of the inhomogenous Lorentz group', Ann. Math. 40 (1939), 149.

S. Albeverio H. Gottschalk MSC 1991: 81Txx, 81T05

MASSLESS KLEIN-GORDON EQUATION- The e q u a t i o n [6], [2], [7]

Klein-Gordon

0=

-

-i

2

¢

+

C4?Tt2 ]

for the case where the mass parameter m is equal to zero. The constant c stands for the speed of light, e is the charge of the positron, h = h/27r where h is the P l a n c k c o n s t a n t , (t,x) are the time, respectively space, variables, and i is the imaginary unit. The (complex-valued) solution ~ describes the wave function of a relativistic spinless and massless particle with charge qe in the exterior electro-magnetic field (¢, A). It is a second-order, 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 . Solutions are being studied in, e.g., [3], [4]. If the outer field is zero, (¢, A) = 0, or the coupling of the spin to the magnetic potential A can be neglected, the massless Klein-Gordon equation also can be used for the description of massless spin-carrying particles, such as e.g. photons. In the case without outer fields the massless Klein-Gordon equation becomes equivalent to the wave e q u a t i o n with wave speed c and is independent of the magnitude of Planck's constant h. This explains, why the wave nature of massless particles, such as e.g. photons ('light'), can also be observed on a macroscopic scale - - in contrast with the wave nature of massive particles (cf.also M a s s l e s s field; M a s s i v e field). The interpretation of the wave function ~ as a quantum mechanical 'probability amplitude' (similarly as in the case of the S c h r S d i n g e r e q u a t i o n ) , however, is not 254

consistent, since the quantity f a 8 [¢ (t, x)l 2 dx in general depends on the time parameter t. Furthermore, the exisfence of negative frequency solutions is in contrast with the required lower boundedness of the energy ('stability of matter'). These problems are resolved through a reinterpretation of ¢ ( t , x ) as a quantum field (cf. Q u a n t u r n field t h e o r y ) , see e.g. [5], [8]. In recent time (as of 2000) solutions of the KleinGordon equation on Lorentzian manifolds have attracted increasing attention in connection with the theory of quantized fields on curved space-time, cf. e.g. [1]. References [1] FULLING, S.A.: Aspects of quantum field theory in curved space-time, Cambridge Univ. Press, 1989. [2] GORDON, O.: 'Der Comptoneffekt nach der SchrSdingersehen Theorie', Z. f. Phys. 40 (1926), 117. [3] GROSS, L.: 'Norm invariance of mass zero equations under the conformal group', J. Math. Phys. 5 (1964), 687-695. [4] JAGER, E.M. DE: 'The Lorentz-invariant solutions of the Klein-Gordon equation I-II', Indag. Math. 25 (1963), 515531; 546-558. [5] JOST, R.: The general theory of quantised fields, Amer. Math. Soc., 1965. [6] KLEIN, O.: 'Quantentheorie und fiinfdimensionale Relativit~tstheorie', Z. f. Phys. 37 (1926), 895. [7] SCHabDINGER, E.: 'Quantisierung als Eigenwertproblem IV', Ann. Phys. 81 (1926), 109. [8] WEINBERG, S.: The quantum theory of fields, Vol. I, Cambridge Univ. Press, 1995.

S. Albeverio H. Gottschalk MSC1991: 81Q05, 81Txx, 81T20

MATCHING POLYNOMIAL OF A G R A P H - A matching cover (or simply a matching) in a g r a p h G is taken to be a subgraph of G consisting of disjoint (independent) edges of G, together with the remaining nodes of G as (isolated) components. A matching is called a k-matching if it contains exactly k edges. If G contains p nodes, then the extreme cases are: i) p is even and k = p/2; in this case, all the nodes of G are covered with edges (a perfect matching); and ii) k = 0; in this case, none of the nodes of G are covered by edges (the empty graph). If a matching contains k edges, then it will have p - 2k component nodes. Now assign weights (or indetermihates over the complex numbers) Wl and w2 to each node and edge of G, respectively. Take the weight of a matching to be the product of the weights of all its components. Then the weight of a k-matching will be k w p-2k 1 w 2. The matching polynomial of G, denoted by M(G; w), is the sum of the weights of all the matchings in G. Setting wl : w2 : w, then the resulting polynomial is called the simple matching polynomial of G.

MATERIAL DERIVATIVE METHOD The matching polynomial was introduced in [1]. Basic algorithms for finding matching polynomials of arbitrary graphs, basic properties of the polynomial, and explicit formulas for the matching polynomials of many well-known families of graphs are given in [1]. The coefficients of the polynomial have been investigated [7]. The analytical properties of the polynomial have also been investigated [8]. Various polynomials used in statistical physics and in chemical thermodynamics can be shown to be matching polynomials. The matching polynomial is related to many of the well-known classical polynomials encountered in combinatorics. These include the C h e b y s h e v p o l y n o m i a l s , the H e r m i t e p o l y n o m i a l s and the Lag u e r r e p o l y n o m i a l s . An account of these and other connections can be found in [16], [14]. The classical rook polynomial is also a special matching polynomial; and in fact, rook theory can be developed entirely through matching polynomials (see [5], [4]). The matching polynomial is also related to various other polynomials encountered in graph theory. These include the chromatic polynomial (see [13]), the characteristic polynomial and the acyclic polynomial (see [15] and [3]). The matching polynomial itself is one of a general class of graph polynomials, called F-polynomials (see [2]). Two graphs are called co-matehin 9 if and only if they have the same matching polynomial. A graph is called matchin 9 unique if and only if no other graph has the same matching polynomial. Co-matching graphs and matching unique graphs have been investigated (see [6], [10]). It has been shown that the matching polynomial of certain graphs (called D-graphs) can be written as determinants of matrices. It appears that for every graph there exists a co-matching D-graph. The construction of co-matching D-graphs is one the main subjects of current interest in the area (see [9], [11], [12]). References

[1] FARRELL, E.J.: 'Introduction to matching polynomials', J. Combin. Th. B 27 (1979), 75-86. [2] FARRELL, E.J.: 'On a general class of graph polynomials', J. Combin. Th. B 26 (1979), 111-122. [3] FARRELL, E.J.: 'The matching polynomial and its relation to the acyclic polynomial of a graph', Ars Combinatoria 9 (1980), 221-228. [4] FARRELL, E.J.: 'A graph-theoretic approach to Rook theory', Caribb. J. Math. 7 (1988), 1-47. [5] FARRELL, E.J.: 'The matching polynomial and its relation to the Rook polynomial', J. Franklin Inst. 325, no. 4 (1988), 527-536. [6] FARRELL, E.J., AND GUO, J.M.: 'On the characterizing properties of the matching polynomial', Vishwa Internat. J. Graph Th. 2, no. 1 (1993), 55-62. [7] FARRELL, E.J., GUO, J.M., AND CONSTANTINE, G.M.: 'On the matching coefficients', Discr. Math. 89 (1991), 203-210.

[8] FARRELL,E.J., AND WAHID, S.A.: 'Some analytical properties of the matching polynomial of a graph': Proc. Fifth Caribb. Conf. in Comb. and Graph Th., Jan.5-8, 1988, pp. 105-119. [9] FARRELL, E.J., AND WAHID, S.A.: 'Matching polynomials: A matrix approach and its applications', J. Franklin Inst. 322 (1986), 13-21. [10] FARRELL, E.J., AND WAmD, S.A.: 'Some general classes of comatcMng graphs', Internat. Y. Math. Math. Sei. IO, no. 3 (1987), 519-524. [11] FARRELL, E.J., AND WAHID, S.A.: 'D-graphs I: An introduction to graphs whose matching polynomials are determinants of matrices', Bull. ICA 15 (1995), 81-86. [12] FARRELL, E.J., AND WAHID, S.A.: 'D-graphs Ih Constructions of D-graphs for some families of graphs with even cycles', Utilitas Math. 56 (1999), 167-176. [13] FAHRELL, E.J., AND WHITEHEAD, E.G.: 'Connections between the matching and chromatic polynomials', Internat. J. Math. Math. Sci. 15, no. 4 (1992), 757-766. [14] GODSIL, C.D., AND GUTMAN, I.: 'On the theory of the matching polynomial', J. Graph Th. 5 (1981), 137-145. [15] GUTMAN, I.: 'The acyclic polynomial of a graph', Publ. Inst. Math. Beograd 22 (36) (1977), 63-69. [16] GUTMAN, I.: 'The matching polynomial', M A T C H , no. 6 (1979), 75-91.

E.J. Farrell MSC 1991: 05Cxx, 05D15 M A T E R I A L D E R I V A T I V E M E T H O D - In the study of m o t i o n in continuum mechanics one deals with the time rates of changes of quantities that vary from one particle to the other. Such quantities include displacement, velocity and acceleration. These quantities may be expressed as functions described in the material form or the spatial form, and the meaning of the time rate of their change depends on the nature of the description.

M a t e r i a l t i m e d e r i v a t i v e . Consider a real-valued function f = f ( x °, t) that represents a scalar or a component of a v e c t o r or tensor. The point x ° determines a continuum particle uniquely, namely the one located at x °. With this notation, the function f = f ( x °, t) can be interpreted as the value of f experienced at time t by the particle x °. The time d e r i v a t i v e of f with respect to time t, with x ° held fixed, is interpreted as the time rate of change of f at the particle x °. This derivative is usually called the particle or material time derivative of f, denoted by D f / D t and defined by

Dr-

(Of(~t't))

,

(1)

where the subscript x ° accompanying the vertical line indicates that x ° is kept constant in the differentiation of f. Note that, like f, Dr~Dr is a function of x ° and t by definition. In other words, D f / D t defined above is a

function in the material form. L o c a l t i m e d e r i v a t i v e . In order to define the local time derivative, one considers a real-valued function ¢ = ¢(x, t) that represents a scalar or a component of a 255

MATERIAL DERIVATIVE METHOD vector or tensor. Since x is point in the current configuration of a continuum, ¢(x, t) can be interpreted as the value of ¢ at the point x at time t. The p a r t i a l d e r i v a t i v e of ¢ with respect to time t, with x held fixed, is interpreted as the time rate of change of ¢ at the particle located at x. This derivative is called the local time derivative of ¢, denoted by the usual partial derivative symbol O¢/Ot and defined by

0¢

{O¢(x,t)

or- \

) x

(2)

It is noted that, like ¢, O¢/Ot is a function of x and t, and is a function in the spatial form. The distinction between the material time derivative and the local time derivative should be emphasized. While b o t h are partial derivatives with respect to t, the former is defined for a function of x ° and t whereas the latter is defined for a function of x and t. Physically, the local time derivative of a function represents the rate at which the function changes with time as seen by an observer currently (momentarily) stationed at a point, whereas the material time derivative represents the rate at which the function changes with time as seen by an observer stationed at a particle and moving with it. The material time derivative is therefore also called the mobile time derivative or the derivative following a particle. For brevity, the material time derivative will be referred to as the material derivative or material rate, and the local time derivative as the local derivative or local rate. V e l o c i t y a n d a c c e l e r a t i o n . Since x is a function of x ° and t in the material description of motion, the material derivative f is denoted by v and is defined by

v-

Dt

37

Evidently, v represents the sition of the particle x ° at velocity of the particle x ° at nents of v, then the velocity x ° at time t take the form

.

(3)

time rate of change of potime t. This is called the time t. If vi are the compoc o m p o n e n t s of the particle

c o m p o n e n t form

vi-

Dui Dt"

It m a y be pointed out t h a t , in solid mechanics, the deformation and m o t i o n are generally described in terms of the displacement vector. In fluid mechanics, the motion is generally described in terms of the velocity vector. W h e n a m o t i o n is described in terms of velocity, it is c o m m o n l y referred to as a flow. Since v is a function o f x ° and t by definition, the material derivative of v, namely, D v / D t , can be defined. This derivative is called the acceleration of the particle x ° at time t. One often writes ~ for D v / D t . Thus, the acceleration of a particle at time t is the rate of change of velocity of t h a t particle at time t. T h e components of the acceleration are d e n o t e d by D v i / D t or ~?i. It is to be emphasized t h a t the velocity and acceleration are defined with reference to a particle and are basically functions of x ° and t. In the spatial description of motion, x ° is a function of x and t. Hence, like the displacement, velocity and acceleration can also be expressed as functions of x and t. W h e n v is expressed as a function of x and t, v ( x , t) is referred to as the instantaneous velocity at the point x. This actually means t h a t v ( x , t) is the velocity at time t of the particle currently located at the point x. Similar terminology is used in respect of acceleration also. Next, one can deduce a formula enabling one to compute the instantaneous acceleration from the instantaneous velocity. M a t e r i a l d e r i v a t i v e i n s p a t i a l f o r m . Consider again the function ¢ = ¢(x, t) for which the local derivative was defined by (2). This function can be expressed as a function of xiro and t, as explicitly indicated in the following: ¢ = ¢(x~, t) = ¢ ( x i ( x °, t), t).

Ot -

"

(7)

Consequently, the material derivative of ¢ can also be defined. B y the chain rule of partial differentiation, we obtain from (7)

= vi -

(6)

\otjxo

(s)

(4) In view of (1), (2) and (4), it follows t h a t

The displacement vector u of the particle x ° is defined as u = x - x °. Thus, u m a y be regarded as a function of x ° and t, or of x and t. Treating u as a function of x ° and t, it follows from the above t h a t v=

~

(x ° + u )

lxo =

~

= Dt"

(5)

Thus, the velocity of a particle at time t is precisely the rate of change of displacement of t h a t particle at time t. The above definition of velocity v assumes the 256

37

= D-7'

37

=o-7'

\ ot j (o)

Hence, denoting (O¢/Oxi)lt as just O¢/Oxi = ¢,i, (8) can be rewritten as De

0¢

Dt = O-7 +

0¢

= 37 + (v. V)¢.

(10)

W h e n v is known as a function of x and t, expression (10) enables one to c o m p u t e D ¢ / D t as a function of x and t. As such, (10) serves as a formula for the material

MATRIX T R E E T H E O R E M derivative in the spatial form. Note that the first term on the right-hand side of this formula, namely c9¢/0t, represents the local rate of change of ¢, and the second term, namely vi¢,i = (v - V)¢, is the contribution due to the motion. The second term is referred to as the convective rate of change of ¢. It can be easily verified that the material derivative operator D 0 0 D t - Ot + vi('),i = ~ + v . V

(11)

which operates on functions represented in spatial form, satisfies all the rules of partial differentiation. The concept of the material derivative and formula (11) are attributed to L. Euler (1770) and J. Lagrange (1783). A c c e l e r a t i o n an s p a t i a l f o r m . Taking ¢ = vi in (10) gives the following expression for the acceleration: Dvi Ovi D t - Ot + vkvi,k

(12)

Dv 0v D t - Ot + ( v . V)v.

(13)

or, equivalently,

When v is known as a function of x and t, expression (13) determines D v / D t directly in terms of x and t; this expression therefore serves as a formula for acceleration in the spatial form. By using the standard vector identity, (13) can be put in the following useful form: Dv 0v 1 2 D~ - Ot + ~ v v + (curlv) x v .

(14)

From (13) and (14), one notes that the acceleration vector is made up of two parts, namely, ( v . V)v = 1Vv2 + (curly) x v. Evidently, the second part is quadratically non-linear in nature. Thus, the acceleration depends quadratically on the velocity field, and a given motion cannot be viewed as a superposition of two independent motions in general. References

[1] CHANDRASEKHARIAH, D.S., AND DEBNATH, L.: C o n t i n u u m mechanics, Acad. Press, 1994. [2] FUI,~G, Y.C.: F o u n d a t i o n s of solid mechanics, Prentice-Hall, 1965.

Lokenath Debnath

MSC 1991: 76Axx, 73Bxx M A T R I X ELEMENT, matrix entry - Any of the a~j of an (n x m)-matrix A = (aij), i = 1 , . . . , n , j = 1,...,m. MSC 1991: 15-XX

MATRIX

TREE

THEOREM

-

Let

G

=

(V,E)

be a g r a p h with ~, vertices { v l , . . . , v , } and e edges { e l , . . . , e~}, some of which may be oriented. The incidence matrix of G is the ( , x e)-matrix M = [rnij] whose entries are given by mij = 1 if ej is a non-oriented link (i.e. an edge that is not a loop) incident to vi or if ej is an oriented link with head vi, mij = - 1 if ej is an oriented link with tail vi, mij = 2 if ej is a loop (necessarily non-oriented) at vi, and mij = 0 otherwise. The mixed Laplacian matrix of G is defined as L = [lij] = M M T. It is easy to see that the diagonal entries of L give the degrees of the vertices with, however, each loop contributing 4 to the count, and the off-diagonal entry lij gives the number of non-oriented edges joining vi and vj minus the number of oriented edges joining them. Let r ( G ) denote the number of spanning trees of G, with orientation ignored. The matrix tree theorem in its classical form, which is already implicit in the work of G. Kirchhoff [9], states that if L is the Laplacian of any orientation of a loopless undirected graph G and L* is the matrix obtained by deleting any row s and column t of L, then T(G) = ( - 1 ) s+t det(L*); that is, each c o f a c t o r of L is equal to the tree-number of G. If adj(L) denotes the adjoint of the matrix L and J denotes the matrix with all entries equal to 1, then adj(L) = ~-(G)J. The proof of this theorem uses the Binet-Cauchy theorem to expand the cofactor of L together with the fact that every nonsingular (u - 1) x (• - 1)-minor of M (cf. also M a n o r ) comes from a spanning tree of G having value 4-1. In the case of the complete graph Kv (with some orientation), L = v I - J, and it can be seen that ~-(K,) = /2 v - 2 , which is Cayley's formula for the number of labelled trees o n , vertices [4]. Temperley's result [3, Prop. 6.4] avoids using the cofactor notation in the following form: v2T(G) = d e t ( J + L). It is interesting to note that this determinantal way of computing T(G) requires v3 operations rather than the 2" operations when using recursion [17, p. 66]. For a loopless directed graph G, let L - = D - - A ~ and L + = D + - A ~, where D - and D + are the diagonal matrices of in-degrees and out-degrees in G, and the /j-entry of A ~ is the number of edges from vj to vi. An out-tree is an orientation of a tree having a root of indegree 0 and all other vertices of in-degree 1. An in-tree is an out-tree with its edges reversed. W.T. Tutte [16] extended the matrix tree theorem by showing that the number of out-trees (respectively, in-trees) rooted at vi is the value of any cofactor in the ith row of L - (respectively, ith column of L+). In fact, the principal minor of L obtained by deleting rows and columns indexed by vii, • • •, vik equals the number of spanning forests of G having precisely k out-trees rooted at vii,. •., vi~. 257

MATRIX TREE THEOREM In all the approaches it is clear t h a t the significant p r o p e r t y of the Laplacian L is t h a t ~ j l i j = 0 for 1 _~ i _~ u. By allowing lij to be indeterminates over the field of rational numbers, the generating function version of the matrix tree theorem is obtained [8, Sect. 3.3.25]: The n u m b e r of trees rooted at r on the vertex set { 1 , . . . , ~,}, with m i j occurrences of the e d g e / ~ (directed away from the root), is the coefficient of the m o n o m i a l I]i,j li~ ~j in the (r, r ) t h cofactor of the matrix [(~ijcti -- lij]uxu, where ?7~ij E {0, 1} and ai is the sum of the entries in the ith row of L, for i = 1 , . . . , zJ. Several related identities can be found in work by J.W. M o o n on labelled trees [14]. For various proofs of Cayley's formula, see [13]. A n o t h e r direction of generalization is to interpret all the minors of the Laplacian rather t h a n just the principal ones. Such generalizations can be found in [5] and [1], where a r b i t r a r y minors are expressed as signed sums over non-singular substructures t h a t are more complicated t h a n trees. T h e edge version of the Laplacian is defined to be the (e x e)-matrix K = M T M . The connection of its cofactors with the Wiener index in applications to chemistry is presented in [11]. The combinatorial description of the arbitrary minors of K when G is a tree is studied in [2]. Applications are widespread. Variants of the matrix tree theorem are used in the topological analysis of passive electrical networks. The n o d e - a d m i t t a n c e matrix considered for this purpose is closely related to the Laplacian m a t r i x (see [10, Chap. 7]). A b u n d a n c e of forests suggests greater accessibility in networks. Due to this connection, the m a t r i x tree theorem is used in developing distance concepts in social networks (see [6]). T h e C - m a t r i x which occurs in the design of statistical experiments (cf. also D e s i g n o f e x p e r i m e n t s ) is the Laplacian of a g r a p h associated with the design. In this context the matrix tree t h e o r e m is used to s t u d y Doptimal designs (see [7, p. 67]). Finally, the m a t r i x tree theorem is closely related to the P e r r o n - F r o b e n i u s t h e o r e m . If A is the transition m a t r i x of an irreducible M a r k o v c h a i n , then by the P e r r o n - F r o b e n i u s t h e o r e m it admits a unique s t a t i o n a r y distribution. This fact is easily deduced from the m a t r i x tree theorem, which in fact gives an interpretation of the components of the stationary distribution in terms of tree-counts. This observation is used to a p p r o x i m a t e the s t a t i o n a r y distribution of a countable Markov chain (see [15, p. 222]). An excellent survey of interesting developments related to Laplacians m a y be found in [12].

References [1] BAPAT,R.B., GaOSSMAN,J.W., AND KULKARNI,D.M.: 'Generalized matrix tree theorem for mixed graphs', Linear and Multilinear Algebra 46 (1999), 299 312. 258

[2] BAPAT, R.B., GROSSMAN,

J.W., AND KULKARNI, D.M.: 'Edge version of the matrix tree theorem for trees', Linear and Mul-

tilinear Algebra 47 (2000), 217 229. [3] BIGGS, N.: Algebraic graph theory, second ed., Cambridge Univ. Press, 1993. [4] CAYLEY,A.: 'A theorem on trees', Quart. J. Math. 23 (1889), 376-378. [5] CHAIKEN, S.: 'A combinatorial proof of the all minors matrix tree theorem', SIAM J. Algebraic Discr. Math. 3, no. 3 (1982), 319-329. [61 CHEBOTAREV,P.Yu., AND SHAMIS, E.V.: 'The matrix-forest theorem and measuring relations in small social groups', Automat. Remote Control 58, no. 9:2 (1997), 1505-1514. [7] CONSTANTINE, G.M.: Combinatorial theory and statistical design, Wiley, 1987. [8] GOULDEN,I.P., AND JACKSON,D.M.: Combinatorial ChUrneration, Wiley, 1983. [9] KIRCHHOFF, G.: @ber die AuflBsung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str5me geffihrt wird', Ann. Phys. Chem. 72 (1847), 497-508. [10] MAYEDA,W.: Graph theory, Wiley, 1972. [11] MERRIS, R.: 'An edge version of the matrix-tree theorem and the Wiener index', Linear and Multilinear Algebra 25 (1989), 291-296. [12] MERRIS, R.: 'Laplacian matrices of graphs: a survey', Linear Alg. ~ Its Appl. 197/198 (1994), 143-176. [13] MOON, J.W.: 'Various proofs of Cayley's formula for counting trees', in F. HARARY(ed.): A Seminar on Graph Theory, Holt, Rinehart & Winston, 1967, pp. 70-78. [14] MOON, J.W.: Counting labeled trees, Vol. 1 of Canad. Math. Monographs, Canad. Math. Congress, 1970. [15] SENETA, E.: Non-negative matrices and Markov chains, second ed., Springer, 1981. [16] TUTTE, W.T.: 'The disection of equilateral triangles into equilateral triangles', Proc. Cambridge Philos. Soc. 44 (1948), 463-482. [17] WEST, D.B.: Introduction to graph theory, Prentice-Hall, 1996. Ravindra B. Bapat Jerrold W. Grossman Devadatta M. Kulkarni MSC 1991:05C50

MEAN-VALUE CHARACTERIZATIONH a r m o n i c functions. Let S(x, r) denote the sphere of radius r and centre x in R n and let dar be the normalized L e b e s g u e m e a s u r e on S ( x , r). One version of the classical converse of Gauss' mean-value theorem for harmonic functions asserts t h a t a function f E C ( R n) which satisfies Is

(~,~)

f(y) dar(y)=f(x),

xeR

n,

rER

+,

(1)

is h a r m o n i c in R n (cf. also H a r m o n i c f u n c t i o n ) . In fact, one need only require t h a t (1) holds for 0 < r < p(x), where p is an a r b i t r a r y positive function of x. A corresponding 'local' result holds for continuous functions defined on an a r b i t r a r y domain in R n.

MEAN-VALUE C H A R A C T E R I Z A T I O N Remarkably, for the harmonicity of f it suffices that (1) holds only for two distinct values of r (and all x), so long as the radii are not related in a special way. Specifically, let

differential equation P(D)u = 0 if and only if it satisfies the generalized mean-value condition

f u(x + rt) d#(t) = O,

x c n ~,

r C R +,

(2)

( ~ ) (~-2)/2j(~_2)/2(~), where Jk is the Bessel function of the first kind of order k (cf. also Bessel functions), and let Hn be the set of positive quotients of zeros of Jn(~) - 1. J. Delsarte proved t h a t if (1) holds for r = rl and r = r2 and ri/r2 ¢ Hn, then f is harmonic in R ~ [11], cf. [20]. (In fact, Ha = {1}, so any two distinct radii are sufficient in dimension 3.) In [10], Delsarte's theorem is extended to non-compact irreducible symmetric spaces of rank 1. There is also a local version of this result [9], [21]. Let BR be the ball of radius R centred at 0 in R ~. Now, if f E C(BR) satisfies (1) for r = rl,r2 (rl/r2 f[ [In) and x such t h a t Ix] + rj < R, then f is harmonic on BR so long as rl + re < R. In this connection, one should also mention Littlewood's one-circle problem, solved by W. Hansen and N. Nadirashvili [14]. Let f be a bounded continuous fnnction on the open unit disc U in R 2. Suppose that for each point in U there exists an r = r(x) such that the mean-value condition of (1) holds. Must f be harmonic? The answer turns out to be 'no' [14]. On the other hand, the one-radius condition obtained by replacing the pcripheral mean in (1) by the (areal) average over the disc of radius r(x) does imply harmonicity [13]. This last result extends to functions defined on arbitrary bounded domains in R ~ (and m a n y unbounded domains as well); one can also weaken the boundedness assumption on f to If] < h for some positive harmonic function h. For a survey of these and related results, see [12]. Interesting new phenomena arise when one allows the integration to extend over the full space on which f is defined. Consider, for instance, functions integrable with respect to the (normalized) Lebesgue measure m on the unit ball B in C ~. If f is harmonic with respect to the invariant Laplacian [17, 4.1], then

where # is an appropriate complex measure supported on the unit ball of R ~ and D = (O/OXl,...,O/Oxn). (The choice d# = d O l - (50 corresponds to (1).) The local version of this result requires t h a t (2) holds for all x E D C R ~ and all 0 < r < dist(x,0D). Solutions of P ( D ) u = 0 are also characterized by two-radius theorems of Delsarte type [23], [22], cf. [19].

Pluriharmonic and separately harmonic functions. Mean-value characterizations of pluriharmonic functions (i.e., real parts of holomorphic functions, cf. also Pluriharmonic function) and separately harmonic functions (i.e., functions harmonic with respect to each variable zj, 1 < j < n) are studied in [3]. Let 1

n

k--1 --

(~k - ak) d-~[k] A d~;

u(~ - a) - (2rci) ~ E ( - 1 )

k=l

here d~[k] = d~l A - . . A d-~k_1 A A - . . A d~n , dC = d~x A ... A dC~. If D c C ~ is a complete bounded Reinhardt domain with centre at the point a and f is separately harmonic in 7? and continuous in 29, then zcn f0 n vol(D)

f(C)u(~ - a) = f(a).

(3)

Take for 29 the n-circular ellipsoids with centre at the point a, ~Dj,k(a ) = {z: b} [ Z l - a l l

2 q- " " -[- b~ [zn - anl 2 ~ r 2 k } ,

where k = 1,2, j = 1 , . . . , n , and all b} > 0. Then the following result holds. Let f C C ( C n) be such that for each a C C n the 2n conditions obtained by setting in (3) 29 = 29j,k(a), j = 1 , . . . , n , and k = 1,2 hold. If no rj,1/rj,2 belongs to H2n and if

B(f o ¢) d m = f ( ¢ ( 0 ) ) for every ~b in Aut(B). The converse holds if and only if n < 12 [1], cf. [7] and, for a Euclidean analogue, [6]. Asymptotic mean-value conditions for (non-integrable) functions on R ~ are studied in [8]. Finally, for a detailed overview of the whole subject, see [15].

Generalization. The extent to which mean-value theorems and their converses generalize to differential equations other than Au = 0 is explored in [23]. There it is shown that if P ( ~ I , . . . , ~ ) is a homogeneous polynomial, then u E C ( R ~) is a (weak) solution of the

det

@

¢ 0,

then f is separately harmonic in C n. Similarly, if 291 C C n is a complete bounded circular (Cartan) domain with centre at the point a (cf. also Reinhardt d o m a i n ) and f is pluriharmonic in 791 and continuous in 291, then zrn fo f(¢)4I n vol(Vl) vl

- a) = f ( a ) .

(4)

259

MEAN-VALUE CHARACTERIZATION Consider now circular ellipsoids with centre at the point a:

In [4], the following criteria are proved for functions that are (n - 1)-times continuously differentiable on C ~. • A function f is holomorphic in C ~ if and only if (5) holds with

=

~:

bj I ~ i ( z x - a l ) + . . .

+ ¢ ~ ( Z n -- a~)l ~ < ~j,~

,

n

/=1

b~ > 0 ;

(Ff)(z) = j=l,...,n;

k=1,2;

p=l,...,n.

Let tld~r~ll (1,m = 1 , . . . , n )

be the inverse matrix of

II~ll

Ilqp~,i*lt ( ; , s

for p fixed. Let Q =

=

1,

,~;

• A function f is anti-holomorphic on C ~ if and only if (5) holds with

i, l = 1 , . . . , n) be the (n 2 x n2)-matrix with entries

(F f)(z) = ~

p -~p qps,il = d i s d l s .

Holomorphic and pluriharmonic functions. In certain situations, Temlyakov-Opial-Siciak-type meanvalue theorems (see [2], [16], [18]) can be used to characterize holomorphic and pluriharmonic functions. For (n - 1)-times continuously differentiable functions f on C ~, the integral representation under discussion can be written as

f(z) = (Lf)(z) = (LF~f)(z) =

(5)

dt/s(Fnf)× n

x

(1-t2 .....

t~)(z,()-~

,~), • .., ~n(Z, ~)

,

where An = { ( t 2 , . . . , t ~ ) : t2,...,t,~ >_ O, t2 + " ' + t n <_ 1} is the unit simplex in the real Euclidean (n - 1)dimensional space, S = {(: Kjl = 1, j = 2 , . . . , n } ,

dt = dt2 A ... A dt,~, d¢/( = d~2/(2 A ... A d(,~/(,~, and

Z

=

(Zl,...,Zn)

~

C n, ;

=

(1,@,...,(~),

(z,() = Zl + z2G + "'" + Zn(~. Let F~ denote a certain differential operator of order n - 1, which will be specified separately for holomorphic functions, for pluriharmonic functions, and also for anti-holomorphic functions (that is, functions holomorphic with respect to = ( g l , - . . , g~)). More precisely, n-J-

F~f=

I[I(F+j

) f,

L/=I with the first-order differential operator to be specified, as mentioned above. 260

n

-5j

j

Then the following result holds. Let f E C(C n) be such that for every a E C ~ the conditions (4) hold for :D1 = :D~,k(a), j = 1 , . . . , n , k = 1,2, p = 1 , . . . , n (2n 2 conditions). If rj,x and rj,2 are such that no rj,1/rj,2 belongs to H2~, det [ll/b~l[ # 0 and d e t Q ~ 0, then f is pluriharmonic. Local versions of the above-mentioned results are given also in [3], as well as mean-value characterizations of pluriharmonic functions and separately harmonic functions by integration over the distinguished boundaries of poly-discs.

= (2=i)1-~/~

Of(z) zj Ozy

Of(z) 02j

• A function f is pluriharmonic on C n if and only if (5) holds with

°s(z) ¢=1 \ ~ ozj

os(z) + ~j O-~j ]"

These results remain true without the assumption of smoothness; in this case, derivatives being understood in the distributional sense [5]. References [1] AHERN, P., FLORES, M., AND RUDIN, W.: 'An invariant volume-mean-value property', J. Funct. Anal. 11 (1993), 380-397. [2] AIZENBERG, L.A.: 'Pluriharmonic functions', Dokl. Akad. Nauk. S S S R 124 (1959), 967-969. (In Russian.) [3] AIZENBERG, L.A., BERENSTEIN, C.A., AND WERTtIEIM, L.: 'Mean-value characterization of pluriharmonic and separately harmonic functions', Pacific J. Math. 175 (1996), 295-306. [4] AIZENBERG, L., AND LIFLYAND, E.: 'Mean-value characterization of holomorphic and pluriharmonic functions', Complex Variables 32 (1997), 131-146. [5] AIZENBERG, L., AND LIFLYAND, E.: 'Mean-value characterization of holomorphic and pluriharmonic functions, II', Complex Variables 39 (1999), 381-390. [6] BEN NATAN, Y., AND WEIT, Y.: 'Integrable harmonic functions on R ~', J. Funct. Anal. 150 (1997), 471-477. [7] BEN NATAN, Y., AND WEIT, Y.: 'Integrable harmonic functions and symmetric spaces of rank one', J. Funct. Anal. 160 (1998), 141-149. [8] BENYAMINI, Y., AND WELT, Y.: 'Functions satisfying the mean value property in the limit', J. Anal. Math. 52 (1989), 167-198. [9] BERENSTEIN, C.A., AND GAY, R.: ~A local version of the twocircles theorem', Israel J. Math. 55 (1986), 267-288. [10] BERENSTEIN, c . n . , AND ZALCMAN, L.: 'Pompeiu's problem on symmetric spaces', C o m m e n t . Math. Helv. 55 (1980), 593621. [11] DELSARTE, J.: Lectures on topics in m e a n periodic f u n c t i o n s and the two-radius theorem, Tata Institute, Bombay, 1961. [12] HANSEN, W.: 'Restricted mean value property and harmonic functions', in J. KRAL ET AL. (eds.): P o t e n t i a l T h e o r y - I C P T 93 (Proc. Intern. Conf., K o n t y ) , de Gruyter, 1996, pp. 67-90. [13] HANSEN, W., AND NADIRASHVILI, N.: 'A converse to the mean value theorem for harmonic functions', Acta Math. 171 (1993), 139-163. [14] HANSEN, W., AND NADIRASHVILI, N.: 'Littlewood's one circle problem', J. L o n d o n Math. Soe. 50 (1994), 349-360.

MSBIUS INVERSION [15] NETUKA, I., AND VESELY, J.: 'Mean value property and harmonic functions', in K. C-OWRISANKARAN ET AL. (eds.): Classical and Modern Potential Theory and Applications, Kluwer Acad. Publ., 1994, pp. 359-398. [16] OPIAL, Z., AND SICIAK, J.: 'Integral formulas for function holomorphic in convex n circular domains', Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 9 (1963), 67-75. [17] RUmN, W.: Function theory in the unit ball of C ~, Springer, 1980. [18] TEMLYAKOV, A.A.: 'Integral representation of functions of two complex variables', Izv. Akad. Nauk. SSSR Ser. Mat. 21 (1957), 89-92. (In Russian.) [19] VOLCHKOV,V.V.: 'New theorems on the mean for solutions of the Helmholtz equation', Russian Acad. Sci. Sb. Math. 79 (1994), 281-286. [20] VOLCHKOV, V.V.: 'New two-radii theorems in the theory of harmonic functions', Russian Acad. Sci. Izv. Math. 44 (1995), 181-192. [21] VOLCHKOV, V.V.: 'The final version of the mean value theorem for harmonic functions', Math. Notes 59 (1996), 247-252. [22] ZALCMAN, L.: 'Mean values and differential equations', Israel J. Math. 14 (1973), 339-352. [23] ZALCMAN, L.: 'Offbeat integral geometry', Amer. Math. Monthly 87 (1980), 161-175.

MSC 1991: 31A05, 46F10, 60Y65

31B05,

31C10,

L. Aizenber9 L. Zalcman 31C35, 32A10,

MOBIUS INVERSION - A method for inverting sums over partially ordered sets (or posets; cf. also P a r t i a l l y o r d e r e d set). The theory of M6bius inversion matured in the classic paper [4] of G.-C. Rota and is a cornerstone of algebraic combinatorics (cf. also C o m binatorics). Let P be a locally finite partially ordered set, that is, a poset in which every interval {z: x < z < y} is finite. The M5bius function # of P is the function on pairs of elements in P defined by the following conditions: #(x, y) = 0 if x ~ y, Ix(x, x) = 1 for all x, and the recursive relation E

#(x,z)=0

ifx
z:x
When P is finite, the function value Ix(x,y) is the xyentry in the inverse of the incidence matrix of the partial order relation on P. Let f , g be functions defined from P to a c o m m u t a t i v e r i n g R with unity. The MSbius inversion formula states: 9(x) = Z

f ( y ) ¢* f ( x ) =

y:y<_x

~

9(y)Ix(y,x)

y:y<_x

or, dually,

MILNOR UNKNOTTING CONJECTURE, Kronheimer-Mrdwka theorem - The unknotting number of the t o r u s k n o t of type (p, q) is equal to ( p - 1)(q

-

1)

2 The conjecture was proven by P.B. Kronheimer and T.S. Mrdwka [1] and generalized to positive links (cf. also P o s i t i v e link). See also Link. References [1] KRONHEIMER, P.B., AND MROWKA, T.S.: 'Gauge theory for embedded surfaces I', Topology 32, no. 4 (1993), 773-826.

Jozef Przytycki MSC 1991:57P25

MIMD,

multiple-instruction multiple-data A phrase denoting that, in a parallel machine, each processor may execute different instructions and operate on different data simultaneously. MSC 1991: 68Mxx

MIMO SYSTEM, multiple-input multiple-output system - A (dynamical) control system with multiple inputs and multiple outputs; see A u t o m a t i c c o n t r o l theory. MSC1991:93A25

g(x) = ~_, f ( y ) ¢* f ( x ) = y:y>_x

~

#(x,y)g(y).

y:y>x

For the B o o l e a n a l g e b r a 2 s of all subsets of a set S, IX(A,B) = (_I)[BI-IAI when A C_ B. When S is thought of as a set of 'properties', the MSbius inversion formula yields the principle of inclusion-exclusion (cf. I n c l u s i o n - a n d - e x c l u s i o n p r i n c i p l e ) . When P is the d i s t r i b u t i v e l a t t i c e of positive integers ordered by divisibility, the MSbius function #(m, n) equals 0 unless m divides n and n / m is square-free, in which case it equals ( - 1 ) e, where e is the number of primes in the prime factorization of n / m . The classical or number-theoretic MSbius inversion formula states:

din

din

For example, this yields the following formula for the Euler totient function ¢(n), the number of positive integers not exceeding n that are relatively prime to n (cf. also E u l e r f u n c t i o n ) :

din

The MSbius function satisfies many identities. Four often-used identities are: a) The P. Hall identity:

#(x,y) = -C1 + C~ - C3 + "" , 261

MSBIUS I N V E R S I O N where Ci is the number of (strict) chains x = x0 < xl < " < xi-1 < xi = y with i links starting at x and ending at y (cf. also C h a i n ) • b) The Weisner theorem: Let L be a finite l a t t i c e with minimum 0. If x and y are elements of L such t h a t x _> y > 0, then •

x)

=

u

the sum ranging over all elements u such t h a t u # x and

uVy:x. c) The Crapo complementation theorem: An element y is a complement of an element x in a lattice L with minimum 0 and m a x i m u m 1 if y V x = 1 and y A x = O. For any element x in a finite lattice L, #(0, y)p(z, 1),

p(O, 1) = E

y~z the sum ranging over all pairs y and z such that y < z and both y and z are complements of x. d) The Boolean expansion lemma: If A ~-~ A is a closure operator on a set S (cf. also C l o s u r e s p a c e ) , then for a closed set X in the lattice of closed sets,

#(O,X)

=

E

(--1)]AI'

A:-A=X The Boolean expansion l e m m a is a special case of the Galois connection theorem. It is also a special case of the cross-cut theorem. A cross-cut C in a finite lattice L is a set of elements of L satisfying: 1) C does not contain the minimum 0 or the maxim u m 1; 2) no pair of elements of C is comparable; 3) any maximal chain from 0 to 1 has non-empty intersection with C. A subset S of elements of L is spanning if both the join of all the elements in S is 1 and the meet of all the elements in S is 0. If L is a finite lattice with more than two elements and C is a cross-cut in L, then

p(0, 1) = q2

-

q3 +

q4

....

,

where qk is the number of spanning subsets of C with k elements. Besides order-theoretic, homological and counting proofs of these results, there are also proofs using the MSbius algebra, a generalization of the Burnside algebra of a group. Much work has been done on calculating MSbius functions of specific partially ordered sets. For exampie, if U and V are subspaees in the lattice of subspaces of a finite-dimensional vector space over a finite field of order q and U C_ V, then

#(U, V) = (--1)dq d(d-1)/2, where d is the difference dim V - dim U. 262

There are m a n y results relating structural properties and properties of the MSbius function. T w o examples follow. For an element x in a lattice, #(0, x) # 0 only if the element x is a join of atoms. (Atoms are elements covering the minimum 0; cf. also A t o m . ) If L is a finite lattice in which #(x, 1) is non-zero for all elements x, then there exists a bijection O' from L to itself such that for every x, 0'(x) V x is the m a x i m u m 1. M6bius functions occur in m a n y proofs. For example, they are heavily used in the original proof of Dilworth's theorem t h a t in a finite m o d u l a r l a t t i c e , the number of elements covering k or fewer elements equals the number of elements covered by k or fewer elements, and its extension, t h a t the incidence matrix or combinatorial Radon transform between these two sets of elements is invertible. The MSbius function has a homological interpretation. The value #(0, 1) + 1 for a partially ordered set P with minimum 0 and m a x i m u m 1 is the E u l e r c h a r a c t e r i s t l e of the order complex of P, the s i m p l i c l a l c o m p l e x whose simplices are the chains of P \ { 0 , 1}, the partially ordered set P with 0 and 1 deleted. Because polytopes are topologically spheres, this yields the following theorem. If E and F are faces in the face lattice of a polytope and E C_ F, then #(E, F ) = ( - 1 ) d, where d is the difference dim F - dim E. Taking the nerve of a covering of the order complex (cf. also N e r v e o f a f a m i l y o f sets), one obtains a homology based on a cross-cut and the cross-cut theorem. The homological interpretation is especially interesting for a geometric lattice L. In this case, the only nontrivial homology groups (cf. H o m o l o g y g r o u p ) are Ho and H a - 2 , where n is the rank of L, and [#(0,1)1 is the rank or dimension of the top homology group H~-2. Rota has proved the following sign theorem: If X is a rank-k flat in a geometric lattice, then ( - 1 ) k # ( 0 , X) is positive. Indeed, ( - 1 ) k # ( 0 , X ) counts certain subsets in the broken-circuit complex defined by H. Whitney. The characteristic polynomial x ( L ; t ) of a ranked partially ordered set L is the polynomial

E

p(O,X)/~rank(L)-rank(X)

X:X6L

in the variable A. With simple modifications, one can obtain from the characteristic polynomial of a geometric lattice the Poinca% polynomial of an a r r a n g e m e n t o f h y p e r p l a n e s and the chromatic polynomial of a graph (cf. also G r a p h c o l o u r i n g ) . The characteristic polynomial is an essential tool in the critical problem for matroids (cf. also M a t r o l d ) . Related to the characteristic polynomial is the Eulerian function ¢(G;s) of a

MOMENT MATRIX

finite gwup G, defined to be the Dirichlet polynomial

P(H,G) IHI ~ , H:H
where the sum ranges over all subgroups in the subgroup lattice of G. When s is a positive integer, ¢(G; s) is the number of s-tuples of group elements whose underlying set generate G. The Mhbius invariant #(M) of a matroid M is the integer I#(0, 1)1 with it calculated in its lattice of fiats if M has no loops, and 0 otherwise. Except when the element a is an isthmus, #(M) satisfies the relation

It(M) = It(M \ a) - p(M/a), where M \ a is M with a deleted and M / a is M contracted at a. The triple (M \ a , M , M / a ) is an analogue of a short exact sequence (cf. also E x a c t s e q u e n c e ) . Many other invariants (including the characteristic polynomial) satisfy contraction-and-deletion relations. In addition, inductive arguments involving contractions and deletions are often used in proofs. References

[1] BARNABEI, M., BRINI, A., AND ROTA, G.-C.: 'Theory of Mhbius functions', Russian Math. Surveys 3 (1986), 135-188. [2] BJORNER, A.: 'Homology and shellability of matroids and geometric lattices', in N.L. WHITE (ed.): Matroid Applications, Cambridge Univ. Press, 1992, pp. 226-283. [3] KUNG, J.P.S.: 'Radon transforms in combinatorics and lattice theory', in I. RIVAL (ed.): Combinatorics and Ordered Sets, Amer. Math. Soc., 1986, pp. 33 74. [4] ROTA, G.-C.: 'On the foundations of combinatorial theory I: Theory of Mhbius functions', Z. Wahrsch. Verw. Gebiete 2 (1964), 340-368. Joseph P.S. K u n 9

MSC1991: 06A07, 05E25, 05Exx, 05B35, 11A25 A m a t r i x containing the moments of a p r o b a b i l i t y d i s t r i b u t i o n (cf. also M o m e n t ; M o m e n t s , m e t h o d o f (in p r o b a b i l i t y t h e o r y ) ) . For example, if ¢ is a probability distribution on a set I C C, then m k = f1 xk de(x) is its kth order moment. If ¢ and thus the moments are given, then a l i n e a r f u n c t i o n a l L is defined on the set of polynomials by L(x k) = ink, k = 0, 1 , . . . . The inverse problem is called a moment problem (cf. also M o m e n t p r o b l e m ) : Given the sequence of moments ink, k = O, 1 , . . . , find the necessary and sufficient conditions for the existence of and an expression for a positive distribution (a nondecreasing function with possibly infinitely m a n y points of increase) that gives the integral representation of that linear functional. A positive distribution can only exist if L(p) > 0 for any polynomial p that is positive on 1. For the Hamburger moment problem (cf. also C o m p l e x m o m e n t p r o b l e m , t r u n c a t e d ) , I is the real axis and the polynomials are real, so the functional L is positive if L(p2(x)) > 0 for any non-zero polynomial p and MOMENT

MATRIX

-

this implies that the m o m e n t matrices, i.e., the Hankel n matrices of the m o m e n t sequence, Mn = [7rt ~+J]<j=0, are positive definite for all n = 0, 1 , . . . (cf. also H a n k e l m a t r i x ) . This is a necessary and sufficient condition for the existence of a solution. For the trigonometric moment problem, I is the unit circle in the complex plane and the polynomials are complex, so that 'positive definite' here means that L(Ip(z)l 2) > 0 for all non-zero polynomials p. The linear functional is automatically defined on the space of Laurent polynomials (cf. also L a u r e n t series) since m - k = L(z -k) = L(z k) = ink. Positive definite now corresponds to the Toeplitz m o m e n t matrices Mn = [mi_y]i~j=0 being positive definite for all n = 0, 1, 2 , . . . (cf. also T o e p l i t z m a t r i x ) . Again this is the necessary and sufficient condition for the existence of a (unique) solution to the m o m e n t problem. Once the positive-definite linear functional is given, one can define an i n n e r p r o d u c t on the space of polynomials as If, g) = L (f(x)g(x)) in the real case or as if, g) = L(f(z)g(z)) in the complex case. The moment matrix is then the G r a m m a t r i x for the standard basis

mi+j = (x ~, xh) or m~_j = (z ~, xJ). Generalized moments correspond to the use of nonstandard basis functions for the polynomials or for possibly other spaces. Consider a set of basis functions fo, f l , . . , that span the space £. The modified or generalized moments are then given by m k = L(fk). The moment problem is to find a positive distribution function ¢ that gives an integral representation of the linear functional on £. However, to define an inner product, one needs the functional to be defined on 7~ = £ - £ (in the real case) or on 7~ = £ . £ (in the complex case). This requires a doubly indexed sequence of 'moments' mij = (fi, fj). Finding a distribution for an integral representation of L on ~ is called a strong moment problem. The solution of m o m e n t problems is often obtained using an orthogonal basis. If the fk are orthonormalized to give the functions ¢0, ¢ 1 , . . . , then the moment matrix n Mn = [?Tt~J]i,j=o can be used to give explicit expressions; namely ¢~(z) = AA~(z)/v/jk4n_~J~A ~ where 2 ~ - 1 = 0, AAo(z) = fo(z) and for n > 1, -/~n = d e t M n with

M~(z) =

lio,.io)

-..

(fn-1, fo) iO(Z)

... • ""

(f0,.A) ] . (A_I, A ) ] fn(z) I

The leading coefficient in the expansion Ca(z) ~ n A ( z ) + . . . satisfies [~nl 2 = ]~/[n_l/fi/[ n.

=

References

[1] AKHIEZER, N.I.: The classical moment problem, Oliver & Boyd, 1969. (Translated from the Russian.)

263

MOMENT MATRIX [2] SHOHAT, J.A., AND TAMARKIN, J.D.: The problem of moments, Vol. 1 of Math. Surveys, Amer. Math. Soc., 1943. (Translated from the Russian.)

A. Bultheel M S C 1991: 44A60, 47A57 MOMENTUM M A P P I N G - The m o m e n t u m m a p ping is essentially due to S. Lie, [5, pp. 300-343]. The modern notion is due to B. Kostant [3], J.M. Souriau [9] and A.A. Kirillov [2]. The setting for the m o m e n t u m mapping is a smooth s y m p l e c t i e m a n i f o l d (M,w) or even a Poisson manifold ( M , P ) (cf. also P o i s s o n a l g e b r a ; S y m p l e c t i c s t r u c t u r e ) with the P o i s s o n b r a c k e t s on functions {f, g} = P(df, dg) (where P = w -1 : T*M --+ T M is the Poisson tensor). To each function f there is the associated Harniltonian vector field H/ = P(df) E Y(M, P), where Y(M, P ) is the Lie a l g e b r a of all locally Harniltonian vector fields Y E Y(M) satisfying £ r P = 0 for the Lie d e r i v a t i v e . The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie algebra homomorphisms:

0 --+ H°(M) --+ C°°(M) • 2~(M,w) --~ H I ( M ) -+ 0, where 7(Y) = [iya;], the d e R h a m e o h o m o l o g y class of the contraction of Y into w, and where the brackets not yet mentioned are all 0. A Lie g r o u p G can act from the right on M by a : M x G --+ M in a way which respects w, so t h a t one obtains a h o m o m o r p h i s m a~: g -+ ff(M,c~), where g is the Lie algebra of G. (For a left action one gets an anti-homomorphism of Lie algebras.) One can lift a t to a linear mapping j : 9 -+ Coo(M) if 7 o a I = 0; if not, one replaces ~ by its Lie subalgebra ker(7 o a ' ) C g. The question is whether one can change j into a homomorphism of Lie algebras. The mapping 9 ~ X, Y ~-~ {jX, j Y } - j([X, Y]) then induces a Chevalley 2-cocycle in H2(g, H°(M)). If it vanishes one can change j as desired. If not, the cocycle describes a central extension of g on which one may change j to a homomorphism of Lie algebras. In any case, even for a Poisson manifold, for a homomorphism of Lie algebras j : g -+ Coo(M) (or more generally, if j is just a linear mapping), by flipping coordinates one gets a momentum mapping J of the l~-action c~' from M into the dual ~* of the Lie algebra g, J : M -+ g*, (J(x), X> = j(X)(x), Hi(x) =

x E M,

XEg,

where (.,-) is the duality pairing. 264

For a particle in Euclidean 3-space and the rotation group acting on T * R 3, this is just the angular rnornenturn, hence its name. The m o m e n t u m mapping is infinitesimally equivariant for the g-actions if j is a homomorphism of Lie algebras. It is a Poisson morphism for the canonical Poisson structure on g*, whose symplectic leaves are the co-adjoint orbits. The m o m e n t u m mapping can be used to reduce the number of coordinates of the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems. [6], [4] and [7] are convenient references; [7] has a large and updated bibliography. The m o m e n t u m mapping has a strong tendency to have a convex image, and is important for representation theory, see [2] and [8]. There is also a recent (1998) proposal for a group-valued m o m e n t u m mapping, see [1]. References

[1] ALEKSEEV,A., MALKIN, A., AND MEINRENKEN,E.: 'Lie group valued moment maps', J. Diff. Geom. 48 (1998), 445-495. [2] KIRILLOV, A.A.: Elements of the theory of representations, Springer, 1976. [3] KOSTANT,B.: 'Orbits, symplectic structures, and representation theory': Proc. United States-Japan Sere. Diff. Geom., Nippon Hyoronsha, 1966, p. 71. [4] LIBERMANN,P., AND MARLE, C.M.: Symplectic geometry and analytic mechanics, Reidel, 1987. [5] LIE, S.: Theorie der Transformationsgruppen, Zweiter Abschnitt, Teubner, 1890. [6] MARMO, a., SALETAN, E., SIMONI, A., AND VITALE, B.: Dynamical systems. A differential geometric approach to symmetry and reduction, Wiley/Interscience, 1985. [7] MARSDEN, J., AND RATIU, T.: Introduction to mechanics and symmetry, second ed., Springer, 1999. [8] NEEB, K.-H.: Holomorphy and convexity in Lie theory, de Gruyter, 1999. [9] SOURIAU, J.M.: 'Quantification g~om~trique', Commun. Math. Phys. 1 (1966), 374-398.

Peter W. Michor M S C 1991: 37J15, 53D20, 70H33

MONTESINOS-NAKANISHI

CONJECTURE

-

A n y link can be reduced to a trivial link by a sequence

of 3-rnoves (that is, moves which add three half-twists into two parallel arcs of a link). The conjecture has been proved for links up to 12 crossings, 4-bridge links and five-braid links except one family represented by the square of the centre of the 5-braid group. This link, which can be reduced by 3moves to a 20-crossings link, is the smallest known link for which the conjecture is open (as of 2001). The conjecture has its stronger version t h a t any ntangle can be reduced by 3-moves to one of g(n) ntangles (with possible additional trivial components), where g(n) : H in-1 = I ( 3i @ 1).

MOONSHINE References [1] CHEN, Q.: 'The 3-move conjecture for 5-braids': Knots in Hellas '98 (Proc. Internat. Conf. Knot Theory and Its Ramifications, Vol. 24 of Knots and Everything, 2000, pp. 36-47. [2] KIRBY, P~.: 'Problems in low-dimensional topology', in W. KAZEZ (ed.): Geometric Topology (Proc. Georgia Internat. Topol. Conf. 1993), Vol. 2:2 of Stud. Adv. Math., Amer. Math. Soc./IP, 1997, pp. 35-473. [3] MORTON, H.R.: 'Problems', in J.S. BIRMAN AND A. LIBGOBER (eds.): Braids (Santa Cruz, 1986), Vol. 78 of Contemp. Math., Amer. Math. Soc., 1988, pp. 557-574. [4] PRZYTYCKI, J.H., AND TSUKAMOTO, T.: 'The fourth skein module and the Montesinos-Nakanishi conjecture for 3algebraic links', J. Knot Th. Ramifications t o a p p e a r

CONJECTURES

Griess algebra [12] extended by an identity element. The automorphism group of V ~ is M . The monstrous moonshine conjectures, and, in particular, the identification of V with V ~, were proved in [4], which also defined vertex operator algebras and a generalization of K a c - M o o d y algebras called Borcherds or generalized K a c - M o o d y algebras [3] (ef. also K a c M o o d y algebra; B o r c h e r d s Lie a l g e b r a ) . The series Tg were generalized by S.P. Norton to commuting pairs (g, h) E M x M [14]. In particular, to each such pair there is a m o d u l a r f u n c t i o n Z(g, h; z), invariant under a genus-0 group, such that

(g , h ; ~az+b j

(2ooi).

Z(gah c , g b h d ; z ) = a z

Jozef Przytycki

MSC 1991:57P25 for some root of unity a, for any M O O N S H I N E CONJECTURES - In 1978, J. McKay observed that 196 884 = 196 883 + 1. The number on the left is the first non-trivial coefficient of the jfunction, and the numbers on the right are the dimensions of the smallest irreducible representations of the Fischer-Griess Monster M [12] (cf. a l s o . S p o r a d i c simple g r o u p ) . On the one side stands a m o d u l a r function; on the other, a finite s p o r a d i c s i m p l e g r o u p . Moonshine is the explanation and generalization of this unlikely connection. Monstrous moonshine [7] conjectured that there is an infinite-dimensional graded v e c t o r s p a c e V = V-1 ® V1 ® V2 ® ' " , with the following properties. Each Vk carries a finite-dimensional representation of M; write Xk for its character. For each g E M, define the T h o m p s o n - M c K a y series Tg(z) = ~ = - 1 X k ( 9 ) qk, where q = exp(27riz). Then Tg is a generator ('Hauptmodul') of the field of modular functions for some genus-0 g r o u p Gg < SL2 (R). The group Gg contains Fo(N) as a n o r m a l s u b g r o u p , where N divides o(9) gcd(24, o(g)) (o(g) is the order of g). There are 171 distinct Thompson series (M has only 194 conjugacy classes). For example, G~ = SL2(Z) and T~ = j - 744, where e E M is the identity. M has two order-2 conjugacy classes, corresponding to the modular groups F0(2) and

r0(2) +

C :)}

Let Pd denote the d-dimensional irreducible represent a t i o n of M; t h e n V-1 = p l , P l = Pl @ fl196883, a n d V2 = / 9 1 @/9196883 @/921296876.

Central to these conjectures is the moonshine roodule V ~, constructed in [11]. It is an important example of a v e r t e x o p e r a t o r a l g e b r a (VOA) [2], [11], and as such possesses infinitely many heavily constrained bilinear products. One of these products makes V1 into the

This action of SL2(Z) is related to its natural action on the fundamental group Z 2 of the torus. The coefficients of the q-expansion of Z(g, h; z) are characters of the centralizer CM(g) evaluated at h. Simultaneous conjugation of g, h leaves Z unchanged: Z(aga -1, aha-1; z) = Z(g, h; z). The T h o m p s o n - M c K a y series are recovered by the specialization g = e: Z(e, h; z) = Th(z). Only special cases of these generalized moonshine conjectures have been proven. There are several other conjectures. For example, the series Tg~ were conjecturally related by the replication formulas [7], [1]:

(1) ad=n, O<_b
where

Qn

is

the

unique

polynomial

for

which

Qn(Tg (z)) - q - n is a power series with only strictly positive powers of q. The resemblance of (1) with Hecke operators is not accidental. These formulas are related to identities such as

p--1 H

(1 - prnqn)am~

-~

j(w) - j(z)

(2)

re>O, nEZ

for p = exp(27riw), where j ( z ) - 744 = ~ k akq k, as well as to the existence of several modular equations obeyed by the Tg. These identities played a large role in the proof of moonshine. R. Borcherds proved that the Tg obey (1) by using the denominator identities of Borcherds algebras; the resulting modular equations can be used to prove the Hauptmodul property [8]. It would be very nice to see other direct connections between the group operation in M and the series Tg: for instance, are there other relations among the Tg which correspond to more general products of conjugacy classes in M ? 265

MOONSHINE CONJECTURES T h e Hirzebruch prize question [13] asks for a compact, differentiable, 24-dimensional manifold on which M acts by diffeomorphisms and whose twisted W i t t e n genus is Tg. It is still open (as of 2000). Vertex o p e r a t o r algebras are a key c o m p o n e n t in moonshine. T h e r e will be some form of moonshine for finite subgroups of the a u t o m o r p h i s m group of any rational vertex o p e r a t o r algebra obeying a technical (and p r o b a b l y r e d u n d a n t ) 'C2 condition' (see e.g. [19]). The resulting functions, however, will not be H a u p t m o d u l s in general. There is no complete understanding yet (2000) of why H a u p t m o d u l s are associated to M , a l t h o u g h it has been conjectured to do with the '6-transposition p r o p e r t y ' of M [14], or the uniqueness of V ~ [18]. Moonshine is related to conformal field t h e o r y (e.g. g ~ is the chiral algebra of a holomorphic c = 24 theory), in which context m a n y of moonshine's special features appear natural [9], [18], and this has motivated some of the developments (most significantly the definition of vertex operator algebras). It still (as of 2000) is unclear why certain H a u p t moduls appear in moonshine and others do not. W h a t form does moonshine (e.g. the replication formulas) take in genus > 0 (see e.g. [17])? How a b o u t moonshine for other groups (see e.g. [15])? In [16], a variant called modular moonshine was established, where V ~ is replaced with vertex algebras over characteristic p, and where the role of M is taken by centralizers in M of order p elements. For his work in monstrous moonshine and related topics, Borcherds was awarded a Fields Medal in 1998 [6].

References [1] ALEXANDER,D., CUMMINS, C., MCKAY, J., AND SIMONS, C.: 'Completely replicable functions': Groups, Combinatorics and Geometry, Cambridge Univ. Press, 1992, pp. 87 98. [2] BORCHERDS, R.E.: 'Vertex algebras, Kac-Moody algebras, and the Monster', Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071. [3] BORCHERDS, R.E.: 'Generalized Kac-Moody algebras', J. Algebra 115 (1988), 501-512. [4] BORCHERDS,R.E.: 'Monstrous moonshine and monstrous Lie superalgebras', Invent. Math. 109 (1992), 405-444. [5] BORCHERDS,R.E.: 'Modular moonshine III', Duke Math. J. 93 (1998), 129-154. [6] BORCHERDS,R.E.: 'What is Moonshine?': Proc. ICM, Berlin, DMV, 1998, pp. 607-615. [7] CONWAY,J.H., AND NORTON, S.P.: 'Monstrous moonshine', Bull. London Math. Soc. 11 (1979), 308-339. [8] CUMMINS,C., AND GANNON,T.: 'Modular equations and the genus 0 property of Moonshine functions', Invent. Math. 129 (1997), 413-443. [9] DIXON,L., GINSPARO,P., AND HARVEY,J.: 'Beauty and the Beast: superconformal symmetry in a Monster module', Commun. Math. Phys. 119 (1988), 221-241. [10] DONO, C., LI, H., AND MASON, G.: 'Modular invariance of trace functions in orbifold theory', preprint q-aiM/9703016

(1997). 266

[11] FRENKEL, I.B., LEPOWSKY, J., AND MEURMAN, A.: Vertex operators and the monster, Acad. Press, 1988. [12] GRIESS, R.L.: 'The friendly giant', Invent. Math. 69 (1982), 1-102. [13] HIRZEBRUCH,F., BERGER, T., AND JUNO, R.: Manifolds and modular forms, second ed., Aspects of Math. Vieweg, 1994. [14] NORTON, S.P.: 'Generalized moonshine', Proc. Syrup. Pure Math. 47 (1987), 208-209. [15] QUEEN, L.: 'Modular functions arising from some finite groups', Math. Comput. 37 (1981), 547 580. [16] RYBA, A.J.E.: 'Modular moonshine?': Moonshine, the Monster, and Related Topics, Vol. 193 of Contemp. Math., Amer. Math. Soc., 1996, pp. 307-336. [17] SMITH, G.W.: 'Replicant powers for higher genera': Moonshine, the Monster, and Related Topics, Vol. 193 of Contemp. Math., Amer. Math. Soc., 1996, pp. 337-352. [18] TUITE, M.P.: 'On the relationship between Monstrous moonshine and the uniqueness of the Moonshine module', Commun. Math. Phys. 166 (1995), 495 532. [19] ZHU,Y.: 'Modular invariance of characters of vertex operator algebras', J. Amer. Math. Soc. 9 (1996), 237 302. Terry Gannon M S C 1 9 9 1 : l l F l l , 20D08, 17B67, 81T10

MORI THEORY OF EXTREMAL R A Y S - Let f : X --+ S be a projective m o r p h i s m of algebraic varieties over a field k of characteristic 0 (cf. also A l g e b r a i c v a r i e t y ) . A relative R - l - c y c l e is a formal linear combination Z = ~-~t~=l r j C j of a finite n u m b e r of curves Cj (reduced irreducible 1-dimensional closed subschemes) on X with real n u m b e r coefficients rj such t h a t f ( C j ) are points on S. (If S = Spec k, then the word 'relative' is dropped.) T w o relative R - l - c y c l e s Z1 and Z2 are said to be n u m e r i c a l l y equivalent if their intersection numbers are equal, ( D - Z1) = ( D . Z2) E R for any Cartier divisor D on X (cf. also D i v i s o r ; I n t e r s e c t i o n i n d e x (in a l g e b r a i c g e o m e t r y ) ) . The set N I ( X / S ) of all the equivalence classes of relative R - l - c y c l e s with respect to the numerical equivalence becomes a finitedimensional real v e c t o r s p a c e . T h e closed cone of curves (the K l e i m a n - M o r i cone) N E ( X / S ) is defined to be the closed convex cone in N1 ( X / S ) generated by the classes of curves on X which are m a p p e d to points on S by f . A half-line R = R_>0v C N E ( X / S ) is called an extremal ray if the inequality ( ( K x + B ) . v ) < 0 holds and if the equality v = vl + v2 for Vl, v2 E N E ( X / S ) implies vl, v2 C R. Cone theorem. Let X be a n o r m a l algebraic variety and B an effective Q-divisor such t h a t the pair ( X , B ) is weakly log t e r m i n a l (cf. K a w a m a t a r a t i o n a l i t y t h e o r e m ) . Let f : X -+ S be a projective m o r p h i s m to another algebraic variety. T h e n there exist at most countably m a n y extremal rays Rj = a_>0vj (j E J) satisfying the following conditions:

• For any v C N E ( X / S ) , there exist an element v ~ E NE(X/S) and n u m b e r s rj C R>_0, which are zero

MORI T H E O R Y OF E X T R E M A L RAYS except for finitely many j, such that ((Kx + B). v') >_0 and v = v' + ~ j rjvj. • (discreteness) For any closed convex cone E in N I ( X / S ) such that ((Kx+B).v) < 0 for any v e E\{O}, there exist only finitely many j E J such that vj E E.

Contraction theorem. Let R be an extremal ray as above. Then there exists a morphism ¢: X -+ Y, called a contraction morphism, to a normal algebraic variety Y with a morphism g : Y + S which is characterized by the following properties: • go¢=f; • ¢,Ox

= Of;

• any curve C which is mapped to a point by f is mapped to a point by ¢ if and only if its numerical class belongs to R. Two methods of proofs for the cone theorem are known. The first one [4] uses a deformation theory of morphisms over a field of positive characteristic and applies only in the case where X is smooth. It is important to note that this is the only known method in mathematics to prove the existence of rational curves (as of 2000). The second approach [1] uses a vanishing theorem of cohomology groups (cf. K a w a m a t a - V i e h w e g vani s h i n g t h e o r e m ) which is true only in characteristic 0. This method of proof, which is obtained via a rationality theorem (cf. K a w a m a t a r a t i o n a l i t y t h e o r e m ) , applies also to singular varieties and easily extends to the logarithmic version as explained above. The contraction theorem has been proved only by a characteristic-0 method (cf. [3]). In the following it is also assumed that the variety X is Q-factorial, that is, for any prime divisor D on X there exists a positive integer m, depending on D, such that mD is a Cartier divisor. Then the contraction morphism ¢ is of one of the following types:

• (Fano-Mori fibre space) dim Y < dim X. • (divisorial contraction) There exists a prime divisor E of X such that eodim ¢(E) _> 2 and ¢ induces an isomorphism X \ E -+ Y \ ¢(E). • (small contraction) ¢ is an isomorphism in cod# mension 1, in the sense that there exists a closed subset E of codimension _> 2 of X such that ¢ induces an isomorphism X \ E -+ Y \ ¢(E).

Flip conjectures. The first flip conjecture is as follows: Let ¢: X -+ Y be a small contraction. Then there exists a birational morphism from a Q-factorial normal algebraic variety ¢+: X + -+ Y which is again an isomorphism in codimension 1 and is such that the pair ( X + , B +) with B + = ( ¢ +, ) -1 ¢ , B is weakly log terminal and Kx+ + B + is a ¢+-ample Q-divisor (cf. also Divisor). The diagram X --+ Y +- X + is called a flip (or log flip). Note that - ( K x + B) is C-ample.

The second flip conjecture states that there does not exist an infinite sequence of consecutive flips. There is no small contraction if dim X < 2. The flip conjectures have been proved for dim X = 3 (see [5], [7] for the first flip conjecture, and [6], [2] for the second). The proofs depend on the classification of singularities and it is hard to extend them to a higher-dimensional case.

Minimal model program (MMP). Fix a base variety S and consider a c a t e g o r y whose objects are a pair (X, B) and a projective morphism f : X --+ S such that X is a Q-factorial normal algebraic variety and B is a Qdivisor such that (X, B) is weakly log terminal. A morphism from ((X, B), f ) to ((X', B'), f ' ) in this category is a b i r a t i o n a l m a p p i n g a : X .. ~ X ' which is surjective in codimension 1, in the sense that any prime divisor on X ' is the image of a prime divisor on X, and such that B' = a . B and f ' o a = f . The minimal model program is a program which works under the assumption that the flip conjectures hold. It starts from an arbitrary object ( ( X , B ) , f ) and constructs a morphism to another object ((X', B'), f ' ) such that one of the following holds: • X ' has a Fano-Mori fibre space structure ¢ : X ' -+ Y' over S. • X ' is minimal over S in the sense that Kx, + B' is f'-nef , i.e., an inequality ((Kx, + B') • C) >_0 holds for any curve C on X ' such that f(C) is a point on S. Construct objects ((Xn,B,O,fn) inductively as follows. Set ((Xo,Bo), fo) = ( ( X , B ) , f). Suppose that ((X,~, Bn), fn) has already been constructed. If Kx~ + B , is fi~-nef, then a minimal model is obtained. If not, then, by the cone theorem, there exists an extremal ray and one obtains a contraction morphism ¢: X , --+ Y by the contraction theorem. If dim Y < dimX~, then a Fano-Mori fibre space is obtained. If ¢ is a divisorial contraction, then one sets ( ( X n + l , B n + l ) , fn+l) = ((Y, ¢ . B ~ ) , f n o ¢-1). If ¢ is a small contraction and if the first flip conjecture is true, then take the flip ¢+: X + -+ Y and set ((X,~+l,Bn+l),fn+l) = ((X +, ( ¢ + ) - ] ¢ , B n ) , fn o ¢-1 o ¢+). If the second flip conjecture is true, then this process stops after a finite number of steps. A normal algebraic variety X is said to be terminal, or it is said that X has only terminal singularities, if the following conditions are satisfied: 1) The canonical divisor K x is a q-Cartier divisor. 2) There exists a projective birational morphism #: Y -+ X from a smooth variety with a normal cross$ ing divisor D = ~ k = l Dk such that one can write #*Kx : K y + ~ k dkDk with dk < 0 for all k. 267

MORI T H E O R Y OF E X T R E M A L RAYS As a special case of the minimal model program, if one assumes that X has only terminal singularities and B = 0, then any subsequent pair satisfies the same condition that X~ has only terminal singularities and Bn = 0. This is the 'non-log' version. It is expected that the minimal model program works also over a field of arbitrary characteristic, although the cone and contraction theorems are conjectural in general.

R e l a t i o n s b e t w e e n a ) - f ) , a) and b) are equivalent representations. Indeed, they can be written as

References

respectively. The remaining systems involve strict inequalities or non-trivial solutions. For example, d) and e) concern the existence of non-trivial solutions and positive solutions, respectively, for the system

[1] KAWAMATA,Y.: 'The cone of curves of algebraic varieties', Ann. of Math. 119 (1984), 603-633. [2] KAWAMATA, Y.: 'Termination of log-flips for algebraic 3folds', Internat. J. Math. 3 (1992), 653-659. [3] KAWAMATA, Y., MATSUDA, K., AND MATSUKI, K.: 'Introduction to the minimal model problem', Adv. Stud. Pure Math. I0 (1987), 283-360. [4] MOal, S.'. 'Threefolds whose canonical bundles are not numerically effective', Ann. of Math. 116 (1982), 133-176. [5] MORI, S.: 'Flip theorem and the existence of minimal models for 3-folds', J. Amer. Math. Soc. 1 (1988), 117-253. [6] SHOKUROV, V.: 'The nonvanishing theorem', Izv. Akad. Nauk.

SSSR 49 (1985), 635-651.

[7] SHOKUROV, V.: '3-fold log flips', Izv. Russian Akad. Nauk. 56 (1992), 105-203.

Yujiro Kawamata MSC 1991: 14Exx, 14Jxx, 14E30

MOTZKIN TRANSPOSITION T H E O R E M - The thesis of T.S. Motzkin, [6], in particular his transposition theorem, was a milestone in the development of linear inequalities and related areas. For two vectors u = (ui) and v = (vi) of equal dimension one denotes by u >_ v and u > v that the indicated inequality holds componentwise, and by u > v the fact u_> v and u 7~ v. Systems of linear inequalities appear in several forms; the following examples are typical:

(A,-A,I)

=b

Ax = 0,

(x:)

and

>__0,

x _> 0.

Taking B = O and c > 0 in c) gives a), showing that a) and b) are special cases of c). Similarly, the systems d) and e) are special cases of f), which itself is a special case of c) with b = 0, c = 0. In fact, every system of linear inequalities can be written as c). The following two versions of Motzkin's transposition theorem, [6], concern systems c) and f): • (solvability of c)) Given matrices A, B and vectors b, c, the following are equivalent: cl) the system Ax < b, B x < c has a solution x; c2) for all vectors y >_ 0, z _> 0,

A T y + B T z = 0 ==~ b T y + c T z > 0 and

ATy+BTz=O, z¢0

=~ b T y + c T z

> 0.

• (solvability of f)) Let A, B, C be given matrices, with A non-vacuous. Then the following are alternatives: fl) Ax > 0, B x > 0, C x = 0 has a solution x; f2) A T y l -~ B T y 2 ~- C T y 3 = 0, Yl ~ 0, Y2 _> 0 has s o l u t i o n s Yl, Y2, Y3.

a) b) c) d) e) f)

Ax _< b; Ax=b, Ax_0,

x_> 0; Bx0; x>0; Bx>0,

Special cases of Motzkin's theorem include the following theorems.

Cx=0.

In each of these so-called primal systems the existence of solutions is characterized by means of a dual system, using the transposes of matrices in the primal system. Hence the name 'transposition theorem'. The relation between the primal and dual systems is sometimes given as a 'theorem of alternatives', listing alternatives, i.e. statements P, Q satisfying P ¢=~ -,Q (where -~ denotes negation), in words: either P or Q but never both. 268

F a r k a s t h e o r e m . (See also [2].) Let A be a given matrix and b a given vector. Farkas' theorem for system a) says that the following are equivalent: al) the system Ax _O =:~bTy>_0. Farkas' theorem for system b) says that the following are equivalent: bl) the system Ax = b, x _> 0 has a solution x; b2) AZy >_ 0 =~ b T y >_ 0. The positively homogeneous systems d) and e) are covered by the following two theorems.

MOTZKIN T R A N S P O S I T I O N T H E O R E M G o r d a n ' s t h e o r e m . (See also [3].) Given a matrix A, the following are alternatives:

The dual (or polar) S* of a non-empty set S C R n is defined as S* : - - - - { y : s e S

dl) Ax = 0, x ~ 0 has a solution x; d2) ATy > 0 has a solution y.

=~ y T s > 0 } ;

(3)

it is a closed convex cone. In particular,

S t i e m k e ' s t h e o r e m . (See also [10].) Given a matrix A, the following are alternatives:

(R+(A)) * = {y: ATy ~_ 0}

(4)

is a polyhedral cone. Farkas' theorem bl) states that the vector b is in the cone R + (A). The equivalent statement b2) says that b cannot be separated from R+(A) by a hyperplane: such a separating hyperplane would have a normal y satisfying

el) Ax = 0, x > 0 has a solution x; e2) ATy ~ 0 has a solution y.

b T y < 0,

vTy > 0

+ (A) for all v E R+(A) (see e.g. Fig. 1), which by (4) is a negation of b2). Farkas' theorem for system b) states that for any matrix A, R + ( A ) = (R+(A)) **

Fig. 1: A hyperplane with normal y separating b and R+(A). .:.;.:..

......

K

L Fig. 2: Illustration of the alternatives (6): L n C = {0}, L ± 2lint C* ~ 9. S e p a r a t i o n t h e o r e m s . The above results are separation theorems, or statements about the existence of hyperplanes separating certain disjoint convex sets. First, some terminology. A set P C R n is polyhedral (and necessarily convex) if it is the intersection of finitely many closed half-spaces, say

P:={x: Bx
In general, a set C C R n is a closed convex cone if and only if C = C**. Farkas' theorem for system b) also implies that a cone in R '~ is polyhedral if and only if it is finitely generated (the Farkas-Minkowski-Weyl theorem, [9, Corol. 7.1a]). More generally, a set S C R n is polyhedral if and only if it is the sum of a finitely generated cone and the convex hull of finitely many points (the Minkowski-Steinitz-Weyl theorem, [9, Corol. 7.1b]). The theorems of Stiemke and Gordan can be interpreted as geometric statements about intersections CML of a pointed closed convex cone C and a subspace L in R n. Let R~_ denote the non-negative orthant in R% Thus, Gordan' theorem dl) says that R~_MN(A) ~ 0, where N(A) = {x: Ax = 0} is the null space of A. And Stiemke's theorem el) says that int(R~_) M N(A) ~ 9, where int(R~_) = {x e a n : x > 0}. In each case, the dual system uses the intersection C* M L ±, where L ± is the orthogonal complement of L. For example, the statements 30¢xECML

and

3yE (intC*)ML ±

(6)

are mutually exclusive (see e.g. Fig. 2), for otherwise xTy ~= 0

[>

0

sincex 3_ Y, since 0

xEC,

yEint

C*.

(1)

for some matrix B and vector b.

A finitely generated cone is the set of non-negative linear combinations of finitely many vectors (generators). An example is the cone generated by the columns of a matrix A: R+(A) := {Ax: x > 0}.

(5)

(2)

To make the statements in (6) alternatives, one has to show that one of them occurs, the hard part of the proof. Returning to the theorems of Gordan and Stiemke, recall that (R~)* = R~_ and N(A) ± = R(AT).

Then Gordan's theorem dl), R$ M N(A) 7~ O, and d2), int(R~_) 21R(A T) ~ 9, are alternatives. Likewise, Stiemke's theorems el), int(R$) M N(A) 7~ 9, and e2), R~_ M R(A T) ~ 0, are alternatives. 269

MOTZKIN TRANSPOSITION THEOREM For history, see [9, pp. 209 228]. For theorems of alternatives, see [5, pp. 2~37]. Generalizations can be found in [11], [1], [8, Sec. 21-22, especially Thm. 21.1; 22.6]. Finally, see [7], [5, p.100] for applications. References [1] FAN, K.: 'Systems of linear inequalities', in H.W. KUHN AND

A.W. TUCKEa (eds.): Linear Inequalities and Related Systems, Vol. 38 of Ann. of Math. Stud., Princeton Univ. Press, 1956, pp. 99-156. [2] FAR!4AS,J.: @ber die Theorie der einfachen Ungleichungen', J. Reine Angew. Math. 124 (1902), 1-24. [3] GORDAN,P.: '0ber die AuflSsungen linearer Gleighungen mit reelen Coefficienten', Math. Ann. 6 (1873), 23-28. [4] KUHN,H.W., AND TUCKER, A.W. (eds.): Linear inequalities and related systems, Vol. 38 of Ann. of Math. Stud., Princeton Univ. Press, 1956. [5] MANOASARIAN,O.L.: Nonlinear programming, McGraw-Hill, 1969.

[6] MOTZ~(IN,T.S.: 'Beitrgge zur Theorie der linearen Ungleichungen', Inaugural Diss. (Basel, Jerusalem) (1936). [7] MOTZKIN, T.S.: 'Two consequences of the transposition theorem on linear inequalities', Econometrica 19 (1951), 184-185. [8] ROCKAFELLAR, R.T.: Convex analysis, Princeton Univ. Press, 1970. [9] SCHRIJVER, A.: Theory of linear and integer programming, Wiley/Interscience, 1986. [10] STIEMKE, E.: '0her positive Lbsungen homogener linearer Gleichungen', Math. Ann. 76 (1915), 340 342. [11] TUCKER, A.W.: 'Dual systems of homogeneous linear relations', in H.W. KUHN AND A.W. TUCKER (eds.): Linear Inequalities and Related Systems, Vol. 38 of Ann. of Math. Stud., Princeton Univ. Press, 1956, pp. 3-18.

Adi Ben-Israel MSC 1991: 15A39, 90C05 MULTIPLICATION OF DISTRIBUTIONS, multiplication of generalized functions - Let f~ be an open subset of R ~. Following L. Schwartz [7], a distribution, or g e n e r a l i z e d f u n c t i o n , u ff 73'(f~) can be multiplied by a smooth function f E C°°(f~), the result being defined by its action on a test function qo C 73(f~): (fu, qo) = (u, fg~). The example of

O = ( ( ~ ( x ) x ) v p lx v7ke$(x)( p l ) - x

on subspaces of 73'(f~) or for certain individual distributions. The first approach is summarized under the heading g e n e r a l i z e d f u n c t i o n a l g e b r a s . By common usage of the term, 'multiplication of distributions' refers to the second approach. Here again one may distinguish multiplier theory (multiplication as a continuous bilinear mapping on linear topological subspaces of 73'(f~)) and methods producing individual distributional products (without continuity at large of the operations). M u l t i p l i e r t h e o r y . Typical examples are provided by the continuous multiplication mapping on the spaces of integration theory ( f , g) -+ f g: LP(F~)xLq(f~) -+ Ll(f~), 1/p + 1/q = 1, or the Sobolov spaces HS(f~) (cf. also S o b o l e v classes ( o f f u n c t i o n s ) ) , which form an algebra when s > n/2. By duality, a multiplication mapping HS(f~) x H-*(f~) -+ H-*(f~) can be defined. For multiplier theory in Sobolev-Besov spaces, see [8]. Another example arises from the convolution algebra 8 ~ ( R n) of tempered distributions with support in an acute cone F C R ~. The inverse image of 8~,(R n) under the F o u r i e r t r a n s f o r m F is the algebra of retarded distributions, on which the product, defined by uv = F - l ( F u . Fv), is a sequentially continuous bilinear mapping. I n d i v i d u a l d i s t r i b u t i o n a l p r o d u c t s . Product mappings will be defined on certain subsets Ad (f~) C 73' (f~) x 73' (f2) with values in 7P' (f~). The product will be bilinear, when applicable, commutative and partially associative: If (u,v) ff M ( f t ) and f C U°°(f~), then both (fu, v) and (u, fv) belong to M(f~) and (fu)v = u(fv) = f(uv). With these properties, localization is possible, that is, the product mapping is uniquely defined by its restrictions to open neighbourhoods of points in fL Equivalently, it suffices to define the products (qou)(qov) for every qo ff 73(f~) to specify uv. The following definitions are instances of such products of increasing generality.

=(~(x) {pairs of distributions with disjoint singular support}. This is the localized version of the product of a distribution and a smooth function. Note that (d(x),vp 1/x) f[ M I ( R ) . b) M 2 ( R n) = {pairs of distributions (u, v) such that the 8'-convolution of F(qou) and F(9~v) exists for all 9~ ff 73(R'0}- The definition of the S'-convolution is a generalization of the convolution in 8~ ( R n) not requiring the support property, see [3]. The product is defined locally by (9~u)(9)v) = F - l (F(9~u)*F(qov) ). The product of retarded distributions is a special case, as is the wave front set criterion of L. Hbrmander [4] (cf. also W a v e front): If for all (x,~) E R n x S ~-1, (x,~) C WF(v) implies ( x , - { ) ¢ WF(u), then (u,v) belongs to M2(R~). a) . A d l ( R n ) =

shows that this product is not associative (5(x) denotes the Dirac measure, vp 1_ the principal value distribution, X cf. G e n e r a l i z e d f u n c t i o n ; G e n e r a l i z e d f u n c t i o n s , p r o d u c t of). There are further limitations on defining products of distributions. Schwartz [6] proved that whenever an associative d i f f e r e n t i a l a l g e b r a (A, O, o) contains 2)'(ft), the operations (0, o) in A cannot simultaneously be faithful extensions of the distributional derivatives and the pointwise product of continuous functions. Thus, a multiplication of distributions can either be defined by imbedding the space of distributions into algebras, but giving up one or the other of the consistency properties above, or else can be defined only 270

M U L T I P L I E R S OF C * - A L G E B R A S c) Regularization and passage to the limit. A strict delta-net is a net (cf. also N e t ( d i r e c t e d s e t ) ) of test functions (P~)e>o C 7?(R n) such that the supports of the functions p~ shrink to {0} as e --~ 0, f p ~ ( x ) dx = 1 and f tp~(x)l dx is bounded independently of c. A model delta-net is a net of the form p~(x) = e - ~ p ( x / c ) with p E T)(R n) fixed. Then - / t 4 3 ( R n) = {pairs of distributions (u,v) such that lim~-,0(u * p~)(v * at) exists for all strict delta-nets (P~)~>0 and (a~)e>0}; - f144(R ~) = {pairs of distributions (u,v) such that l i m ~ 0 ( u * p~)(v * Pc) exists for all model delta nets (P~)~>0 and does not depend on the net chosen}. The product of u and v is defined by the respective limit. Various other classes of delta nets are in use as well. d) Harmonic regularization. Every distribution u E 77t(R ~) can be represented as the boundary value as c -+ 0 of a h a r m o n i c f u n c t i o n u ( x , c ) in the variables (x, e) E R ~ x (0, oc), obtained by convolution with the Poisson kernel (locally; cf. also P o i s s o n i n t e g r a l ) . Then - /tdh(R ~) = {pairs of distributions (u,v) such that lim~--,o u(-, c)v(., c) exists}. The product by analytic regularization in dimension n = 1 is a special case. It holds that ~ 4 i ( R ~) C A d i + I ( R n) for all i, and the products coincide when they exist, see [1], [5]. Every inclusion is strict. The products defined in multiplier theory are special cases of A/13. A short review of further definitions, which may produce results not consistent with ~45, can be found in [5]. The products Adl-Ad5 can be used to define restrictions of distributions to submanifolds or to compute convolutions, for exmnple. Generally (with exceptions), they cannot be used to define multiplications arising in non-linear partial differential equations because they are not stable with respect to perturbations, due to lack of continuity. In non-linear partial differential equations, either g e n e r a l i z e d f u n c t i o n a l g e b r a s or multiplier theory are applicable. A typical example for the latter is a conservation law like Otu(x, t) + O~ (u "~(x, t)) = 0 where the multiplication is done in L ~ and the derivatives are computed in ~ . Related to multiplier theory, introduced to derive estimates in non-linear (pseudo-)differential equations, is the paraproduct of J.M. Bony [2]. Given v E L ~ ( R '~) with compact support, the paramultiplication by v is a l i n e a r o p e r a t o r T~ mapping the Sobolev space H ~ ( R n) into itself for any s C R. The paraproduct does not reproduce the pointwise product (when defined by multiplier theory, for example) but serves to control non-linear terms up to some more regular deviation. For

example, if u, v belong to H ~ ( R ~) with s > n / 2 , then uv - (TuV + T~u) E H ~ ( R ~) for every r < 3 n / 2 . See also G e n e r a l i z e d f u n c t i o n a l g e b r a s . References [1] BOLE, V.: 'Multiplication of distributions', Comment. Math. Univ. Carolinae 39 (1998), 309-321. [21 BONY, J.M.: 'Calcul symbolique et propagation des singularit~s pour les ~quations aux d~riv~es partielles non linfiaires', Ann. Sci. t~cole Norm. Sup. Sdr. 4 14 (1981), 209246. [3] DmROLF, P., AND VOIGT, J.: 'Convolution and S ~convolution of distributions', Collect. Math. 29 (1978), 185196. [4] H6RMANDER,L.: 'Fourier integral operators I', Aeta Math. 127 (1971), 79-183. [5] OBERGUGGENBERGER,M.: Multiplication of distributions and applications to partial differential equations, Longman, 1992. [6] SCHWARTZ, L.: 'Sur l'impossibilit~ de la multiplication des distributions', C.R. Acad. Sci. Paris 239 (1954), 847-848. [7] SCHWARTZ,L.: Thdorie des distributions, nouvelle ed., Hermann, 1966. [8] TRIEBEL, H.: Theory of function spaces, Birkh~user, 1983. Michael Oberguggenberger M S C 1991:46F10 MULTIPLIERS OF C*-ALGEBRAS - A C*a l g e b r a A of operators on some H i l b e r t s p a c e 74 may be viewed as a non-commutative generalization of a function algebra Co (f~) acting as multiplication operators on some L2-space associated with a measure on the locally compact space t2. The space f~ being compact corresponds naturally to the case where the algebra A is unital. In the non-unital case any embedding of A as an essential ideal in some larger unital C*-algebra B (i.e., the annihilator of A in B is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space 12. Thus, the one-point compactification fl O {oc} of fl corresponds to the unitization .4 = A O C of the algebra A. The analogue of the maximal compactification - - the S t o n e - ( ~ e c h c o m p a c t i f i c a t i o n - - is the algebra M ( A ) of multipliers of A, defined by R.C. Busby in 1967 [4] and studied in more detail in [2]. It is defined simply as the idealizer of A in B(74) (assuming t h a t AN = 7l or, equivalently, t h a t no non-zero vector in 74 is annihilated by A). Linear operators A and p on A are called left and right centralizers if A(xy) = A(x)y and p(xy) = xp(y) for all x, y in A. They are automatically bounded. A double centralizer is a pair (1, p) of left, right centralizers such that x t ( y ) = p ( x ) y (whence Iit]1 = ]]PI]), and the closed linear spaces of double centralizers becomes a C*-algebra when product and involution are defined by ( t l , pl)(t2, P2) = (11t2, P2Pl) and (t, p)* = (p*, 1") (where t*(x) = (A(x*))*). As shown by B.E. Johnson, 271

MULTIPLIERS OF C*-ALGEBRAS [8], there is an isomorphism between the abstractly defined C*-algebra of double centralizers of A and the concrete C*-algebra M(A). This, in particular, shows that M ( A ) is independent of the given representation of A on 7/.

The strict topology on M(A) is defined by the seminorms x --+ ]taxll + Ilaxll on B(7/) with a in A, [4]. It is used as an analogue of u n i f o r m c o n v e r g e n c e on compact subsets of f~ in function algebras. Thus, it can be shown that M ( A ) is the strict completion of A in B(7/) and that the strict dual of M ( A ) equals the norm dual of A, [16]. If 7 / i s the universal Hilbert space for A (the orthogonal sum of all Hilbert spaces obtained from states of A via the Gel'fand-NaYmark-Segal construction), then M ( A ) has a more constructive characterization: Let (Asa) "~ denote the space of self-adjoint operators in B(7/) that can be obtained as limits (in the s t r o n g t o p o l o g y ) of some increasing net of self-adjoint elements from the unitized algebra A (cf. also N e t (dir e c t e d set); S e l f - a d j o i n t o p e r a t o r ) . Similarly, let (Asa)m = --((fi"sa) m) be the space of limits of decreasing nets. Then

M ( A ) ~ = (A~a) '~ n (A~),n. Thus, for every self-adjoint multiplier x there are nets (ax) and (b,) in fi-~a, one increasing, the other decreasing, such that a~ /~ x Z b,. If A is a-unital, i.e. contains a countable approximate unit, in particular if A is separable (cf. also S e p a r a b l e a l g e b r a ) , these nets can be taken as sequences, [2], [12, p. 12]. In the commutative case, where A = C0(f/), whence M(A) = Cb(f~), this expresses the well-known fact that a bounded, real function on f/ is continuous precisely when it is both lower and upper semi-continuous. For any C*-algebra X containing A as an ideal there is a natural morphism (i.e. a *-homomorphism) cr : X --+ M(A), defined by ~r(x)a = xa, that extends the identity mapping of A C X onto A C M(A). If A is essential in X, one therefore obtains an embedding X C M(A). Any morphism a : A --+ B between C*-algebras A and B extends uniquely to a strictly continuous morphism -~: M(A) -+ M(B), provided that c~ is proper (i.e. maps an approximate unit for A to one for B). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If A is a-unital and c~ is a quotient morphism, i.e. surjective, then ~ is also surjective. This result may be viewed as a non-commutative generalization of the Tietze extension theorem, [2], [13] (cf. also E x t e n s i o n t h e o r e m s ) . The corona of a C*-algebra A is defined as the quotient C*-algebra Q(A) = M ( A ) / A , [13]. The commutative analogue is the compact Hausdorff space/3f~ \ ft (the 272

corona of the locally compact space f~, [7]), but the preeminent example of such algebras is the Calkin algebra B(7/)/K(7-t), obtained by taking A as the algebra K(7/) of compact operators on 7/ (whence M ( A ) = B(7/)). Corona C*-algebras are usually non-separable and cannot even be represented on separable Hilbert spaces, [14]. Nevertheless, they have important roles in the formulation of G. Kasparov's KK-theory and the later variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of C*-algebras A and B is any C*-algebra X that fits into a short exact sequence (cf. also E x a c t s e q u e n c e )

O--+ A--+ X -~ B ~ O. Thus, X contains A as an ideal, and zr is simply the quotient morphism. In particular, M ( A ) may be regarded as an extension of A by Q(A), and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram 0

~

A

0

--+ A

--+

X

--+ M ( A )

-~

B

--+ 0

4

Q(A)

-+ 0

Here ~: X --+ M ( A ) is the morphism defined above and the induced morphism T: B --+ Q(A) is known as the Busby invariant for X. This invariant determines X up to an obvious equivalence, because the right square in the diagram above describes X as the pull-back of B and M(A) over Q(A), i.e.

X = M ( A ) ®Q(A) B = = {(re, b) C M ( A ) ® B : 7c(rn) = T(b)}. One therefore has the identification E x t ( A , B ) = Hom(B,Q(A)), [4], [5], [15]. For any quotient morphism ~r: X -+ B between C*algebras one may ask whether an element b in B with specific properties is the image of some x in X with the same properties. This is known as a lifting problem, and is the non-commutative analogue of extension problems for functions. Many lifting problems have positive (and easy) solutions: If b = b* or b _> 0 or Ilbl[ < 1, one can find counter-images in X with the same properties. However, the properties b2 = b (being idempotent) and b*b = bb* (being normal) are not liftable in general. It follows that the more general commutator relation bib2 = b2bl is not liftable either. But the orthogonality relation bib2 = 0 is liftable (even in the n-fold version bl -.. bn = 0). Using this one may show that the nilpotency relation b~ = 0 is liftable, [1], [11], [9]. As advocated by T.A. Loring, lifting problems may with advantage be replaced by C*-algebra problems concerning projectivity. A C*-algebra P is projective if any morphism a : P --+ B into a quotient C*-algebra

M U L T I P L I E R S OF C * - A L G E B R A S B = 7r(X) can be factored as a = 7r o ~ for some morphism ~: P --+ X , [3]. This m e a n s t h a t one is lifting a whole C * - s u b a l g e b r a and not just some elements. Projective C * - a l g e b r a s are the n o n - c o m m u t a t i v e analogues of topological spaces t h a t are absolute retracts, but since the category of C * - a l g e b r a s is vastly larger t h a n the category of locally c o m p a c t Hausdorff spaces, projectivity is a rare p h e n o m e n o n . However, the cone over the n x nmatrices, i.e. the a l g e b r a C M ~ = Co (]0, 1]) ® M s is always projective. This m e a n s t h a t although m a t r i x units cannot, in general, be lifted from quotients, there are lifts in the ' s m e a r e d ' form given by C M n , [10], [9]. C o r o n a C*-algebras form an indispensable tool for m o r e complicated lifting problems, because by B u s b y ' s theory, m e n t i o n e d above, it suffices to solve the lifting for quotient m o r p h i s m s of the form 7r: M(A) ~ Q(A). Thus, one m a y utilize the special properties t h a t corona algebras have. A brief outline of these follows. C o r o n a a l g e b r a s . In topology, a c o m p a c t H a u s d o r f f s p a c e is called sub-Stonean if any two disjoint, open, ac o m p a c t sets have disjoint closures. Exotic as this m a y sound, it is a p r o p e r t y t h a t any corona set/3f~ \ ft will have, if ~ is locally c o m p a c t and a - c o m p a c t . In such a space, every open, or-compact subset is also regularly erabedded, i.e. it equals the interior of its closure in/3£t \ ft, [7]. T h e n o n - c o m m u t a t i v e generalization of this is the fact t h a t if A is a ~-unital C*-algebra, then every aunital h e r e d i t a r y C * - s u b a l g e b r a t3 of its corona algebra Q(A) equals its double annihilator, i.e. B = ( B ± ) ±, [13]. T h e analogue of the s u b - S t o n e a n property, sometimes called the SAW*-condition, is even m o r e striking: For any two orthogonal elements x and y in Q(A) (say xy = 0) there is an element e in Q(A) with 0 < e < 1, such t h a t xe = x and ey = 0. Even better, if C and N are separable subsets of Q(A) such t h a t x c o m m u t e s with C and annihilates N , then the element e can be chosen with the same properties, [11], [14]. Note t h a t if e could be taken as a projection, e.g. the range projection of e, this would be a familiar p r o p e r t y in v o n N e u m a n n a l g e b r a theory. T h e fact t h a t corona algebras will never be von N e u m a n n algebras (if A is non-unital and cr-unital) indicates t h a t the p r o p e r t y (first established by G. K a s p a r o v as a 'technical l e m m a ' ) is useful. Actually, a potentially stronger version is true: If x~ and Yn are m o n o t o n e sequences of self-adjoint elements in Q(A), one increasing, the other decreasing, such t h a t x~ _< Yn for all n, and if C and N are separable subsets

of Q(A), such t h a t all xn c o m m u t e with C and annihilate N , then there is an element z in Q(A) such t h a t xn _< z < y,~ for all n, and z c o m m u t e s with C and annihilates N , [11]. This has as a consequence t h a t if B is any a - u n i t a l C * - s u b a l g e b r a of Q(A), c o m m u t i n g with C and annihilating N , as above, t h e n for any multiplier x in M ( B ) there is an element z in the idealizer I(B) of B in Q(A), still c o m m u t i n g with C and annihilating N , such t h a t zb = xb for every b i n / 3 , [5], [15]. In other words, the n a t u r a l m o r p h i s m cr : I(B) n C' 3 N ± -+ M ( B ) (with k e r n = B ± C3C ~ A N ±) is surjective. This indicates the size of corona algebras, even c o m p a r e d with large multiplier algebras. References [1] AKEMANN,CH.A., AND PEDERSEN, G.K.: 'Ideal perturbations of elements in C*-algebras', Math. Scan& 41 (1977), 117139. [2] AKEMANN, CH.A., PEDERSEN, G.K., AND TOMIYAMA, J.: 'Multipliers of C*-algebras', J. Funct. Anal. 13 (1973), 277301. [3] BLACKADAR, B.: 'Shape theory for C*-algebras', Math. Scan& 56 (1985), 249-275. [4] BUSBY, R.C.: 'Double centralizers and extensions of C*-

algebras', Trans. Amer. Math. Soc. 132 (1968), 79-99. [5] EmERS, S., LORINO, T.A., AND PEDERSEN, G.K.: 'Morphisms of extensions of C*-algebras: Pushing forward the Busby invariant', Adv. Math. 147 (1999), 74-109. [6] GROVE, K., AND PEDERSEN, G.K.: 'Diagonal±zing matrices over C(X)', Z. Funct. Anal. 59 (1984), 65-89. [7] GROVE, K., AND PEDERSEN, G.K.: 'Sub-Stonean spaces and corona sets', J. Funct. Anal. 56 (1984), 124-143. [8] JOHNSON, B.E.: 'An introduction to the theory of centralizers', Proc. London Math. Soc. 14 (1964), 299-320. [9] LORING, T.A.: Lifting solutions to perturbing problems in C*-algebras, Vol. 8 of Fields Inst. Monographs, Amer. Math. Soc., 1997. [10] LOmNG, T.A., AND PEDERSEN, G.K.: 'Projectivity, transit±vity and AF telescopes', Trans. Amer. Math. Soc. 350 (1998), 4313-4339. [11] OLSEN, C.L., AND PEDERSEN, G.K.: 'Corona C*-algebras and their applications to lifting problems', Math. Scand. 64 (1989), 63-86. [12] PEDERSEN, G.K.: C*-algebras and their automorphism groups, Acad. Press, 1979. [13] PEDERSEN, G.K.: 'SAW*-algebras and corona C*-algebras, contributions to non-commutative topology', Y. (?per. Th. 4 (1986), 15-32. [14] PEDERSEN, G.K.: 'The corona construction', in J.B. CONWAY AND B.B. MORREL (eds.): Proc. 1988 GPOTS-Wabash Conf., Longman Sci., 1990, pp. 49-92. [15] PEDERSEN, G.K.: 'Extensions of C*-algebras', in S. DOt'LICHER ET AL. (eds.): Operator Algebras and Quantum Field Theory, Internat. Press, Cambridge, Mass., 1997, pp. 2-35. [16] TAYLOR, D.C.: 'The strict topology for double centralizer algebras', Trans. Amer. Math. Soc. 150 (1970), 633-643.

Gert K. Pedersen M S C 1 9 9 1 : 46L80, 46J10, 46L85, 46L05

273

N Let X be a regular, strongly countably complete t o p o l o g i c a l s p a c e (cf. also S t r o n g l y c o u n t a b l y c o m p l e t e t o p o l o g i c a l space), let Y be a locally compact and or-compact space, let Z be a p s e u d o - m e t r i c s p a c e , and let f : X × Y -4 Z be an arbitrary separately continuous function (cf. also Separate and joint continuity). I. Namioka [10] proved that NAMIOKA

SPACE

-

N) there is a dense Gs-set A contained in X such that A x Y is contained in C ( f ) , the set of points of (joint) continuity of f (cf. also Set o f t y p e F~ (Gs)). This is known as the N a m i o k a t h e o r e m . Following [3], one says that a (Hausdorff) space X is a Namioka space if for any compact space Y, any metric space M and any separately continuous function f : X x Y --+ M, assertion N) holds. J. Saint-Raymond [11] proved that separable Baire spaces are Namioka and all Tikhonov Namioka spaces are Baire; he also showed that in the class of metric spaces, Namioka and Baire spaces coincide (cf. also B a i r e space). M. Talagrand [12] constructed an c~-favourable (hence, Baire) space that is not Namioka. It has been shown that cr-/~-defavourable spaces [11] and Baire spaces having dense subsets that are countable unions of K-analytic subsets [4] are Namioka. The Sorgenfrey line is Namioka (cf. also S o r g e n f r e y t o p o l o g y ) , although it is a-favourable. Many permanence properties of Namiolca spaces are known. In view of Saint Raymond's result, the Cartesian product of two (metric) Namioka spaces need not be Namiokn. Also, Namioka spaces are not preserved, even in the metric case, by continuous perfect mappings (cf. also B l u m b e r g t h e o r e m ) . Following G. Debs [4], one says that a compact space Y is co-Namioka, or has the Namioka property N* (or belongs to the class iV*) if for every Baire space X and for every semi-continuous function f : X x Y --+ R, the

conclusion of Namioka's theorem holds. It was shown that N* holds for many compact-like spaces appearing in functional analysis; among them are Eberlein compact spaces [6], Corson compact spaces [5], Valdivia compact spaces [7], and, more generally, all compact spaces Y such that Cp(Y) is cr-fragmentable [9]. It was shown by R. Deville [6] that ~ N ~ iV*. Recently (1999), A. Bouziad [2] showed that N* holds for all scattered compact spaces that are hereditarily submetacompact. Certain permanence properties of co-Namioka spaces have been studied. For example, it is known that the class iV* is closed under continuous images, arbitrary products [1] and countable unions [8]. References [1] BOUZIAD, A.: 'The class of co-Namioka compact spaces is stable under products', Proc. Amer. Math. Soc. 124 (1996), 983-986. [2] BOUZIAD, A.: 'A quasi-closure preserving sum theorem about the Namioka property', Topol. Appl. 81 (1997), 163-170. [3] CHRISTENSEN,J.P.R.: 'Joint continuity of separately continuous functions', Proc. Amer. Math. Soc. 82 (1981), 455-461. [4] DEsS, G.: 'Points de continuitfi d'une fonction s@arfiment continue', Proc. Amer. Math. Soc. 97 (1986), 16~176. [5] DEsS, G.: 'Pointwise and uniform convergence on a Corson compact space', Topol. Appl. 23 (1986), 299-303. [6] DEVILLE, R.: 'Convergence ponctuelle et uniforme sur un espace compact', Bull. Acad. Polon. Sci. 37 (1989), 7-12. [7] DEVILLE, R., AND GODEFROY, G.: 'Some applications of projective resolutions of identity', Proe. London Math. Soc. 22 (1990), 261-268. [8] HAYDON, R.: 'Countable unions of compact spaces with Namioka property', Mathematika 41 (1994), 141-144. [9] JAYNE, J.E., NAMIOKA, I., AND ROGERS, C.A.: 'orfragmentable Banach spaces', Mathematika 41 (1992), 161 188; 197 215. [10] NA~IOKA, I.:'Separate and joint continuity', Pacific J. Math. 51 (1974), 515-531. [11] SAINT-RAYMOND, J.: 'Jeux topologiques et espaces de Namioka', Proc. Arner. Math. Soc. 87 (1983), 499-504. [12] TALAGRAND, M.: 'Propri~t~ de Baire et propri~t~ de Namioka', Math. Ann. 270 (1985), 159-174.

Z. Piotrowski MSC 1991: 54C05, 26A15

NATURAL F R E Q U E N C I E S Let X be a regular, strongly countably complete t o p o l o g i c a l s p a c e (cf. also S t r o n g l y c o u n t a b l y c o m p l e t e t o p o l o g i c a l space), let Y be a locally compact and a-compact space (cf. also C o m p a c t s p a c e ) and let Z be a p s e u d o m e t r i c s p a c e . In 1974, I. Namioka [8] proved that for every separately continuous function f : X × Y -+ Z there is a dense Gb-subset A of X such that the set A × Y is contained in C ( f ) , the set of points of continuity of f (cf. also S e t o f t y p e Fz (Gb); S e p a r a t e a n d joint continuity). The original proof of this theorem starts with an interesting reduction to the case when Y is compact. Next, using purely topological methods, such as, e.g., the Arkhangel'skiY-Frollk covering theorem and Kuratowski's theorem on dosed projections, Namioka shows that, given that the set Oe is the union of all open subsets 0 of X × Y such that d i a m f ( 0 ) < ~, the set A~ = {x: {x} x Y C O~} is dense in X. For X = Y = Z = R (the real numbers), such a result was known already to R. Baire [2] (cf. S e p a r a t e and joint continuity). If X is complete metric, Y is compact metric and Z = R, Namioka's theorem was shown by H. Hahn [7] NAMIOKA

THEOREM

-

(see also [121). The question whether the completeness of Y suffices in Hahn's result was asked, independently, in [I] and [5]. The following example, due to J.B. Brown [9] shows that completeness does not suffice and proves the necessity of compactness of Y. In fact, let X = [0, 1], Y = U~c[0,1]Y~ , where Y~ = [0,1] and U denotes the free union of, in fact, c many copies of [0, 1]. Let f : X x Y ~ R be separately continuous on every 'square' X x Y~ and having a point of discontinuity along the line x = c~. Then, clearly, the set A mentioned in Namioka's theorem is empty. Answering a problem of Namioka, it was shown [13] that Namioka's theorem fails for all Baire spaces X (cf. also B a i r e space). Still, the theorem holds for certain Banach-Mazur game-defined spaces (cf. also B a n a c h M a z u r g a m e ) , namely for cr-fl-defavourable spaces [3], [11] and for Baire spaces having dense subsets that are countable unions of K-analytic subsets [6]. The importance of Namioka's theorem lies in the fact that both X and Y are neither metrizable nor having any kind of countability of basis. If Y has a countable base, then Namioka's theorem holds for all Bake spaces X, see [4] and [10]. For further information, see N a m i o k a space. References

[1] ALEXIEWICZ, A., AND ORLICZ, W.: 'Sur la continuit4 et la classification de Baire des fonctions abstraites', Fundam. Math. 35 (1948), 105-126.

[2] BAIRE, R.: 'Sur les fonctions des variables r6elles', Ann. Mat. Pura Appl. 3 (1899), 1-122. [3] BOUZIAD,A.: 'Jeux topologiques et point de continuit6 d'une application s6par6ment continue', C.R. Acad. Sci. Paris 310 (1990), 359-361. [4] CALBRIX, J., AND TROALLIC, J.P.: 'Applications s6par6ment continue', C.R. Acad. Sci. Paris Sdr. A 288 (1979), 647-648. [5] CHRISTENSEN,J.P.R.: 'Joint continuity of separately continuous functions', Proc. Amer. Math. Soc. 82 (1981), 455-461. [6] DEBS, G.: 'Points de continuit6 d'une fonction sdpar6ment continue', Proc. Amer. Math. Soc. 9T (1986), 167-176. [7] HAHN, H.: Reelle Funktionen, Leipzig, 1932, pp. 325-338. [8] NAMIOKA,I.: 'Separate and joint continuity', Pacific J. Math. 51 (1974), 515-531. [9] PIOTROWSKI, Z.: 'Separate and joint continuity', Real Analysis Exchange 11 (1985/86), 293-322. [10] PIOTROWSKI,Z.: 'Topics in separate and joint continuity', in preparation (2001). [11] SAINT-RAYMOND, J.: 'Jeux topologiques et espaces de Namioka', Proc. Amer. Math. Soc. 87 (1983), 499-504. [12] SIKORSKI,R.: Funkcje rzeczywiste, Vol. I, PWN, 1958, p. 172; Problem (6fl). (In Polish.) [13] TALAGRAND, M.: 'Propri6t6 de Baire et propri6t6 de Namioka', Math. Ann. 270 (1985), 159-174.

Z. Piotrowski MSC 1991: 54C05, 26A15 Resonances, vibrations, together with natural frequencies, occur everywhere in nature. For example, one associates natural frequencies with musical instruments, with response to dynamic loading of flexible structures, and with spectral lines present in the light originating in a distant part of the Universe. NATURAL

FREQUENCIES

-

The simplest case of natural frequencies is illustrated by the vibration of a string. Its deflection u(x, t) satisfies boundary conditions, u(0, t) = u0r , t) = 0, and an initial condition, u(x,O) = uo(x). Its motion is described by the equation ~'uxx = putt. Separation of variables u(x, t) = v(x)w(t) leads to a pair of equations vxx -- Av, Wtt -~- )~W.

In equations of the type A¢ = A¢, where A is an operator whose domain is a certain class of functions, the number A is called an eigenvalue (cf. E i g e n value), and ¢ is the corresponding eigenfunction. A (possibly complex) number # is said to belong to the spectrum a(A) of A (cf. also S p e c t r u m o f a n o p e r a t o r ) if the 'resolvent' operator ( A - # I ) -1 does not exist (cf. also R e s o l v e n t ) . # = A is an eigenvalue if it is a pole of ( A - # I ) -1, where I denotes the identity operator. In the equation of a vibrating string, the boundary conditions are satisfied only if A is the square of a natural number n, Am = n 2, n = 1, 2 , . . . . The natural frequencies wn are square roots of the eigenvalues: w~ = n. The corresponding natural modes ¢(t) are the trigonometric functions cos nt, sin nt. The E u l e r f o r m u l a exp(ia) = c o s a + i s i n a , with 275

NATURAL F R E Q U E N C I E S

i2

= - 1 , simplifies many arguments and offers a better insight into vibration and resonance, among others. The eigenvalues are real because in this case the operator A is self-adjoint, meaning that for any pair f, g in the domain of A, the inner product has the following symmetry: (A f, g) = (f, Ag} (cf. also S e l f - a d j o i n t ope r a t o r ) . In simple cases these products can be written as integrals. As an example of this abstract theory, consider a free vibration of a membrane occupying a region ft. It is modelled by an eigenvalue equation (cf. also

Neumann eigenvalue; Rayleigh-Faber-Krahn ine q u a l i t y ) . Let -TAw(x,y) be denoted by Aw, and p(x,y)w(x,y) by Bw. A is the L a p l a c e o p e r a t o r (02/0X 2 -t-02/0y2). The deflection w(x,y) = 0 on the boundary Oft of ft for all t > 0. (Here, T denotes the uniform membrane tension, p is mass per unit length.) Then Aw = ABw is the disguised equation of motion, with eigenvalue A = w 2, where a~ is the natural frequency of vibration. One can introduce the following product for arbitrary functions satisfying boundary conditions whose gradieAts are square integrable in ft: v} = .f/o[AW(x, y)]v(x, y) dx dy =

=

f£ w(x,y)[dv(x,y)] dx dy.

Putting v = w, one obtains an energy equation for a freely vibrating membrane. It connects the two basic energy forms: potential and kinetic. Rayleigh's principle relates the value of the smallest (fundamental) natural frequency of the system to the minimum, attained over all possible forms of vibration, of the ratio of the average kinetic energy over average potential energy, computed over a single cycle of vibration. Note that A being self-adjoint implies conservation of energy. So the problem with self-adjoint operators is not very realistic: The vibrating string does not know how to stop vibrating. It will go on forever with the same frequency and the same amplitude. However, a real string will insist on dissipating some of its energy and this has to be reflected in the properties of the operator. Generally, a correction is made by inserting firstorder differential terms into the differential equation, but not always. In the example of a vibrating elastic shaft one has (with suitable boundary conditions) a self-adjoint

Euler-Bernoulli equation: 02( 02u ) c92u Lu= ~ EI(x)~ +pA(x)-~. S. Timoshenko suggested a fifth-order derivative correction term: G(x)O5/Ox4cgt or similar, taking care of small 276

dissipative effects caused by rotational inertia deforming the shape of the cross-sectional area. With this term, and also possible first-order derivative terms included, the operator L is no longer self-adjoint, and the eigenvalues become complex numbers. Ignoring small damping terms, the equation (L - Re(AI)u = f can be solved. Here A is an eigenvalue of L, and Re(A) is the real part of A. Then the approximate dissipationfree solution can be written as ~ = (L - R e ( A ) I ) - l f . Observe that the indicated inverse exists, since Re(A) is not an eigenvalue. If f is close to an eigenfunction corresponding to the 'true' A and Re(A) is very close to the pole of the resolvent, the response ~ may become very large. This is a classical example of natural frequency re8onance.

In the energy-conserving problems described above, the domain of the operator L is compact, the inverse L -1 is a c o m p a c t o p e r a t o r , the spectrum of L consists of real eigenvalues only, and the only accumulation point for the eigenvalues is at infinity. Complications arise in quantum physics, where, in general, the domain of the operator is not compact and a continuous spectrum is superimposed on the true eigenvalues. Consider the Schr6dinger operator - h A + V(x), where V is a potential (cf. also SchrSdinger equat i o n ) . Since boundary values are absent, the spectrum of - h A is the positive part of the real line. Since solutions cannot be contained in a compact set, barriers set by the potential which produce a well of minimum energy surrounded by 'hills' are not respected. In fact, there is a unique (meromorphic) continuation of the resolvent, whereby the solutions tunnel through the obstacles. This refutes the classical laws of physics, under which particles can be trapped at the bottom of a potential well (corresponding to a minimal energy level).

References [1] ARFKEN, G.: Mathematical methods for physicists, third ed., Acad. Press, 1985, particularly Chapt. 9. [2] COURANT, R., AND HILBERT, D.: Methods of mathematical physics, Interscience, 1953, particularly Chapts. 6-7. [3] GOLDSTEIN, H.: Classical mechanics, Addison-Wesley, 1950, particularly Chapt. 10. [4] KELLER, J.B., AND ANTMAN, S.: Bifurcation theory and nonlinear eigenvalue problesm, Lecture Notes Courant Inst. Math. Sci. New York Univ., 1968. [5] LANDAU, L.D., AND LIFSHITZ, E.M.: A course in theoretical physics, Vol. 1: Mechanics, Pergamon & Addison-Wesley, 1960. (Translated from the Russian.) [6] TITCHMARSH, E.C.: Eigenfunction expansions associated with second order differential equations, second ed., Vol. 1, Oxford Univ. Press, 1958.

V. Komkov

MSC 1991: 70Jxx, 70Kxx, 73Dxx, 73Kxx

NATURAL L A N G U A G E P R O C E S S I N G

NATURAL LANGUAGE PROCESSING, N L P Natural language processing is concerned with the computational analysis or synthesis of natural languages, such as English, French or German (cf. [1], [8], [10] for surveys). Natural language analysis proceeds from some given (written or spoken) natural language utterance and computes its grammatical structure or meaning representation. The reverse procedure, natural language synthesis (or generation), takes some grammatical or meaning representation as input and produces (written or spoken) natural language surface expressions as output. A working hypothesis in this field is that natural languages should be studied from a formal language perspective (cf. F o r m a l i z e d l a n g u a g e ) . Though apparent parallels exist, there is also striking evidence which makes natural languages a particularly hard case for a formal language approach (for a survey of linguistic research, cf. [2]): • Unlike formal languages, natural languages are dynamic, by nature. Rule systems and vocabularies of natural languages continuously change over time, the lexical system in particular. This change behaviour and, furthermore, the sheer size of the required rule set and number of lexical items has up to now (2000) prevented linguists from providing a reasonably complete grammar for any natural language. Even worse, natural languages have productive mechanisms to enlarge their lexical repertoires on the fly (e.g., by deriving or composing new words from already known basic forms). • Compared with formal languages, natural languages exhibit an almost excessive degree of ambiguity. A distinction is made between sense ambiguities, i.e., different meanings of a word (e.g., 'bank' as an object to sit on vs. a financial institution), and structural ambiguities such as various parts of speech for one lexical item (e.g., 'orange' as a noun or an adjective) or alternative syntactic attachments (e.g., 'He saw [the man [with a telescope]object]' vs. 'He saw [the man] [with a telescope]instrument'). The ambiguity potential of syntactic structures like the attachment of prepositional or noun phrases, conjunctions, etc. can be described in terms of a well-known combinatoric series, the Catalan numbers, as characterized by C~ = (2~) - (~-1), 2~ where gn is the number of ways to parenthesize a formula of length n [5]. • H u m a n s process natural languages with a remarkably high degree of robustness when faced with ungrammatical input, i.e., ill-formed natural language utterances violating syntactic, semantic or lexical constraints. In addition, computing devices have to cope with the problem of extra-grammatical language, i.e., the processing of well-formed natural language utterances for

which, however, no grammar rules or lexical items exist at the representational level of the natural language processor. Extra-grammatical language is a particularly pressing issue for NLP. Although grammars for real-world data tend to be large already, their coverage is by no means sufficient to account for all relevant natural language phenomena. Hence, either the analyzer has to degrade gracefully in terms of its understanding depth relative to the amount of missing grammatical or lexical specifications, or grammars and lexicons have to be automatically learned in order to improve the effectiveness of future analyses (cf. M a c h i n e l e a r n i n g ) . • In contrast to formal languages, natural languages are often underconstrained with respect to unique specifications. This can be observed at the syntax level already, where so-called free-word-order languages allow for an (almost) unrestricted way of positioning syntactic entities in the sequential ordering of a sentence. Similar phenomena occur at the level of semantics, e.g., in terms of pronouns (which per se have no conceptual meaning, though they refer to other concepts), or imprecise, vague or fuzzy concepts (e.g., 'he wins quite often', 'a large elephant' vs. 'a large mouse'), or varieties of figurative speech such as metaphors. • Understanding natural languages is dependent on reference to particular domains of discourse, such as the language-independent knowledge about the commonsense world or highly specialized science domains. In any case, a corresponding knowledge repository (ontology, domain knowledge base, etc.) must be supplied, which complements language-specific grammatical and textual specifications. • The communicative function of natural languages (e.g., whether an utterance is to be interpreted as a command, a question or a plain factual statement) is dependent on the discourse or situational context in which an utterance is made. Unlike syntax and sematics, this level of pragmatics of natural language usage is entirely missing in formal languages. While formal languages can completely be described in terms of their syntax and semantics only (cf. Form a l i z e d l a n g u a g e ) , natural languages, due to their inherent complexity, require a more elaborate staging of description levels in order to properly account for combinatorial and interpretation processes at the lexical level (single words), at the phrasal and clausal level (single sentences) and the discourse level (texts or dialogues). Phonology, the most basic level of investigation of a spoken language, is concerned with the different types and articulatory features of single, elementary sounds, which are represented as phonemes. While phonemes are abstract description units, the link to concrete speech is 277

NATURAL LANGUAGE PROCESSING made in the field of phonetics, where spoken language has to be related to phonological descriptions. NLP considers various applications aiming at speech recognition and speech synthesis. The dominant methodologies used in this branch of NLP are probabilistic finite-state automata, hidden Markov processes in particular [9] (cf. also A u t o m a t o n , finite; A u t o m a t o n , probabilistic; M a r k o v process). At the level of morphology, phonemes are concatenated in terms of morphemes, i.e., either contentbearing units (syllables, stems) or grammatical elements (prefixes, infixes or suffixes such as past tense or plural markers). Content-bearing and grammatical items are combined to form lexical items which closely resemble our naTve intuition of words. Morphology accounts for phenomena which range from inflection, such as with 'swim®s' or 'swim[m]®ing', and derivation (as in 'swim[m]®er') to complex composition (as with 'swim[m]®ing (?) pool'). Morphological analysis within the NLP framework is mainly performed using a twolevel, finite-state automaton approach [16]. The level of syntax deals with the formal organization of phrases, clauses and sentences in terms of linguistically plausible constituency or dependency structures (cf. S y n t a c t i c s t r u c t u r e ) . Starting from the introduction of formal grammars into the linguistic research paradigm (by N. Chomsky in the late 1950s; cf. G r a m m a r , generative), and his claim that any finitestate device is unable to adequately account for basic syntactic phenomena (e.g., centre embedding of relative clauses, a pattern that can formally be characterized by the context-free mirror language anbn), linguistic theorists have continuously elaborated on this paradigm. Within NLP, Chomsky's transformational grammar (cf. G r a m m a r , t r a n s f o r m a t i o n a l ) was early rejected as a suitable analytic device due to its inherent computational intractability (the word, or membership, problem cannot be decided for transformational grammars, since they are essentially type-0 grammars; cf. F o r m a l languages and a u t o m a t a ) . Formal considerations relating to the generative power, computational complexity and analytic tractability of different types of generative grammars have since then always played a prominent role in NLP research [12], [3], [14]. Today (2000), two paradigms of syntactic analysis are dominating the NLP scene. On the one hand, featurebased unification grammars (such as lexical-functional grammar, head-driven phrase structure grammar) combine rule-oriented descriptions with a variety of phonological, syntactic and semantic features [15]. The basic operation besides rule application is feature unification, which has its roots in the logic p r o g r a m m i n g paradigm. Unification grammars are descriptively powerful 278

but their parsers tend to face serious complexity problems, since unconstrained unification is 2kf79-complete (cf. Complexity theory). On the other hand, carefully crafted 'mildly context-sensitive' grammars (cf. Grammar, context-sensitive), such as tree adjoining grammars (TAGs), use adjunction, a simple tree manipulation operation for syntactic analysis (elementary trees are embedded into derived trees by substitution of a single nonterminal node). TAG parsers stay clearly within feasibility regions, the most efficient ones are characterized by time complexity O(n 4) for sentence length n. While the unification paradigm is still heavily influenced by theoretically motivated claims about the proper formal description of natural languages, rapidly emerging requirements for processing large amounts of real-world natural language data have spurred the search for linguistically less sophisticated, performancedriven finite-state devices [13]. This has also led to a renaissance of statistical methodologies in language research (cf. the survey in [11]). As with phonology and morphology, Markov models (cf. M a r k o v process) play a major role here, together with probabilistic grammars, mostly probabilistic context-free grammars (though hybrid mergers with more advanced unification grammars and tree adjoining grammars also exist), where derivations are controlled by probabilistie weights assigned to single rules. Within the NLP community, a commonly shared belief is held that, by and large, natural languages have a significant context-free kernel, with only few extensions towards context-sensitivity (for a discussion of this issue, cf. [14]). Hence, the Earley algorithm for efficiently parsing context-free languages with time complexity of O(n 3) (cf. G r a m m a r , context-free) has been adopted as the fundamental parsing procedure for NLP and has been reformulated as the active chart parsing procedure (for a survey of natural language parsing techniques, cf. [17]). The field of (formal) semantics of natural languages has been dominated by logic approaches since the seminal work of R. Montague. He already advocated typed higher-order logics as an appropriate framework for semantic description. Logic semanticists agree on the finding that pure first-order p r e d i c a t e calculus is not expressive enough to capture major semantic phenomena of natural languages such as temporal or modal expressions (belief or normative statements), hypotheticals, distributive (individual) vs. collective (set) readings of plurals ('three men moved the piano'), generalized quantifiers ('the majority of ...', 'three out of five'). Hence, consensus has been reached to focus on Kripke-style higher-order modal logics and a strong typing discipline (ef. T y p e s , t h e o r y of) in order to adequately describe

NET (IN FINITE GEOMETRY) semantic phenomena in natural languages (for a survey, cf. [41). While this may be the appropriate answer from a theoretical point of view, such highly expressive formalisms pose serious computational problems. Since first-order predicate logic is only semi-decidable, and all higher-order logics have even worse decision properties, this raises a fundamental question to NLP: Should intractable formalisms be cut down to less expressive ones, which, as a consequence, then are tractable (e.g., monadic predicate logic)? Or should one still subscribe to those expressive and computationa]ly expensive models but impose limitations on the consumption of computation resources? There are, indeed, proposals that trade computation time against solution quality during the run-time of an algorithm (e.g., anytime algorithms). Alternatively, computationally hard problems can be segmented into 'cheap' and 'expensive' solution regions (e.g., by models of phase transitions). Strategies then have to be defined to circumvent the expensive solution regions that exhaust computation resources excessively. All these attempts aim at keeping control of resource consumption in a resource-greedy computing environment. While syntax and semantics have already wellestablished formal foundations, this is not so true for the broad field of pragmatics, where linguists investigate the regularities of language use in the discourse context. Though some formalizations for speech acts (rules of adequate interaction behaviour when talking to each other such as being informative, being as precise as possible and as necessary), communicative intentions, or assumption-based planning (e.g., for text generation) have already been developed, a homogeneous and widecoverage methodology (such as generative grammars for syntax) is still missing. As a consequence, NLP suffers from only few and incoherent attempts at computing appropriate pragmatic behaviour for language understanding (for a state-of-the-art survey as of 2000, cf. [6]). The applied side of NLP is concerned with the construction of natural language systems that exhibit a welldefined functionality (for a survey, cf. [7]). Three major application areas can be distinguished: systems which support natural language interaction with computer systems, either in a written or spoken mode (so-called natural language interfaces), systems for machine translation (cf. A u t o m a t i c t r a n s l a t i o n ) , and systems for automatic text analysis and text understanding, which deal with information retrieval tasks (automatic indexing, classification and document retrieval), information extraction from texts or text summarization. The field of language technology also benefits from the increasing availability of (annotated) corpora (text

and speech databases, parse tree banks, etc.), off-theshelf knowledge sources (such as machine-readable dictionaries or large-scale ontology servers), and standardized analysis tools (taggers, parsers, etc.). These resources are crucial for any serious attempt to properly evaluate the efficiency and effectiveness of natural language processors under realistic and experimentally valid conditions. These emperical considerations thus complement the focus on formal issues of natural language analysis and synthesis, which was prevailing in the past. References [1] ALLEN, J.: Natural language understanding, 2nd ed., Benjamin/Cummings, 1995. [2] ASHER, R.E., AND SIMPSON, J. (eds.): The encyclopedia of language and linguistics, Pergamon, 1994. [3] BARTON, JR., E.G., BERWICK, R.C., AND RISTAD, E.S.: Computational complexity and natural language, MIT, 1987. [4] CARPENTER, B.: Type-logical semantics, MIT, 1997. [5] CHURCH, K., AND PATIL, R.: 'Coping with syntactic ambiguity or how to put the block in the box on the table', Amer. J. Comput. Linguistics 8, no. 3/4 (1982), 139-149. [6] COHEN, P.R., MORCAN, J., AND POLLACK, M.E. (eds.): Intentions in communications, MIT, 1990. [7] COLE, R., MAHIANI, J., USZKOREIT, H., ZAENEN, A., AND ZUE, V. (eds.): Survey of the state of the art in human language technology, Cambridge Univ. Press and Giardini Ed., 1997. [8] GAZDAR, G., AND MELLISH, C.: Natural language processing in Lisp. An introduction to computational linguistics, Addison-Wesley, 1989. [9] JELINEK, F.: Statistical methods for speech recognition, MIT, 1998. [10] JURAFSKY, D., AND MARTIN, J.A.: Speech and language processing. An introduction to natural language processing, computational linguistics, and speech recognition, Prentice-Hall, 2000. [11] MANNING, C.D., AND SCHOTZE, H.: Foundations of statistical natural language processing, MIT, 1999. [12] PERRAULT, C.R.: 'On the mathematical properties of linguistic theories', Comput. Linguistics 10, no. 3/4 (1984), 165176. [13] ROCHE, E., AND SCHABES,Y. (eds.): Finite-state natural language processing, MIT, 1997. [14] SAVITCH, W.J., BACH, E., MARSH, W., AND SAFRAN-NAVEH, G. (eds.): The formal complexity of natural language, Reidel, 1987. [15] SHIEBER, S.M.: An introduction to unification-based approaches to grammar, Vol. 4 of CSLI Lecture Notes, Stanford Univ., 1986. [16] SPROAT, R.: Morphology and computation, MIT, 1992. [17] WINOGRAD, T.: Language as a cognitive process, Vol. 1: Syntax, Addison-Wesley, 1983.

Udo Hahn

MSC 1991:68S05 NET (IN FINITE GEOMETRY) (update) - I n the language of design theory (cf. Block d e s i g n and the links given there), a net of order s, degree r and index 279

N E T (IN F I N I T E GEOMETRY) # (for short, an (s, r; #)-net) is the same as an afSne resolvable 1 - (s:p, s#, r)-design (see T a c t i c a l c o n f i g u r a tion; A f f i n e d e s i g n ) . Thus, it is an incidence structure 7) = (V, B) for which the set of blocks B is partitioned into parallel classes each of which in turn partitions the point set V, and such that any two non-parallel blocks intersect in exactly it points. Moreover, there are r > 3 parallel classes each consisting of s blocks; then each block has k = sit points. The dual of an (s, r; it)-net is called a transversal design (denoted a TD~[r, s]); in this setting, the parallel classes of blocks of a net become the point classes of the transversal design. In a more combinatorial language, a net is also equivalent to an o r t h o g o n a l a r r a y of strength t -- 2. For detailed studies of nets, transversal designs and orthogonal arrays, see [1] and [3]. Any net 7) satisfies the inequality r _< s2it - 1 it- 1 '

(1)

and equality holds if and only the net is an (affine) 2design; then 7) is also called a complete net. If the dual transversal design is also resolvable (that is, if 'not being joined' induces an equivalence relation on the point set of T)), a stronger bound holds, namely r _< sp; in the case of equality (in which case the dual of 7) is again an (s, sit;it-net), it is referred to as a symmettic net (or a symmetric transversal design). The classical examples for complete nets are the affine designs AGd-l(d,q) formed by the points and hyperplanes of the d-dimensional finite affine spaces AG(d, q) (cf. also Afflne space) over the G a l o i s field GF(q) of order q (so q is a prime power here); and the classical symmetric nets can be obtained from the complete ones by omitting all hyperplanes parallel to some selected line. As of 2001, the outstanding problem in this area is the determination of the triples (s,r, it) for which an (s, r; it)-net exists. This problem is exceedingly difficult; for instance, it includes the famous problem of deciding whether or not a p r o j e c t i v e p l a n e of order not a prime power exists, and, more generally, the existence problem for affine designs (cf. also Affine design). Many constructions and a thorough discussion of the existence problem can be found in [1], and an extensive set of tables is given in [2]. In general, a net is not characterized just by its parameters. For instance, the number of non-isomorphic nets with the same parameters as AGg_I (d, q) grows exponentially with a growth rate of at least e kln k, where k = qd-1 and a similar assertion holds for symmetric nets. Hence, it is desirable to characterize the classical examples among the complete and symmetric nets. For instance, a symmetric net 7) with it > 1 and s > 2 in 280

which every line (that is, the intersection of all blocks through two given points) has cardinality s is isomorphic to a classical example; see A f f i n e d e s i g n for sireilar results in the complete case. There is also considerable interest in nets with 'nice' automorphism groups, for instance in translation nets, a generalization of the well-known translation planes (cf. Plane). As a further example, any generalized Hadamard matrix (see Hadamard matrix) is equivalent to a 'class-regular' symmetric net. These topics are discussed in detail in [1], see also [4]. The case # = 1 has received particular attention. An (s, r; ])-net is often called a Bruck net and is simply referred to as an (s, r)-net. Here the dual structure is denoted by TD[r, s] (see also T r a n s v e r s a l s y s t e m ) and the corresponding orthogonal arrays are equivalent to sets of mutually o r t h o g o n a l L a t i n s q u a r e s . The Bruck nets satisfying the bound (1) with equality are precisely the affine planes of order s (see also Affine space; P l a n e ) . An (s, r)-net is called maximal if it cannot be extended to an ( s , r + 1)-net by adding a new parallel class of lines. Any candidate for a new line necessarily is an s-set of points which meets every existing line in a unique point; such a set is called a transversal. Many known constructions of maximal nets actually yield transversal-free nets. A related problem is deciding whether or not a given net is maximal, and to find conditions guaranteeing that nets may be extended to larger nets. There is also considerable interest in determining the spectrum of all pairs (s, r) for which a maximal (s,r)-net exists, a problem even harder than the existence problem discussed above and in O r t h o g o n a l L a t i n s q u a r e s . See [1] for a detailed study of all these topics and [2] for an extensive set of tables. For a survey emphasizing the geometric properties of nets as well as their automorphism groups, see [4]. References

[1] BETH, T., JUNGNICKEL, D., AND LENZ, H.: Design theory, second ed., Cambridge Univ. Press, 1999. [2] COLBOURN, C.J., AND DINITZ, J.H.: The CRC handbook of combinatorial designs, CRC, 1996. [3] HEDAYAT,A.S., SLOANE, N.J.A., AND STUFKEN, J.: Orthogonal arrays, Springer, 1999. [4] JUNGNICKEL, D.: 'Latin squares, their geometries and their groups', in D.K. RAY-CHAUDHURI(ed.): Coding Theory and Design Theory Part II, Springer, 1990, pp. 166-225.

Dieter Jungnickel MSC 1991: 05Bxx NEUMANN EIGENVALUE - Consider a bounded domain ~ C R n with a piecewise smooth boundary cOgt. A number # is a Neumann eigenvalue of f / i f there exists a function u C C2(t2) A C°(~) (a Neumann eigenfunction) satisfying the following Neumann boundary value

NON-ADDITIVE MEASURE problem (cf. also N e u m a n n -Au

boundary

= #u

conditions):

inf,,

(1)

Ou = 0 incgf~, (2) On where A is the L a p l a c e o p e r a t o r (i.e., A = ~i~=1 02/Ox~). For more general definitions, see [8]. Neum a n n eigenvalues (with n = 2) appear naturally when considering the vibrations of a free membrane (cf. also N a t u r a l f r e q u e n c i e s ) . In fact, for n = 2 the non-zero Neumann eigenvalues are proportional to the square of the eigenfrequencies of the m e m b r a n e with free boundary. Provided f~ is bounded and the boundary cgf~ is sutficiently regular, the Neumann Laplacian has a discrete spectrum of infinitely m a n y non-negative eigenvalues with no finite accumulation point: --

0 = # 1 ( a ) < #2(a) ~ ' - "

(3)

(#k ~ ec as k --~ oc). The Neumann eigenvalues are characterized by the max-rain principle [3]: #k = s u p i n f

f (w)

fa u2 dx

k=l,2,....

'

k=0,1,...,

(8)

47r2k2/~ Pk+~ <_ ( C ~ l a l ) 2 / n ,

k--0,1,....

(9)

The most significant result towards the proof of Pfilya's conjecture for N e u m a n n eigenvalues is the result by P. KrSger [5]: k

< -

n n+

47r2k 2/n 2 (C~lf~l)2/~'

k = 1,2,

""

(4)

(5)

7r2

(6)

where d is the diameter of the domain. There are many more isoperimetric inequalities for Neumann eigenvalues (see R a y l e i g h - F a b e r - K r a h n inequality). For large values of k, H. Weyl proved [9] 47r2k2/~

#k+l ~ (C~lf~l)2/~,

4~rk A '

and conjectured the same bound for any bounded domain in R 2. This is equivalent to saying that the Weyl asymptotics of #k is an upper bound for #k. The analogous conjecture in dimension n is

i=1

How far the first non-trivial Neumann eigenvalue is from zero for a convex domain in R 2 is given through the optimal inequality [6]

#1 > d~,

#~+1_<

EPi

where the inf is taken over all u C H l(ft) orthogonal to ~ 1 , . . . ,Pk-1 E Hl(f~), and the sup is taken over all the choices of { ~ i } ik-1 = l . For simply-connected domains the first eigenfunction ul, corresponding to the eigenvalue #1 = 0 is constant throughout the domain. All the other eigenvalues are positive. While Dirichlet eigenvalues satisfy stringent constraints (e.g., A2/AIcannot exceed 2.539... for any bounded domain in R 2, [1]; see also D i r i c h l e t e i g e n v a l u e ) , no such constraints exist for Neumann eigenvalues, other than the fact that they are non-negative. In fact, given any finite sequence #1 = 0 < ... < #N, there is an open, bounded, smooth, simply-connected domain of R 2 having this sequence as the first N Neumann eigenvalues of the Laplacian on that domain [2]. Though it is obvious from the variational characterization of both Dirichlet and Neumann eigenvalues (see (4)) t h a t #k < Ak, L. Priedlander [4] proved the stronger result #k+l < A k ,

where ]~tl and Cn = 7rn/2/F(n/2 + 1) are, respectively, the volumes of f~ and of the unit ball in R ~. For any plane-covering domain (i.e., a domain t h a t can be used to tile the plane without gaps, nor overlaps, allowing rotations, translations and reflections of itself), G. Pdlya [7] proved t h a t

(7)

A proof of Pdlya's conjecture for both Dirichlet and Neum a n n eigenvalues would imply Friedlander's result (5). References

[1] ASHBAUGH,M.S., AND BENGURIA, R.D.: 'A sharp bound for

[2]

[3]

[4]

[5]

[61

the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions', Ann. of Math. 135 (1992), 601 628. COLINDE ViRDmRE, Y.: 'Construction de laplaeiens dont une partie finie du spectre est donn6e', Ann. Sci. l~cole Norm. Sup. 20, no. 4 (1987), 599-615. COURANT,R., AND HILBEaT, D.: Methoden der mathematischen Physik, Vol. I, Springer, 1931, English transl.: Methods of Mathematical Physics, vol. I., Interscience, 1953. FRIEDLANDER, L.: 'Some inequalities between Dirichlet and Neumann eigenvalues', Arch. Rational Mech. Anal. 116 (1991), 153-160. KRbGEa, P.: 'Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean Space', J. Funct. Anal. 106 (1992), 353 357. PAYNE, L.E., AND WEINBERGER,H.F.: 'An optimal Poincar~ inequality for convex domains', Arch. Rational Mech. Anal.

5 (1960), 286-292. G.: 'On the eigenvalues of vibrating membranes', Proc. London Math. Soc. 11, no. 3 (1961), 419-433. [8] REED, M., AND SIMON, B.: Methods of modern mathematical physics IV: Analysis of operators, Acad. Press, 1978. [9] WEYL, H.: 'Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen', Math. Ann. 71 (1911), 441-479.

[7] POLYA,

Rafael D. Benguria M S C 1991:35J05 N O N - A D D I T I V E MEASURE - Non-additive set functions, as for example outer measures and semivariations of vector measures, appeared early in classical measure theory concerning countable additive set functions (cf. also S e t f u n c t i o n ; M e a s u r e ) or, more 281

NON-ADDITIVE MEASURE general, concerning finite additive set functions. The pioneer in the theory of non-additive set functions was G. Choquet [3] with his theory of capacities (cf. also C a p a c i t y ) . This theory had influences on m a n y parts of mathematics and different areas of sciences and technology ([4], [7], [8], [9], [11], [12J). One can compare additive set functions (which are the basis for classical measure theory) and non-additive set functions in the following simple way. For a fixed set A from a or-algebra E, a classical measure #: E --+ [0, +oe] gives t h a t for every set B from E such that A n B = (3 one has that #(A U B) - #(B) is always equal to a constant, #(A), i.e., it is independent of B. In contrast, for a non-additive set function m the difference m ( A U B ) - r n ( B ) depends on B and can be interpreted as the effect of A joining B. Non-additive set functions are extensively used in decision theory ([7], [12]), mathematical economy, social choice problems, with early traces by R.J. A u m a n n and L.S. Shapley in [2]. Many authors have investigated various kinds of non-additive set functions, such as subadditive and superadditive set functions, submeasures, ktriangular set functions, t-conorm and pseudo-addition decomposable measures (cf., e.g., I d e m p o t e n t a n a l y sis), null-additive set functions, and many other types. Although in m a n y results the monotonicity of the observed set functions was supposed, there are some results concerning certain classes of set functions which include also non-monotone set functions (for example superadditive set functions, k-triangular set functions; [2], [9]). On the other hand, 'fuzzy measures' regarded as monotone and continuous set functions were investigated by M. Sugeno in [10] with the purpose to evaluate non-additive quantities in systems engineering. This notion of 'fuzziness' is different from the one given by L.A. Zadeh (cf. also N o n - p r e c i s e d a t a ) . Namely, instead of taking membership grades of a set, one takes (in the fuzzy measure approach) the measure that a given unlocated element belongs to a set. Many important types of non-additive set functions occur in various branches of mathematics, such as potential theory ([3]), harmonic analysis, fractal geometry ([6]), functional analysis, the theory of nonlinear differential equations, the theory of difference equations, and in optimization ([9], [12]). Also, interest in non-additive set functions is growing. In artificial intelligence, belief functions have been applied to model uncertainty, [12]. Belief functions, corresponding plausibility measures and other kinds of nonadditive set functions are used in statistics, [8]. Nonadditive expected utility theory has been applied, for example, in multi-stage decision and economics, [7]. 282

The unification of m a n y different kinds of nonadditive set functions is achieved by the wide class of so-called null-additive set functions. A set function rn defined on a family 7? of sets such t h a t it is closed with respect to the union and (3 E ~D, and with values in [ - o c , oe] (or more general in a s e m i - g r o u p with a neutral element 0) and such that r n ( B ) = 0 implies m ( A U B ) = re(A) whenever A , B E E, A n B = (3, is called a null-additive set f u n c t i o n . These set functions have very nice properties with respect to the usual notions in measure theory and occur naturally in the theories of integrals ([9], [12]). The origins of null-additive set functions are in papers of V.N. Aleksyuk [1] (under the name 'quasi-measures') and I. Dobrakov [5] (under the name 'submeasures'). References

[1] ALEKSJUK, V.N.: :Two theorems on existence of quasibase for a family of quasimeasures', Izv. Visch. Uchebn. Zav. 6, no. 73 (1968), 11-18. (In Russian.) [2] AUMANN,R.J., AND SHAPLEY, L.S.: Values of non-atomic games, Princeton Univ. Press, 1974. [3] CHOQUET, G.: 'Theory of capacities', Ann. Inst. Fourier (Grenoble) 5 (1953), 131-295.

[4] DENNEBEaG,D.: Non-additive measure and integral, Kluwer Acad. Publ., 1994. [5] DOBRAKOV,I.: 'On submeasures I', Diss. Math. 112 (1974). [6] FALCONER,K.: Fractal geometry, Wiley, 1990. [7] GRABISCH,M., NGUYEN, H.T., AND WALKER, E.A.: Fundamentals of uncertainity calculi with application to fuzzy inference, Kluwer Acad. Publ., 1995.

[8] HUBER, P.J.: 'The use of Choquet capacities in statistics', Bull. Int. Inst. Statist. 45 (1973), 181-191. [9] PAP, E.: Null-additive set functions, Kluwer Acad. Publ./Ister, 1995. [10J SUGENO, M.: 'Theory of fuzzy integrals and its applications', PhD Thesis Tokyo Inst. Technol. (1974). [11] SUGENO,M., ANDMUROFUSm,T.: 'Pseudo-additive measures and integrals', J. Math. Anal. Appl. 122 (1987), 197-222. [12] WANG, Z., , AND KLIR, G.: Fuzzy measure theory, Plenum, 1992.

E. Pap

M S C 1991: 28-XX N O N - C O M M U T A T I V E ANOMALY - Many fundamental calculations of q u a n t u m field t h e o r y reduce, in essence, to the computation of the d e t e r m i n a n t of some operator. One could even venture to say that, at one-loop order, any such theory reduces to a theory of determinants. It has been realized recently (2000) that the calculation of effective actions through the corresponding determinants is not free from dangers (which were overlooked till very recently by the physics community) stemming from the existence of a multiplicative anomaly (also n o n - c o m m u t a t i v e anomaly or determ i n a n t anomaly), i.e., the fact that the determinant of a product of operators is not equal, in general, to the product of the determinants. The difference proves to be

NORMAL BASIS T H E O R E M physically relevant, since only in some particular cases can it be simply absorbed into the renormalized constants. The operators involved are 'differential' ones, as the normal physicist would say. In fact, properly speaking, they are pseudo-differential operators (~DOs; cf. also 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 ) , that is, in loose terms 'some analytic functions of differential operators' (such as x/1 + D or log(1 + D), but not logD). This is explained in detail in [1]. Important as the concept of determinant of a differential or pseudo-differential operator may be for theoretical physicists, it is surprising that this seems not to be a subject of study among function analysts or mathematicians in general. The question of regularizing infinite determinants was already addressed by K. Weierstrass in a way that, although it has been pursued by some theoretical physicists with success, is not without problems (as a general method) since it ordinarily leads to non-local contributions that cannot be given a physical meaning in quantum field theory. One should mention, for completion, that there are, since long ago, well-established theories of determinants for degenerate operators, for trace-class operators in a Hilbert space, for Fredholm operators, etc. but, again, these definitions of determinant do not fulfil all the needs mentioned above which arise in quantum field theory. Take the determinant of the Dirac operator. It is defined roughly as det D = ]--[ Ai, i

where the infinite product is regularized with (for example) zeta-function or Pauli-Villars regularization (cf. also Z e t a - f u n c t i o n m e t h o d for r e g u l a r i z a t i o n ) . The zeta-function definition of the determinant det; 29 = exp [ - ~ ( 0 ) ] , is possibly the one that has more firm mathematical grounds [3], [4]. In spite of starting from the identity log det = tr log, it is known to develop a multiplicative anomaly: the determinant of the product of two operators is not equal, in general, to the product of the determinants (even if the operators commute!). This happens already with very simple operators (as two onedimensional harmonic oscillators only differing in a constant term, Laplacians plus different mass terms, etc.) [2]. Given pseudo-differential operators A, B and AB, even if their zeta-functions ~A, @ and ~AB exist it turns

out that, in general, d e t i ( A B ) ¢ detg Adet¢ B. The non-commutative anomaly is defined as:

[ det(Am ] :

5(A, B) = In [det¢ A det¢ BJ

--¢AB(0)+¢A(0)+¢b(0)'

The Wodzicki formula for the multiplicative anomaly is very useful. It reads [2]: res {[ln or(A, B)] 2 }

5(A,B) = 2 o r d A o r d B ( o r d A + o r d B ) ' a(A,B) := A°raBB -°rdA, in terms of the W o d z i e k i r e s i d u e . References

[1] EL:ZALDE, E.: 'Zeta functions: formulas and applications', in H.M. SRIVASTAVA(ed.): Higher Transcendental Functions and their Applications, 2000, Special Issue J. Computational Applied Math. 118 (2000), 125. [2] ELIZALDE, E., VANZO, L., AND ZERBINI, S.: 'Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue', Commun. Math. Phys. 194 (1998), 613-630. [3] RAY, D.B., AND SIr~GER, I.M.: 'Zeta function regularization, the multiplicative anomaly and the Wodzicki residu', Adv. Math. 7 (1971), 145. [4] RAY, D.B., AND SINGER, I.M.: 'Analytic torsion for complex manifolds', Ann. of Math. 98 (1973), 154-177.

E. Elizalde MSC 1991:81T50 NORMAL

BASIS

THEOREM

- Let E be a (finite-

dimensional) G a l o i s e x t e n s i o n of a field F. Then there exists a normal basis for E / F , that is, a basis consisting of an orbit of the G a l o i s g r o u p G -- Gal(E/F). Thus, an element z E E generates a normal basis if and only if its conjugates z s, a E G, are linearly independent over F; see, e.g., [3]. The element z is called a normal basis generator or a free element in E / F . A far-reaching strengthening of the normal basis theorem is due to D. Blessenohl and K. Johnsen [1]: There exists an element w C E that is simultaneously free in E / K for every intermediate field K . Such an element is called completely free (or completely normal). For the important special case where E is a G a l o i s field, a constructive treatment of normal bases and completely free elements can be found in [2]. References [1] BLESSENOHL, D., AND JOHNSEN, K.: 'Eine Verschgrfung des Satzes yon der Normalbasis', J. Algebra 103 (1986), 141-159. [2] HACHENBERGER, D.: Finite fields: Normal bases and completely free elements, Kluwer Acad. Publ., 1997. [3] JACOBSON, N.: Basic algebra, second ed., Vol. I, Freeman, 1985.

Dieter Jungnickel MSC 1991:11R32

283

O R 3 be a bounded domain with boundary S. A large amount of literature, going back to the mid-1930s, deals with wave scattering by obstacles when S is smooth, for example, a C1A-surface, 0 < ~ _< 1. (See [3], [14], [2].) The obstacle scattering problem consists of finding the solution to the equation OBSTACLE

SCATTERING

(V 2 + k 2 ) u = 0

eikc~'x + v,

Let

inD':=Ra\D, Pu=0

n

-

lim f ,

D

C

k>0,

onS,

Ov - ikv 2

r :=

[xl --+ o o ,

A(a', a, k) - A(a, a', k) 2i = 4-7

(2) = 0. (3)

(4)

X o~l :~- - . r

The function A(a', a, k) is called the scattering amplitude. This function has the following properties [14]: i) realness: A(a', a , - k ) = A(a', a , - k ) , k > 0, the bar stands for complex conjugation;

2

f(a',/3, k)f(c~, fl, k) d/3,

and its consequence, which is called the optical theorem:

k £ 2 ]f(a'Z'k)le

(1)

Here r u = u ( the Dirichlet condition), or Pu = UN (the Neumann condition) or Fu = uN+h(s)u (the Robin condition), where N is the unit normal to S pointing into D' and h(s) is a c o n t i n u o u s f u n c t i o n (cf. also Dirichlet boundary conditions; Neumann bounda r y c o n d i t i o n s ) . Condition (3) is the radiation condition, which selects a unique solution to problem (1)-(3). In (3), a C 5'2 is a given unit vector, the direction of the incident plane w a v e e ika'x, and k > 0 is the wave number. The scattering problem (1)-(3) has a solution and the solution is unique. This basic result was proved originally by the integral equations method [3]. There are many different types of integral equations which allow one to study problem (1)-(3) (see [14], where most of these equations are derived). The scattering field v in (3) has the following asymptotics:

v(x'a'k): eikrA(ce"a'k)+°(1)

ii) reciprocity: A(a', a, k) = A ( - a , - a ' , k); iii) unitarity:

Imd(a,a,k) = ~

d/3 . -

k~(a) 4~'

where a(a) := fs2 If(a,/3, k)l 2dfl is called the cross-

section. The function A(a',a, k) is analytic with respect to k in C+ := {k: I m k > 0} and meromorphic in C; it is analytic with respect to a' and a on the variety := {e: 0 e c 3, 0 . e = 1}, where 0 . w := E

=I 05w ,

see [14], [16] (ef. also A n a l y t i c f u n c t i o n ; M e r o m o r phic function). Necessary and sufficient conditions for a scatterer to be spherically symmetric is: A(a', a, k) = A ( a ' . a, k), where a ' . a is the dot product [16], [15]. The solution u(x, a, k) to (1) (3) is called the scattering solution. Any f(x) C L2(D ') can be expanded with respect to scattering solutions:

f(x) - (2~r)3/2

3 f(g)u(x,g) dG

1/o

f({) - (2rr)3/~

~ := ka,

, f(x)u(x, ~) dx := S f .

The operator 5c: L2(D ') --+ L2(R 3) is unitary: ][Yf[[L2(Ra ) = Hf[IL2(D,), F* = 5 -1, see [14] (cf. also Unitary operator). The above results hold in R n with odd n. In R '~ with even n the scattering amplitude A(a',a, k) as a function of complex k has a l o g a r i t h m i c b r a n c h p o i n t at k=0. The scattering problem with minimal assumptions on the smoothness of the boundary S is studied in [23]. If Pu = u, then existence and uniqueness of the scattering

OBSTACLE S C A T T E R I N G solution have been proved without any assumption on the smoothness of the boundary S of a bounded domain D. In this case a weak formulation of problem (1)-(3) is considered and the l i m i t - a b s o r p t i o n p r i n c i p l e has been proved. If Fu = u s , then again a weak formulation of (1)-(3) is considered and the only assumption on the smoothness of the boundary S is compactness of the embedding H I ( D ~ ) into L2rD'tnJ, ~ where D~ := D ~ N BR and BR = {x: Ix[ _< R} is a ball which contains D. Existence and uniqueness of the scattering solution have been proved and the limiting-absorption principle has been established. Finally, if Fu = UN + hu, then the same results are obtained under the assumptions of compactness of the embeddings i1: H I ( D ~ ) --+ L2(D~) and ~ -+ L2(S), where S is equipped with the is: H 1 (DR) (n - 1)-dimensional H a u s d o r f f m e a s u r e and H1 (DR) is the Sobolev space. For example, the embedding il is compact for Cdomains, that is, domains whose boundary can be covered by finitely many sets open in R 3 and on each of these sets the equation of S in a local coordinate system can be written as x3 = f ( x ' ) , x' = ( X l , x 2 ) , where f ( x ' ) is a c o n t i n u o u s

function.

The scattering problem for one obstacle, small in comparison with the wavelength (ka << 1, where a is the diameter of the small obstacle), for many such bodies (the number J of bodies of order 20), and in a medium consisting of many such bodies randomly distributed in the space, has been studied in [11] and later in [12]. Formulas for the scattering amplitude for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes are derived in [12]. These formulas for acoustic wave scattering on a single small body, containing the origin, are

A(a~'a'k)-

C 47r'

ifFu=u,

ka << l,

where C is the electrical capacitance of the body D;

A(a', a, k) ..~

k2V ~ ("1 "t-/3pqOZqO~p) ifFu ~-- UN, ka << 1,

where over the repeated indices one sums up from 1 to 3, v is the volume of D and ~pq =/3qp is the magnetic polarizability tensor;

A(a', a, k) if Pu = UN + hu,

hlSr 4~-(1 + hlSIC -1) ka << l,

h=const,

where IS[ is the area of the boundary S.

The S-matrix for electromagnetic wave scattering by a small homogeneous body of arbitrary shape is:

47c

CD

'

A = #0/311 + c~22cos 0 - a32 sin 0, B = a : l cos 0 - a31 sin 0 - #0/312, C = a12 - #0/321 cos0 + po/3al sin0, D = a l l + #0/322 cos0 - #0/332 sin0, where/3ij and aij are tensors of magnetic and electric polarizability and 0 is the angle between the direction e3 of the incident field and the direction of the scattered field; #0 is the magnetic permeability of the medium in which the small body is embedded (cf. also S c a t t e r i n g matrix). Formulas for the tensors aij and /3ij are derived in [11] and [12]. One can derive an equation for the average field in the medium which consists of many small obstacles (particles) randomly distributed in a region. This done in [12] and [5]. Scattering by random surfaces is studied in [1]. I n v e r s e o b s t a c l e s c a t t e r i n g . Three inverse obstacle scattering problems are of interest: ISP1) Given A(a~,ao,k) for all a r E S 2 and all k > 0, a0 E S 2 being fixed, find S and the boundary condition on S.

ISP2) Given A ( a ~, a, ko) for all a ~, a C S 2, k0 > 0 being fixed, find S and the boundary condition on S. ISP3) Given A(a~,ao,ko) for all a ~ C S2,ao E S 2 and k0 > 0 being fixed, find S. Uniqueness of the solution to ISP1) (for Fu = 0) was first proved by M. Schiffer (1964), whose argument is given in [14]. Uniqueness of the solution to ISP2) was first proved by A.G. Ramm (1985) and his proof is given in [14]. A uniqueness theorem for ISP3) has not yet (2000) been proved: it is an open problem to prove (or disprove the existence of) such a theorem. One can consider inverse obstacle scattering for penetrable obstacles [22]. Schiffer's proof of the uniqueness theorem is based on a result saying that the spectrum of the Dirichlet Laplacian in any bounded domain is discrete. This result follows from the compactness of the embedding i: H i ( D ) -+ L2(D) for any bounded domain (without any assumptions on the smoothness of its boundary S), H i ( D ) is the S o b o l e v s p a c e which is the closure in H i ( D ) of C ~ ( D ) . It is known [6] that i: H i ( D ) --+ L2(D) is not compact for rough domains (it is compact for Lipschitz domains, for domains satisfying the 285

OBSTACLE SCATTERING cone c o n d i t i o n , for C - d o m a i n s , a n d for E - d o m a i n s , i.e. d o m a i n s for which a b o u n d e d e x t e n s i o n o p e r a t o r f r o m H i ( D ) into H I ( R 3) exists, see [6]). T h e r e f o r e t h e s p e c t r u m of a N e u m a n n L a p l a c i a n in such a r o u g h d o m a i n for which t h e i m b e d d i n g i: H i ( D ) --+ L 2 ( D ) is n o t c o m p a c t is n o t discrete. One way o u t is given in [21] a n d a n o t h e r one in [18]. S u p p o s e t h a t A1 (cd, a , k0) a n d A 2 ( a ~, a, ko) are scatt e r i n g a m p l i t u d e s a t a fixed k = k0 > 0 for two o b s t a c l e s a n d let sup~,,~es~ lax - A2I < 5. A s s u m e t h a t t h e b o u n d a r i e s of t h e two o b s t a c l e s are C 2'x, 0 < A _< 1, t h a t is, in local c o o r d i n a t e s these b o u n d a r i e s S,~, m = 1,2, have e q u a t i o n s x3 = fm(xi,z2),

const

Ilfmllc~,~ -< c0 =

where f E C 2,x, m = 1,2,

> 0.

Let p d e n o t e t h e H a u s d o r f f distance b e t w e e n $ t a n d $2: fl = SUPxGS1 infyes2 Ix - Yl. T h e b a s i c s t a b i l i t y result [20] is: p _< Cl

,

where cl a n d c2 a r e p o s i t i v e c o n s t a n t s . In [20] a yet o p e n p r o b l e m (as of 2000) is f o r m u l a t e d : Derive an inversion f o r m u l a for finding S, given t h e d a t a A ( a ' , a ) : = A ( a ' , a, ko), Va', a • S 2. T h e existence of such a f o r m u l a is p r o v e d in [20]: if Z ( x ) : = XD(X) is t h e c h a r a c t e r i s t i c f u n c t i o n of D a n d ;~(~) is its F o u r i e r t r a n s f o r m , t h e n t h e r e exists a function v ~ ( a , 0 ) E L 2 ( S 2) such t h a t ~(~) = i

lim

A(O', a)v~ (a, 0) da,

where 0, 0 ~ E M , 0 ~ - 0 = ~, ~ E R a is an a r b i t r a r y vector. A f o r m u l a for c a l c u l a t i n g A(O~,a), 0 ~ E M , given A ( a ' , a ) , Ya', a E S 2, is d e r i v e d in [20]. T h e p r o b l e m is to c o n s t r u c t v~(a, O) from t h e d a t a A ( a ~, a) a l g o r i t h mically. For inverse p o t e n t i a l s c a t t e r i n g this is done in [17]. References [1] BASS, F., AND FUKS, I.: Wave scattering from statistically rough surfaces, Pergamon, 1979. [2] COLTON, D., AND KRESS, R.: Integral equations methods in scattering theory, Wiley, 1983. [3] KUPRADZE, V.: Bandwertaufgaben der Schwingungstheorie und Integralgleichungen, DVW, 1956. [4] LEIS, R.: Initial boundary value problems in mathematical physics, New York, 1986. [5] MARCHENKO, V., AND KHRUSLOV, E.: Boundary value problems in domains with granulated boundary, Nauk. Dumka, Kiev, 1974. (In Russian.) [6] MAZ'JA, V.: Sobolev spaces, Springer, 1985. [7] RAMM, A.G.: 'Spectral properties of the Schroedinger operator in some domains with infinite boundaries', Soviet Math. Dokl. 152 (1963), 282-285. [8] RAMM, A.G.: 'Reconstruction of the domain shape from the scattering amplitude', Radioteeh. i Electron. 11 (1965), 2068 2070. 286

[9] RAMM, A.G.: 'Approximate formulas for tensor polarizability and capacitance of bodies of arbitrary shape and its applications', Soviet Phys. Dokl. 195 (1970), 1303-1306. [10] R.AMM,A.G.: 'Electromagnetic wave scattering by small bodies of an arbitrary shape', in V. VARADAN(ed.): Acoustic, Electromagnetic and Elastic Scattering: Focus on T-Matrix Approach, Pergamon, 1980, pp. 537-546. [11] RAMM, A.G.: Theory and applications of some new classes of integral equations, Springer, 1980. [12] P~AMM, A.G.: Iterative methods for calculating the static fields and wave scattering by small bodies, Springer, 1982. [13] RAMM,A.G.: 'On inverse diffraction problem', J. Math. Anal. Appl. 103 (1984), 139-147. [14] RAMM, A.G.: Scattering by obstacles, Reidel, 1986. [15] RAMM, A.G.: 'Necessary and sufficient condition for a scattering amplitude to correspond to a spherically symmetric scatterer', Appl. Math. Lett. 2 (1989), 263-265. [16] RAMM, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992. [17] RAMM, A.G.: 'Stability estimates in inverse scattering', Acta Applic. Math. 28, no. 1 (1992), 1-42. [18] RAMM, A.G.: 'New method for proving uniqueness theorems for obstacle inverse scattering problems', Appl. Math. Lett. 6, no. 6 (1993), 19-22. [19] RAMM, A.G.: 'Stability estimates for obstacle scattering', J. Math. Anal. Appl. 188, no. 3 (1994), 743-751. [20] RAMM, A.G.: 'Stability of the solution to inverse obstacle scattering problem', Y. Inverse Ill-Posed Probl. 2, no. 3 (1994), 269-275. [21] RAMM, A.G.: 'Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains', Applic. Anal. 59 (1995), 377-383. [22] RAMM, A.G., PANG, P., AND YAN, G.: 'A uniqueness result for the inverse transmission problem', Internat. J. Appl. Math. 2, no. 5 (2000), 625-634. [23] RAMM, A.G., AND SAMMARTINO,M.: 'Existence and uniqueness of the scattering solutions in the exterior of rough domains', in A.G. RAMM, P.N. SHIVAKUMAR, AND A.V. STRAUSS (eds.): Operator Theory and Its Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc., 2000, pp. 457-472. [24] URSELL, F.: 'On the exterior problems of acoustics', Proc. Cambridge Philos. Soc. 74 (1973), 117-125, See also: 84 (1978), 545-548. [25] VEKUA, I.: 'Metaharmonic functions', Trudy Tbil. Math. Inst. 12 (1943), 105-174. (In Russian.) A.G. Ramm MSC 1991:35P25

ODLYZKO

BOUNDS

-

Effective lower b o u n d s for

M ( r z , r 2 ) , t h e m i n i m a l value of t h e d i s c r i m i n a n t ]d(K)] of a l g e b r a i c n u m b e r fields K h a v i n g s i g n a t u r e @1, r2) (i.e. h a v i n g r l real a n d 2r2 n o n - r e a l conjugates), o b t a i n e d in 1976 by A . M . O d l y z k o . See also A l g e b r a i c number; Number field. T h e first such b o u n d was p r o v e d in 1891 by H. Minkowski [5], who showed

(±)2 (1)

OKUBO ALGEBRA with n = rl + 2r2. He o b t a i n e d it using methods from the g e o m e t r y o f n u m b e r s ; the same m e t h o d was used later by several a u t h o r s to improve (1) (see [6] for the strongest result o b t a i n e d in this way). In 1974, H.M. Stark ([12], [13]) observed t h a t H a d a m a r d factorization of the D e d e k i n d zetaf u n c t i o n CK (s) leads to a formula expressing log Id(K) l by the zeros of ~K (s) and the value of its logarithmic derivative at a complex n u m b e r So ~ 0, 1 with 4K(sO) ¢ O. He used this formula with a proper choice of so to deduce lower b o u n d s for M ( r l , r2) which were essentially stronger t h a n Minkowski's bound, but did not reach the bounds o b t a i n e d by geometrical methods. In 1976, Odlyzko [8] (cf. [11]) modified Stark's formula and o b t a i n e d the following i m p o r t a n t improvement of (1): M ( r l , r 2 ) 1/n >_ 60~'/n22 rz/n - e(n)

(2)

with limn-~ec e(n) = 0. In particular, one has D = lim inf M ( r l , r2) 1/'~ > 22. n--+ oo

If the extended R i e m a n n hypothesis is assumed (cf. also R i e m a n n h y p o t h e s e s ; Z e t a - f u n c t i o n ) , then the constants 60 and 22 in (2) can be replaced by 180 and 41, respectively. For small degrees the b o u n d (2) can be improved (see [3], [10]) and several exact values of M ( r l , r2) are known. On the other hand, it has been shown in [1], as a consequence of their solution of the class field tower problem (cf. also T o w e r o f fields; C l a s s field t h e o r y ) , t h a t D is finite. The best explicit u p p e r b o u n d for it, D < 92.4, is due to J. M a r t i n e t [2], who obtained this as a corollary of his constructions of infinite 2-class towers of suitable fields. For surveys of this topic, see [11], [4] and [9]. References

[9] ODLYZKO, A.: 'Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results', Sdm. de Thdorie des Nombres, Bordeaux 2 (1990), 119-141. [10] POITOU, G.: ~Sur les petits discriminants', Sdm. DelangePisot-Poitou 18, no. 6 (1976/77). [11] POlTOU, G.: 'Minoration de discriminants (d'aprds A.M. Odlyzko)': Sdm. Bourbaki (1975/76), Vol. 567 of Lecture Notes in Mathematics, Springer, 1977, pp. 136-153. [12] STARK,H.M.: 'Some effective cases of the Brauer-Siegel theorem', Invent. Math. 23 (1974), 135-152. [13] STARK,H.M.: 'The analytic theory of numbers', Bull. Amer. Math. Soc. 81 (1975), 961-972.

Wtadystaw Narkiewicz

MSC 1991:11R29

OKUBO A L G E B R A - Discovered by S. O k u b o [5] when searching for an algebraic structure to model su(3) particle physics. O k u b o looked for an algebra t h a t is 8-dimensional over the complex numbers, powerassociative and, unlike the o c t o n i o n algebra, has the L i e a l g e b r a A2 as b o t h its derivation algebra and minus algebra. His algebra provides an i m p o r t a n t example of a d i v i s i o n a l g e b r a t h a t is 8-dimensional over the real numbers with a n o r m p e r m i t t i n g composition t h a t is not alternative. For m o r e information on these algebras, their generalizations and the physics, see [2], [3], [4], [6], and [7].

Following Okubo, [6], let M be the set of all 3 x 3 traceless Hermitian matrices. T h e O k u b o algebra Ps is the v e c t o r s p a c e over the complex numbers spanned by the set M with p r o d u c t * defined by

X * Y = #XY

+ uYX

+ 1 Tr(XY),

[1] GOLOD, t~.S., AND SHAEAREVICH, I.R.: 'On the class-field tower', Izv. Akad. Nauk. SSSR 28 (1964), 261-272. (In Rus-

sian.) [2] MARTINET, J.: 'Tours de corps de classes et estimations de discriminants', Invent. Math. 44 (1978), 65-73. [3] MARTINET, J.: 'Petits discriminants', Ann. Inst. Fourier (Grenoble) 29, no. fasc.1 (1979), 159 170. [4] MARTINET, J.: 'Petits discriminants des corps de nombres': Journ. Arithm. 1980, Cambridge Univ. Press, 1982, pp. 151 193. [5] MINKOWSKI,H.: 'Th~or~mes arithmfitiques', C.R. Acad. Sci. Paris 112 (1891), 209-212. [6] MULHOLLAND,H.P.: 'On the product of n complex homogeneous linear forms', J. London Math. Soe. 35 (1960), 241250. [7] ODLYZKO,A.: 'Some analytic estimates of class numbers and discriminants', Invent. Math. 29 (1975), 275-286. [8] ODLYZKO, A.: 'Lower bounds for discriminants of number fields', Acta Arith. 29 (1976), 275-297.

where X Y denotes the usual m a t r i x p r o d u c t of X and Y, T r ( X Y ) is the trace of the m a t r i x X Y (el. also T r a c e o f a s q u a r e m a t r i x ) and the constants # and u satisfy 3#u = # + u = 1, t h a t is, # = V = (3 + i x / 3 ) / 6 . In the discussion below, /z = (3 + i v Y ) ~ 6 . The algebra Ps is not a division algebra; however, it contains a division algebra. T h e real vector space spanned by the set M is a subring Ps of Ps u n d e r the p r o d u c t , and is a division algebra over the real numbers. Both the algebras Ps and Ps are 8-dimensional over their respective fields of scalars. An explicit construction of the algebra Ps can be given in terms of the following basis of 3 x 3 traceless 287

OKUBO ALGEBRA Hermitian matrices, introduced by M. Gell-Mann [1]:

[7] OKUBO, S., AND MYUNG, H.C.: 'Some new classes of division algebras', J. Algebra 6 7 (1980), 479-490.

G.P. Wene A1 =

0 0

A3 =

-1 0 0

A5 =

A7 =

,

0 i

A2 =

,

0 0 0

,

MSC1991: 17A35, 17D25, 83C20

,

ONSAGER-MACHLUP FUNCTION, MachlupOnsager function - A function having its origin in

0 0

A4 =

°Z)

0

i

--i

0 0

0

A6 =

1

,

As =

~ 0

•

The elements ej --- v/-3Aj (j = 1 , . . . , 8 ) form an orthonormal basis; the multiplication follows from 8 /----1

The constants djkt and fjkL must satisfy 1

fjkl = --4---iTr [(AjAk - AkAj)AI] A partial tabulation of the values of djkl and f y can be found in [1]. The norm N ( X ) of X = ~ s = l X j e j is N ( X ) = E ~ = l Xy. In the case of the algebra Pss, all the Xj are real and N ( X ) = 0 if and only if X = 0. The elements

Yj = -x/3Aj

(j = 1, 2, 3),

physics and arising in a particular description of the dynamics of macroscopic systems. In this description the starting point is the calculation of a probability density for observing a complete path of a system in phase space spanned by the macroscopic variables. This approach was pioneered by L. Onsager and S. Machlup in [5], who used this to develop a theory of fluctuations in (non-) equilibrium thermodynamics. Their work was restricted to the linear Gaussian case, which was subsequently extended to non-linear equations. This probability density can be expressed, apart from a normalizing factor, by means of a functional integral over paths of the process. The corresponding integrand has the form of the Lagrangian and has been called the Onsager-Machlup function by physicists. R.L. Stratonovich [6] first calculated this OnsagerMachlup function from a probabilistic viewpoint. The idea is to fix a smooth path in the state space, form a tube of small radius around this path and calculate asymptotically the probability of the sample paths of a diffusion lying within this tube. The most general result in this direction may be found in [3]. Consider a R i e m a n n i a n m a n i f o l d M and suppose that a non-singular d i f f u s i o n p r o c e s s X(.) is generated on M by 1 A= ~A+b,

Y4 = V~A8

generate a 4-dimensional subalgebra, denoted by P4. Likewise, any non-identity element ~ will generate a 2dimensional subalgebra. In addition to the above properties, each algebra will be flexible, power associative and Lie-admissible (cf. also Lie-admissible algebra; Algebra with associative p o w e r s ) ; none of these algebras will have a unit element.

where A is the Laplace-Beltrami operator (cf. L a p l a c e B e l t r a m i e q u a t i o n ) and b is a v e c t o r field. Let P~ : = P x {a~: p(Xt(co),¢(t)) <_ e f o r every t E [0, T]}, where ¢: [0, T] -+ M, ¢(0) = x, is a smooth curve on M and p(x, y) is the Riemannian distance. For any two smooth curves ¢ and ~, lim #~ (¢)

References [1] GELL-MANN, M.: 'Symmetries of baryons and mesons', Phys. Rev. 125 (1962), 1067-1084. [2] OKUBO, S.: 'Deformation of the Lie-admissible pseudooctonion algebra into the octonion algebra', Hadronic J. 1 (1978), 1383 1431. [3] OKUBO, S.: 'A generalization of Hurwitz theorem and flexible Lie-admissible algebras', Hadronic J. 3 (1978), 1-52. [4] OKUBO, S.: 'Octonion as traceless 3 x 3 matrices via a flexible Lie-admissible algebra', Hadronic J. 1 (1978), 1432-1465. [5] OKUBO, S.: 'Pseudo-quaternion and psuedo-octonion algebras', Hadronic J. I (1978), 1250-1278. [6] OKUBO, S.: Introduction to octonion and other nonassociative algebras in physics, Cambridge Univ. Press, 1995.

288

exists and can be expressed as

exp

[/;

L(¢(s), ¢(s)) as -

/;

c(¢(s), ¢(s)) as

]

for some functional L(~, x) on the tangent bundle T M . This function L, which has the form of a L a g r a n g i a n (cf. also L a g r a n g e f u n c t i o n ) , is the 0nsager-Machlup function. A detailed calculation for L(~, x) when M = R e may be founded in [4]. The Onsager-Machlup function has recently (1990s) been applied to the non-linear filtering problem. Along

O P E R A T O R COLLIGATION with the development of the usual theory using conditional mean, various attempts have been made in the 19608 to heuristically calculate the conditional mode in the non-linear filtering problem (maximum a posteriori estimator). The mathematical difficulty arose due to the lack of a translation-invariant measure in function spaces. The first correct probabilistic interpretation of this filter has been given by A. Dembo and D. Zeitouni [2] using the Onsager-Machlup function. Maximizing this function may be interpreted as giving the most probable trajectory, which is a natural generalization of the maximum-likelihood principle in statistics (cf. also M a x i m u m - l i k e l i h o o d m e t h o d ) . An extension of this approach to the estimation (smoothing) problem for random fields (cf. also R a n d o m field) may be found in [1]. References [1] AmARA, S.I., AND BACCm, A.: 'Nonlinear s m o o t h i n g for random fields', Stochastic Processes Appl. 55 (1995), 143-158. [2] DEMBO, A., AND ZEITOUNI, O.: 'A m a x i m u m a posteriori est i m a t o r for trajectories of diffusion processes', Stochastics 20 (1987), 221-246. [3] FUJITA, T., AND KOTANL S.: ' T h e O n s a g e r - M a c h l u p function for diffusion processes', J. Math. Kyoto Univ. 22 (1982), 115130. [4] IKEDA, N., AND WATANABE, S.: Stochastic differential equations and diffusion processes, second ed., North-Holland, 1989. [5] ONSAGER~ L., AND MACHLUP, 9.: 'Fluctuations and irreversible proceses, I-IF, Phys. Ray. 91 (1953), 1505-1512; 1512-1515. [6] STaATONOWCH,R.L.: ' O n the probability functional of diffusion processes', Selected Transl. Math. Statist. Prob. 10 (1971), 273 286.

A. Bagchi MSC 1991: 82B35, 82C35 OPERATOR COLLIGATION, colligation of operators, node of operators, operator node - An aggregate of spaces and operators, to which is associated a characteristic operator-valued function. This characteristic function reflects properties of the colligation; for instance, multiplication of colligations results in multiplication of the corresponding characteristic functions. Colligations are used to build models for individual operators, but they are also employed in extension theory, interpolation theory, factorization problems, scattering theory, and system theory. They appear in different forms; the most frequently encountered forms are discussed below. Let T be a bounded linear mapping in a H i l b e r t s p a c e ~, and assume that the imaginary part I m T = ( T - T * ) / 2 i of T is represented as Im T = K J K * , where K is a bounded linear mapping from a Hilbert space into ~ and J is a bounded, self-adjoint and unitary operator in ~ (cf. also L i n e a r o p e r a t o r ; S e l f - a d j o i n t ope r a t o r ; U n i t a r y o p e r a t o r ) . The aggregate of spaces

and operators

is called an operator eolligation, and the corresponding operator-valued function We (z) acting in ~ and defined by We(z) = I - 2iK*(T - zI)-lKJ, is called the characteristic operator-valued function of the colligation @. Clearly, the characteristic function satisfies the identity J - W e ( z ) J W e (w)* =

2iK*(r

-

zI)-l(r

* -

Z--W

which exhibits many properties of W o ( z ) . The present notion of colligation is due to M.S. Brodskff and M.S. Livgic [8], while the characteristic function was introduced, in a slightly different form, by Livgic in [10]. The characteristic function is a unitary invariant of the n o n - s e l f - a d j o i n t o p e r a t o r T, and the singular points of this function coincide with the spectrum of the operator T (cf. also S p e c t r u m o f a n o p e r a t o r ) . The function We (z) is a powerful tool for the investigation of the spectral properties of T (invariant subspaces, triangular and functional models, Jordan representations, characterization of spectra, similarity), but it also plays a role in complex analysis (factorization into BlaschkePotapov factors, interpolation problems); see [6], [16]. In system theory, the characteristic function is interpreted as the operator-valued t r a n s f e r f u n c t i o n of the system (conservative in the sense that I m T = K J K * ) of the form (T- zI)x = KJ~_, ~+ = ~_ - 2iK*x, where ~_ E ~ is an input vector, W+ E ~ is an output vector and x is a state space vector in ~, so that ~+ = W e ( z ) ~ - . The function W o ( z ) is determined through the imaginary part of the non-self-adjoint operator T, and therefore this function is responsible for the spectral analysis of operators 'close' to bounded selfadjoint operators. In the case where the operator T is unbounded, the Hilbert space ~ is assumed to be rigged, i.e. there exists a triplet of Hilbert spaces ~ + C ~) C Y)- (cf. also R i g g e d H i l b e r t space). Let A be a bounded linear operator from ~ + into ~ _ such that T C A and T* C A*. In this case, assume that I m A = K J K * , where K is a bounded linear operator from a Hilbert space ~ into J)_ and J is a bounded, self-adjoint, and unitary operator in ~. Now the aggregate of spaces and operators

is called a rigged operator colligation and the corresponding characteristic function W e ( z ) , acting in ~, is 289

OPERATOR COLLIGATION defined by We(z) = I - 2iK*(A - zI)-lKJ.

In this triplet setting, the system ( A - z I ) x = K J ~ _ and ~+ = ~ _ - 2 i K * x (conservative in the sense that I m , 4 = K J K * ) with an input vector ~ _ 6 ~, an output vector ~+ E ff and a state space vector x E -9+, leads to ~+ = W o ( z ) ~ - . Clearly, the definition for the unbounded case preserves the algebraic formalism developed for the bounded case. Rigged operator colligations and their characteristic operator-valued functions have been introduced by E.R. Tsekanovskff and Yu.L. Shmulyan [19]. These types of colligations (with bounded or unbounded operator T) appear for instance in the theory of circuits, systems with distributed parameters and in scattering theory, see [11], [13]. Different definitions of characteristic functions (without colligations) for unbounded non-self-adjoint operators have been introduced and studied in [9], [15], [18]. The above colligations are associated to operators T which are 'close' to being self-adjoint. Similar notions can be developed for operators which are 'close' to being unitary. Let T be a bounded operator in a Hilbert space .9. Now, let ~' and ~5 be Hilbert spaces, and let F : ~ ~ -9, G: .9 ~ (9 and H : ~ ~ O be bounded linear operators. The aggregate of spaces and operators A = (~j,~,®;T,F,G,H)

is called a unitary colligation if the operator matrix

their characteristic functions, see [7]. The colligation A is called isometric, or co-isometric, if the operator matrix U is isometric, respectively co-isometric. Unitary, isometric, or co-isometric colligations and their characteristic functions are being used in model theory [5], [14], and interpolation problems in Schur classes (cf. S e h u r f u n c t i o n s in c o m p l e x f u n c t i o n t h e o r y ) ; they also appear in extension theory, scattering theory and syst e m theory [2]. For instance, a discrete-time system of the form X n + l = T x n -[- F U n ,

vn : G x n + H u n ,

with x,~ E ~, un E ~ and vn C (9, n E N, is called conservative if the corresponding operator colligation A = (-9,5, ~5; T, F, G, H ) is unitary. Solving this system by means o f x n = x / z ~, un = u / z ~ and v~ = v / z ~, with x E ~), u C ;~, v E qh, and z E D, leads to v = Ozx(z)u. In all types of colligations questions arise concerning the dimension of the state space ~3 and the minimality of the representation. Likewise, there are various constructions to build state spaces by means of characteristic functions (model theory). Operator colligations have been considered in the setting of Banach spaces [4] and in the setting of indefinite inner product spaces (see for instance [1] for Pontryagin spaces (cf. also P o n t r y a g i n s p a c e ) , and [1, p. 205] for references for the case of KreYn spaces, cf. also K r e ~ n s p a c e ) . Recently (1999) there is interest in colligations associated with commuting operators [12] and in colligations with several variables [3]. References

is a unitary operator from .9 ® ~ onto .9 ® ~5. The corresponding characteristic operator-valued function ®zx (z), acting from ~ to ~5, is defined on the unit disc D by ®A(z) = g + zG(I-

zT)-IF,

whose values are contractive inside D and unitary on the boundary of D, as can be seen from identities such as z

-

=

a(I

-

-

z~ The operator T is called the main operator of the colligation A and it is a c o n t r a c t i o n . Conversely, any contraction T in -9 is the main operator of some unitary colligation, since the operator matrix 1 -

T

(I-TT*)U2~

(I - T ' T ) 1/2

T*

J

is unitary. The spectral theory of operators 'close' to unitary operators has been developed by B. Sz.-Nagy and C. Foia~ [17] through the characteristic functions of these augmented contractions. Later the theory was reformulated in terms of unitary operator colligations and 290

[1] ALPAY, D., DIJKSMA, A., ROVNYAK, J., AND SNOO, H.S.V. DE: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Vol. 96 of Oper. Th. Adv. Appl., Birkh~user, 1997. [2] AROV, D.Z., AND GROSSMAN, L.Z.: 'Scattering matrices in the theory of extensions of isometric operators', Math. Nachr. 157 (1992), 105-123. [3] BALL, J.A., AND TRENT, T.T.: 'Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables', J. Funct. Anal. 157 (1998), 1-61. [4] BAaT, H., GOHBEae,I., AND KAASHOEK, M.A.: Minimal fae-

torization of matrix and operator functions, Vol. 1 of Oper. Th. Adv. Appl., Birkh~iuser, 1979. [5] BaANGES,L. DE, AND ROVNYaK,J.: Square summable power series, Holt, Rinehart &: Winston, 1966. [6] BRODSKI]', M.S.: Triangular and Jordan representations of linear operators, Vol. 32 of Transl. Math. Monographs, Amer. Math. Soc., 1971. (Translated from the Russian.) [7] BRODSKff, M.S.: 'Unitary operator colligations and their characteristic functions', Russian Math. Surveys 33, no. 4 (1978), 159-191. ( Uspekhi Mat. g a u l 33, no. 4 (202) (1978), 141-1680 [8] BRODSKII, M.S., AND LIVSIC, M.S.: 'Spectral analysis of nonselfadjoint operators and intermediate systems', Amer. Math. Soc. Transl. 13, no. 2 (1960), 265-346. (Uspekhi Mat. Nauk. 13, no. 1 (79) (1958), 3-85.)

O P E R A T O R VESSEL [9] KUZHEL, A.: Characteristic functions and models of non-

selfadjoint operators, Kluwer Acad. Publ., 1996. [10] LIV~IC, M.S.: 'On the spectral decomposition of linear nonselfadjoint operators', Amer. Math. Soc. Transl. 5, no. 2 (1957), 67-114. (Mat. Sb. 34, no. 76 (1954), 145-199.) [11] LIV~IC, M.S.: Operators, oscillations, waves, Vol. 34 of Transl. Math. Monographs, Amer. Math. Soc., 1973. (Translated from the Russian.) [12] LivsIc, M.S., KRAVITSKY,N., MARKUS, A.S., AND VINNIKOV, V.: Theory of commuting nonselfadjoint operators, Kluwer

Acad. Publ., 1995. [13] L~v~c, M.S., AND YANTSEVICH, A.A.: Operator colligations in Hilbert spaces, Winston, 1979. (Translated from the Russian.) [14] NIKOLSKI, N., AND VASYUNIN, V.: 'Elements of spectral theory in terms of the free function model L Basic constructions': Holomorphic spaces (Berkeley, CA, 1995), Cambridge Univ. Press, 1998, pp. 211-302. [15] PAVLOV, B.S.: 'Spectral analysis of a singular SchrSdinger operator in terms of a functional model': Partial Differential Equations VIII, Springer, 1995, pp. 89-153.

[16] SAKHNOVICH,L.A.: Interpolation theory and its applications,

Kluwer Acad. Publ., 1997. [17] Sz.-NAGY, B., AND FOIA~, C.: Harmonic analysis of operators on Hilbert space, North-Holland, 1970. [18] ST.aAUSS,A.V.: 'Characteristic functions of linear operators', Amer. Math. Soc. Transl. 40, no. 2 (1964), 1-37. (Izv. Akad. Nauk. SSSR Ser. Mat. 24 (1960), 43-74.) [19] TSnKANOVSKff,E.R., AND SHMULYAN,Yu.L.: 'The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions', Russian Math. Surveys 32, no. 5 (1977), 73-131. (Uspekhi Mat. Nauk. 32, no. 5 (1977), 69-124.) Henk de Snoo Eduard Tsekanovski~ MSC 1991: 30E05, 47A48, 47Bxx, 47A57, 47A65, 47N50, 47N70 OPERATOR VESSEL - The theory of operator vessels provides a framework for the s p e c t r a l a n a l y s i s and synthesis of tuples of commuting non-self-adjoint (or non-unitary) operators, especially for operators that are not 'too far' from being self-adjoint (or unitary). It reveals deep connections with a l g e b r a i c g e o m e t r y , especially with function theory on a compact real Riemann surface (i.e., a compact Riemann surface endowed with an anti-holomorphic involution, cf. also R i e m a n n s u r f a c e ) , and with the theory of multi-dimensional systems. The theory of operator vessels can be also generalized for the study of tuples of non-commuting nonself-adjoint (or non-unitary) operators (cf. also N o n s e l f - a d j o i n t o p e r a t o r ) , especially for non-self-adjoint representations of Lie algebras (or non-unitary representations of discrete groups). Let A1, A2 be a pair of commuting bounded linear operators in a H i l b e r t s p a c e 7-/. A quasi-Hermitian commutative two-operator vessel ill is a collection

i f / = (A1, A2,7/, (I), $, a~, ae, %~).

(1)

Here, £ is an auxiliary Hilbert space called the external space of the vessel (7-/is called the internal space), (I): 7-/ --~ $ is a bounded linear mapping, al, ~2, 7, are bounded self-adjoint operators in $ (cf. also Selfa d j o i n t o p e r a t o r ) , such that 1 (A1 - A~) = ~ * a l O , _1 (A2 - A~) = (~*cr2~,

(2)

al OA~ - a2OA~ = ~ ,

(3)

al 4)A2 - a2(I)A1 = ~(I),

(4)

-

=

-

(5)

The equations (2) are so-called colligation conditions (or node conditions) well-known from the spectral analysis of a single non-self-adjoint operator (cf. also O p e r a t o r c o l l i g a t i o n ) ; they allow one to 'isolate' the nonHermitian parts of the operators A1 and A2. The equations (3), (4) and (5) are deeper; the self-adjoint operators 7 and ~ carry information about the interaction of A1 and A2. Notice that a given pair A1, A2 of commuting operators in 7-/can be always embedded in a quasi-Hermitian commutative vessel by setting E=(A1-A~)7/+(A2-A~)7/, ~ 1 = ~ ( A 1 - A~) E, 7 = 1 ( A I A ~ - 2 AA~ ) * e '

~=PE,

or2= ~(A2 - A~) E, ~= ~ (A~A1-A~A2) e

(Pc denotes the orthogonal projection on the subspace £; it is an easy consequence of the commutativity of A1 and A2 that g is invariant under A1A~ - A2A~ and A~A1 - A~ A2). The notion of an operator vessel allows one to construct commuting operators with more complicated spectral data out of commuting operators with simpler spectral data while controlling the non-Hermitian parts. Let ~ t ~ ( A t1, A~, 7/', ~t, ~, o'1, o'2, "/', ~'),

~ , = ( A 1it , A 2, , 7 / , , ~ , , £, al, o-2,71', ~'1) be two quasi-Hermitian commutative vessels with the same (g, oh, ~r2). Using the coupling construction familiar in the spectral analysis of a single non-self-adjoint operator, one sets 7i = 7i' ® 7-/" and defines operators A1, A2 : 7 / ~ 7'/t and • : 7-/--+ g by

\i(I)"ak ~'

(the operators being written in block form with respect to the direct sum decomposition 7-/ = 7i I • 7/"). It is 291

O P E R A T O R VESSEL clear that A1, A2 satisfy the colligation conditions:

l(&

A~)

¢,'0.k,I,,

but in general they do not commute. It turns out that A l A s = A2Ax exactly when 7' = 7"; more precisely,

= (Ax,As,?-l,i~,g,0.1,0.2,7,~/) (with 7 = 7' and ~ = ~") is a quasi-Hermitian commutative vessel if and only if ~' = 7" (the matching theorem). Assume now that the external space g of a vessel as in (1) is finite dimensional; one defines a polynomial in two complex variables A1, A 2 by setting p(A1, A2) : det(Al0.2 - A20.1 -[- "~).

(7)

It is assumed that det({10.1 + {20.2) ~ 0, so that p(A1,A2) ~ 0 and p(A1,As) is a polynomial with real coefficients of degree M = dimg. One calls p(A1,As) the discriminant polynomial of the vessel ~ , and the projective p l a n e r e a l a l g e b r a i c c u r v e with an a n n e equation p(A1, As) = 0 - - the discriminant curve. A generalized Cayley-Hamilton theorem holds: p(A1,As) = 0 (under the natural minimality assumption H = V kcc l , k 2 = 0 ~Ak~ l ak2 ~2 ~ . g ) . It follows that the joint spectrum of A1, As (cf. also T a y l o r j o i n t s p e c t r u m ) lies on the (aNne part of the) discriminant curve. The following remarkable equality also holds: det(A10.2 - A20.1 ~- "}/) : det(A10.2 - A2Ol "Jr-~/).

(8)

Thus, associated to the vessel ~ there is the discriminant polynomial p(A1, A2), and two self-adjoint determinantal represent ations of it, A10.2- A20.1 + 3' and A10.2- A20.1 + 7, called (for system-theoretic reasons, see below) the input and the output determinantal representations, respectively. Consider now the inverse problem of constructing, up to unitary equivalence, all (minimal) quasi-Hermitian commutative two-operator vessels with given discriminant polynomial p(A1, A2), given input determinantal r e p r e s e n t a t i o n s A10. 2 - A20.1 -~- 7, and with the operators A1, As in the vessel having given joint spectrum ®, which is a subset of the a n n e part of the real projective plane curve C defined by p(A1,A2). Here, two vessels ~(0 = (A~0, A~t), ~ ( 0 , ¢(0, g, 0.1,02, ~/, 7) (l ----- 1, 2) are said to be unitarily equivalent if there is an i s o m e t r i c m a p p i n g U from ~(1) onto ~(2) such that d(k s) = U A ~ I ) u - 1

( k ~- 1,2),

(9)

~5(s) = ~0)U. If p(A1, As) is an i r r e d u c i b l e p o l y n o m i a l and C is a smooth irreducible curve (these assumptions can be relaxed), a complete and explicit solution of the inverse problem stated above has been obtained. This solution 292

leads to triangular models for the corresponding pair of operators A1, A2 with finite non-Hermitian ranks, similar to the well-known triangular models for a single non-self-adjoint operator. The solution is based on first constructing elementary objects - - vessels with onedimensional internal space corresponding to the points of the joint spectrum - - and then coupling them using the matching theorem. It follows from (5) that in a vessel with one-dimensional internal space the output determinantal representation is determined by the input determinantal representation and the spectral data; the successive matching of output and input determinantal representations in the matching theorem then gives a system of non-linear difference (for the discrete part of the spectrum) and differential (for the continuous part of the spectrum) equations for self-adjoint determinantal representations of the polynomial p(A1, A2). The algebro-geometrie assumptions on the polynomial p(A1, A2) and the curve C imply that self-adjoint determinantal representations are naturally parametrized by certain points in the J a c o b i v a r i e t y of C; and it turns out that passing from a self-adjoint determinantal representation to the corresponding point in the Jacobi variety linearizes the systems of non-linear difference and differential equations alluded to above. Actually, the system can even be solved explicitly using theta-functions, yielding explicit formulas for the operators A1, A2 in a triangular model. The fundamental interplay between the spectral theory of a pair of commuting non-self-adjoint operators with finite non-Hermitian ranks and function theory on a compact real R i e m a n n s u r f a c e is based on the notion of the joint characteristic function. Let ~ be a quasiHermitian commutative vessel as in (1), with discriminant polynomial p(A1, A2) and discriminant curve C. For each a n n e point A = (A1, A2) on C, one may define two non-trivial subspaces of the external space g: e(A) : ker(Ax0.2 - A20.1 + 7),

(10)

~(A) = ker(A10.2 - A20.1 + 7).

(11)

Then for arbitrary complex numbers (1, (2 (such that (1A1 -~-(2A2is outside the spectrum of (1A1 + (2A2), the operator on g,

Ic - iO(~lA1 + ~sA2 - ~1A1 --

(2),s) --1(1)*((1 (7i + (20.2),

maps ~(A) into ~(A) and the restriction of this operator to ~(A) is independent of (1, ~2. The joint characteristic function of the vessel gJ is defined by S(A) = Ig - i~(~IA1 + ~2As - ~tA1 - ~sAs) -1 '

(12)

• 1I)*(~10.1 -~- ~20"2)]~(A) : e(A) -"} ~(A), where A = (A1, A2) is an affine point on C outside the joint spectrum of A1, As.

O P E R A T O R VESSEL The joint characteristic function is thus a mapping of certain sheaves on the discriminant curve (cf. also S h e a f ) . For simplicity, assume that the discriminant polynomial P()~I, A2) has only one, possibly multiple, irreducible factor; thus p(A1, Ae) = (f(A1, Ae)) ~ for some r > 1, where f(A1, Ae) = 0 is the irreducible affine equation of the discriminant curve C. Assume also that both the input and the output determinantal representations ofp(A1, Ae) are maximal, meaning that for every point # on C the subspaces ~(#) and ~(#) have maximal possible dimension (which is equal to r times the multiplicity of # on C; notice that all these assumptions are trivially satisfied when the discriminant polynomial is irreducible, i.e., r = 1, and C is a smooth irreducible curve). It follows then that the subspaces ~(#) and ~(#) for different points # on C (including, of course, the points at infinity) fit together to form two complex holomorphic rank-r vector bundles ~ and ~ on a compact Riemann surface X which is the desingularization of C (cf. also Resolution of singularities). The joint characteristic function S: ~ -+ ~ (naturally extended to be identity at the points of C at infinity) is simply a bundle mapping, holomorphic outside the joint spectrum of At, A2. Notice that since C is a real curve, X is a real Riemann surface, that is, a Riemann surface equipped with an anti-holomorphic involution (the complex conjugation

on C). Assuming the maximality of the input and the output determinantal representations, the joint characteristic function of a (minimal) vessel determines the vessel uniquely up to unitary equivalence. The joint characteristic function is expansive with respect to certain naturally defined scalar products on the vector bundles and ~. Conversely, given any bundle mapping between the kernel vector bundles corresponding to the given two maximal self-adjoint determinantal representations, which is expansive with respect to the corresponding scalar products, this bundle mapping can be realized as the joint characteristic function of a quasi-Hermitian commutative vessel with these input and output determinantal representations. Kernel vector bundles corresponding to maximal selfadjoint determinantal representations are isomorphic (up to an inessential twist) to vector bundles of multiplicative half-order differentials, i.e., to vector bundles of the form Vx ® A; here A ® A ~ K x , the canonical line bundle (the line bundle of holomorphic differentials), and Vx is a flat vector bundle associated to some representation X of the fundamental group of X. Using this isomorphism one may replace the joint characteristic function by the so-called normalized joint characteristic function, which is simply a mapping of flat vector bundles on X, i.e., a multiplicative multi-valued matrix

function on X (with appropriate matrix multipliers on the left and on the right). The normalized joint characteristic function is usually more convenient for analytic investigations. There are also functional models for the corresponding pair of operators A1, A2 with finite nonHermitian ranks, similar to the well-known functional models of Sz.-Nagy-Foias and de Branges-Rovnyak for a single operator; the model space is an appropriately defined space of multiplicative half-order differentials on X, and the model operators are certain 'compressed multiplication operators' by the affine coordinate functions A1, )~2. Like the notion of colligation in the spectral theory of a single non-self-adjoint operator (cf. also O p e r a t o r c o l l i g a t l o n ) , the notion of a vessel has a systemtheoretic significance. Given a quasi-Hermitian commutative two-operator vessel ~3 as in (1), one writes a linear shift-invariant continuous two-dimensional system • Of

~-5~1 + A l l = ~*~1~,

(13)

• Of z-~2 + A 2 f = ~*aeu,

(14)

v = u - i~f.

(15)

Here, f = f ( t l , t e ) is the state with values in the internal space 7{, u = u(tl, te) and v = v(tl, re) are, respectively, the input and the output with values in the external space E, and (tl, te) E R e. The colligation conditions (2) imply that the system (13)-(15) satisfies the

energy balance law:

(o ~1~

o)

+ ~e ~

(f, I)~ =

(16)

for any direction (~1, ~2) in R e. Unlike the usual onedimensional systems, the system (13)-(15) is overdetermined (cf. also O v e r d e t e r m i n e d s y s t e m ) , the compatibility conditions arising from the equality of mixed partial derivatives:

Oe f oe f Oh Ote OteOtl The commutativity AIA2 = A2A1 means precisely that the system is consistent for an arbitrary initial state f ( 0 , 0 ) and the identically zero input. The vessel condition (3) implies that a sufficient (and under some assumptions also a necessary) condition for the input signal to be compatible is given by or2

-a1~+7

u=0.

(17)

The vessel conditions (4), (5) imply that the corresponding output satisfies

~e

- ~ + ~

~=0.

(is) 293

O P E R A T O R VESSEL The joint characteristic function of the vessel ~ is the so-called joint t r a n s f e r f u n c t i o n of the overdetermined system (13) (15) together with the compatibility partial differential equations (17) and (18) at the input and at the output, respectively. The notion of a quasi-Hermitian commutative twooperator vessel is the simplest and the best studied; it can be successfully generalized in various directions, like: 1) Quasi-Hermitian commutative d-operator vessels for any d, which give a framework for the spectral analysis of d-tuples of commuting non-self-adjoint operators (especially with finite non-Hermitian ranks). 2) Quasi-unitary commutative operator vessels, which give a framework for the spectral analysis of tuples of commuting non-unitary operators (especially with finite defects); they are related to discrete conservative multi-dimensional systems (rather than continuous).

3) 'Non-metric' commutative operator vessels, which correspond to overdetermined multi-dimensional systerns together with compatibility partial differential equations at the input and at the output, but without any energy balance laws. 4) Non-commutative generalizations, in particular (quasi-Hermitian) 'Lie algebra' vessels, where one replaces a tuple of commuting operators by a representation of a given Lie a l g e b r a g. Such vessels provide a framework for the spectral analysis of non-selfadjoint representations of ft. The associated (conservative) multi-dimensional system evolves on a Lie g r o u p G having the Lie algebra g. The theory of operator vessels was initiated by M.S. Liv~ic [1], [2]. The term 'vessel' was coined in the book [3]; earlier papers use the term 'regular colligation'. The book [3] provides a comprehensive treatment of the subject. A shorter survey, containing also the more recent results, is [4].

In the financial world, an option right is a right to choose between several possible trades at a time in the future that may be determined in advance or that may be subject to choice. An option is a contract in which an option right is sold. For example, consider a contract that gives the holder the right, but not the obligation, to exchange one million euros for one million American dollars at a given time T in the future. Such a contract may be useful for a European company that will have to make a payment in American dollars at a known time. The contract allows the company to choose at time T whether it will buy American dollars at the exchange rate 1 : 1 or whether it will not do so; in the latter case the company may of course still buy American dollars directly in the market. The company's decision will depend on the actual exchange rate at time T. Because this exchange rate is not known at the time the contract is entered, it is not obvious on which principle the pricing of the contract can be based. An approach to this problem, which holds for options in general, was developed by F. Black, M. Scholes and R.C. Merton in the early 1970s [3], [9] and is now generally accepted. The Black-Scholes-Merton method is based on the observation that an institution that confers an option (say on the euro-dollar exchange rate) may modify the risk involved in the option by buying and selling dollars against euros during the life-time of the contract. Under appropriate assumptions it is in fact possible to eliminate risk completely, so that there is a unique price for the option that does not depend on the risk preferences of any of the parties involved in the contract. The Black-Scholes-Merton option pricing methodology uses a fairly elaborate mathematical framework. The behaviour of the underlying variables is modelled by means of stochastic differential equations (cf. also Stoc 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 ) . In the original paper by Black and Scholes, [3], stock prices are modelled by the geometric Brownian motion OPTION

PRICING

-

References

[1] LIVgIC, M.S.: 'Operator waves in Hilbert space and related partial differential equations', Integral Eq. Oper. Th. 2, no. 1 (1979), 25 47. [2] LIv~IC, M.S.: 'A method for constructing triangular canonical models of commuting operators based on connections with algebraic curves', Integral Eq. Oper. Th. 3, no. 4 (1980), 489507. [3] LIVSIC, M.S., KRAVITSKY~ N., 1V[ARKUS~A.S.~ AND VINNIKOV~ V.: Theory of commuting nonselfadjoint operators, Kluwer Acad. Publ., 1995. [4] VmNIKOV, V.: 'Commuting operators and function theory on a Riemann surface', in S. AXLER, J. MCCARTHY, AND D. SARASON (eds.): Holomorphic Spaces and Their Operators, Vol. 33 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, 1998, pp. 445-476.

Victor Vinnikov MSC 1991: 47A48, 47A45, 47A65, 47N70, 47D40 294

dSt -~ # S t dt + aSt dwt,

where # and a are constants and wt is standard B r o w n ian m o t i o n ; later on, researchers have used a variety of other diffusion models (cf. also D i f f u s i o n e q u a tion) to describe the behaviour of financial indicators such as interest rates and exchange rates. In the general Black-Scholes-Merton framework one works with models in which there are several tradeable assets and in which a vector-valued Brownian motion enters. It is assumed that continuous trading is possible, so that portfolios may be formed of tradeable assets with continually adjusted weights (cf. also P o r t f o l i o o p t i m i z a t i o n ) . In general, the weights may follow processes that are adapted to a filtration associated with the process

ORDINARY D I F F E R E N T I A L EQUATIONS, P R O P E R T Y C F O R of the underlying variables. Weight processes are usually subjected to integrability conditions and moreover constrained to be self-financing, which means that no funds are added or withdrawn; thus, any change in value of the portfolio is due to price changes of the assets. Consider the random variables (cf. also R a n d o m variable) that arise as portfolio values at time T corresponding to such portfolio strategies that are followed during an interval [0, T] and that start from a portfolio with some given value at time O. If any random variable with finite variance can be produced in this way, then the model under consideration is said to represent a complete market. Roughly speaking, markets are complete when the number of independent tradeable assets is at least one larger than the dimension of the vector of Brownian motions appearing in the model. In particular, in a complete market any option can be replicated, that is, reproduced by a suitable trading strategy. Under the assumption that the given model allows no arbitrage (i.e. no riskless profits), there can be only one initial portfolio value corresponding to a replicating portfolio for a given option. Again under the no-arbitrage assumption, this must then be the price of the option at time O. A powerful tool in the pricing of options is the replacement of the probability measure in the given model by an equivalent martingale measure under which all price processes, after discounting, are martingales (cf. also Martingale). Under suitable hypotheses it can be shown that absence of arbitrage implies the existence of an equivalent martingale measure, and that at most one equivalent martingale measure can exist in a complete market. If a unique equivalent martingale measure exists, the price of an option can be computed as the expected value (with respect to this measure) of its discounted pay-off. The transformation to an equivalent martingale measure can often be simply achieved by a change of the drift term in the given stochastic differential equations (the Cameron-Martin-Girsanov theorem). For instance, the well-known B l a c k - S c h o l e s formula can be obtained in this way. Options that have a fixed time of expiry are called European options. In the financial markets one also trades contracts in which the holder is free to choose the time at which the option is exercised. Such contracts are called American options. Even in a complete and arbitragefree model, the pricing of such options cannot be based on an arbitrage argument alone. Usually, the price of an American option is defined by maximizing its value over all exercise strategies; the pricing problem then becomes an optimal stopping problem (cf. also S t o p p i n g t i m e ) . For computational purposes, it is often useful to reformulate such problems as free boundary problems for a

related partial differential equation (cf. also Differen-

tial equation, partial, free boundaries). More information about option pricing can be found in, for instance, [1], [2], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14].

References [1] BINOHAM,N.H., ANDKIESEL,R.: Risk-neutral valuation: The pricing and hedging of financial derivatives, Springer, 1998. [2] BJORK, T.: Arbitrage theory in continuous time, Oxford Univ. Press, 1998. [3] BLACK, F., AND SCHOLES,M.: 'The pricing of options and corporate liabilities', J. Political Economy 81 (1973), 637659. [4] ELLIOTT, ]~.J., AND KOPP, E.: Mathematics of financial markets, Springer, 1999. [5] KARATZAS,I., AND SHREVE, S.E.: Methods of mathematical finance, Springer, 1998. [6] KWOK, Y.-K.: Mathematical models of financial derivatives, Springer, 1997. [7] LAMBERTON,D., AND LAPEYRE, B.: Introduction to stochastic calculus applied to finance, Chapman and Hall, 1996. [8] LUENBERGER,D.G.: Investment science, Oxford Univ. Press, 1997. [9] MERTON, R.C.: 'Theory of rational option pricing', Bell J. Economics and Management Sci. 4 (1973), 141-183. [10] MUSIELA, M., AND RUTKOWSKI, M.: Martingale methods in financial modeling. Theory and applications, Springer, 1997. [11] NIELSEN, L.T.: Pricing and hedging of derivative securities, Oxford Univ. Press, 1999. [12] PLISKA, S.R.: Introduction to mathematical finance. Discrete time models, Blackwell, 1997. [13] SHmYAEV,A.N.: Essentials of stochastic finance, World Sci., 1999. [14] WILMOTT,P.: Derivatives. The theory and practice of financial engineering, Wiley, 1998.

J.M. Schumacher

M S C 1991:90A09

ORDINARY DIFFERENTIAL PROPERTY C F O R - Let lmu =

- ~ x 2 + qm(X)

m = 1,2,

EQUATIONS~

)

u,

x E R + := [O, oc),

and let q~(x) be a real-valued function, qm(X) e L I , I ( R + ) :=

{// q:

xIq(x)l dx < ec

}

.

Consider the problem - k 2)

xcR+,

= o,

f,~(x,k)=eik~ +o(1)asx ++oc.

This problem has a unique solution, which is called the Jost function. Define also the solutions to the problem (e~ - k 2) ~ ( x , k ) x e R+,

tOm(0,k) = 0,

= 0, qo'(0, k) = 1, 295

ORDINARY DIFFERENTIAL EQUATIONS, PROPERTY C FOR and to the problem (era - k 2) ~ ( x , k ) xcR+,

¢,~(0, k) = 1 ,

= 0, ¢~(0, k) = 0 .

Assume h(x) • L2(R+) and

o~h(x)fl(x,k)f2(x,k)dx=O,

Vk>0.

(1)

If (1) implies f(x) - O, then one says that the pair {~1, g2} has property C+. Let b > 0 be an arbitrary fixed number, let h(x) • L I(R+) and assume

obh(x)~l(x,k)~2(x,k)dx=O,

Vk>0.

(2)

If (2) implies h(x) - O, then one says that the pair {ll,/2} has property C~. Similarly one defines property C¢. It is proved in [3] that the pair {ll, 12} has property C+ ifqm • L1,1, m = 1,2. It is proved in [4] that the pair {ll, 12} has properties C v and C¢. However, if b = ec, then, in general, property C~ fails to hold for a pair {11,12}. This means that there exist a function h(x) ~ O, h • L I ( R + ) , and two potentials ql,q2 • LI,i, such that (1) holds for all k > 0. In [4] many applications of properties C+, C~ and C¢ to inverse problems are presented. For instance, suppose that the I-function, defined as I(k) := if(0, k ) / f ( k ) , is known for all k > 0, f(k) := f(0, k) and f(x, k) is the Jost function corresponding to a potential q(x) • LI,1. The function I(k) is known as the impedance function [1], and it can be measured in some problems of

296

electromagnetic probing of the Earth. The inverse problem (IP) is: Given I(k) for all k > 0, can one recover q(x) uniquely? This problem was solved in [1], but in [3] and [4] a new approach to this and many other inverse problems is developed. This new approach is sketched below. Suppose that there are two potentials, ql(x) and q2 (x), which generate the same data I(k). Subtract from the equation (/1- k2)fl = 0 the equation (12- k2)f2 = O, and denote fx - f 2 := f , q2 - q z := p(x), to get (11 - k 2 ) f = p f2. Multiply this equation by fl(x,k), integrate over (0, oc) and then by parts. The assumption I1 (k) - f; k (0, ) fl(k)

_ f~ (0, k) _ / 2 (k) f2(k)

implies f o p(x)fl (x, k)f2 (x, k) dx = O, Vk > O. Using property C+ one concludes p(x) =- 0, that is, ql (x) = q2(x). This is a typical scheme for proving uniqueness theorems using property C. References [1] RAMM, A.G.: 'Recovery of the potential from /-function', Math. Rept. Acad. Sci. Canada 9 (1987), 177-182. [2] RAMM, A.G.: 'Inverse scattering problem with part of the fixed-energy phase shifts', Comm. Math. Phys. 207, no. 1 (1999), 231-247. [31 RAMM, A.G.: 'Property C for ODE and applications to inverse scattering', Z. Angew. Anal. 18, no. 2 (1999), 331-348. [4] RAMM, A.G.: 'Property C for ODE and applications to inverse problems', in A.G. RAMM, P.N. SHIVAKUMAR, AND A.V. STRAUSS(eds.): Operator Theory A n d Its Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc., 2000, pp. 15 75.

A. G. Ramm MSC 1991: 34A55, 34L25

P P - P O I N T - As defined in [1], a point in a c o m p l e t e l y - r e g u l a r s p a c e X at which any p r i m e i d e a l of the ring C(X) of real-valued continuous functions is maximal (cf. also C o n t i n u o u s f u n c t i o n ; M a x i m a l ideal). A prime ideal P is 'at x' if f(x) = 0 for all f ¢ P; thus x is a P-point if and only if Mx = { f : f(x) = O} is the only prime ideal at x. Equivalent formulations are:

References

1) if f is a continuous function and f(x) = 0, then f vanishes on a neighbourhood of x; and 2) every countable intersection of neighbourhoods of x contains a neighbourhood of x.

M S C 1991:54G10

The latter is commonly used to define P-points in arbit r a r y topological spaces. Of particular interest are P-points in the space N* = f i n \ N, the remainder in the S t o n e - C e c h c o m p a c t i f i c a t i o n of the space of natural numbers. This is so because W. Rudin [2] proved that the space N* has P points if the c o n t i n u u m h y p o t h e s i s is assumed; this showed t h a t N* cannot be proved homogeneous (cf. also H o m o g e n e o u s s p a c e ) , because not every point in an infinite compact space can be a P-point. Points of N* are identified with free ultrafilters on the set N (cf. also U l t r a f i l t e r ) . A point or ultrafilter u is a P-point if and only if for every sequence (U~}~ of elements of u there is an element U of u such that U C_* U~ for all n, where A C C_* B means that A \ B is finite. Equivalently, u is a P-point if and only if for every partition {An}~ of N either there is an n such that A~ E u or there is a U E u such that U ~ An is finite for all n. S. Shelah [3] constructed a model of set theory in which N* has no P-points, thus showing that Rudin's theorem is not definitive. There is contimmd interest in P-point ultrafilters because of their combinatorial properties; e.g., u is a Ppoint if and only if for every function f : N ~ R there is an element U of u such that flU] is a converging sequence (possibly to oc or - c o ) .

[1] GILLMAN,L., AND HENRIKSEN, M.: 'Concerning rings of continuous functions', Trans. Amer. Math. Soc. 77 (1954), 340362. [2] RUDIN, W.: 'Homogeneity problems in the theory of Cech compactifications', Duke Math. J. 23 (1956), 409 419; 633. [3] WIMMERS, E.: 'The Shelah P-point independence theorem', Israel J. Math. 43 (1982), 28-48.

K.P. Hart

P-SPACE P - s p a c e in t h e s e n s e o f G i l l m a n H e n r i k s e n . A Pspace as defined in [2] is a c o m p l e t e l y - r e g u l a r s p a c e in which every point is a P - p o i n t , i.e., every fixed prime ideal in the ring C(X) of real-valued continuous functions is maximal (cf. also M a x i m a l ideal; P r i m e ideal); this is equivalent to saying that every Gb-subset is open (of. also S e t o f t y p e F~ (Gb)). The latter condition is used to define P-spaces among general topological spaces. In [6] these spaces were called Rl-additive, because countable unions of closed sets are closed. Non-Archimedean ordered fields are P-spaces, in their order topology; thus, P-spaces occur in nonstandard analysis. Another source of P-spaces is formed by the w~-metrizable spaces of [6]. If w~ is a regular cardinal number (ef. also C a r d i n a l n u m b e r ) , then an w~-metrizable space is a set X with a mapping d from X x X to the ordinal w~ + 1 that acts like a m e t r i c : d(x,y) = 0J, if and only if x = y; d(x,y) = d(y, x) a n d d(x, z) ~ 1Tlin{d(x, y), d(y, z)}; d is called an w,-metric. A topology is formed, as for a m e t ric s p a c e , using d-balls: B(x,a) = {y: d(x,y) > a}, where a < w,. The w0-metrizable spaces are exactly the strongly zero-dimensional metric spaces [3] (cf. also Z e r o - d i m e n s i o n a l s p a c e ) . If w~ is uncountable, then (X, d) is a P-space (and conversely). One also employs P-spaces in the investigation of box products (cf. also T o p o l o g i c a l p r o d u c t ) , [8]. If

P-SPACE i=1 x i is endowed with the box topola product X = I-I °° ogy, then the equivalence relation x -= y defined by {i: xi 7~ yi} is finite and defines a quotient space of X, denoted Vi=IX,, that is a P-space. The quotient mapping is open and the box product and its quotient share many properties.

P - s p a c e in t h e sense of Morita. A P-space as defined in [4] is a topological space X with the following covering property: Let f~ be a set and let { G ( a i , . . . , a n ) : a l , . . . , a n E f~} be a family of open sets (indexed by the set of finite sequences of elements of f~). Then there is a family {F(c~I,..., an) : ctl,...,an E f~} of closed sets such that F ( a i , . . . , an) C_ G ( a i , . . . , an) and whenever a seoo quence (O! i)i=l satisfies U~_IG(Ozl , . . . , a n ) = X, then also U ~ = l F ( a l , . . . , a n ) = X. K. Morita introduced Pspaces to characterize spaces whose products with all metrizable spaces are normal (cf. also N o r m a l space): A space is a normal (paracompact) P-space if and only if its product with every metrizable space is normal (paracompact, cf. also P a r a c o m p a c t space). Morita [5] conjectured that this characterization is symmetric in that a space is metrizable if and only if its product with every normal P-space is normal. K. Chiba, T.C. Przymusifiski and M.E. Rudin [1] showed that the conjecture is true if V = L, i.e. GSdel's axiom of constructibility, holds (cf. also G g d e l constructive set). These authors also showed that another conjecture of Morita is true without any extra set-theoretic axioms: If X x Y is normal for every normal space Y, then X is discrete. There is a characterization of P-spaces in terms of topological games [7]; let two players, I and II, play the following game on a topological space: player I chooses open sets U i , U 2 , . . . and player II chooses closed sets Fi, F2,..., with the proviso that F,~ C Ui
298

[7] TELGA.RSKI, R.: 'A characterization of P-spaces', Proc. Japan Acad. 51 (1975), 802-807. [8] WmLIAMS, S.W.: 'Box products', Handbook of Set Theoretic Topology, in K. KUNEN AND J.E. VAUGHAN (eds.). NorthHolland, 1984, pp. Chap. 4; 169-200.

K.P. Hart MSC 1991:54G10

PARTIAL DIFFERENTIAL EQUATIONS~ PROPERTY C F O R - P r o p e r t y C stands for 'completeness' of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [11] and used in [1], [2], [3], [5], [4], [6], [8], [7], [9], [10], [12], [13] as a powerful tool for proving uniqueness results for many multi-dimensional inverse problems, in particular, inverse scattering problems (cf. also Inverse scattering, m u l t i - d i m e n s i o n a l case). Let D be a bounded domain in R n, n _> 2, let J L,~u(x) := ~lJl=0 aJ m(x)Dju(x)' where j is a multiindex, DJu = OIJlu/ Ox{ 1... cqxJnn, derivatives being understood in the distributional sense, the aj,~(x), m = 1,2, are certain L°°(D) functions, Nm := {w: Lmw = 0 in D} is the null-space of the formal d i f f e r e n t i a l ope r a t o r L,~, and the equation Lmw = 0 is understood in the distributional sense. Consider the subsets N1 E N2 and N2 E Nn for which the products wiw2 are defined, Wl E N1, w2 E N2. The pair {L1, L2} has property Cp if and only if the set {WlW2}wme~m is total (complete) in LP(D), (p >_ 1 is fixed), that is, if f ( x ) E LP(D) and D f ( X ) W l (X)W2(X dx = O,

VWl ~ N1,

Vw2 ~ N2,

then f(x) =- O. By property C one often means property C2 or Cp with any fixed p > 1. Is property C generic for a pair of formal partial differential operators Li and L2? For the operators with constant coefficients, a necessary and sufficient condition is given in [9] for a pair {L1, L2 } to have property C. For such operators it turns out that property C is generic and holds or fails to hold simultaneously for all p E [1, oo): Assume aim(X) = J 0 aim z3,• z E C n" aim = c o n s t . D e n o t e Lm(z) := ~-'~qjl= n Note that Lm(e zx) = eZ'XLm(z), z . x := ~-~j=l zjxj. Therefore e z'x E 2Vm if and only if Lm(z) = O. Define the algebraic varieties (cf. also A l g e b r a i c variety) /2m : = {Z: Z E C n,

nm(z) = 0}.

One says that/21 is transversal to/22, and writes/21 }{ /22, if and only if there exist a point ~ E/21 and a point E /22 such that the tangent space Ti to/21 (in C n) at

PASCH the point ~ and the tangent space T2 to g2 at the point are transversal (cf. Transversality). The following result is proved in [Ii]: The pair {LI,L2} of formal partial differential operators with constant coemcients has property C if and only if/:i

/:2. Thus, property C fails to hold for a pair {LI, L2} of formal differential operators with constant coefficients if and only if the variety I:i U 1:2 is a union of parallel hyperplanes in C n. Therefore, property C for partial differential operators with constant coefficients is generic. If L1 = L2 = L and the pair {L, L} has property C, then one says that L has property C. E x a m p l e s . Let n > 2, L = V 2 := ~ j----1 a2/Ox~. Then -L = {z: z e C ~, z~ + . . . + z~ = 0). It is easy to check that there are points ~ E £ and ~ C £ at which the tangent hyperplanes to £ are not parallel. Thus L -- ~72 has property C. This means that the set of products of harmonic functions in a bounded domain D C R ~ is complete in LP(D), p > 1 (cf. also H a r m o n i c function). Similarly one checks that the operators

L-

0 Ot

V2'

L-

02 Ot 2

V 2,

0 L = i-~ - ~72

have property C. Numerous applications of property C to inverse problems can be found in [11]. Property C = Cz holds for a pair of Schrhdinger operators with potentials q,~(x) E L2(Rn), n > 3, where L02(Rn) is the set of L2(R n) functions with compact support (cf. also S c h r h d i n g e r e q u a t i o n ) . If Um(X,a,k), m = 1,2, a C S ~-1, k = const > 0, S ~-1 is the unit sphere in R ~, are the scattering solutions corresponding to the Schrhdinger operators lm = - V 2 + q,~(x) - k 2, qm(x) e L02(R~), n > 3, then the set of products {ul (x, a, k)u2 (x,/3, k)}w,2es~-~, k = const > 0 is fixed, is complete in L 2 (D), where D C R n is an arbitrary fixed bounded domain [11]. The set {urn(x, ~, k)}we8~-~, where k > 0 is fixed, is total in the set Nm := {w: ImW = 0 i n D , w e H2(D)}, where H2(D) is the S o b o l e v s p a c e [11].

References [1] RAMM, A.G.: Scattering by obstacles, Reidel, 1986. [2] RAMM, A.G.: 'Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering', Inverse Probl. 3 (1987), L77-L82. [3] RAMM, A.G.: 'Multidimensional inverse problems and completeness of the products of solutions to PDE', J. Math. Anal. Appl. 134, no. 1 (1988), 211-253, Also: 139 (1989), 302. [4] RAMM, A.G.: 'Multidimensional inverse problems: Uniqueness theorems', Appl. Math. Lett. 1, no. 4 (1988), 377-380. [5] RAMM, A.G.: 'Recovery of the potential from fixed energy scattering data', Inverse Probl. 4 (1988), 877-886.

CONFIGURATION

[6] RAMM, A.G.: 'Multidimensional inverse scattering problems and completeness of the products of solutions to homogeneous PDE', Z. Angew. Math. Mech. 69, no. 4 (1989), T13-T22. [7] R~,MM, A.G.: 'Completeness of the products of solutions of PDE and inverse problems', Inverse Probl. 6 (1990), 643-664. [8] RAMM, A.G.: 'Property C and uniqueness theorems for multidimensional inverse spectral problem', Appl. Math. Lett. 3 (1990), 57-60. [9] RAMM, A.G.: 'Necessary and sufficient condition for a PDE to have property C', J. Math. Anal. Appl. 156 (1991), 505509. [10] RAMM, A.G.: 'Property C and inverse problems': ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci., Springer, 1991, pp. 139-144. [11] RAMM, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992. [12] RAMM, A.G.: 'Stability estimates in inverse scattering', Acta Applic. Math. 28, no. 1 (1992), 1-42. [13] RAMM, A.G.: 'Stability of solutions to inverse scattering problems with fixed-energy data', Rend. Sere. Mat. e Fisieo

(2001), 135-211.

A.G. Ramm MSC 1991:35P25 P A S C t t CONFIGURATION, quadrilateral - A collection of four triples isomorphic to {a, b, c}, {a, y, z}, { x, b, z }, { x, y, e}. easch configurations have been studied extensively in the context of Steiner triple systems. A Steiner triple system of order v, STS(v), is an ordered pair (V, B) where V is a set of cardinality v, called elements or points, and B is a collection of triples, also called lines or blocks, which collectively have the property that every pair of distinct elements of V occur in precisely one triple. STS(v) exist if and only if v -- I or 3 (rood 6), [10] (cf. also S t e i n e r s y s t e m ) . To within isomorphism, the Steiner triple systems of orders 7 and 9 are unique, but for all greater orders the structure is not unique. A (19,1)-configuration in a Steiner triple system is a collection of I lines whose union contains precisely p points. A configuration whose number of occurrences in an STS(v) depends only upon the order v and not on the structure of the STS(v) is called constant and otherwise variable. There are two configurations with l = 2 and five with 1 = 3, all of which are constant. There are 16 configurations with l = 4, of which the Pasch configuration or quadrilateral is the unique (6, 4)-configuration and the one containing the least number of points. Five of the 4-line configurations are constant but the Pasch configuration is variable. It was shown in [5] that the number of occurrences of all the other variable 4-line configurations can be expressed in terms of the order v and the number c of Pasch configurations in the STS(v). The above gives motivation to the problem of constructing STS(v) containing no Pasch configurations, so-called anti-Pasch or quadrilateral free Steiner triple systems. A solution for v = 3 rood 6 was first given 299

PASCH C O N F I G U R A T I O N by A.E. Brouwer ([1], see also [9]) and it was a longstanding conjecture that anti-Pasch STS(v) also exist for all v = 1 mod 6, v # 7 or 13. This was settled in the affirmative in two papers, [11] and [8], published in 2000. The proof resolves the first case of a conjecture by P. Erd6s, [3], that for every m _> 4 there is an integer vm so that for every v >_ Vm, v -- 1 or 3 (mod 6), there is an STS(v) avoiding (I + 2,/)configurations for 4 < I < m. Anti-Pasch STS(v) have application to erasure-correcting codes, [2]. The theoretical maximum number of Pasch configurations in an STS(v) is v(v - 1)(v - 3)/24 but this is achieved only in the point-line designs obtained from the projective spaces PG(n, 2), [12]. The Pasch configuration is an example of a so-called trade. A pair of distinct collections of blocks (T1,T2) is said to be mutually t-balanced if each t-element subset of the base set V is contained in precisely the same number of blocks of T1 as of T2. Each collection T1, T2 is then referred to as a trade. The Pasch configuration is the smallest trade that can occur in a Steiner triple system. If T1 is the collection {a, b, c}, {a, y, z}, {x, b, z}, {x, y, c}, then, by replacing each triple with its complement, a collection T2, {x, y, z}, {x, b, c}, {a, y, c}, {a, b, z}, is obtained which contains precisely the same pairs as T1. This transformation is known as a Pasch switch, and when applied to a Steiner triple system yields another, usually non-isomorphic, Steiner triple system. There are 80 non-isomorphic STS(15)s, of which precisely one is anti-Pasch. It was shown in [4] that all of the remaining 79 systems can be obtained from one another by successive Pasch switches. Other relevant papers in this area are [6] and [7]. The number of Pasch configurations and their distribution within a Steiner triple system is an invariant and provides a simple and useful test to help in determining whether two systems are isomorphic.

[7] (]RANNELL,M.J., GRIGGS, T.S., AND MURPHY, J.P.: 'Switching cycles in Steiner triple systems', Utilitas Math. 56 (1999), 3 21. [8] GRANNELL, M.J., GRIGGS, W.S., AND WHITEHEAD, C.A.: 'The resolution of the anti-Pasch conjecture', J. Combin. Designs 8 (2000), 300-309. [9] GRIGGS, T.S., MURPHY, J.P., AND PHELAN, J.S.: 'Anti-Pasch Steiner triple systems', J. Combin. lnform, f3 Syst. Sci. 15 (1990), 79-84. [10] KIRKMAN, T.P.: 'On a problem in combinations', Cambridge and Dublin Math. J. 2 (1847), 191-204. [11] LINe, A.C.H., COLBOURN, C.J., GRANNELL, M.J., AND C-RIGGS, T.S.: 'Construction techniques for anti-Pasch Steiner triple systems', J. London Math. Soc. (2) 61 (2000), 641-657. [12] STINSON, D.R., AND WEI, Y.J.: 'Some results on quadrilaterals in Steiner triple systems', Discr. Math. 105 (1992), 207219.

M.J. GranneU T.S. Griggs

MSC 1991: 05B07, 05B30 P A U L I A L G E B R A - The 23-dimensional real Cliff o r d a l g e b r a generated by the P a u l i m a t r i c e s [2]

=

=

zwl04/rr

(1977).

[2] COLBOURN, C.J., DINITZ, J.H., AND STINSON, D.R.: 'Applications of combinatorial designs to communications, cryptography and networking', London Math. Soc. Lecture Notes 267 (1999), 37-100. [3] ERD6S, P.: 'Problems and results in combinatorial analysis', Creation in Math. 9 (1976), 25. [4] OIBBONS, P.B.: 'Computing techniques for the construction and analysis of block designs', Techn. Rept. Dept. Computer

Sci. Univ. Toronto 92 (1976). [5] GRANN~LL,M.J., GHIGGS,T.S., AND MENDELSOHN,E.: 'A small basis for four-line configurations in Steiner triple systems', J. Combin. Designs 8 (1995), 51-59. [6] GRANNELL, M.J., GRIGGS, T.S., AND MURPHY, J.P.: 'Twin Steiner triple systems', Discr. Math. 167"/8 (1997), 341-352.

300

az =

--1

'

where i is the complex unit x/Z1. The matrices ¢,, ay and ~z satisfy ~x2 = ~y2 = ~z2 = 1 and the anticommutative relations: a~aj + a s r i = 0

fori,j C {x,y,z}.

These matrices are used to describe angular momentum, spin-l/2 fermions (which include the electron) and to describe isospin for the neutron, proton, mesons and other particles. The angular m o m e n t u m algebra is generated by elements {&, J2, Ja } satisfying J1J2 - J2J1 = iJa,

&Ja - J a & =i&, Ja& - & J a =i&.

References

[1] BROUWER, A.E.: 'Steiner triple systems without forbidden subconfigurations', Rept. Math. Centrum Amsterdam

,

The Pauli matrices provide a non-trivial representation of the generators of this algebra. The correspondence 1~

(:

01) ,

I+-~icrl,

J++ic*2,

K++icr3

leads to a realization of the quaternion division algebra (cf. also Q u a t e r n i o n ) as a subring of the Pauli algebra. See [1], [3] for algebras with three anti-commuting elements. References

[1] ILAMED, Y., AND SALINGAaOS, N.: 'Algebras with three anticommuting emements I: spinors and quaternions', d. Math. Phys. 22 (I981), 2091-2095. [2] PAULI, W.: 'Zur Quantenmechanik des magnetischen Elektrons', Z. f. Phys. 43 (1927), 601-623.

PLURIPOTENTIAL THEORY [3] SALINGAROS, N.: 'Algebras with three a n t i c o m m u t i n g elem e n t s II', J. Math. Phys. 22 (1881), 2096-2100.

G.P. Wene M S C 1991: 15A66, 81R05, 81R25 PEARSON

PRODUCT-MOMENT

. ?2 from a bivariate For a r a n d o m sample {( X i,Y~)}i=l population, p is estimated by the sample correlation coefficient (cf. also C o r r e l a t i o n c o e f f i c i e n t ) r, given by

r~

CORRELATION

n

-2

-- x )

E

=l(y

-

C O E F F I C I E N T - While the modern theory of c o r r e l a t i o n and r e g r e s s i o n has its roots in the work of F.

Galton, the version of the product-moment correlation coefficient in current use (2000) is due to K. Pearson [2]. Pearson's p r o d u c t - m o m e n t correlation coefficient p is a measure of the strength of a linear relationship between two r a n d o m variables X and Y (cf. also R a n d o m v a r i a b l e ) with means #~ = E(X), #y = E(Y) and finite variances ~ 2 = var(X), a u2 = var(Y): p = corr(X, Y) - coy(X, Y ) ,

#y)] = E ( X Y ) - #~#y.

It readily follows t h a t - 1 ___p < +1, and that p is equal to - 1 or +1 if and only if each of X and Y is almost surely a linear function of the other, i.e., Y = a + fiX (/3 ~ 0) with probability 1 (furthermore, p and/~ have the same sign). If p = 0, X and Y are said to be uncorrelated. Independent random variables are always uncorrelated, however uncorrelated random variables need not be independent (cf. also I n d e p e n d e n c e ) . The term ' p r o d u c t - m o m e n t ' refers to the observation t h a t p = #11/~V/-~,02, where #ij = E[(X - #x)i(Y #y)J] denotes the ( i , j ) t h product m o m e n t of X and Y about their means. The coefficient p also plays a role in linear regresSion (cf. also R e g r e s s i o n a n a l y s i s ) . If the regression of Y on X is linear, then y = E ( Y ] X = x) = #y + p ( e y / a ~ ) ( x - #x), and if the regression of X on Y is linear, then x = E ( X I Y = y) = #~ + p ( ~ = / % ) ( y - #y). Note that the product of the two slopes is p2. When X and Y have a bivariate normal distribution (cf. also N o r m a l d i s t r i b u t i o n ) , p is a p a r a m e t e r of the joint density function -

exp

2(1 -

Q

Q= /

\

ax

/ \

ay

Further interpretations of r can be found in [3]. For details on the use of r in hypothesis testing, and for largesample theory, see [1].

[2]

K.: 'Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia', Philos. Trans. Royal Soc. London Set. A 187 (1896), 253-318. PEARSON,

[3] RODGERS, J.L., AND NICEWANDER, W.A.: 'Thirteen ways to

look at the correlation coefficient', The Amer. Statistician 42 (1988), 59-65. R.B. Nelsen M S C 1991:62H20 PLURIPOTENTIAL

T H E O R Y - The natural brand

of p o t e n t i a l t h e o r y in the setting of function theory of several complex variables (cf. also A n a l y t i c f u n c t i o n ) . The basic objects are plurisubharmonic functions (cf. also P l u r i s u b h a r m o n i c f u n c t i o n ) . These are studied much from the same perspective as subharmonic functions (cf. also S u b h a r m o n i c f u n c t i o n ) are studied in potential theory on R '~. General references are [1], [10], [16], [23]. A function u on a domain D C C ~ is called plurisubharmonic if it is subharmonic on D, viewed as a domain in R 2n, and if the restriction of u to every complex line in D is subharmonic (cf. also P l u r i s u b h a r m o n i c f u n c t i o n ; S u b h a r m o n i c f u n c t i o n ) . If u is C 2 on a domain D C C ~, then u is plurisubharmonic if and only if

OziO-2j /

with

ax

- cos 0.

'

- c ~ < x , y < oc,

\

IxllyI

[1] DUNN, O.J., AND CLARK, V.A.: Applied statistics: analysis of variance and regression, Wiley, 1974.

where coy(X, Y) is the c o v a r i a n c e of X and Y,

Y) =

x.y r - - -

References

O"x O'y

coy(X, Y) = E [ ( X - # ~ ) ( Y -

If x and y denote, respectively, the vectors (xl 7 , . . . , x n - 7) and (Yl - Y , . . . , Y ~ - Y), and 0 denotes the angle between x and y, then

/

+ ( y _ #_____yy~2. \ ay /

Unlike the general situation, uncorrelated random variables with a bivariate normal distribution are independent.

is a non-negative H e r m i t i a n m a t r i x on D. One denotes the set of plurisubharmonic functions on a domain D C C ~ by P S H ( D ) . Plurisubharmonic functions can be defined on domains in complex manifolds via local coordinates (cf. also A n a l y t i c m a n i f o l d ) . Plurisubharmonic functions are precisely the subharmonic functions invariant under a holomorphic change of coordinates. If f is holomorphic on a domain D in C n 301

P L U R I P O T E N T I A L THEORY (cf. also A n a l y t i c f u n c t i o n ) , then log If[ is plurisubharmonic on D. Moreover, every plurisubharmonic function can locally be written as

Then the function 0 U(Zl,Z2)

=

max

iflzll2, lz2]2 < ~, Z

for suitable holomorphic functions fj, see [7]. Plurisubharmonic functions were formally introduced by P. Lelong, [19], and K. Oka, [22], although related ideas stem from the end of the nineteenth century. The analogue of the L a p l a c e o p e r a t o r on domains in C is the Monge-Amp@re operator: { 02f

M f = det \ Oz~O2j] " This operator is originally only defined for C 2 plurisubharmonic functions (cf. also M o n g e - A m p ~ r e e q u a t i o n ) . Due to the non-linearity of M it is impossible to extend it to a well-defined operator on all plurisubharmonic functions on a domain D in such a way that lim,_+~ M(u,) = M(u) if {Un} is a decreasing sequence of phrisubharmonie functions with limit u, see [9]. Nevertheless, the domain of M can be enlarged to include all bounded plurisubharmonic functions, [3]. The most recent result (as of 2000) in this direction is in [11]. On strongly pseudo-convex domains D (cf. also P s e u d o - c o n v e x a n d p s e u d o - c o n c a v e ) , the following D i r i c h l e t p r o b l e m for the Monge-Amp~re operator was solved by E. Bedford and B.A. Taylor [3]: Given f continuous on OD and ¢ continuous on D, there exists a continuous plurisubharmonic function u on D, continuous up to the boundary of D, such that onm,

ubD = / .

(1)

This result has been extended by weakening the conditions on D, and replacing ¢ by certain positive measures; see e.g. [5], [18]. In [11], large classes of plurisubharmonic functions on which the Monge-Amp~re operator is well defined are determined and necessary and sufficient conditions on a positive measure ¢ are given, so that the problem (1) has a solution within such a class. The regularity of this Dirichlet problem is quite bad. The following example is due to T. Gamelin and N. Sibony: Let D be the unit ball in C 2,

(zl, z2) ~ 019. 302

1

2

, (Iz212-

1

2

elsewhere on D,

lira sup 1 log]&], j--+c~ ?

M(u)=¢

2

(I 11 -

satisfies Mu --- 0 on D, UIOD = f . However, if f and ¢ are both smooth and ¢ > 0 on D, then was shown in [8] that there exists a smooth u satisfying (1). There have been defined several capacity functions (cf. also C a p a c i t y ; C a p a c i t y p o t e n t i a l ) on C ~ that all share the property that sets of capacity 0 are precisely the pluripolar sets, i.e. sets that are locally contained in the - o o locus of plurisubharmonic functions. See [4], [10], [23], [24]. Firstly, the classical construction of l o g a r i t h m i c c a p a c i t y carries over: Let £ = {u e PSH(C~): u - log(1 + Iz[) = O(1) (z --+ ~ ) } . For a bounded set E in C ~, define the Green function with pole at infinity by

LE(z) -- sup {v(z) : v e £, v < 0 on E ) . Set L*E(z)

-= limsuPw__+zLE(w), the upper semicontinuous regularization of LE. Then either L~ - oo or L E PSH(C~). For u E £ one defines the Robin function on C ~ by

p~(z) = lira sup(u(tz) - log [tzl). tGC

Next the logarithmic capacity of E is defined as Cap(E) = exp (-zcc-SUp pLE(Z)). It is, however, a non-trivial result that Cap is a Choquet capacity (cf. C a p a c i t y ) , see [17]. Another important (relative) capacity is the Monge-Amp@re capacity introduced by Bedford and Taylor, [4]. It is defined as follows: Let ~ be a strictly pseudo-convex domain in C ~ and let K be a compact subset of ~. The Monge-Amp~re capacity of K relative to 9t is

C(K, a) = =sup{~

M(u) dV: uCPSH(~), 0 < u <

1}.

If E C D is an arbitrary subset, one defines C(m~) = sup{C(K): K C ~}. It is shown in [4] that plurisubharmonic functions are

quasi-continuous, i.e. continuous outside an open set of arbitrarily small capacity. Another application is a new proof of the following Josefson theorem [14]: If E C C n is pluripolar, then there exists a u 6 PSH(C n) with ~[E = --00.

PLURIPOTENTIAL

THEORY

Although there is no analogue of the Riesz decomposition theorem (cf. also R i e s z t h e o r e m ; R i e s z d e c o m p o s i t i o n t h e o r e m ) , there are notions of Green functions.

ery bounded hyperconvex domain is complete in the Bergman metric (cf. B e r g m a n spaces). A more elementary proof is given in [13].

1) The (Klimek or pluricomplex) Greenfunction on a domain ft C C n with pole at w E f t is the function

References

a ( z , w) =

{ =sup

h EPSH(f~),h<0, } h ( ( ) - l o g l l ( - w H = O ( 1 ) ((--+w) "

h(z):

If ft is hyperconvex, i.e. pseudo-convex and admitting a bounded plurisubharmonic exhaustion function, then G(z, w) is negative and, for w fixed, tends to 0 if z --+ Oft. Moreover, M(G(z,w)) = (2rr)*~aw, where 5w is the D i r a e d i s t r i b u t i o n at w; see [12], [15] for more details. 2) The symmetric Greenfunction on a domain f~ C C n is the function

W(z, w) = sup h(z, w) where the supremum is taken over h E SPSH(ft x ft), h < 0,

h(z, w) - log IIz - w l l _< < - log(max{dist(z, 0fl), dist(w, Oft)}). Here, S P S H ( f / x f~) stands for the functions f(z, w) on ft x f~ that are plurisubharmonic in each of the variables z, w separately, when the other is kept fixed. On strictly pseudo-convex domains f~, the symmetric Green function is negative and, for w fixed, tends to 0 as z -+ OFt. In general W < G, and there need not be equality, see [2]. In particular, G need not be symmetric and W need not be a f u n d a m e n t a l s o l u t i o n of M. However, on bounded convex domains G = W. This is based on work of L. Lempert [20], [21] showing that on bounded convex domains in C n the Kobayashi distance K(z, w) (cf. H y p e r b o l i c m e t r i c ) , the Lempert functional 5(z,w) and the Carath@odory distance C(z, w) (cf. also G r e e n f u n c t i o n ) coincide. The relation between these objects and the Green functions on a domain f~ is (see e.g. [10]) log tanh

C(z, w) <_W(z, w) <

<_G(z, w) <_log tanh 5(z, w), where ~ is the

Lempertfunctional

a(z,w) = )~f{logl,~l: f(~) = z, f ( 0 ) =

w},

with ~ the family of holomorphic mappings from the unit disc in C to ft. The Green function is instrumental in the following result of Z. Btocki and P. Pflug, [6], which is one of the first applications outside pluripotential theory: Ev-

[1] BEDFORD, E.: 'Survey o f pluri-potential theo@': Several Complex Variables (Stockholm, 1987/8), Vol. 38 of Math. Notes, Princeton Univ. Press, 1993, pp. 48-97. [2] BEDFORD, E., AND DEMAILLY, J.P.: 'Two counterexamples concerning the pluri-complex Green function in C ~', Indiana Univ. Math. J. 37 (1988), 865-867. [3] BEDFORD, E., AND TAYLOR, B.A.: 'The Dirichlet problem for a complex Monde-Ampere equation', Invent. Math. 37 (1976), 1-44. [4] BEDFORD, E., AND TAYLOR, B.A.: 'A new capacity for plurisubharmonic functions', Acta Math. 149 (1982), 1-40. [5] BLOCKI, Z.: 'The complex Monge-Amp~re equation in hyperconvex domain', Ann. Scuola Norm. Sup. Pisa 23 (1996), 721-747. [6] BLOCKI, Z., AND PFLUG, P.: 'Hyperconvexity and Bergman completeness', Nagoya Math. J. 151 (1998), 221-225. [7] BREMERMANN, H.: 'On the conjecture of equivalence of plurisubharmonic functions and Hartogs functions', Math. Ann. 131 (1956), 76-86. [8] CAFFARELLI, L., KOHN, J.J., NIRENBERG, L., AND SPRUCK, J.: 'The Dirichlet problem for nonlinear second order elliptic equations. II. Complex Monde-Ampere, and uniform elliptic, equations', Commun. Pure Appl. Math. 38 (1985), 209-252. [9] CEGRELL, U.: 'Discontinuit@ de l'op@rateur de Monge Ampere complexe', C.R. Acad. Sei. Paris Sdr. I Math. 296 (1983), 869-87!. [10] CEGRELL, U.: Capacities in complex analysis, Viewed, 1988. [11] CEGRELL, U.: 'Pluricomplex energy', Acta Math. 180 (1998), 187-217. [12] DEMAILLY, J.P.: 'Mesures de Monge-Amp~re et mesures pluriharmoniques', Math. Z. 194 (1987), 519-564. [13] HERBORT, G.: 'The Bergman metric on hyperconvex domains', Math. Z. 232 (1999), 183-196. [14] JOSEFSON, B.: 'On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on C n~, Ark. Mat. 16 (1978), 109-115. [15] KLIMEK, M.: 'Extremal plurisubharmonic functions and invariant pseudodistances', Bull. Soc. Math. Prance 113 (1985), 231-240. [16] KLIMEK, M.: Pluripotential theory, Clarendon Press/Oxford Univ. Press, 1991. [17] KO~.ODZmJ, S.: 'The logarithmic capacity in C n', Ann. Polon. Math. 48 (1988), 253-267. [18] KOLODZIEJ, S.: 'The complex Monde-Ampere equation', Acta Math. 180 (1998), 69-117. [19] LELONC, P.: 'Les fonctions plurisousharmonique', Ann. Sci. l~cole Norm. Sup. 62 (1945), 301-338. [20] LEMPERT, L.: 'La m@trique de Kobayashi et la repr@sentation des domaines sur la boule', Bull. Soc. Math. Prance 109 (1981), 427-474. [21] LEMFERT, L.: 'Holomorphic retracts and intrinsic metrics in convex domains', Anal. Math. 8, no. 4 (1982), 257-261. [22] OKA, K.: 'Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes', T6hoku Math. J. 49 (1942), 15-52. [23] SADULLAEV, A.: 'Plurisubharmonic measures and capacities on complex manifolds', Russian Math. Surveys 36 (1981), 61-119. (Uspekhi Mat. Nauk. 36 (1981), 53-105.)

303

PLURIPOTENTIAL

THEORY

[24] SICIAK, J.: 'Extremal functions and capacities in C n', Sophia Kokyuroku Math. 14 (1982).

Jan Wiegerinek

For each ~ • 0f~ the function Pa (', ~) is positive and harmonic in fl, and for each x E f l the measure

#~ = P•(x,¢) da(¢)

M S C 1991: 31C10, 32F05

is a probability, called the harmonic measure for ~ at P O I N C A R I ~ M A P P I N G , Poincard map See P o i n e a r ~ r e t u r n m a p and P o i n c a r ~ r e t u r n t h e o -

X.

The Poisson kernel has the properties (rl • 0f~)

rem.

lira P a ( x , ~ ) =

x-+r]

M S C 1991: 58Fxx

0,

r]~,

and POISSON

FORMULA

FOR H A R M O N I C

T I O N S - Consider a h a r m o n i c

f u n c t i o n f : D --+ R defined in a domain D in a Euclidean space R n, n 2 2. Let B(xo, r) denote the open ball B(

0,r) = { x •

Ix-x0l

lim #~ = 5~,

FUNC-

with centre x0 and radius r > 0. Assume that the closure of this ball is contained in D. T h e classical Poisson formula expresses that f can be recovered inside the ball by the values of f on the boundary of the ball integrated against the Poisson kernel P

x-+r]

where the last limit is in the weak topology for probability measures on 0 ~ (cf. also W e a k c o n v e r g e n c e o f p r o b a b i l i t y m e a s u r e s ) and 57 is the D i r a e d i s t r i b u t i o n at r1.

There are only a few cases where the Poisson kernel can be given in closed form as for the ball. The Poisson formula for a domain f~ is related to the solution of the D i r i c h l e t p r o b l e m : For a function f : 0f~ -~ R, the harmonic continuation in f~ is (under suitable assumptions) given as

for the ball,

x ~+ foa f d,~.

- Ix - x01

P(x,()

nrlx-

=

z • B(zo,r),

'

• OB(xo,r),

where

a~_ • { ( x , t ) :

27rn/2 (.oft

--

B(~o,~)

&(¢),

where a is the surface measure of the ball, the total mass of which is w~r n-1. For x = x0 the formula reduces to the mean-value theorem for harmonic functions, stating that the value at the centre of the ball is the average over the boundary of the ball. The same type of formula holds when the ball is replaced by a bounded domain f~ with a sufficiently smooth boundary and such t h a t the closure of t2 is contained in D. The Poisson kernel (1) is replaced by the Poisson kernel Pn (x, ~) for f~ and a is replaced by the surface measure on the boundary of Ft. The Poisson kernel, defined on f~ x Off, is given as Pa(x,

) = °

where the inward normal derivative of the G r e e n f u n c t i o n Ga (x, y) for f~ with respect to the second variable y is used. 304

x e a n - l , t > 0}.

The formula is

is the (n - 1)-dimensional surface area of the unit ball in R n. The Poisson formula is f(x)=

There is also a Poisson formula for unbounded domains, the simplest of which is for the upper half-space

tf(y, o) f(x,t) = ~2 f R ~-1 (Ix 2-~+7~)~/2 dy, and it is valid for a harmonic function f(x, t) in the upper half-space provided it has a continuous extension to the closure and satisfies some growth condition. See Hardy spaces. References [1] DOOB, J.L.: Classical potential theory and its probabilistic counterpart, Springer, 1984.

Ch. Berg MSC1991: 31A05, 31A10 P O L Y N O M I A L C O N V E X I T Y - Let 5° denote the set of holomorphic polynomials on C ~ (cf. also A n a l y t i c f u n c t i o n ) . Let K be a compact set in C n and let IIP[IK = maxzeK IF(z)] be the sup-norm of P • 79 on K . The set R = {z • Cn:

IP(z)F <_ IIP[IK, VP • 79},

is called the polynomially convex hull of K . If K = K one says t h a t K is polynomially convex. An up-to-date (as of 1998) text dealing with polynomial convexity is [3], while [13] and [27] contain some sections on polynomial convexity, background and older

POLYNOMIAL C O N V E X I T Y results. The paper [25] is an early study on polynomial convexity. Polynomial convexity arises naturally in the context of function algebras (cf. also A l g e b r a o f f u n c t i o n s ) : Let P ( K ) denote the u n i f o r m a l g e b r a generated by the holomorphic polynomials on K with the sup-norm. The maximal ideal space M of P ( K ) is the set of homomorphisms mapping P ( K ) onto C, endowed with the topology inherited from the dual space P(K)*. It can be identified with K via Z C ~[

++

P ~+ P(z),

mz,

P ~ 7).

Moreover, if A is any finitely generated function algebra on a compact Hausdorff space, then A is isomorphic to P ( K ) , where for K one can take the joint spectrum of the generators of A (cf. also S p e c t r u m o f a n o p e r a tor). By the Riesz representation theorem (el. R i e s z t h e o r e m ) there exists for every z E h" at least one representing measure #z, that is, a p r o b a b i l i t y m e a s u r e #z on K such that

P(z)

mz(P) = £ P(¢) + z ( ¢ ) ,

P e 7).

One calls #z a Jensen measure if it has the stronger property loglP(z)[ __ £1oglP(¢)l

P e 7).

It can be shown that for each z E K there exists a Jensen measure #~. See e.g. [27]. For compact sets K in C one obtains K by 'filling in' the holes' of K, that is, h" = C \ f~oo, where f~oo is the unbounded component of C \ K. In C ~, n > 1, there is no such a simple topological description. Early results on polynomial convexity, cf. [13], are

• Oka's theorem: If K is a polynomially convex set in C n and f is holomorphic on a neighbourhood of K , then f can be written on K as a uniform limit of polynomials. Cf. also O k a t h e o r e m s . • Browder's theorem: If K is polynomially convex in C ~, then HP(K, C) = 0 for p > n. Here, HP(K, C) is the pth C e c h c o h o m o l o g y group. More recently (1994), the following topological result was obtained, cf. [10], [3]:

• Forstnerid' theorem: Let K be a polynomially convex set in C n, n _> 2. Then Hk(C n \ K ; G ) = O ,

l
7rk(Cn \ K) = O,

l < k < n-1.

and

Here, H k ( X , G ) denotes the kth h o m o l o g y g r o u p of X with coefficients in an Abelian group G and 7rk(X ) is the kth homotopy group of X. One method to find b~" is by means of analytic discs. Let A be the unit disc in C and let T be its boundary. An analytic disc is (the image of) a holomorphic mapping f : A --+ C ~ such that f is continuous up to T. Similarly one defines an H a-disc as a bounded holomorphic mapping f : A ~ C '~. Its components are elements of the usual Hardy space H ° ° ( A ) (cf. H a r d y spaces). Now, let K be compact in C n and suppose that f ( T ) C K for some analytic disc f . Then f ( A ) C K by the m a x i m u m p r i n c i p l e applied to P o f for polynomials P E 7). The same goes for H a - d i s c s whose boundary values are almost everywhere in K . One says that the disc f is glued to K . Next, one says that K has analytic structure at p E ~" if there exists a non-constant analytic disc f such that f(0) = p and the image of f is contained in K. It was a major question whether _~ \ K always has analytic structure. Moreover, when is K obtained by glueing discs to K ? One positive result in this direction is due to H. Alexander [1]; a corollary of his work is as follows: If K is a rectifiable curve in C n, then either K = K and P ( K ) = C(K), or ~" \ K is a pure i-dimensional analytic subset of C ~ \ K (cf. also A n a l y t i c set). If K is a rectifiable arc, K is polynomially convex and P ( K ) = C(K). See [1] for the complete formulation. Alexander's result is an extension of pioneering work of J. Wermer, cf. [30], E. Bishop and, later, G. Stolzenberg [26], who dealt with real-analytic, respectively C 1 , curves. Wermer [29] gave the first example of an arc in C 3 that is not polynomially convex, cf. [3]. However, Gel'fand's problem (i.e., let ~, be an arc in C n such that ~ = 7; is it true that P ( 7 ) = C(@?) is still open (2000). Under the additional assumption that its projections into the complex coordinate planes have 2-dimensional H a u s d o r f f m e a s u r e 0, the answer is positive, see [3]. F.R. Harvey and H.B. Lawson gave a generalization to higher-dimensional K , cf. [12], which includes the following. Let p _> 1. If K is a C 2 (2p+ 1)-dimensional submanifold of C n and at each point of K the tangent space to K contains a p-dimensional complex subspace, then K is the boundary of an analytic variety (in the sense of Stokes' theorem). Another positive result is contained in the work of E. Bedford and W. Klingenberg, cf. [4]: Suppose F C C 2 is the graph of a C2-function ¢ over the boundary of a strictly convex domain f~ C C x R. Then F is the graph of a Lipschitz-continuous extension q~ of ¢ on f~. 305

POLYNOMIAL CONVEXITY

Moreover, F is foliated with analytic discs (cf. also Foliation). The work of Bedford and Klingenberg has been generalized in various directions in [16], [21] and [7]. One ingredient of this theorem is work of Bishop [5], which gives conditions that guarantee locally the existence of analytic discs with boundary in real submanifolds of sufficiently high dimension. See [11], [32] and [15] for results along this line. A third situation t h a t is fairly well understood is when K C C ~+1 is a compact set fibred over T, that is, K is of the form K = {(z,w): z C T, w C Kz}, where Kz is a compact set in C n depending on z. In this case the following is true: Let K C C 2 be a compact fibration over the circle T and suppose that for each z the fibre Kz is connected and simply connected. Then i ~ \ K is the union of graphs El, where f 6 H °~ (A) and the boundary values f* (z) are in Kz for almost all

real at p E M if the tangent space in p does not contain a complex line (cf. also C R - s u b m a n i f o l d ) . The HSrmander-Wermer theorem is as follows, cf. [14]: Let M be a sufficiently smooth real submanifold of C n and let K0 be the subset of M consisting of points that are not totally real. If K C M is a compact polynomially convex set that contains an M-neighbourhood of K0, then P ( K ) contains all continuous functions on K that are on K0 the uniform limit of functions holomorphic in a neighbourhood of K0.

zcT.

References

Of course, it is possible that K \ K is empty. The present theorem is due to Z. Slodkowski, [22], earlier results are in [2] and [9]. Slodkowski proved a similar theorem in C n+l under the assumption that the fibres are convex, see [23]. Despite these positive results, in general h" \ K need not have analytic structure. This has become clear from examples by Stolzenberg [24] and Wermer [31]. Presently (2000) it is not known w h e t h e r / ~ has analytic structure everywhere if K is a (real) submanifold of C n, nor is it known under what conditions K is obtained by glueing discs to K . However, it has been shown that in a weaker sense there is always a kind of analytic structure in polynomial hulls. Let dO denote L e b e s g u e m e a s u r e on the circle T and let f'dO denote the push-forward of dO under a continuous mapping f : T -+ C ~. Let also K be a compact set in C n. The following are equivalent: 1) z C K and #~ is a Jensen measure for z supported on K; 2) There exists a sequence of analytic discs fj : A C ~ such that fj(O) ~ z and f~dO/27r -+ #z in the weak* sense (cf. also W e a k t o p o l o g y ) . This was proved in [6]; [8] and [20] contain more information about additional nice properties that can be required from the sequence of analytic discs. Under suitable regularity conditions on K , it is shown in [19] that / ( consists of analytic discs f such that f - l ( K ) N T has Lebesgue measure arbitrary close to 21r. Another problem is to describe P ( K ) assuming that K = K and given reasonable additional conditions on K. In particular, when can one conclude that P ( K ) = C ( K ) ? Recall that a real submanifold M of C ~ is totally 306

See [17] for a variation on this theme. One can deal with some situations where the manifold is replaced by a union of manifolds; e.g., B.M. Weinstock [28] gives necessary and sufficient conditions for any compact subset of the union of two totally real n-dimensional subspaces of C ~ to be polynomially convex; then also P ( K ) = C(K). See also [18].

[1] ALEXANDER,H.: 'Polynomial approximation and hulls in sets of finite linear measure in C n', A m e r J. Math. 62 (1971), 6574. [2] ALEXANDER, H., AND WERMER, J.: 'Polynomial hulls with convex fibres', Math. A n n . 281 (1988), 13-22. [3] ALEXANDER,H., AND WERMER, J.: Several complex variables and Banaeh algebras, Springer, 1998. [4] BEDFORD, E., AND KLINGENBERG JR., W.: 'On the envelope of holomorphy of a 2-sphere in C 2', J. A m e r . Math. Soc. 4 (1991), 623-646. [5] BISHOP, E.: 'Differentiable manifolds in Euclidean space', Duke Math. J. 32 (1965), 1-21. [6] Bu, S., AND SCHACHERMAYER,W.: 'Approximation of Jensen measures by image measures under holomorphic functions and applications', Trans. A m e r . Math. Soc. 331 (1992), 585608. [7] CHIRKA, E.M., AND SHCHERBINA,N.V.: 'Pseudoconvexity of rigid domains and foliations of hulls of graphs', A n n . Scuola N o r m . Sup. Pisa 22 (1995), 707-735. [8] DUVAL, J., AND SIBONY, N.: 'Polynomial convexity, rational convexity and currents', D u k e Math. J. 79 (1995), 487-513. [9] FORSTNERIC, F.: 'Polynomial hulls of sets fibered over the circle', I n d i a n a Univ. Math. J. 37" (1988), 869-889. [10] FORSTNERIC, F.: 'Complements of Runge domains and holomorphic hulls', Michigan Math. d. 41 (1994), 297-308. [11] FORSTNERIC, F., AND STOUT, E.L.: 'A new class of polynomially convex sets', Ark. Mat. 29 (1991), 51-62. [12] HARVEY, F.R., AND LAWSON JR., H.B.: 'On boundaries of complex analytic varieties I', A n n . of Math. 102 (1975), 223290. [13] HORMANDER,L.: A n introduction to complex analysis in several variables, North-Holland, 1973. [14] HORMANDER, L., AND WERMER, J.: 'Uniform approximation on compact sets in C n', Math. Scan& 23 (1968), 5-21. [15] JORICKE, B.: 'Local polynomial hulls of discs near isolated parabolic points', Indiana Univ. Math. J. 46, no. 3 (1997), 789-826. [16] KRUZHILIN, N.G.: 'Two-dimensional spheres in the boundaries of strictly pseudoconvex domains in C 2', Math. USSt~ Izv. 39 (1992), 1151-1187. (Translated from the Russian.)

P O R T F O L I O OPTIMIZATION [17] O'FARRELL, A.G., PRESKENIS, K.J., AND WALSH, D.: 'Holomorphic approximation in Lipschitz norms': Proc. Conf. Banach Algebras and Several Complex Variables (New Haven, Conn., 1983), Vol. 32 of Contemp. Math., 1983, pp. 187-194. [18] PAEPE, P.J. DE: 'Approximation on a disk I', Math. Z. 212 (1993), 145-152. [19] POLETSKY, E.A.: 'Holomorphic currents', Indiana Univ. Math. J. 42 (1993), 85-144. [20] POLETSKY, E.A.: 'Analytic geometry on compacta in C n', Math. Z. 222 (1996), 407-424. [21] SHCHERBINA,N.: 'On the polynomial hull of a graph', Indiana Univ. Math. J. 42 (1993), 477-503. [22] SLODKOWSKI,Z.: 'Polynomial hulls with convex convex sections and interpolating spaces', Proc. Amer. Math. Soc. 96

(1986), 255-260. [23]

[24] [25]

[26] [27]

[28]

[29] [30] [31]

[32]

SLODKOWSKI,Z.: 'Polynomial hulls in C 2 and quasi circles', Ann. Scuola Norm. Sup. Pisa 16 (1989), 367-391. STOLZENBERG,G.: 'A hull with no analytic structure', J. Math. Mech. 12 (1963), 103-112. STOLZENBERG,G.: 'Polynomially and rationally convex sets', Acta Math. 109 (1963), 259-289. STOLZENBERG, G.: 'Uniform approximation on smooth curves', Acta Math. 115 (1966), 185-198. STOUT, E.L.: The theory of uniform algebras, Bogden and Quigley, 1971. WEINSTOCK,B.M.: 'On the polynomial convexity of the union of two maximal totally real subspaces of C n ', Math. Ann. 282 (1988), 131-138. WERMER, J.: 'Polynomial approximation on an arc in C 3', Ann. of Math. 62 (1955), 269-270. WERMER, J.: 'The hull of a curve in C n', Ann. of Math. 68 (1958), 550-561. WERMER, J.: 'On an example of Stolzenberg': Symp. Several Complex Variables, Park City, Utah, Vol. 184 of Lecture Notes in Mathematics, Springer, 1970. WIEGERINCK, J.: 'Local polynomially convex hulls at degenerated CR singularities of surfaces in C 2', Indiana Univ. Math. J. 44 (1995), 897-915. Jan Wiegerinck

MSC 1991:32E20 PORTFOLIO OPTIMIZATION - The problem of optimally choosing a distribution of available wealth over various investment opportunities may be studied in several settings. One may consider static problems, in which a decision is made once and for all, or dynamic problems, in which rearrangements are possible in the course of time. The latter may be formulated either in discrete time or in continuous time, on a finite interval or on an infinite interval. The characteristics of the available investment opportunities are typically described in stochastic terms, and here many options are open with respect to the distributions that are used and the behaviour in time; in continuous time one may use models in which asset prices follow continuous paths or models in which prices can be subject to jumps. One may or may not include transaction costs, position limits, and other frictions and side constraints in the problem formulation. Finally, a specification has to be given of

the criterion that will be used to compare investment strategies. A frequently used criterion is expected utility, that is, the expected value of a utility function of portfolio value, summed or integrated over time as appropriate. Other criteria may be applied as well however, for instance relating to asymptotic properties of portfolio value, or to worst-case behaviour with respect to a given class of probability measures. The work of H. Markowitz [8] is usually viewed as the starting point of modern portfolio theory. Markowitz considered static problems and called a portfolio meanvariance efficient if it achieves a given mean with minimal variance (cf. also D i s p e r s i o n ; A v e r a g e ) . Such portfolios may be found by solving a q u a d r a t i c p r o g r a m m i n g problem. Early results on portfolio problems in continuous time with criteria of the expected utility type were obtained by R.C. Merton [9], [10], who used the method of d y n a m i c p r o g r a m m i n g . Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. This martingale approach was developed in [11], [4], [1]. An important technical achievement, needed to overcome difficulties associated with limited regularity of value functions, has been the introduction of v i s c o s i t y s o l u t i o n s [6], [7].

See also [2], [3], [5] References [1] Cox, J., AND HUANG, C.F.: 'Optimal consumption and portfolio policies when asset prices follow a diffusion process.', J. Economic Th. 49 (1989), 33-83. [2] FLEMING, W.H., AND RISHEL, R.W.: Deterministic and stochastic optimal control, Springer, 1975. [3] FLEMING, W.H., AND SONER, H.M.: Controlled Markov processes and viscosity solutions, Springer, 1993. [4] KARATZAS, I., LEHOCZKY, J.P., AND SHREVE, S.E.: 'Optimal portfolio and consumption decisions for a small investor on a finite horizon', S I A M J. Control Optim. 27 (1987), 11571186. [5] KORN, R.: Optimal portfolios. Stochastic models for optimal investment and risk management in continuous time, World Sci., 1997. [6] LIONS, P.L.: 'Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications', Commun. Partial Diff. Eqs. 8 (1983), 1101-1174. [7] LIONS, P.L.: 'Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 2: Viscosity solutions and uniqueness', Commun. Partial Diff. Eqs. 8 (1983), 1229-1276. [8] MARKOWITZ, H.: 'Portfolio selection', J. Finance 7 (1952), 77-91. [9] MERTON, R.C.: 'Lifetime portfolio selection under uncertainty: The continuous case', Rev. Economical Statist. 51 (1969), 247-257.

307

PORTFOLIO

OPTIMIZATION

[10] MERTON, m.c.: 'Optimum consumption and portfolio rules in a continuous time model', J. Economic Th. 3 (1971), 373 413. [11] PLISKA, S.R.: 'A stochastic calculus model of continuous trading: Optimal portfolios', Math. Operat. Res. 11 (1986), 371-382. J.M. Schumaeher

MSC 1991:90A09 P O S I T I V E LINK - An oriented link t h a t has a dia g r a m with all positive crossings (cf. also L i n k ) . More generally, a link is m-almost positive if it has a d i a g r a m with all but m of its crossings being positive. T h e unknotting number (Gordian number) of a positive link is equal to

l ( c ( D ) - s(D) + corn(D)),

where D is a positive d i a g r a m of the link, c(D) is the n u m b e r of crossings, s(D) is the n u m b e r of Seifert circles of D, and corn(D) is the n u m b e r of c o m p o n e n t s of the link (this generalizes the M i l n o r u n k n o t t i n g c o n j e c t u r e , 1969, and the Bennequin conjecture, 1981). Furthermore, for a positive knot the u n k n o t t i n g n u m b e r is equal to the 4-ball genus of the knot, to the genus of the knot (cf. also K n o t t h e o r y ) , to the planar genus of the knot (from the Seifert construction), to the minimal degree of the Jones polynomial, and to half the degree of the Alexander polynomial. One can define a relation > on links by L1 > L2 if and only if L2 can be obtained from L1 by changing some positive crossings of L1. This relation allows one to express several f u n d a m e n t a l properties of positive (and m - a l m o s t positive) links: 1) If K is a positive knot, then K > (5, 2) positive torus knot unless K is a connected sum of pretzel knots L(pl,P2,P3), where Pl, P2 and P3 are positive odd numbers; a) if K is a non-trivial positive knot, then either the signature a ( K ) < - 4 or K is a pretzel knot L(pl,p2,p3) (and then a ( K ) = - 2 ) ; b) if a positive knot has u n k n o t t i n g n u m b e r one, then it is a positive twist knot. 2) Let L be a non-trivial 1-almost positive link. T h e n L _> right-handed trefoil knot (plus trivial components), or L >__right-handed H o p f link (plus trivial components). In particular, L has a negative signature. 3) If K is a 2-almost positive knot, then either i) K > right handed trefoil; or ii) K > mirror image of the 62-knot (G3a~lcrla~ 1 in the braid notation); or iii) K is a twist knot with a negative clasp. 4) If K is a 2-almost positive knot different from a twist knot with a negative clasp, then K has negative signature and K ( 1 / n ) (i.e. 1/n surgery on K , n > 0; 308

cf. also S u r g e r y ) is a h o m o l o g y 3-sphere t h a t does not b o u n d a c o m p a c t , s m o o t h h o m o l o g y 4-ball, [2], [7]; 5) if K is a non-trivial 2-almost positive knot different from the Stevedore knot, t h e n K is not a slice knot; 6) if K is a non-trivial 2-almost positive knot different from the figure eight knot, then K is not amphicheiral. 7) Let K be a 3-almost positive knot. T h e n either K > trivial knot or K is the left-handed trefoil knot (plus positive knots as connected s u m m a n d s ) . In particular, either K has a non-positive signature or K is the left-handed trefoil knot. References [1] BUSKIRK, J.M. VAN: 'Positive knots have positive Conway polynomials': Knot Theory And Manifolds (Vancouver, B.C., 1983), Vol. 1144 of Lecture Notes in Mathematics, Springer, 1985, pp. 146-159. [2] COCHRAN, T., AND GOMPF, E.: 'Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and property P', Topology 27, no. 4 (1988), 495-512. [3] KRONHEIMER,P.B., AND MROWKA, T.S.: 'Gauge theory for embedded surfaces. I', Topology 32, no. 4 (1993), 773-826. [4] MENASCO, W.W.: 'The Bennequin-Milnor unknotting conjectures', C.R. Acad. Sci. Paris Sdr. I Math. 318, no. 9 (1994), 831-836. [5] NAKAMURA,T.: 'Four-genus and unknotting number of positive knots and links', Osaka J. Math. 37 (2000), to appear. [6] PRZYTYCKI, J.H.: 'Positive knots have negative signature', Bull. Acad. Polon. Math. 37 (1989), 559-562. [7] PRZYTYCKI, J.H., AND TANIYAMA, K.: 'Almost positive links have negative signature', preprint (1991), See: Abstracts Amer. Math. Soc., June 1991, Issue 75, Vol. 12 (3), p. 327, .91T-57-69. [8] RUDOLPH, L.: 'Nontrivial positive braids have positive signature', Topology 21, no. 3 (1982), 325-327. [9] RUDOLPH,L.: 'Quasipositvity as an obstruction to sliceness', Bull. Amcr. Math. Soc. 29 (1993), 51-59. [10] RUDOLPH,L.: 'Positive links are strongly quasipositive': Proc. Kirbyfest, Vol. 2 of Geometry and Topology Monographs, 1999, pp. 555-562. [11] TANIYAMA,K.: 'A partial order of knots', Tokyo J. Math. 12, no. 1 (1989), 205-229. [12] THAeZYK, P.: 'Nontrivial negative links have positive signature', Manuscripta Math. 61, no. 3 (1988), 279-284. Jozef Przytycki

MSC 1991:57M25 P R O J E C T I V E R E P R E S E N T A T I O N S OF S Y M M E T -

T h e classification of the projective representations of a f i n i t e g r o u p G (eft also P r o j e c t i v e r e p r e s e n t a t i o n ) was obtained by I. Schur [9], [10], who showed t h a t over the complex field C the p r o b l e m of determining all projective representations of G can be reduced to determining the linear representations of stem extensions G of G, called representation groups, by its Schur multiplier M ( G ) (cf. also S c h u r m u l t i p l i c a t o r ) . A s t a n d a r d reference is [5]. RIC A N D A L T E R N A T I N G G R O U P S -

P R O J E C T I V E REPRESENTATIONS OF SYMMETRIC AND ALTERNATING GROUPS In the case of the symmetric groups S~ and the alternating groups An (cf. also S y m m e t r i c group; Alt e r n a t i n g g r o u p ) , Schur [11] further showed that

M(S~) - ~Z2

M(A~)--

{

[ {e}

i f n > 4, ifn < 4,

Z2

ifn >_ 4, n 7~ 6,7,

Z6

if n = 6 , 7 ,

{e}

i f n < 4.

where (~ is the k~ is the order ing power-sum according as n then

value of ~x at the class of cycle-type % of that class and p~ is the correspondsymmetric function and e(A) = 0 or 1 - r(A) is even or odd. If ,~ E S P - ( n ) ,

= i(n-r(~)+l)/2

and ~ =0

i f # 7 ~A, # E S P - ( n ) .

Schur also determined the dimension formula dimT~=2[(n_r(A))/2]

The representation groups are not unique, for n _> 4 there are two for Sn; however, to determine the projective representations of Sn it suffices to consider one of these, which will be denoted by Sn; similarly, A~ is a representation group of An. The non-linear representations of Sn and J,n, that is, those representations T for which T(z) = - I r a n = dimT, where z is the generator of Z2 are called spin representations. Schur [10] classified the complex irreducible spin representations of Sn and J,n, n _> 4 (and also the remaining non-linear projective representations for J*6 and J*7). Although more complicated, the classification of the spin representations follows the corresponding results for the linear representations of these groups. (cf. R e p r e s e n t a t i o n of t h e s y m m e t r i c groups). A standard reference is [4], but see also [12]. In this case, the irreducible spin representations are parametrized by the set SP(n) of strict partitions k = (kl,-..,k~(a)) of n, where ~1 ) ' ' ' ) "~r(A) ) 0. If SP+(n) (respectively, S P - ( n ) ) denotes the subset of SP(n) where the number of even parts is even (odd), then a complete list of irreducible spin representations is: {T;~: ~ C SP+(n)} tO {T),,T'), = sgn-T;~: ~ C S P - ( n ) } , where sgn is the sign representation of Sn. The characters of these representations, called spin characters and denoted by ~A and ~ , can take only non-zero values on the classes of Sn which are of cycle-type corresponding to partitions in O(n), with all parts odd, and in SP-(n). The values of the spin characters can be given explicitly in the case SP-(n), but for O(n) can be determined from a class of symmetric functions introduced for this purpose by Schur and now called Schur Q-functions (cf. Schur Q-function) - - these play an analogous role to that of Schur functions for linear representations of S~ (cf. Schur f u n c t i o n s in a l g e b r a i c c o m b i n a t o r i c s ) . For each )~ C SP(n), let Qx denote the corresponding Schur Q-function; then 1

Q~ = 7. ~ 2(~(x)+~(~)+~(~))/2k~ffP~' ~CO(n)

~/(/~1"'" )~r()0)/2

n!

/
The spin representations of J*n are now easily determined; if A E S P - ( n ) , then Ta $ 2 = T~ $2" is an irreducible spin representation and if )~ E SP + (n), then Tx splits into two conjugate irreducible spin representations T + and T~- of equal dimension and =

-

=

i(

All these results appeared in Schur's 1911 paper [11] the subject then lay dormant until the appearance of papers by A.O. Morris in the early 1960s [6], [7], where the combinatorial concepts of bars and bar lengths were introduced (cf. Schur Q - f u n c t i o n ) ; these correspond to the concepts of hooks and hook lengths in the linear case. Thus, the above dimension formula can be interpreted in terms of bar lengths: -

dim T~ = 2 [(n-'(~))/2]

n[

1-I(ij) bij'

where bij denotes the bar length at the (i,j)th node in the Y o u n g d i a g r a m corresponding to ~. Also, a recursion formula for calculating the irreducible spin characters analogous to the Murnaghan-Nakayama formula in the linear case was obtained in terms of these concepts. In all these formulas, as in the above dimension formula, the real difference is the complication added by the powers of 2 which appear. Totally lacking until the 1990 work of M.L. Nazarov [8] were explicit methods for constructing the irreducible spin matrix representations corresponding to each partition ~ E SP(n) - - these generalize the ones given by Schur for the partition (n). The method is comparable to the classical construction of the semi-normal form given by A. Young (cf. R e p r e s e n t a t i o n of t h e s y m m e t ric groups). More recently, Nazarov has generalized Young's symmetrizer to the spin case. However, there are presently (2000) no analogues developed to Specht modules (cf. S p e c h t m o d u l e ) . Some progress has been made on the modular spin representations of these groups. In 2001, the two papers [3] and [2] by J. Brundan and A. Kleshchev completely overturned the position. A conjecture corresponding to 309

P R O J E C T I V E R E P R E S E N T A T I O N S OF SYMMETRIC AND ALTERNATING G R O U P S the classical Nakayama conjecture on the distribution of the spin characters into their p-blocks has been proved -- but, in general, the position here is even less understood than in the case of the modular ordinary representations. See [1] for the most recent developments. References [1] BESSENRODT, C.: 'Algebra and combinatorics', Progress in Math. 168 (1998), 64-91. [2] BRUNDAN, J., AND KLESHCHEV, A.: 'Heeke-Clifford superalgebras, crystals of type A ~ ) and modular branching rules for

It is not possible, in general, to find f(xo) from the local tomographic data [2]. W h a t practically useful information about f(x) can one get from these data? Information, very useful practically, is the location of discontinuity curves of f(x) and the sizes of the jumps of f(x) across these curves. Pseudo-local tomography solves the problem of finding the above information from the local tomographic data. This is done by computing the pseudo-local tomography function, introduced in [2]:

S r ~ ~,

[3] BRUNDAN, a., AND KLESHCHEV, A.: 'Projective representations of the symmetric group via Sergeev duality', Math. Z. (to appear). [4] HOFFMAN, P.N., AND HUMPHREYS, J.F.: Projective representations of the symmetric groups, Oxford Univ. Press, 1992. [5] KARPILOVSKY,G.: Projective representations of finite groups, M. Dekker, 1995. [6] MORRIS, A.O.: 'The spin representation of the symmetric group', Proc. London Math. Soc. 12, no. 3 (1962), 55-76. [7] MORRIS, A.O.: 'The spin representation of the symmetric group', Canad. J. Math. 17 (1965), 543-549. [8] NAZAROV,M.L.: 'Young's orthogonal form of irreducible projective representations of the symmetric group', J. London Math. Soc. 42, no. 2 (1990), 437-451. [9] SCHUR, I.: 'lJber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen', J. Reine Angew. Math. 127 (1904), 20-50. [10] SCHUR, I.: 'Untersuchungen fiber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen', J. Reine Angew. Math. 132 (1907), 85-137. [11] SCHUR, I.: '/Jber die Darstellung der symmetrischen und der alternierenden Gruppe dutch gebrochene lineare Substitutionen', J. Reine Angew. Math. 139 (1911), 155-250. [12] STEMBRIDGE,J.R.: 'Shifted tableaux and projective representations of symmetric groups', Adv. Math. 74 (1989), 87-134.

PSEUDO-LOCAL TOMOGRAPHY - Let f(x) be a piecewise smooth, compactly supported function, x • R 2. The R a d o n t r a n s f o r m f(a,p) := R f is defined by the formula f(a,p) = fg~v f(x) ds, where gap := {x: a • x = p} is a straight line parametrized by a unit vector a • S i, S 1 is the unit circle in R 2 a n d p • R + := [0, oc). By definition, f ( - a , - p ) = f(a,p). By local tomographic data one means the values of f(a,p) for a and p satisfying the condition ]a.Xo -Pl < P, where x0 is a given point and p > 0 is a small number. Thus, local tomographic data are the line integrals of f(x) for the lines intersecting the 'region of interest', the disc centred at x0 of radius p (cf. also L o c a l t o m o g r a p h y ; Tomography). 310

(1)

where fp := OflOP. The inversion formula reads: 1 ~

f(x) = 4--7E

/j ~

L(a'p) oo a - x T P

dp da,

(2)

so that (1) is based on the following idea: Keep a small neighbourhood of the singular point p = a . x in (1) and neglect the rest of the C a u c h y i n t e g r a l in (1). By definition, one needs only the local tomographic data to calculate the pseudo-local tomography function

fp(x). The basic result is: f(x) - fz(X) • C ( R 2) is a conf u n c t i o n [2], [1]. Therefore, f(x) and fp(x) have the same discontinuity curves and the same sizes of the jumps across discontinuities. It is also proved in [2] that if f • c a ( u ) , where U C R 2 is an open set, then the function fC(x) := f(x) - fp(x) has the following properties: tinuous

I f C ( x ) - f ( x ) l =O(p)

A.O. Morris MSC 1991:20C25

z ~a.x-p c ~ T x Z P dpdx,

fp(x) := 4rr2

asp--~0,

x•U,

(3)

and the convergence in (3) is uniform on compact subsets of U. If xo E S, where S is a smooth discontinuity curve of f(x), then

fC(xo) _ f+(xo) + f-(xo)

= O(plnp)

asp--+ 0.

2 (4)

Here, f--(xo) are the limiting values of f(x) as x -4 x0 from different sides of S along a path non-tangential to S. If no is a unit vector normal to S at the point x0, then for an arbitrary 7 E R, 7 ~ 0, one has lim [f(x0 + 7pno) - fC(xo + 7pn0)] = D(x0)~b(7),

p-~0

where D(.o)

:=

+ trio) - f ( x -

PSEUDO-LOCAL TOMOGRAPHY and

Other properties of fp can be found in [2], which also contains a general method for constructing a family

2 / min(l'l/'T) arccos(vt )

x / 1 - t 2 dt,

¢(7):=~j0

¢ ( - ~ ) := ¢(~),

3'>0;

~ > 0.

The function ¢(@ > O, is monotonically decreasing on R+, ¢(+0) = 1/2, ¢ ( v ) = ~2---# + o

as v -~ +oo.

If f E Ck-l(Up), for k _> 1 the kth order derivatives of f(x) exist in Uo, some of them being discontinuous across S, and S is piecewise-smooth in Up, then

fy c ck(u).

of pseudo-local tomography functions, that is, functions which are computable from local tomographic data and having the same discontinuities and the same sizes of the jumps as f(x). References [1] RAMM, A.G., AND KATSEVICH,A.: 'Pseudolocal tomography', S I A M 3. Appl. Math. 56, no. 1 (1996), 167-191. [2] RAMM, A.G., AND KATSEVICH, A.: The Radon transform and local tomography, CRC, 1996.

A.G. Ramm MSC1991: 44A12, 92C55, 65R10

311

Q A complete sup-lattice 62 together with an associative product & satisfying the distributive laws QUANTALE

-

a & ( y b i ) =Va&bi'i

for all a, bi E 62 (cf. also L a t t i c e ; D i s t r i b u t i v i t y ; Associativlty). The name 'quantale' was introduced by C.J. Mulvey [7] to provide a non-commutative extension of the concept of locale. The intention was to develop the concept of non-commutative topology introduced by R. Giles and H. Kummer [4], while providing a constructive, and non-commutative, context for the foundations of quantum mechanics and, more generally, non-commutative logic. The observation that the closed right ideals of a C * - a l g e b r a form a quantale satisfying the conditions that each element is right-sided (a&lQ _< a) and idernpotent (a&a = a) led certain authors to restrict the term 'quantale' to mean only quantales of this kind [14], but the term is now applied only in its original sense. The realization by J. Rosickj~ [15] that the development of topological concepts such as regularity required additional structure led [8] to the consideration of involutive quantales, and of the spectrum M a x A of a C*algebra A (cf. also S p e c t r u m o f a C * - a l g e b r a ) as the quantale of closed linear subspaces of A, together with the operations of join given by closed linear sum, product given by closed linear product of subspaces, and involution by involution within the C*-algebra. The rightsided elements of the spectrum M a x A are the closed right ideals of the C*-algebra A (cf. [4], [1]). By the existence of approximate units, each element a E R(Max A) of the sup-lattice of right-sided elements satisfies the condition that a&a*&a = a. By a Gel'land quantale 62 is meant an involutive unital quantale in which the

right-sided (equivalently, left-sided) elements satisfy this condition. Generalizing an observation in [15], the right-sided elements of any involutive quantale Q may be shown to admit a psendo-orthocomplement, defined by a ± = Va*~=0Q b. In any Gel'fand quantale Q, the right-sided elements are idemp0tent, and the two-sided elements form a locale. Observing that relations on a set X forming a quantale with respect to arbitrary union and composition is applied implicitly by C.A. Hoare and He Jifeng when considering the weakest pre-specification of a program [5], and noting that the quantale Q(X) in question is exactly that of endomorphisms of the sup-lattice P(X) of subsets of X x X, led to the consideration [10] of the quantale Q(S) of endomorphisms of any orthocomplemented sup-lattice S, in which the involute a* of a suppreserving mapping c~ is defined by sa* = (Vt~<~± t) ± for each s C S. In the quantale Q(X) of relations on a set X, this describes the reverse of a relation in terms of complementation of subsets. Observing that the quantale Q(H) of endomorphisms of the orthocomplemented sup-lattice of closed linear subspaces of a H i l b e r t s p a c e H provides a motivating example for this quantization of the calculus of relations, the term Hilbert quantale was introduced for any quantale isomorphic to the quantale Q(S) of an orthocomplemented sup-lattice S. Noting that the weak spectrum Maxw B of a y o n N e u m a n n a l g e b r a B is a Gel'land quantale of which the right-sided elements correspond to the projections of B and on which the right pseudo-orthocomplement corresponds to orthocomplementation of projections, a Gel'land quantale Q is said to be a yon Neurnann qnantale if a ±± = a for any right-sided element a E Q. For any yon Neumann quantale Q, the locale I(Q) of twosided elements is a complete B o o l e a n a l g e b r a . Any Hilbert quantale 62 is a yon Neumann quantale, and a yon Neumann quantale 62 is a Hilbert quantale exactly

QUANTALE if the canonical homomorphism #Q: Q --+ Q(R(Q)), assigning to each a C Q the sup-preserving mapping b E R(Q) ~-+ a*&b E R(Q) on the orthocomplemented sup-lattice R(Q) of right-sided elements of Q, is an isomorphism [10]. Any Hilbert quantale Q is a yon Neumann factor quantale in the sense that I(Q) is exactly 2. The weak spectrum Maxw B of a yon Neumann algebra B is a factor exactly if B is a factor [13] (cf. also von Neumann

algebra).

A homomorphism ~: Q --+ Q(S) from a Gel'fand quantale Q to the Hilbert quantale Q(S) of an orthocomplemented sup-lattice S is said to be a representation of Q on S [11]. A representation is said to be irreducible provided that s C S invariant (in the sense that spa < s for all a C Q) implies s = 0s or s = ls. The irreducibility of a representation p: Q --+ Q(S) is equivalent to the homomorphism being strong, in the sense that ~(1Q) = 1Q(s). A homomorphism Q/_+ Q of Gel'fand quantales is strong exactly if QI ~ Q --+ Q(S) is irreducible whenever Q --+ Q(S) is irreducible. A representation p: Q -+ Q(S) of Q on an atomic orthocomplemented sup-lattice S is said to be algebraically irreducible provided that for any atoms x , y E S there exists an a E Q such that Xpa = y (eft also A t o m i c lattice). Any algebraically irreducible representation is irreducible: the algebraically irreducible representations are those for which every atom is a cyclic generator. An algebraically irreducible representation p: d2 --+ Q(S) on an atomic orthocomplemented sup-lattice S is said to be a point of the Gel'fand quantale Q. The points of the spectrum Max A of a C*-algebra A correspond bijectively to the equivalence classes of irreducible representations of A on a Hilbert space [11]. (Presently (2000), this is subject to the conjecture that every point of M a x A is non-trivial in the sense that there exists a pure state that maps properly. For a discussion of the role of pure states in this context, see [11].) In particular, the spectrum M a x A is an invariant of the C*-algebra A. It may be noted that the Hilbert quantale Q(S) of an atomic orthocomplemented sup-lattice has, to within equivalence, a unique point; moreover, the reflection of such a Gel'fand quantale into the category of locales is exactly 2. In particular, the points of any locale are exactly its points in the sense of the theory of locales. A yon Neumann quantale Q is said to be atomic provided that the orthocomplemented sup-lattice R(Q) of its right-sided elements is atomic. For any atomic yon Neumann quantale Q the complete Boolean algebra I(Q) of two-sided elements is atomic. Moreover, the canonical homomorphism pQ: Q --+ Q(R(Q)) is algebraically irreducible exactly if Q is a yon Neumann factor quantale. A Gel'land quantale Q is said to be discrete provided that it is an atomic von Neumann quantale

that admits a central decomposition of the unit eQ C Q, in the sense that the atoms of the complete Boolean algebra I(Q) majorize a family of central projections with join eQ E Q. For any atomic yon Neumann algebra B, the weak spectrum Maxw B is a discrete yon Neumann quantale. A locale L is a discrete yon Neumann quantale exactly if it is a complete atomic Boolean algebra, hence the power set of its set of points. A homomorphism X -+ Q of Gel'fand quantales is said to be: • algebraically strong if X --+ Q --+ Q(S) is algebraically irreducible whenever Q --+ Q(S) is an algebraically irreducible representation of Q on an atomic orthocomplemented sup-lattice S; • a right embedding if it restricts to an embedding R(X) -+ R(Q) of the lattices of right-sided elements; • discrete if it is an algebraically strong right embedding. A Gel'fand quantale X is said to be spatial if it admits a discrete homomorphism X --~ Q into a discrete von Neumann quantale Q [12]. For any C*-algebra A, the canonical homomorphism Max A --+ Maxw B of its spectrum M a x A into the weak spectrum of its enveloping atomic yon Neumann algebra B is discrete, hence Max A is spatial. Similarly, a locale L is spatial as a Gel'fand quantale exactly if its canonical homomorphism into the power set of its set of points is discrete. More generally, a Gel'land quantale Q is spatial exactly if it has enough points, in the sense that if a, b E R(Q) are distinct, then there is an algebraically irreducible representation ~: Q -+ Q(S) on an atomic orthocomplemented sup-lattice S such that ~a, Pb C R(Q(S)) are distinct [12]. In other important directions, Girard quantales have been shown [16] to provide a semantics for noncommutative linear logic, and Foulis quantales to generalize the Foulis semi-groups of complete orthomodular lattices [6]. The concepts of quantal set and of sheaf have been introduced [9] for the case of idempotent right-sided quantales, generalizing those for any locale. These concepts may be localized [2] to allow the construction of a fibration from which the quantale may be recovered directly. The representation of quantales by quantales of relations has also been examined [3]. References [1] AKEMANN, C.A.: 'Left ideal structure of C*-algebras', Y. Funct. Anal. 6 (1970), 305-317. [2] BERNI-CANANI, U., BORCEUX, F., AND SUCCI-CRUCIANI,R.: 'A theory of quantale sets', J. Pure Appl. Algebra 62 (1989), 123-136. [3] BROWN, C., AND GURR, D.: 'A representation theorem for quantales', J. Pure Appl. Algebra 85 (1993), 27-42.

313

QUANTALE [4] GILES, R., AND KUMMER, H.: 'A non-commutative generalization of topology', Indiana Univ. Math. J. 21 (1971), 91 102. [5] HOAaE, C.A.R., AND JIFENG, HE: 'The weakest prespecification', Inform. Proc. Lett. 24 (I987), 127-132. [6] MULVEY, C.J.: 'Foulis quantales', to appear. [7] MULVEY, C.J.: '&', Rend. Circ. Mat. Palermo 12 (1986), 99 104. [8] MULVEY, C.J.: 'Quantales': Invited Lecture, Summer Conf. Locales and Topological Groups, Curafao, 1989. [9] MULVEY, C.J., AND NAWAZ, M.: 'Quantales: Quantal sets': Non-Classical Logics and Their Application to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory, Kluwer Acad. Publ., 1995, pp. 159-217. [10] MULVEY, C.J., AND PELLETIER, ,J.W.: 'A quantisation of the calculus of relations': Category Theory 1991, CMS Conf. Proc., Vol. 13, Amer. Math. Soc., 1992, pp. 345-360. [11] MULVEY, C.J., AND PELLETIER, J.W.: 'On the quantisation of points', J. Pure Appl. Algebra 159 (2001), 231-295. [12] MULVEY, C.J., AND PELLETIEa, J.W.: 'On the quantisation of spaces', J. Pure Appl. Math. (to appear). [13] PELLETIER, J.W.: 'Von Neumann algebras and Hilbert quantales', Appl. Cat. Struct. 5 (1997), 249 264. [14] ROSENTHAL, K.I.: Quantales and their applications, Vol. 234 of Pitman Research Notes in Math., Longman, 1990. [15] I~OSICK% J.: 'Multiplicative lattices and C*-algebras', Cah. Topol. Gdom. Diff. Cat. 30 (1989), 95-110. [16] YETTER, D.: 'Quantales and (non-commutative) linear logic', J. Symbolic Logic 55 (1990), 41 64.

C.J. Mulvey

MSC 1991: 06D99, 03G25

QUANTUM COMPUTATION~ THEORY OF- The study of the model of computation in which the state space consists of linear superpositions of classical configurations and the computational steps consist of applying local unitary operators and measurements as permitted by quantum mechanics. Quantum computation emerged in the 1980s when P. Benioff and R. Feynman realized that the apparent exponential complexity in simulating quantum physics could be overcome by using a sufficiently well controlled quantum mechanical system to perform a simulation. Quantum Turing machines were introduced by D. Deutsch in 1985 (cf. also T u r i n g m a c h i n e ) . Initial work focused on how quantum mechanics (cf. also Q u a n t u m field t h e o r y ) could be used to implement classical computation (computation in the sense of A. Church and A.M. Turing), and on analyzing whether the quantum Turing machine model provided a universal model of computation. In the early 1990s, Deutsch and R. Jozsa found an oracle problem that could be solved faster on an error-free quantum computer than on any deterministic classical computer. E. Bernstein and U. Vazirani then formalized the notion of quantum complexity from a theoretical computer science point of view, and showed that with respect to oracles that reversibly compute classical functions, quantum computers are super-polynomially more efficient than classical 314

computers. The gap was soon improved to an exponential one. This work culminated in the discovery, by P. Shor, of an efficient (that is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete logarithms. It implied that widely-used public-key cryptographic systems would be insecure if quantum computers were available. Subsequently, L. Grover found an algorithm which permitted a square-root speed-up of unstructured search. Finding new algorithmic improvements achievable with quantum computers which are not reducible to Shor's or Grover's algorithm is currently (2000) an active research area. Also of great current interest is understanding how the problem of simulating quantum systems, known to be tractable on a quantum computer, relates to the problems conventionally studied within classical computational complexity theory (cf. also C o m p l e x i t y t h e ory; C o m p u t a t i o n a l c o m p l e x i t y classes). Comprehensive introductions to quantum computation and the known quantum algorithms may be found in [4], [3]. The algorithmic work described above firmly established the field of quantum computation in computer science. However, it was initially unclear whether quantum computation was a physically realizable model. Particularly worrisome was the fact that, in nature, quantum effects are rarely observable because physical noise processes tend to rapidly remove the necessary phase relationships. To solve the problem of quantum noise, Shor and A. Steane introduced quantum error-correcting codes (cf. also E r r o r - c o r r e c t i n g code). This idea was expanded and applied by several research groups to prove that under physically reasonable assumptions, fault tolerant quantum computation is possible. Among the assumptions are the requirements that quantum noise is sufficiently weak (below some constant threshold error per quantum bit and operation), and that the basic operations can be performed in parallel. As a result, there are now many intense experimental efforts devoted toward realizing quantum computation in a wide and increasing variety of physical systems. Progress to date (2000) has been modest, with existing systems limited to just a few qubits (quantum bits), and on the order of one hundred operations [6]. Models of quantum computation largely parallel and generalize the classical models of computation. In particular, for formal studies of complexity, many researchers use various versions of quantum Turing machines (cf. also T u r i n g m a c h i n e ) , while quantum random access machines or quantum networks (also known as quantum circuits) are preferred for describing and investigating specific algorithms. To obtain a quantum version of a classical model of deterministic computation, one begins with the classical model's state space. The classical state

QUANTUM COMPUTATION, T H E O R Y OF

space usually consists of an enumerable set of configurations ¢i, with index i often constructed from strings of symbols. The quantum model associates to each ¢i a member of a standard orthonormal basis li) (called classical or logical states) of a H i l b e r t s p a c e 7{. The states of the quantum model are given by 'superpositions' of these basis states, which are unit vectors in 7/. The classical model's initial state ¢0 becomes the quantum model's initial state 10), and the classical model's transition function is replaced by a u n i t a r y o p e r a t o r U acting on ?-/. U has to satisfy certain locality restrictions that imply, for example, that Uli ) must be a superposition of classical states that are accessible by an allowed classical transition function in one step from ¢iThe computation's answer can be obtained by measuring the state after each step. In the simplest case, the classical computation's answer is determined by whether the configuration is an 'accepting' one. Accepting configurations form a set ,4 which may be associated with the closed subspace of ~/ spanned by the corresponding classical states. Let P be the projection operator onto this subspace. If the state of the quantum model is [¢), measurement has two possible outcomes. Either the new state is Pl¢)/I]Pl¢)]l with probability p = ]lP]¢)]l 2, in which case the computation 'accepts', or the state is (1 - P)[¢)/1[ (1 - P)I¢)11 with probability 1 - p , in which case the computation continues. The possible measurement outcomes can be expanded by adding a set of 'rejecting' states. In the early days of quantum computation there were lively discussions of how quantum Turing machines should halt, implying different rules about when measurements are applied during a computation. The method outlined above for obtaining a quantum model of computation from a classical model yields a generalization of the classical model restricted to reversible transition functions. This implies that quantum complexity classes do not necessarily enlarge the classical analogues, particularly for the low-lying classes or when restricted models of computation (for example, finite state automata) are involved. To obtain a generalization of the usual model of computation it suffices to extend the set of transition operators with suitable irreversible ones. One way to do that is to allow transition operators which are the composition of a measurement (satisfying an appropriate locality constraint) followed by unitary operators depending on the measurement outcome. A different approach which works well for random access machines (RAMs) is to enhance the RAM by giving it access to an unbounded number of quantum bits which can be controlled by applying quantum gates (cf. Q u a n t u m i n f o r m a t i o n p r o c e s s ing, s c i e n c e of). This is in effect how existing quantum algorithms are described and analyzed.

As in classical complexity studies, resources considered for quantum complexity include time and space (cf. also C o m p l e x i t y t h e o r y ) . In the context of irreversible processes, an additional resource that may be considered is e n t r o p y generated by irreversible operations. When analyzing algorithms based on quantum RAMs, it is also useful to separately account for classical and quantum resources. It is important to realize that if the complex coefficients of the unitary transition operators are rational (or, in general, computable complex numbers), then there is no difference between classical and quantum computability. Thus, the functions computable by quantum Turing machines are the same as those computable by classical Turing machines. An important issue in studying quantum models of computation is how to define the computation's 'answer' given that the output is intrinsically probabilistic. How this is defined can affect complexity classes. Guidance comes from studies of probabilistic (or randomized) computation, where the same issues arise. Since quantum computation with irreversibility can be viewed as a generalization of probabilistic computation, most comparisons of the quantum and classical complexity of algorithmic problems use bounds on the efficiency of probabilistic algorithms. The best known quantum complexity class is the class of bounded-error quantum polynomial-time computable languages ( B Q P ) . This is the class of languages decided in polynomial time with probability > 2/3 (acceptance) and < 1/3 (rejection) by a quantum Turing machine. Based on the oracle computing studies, the quantum factoring algorithm, and the difficulty of classically simulating quantum physics, it is conjectured that B Q P strictly contains B P P (the class of bounded-error polynomialtime computable languages for the model of probabilistic classical computation). B Q P is contained in P # ~ (the class of languages decidable in polynomial time on a classical Turing machine given access to an oracle for computing the permanent of 0-1 matrices - - this class is contained in the class P S P A C E of languages computable using polynomial working space). Thus, a proof of the important conjecture that B Q P is strictly larger than B P P will imply the long-sought result in classical computational complexity that B P P ~ P S P A C E . The relationship of B Q P to AfT) (the class of nondeterministic polynomial-time languages) is not known, though it is conjectured that A;7) ¢ B Q P . If this is not the case, it would have immense practical significance, as many combinatorial optimization problems are in YtP (cf. also AFT)). One reason for thinking that H 7 ) ¢ B Q P is the fact that Grover's algorithm provides the optimal speedup for unstructured quantum search, and it is widely believed that the reason for the 315

QUANTUM COMPUTATION, THEORY OF difficulty of solving A?7)-complete problems is that it is essentially equivalent to searching an unstructured search space. A generalization of unstructured search involves determining properties of (quantum) oracles by means of queries. In classical computation, an oracle is a function f with values in {0, 1}. The corresponding quantum oracle applies the unitary operator f" defined on basis states by ~ z , O , w ) -+ Ix, f ( z ) , w ) and ~ x , l , w } --+ Ix, l - f ( x ) , w ) . To query the oracle, one applies f" to the current state. Grover's algorithm can be cast in terms of an oracle problem. The observation that this algorithm is optimal has been extended by using the method of polynomials [1] to show that when no promise is made on the behaviour of the oracle, quantum computers are at most polynomially more efficient than classical computers. An area where there are provable exponential gaps between the efficiency of quantum and classical computation occurs when communication resources are taken into consideration. This area is known as quantum cornmunication complezity (introduced by A. Yao in 1993) arid considers problems where two parties with quantum computers and a quantum channel between them

(cf. Q u a n t u m information processing, science of) jointly compute a function of their respective inputs and wish to minimize the number of quantum bits communicated. The exponential gaps between quantum and classical communication complexity are so far confined to problems where the inputs to the function computed are constrained by a 'promise' [5]. The best known gap without a promise is a quadratic separation between classical and quantum protocols with bounded probability of error [2]. Several research groups have developed techniques for proving lower bounds on quantum communication complexity, mostly variations of the logrank lower bound also used in classical communication complexity. These results show that for some problems (for example, computing the inner product modulo two of bit strings known to the respective parties) there is little advantage to using quantmn information processing.

References [1] BEALS, R., BUIIRMAN, H., CLEVE, R., MOSCA, M., AND WOLF, R. DE: ' Q u a n t u m lower bounds by polynomials': Proc. 39th Ann. Syrup. Foundations of Computer Sci., IEEE Press, 1998, pp. 352-361. [2] BUHRMAN, H., CLEVE, R., AND WIGDERSON, A.: ~Quantum vs. classical communication and computation': Proc. 30th Ann. ACM Syrup. Theory of Computation, ACM Press, 1998, pp. 63 68. [3] GRUSKA, J.: Quantum computing, McGraw-Hill, 1999. [4] NIELSEN, M.A., AND CHUANG, I.L.: Quantum computation and quantum information, Cambridge Univ. Press, 2000.

316

[5] RAZ, R.: 'Exponential separation of q u a n t u m and classical communication complexity': Proc. 31st Ann. ACM Syrup. Theory of Computing (STOCEQT99), 1999, pp. 358-367. [6] SPECIAL FOCUS ISSUE: 'Experimental proposals for quantum computation', Fortschr. Phys. 48 (2000), 767 1138.

E.H. Knill M.A. Nielsen MSC 1991: 81Pxx, 68Q05, 68Q10, 68Q15, 68Q25

QUANTUM INFORMATION PROCESSING~ SCIENCE OF - The theoretical, experimental and technological areas covering the use of quantum mechanics for communication and computation. Quantum information processing includes investigations in quantum information theory, quantum communication, quantum computation, quantum algorithms and their complexity, and quantum control. The science of quantum information processing is a highly interdisciplinary field. In the context of mathematics it is stimulating research in pure mathematics (e.g. coding theory, .-algebras, quantum topology) as well as requiring and providing many opportunities for applied mathematics. The science of quantum information processing emerged from the recognition that usable notions of information need to be physically implementable. In the 1960s and 1970s, researchers such as R. Landauer, C. Bennett, C. Helstrom, and A. Holevo realized that the laws of physics give rise to fundamental constraints on the ability to implement and manipulate information. Landauer repeatedly stated that 'information is physical', providing impetus to the idea that it should be possible to found theories of information on the laws of physics. This is in contrast to the introspective approach which led to the basic definitions of computer science and information theory as formulated by A. Church, A.M. Turing, C. Shannon and others in the first half of the 20th century (cf. also T u r i n g machine). Early work in studying the physical foundations of information focused on the effects of energy limitations and the need for dissipating heat in computation and communication. Beginning with the work of S. Wiesner on applications of quantum mechanics to cryptography in the late 1960s, it was realized that there may be intrinsic advantages to using quantum physics in information processing. Quantum cryptography and quantum communication in general were soon established as interesting and non-trivial extensions of classical coinmmfication based on bits. T h a t quantum mechanics may be used to improve the efficiency of algorithms was first realized when attempts at simulating quantum mechanical systems resulted in exponentially complex algorithms compared to the physical resources associated with the system simulated. In the 1980s, P. Benioff and R. Feynman introduced the idea of a quantum computer

QUANTUM INFORMATION PROCESSING, SCIENCE OF for efficiently implementing quantum physics simulations. Models of quantum computers were developed by D. Deutsch, leading to the formulation of artificial problems that could be solved more efficiently by quantum than by classical computers. The advantages of quantum computers became widely recognized when P. Shor (1994) discovered that they can be used to efficiently factor large numbers - - a problem believed to be hard for classical deterministic or probabilistic computation and whose difficulty underlies the security of widelyused public-key encryption methods. Subsequent work established principles of quantum error-correction to ensure that quantum information processing was robustly implementable. See [5], [2] for introductions to quantum information processing and a quantum mechanics tutorial. In the context of quantum information theory, information in the sense of Shannon is referred to as classical information. The fundamental unit of classical information is the bit, which can be understood as an ideal system in one of two states or configurations, usually denoted by 0 and 1. The fundamental units of quantum information are qubits (short for quantum bits), whose states are identified with all 'unit superpositions' of the classical states. It is common practice to use the bra-ket conventions for denoting states. In these conventions, the classical configurations are denoted by 10) and I1), and superpositions are formal sums al0 } +/311>, where a and/3 are complex numbers satisfying I(~l2 + 1/312 = 1. The states 10} and I1> represent a standard orthonormal basis of a two-dimensional H i l b e r t space. Their superpositions are unit vectors in this space. The state space associated with n > 1 qubits is formally the tensor product of the Hilbert spaces of each qubit. This state space can also be obtained as an extension of the state space of n classical bits by identifying the classical configurations with a standard orthonormal basis of a 2*~-dimensional Hilbert space. Access to qubit states is based on the postulates of quantum mechanics with the additional restriction that they are local in the sense that elementary operations apply to one or two qubits at a time. Most operations can be expressed in terms of standard measurements of a qubit and two-qubit quantum gates. The standard qubit measurement has the effect of randomly projecting the state of the qubit onto one of its classical states; this state is an output of the measurement (accessible for use in a classical computer if desired). For example, using the tensor product representation of the state space of several qubits, a measurement of the first qubit is associated with the two projection operators p(1) = Po®I®. • • and 1 - P ( * ) , where PolO) = I0) and Poll) = IO). If

~b is the initial state of the qubits, then the measurement outcome is 0 with probability P0 = IIP0~bll2, in which case the new state is Po~b/po, and the outcome is 1 with probability 1 - P 0 = IIPz~bll2, with new state Pl~b/(1 - Po). This is a special case of a yon N e u m a n n measurement. A general two-qubit quantum gate is associated with a u n i t a r y o p e r a t o r U acting on the state space of two qubits. Thus, U may be represented by a (4 × 4)-dimensional unitary matrix in the standard basis of two qubits. The quantum gate may be applied to any two chosen qubits. For example, if the state of n qubits is ¢ and the gate is applied to the first two qubits, then the new state is given by (U ® I @ - . . ) ¢ . Another important operation of quantum information processing is preparation of the 10} state of a qubit, which can be implemented in terms of a measurement and subsequent applications of a gate depending on the outcome. Most problems of theoretical quantum information processing can be cast in terms of the elementary operations above, restrictions on how they can be used and an accounting of the physical resources or cost associated with implementing the operations. Since classical information processing may be viewed as a special case of quantum information processing, problems of classical information theory and computation are generalized and greatly enriched by the availability of quantum superpositions. The two main problem areas of theoretical quantum information processing are quantum computation and quantum communication. In studies of q u a n t u m computation (cf. Q u a n t u m c o m p u t a t i o n , t h e o r y of) one investigates how the availability of qubits can be used to improve the efficiency of algorithmic problem solving. Resources counted include the number of quantum gates applied and the number of qubits accessed. This can be done by defining and investigating various types of quantum automata, most prominently quantum Turing machines, and studying their behaviour using approaches borrowed from the classical theory of automata and languages. It is convenient to combine classical and quantum automata, for example by allowing a classical computer access to qubits as defined above, and then investigating the complexity of algorithms by counting both classical and quantum resources, thus obtaining tradeoffs between the two. Most of the complexity classes for classical computation have analogues for quantum computation, and an important research area is concerned with establishing relationships between these complexity classes (cf. also C o m p l e x i t y t h e o r y ) . Corresponding to the classical class 7) of polynomially decidable languages is the class of languages decidable in bounded-error quantum polynomial time, B Q P . While it is believed that 79 is 317

QUANTUM INFORMATION PROCESSING, SCIENCE OF properly contained in B Q P , whether this is so is at present (2000) an open problem. B Q P is known to be contained in the class P # ~ (languages decidable in classical polynomial time given access to an oracle for computing the permanent of 0-1 matrices), but the relationship of B Q P to the important class of nondeterministic polynomial time languages A;7) is not known (cf. also In quantum communication one considers the situation where two or more entities with access to local qubits can make use of both classical and quantum (communication) channels for exchanging information (cf. also Q u a n t u m c o m m u n i c a t i o n c h a n n e l ) . The basic operations now include the ability to send classical bits and the ability to send quantum bits. There are two main areas of investigation in quantum communication. The first aims at determining the advantages of quantum communication for solving classically posed communication problems with applications to c r y p t o g r a p h y and to distributed computation. The second is concerned with establishing relationships between different types of communication resources, particularly with respect to noisy quantum channels, thus generalizing classical communication theory (cf. also S h a n n o n theorem). Early investigations of quantum channels focused on using them for transmitting classical information by encoding a source of information (cf. I n f o r m a t i o n , s o u r c e of) with uses of a quantum channel (cf. Q u a n t u m c o m m u n i c a t i o n c h a n n e l ) . The central result of these investigations is Holevo's bound (1973) on the amount of classical information that can be conveyed through a quantum channel. Asymptotic achievability of the bound (using block coding of the information source) was shown in the closing years of the 20th century. With some technical caveats, the bound and its achievability form a quantum information-theoretic analogue of Shannon's capacity theorem for classical communication channels. Quantum cryptography, distributed quantum computation and quantum memory require transmitting (or storing) quantum states. As a result it is of great interest to understand how one can communicate quantum information through quantum channels. In this case, the source of information is replaced by a source of quantum states, which are to be transmitted through the channel with high fidelity. As in the classical case, the state is encoded before transmission and decoded afterwards. There are many measures of fidelity which may be used to evaluate the quality of the transmission protocol. They are chosen so that a good fidelity value implies that with high probability, quantum information processing tasks behave the same using the original or the 318

transmitted states. A commonly used fidelity measure is the Bures-Uhlmann fidelity, which is an extension of the Hilbert space norm to probability distributions of states (represented by density operators). In most cases, asymptotic properties of quantum channels do not depend on the details of the fidelity measure adopted. To improve the reliability of transmission over a noisy quantum channel, one uses quantum error-correcting codes to encode a state generated by the quantum information source with multiple uses of the channel (cf. also E r r o r - c o r r e c t i n g c o d e ) . The theory of quantum codes can be viewed as an extension of classical coding theory. Concepts such as minimum distance and its relationship to error correction generalize to quantum codes. Many results from the classical theory, including some linear programming upper bounds and the Gilbert-Varshamov lower bounds on the achievable rates of classical codes, have their analogues for quantum codes. In the classical theory, linear codes are particularly useful and play a special role. In the quantum theory, this role is played by the stabilizer or additive quantum codes, which are in one-to-one correspondence with self-dual (with respect to a specific symplectic inner product) classical GF2linear codes over GF4 (cf. F i n i t e field). The capacity of a quantum channel with respect to encoding with quantum codes is not as well understood as the capacity for transmission of classical information. The exact capacity is known only for a few special classes of quantum channels. Although there are information-theoretic upper bounds, they depend on the number of channel instances, and whether or not they can be achieved is an open problem (as of 2000). A further complication is that the capacity of quantum channels depends on whether one-way or two-way classical communication may be used to restore the transmitted quantum information [1]. The above examples illustrate the fact that there are many different types of information utilized in quantum information theory, making it a richer subject than classical information theory. Another physical resource whose properties appear to be best described by information-theoretic means is quantum entanglement. A quantum state of more than one quantum system (e.g. two qubits) is said to be entangled if the state cannot be factorized as a product of states of the individual quantum systems. Entanglement is believed to play a crucial role in quantum information processing, as demonstrated by its enabling role in effects such as quantum key distribution, superdense coding, quantum teleportation, and quantum error-correction. Beginning in 1995, an enormous amount of effort has been devoted to understanding the principles governing the behaviour of

QUANTUM INFORMATION PROCESSING, SCIENCE OF entanglement. This has resulted in the discovery of connections between quantum entanglement and classical information theory, the theory of positive mappings [3] and majorization [4] (cf. also M a j o r l z a t l o n o r d e r i n g ) . The investigation of quantum channel capacity, entanglement and many other areas of quantum information processing involves various quantum generalizations of the notion of e n t r o p y , most notably the von Neumann entropy. The yon Neumann entropy is defined as H(p) = Tr p log 2 (p) for density operators p (p is positive Hermitian and of trace 1). It has many (but not all) of the properties of the classical information function H(.) (cf. I n f o r m a t i o n , a m o u n t of). Understanding these properties has been crucial to the development of quantum information processing (see [5], [8], [6] for reviews). Probably the most powerful known result about the yon Neumann entropy is the strong subadditivity inequality. Many of the bounds on quantum communication follow as easy corollaries of strong subadditivity. Whether still more powerful entropic inequalities exist is not known (as of 2000). An important property of both classical and quantum information is that although it is intended to be physically realizable, it is abstractly defined and therefore independent of the details of a physical realization. It is generally believed that qubits encapsulate everything that is finitely realizable using accessible physics. This belief implies that any information processing implemented by available physical systems using resources appropriate for those systems can be implemented as efficiently (with at most polynomial overhead) using qubits. It is noteworthy that there is presently (2000) no proof that information processing based on quantum field theory (cf. Q u a n t u m field t h e o r y ) is not more efficient than information processing with qubits. Furthermore, the as-yet unresolved problem of combining quantum mechanics with general relativity in a theory of quantum gravity prevents a fully satisfactory analysis of the information processing power afforded by fundamental physical laws. Much effort in the science of quantum information processing is being expended on developing and testing the technology required for implementing it. An important task in this direction is to establish that quantum information processing can be implemented robustly in the presence of noise. At first it was believed that this was not possible. Arguments against the robustness of quantum information were based on the apparent relationship to analogue computation (due to the continuity of the amplitudes in the superpositions of configurations) and the fact that it seemed difficult to observe quantum superpositions in nature (due to the rapid loss of phase relationships, called decoherence). However, the

work on quantum error-correcting codes rapidly led to the realization that, provided the physical noise behaves locally and is not too large, it is at least in principle possible to process quantum information fault tolerantly. Research in how to process quantum information tellably continues; the main problem is improving the estimates on the maximum amount of tolerable noise for general models of quantum noise and for the types of noise expected in specific physical systems. Other issues include the need to take into consideration restrictions imposed by possible architectures and interconnection networks. There are many physical systems that can potentially be used for quantum information processing [7]. An active area of investigation involves determining the general mathematical features of quantum mechanics required for implementing quantum information. More closely tied to existing experimental techniques are studies of specific physical systems. In the context of communication, optical systems are likely to play an important role, while for computation there are proposals for using electrons or nuclei in solid state, ions or atoms in electromagnetic traps, excitations of superconductive devices, etc. In all of these, important theoretical issues arise. These issues include how to optimally use the available means for controlling the quantum systems (quantum control), how to best realize quantum information (possibly indirectly), what architectures can be implemented, how to translate abstract sequences of quantum gates to physical control actions, how to interface the system with optics for communication, refining the theoretical models for how the system is affected by noise and thermodynamic effects, and how to reduce the effects of noise. References [1] BENNETT, C.H., DIVINCENZO, D.P., SMOLIN, J.A., AND WOOTTERS, W.Z.: 'Mixed state entanglement and quantum error-correcting codes', Phys. Rev. A 54 (1996), 3824-3851. [2] GRUSKA,3.: Quantum computing, McGraw-Hill, 1999. [3] HORODECKI, M., HORODECKI, P., AND HORODECKI, R.: 'Separability of mixed states: necessary and sufficient conditions', Phys. Lett. A 223, no. I-2 (1996), I-8. [4] NIELSEN, M.A.: 'A partial order on the entangled states', quant-ph/9811053 (1998). [5] NIELSEN, M.A., AND CHUANa, I.L.: Quantum computation and quantum information, Cambridge Univ. Press, 2000. [6] OHYA, M., AND PETZ, D.: Quantum entropy and its use, Springer, 1993. [7] SPECIAL FOCUS ISSUE: 'Experimental proposals for quantum computation', Fortschr. Phys. 48 (20OO), 767-1138. [8] WEHRL, A.: 'General properties of entropy', Rev. Mod. Phys. 50 (1978), 221.

MSC 1991: 81Pxx, 94Axx, 81P15

68Q05,

68Q10,

E.H. KniIl M.A. Nielsen 68Q15, 68Q25,

319

QUASI-REGULAR MAPPING QUASI-REGULAR MAPPING, mapping with bounded distortion - Let G be an open connected subset of the Euclidean space R n, n > 2, and let f : G -+ R n l~n be a continuous mapping of Sobolev class Wlo c (G) (cf. also S o b o l e v space). Then f is said to be quasi-regular if there is a n u m b e r / ( E [1, oo) such that the inequality

If'(x)l ~ _
= m a x {lf'(x)hl

: Ihl = 1}

and Jf(x) is the Jacobian determinant (cf. also J a c o b i a n ) of f at x. The smallest / ( _> 1 in the above inequality is called the outer dilatation of f and is denoted b y / ( o (f)- If f is quasi-regular, then the inner dilatation Ki(f) is the smallest constant K _> 1 in the inequality

J~(x) < _ / ( e ( f ' ( x ) F ,

e(f'(x))

= min {If'(x)h] : Ih} = 1}.

maximal dilatation is the number / ( ( f ) = m a x { / ( o ( f ) , / ( i ( f ) } , and one says that f is /(-quasiregular i f / ( ( f ) <_/(.

The

P a r t i c u l a r cases. For the dimension n = 2 a n d / ( = 1, the class of/(-quasi-regular mappings agrees with that of the complex-analytic functions (cf. also A n a l y t i c f u n c t i o n ) . Injective quasi-regular mappings in dimensions n >_ 2 are called quasi-conformal (see Q u a s i c o n f o r m a l m a p p i n g ; [4], [8], [10]). For n = 2 and / ( > 1, a /(-quasi-regular mapping f : G --+ R 2 can be represented in the form ~ o w, where w : G --+ G' is a K-quasi-conformal h o m e o m o r p h i s m and qo: G' -+ R 2 is an analytic function (StoRow's theorem). There is no such representation in dimensions n _> 3. Note that for all dimensions n _> 2 a n d / ( > 1 the set of those points where a/(-quasi-conformal mapping is not differentiable may be non-empty and the behaviour of the mapping may be very curious at the points of this set, which has also L e b e s g u e m e a s u r e zero. Thus, there is a substantial difference between the two c a s e s / ( > 1 a n d / ( = 1. For higher dimensions n _> 3 the theory of quasiregular mappings is essentially different from the plane case n = 2. There are several reasons for this: a) there are neither general existence theorems nor counterparts of power series expansions in higher dimensions; b) the usual methods of function theory are not applicable in the higher-dimensional setup; c) in the plane case the class of conformal mappings is very rich (cf. also C o n f o r m a l m a p p i n g ) , while in higher dimensions it is very small (J. Liouville proved 320

that for n _> 3 a n d / ( = 1, sufficiently smooth K-quasiconformal mappings are restrictions of MSbius transformations, see Q u a s i - c o n f o r m a l m a p p i n g ) ; d) for dimensions n _> 3 the branch set (i.e. the set of those points at which the mapping fails to be a local homeomorphism) is more complicated than in the two-dimensional case; for instance, it does not contain isolated points (see also Z o r i c h t h e o r e m ) . In spite of these difficulties, many basic properties of analytic functions have their counterparts for quasiregular mappings. In a pioneering series of papers, Yu.G. Reshetnyak proved in 1966-1969 that these mappings share the fundamental topological properties of complex-analytic functions: non-constant quasi-regular mappings are discrete, open and sense-preserving. He also proved important convergence theorems for these mappings and several analytic properties: they preserve sets of zero Lebesgue n-measure, are differentiable almost everywhere and are HSlder continuous (cf. also H S l d e r c o n d i t i o n ) . The Reshetnyak theory (which uses the phrase mapping with bounded distortion for 'quasi-regular mapping') makes use of Sobolev spaces, potential theory, partial differential equations, calculus of variations, and differential geometry. These results led to the discovery of the connection between partial differential equations and quasi-regular mappings. One of the main results says that if f is quasi-regular and nonzero, then log If(x)] is a solution of a certain non-linear elliptic 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 . For n = 2 and K = 1 these equations reduce to the familiar L a p l a c e e q u a t i o n and the corresponding potential theory is linear (see [5]). The word 'quasi-regular' was introduced in this meaning in 1969 by O. Martio, S. Rickman and J. Vgisglg, who found another approach to quasi-regular mappings. Their approach uses techniques and tools from the theory of spatial quasi-conformal mappings and is based, in part, on Reshetnyak's fundamental results. The tools they use in their 1969-1972 work involve the modulus of a family of curves (the e x t r e m a l l e n g t h ) and capacities of condensers. Their work showed the power of the method of the modulus of a family of curves, made these mappings more widely known, and led to a series of important results (cf. below). Some of the topics considered for quasi-regular mappings include: i) stability theory, which refers to the study o f / ( quasi-regular mappings with d i l a t a t i o n / ( > 1 close to one, see [6]; ii) v a l u e - d i s t r i b u t l o n t h e o r y , such as the analogue of the P i c a r d t h e o r e m and its refinements, see [7];

QUASI-SYMMETRIC F U N C T I O N OF A C O M P L E X VARIABLE iii) non-linear potential theory (cf. also P o t e n t i a l t h e o r y ) and connection of quasi-regular mappings to partial differential equations, see [2]; iv) geometric analysis, differential forms, and applications to elasticity theory, see [3]; v) conformal invariants and quasi-regular mappings, see [9], [1]. References [1] ANDERSON, G.D., VAMANAMURTHY,M.K., aND VUOaINEN, M.: Conformal invariants, inequalities and quasiconformal mappings, Wiley, 1997, book with a software diskette. [2] HEINONEN, J., KILPELAINEN, T., AND MARTIO, O.: Nonlinear potential theory of degenerate elliptic equations, Oxford Math. Monographs. Oxford Univ. Press, 1993. [3] IWANIEC, T.: 'p-Harmonic tensors and quasiregular mappings', Ann. of Math. 136 (1992), 39-64. [4] LEHTO, O., AND VIRTANEN, K.I.: Quasiconformal mappings in the plane, second ed., Vol. 126 of Grundl. Math. Wissenschaft., Springer, 1973. [5] RESHETNYAK: Yu.G.: Space mappings with bounded distortion, Vol. 73 of Transl. Math. Monogr., Amer. Math. Soc., 1989. (In Russian.) [6] I~ESHETNYAK, Yu.G.: Stability theorems in geometry and analysis, Kluwer Acad. Publ., 1994. [7] RICKMAN, S.: Quasiregular mappings, Vol. 26 of Ergeb. Math. Grenzgeb., Springer, 1993. [8] VXISXLX,J.: Lectures on n-dimensional quasiconformal mappings, Vol. 229 of Lecture Notes in Mathematics, Springer, 1971. [9] VUORINEN, M.: Conformal geometry and quasiregular mappings, Vol. 1319 of Lecture Notes in Mathematics, Springer, 1988. [10] VUORINEN, M. (ed.): Quasicoi~formal space mappings: A collection of surveys, 1960-1990, Vol. 1508 of Lecture Notes in Mathematics, Springer, 1992. M. Vuorinen

MSC 1991: 30C62, 30C65, 26B99 QUASI-SYMMETRIC F U N C T I O N OF A COMPLEX VARIABLE - An a u t o m o r p h i s m h of the real axis R (i.e. a sense-preserving h o m e o m o r p h i s m h of R onto itself) is said to be M-quasi-symmetric on R (notation: h E M - Q S ( R ) ) if

M - 1 < h(x + t) - h(x) < M -

-

-

t)

-

holds for all x E R and all t > 0. An automorphism h of R is quasi-symmetric (notation: h E QS(R)) if h E M-QS(R) for some M _> 1. A. Beurling and L.V. Ahlfors established a close relation between h E QS(R) and quasi-conformal mappings of the upper half-plane H onto itself (cf. also Q u a s i - e o n f o r m a l m a p p i n g ) , cf. statements A), B) below. The term 'quasi-symmetric' was proposed in [2]. A) Any K-quasi-conformal automorphism f of H normalized by the condition f(oc) = oo admits a homeomorphic extension to the closure of H and generates

in this way h E M - Q S ( R ) , where M = )~(K) := - 1, cf. [1], [6]. Here #(r), 0 < r < 1, is the module of the ring domain D \ [ 0 , r], D = {z E C: Iz] < 1} (cf. also M o d u l u s o f a n a n n u l u s ) . The bound for M is sharp. B) Conversely, for any M k 1 there exists a constant K ( M ) such that an arbitrary h E M - Q S ( R ) has a quasiconformal extension f to H with f ( o c ) = oc whose maximal dilatation K[f] satisfies K[f] < K ( M ) , cf. [1], [6]. The best value of K ( M ) known today (2000) is min{M a/2, 2 M - 1}, cf. [5]. Quasi-symmetric functions on R satisfy the following: If h E QS(R), so does h - l ; if h~,h2 E QS(R), so does hi o h2. However, there exist singular functions on R that are also quasi-symmetric [1]. One may also distinguish the class M - Q S ( T ) of Mquasi-symmetric automorphisms h of the unit circle T = 0D. To this end, let lal denote the length of an open arc a C T. Then h E M - Q S ( T ) if there is an M > 1 such that for any pair (~, ~ of open disjoint subarcs of T with a common end-point Ih(~)l < M. Ih(~)l The class QS(T) = U M > I M - Q S ( T ) has some nice properties: no boundary point of D is distinguished, Hhlder continuity is global on T and any h E QS(T) may be represented by an absolutely convergent F o u r i e r series, cf. [3], [4]. Quasi-symmetric automorphisms of R or T are intimately connected with quasi-circles, i.e. image curves of a circle under a quasi-conformal automorphism of C. Let ff be a J o r d a n c u r v e in the finite plane C and let f (or F ) be a c o n f o r m a l m a p p i n g of the inside (or outside) domain of ,7 onto D (respectively, D* = C \ D). Then h = F o f - 1 is an automorphism of T and h E QS(T) is equivalent to J being a quasi-circle [6], [7]. A sense-preserving homeomorphism h : T ~ C is said to be an M-quasi-symmetric function on T (notation: h E M - Q S ( T , C)) if for any triple q , z2, za E T, z2 # z3, A

_

_

Ih(z ) - h(z2)l < M. Ih(z2) - h(za)l -

Obviously, M - Q S ( T ) C M - Q S ( T , C). One defines h to be a quasi-symmetric function on T if h E QS(T, C) := UM>IM-QS(T,C). For any h E Q S ( T , C ) the Jordan curve h(W) is a quasi-circle, cf. [8]. The following characterization of QS(T, C) was given by P. ~hkia and J. Vgisgl/i in [9]: For a,b,x E T with b ¢ x, put p = ] a - x l / I b - x I. Then h E Q S ( T , C ) if and only if there is an automorphism rl of [0, +oc) such that I h ( a ) - h(x)l/lh(b ) - h ( x ) l < rl(p) for all admissible triples a, b, x. 321

QUASI-SYMMETRIC

FUNCTION

OF A COMPLEX

References [1] BEURLING, A., AND AHLFORS, L.V.: 'The boundary correspondence under quasiconformM mappings', Acta Math. 96 (1956), 125-142. [2] KELINGOS, J.A.: Contributions to the theory of quasiconformal mappings, Diss. Univ. Michigan, 1963. [3] KRZY2, J.G.: 'Quasicircles and harmonic measure', Ann. Acad. Sci. Fenn. Ser. A.I. Math. 12 (1987), 19-24. [4] KRzY~, J.G., AND NOWAK, M.: 'Harmonic automorphisms of the unit disk', J. Comput. Appl. Math. 105 (1999), 337 346. [5] LEHTINEN, M.: 'Remarks on the maximal dilatations of the Beurling-Ahlfors extension', Ann. Acad. Sci. Fenn. Ser. A.I. Math. 9 (1984), 133-139.

322

VARIABLE [6] LEHTO, O., AND VIRTANEN, K.I.: Quasiconformal mappings in the plane, Springer, 1973. [7] PARTYKA,D.: 'A sewing theorem for complementary Jordan domains', Ann. Univ. Mariae Curie-Sktodowska Sect. A 41 (1987), 99-103. [8] POMMERENKE, CH.: Boundary behaviour of conformal maps, Springer, 1992. [9] TUKIA, P., AND VAISALA, J.: 'Quasisymmetric embeddings of metric spaces', Ann. Acad. Sci. Fenn. Ser. A.I. Math. 5 (1980), 97-114. Jan G. K r z y i

MSC1991:

30C62, 30C99

R RABINOWITSCtt TRICK - This 'trick' deduces the general Hilbert Nullstellensatz (cf. H i l b e r t t h e o r e m ) from the special case t h a t the polynomials have no common zeros. Indeed, let f, f l , . . . , f ~ E R := k [ x l , . . . , x ~ ] , where k is a field. If f vanishes on the common zeros of f l , - . . , fro, then there are polynomials ao, a l , . . . , a,~ E R[Xo] such t h a t

ao(1 - xof) + air1 + "'" + a,~f,~ = 1. Substitution of x0 = 1 / f into this identity and clearing out the denominator shows t h a t

blfl + " " + bmfm = f ' , where p := maxdegxo ai and bj = aj[xo=l/ff'. This ingenious device was published in the one(!) page article [1]. References

[1] RABINOW~TSCH, J.L.: 'Zum Hilbertschen Nullstellensatz', Math. Ann. 102 (1929), 520.

W. Dale Brownawell M S C 1991: 14Axx RADIAL BASIS FUNCTION - The radial basis function method is a multi-variable scheme for function interpolation, i.e. the goal is to approximate a c o n t i n u o u s f u n c t i o n f by a relatively simple interpolant s which meets f at a certain number (usually finite) of prescribed points (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 ; I n t e r p o l a t i o n ) . In the n-dimensional real space R n, given a continuous function f : R ~ --> R and socalled ccntres xj E l:t n, j = 1, 2 , . . . ,ra, the interpolant to f at the centres reads m

s(x) = E

Aj¢ ( l l x - xjll) ,

x e R ~,

j=l

where ¢ : R + --+ R is the radial basis function, the norm is the n-dimensional Euclidean norm and the real coefficients ~j are fixed through the interpolation conditions

s(xj) = f ( x j ) ,

j= l,...,m.

Norms I['ll other t h a n Euclidean are possible in principle, but rarely used. In particular, the remarkable existence properties described below are usually no longer guaranteed if the norm is not Euclidean. Examples of radial basis functions are the multi-quadric function ¢(r) = x / ~ + c 2, c a positive p a r a m e t e r [7], which is known to be particularly useful in applications, the thinplate spline ¢(r) = r 2 log r [6], the aaussian function ¢(r) = e x p ( - c 2 r 2 ) , and the linear radial basis function ¢(r) = r. For the thin-plate spline and several other radial basis functions, a linear (generally, loworder) polynomial has to be added to s with side conditions ~ j = l Aj = ~ j = l ) t j X j -~ 0 in order to be able to solve the interpolation equations uniquely. In that case, the centres must not lie on a straight line, but may otherwise be arbitrarily distributed ('scattered'). For multi-quadrics, Gaussian and linear radial functions, a m o n g others, the extra geometric condition is not needed: the interpolation linear system defined through the above interpolation conditions is uniquely solvable for all m > 1 and n if the centres are distinct [9]. This is one of the most striking and useful features of radial basis function interpolation. In fact, for large classes of radial basis functions, which contain all the examples mentioned, the m a t r i x which defines the coefficients through the interpolation conditions is conditionally positive definite (or conditionally negative definite) [9], which means t h a t it is positive (negative) definite on a subspace of R m with small co-dimension. See, for instance, [10] or [5] for reviews of this method. For the history of the m e t h o d see [7]. Besides the question of existence and uniqueness outlined above, the question of (uniform) convergence (cf. also U n i f o r m c o n v e r g e n c e ) of s to f when the centres become dense in a domain or on R ~ is of central importance. J. Duchon [6] has studied this issue for scattered centres xj in a Lipschitz domain ft C R n for thinplate splines and proved uniform convergence provided cgf~ satisfies a cone condition, the xj become dense in f~

RADIAL BASIS FUNCTION and f is sufficiently smooth. His work was generalized to multi-quadrics, Gaussians and others (see, for instance [13], [8]), while the question of uniform convergence and approximation order on infinite square grids of spacing h > 0 was settled in [2]. Estimates for the interpolation error when h ~ 0 have been given (see [2]) and provide error bounds of order O(h n+l) in n dimensions for the linear radial basis function ¢(r) = r, for example, if f is sufficiently smooth.

[7"] HARDY, R.L.: 'Theory and applications of the multiquadricbiharmonic method', Computers Math. Appl. 19 (1990), 163208. [8] MADYCH,W.R., AND NELSON, S.A.: 'Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation', J. Approx. Th. 70 (1992), 94-114. [9] MICCHELLI, C.A.: 'Interpolation of scattered data: distance matrices and conditionally positive definite functions', Constructive Approx. 1 (1986), 11-22. [10] POWELL, M.J.D.: 'The theory of radial basis function approximation', in W.A. LIGHT (ed.): Advances in Numerical

The remarkable convergence orders which occur, together with the above existence theorems, make the radial basis function method attractive if n is large, especially when the centres are scattered, because in that case other schemes, such as polynomial interpolation (cf. e.g. A l g e b r a i c p o l y n o m i a l o f b e s t a p p r o x i m a t i o n ) , are often ruled out.

Analysis II. Wavelets, Subdivision, and Radial Functions,

Since most of the radial basis functions are globally supported (however, see [12] or [4] for compactly supported ones), special attention is needed in the computation of the approximants, in particular if m is large. Major contributions to this aspect can be found in [11] and [1], which include working software admitting efficient computation of the desired coefficients Aj for m = 50000 and larger. Thin-plate splines and multiquadrics for n = 2, 3, 4 have also received consideration in implementations. Given the accuracy and availability of the methods for arbitrary n and m, other approximation schemes (not interpolation) such as wavelet schemes [3], quasiinterpolation or least-squares approaches have been studied and used successfully, but the real advantage of the scheme remains in its availability for multi-variable interpolation to scattered data. The applications range from modelling the Earth's surface [7] to optimization problems and applications in the numerical solutions of partial differential equations in high dimensions.

Oxford Univ. Press, 1992, pp. 105-210. [11] POWELL, M.J.D.: 'A new iterative method for thin plate spline interpolation in two dimensions', Ann. Numer. Math. 4 (1997), 519 527. [12] WENDLAND, H.: 'Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree', Adv. Comput. Math. 4, no. 10 (1995), 389-396. [13] Wu, Z., AND SCHABACK, R.: 'Local error estimates for radial basis function interpolation of scattered data', IMA J. Numer. Anal. 13 (1993), 13-27.

Martin Buhmann MSC 1991: 41A05, 41A63, 41A30 RATIONAL TANGLES - A family of 2-tangles (cf. T a n g l e ) classified by J.H. Conway. The 2-tangle of Fig. 1 is called a rational tangle with Conway notation T ( a l , . . . , as). It is a rational p/q-tangle if p 1

-=an-t-

q

1 •

a,~-i + ' " + a-Y

The fraction p/q is called the slope of the tangle and can be identified with the slope of the meridian disc of the solid torus that is the branched double covering of the rational tangle.

a3

References

[1] BEATSON, R.K., AND GREENGARD, L.: 'A short course on fast multiple methods', in M. AINSWORTH,J. LEVESLEY, M. MARLETTA, AND W. LIGHT (eds.): Wavelets, Multilevel Methods and Elliptic PDEs, Oxford Univ. Press, 1997, pp. 1-37. [2] BUHMANN, M.D.: 'Multivariate cardinal-interpolation with radial-basis functions', Constructive Approx. 6 (1990), 225255. [3] BUHMANN, M.D.: 'Multiquadric pre-wavelets on non-equally spaced knots in one dimension', Math. Comput. 64 (1995), 1611-1625. [4] BUHMANN, M.D.: 'Radial functions on compact support', Proc. Edinburgh Math. Soc. 41 (1998), 33 46. [5] BUHMANN, M.D.: 'Radial basis functions', Acta Numeriea 9

an_1

Y A J.

n is odd

a3!

a n _ l ~ ~

(2000), 1-38. [6] DUCHON, J.: 'Splines minimizing rotation-invariante seminorms in Sobolev spaces', in W. SCHEMPP AND K. ZELLER (eds.): Constructive Theory of Functions of Several Variables, Springer, 1979, pp. 85-100.

324

n is even

Fig. 1.

"

RAYLEIGH-FABER-KRAHN INEQUALITY Conway proved t h a t two rational tangles are mnbient isotopic (with boundary fixed) if and only if their slopes are equal. A rational n-tangle (also called an n-bridge n-tangle) is an n-tangle that can be obtained from the identity tangle by a finite number of additions of a single crossing. 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. [2] KAWAUCHI,A.: A survey of knot theory, Birkhguser, 1996.

Jozef Przytycki MSC 1991:57M25 RAYLEIGH-FABER-KRAHN INEQUALITY - An inequality concerning the lowest eigenvalue of the L a p l a c e o p e r a t o r , with Dirichlet boundary condition, on a bounded domain in R n (n > 2). Let 0 < AI(Q) < A2(~) < A3(fl) < -.- be the Diriehlet eigenvalues of the Laplaeian in ~ C R n, i.e.,

-Au = tu u = 0

in f~,

on the boundary of f~.

(1) (2)

(Cf. also D i r i c h l e t b o u n d a r y c o n d i t i o n s ; D i r i c h l e t e i g e n v a l u e . ) Here, A is the Laplace operator and f~ is an open bounded subset of R n (n >_ 2). If n = 2, the Dirichlet eigenvalues are proportional to the square of the eigenfrequencies of an elastic, homogeneous, vibrating membrane with fixed boundary (cf. also N a t u r a l frequencies). The Rayleigh-Faber-Krahn inequality for the membrane (i.e., n = 2) states that A1 -> -~Jg,1 X-'

(3)

where jo,1 = 2.4048... is the first zero of the Bessel function of order zero (cf. also B e s s e l f u n c t i o n s ) and A is the area of the membrane. Equality is attained in (3) if and only if the membrane is circular. In other words, among all membranes of given area, the circle has the lowest fundamental frequency. This inequality was conjectured by Lord Rayleigh [14], based on exact calculations for simple domains and a variational argument for near circular domains. In 1918, R. Courant [5] proved the weaker result that among all membranes of the same perimeter L, the circular one yields the least lowest eigenvalue, i.e.,

L2

m e t r i c i n e q u a l i t y in dimension n, ( 1)2/~

,

~/~ •

Ffi

(5)

was proven by Krahn [8]. In (5), jm,1 is the first positive zero of the Bessel function J,~, If~] is the volume of the domain and C , = rcn/2/F(n/2 + 1) is the volume of the n-dimensional unit ball. Equality is attained in (5) if and only if f~ is a ball. The proof of the Rayleigh-Faber-Krahn inequality rests upon two facts: a variational characterization for the lowest Dirichlet eigenvalue and the properties of symmetric decreasing rearrangements of functions. The variational characterization of the lowest eigenvalue is given by AI(•) :

inf fa(Vu)2dx ~eH~(a) fa u2 dx

(6)

Concerning decreasing rearrangements, let ~ be a measurable subset of R =, then the symmetrized domain ~* is a ball with the same measure as ~. If u is a realvalued measurable function defined on a bounded domain ~ C R n, its spherical decreasing rearrangement u* is a function defined on the ball ~* centred at the origin and having the same measure as ~, such that u* depends only on the distance from the origin, is decreasing away from the origin and is equi-measurable with u. See [13], [18], [4] for properties of rearrangements of functions. Since the function u and its spherical decreasing rearrangement are equi-measurable, their L2-norms are the same. What Faber and Krahn actually proved is that the L2-norm of the gradient of a function decreases under rearrangements (see [18] for details, and also [9] for a different approach to this fact). The fact that the L2-norm of the gradient of a function decreases under rearrangements, combined with the variational characterization (6), immediately gives the Rayleigh-FaberKrahn inequality. I s o p e r i m e t r i c i n e q u a l i t i e s for t h e lowest e i g e n value. There are several isoperimetric inequalities for the lowest eigenvalue of boundary value problems, similar to the Rayleigh-Faber-Krahn inequality. The lowest non-trivial N e u m a n n e i g e n v a l u e also satisfies an isoperimetric inequality. Let 0 = #1(~) < p2(~) < P3 (~) < " " be the Neumann eigenvalues of the Laplace operator in ~ C R =, i.e.,

-Au=pu 0u 0--n = 0

4re 2"2 30,1 A1 ~

Faber [6] and E. Krahn [7]. The corresponding i s o p e r i -

in~,

on the boundary of ~.

(7) (8)

(4)

If n = 2, G. Szeg5 [17] proved with equality if and only if the membrane is circular. Rayleigh's conjecture was proven independently by G.

P2 (f~) < 7rp2

_ --A--,

(9)

325

RAYLEIGH-FABER-KRAHN INEQUALITY

where Pl = 1 . 8 4 1 2 . . . , with equality if a n d only if ft is

left-hand side on either (3), (4) or (13) is n o t too differ-

a circle. T h e c o r r e s p o n d i n g result for d i m e n s i o n n,

ent from its c o r r e s p o n d i n g i s o p e r i m e t r i c value, t h e n is a p p r o x i m a t e l y a ball).

P 2 ( ~ ) <- ( ~ [ )

2 / n ' ' 2 / n Pn/2,1,

(10)

was p r o v e n by H.F. W e i n b e r g e r [19], with equality if a n d only if ft is a ball. Here, C,~ is the v o l u m e of the u n i t ball in d i m e n s i o n n. I n (9) a n d (10), P,~,I denotes the first positive zero of the derivative of the Bessel f u n c t i o n Jr~. For n = 2, W e i n b e r g e r [19] also proved 1 1 #2(f~----~+ ~

2A > 7rp--~'

(II)

with equality if and only if ~ is a circle. There is also an analogue of the Rayleigh-FaberKrahn inequality for domains in spaces of constant curvature [15]. The optimal Rayleigh-Faber-Krahn inequalities for domains in S ~ was proven by E. Sperner

[16]. In [14], Lord Rayleigh also conjectured an isoperimetric inequality for the lowest eigenvalue, At, of the c l a m p e d plate. T h e eigenvalue problem f o r the clamped

plate is given by A2ul = Alul

References [1] ASHBAUGH,M.S., AND BENGURIA,R.D.: 'A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions', Ann. of Math. 135 (1992), 601 628. [2] ASHBAUGI-I,M.S., AND BENGURIA,R.D.: 'On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions', Duke Math. J. 78 (1995), 1-17. [3] ASHUAUGH,M.S., AND LAUOESEN,R.S.: 'Fundamental tones and buckling loads of clamped plates', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (Ser. IV) X X I I I (1996), 383-402. [4] BANDLE: C.: Isoperimetric inequalities and applications, Adv. Publ. Program. Pitman, 1980. [5] COURANT, R.: 'Beweis des Satzes, dass yon allen homogenen Membranen gegebenen Umfanges und gegebener Spannung die Kreisfbrmige den tiefsten Grundton besitzen', Math. Z. 1

(1918), 321-328. [6] FABER, G.: 'Beweis, dass unter allen homogenen Membranen yon gleicher Fl~che und gleicher Spannung die kreisfbrmige den tiefsten Grundton gibt', Sitzungsber. Bayer. Akad. Wiss. Miinchen, Math.-Phys. Kl. (1923), 169 172. [7] KRAHN, E.: 'f)ber eine yon Rayleigh formulierte Minimaleigensehaft des Kreises', Math. Ann. 94 (1925), 97-100. [8] KRAHN, E.: '0ber Minimaleigenschaft der Kugel in drei und

in ft

with [9] ul =

=0

in the b o u n d a r y o f ~ .

Here, 9 is a b o u n d e d o p e n subset of R 2. Rayleigh %' con-

jecture f o r the clamped plate reads

[11]

A1(f~) >_ AI(fF),

(12)

where f~* is a ball of the same area as f~. Rayleigh's conjecture was proven by N. Nadirashvili [12]. Equality is attained in (12) if and only if f~ is a circle. For dimension 3, the corresponding isoperimetric inequality was proven by M.S. Ashbaugh and R.D. Benguria [2]. To prove the analogous result for dimensions 4 and higher is still an open problem (as of 2000, see [3] however). Back in the membrane problem, if one goes beyond the lowest eigenvalue, there are several isoperimetric inequalities as well as a number of open problems. The simplest c o m b i n a t i o n ,~2(~)/Al(~~) satisfies the following i n e q u a l i t y [1]:

[12]

[13]

[14]

[15]

[16] [17] [18]

j2 ~,2(a___))< n/~,___!_~ Az(f~)

.2

3n/2-1,1

,

(13)

in n dimensions, where equality is obtained if and only if f~ is a ball. Stability results for both the RayleighFaber-Krahn inequality (3), (4) and inequality (13) have been obtained by A.D. Melas [ii] (in simple words, 'stability' means that if f~ is convex and the appropriate 326

[10]

[19]

mehr Dimensionen', Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1-44, English transl.: i). Lumiste and J. Peetre (eds.), Edgar Krahn, 1894-1961, A Centenary Volume, IOS Press, 1994, Chap. 6, pp. 139-174. LmB, E.H.: 'Existence and uniqueness of the minimizing solution of Chocquard's nonlinear equation', Stud. Appl. Math. 57 (1977), 93-105. LUMISTE, fJ., AND PEETRE, J.: Edgar Krahn, 1894-1961, A Centenary Volume, IOS Press, 1994, p. Chap. 6. MELAS, A.D.: 'The stability of some eigenvalue estimates', Y. Diff. Geom. 36 (1992), 19 33. NADIRASHVILI,N.S.: 'Rayleigh's conjecture on the principal frequency of the clamped plate', Arch. Rational Mech. Anal. 129 (1995), 1-10. POLYA, G., AND SZEGO, G.: Isoperimetric inequalities in mathematical physics, Vol. 27 of Ann. of Math. Stud., Princeton Univ. Press, 1951. RAYLEIGH,J.W.S.: The theory of sound, second ed., London, 1894/96, pp. 339-340. SCHMIDT, E.: 'Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sf£rischen Raum jeder Dimensionzahl', Math. Z. 49 (1943), 1-109. SPERNER, E.: 'Zur Symmetrisierung yon Funktionen auf Sph~ren', Math. Z. 134 (1973), 317-327. SZEG6, G.: 'Inequalities for certain eigenvalues of a membrane of given area', J. Rat. Mech. Anal. 3 (1954), 343-356. TALENTI,G.: 'Elliptic Equations and Rearrangements', Ann. Scuola Norm. Sup. Pisa 3, no. 4 (1976), 697 718. WmNBEaCER, H.F.: 'An isoperimetric inequality for the Ndimensional free membrane problem', Y. Bat. Mech. Anal. 5 (1956), 633-636.

Rafael D. Benguria MSC 1991:35P15

REIDEMEISTER THEOREM REGULAR G R O U P - There are several (different) notions of regularity in group theory. Most are not intrinsic to a group itself, but pertain to a group acting on something.

R e g u l a r g r o u p o f p e r m u t a t i o n s . Let G be a finite group acting on a set Ft, i.e. a permutation group (group of permutations). The permutation group G is said to be regular if for all a C ft, Ga = {g E G: ga = a}, the stabilizer subgroup at a, is trivial. In the older m a t h e m a t i c a l literature, and in physics, a slightly stronger notion is used: G is transitive (i.e., for all a,b E ~t there is a g E G such that ga = b) and degree(G, ~t) = order(G), where degree(G, ~) is the number of elements of ~ and order(G) is, of course, the number of elements of G. It is easy to see that a transitive regular permutation group satisfies this condition. Inversely, a transitive permutation group for which degree(G, ~) = order(G) is regular. A permutation is regular if all cycles in its canonical cycle decomposition have the same length. If G is a transitive regular p e r m u t a t i o n group, then all its elements, regarded as permutations on ~, are regular permutations. An example of a transitive regular permutation group is the Klein 4-group G = V4 = {(1), (12)(34), (13)(24), (14)(23)} of permutations of ~ = {1,2,3,4}. The regular permutation representation of a group G defined by left (respectively, right) translation g: h gh (respectively, g: h ~-~ hg -1) exhibits G as a regular permutation group on ~ = G. R e g u l a r g r o u p o f a u t o m o r p h i s m s . Let G act on a group A by means of automorphisms (i.e., there is given a homomorphism of groups G --~ Aut(A), a ~ a g, a E A). G is said to act fixed-point-flee if for all a E A there is a g E G such t h a t a g ~ a, i.e. there is no other global fixed point except the obvious and necessary one 1 C A. There is a conjecture t h a t if G acts fixed-pointfree on A and (IGI, [AI) = 1, then A is solvable, [6]; see also F i t t i n g l e n g t h for some detailed results in this direction. G is said to be a regular group of automorphisms of A if for all 1 ~ g E G only the identity element of A is left fixed by g, i.e. CA(g) = {a C A: ag = a} = {1} for all g ~ 1. Some authors use the terminology 'fixedpoint-free' for the just this property. R e g u l a r p - g r o u p . A p - g r o u p is said to be regular if (xy) p = xPyPz, where z is an element of the c o m m u t a t o r subgroup of the subgroup generated by x and y, i.e. z is a product of iterated commutators of x and y. See [1].

References

[1] CARMICHAEL, R.D.: Groups of finite order, Dover, reprint, 1956, p. 54ff. [2] DOERK, K., AND HAWKES, T.: Finite soluble groups, de Gruyter, 1992~ p. 16. [3] DORNHOFP, L.: Group representation theory. Part A, M. Dekker, 1971, p. 65. [4] HALL JR., M.: The theory of groups, Macmillan, 1963, p. 183. [5] HAMERMESIt, M.: Group theory and its applications to physical problems, Dover, reprint, 1989, p. 19. [6] HUPPERT, B., AND BLACKBURN, N.: Finite groups III, Springer, 1982, p. Chap. X. [7] LEDERMANN,W., AND WEIR, A.J.: Introduction to group theory, second ed., Longman, 1996, p. 125. M. H a z e w i n k e l

MSC1991: 20-XX REIDEMEISTER T H E O R E M - Two link diagrams represent the same ambient isotopy class of a link in S 3 if and only if they are related by a finite number of Reidemeister moves (see Fig. 1) and a plane isotopy.

S R3

Fig. 1. Proofs of the theorem were published in 1927 by K. Reidemeister [3], and by J.W. Alexander and G.B. Briggs [1]. The theorem also holds for oriented links and oriented diagrams, provided t h a t Reidemeister moves observe the orientation of diagrams. It holds also for links in a manifold M = F × [0, 1], where F is a surface. The first formalization of knot theory was obtained by M. Dehn and P. Heegaard by introducing lattice knots and lattice moves [2]. Every knot has a lattice knot representation and two knots are lattice equivalent if and only if they are ambient isotopic. The Reidemeister approach was to consider polygonal knots up to A-moves. (A A-move replaces one side of a triangle by two other sides or vice versa. A regular projection of a A-move can be decomposed into Reidemeister moves.) This approach was taken by Reidemeister to prove his theorem. 327

REIDEMEISTER THEOREM References

[1] ALEXANDER, J.W.~ AND BRIGGS, G.B.: 'On types of knotted curves', Ann. of Math. 28, no. 2 (1927/28), 563-586. [2] DEHN, M.~ AND HEEGAARD, P.: 'Analysis situs': Encykl. Math. Wiss., Vol. III AB3, Leipzig, 1907, pp. 153 220. [3] REIDEMEISTER, K.: 'Elementare Begrundung der Knotentheorie', Abh. Math. Sere. Univ. Hamburg 5 (1927), 24-32.

Jozef Przytycki MSC1991:57M25 Consider an abstract set E and a linear set F of functions f : E ~ C. Assume that F is equipped with an i n n e r p r o d u c t (f, g) and F is complete with respect to the norm Ilfl[ = (f, f)1/2. Then F is a H i l b e r t space. A function K(x,y), x,y • E, is called a reproducing kernel of such a Hilbert space H if and only if the following two conditions are satisfied: REPRODUCING

KERNEL

This definition is given in [1]; see also [6]. Some properties of reproducing kernels are: 1) If a reproducing kernel K(x,y) exists, then it is unique. 2) A reproducing kernel K(x, y) exists if and only if If(Y)l < e(y)llf[I, Vf • H, where c(y) = [IK(.,y)tl . 3) K(x, y) is a non-negative-definite kernel, that is,

Vxi,Yj • E,

Vt • C n,

where the overbar stands for complex conjugation. In particular, 3) implies:

K(x,y) = K(y,x), IK(x,v)l 2 <

K(x,x) >_O,

K(x,x)K(>V).

Every non-negative-definite kernel K(x, y) generates a Hilbert space HE for which K(x, y) is a reproducing kernel (see also R e p r o d u c i n g - k e r n e l H i l b e r t space). If K(x,y) is a reproducing kernel, then the operator K f := (K f)(.) := (f,K(x,.)) = f(-) is injective: K f = 0 implies f = 0, by reproducing property ii), and K : H + H is surjective (cf. also I n j e c t i o n ; S u r j e c tion). Therefore the inverse operator K -1 is defined on R(K) = H, and since K f = f, the operator K is the identity operator on HK, and so is its inverse. E x a m p l e s o f r e p r o d u c i n g kernels. Consider the Hilbert space H of analytic functions (cf. A n a l y t i c f u n e t i o n ) in a bounded s i m p l y c o n n e c t e d d o m a i n D of the complex z-plane. If f(z) is analytic in D, zo • D, and the disc Dzo,~ := {z: I z - z 0 l _< r} c D, then if(zo)12 < 1 ~ -- 7rr2 328

zo.~

CJ (z)¢j (4) If w = f(z, zo) is the c o n f o r m a l m a p p i n g of D onto the disc [w I < pD, such that f(z, Zo) = O, f'(zo, zo) = 1, then [2]:

-

i) for every fixed y • E, the function K(x, y) • H; ii) (f(x),K(x,y)) = f(y), Vf • H.

~ K ( x i , x j ) t j t i > O, i,j=l

Therefore H is a reproducing-kernel Hilbert space. Its reproducing kernel Ko(z, ¢) is called the Bergman kernel (cf. also B e r g m a n k e r n e l f u n c t i o n ) . If {¢j(z)} is an orthonormal basis of H (cf. also O r t h o g o n a l s y s t e m ; Basis), Cj < H , then KD(Z, 4) =

If(¢)l 2 d x d y <

1 (f,f)L~(.) -- 7rr2 •

1 jfz z f ( z , Zo) -- t ( D (Zo, Zo ) I ( D (t, zo) dt. 0

Let T be a domain in R ~ and h(t,p) C L2(T, dm) for every p 6 E. Here re(t) > 0 is a finite m e a s u r e on T. Define a linear mapping L: L 2(T, din) -+ F by

f(p) = Lg :=

frg(t)h(t,p)din(t).

(1)

Define the kernel

K(p,q) : = / T h ( * , q ) h ( t , p ) din(t),

p,q • E.

(2)

This kernel is non-negative-definite:

K(pi,pj)~j~i = I T ~.~=1~jh(t'pj) 2din(t) > 0 i,j+l if~ # 0, provided that for any set {Pl,...,P~} • E the set of functions {h(t, pj)}l<_j<_~ is linearly independent in L2(T, din) (cf. L i n e a r i n d e p e n d e n c e ) . In this case the kernel K(p, q) generates a uniquely determined reproducing-kernel Hilbert space HK for which K(p, q) is the reproducing kernel. In [6] it is claimed that a convenient characterization of the range R(L) of the linear transformation (1) is given by the formula R(L) = HK. In [4] it is shown by examples that such a characterization is often useless in practice: in general the norm in HK can not be described in terms of the standard Sobolev or I-Ihlder norms, and the assumption in [6] that HK can be realized as L2(E, d#) is not justified and is not correct, in general. However, in [6] there are some examples of characterizations of HK for some special operators L and in [3] a characterization of the range of a wide class of multidimensional linear transforms, whose kernels are kernels of positive rational functions of self-adjoint elliptic operators, is given. Reproducing kernels are discussed in [3] for rigged triples of Hilbert spaces (cf. also R i g g e d H i l b e r t space). If H0 is a Hilbert space and A > 0 is a linear c o m p a c t o p e r a t o r defined on all of H, then the closure of Ho in the norm (Au, u) 1/2 = IlA1/2ull is a Hilbert space H _ D H0. The space dual to H_, with

R E P R O D U C I N G - K E R N E L H I L B E R T SPACE respect to H0, is denoted by H + , H+ C H0 C H _ . The inner product in H + is given by the formula (u, v)+ = (A-1/2u, A-1/2v)o. The space H+ = R(A1/2), equipped with this inner product, is a Hilbert space. Let Ay)j = Aj~j, where the eigenvalues Aj are counted according to their multiplicities and (~j,~,~)o = 5jm, where 5jm is the Kronecker delta. Assume t h a t I~j(x)[ < c for all j and all x, and A2 : = E j : I

)~J < oc.

Then H+ is a reproducing-kernel Hilbert space and 0(3 its reproducing kernel is K(x, y) = ~j=l )~J~J(Y)~J (x). To check t h a t K(x,y) is indeed the reproducing kernel of H + , one calculates (A-1/2u, A-1/2K)o = (u,A-1K)o = u(y). Indeed, A - ~ K = I is the identity OO operator because Au = Z j = I A j ( u , ~ J ) ~ J ( x ) , so that K(x, y) is the kernel of the operator A in H0. The value u(y) is a linear functional in H+, so that H + is a reproducing-kernel Hilbert space. Indeed, if u E H+, then v := A-1/2u C Ho. Therefore, denoting vj := (v, wj)0 and using the C a u c h y i n e q u a l i t y and P a r s e v a l e q u a l i t y one gets: [u(y)] =

j=~a.~/2 Aj vj~j (x)

< cAllvI]0 = callull+,

as claimed. From the representation of the inner product in the reproducing-kernel Hilbert space H+ by the formula (u,v)+ = (A-1/2u, A-1/2v)o it is clear that, in general, the inner product in H+ is not an inner product in

L 2(E, d,). The inner product in H+ is of the form

(u, v)+ =/r~ fr) B(x,y)u(y)v(x) dy dx if H0 = L2(D), the distributional kernel B(x, y) = ~j°°=l A~lpj(x)~j(y) acts on u C R(A) by the formula

where

YD J~(X, y)u(y) dy

=

E j % I )~;1 (u, ~)j)o)Pj (x), w h e r e

(u, pj)0 := fDu(y)~y(y)dy is the Fourier coefficient of u (cf. also F o u r i e r coefficients). If u C R(A), then u = Aw for some w E Ho, and (u,~j) = Ajwi. Thus, the O~ series E j % I )~;1 (u, ~j)oqDj(x) : E j = I Wj~j(~) = W(X) converges in H0 = L~(D).

References [1] ARONSZAJN, N.:

'Theory

of

reproducing

kernels',

Trans. Amer. Math. Soc. 68 (1950), 337-404. [2] BEa~MAN, S.: The kernel function and conformal mapping, Amer. Math. Soc., 1950. [3] RAMM, A.G.: Random fields estimation theory, Longman/Wiley, 1990. [4] RAMM, A.G.: 'On Saitoh's characterization of the range of linear transforms', in A.G. RAMM (ed.): Inverse Problems, Tomography and Image Processing, Plenum, 1998, pp. 125 128. [5] RAMM, A.G.: 'On the theory of reproducing kernel Hilbert spaces', J. Inverse Ill-Posed Probl. 6, no. 5 (1998), 515-520.

[6] SAITOH, S.: Integral transforms, reproducing kernels and their applications, Pitman Res. Notes. Longman, 1997. [7] SCHWARTZ, L.: 'Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associ~s (noyaux reproduisants)', J. Anal. Math. 13 (1964), 115-256.

A.G. Ramm M S C 1991:46E22

REPRODUCING-KERNEL HILBERT SPACE- Let H be a H i l b e r t s p a c e of functions defined on an abstract set E. Let (f,g) denote the i n n e r p r o d u c t and let IIfll = ( f , f ) l / 2 be the norm in H . The space H is called a reproducing-kernel Hilbert space if there exists a function K(x, y), the r e p r o d u c i n g k e r n e l , on E x E such that: 1) K(z,y) C H for any y E E; 2) (f(.), K(., y)) = f(y) for all f C H (the reproduc-

ing property). From this definition it follows that the value f(y) at a point y E E is a continuous l i n e a r f u n c t i o n a l in H:

If(y)[ _< c(y)Ilfll,

c(y) := IIK(,y)II.

The converse is also true. The following theorem holds: A Hilbert space of functions on a set E is a reproducingkernel Hilbert space if and only if if(Y)[ < c(y)llf[[ for all y E E. By the R i e s z t h e o r e m , the above assumption implies the existence of a linear functional K ( . , y ) such that f(y) = (f, K(., y)). By the construction, the kernel K(x, y) is the reproducing kernel for H . An example of a construction of a reproducing-kernel Hilbert space is the rigged triple of Hilbert spaces H+ C H0 C H _ , which is defined as follows [3] (cf. also R i g g e d H i l b e r t s p a c e ) . Let H0 be a Hilbert space of functions, let A > 0 be a linear densely defined selfa d j o i n t o p e r a t o r on Ho, A~j = Aj~j (the eigenvalues Ai > 0 are counted according to their multiplicities) and assume that

, 2 :=

<

I

(x)l < c,

Vi, x.

j=l

Define H_ D /4o to be the Hilbert space with inner product (u,v)_ = (A1/2u, A1/2v)o . H_ is the completion of Ho in the norm l[ull :-- (u, u)~ 2. Let H+ C H0 be the dual space to H_ with respect to H0. Then the inner product in H+ is defined by the formula (u, v)+ := (A-i/2u, A-i/2v)o, and H+ = R(AI/2), equipped with the inner product (% v)+, is a Hilbert space. Define B (x, y) = ~j=1 Aj ~j (x) ~j (y), where the overline stands for complex conjugation. For any y, one has 329

REPRODUCING-KERNEL

HILBERT SPACE

B(x, y) • H+. Indeed,

dense in H i : If f C H1, 0 = (f,K(x,y))H~ = f(y) for all y C E, then f - 0. Using this and the equality (f, .q)I-±~= (f, g)H for all f, g C H °, one can check t h a t H C H i and vice versa, so H = H I , t h a t is, H and H I consist of the s a m e set of elements. Moreover, the n o r m s in H and H1 are equal. Indeed, take an a r b i t r a r y f E H1 and a sequence f~ C H °, []f~ - fill -+ 0. T h e n

OG

liB(x, y)ll+ < c3~ II~,~¢(x)ll+ = j=l oo

= c ~(A~;,

~j)0 = cA ~ < oo.

j=l

Furthermore,

Infll~ = ~ i m IIAII7 =

(u, B ( x , y))+ = (u, A - ~ B ) = u(y), so t h a t B(x,y) is the r e p r o d u c i n g k e r n e l in H + . Moreover lu(y)l <_ c(y)tlull+, where c(y) > 0 is a constant independent of u • /4+. Indeed, if uj := (u, ~j)0 and u • H + , then u =

A1/2v, v • Ho vjA~/2 = uj, and

lu(Y)l < Ej%~ luj~J(y)l < cAIIvll = c i l l u L . Thus H + is a reproducing kernel Hilbert space with the reproducing kernel B(x, y) defined above. If K(x, y) is a function on E x E such t h a t

~

K(z~,x~)tf > o, vt • c ~, vx~ ~ E,

(~)

i,j:l

then one can define a p r e - H i l b e r t tions of the form

s p a c e H ° of func-

K(x, Yi~)ci~, E

= lim

n-+ O jn=l

K(x, y,~)Cmn

m~=l

= ]

1

Jn

=

E I~(Y~n'Yjn)CjnCT~'~ = jn ~rO~n =

lira ( A , A )

= II/H 2 •

Thus, the n o r m s in H1 and H are so are the inner p r o d u c t s (by the Define a l i n e a r o p e r a t o r L : where H = L2(T, din) and H is which will be e q u i p p e d with the space below:

equal, as claimed, and polarization identity). 7/ -+ H, D(L) = 7/, the range R(L) of L, structure of a Hilbert

J

f(~) := ~ K(~, ~)c~,

f(x) = LF

c~ =const.

:=

(2)

IT F(t)h(t, x) din(t).

j=l

T h e inner p r o d u c t of two functions from H ° is defined by

(/,~) :=

~(x, yj)c~,

~(~,z,~)9,~ m=l

= ~

~(~,

=

Here, T is a d o m a i n in R ~ and m is a positive m e a s u r e on T, re(T) < co, h(t,x) E 7-{ for all x E E , and it is assumed t h a t L is injective, t h a t is, the system {h(t, x ) } v x e z is total in 7t (cf. also T o t a l set). Define

/

K(x,y) := ./~r h(t,y)h(t,x) din(t) =

~)e~.

= (h(., y), h(., x))~.

j,m

This definition makes sense because of (1) and because of reproducing p r o p e r t y 2). In particular, (f, f ) _> 0, as follows from (1), and if (f, f ) = 0 then f = 0, as follows from p r o p e r t y 2). Indeed,

J

= ~

K(y, ~j)~j = f(y),

Vy • E.

j=l

Thus, if ( f , f ) = 0, then IIf[I = 0 and If(Y)I < IIfll][K(x,y)l[ = 0, so f(y) = 0 as claimed. Denote by H the completion of H ° in the n o r m Ilfll. Then H is a reproducing-kernel Hilbert space and K(x, y) is its reproducing kernel. A reproducing-kernel Hilbert space is uniquely defined by its reproducing kernel. Indeed, if H i is another reproducing-kernel Hilbert space with the same reproducing kernel K(x,y), then H ° C H I and H ° is 330

(3)

This kernel clearly satisfies condition (1) and therefore is a reproducing kernel for the reproducing-kernel Hilbert space HK which it generates. Clearly K(x, y) E H for all y C E. If f E H , t h a t is, f = LF, f E ~, then

(f(.),K(.,y)) H = (LF, K(.,y)) g = = ((y(.), h(., x))~, (h(.., y), h(-., ~ ) ) ~ ) , = = (F(.), (h(.., ~), (h(., z), h(.., ~ ) ) , ) ~ ) ~ = = (F(.), h(., y))~ = / ( y ) , if one equips H

with the inner p r o d u c t

such t h a t

(f, g)H = (F, G)~. This requirement is formally equivalent to the following one: (h(s, x), h(t, X)) H = 5~(t - s), where (h(s, y), 5,~(t-s))~ = h(t, y), so t h a t the distributional kernel d,~ (t - s) is not the usual d e l t a - f u n c t i o n , but the one which acts by the rule

T dm(t)F(t) /T dm(s)G(s)hm (t - s) = = IT dm(t)F(t)G(t),

RIDGE FUNCTION and formally one has fT dm(s)G(s)~,~ (t - s) = G(t). With the inner product (f, g)H, the linear set R(L) becomes a Hilbert space: (f, g)H = (LF, LG)H =

(4)

= IT IT din(t) dm(s)F(t)G(s)(h(s,x),h(t,x))g = = /T dm(t)F(t)G(t) = (F, G)n. Thus, this inner product makes L an i s o m e t r i c o p e r a t o r defined on all of 7 / a n d makes H = R(L) a (complete) Hilbert space, namely H = /arK, a reproducingkernel Hilbert space. Since L is assumed injective, it follows that L -1 is defined on all of R(L) = H and, since H is complete in the norm IIfll = (f, f ) ~ 2 , one concludes t h a t L -1 is continuous (by the Banach theorem). Consequently, L is a co-isometry, t h a t is, L* = L -1, where L* is the a d j o i n t o p e r a t o r to L. If L* = L -1, then one can write an inversion formula for the linear transform L similar to the well-known inversion formula for the F o u r i e r t r a n s f o r m . Formally one has:

f(x) = (F(t),h(t,x))~,

(f(x),h(s,x)) H = F(s).

The space H = HK is the reproducing-kernel Hilbert space generated by kernel (3) which is the reproducing kernel for H . The above formal inversion formulas may be of practical interest if the norm in H is a standard one. In this case the second formula should be suitably interpreted, since F(s) is defined at m-almost all s. In [6] it is claimed t h a t the characterization of the range of the linear operator L, defined in (3), can be given as follows: R(L) = HK, where H K is the reproducing-kernel Hilbert space generated by kernel (3). However, in fact such a characterization does not give, in general, practically useful necessary and sufficient conditions for f(x) E R(L) because the norm in HK is not defined in terms of standard norms such as Sobolev or HSlder ones (see [5], [4], [3]). However, when the norm in HK is equivalent to a standard norm, the above characterization becomes efficient (see [5], [4], [3], and also [6]). Many concrete examples of reproducing-kernel Hilbert spaces can be found in [1], [2] and [6]. The papers [1] and [7] are important in this area, the book [6] contains m a n y references, while [2] is an earlier book important for the development of the theory of reproducing-kernel Hilbert spaces. References [1] ARONSZAJN, N.: 'Theory of reproducing kernels', Trans. Amer. Math. Soc. 68 (1950), 337-404. [2] BERGMAN, S.: The kernel function and eonformal mapping, Amer. Math. Soc., 1950.

[3] RAMM, A.G.: Random fields estimation theory, Longman/Wiley, 1990. [4] RAMM, A.G.: 'On Saitoh's characterization of the range of linear transforms', in A.G. RAMM (ed.): Inverse Problems, Tomography and Image Processing, Plenum, 1998, pp. 125128. [5] RAMM, A.G.: 'On the theory of reproducing kernel Hilbert spaces', J. Inverse Ill-Posed Probl. 6, no. 5 (1998), 515 520. [6] SAITOH, S.: Integral transforms, reproducing kernels and their applications, Pitman Res. Notes. Longman, 1997. [7] SCHWARTZ,L.: 'Sous-espaces hilbertiens d'espaces vectoriels topologique et noyaux associes', Anal. Math. 13 (1964), 115256. A.G. Ramm

MSC1991:46E22 R I D G E FUNCTION, plane wave - In its simplest form, a ridge function is a multivariate function f:Rn~R of the form f(xl,...,Xn)

~- g ( a l x l - ~ - - " -/- a n X n ) : g ( a - x),

where g: R --~ R and a = (al,..., an) e R n \ {0}. The vector a E R n \ {0} is generally called the direction. In other words, a ridge function is a multivariate function constant on the parallel hyperplanes a . x = c, c C R. Ridge functions appear in various areas and under various guises. In 1975, B.F. Logan and L.A. Shepp coined the name 'ridge function' in their seminal paper [6] in computerized tomography. In t o m o g r a p h y , or at least in t o m o g r a p h y as the theory was initially constructed in the early 1980s, ridge functions were basic. However, these functions have been considered for some time, but under the name of plane waves. See, for example, [5] and [1]. In general, linear combinations of ridge functions with fixed directions occur in the study of hyperbolic partial differential equations with constant coefficients. Ridge functions and ridge function approximation are studied in statistics. There they often go under the name of projection pursuit, see e.g. [3], [4], [2]. Projection pursuit algorithms approximate a function of n variables by functions of the form

~ gi(a ~' x), i=1

where the a i and gi are the variables. The idea here is to 'reduce dimension' and thus bypass the c u r s e o f dim e n s i o n . The vector a i. x is considered as a projection of x. The directions a are chosen to 'pick out the salient features'. One of the popular models in the theory of neural nets is t h a t of a multi-layer feedforward neural net with input, hidden and output layers (cf. also N e u r a l n e t w o r k ) . The simplest case (which is that of one hidden 331

RIDGE F U N C T I O N layer, r processing units and one output) considers, in mathematical terms, functions of the form r

i • x + 00 i=1

where a: R -+ R is some given fixed univariate function. In this model, which is just one of many, one is in general permitted to vary over the w i and 0i, in order to approximate an unknown function. Note that for each 0 E R and w E R n \ {0} the function x + 0)

is also a ridge function, see e.g. [8] and references therein. For a survey on some approximation-theoretic questions concerning ridge functions, see [7] and references therein. References [1] COURANT~ i%., AND HILBERT, D.: Methods of mathematical physics, Vol. II, Interscience, 1962. [2] DONOHO, D.L., AND JOHNSTONE, I.M.: 'Projection-based approximation and a duality method with kernel methods', Ann. Statist. 17 (1989), 58-106. [3] FRIEDMAN, J.H., AND STUETZLE, W.: 'Projection pursuit regression', J. Amer. Statist. Assoc. 76 (1981), 817-823. [4] HUBER, P.J.: 'Projection pursuit', Ann. Statist. 13 (1985), 435-475. [5] JOHN, F.: Plane waves and spherical means applied to partial differential equations, Interscience, 1955. [6] LOCAN, B.F., AND SHEPP, L.A.: 'Optimal reconstruction of a function from its projections', Duke Math. Y. 42 (1975), 645 659. [7] PINKUS, A.: 'Approximating by ridge functions', in A. LE Mt~HAUTI~, C. RABUT, AND L.L. SCHUMAKER (eds.): Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, 1997, pp. 279 292. [8] PINKUS, A.: 'Approximation theory of the MLP model in neural networks', Acta Numerica 8 (1999), 143-195. Allan Pinkus

MSC1991: 41A30, 92C55 finitedimensional a l g e b r a A over an a l g e b r a i c a l l y c l o s e d field k is called self-injeetive if A, considered as a right A-module, is injective (cf. also I n j e c t i v e m o d u l e ) . Well-known examples for self-injective algebras are the group algebras kG for finite groups G (cf. also G r o u p a l g e b r a ) . An arbitrary finite-dimensional algebra A is said to be representation-finite provided that there are only finitely many isomorphism classes of indecomposable finite-dimensional right A-modules. C. Riedtmann made the main contribution to the classification of all self-injective algebras that are representation-finite. Her key idea was not to look at the algebra A itself, but rather at its Auslander-Reiten quiver FA. (Quiver is an abbreviation for directed graph, see Quiver.) The vertices of the Auslander-Reiten RIEDTMANN

332

CLASSIFICATION

-

A

quiver (see also R e p r e s e n t a t i o n o f a n a s s o c i a t i v e alg e b r a ) are the isomorphism classes of finite-dimensional A-modules. The number of arrows from the isomorphism class of X to the isomorphism class of Y is the dimension of the space radA(X,Y)/rad2A(X,Y), where rag is the J a c o b s o n r a d i c a l of the category of all finitedimensional A-modules. The Auslander-Reiten quiver is a translation quiver, which means that it carries an additional structure, namely a translation TA mapping the non-projective vertices bijectively to the non-injective vertices. The translation is induced by the existence of almost-spit sequences 0 --+ X --+ Y --+ Z --+ 0 (see also R e p r e s e n t a t i o n o f a n a s s o c i a t i v e a l g e b r a ; A l m o s t - s p l i t s e q u e n c e ) and sends the isomorphism class of a non-projective indeeomposable module Z to the starting term X. The stable part (FA)s of the Auslander-Reiten quiver FA of A is the full subquiver of FA given by the modules that cannot be shifted into an injective or projective vertex by a power ~-~ for some integer j. In [3], Riedtmann succeeded to prove that for any connected representation-finite finite-dimensional A the stable part (FA)s of the Auslander-Reiten quiver is of the shape Z A / G , where A is a quiver whose underlying graph A is a D y n k i n d i a g r a m A~ (n C N), D~ (n C N, _> 4), or E~ (n = 6, 7, 8) and G is an infinite cyclic g r o u p of automorphisms of the translation quiver ZA. The vertices of ZA are the pairs (i, x) such that i is an integer and x a vertex of A. From (i, x) to (i, y) there are the arrows (i, c~) with c~: x --+ y an arrow of A. In addition, from (i + 1, x) to (i, y) there exist the arrows (i, c~)' with c~: y ~ x an arrow of/~. The translation maps (i, x) to (i + 1, x).

For a self-injective algebra A, the only vertices of the Auslander-Reiten quiver that do not belong to (FA)s are the isomorphism classes of the indecomposable projective (and injective) modules. Thus, one can reconstruct FA from (FA)s by finding in (FA)s the starting points of arrows of FA ending in projective vertices. These combinatorial data are called a configuration. This shows that for finding all possible Auslander-Reiten quivers FA of all connected representation-finite self-injective algebras A one has to classify the infinite cyclic automorphism groups G of Z/~ and the G-invariant configurations of Z/~ for all Dynkin diagrams. For the Dynkin diagrams A~ and D,~ this classification was carried out in [4] and [5]. The classification of the possible configurations for the exceptional Dynkin diagrams E6, E7, Es turned out to be more difficult. Fortunately, the development of t i l t i n g t h e o r y offered a convenient way for a solution. Namely, it was observed in [1] and [2] that in order to equip ZZX with all possible configurations, one has

RIEMANN ~-FUNCTION to form the Auslander-Reiten quivers of the repetitive algebras of the tilted algebras of representation-finite hereditary algebras of type A (cf. also T i l t e d algebra). Nevertheless, the full classification of all these repetitive algebras eventually obtained in [7] required the use of a computer for handling the huge amount of structures appearing in the case Es. If one finally wants to return from the Auslander Reiten quiver FA to the algebra A itself, one considers the factor of the free k-linear category of FA by the mesh relations induced by the almost-split sequences. This factor is called the mesh category of FA. Forming the endomorphism algebra of the direct sum of all projective objects in this mesh category yields A (up to Morita equivalence), provided that A is standard (i.e. the mesh category is equivalent to the category of indecomposable finite-dimensional A-modules). Nonstandard algebras appear only if the characteristic of the field k is 2 and A is of type Dn. They were classified in [6] and [9]. It is worth noting that the approach using repetitive algebras was generalized in order to classify the representation-tame self-injective standard algebras of polynomial growth in [8]. In this case tilted algebras of representation-tame hereditary and canonical algebras replace the tilted algebras of representation-finite hereditary algebras. References [1] BRETSCHER, O., LASER, C., AND RIEDTMANN, C.: 'Selfinjective and simply connected algebras', Manuscripta Math. 36 (1981/82), 253-307. [2] HUGHES, D., AND WASCHBUSCH, J.: 'Trivial extensions of

tilted algebras', Proc. London Math. Soe. 46 (1983), 347364.

[3] RIEDTMANN, C.: ~Algebren, Darstellungen, Uberlagerungen und zurfick', Comment. Math. Helv. 55 (1980), 199 224. [4] RIEDTMANN, C.: 'Representation-finite selfinjective algebras of class An': Representation theory II, VoI. 832 of Lecture Notes in Mathematics, Springer, 1981, pp. 449-520. [5] RIEDTMANN, C.: 'Configurations of ZD~', d. Algebra 82

(1983), 309-327. [6] RIEDTMANN, C.: 'Representation-finite self-injective algebras of class Dn', Compositio Math. 49 (1983), 231-282. [7] ROGGON, B.: Sel]injective and iterated tilted algebras of type E6, ET, Es, Vol. 343 of E 95-008 SFB, Bielefeld, 1995. [8] SKOWROJSKI, A.: 'Selfinjective algebras of polynomial growth', Math. Ann. 285 (1989), 177-199. [9] WASCHBI)SCH, J.: 'Symmetrische Algebren vom endlichen Modultyp', J. Reine Angew. Math. 321 (1981), 78 98.

Peter Dr~xler MSC 1991:16G70 RIEMANN E-FUNCTION, ~-function - In 1859, the newly elected member of the Berlin Academy of Sciences, B.G. Riemann published an epoch-making ninepage paper [5] (see also [1, p. 299]). In this masterpiece,

Riemann's primary goal was to estimate the number of primes less than a given number (cf. also de la Vall6eP o u s s i n t h e o r e m ) . Riemann considers the Euler zetafunction (also called the R i e m a n n zeta-function or Zeta-function) 1

¢(s) :=

E -n =~ I Ip n----1

1 1

i

(I)

P~

for complex values of s = cr + it, where the product extends over all prime numbers and the D i r i c h l e t series in (1) converges for ¢ > 1 (cf. also Z e t a - f u n c t i o n ) . His investigation leads him to define a function, called the Riemann ~-function, 1

~(s) := ~ s ( s - 1)Tr-S/2F ( 2 ) ¢ ( s ) ,

(2)

where F denotes the g a m m a - f u n c t i o n . The function ~(s) is a real entire function of order one and of maximal type and satisfies the functional equation ~(s) = 4(1 - s) [6, p. 16]. By the Hadamard factorization theorem (cf. also H a d a m a r d t h e o r e m ) , ~(s)=~(O)II(l-p)e

s/p,

p

where p ranges over the roots of the equation ~(p) = 0. These roots (that is, the zeros of the Riemann ~function) lie in the strip 0 < a < 1. The celebrated Riemann hypothesis (one of the most important unsolved problems in mathematics as of 2000) asserts that all the roots of ~ lie on the critical line R e s = a = 1/2 (cf. [2], [1], [3], [6]; cf. also R i e m a n n h y p o t h e s e s ) . The appellation 'Riemann ~-function' is also used in reference to the function E(t):=~

~+it

.

(In [5], Riemann uses the symbol ~ to denote the function which today is denoted by E.) In fact, Riemann states his conjecture in terms of the zeros of the F o u r i e r transform [4, p. 11]

(;) :=g Z I

~(u) cos(ut) d~,

where C~

~(U) : : E

7rn2 ( 271-n2e4u -- 3 ) e x p ( h u - - 71n2e4u) .

rt:l

The Riemann hypothesis is equivalent to the statement that all the zeros of E(t) are real (cf. [6, p. 255]). Indeed, Riemann writes '[.-.] es ist sehr wahrscheinlich, dass alle Wurzeln reell sind.' (That is, it is very likely that all the roots of E are real.) References

[1] EDWARDS,H.M.: Riemann's zeta function, Acad. Press, 1974. [2] IvId, A.: The Riemann zeta-function, Wiley, 1985. 333

RIEMANN ~-FUNCTION [3] KARATSUBA, A.A., AND VORONIN, S.M.: The Riernann zetafunction, de Gruyter, 1992. [4] PdLYA, G.: @ber die algebraisch-funktionentheoretischen Untersuchungen yon J.L.W.V. Jensen', Kgl. Danske Vid. Sel. Math.--Fys. Medd. 7 (1927), 3 33. [5] RIEMANN, B.: 'Ueber die Anzahl der Primzahlen unter einer gegebenen GrSsse', Monatsber. Preuss. Akad. Wiss. (1859), 671 680. [6] TITCHMARSH, E.C.: The theory of the Riernann zetafunction, second ed., Oxford Univ. Press, 1986, (revised by D.R. Heath-Brown).

George Csordas MSC 1991:11M06 R I E S Z D E C O M P O S I T I O N P R O P E R T Y - Let (E, C)

be a partially ordered vector space, [5], i.e. E is a real v e c t o r s p a c e with a convex cone C defining the p a r t i a l o r d e r by x >- y if and only if x - y C C. For x -< y, the corresponding interval is [x, y] = {u C E : x -< u -< y}. The (partially) ordered vector space (E, C) has the Riesz decomposition property if [0, u] + [0, v] = [0, u + v] for all u, v E C, or, equivalently, if Ix1, Yl] + Ix2, y2] = Ix1 +x2,yl +Y2] for all xl -~ yl, x: -~ Y2. A R i e s z s p a c e (or v e c t o r l a t t i c e ) automatically has the Riesz decomposition property. Terminology on this concept varies a bit: in [2] the property is referred to as the dominated decomposition property, while in [3] it is called the decomposition property of F. Riesz. The Riesz decomposition property and the R i e s z dec o m p o s i t i o n t h e o r e m are quite different (although there is a link) and, in fact, the property does turn up as an axiom used in axiomatic potential theory (see also P o t e n t i a l t h e o r y , a b s t r a c t ) , see [1], where it is called the axiom of natural decomposition. There is a natural non-commutative generalization to the setting of C*-algebras, as follows, [4]. Let x, y, z be elements of a C * - a l g e b r a A. If x*x <_ yy* + zz*, then there are u, v E A such that u*u <_y'y, v*v <_x*x and XX* ~ ~t~t* -~ VV*.

References [1] CONSTANTINESCU,C., AND CORNEA, A.: Potential theory on harmonic spaces, Springer, 1972, p. 104. [2] LUXEMBURG, W.A.J., AND ZAANEN, A.C.: Riesz spaces, Vol. I, North-Holland, 1971, p. 73. [3] MEYER-NmBERG, P.: Banach lattices, Springer, 1971, pp. 3, Thm. 1.1.1. [4] PEDERSEN, G.K.: C*-algebras and their autornorphism groups, Acad. Press, 1979, p. 14. [5] WONO, Y.-CH., AND NO, K.-F.: Partially ordered topological vector spaces, Oxford Univ. Press, 1973, p. 9.

Riesz decomposition theorem

f o r s u p e r - or s u b h a r m o n i c f u n c t i o n s . Roughly speaking, this asserts that a super- or subharmonic function is the sum of a potential and a harmonic function. For precise statements, see S u b h a r m o n i c f u n c t i o n (where it is called the Riesz local representation theorem), and R i e s z t h e o r e m (where it is simply called the Riesz theorem), [12], [20]. See also [8, 1.IV.S-9, 1.IX.11, 1.XIV.9], and [8, 1.XV.7, 1.XVII.7] for a corresponding result for superparabolic functions. In [4] the decomposition formula is called the Riesz integral representation and Riesz representation of a superharmonic function There is also an abstract version (see also P o t e n t i a l t h e o r y , a b s t r a c t ) , dealing with harmonic spaces, which states (see [5, Thin. 2.2.2, p. 38]) that every superharmonic function u on a harmonic space which has a subharmonic minorant may be written uniquely as the sum of a potential and a harmonic function. This harmonic function is the greatest hypo-harmonic minorant of u and is the infimum of any Perron set generated by An immediate consequence is the Brelot-Bauer theorem ([5, Corol. 2.2.1, p. 38]) that the real vector space of differences of positive harmonic functions on a harmonic space is a conditionally complete v e c t o r l a t t i c e (Riesz space) with respect to the natural order (i.e., pointwise comparison). This gives a link with the R i e s z decomposition property. There is also a converse Riesz decomposition theorem, [11]. In the mid-1950s, the pioneering work of J.L. Doob and G.A. Hunt, [7], [14], [15], [16], showed a deep connection between potential theory and stochastic processes. Correspondingly, there are probabilistic Riesz decomposition theorems on decompositions of excessive functions, excessive measures and super-martingales. See [3], [9], [8, 2.III.21] for precise formulations. There are also versions of these on commutative and non-commutative groups, [1], [2], [6]. R i e s z d e c o m p o s i t i o n t h e o r e m for o p e r a t o r s . This

theorem is also called the Riesz splittin 9 theorem and deals with splitting the spectrum of an operator. Following [10, p. 9ff], let A be a bounded l i n e a r o p e r a t o r on a B a n a c h s p a c e X with spectrum or(A). Let G C G(A) be an isolated part of G(A), i.e. cr and ~- = or(A) \ G are both closed in G(A). Let P~ = 2 ~

(~ - A ) - I d~,

M. Hazewinkel MSC 1991: 46A40, 06F20, 46L05, 31D05 RIESZ D E C O M P O S I T I O N T H E O R E M - There are

two different theorems that go by this name. 334

where F is a contour in the resolvent set of A with a in its interior and separating a from T. Then P~ is a projection (i.e. P~ = P~), called the Riesz projection or Riesz projector (cf. also S p e c t r a l s y n t h e s i s (for

RIESZ O P E R A T O R a single point) and K r e ~ n space). Put M = Im(P~), L = Ker(Pz). Then X = M • L, both M and L are invariant under A, and a(A[M) = cr, a(AIL) = 7-. If, moreover, a(A) is the disjoint union of two closed subsets a and % then P , + P , = id, P~P~ = 0 = P~P~. For more general results (for closed linear operators), see [10, p. 326f~. See also F u n c t i o n a l c a l c u l u s (particularly the part dealing with the Riesz-Dunford functional calculus) and, e.g., [13]. F. Riesz himself, to whom the original result is due, called it the Zerlegungssatz. References

[1] BANALESCU,M.: 'On the Riesz decomposition property', Rev. Roum. Math. Pures Appl. 36 (1991), 107-114. [2] BERG, CH., AND FROST, G.: Potential theory on locally compact Abelian groups, Springer, 1975, p. 148. [3] BLUMENTHAL,R.M., AND GETOOR, R.K.: M a r k o v p r o c e s s e s and potential theory, Acad. Press, 1968, pp. 272, Thin. 2.11. [4] BRELOT, M.: On topologies and boundaries in potential theory, Springer, 1971, p. 93; 45. [5] CONSTANTINESCU,C., AND CORNEA, A.: P o t e n t i a l theory o n harmonic spaces, Springer, 1972. [6] DENY, J.: 'Noyaux de convolution de Hunt et noyaux associes une famille fondamentale', Ann. Inst. Fourier (Grenoble) 12 (1962), 643-667. [7] DOOB, J.L.: ~Semimartingales and subharmonic functions', Trans. Amer. Math. Soc. 77 (1954), 86-121. [8] DOOB, J.L.: Classical potential theory and its probabilistic counterpart, Springer, 1984. [9] GETOOR, R.K., AND (]LOVER, J.: 'Riesz decompositions in Markov process theory', Trans. Amer. Math. Soc. 285 (1984), 107-132. [10] GOHBERG, I., GOLDBERG, S., AND KAASHOEK, M.A.: C l a s s e s of linear operators, Vol. I, Birkh~iuser, 1990. [11] GOLDSTEIN, M., AND OW, W.H.: 'A converse of the Riesz decomposition theorem for harmonic spaces', Math. Z. 173 (1980), 105-109. [12] HAYMAN, W.K., AND KENNEDY, P.B.: S u b h a r m o n i c functions, Vol. I, Acad. Press, 1976, p. Sect. 3.5. [13] HILLE, E.: Methods in classical and functional analysis, Addison-Wesley, 1972, pp. 349-350. [14] HUNT, (].A.: 'Markoff processes and potentials I', Illinois J. Math. 1 (1957), 44-93. [15] HUNT, G.A.: 'Markoff processes and potentials II', Illinois J. Math. 1 (1957), 316-369. [16] HUNT, G.A.: 'Markoff processes and potentials III', Illinois J. Math. 2 (1958), 151-213. [17] RIESZ, F.: 'Sur les fonctions subharmoniques et leur rapport la theorie du potentiel I', Acta Math. 48 (1926), 329-343. [18] RIESZ, F.: 'Sur les fonctions subharmoniques et leur rapport la theorie du potentiel II', Acta Math. 54 (1930), 321-360. [19] RIESZ, F.: 'Uber die linearen Transformationen des komplexen Hilbertschen Raumes', Acta Sci. Math. (Szeged) 5

(1930/32), ~3-54. [20] SAFE, E.B., AND TOTIK, V.: Logarithmic potentials a n d ext e r n a l fields, Springer, 1997, p. 100. M. Hazewinkel

MSC1991: 31A10, 31D05, 47A60, 47A10, 47A15

R I E S Z O P E R A T O R - The Riesz operators on a Banach space are, roughly speaking, those bounded linear operators that have a Riesz spectral theory, i.e. that have a spectral theory like that of compact operators, [3] (see also S p e c t r a l t h e o r y o f c o m p a c t o p e r a t o r s ) . The precise definition is as follows ([2], [7]). Let R be a bounded operator on a B a n a c h s p a c e E, and let a(R) be the spectrum of R. A point )~ E a(R) is isolated if a ( R ) \ ~ is closed in a(R), i.e. if there is an open subset U C C such that UN~(R) = {A}. A point A E a(R) is a Riesz point if it is isolated and E is the direct sum of a closed subspace F(A) and a finite-dimensional subspace N(A), both invariant under R and such that R - A is nilpotent on N(A) and a homeomorphism on F(A). A bounded operator R is a Riesz operator if all points A E or(R) \ {0} are Riesz points. Every compact operator is a Riesz operator (the Riesz theory of compact

operators). A bounded operator T on E is called quasi-nilpotent if l i m n - ~ []Tnl[ 1/n = 0 (which is equivalent to a(T) =

{0)). A bounded operator R is a Riesz operator if and only if, [4]: lim { i n f l I R n - C l l U ~ } = O , where C runs over all compact operators (see C o m p a c t operator). It is a long-standing question (still open as of 2000) whether every Riesz operator splits as the sum of a quasi-nilpotent operator and a compact operator. Such a decomposition is called a West decomposition, after T.T. West, who proved this for the case that E is a H i l b e r t space, [6]. Further results can be found in [1],

[8]. There is another, quite different, notion in which the phrase 'Riesz operator' occurs, viz. the parametrized family of multiplier operators

f(x) ~ (S~f)(x) = f

f'(~)(1 - 1~12)~e2~i~'~ d~, I<1 called the Bochner-Riesz operator, [5]. They are important in Bochner-Riesz summability (see also R i e s z summation method). References [1] DAVIDSON,K.R., AND HERRERO, D.A.: 'Decomposition of Banach space operators', Indiana Univ. Math. J. 35 (1986), 333-343. [2] DIEUDONN~, J.: Foundations of modern analysis, Acad. Press~ 1960, p. 323. [3] DOWSON, H.R.: Spectral theory of linear operators, Acad. Press, 1978, p. 67ff. [4] RUSTON, A.F.: 'Operators with a Fredholm theory', J. London Math. Soc. 29 (1954), 318-326. [5] STEIN, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993, p. 389.

335

RIESZ O P E R A T O R [6] WEST,

T.T.:

'The

decomposition

of Kiesz

operators',

Proc. L o n d o n Math. Soc. 16 (1966), 737-752.

[7] WEST, T.T.: 'Riesz operators in Banach spaces', Proc. London Math. Soc. 16 (1966), 131-140. [8] ZHONG, H.: ' O n B-convex spaces and West decomposition of Riesz operators on t h e m ' , Acta Math. Sinica 37 (1994), 563-569. M. H a z e w i n k e l

MSC 1991:47B06 ROBBINS EQUATION - Researchers have long been interested in finding simple axiomatizations of theories, including B o o l e a n a l g e b r a . Boolean algebra can be axiomatized in many different ways using various sets of operations. The ordinary operations of Boolean algebra are disjunction (+), conjunction (,), negation ('), zero (0), and one (1). One of the first simple axiomatizations was presented by E.V. Huntington in 1933 [2]: • (Commutativity) x + y = y + x; • (Associativity) (x + y) + z = x + (y + z); • (Huntington) ((x' + y)' + (x' + y')') = x. From these three equations, one can prove the existence of a zero and a one, and when conjunction is defined (as x * y - ( x ~ + y')~), one can prove all of the properties of Boolean algebra. Shortly thereafter, Huntington's student H. Robbins conjectured that the Huntington equation can be replaced by the following equation, which is simpler by one occurrence of the negation symbol: • (Robbins) ((x + y)' + (x' + y)')' = y. Neither Robbins nor Huntington could find a proof or a counterexample. The theory given by the three equations {Commutativity, Associativity, Robbins} became known as R o b b i n s a l g e b r a , and the question whether {Commutativity, Associativity, Robbins} would imply Huntington became known as the R o b b i n s p r o b l e m . The problem became a favourite of A. Tarski, who gave it to many of his students and colleagues, but it remained unsolved until 1996. It is not difficult to show that every finite Robbins algebra satisfies the Huntington equation and that every Robbins algebra satisfying x" = x satisfies the Huntington equation. A bit more difficult is showing that any of the following properties of Boolean algebra is also sufficient: • X+X-~X;

• 30(x+0=x); • ~l(x+l=l).

An important step was taken in 1982, when S. Winker showed (by hand) that every Robbins algebra satisfying the (very weak) condition 3 c 3 d ( c + d) ~ = c ~ also satisfies the Huntington equation [6]. 336

From 1980 through 1996, many attempts were made, with and without computers, to solve the problem. The attempts with computers relied on automated theoremproving programs, such as OTTER, for first-order logic with equality. A new theorem prover EQP [3] featuring associative-commutative unification was written in the early 1990s. Associative-commutative unification (AC unification) [5] builds the properties of associativity and commutativity of binary functions into the inference processes so that those properties need not be used explicitly to make inferences. The main advantage of AC unification is that expressions can be stored and used in a canonical form rather than in various commuted and associated forms. The main disadvantage is that a pair of expressions can be AC-unified in an enormous number of ways. To address this particular problem, a heuristic was developed that uses only the simplest AC unifiers, reducing the number of inferences from a pair of equations, in some cases from millions to tens. The heuristic is incomplete (i.e., it can block all paths to a proof), but it is valuable in practice. A second important feature of EQP is the 'basic' restriction on paramodulation [I], which reduces redundancy in the search for a proof. In 1996, a series of experiments was designed to attack the Robbins problem with EQP. Proof searches were conducted with various combinations of parameters to the program, including use of the AC heuristic and the 'basic' restriction, limits on the size of equations, and strategies for selecting the next equations for making inferences. After fourteen multi-day searches, using a total of about five CPU-weeks of computer time, a proof was found. The successful search, which took about eight CPU-days, produced a proof of Winker's condition 3c3d(c + d) I = c ~ from {Commutativity, Associativity, Robbins}. Because Winker's condition is sufficient to derive the Huntington equation, the Robbins problem had been solved. In subsequent searches, EQP was able to derive the Huntington equation directly. Details of the work and a proof can be found in [4]. The proof was accepted as correct after being checked by hand and by several independent proof-checking programs. Mathematicians and logicians have carefully studied EQPs proofs, but little insight into the nature of the problem has been gained. All presently known proofs (as of 2000) of the Robbins conjecture are based on EQPs proofs. References [I] BACI-IMAIR, L., GANZINGEI.I, H., LYNCH, C., AND SNYDER, W.: 'Basic paramodulation and superposition', in D. KAPUR (ed.): Proc. 11th Internat. Conf. A u t o m a t e d Deduction,

ROTOR

[2]

[3]

[4] [5]

[6]

Vol. 607 of Lecture Notes in Artificial Intelligence, Springer, 1992, pp. 462-476. HUNTINGTON, E.V.: 'New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia Mathematica', Trans. Amer. Math. Soc. 35 (1933), 274 304. MCCUNE, W.: '33 basic test problems: A practical evaluation of some paramodulation strategies', in R. VEROFF (ed.): Automated Reasoning and its Applications: Essays in Honor of Larry Wos, MIT, 1997, pp. 71-114. McCUNE, W.: 'Solution of the Robbins problem', J. Automated Reasoning 19, no. 3 (1997), 263-276. STICKEL, M.: 'A unification algorithm for associativecommutative functions.', J. A C M 28, no. 3 (1981), 423-434. WINKER, S.: 'Absorption and idempotency criteria for a problem in near-Boolean algebras', J. Algebra 153, no. 2 (1992), 414 423.

William McCune MSC 1991: 68T15, 06Exx

Replacing the single differential algebra 7~ -C ~ ( f / ) / l : s with chains of algebras 7~~ ~ ... -+ 74TM -+ • -. --+ T4° using the spaces cm(fl) in the place of C°°(fI) at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of cm-functions as well as the derivative of C'~+1 -+ C"~ at each fixed level m. See also G e n e r a l i z e d f u n c t i o n a l g e b r a s . References [1] MALLIOS, A., AND ROSINGER, E.E.: 'Space-time foam dense singularities and de Rham cohomology', Acta Applic. Math. (to appear).

[2] O~EaGUGGENBEaGEa,M., AND ROSlNGER, E.E.: Solution

[3] [4] [5]

ROSINGER

NOWHERE-DENSE

GENERALIZED

A L G E B R A - In the general framework of g e n e r a l i z e d f u n c t i o n a l g e b r a s developed by E.E. Rosinger [3], [4], [5], [6], a distinguished role is played by ideals in the sequence algebra C~ (f~)N, ft an open subset of R n, which are defined by vanishing properties. Given a family S of subsets of fl, stable under finite unions, one considers the ideal Zs in C~(f~) N determined by those (uj)jcN for which there is a F E S such that for all x C t i \ F , (uj)jeN vanishes near x eventually, that is, there are a j0 and a neighbourhood V C ft \ F of x such that ujIv =- 0 for j > j0. The nowhere-dense generalized function algebra /r~nd(~ ) : C e ° ( ~ ) N / ~ n d is obtained when S is the class of nowhere-dense, closed subsets of ft. The space T~nd(f~) contains C~(f~) via the constant imbedding. It has two distinguishing features. First, the family {7~d(f~): ~ open} forms a f l a b b y s h e a f , and in a certain sense the smallest flabby sheaf containing C~(f/), see [2]. Secondly, the algebra C,~(ft) of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of f~ can be imbedded into ~]-~nd(Q)' In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets F (no growth restrictions near F) can be interpreted as global solutions in ~-~nd(~) by means of a suitable regularization method. The space of distributions :D~(f/) (cf. also G e n e r a l i z e d f u n c t i o n s , s p a c e of) is imbedded in any algebra of the form C ~ ( f / ) / Z z by a general procedure [4] using an algebraic basis. Further generalizations of the ideal ~nd to include larger exceptional sets as well as applications to nonsmooth differential geometry can be found in [1]; nonlinear Lie group actions on generalized functions using the framework of "~nd (~) are studied in [7]. FUNCTION

[6] [7]

of continuous nonlinear PDEs through order completion, North-Holland, 1994. ROSINGER, E.E.: Distributions and nonlinear partial differential equations, Springer, 1978. ROSINGER, E.E.: Nonlinear partial differential equations. Sequential and weak solutions, North-Holland, 1980. ROSINGER, E.E.: Generalized solutions of nonlinear partial differential equations, North-Holland, 1987. ROSINGER, E.E.: Nonlinear partial differential equations, an algebraic view of generalized solutions, North-Holland, 1990. ROSINGER, E.E.: Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem, Kluwer Acad. Publ., 1998.

Michael Oberguggenberger MSC1991:46F30 ROTOR -

in g r a p h t h e o r y . The n-rotor of a g r a p h is the part of the graph that is invariant under the action of the cyclic g r o u p Zn; [7], [8]. Rotor

R o t o r i n k n o t t h e o r y . The n-rotor of a link diagram

(cf. K n o t a n d link d i a g r a m s ) is the part of the link diagram that is invariant under rotation by an angle of

2~r/n. If one modifies the rotor part of a link diagram by rotation of the rotor along an axis of symmetry of an n-don in which the rotor is placed, one obtains a rotant of the original diagram. A link diagram and its rotant share, in some cases, polynomial invariants of links: the Jones polynomial for n < 5, the J o n e s - C o n w a y p o l y n o m i a l for n < 4 and the K a u f f m a n b r a c k e t p o l y n o m i a l for n < 3. Also, the problem for which n and p a link and its n-rotant share the same space of Fox p-colourings (cf. Fox n - c o l o u r i n g ) has been solved for n not divisible by p, or n = p. Rotors can be thought of as generalizing the notion of mutation [1]. It is an open problem (as of 2000) whether the Alexander polynomial is preserved under rotation for any n, [3]. P. Traczyk has announced (in March 2001) the affirmative answer to the problem. There is a relation of rotors with statistical mechanics (cf. also Stat i s t i c a l m e c h a n i c s , m a t h e m a t i c a l p r o b l e m s in), 337

ROTOR

w h e r e a t a n g l e p l a y s t h e r o l e of s p e c t r a l p a r a m e t e r in

[5] PRZYTYCKI, J.H.: 'Search for different links with the same

the Yang-Baxter

Jones' type polynomials: Ideas from graph theory and statistical mechanics': Panoramas of Mathematics, Vol. 34 of Banach Center Publ., Banach Center, 1995, pp. 121 148. [6] TRACZYK, P.: 'A note on rotant links', J. Knot Th. Ramifications 8, no. 3 (1999), 397-403. [7] TUTTE, W.T.: 'Codichromatic graphs', J. Combin. Th. B 16 (1974), 168-174. [8] TUTTE, W.T.: 'Rotors in graph theory', Ann. Discr. Math. 6 (1980), 343-347.

equation,

[4], [2], [5], [6].

References [1] ANSTEE, R.P., PRZYTYCKI, J.H., AND ROLFSEN, D.: 'Knot polynomials and generalized mutation', Topol. Appl. 32 (1989), 237-249. [2] HOSTE, J., AND PRZYTYCKI, J.H.: 'Tangle surgeries which preserve Jones-type polynomials', Internat. J. Math. 8 (1997), 1015-1027. [3] JIN, G.T., AND ROLFSEN, D.: 'Sortie remarks on rotors in link theory', Canad. Math. Bull. 34 (1991), 480-484. [41 JONES, V.F.R.: 'Commuting transfer matrices and link polynomials', Internat. J. Math. 3 (1992), 205-212.

338

J o z e f Przytycki

MSC 1991:57M25

S S - I N T E G E R - As a simple example, let S = { P l , . . . ,Pn} be a finite set of rational prime numbers. The rational integers a/b, a, b E Z, relatively prime (cf. also M u t u a l l y - p r i m e n u m b e r s ) , such that the set of prime divisors of b (possibly empty) is contained in S are the so-called S-integers (corresponding to the specific set S). Clearly, this is a subring Rs of Q. Let R ) denote the group of units of R s , i.e. the group of multiplicatively invertible elements of R s (the S-units). Clearly, these are ± and the rational numbers x in the prime decomposition of which only prime numbers from the set S appear. These notions can be defined in a more sophisticated way, the advantage of which is that it can be generalized to the more general case of a n u m b e r field. For this the notion of absolute value on a number field is needed. Unfortunately, there is no general agreement on the definition of this notion. Below, this 'absolute value' is taken in the sense of a metric as in [1, Chap. 1, Sect. 4; Chap. 4, Sect. 4]; equivalently, an absolute value is a function pV(-), where p is a fixed, conveniently chosen positive real number < 1 and v is a valuation, as defined and used in [2, Chap. 1, §2; Chap. 3 §1] (cf. also V a l u a t i o n , which gives a slightly different definition). In the special case above, every rational prime number p gives rise to a p-adic absolute value and all possible absolute values of Q are (up to topological equivalence) the p-adic ones (non-Archimedean), denoted by I'lp, and the usual absolute value (Archimedean), denoted by I'l~o. Let M q denote the set of absolute values (more precisely, the set of equivalence classes of absolute values (i.e. places) of Q; cf. also P l a c e o f a field). Thus, every element of this set is of the form I'l., where v is either a rational prime number or the symbol oc. One now modifies the definition of the set S above as the subset of M q containing the absolute values (i.e. places) J'lv, where v e {Pl,...,P,~,oo}. Then R s = {x E Q: Ixl~ < 1, Vl.lv ¢ S} and R~ = {x E Q: [xt~ = 1, vl.l~ ¢ s}.

Consider now the more general situation, where a number field K is taken in place of Q and its ring of integers OK is taken in place of Z. Let MK be the set of absolute values of K (more precisely, the set of equivalence classes of absolute values, i.e. places, of K). These are divided into two categories, namely, the non-Archimedean ones, which are in one-to-one correspondence with the prime ideals (or, what is essentially the same, with the prime divisors) of K and the Arehimedean ones, which are in one-to-one correspondence with the isomorphic embeddings K ~-> C (complex-conjugate embeddings giving rise to the same absolute value). As before, let S be a finite subset of M K containing all Archimedean valuations of K . Then, the set R s of S-integers and the set R ) of S-units are defined exactly as in the case of rational numbers (see the definitions above), where now Q is replaced by K. Many interesting problems concerning the solution of D i o p h a n t i n e e q u a t i o n s are reduced to questions about S-integers of 'particularly simple form' (e.g. linear forms in two unknown parameters), which are S-units, and then results are obtained by applying a variety of relevant results on S-integers and S-units. References

[1] BOREVICH, Z.I., AND SHAFAREVICH, I.R.: Number Theory, Acad. Press, 1966. (Translated from the Russian.) [2] NARI(IEWICZ, W.: Elementary and analytic theory of algebraic numbers, PWN/Springer, 1990.

N. Tzanakis

MSC 1991: 12J10, 12J20, 13A18, 16W60 SANTALO

FORMULA

- A formula describing the

Liouville measure on the unit tangent bundle of a R i e m a n n i a n m a n i f o l d in terms of the g e o d e s i c flow and the measure of a codimension-one submanifold (see [5] and [6, Chap. 19]). Let M be an n-dimensional Riemannian manifold, let 7r : U M -+ M be the unit tangent bundle of M, let du be the Liouville measure on U M , and let gt : U M -+ U M be the geodesic flow. One way to define du is to start

SANTAL0 FORMULA with the standard contact form a (cf. C o n t a c t s t r u c t u r e ) and define du = a A d(~n-1. Liouville's theorem says that du is invariant under the geodesic flow gt (since is). Locally, du is just the product measure dm x dv where dm is the Riemannian volume form and dv is the standard volume form on the unit (n - 1)-sphere. For any (locally defined) codimension-one submanifold N C M, let dx be the Riemannian volume element of the submanifold. Let S N = lr -1 (N) C U M , and, for each x E N , let Nx be a unit normal to N at z. Then there is a smooth m a p p i n g G: S N x R --+ U M , given by G(v, t) = gt(v). Santald's formula says: C*(du) = I(v, Nx)l d t d v dx. The formula is used to convert integrals over subsets Q c U M of the unit tangent bundle to iterated integrals, first over a fixed unit-speed geodesic (say parametrized on I(7) C R) and then over the space F of geodesics which are parametrized by their intersections with a fixed codimension-one submanifold and endowed with the measure d 7 = I(v, N~)l dv dx, i.e.

fQf(U)du=fcr f~(~)f@'(t))dtdT" One of the most i m p o r t a n t applications is to the study of Riemannian manifolds with smooth boundary. In this case N = OM, Nx is the inwardly pointing unit normal vector and U + O M = {v E S N : (v,N~) > 0}. For any u E U M , set l(u) = sup{t _> 0: gt(u)is defined}. Note that l(u) = ec means that 9t(u) is defined for all t > 0. Let U M = {u C U M : l ( - u ) < oc} U U+OM, i.e. u E U M means u = gt(v) for some some v C U+OM and some t _> 0. In this setting, Santald's formula takes the form:

L

f ( u ) du =

2 f,v, +OM

f(gt(v)) dt (v, Nx) dv dx.

One immediate application, by simply putting f ( u ) = 1, is:

Vo1(UM) = Cl(n)

fv +oM

Since the Liouville measure is locally a product measure, in the special case U M = U M this says Vol(M) =

fv+oMl(v)(v,

dv dx.

The formula is often used to prove isoperimetric and rigidity results. A sample of such applications can be found in the references. See [1] for Santald's formula for time-like geodesic flow on Lorentzian surfaces. References

[1] ANDERSSON, L., DAHL, M., AND HOWARD, R.: 'Boundary and lens rigidity of Lorentzian surfaces', Trans. Amer. Math. Soc. 348, no. 6 (1996), 2307-2329. [2] CaOKE, C.: 'Some isoperimetric inequalities and eigenvaiue estimates', Ann. Sci. t~cole Norm. Sup. 13, no. 4 (1980), 419-435.

340

[3] CROKE, C.: 'A sharp four-dimensional isoperimetric inequality', Comment. Math. Helv. 59, no. 2 (1984), 187-192. [4] CROKE, C., DAIRBEKOV, N., AND SHARAFUTDINOV, V.: 'Local boundary rigidity of a compact Riemannian manifold with curvature bounded above', Trans. Amer. Math. Soe. (to appear). [5] SANTALO, L.A.: 'Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces', S u m m a Brasil. Math. 3 (1952), 1-11. [6] SANTALO, L.A.: Integral geometry and geometric probability (With a foreword by Mark Kac), Vol. l of Encyclopedia Math. Appl., Addison-Wesley, 1976.

C. Croke M S C 1991: 53C20, 53C22

SAS, statistical

analysis software A widely used commercial software package for statistics and optimization. M S C 1991:62-04 A type of compactification arising from work of I. Satake on the compactification of quotients of symmetric spaces by arithmetically-defined groups ([9], [11], [10]). Below, the simplest case of this is presented first, to help suggest its generalization. Let H be the upper half-plane, the s y m m e t r i c s p a c e of non-compact type for G = SL(2, R). For any subgroup F of finite index in SL(2, Z) - - these are arithmetic groups (cf. also A r i t h m e t i c g r o u p ) - - the quotient space X = F \ H is a R i e m a n n s u r f a c e , a m o d u l a r c u r v e . A c o m p a e t i f i c a t i o n X* of X is obtained by first taking the SL(2, Q)-invariant set SATAKE

COMPACTIFICATION

-

H* = H U P l ( Q ) C P I ( C ) , with an SL(2, Q)-equivariant topology t h a t is the given one on H and makes p l ( Q ) discrete; a deleted neighbourhood base for oc C H* is given by H L = {z C H : I m z > L}

for L > 0.

Then X* is taken to be F \ H * . It is a compact Riemann surface (thus automatically an a l g e b r a i c c u r v e ) . An important ingredient, b o t h here and in Satake's generalization, is reduction theory. Relative to the point oo E H*, it asserts t h a t if z E H and 3' C F satisfy I m z > 1 and Im(Tz ) > 1, then 7 lies in the group of real translations (equivalently, in the parabolic group P of upper-triangular matrices, which is the stabilizer of oc). This gives an embedding of the punctured disc (F A P ) \ H 1 in X, and one is inserting the missing origin by adjoining ec to H . There was great interest in doing something similar for Ag, the moduli space of Abelian varieties (cf. also M o d u l i t h e o r y ; A b e l i a n v a r i e t y ) , which is the quotient Xg = Sp(2g, Z ) \ H g ; here, H9 is the Siegel upper

S ATAKE COMPACTIFICATION half-space of genus g, which is the symmetric space for G = Sp(2g, R), the rank-g group of (2g) x (2g) symplectic matrices. For g = 1 one has H1 = H (from the preceding paragraph). Satake first observed, in [9], that X~ = [_]~_
The applications of Satake compactifications cover a range of other areas. The stable cohomology of 'the' Satake compactification was determined by R. Charney and R. Lee [5] for application in K - t h e o r y . NonHermitian Satake compactifications occur in the combinatorial data of the toroidal compactifications of [1] (see [8, §2]), which provide resolution of the singularities of X* (cf. also R e s o l u t i o n o f s i n g u l a r i t i e s ) .

Satake compactifications, in the sense of [13] (after [10] and [3]), are defined from the following setting [11]. Let D be the symmetric space of non-compact type for the real semi-simple Lie group G (cf. also Lie g r o u p , s e m i - s i m p l e ) . Each faithful finite-dimensional representation of G (cf. also R e p r e s e n t a t i o n o f a Lie algebra) determines an embedding of D in some real projective space, so let D be the closure of D. The boundary of D consists of pieces, called boundary components, that are homogeneous under a class of parabolic subgroups of G. If r is the (real) rank of G, there are only 2 ~ - 1 distinct such spaces D up to homeomorphism, corresponding to the non-empty subsets S of a set of simple roots, and as such they form a semi-lattice; if one then writes D = Ds, the identity mapping of D extends to a continuous mapping D s -+ DT whenever S D T.

Another compactification, the reductive Borel-Serre, a simple quotient of the manifold-with-corners constructed in [4] (see, e.g., [7]), dominates all Satake compactifications, and often coincides with the unique maximal one. It has played an increasing role in the theory of automorphic forms. It is the natural place to study the LP-cohomology [12] and to define weighted cohomology [6]. The spaces D (also defined in a different manner by H. Furstenberg) themselves have played an important role in harmonic analysis, rigidity theory and potential theory.

When G is the real Lie g r o u p associated to an algeb r a i c g r o u p defined over Q, a boundary component is said to be rational when its normalizing p a r a b o l i c s u b g r o u p is defined over Q. Likewise, the structure over Q determines the class of arithmetic subgroups F of G. Take D* to be the union of D and its rational boundary components. Then, with a suitable topology on D*, X* = F\D* is, under mild hypotheses, a compactification of X = F \ D . The collection of these inherit the semi-lattice structure from the above. There is a precise sense in which the topology of X* is induced from that of the closure of a Siegel set in D. The B a i l y - B o r e l c o m p a c t i f i c a t i o n is one of the minimal Satake compactifications in the case where D is Hermitian, i.e., has a G-invariant complex structure (cf. also H e r m i t i a n s y m m e t r i c space). This includes the case of X~ above. Here, D gets embedded as a bounded symmetric domain. By means of automorphic forms of sufficiently high weight, X* gets embedded as a normal algebraic subvariety of complex projective space [2]. This fact has rather strong consequences, for the singular locus of D* has 'high' codimension in general, and one can invoke general results from a l g e b r a i c g e o m e t r y . It implies the existence of big families of Abelian varieties that do not degenerate. This compactification also enters into the topological interpretation of the L 2cohomology of X, as conjectured in [12] and proved by E. Looijenga, and L. Saper and M. Stern.

References [1] ASH, A., MUMFORD, D., RAPOPORT, M., AND TAI, Y.S.: Smooth compactification of locally symmetric varieties, Math. Sci. Press, 1975. [2] BAILY, W., AND BOREL, A.: 'Compactification of quotients of bounded symmetric domains', Ann. of Math. 84 (1966), 442-528. [3] BOREL, A.: 'Ensembles fondamentaux pour les groupes arithm~tiques', Colloq. Thdorie de Groupes Algdbriques, Bruxelles (1962), 23-40. [4] BOREL, A., AND SERRS, J.-P.: 'Corners and arithmetic groups', Comment. Math. Helv. 48 (1973), 436-491. [5] CHARNEY,R., AND LEE, R.: 'Cohomology of the Satake compactification', Topology 22 (1983), 389-423. [6] GORESKY, M., HARDER, G., AND MAePHERSON, R.: 'Weighted cohomology', Invent. Math. 116 (1994), 139-213. [7] GORESKY, M., AND TAI, Y.-S.: 'Toroidal and reductive BorelSerre compactifications of locally symmetric spaces', Amer. J. Math. 121 (1999), 1095-1151. [8] HARRIS, M., AND ZUCKER, S.: 'Boundary cohomology of Shimura varieties, II: Hodge theory at the boundary', Invent. Math. 116 (1994), 243-307. [9] SATAKE, I.: 'On the compactification of the Siegel space', J. Indian Math. Soc. 20 (1956), 259-281. [10] SATAKE, I.: 'On compactifications of the quotient spaces for arithmetically defined discontinuous groups', Ann. of Math.

r2 (1960), 555-58o. [11] SATAKE, I.: 'On representations and compactifications of symmetric Riemannian spaces.', Ann. of Math. 71 (1960), 77-110. [12] ZUCKER, S.: 'L 2 cohomology of warped products and arithmetic groups', Invent. Math. 70 (1982), 169-218. [13] ZUCKER, S.: 'Satake compactifications', Comment. Math. Helv. 58 (1983), 312-343.

Steven Zueker MSC 1991: l l F x x

341

SBIBD

SBIBD,

symmetric balanced incomplete block design

- See B l o c k design. MSC 1991:05B05 SCHNEIDER METHOD - The name 'Schneider method' arises from the solution, by Th. Schneider [4] in 1934, to Hilbert's seventh problem (cf. also H i l b e r t p r o b l e m s ) : If c~ is a non-zero a l g e b r a i c n u m b e r , log c~ a non-zero logarithm of c~ and ~ an irrational algebraic number, then the number ct~ = exp{fl log a} is transcendental (cf. also T r a n s c e n d e n t a l n u m b e r ) . One main idea in Schneider's proof is to investigate values of a function F(z) = P(z, a z) at points Sl + s2fl ((Sl, s2) C Z~), where P is a polynomial with algebraic coefficients. Assuming a ~ is algebraic, a non-zero polynomial P is constructed so that F vanishes at many such points sl + s2fl; this construction is based on Dirichlet's box principle or pigeon hole principle (the Thue-Siegel lemma, cf. also T r a n s e e n d e n e y , m e a s u r e of; D i r i e h let p r i n c i p l e ) . A clever computation of a determinant is an essential tool of Schneider's proof. A slight modification of the same argument yields the so-called six exponentials theorem [1], [2]: If xl, x2 are two complex numbers that are linearly independent over Q (cf. also L i n e a r i n d e p e n d e n c e ) and if Yl, y2, Y3 are three complex numbers that are linearly independent over Q, then at least one of the six numbers eXlyl,

eXlY 2 '

exly3

eX2yl

ex2Y 2 '

eX2y3

is transcendental. In relation with this statement, one of the simplest open (as of 2000) problems which would follow from the conjecture that 'linearly independent logarithms of algebraic numbers are algebraically independent' is the four exponentials conjecture: If xl, x2 are two complex numbers that are linearly independent over Q and if yl, y2 are also two complex numbers that are linearly independent over Q, then at least one of the four numbers e xlyl ,

e ccly2 ,

e x2yl ,

e x2y2

is transcendental. Both S. Lang [1] and K. Ramachandra [2] have given general statements concerning the simultaneous algebraic values of analytic functions by means of Schneider's method. Typically, these statements are efficient for functions satisfying functional equations, for instance when f(zl + z2), f(zl) and f(z2) are algebraically dependent. This method yields transcendence results as well as results of algebraic independence on values of exponential, elliptic or Abelian functions; more generally it applies to the arithmetic study of commutative algebraic groups. 342

Schneider's method can be extended to several variables, and then yields partial results related to the Leopoldt conjecture on the p-adic rank of the units of an algebraic n u m b e r field [5]. It also gives lower bounds for the ranks of matrices whose entries are linear combinations with algebraic coefficients/~o + f~x log C~1 -F- " " • -]3n log C~ of logarithms of algebraic numbers [3], a special case of which is Roy's strong six exponentials theorem: If M is a (d x g)-matrix, with df > d + f, whose entries are linear combinations of logarithms of algebraic numbers, with rows linearly independent over the field of algebraic numbers and with columns linearly independent over the field of algebraic numbers, then the rank of M is at least 2. An extension of Schneider's method also provides sharp measures for linear independence of logarithms of algebraic numbers [6]. References [1] LANG, S.: Introduction to transcendental numbers, AddisonWesley and Don Mills, 1966. [2] RAMAeHANDHA, K.: ' C o n t r i b u t i o n s to the theory of transcendental n u m b e r s I-II', A c t a Arith. 14 (1967/68), 65-72; 73-88. [3] ROY, D.: ~Matriees whose coefficients are linear forms in logarithms', J. N u m b e r T h e o r y 41, no. 1 (1992), 22-47. [4] SCHNEIDER,TH.: ' T r a n s z e n d e n z u n t e r s u c h u n g e n periodischer Punktionen I', J. R e i n e A n g e w . Math. 172 (1934), 65-69. [5] WALDSCHMIDT, M.: 'Transcendance et exponentielles en plusieurs variables', I n v e n t . Math. 63, no. 1 (1981), 97-127. [6] WALDSCHMIDT,M.: D i o p h a n t i n e approximation on linear algebraic groups. Transcendence properties of the exponential f u n c t i o n in several variables, Vol. 326 of Grundl. Math. Wissenschaft., Springer, 2000. Michel Waldschmidt

MSC1991:11J81 SCHRODER

FUNCTIONAL

EQUATION

-

The

equation

¢(/(x))

=

(1)

where ¢ is the unknown function and f(x) is a known real-valued function of a real variable x. I.e. one asks for the eigenvalues and eigenfunctions of the composition operator (substitution operator) ¢ ~-~ ¢ o f . Sometimes A is allowed to be a function itself. One also considers the non-autonomous Schrhder

functional equation ¢ ( f ( x ) ) = g(x)C(x) + h(x). The Schrhder and Abel functional equations (see also F u n c t i o n a l e q u a t i o n ) have much to do with the trans-

lation functional equation ¢(¢(s, u), v) = ¢(s, sCS,

u, v E H ,

• v),

¢:S×H-->S,

where H is a s e m i - g r o u p , which asks for something like a right action of H on S, [1], [5].

SCHUBERT CALCULUS The equation was formulated by E. SchrSder, [4], and there is an extensive body of literature. References [1] AeZl~L, J.: A short course on functional equations, Reidel, 1987. [2] KUCZMA, M.: On the Schr5der operator, PWN, 1963. [3] KUCZMA, M.: Functional equations in a single variable, P W N , 1968. [41 SCHR6DER, E.: 'Uber iterierte Funktionen III', Math. Ann. 3 (1970), 296 322. [5] TARGONSKI, G.: Topics in iteration theory, Vandenhoeck and Ruprecht, 1981, p. 82ff. [6] WALORSKI,J.: 'Convex solutions of the SchrSder equation in Banach spaces', Proc. A m e r . Math. Soe. 125 (1997), 153158.

M. Hazewinkel

where 17. : V0 C ... C Vn = P ~ is a flag of linear subspaces with dim Vj = j. The S c h u b e r t c y c l e Gao.....as is the cohomology class Poinca% dual to the fundamental homology cycle of fl~ o..... ~ 17. (cf. also H o m o l o g y ) . The basis theorem asserts that the Schubert cycles form a basis of the Chow ring A*Gm,n (when k is the complex number field, these are the integral cohomology groups H*G,~,~) of the Grassmannian with a~o .....~m C A('~+l)(n+])-(:++:) -~° . . . . . . . Gin,n, (see also G r a s s m a n n m a n i f o l d ) . The duality theorem asserts that the basis of Schubert cycles is self-dual under the intersection pairing (a,/~) E H*Gm,~ ® H*G,~,~ --+ deg(a •/3) = f _

MSC 1991: 39B12, 39B05

JC4 m ,

SCHUBERT CALCULUS, Schubert enumerative calculus - A formal calculus of symbols representing geo-

metric conditions used to solve problems in enumerative geometry. This originated in work of M. Chasles [3] on conics and was systematized and used to great effect by H. Schubert in [13]. The justification of Schubert's enumerative calculus and the verification of the numbers he obtained was the contents of Hilbert's 15th problem (cf. also H i l b e r t p r o b l e m s ) . Justifying Schubert's enumerative calculus was a major theme of twentieth century a l g e b r a i c g e o m e t r y , and i n t e r s e c t i o n t h e o r y provides a satisfactory modern framework. E n u m e r a t i v e geometry deals with the second part of Hilbert's problem. See [6] for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see [9], [10]. The Schubert calculus also refers to mathematics arising from the following class of enumerative geometric problems: Determine the number of linear subspaces of projective space that satisfy incidence conditions imposed by other linear subspaces. For a survey, see [11]. For example, how many lines in projective 3-space meet 4 given lines? These problems are solved by studying both the geometry and the cohomology or Chow rings of Grassmann varieties (cf. also C h o w ring; G r a s s m a n n m a n i f o l d ) . This field of Schubert calculus enjoys important connections not only to algebraic geometry and a l g e b r a i c t o p o l o g y , but also to algebraic combinatorics, representation theory, differential geometry, linear algebraic groups, and symbolic computation, and has found applications in numerical homotopy continuation [8], linear algebra [7] and systems theory [2]. The Grassmannian G, .... of m-dimensional subspaces (m-planes) in P~ over a field k has distinguished Schubert varieties

a~o ..... ~ v. := { w c a,~,~:

wnv~, >j},

a./~ n

with Gao,...,a,~ dual t o G n - a . . . . . . n - d o . Let ~-b := c~-,~-b,~-m+l .....~ be a special Schubert cycle (cf. S c h u b e r t cycle). Then Gao,...,a m " T b ~ E

Gco,...,cm,

the sum running over all (Co,..., cm) with 0 _< co G ao < cl _< al _< -'. _< c,~ G a,~ and b = ~ ( a i G). This Pieri f o r m u l a determines the ring structure of cohomology; an algebraic consequence is the Giambelli formula for expressing an arbitrary Schubert cycle in terms of special Schubert cycles. Define Tb = 0 if b < 0 or b > m, and To = 1. Then Giambelli's formula is ~rao..... ~m = det[~-~-m+d-~]i,j=0 ..... ~. These four results enable computation in the Chow ring of the Grassmannian, and the solution of many problems in enumerative geometry. For instance, the number of m-planes meeting (m + 1)(n - m) general (n - m - 1)-planes non-trivially is the coefficient of or0.....,~ in the product (~-l)("~+l)(n-'~), which is [14] 1!... (n - m - 1)!. [(m + 1 ) ( n - m)]! ( n - m)! (n - m + 1 ) ! . . . (n! - 1)! These four results hold more generally for cohomology rings of flag manifolds G / P ; Schubert cycles form a self-dual basis, the Chevalley formula [4] determines the ring structure (when P is a B o r e l s u b g r o u p ) , and the Bernshte~n-Gel'fand-Gel'fand formula [1] and Demazure formula [5] give the analogue of the Giambelli formula. More explicit Giambelli formulas are provided by S c h u b e r t p o l y n o m i a l s . One cornerstone of the Schubert calculus for the Grassmannian is the Littlewood-Richardson rule [12] for expressing a product of Schubert cycles in terms of the basis of Schubert cycles. (This rule is usually expressed in terms of an alternative indexing of Schubert cycles using partitions. A sequence (a0,... ,a,~) corresponds to the partition (n - m - a o , n - m + 1 - a l , . . . , n - am); cf. S c h u r f u n c t i o n s in a l g e b r a i c c o m b i n a t o r i c s . ) 343

SCHUBERT CALCULUS T h e analogue of the L i t t l e w o o d - R i c h a r d s o n rule is not k n o w n for most other flag varieties G / P .

References [1] BERNSHTEIN~

[2]

[3]

[4]

[5]

[6] [7]

I.N.,

GEL'FAND,

I.M.,

AND G E L ' F A N D ,

S.I.:

'Schubert cells and cohomology of the spaces G/P', Russian Math. Surveys 28, no. 3 (1973), 1-26. BYRNES,C.I.: 'Algebraic and geometric aspects of the control of linear systems', in C.I. BYRNES AND C.F. MARTIN(eds.): Geometric Methods in Linear systems Theory, Reidel, 1980, pp. 85-124. CHASLES, M.: 'Construction des coniques qui satisfont /~ claque conditions', C.R. Aead. Sci. Paris 58 (1864), 297308. CHEVALLEY, C.: 'Sur les d~compositions cellulaires des espaces G/B', in W. HABOUSH (ed.): Algebraic Groups and their Generalizations: Classical Methods, Vol. 56:1 of Proc. Syrup. Pure Math., Amer. Math. Soc., 1994, pp. 1-23. DEMAZURE,M.: 'D~singularization des varifit~s de Schubert g~n6ralis~es', Ann. Sci. t~cole Norm. Sup. (~{) 7 (1974), 53 88. FULTON,W.: Intersection theory, second ed., Vol. 2 of Ergebn. Math., Springer, 1998. FULTON, W.: 'Eigenvalues, invariant factors, highest weights, and Schubert calculus', Bull. Amer. Math. Soc. 37 (2000), 209-249.

[8] H U B E R , B . ,

SOTTILE, F.,

AND STURMFELS, B . : ' N u m e r i c a l

Schubert calculus', or. Symbolic Comput. 26, no. 6 (1998), 767-788. [9] KLEIMAN, S.: 'Problem 15: Rigorous foundation of Schubert's enumerative calculus': Mathematical Developments arising from Hilbert Problems, Vol. 28 of Proe. Syrup. Pure Math., Amer. Math. Soc., 1976, pp. 445-482. [10] KLEIMAN,S.: 'Intersection theory and enumerative geometry: A decade in review', in S. BLOCH (ed.): Algebraic Geometry (Bowdoin, 1985), Vol. 46:2 of Proc. Syrup. Pure Math., Amer. Math. Soc., 1987, pp. 321-370. [11] KLEIMAN, S.L., AND LAKSOV,D.: 'Schubert calculus', Amer. Math. Monthly 79 (1972), 1061-1082. [12] L I T T L E W O O D , D.E., AND RICHARDSON, A . R . : 'Group characters and algebra', Philos. Trans. Royal Soe. London. 233 (1934), 99 141. [13] SCHUBERT,H.: Kiilkul der abz~ihlenden Geometric, Springer, 1879, Reprint (with an introduction by S. Kleiman), 1979. [14] SCHUBERT,H.: 'Anzahl-Bestimmungen f/Jr lineare RSume beliebiger Dimension', Acts Math. 8 (1886), 97-118. Frank SottiIe M S C 1 9 9 1 : 14N15, 14M15, 14C15, 20G20, 57T15

SCHUBERT C E L L - T h e orbit of a B o r e l s u b g r o u p B C G on a flag variety G / P [1, 14.12]. Here, G is a semi-simple l i n e a r a l g e b r a i c g r o u p over an a l g e b r a i c a l l y c l o s e d field k and P is a p a r a b o l i c s u b g r o u p of G so t h a t G / P is a complete homogeneous variety• Schubert cells are indexed by the cosets of the W e y l g r o u p W p of P in the Weyl group W of G. Choosing B C P , these cosets are identified with Tfixed points of G / P , where T is a m a x i m a l t o r u s of G and T C B. T h e fixed points are conjugates P ' of P containing T. T h e orbit B w W p ~- A e(wWp), the a f f i n e s p a c e of dimension equal to the length of the shortest 344

element of the coset w W p . W h e n k is the complex number field, Schubert cells constitute a C W - d e c o m p o s i t i o n of G / P (cf. also C W - c o m p l e x ) . Let k be any f i e l d and suppose G / P is the Grassm a n n i s h Gm,n of m-planes in k n (cf. also G r a s s m a n n m a n i f o l d ) . Schubert cells for Gm,n arise in an element a r y manner• A m o n g the m by n matrices whose row space is a given H E G,~,n, there is a unique echelon matrix (E0

<

...

where

•

"..

,

E1

---__

•

°

.

(i °

.

°.

°. . •

0

i)

°•.

-••

•

••°

... • .°

>~

•

.

.

where • represents an a r b i t r a r y element of k. This echelon representative of H is c o m p u t e d from any representative by Gaussian elimination (cf. also E l i m i n a t i o n t h e o r y ) • T h e column n u m b e r s al < " • < a,~ of the leading entries (ls) of the rows in this echelon representative determine the t y p e of H . Counting the u n d e t e r m i n e d entries in such an echelon matrix shows t h a t the set of H E Gm,n with this t y p e is isomorphic to A m n - E ( a i + i - 1 ) . This set is a Schubert cell of Grn,n.

References [1] BOREL, A.: Linear algebraic groups, second ed., Vol. 126 of Grad. Texts Math., Springer, 1991. Frank Sottile

M S C 1991: 14M15, 14L35, 20G20 S C H U B E R T CYCLE, Schubert class - The cycle class of a S c h u b e r t v a r i e t y in the c o h o m o l o g y r i n g of a complex flag manifold G / P (cf. also F l a g s t r u c t u r e ) , also called a Schubert class• Here, G is a semi-simple l i n e a r a l g e b r a i c g r o u p and P is a p a r a b o l i c s u b g r o u p • Schubert cycles form a basis for the cohomology groups [4], [1, 14.12] of G / P (cf. also C o h o m o l o g y g r o u p ) • T h e y arose [4] as representatives of Schubert conditions on linear subspaces of a vector space in the S c h u b e r t c a l c u l u s for enumerative g e o m e t r y [3]. T h e justification of Schubert's calculus in this context by C. E h r e s m a n n [2] realized Schubert cycles as cohomology classes Poincar6 dual to the f u n d a m e n t a l h o m o l o g y cycles of Schubert varieties in the Grassmannian. While Schubert, E h r e s m a n n and others worked primarily on

S C H U B E R T POLYNOMIALS the Grassmannian, the pertinent features of the Grassmannian extend to general flag varieties G/P, giving Schubert cycles as above. More generally, when G is a semi-simple linear algebraic group over a field, there are Schubert cycles associated to Schubert varieties in each of the following theories for G/P: singular (or de Rham) cohomology, the Chow ring, K-theory, or equivariant or quantum versions of these theories. For each, the Schubert cycles form a basis over the base ring. For the cohomology or the Chow ring, the Schubert cycles are universal characteristic classes for (flagged) G-bundles. In particular, certain special Schubert cycles for the Grassmannian are universal Chern classes for vector bundles (cf. also C h e r n class). References [1] BOREL, A.: L i n e a r algebraic groups, second ed., Vol. 126 of Grad. T e x t s M a t h . , Springer, 1991. [2] EHaESMANN, C.: 'Sur la topologie de certains espaces homog~nes', A n n . M a t h . 35 (1934), 396-443. [3] SCHUBERT,H.: K i i l k u l d e r a b z i i h l e n d e n G e o m e t r i e , Springer, 1879, Reprinted (with an introduction by S. Kleiman): 1979. [4] SCHUBERT, H.: 'Losiing des Charakteristiken-Problems fiir lineare Rgume beliebiger Dimension', M i t t . M a t h . Ges. H a m burg (1886), 135-155.

Frank Sottile MSC 1991: 14M15, 14C15, 14C17, 20G20, 57T15 SCHUBERT POLYNOMIALS - Polynomials introduced by A. Lascoux and M.-P. Schfitzenberger [18] as distinguished polynomial representatives of Schubert cycles (cf. also S c h u b e r t cycle) in the c o h o m o l o g y r i n g of the manifold Fg~ of complete flags in C ~. This extended work by I.N. Bernshtein, I.M. Gel'fand and S.I. Gel'land [3] and M. Demazure [8], who gave algorithms for computing representatives of Schubert cycles in the co-invariant algebra, which is isomorphic to the cohomology ring of Fg~ [6]:

H*(Fg~, Z) _~ Z[Xl,... , X n ] / Z ÷ [ X l , . . . ,Xn] S~ • Here, Z + [ Z l , . . . , Xn]$~ is the ideal generated by the nonconstant polynomials that are symmetric in Xl, • • •, x~. See [19] for an elegant algebraic treatment of Schubert polynomials, and [12] and [20] for a more geometric treatment. For each i = 1 , . . . , n - 1, let si be the transposition (i, i + 1) in the s y m m e t r i c g r o u p Sn, which acts on Z [ x l , . . . , z ~ ] . The divided difference operator cgi is defined by f - s~f &f

-

x i -- x i + 1 "

These satisfy = o,

a oj

= ajo~

if li - Jl > 1,

(1)

t O i O i + l Oi = 0 i + 1 0 i O i + l "

If f,w E Z [ x l , . . . , x ~ ] is a representative of the Schub e r t c y c l e G~, then

Oifw = ~0 t fs, w

if~(siw) > ~(w), ifg(siw) < g(w),

where g(w) is the length of a permutation w and f~w represents the Schubert cycle a,{~. Given a fixed polynomial representative of the Schubert cycle cr~ (the class of a point as w,~ E $~ is the longest element), successive application of divided difference operators gives polynomial representatives of all Schubert cycles, which are independent of the choices involved, by (1). The choice of the representative @~n = X n-1 1 X 2n--2 • . . X n _ 1 for a~. gives the Schubert polynomials. Since cgn.-.c91Gw~+l = ® ~ , Schubert polynomials are independent of n and give polynomials G~ E Z [ x l , x 2 , . . . ] for w E S ~ = UN~. These form a basis for this polynomial ring, and every Sehur polynomial is also a Schubert polynomial. The transition formula gives another recursive construction of Schubert polynomials. For w E $ ~ , let r be the last descent of w (w(r) > w(r + 1) < w(r + 2) < ...) and define s > r by w(s) < w(r) < w(s + 1). Set v = w(r, s), where (r, s) is the transposition. Then

the sum over all q < r with g(v(q, r)) = ~(v) + 1 = g(w). This formula gives an algorithm to compute ~ as the permutations that appear on the right-hand side are either shorter than w or precede it in reverse lexicog r a p h i c o r d e r , and the minimal such permutation u of length rn has ®~ = x ~ . The transition formula shows that the Schubert polynomial ®~ is a sum of monomials with non-negative integral coefficients. There are several explicit formulas for the coefficient of a monomial in a Schubert polynomial, either in terms of the weak order of the symmetric group [5], [1], [11], an intersection number [15] or the Bruhat order [2]. An elegant conjectural formula of A. Kohnert [16] remains unproven (as of 2000). The Schubert polynomial ®~ for w E $~ is also the normal form reduction of any polynomial representative of the Schubert cycle ~ with respect to the degree-reverse lexicographic term order on Z [ x l , . . . , x ~ ] with xl < ... < x~. The above-mentioned results of [6], [3], [8] are valid more generally for any flag manifold G / B with G a semisimple r e d u e t i v e g r o u p and B a B o r e l s u b g r o u p . When G is an orthogonal or s y m p l e c t i e g r o u p , there 345

SCHUBERT POLYNOMIALS are competing theories of Schubert polynomials [4], [10], [17], each with own merits. There are also double Schubert polynomials suited for computations of degeneracy loci [14], quantum Schubert polynomials [9], [7] and universal Schubert polynomials [13].

[20] MANIVEL, L.: 'Fonctions sym6triques, polyn6mes de Schubert et lieux de dfig~nfirescence', Cours Spdcialisds Soc. Math. France 3 (1998). Frank Sottile

MSC 1991: 05E05, 20G20, 57T15 SCHUR

References

14N15,

ALGEBRA

-

14M15,

14C15,

13P10,

See S c h u r r i n g .

MSC 1991: 20Bxx, 20B15, 20C30 [1] BERGERON,•.: 'A combinatorial construction of the Schubert polynomials', J. Combin. Th. A 60 (1992), 168-182. [2] BERGERON, N., AND SOTTILE, F.: 'Skew Schubert functions and the Pieri formula for flag manifolds', Trans. Amer. Math. Soc. (to appear). [3] BERNSHTEIN, I.N., GEL'FAND, I.M., AND GEL'FAND, S.I.: 'Schubert cells and cohomology of the spaces G / P ' , Russian Math. Surveys 28, no. 3 (1973), 1-26. [4] BILLEY, S., AND HAIMAN, M.: 'Schubert polynomials for the classical groups', J. Amer. Math. Soc. 8, no. 2 (1995), 443482. [5] BILLEY, S., JOCKUSIt, W., AND STANLEY, R.: 'Some combinatorial properties of Schubert polynomials', J. Algebraic Combin. 2, no. 4 (1993), 345-374. [6] BOREL, A.: 'Sur la cohomologie des espaces fibres prineipaux et des espaces homog~nes des groupes de Lie compacts', Ann. Math. 57 (1953), 115-207. [7] CIOCAN FONTANINE, I.: 'On quantum eohomology rings of partial flag varieties', Duke Math. J. 98, no. 3 (1999), 485524. [8] DEMAZURE, M.: 'D~singularization des vari6t6s de Schubert g6n~ralis6es', Ann. Sei. t~cole Norm. Sup. (4) 7 (1974), 53 88. [9] FOMIN, S., GELFAND, S., AND POSTNmOV, A.: 'Quantum Schubert polynomials', J. Amer. Math. Soc. 10 (1997), 565596. [10] FOMIN, S., AND KIRILLOV, A.N.: 'Combinatorial Bn-analogs of Schubert polynomials', Trans. Amer. Math. Soc. 348 (1996), 3591-3620. [11] FOMIN, S., AND STANLEY, R.: 'Schubert polynomials and the nilCoxeter algebra', Adv. Math. 103 (1994), 196 207. [12] FULTON, W.: Young tableaux, Cambridge Univ. Press, 1997. [13] FULTON, W.: 'Universal Schubert polynomials', Duke Math. J. 96, no. 3 (1999), 575-594. [14] FULTON, W., AND PRAGACZ, P.: Schubert varieties and degeneracy loci, Vol. 1689 of Lecture Notes in Mathematics, Springer, 1998. [15] KIRILLOV, A., AND MAENO, T.: 'Quantum double Schubert polynomials, quantum Schubert polynomials, and the VafaIntriligator formula', Diser. Math. 217, no. 1-3 (2000), 191223, Formal Power Series and Algebraic Combinatorics (Vienna, 1997). [16] KOHNEHT, A.: 'Weintrauben, polynome, tableaux', Bayreuth Math. Schrift. 38 (1990), 1-97. [17] LASCOUX, A., PaACACZ, P., AND RATAJSKI, J.: 'Symplectic Schubert polynomials /~ la polonaise, appendix to operator calculus for Q-polynomials and Schubert polynomials', Adv. Math. 140 (1998), 1-43. [18] LASCOUX, A., AND SCH/)TZENBEaGER,M.-P.: 'PolynSmes de Schubert', C.R. Acad. Sci. Paris 294 (1982), 447-450. [19] MACDONALD, I.G.: 'Notes on Schubert polynomials', Lab. Combin. et d'Inform. Math. (LACIM) Univ. Qudbec (1991).

346

Let F be a field. The Schur group S(F) of F is the subgroup of the B r a u e r g r o u p B ( F ) consisting of those classes of centrally simple Falgebras that occur in the g r o u p a l g e b r a FG of some f i n i t e g r o u p G. Since the Schur indices for G are trivial in prime characteristic (Wedderburn's theorem; cf. also S c h u r i n d e x ) , one may assume that char(F) -0. By Brauer's theorem (cf. S c h u r i n d e x ) , the field Q(exp(G)) of exp(G)th roots of unity is a splitting field for G. Thus, the study of S(F) essentially is reduced to the cases where F is an algebraic number field (finite over the rational numbers; cf. also A l g e b r a i c n u m b e r ; N u m b e r field) or a completion of such an F with regard to an (infinite or finite) prime. Considering direct products of groups and groups with opposite multiplication shows that S(F) is indeed a subgroup of B(F). The celebrated B r a u e r - W i t t theorem implies that the elements of S(F) are represented by cyclotomic algebras. These are crossed products of fields with Galois groups (cf. also G a l o i s g r o u p ) , in the sense of E. Noether, where the factor sets have finite order. Schur indices over the real numbers are computed by means of the Frobenius-Schur count of involutions. Let L]F be a G a l o i s e x t e n s i o n of p-adic number fields for some prime number p, with group F, and let e = e(L]F) be the ramification index and let UL be the group of units in L. Then H2(F, UL) is the (cyclic) subgroup of B ( F ) of order e. It follows that S(F) has order dividing e(F(p)lF ) when p is odd and dividing e ( F ( 4 ) I F ) otherwise. More detailed investigations of the underlying cyclotomic algebras have been carried out in [3]. An alternative approach can be found in [1]. Explicit generators for S(F) in terms of 'Schur groups' have been given in SCHUR

GROUP

-

[2]. These 'Schur groups' occur as terminal reduction steps by repeated application of the Brauer Witt theorem over local fields. They are the smallest groups admitting characters with non-trivial Schur index, and they appear as sections in any finite group having irreducible characters with non-trivial Schur index. In fact, one has a substitute for the concept of a cyclotomic algebra from the point of view of group representations and

SCHUR Q - F U N C T I O N characters. There is also a close relationship to C l i f f o r d t h e o r y (of simple modules). The theory has numerous applications concerning the behaviour of the Schur index m(x) of an irreducible character X of some finite group (over the rational numbers; cf. also C h a r a c t e r o f a g r o u p ) . For example, the Benard-Schacher theorem states that re(X) is a divisor of the number of roots of unity in the value field Q(X)This leads to interesting block-theoretic consequences (the Feit-Solomon theorem), and to a similar result in Clifford theory. References [1] ADEM, A., AND MILGRAM, R.J.: Cohomology of finite groups, Springer, 1994. [2] RIESE, U., AND SCHMID, P.: 'Schur indices and Schur groups, II', J. Algebra 182 (1996), 183-200. [3] YAMADA, T.: The Schur subgroup of the Braucr group, Vol. 397 of Lecture Notes in Mathematics, Springer, 1974. Peter Schmid

An alternative purely combinatorial definition has been given by J.R. Stembridge [7] in terms of shifted (Young) diagrams. These differ from Young diagrams (cf. Y o u n g d i a g r a m ) in that there is an indentation along the diagonal. Young tableaux are replaced by marked shifted tableaux which are defined as follows. Let P ' denote the ordered alphabet {1' < 1 < 2' < 2 < .--}; then a marked shifted tableau T is a labelling of the nodes of the shifted diagram of shape ~ such that i) the labels weakly increase along each row and down each column; ii) each column contains at most one k, for each k _> 1; and iii) each row contains at most one k', for each k _> 1. If 7k denotes the number of nodes labelled either k or k', then 3' = (~'1,"/2,...) is the content of T and if x T = x~lx~ ~ ..., then

MSC1991: 11R34, 20C05, 16S35, 12G05, 13A20

Q~ = E

xT'

T

SCHUR Q - F U N C T I O N - A symmetric function introduced by I. Schur [6] in 1911 in the construction of the irreducible spin characters of the symmetric groups Sn (cf. P r o j e c t i v e r e p r e s e n t a t i o n s o f s y m m e t r i c a n d a l t e r n a t i n g g r o u p s ) . Schur Q-functions are analogous to the Schur functions, which play the same role for linear characters (cf. S c h u r f u n c t i o n s in algebraic c o m b i n a t o r i c s ) . In fact, both are special cases of Hall-Littlewood functions discovered by D.E. Littlewood [3], but see [4] for a description of their development and subsequent generalizations, for example, Macdonald polynomials. There are by now (as of 2000) several other definitions; the original by Schur [6] was in terms of Pfaffians (cf. Pfafilan), a modern version of his work is [2]. Let x = { x l , . . . , xl} be a set of variables (1 _> n); then

Q(t) = H 1 + xit 1--xit

-

i

Eqrt~"

(1)

r>_O

For r, s _> 0, define 8

Q(r,8) = q~q8 + 2 E ( -

1 ) i q~+iqs_i,

i=1

and then Q(~,r) = -Q(r,8), because ~ino(--1)iqiqn_i -: 0, as follows directly from (1). If A = (A1,..., A2m) is a strict partition of n, where A1 > " " > )~2m _> 0, then the matrix

M~ = (Q(~,,~)) is skew-symmetric, and the Schur Q-function Q~ is defined as Q~ = Pf(Mx), where P f stands for the Pfaffian.

summed over all marked shifted tableaux of shape ~. It is a non-trivial task to prove that this is the Schur Q-function. For example, if ~ = (4, 2, 1), then the corresponding shifted diagram and a possible marked shifted tableau are • ~=

and x T

• •

• • •

•

1' T=

1 2'

1 2 2

2

~- Xl3X 4 2.

This combinatorial definition has been a rich source of significant combinatorial results, for example, Stembridge [7] has proved an analogue of the LittlewoodRichardson rule that describes the Schur Q-function expansion of Q x Q , and also gives a purely combinatorial proof for the Murnaghan-Nakayama rule for computing the irreducible spin characters of Sn (el. R e p r e s e n t a t i o n o f t h e s y m m e t r i c g r o u p s ) . All of this is based on a shifted version of the Robinson-Schensted-Knuth correspondence given independently by B.E. Sagan [5] and D.R. Rowley (cf. also R o b i n s o n - S e h e n s t e d correspondence). Schur Q-functions also arise naturally in other contexts, for example, the characters of irreducible representations of the queer Lie super-algebra Q(n), the cohomology classes dual to Schubert cycles in is•tropic Grassmannians and in polynomial solutions of the BKPhierarchy of partial differential equations. References [1] HOFFMAN, P.N., AND HUMPHREYS, J.F.: Projective representations of the s y m m e t r i c groups, Oxford Univ. Press, 1992. [2] JOZEFIAK, T.: 'Characters of projective representations of symmetric groups', Exp. Math. 7 (1989), 193-247.

347

SCHUR Q - F U N C T I O N [31 LITTLEWOOD, D.E.:

'On certain symmetric functions',

Proc. London Math. Soc. 11, no. 3 (1961), 485-498. [4] MACDONALD,I.G.: Symmetric functions and Hall polynomials, second ed., Oxford Univ. Press, 1997.

p = s t r s 1 -- $2tr $2, where [3]:

an-2.

[5] SAGAN,B.E.: 'Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley', J. Combin. Th. A 45 (1987), 62-103. [6] SCHUR, I.: @ber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutiohen', J. Reine Angew. Math. 139 (1911), 155-250. [7] STEMBaIDCE, J.R.: 'Shifted tableaux and projective representations of symmetric groups', Adv. Math. 74 (1989), 87-134.

i'

S1 =

t = O, 1, 2, . . . ,

where xt E R n and A = (aij), i , j = 1 , . . . , n , is an (n x n)-matrix with real coefficients. Let w ( z ) = aoz ~ + .•. + a ~ - l z + an = d e t ( z I - A) be the c h a r a c t e r i s t i c p o l y n o m i a l for the dynamical system. The polynomial w ( z ) (or, equivalently, the m a t r i x A) is said to be stable if all its roots are inside the unit circle on the complex plane. Similarly, the dynamical system is said to be asymptotically stable if its characteristic polynomial w ( z ) is stable [3]. Asymptotic stability of the polynomial or dynamical system is strongly connected with Schur matrices and Schur's theorem. A Sehur matrix is a square matrix with real entries and with eigenvalues (cf. also E i g e n v a l u e ) of absolute value less than one [1], [2]. Schur's theorem states that every matrix is unitarily similar to a triangular matrix• It has been noted that the triangular matrix is not unique [1]. A consequence of this theorem is the following. Let a matrix A have eigenvalues s l , . . . , s~. Then

~ lskl2 ~ ~ k=l

a0 / a2ai/

i ''"

0

and the symbol tr denotes transposition. Therefore, the matrix P = (Pij), i, = 1 , . . . , n, where i-1

Pij = E ( a i - t - - l a j - - t - - 1 t=0

-- a ~ + t - i + l a n + t - j + l ) ,

j >_ i.

The following main stability theorem holds [3]: The polynomial w ( z ) is asymptotically stable if and only if the matrix P is positive definite, i.e. Pk > 0 for k = 1 , . . . , n, where P1 = P n , Pll •..,Pk =

P2 = P n P21 '"

Plk.

' IPkl

P12 , . . . P22

,...,P~=detP. "'"

Pkkl

Using this theorem, one can prove [3] t h a t if Pk 7~ 0 for k = 1 , . . . , n, then the characteristic polynomial w ( z ) has m roots inside and n - m roots outside the unit circle, where m = n - v(1, P 1 , . . . , P~) and v denotes the number of sign changes in the sequence 1, P 1 , . . . , pn. Moreover, it should be pointed out t h a t Schur's matrix and Schur's theorem can be also used in the solution of the p o l e a s s i g n m e n t p r o b l e m for linear control systems [4].

References

la~jl,

i,j=l

with equality if and only if A is normal (cf. also N o r m a l m a t r i x ) . This leads to the estimate Isk] < n m a x l a i j J ,

which can be directly used in asymptotic stability investigations for the dynamical system. However, it should be stressed that it is possible to use also a different method in asymptotic stability considerations. Namely, it is possible to associate to the characteristic polynomial w ( z ) the symmetric matrix 348

0

$2 =

M S C 1991: 05E99, 05E10, 20C25

xt+l = A x t ,

an-2] ,

a2.

A. O. Morris

SCHUR STABILITY OF POLYNOMIALS AND MATRICES - Consider the linear discrete-time d y n a m i c a l s y s t e m described by the difference equation

an-l~

[1] BHATIA, R.: Matrix analysis, Springer, 1997. [2] Comprehensive dictionary of electrical engineering, CRC, 1999. [3] KACZOREK, T.: Theory of control and systems, PWN, 1993. (In Polish.) [4] VAROA, A.: 'A Schur method for pole assignment', IEEE Trans. Autom. Control AC-26, no. 2 (1981), 517-519. J. Klamka MSC1991: 15A18, 93C05, 93D15

SCHWARZSCHILD Schwarzschild metric.

GEOMETRY

M S C 1991: 53B30, 53B50, 83C20, 83F05

See

SEGAL-SHALE-WEIL REPRESENTATION SCHWARZSCHILD SOLUTION - The same as the Schwarzschild metric. MSC 1991: 53B30, 53B50, 83C20, 83F05

SEGAL-StIALE-WEIL

REPRESENTATION

-

A

representation of groups arising in both number theory and in physics. For number theorists, the seminal paper is that of A. Weil, [10]. He cites earlier papers of I. Segal and D. Shale as precedents, and the deep work of C.L. Siegel on theta-series as inspiration. Let H be a g r o u p with centre Z such that A = H / Z is Abelian, and let X be a unitary character of Z (cf. also C h a r a c t e r o f a g r o u p ) . If g,b E A, choose representatives a,b E H and note that (g,b) = x(aba-lb -1) is independent of the choice of representatives. This is a skew-symmetric bilinear pairing A x A -~ C × . One assumes that this pairing is non-degenerate. The Stone yon Neumann theorem asserts that H has a unique irr e d u c i b l e r e p r e s e n t a t i o n 7r with central character XFurthermore, the representation may be constructed as follows. Let L be a Lagrangian subgroup, that is, any subgroup of H containing Z such that L / Z is a maximal subgroup of A on which the form (., .) is trivial. Extend X to L in an arbitrary manner, then induce. This gives a model for ~r. Let G be a group of automorphisms of H which acts trivially on Z (cf. also A u t o m o r p h i s m ) . If g E G, the Stone-yon Neumann theorem implies that grc ~ 7r. Let w(g): 7r -+ gTr be an intertwining mapping, well defined up to constant multiple (cf. also I n t e r t w i n i n g o p e r a tor). Then w is a p r o j e c t i v e r e p r e s e n t a t i o n of G. For example, let F be a local field and let W be a v e c t o r s p a c e over F endowed with a non-degenerate skew-symmetric bilinear form (., .). Its dimension 2n is even, and the automorphism group of the form is the s y m p l e c t i c g r o u p Sp(2n, F). One can construct a 'Heisenberg group' H = W • F with the multiplication (w,x)(w',x') = (w + w',x + x' + (w,w')). Choosing any non-trivial additive character X0 of F, let X(w,x) = X0(x). Then the hypotheses of the Stone von Neumann theorem are satisfied. As the Lagrangian subgroup of H one may take V®F, where V is any maximal isotropic subspace of H. Then the induced model of 7c described above may be realized as the Schwartz space S(V). The Segal-Shale-Weil representation is the resulting projective representation of Sp(2n, F). It may be interpreted as a genuine representation of a covering group Sp(2n, F), the so-called metaplectie group. Now let F be a g l o b a l field, A its addle ring (cf. also Adgle), and let V and W be as before. Then one may construct a similar representation w of Sp(2n, A) on the Schwartz space S(A ® V). If • E S(A ® V), let

A(~) = ~ v c v ~(v). This linear form is invariant under the action of Sp(2n, F), generalizing the P o i s s o n s u m m a t i o n f o r m u l a . This implies that the representation w is automorphic. The corresponding automorphic forms are theta-functions (of. T h e t a - f u n e t i o n ) , having their historical origins in the work of C.G.J. Jacobi and Siegel. As Weil observed, the automorphicity of this representation is closely related to the q u a d r a t i c r e c i p r o c i t y law. Later authors, notably R. Howe [3], have emphasized the theory of dual reductive pairs. When a pair of reductive groups G1 X G2 embeds in Sp(2n), each being the centralizer of the other (cf. also C e n t r a l i z e r ) , then w sets up a correspondence between representations of G1 and representations of G2. This works at the level of automorphic forms and gives instances of Langlands functoriality, including some historically important ones such as quadratic base change (cf. also B a s e c h a n g e ) . See [7]. The use of the Weil representation in [4] to construct automorphic forms and representations may be understood as arising from the dual reductive pairs 0(2) x SL(2) and 0(4) x SL(2). The dual pair 0(3) x SL(2) underlies the important work of J.-L. Waldspurger [9] on automorphic forms of half-integral weight. In recent years (as of 2000) it has been noted that since the Segal-Shale-Weil representation is the minimal representation of Sp(2n), that is, the representation with smallest Gel'fand-Kirillov dimension, minimal representations of other groups can play a similar role. Many interesting examples may be found in the exceptional groups (cf. also Lie a l g e b r a , e x c e p t i o n a l ) . The possibly first paper where this phenomenon was noted was [5]. Many interesting examples come from the exceptional groups. There is much current literature on this subject, but for typical papers see [1] and [2]. Dual pairs in the exceptional groups were classified in [8]. For further references see [6]. References

[i] GINZBURG, D., RALLIS, S., AND SOUDRY, D.: 'A tower oftheta correspondences for G2', Duke Math. Y. 88 (1997), 537 624. [2] GROSS, B.H., AND gAVIN, G.: 'The dual pair PGLa x G2', Canad. Math. Bull. 40, no. 3 (1997), 376-384. [3] HOWE, R.E.: '0-series and invariant theory': Automorphic forms, representations and L-functions, Vol. 33:1 of Proc. Syrup. Pure Math., Amer. Math. Soc., 1977. [4] JACQUET, H., AND LANGLANDS, R.P.: Automorphic forms on GL(2), Vol. 114 of Lecture Notes in Mathematics, Springer, 1970. [5] KAZHDAN, D.: 'The minimal representation of D4': Operator

Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Birkhguser, 1990, pp. 125158.

349

SEGAL SHALE-WEIL REPRESENTATION [6] PRASAD, D.: 'A brief survey on the theta correspondence':

Number theory, Vol. 210 of Contemp. Math., Amer. Math. Soc., 1998, pp. 171 193. [7] RALLIS, S.: 'Langlands' functoriality and the Weil representation', Amer. J. Math. 104, no. 3 (1982), 469-515. [8] RUBENTHALER,H.: 'Lee paires duales dane lee alg~bres de Lie r~ductives', Astdrisque 219 (1994). [9] WaLDSPUROER, J.-L.: 'Sur lee coefficients de Fourier des formes modulaires de poids demi-entier', J. Math. Puree Appl. 60 (1981), 375-484. [10] WEIL, A.: 'Sur certains groupes d'opfirateurs unitaires', Acta Math. 111 (1964), 143 211, Also: Collected Works, Vol. 3. D. Bump M S C 1 9 9 1 : 11F27, 11F70, 20G05, 81R05 SEGRE

CHARACTERISTIC

OF A SQUARE

MA-

T R I X - Let A be a square m a t r i x over a field F and let

a C F , the algebraic closure of F , be an eigenvalue (of. E i g e n v a l u e ) of A. Over F the matrix A can be put in J o r d a n normal form (see J o r d a n m a t r i x ) . T h e Segre characteristic of A at the eigenvalue c~ is the sequence of sizes of the J o r d a n blocks of A with eigenvalue a in non-increasing order. T h e Segre characteristic of A consists of the complete set of d a t a describing the J o r d a n n o r m a l form comprising all eigenvalues a s , . . . , a r and the Segre characteristic of A at each of the c~i. References [1] CULLEN, CH.G.: Matrices and linear transformations, Addison-Wesley, 1972, p. Chap. 5. [2] TURNBULL,H.W., AND AITKEN, A.C.: An introduction to the theory of canonical matrices, Blackie, 1932, p. Chapt. VI. M. Hazewinkel M S C 1 9 9 1 : 15A18, 15A21 S E I F E R T C O N J E C T U R E - T h e assertion t h a t every non-singular (i.e. everywhere non-zero) C 1 v e c t o r field on the three-dimensional sphere S s possesses a circular orbit. The conjecture is a three-dimensional analogue of the well-known hairy ball theorem, stating t h a t there is no continuous non-singular vector field on the twodimensional sphere S 2. Integrating a C 1 vector field results in a flow, which on a closed manifold M is a dynamical system, i.e. a m a p p i n g ~ : R x M --+ M with the properties:

1) q~(0,p) = p; and

2)

=

+ 8,p)

(the p a r a m e t e r t is usually interpreted as time; cf. also Dynamical system; Flow (continuous-tlme dynamical system)). An orbit, or a trajectory, of a point p E M is the set ~b(R x {p}). If an orbit is simple closed curve, then it is called circular, closed or periodic. The H o p f f i b r a t i o n is an e s s e n t i a l m a p p i n g from S 3 onto S 2 whose fibres, the inverse images of single 350

points, are simple closed curves. T h e Seifert conjecture has its roots in a 1950 p a p e r of H. Seifert [8], who proved t h a t a C 1 non-singular vector field on S 3 possesses a periodic orbit if it is 'almost parallel' to the fibres of the Hopf fibration. T h e even-dimensional spheres do not a d m i t nonsingular vector fields, and a higher-dimensional version of the Seifert conjecture for the odd-dimensional spheres has been established in 1966 by F . W . Wilson [9] as follows: A n y non-singular vector field on a s m o o t h ndimensional manifold M , n _> 3, can be modified to a vector field with a set of isolated invariant (n - 2)-tori, S lx...xS 1, so t h a t for p E M : a) in b o t h cases as t -+ oc and as t --+ - e c , the orbit q~(t,p) limits on one of the tori; and b) every orbit contained in one of the tori is dense in t h a t torus. Thus, each of the spheres $ 5 , $ 7 , . . . admits a nonsingular vector field with no circular orbits. For his construction, Wilson introduced a plug, a special non-singular vector field on the n-dimensional disc D n = I x D n - l , where I is the unit interval. T h e plug is constant and parallel to I x {p} on the b o u n d a r y of D n, and satisfies the t r a p p e d - o r b i t condition and the matched-ends condition (see below). T h e plug can be inserted in a non-singular vector field on an n-dimensional manifold (the m e c h a n i s m of insertion is illustrated in Fig. 1).

////// Fig. 1: Inserting a plug. T h e trapped-orbit condition guarantees t h a t at least one orbit enters the disc D ~ at the b o t t o m , {0} x D n-~, but never leaves D n. T h e matched-ends condition means t h a t if an orbit enters the disc D n at the b o t t o m and leaves D ", then the exit point is the point on {1} x D n-1 exactly above the entry point. B y appropriately inserting a n u m b e r of copies of a plug in a vector field on a manifold, Wilson changed the flow so t h a t each orbit starts inside a plug and ends inside one, too. In dimension three, Wilson's t h e o r e m yields isolated circular orbits and does not resolve the Seifert conjecture. T h e conjecture remained unsolved until a remarkable construction by P.A. Schweitzer in 1972. His 1974 paper [7] describes a three-dimensional plug without periodic orbits, which Schweitzer used to break the isolated periodic orbits, see Fig. 2 and Fig. 3. Inside the plug,

SELBERG CONJECTURE instead of circular orbits, there are invariant Denjoy sets to trap the entering orbits. This, initially C 1, construction was later improved to C 2+~ by J.M. Harrison [2].

Fig. 2: Schweitzer's plug.

Fig. 3: Breaking an orbit. Significant changes to the status of the Seifert conjecture came about in 1993 when H. Hofer [3] proved t h a t the Seifert conjecture holds for the Reeb vector field of a contact form on S 3 (cf. also C o n t a c t s t r u c t u r e ) . It was the next, after Seifert, advancement in the spirit of the conjecture. On the other hand, a C ~ counterexample to the Seifert conjecture (in its original formulation) was found by K. Kuperberg [6] the same year. This aperiodic vector field on S 3 also employs a plug. A partial self-insertion performed on a Wilson-type plug breaks the periodic orbits in the plug itself in a recursive process, see Fig. 4.

the gap between the counterexamples rem.

and Hofer's theo-

The above constructions generalize to higher dimensions, but counterexamples with stronger properties exist in dimensions above three. The Hamiltonian version of the Seifert conjecture is false for S 2n+a for n _> i, as V.L. Ginzburg [I] proved that there is a smooth function H : R 2n --+ R, n > 3, such t h a t the Hamiltonian flow of H on { H = 1} has no closed orbits (cf. also Hamiltonian system). M o d i f i e d S e i f e r t c o n j e c t u r e . A minimal set of a dynamical system is an invariant, non-empty, compact set containing no proper invariant, non-empty, compact subsets. The modified Seifert conjecture [7], [9] asserts t h a t every non-singular C 1 vector field on an odd-dimensional sphere S 2n+l, n > 1, has a minimal set of codimension at least two, i.e. of dimension at most 2n - 1. The invariant sets in the three-dimensional plugs of Wilson and Schweitzer are one-dimensional. In 1996 it was shown [5] t h a t the modified Seifert conjecture is false for real-analytic as well as for piecewiselinear flows, for all odd-dimensional spheres: Every nonsingular vector field on any manifold can be modified in the given smoothness category so t h a t every minimal set is of codimension one. References

[1] GINZBURG,V.L.: 'A smooth counterexample to the HamiltonJan Seifert conjecture in R 6', Internat. Math. Res. Notices 13

(1997), 641-650. [2] HARRISON,J.: 'C 2 counterexamples to the Seifert conjecture', Topology 27 (1988), 249-278. [3] HOFER, H.: 'Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three', Invent. Math. 114 (1993), 515-563. [4] KUPERBERG, G.: 'A volume-preserving counterexample to the Seifert conjecture', Comment. Math. Helv. 71 (1996), 70-97. [5] KUPERBERG, G., AND KUPERBERG, K.: 'Generalized counterexamples to the Seifert conjecture', Ann. of Math. 144 (1996), 239-268. [6] KUPERBERG, K.: 'A smooth countere×ample to the Seifert conjecture', Ann. of Math. 140 (1994), 723-732. [7] SCHWEITZER, P.A.: 'Counterexamples to the Seifert conjecture and opening closed leaves of foliations', Ann. of Math. 100 (1974), 386-400. [8] SEIFERT, H.: 'Closed integral curves in 3-space and isotopic two-dimensional deformations', Proc. Amer. Math. Soc. 1 (1950), 287-302. [9] WILSON, F.W.: 'On the minimal sets of non-singular vector fields', Ann. of Math. 84 (1966), 529-536. K.M. Kuperberg

Fig. 4: The K-plug - - a self-inserted Wilson plug. The following year, G. Kuperberg [4] modified Schweitzer's vector field to a volume-preserving counterexample to the Seifert conjecture, thereby narrowing

M S C 1991: 58F22, 58F25

SELBERG CONJECTURE - Let 7-/ denote the upper half-plane, SL(2, Z) the group of integer matrices of

351

SELBERG CONJECTURE determinant one and Fo(N)= {(;

bd) ESL(2, Z):c=_O

(moaN)}.

Following H. Maass [9], let W, (P0 (N)) denote the space of bounded functions f on r0(N) \ 7 / t h a t satisfy

A f = (14s--~2) f for

~Tz + the Laplace-Beltrami operator (ef. also L a p l a c e ope r a t o r ) . Such eigenfunctions f are called Maass wave forms. Since A in this context is essentially self-adjoint and non-negative (cf. also S e l f - a d j o i n t o p e r a t o r ) , it follows that (1 - s2)/4 is real and # 0. A. Selberg conjectured [12] that there is a lower bound gl (N) for the smallest (non-zero) eigenvalue: For N¢1, 1

of p-adic groups GL(2, Qp) inside a cuspidal representation of GL(2, A) (see below), Selberg's conjecture will follow as a statement for GL(2, R). Indeed, first let Qp denote the completion of the rational field Q with respect to the p-adic absolute value I'l;, p < oc, and view R as the completion with respect to l'loo = I'l. By the addles, denoted A, one means the I 'restricted' direct product I-Ip
is the centre of GA,

fz

el(N) >_ ~.

This innocent looking conjecture is (cf. [10]) one of the fundamental unsolved questions in the theory of modular forms (as of 2000; cf. also M o d u l a r form). For SL(2, Z) and for small values of N, it has been known for some time (Selberg, W. Roelcke). In general, it has many applications to classical number theory (see [4] and [12], for example). To back up his conjecture, Selberg also proved the following assertion: 3 Selberg's approach was to relate this problem to a purely arithmetical question about certain sums of exponentials, called Kloosterman sums (cf. also E x p o n e n t i a l s u m e s t i m a t e s ; T r i g o n o m e t r i c sum). This allowed him to invoke results from arithmetic geometry. The key ingredient giving the estimate is a (sharp) bound on Kloosterman sums due to A. Weil [13]. This bound, in turn, is a consequence of the Riemann hypothesis for the z e t a - f u n c t i o n of a curve over a finite field, which he had proven earlier (cf. also R i e m a n n h y p o t h e s e s ) . On the other hand, to go further than the theorem by this approach one needs to detect cancellations in sums of such Kloosterman sums, and arithmetic geometry offers nothing in this direction. This is the reason that the approach through Kloosterman sums has a natural barrier at 3/16. It is interesting that H. Iwaniec [5] has given a proof of Selberg's theorem which, while still being along the lines of Kloosterman sums, avoids appealing to Weil's bounds. Presently (2000), Selberg's conjecture is part of the 'Ramanujan-Petersson conjecture at infinity'. In other words, if interpreting the Ramanujan-Petersson conjecture as a statement about the irreducible representations 352

AGQ\GA

I¢(g)l 2

dg < o%

and

for almost every 9. Now, assuming f E Ws(Fo(N)) is an eigenfunction of all Hecke operators T(p), one can define in a one-to-one way, a function Cf C L~(ZAGQ \ GA) such that the GA-module 7rf = @71-p generated by the right GA-translates of Cf is an irreducible subrepresentation of L~(ZAGQ \ GA). Then Selberg's conjecture states that the representation (~r)oc of GL(2, R) is a principal series 7r(itl, it2) with trivial central character and 1-s 2 _ 1-[2i(tl-t2)] 2 l+4(t1-t2) 2 > 1 4 4 4 -4 In other words, complementary series ~r(sl,s2) with sl - s2 = 2s between 0 and 1 (and 1 - s2/4 between 0 and 1/4) should not occur (cf. [3]; see also I r r e d u c i b l e r e p r e s e n t a t i o n ; P r i n c i p a l series). In this context the Ramanujan-Petersson conjecture says that for (ahnost every) p the same conclusion holds, i.e., for p a classone representation, that is, p /~N, re; = rc(sl,;,-sl,p) satisfies (cf. [11]) Re(Sl,p) = 0.

(1)

P. De]igne proved the original Ramanujan conjecture [1] when 7ro~ is a holomorphic discrete series of weight k. For example, when f equals Ramanujan's A(z) = }-~=1 ~-(n)e2~i~z, the condition (1) implies [z-(p) < pll/2(pSl,v + pS2,p) < 2plU2, the famous Ramanujan inequality. In general, Deligne was able to exploit algebraic-geometric interpretations of the classical Ramanujan-Petersson identities. Note

SEMI-FREDHOLM OPERATOR that for Selberg's conjecture one again assumes that 7r~ is of class-one and Irzc = 7r(s1,~, s2,~) with Re(si,cc) = 0. It was with this modern representation-theoretic point of view that progress was made on Selberg's theorem. First, consider the mapping Sym "~ : GL(2, C) -+ GL(m

+ I, C)

defined by action of GL(2, C) on symmetric tensors of rank m. It was conjectured by R.P. Langlands [7] that there should be a corresponding mapping Sy m m, that (roughly) maps cuspidal automorphic representations of GL(2) to those 'of OL(m + 1)'; moreover, whenever 7~ corresponds to class-one representations indexed by

(o 0) (including possibly p -- oc), Sym m (Tr) should correspond to S y m

TM

(Op

~p 0 ) " This conjecture, for all m, imp lies

both the Selberg and the Ramanujan-Petersson conjecture. In 1978, S. Gelbart and H. Jacquet [2] proved Langlands' conjecture for m = 2; for Selberg's conjecture, this simply replaced the equality in his theorem by an inequality. Then, in 1994 [8], W. Luo, Z. Rudnick and P. Sarnak used Sym2. and analytic properties of L-functions to go well beyond Selberg's conjecture: 171 ~I(N) _> 7S---4= 0 . 2 1 8 1 . - . . And in 2000, H. Kim and F. Shahidi [6] proved Langlands' conjecture for m = 3 and established ~I(N) _> 0.22837, i.e., 5

IRe(sp,i)l _<

Either Selberg's conjecture will continue to be proved along the lines of Langlands' conjecture, or by entirely new ideas. There is a far-reaching generalization of Selberg's conjecture to GL(n): If 7r = ®p<_~Trp is an irreducible cuspidal automorphic representation of GL(n,.4), then every class-one local representation of GL(n, Qp) is 'tempered'. References [I] DELIGNE, P.: 'La conjecture de Well I', Publ. Math. IHES 43 (1974), 273-307. [2] GELBART,

S., AND

JACQUET,

H.: 'A relation between

au-

tomorphic representations of GL(2) and GL(3)', Ann. Sci. Ecole Norm. Sup. 11 (1978), 471-552. [3] GEL'FAND, I., GRAEV, iV[., AND PIATETSKI-SHAPIRO,I.: Representation theory and automorphic functions, W.B. Saunders, 1969. [4] HEJHAL, D.: The Selberg trace formula II, Vol. 1001 of Lecture Notes in Mathematics, Springer, 1983.

[5] IWANIEC, H.: 'Selberg's lower bound for the first eigenvalue of congruence groups': Number Theory, Trace Formula, Discrete Groups, Acad. Press, 1989, pp. 371-375. [6] KIM, H., AND SHAHIDI, F.: 'Functorial products for GL(2) × GL(3) and functorial symmetric cube for GL(2)', C.R. Acad. Sci. Paris 331, no. 8 (2000), 599-604. [7] LANGLANDS, R.P.: 'Problems in the theory of automorphic forms': Lectures in Modern Analysis and Applications, Vol. 170 of Lecture Notes in Mathematics, Springer, 1970, pp. 18-86. [8] LUO, W., RUDNICK, Z., AND SARNAK,P.: 'On Selberg's eigenvalue conjecture', Geom. Funct. Anal. 5 (1995), 387-401. [9] MAASS, H.: 'Nichtanalytishe Automorphe Funktionen', Math. Ann. 121 (1949), 141-183. [10] SARNAK,P.: 'Selberg's eigenvalue conjecture', Notices Amer. Math. Soc. 42, no. 4 (1995), 1272-1277. [11] SATAKE, I.: 'Spherical functions and Ramanujan's conjecture': Algebraic Groups and Discontinuous Subgroups, Vol. IX of Proe. Symp. Pure Math., Amer. Math. Soc., 1966, pp. 258-264. [12] SELBERG, i . : 'On the estimation of Fourier coefficients of modular forms': Proc. Syrup. Pure Math., Vol. VIII, Amer. Math. Soc., 1965, pp. 1-15. [13] WEIL, A.: 'On some exponential sums', Proc. Nat. Acad. Sci. 34 (1948), 204-207. S. Gelbart

MSC 1991: l l F 0 3 , l l F 7 0 S E M I - F R E D t t O L M O P E R A T O R - Let X and Y be two Banach spaces and let L(X, Y) denote the Banach space of all continuous (bounded) operators from X into Y (cf. also B a n a c h space; C o n t i n u o u s o p e r a t o r ) . For an operator T in L(X, Y), let k e r T be the set of all x E X such that Tx = 0 and let cokerT be the quotient space Y/TX, where TX denotes the range of T. By definition, T is a semi-Fredholm operator if TX is closed (i.e. it is a n o r m a l l y - s o l v a b l e o p e r a t o r ) and at least one of the vector spaces ker T and coker T is of finite dimension. (The definition is partially redundant, since if the dimension of coker T is finite, TX is closed.) For a semi-Fredholm operator T, its index, i.e.

dim ker T - dim coker T, is uniquely determined either as an integer, or as plus or minus infinity. In the first case T is a F r e d h o l m ope r a t o r . Cf. also I n d e x o f a n o p e r a t o r . The set SF(X, Y) of all semi-Fredholm operators in L(X, Y) is open in L(X, Y) and the index is constant on each connected component of SF(X, Y). Moreover, if K is a c o m p a c t o p e r a t o r in L(X,Y) and T is in SF(X, Y), then T + K is also in SF(X, Y) and its index equals that of T. Due to these properties, semi-Fredholm operators play an important role in linear and non-linear analysis. They were first explicitly considered by I.C. Gohberg and M.G. KreYn [1] and T. Kato [2], who also treated the case when T is unbounded. 353

SEMI-FREDHOLM O P E R A T O R References

and

[1] GOHBERG, I.C., AND KREIN, M.G.: 'The basic propositions on defect numbers, root numbers and indices of linear operators', Transl. Amer. Math. Soc. (2) 13 (1960), 185-264. (Uspekhi Mat. Nauk. 12 (1957), 43 118.) [2] KATO, T.: 'Perturbation theory for nullity, deficiency and other quantities of linear operators', Y. d'Anal. Math. 6

(1958), 261 322. C. Foias

MSC 1991:47A53

lira S(t)(x) = x,

t-+0 +

x C D,

(2)

where the limit is taken with respect to the s t r o n g t o p o l o g y of X. Differentlability of seml-groups with respect to t h e p a r a m e t e r . Let {8(t)}, t E (0, T), be a continuous semi-group defined on D. If the strong limit

g(x) = lira -1 (x - ,S(t)(x))

(3)

t-+0+ t

S E M I - G R O U P OF H O L O M O R P H I C M A P P I N G S Non-linear semi-group theory is not only of intrinsic

interest, but is also important in the study of evolution problems (cf. also E v o l u t i o n e q u a t i o n ) . In recent years (as of 2000) many developments have occurred, in particular in the area of non-expansive semigroups in Banach spaces. As a rule, such semi-groups are generated by accretive operators (cf. also A c c r e t i v e m a p p i n g ) and can be viewed as non-linear analogues of the classical linear contraction semi-groups (aft also C o n t r a c t i o n s e m i - g r o u p ) . Another class of nonlinear semi-groups consists of the semi-groups generated by holomorphic mappings. Such semi-groups appear in several diverse fields, including, for example, the theory of Markov stochastic branching processes, KreYn spaces, the geometry of complex Banach spaces, control theory, and optimization. These semi-groups can be considered natural non-linear analogues of semi-groups generated by bounded linear operators (cf. also S e m i g r o u p o f o p e r a t o r s ) . These two distinct classes of nonlinear semi-groups are related by the fact that holomorphic self-mappings are non-expansive with respect to Schwarz-Pick pseudo-metrics (see, for example, [S],

[7], [6]) Recall that a function h defined on a domain (open connected subset) D in a complex B a n a c h s p a c e X and with values in a complex Banach space is said to be holomorphic in D if for each x • D the F r d c h e t d e r i v a t i v e of h at x (denoted by Dh(x) oi" h'(x)) exists as a bounded complex-linear mapping of X into the Banach space containing the values of h. (Cf. also B a n a c h s p a c e o f a n a l y t i c f u n c t i o n s w i t h infinitedimensional domains.) If D and ~ are domains in complex Banach spaces X and Y, respectively, then the set of holomorphic mappings from D into ~ is denoted by Hol(D, f~). The notation Hol(D) is used to denote the set H o l ( D , D ) of holomorphic self-mappings of D. A family {S(t)} C Hol(D), where t E (0, T), T > 0, is called a (one-parameter) continuous semi-group if S ( s + t) = s ( s ) o s ( t ) , 354

s, t, s + t • (0, T),

(1)

exists for each m C D, then g E H o l ( D , X ) is called the (infinitesimal) generator of the semi-group {3(t)}. In this case the semi-group {$(t)}, t E (0, T), is said to be differentiable (or generated). For the finite-dimensional case, M. Abate proved in [1] that each continuous semi-group of holomorphic mappings is everywhere differentiable with respect to its parameter, i.e., it is generated by a holomorphic mapping. In addition, he established a criterion for a holomorphic mapping to be a generator of a one-parameter semi-group. Earlier, for the one-dimensional case, similar facts were presented by E. Berkson and I-I. Porta in their study [4] of linear C0-semi-groups of composition operators on Hardy spaces. E. Vesentini investigated semi-groups of fractional-linear transformations that are isometries with respect to the infinitesimal hyperbolic metric on the unit ball of a Banach space [18]. He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Note that, generally speaking, such semigroups are not everywhere differentiable in the infinitedimensional case. In fact, it can be shown (see for example, [14]), that a continuous semi-group $(t) of holomorphic selfmappings of a domain D in X is generated if and only if the convergence in (2) is locally uniform on D (cf. also Uniform convergence). Moreover, if D is hyperbolic (in particular, bounded; cf. also H y p e r b o l i c m e t r i c ) , then S(t) can be continuously extended to all of R + = [0, co) as the solution of the C a u c h y p r o b l e m

ot +g(u(t,x)) = 0, u(O, x) = x,

(4)

where x C D and t C R +, i.e.,

u(t,x)=S(t)x,

x•D,

t • [ 0 , oo)

(5)

(see, for example, [12], [13]). Thus, there is a one-toone correspondence between locally uniformly continuous semi-groups and their generators. If $(t) has a continuous extension to all of R = ( - o o , oo), then it is actually a one-parameter group of automorphisms of D.

SEMI-GROUP OF HOLOMORPHIC MAPPINGS E x p o n e n t i a l a n d p r o d u c t f o r m u l a s . A holomorphic v e c t o r field

Tg = g(x)

(6)

on a domain D is determined by a holomorphic mapping g E H o l ( D , X ) and can be regarded as a linear operator T mapping Hol(D,X) into itself, where Tgf E Hol(D, X) is defined by

(Tgf) (x) = Df(x)g(x),

x E D.

~;(t)In = S(t)

(11)

TgID = g,

(12)

and

(7)

The set of all holomorphic vector fields on D is a Lie a l g e b r a under the commutator bracket

[Tg,Th] = [g(x) O,h(x) O l :=

Thus, a holomorphic vector-field T~ is semi-complete (respectively, complete) if and only if it is the Lie generator of a linear semi-group (respectively, group) of composition operators on Hol(D, X). This follows from the observation that

(8)

where ID is the restriction of the identity operator to

D. Moreover, using the exponential formula representation for the linear semi-group, C~

0

£(t)f = E (--1)ktkT~fk!

= (Dg(x)h(x) - Dh(x)g(x)) ~x

= exp[-tTg]f

(13)

k=O

(see, for example, [9], [15], [6]). Furthermore, each vector field (6) is locally integrable in the following sense: for each x E D there exist a neighbourhood f~ of x and a 5 > 0 such that the Cauchy problem (4) has a unique solution {u(t, x)} C D defined on the set {Itl < 5 } x • E R x D . A holomorphic vector field Tg defined by (6) and (7) is said to be (right) semi-complete (respectively, complete) on D if the solution of the Cauchy problem (4) is well-defined on all of R + x D (respectively, R x D), where R + = [0, ~ ) (respectively, R = (-oe, ~ ) ) . Thus, if D is hyperbolic, then Tg is semi-complete (respectively, complete) if and only if g is the generator of a one-parameter continuous semi-group (respectively, group). On the other hand, if D is bounded and Hol(D, X) is the subspace of Hol(D,X) consisting of all f E Hol(D,X) that are bounded on each ball strictly inside D, then a semi-group (group) {3(t)}, t E R + (respectively, t E R), induces a linear semi-group (group) {£(t)} of linear mappings £(t): Hol(D,X) --+ Hol(D, X), defined by

(£(t)f) (x) := f($(t)x),

+ Tg(£(t)f) = O,

$(t) = E~ (--1)ktk~ lg~kTiD= exp [-tTg]ID.

(io)

(0)f = f,

for all f e Hol(D, X), where g = -dS(t)/dt]t=o. In other words, a holomorphic vector field Tg, defined by (6) and (7), and considered as a linear operator on Hol(D,X), is the infinitesimal generator of the semigroup {£(t)}. It is sometimes called the Lie generator.

(14)

k=0

So, a locally uniformly continuous semi-group of holomorphic self-mappings can be represented in exponential form by the holomorphic vector field induced by its generator. Another exponential representation on a hyperbolic convex domain can be given by using the so-called nonlinear resolvent of g. More precisely, let D be a bounded (or, more generally, hyperbolic) convex domain. Then it was shown in [12] and [13] that g E Hol(D,X) is a generator if and only if for each r > 0 the mapping (I + rg)-i = J ( r ) is a well-defined holomorphic self-mapping of D. Furthermore, if {G(r)}, r _ 0, is any continuous family of holomorphic self-mappings of D such that the limit

g(x) = lira l r~o+ r

(x _ V(r)x)

exists, then g is a generator and the semi-group generated by g can be defined by the product formula

(9)

wheretER + (tER) andxED. This semi-group is called the semi-group of composition operators on Uol(D,X). If {8(t)}, t E R + (t E R), is T-continuous, (that is, differentiable), then g E Hol(D, X), {£(t)}, t C R + (t E R), is also differentiable and

{~

(see, for example, [19], [9], [12]), one also has

S(t)=

lim

Gn(t~.

(15)

I+

(16)

\n/

n-~(~

In particular,

S(t)= lim

g

(exponential formula), where the limits in (15) and (16) are taken with respect to the locally uniform topology on Hol(D, X). F l o w - i n v a r i a n c e c o n d i t i o n s . Let D be a convex subset of a Banach space X and let g : D -+ X be a continuous mapping on D, the closure of D. Then the following

tangency condition of flow invariance lira d i s t ( x - h g ( x ) ' D ) h-+0 +

=0,

xeD,

(17)

h

355

S E M I - G R O U P OF H O L O M O R P H I C M A P P I N G S

is a necessary condition for the solvability of the evolution equation (4). A result of R.H. Martin [ii] shows that if g: D --~ X is a continuous accretive mapping on D, then (17) is also sufficient for the existence of solutions to the Cauchy problems (4). These solutions yield a continuous semi-group of contraction mappings on D. For the class of holomorphie mappings, an analogue of Martin!s theorem was given in [3]; namely, if 9 E Hol(D, X ) has a uniformly continuous extension to D, then it is a semi-complete vector field if and only if it satisfies the b o u n d a r y flow invariance condition (17). However, there are m a n y examples of semi-complete vector fields t h a t have no continuous extension to D. In particular, if F E Hol(D), then 9 = I - F is semicomplete (see [12]). For absolutely convex domains, interior flow invariance conditions can be given in terms of their support functionals. Let X I be the dual of X (cf. also D u a l i t y ; A d j o i n t s p a c e ) . For x E X and x I E X ~, the pairing (x,x I} will denote x'(x). The duality mapping J: X --+ 2 x' is defined by

J(x)

c x,: Re

:=

<x,x'l = 11 ll2 = IIx'll 2}

for each x E X. If D is the open unit ball in X and g maps D into X, then (17) is equivalent to the condition inf

z'~J(x)

R e ( g ( x ) , x ' } > 0,

For the Euclidean ball D dition in this direction was Namely, he proved t h a t g complete vector field if and timate

x E 0D.

(18)

in X = C ~, a certain conestablished by Abate [1]. E H o l ( D , C ~) is a semionly if it satisfies the es-

2 [llr(x)ll 2 -I(r(x),x>l

2]Re
+

(19)

where

x.

For n = 1 this condition becomes 1 R e g ( z ) ~ > - 2 R e g ' ( z ) ( 1 - Iz]2),

(20)

where z E A, the open unit disc in the complex plane C, and g E Hol(A, C). Despite the usefulness and simplicity of condition (20) it is not clear how (18) can be derived from (20) when g has a continuous extension to A. Note also that in the one-dimensional case it follows from the m a x i m u m p r i n c i p l e for harmonic functions that (18) implies the following interior condition: Reg(z)~ > Reg(0)~(1-Izl2), 356

z e Zx.

a) For each x E D there exists an x' E J(x) such t h a t Re__0; b) inf~,cz(x ) Re(2llxll2g(x) + (1 - [Ixil2Dg'(x)x,x ') >_ 0, x E D; c) For each x E D and for each x ~ E J(x),

1 -IlXllg,(O)z + Re --4---1 Ilxll

(1 -Ifxll2)g(O),x'}

(21)

<

_< Re (g(x), x'> <_

< Re

(l÷llxil

1-- [-~g iu)x ÷

(1 - Ilxl12)g(0),x' }

Furthermore, equality in one of the conditions a), b) or c) holds if and only if it holds in the other conditions and f is complete. Also, if 9 E Hol(D, X ) is a semi-complete vector field, then g is, in fact, complete if and only if its derivative at zero, 91(0), is a conservative linear operator, i.e.,

Re

= o

for all x E X and x I E J(x) (see [10], [17]). Parametric representations of well-known t h a t a complete vector unit ball D in a Banach space X degree at most 2 (see, for example, cisely, g has the form

g e n e r a t o r s . It is field g on the open is a polynomial of [15], [6]). More pre-

g(x) = a + Ax + Pa(x),

+(1 - Ilxl12) Re (gl(x)g(x),r(x)> > O, r(x) = (1 -Ilxll2)g(x) +

Conversely, it is clear t h a t (18) does result from (21) if g has a continuous extension to all of A. It turns out that an analogue of (21) is a necessary and sufficient condition for g to be semi-complete [2]: Let D be the open unit ball in a complex Banach space X. Then g E H o l ( D , X ) is a semi-complete vector field on D if and only if it is bounded on each subset strictly inside D and one of the following conditions holds:

(22)

where a is an element of X , A is a conservative operator on X and Pa is a homogeneous form of the second degree such t h a t Pin = iP~. Suppose now t h a t a complex Banach space X is a so-called JB* triple system. This is equivalent to saying that its open unit ball D is a homogeneous domain, i.e., for each pair x, y E D there exists a holomorphic automorphism F of D such t h a t F(x) = y (see, for example, [15], [6]). Then it is well-known that for each a E X there exists a homogeneous polynomial Pa (x) such that Pi~ = iP~ and the mapping g: D --+ X defined by = a - P

(x)

(23)

is a complete vector field on D, which is called a transvection of D (cf. also T r a n s v e c t i o n ) .

SEMI-GROUP OF H O L O M O R P H I C M A P P I N G S The cone 6 of semi-complete vector fields on D admits the decomposition 6 = 60 O

6+,

(24)

where 60 is the real Banach subspace of Hol(D, X) consisting of transvections and 6+ is the subcone of 6 such that for each h E 6+, inf R e ( h ( x ) , x ' ) > 0 , x'~J(x)

f = g + h,

(25)

where g = f(0) - P f ( o ) ( X ) is complete, h e 6+ and

h(0) = 0. The natural examples of JB* triple systems are a complex Hilbert space H , the space L(H) of bounded linear operators on H, and its subspaces J such that A E J if and only if A A * A E J (such subspaces are usually called J*-alyebras ). In the latter case the general form of transvections on D is g(x) = a - xa*x, where a E J and a* is its conjugate. Thus, each semi-complete vector field on the open unit ball of a J*-algebra has the form f ( x ) = f(O) - xf*(O)x + h(x), (26) where h E 6+ and h(0) = 0. In particular, when X = C is the complex plane and D = A, the open unit disc in C, (26) becomes

f ( z ) = f(0) - f ( 0 ) z 2 + zp(z),

(27)

where p(z) E Hol(A, C) and z E A.

(28)

In 1978 E. Berkson and H. Porta [4], solving an entirely different problem, gave a parametric representation of generators on the unit disc A in the complex plane. More precisely, g C 6 if and only if for some 7- E A, g has the representation g(z)

= (z -

Re ((x, f(x)) + (y, f(y))

for a l l x C D .

In other words, f C 6 admits a unique representation

Rep(z) > 0,

was shown in [13] that if H is separable and f : B -+ H is a bounded continuous mapping, then f is p-monotone if and only if it generates a semi-group of p-non-expansive self-mappings of B. Note also that p-monotonicity can be equivalently described as follows:

-

z

)p(z)

with Rep(z) >_ 0 everywhere. This point 7 C A is exactly the limit point of the semi-group generated by g (that is, its Denjoy-Wolff point, cf. D e n j o y - W o l f f t h e o r e m ) . The Berkson-Porta formula has also been successfully exploited in other fields; for example, in the classical functional equations of E. Sehrhder and N.H. Abel (see [5] and Functional equation; S c h r h d e r

functional equation). Let H be a complex Hilbert space with inner product (.,-) and let B be its open unit ball. Let p denote the Poincar~ h y p e r b o l i c m e t r i c on B [7] (cf. also P o i n c a r ~ m o d e l ) . A mapping f : B --4 H is said to be p-monotone if for each pair x , y E B and positive r the following condition holds: p(x + r f ( x ) , y + r f ( y ) ) > p(x, y) whenever x + r f (x) and y + r f (y) belong to B. It

>Re

((f(x),y/+ (x, -1- -(x:~

,

> x, y e B.

For a bounded holomorphic mapping f : B -~ H and for an arbitrary H the latter condition is a criterion for f to be semi-complete. For the one-dimensional case, if f ( y ) = 0, then this condition becomes the BerksonPorta representation of semi-complete vector fields.

References [1] ABATE, M.: 'The infinitesimal generators of semi-groups of holomorphic maps', Ann. Mat. Pura Appl. 161 (1992), 167180. [2] AHARONOV,D., REICH, S., AND SHOIKHET, D.: 'Flow invariance conditions for holomorphic mappings in Banach spaces', Math. Proc. Royal Irish Acad. 9 9 A (1999), 93-104. [3] AIZENBERG,L., REICH, S., AND SHOIKHET, D.: 'One-sided estimates for the existence of null points of holomorphic mappings in Banach spaces', J. Math. Anal. Appl. 203 (1996), 38-54. [4] BERKSON, E., AND FORTA, I-I.:'Semi-groups of analytic functions and composition operators', Michigan Math. J. 25 (1978), 101-115. [5] COWEN, C.C., AND MACCLUER, B.D.: Composition operators on spaces of analytic functions, CRC, 1995. [6] DINEEN, S.: The Schwartz lemma, Clarendon Press, 1989. [7] GOEBEL, K., AND REICH, S.: Uniform convexity, hyperbolic geometry and nonexpansive mappings, M. Dekker, 1984. [8] HARRIS, L.A.: 'Schwarz-Pick systems of pseudometrics for domains in normed linear spaces': Advances in Holomorphy, North-Holland, 1979, pp. 345-406. [9] ISIDRO, J.M., AND STACHO, L.L.: Holomorphic automorphism groups in Banach spaces: A n elementary introduction, North-Holland, 1984. [10] KREIN, S.G.: Linear differential equations in Banaeh spaces, Amer. Math. Soc., 1971. [11] MARTIN JR., R.H.: 'Differential equations on closed subsets of a Banach space', Trans. Amer. Math. Soc. 179 (1973), 399 414. [12] REICH, S., AND SHOIKHET, D.: 'Generation theory for semigroups of holomorphie mappings in Banach spaces', Abstr. Appl. Anal. 1 (1996), 1-44. [13] REICH, S., AND SHOIKHET, D.: 'Semi-groups and generators on convex domains with the hyperbolic metric', Atti Accad. Naz. Lincei 8 (1997), 231-250. [14] REICH, S., AND SHOIKHET, D.: 'Metric domains, holomorphic mappings and nonlinear semi-groups', Abstr. Appl. Anal. 3 (1998), 203-228. [15] UPMEIER, H.: Jordan algebras in analysis, operator theory and quantum mechanics, Vol. 67 of C B M S - N S F Reg. Conf. Ser. in Math., Amer. Math. Soc., 1987. [16] VESENTINI, E.: 'semi-groups of holomorphic isometries', Adv. Math. 65 (1987), 272-306.

357

SEMI-GROUP OF H O L O M O R P H I C MAPPINGS

[17] VESENTINI, E.: 'Krein spaces and holoinorphic isometrics of Cartan domains', in S. COEN (ed.): Geometry and Complex Variables, M. Dekker, 1991, pp. 409-413. [18] VESENTINI, E.: 'Semi-groups of holomorphic isometrics', in S. COEN (ed.): Complex Potential Theory, Kluwer Acad. Publ., 1994, pp. 475-548. [19] YOSIDA, K.: Functional analysis, Springer, 1968.

Simeon Reich David Shoikhet MSC 1991: 32H15, 34G20, 46G20, 47D06, 47H20

SEQUENTIAL PROBABILITY RATIO TEST, SPRT - L e t X 1 , X 2 , . . . be a sequence of independent random variables with common discrete or continuous probability density function f (cf. also R a n d o m v a r i a b l e ; D i s t r i b u t i o n f u n c t i o n ) . In testing the hypotheses H0: f = f0 versus H i : f = fi (cf. also Statistical hypotheses, v e r i f i c a t i o n of), the probability ratio or likelihood ratio after n observations is given by (cf. also

Likelihood-ratio test) AN =

fl(Xi) {=i fo(Xi)"

(1)

In the Neyman-Pearson probability ratio test, one fixes a sample size n, chooses a value k and decides to accept H1 if A~ > k and decides to accept H0 if A < k. In the sequential probability ratio test introduced by A. Wald [4], the sample size is not chosen ahead of time. Instead, one chooses two positive constants A B, one stops and decides to accept Hi. Otherwise, another observation is taken. The number of observations taken is thus a random variable N, which can be shown to be finite with probability one. Denote the error probabilities of this procedure by c~ = P0(accept HI) = P0(AN _> B) and /9 = Pi(accept H0) = PI(AN _< A). It then follows that /9 = f A N dP0 _< A(1 - a), JAN<_A

1--/9=/

JAN>_B

(2)

ANdP0>_Bct.

If the likelihood ratio always hits the boundary when the test stops, so that AN = A or AN = B, then these inequalities become equalities. Otherwise, the inequalities become close approximations in the standard cases. The logarithm of the likelihood ratio as given in (1) is a sum of independent, identically distributed random variables Zi = l o g f l ( X i ) / f o ( X i ) . It then follows from Wald's Iemina that E(Zi)E(N) = E(lOgAN). Using the same type of approximations as above, this gives the following formulas for the a v e r a g e s a m p l e n u m b e r of 358

the test: a log ( L ~ ) +

Eo(N) ~

(1 - c0 log (1_--~)

Eo (Zi)

,

(3)

El(N) If the likelihood ratio always hits the boundary when the test stops, these approximations become equalities. Wald and J. Wolfowitz [5] proved a strong optimality property for the sequential probability ratio test. It states that among all sequential tests with error probabilities no bigger than that of a given sequential probability ratio test, the sequential probability ratio test has the smallest average sample number under both Ho and Hi. Indeed, the average savings in sampling relative to the Neyman Pearson test with the same error probabilities is 50% or more in many cases (see [2] for details). In most realistic situations the hypotheses to be tested are composite, the probability distributions are parametrized by some parameter 0, and one is interested in testing H0 : 0 _< 00 versus H i : 0 >__01. In such a case one can perform the sequential probability ratio test for the simple hypotheses H0 : 0 = 00 versus H1 : 0 = 01. In the most important cases one can apply the fundamental identity of s e q u e n t i a l a n a l y s i s (see [2] for details) to find the approximate power functions and average sample number functions for such procedures. However, even when such tests achieve specified error probabilities for all values of the parameter, the Wald-Wolfowitz optimality will not carry over to values of 0 other than 00 and 0i. Indeed, the expected sample size may even exceed that of the corresponding fixed sample size test at some values of 0 between 0o and O1. It is because of this phenomenon that J. Kiefer and L. Weiss [6] raised the question of finding the sequential test with given error probabilities at 00 and 01, which minimizes the maximum expected sample size over all 0. To solve this problem, G. Lorden [3] introduced a test based on performing two one-sided sequential probability ratio tests simultaneously. First, a third value 0* is chosen. Test one is then a sequential probability ratio test of H 0 : 0 = 00 versus H i : 0 = 0", where the constant A = 0. Test two is a sequential probability ratio test of H 0 : 0 = 01 versus H i : 0 = 0", where A = 0. The 2-sequential probability ratio test stops and makes its decision as soon as either one of the individual sequential probability ratio tests stops. The decision is to accept H1 if test one stops first and the decision is to accept H0 if the second test stops first. It can be shown that for the proper choice of 0* this test does asymptotically solve the Kiefer-Weiss problem (see [1] for a simple way to select 0").

SERRE THEOREM IN GROUP COHOMOLOGY The sequential probability ratio test can also be applied in situations where the observations are not independent and identically distributed. In such a situation the likelihood ratio ,kn can be more difficult to compute at each stage. The inequalities (2) continue to hold for such a sequential probability ratio test, but the formulas (3) for the average sample number are no longer valid. The sequential probability ratio test has also been studied where the observations form a continuous-time s t o c h a s t i c p r o c e s s . In fact, parts of the theory simplify in such a situation, since the likelihood ratio process often becomes a process with continuous sample paths and thus always hits the b o u n d a r y when it stops.

References [1] EISENBERG, B.: 'The asymptotic solution of the Kiefer-Weiss problem', Sequential Anal. I (1982), 81-88.

[2] GHOSH, B.K.: Sequential tests of statistical hypotheses, Addison-Wesley, 1970. [3] LORDEN,G.: '2-SPRT's and the modified Kiefer-Weiss problem of minimizing an expected sample size', Ann. Statist. 4 (1976), 281-291. [4] WALD, A.: Sequential analysis, Wiley, 1947. [5] WALD, A., AND WOLFOWITZ, J.: 'Optimum character of the sequential probability ratio test', Ann. Math. Stat. 19 (1948), 326-339. [6] WEISS, L.: 'On sequential tests which minimize the maximum expected sample size', J. Amer. Statist. Assoc. 57 (1962), 551-566.

Bennett Eisenber9 M S C 1991:62L10

SERRE THEOREM IN GROUP COHOMOLOGYA theorem proved by J.-P. Serre in 1965 about the cohomology of pro-p-groups which has important consequences in group cohomology and representation theory (cf. also P r o - p - g r o u p ; Cohomology of g r o u p s ) . The original proof appeared in [7], a proof in the context of finite group cohomology appears in [1]. Let p denote a fixed prime number and G a pro-pgroup, that is, an inverse limit of finite p-groups (cf. also p - g r o u p ) . Assume that G is not an elementary Abelian p-group (i.e. it is not isomorphic to (Z/p) [ for some indexing set I, where Z/p is cyclic of order p). Then Serre's theorem asserts that there exist non-trivial m o d p cohomology classes V l , . . . , v k E H I ( G , Z / p ) such that the product ~(vl)'"/~(vk) = 0, where /~: HI(G,Z/p) --~ H 2(G, Z/p) is the Bockstein operation associated to the exact coefficient sequence 0 --+ Z/p --+ Z/p 2 -~ Z/p --+ 0 (see [9] and Cohomology operation). Note t h a t for p = 2 this is simply the squaring operation. For a finite p-group, this can be made more explicit as follows. Each cohomology class vi corresponds to a (non-zero) h o m o m o r p h i s m ¢i: G ~ Z/p and hence an index-p subgroup Gi C G. The class /~(vi) C Ext~/p[G] (Z/p, Z/p) can be represented as an extension

class

Z/p

0

Z/p[a/cd

z/p[c/ai]

Z/p

0,

where Z/p[G/Gi] denotes the usual permutation module obtained by induction. When concatenated together, one obtains a representation of the product, which is an

element in Ext2~p[G]

0

(Z/p, Z/p),

as

Z/p -+ Z / p [ a / a d -+ Z / p [ a / c d z/p[G/a

] -+ Z/p[a/c

... -+

] -+ Z / p -+ o,

which the theorem asserts to be the trivial extension class. The original application of Serre's result was for proving that if G is a profinite group without elements of order p, then the p-cohomological dimension of G is equal to the p-cohomological dimension of U for any open subgroup U C G (see [8] for more on this; cf. also

Cohomological dimension). However, it is also a basic technical result used in proving the landmark result (see [5] and [6]) that the Krull dimension (cf. D i m e n s i o n ) of the m o d p cohomology of a f i n i t e g r o u p G is equal to the rank of the largest elementary Abelian p-subgroup in G. More precisely, Serre's theorem can be used to verify that for a finite non-Abelian p-group G, the Krull dimension of H* (G, Z/p) (the maximal rank of a polynomial subalgebra) is determined on maximal proper subgroups, hence leading to an inductive argument which can be reduced to elementary Abelian subgroups. This, in turn, can be extended to arbitrary finite groups and to cohomology with coefficients in a modular representation. Indeed, it is a basic result in the theory of complexity and cohomological varieties in representation theory. This is explained [2], [3] and [4].

References [1] ADEM, A., AND MILGRAM, R.J.: Cohomology of finite groups, Vol. 309 of Grundlehren, Springer, 1994. [2] BENSON, D.J.: Representations and cohomology II: Cohomology of groups and modules, Vol. 32 of Studies in Advanced Math., Cambridge Univ. Press, 1991. [3] CARLSON, J.F.: Modules and group algebras, ETH Lect. Math. Birkh~user, 1994. [4] EVENS, L.: Cohomology of groups, Oxford Univ. Press, 1992. [5] QUILLEN, D.: 'The spectrum of an equivariant cohomology ring I-II', Ann. of Math. 94 (1971), 549-602. [6] QUILLEN, D., AND VENKOV, B.: 'Cohomology of finite groups and elementary Abelian subgroups', Topology 11 (1972), 317-318. [7] SERRE, J.-P.: 'Sur la dimension cohomologique des groupes profinis', Topology 3 (1965), 413-420. [8] SERRE~ J.-P.: Cohomologie Galoisienne, fifth ed., Vol. 5 of Lecture Notes in Mathematics, Springer, 1994. [9] SPANIER, E.: Algebraic topology, Springer, 1989.

Alejandro Adem M S C 1991:20J06

359

SHAFAREVICH CONJECTURE SHAFAREVICH CONJECTURE in inverse Galois theory - The absolute Galois group GQab := G a l ( Q / Q ab) of Qab (cf. also Galois g r o u p ) is a free p r o f i n i t e g r o u p of countable rank. Here, Qab is the maximal Abelian extension of Q, or, equivalently (by the Kronecker Weber theorem), the maximal cyclotomic extension of Q. I.R. Shafarevich posed this assertion as an important problem during a 1964 series of talks at Oberwolfach on the solution to the class field tower problem (cf. Tower of fields; Class field t h e o r y ) . The conjecture would imply an affirmative answer to the inverse Galois problem over Q~b, i.e. that every finite g r o u p is a Galois group over Qab (cf. also G a l o i s t h e o r y , inverse probl e m of). By the Iwasawa theorem [7, p. 567] (see also [2, Cor. 24.2]), a profinite group II of countable rank is free (as a profinite group) if and only if every finite embedding problem for II has a proper solution. Thus, the Shafarevich conjecture is equivalent to the assertion that if H is a quotient of a finite group G, then every HGalois field extension of Qab is dominated by a G-Galois field extension of Qab. A weakening of this assertion is known: that the profinite group GQ~b is projective, i.e. every finite embedding problem for GQ~ has a weak solution (cf. also P r o j e c tive group). Projectivity is equivalent to the condition of c o h o m o l o g i c a l d i m e n s i o n _< 1 [12, Chap. 1; Props. 16, 45], and this holds for GQob by [12, Chap. 2; Prop. 9]. On the other hand, the absolute Galois group GQ is not projective, since the surjection GQ --+ Z/2Z corresponding to the extension Q ( i ) / Q does not factor through Z/4Z. Thus, the analogue of the Shafarevich conjecture does not hold for Q. E v i d e n c e for t h e c o n j e c t u r e . Many finite groups, including 'most' simple groups, have been realized as Galois groups over Qab [9, Chap. II, Sec. 10]. These realizations provide evidence for the inverse Galois problem over Qab and hence for the Shafarevich conjecture. Typically, these realizations have been achieved by constructing Galois branched covers of the projective line over Qab. Since Q~b is Hilbertian [13, Cor. 1.28], such a realization implies that the covering group is a Galois group of a field extension of Qab. Most of these branched covers have been constructed by means of rigidity; cf. [9] and [13] for a discussion of this approach. (Some of these covers are actually defined over the Q-line, and their covering groups are thus Galois groups over Q.) The rigidity approach also suggests a possible way of proving the Shafarevich conjecture. B.H. Matzat introduced the notion of GAR-realizability of a group, this being realizability as the Galois group of a branched cover with certain additional properties (cf. [9, Chap. 4, 360

Sec. 3.1]). Many simple groups have been GAR-realized over Qab and the Shafarevich conjecture would follow if it were shown that every finite simple group has a GAR-realization over Qab. See [9, Chap. 4; See. 3, 4]. The solvable case of the Shafarevich conjecture has been proven: K. Iwasawa [7] showed that the maximal pro-solvable quotient of GQ~b is a free pro-solvable group of countable rank. In particular, every finite solvable group is a Galois group over Qab, and every embedding problem for GQ~b with finite solvable kernel has a proper solution. Iwasawa's result also holds for the maximal Abelian extension K ab of any global field K, and for the maximal cyclotomic extension K cycl of any global field K [7, Thm. 6, 7]. G e n e r a l i z a t i o n s . The Shafarevich conjecture can be posed with Q replaced by any g l o b a l field K. In this generalized form, it asserts that the absolute Galois group of [(cycl is free of countable rank (as a profinite group). This conjecture remains open (as of 2001) in the number field case, but has been proven by D. Harbater [6, Cor. 4.2] and F. Pop [10] in the case that K is the function field of a curve over a finite field k. (See also [5, Cor. 4.7] and [9, Sec. V.2.4].) Since k cyd = Fp if k is a finite field of characteristic p, this assertion is equivalent to stating that the absolute Galois group of K is free of countable rank if K is the function field of a curve over Fp. This result is shown by using patching methods involving formal schemes or rigid analytic spaces, in order to show that all finite embedding problems for GK have a proper solution - - i.e. that every connected H-Galois branched cover of the curve is dominated by a connected G-Galois branched cover, if H is a quotient of the finite group G. By Iwasawa's theorem [7, p. 567], the result follows. The proof also shows that if C is a curve over an arbitrary a l g e b r a i c a l l y closed field of cardinality ~, and if K is the function field of C, then every finite embedding problem for GK has exactly ~ proper solutions. By the Mel'nikov Chatzidakis theorem [8, Lemma 2.1], it follows that GK is free profinite of rank ~, generalizing the geometric case of the Shafarevich conjecture (see [6, Thm. 4.4], [10, Cor. to Thm. 1]). As another proposed generalization of the Shafarevich conjecture (which would subsume the above case of global fields), M. Fried and H. VSlklein conjectured [3, p. 470] that if K is a countable Hilbertian field whose absolute Galois group GK is projective, then GK is free of countable rank. They proved a special case of this [3, Thin. A], viz. that G~c is free of countable rank if K is a countable Hilbertian pseudo-algebraically closed field (a PAC field) of characteristic 0. For example, this apf~lies to the field K = Qtr(vrX-f), where Qtr is the field of totally real algebraic numbers, by results of R. Weissauer

SHIFT R E G I S T E R S E Q U E N C E and Pop; see [13, p. 151], [9, p. 286]. Later, Pop [11, Thm. 1] removed the characteristic 0 hypothesis from the above result. This solves a problem in [2, Problem 24.41]. (See also [4].) Since Qab is not PAC (as proven by G. Frey [2, Cor.10.15]), this result does not prove the Shafarevich conjecture itself. But it does imply that GQ has a free normal subgroup of countable rank for which the quotient is of the form 1-I~=2 S~ [3] (instead of the form Z* = G a l ( Q a b / Q ) as in the Shafarevich conjecture). The above Fried Vhlklein conjecture holds i f / C is Galois over k(x), for k an algebraically closed field ([8, Prop. 4.4], using the geometric case of the Shafarevich conjecture [61, [10]). More generally, it holds if is large in the sense of Pop [11, Thin. 2.1]; cf. also [9, Sec. V.4]. A solvable case of the conjecture holds, extending Iwasawa's result: For K Hilbertian with GK projective, every embedding problem for G K with finite solvable kernel has a proper solution [13, Cor.8.25]. References [1] FRIED, M. (ed.): Recent developments in the inverse Galois problem, Vol. 186 of Contemp. Math., Amer. Math. Soc., 1995. [2] FRIED, M., AND JARDEN, M.: Field arithmetic, Springer, 1986. [3] FRIED, M., AND VOLKLEIN, H.: 'The embedding problem over a Hilbertian PAC field', Ann. of Math. 135 (1992), 469-481. [4] HARAN, D., AND JARDEN, M.: 'Regular split embedding problems over complete valued fields', Forum Math. 10 (1998), 329-351. [5] HARAN, D., AND V()LKLEIN, H.: 'Galois groups over complete valued fields', Israel J. Math. 93 (1996), 9-27. [6] HARBATER, D.: 'Fundamental groups and embedding problems in characteristic p', in M. FRIED (ed.): Recent Developments in the Inverse Galois Problem, Vol. 186 of Contemp. Math., Amer. Math. Soc., 1995, pp. 353-370. [7] IWASAWA, K.: 'On solvable extensions of algebraic number fields', Ann. of Math. 58 (1953), 548-572. [8] JARDEN, M.: 'On free profinite groups of uncountable rank', in M. FRIED (ed.): Recent Developments in the Inverse Galois Problem, Vol. 186 of Contemp. Math., Amer. Math. Soe., 1995, pp. 371-383. [9] MALLE, G., AND MATZAT, B.H.: Inverse Galois theory, Springer, 1999. [10] POP, F.: 'l~tale Galois covers over smooth affine curves', Invent. Math. 120 (1995), 555-578. [11] PoP, F.: 'Embedding problems over large fields', Ann. of Math. 144 (1996), 1-34. [12] SnaRE, J.-P.: Cohomologie Galoisienne, Vol. 5 of Lecture Notes in Mathematics, Springer, 1964. [13] VSLiiLEIN, H.: Groups as Galois groups, Vol. 53 of Studies in Adv. Math., Cambridge Univ. Press, 1996.

David Harbater

MSC 1991:11R32

'shift register sequence' stems from the engineering literature; in mathematics, the terms r e c u r s i v e s e q u e n c e or recurrent sequence are more common. The classical reference on shift register sequences is [1]; see also [2] or [3] for expositions. A linear feedback shift register o f length n (LFSR) is a time-dependent device (running on a clock) of n cells each capable of holding a value from some field F, such that with each clock cycle the contents of the cells are shifted cyclically by one position (to the right, say). While the LFSR discards (or outputs) the rightmost entry b0 (and replaces it by bl), it computes the linear function e ] b n - 1 + " " + cnbo

of the present state vector ( b o , . . . , bn-1) and the feedback coefficients ( c l , . . . , c ~ ) , see Fig. 1. Thus, the box with the entry 'ADD' stands for an adder over F , and the circle with entry ci indicates multiplication by ci E F. (The question of how this might be realized in hardware is not addressed here; see [5], [6].) In practice, the case of the binary field GF(2) is by far the most iraportant one, but the general notion of an LFSR serves as a good intuitive way of modelling recursive sequences.

++ ++ I

"°°

Fig. 1: A linear feedback shift register. Given the initial conditions ( a 0 , . . . , a ~ - l ) , after t clock cycles the LFSR will hold the state vector a (t+l) = ( a t , . . . , a t + n - l ) , where a t + ~ - i = cla~+~-2 + " " + chat-1.

(1)

Thus, the shift register sequence a = (ak) produced by the LFSR will satisfy a linear recurrence relation of order n; namely, for k > n: ak = ~

(2)

c~ak-i.

i=1

With the convention Co = - 1 , one defines the feedback polynomial of the LFSR as

f(x) SHIFT REGISTER SEQUENCE, recursive sequence, recurrent sequence A sequence which can be obtained as the output of a linear feedback shift register. The term

I

= -t0

.....

(3)

its reciprocal polynomial i f ( x ) = x ~ - e l x n-1 . . . . .

e ~ - l x - e~

(4) 361

SHIFT REGISTER SEQUENCE is called the characteristic polynomial of the LFSR. Using its companion matrix

/0 0

A=

1

0

0

1

0

c~

0

Ca_ 1

0

c2

1

cl ]

Let a = (ak) be a shift register sequence over a G a lois field F = GF(q) with minimal polynomial m of degree n. Then a is ultimately periodic with least period r0 _< qn _ 1 (cf. U l t i m a t e l y p e r i o d i c s e q u e n c e ) . Conversely, any ultimately periodic sequence over a Galois field is in fact a shift register sequence.

! 0 :

:

•.

\0 0

the recursion (2) can be rewritten in terms of the state vectors as a (t+l) = a ( t ) A f o r t > 0 . A is usually calIed the feedback matrix of the LFSR, and it satisfies the equation mA = XA = f*, where XA and m a denote the characteristic and the minimal polynomial of A, respectively. One may characterize the shift register sequences over F by associating an arbitrary sequence a = (ak) over F with the f o r m a l p o w e r s e r i e s eND

a(x) = ~

ak xk E F[[x]].

k=0

Then a is a shift register sequence if and only if a(x) belongs to the field F ( x ) of rational functions over F. More precisely, a can be obtained from the LFSR of length n with feedback polynomial f E F[x] if and only if a(x)-

g(x)

f(x)

(5)

for a suitable polynomial g E FIx] with degg < n, and this correspondence between shift register sequences a belonging to f and polynomials g is a bijection. For instance, the Fibonacci sequence, defined by the recursion ak = ak-1 -kak-2 with initial conditions (ao, a l ) = (1, 1) over the rational numbers, belongs to the feedback polynomial f ( x ) = 1 - x - x 2, and the polynomial g(x) is simply g(x) = 1. Thus, the formal power series describing a is 1 a(x) - 1 - x - x 2 - l + x + 2x 2 + 3 x 3 + 5 x 4 + 8 x 5 +

+13x 6 + 21x 7 + 34x s + ... (cf. F i b o n a c c i n u m b e r s ) • There exists a uniquely determined polynomial m such that a given shift register sequence a can be obtained from the LFSR with characteristic polynomial f* if and only if f* is a multiple of m; this polynomial is called the minimal polynomial of the shift register sequence a. In other words, m is the characteristic polynomial of the linear recurrence relation of least order that is satisfied by a. If a = (ak) belongs to an LFSR of length n with characteristic polynomial f*, then f* is actually the minimal polynomial of a if and only if 362

the first n state vectors a(°) , . . . , a (n-l) are linearly independent.

If a = (ak) belongs to the L F S R with feedback polynomial (3), where c n ¢ 0, then a is actually periodic and the feedback matrix A is invertible. The particular shift register sequence d determined by the initial conditions ( 0 , . . . , 0, 1) is called the impulse-response sequence for the given LFSR. This name is motivated by thinking of the LFSR of Fig. 1 as being started by sending the 'impulse' 1 through the left-most cell, where initially each cell is 'empty'. The sequence d is periodic with least period r0 equal to the order of A (that is, r0 equals the least positive integer e such t h a t A ~ = I). Moreover, the least period of any shift register sequence a which can be obtained from the given L F S R divides r0. In particular, r0 = q~ - 1 if and only if f is a primitive polynomial for F (cf. G a l o i s field s t r u c t u r e ) . Hence, there exists a periodic shift register sequence with least period q~ - 1 belonging to an L F S R of length n over F = GF(q). Any such sequence is called a m a x i m a l period sequence (for short, an m-sequence) or a pseudo-noise sequence (for short, a PN-sequence). The latter name stems from the fact that these sequences can be used as pseudo-random sequences for certain engineering applications; indeed, they satisfy the axioms formulated by S.W. Golomb [1], cf. also [2] and P s e u d o - r a n d o m n u m b e r s • The impulse response sequences belonging to LFSRs with primitive feedback polynomials are essentially (up to cyclical equivalence) the only m-sequences. In the special case of an irreducible feedback polynomial f over F = G F ( q ) there is an easy explicit description of the associated shift register sequences in terms of the trace function, el. G a l o i s field s t r u c t u r e . For this, let c~ be a root of f* in the extension field E = GF(qn). Then the shift register sequences belonging to the given LFSR are precisely the sequences s = (Sk) of the form

sk = TrE/F(OC~k),

k > O,

where 0 is an arbitrary element of E; moreover, the element 0 is uniquely determined by the sequence s. Except for the trivial sequence 0 belonging to 0 = 0, the sequences s are periodic with least period ro equal to the order of c~ (that is, the least positive integer e such that a~ = 1) and split into (qn _ 1)/r0 equivalence classes of r0 sequences each.

SHIMURA C O R R E S P O N D E N C E While shift register sequences per se are too weak for use in c r y p t o g r a p h y , suitable (non-linear) combinations of such sequences have been studied in this context, see, e.g., [4]. References [1] GOLOMB, S.W.: Shift register sequences, Aegean Park Press, 1982. [2] JUNGNICKEL, D.: Finite fields: Structure and arithmetics, Bibliographisches Inst. Mannheim, 1993. [3] LIDL, R., AND NIEDERREITER, H.: Introduction to finite fields and their applications, Cambridge Univ. Press, 1994. [4] t=~UEPPEL, R.: Analysis and design of stream ciphers, Springer, 1986. [5] TIETZE, U., AND SCHENK, C.: Electronic circuits: Design and applications, Springer, 1991. [6] WESTE, N., AND ESHRAGHIAN, K.: Principles of C M O S V L S I design, Addison-Wesley, 1985.

Dieter Jungnickel MSC1991: 93C05, 11T71, 11B37 SHIMURA CORRESPONDENCE - By a modular f o r m of weight k one understands a function f on the

upper half-plane satisfying f(Tz) = X(7)(cz + for some suitable function X: P --+ CX when

d)kf(z)

is an element of some congruence subgroup of SL(2, Z) (cf. also M o d u l a r f u n c t i o n ) . If k is an integer, E. Hecke defined operators T~ for every integer n, and showed they could be simultaneously diagonalizable (cf. also H e c k e o p e r a t o r ) . Tile Lseries of a simultaneous eigenfunction (cf. also D i r i c h l e t L - f u n c t i o n ) is then an E u l e r p r o d u c t . Modular forms of half-integral weight arise naturally, for example as t h e t a - s e r i e s . A theta-series in r variables is a modular form of weight r/2. If k is a half-integer, T~ can only be defined if n is a square on forms of weight k, and there is not enough information in the Hecke eigenvalues to determine the F o u r i e r c o e f f i c i e n t s . The coefficients are not multiplicative, so the L-series is not an Euler product. Using the Rankin Selberg method and a converse theorem, G. Shimura [12] showed that if f is a modular form of weight k + 1/2, then there is a corresponding modular form of weight 2k such that the T,~2 Hecke eigenvalue on f agrees with the T~ Hecke eigenvalue of

f. This result was complemented by the important theorem of J.-L. Waldspurger [14], showing that the D t h Fourier coefficient of f agrees with L(k/2, f, XD). Waldspurger also gave interpretations of these special values as periods of f (integrals over over geodesics). W. Kohnen and D. Zagier [8] gave a particularly useful

treatment of a special case. Also useful is [9]. P. Sarnak and S. Katok [10] found similar results for Maass forms. Given Waldspurger's theorem, the case where k = 1 becomes particularly interesting, since if f is the modular form of weight two associated with an e l l i p t i c c u r v e , L(1, f, XD) has an interpretation in terms of the Birch-Swinnerton-Dyer conjecture. The period interpretation of the special values is then connected with the work of B.H. Gross, Kohnen and Zagier [6] on heights of Heegner points. A beautiful application of this connection with the Birch-Swinnerton-Dyer conjecture to the classical problem of computing the set of areas of rational right triangles was given in [13]. An interesting approach to the Shimura correspondence and Waldspurger's theorem is offered by the theory of Jacobi forms, in which both f and its correspondent f may be related to automorphic forms on the Jacobi group. See [2] and [5]; cf. also A u t o m o r p h i e f o r m . A. Weil realized that (Siegel) modular forms, particularly theta-series, should be interpreted as automorphic forms not on Sp(2n), but on a certain double cover Sp(2n), the so-called metaplectic group. If n = 1, then Sp(2n) = SL(2), and this is the proper framework for understanding the classical Shimura correspondence, which can be regarded as a lifting from either SL(2) to PGL(2) = O(3), oi" from GL(2) to GL(2). T. Kubota and K. Matsumoto constructed metaplectic covers of more general groups, provided the ground field contains sufficiently many roots of unity. The Shimura correspondence in this context is a lifting from automorphic forms on the covering group to automorphic forms on G or (sometimes) its dual, obtained by reversing the long and short roots and interchanging the fundamental group with the dual of the centre. See [7], [3], [4], [1], [11] for the Shimura correspondence on higher covers of higher rank groups. Finding analogues of Waldspurger's theorem in this context is an important open problem (as of 2000). References [1] BUMP, D., AND HOFFSTEIN, J.: 'On Shimura's correspondence', Duke Math. J. 55 (1987), 661-691. [2] EICHLER, M., AND ZAGIER, D.: Jacobi forms, Birkhguser, 1985. [3] FLICKER, Y.Z.: 'Automorphie forms on covering groups of GL(2)', Invent. Math. 57, no. 2 (1980), 119-182. [4] FLICKER, Y.Z., AND KAZHDAN, D.: 'Metaplectic correspondence', Publ. Math. IHES 64 (1986), 53-110. [5] GINZBURG, D., RALLIS, S., AND SOUDRY, D.: 'A new construction of the inverse Shimura correspondence', Internat. Math. Res. Notices 7 (1997), 349-357. [6] GRoss, B.H., KOHNEN, W., AND ZAGIER, D.: 'Heegner points and derivatives of L-series II', Math. Ann. 278 (1987), 497562.

363

SHIMURA C O R R E S P O N D E N C E [7] KAZHDAN, D., AND PATTERSON, N.J.: 'Towards a generalized Shimura correspondence', Adv. Math. 60 (1986), 161-234. [8] KOHNEN, W., AND ZAGIER, D.: 'Values of L-series of modular forms at the center of the critical strip', Invent. Math. 64 (1981), 175 198. [9] PIATETSKI--SHAPIRO, I.: 'Work of W'aldspurger': Lie Group Representations II, Vol. 1041 of Lecture Notes in Mathematics, Springer, 1984. [10] SARNAK, P., AND KATOK, S.: 'Heegner points, cycles and Maass forms', Israel J. Math. 84 (1993), 193-227. [11] SAVIN, D.: 'Local Shimura correspondence', Math. Ann. 280

(1988), 185-190. [12] SHIMURA, G.: 'On modular forms of half integral weight', Ann. of Math. 97 (1973), 440-481. [13] TUNNELL, J.B.: 'A classical Diophantine problem and modular forms of weight 3/2', Invent. Math. 72 (1983), 323-334. [14] WALDSPURCER, J.-L.: 'Sur les coefficients de Fourier des formes modulaires de poids demi-entier', d. Math. Pures Appl. 60 (1981), 375-484. D. B u m p

MSC1991: l l F l l , 11F12 SIEGEL-SHIDLOVSKI[ METHOD, See Siegel m e t h o d .

Shidlovskif-

Siegel method -

MSC 1991:11R99 SIERPII~SKI GAME - Let Y be a t o p o l o g i c a l s p a c e and X an uncountable subset of Y. Two players alternatively select subsets of X. Player I selects some uncountable subset A1 of X. Player II answers by picking up an uncountable subset B1 C A1. Then again player I selects some uncountable set A2 C B1 and player II responds by selecting some uncountable subset B2 C A2. Playing this way the two players generate a decreasing sequence p = (Ai, Bi)i_>l of uncountable sets, which is called a play. By definition, player II wins this play if the intersection nBi of the closures of B i (in Y) is contained in X. Otherwise the play is won by player I. A given 'rule' of selecting the moves of player II is called a winning strategy for player II if every play generated by this rule is won by this player. If Y is a Polish space (a completely metrizable and separable space, cf. also V a g u e t o p o l o g y ; D e s c r i p t i r e set t h e o r y ; C o m p l e t e m e t r i c space; S e p a r a ble space), then the existence of a winning strategy for player II implies that X contains the C a n t o r disc o n t i n u u m (and therefore contains continuum many points). On the other hand, if X is a Suslin subset of Y (cf. also D e s c r i p t i v e set t h e o r y ) , then player II has a winning strategy ([3]). Thus, every uncountable Suslin subset of a Polish space contains the C a n t o r d i s c o n t i n u u m . For Borel subsets of the unit segment this was proved by P.S. Aleksandrov ([1] and B o r e l set) and F. Hausdorff ([2]) when they were verifying the truth of the c o n t i n u u m h y p o t h e s i s for such subsets of the unit 364

segment. W. Sierpifiski ([5]) gave another proof of the same result. It was this proof of Sierpifiski that made R. Telg~irsky ([6]) introduce the above game and name it after Sierpifiski. Further information concerning the game of Sierpifiski can be found in [3], [4] and [7]. References [1] ALEXANDROV, P.S.: 'Sur la puissance des ensembles mesurables B', C.R. Acad. Sci. Paris 162 (1916), 323-325. [2] HAUSDORFF, F.: 'Die M~chtigkeit der Borelschen Mengen', Math. Ann. 77 (1916), 430-437. [3] KUB~CKI, G.: 'On a game of Sierpifiski', Colloq. Math. 54 (1987), 179 192. [4] KUBICKI, G.: 'On a modified game of Sierpifiski', Colloq. Math. 53 (1987), 81-91. [5] SmRPI~SKI, W.: 'Sur le puissance des ensembles mesurables (B)', Fundam. Math. 5 (1924), 166 171. [6] TELGJ~RSKI, R.: 'On some topological games': Proc. Fourth Prague Topological Syrup. 199"6, Part B: Contributed papers,

Soc. Czech. Math. and Physicists, 1977, pp. 461-472. [7] TELG~.RSKI, R.: 'Topological games: On the 50th anniversary of the Banach-Mazur game', Rocky Mount. J. Math. 17 (1987), 227-276. P . S . Kenderov

MSC 1991: 03E50, 54-XX, 90D80 SIERPII~SKI GASKET, t a m i s de Sierpidski - The Sierpifiski gasket (in French: 'tamis de Sierpifiski') - along with its companion, the Sierpifiski carpet, or 'tapis de Sierpifiski' - - belongs to the toolkit of every fractal geometer. It adorns many articles and books on the subject and is frequently used as an example or test case in various mathematical and physical studies of self-similarity. Although it is geometricaIly more complex than the classic C a n t o r set, it is still one of the simplest interesting f r a c t a l s . It was introduced in 1915 [39] by the Polish mathematician W. Sierpifiski, about forty years after the discovery of the Cantor set. Like other self-similar fractals, the Sierpifiski gasket is constructed iteratively. Beginning with an equilateral triangle, an inverted triangle with half the side-length of the original is removed. This process is then repeated with each of the remaining triangles ad infinitum (see Fig. 1).

AAAA

,A

Fig. 1: The Sierpifiski pre-gaskets (left) and the Sierpifiski gasket (right). Unlike the ternary Cantor set, which is a totally disconnected and compact subset of the real line (and hence has topological d i m e n s i o n zero [11], ef. also Z e r o d i m e n s i o n a l space; T o t a l l y - d i s c o n n e c t e d space), the Sierpifiski gasket is a connected compact subset of the Euclidean plane R 2 (cf. also C o n n e c t e d space). In fact, it can be viewed as a simple, continuous and closed plane curve (i.e., a Jordan curve); see [39], [43,

SIERP1NSKI GASKET §3.7]. Hence, it has topological dimension one [11]. In addition, it is non-rectifiable (i.e., it is a curve of infinite length, cf. also P e a n o curve; R e c t i f i a b l e curve). The gasket is strictly self-similar in the sense that it can be written as a finite union of scaled copies of itself; namely, as a union of three Sierpifiski gaskets, each with a side-length equal to half that of the original (see Fig. 2). More precisely, the Sierpifiski gasket is the unique non-empty compact subset G of the plane such that G = O~=ISj(G), where Sj is the similarity transformation of R ~ with contraction ratio 1/2 and with fixed point vj, the j t h vertex of the initial triangle in the construction of the gasket: Sj(z)=

1

- vj)+vj,

for z E R 2 and j = 1, 2, 3. Moreover, the Hausdorff and Minkowski (or box) dimensions of the Sierpifiski gasket are both equal to its similarity dimension: log(3)/log(2) (cf. also H a u s d o r f f d i m e n s i o n ) . This common value, log(3)/log(2) ~ 1.58, is often referred to as the fractal dimension of G. Here, 2 is the reciprocal of the contraction ratio and 3 is the number of parts of G similar to the whole in that ratio (see, for example, [8, Chapt. 9], [12], [33], [28, Plate 141]). vl

v3

Recently (2000), it has been suggested that the oscillations intrinsic to the geometry of G (and other selfsimilar fractals) can be described via suitable numbertheoretic explicit formulas [24, Chapt. 4] by means of a set of 'complex dimensions' having maximum real part equal to the (real) fractal dimension. Here, the complex dimensions of G with real part log(3)/log(2) are of the form log(3)/log(2) + 27tin~ log(2), where n E Z and i = x/L-1 (see [24, Chapts. 2-6 and 10], where a mathematical theory of complex dimensions is developed in the one-dimensional case). It is noteworthy that the self-similarity equation is analogous to that satisfied by an a l g e b r a i c n u m b e r ; for example, ~ is a solution of the quadratic equation x 2 - 2 = 0 [33, §3.4]. Thus, in a sense, fractal geometries can be viewed as extensions of ordinary (Euclidean) geometries, much like algebraic number fields are extensions of the field of rational numbers. A more precise analogy is developed in [24, Chapt. 2] (and the relevant references therein), where the corresponding dichotomy algebraic versus transcendental (or rational versus irrational) is given a concrete meaning (see also [26], [25]). The Sierpifiski gasket is a prototypical example of a 'finitely ramified fractal'. Roughly speaking, this means that it may be disconnected by removing only finitely many points; see Fig. 4. Indeed, every point of G has a finite order of ramification, namely, either 2, 3 or 4; see [39], [28, Chapt. 14] or [33, p. 118]. This topological notion has been abstracted in [15], where G is viewed as an example of a post-critically finite (p.c.f.) self-similar set.

Disconnected after removing only two points.

v~

Fig. 2: Self-similarity of the SierpiIiski gasket. It follows from the above self-similarity equation that G is the basin of attraction of the d y n a m i c a l s y s t e m formed by the mappings $1, $2 and Sa. (See [12] or [8, Chapt. 9].) In particular, every point of G can be written (not necessarily uniquely) as a ternary string of possibly infinite length (see Fig. 3). 1

~

~....~

1222

....

Fig. 4: The Sierpifiski gasket is finitely ramified. Sierpifiski c a r p e t . The Sierpidski carpet C is defined analogously to the gasket. Beginning with a square, a square with one-third the side-length of the original is removed from the centre. This process is then repeated with each of the remaining squares ad infinitum.

2111... Fig. 5: The Sierpifiski pre-carpets (left) and the Sierpifiski carpet (right).

3

2

Fig. 3: Coding of the points of G.

From Fig. 5, it is clear that the medians and diagonals of the original square intersect C in a C a n t o r 365

SIERPINSKI GASKET set; in fact, C can be thought of as a natural analogue of the Cantor set in the plane. The Sierpifiski carpet is also a strictly self-similar fractal: it is the union of eight copies of itself, scaled in the contraction ratio 1/3 (see Fig. 6). Therefore, its fractal dimension is equal to log(8)/log(3) ~ 1.89, the similarity dimension of C (see

[8], [28] or [33]).

s s s s s s s S s

Fig. 6: Self-similarity of the Sierpiriski carpet. Unlike the Sierpifiski gasket, however, the Sierpifiski carpet is infinitely ramified, as can be easily seen from Fig. 5 or Fig. 6. Actually, one needs to remove uncountably many points in order to disconnect C. This topological property has interesting physical consequences. For example, in the context of statistical physics and the theory of critical phenomena, it is expected that a phase transition occurs for Ising models (cf. Ising m o d e l ) on fractal lattices such as the Sierpifiski carpet but not the Sierpifiski gasket (see [28, p. 138]).

[3], [4], [9], [10], [14], [15], [16], [17], [18], [19], [20], [21],

[22], [23], [41], [42]). Brownian motion on the Sierpifiski gasket is defined as a suitably rescaled limit of random walks on the Sierpiriski pre-gaskets (see, for example, [10], [19], [4], [2]). Similarly, Laplacians on the Sierpifiski gasket are defined as suitably rescaled (or renormalized) limits of finite difference operators acting on an increasing sequence of finite graphs approximating G (of. Fig. 1; see also [14], [15]). One can obtain an analogue of Weyl's classic asymptotic formula for the eigenvalue distribution of Laplacians on G and other finitely ramified fractals (see [17] and earlier references therein, including [9]). One can also introduce a corresponding notion of 'spectral dimension', an appropriate analogue of the notion of fractal dimension in this context. Hence, paraphrasing M. Kac [13], one can 'hear' the spectral dimension of the Sierpifiski drum and other 'fraetal drums' [17]. In some sense, one can also hear the volume of G (see [21], [23] and [18]). More precisely, one can introduce a suitable notion of volume measure or 'spectral volume' of G (and other finitely ramified fractals); see [21], [23], where (in particular, for homogeneous mass distributions) it is proposed to be an analogue for this class of fractals of the Riemannian volume measure on a Riemannian manifold. It is then shown in [18] to be a specific selfsimilar measure, which, in the case of the homogeneous gasket, coincides with the natural Hausdorff measure on G. (The proof of this fact given in [18] makes use of the so-called 'decimation method' for computing recursively the eigenvalues and the eigenfunctions of the Laplacian on G; see [34], [35] in the physics literature and [9], [38], [42] in the mathematics literature. It also makes use of the existence of many localized eigenfunctions on the gasket; see, for example, [1] and [16].) The results of [17], [21] and [18] yield a precise form of Weyl's asymptotic law in this context.

From the mathematical point of view, the most striking property of the Sierpifiski carpet is its universality. Sierpifiski proved in his original 1916 paper on the subject [40] that every Jordan curve in the plane can be homeomorphically embedded in the Sierpifiski carpet. Hence, for example, C contains a homeomorphic image of the Sierpifiski gasket G. This remarkable and underappreciated theorem was extended by the Austrian mathematician K. Meager in [29]. For instance, the Menger sponge (the three-dimensional analogue of the Sierpifiski carpet, see [28, Plate 145]) is universal for all compact (metrizable) spaces of topological dimension one, and thus for all Jordan curves in space. In addition to the original references [40] and [29], see [33, §2.7] for a helpful heuristic discussion and [32] for an exposition of the proof of the Sierpiriski-Menger theorem (see also [30, Chapt. 9] and [6, pp. 433; 501]).

These problems have applications in condensed matter physics and solid state physics; for example, in the study of electrical transport in porous or in random media, as well as of heat diffusion or wave propagation on fractals and in disordered systems (see, for instance, [1], [5], [201, [22], [27], [31], [341, [35], [36], [37]).

Finally, a lot of work has been recently (as of 2000) done in order to develop 'analysis on fractals', using the Sierpifiski gasket (and sometimes the Sierpifiski carpet) as a prototypical example. In particular, one can obtain suitable analogues of Laplacians, diffusions (or Brownian motions) and related notions on these self-similar fractals and their generalizations (see, for instance, [2],

A number of other topics from classical harmonic analysis, probability theory, partial differential equations, mathematical physics, spectral geometry, and even number theory have (or are expected to have) interesting counterparts in this context (see, for example, [2], [16], [20], [22], [23], [24], [41], and the relevant references therein). As was mentioned previously, the Sierpifiski

366

SIERPEqSKI GASKET

g a s k e t is o f t e n a t e s t i n g g r o u n d for c o n j e c t u r e s c o n c e r n ing f i n i t e l y r a m i f i e d (or p.c.f.) self-similar fractals. Although several probabilistic results have been obtained for t h e S i e r p i f i s k i c a r p e t

( a n d its h i g h e r - d i m e n s i o n a l

a n a l o g u e s ) [3], [2], t h e r e a l m of i n f i n i t e l y r a m i f i e d fractals r e m a i n s m u c h m o r e e l u s i v e f r o m t h e a n a l y t i c a l p o i n t of v i e w ( e s p e c i a l l y in d i m e n s i o n t h r e e or h i g h e r ) a n d will n o d o u b t b e t h e o b j e c t of m a n y f u r t h e r i n v e s t i g a t i o n s in t h e f u t u r e . T h e a u t h o r is g r a t e f u l t o his s t u d e n t , E . P . J . P e a r s e , for his c o m m e n t s a n d for h e l p w i t h t h e p r e p a r a t i o n of t h e figures. References

[1] ALEXANDER, S., AND ORBACH, R.: 'Density of states on fractals: fractons', J. Physique Lettres 43 (1982), L625-L631.

[2] BARLOW, M.T.: 'Diffusions on fractals', in M.T. BARLOWAND D. NUALART (eds.): Lectures in Probability Theory and Statistics, t~cole d'Etd de Probab. de Saint Flour X X V - - 1 9 9 6 , Vol. 1690 of Lecture Notes in Mathematics, Springer, 1998, pp. 1-121. [3] BARLOW, M.T., AND BASS, R.F.: 'Construction of Brownian motion on the Sierpifiski carpet', Ann. Inst. H. Poincard 25 (1989), 225-257. [4] BARLOW, M.T., AND PERKINS, E.A.: 'Brownian motion on the Sierpifiski gasket', Probab. Th. Rel. Fields 79 (1988), 543-623. [5] BERRY, M.V.: 'Distribution of modes in fractal resonators', in W. G/iTTINGER AND H. EIKEMEIER (eds.): Structural Stability in Physics, Springer, 1979, pp. 51-53. [6] BLUMENTHAL, L.M., AND MENGER, K.: Studies in geometry, Freeman, 1970. [7] EDGAR, G.A. (ed.): Classics on fractals, Addison-Wesley, 1993. [8] FALCONER, K.: Fractal geometry: Mathematical foundations and applications, Wiley, 1990. [9] FUKUSHIMA,M., AND SHIMA, T.: 'On a spectral analysis for the Sierpifiski gasket', Potential Anal. 1 (1992), 1-35. [10] GOLDSTEIN, S.: 'Random walks and diffusions on fractals', in H. KESTEN (ed.): Percolation Theory and Ergodie Theory of Infinite Particle Systems, Vol. 8 of IMA Math.Appl., Springer, 1987, pp. 121-129. [11] HUREWICZ, W., AND xcVALLMAN, H.: Dimension theory, Princeton Univ. Press, 1948. [12] HUTCHINSON, J.E.: 'Fractals and self-similarity', Indiana Univ. Math. J. 30 (1981), 713 747. [13] KAC, M.: 'Can one hear the shape of a drum?', Amer. Math. Monthly 73 (1966), 1-23. [14] KIGAMI, J.: 'A harmonic calculus on the Sierpiflski spaces', Japan J. Appl. Math. 6 (1989), 259-290. [15] KIGAMI, J.: 'Harmonic calculus on p.c.f, self-similar sets', Trans. Amer. Math. Soc. 335 (1993), 721-755. [16] KIGAMI, J.: Analysis on fractals, Cambridge Univ. Press, in press. [17] KIGAMI, J., AND LAPIDUS, M.L.: 'Weyl's problem for the spectral distribution of Laplacians on p.c.f, self-similar fractals', Commun. Math. Phys. 158 (1993), 93-125. [18] KIGAMI, J., AND LAPIDUS, M.L.: 'Self-similarity of volume measures for Laplacians on p.c.f, self-similar fractals', Commum Math. Phys. 217" (2001), 165-180.

[19] KUSUOKA, S.: 'A diffusion process on a fractal', in K. ITO AND N. IKEDA (eds.): Probabilistic Methods in Mathematical Physics, Proc. Taginuchi Internat. Syrup. (Katata and Kyoto, 1985), Kinokuniya, 1987, pp. 251-274. [20] LAPIDUS, M.L.: 'Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl Berry conjecture', in B.D. SLEEMAN AND R.J. JARVIS (eds.): Ordinary and Partial Differential Equations, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, 1992) IV, Vol. 289 of Pitman Research Notes in Math., Longman, 1993, pp. 126-209. [21] LAPIDUS, M.L.: 'Analysis on fractals, Laplacians on selfsimilar sets, noncommutative geometry and spectral dimensions', Topoi. Methods in Nonlin. Anal. 4 (1994), 137-195. [22] LAPIDUS, M.L.: 'Fractals and vibrations: Can you hear the shape of a fractaI drum?', Fractals 3 (1995), 725-736, Proc. Symp. Fractal Geometry and Self-Similar Phenomena in honor of Benoit B. Mandelbrot's 70th Birthday (Curacao, Netherlands Antilles, 1995). [23] LAPlDUS, M.L.: 'Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals', Contemp. Math. 208 (1997), 211-252. [24] LAPIDUS, M.L., AND FRANKENHUYSEN, M. VAN: Fractal geometry and number theory (complex dimensions of ffactal strings and zeros of zeta functions), Research Monograph. Birkh/iuser, 2000. [25] LAPIDUS, M.L., AND FRANKENHUYSEN, M. VAN: 'Complex dimensions of self-similar fractal strings and Diophantine approximation', Preprint (2001). [26] LAPIDUS, M.L., AND FRANKENHUYSEN, M. VAN: 'A prime number theorem for self-similar flows', in M.L. LAPIDUS AND M. VAN FRANKENHUYSEN (eds.): Dynamical, Spectral and Arithmetic Zeta Functions, Contemp. Math., Amer. Math. Soc., 2001. [27] L~u, S.H.: 'Fractals and their applications in condensed matter physics', Solid State Phys. 39 (1986), 207-273. [28] MANDELBROT, B.B.: The fractal geometry of nature, revised and enlarged ed., Freeman, 1983. [29] MENGER, K.: 'Allgemeine R/iume und Cartesische Rgume, Zweite Mitteilung: fiber umfassenste n-dimensionale Mengen', Proc. K. Akad. Wetensch. Amsterdam 29 (1926), 476 482; 1125-1128, reprinted as Chap. 9 in K. Menger, Dimensionstheorie, Tenbner, 1928; English transl.: General spaces and Cartesian spaces, G.A. Edgar (ed.), Classics on fractals, Addison-Wesley, 1993, pp.103-117. [30] MENGER, K.: Dimensionstheorie, Teubner, 1928. [31] NAKAYAMA,T., YAKUBO, K., AND ORBACH, R.L.: 'Dynamical properties of fractal networks: Scaling, numerical simulation, and physical realization', Rev. Mod. Phys. 66 (1994), 381443. [32] PEARSE, E.P.J.: 'Universality of the Sierpifiski carpet', Honors Undergraduate Thesis Math. Univ. California June (1998), Available from: Univ. Honors Program at UC Riverside and at http://web.dreamsoft.com/freakomatic/sierpinski. [33] PEITGEN, H.-O., JIJRGENS, H., AND SAUPE, D.: Chaos and fractals: New frontiers of science, Springer, 1986. [34] RAMMAL, R.: 'Spectrum of harmonic excitations on fractals', J. Physique 45 (1984), 191-206. [35] RAMMAL, P~., AND TOULOUSE, G.: 'Random walks on fractal structures and percolation clusters', Y. Physique Lettres 44 (1983), L13-L22. [36] SAPOVAL,B.: Les ffactales-fractals, Aditech, 1989.

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[37] SCHROEDER, M.R.: Fraetals, chaos, power laws: Minutes from an infinite paradise, Freeman, 1991. [38] SHIMA,T.: 'On eigenvalue problems for Laplacians on p.c.f. self-similar sets', Japan J. Indus. Appl. Math. 13 (1996), 123. [39] SmRPI~SKI,W.: 'Sur une courbe cantorienne dont tout point est un point de ramification', C.R. Acad. Sci. Paris 160 (1915), 302. [40] SIERPII~SKI,W.: :Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donn~e', C.R. Acad. Sci. Paris 162 (1916), 629-632. [41] STRICHARTZ, R.S.: 'Analysis on fractals', Notices Arner. Math. Soc. 46 (1999), 1199-1208. [42] TEPLYAEV,A.: 'Spectral analysis on infinite Sierpifiski gaskets', J. Funct. Anal. 159 (1998), 537-567. [43] TRICOT, C.: Curves and ffactal dimensions, Springer, 1995. Michel L. Lapidus

MSC 1991:28A80 S I M D , single-instruction multiple-data - A phrase denoting that, in a parallel computation, each active processor executes the same instruction, but possibly with different data. MSC 1991:68Q10 S I S O SYSTEM, single-input single-output system A (dynamical) control system with a single input and a single output; see A u t o m a t i c c o n t r o l t h e o r y . MSC 1991: 73Axx SKEIN MODULE, linear skein - An algebraic object associated to a m a n i f o l d , usually constructed as a formal linear combination of embedded (or immersed) submanifolds, modulo locally defined relations. In a more restricted setting, a skein module is a m o d u l e associated to a t h r e e - d i m e n s i o n a l m a n i f o l d by considering linear combinations of links in the manifold, modulo properly chosen (skein) relations (cf. also Link; L i n e a r skein). It is the main object of a l g e b r a i c t o p o l o g y b a s e d o n k n o t s . In the choice of relations one takes into account several factors: i) Is the module obtained accessible (computable)? ii) How precise are the modules in distinguishing three-dimensional manifolds and links in them? iii) Does the module reflect the topology/geometry of a three-dimensional manifold (e.g. surfaces in a manifold, geometric decomposition of a manifold)? iv) Does the module admit some additional structure (e.g. filtration, gradation, multiplication, Hopf algebra structure)? One of the simplest skein modules is a q-deformation of the first h o m o l o g y g r o u p of a three-dimensional manifold M, denoted by $2(M; q). It is based on the skein 368

relation (between non-oriented framed links in M) L+ = qLo.

Already this simply defined skein module 'sees' nonseparating surfaces in M. These surfaces are responsible for the torsion part of this skein module. There is a more general pattern: most of the skein modules analyzed reflect various surfaces in a manifold. The best studied skein modules use skein relations which worked successfully in classical knot theory (when defining polynomial invariants of links in R 3, cf. also

Link). 1) The Kauffman bracket skein module is based on the Kauffman bracket skein relation L+ = A L _ + A-1Lo~, and is denoted by S2,~(M). Among the Jonestype skein modules it is the one best understood. It can be interpreted as a quantization of the coordinate ring of the character variety of SL(2, C) representations of the f u n d a m e n t a l g r o u p of the manifold M, [2], [4], [17]. For M = F x [0, 1], the Kauffman bracket skein module is an a l g e b r a (usually non-commutative). It is a finitely-generated algebra for a compact F [3], and has no zero divisors [17]. Incompressible tori and twodimensional spheres in M yield torsion in the Kauffman bracket skein module; it is a question of fundamental importance whether other surfaces can yield torsion as well. 2) Skein modules based on the Jones Conway relation (Homflypt relation) are denoted by $3 (M) and generalize skein modules based on the Conway relation which were hinted at by J.H. Conway. For M = F x [0, 1], S3(M) is a H o p f a l g e b r a (usually neither commutative nor co-commutative), [20], [11]. S 3 ( F x [0, 1]) is a free module and can be interpreted as a quantization [6], [19], [12], [20] (cf. also D r i n f e l ' d - T u r a e v q u a n t i z a t i o n ) . S 3 ( M ) is related to the algebraic set of SL(n, C) representations of the fundamental group of the manifold M, [18]. 3) The skein module based on the Kauffman polynomial relation is denoted by $3,~ and is known to be free forM=Fx[0,1]. 4) In homotopy skein modules, L+ = L _ for selfcrossings. The best studied example is the q-homotopy skein module with the skein relation q - l L + - q L _ = zLo for mixed crossings. For M = F x [0, 1] it is a quantization, [7], [20], [16], and as noted by U. Kaiser they can be ahnost completely understood using Lin's singular tori technique [9]. 5) The only studied skein module based on relations deforming n-moves to date (2000) is the fourth skein module $4 (M) = R g / ( boLo + biLl + b2L2 + b3La), with possible additional framing relation. It is conjectured

SKOLEM-NOETHER THEOREM that in S 3 this module is generated by trivial links. Motivation for this is the Montesinos-Nakanishi three-move

conjecture (cf. M o n t e s i n o s - N a k a n i s h i conjecture). 6) Extending the family of knots, ]C, by singular knots, and resolving singular crossing by Kc~ = K+ K _ allows one to define the Vassiliev-Gusarov filtration: . . . c Ca c . . . c C2 c . . . c C , c

... c Co = RK.,

where Ck is generated by knots with k singular points. The kth Vassiliev-Gusarov skein module is defined to be a quotient:

Wk (M) = RY./Ck+I. The completion of the space of knots with respect to the Vassiliev-Gusarov filtration, R]C, is a H o p f algebra (for M = Sa). Functions dual to Vassiliev-Gusarov skein modules are called finite type or Vassiliev invariants of knots, [13]. Skein modules have their origin in the observation by J.W. Alexander [1] that his polynomials of three links, L+, L_ and L0 in R a, are linearly related. They were envisioned by Conway (linear skein) [5] and the outline of the theory was given first in the spring of 1987 [10] after Jones' construction of his polynomial (the Jones polynomial) in 1984; see [8], [14], [15] for the history of the development of skein modules. V.G. Turaev pointed out the importance of skein modules as quantizations,

[20] (cf. also Drinfel'd-Turaev quantization).

[11] PRZYTYCKI, J.H.: ' Q u a n t u m group of links in a handlebody', in M. GERSTENHABEK AND J.D. STASHEFF (eds.): Contemporary Math.: Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Vol. 134, 1992, pp. 235-245. [12] PRZYTYCKI, J.H.: 'Skein module of links in a handlebody', in B. APANASOV, W.D. NEUMANN, A.W. REID, AND L. SIEBENMANN (eds.): Topology 90, Proc. Research Sem. Low Dimensional Topology at OSU, de Gruyter, 1992, pp. 315-342. [13] PRZYTYCKI, J.H.: 'Vassiliev Gusarov skein modules of 3manifolds and criteria for periodicity of knots', in K. JOHANNsON (ed.): Low-Dimensional Topology (Knoxville, 1992), Internat. Press, Cambridge, Mass., 1994, pp. 157-176. [14] PRZYTYCKI, J.H.: 'Algebraic topology based on knots: an introduction', in S. SUZUKI (ed.): Knots 96, Proc. Fifth Internat. Research Inst. M S J, World Sci., 1997, pp. 279-297. [15] PRZYTYCKI, J.H.: 'Fundamentals of Kauffman bracket skein modules', Kobe Math. J. 16, no. 1 (1999), 45-66. [16] PRZYTYCKI, J.H.: 'Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs': Proe. Conf. Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, New York, March, 14-15, 1998), 2001. [17] PRZYTYCKI, J.H., AND SIKORA, A.S.: 'On skein algebras and S12(C)-character varieties', Topology 39, no. 1 (2000), 115148. [18] SrKOaA, A.S.: 'PSLn-character varieties as spaces of graphs', Trans. Amer. Math. Soc. 353 (2001), 2773-2804. [19] TURAEV, V.G.: 'The Conway and Kauffman modules of the solid torus', J. Soviet Math. 52 (1990), 2799-2805. (Zap. Nauchn. Sere. L O M I 167 (1988), 79-89.) [20] TURAEV, V.G.: 'Skein quantization of Poisson algebras of loops on surfaces', Ann. Sci. l~cole Norm. Sup. 4, no. 24 (1991), 635-704. Jozef Przytycki

MSC1991: 57M25, 57Mxx

References [1] ALEXaNDEa, J.W.: 'Topological invariants of knots and links', Trans. Amer. Math. Soc. 30 (1928), 275-306. [2] BULLOCK, D.: 'Rings of Sl2(C)-characters and the Kauffman bracket skein module', Comment. Math. Helv. 72 (1997), 521-542. [3] BULLOCK, D.: 'A finite set of generators for the Kauffman bracket skein algebra', Math. Z. 231, no. 1 (1999), 91-101. [4] BULLOCK, D., FROHMAN, C., AND KANIA--BARTOSZYI~SKA, J.: 'Understanding the Kauffman bracket skein module', J. Knot Th. Ramifications (1999). [5] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): Computational Problems in Abstract Algebra, Pergamon, 1969, pp. 329-358. [6] HOSTE, J., AND KIDWELL, M.: 'Dichromatic link invariants', Trans. Amer. Math. Soc. 321, no. 1 (1990), 197-229. [7] HOSTE, J., AND PRZYTYCKI, J.H.: 'Homotopy skein modules of oriented 3-manifolds', Math. Proc. Cambridge Philos. Soc. 108 (1990), 475-488. [8] HOSTE, J., AND PRZYTYCKI, J.H.: 'A survey of skein modules of 3-manifolds', in A. KAWAUCHI (ed.): Knots 90, Proc. Internat. Conf. Knot Theory and Related Topics (Osaka, Japan, August 15-19, 1990), de Gruyter, 1992, pp. 363-379. [9] KAISER, V.: 'Presentations of homotopy skein modules of oriented 3-manifolds', J. Knot Th. Ramifications 10, no. 3 (2001), 461-491. [10] PRZYTVeKI, J.H.: 'Skein modules of 3-manifolds', Bull. Polish Acad. Sei. 39, no. 1-2 (1991), 91-100.

S K O L E M - N O E T H E R THEOREM - In its classical form, the Skolem-Noether theorem can be stated as follows. Let A and B be finite-dimensional algebras over a field k, and assume that A is simple and B is central simple (cf. also S i m p l e algebra; Central algebra; Field). If f,g: A -+ B are k-algebra homomorphisms, then there exists an invertible u E B such that

f(a) = u-lg(a)~ for all a E A. A proof can be found, for example, in [5, p. 21] or [4, Chap, 4]. In particular, every k-algebra automorphism of a central simple algebra is inner (cf. also Inner a u t o m o r p h i s m ) . This can be generalized to an Azumaya algebra A over a commutative ring R (cf. also Separable algebra): There is an exact sequence, usually called the Rosenberg-Zelinsky exact sequence: 0 -+ Inn(A) --+ aut(A) + Pie(R), where Pie(R) is the P i e a r d g r o u p of R, Aut(A) is the group of k-algebra automorphisms of A and Inn(A) is the subgroup consisting of inner automorphisms. The proof is an immediate application of the categorical 369

SKOLEM-NOETHER THEOREM

characterization of Azumaya algebras: An R-algebra A is Azumaya if and only if the categories of R-modules and A-bimodules are equivalent via the functors sending an R-module N to A ® N, and sending an A-bimodule M to M A = {m E M: am = m a f o r alla E A} (see, e.g., [6, IV.l] for details). The Skolem-Noether theorem plays a crucial role in the theory of the B r a u e r g r o u p ; for example, it is used in the proof of the Hilbert 90 theorem (cf. also H i l b e r t t h e o r e m ) and the cross p r o d u c t theorem. There exist versions of the Skolem-Noether theorem (and the Rosenberg-Zelinsky exact sequence) for other generalized types of Azumaya algebras; in particular, for Azumaya algebras over schemes [3], Azumaya algebras relative to a torsion theory [7, III.3.26] and Long's Hdimodule Azumaya algebras [1], [2].

• Bt is a one-dimensional Brownian motion starting at 0 and independent of Y0; • Yt > 0 f o r a l l t > _ 0 ; • ~t is increasing in t >_ 0 with 60 = 0 and ±(o)(Ys) des =

In fact, the solution Yt of this Skorokhod equation can be described uniquely and deterministically by the given Brownian motion Bt as

Yt = B t -

a formula due to P. L6vy in case that Y0 = 0. Further, gt is twice the L6vy local time of Bt at the origin. The Skorokhod equation has been extended to the higher-dimensional case R d, d > 2, to describe a norreally reflecting Brownian motion Yt on the closure D of a domain D C R d. In this case, the equation takes the form = Y0 + B , +

References [1] BEATTIE, M.: 'Autornorphisms of G-Azumaya algebras', Canad. J. Math. 3~" (1985), 1047-1058. [2] CAENEPEEL, S.: Brauer groups, Hopf algebras and Galois theory, Vol. 4 of K-Monographs Math., Kluwer Acad. Publ., 1998. [3] GROTtIENDIECK, A.: Le groupe de Brauer I, North-Holland, 1968. [4] HERSTEIN, I.N.: Noncommutative rings, Vol. 15 of Carus Math. Monographs, Math. Assoc. Amer., 1968. [5] KERSTEN, I.: Brauergruppen von KSrpern, Vol. D6 of Aspekte der Math., Vieweg, 1990. [6] KNUS, M.A., AND OJANGUREN, M.: Thdorie de la descente et alg~bres d'Azumaya, Vol. 389 of Lecture Notes in Mathematics, Springer, 1974. [7] OYSTAEYEN, F. VAN, AND VERSCHGREN, A.: Relative invariants of rings I, Vol. 79 of Monographs and Textbooks in Pure and Appl. Math., M. Dekker, 1983.

S. Caenepeel MSC1991: 13-XX, 16-XX, 17-XX SKOROKHOD EQUATION, Skorohod equation- A stochastic equation describing a reflecting Brownian motion. Given a one-dimensional B r o w n i a n m o t i o n Xt on R 1 = ( - o c , oo), the reflecting Brownian motion X + is defined by = lX l ,

t >_ o,

which is a M a r k o v p r o c e s s on [0, oo) with continuous sample paths. A.V. Skorokhod discovered that the reflecting Brownian motion X +, t _> 0, is identical in law with the solution Yt, t _> 0, of the equation

Yt = Yo + Bt + gt,

t > O,

where the triple {Yt, Bt, ft} is a system of real continuous stochastic processes (cf. also S t o c h a s t i c p r o c e s s ) required to have the following properties: 370

min B ~ A 0 ,

0<s
/0'

n(Y,)d&,

t>_O,

where Bt is a d-dimensional Brownian motion starting at the origin, n(x), x C O D , is the inward normal vector field on the boundary OD and ~ is a real increasing process such that /ta IoD(Ys ) d G = ~t, t >_ O. The third term at the right-hand side of the equation expresses a singular drift, keeping the process Yt inside D against the isotropic nature of the Brownian motion Bt. For a bounded convex domain in R d the Skorokhod equation has a unique solution. For other domains, the Skorokhod equations are studied not only from the point of view of stochastic differential equations, but also in relation to other principles, e.g. submartingale problems or Dirichlet forms. Obliquely reflecting Brownian motions, where the vector fields n in the Skorokhod equations are different from the normal vector field, also arise naturally in the diffusion approximation in stochastic network theory. References [1] IKEDA, N., AND WATANABE, S.: Stochastic differential equations and diffusions, second ed., North-Holland, 1989. Masatoshi Fukushima

MSC 1991: 60Hxx, 60J55, 60J65 SKOROKHOD SPACE - Let Z? = ~[0,1] be the space of real-valued functions x on [0, 1] that are rightcontinuous and have left-hand limits, i.e.

x(t+) --- limx(s) sSt

x(t+) = x ( t ) x ( t - ) = limx(s) set

exists,

for a I 1 0 < t <

1,

exists for all0 < t < 1.

(In probabilistic literature, such a function is also said to be a cadlag function, 'cadlag' being an acronym for the French 'continu £ droite, limites 5. gauche'.) Introducing

SLOBODNIK P R O P E R T Y a n o r m on 79 by setting ILxll = sup0
sup

Ilog{(t-s)-l(~(t)-)~(s))}l.

0_<s
Then for x, y E 79 one defines

d(x,y) = kEA inf, max {11~1,, suP0
In his fundamental paper [13], Skorokhod introduced four topologies M1, M2, J1, J2; the topology J1 became the most famous one and now bears his name (el. also S k o r o k h o d t o p o l o g y ) . At the end of the 1980s it was found that in certain problems the other topologies introduced by Skorokhod in the space of cadlag functions can be useful (see, for example, [8], [1], [2]). References [11 AVRAM, F., AND TAQQU, M.: 'Probability bounds for M-Skorokhod oscillations', Stochastic Processes Appl. 33 (1989), 63-72. [21 A_VRAM, F., AND TAQQU, M.: 'Weak convergence of sums of moving averages in the c~-stable domain of attraction', Ann. Probab. 20 (1992), 483-503. [3] BILLINGSLEY, P.: Convergence of probability measures, Wiley, 1968. [41 BILLINGSLEY, P.: Convergence of probability measures, second ed., Wiley, 1999. [5] BLOZNELIS, M., AND PAULAUSKAS, V.: 'On the central limit theorem for multiparameter stochastic processes', in J. HOFFMANN-JORGENSEN, ]~/I. M~ARCUS, AND J. KUELBS (eds.): Probability in Banaeh spaces 9 (Proe. Conf.), Birkhg~user, 1994, pp. 155-172. [6] CHENTSOV, N.N.: 'Weak convergence of stochastic processes whose trajectories have no discontinuities of the second order', Th. Probab. Appl. 1-3 (1956), 140 143. [7] ETHIER, S.N., AND KURTZ, T.G.: Markov processes: Characterization and convergence, Wiley, 1986. [8] JAKUBOWSKI, A.: 'A non-Skorohod topology on the Skorohod space', Electron. J. Probab. 2, no. 4 (1997), 1-21. [91 IKOLMOGOROV, A.N.: 'On Skorohod convergence', Th. Probab. Appl. 1-3 (1956), 213-222. [10] NEUItAUS, G.: 'On weak convergence of stochastic processes with multidimensional time parameter', Ann. Math. Stat. 42 (1971), 1285-1295. [11] POLLARD, D.: Convergence of stochastic processes, Springer, 1984. [12] P•OKHOROV, Yu.V.: 'Convergence of random processes and limit theorems in probability theory', Th. Probab. Appl. 1-3 (1956), 157-214. [13] SKOROKHOD, A.V.: 'Limit theorems for stochastic processes', Th. Probab. Appl. 1-3 (1956), 261-290. [14] STRAF, M.L.: 'Weak convergence of stochastic processes with several parameters', Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 (1972), 187-221.

Vygantas Paulauskas MSC 1991: 60G05, 60B10 P R O P E R T Y - Recall that a B a i r e is a t o p o l o g i c a l s p a c e in which every non-empty open subset is of the second category in itself (cf. also C a t e g o r y o f a set). A space X is Baire if and only if the intersection of each countable family of dense open sets in X is dense (cf. also D e n s e set). In what follows, consider a space X equipped with two topologies, p and T, and assume that ~- is finer than p. A topology ~- has the Slobodnik property if the intersection of each countable family of T-open p-dense sets in X is p-dense. If ~- has the Slobodnik property, then SLOBODNIK

space

371

SLOBODNIK PROPERTY (X,p) is a Baire space. Following a definition of A.R. Todd from [3], the topologies p and T are S-related if for any subset A of X , Intp A (the interior of A with respect to the topology p) is non-empty if and only if Int~ A is non-empty. If the topologies 7 and p are Srelated, then 7 has the Slobodnik property if and only if (X, p) is a Baire space, and this is the case if and only if (X, T) is a Baire space. Let f be a function on X of the first Baire class in the topology T, i.e. f is a pointwise limit of a sequence of T-continuous functions (cf. also B a i r e classes). A very general problem emerges: How large can the set of all p-continuity points o f f be? If T has the Slobodnik property, then f is p-continuous at all points of X except at a set of p-first category. This theorem generalizes Slobodnik's theorem from [2]: Any limit of a sequence of separately continuous functions on the Euclidean space R 2 is continuous on R 2, except at a set of the first category. Notice that separately continuous functions are of the first Baire class on R 2, that a function f is separately continuous on R 2 exactly when it is continuous in the finer crosswise topology on R 2 (a set G C R 2 is open in this topology if for any z = (zl, z2) C G there is a 5 > 0 such that the 'cross'

K(z,5):=

(tl,t2):

Izj-tjl<5, i , j = l,2, i ¢ j

is a subset of G), and that the crosswise topology has the Slobodnik property. A more detailed investigation of the Slobodnik property and related notions can be found in [1]. References [1] LUKES, J., MAL'), J., AND ZAJfCEK, L.: Fine topology methods in real analysis and potential theory, Vol. 1189 of Lecture Notes in Mathematics, Springer, 1986. [2] SLOBODNIK,S.G.: 'Expanding system of linearly closed sets', Mat. Zametki 19 (1976), 61-84. (In Russian.) [3] TODD, A.R.: 'Quasiregular, pseudocomplete, and Baire spaces', Pacific J. Math. 95 (1981), 233-250.

J. Lukeg MSC1991: 54E55, 26A21 S M A R A N D A C H E F U N C T I O N - Given a natural number n, the value of the Smarandache function r/ at n is the smallest natural number m such that n divides m!. An elementary observation is that rT(n) _< n, and that rl(n) = n if and only if n is a prime number or equal to 4. References [1] SMARANDACHE,F.: 'A function in number theory', Smarandache Function J. 1 (1990), 3 65.

M. Hazewinkel M S C 1991:11A05

372

S M I T H T H E O R Y OF G R O U P A C T I O N S A collection of techniques and results first obtained by P.A. Smith around 1940 (see [5], [6], [7]) in the area of finite transformation groups. Smith theory is now (2000) best understood via cohomological methods, following an approach introduced by A. Borel (see [2], [3]). The main goal of Smith theory is to study actions of finite p-groups on familiar and accessible spaces such as polyhedra or manifolds (cf. also A c t i o n o f a g r o u p o n a m a n i f o l d ; p - g r o u p ) . However, it can easily be adapted to a very large class of spaces, the so-called finitistic spaces. These are spaces such that every coyering has a finite-dimensional refinement (see [4, p. 133] for details; see also G e n e r a l t o p o l o g y ; C o v e r i n g ( o f a set)). The most i m p o r t a n t examples are compact spaces and finite-dimensional spaces. The spaces occurring below are assumed to be of this type. Let X be such a finitistic space and let P be a finite p-group acting on it (here p is a fixed prime number). Let X g be the fixed-point set of the action, that is, X P = {x e X: gx : x, Vg E P}.

The two basic theorems of Smith theory are as follows: a) If X has the m o d p homology of a point (cf. also H o m o l o g y ) , then the fixed-point set X P also has the m o d p homology of a point; in particular, it is nonempty. b) If X has the m o d p homology of a sphere, then the fixed-point set X p (possibly empty) also has the m o d p homology of a sphere. Of course, the main examples here are when X -~ D ~, the n-dimensional disc, and when X ~ S m, the mdimensional sphere. However, the mod p homological nature of the results are important, as they can fail at other prime numbers. Homological methods building on Smith's original approach can be used to verify very general restrictions associated to actions of finite p-groups. For example, if X satisfies the additional hypothesis that its total m o d p c o h o m o l o g y is finite, then there is an inequality arising from an action of a finite p-group P on X: E i=0

dim Hi(X, Z/p) > E

dim H i ( X P, Z/p).

i=0

Note t h a t this implies t h a t the fixed-point set X P has finitely m a n y components and that each of them has finite m o d p cohomology. The two previous results can be derived from this inequality. Another important result which follows from Smith theory is the fact t h a t if G is a finite g r o u p acting on a space X which is finitistic and acyclic (i.e. has the integral h o m o l o g y of a point), then the orbit space X / G is also acyclic.

SOBOLEV INNER PRODUCT Smith theory can be considered a precursor to the general cohomological theory of transformation groups (cf. also T r a n s f o r m a t i o n g r o u p ) . Given a finite group G acting on a space X , one constructs a space, called the Borel construction on X, as follows: X x a EG = (X x EG)/G, where EG is a free, contractible G-space. The projection induces a bundle mapping X x a EG -+ BG, where BG = E G / G is the so-called c l a s s i f y i n g s p a c e of G, an E i l e n b e r g - M a c L a n e s p a c e of type K(G, 1). The analysis of this bundle and related constructions is the basic tool in this area. In particular, the main results from Smith theory follow from considering the case G = Z/p; if X is an n-dimensional complex with a G-action, then the inclusion X a ~-+ X induces an isomorphism

HJ(X x a EG, Z/p) -+ HJ(X a x BG, Z/p) provided j > n. This fact, combined with the s p e c t r a l sequence in m o d p cohomology associated to the fibration

X xa EG --+ BG, are the two main elements used in this reformulation of Smith theory. See [1], [4] and [8] for excellent references regarding Smith theory and transformation groups. References [1] ALLDAY, C., AND PUPPE, V.: Cohomological methods in transformation groups, Vol. 32 of Studies Adv. Math., Cambridge Univ. Press, 1993. [2] BOREL, A.: 'Nouvelle d ~ m o n s t r a t i o n d ' u n th6or~me de P.A. Smith', Comment. Math. Helv. 29 (1955), 27-39. [3] BOREL, A.: Seminar on transformation groups, Vol. 46 of Ann. of Math. Stud., P r i n c e t o n Univ. Press, 1960. [4] BREDON, G.E.: Introduction to compact transformation groups, Acad. Press, 1972. [5] SMIT~, P.A.: ' T r a n s f o r m a t i o n s of finite period', Ann. of Math. 39 (1938), 127-164. [6] SMIT~, P.A.: ' T r a n s f o r m a t i o n s of finite period II', Ann. of Math. 49 (1939), 690-711. [7] SMITH, P.A.: 'Fixed point t h e o r e m s for periodic transformations', Amer. J. Math. 63 (1941), 1-8. [8] TOM-DIECK, T.: Transformation groups, Vol. 8 of Studies in Math., de Gruyter, 1987.

Alejandro Adem M S C 1991: 54H15, 55R35, 57S17 SOBOLEV

INNER

PRODUCT

- Let 7) be the linear

space of polynomials in one variable with real coefficients and let {#i}i=0 N be a set of positive Borel measures supported in the real line (cf. also B o r e l m e a sure; Polynomial). One introduces an i n n e r p r o d u c t in P N

t~

(p, q}s = E Ai [_p(i)q(i) dpi, i=0

(1)

such t h a t the integrals are convergent for all p, q C 7) and Ai E R +. Here, p(i) is the ith derivative of p. As usual, the associated n o r m is N

N

i=0

i=0

i

Inner products such as (1) a p p e a r in least-square approximation when smooth conditions are involved both in the approximation and in the functions to be approximated. See [4] for an introduction to this. One says that (1) is a Sobolev inner product in 7). In a pioneer work, P. A l t h a m m e r [1] considered the so-called Legendre-Sobolev inner products, when N = 1 and #0 = #1 is the L e b e s g u e measure supported on [ - 1 , 1]. Most of the tools of the standard case ( N = 0) are not useful for N _> 1 since a basic property concerning the s y m m e t r y of the shift operator is lost for (1). This is the reason why further work focused initially on some particular cases of (1) when N = 1. In [7], the case #0 = #1 = the Gegenbauer weight function and A0 = 1 is considered with some detail. In such a situation, there exists a linear differential operator £ of second order such t h a t (£p, q)8 = (P, £q)8. This fact leads to the study of the algebraic properties of the so-called Gegenbauer-Sobolev orthogonal polynomials, with a special emphasis on the location of their zeros as well as their strong asymptotics (see [11]; cf. also O r thogonal polynomials). A similar approach was m a d e in [8] for #0 = #1 = the Laguerre weight function and A0 = 1. Thus, the Laguerre-Sobolev orthogonal polynomials are introduced. Some estimates for them, as well as their relative asymptotics with respect to Laguerre polynomials off the positive real semi-axis, are given in [6]. Beyond these two examples, an approach to a general theory was started in [3], where the concept of a coherent pair of measures is introduced. The main idea consists in the assumption of a kind of correlation between the measures #o and #1. Consider an inner product

(p,q) = ~ p q d p o + A £ p ' q ' d#l,

(2)

with A E R +, and let (Pn) and (Tn) be sequences of monic polynomials orthogonal with respect to #o and #1, respectively. Then (#o, #1) is called a k-coherent pair of measures if =

(x),

n > k + 1,

j=n--k

with b~,~+l = 1 and b~,~-k # 0. If (Q~) denotes the sequence of monic polynomials orthogonal with respect to (2) and (#0, #1) is a k-coherent 373

SOBOLEV INNER PRODUCT pair, then

References n+l

n+l

E

bn,jPj(x)=

j=n-k

E

fln+l,jQj(x).

j=n--k

Thus, analytic properties of (Q~) can be studied in t e r m s of analytic properties of (P~). T h e first p r o b l e m is to classify the set of k-coherent pairs of measures. This was described in [13] for k = 0 (see Table 1). Note t h a t one of the measures must be the Jacobi or the Laguerre weight function. This m e a n s t h a t the concept is very restrictive from the point of view of a general theory. T h e study of the general case k >_ 1 remains open (as of 2000). parameters

d/t0

a,~>O

( 1 - x ) a - l (l + x ) f l - l dx

d/~l (1-~) a (l+x) ¢~dx

+

MS(~), I~11 _> 1, M_> 0 a,~>0 a > 0 ~>0 O~

> O

a >0

Ix-~2l(1-x) a-1. (1 ÷ x) ~ - 1 dx (1-x)~-ldx+MS(-1) (l + x ) ~ - l dx + MS(1)

(1-x)a(l+x)Zdx, 1~21 > 1 (1 - x) a dx, M > 0 (l + x)~ dx, M >_ O

~_~ dx + M~(~), ~<_O,M>_O xae - x dx, ~ < 0 e -~ dx, M k O xa~ -x

x ~ - l e -m d x

(x -- ~)xa-le - x dx e -~ dx + MS(O)

Table 1. Nevertheless, in [10] a first a p p r o a c h is given when pl is the Jacobi weight function. Let d#l = (1 - x)~(1 + x) ~ dx, a , f l > 0, s u p p o r t e d on [ - 1 , 1]. T h e m e a s u r e #0 is said to be admissible with respect to #1 if i) It0 belongs to the Szeg5 class, i.e.,

/ ii)

t ln#~( x ) 1 dz >

pn(a-1 'Z-l) ,o = o(n), n ~ oc, where (p(~,9)) de-

[1] ALTHAMMER, P.: 'Eine Erweiterung des Orthogonalit~tsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation', J. Reine Angew. Math. 211 (1962), 192-204. [2] C-AUTSCHI, W., AND ZHANG, M.: 'Computing orthogonal polynomials in Sobolev spaces', Numer. Math. 71 (1995), 159-184. [3] ISERLES, A., KOCH, P.E., NORSETT, S.P., AND SANZ-SERNA, J.M.: 'On polynomials orthogonal with respect to certain Sobolev inner products', Y. Approx. Th. 65 (1991), 151-175. [4] LEWIS, D.C.: 'Polynomial least square approximations', Amer. J. Math. 69 (1947), 273-278. [5] MARCELLAN, F., ALFARO, M., AND REZOLA, M.L.: 'Orthogonal polynomials on Sobolev spaces: Old and new directions', J. Comput. Appl. Math. 48 (1993), 113 131. [6] MARCELLJ~N, F., MEIJER, H.G., P~REZ, T.E., AND PIt~AR, M.A.: 'An asymptotic result for Laguerre-Sobolev orthogohal polynomials', J. Comput. Appl. Math. 87 (1997), 87-94. [7] MARCELL~N, F., P~REZ, T.E., AND PI~AR, M.A.: 'Gegenbauer Sobolev orthogonal polynomials', in A. CUYT (ed.): Proc. Conf. Nonlinear Numerical Methods and Rational Approximation IT, Kluwer Acad. Publ., 1994, pp. 71-82. [8] MARCELL£N, F., P~REZ, T.E., AND P;~AR, M.A.: 'LaguerreSobolev orthogonal polynomials', J. Comput. Appl. Math. 71 (1996), 245-265. [9] MARTINEZ-FINKELSHTEIN, A.: 'Bernstein-Szeg6's theorem for Sobolev orthogonal polynomials', Constructive Approx. (2000), 73-84. [10] MARTfNEZ-FINKELSHTEIN,A., AND MORENO-BALC£ZAR,J.J.: 'Asymptotics of Sobolev orthogonal polynomials for a Jaeobi weight', Meth. Appl. Anal. 4 (1997), 430-437. [11] MARTfNEZ-FINKELSHTEIN,A., MORENO-BALCXZAR,J.J., AND PIJEIRA, H.: 'Strong asymptotics for Gegenbauer Sobolev orthogonal polynomials', J. Comput. AppL Math. 81 (1997), 211-216. [12] MEIJER, H.G.: 'A short history of orthogonal polynomials in a Sobolev space I: The non-discrete case', Nieuw Arch. Wisk. 14 (1996), 93-112. [13] MEIJER, H.G.: 'Determination of all coherent pairs of functionals', J. Approx. Th. 89 (1997), 321-343.

F. MarceIldn

M S C 1991: 33Exx, 33C45, 46E35

notes the sequence of o r t h o n o r m a l Jacobi polynomials. In such a case one obtains the following relative a s y m p totics: for z C C \ [ - 1 , 1],

@dz)

2 ¢'¢)'

where ¢(z) = z + v / ~ - 1, with ~ - 1 > 0 when z > 1. This result has been extended [9] to the case when #0 and #1 are 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 supported in [ - 1 , 1] and belong to the Szeg5 class. In fact, Q , ( z ) / T n ( z ) = 2 / ¢ ' ( z ) , z E C \ [ - 1 , 1]. From a numerical point of view, [2] is a nice survey a b o u t the location of zeros of polynomials orthogonal with respect to (1) when N = 1. For more information a b o u t Sobolev inner products, see the surveys [5] and [12] 374

SOR M E T H O D , successive overrelaxation method See A c c e l e r a t i o n m e t h o d s ; R e l a x a t i o n m e t h o d .

-

MSC 1991:65F10

SORGENFREY TOPOLOGY, right half-open interval topology - A t o p o l o g y ~- on the real line R (cf. also T o p o l o g i c a l s t r u c t u r e ( t o p o l o g y ) ) defined by declaring t h a t a set G is open in 7- if for any x C G there is an ex > 0 such t h a t Ix, x + ex) C G. R endowed with the t o p o l o g y ~- is t e r m e d the Sorgenfrey line, and is denoted by R s. T h e Sorgenfrey line serves as a counterexample to several topological properties, see, for example, [3]. For example, it is not metrizable (cf. also M e t r i z a b l e

SPEARMAN RHO METRIC space) but it is Hausdorff and perfectly normal (cf. also H a u s d o r f f space; P e r f e c t l y - n o r m a l space). It is first countable but not second countable (cf. also F i r s t a x i o m of c o u n t a b i l i t y ; S e c o n d a x i o m of c o u n t a b i l ity). Moreover, the Sorgenfrey line is hereditarily Lindel6f, zero dimensional and paracompact (cf. also LindelSf space; Z e r o - d i m e n s i o n a l space; P a r a c o m p a c t space). Any compact subset of the Sorgenfrey line is countable and nowhere dense in the usual Euclidean topology (cf. N o w h e r e - d e n s e set). The Sorgenfrey topology is neither locally compact nor locally connected (cf. also L o c a l l y c o m p a c t space; L o c a l l y c o n n e c t e d space). Consider the Cartesian product X := R s x R s equipped with the product topology (cf. also Topological p r o d u c t ) , which is called the So~yenfrey halfopen square topology. Then X is completely regular but not normal (cf. C o m p l e t e l y - r e g u l a r space; N o r m a l space). It is separable (cf. S e p a r a b l e space) but neither LindelSf nor countably paracompact. Many further properties of the Sorgenfrey topology are examined in detail in [1]. Namely, the Sorgenfl'ey topology is a fine t o p o l o g y on the real line, and R equipped with both the Sorgenfrey topology and the Euclidean topology serves as an example of a bitopological space (that is, a space endowed with two topological structures). The Sorgenfrey topology satisfies the condition (tFL) when studying fine limits (if a realvalued function f has a limit at the point x with respect to the Sorgenfrey topology T it has the same limit at x with respect to the Euclidean topology when restricted to a T-neighbourhood of x). It has also the Gsinsertion property (given a subset A of R, there is a Gs-subset G of R such that G lies in between the Tinterior and the T-closure of A). The Sorgenfrey topology satisfies the so-called essential radius condition: For any point x and any T-neighbourhood U~ of x there is an 'essential radius' r(x, Ux) > 0 such that whenever the distance of two points x and y is majorized by min(r(x, Ux),r(y, Uy)), then Ux and Uy intersect. The real line R equipped with the Sorgenfrey topology and the Euclidean topology is a binormal bitopological space, while R with the Sorgenfrey and the d e n s i t y t o p o l o g y is not binormal. See [1] for answers to interesting questions concerning the class of continuous functions in the Sorgenfrey topology and for functions of the first or second B a i r e classes. References [1] LUKES, J., MAL'/, J., AND ZAJiCEK, L.: Fine topology methods in real analysis and potential theory, Vol. 1189 of Lecture Notes in Mathematics, Springer, 1986. [2] SORGENFREY,R.H.: 'On the topological product of paracompact spaces', Bull. Amer. Math. Soc. 53 (1947), 631-632.

[3] STEEN, A.S., AND SEEBACH JR., J.A.: Counterexamples in topology, Springer, 1978.

J. Luke5 MSC 1991: 54G20, 54E55, 26A21 SPANIER-WHITEHEAD DUALITY, WhiteheadSpanier duality Let X be a CW-spectrum (see Spect r u m of spaces) and consider

[w A X, S]0, where W is another CW-spectrum, W A X is the smash product of W and X (see [1, Sect. III.4]), S is the sphere spectrum, and [, ]0 denotes stable homotopy classes of mappings of spectra. With X fixed, this is a contravariant functor of W which satisfies the axioms of E.H. Brown (see [2]) and which is hence representable by a spectrum D X , the Spanier-Whitehead dual of X. X ~-+ D X is a contravariant functor with many duality properties. E.g., i) [W, Z A D X ] . ~ _ [ W A X , Z].; ii) w. ( D X A Y) ~- IX, Y]. ; iii) [X, Z], _~ [DY, DX], ; iv) D D X ~_ X; v) for a (generalized) homology theory E , there is a natural isomorphism between Ek (X) and E -k (DX). In many ways X ~-~ D X is similar to the linear duality functor V ~-~ HOmk (V, k) for finite-dimensional vector spaces over a field k. For X C S N, the N-dimensional sphere, the classical Alexander duality theorem says that Hk(X) is isomorphic to H N - I - k ( S '~ \ X), and this forms the basic intuitive geometric idea behind Spanier-Whitehead duality. For more details, see [1, Sect. II.5], and [4, Sect. 5.2]. For an equivariant version, see [3, p. 300It]. References [1] ADAMS, J.F.: Stable homotopy and generalised homology, Chicago Univ. Press, 1974. [2] BROWN, E.H.: 'Cohomology theories', Ann. of Math. 75 (1962), 467-484. [3] GREENLEES, J.P.C., AND MAY, J.P.: 'Equivariant stable homotopy theory', in I.M. JAMES (ed.): Handbook of Algebraic Topology, Elsevier, 1995, pp. 227 324. [4] RAVENEL, D.C.: 'The stable homotopy theory of finite complexes', in I.M. JAMES (ed.): Handbook of Algebraic Topology, Elsevier, 1995, pp. 325-396.

M. Hazewinkel MSC 1991:55P25 SPEARMAN RHO METRIC, Spearman rho - The non-parametric c o r r e l a t i o n coefficient (or measure of association) known as Spearman's rho was first discussed by the psychologist C. Spearman in 1904 [4] as a coefficient of correlation on ranks (cf. also C o r r e l a t i o n coefficient; R a n k s t a t i s t i c ) . In modern use, the 375

SPEARMAN RHO METRIC term 'correlation' refers to a measure of a linear relationship between variates (such as the P e a r s o n p r o d u c t m o m e n t c o r r e l a t i o n c o e f f i c i e n t ) , while 'measure of association' refers to a measure of a monotone relationship between variates (such as the K e n d a l l t a u m e t r i c and Spearman's rho). For an historical review of Spearman's rho and related coefficients, see [2]. Spearman's rho, denoted rx, is computed by applying the Pearson product-moment correlation coefficient procedure to the ranks associated with a sample {(xi,yi)}n=l . Let Ri = rank(x/) and Si = rank(yi); then computing the sample (Pearson) correlation coefficient r for {(Ri,Si)}~=I yields E i = x ( R ~ - R)(s~ - 8) rs=

~/ Y

n

Ei=I(

R

/-R)2

.

n

E/=I(S/-S)

2

6 E "/=1 ( R / - &)~ n ( n 2 - 1) ' n

n

where R = ~ i = l R i / n = (n + 1)/2 = ~ i = 1 S i / n = -S. When ties exist in the data, the following adjusted formula for rs is used: n(n 2 rs

=

1) - 6 2n=1(./~i - Si) 2 - 6(T -~- U)

-

v/n(n

2 -

1) -

12Tv/n(n

2 -

1) - 12U

where T = ~ t t ( t2 - 1)/12 for t the number of X observations that are tied at a given rank, and U = ~ u u( u2 - 1)/12 for u the number of Y observations that are tied at a given rank. For details on the use of r s in hypothesis testing, and for large-sample theory, see [1]. If X and Y are random variables (cf. R a n d o m variable) with respective distribution functions F x and F y , then the population parameter estimated by rs, usually denoted Ps, is defined to be the Pearson productmoment correlation coefficient of the random variables F x ( X ) and F y ( Y ) : PS = c o r r [ F x (X), F y (Y)] = = 12E[Fx(X)Fy(Y)]

- 3.

Spearman's ps is occasionally referred to as the grade correlation coefficient, since F x ( X ) and F v ( Y ) are sometimes called the 'grades' of X and Y. Like Kendall's tau, Ps is a measure of association based on the notion of concordance. One says that two pairs (xz,yl) and (x2,y2) of real numbers are concordant if Xl < x2 and Yl < Y2 or if xl > x2 and Yl > Y2 (i.e., if (xl -- x2)(yl -- Y2) > 0); and discordant if xi < x2 and yl > y2 or if xl > x2 and Yl < Y2 (i.e., if (xl - x 2 ) ( y l - y 2 ) < 0). Now, let (X1,Y1), (X2, Y2) and (X3, Y3) be independent random vectors with the same 376

distribution as (X, Y). Then flS = 3[:)[(Zl - 22)(]11 - Y3) > 0]+

-3P[(X1

- X2)(Y1 - ]73) < 0],

that is, Ps is proportional to the difference between the probabilities of concordance and discordance between the random vectors (X1, Y1) and (X2, Y3) (clearly, (X2, Y3) can be replaced by (X3, !/2)). When X and Y are continuous, Ps = 12

/01/0

uv d C x , y ( u , v) - 3 =

= 12 ~01~01 [ C x , y (u, v) - up] dudv, where C x , y is the c o p u l a of X and Y. Consequently, Ps is invariant under strictly increasing transformations of X and Y, a property Ps shares with Kendall's tau but not with the Pearson product-moment correlation coefficient. Note that p s is proportional to the signed volume between the graphs of the copula C x , y (u, v) and the 'product' copula II(u, v) = up, the copula of independent random variables. For a survey of copulas and their relationship with measures of association, see [3]. Spearman [5] also proposed an L1 version of rs, known as Spearman's footrule, based on absolute differences IRi - Si[ in ranks rather than squared differences: fs =1-

3 Ei%~ IR~ - s~l n2_l

The population parameter Cs estimated by f s is given by ¢ s = 1 - 3 jf0i jr01 lu - v[ d C x , y ( u , v ) =

---=6

]o.1 C x , y ( u ,

@ d u - 2.

References

[1] GIBBONS, J.D.: Nonparametric methods for quantitative analysis, Holt, Rinehart & Winston, 1976. [2] KRUSKAL, W.H.: 'Ordinal measures of association', J. Amer. Statist. Assoc. 53 (1958), 814-861. [3] NELSEN, R.B.: A n introduction to copulas, Springer, 1999. [4] SPEARMAN,C.: 'The proof and measurement of association between two things', Amer. J. Psychol. 15 (1904), 72-101. [5] SPEARMAN, C.: 'A footrule for measuring correlation', Brit. J. Psychol. 2 (1906), 89-108. R.B. Nelsen MSC 1991:62H20 SPECHT PROPERTY - A variety of some universal algebras (e.g. groups, semi-groups, associative, Lie,

Jordan, etc., rings and algebras; cf. also V a r i e t y o f univ e r s a l a l g e b r a s ; U n i v e r s a l a l g e b r a ) is the class of all algebras satisfying a given system of identical relations (polynomial identities in the case of rings and algebras over a field). The description of the identities of concrete

SPECTRAL THEORY OF COMPACT OPERATORS varieties and algebras is one of the central problems in the theory. A variety is finitely based (or has a finite basis for its identities) if it can be defined by a finite number of identities. A variety satisfies the Specht property if it itself and all its subvarieties are finitely based. The problem of existence of infinitely based varieties of groups was raised by B.H. Neumann in his thesis in 1935, see also [12], and for associative algebras by W. Specht [17] in 1950. Nowadays (2001), the finite basis problem for all main classes of universal algebras is known also as the Specht problem. The investigations are in two directions: to show that classes of varieties satisfy the Specht property and to construct counterexamp]es. For comments and results for groups, semi-groups and algebras see [1], [7], [8], [13] and [21, [9]. The positive results include the Specht property for varieties generated by finite objects with reasonable good structure (e.g. groups, associative, Lie, Jordan rings and algebras over finite fields), classes of groups, rings and algebras satisfying some specific identities (e.g. nilpotent or metabelian groups and Lie algebras). One of the most important results in this direction is the positive solution by A.R. Kemer of the Specht problem for associative algebras over a field of characteristic zero, see [8]. It is relatively easy to construct counterexamples to the Specht problem for sufficiently general algebras. There exist also finite semi-groups [15] and finite nonassociative rings [16] without finite bases for their identities. The first counterexample to the finite basis problem for groups was given by A.Yu. Ol'shanskiY [14] in 1970. The simplest example is due to Yu.G. Zle~man [10], [11] and R.M. Bryant [5], who showed that the system of group identities (xl2 . . . x 2 ) 4 = 1, n = 1,2,..., does not follow from any of its finite subsystems. The first example of a Lie a l g e b r a without a finite basis for its identities was given by M.R. Vaughan-Lee [18] in characteristic two, and then generalized to any field of positive characteristic by V. Drensky [6] and KleYman (unpublished). The variety of Vaughan-Lee is defined by the centre-by-metabelian identity [[[xl, x2], [x3, x4]], xs] = 0 and the identities [[... [[xl, x2], x a ] , . . . , xn], Ix1, x2]] = 0, n = 3, 4 , . . . . He also showed that over an infinite field of characteristic two the Lie a l g e b r a of all (2 × 2)-matrices has no finite basis of its polynomial identities. Recently (1999), A.Ya. Belov [3], see also [4], constructed an example of a non-finitely based variety of associative algebras over any field of positive characteristic. Presently (2001), the Specht problem is still open for Lie algebras over a field of characteristic zero. Many questions concerning finite bases of polynomial identities are naturally connected also with other problems

at the meeting point of algebra and logic, in particular with various algorithmic problems, see [9]. References [1] BAHTURIN, Yu.A.: Identical relations in Lie algebras, VNU Press, 1987. (Translated from the Russian.) [2] BAHTURIN, Yu.A., AND OLSHANSKII, A.Yu.: 'Identities', in A.I. KOSTRIKIN AND I.R. SHAEAREVICH (eds.): Algebra II, Vol. 18 of Encyclopedia Math. Sci., Springer, 1991, pp. 107221.

[3] BELOV, A.YA.: 'On nonspechtian varieties', Fundam. i Prikladn. Mat. 5, no. 1 (1999), 47-66. (In Russian.) [4] BELOV, A.YA.: 'Counterexamples to the Specht problem', Sb. Math. 191 (2000), 329-340. (Mat. Sb. 191 (2000), 13-24.) [5] BRYANT, R.M.: 'Some infinitely based varieties of groups', J. Austral. Math. Soc. 16 (1973), 29-32. [6] DRENSKY, V.: 'Identities in Lie algebras', Algebra and Logic 13 (1974), 150-165. (Algebra i Logika 13 (1974), 265-290.) [7] DRENSKY, V.: Free algebras and PI-algebras, Springer, 1999. [8] KEMER, A.R.: Ideals of identities of associative algebras, Vol. 87 of Transl. Math. Monographs, Amer. Math. Soc., 1991. [9] KHARLAMPOVICH, O.G., AND SAPIR, M.V.: 'Algorithmic problems in varieties', Internat. J. Algebra Comput. 5 (1995), 379-602. [10] KLEYMAN,Yu.G.: 'The basis of a product variety of groups I', Math. USSR Izv. 7 (1973), 91-94. (Izv. Akad. Nauk. S S S R Ser. Mat. 37 (1973), 95-97.) [11] KLEI'MAN,YU.G.: 'The basis of a product variety of groups II', Math. USSR Izv. 8 (1974), 481-489. (Izv. Akad. Nauk. SSSR Set. Mat. 38 (1974), 475-483.) [12] NEUMANN,B.H.: 'Identical relations in groups I', Math. Ann. 114 (1937), 506-525. [13] NEUMANN,H.: Varieties of groups, Springer, 1967. [14] OLSHANSKII,A.Yu.: 'On the problem of a finite basis of identities in groups', Math. USSR Izv. 4 (1970), 381-389. (Izv. Akad. Nauk. S S S R Set. Mat. 34 (1970), 376-384.) [15] PERKINS, P.: 'Decision problems for equational theories of semigroups and general algebras', PhD Thesis Univ. California at Berkeley (1966). [16] POLIN, S.V.: 'Identities of finite algebras', Sib. Math. J. 17 (1976), 992-999. (Sibirsk. Mat. Zh. 17 (1976), 1356-1366.) [17] SPECHT, W.: 'Gesetze in Ringen I', Math. Z. 52 (1950), 557589. [18] VAUGHAN-LEE, M.R.: 'Varieties of Lie algebras', Quart. J. Math. Oxford Set. 2 21 (1970), 297-308.

V. Drensky MSC1991: 08Bxx, 16R10, 17B01, 20El0 SPECTRAL THEORY OF COMPACT OPERATORS, Riesz theory of compact operators - Let X be a complex B a n a c h s p a c e and T a c o m p a c t opera t o r on X. Then a(T), the spectrum of T, is countable and has no cluster points except, possibly, 0. Every 0 # t 6 a(T) is an eigenvalue, and a pole of the resolvent function A ~+ ( T - AI) -1. Let v(A) be the order of the pole A. For each n 6 N, ( T - I I ) n X is closed, and this range is constant for n > v(A). The null space N ( ( T - )~I) n) is finite dimensional and constant for n > v(A). The spectral projection E(A)

377

S P E C T R A L T H E O R Y OF COMPACT OPERATORS (the Riesz projector, see R i e s z d e c o m p o s i t i o n t h e o r e m ) has non-zero finite-dimensional range, equal to N ( ( T - M)'(~)), and its null space is (T - ;~I)'(X)X. Finally, dim(E(A)X) >_ u(A) _> 1. The respective integers u(A) and d i m ( E ( ~ ) X ) are called the index and the algebraic multiplicity of the eigenvalue I # 0. References

[1] DowsoN, H.R.: Spectral theory of linear operators, Aead. Press, 1978, p. 45ff. [2] DUNFORD,N., AND SCHWARTZ,J.T.: Linear operators I: Gen-

theory, Interscience, 1964, p. Sect. VII.4. M. Hazewinkel MSC1991: 47A10, 47B06 eral

S P E N C E R C O H O M O L O G Y - The d e R h a m coh o m o l o g y and Dolbeault cohomology (cf. D u a l i t y in c o m p l e x a n a l y s i s ) can be viewed as cohomologies with coefficients in the s h e a f of locally constant, respectively harmonic, functions. Spencer cohomology is a generalization of these two cohomologies for the case of the solution sheaf of an arbitrary l i n e a r d i f f e r e n t i a l operator. Namely, let a : E(a) ~ M and /3: E(/~) -+ M be smooth vector bundles (cf. also V e c t o r b u n d l e ) and let D: r(a) ~ F(Z) be a linear differential operator acting from the module F(a) of smooth sections of a to the module F(~). Denote by ®D the sheaf of solutions of Da = 0. To find the cohomology of M with coefficients in ~D one needs a r e s o l v e n t of the sheaf. Spencer cohomology appears as a result of constructing a resolvent by a locally exact complex of differential operators

F(a0) ~ r ( a l ) ~ F(a2) -~ . . . , where a = a0, ctl = /3, D = Do. The condition that the complex be locally exact is too strong, and therefore D. Spencer proposed the weaker condition that the complex should be 'formally exact'. In this setting, there exists for a formally integrable differential operator D a canonical construction ([5], [6], [1]) of a complex, called the second (or sophisticated) Spencer complex. In this complex, Co @

... ,

the vector bundles Ck have the form Ck = AkT*M ® R,~/5(Ak-IT*M ® g,~+l), where R,~ C Jm(a) are prolongations of the differential equation corresponding to D (cf. also P r o l o n g a t i o n o f s o l u t i o n s o f d i f f e r e n t i a l e q u a t i o n s ) and g,~ are the symbols of these prolongations (cf. also S y m b o l o f a n o p e r a t o r ) . The differential operators Dk are first-order partial differential operators whose symbols are induced by the exterior multiplication. 378

The 5-Poincar~ lemma [6] shows that the c o h o m o l o g y of the complex does not depend on m when m is large enough. The stable cohomology H } ( D ) is called the Spencer cohomology of the differential operator D. In general, the second Spencer complex does not produce a resolvent of GD; however, it does in certain special cases, e.g. when D is analytical operator [6]. Almost-all cohomologies encountered in applications are of Spencer type. For example, de Rham cohomology corresponds to the differential D = d: C°~(M) --+ f t l ( M ) , and the Dolbeault cohomology corresponds to the Cauchy Riemann O-operator 0: ~P'°(M) --+ ~p,1(M). If D is a determined operator such that not all covectors are characteristic, then H~(D) = kerD, H } ( D ) = cokerD and H b ( D ) = 0 for i > 2. In general, H ° ( D ) = k e r D for each formally integrable operators D. In the case of Lie equations and corresponding geometrical structures (see [2]), the first Spencer cohomology gives an estimate of the set of deformations of the structure. If D is an elliptic partial differential operator (cf. 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 ) and M is a compact m a n i f o l d , then dim Hb(D) < oc and the E u l e r c h a r a c t e r i s t i c x(D) = E(-1)idimHb(D)of the Spencer complex is called the index of D (cf. also I n d e x f o r m u l a s ; I n d e x t h e o r y ) . For elliptic Lie equations the index can be expressed in terms of characteristic classes corresponding to the geometrical structure ([3]). As is well-known, there are two main methods for calculating the de Rham cohomology: the Leray-Serre spectral sequence (cf. also S p e c t r a l s e q u e n c e ) and the theorem on coincidence of de Rham cohomology with invariant cohomology on homogeneous manifolds. These methods also apply to Spencer cohomology, provided the operator D satisfies certain extra conditions. Thus, if the base manifold M is the total space of a smooth bundle 7r: M --+ B over a simply-connected manifold B and if the fibres of 7r are not characteristic for D, then there exists a spectral sequence k(E rpq , d pq~ r ] converging to the Spencer cohomology H~(D); its second term is E pq = HP(B) ® Has(Dr), where D~ is the fibrewise differential operator corresponding to D [4]. If M = G/Go is a homogeneous manifold and the structure group G is a compact connected Lie g r o u p of symmetries of D, then [4] the Spencer cohomology H~ (D) coincides with the cohomology of the G-invariant Spencer complex if the non-trivial characters of (G, Go) are non-characteristic. References [1] GOLDSCHMIDT, H.: 'Existence theorems for analytic linear

partial differential equations', Ann. Math. 86 (1967), 246270.

SPERNER THEOREM [2] KUMPERA, A., AND SPENCER, D.: 'Lie equations', Ann. Math. Studies 73 (1972). [3] LYCHAGIN, V., AND RUBTSOV, V.: 'Topological indices of Spencer complexes that are associated with geometric structures', Math. Notes 45 (1989), 305-312. [4] LYCHAGIN, V., AND ZILBERGLEIT, L.: 'Spencer cohomologies and symmetry groups', Acta Applic. Math. 41 (1995), 227245. [5] QUmLEN, D.G.: 'Formal properties of overdetermined systerns of linear partial differential equations', Thesis Harvard Univ. (1964). [6] SPENCER, D.: 'Overdetermined systems of linear partial differential operators', Bull. Amer. Math. Soc. 75 (1969), 179239.

Valentin Lychagin MSC 1991: 55N35, 53C15 Let P be a finite p a r t i a l l y o r d e r e d set (abbreviated: poset) which possesses a rank function r, i.e. a function r: P -~ N such that r(p) = 0 for some minimal element p of P and r(q) = r(p) + 1 whenever q covers p, i.e. p < q and there is no element between p and q. Let Nk := {p e P : r(p) = k} be its kth level and let r(P) := max{r(p): p E P } be the rank of P. An anti-chain or Sperner family in P is a subset of pairwise incomparable elements of P. Obviously, each level is an anti-chain. The width (Dilworth number or Sperner number) of P is the maximum size d(P) of an anti-chain of P. The poset P is said to have the Sperner property if d(P) = maxk INk]. E. Sperner proved in 1928 the Sperner property for Boolean lattices (cf. also Sperner theorem). More generally, a k-family, k = 1 , . . . , r ( P ) , is a subset of P containing no chain of k + 1 elements in P, and P has the strong Sperner property if for each k the largest size of a k-family in P equals the largest size of a union of k levels. There exist several classes of posers having the strong Sperner property: SPERNER

PROPERTY

-

• L Y M posets, i.e. posets P satisfying the L Y M inequality (cf. also S p e r n e r t h e o r e m )

I:r n Nk[

• Peck posets, i.e. ranked posets P such that INkl = INT(p)_al for all k and there is a linear operator V on the vector space having the basis {~: p C P} with the following properties: ~(P~ = ~q:q coversp c(p, q)~ with some numbers -

e(p, q), - t h e subspaee N generated by {~: p E Ni} is mapped via V j - i to a subspace of dimension rain{]Nil, INjl} for all 0 < i < j < r(P). If P and Q are posets from one class, then also the direct product P × Q (ordered componentwise) belongs to that class, where in the case of LYM posers an additional condition must be supposed: [Nkl 2 > ]Nk_l]lNk+ll for all k (so-called logarithmic concavity). Moreover, quotient theorems have been proved for LYM posets with weight functions and Peck posers. Every LYM poser with the symmetry and unimodality property IN01 = INr(p)] ~ ]N1] = INr(p)-l] < " " is a symmetric chain order and every symmetric chain order is a Peck poset. Standard examples of posets belonging to all these three classes are the lattice of subsets of a finite set, ordered by inclusion (the Boolean lattice), the lattice of divisors of a natural number, ordered by divisibility, the lattice of all subspaces of an n-dimensional vector space over a finite field, ordered by inclusion. The poser of faces of an n-dimensional cube, ordered by inclusion, belongs only to the class of LYM posets. The lattice of partitions of a finite set, ordered by refinement, even does not have the Sperner property if n is sufficiently large. Details can be found in [1]. References

[1] ENGEL, K.: Sperner theory, Cambridge Univ. Press, 1997.

K. Engel MSC 1991: 05D05, 06A07 SPERNER THEOREM - Let [n] := { 1 , . . . , n } . A family ~ of subsets of [n] that are pairwise unrelated with respect to inclusion is called a Sperner family (or Sperner system) on [n]. Examples are the families

k=0

for every anti-chain F in P or, equivalently,

Iv(A)l

IAI

[Nk+~l -> [Nkl for a l l A C Nk, k = O , . . . , r ( P ) - l , where ~7(A) := {q C Nk+l : q > P for some p E A}. This equivalent property is called the normalized matching property of P. • Symmetric chain orders, i.e. ranked posets P which can be decomposed into chains of the form (P0 < • "" < Ph) where r(pi) = r(po) + i, i = O , . . . , h , and

r(po) + r(ph) = r(P).

Since the binomial coefficients satisfy the inequalities <...<

n/2

_--

n/2

>...>

f [n] ~aswell in these examples ~n/2] ~ in] ~, if n is even, and ~(u-1)/2] as (~(n+l)/2J, [~] ~ if n is odd, have maximum size. Sperner's theorem from 1928 states that these best examples have even maximum size among all Sperner families on [n] and that they are the only optimal families. 379

SPERNER THEOREM

Given a Sperner family 5 , let :Fk := ([~]) N )c and

fk := IFkl. In his original proof, E. Sperner used a shifting technique: Consider the smallest l with ~cl ¢ ~ and replace A := -Pl by its upper shadow V(A) := { Y ¢ (l[~]l): Y D X f o r s o m e X E A }. Double-counting easily yields and, equivalently,

IAI(n -

IV(~4)------!> I~4J

l) _< IV(A) I(1 + 1) (1)

(l+l) - ( ; ) Thus, each Sperner family can be shifted from below to the 'middle' and, analogously, from above to the 'middle' and thereby increasing its size. The inequality (1) holds for all A c_ ([~1) and all l, and this property is called the normalized matching property of the lattice of subsets of [n]. If ]A I and l are fixed, the best possible estimate of the upper shadow, and, dually, of the lower shadow (replace l + 1 by 1 - 1 and superset by subset), is given by the K r u s k a l - K a t o n a t h e o r e m . Sperner's theorem follows also easily from the inequality

k=0

which can be obtained by counting in two different ways the number of pairs (X, 7r) where X C ~ , ~ is a permutation of [n] and X = {;r(1),..., ~(IXl)}. This inequality was proved independently by D. Lubell, S. Yamamoto and L. Meshalkin, and is hence called the LYM inequality; a more general form of it was given by B. Bollob~s. An essential part of Sperner theory consists of the study of other partially ordered sets having analogous properties, e.g. LYM posets and Peck posers (cf. Sperner property). Details can be found in [1].

function was referred to as the Grundy function, [7]. 0 n l y later the more obscure but earlier reference [10] became known, whence the name changed to SpragueGrundy function, or g-function. A digraph is locally walk-bounded if for every vertex ui there is a bound bi C Z ° such that the length of every (directed) walk emanating from ui does not exceed hi. Every locally walk-bounded digraph has a unique gfunction. Moreover, g(ui) < 0}, where 7) is the set of all P-positions of the game and A/" is the set of all its N-positions. Informally, a P-position is any game position u from which the 'p'revious player can force a win, that is, the opponent of the player moving from u. An N-position is any position v from which the 'n'ext player can force a win, that is, the player who moves from v. More precisely, suppose one is given a finite or infinite game with game-graph G = (V, U), which may be cyclic. Denote by (9 the set of all non-negative ordinals not exceeding IVI. By recursion on n E (9, define

Pn={uCV:n=minm'F(u)

C- U<mNi}'i

References [1] ENGEL, K.: Sperner theory, Cambridge Univ. Press, 1997. [2] SPERNER, E.: 'Ein Satz fiber Untermengen einer endlichen Menge', Math. Z. 27 (1928), 544 548.

K. Engel

MSC 1991: 05D05, 06A07

SPRAGUE-GRUNDY FUNCTION, Grundy Sprague function, Grundy function - The function g: V ~ Z ° from the vertex set V of a digraph G = (V,E) into the non-negative integers Z ° defined inductively by: g = mexg(F(u)), where for any subset S C Z °, S # Z °, m e x S = r a i n s = least non-negative integer not in S, F(u) = {v C V: (u,v) C E} is the set of (directed) followers of u and g(F(u)) = {g(v): v C F(u)}. Informally, g(u) = least non-negative integer not among the values g(F(u)). Note that g(u) = 0 if u is a leaf of G, i.e., if F(u) = ~. In the earlier literature the 380

and 79 = Un~o Pn and Af = U , ~ o Nn. If G is finite, acyclic and connected, there is a depthfirst O(]EI) algorithm for computing g. However, there is an algorithm of the same complexity for computing 79 and A; directly, so who needs 9? (Cf. also C o m p l e x i t y theory.) The answer lies in the important concept of a sum of games. Let { F 1 , . . . , r , ~ } be a finite disjoint collection of games with game-graphs {G1 = (V1, E l ) , . . . , Grn = (Vr~,E,~)}, which may have cycles or loops, or may be infinite. Then the sum-game r = F1 + ... + F,~ is the two-player game in which a position has the form ( u l , . . . , u,~) with ui C If/ for all i, and a move consists

SPRAGUE-GRUNDY FUNCTION of selecting some Fi and making a legal move ui --+ vi in it ((ui,vi) E El). The sum-graph G = G1 + ".. + Gm is the digraph G = (V, E) defined as follows:

a) k = ~ < o o ; b) k = co(K),/~ = oc(L) and K = L. One also uses the notations v : = {u c v :

V=

{ ( U l , . . . , U m ) : ui • Vi, i • { 1 , . . . , m } } ;

if u = ( U l , . . . , u , ~ ) , v = ( v l , . . . , V m ) • V, then ( u , v ) • E if there is some j • { 1 , . . . , m } such that vj • F(uj), that is, (uj,vj) • Ej, and ui = vi for all i # j. Informally, the sum of games is a game in which a move consists of selecting one of the Fi and making a move in it. For example, Nim is the sum of its heaps, and sums arise naturally in many games. The gamegraph of a sum is normally exponential in the size of the Fi, so computing P , A/ on it involves an exponentiM computation. But g enables one to formulate a polynomial algorithm: For u = (Ul,...,Um) • V, let or(u) = g(ul) ® " " ® g(um), where @ denotes Nim addition, also known as Xor (i.e. exclusive or), or addition over GF(2). Then g(u) = a(u). For normal play one then has, in view of the above result, 7) = {u • V : ~(u) = 0}, A/: {. • v:

> 0}.

<

V ~ = V \ V:, = mexT(F(u)).

Further, one associates a counter function with 7, in order to enable the winner to realize a win rather than merely maintaining a non-losing status in cycles. Given a cyclic digraph G = (V, E), a function '7 : V -9 Z°U{oo} is a 7-function with counter function c: V f -+ J, where J is any infinite well-ordered set, if the following three conditions hold: A) If 7(u) < 0% then ,7(u) = "7'(u). B) If v E F(u) with ,7(v) > ,7(u), then there exists a w E F(v) satisfying ,7(w) = ,7(u) and c(w) < c(u). C) If,7(u) = oo, then there is a v E F(u) with ,7(v) = oo(K) such that '7'(u) ¢ K . The generalized Nim-sum is defined as the Nim-sum above, augmented by: k ® oo(Z)

=

oo(n) @ k

=

oo(L @ k),

The polynomiality of the computation is valid for a standard game graph with input size O(IV [ + IEI). But many of the more interesting games are succinct, i.e., have input size O(log(]V/ + IEI)), and for them some additional property is needed to establish polynomiality. For Nim it is the fact that the g-values form an arithmetic sequence (cf. A r i t h m e t i c p r o g r e s s i o n ) ; for many octM games [8] g is ultimately polynomial, and for some other games special numeration systems can be exploited to recover polynomiality [3]. If the game-graph is cyclic, the game's outcome may be a draw, i.e., no player can force a win, but each has a non-losing next move. Two properties of g collapse when G has cycles:

w h e r e k E Z° , L C z ° , L ~ Z ° , L ® k = { g @ k : ~ E L } . The generalized Nim-sum of oo(L1) and oo(L2), for any subsets L1,L2 C Z °, L1,L2 ~ Z °, is defined by

i) it may not exist or not exist uniquely; in fact, the question of the existence of g is A/P-complete [2]; and ii) it may not determine the strategy.

where i9 is the set of all 'd'raw positions. For a finite connected digraph G = (V,E), 7 can be computed in O(IVI]EI) steps, which is polynomial in the size of a standard digraph. Many applications of the g-function to games appear in [1], and some of the results mentioned above are taken from [4].

Fortunately, however, there is a generalized SpragueGrundy function '7: V -+ Z ° U {oo}, which exists uniquely on all finite and some infinite digraphs [9], [6], [5], where the symbol oo indicates a value larger than any natural number. One can define ,7 also on certain subsets of vertices. Specifically:

,7(F(u)) = {'7(v) < c~: v • F(u)} ; if ,7(u) : cc and ,7(F(u)) = K , one also writes '7(u) = Equality of ,7(u) and ,7(v): If ,7(u) = k and ,7(v) = ~, then ,7(u) = ,7(v) if one of the following holds:

oo(nl) • oo(L2)

:

oo(L2) @ c~(L1) = oo(0).

To handle sums of games, one sets, analogously to the above Nim addition, a(u) = 7(ui) ®"" ® 7(urn), where now @ denotes generalized Nim addition. For normal play one then has p = {. E v:

= 0},

:D = {u E V: or(u) = oo(K), 0 ~ K } , A / = {u e v : 0 < U ( u e V:

=

< 0 e K),

References [1] BERLEKAMP,F,.R., CONWAY,J.H., AND GUY, ]~.K.: Winning ways for your mathematical plays, Vol. I-II, Acad. Press, 1982. [2] FRAENKEL, A.S.: 'Planar kernel and Grundy with d _ 3, dour ~ 2, din ~ 2 are NP-complete', Discr. Appl. Math. 3 (1981), 257-262. [3] FRAENKEL, A.S.: 'Heap games, numeration systems and sequences', Ann. Combinatorics 2 (1998), 197-210.

381

S P R A G U E - G R U N D Y FUNCTION [4] FRAENKEL, A.S.: Adventures in games and computational complexity, Graduate Studies in Mathematics. Amer. Math. Soc., ~o appear. [5] FRAENKEL, A.S., AND RAHAT, O.: 'Infinite cyclic impartial games', Theoret. Computer Sci. 252 (2001), 13-22, Special issue on Computer Games '98. [6] FRAENKEL, A.S., AND YESHA, Y.: 'Theory of annihilation games I', J. Combin. Th. B 33 (1982), 60-86. [7] GRUNDY, P.M.: 'Mathematics and games', Eureka 27 (1964), 9-11, Reprint; originally: ibid. 2 (1939), 6-8. [8] GUY, R.K., AND SMITH, C.A.B.: 'The G-values of various games', Proc. Cambridge Philos. Soc. 52 (1956), 514 526. [9] SMITH, C.A.B.: 'Graphs and composite games', J. Combin. Th. 1 (1966), 51-81. [10] SPRAGUE, R.: @ber mathematische Kampfspiele', Tdhoku Math. J. 41 (1935/36), 438-444.

Aviezri S. Fraenkel MSC 1991:90D05 S T A N L E Y - R E I S N E R RING, Stanley-Reisner face ring, face ring - The Stanley-Reisner ring of a s i m p l i cial c o m p l e x A over a field k is the quotient ring

k[zx] := k[xl,...,

The mapping from A to k[A] allows properties defined for rings to be naturally extended to simplicial complexes. The most well-known and useful example is Cohen-Macaulayness: A simplicial complex A is defined to be Cohen-Macaulay (over the field k) when k[A] is Cohen-Macaulay (cf. also C o h e n - M a c a u l a y ring). The utility of this extension is demonstrated in the proof that if (the geometric realization of) a simplicial complex is homeomorphic to a sphere, then its f-vector satisfies a condition called the upper bound conjecture (for details, see [1, Sect. II.3,4]). The statement of this result requires no algebra, but the proof relies heavily upon the Stanley-Reisner ring and Cohen-Macaulayness. Many other applications of the Stanley-Reisner ring may be found in [1, Chaps. II, III]. Finally, there is an anti-commutative version of the Stanley-Reisner ring, called the exterior face ring or indicator algebra, in which the polynomial ring k[xl,..., x~] in the definition of k[A] is replaced by the e x t e r i o r a l g e b r a k(Xl,. . . , Xn). References

where { X l , . . . , x ~ } are the vertices of A, k[xl,...,x~] denotes the polynomial ring over k in the variables x l , . . . , x ~ , and Izx is the i d e a l in k[Xl,...,x~] generated by the non-faces of A, i.e.,

IA ~- <Xil '''Xij: {Xil,... , Xij } ~- /~>.

xi"

xiEF

One may thus write Izx more compactly as IA

=

<S: F ¢ A). It is easy to verify that the Krull dimension of k[A] (cf. also D i m e n s i o n ) is one greater than the dimension of A (dim k[A] = (dim A 1 + 1). Recall that the Hilbert series of a finitely-generated Z-graded module M over a finitely-generated k-algebra is defined by F(M, A) := ~ i E z dimk MiA/. The Hilbert series of k[A] may be described from the combinatorics of A. Let d i m A = d - l , let f~ := I{F E A: d i m F = i}l , and call ( f - x , . . . , fd-1) the f-vector of A. Then d-1

r(k[A],~) = E

fiAi+l

ho + h i A + "'" + hd Ad

(1:-~-)i+]-

(1-A)~

'

i=--1

where the sequence ( h 0 , . . . , hd), called the h-vector of A, may be derived from the f-vector of A (and vice versa) by the equation

382

d

d

i=0

i=0

Art Dural MSC1991: 55U10, 05Exx, 13C14 Let G = GL~(Fq), the group of all invertible (n x n)-matrices over the finite field Fq with q elements and characteristic p, let B be the subgroup of all superdiagonal elements, let U be the subgroup of elements of B whose diagonal entries are all 1, and let W be the subgroup of permutation matrices. In the g r o u p a l g e b r a k[G] of G over any field k of characteristic 0 or p, the element STEINBERG

The support of any monomial in k[A] is a face of A. In particular, the square-free monomials of k[A] correspond bijectively to the faces of A, and are therefore called the faee-monomials

xF :~ E

[1] STANLEY, R.: Combinatorics and commutative algebra, second ed., Birkhguser, 1996.

e =

MODULE

-

IYl

is an idempotent, called the Steinberg idempotent, and the G-module that it generates in k[G] by right multiplication is called the Steinberg module (see [8]) and is commonly denoted St (as are all modules isomorphic to it). A similar construction holds for any finite g r o u p G of Lie type (and for any BN-pair, which is an axiomatic generalization due to J. Tits) defined over a field of characteristic p with B replaced by a B o r e l s u b g r o u p (which is a certain kind of solvable subgroup), U by a maximal unipotent subgroup (cf. U n i p o t e n t g r o u p ) of 13 (which is also a Sylow p-subgroup of G; cf. also S y l o w s u b g r o u p ; p - g r o u p ) and W by the corresponding W e y l g r o u p . St is always irreducible and it has {eu: u E U} as a basis, so that its dimension is IUI (see [8]). Its character values are given as follows [3]. If x E G has order prime to p, then X(X) equals, up to

S T E I N B E R G SYMBOL a sign which can be determined, the order of a Sylow p-subgroup of the centralizer of x; otherwise it equals 0. In case the characteristic of k equals p, St has the following further properties [5]. It is the only module (for G) which is both irreducible and projective. As an irreducible module it is the largest (in dimension), and as a projective module it is the smallest since it is a tensor factor of every projective module. It follows that it is also self-dual and that every projective module is also injective and vice versa. Because of these remarkable properties, St plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible G-modules (with characteristic k still equal to p), or equivalently, as it turns out, of determining all of the irreducible rational G-modules, where G is the a l g e b r a i c g r o u p obtained from G by replacing Fq by its algebraic closure ffq, i.e., where G is any simple affine algebraic group of characteristic p (see [6]). This equivalence comes from the fact that every irreducible G-module extends to a rational G-module. In particular, St extends to the G-module with highest weight q - 1 times the sum of the fundamental weights, which is accordingly also denoted St, or Stq since there is one such G-module for each q = p, p 2 , p 3 , . . . . These modules are ubiquitous in the module theory of G and figure prominently, for example, in the proofs of many cohomological vanishing theorems and in W. Haboush's proof of the M u m f o r d h y p o t h e s i s (see [4]). Back in the finite case, some other constructions of St, with the characteristic of k now equal to 0, are as follows. According to C.W. Curtis [2] St = E + I ~ , P

in which P runs through the 2 r (r equal to the rank of G) (parabolic) subgroups of G containing B, 1~ is the G-module induced by the trivial P-module, and the + or - is used according as the rank r p of P is even or odd. For G = GL~(Fq), for example, there is one P for each solution of n = al + ..- + as (1 <__ s < n, each ai ~ 1); it consists of all of the elements of G that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [7], yields St as the top homology space H r - l ( C ) for the Tits simpIicial complex or T i t s b u i l d i n g C of G, formed as follows: corresponding to each parabolic subgroup P there exists an (r - r p - 1)-simplex Sp in C, and S p is a facet of SQ just when P contains Q. These three constructions are, in fact, closely related to each other (see [9]). In particular, the idempotent e used at the start can be identified with an (r - 1)-sphere in the Tits building, the sum over W corresponding to a decomposition of the sphere into simplexes: in the usual

action of W o n S r - 1 C R r the reflecting hyperplanes divide S r-1 into [W[ oriented spherical simplexes, each of which is a f u n d a m e n t a l d o m a i n for W. Finally, St has a simple presentation (as a linear space). It is generated by the Borel subgroups of G subject only to the relations that for every parabolic subgroup of rank I the sum of the Borel subgroups that it contains is 0. There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups - - real, complex or p-adic - - such as GLn. The p-adic case most closely resembles the finite case. There, the affine Weyl group and a certain compact-open subgroup, called an Iwahori subgroup, come into play (in place of W and B), and the three constructions agree. In [1] several types of buildings, Curtis' formula and the Steinberg idempotent, in the guise of a homology cycle, all appear. In the infinite case the constructed object is sometimes called the Steinberg representation, sometimes the special representation. References [5] and [9] are essays on St. References [1] BOaEL, A., AND SERRE, J-P.: 'Cohomologie d'immeubles et de groupes S-arithm~tiques', Topology 15 (1976), 211-232. [2] CURTIS, C.W.: 'The Steinberg character of a finite group with BN-pair', J. Algebra 4 (1966), 433-441. [3] CURTIS, C.W., LEHRER, G.I., AND TITS, J.: 'Spherical buildings and the character of the Steinberg representation', Invent. Math. 58 (1980), 201-220. [4] HABOUSH, W.: 'Reductive groups are geometrically reduc-

tive', Ann. of Math. 102 (1975), 67-83. J.E.: 'The Steinberg representation', Bull. Amer. Math. Soc. (N.S.) 16 (1987), 237-263. [6] JANTZEN, J.C.: Representations of algebraic groups, Acad. Press, 1987. [7] SOLOMON,L.: 'The Steinberg character of a finite group with BN-pair': Theory of Finite Groups (Harvard Syrup.), Benjamin, 1969, pp. 213-221. [8] STEINBERG, R.: 'Prime power representations of finite linear groups II', Canad. g. Math. 9 (1957), 347-351. [9] STEINBERG, R.: Collected Papers, Amer. Math. Soc., 1997, pp. 580-586. Robert Steinberg MSC 1991:20G05 [5] HUMPHREYS,

Let G be the g r o u p SLn(F) (n >_ 3, F any field). (Much of what follows holds for arbitrary simple algebraic groups, not just for SLn.) For i , j = 1 , . . . , n , i ~ j , a C F, let Xiy(a) denote the element of G which differs from the identity matrix only in the (i, j)-entry, which is a rather than 0. The following relations hold for all (i, j) as above and a, b E F: STEINBERG

SYMBOL

a) z~j(a)x~j(b) = x~j(a + b) (xij(a), xae(b)) =

-

b);

1 xie(ab)

ifiCe, ifi ¢ g,

jCk, j = k.

383

S T E I N B E R G SYMBOL Here, (x, y) denotes the commutator x y x - l y -1. R. Steinberg [4] proved that if H denotes the abstract group defined by these generators and relations and 7r is the resulting h o m o m o r p h i s m of H onto G, then 7r : H --+ G is a universal central extension of G: its kernel is central and it covers all central extensions uniquely (cf. also E x t e n s i o n o f a g r o u p ) . It follows that every p r o j e c t i v e r e p r e s e n t a t i o n of G lifts uniquely to a line a r r e p r e s e n t a t i o n of H, and, at least when F is finite, that Ker 7r is just the S c h u r m u l t i p l i c a t o r of G, which was the motivation for Steinberg's study. Now, in the group H, let x(a) = x12(a), y(a) =

x21(a), w(a) = x(a)y(-a-1)z(a), h(a) = w(a)w(1) -1 and finally {a, b} = h(ab)h(a)-lh(b) -1 for all a, b E F*, the group of units of F. Since 7rh(a) works out to the matrix diag(a, a -1, 1, 1,...), it follows that {a, b} is always in Ker 7r. As is mostly shown in [4], these elements generate Ker 7r and they satisfy: c) {a, b} is multiplicative as a function of a or of b; d) {a,b} = 1 i f a + b = 1 (and a,b E F*).

Matsumoto's theorem [2] states that c) and d) form a presentation of KerTr. Thus, KerTc is independent of n _> 3 and hence may be (and will be) written K~F. The symbol {-,.} is called the Steinberg symbol, as is also any symbol in any A b e l i a n g r o u p A for which c) and d) hold (which corresponds to a homomorphism of K2F into A). As a first example, if F is finite, then K2F is trivial, with a few exceptions (see [4]). Hence a) and b) form a presentation of SLy(F) (n > 3) and ~r, as above, is an isomorphism. If F is a d i f f e r e n t i a l field, then {a, b} = da/a A db/b defines a symbol into A2F. Consider next the field Q and its completions R and Qp (one for each prime number p), which are topological fields (cf. also T o p o l o g i c a l field). According to J. Tate (see [3]), K2Q = H #P' p

where pp is the group of roots of unity in Q;, which is cyclic, of order 2 if p = 2 and of order p - 1 if p is odd. The factor for p odd arises from the symbol {a,b}p = ( - 1 ) ~ r ~ s ~ on Q;, and hence also on Q, in which a, b = p~r, pgs, with r, s units in Zp. Since {., .}; generates the group of continuous symbols on Qp into C* [3], one of the interpretations of this result is that the f u n d a m e n t a l g r o u p of SL~(Qp) is cyclic of order p - 1. And similarly for p = 2. For K 2 R one again gets the group of roots of unity, generated by {a, b}oo, which is - 1 if a and b are both negative and is 1 otherwise. Fitting {a, b}o~ into Tate's formula above is the last step in a beautiful proof by him (see [3]) of Gauss' quadratic 384

reciprocity law (cf. also Q u a d r a t i c r e c i p r o c i t y law). All of these ideas (as well as the norm residue symbol, for which c) and d) also hold) figure in a deep study of the group SLn (and other groups) over arbitrary algebraic number fields and their completions initiated by C. Moore and completed by H. Matsumoto in [2]. The definition of K2 has been extended by J. Milnor [3] to arbitrary commutative rings R as follows. Let G = E(R) denote the group of (oe × ec)-matrices over/~ generated by the matrices xij (') defined earlier, but with no upper bound on i or j. The relations a) and b) continue to hold and they again define a universal central extension, whose kernel is called K2R. The motivation comes from algebraic K-theory, where this definition fits in well with earlier definitions of KoR and K1R (see [3]) via natural exact sequences, product formulas and so on. The Steinberg symbol {a, b} still exists, but only if a and b commute and are in _R*. For some rings there are enough values of {.,-} to generate K2R, e.g., for R = Z (in which case K2R is of order 2 generated by { - 1 , - 1 } ) , or for any semi-local ring or for any discrete valuation ring (in which case R.K. Dennis and M.R. Stein [1] have given a complete set of relations, which include c) and d) above). For other rings, new symbols are needed. The Dennis-Stein symbol is defined by

(a,b)

=

= y ( - b ( 1 + ab)-l)x(a)y(b)x(-(1 + ab)-la)h(1 + ab) -1 for all commuting pairs a, b E R such that 1 + ab C R*. There are various identities pertaining to (-,-) and connecting it to {.,.}. These symbols, and yet others not defined here, have been used to calculate K2R, or at least to prove that it is non-trivial, for many rings arising in K-theory, number theory, algebraic geometry, topology, and other parts of mathematics. References [1] and [3] give good overall views of the subjects discussed. References [1] DENNIS, R.K., AND STEIN, M.R.: 'The functor K2: A survey of computations and problems': Algebraic K-Theory II, Vol. 342 of Lecture Notes in Mathematics, Springer, 1973, pp. 243-280. [2] MATSUMOTO, H.: 'Sur Ies sous-groupes arithm~tiques des groupes semisimples d@loyfis', Ann. Sci. l~cole Norm. Sup.

(4) 2 (1969), 1-62. [3] MILNOR, J.: Introduction to algebraic K-theory, Vol. 72 of Ann. of Math. Stud., Princeton Univ. Press, 1971. [4] STEINBEaG, R.: 'G~n~rateurs, relations et rev~tements de groupes alg~briques': Colloq. Thdorie des Groupes Algdbriques (Bruxelles, 1962), Gauthier-Villars, 1962, pp. 113-127.

Robert Steinberg MSC 1991: 19Cxx

STOKES PARAMETERS STEINER P R O B L E M - See S t e i n e r t r e e p r o b l e m .

MSC1991: 05C35, 51M16 STEINNESS - The property of a manifold or domain to be Stein (cf. S t e i n m a n i f o l d ; S t e i n s p a c e ) .

M S C 1991:32E10 STEP HYPERBOLIC CROSS A summation domain of multiple F o u r i e r series. Like a h y p e r b o l i c cross, it is used for good approximation in the space of functions with bounded mixed derivative (in Lp). Let f ( x ) be an integrable periodic function of n variables defined on T ~. It has a Fourier series expansion ~ k ekeikx, k = ( k l , . . . , k s ) , x = ( x l , . . . , x ~ ) , k . x = klXl +" • " + k~xn. Unlike in the one-dimensional case, there is no natural ordering of the Fourier coefficients, so the choice of the order of summation is of great importance. Let r = ( r l , . . . , r ~ ) E R ~ with all coordinates positive, rj > 0. Let

Am(f) =

E

Ckeikx

STOKES PARAMETERS - To characterize the radiance (intensity) or flux and state of polarization of a b e a m of electromagnetic radiation (cf. also E l e c t r o m a g n e t i s m ) one can use four real parameters which have the same physical dimension. These so-called Stokes parameters were first introduced by G.C. Stokes [7] in 1852. It took about a hundred years before Stokes parameters were used on a large scale in optics and theories of light scattering by molecules and small particles. (See, e.g., [1], [2], [3], [4], [5], [6], [8].) To define the Stokes parameters, I, Q, U, and V, one first considers a monochromatic b e a m of electromagnetic radiation. One defines two orthogonal unit vectors 1 and r such t h a t the direction of propagation of the beam is the direction of the vector product r x 1. The components of the electric field vectors at a point, O, in the b e a m can be written as =

sin(

t -

j~l,...,n

be a dyadic 'block' of the Fourier series. The step hyperbolic partial sums u

where introduced by B. Mityagin [2] for problems in a p p r o x i m a t i o n t h e o r y . They have approximately the same number of harmonics as a hyperbolic cross, but structurally they fit the Marcinkiewicz multiplier theorem (cf. also I n t e r p o l a t i o n o f o p e r a t o r s ) . It implies t h a t the operator of taking step hyperbolic partial Fourier sums is bounded in each L p, 1 < p < oc. This means that step hyperbolic partial sums give the best approximation among all hyperbolic cross trigonometric polynomials in Lp, 1 < p < co. In the limit cases p = 1 and p = co, the L e b e s g u e c o n s t a n t s of step hyperbolic partial sums have only logarithmic growth, while for hyperbolic partial Fourier sums they grow as a power of N. References [1] BELINSKY, E.S.: 'Lebesgue constants of 'step-hyperbolic' partial stuns': Theory of Functions and Mappings, Nauk. Dumka, Kiev, 1989. (In Russian.) [2] MITYAGIN,B.S.: 'Approximation of functions in L p and C spaces on the torus', Mat. Sb. (N.S.) 58 (100) (1962), 397 414. (In Russian.) [3] TEMLYAKOV,V.: Approximation of periodic functions, Nova Sci., 1993. E.S. Belinsky M S C 1991: 42B05, 42B08

sin(

t -

(1)

where w is the circular frequency, t is time, and ~ and ~o are (non-negative) amplitudes. One now defines the Stokes parameters by

2mJ-l
E am(f) Im-rl
=

± = [ o]2 + [ o]2

(2)

Q =

(3)

=

_ ° cos(

-

V = 2~°~ ° s i n ( e / - er).

(4)

(5)

The end point of the electric vector at a point, O, in the b e a m describes an ellipse, the so-called polarization ellipse, whose ellipticity and orientation with respect to 1 and r follow from Q, U and V. If V > 0, the electric vector at O moves clockwise, as viewed by an observer looking in the direction of propagation. Clearly, the following relation holds: i : (Q2 -4- U 2 --~ V2) 1/2,

(6)

where V = 0 for linearly polarized radiation and Q = U = 0 for circularly polarized radiation. In general, electromagnetic waves are not exactly monochromatic, but the amplitudes ~o and ~ , as well as the phase differences et - e ~ , m a y vary slowly in time. In this case the Stokes p a r a m e t e r s are defined as before, with one exception, namely time averages must be taken on the right-hand sides of (2)-(5). The polarization may now be partial and the beam can be decomposed in a completely unpolarized and a completely polarized beam. The orientation and shape of the polarization ellipse of the latter beam is again given by Q, U and V. The identity of (6) is now replaced by the inequality I k ( 0 2 ~- U2 -~- V2) 1/2

(7) 385

STOKES PARAMETERS and t h e ratio p = [Q2 + U 2 + V2]S/2/i is called the degree of polarization. For completely polarized radiation p = 1, for p a r t i a l l y polarized radiation 0 < p < 1, and for unpolarized ( n a t u r a l ) radiation p = 0. T h e Stokes p a r a m e t e r s can be combined into a colu m n vector with e l e m e n t s I, Q, U, and V, called a Stokes vector. Stokes vectors of constituent b e a m s are added to o b t a i n the Stokes vector of a composite b e a m if no interference effects occur. Optical devices and processes like scattering and a b s o r p t i o n can be described by real 4 x 4 (Mueller) m a t r i c e s t h a t t r a n s f o r m the Stokes vectors of p r i m a r y b e a m s into those of secondary beams. T h e Stokes p a r a m e t e r s as defined above are one of m a n y possible r e p r e s e n t a t i o n s of polarized radiation, several of which are only slight modifications of each other (see, e.g., [1]). Stokes p a r a m e t e r s are also used in q u a n t u m mechanics in connection with polarization of e l e m e n t a r y particles•

References [1] BORN, M., AND WOLF, E.: Principles of optics. Electromagnetic theory of propagation, interference and diffraction of light, sixth ed., Pergamon, 1993. [2] C H A N D R A S E K H A R , S.: Radiative transfer, Oxford Univ. Press, 1950, Reprint: Dover 1960. [3] DOLG:NOV, A.Z., GNEDIN, Yu.N., AND SILANT'EV, N.A.: Propagation and polarization of radiation in cosmic media, Gordon & Breach, 1995. [4] HOVENIER,J.W., AND MEE, C.V.M. VAN DER: 'Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere', Astron. Astrophys. 128 (1983), 1-16. [5] H U L S T , H.C. VAN DE: Light scattering by small particles, Wiley, 1957, Reprint: Dover 1981. [6] M:SHCHENKO, M.I., HOVENIER, J.W., AND TRAVlS, L.D.: Light scattering by nonspherical particles. Theory, measurements, and applications, Acad. Press, 2000. [7] STOKES,G.C.: 'On the composition and resolution of streams of polarized light from different sources', Trans. Cambridge Philos. Soc. 9 (1852), 399-416. [8] U L A B Y , F.T., AND ELACHI, C.: Radar polarimetry for geoscience applications, Artech House, Boston, 1990. C.V.M. van der Mee or. W. Hovenier MSC 1991:78A40

Let Ap,q be the c o m p l e x linear space spanned by the set of functions {zJ}~=v with p < q, and define A2,~ = A _ , ..... and h2,~+: = A-(,~+l),m for m = 0, 1 , . . . , and A = Ua~=0An. An element of A is called a Laurent polynomial. For a given sequence { C n } aoo= - o o , a necessary and sufficient condition for the strong Stieltjes m o m e n t p r o b l e m to be solvable is t h a t the l i n e a r o p e r a t o r M defined on the base elements z a of A by

M[z a ] = c a ,

n=0,±1,±2,...,

is positive on (0, e c), i.e. for any L C A such t h a t L(z) > 0 for z e (0, oc) and L(z) ~ O, then M[L] > O. An equivalent condition is t h a t if

H~ rn) = 1,

H~ m) = det(cm+i+j)ki,j-lo

ca =

//

t ~ de(t),

n = 0,-t-1,:t=2,....

(1)

This problem, which generalizes the classical Stieltjes moment problem (where the given sequence is {c,~}n~__0; cf. also K r e ~ n c o n d i t i o n ) , was first studied by W.B. Jones, W.J. T h r o n and H. W a a d e l a n d [3]. 386

(3)

for m = 0, +1, -t-2,..., k = 1, 2 , . . . , are the Hankel determ i n a n t s associated with {ck } (cf. also H a n k e l m a t r i x ) , then H~'~)>0,

m=0,-t-1,+2,...,

k=l,2,....

(4)

O r t h o g o n a l L a u r e n t p o l y n o m i a l s {Qn(z) E A n : n = 0, 1 , . . . } m a y be defined with respect to the i n n e r p r o d u c t (P, Q} = M [ P ( z ) Q ( z ) ] and are given by: C--2a

" " "

C--1

,

Z -n

.

.

z **2a

e--i

" " "

co

•

.

,

(5)

C2n--2 C2n_

.

1

Zn

n= 1,2,..., and C_2n_

O2a+,(z)

-

-1

:

H(-2n)

C_ 1

2n+l

1

• ..

C_ 1

Z -a-1

:

•

go

.

C2n_

.

• ••

: 1

Z n-1

C2n

Z a

(6)

n=0,1,..., and Qo(z) = 1. C o r r e s p o n d i n g associated orthogonal Laurent polynomials {P~} are defined by

= STRONG STIELTJES MOMENT PROBLEM- T h e strong Stieltjes m o m e n t p r o b l e m for a given sequence { C a } aoo= - ~ of real n u m b e r s is concerned with finding real-valued, bounded, m o n o t o n e non-decreasing functions ¢(t) with infinitely m a n y points of increase for 0 < t < ec such t h a t

(2)

,

n=O,1,....

T h e rational functions ( - z ) P n ( - z ) / Q a ( - z ) convergents of the positive T-fraction [5],

F:z

&z

Fa

1 + G~z+ 1 + G2z+ 1 + Gaz+ ( & > 0 , a ~ > O), where

H ( - n~r(-n+3) ) H ( - n + 2 ) rz(-n+: , n--1

Gr~

~*n--1

H(-~)u(-a+2) H(-n4-1) 14(-n+l) ' **n--1

(7) are the

(S)

STURM-LIOUVILLE THEORY which corresponds to the formal pair of power series, Lo = -

c _ k ( - z ) k,

L+ = E

k:l

ck(--z)-k.

(9)

k=0

The T-fraction is equivalent to the c o n t i n u e d f r a c t i o n Z

Z

Z

el + d l z + e 2 + d 2 z + e 3 + d 3 z + " "

'

(10)

where Fn-

1 enen-1

,

d~ Gn=-en

(e0:l),

(11)

A.K. Common

n = 1, 2, . . . .

MSC 1991:44A60

The following result may then be proved [3]: The solution of the strong Stieltjes moment problem (1) is unique if and only if at least one of the series ~ en, ~ d,~ diverges, and then lim++ ( - Z ) Q , ~ ( _ z ) j = z

zTt'

where ¢(t) is this unique solution.The convergence is uniform on every compact subset of R = {z: [arg z I < 7r}. The strong Stieltjes moment problem is said to be determinate when it has a unique solution and indeterminate otherwise. A detailed discussion of the latter case has been given in [6]. A classic example of a strong Stieltjes moment problem is the log-normal distribution, de(t)

.1/2

2

-- : N v ~ C -(lnt/2t~) ,

q = C-2n~.

(13)

(Cf. also N o r m a l d i s t r i b u t i o n . ) The corresponding sequence of moments is {ca}, where cn = q -n-n2~2,

n = O, -t-1, =t=2,...,

(14)

and the strong Stieltjes moment problem in this case is indeterminate [2]. The moments corresponding to the log-normal distribution are related to a subclass of strong Stieltjes moment problems where e - n = Ca,

[4] JONES, W.B., NJ]~STAD, O., AND THRON, W.J.: 'Continued fractions and strong Hamburger moment problems', Proc. London Math. Soc. 47 (1983), 105-123. [5] JONES, W.B., AND THRON, W.J.: Continued fractions: Analytic theory and applications, Vol. 11 of Encycl. Math. Appl., Addison-Wesley, 1980. [6] NJ]~STAD, O.: 'Solutions of the strong Stieltjes moment problem', Meth. Appl. Anal. 2 (1995), 320-347. [7] SRI RANGA, A., ANDRADE, E.X.L. DE, AND MCCABE, J.: 'Some consequences of symmetry in strong distributions', J. Math. Anal. Appl. 193 (1995), 158-168.

n = 1,2,....

(15)

This subclass has been called strong symmetric Stieltjes m o m e n t problems by A.K. Common and J. McCabe, who studied properties of the related continued fractions [1]. Other subclasses have been investigated in [7]. Cf. also M o m e n t p r o b l e m . References

[1] COMMON, A.K., AND MCCABE, J.: 'The symmetric strong moment problem', d. Comput. Appl. Math. 67" (1996), 327341. [2] COOPER, S.C., JONES, W.B., AND THRON, W.J.: 'Orthogonal Laurent polynomials and continued fractions associated with log-normal distributions', J. Comput. Appl. Math. 32 (1990), 39-46. [3] JONES, W.B., NJASTAD, O., AND THRON, W.J.: 'A strong Stieltjes moment problem', Trans. Amer. Math. Soc. 261 (1980), 503-528.

STRONGLY COUNTABLY COMPLETE TOPOLOGICAL SPACE - A topological space X for which

there is a sequence {~4i} of open coverings of X such that a sequence {F/} of closed subsets of X has a nonempty intersection whenever Fi D Fi+l for all i and each F/ is a subset of some member of Ai. Locally countably compact spaces and Cech-complete spaces are strongly countably complete. Every strongly countably complete space is a B a i r e space (but not vice versa). This rather technical notion plays an important role in questions whether separate continuity of a mapping on a product X × Y implies joint continuity on a large subset of X × Y, see N a m i o k a space; N a m i o k a t h e orem; S e p a r a t e a n d j o i n t c o n t i n u i t y ; or [2]. Strongly countably complete topological spaces were introduced by Z. Frolik, [1]. References [1] FROLIK, Z.: 'Baire spaces and some generalizations of complete metric spaces', Czech. Math. J. 11 (1961), 237-248. [2] NAMIOKA, I.: 'Separate continuity and joint continuity', Pacific J. Math. 51 (1974), 515-531.

M. Hazewinkel

MSC 1991: 54C05, 54C08 S T S , Steiner triple system -

See S t e i n e r s y s t e m .

MSC1991: 05B05, 05B07, 51E10 S T U R M - L I O U V I L L E THEORY - Sturm-Liouville problems (cf. S t u r m - L i o u v i l l e p r o b l e m ) have continued to provide new ideas and interesting developments in the spectral theory of operators (cf. also S p e c t r a l theory). Consider the Sturm-Liouville differential equation on the half-line 0 < x < oc, in its reduced form

- y " + q(x)y = ~y,

(1)

where )~ is the complex spectral parameter and the realvalued function q(x) is assumed to be integrable over any finite subinterval of [0, ec). The time-independent 387

STURM-LIOUVILLE THEORY S c h r S d i n g e r e q u a t i o n , at energy A, for a particle having fixed angular momentum quantum numbers moving in a spherically symmetric potential, may be written in the form (1) - - hence there are numerous applications to quantum mechanics ([13], [15]). Suppose the end-point x = +oo is a limit point. This holds in almost all applications and is valid, for example, if either q is bounded or if q satisfies the inequality q(x) >_ - c x 2 for some positive constant e. Let T denote the second-order differential operator T = - d 2 / d x 2 + q(x), defined as a s e l f - a d j o i n t o p e r a t o r in L~(0, ec), subject to the Dirichlet boundary condition y(0) = 0 (cf. also Linear ordinary differential e q u a t i o n o f t h e s e c o n d o r d e r ) . (Other boundary conditions may be considered - - in general, there is a oneparameter family of boundary conditions (cos a)y(0) + (sin oz)y'(0) = 0,

(2)

with the real parameter a varying over the interval 0 <_ a <~-.) The eigenvalues of the Sturm Liouville operator T may be characterized as those A E R for which the differential equation has a (non-trivial) solution y(x, A) satisfying both the boundary condition y(0, A) = 0 and the L 2 condition f o [Y(X,A)[2 dx < oc. The solution y(-, A) will always be locally square-integrable, and the L 2 condition is a restriction on the large-x asymptotic behaviour of this function. It follows, therefore, that the set of discrete points of the spectrum of T (cf. also S p e c t r u m o f a n o p e r a t o r ) is governed by the asymptotic behaviour of appropriate solutions of (1). Such considerations, which link asymptotic behaviour of solutions y(x, A) of (1) to spectral properties of the SturmLiouville operator T, may be extended to other parts of the spectrum of T, and provide a powerful tool of

spectral analysis. As a preliminary straightforward application of this general idea, one may use a Weyl sequence of approximate eigenfunctions of T to show that if q(x) -+ 0 as z ~ co; then the entire positive line [0, oc) belongs to the essential spectrum of T, the essential spectrum consisting of all points of the spectrum of T apart from isolated eigenvalues. In contrast, if q(x) --+ +oc as x -+ oe, then no A > 0 belongs to the essential spectrum. In order to carry out a more detailed spectral analysis of the Sturm-Liouville operator T, one has to consider the spectral measure /A associated with T, as well as its spectral decomposition. One of the most convenient ways to do this is through the Weyl rnfunction for T ([19], [3], [2], [7]), here denoted by rn0(A) (cf. also T i t c h m a r s h - W e y l m - f u n c t i o n ) . The mfunction rno(A) for the DirichletSturm-Liouville operator - d 2 / d x 2 + q(x) on the half-line is uniquely defined 388

for Im A > 0 by the condition that ¢(., A) + rno(A)0(., A) E L2(0, co),

(3)

where ¢, 0 are solutions of (1), subject, respectively, to the conditions ¢}0, A) = 1,

O(O,A) = 0,

¢(O,A) = 0 ,

0'(0, A) = 1 .

(4)

The function rn0(A) is an a n a l y t i c f u n c t i o n of A in the upper half-plane, and has strictly positive imaginary part. Such functions are called Herglotz functions, or Nevanlinna functions. (Corresponding to a general boundary condition, as given by (2), one can define in a similar way an m-function ms(A), which is again a Herglotz function, to which the theory outlined below applies with minor modifications.) As a Herglotz function, too(A) has a representation of the form ([1]; cf. also H e r g l o t z f o r m u l a ) rn0(A) = A +

I_-(1 o~ t - A

' )..o(,I, (,)

t2+1

valid for all A E C with Im A > 0. (Actually, for a general Herglotz function, a term BA, linear in A with B >_ 0, must be added on the right-hand side, but the asymptotics of m-functions imply that here B = 0 [1].) In (6), A = Rern0(i) is a positive constant and P0 is the s p e c t r a l f u n c t i o n for the problem (1) with Dirichlet boundary condition at x = 0. The spectral function may be taken to be non-decreasing and right continuous, in which case P0 is defined by (5) up to an additive constant, for a given m-function rn0(A). The measure # = dpo, defined on Borel subsets of R, is called the spectral measure associated with the Dirichlet problem. The Lebesgue decomposition theorem (cf. L e b e s g u e t h e o r e m ) leads to a decomposition of the spectral measure into the sum of a part absolutely continuous with respect to Lebesgue measure and a singular part, i.e. # = #ac + #s,

(6)

where #s may be further decomposed into its singular continuous and discrete components, thus /AS = /ASC -~- /Ad.

(7)

The R a d o n - N i k o d : ~ m t h e o r e m implies that the absolutely continuous part/Aac of the spectral measure may be described by means of a density function f(A), given at (Lebesgue) almost all A C R by f(A) = dp(A)/dA; thus, for Borel subsets A of R one then has/A~c(A) = fA f(A) dA. The support of the singular component /As will be a set B C_ R having L e b e s g u e m e a s u r e zero. The discrete part/Ad is supported on the set of eigenvalues of the Sturm-Liouville operator T (cf. also Eigen value). These may be characterized as the points A C R

STURM-LIOUVILLE THEORY for which p({A}) > 0, and alternatively as the points of discontinuity of the spectral function p(A). For many physical applications of the S t u r m Liouville problem (1), the spectrum of the associated differential operator with Dirichlet boundary condition is either purely discrete (e.g. if q(x) = x2), or purely absolutely continuous (e.g. if q(x) >_ 0 and q E LI(0, oc)), or a combination of discrete and absolutely continuous spectrum (e.g. if q C Ll(0, oo) and T = - d 2 / d x 2 + q(x) is not a positive operator). However, solution of the inverse Sturm-Liouville problem (cf. also S t u r m L i o u v i l l e p r o b l e m , inverse), which leads to the determination of a function q(x) from its spectral measure #, shows that other types of spectra, including for example combinations of absolutely continuous, singular continuous and discrete spectra, are possible. In view of the generality of types of spectral behaviour, mathematicians have sought ways of further characterizing the spectral properties of Sturm-Liouville operators which will apply to a wide range of cases. The supports of the various components of the spectral measure may be characterized in terms of the boundary behaviour of the m-function m0(A). For (Lebesgue) almost-all A E R, define the boundary value function m+ (A) by m+(A) = lim m(A + ie). e--~0q-

(8)

Here m+(A) exists as a finite limit for almost-all A E R, and one defines m+(A) = ec whenever lim¢~0+ Im m+(A) = oo. Then: i) the set of all A C R at which m+ (A) exists and is real and finite, has zero p-measure; ii) #ac is supported on the set of all A E R at which m+(A) exists finitely, with Imm+(A) > 0; the density function for the measure #ac is then (1/7r) Imm+(A); iii) Ps is supported on the set of all A C R at which m + ( a ) = oc.

subordinate solution of (i) exists, at real spectral parameter A, if and only if either a) m+(A) exists and is real and finite (in which case the solution ¢(., ),) + rn+(A)O(., A) is subordinate); or b) m+(A) = ec (in which case 0(., A) is subordinate). In particular, the singular component Ps of the spectral measure is concentrated on the set of A E R at which the solution 0(., A) is subordinate, and Pac is concentrated on the set of A E R at which there is no subordinate solution. Recent developments (as of 2000) of the idea of subordinacy have led [11] to further refinements of the analysis of singular spectra, in which the H a u s d o r f f d i m e n sion of the spectral support plays a significant role. The use of subordinacy and other techniques of spectral analysis have led to a deeper understanding of spectral properties for Sturm-Liouville operators in terms of the large-x behaviour of solutions of the Sturm-Liouville equation (1). Of course there still remains the problem of analyzing the large-x asymptotics of solutions of (1). However, advances in asymptotic analysis have led to the successful treatment of an ever widening class of Sturm-Liouville spectral problems. Examples of some of the most significant classes of function q(x) that can be handled in this way are as follows.

q integrable plus function of bounded variation. (For this case plus a more general treatment of asymptotics of solutions of systems of differential equations, see [6].) Suppose one can write q(x) = qx(x) + q2(x), where ql is continuous and of bounded variation, and q2(') E L 1 (0, oc). Suppose also, for simplicity, that qi(x) --+ 0 as x --+ oc. Then, for A > 0, the W K B m e t h o d leads to solutions y(x,A) of (1), having the asymptotic behaviour, as x --+ oc, y ~ acos

I x( A - V ~ ( t ) ) l / 2 d t + b s i n F (A-Vz(t))l/2dt. C

These supports can also be characterized in terms of large-x asymptotics of solutions of (1), by using the notion of subordinacy. A non-trivial solution y(x, A), for given A E R, is said to be subordinate if the norm of y(., A) in L2(0, N) is much smaller, in the limit N --~ oc, than that of any other solution of (1) that is not a constant multiple of y(., A). That is, y(., A) is subordinate if, for any other solution v(., A) linearly independent of y(., A), one has (see [9], and [8] for extensions to operators with two singular end-points) lim f°N [y(x, f0N

dx = O. dx

Then the following result holds, linking subordinacy with boundary behaviour of the m-function, and thereby to the spectral analysis of Sturm-Liouville operators: A

JC

Asymptotics for A < 0 lead to exponential growth or decay of solutions. The spectrum is purely absolutely continuous for A > 0 and purely discrete for A < 0.

Example of eigenvalues embedded in continuous spectrum. ([16]) In (1), with A = 1, let sin x

=

1+ (2x - sin2x) 2"

A simple calculation then shows that q(x) -

- 8 sin 2x -

-

+ 0(x -2)

X

as x -+ oo. This solution y(x, A) is an eigenfunction of the Dirichlet operator T, with eigenvalue A = 1. One may verify that the interval [0, oo) belongs to the absolutely continuous spectrum in this example. 389

STURM-LIOUVILLE THEORY

q periodic. ([5]) Suppose that q satisfies q(x + L) = q(x) for some L > 0. Then the absolutely continuous spectrum consists of a sequence of disjoint intervals. The detailed location of these intervals is dependent on the particular function q, though general results can be obtained regarding the asymptotic separation of the intervals for large A.

continuous spectrum. Any subinterval A of [0, oe) will satisfy pat(A) > 0; this does not exclude the possibility of a subset B C_ A having Lebesgue measure zero with #s(B) > 0, and results have been obtained which further characterize the support of #s, for given q. Further extensions of some of these results to the more general case of q square integrable have been obtained (see [4]).

q almost periodic or random. There is an extensive literature (see, for example, [14]) on the spectral properties of - d 2 / d x 2 + q(x) with q either almost periodic or q a random function. Such problems can give rise to a singular continuous spectrum, or to a pure point spectrum which is dense in an interval. As an example of the latter phenomenon, on each interval (n, n + 1], with n = 0, 1,..., set q(x) = q~, where the qn are constant and distributed independently for different n, with (say) uniform probability distribution over the interval [0, 1]. Then, with probability 1, the Sturm-Liouville operator - d 2 / d x 2 + q(x) will have eigenvalues dense in the interval [0, 1].

Numerical approaches. (See, for example, [10] and references contained therein.)

q slowly oscillating. ([18]) A typical function of this type is given by q(x) = g c o s y ~ , where g is a constant. The function cos v/~ oscillates more and more slowly as x increases. One can show that, for almost all g, - d 2 / d x 2 +g cos v/~ has eigenvalues dense in the interval

[-g,g]. q a sparse function. ([17]) A typical function of this type may be defined by q(x) = En~=l f ( z - Xn), where f has compact support and the sequence {x~} is strongly divergent as n --+ oc. Such a function q will give rise to a singular continuous spectrum provided {Xn} diverges sufficiently rapidly. q slowly decaying. ([12]) A challenging problem in the spectral theory of Sturm-Liouville equations has been the analysis of the Dirichlet operator - d 2 / d x 2 + q(x) under the hypothesis that q satisfy a bound for sufficiently large x, of the form [q(x)[ <_ const/x ~, for some fl > 1/2. If additional conditions are imposed, for example appropriate bounds on the derivative of q (assuming q to be differentiable), then such functions q would fall under the category 'integrable plus function of bounded variation' considered above, for which a spectral analysis can be carried out. However, in the absence of further conditions on q, it is already clear from the example of an eigenvalue in the continuous spectrum above that one cannot prove absolute continuity of the spectrum for A > 0. In fact, for various q, a dense point spectrum or singular continuous spectrum may be present. A major advance in understanding this problem has been the proof [12] that, under the hypothesis of q locally integrable and Iq(x)[ < const/x z (fl > 1/2), the entire semi-interval [0, ec) is contained in the absolutely 390

Sophisticated software capable of treating an increasingly wide class of spectral problems has been developed. These numerical approaches, often incorporating the use of interval analysis and leading to guaranteed error bounds for eigenvalues, have been used to investigate a variety of limit point and limit circle problems, and to estimate the m-function and spectral density function for a range of values of A. References [1] AKHIEZER, N.I., AND GLAZMAN, I.M.: Theory of linear operators in Hilbert space, Pitman, 1981. [2] CHAUDHURY, J., AND EVERITT, W.N.: 'On the spectrum of ordinary second order differential operators', Proc. Royal Soc. Edinburgh A 68 (1968), 95-115. [3] CODDINGTON, E.A., AND LEVINSON, N.: Theory of ordinary differential equations, McGraw-Hill, 1955. [4] DEIFT, P., AND KILLIP, R.: 'On the absolutely continuous spectrum of one-dimensional SchrSdinger operators with square-snmmable potentials', Comm. Math. Phys. 203 (1999), 341-347. [5] EASTHAM,M.S.P.: The spectral theory of periodic differential operators, Scottish Acad. Press, 1973. [6] EASTHAM, M.S.P.: The asymptotic solution of linear differential systems, Oxford Univ. Press, 1989. [7] EASTHAM, M.S.P., AND KALF, H.: SchrSdinger-type operators with continuous spectra, Pitman, 1982. [8] GILBERT, D.J.: 'On subordinacy and analysis of the spectrum of SchrSdinger operators with two singular endpoints', Proc. Royal Soe. Edinburgh A 112 (1989), 213-229. [9] GILBERT, D.J., AND PEARSON, D.B.: 'On subordinacy and analysis of the spectrum of one-dimensional SchrSdinger operators', d. Math. Anal. Appl. 128 (1987), 30-56. [I0] HINTON, D., AND SCHAEFER, P.W. (eds.): Spectral theory and computational methods of Sturm-Liouville problems, M. Dekker, 1997. [11] JITOMIRSKAYA, S., AND LAST, Y.: 'Dimensional Hausdorff properties of singular continuous spectra', Phys. Rev. Lett. 76, no. 11 (1996), 1765-1769. [12] KISELEV, A.: 'Absolutely continuous spectrum of onedimensional SchrSdinger operators with slowly decreasing potentials', Comm. Math. Phys. 179 (1996), 377-400. [13] NEWTON, R.G.: Scattering theory of waves and particles, Springer, 1982. [14] PASTUR, L., AND FIGOTIN, A.: Spectra of random and almost periodic operators, Springer, 1991. [15] PRUOOVE~KI, E.: Quantum mechanics in Hilbert space, Acad. Press, 1981. [16] REED, M., AND SIMON, B.: Methods of modern mathematical physics: Analysis of operators, Vol. IV, Acad. Press, 1978.

SZEGI3 LIMIT T H E O R E M S [17] SIMON, B., AND STOLZ, G.: 'Operators with singular continuous spectrum: sparse potentials', Proc. Amer. Math. Soc. 124, no. 7 (1996), 2073-2080. [18] STOLZ, G.: 'Spectral theory for slowly oscillating potentials: Schrgdinger operators', Math. Naehr. 183 (1997), 275-294. [19] TITCHMARSH, E.C.: Eigenfunction expansions, Part 1, Oxford Univ. Press, 1962.

D.B. Pearson MSC 1991: 34B24, 34L40 SYSTEM

OF

PARAMETERS

OF

A

MODULE

Let (A,m) be an rdimensional Noetherian ring (cf. also the section 'Dimension of an associative algebra' in D i m e n s i o n ) . Then there exists an m-primary ideal generated by r elements (cf., e.g., [1, p. 98], [2, p. 27]). If X l , . . . , Xr generate such an m-primary ideal, they are said to be a system of parameters of A. The terminology comes from the situation that (A, m) is the local ring of functions at a (singular) point on an a l g e b r a i c v a r i e t y . The system of parameters x i , . . . , xr is a regular system of parameters if x l , . . . , xr generate m, and in that case (A, m) is OVER

A

LOCAL

RING

-

a regular local ring. More generally, if M is a finitely-generated A-module of dimension s, then there are y l , . . . , y ~ E m such that M / ( y i , . . . , y ~ ) M is of finite length; in that case yl,. •., Ys is called a system of parameters of M. The ideal ( Y i , . . . , Y~) is called a parameter ideal. For a semi-local ring A with maximal ideals m l , . . . , m~, an ideal a is called an ideal of definition if (ml n ' "

n m~) k _c a c_ (ml n - - . nm~)

for some natural number k. If A is of dimension d, then any set of d elements that generates an ideal of definition is a system of parameters of A, [3, Sect. 4.9]. References [1] MATSUMURA,H.: Commutative ring theory, Cambridge Univ. Press, 1989. [2] NAGATA, M.: Local rings, Interseience, 1962. [3] NOTHCOTT, D.G.: Lessons on rings, modules, and multiplicities, Cambridge Univ. Press, 1968.

M. Hazewinkel

For real positive functions a E L I ( T ) for which log a E L I(T), G. Szeg5 [8] has proved that det T~ (a) l i r a det T~-l(a) - G(a),

(1)

with the constant G(a) = exp([loga]0). Here, [loga]k stands for the kth Fourier coefficient of the logarithm of a. A statement of type (1) is referred to as a first Szeg5 limit theorem. Szeg6's result has been considerably extended. In particular, (1) holds for functions that are the exponentials of continuous complex-valued functions defined on the unit circle. The strong Szeg5 limit theorem states that det T~(a) = E(a), nli~Inoo G ( a ) n

(2)

with the constant E(a) defined by

E(a) = exp

- - kElog alkIlog a]_k i. k=l

]

r

Relation (2) was first proved by Szeg6 [9] for positive real functions whose derivatives satisfy a HSlder-Lipschitz condition. This result has been generalized too. For instance, the strong Szeg5 limit theorem holds for functions that are the exponentials of continuous and sufficiently smooth complex-valued functions defined on the unit circle. Such results about the asymptotics of Toeplitz determinants can be used to obtain information about the asymptotic distribution of the eigenvalues {A~n)}~=1 of the matrices T~(a). It turns out that

1 n

f(A~n))

~1 f02~ f(a(ei°)) dO + o(1),

(3)

k=l

as n -+ co, if, for instance, one of the following assumptions is satisfied: • a C L 1(T) is real-valued and f is a c o n t i n u o u s f u n c t i o n on the real line with a compact support [11]; • a is a continuous complex-valued function and f is an a n a l y t i c f u n c t i o n defined on an open neighbourhood of the set

MSC 1991: 13Hxx specT(a) = Ran(a) U {z ~ Ran(a): w i n d ( a - z) ~ 0). S Z E G O LIMIT T H E O R E M S - Let a be a complex-

valued function defined on the complex unit circle T, with F o u r i e r c o e f f i c i e n t s an = - ~1 ~0 2~ a(eiO)e_in 0 dO. Szeg5 limit theorems describe the behaviour of the determinants of the Toeplitz matrices T~ (a) = (aj-k)j,k=0,n-1 as n tends to infinity, for certain classes of functions a (cf. also T o e p l i t z m a t r i x ) .

Here, T(a) = (aj-k)~,k=o stands for the T o e p l i t z ope r a t o r acting on the H i l b e r t s p a c e t 2, spec T(a) refers to its spectrum (cf. also S p e c t r u m o f a n o p e r a t o r ) , Ran(a) stands for the range of the function a, and wind(a - z) denotes the w i n d i n g n u m b e r of the function a(e i°) - z . The asymptotic formula (3) is sometimes also called the first Szeg5 limit theorem or a first-order trace formula. A second-order trace formula, which is the pendant of the strong Szeg5 limit theorem, has also been established [5], [10]. 391

SZEGO LIMIT T H E O R E M S Some work was also done in order to determine the higher-order terms of the a s y m p t o t i c e x p a n s i o n of Toeplitz determinants [3]. Exact formulas for Toeplitz determinants in terms of the Wiener-Hopf factorization (cf. also W i e n e r - H o p f m e t h o d ; W i e n e r - H o p f ope r a t o r ) of the generating function a do also exist (see,

e.g., [2]). H. Widow [10] was the first to give a crystal clear proof of the strong Szeg5 limit theorem, by an elegant application of ideas from operator theory and thereby replacing earlier long-winded proofs. With his approach he was able to generalize this theorem to the case of matrix-valued functions. Under the assumption that a is a sufficiently smooth matrix-valued function defined on the unit circle for which det a is the exponential of a continuous function, (2) still holds, but with constants defined by G(a) = exp([logdeta]0) and E(a) = detT(a)T(a-1). The last expression has to be understood as an operator determinant. In this connection, the identity T(a)T(a -1) = I - H(a)H(~d -1) plays an oc important role, where H(a) = (a l+j+k)j,k=0 is a Hartkel o p e r a t o r and ~d(ei°) = a(e-i°). Note that for sufficiently smooth and invertible matrix functions a the operator H(a)H(~ -1) is a trace-class operator (ef. also N u c l e a r o p e r a t o r ) . An explicit expression for E(a) is not known yet (as of 2000), apart from special cases related to the scalar situation. On the other hand, an operator-valued version of the strong Szeg5 limit theorem has been established [4]. The asymptotic behaviour of Toeplitz determinants changes considerably if the function a is discontinuous. If a possesses zeros, poles, jumps, or certain oscillations, then the asymptotics is predicted by the Fisher-Hartwig conjecture or by the more general Basor Tracy conjecture. Let R

a(ei°) = v(ei°) I-[ r~l

Wc~,~(eiO) = (2 -- 2cosO)C~ei~(O-~r), 0 < 0 < 27r. Then the Fisher Hartwig conjecture [7] asserts that det T~ (a)

,~--+~ G(b),~n a R

- E,

where Ft = y ~ = l ( a ~ -/3~). An explicit, but more complicated expression is known for the constant E. It has turned out that in some cases the Fisher Hartwig conjecture breaks down. However, this conjecture has been proved in all the cases in which it is suspected to apply [5], [6]. It is believed that the Basor-Tracy conjecture 392

S Itl I~(t) l2 dt < oo

Then lira d e t ( I + W~.(k)) = E(a), with the constants G(a) = exp(g(0)) and

E(a) = exp ( fo~tg(t)~(-t) dt) . There are many further results for Wiener-Hopf determinants which are quite similar to those of the discrete

case [3], [5]. Finally, analogues of the Szeg5 limit theorem have also been established for multi-dimensional (i.e., multilevel) Toeplitz and Wiener-Hopf operators, for pseudodifferential operators, and in several abstract settings. Another direction deals with the asymptotic distribution of the singular values of the matrices T~(a), their analogues and generalizations. Results of such a type are called Avram Parter theorems [5]. References

where 0 1 , . . . , 0 R E [0, 27r) are distinct points, b is the exponential of a sufficiently smooth function and a~,/3~ are complex parameters. The function w~,~ is defined as

lim

[1], which is proved so far (2000) only in special cases, gives the correct answer for all cases. The continuous analogue of Toeplitz determinants are the determinants of truncated Wiener-Hopf operators (cf. also W i e n e r - H o p f o p e r a t o r ) . Let k be a complexvalued function in L I ( R ) N L ~ ( R ) defined on the real axis, and denote by k the F o u r i e r t r a n s f o r m of k. The i n t e g r a l o p e r a t o r defined on L2[0, T] with kernel k ( z - y) is called a truncated Wiener Hopf operator and denoted by W, (k). Under the above assumption, W~ (k) is a trace-class operator. The asymptotics of the operator determinants of I + Wr(k), as ~- --+ o% for certain classes of functions k is described by the Akhiezer-Kac formula, which is the continuous pendant of the strong Szeg5 limit theorem. Suppose a = 1 + k = exp(s), where s E L I(R) n L ~ ( R ) such that its Fourier transform belongs to L I(R) and

[1] B a s o a , E.L., AND TRACY, C.A.: 'The Fisher-Hartwig conjecture and generalizations', Phys. A 177 (1991), I67-173. [2] BASOR, E.L., AND WIDOM, H.: 'On a Toeplitz determinant identity of Borodin and Okounov', Integral Eq. Oper. Th. 37, no. 4 (2000), 397-401. [3] BOTTCHER, A., AND SILBERMANN, B.: Analysis of Toeplitz operators, Springer, 1990. [4] BOTTCttER, A., AND SILBERMANN, B.: 'Operator-valued SzegS-Widom limit theorems': Oper. Theory Adv. Appl., Vol. 71, Birkhguser, 1994, pp. 33 53. [5] B()TTCHER, A., AND SILBERMANN, B.: Introduction to large truncated Toeplitz matrices, Springer, 1998. [6] EHRHARDT, T.: 'Toeplitz determinants with several Fisher Hartwig singularities', PhD Thesis Techn. Univ. Chemnitz

(1997) [7] FISHER, M.E., AND HARTWIG, R.E.: 'Toeplitz determinants: Some applications, theorems and conjectures', Adv. Chem. Phys. 15 (1968), 333-353.

SZEGO POLYNOMIAL [8] SZEO6, G.: 'Ein Grenzwertsatz fiber die Toeplitzschen Determinanten einer reellen positiven Funktion', Math. Ann. 76 (1915), 490 503. [9] SZEO6, G.: 'On certain Hermitian forms associated with the Fourier series of a positive function', Comm. Sdm. Math. Univ. Lurid (1952), 228-238.

[10] WIDOM,H.: 'Asymptotic behavior of block Toeplitz matrices and determinants. II', Adv. Math. 21 (1976), 1-29. [11] Z A M A R A S H K I N , N.L., AND T Y R T Y S H N I K O V , E.g.: 'Distribution of eigenvalues and singular numbers of Toeplitz matrices under weakened requirements of the generating function', Mat. Sb. 188 (1997), 83-92. (In Russian.) T. Ehrhardt B. Silbermann

MSC 1991: 47B35, 42A16

1

SZEGO POLYNOMIAL - The Szeg5 polynomials form an orthogonal polynomial sequence with respect to the positive definite HermRian i n n e r p r o d u c t

//

(f, g) =

f(eW)g(e i°) d#(O),

7T

where p is a positive m e a s u r e on I-re, re) (cf. also O r t h o g o n a l p o l y n o m i a l s o n a c o m p l e x d o m a i n ) . The monic orthogonal Szeg5 polynomials satisfy a recurrence relation of the form

¢~+1(z) = ~¢~(~) + p~+a¢~(z), for n > 0, with initial conditions (I)o = 1 and (I)_l (z) = 0. Here, (I)~(z) = ~-~-;=0 b n k z n - k if d2n(Z ) = ~ = 0

bnkZk"

The parameter P~+I = (I)~+1(0) is called a reflection coefficient or Schur or Szeg5 parameter. Szega's extremum problem is to find a~ = minH IIHII,, with [IH[I, the L2(p)-norm and where the minimum is taken over all H ¢ H 2 ( p , D ) (D being the open unit disc) satisfying H(0) = 1. If H is restricted to be a polynomial of degree at most n, then a solution is given by H = ~*. Szegh's theory involves the solution of this extremum problem and related questions such as the asymptotics of ¢~ as n --+ oc. The essential result is that a, equals the g e o m e t r i c m e a n of p', i.e., 6, = exp{%/(4rc)} with c u = f ~ log #' (0) dO. Szegh's condition is that cu > - 0 % and it is equivalent with a, > 0 and with the fact that the system {(~k}~_-0 is not complete in H2(#) (cf. also C o m p l e t e s y s t e m ) . Defining the orthonormal Szeg5 polynomials

¢~(z) then if Szega's condition holds one has lim ¢~(z) = D,(z) -1,

n--+ oo

where the Szeg5 function is defined as D.(z)

= exp

with R(t, z) = (t + z)/(t - z) the Riesz Herglotz kernel (cf. also C a r a t h 6 o d o r y class). The convergence holds uniformly on compact subsets D. The flmction D is an outer function (cf. H a r d y classes) in D with radial limit to the boundary, and a.e. [D~(ei°)[ 2 = #'(0). Therefore it is also called a spectral factor of the weight function #'. Other asymptotic formulas were obtained under much weaker conditions, such as #' > 0 a.e. or the Carleman conditions for the moments of >. Szeg5 polynomials of the second kind are defined inductively as ~o = 1 and, for n _> 1,

{1//

}

log~'(O)R(e ~°, z) dO ,

U~

7r

/

R ( e ~° , z ) [ ¢ n ( ~ i°) - ¢ ~ ( z ) ]

d~(O).

The rational functions E~ = -~b,~/¢~ interpolate the

Riesz-Herglotz transform

1

/

R(~ ~°, z) d~(O)

at zero and infinity. F u is a Carath4odory or positive real function because it is analytic in the open unit disc and has positive real p a r t there. The C a y l e y t r a n s f o r m gives a one-to-one correspondence between F~ and a Schur function (cf. also S c h u r f u n c t i o n s in c o m p l e x f u n c t i o n t h e o r y ) , namely

S.(z) = F~(z) - F.(O) F , (z) 7 F, (0) A Schur function is analytic and its modulus is bounded by 1 in D. I. Schur developed a continued-fraction-like algorithm to extract the reflection coefficients from Sp. It is based on the recursive application of the lemma saying that Sk is a Schur function if and only if Sk(O) 6 D and &+~(~) = _ ~ &(~) - & ( o ) 1 - Sk (O)Sk (z)

is a Schur function. The Sk(0) correspond to reflection coefficients associated with p if So = Su and the successive approximants t h a t are computed for S , are related to the Cayley transforms of the interpolants F~ given above. It also follows that there is an infinite sequence of reflection coefficients in D, unless Su is a rational function, i.e. unless p is a discrete measure. It also implies that, except for the case of a discrete measure, the Szeg5 polynomials have all their zeros in D. All these properties have a physical interpretation and are important for the application of Szeg5 polynomials in linear prediction, modelling of stochastic processes, scattering and circuit theory, optimal control, etc. The polynomials orthogonal on a circle are of course related to polynomials orthogonal on the real line or on an interval, e.g., I = [ - 1 , 1], using an appropriate transformation. Given the polynomials orthogonal for a 393

SZEGO POLYNOMIAL weight function w on an interval I, then the orthogonal polynomials for a rational modification w/p, where p is a polynomial positive on I, can be derived. BernshtefnSzeg5 polynomials are orthogonal polynomials for rational modifications of one of the four classical Chebyshev weights on I, i.e. for w(x) = (1 - x)~(1 + x) 9 with a, f l e { - 1 / 2 , 1/2}. References [1] FREUD, G.: Orthogonal polynomials, Pergamon, 1971. [2] GERONIMUS, YA.: Orthogonal polynomials, Consultants Bureau, 1961. (Translated from the Russian.) [3] STAHL, H., AND TOTIK, V.: General orthogonal polynomials, Encycl. Math. Appl. Cambridge Univ. Press, 1992. [4] SZEG6, G.: Orthogonal polynomials, 3rd ed., Vol. 33 of Colloq. Publ., Amer. Math. Soc., 1967.

A. BuItheel M S C 1991:33C45 SZEGI~ Q U A D R A T U R E - Szeg5 quadrature formulas are the analogues on the unit circle T in the complex plane of the Gauss quadrature formulas on an interval (cf. also G a u s s q u a d r a t u r e f o r m u l a ) . They approxim a t e the integral

I.(f) = fT f(t) dp(t), where T = {z C C : Izl = 1} and # is a positive m e a s u r e on T, by a q u a d r a t u r e f o r m u l a of the form

In(f) =

ankf( k=l

394

k).

One cannot take the zeros of the Szeg5 polynomials qhn as nodes (as in Gaussian formulas), because these are all in the open unit disc D (cf. also S z e g 5 p o l y n o m i a l ) . Therefore, the para-orthogonal polynomials are introduced as Q~(z,T) = Ca(Z) + T¢*(Z), where ~- E T and ¢*~(z) = ZnCn(1/g). These are orthogonal to { z , . . . , z n - l } and have n simple zeros, which are on T. The Szeg5" quadrature formula then takes as nodes the zeros ~nk , k = 1 , . . . ,n, of Qn(z,~-), and as weights the C h r i s t o f f e l n u m b e r s 1 Ank =

n-1

Ej=0 ICj

2

> O.

The result is a quadrature formula with a maximal domain of validity in the set of Laurent polynomials, i.e., the formula is exact for all trigonometric polynomials in s p a n { z - n - i , . . . , z -1, 1, z , . . . , zn-1}, a space of dimension 2 n - 1, which is the maximal dimension possible with a quadrature formula of this form. The Szeg5 quadrature formulas were introduced in [2]. The underlying ideas have been generalized from polynomials to rational functions. See [1]. References [l] BULTHEEL, A., GONZ/~LEZ-VERA, P., HENDRIKSEN, E., AND NJASTAD, O.: 'Quadrature and orthogonal rational functions', J. Comput. Appl. Math. 127 (2001), 67-91. [2] JONES, W.B., NJ~.STAD, O., AND THRON, W.J.: 'Moment theory, orthogonal polynomials, quadrature and continued fractions associated with the unit circle', Bull. London Math. Soc. 21 (1989), 113-152.

A. Bultheel

M S C 1991:65D32

T TACNODE, point of osculation, osculation point, double cusp - The third in the series of Ak-curve singularities. The point (0,0) is a tacnode of the curve X 4 __ y 2 • 0 in R 2. The first of the Ak-curve singularities are: an ordinary double point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp. They are exemplified by the curves X k+l - y2 = 0 for k = 1,2,3,4. The terms 'crunode' and 'spinode' are seldom used nowadays (2000). See also N o d e ; Cusp. References

[1] ABHYANKAR, S.S.: Algebraic geometry for scientists and engineers, Amer. Math. Soc., 1990, p. 3; 60. [2] DIMCA, A.: Topics on real and complex singularities, Vieweg, 1987. [3] GRIFFITHS, PH., AND HARRIS, J.: Principles of algebraic geometry, Wiley, 1978, p. 293; 507. [4] WALKER,R.J.: Algebraic curves, Princeton Univ. Press, 1950, Reprint: Dover 1962. M. Hazewinkel

MSC 1991:14H20 TANGLE, relative link - A one-dimensional manifold properly embedded in a 3-ball, D a. Two tangles are considered equivalent if they are ambient isotopic with their boundary fixed. An n-tangle has 2n points on the boundary; a link is a 0-tangle. The term arcbody is used for a one-dimensional manifold properly embedded in a 3-dimensional manifold. Tangles can be represented by their diagrams, i.e. regular projections into a 2-dimensional disc with additional over- and under-information at crossings. Two tangle diagrams represent equivalent tangles if they are related by Reidemeister moves (cf. R e i d e m e i s t e r t h e orem). The word 'tangle' is often used to mean a tangle diagram or part of a link diagram. The set of n-tangles forms a m o n o i d ; the identity tangle and composition of tangles is illustrated in Fig. 1.

o.o

T1

~ J

TId

T2 I

T 1 * T2 Fig. 1.

Several special families of tangles have been considered, including the r a t i o n a l t a n g l e s , the a l g e b r a i c t a n g l e s and the periodic tangles (see R o t o r ) . The nbraid group is a subgroup of the monoid of n-tangles (cf. also B r a i d e d g r o u p ) . One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is important in developing quantum invariants of links and 3-manifolds (e.g. Reshetikhin-Turaev invariants). Tangles are also used to construct topological quantum field theories. References [1] BONAHON, P., AND SIEBENMANN, L.: Geometric splittings of classical knots and the algebraic knots of Conway, Vol. 75 of Lecture Notes, L o n d o n M a t h . Soc., to appear. [2] CONWAY, J.H.: ' A n e n u m e r a t i o n of knots and links', in J. LEECH (ed.): Computational Problems in Abstract Algebra, P e r g a m o n Press, 1969, pp. 329-358. [3] LOZANO, M.: 'Arcbodies', Math. Proc. Cambridge Philos. Soe. 94 (1983), 253-260.

Jozef Przytycki MSC 1991:57M25 TANGLE M O V E - For given n-tangles 2/"1 and T2 (cf. also Tangle), the tangle move, or more specifically the (T1,T2)-move, is substitution of the tangle T2 in the place of the tangle T1 in a link (or tangle). The simplest tangle 2-move is a crossing change. This can be generalized to n-moves (cf. M o n t e s i n o s - N a k a n i s h i c o n j e c t u r e or [5]), (m, q)-moves (cf. Fig. 1), and p/qrational moves, where a rational 2-tangle is substituted in place of the identity tangle [6] (Fig. 2 illustrates a 13/5-rational move).

TANGLE MOVE

A p/q-rational move preserves the space of Fox pcolourings of a link or tangle (cf. F o x n - c o l o u r i n g ) . For a fixed prime number p, there is a conjecture that any link can be reduced to a trivial link by p/q-rational m o v e s (Iql _< p/2).

Kirby moves (cf. K i r b y c a l c u l u s ) can be interpreted as tangle moves on framed links.

... J~"-J'~"~"" "~'~

q half twists

(m,q)-move

m half twists Fig. 1.

13/5-move

TAU METHOD, r method A method initially formulated as a tool for the approximation of special functions of mathematical physics (cf. also Special functions), which could be expressed in terms of simple differential equations. It developed into a powerful and accurate tool for the numerical solution of complex differential and functional equations. A main idea in it is to approximate the solution of a given problem by solving exactly an approximate problem. L a n c z o s ~ f o r m u l a t i o n o f t h e t a u m e t h o d . In [17], C. Lanczos remarked t h a t truncation of the series solution of a differential equation is, in some way, equivalent to introducing a perturbation t e r m in the right-hand side of the equation. Conversely, a polynomial perturbation t e r m can be used to produce a truncated series, that is, a polynomial solution. Assume one wishes to solve by means of a power series expansion the simple linear differential equation (cf. also Linear differential operator) Dy(x):=y'(x)+y(x)=0,

O<x
Fig. 2.

v(0) = 1,

Habiro Cn-moves [1] are prominent in the theory of Vassiliev Gusarov invariants of links and 3-manifolds. The simplest and most extensively studied Habiro move (beyond the crossing change) is the A-move on a 3tangle (cf. Fig. 3). One can reduce every knot into the trivial knot by A-moves [4].

which defines y(x) = e x p ( - x ) . To find the coefficients of a formal series expansion of y(x), one substitutes the series in the equation and generates a system of algebraic equations for the coefficients: jaj + aj-1 = 0 for j = 1, 2 , . . . , solving it in terms of a0. The value of a0 is fixed using the initial condition. To find a finite expansion, say of order n, one needs to make all coefficients aj with j > n equal to zero. This is achieved by adding a term of the form r x n to the right-hand side of the differential equation. One has (n + 1)an+l + an = % so that a,,+l, and all the coefficients following it, will be equal to zero if one chooses as = r. The same condition follows by substituting a segment of degree n of the series expansion of y(x) = e x p ( - x ) into the equation. If the solution of the perturbed differential equation is regarded as an approximation to that of the original equation with, say, a right-hand side equal to zero, it seems natural to replace it by the best u n i f o r m a p p r o x l m a t i o n of zero over the same interval J, which is a Chebyshev polynomial T2 (x) of degree n, defined over J (cf. also C h e b y s h e v p o l y n o m i a l s ) . Therefore, to find an accurate polynomial approximation of y(x), Lanczos proposed solving exactly the more complex perturbed problem (the tau problem):

-move

Fig. 3.

References [1] HABIRO, K.: 'Claspers and finite type invariants of links', Geometry and Topology 4 (2000), 1-83. [2] HARIKAE, T., AND UCHIDA, Y.: 'Irregular dihedral branched coverings of knots', in M. BOZH/SY/)K (ed.): Topics in Knot Theory, Vol. 399 of N A T O A S I Ser. C, Kluwer Acad. Publ., 1993, pp. 269-276. [3] KIRBY, R.: ' P r o b l e m s in low-dimensional topology', in W. KAZEZ (ed.): Geometric Topology (Proc. Georgia Internat. Topolo9y Conf., 1993), Vol. 2 of Studies in Adv. Math., Amer. Math. Soc./IP, 1997, pp. 35-473. [4] MURAKAMI, H., AND NAKANISHI, Y.: 'On a certain move generating link homology', Math. Ann. 2 8 4 (1989), 75-89. [5] PRZYTYCKI, J.H.: '3-coloring and other elementary invariants of knots': Knot Theory, Vol. 42, B a n a c h Center Publ., 1998, pp. 275-295. [6] UCHTDA, Y., in S. SUZUKI (ed.): Knots '96, Proc. Fifth Internat. Research Inst. of MS J, World Sei., 1997, pp. 109 113.

Jozef Przytycki M S C 1991:57M25

396

D y e ( x ) = rT,~ (x), with the same initial conditions as before. The polynomial y*(x) is called the tau method approximation of y(x) over the given interval J. This tau problem can be solved for the unknown coefficients of y*(x) using several alternative procedures.

TAU One of them is described above, that is, to set up and solve a system of linear algebraic equations linking the unknown coefficients of Dy* (x) with those of 7T~ (x). In this process one can assume that yn(X) itself can be expressed in either powers of x, or in Chebyshev, Legendre or other polynomials. The first choice was Lanczos' original choice, and he explicitly indicated the possibility of choosing the others. The second choice is a tau method, often [8] called the Chebyshev method (or Legendre method) and, also, the spectral method. This last formulation of the tau method has been extensively used and applied, since 1971, to complex problems in fluid dynamics by S.A. Orsag [11]. There are at least three other approaches to the tau method. One of them is to find the coefficients of the approximant through a process of interpolation at the zeros of the perturbation term. This early form of collocation was termed the 'method of selected points' by Lanczos [17]. When the perturbation term is an orthogonal polynomial (such as a Chebyshev, Legendre, or other polynomial), this process is called 'orthogonal collocation'. This is the name by which Lanczos' method of selected points is usually designated today (as of 2000); the name 'pseudo-spectral method' is also often applied to it. Algorithms for these methods have been well developed.

Recurslve formulation of the tau method based o n c a n o n i c a l p o l y n o m i a l s . In his classic [18], Lanczos noted that if a sequence of polynomials Q~(x), n = 0, 1 , . . . , such that D Q n ( x ) := x ~ for all n E N can be found for any linear differential operator with polynomial coefficients D, then, since Tg(x) := c~ + e~x + • .. + c~,x'~ (the coefficients of which are tabulated), the solution of the tau problem would be immediately given by: n

k=0

where the parameter T is fixed using the initial condition. An extension of this approach to a wider range of differential operators than the trivial one, given in the example, has several advantages: canonical polynomials are independent of the interval in which the solution is sought, allowing for easy segmentation of the domain; they are permanent, in the sense that if an approximation of a higher degree is required, the computation does not need to be repeated from scratch; they are also independent of the supplementary conditions of the problem, which can now equally be initial, boundary or multipoint conditions. Furthermore, the tau method does not require a stage of discretization of the given differential operator, as discrete-variable methods do.

METHOD

A sequence of canonical polynomials defined as simply as DQn(x) := x n for all n = 0, 1,..., need not always exist or need not be unique. An algebraic and algorithmic theory of the tau method, initially constructed for elements D of the class D of linear differential operators of arbitrary integer order, with polynomial or rational coefficients (essentially the tools a computer handles) was discussed by E.L. Ortiz in [24]. In this work, canonical polynomials are defined as realizations of classes of equivalence of polynomials, for which the algebraic kernel of the differential operator is the modulus. These classes have gaps in their index sequence. Elements D E D are then uniquely associated with representatives of such classes of canonical sequences. The codimension of the image of the space of polynomials under operators D C D is usually small, and bounded by the order of D plus the height h := maxncN{a~ - n } (where an is the degree of Dx n) of the differential operator. For more general operators than the one used as an example, more than a single ~- term is usually required to satisfy the more elaborate supplementary conditions and, also, internal conditions of the method. In the case of a problem defined by a differential operator D in l?, of order # > i and with non-constant coefficients, the question of the number of 7 terms required for a tau method approximation has been shown to be related to the size of the gap in the canonical sequence, and to the existence of a non-empty algebraic kernel in D. The number of ~- terms can be easily determined in this approach using information on the degree of polynomial (or rational) coefficients and the order of differentiation of the function to which they apply. It was also shown in [24] that canonical sequences can be generated recursively. This approach was used to formulate the first recursire algorithms for the automatic solution of differential equations using the tau method. The theory of canonical polynomials has been discussed and extended by several authors; see [10] and the references given therein. Theoretical error analysis for the tau method [18], [30], [9], [22], [26] have shown that tau method approximations are of the order of best uniform approximations by polynomials defined over the same interval. This connection with best approximation is preserved when a tau method based on rational approximation [18], [21] is used [5]. O p e r a t i o n a l f o r m u l a t i o n o f t h e t a u m e t h o d . There is yet another way in which tau method approximations can be constructed. An operational formulation of the tau method was introduced by Ortiz and H. Samara in [27]. In this formulation, derivatives and polynomial coefficients of operators in 7? are represented in terms of 397

TAU METHOD m u l t i p l i c a t i v e d i a g o n a l m a t r i c e s . F u r t h e r m o r e , t h e differential o p e r a t o r a n d t h e s u p p l e m e n t a r y c o n d i t i o n s are d e c o u p l e d . T h r o u g h a simple a n d s y s t e m a t i c a l g o r i t h m , which t r e a t s t h e differential o p e r a t o r a n d s u p p l e m e n t a r y c o n d i t i o n s with s i m i l a r machinery, this technique t r a n s f o r m s a given differential t a u m e t h o d p r o b l e m into one in l i n e a r algebra. T h e a p p r o x i m a t e s o l u t i o n can be g e n e r a t e d , indistinctively, in t e r m s of powers of the variables or in t e r m s of e l e m e n t s of a m o r e s t a b l e p o l y n o m i a l basis, such as C h e b y s h e v , L e g e n d r e or o t h e r p o l y n o m i als. T h e o p e r a t i o n a l f o r m u l a t i o n f u r t h e r simplified t h e d e v e l o p m e n t of software for t h e t a u m e t h o d . Numerical applications of the tau method. The recursive a n d o p e r a t i o n a l a p p r o a c h e s to t h e t a u m e t h o d have b e e n e x t e n d e d in several directions. To s y s t e m s of linear differential e q u a t i o n s [9], [4]; to n o n - l i n e a r p r o b lems [25], [23], [26]; to p a r t i a l differential e q u a t i o n s [28], [29]; and, in p a r t i c u l a r , to t h e n u m e r i c a l s o l u t i o n of nonlinear s y s t e m s of p a r t i a l differential e q u a t i o n s t h e solution of which has s h a r p spikes, with high g r a d i e n t s , as

in the case of soliton interactions [14], [13]; to the approximate s o l u t i o n of o r d i n a r y a n d p a r t i a l functionaldifferential e q u a t i o n s [25], [20], [15]; a n d to singular p r o b l e m s for p a r t i a l differential e q u a t i o n s r e l a t e d to crack p r o p a g a t i o n [7]. T h e t a u m e t h o d is well a d a p t e d to p r o d u c e a c c u r a t e a p p r o x i m a t i o n s in t h e n u m e r i c a l t r e a t m e n t of differential eigenvalue p r o b l e m s with one or m u l t i p l e s p e c t r a l p a r a m e t e r s , entering either linear or n o n - l i n e a r l y into t h e e q u a t i o n [2], [19]. T h e t a u m e t h o d has been e x t e n s i v e l y used for t h e high-precision a p p r o x i m a t i o n of real- [16] a n d c o m p l e x - v a l u e d functions. A w e a k f o r m u l a t i o n of t h e t a u m e t h o d has b e e n p r o p o s e d a n d a p p l i e d to inverse p r o b l e m s for p a r t i a l differential e q u a t i o n s [1]. Analytical

applications

of the tau

method.

The

t a u m e t h o d has also been used in a t o t a l l y different direction, as a t o o l in t h e discussion of p r o b l e m s in m a t h e m a t i c a l analysis, for e x a m p l e , in c o m p l e x function theory [12]. Possible connections b e t w e e n t h e t a u m e t h o d , collocation, G a l e r k i n ' s m e t h o d , a l g e b r a i c kernel m e t h o d s , a n d o t h e r p o l y n o m i a l or d i s c r e t e - v a r i a b l e techniques have also been e x p l o r e d [31], [13], [6]. T h e t a u m e t h o d has also received s o m e a t t e n t i o n as an a n a l y t i c tool in t h e discussion of equivalence results across n m n e r i c a l m e t h o d s [6]. It has b e e n f o u n d t h a t , with it, it is possible to c o n s t r u c t special ' t a u m e t h o d s ' , which recursively g e n e r a t e solutions n u m e r i c a l l y identical to those of collocation, G a l e r k i n ' s a n d o t h e r weighted residual m e t h o d s , a n d to t h o s e of d i s c r e t e - v a r i a b l e m e t h ods, such as s o p h i s t i c a t e d forms of R u n g e - K u t t a m e t h ods. This work suggests a w a y of unifying a large g r o u p 398

of continuous- a n d d i s c r e t e - v a r i a b l e a p p r o x i m a t i o n techniques. References [1] BANKS, H.T., AND WADE, J.G.: 'Weak tau approximations for distributed parameter systems in inverse problems', Numet. Funct. Anal. Optim. 12 (1991), 1-31.

[2] CHAVES, T., AND ORTIZ, E.L.: 'On the numerical solution of two point boundary value problems for linear differential equations', Z. Angew. Math. Mech. 48 (1968), 415 418. [3] CRISCI, M.R., AND RUSSO, E.: 'A-stability of a class of methods for the numerical integration of certain linear systems of differential equations', Math. Comput. 41 (1982), 431-435. [41 CRISCg M.R., AND RUSSO, E.: 'An extension of Ortiz's recursive formulation of the tau method to certain linear systems of ordinary differential equations', Math. Comput. 41 (1983), 27-42. [5] EL DAOU, M., NAMASIVAYAM,S., AND ORTIZ, E.L.: 'Differential equations with piecewise approximate coefficients: discrete and continuous estimation for initial and boundary value problems', Computers Math. Appl. 24 (1992), 33-47. [6] EL DAOU, M., AND ORTIZ, E.L.: 'The tau method as an analytic tool in the discussion of equivalence results across numericaI methods', Computing 60 (1998), 365-376. [7] EL MISlERY, A.E.M., AND ORTIZ, E.L.: 'Tau-lines: a new hybrid approach to the numerical treatment of crack problems based on the tau method', Computer Methods in Applied Mechanics and Engin. 56 (1986), 265 282. [8] Fox, L., AND PARKER, I.B.: Chebyshev polynomials in numerical analysis, Oxford Univ. Press, 1968. [9] FREILICH,J.G., AND ORTIZ, E.L.: 'Numerical solution of systerns of differential equations: an error analysis', Math. Comput. 39 (1982), 467-479. [10] FROES BUNCHAFT, M.E.: 'Some extensions of the LanczosOrtiz theory of canonical polynomials in the tau method', Math. Comput. 66, no. 218 (1997), 609 621. [11] GOTLIEB,D., AND ORSZAG, S.A.: Numerical analysis of spectral methods: Theory and applications, Philadelphia, 1977. [12] HAYMAN,W.K., AND ORTIZ, E.L.: 'An upper bound for the largest zero of Hermite's function with applications to subharmonic functions', Proc. Royal Soc. Edinburgh 75A (1976), 183-197. [13] HOSSEIM AH-ABAD% M., AND ORTm, E.L.: 'The algebraic kernel method', Namer. Funct. Anal. Optim. 12, no. 3-4 (1991), 339 360. [14] HOSSEINI ALI-ABADI, M., AND ORTIZ, E.L.: 'A tau method based on non-uniform space-time elements for the numerical simulation of solitons', Computers Math. Appl. 22 (1991), 7-19. [15] KHAJAH, H.G., AND ORTIZ, E.L.: 'Numerical approximation of solutions of functional equations using the tau method', Appl. Namer. Anal. 9 (1992), 461-474. [16] KHAJAH, H.G., AND ORTIZ, E.L.: 'Ultra-high precision computations', Computers Math. Appl. 27, no. 7 (1993), 41-57. [17] LANeZOS, C.: 'Trigonometric interpolation of empirical and analytic functions', J. Math. and Physics iT (1938), 123-199. [18] LANCZOS, C.: Applied analysis, New Jersey, 1956. [19] LIu, K.M., AND ORTIZ, E.L.: 'Tau method approximation of differential eigenvalue problems where the spectral parameter enters nonlinearly', Y. Comput. Phys. 72 (1987), 299-310. [2o] LIU, K.M., AND ORTIZ, E.L.: 'Numerical solution of ordinary and partial functional-differential eigenvalue problems with the tau method', Computing 41 (1989), 205-217.

TAYLOR JOINT SPECTRUM [21] LUKE, Y.L.: The special functions and their approximations l-II, New York, 1969. [22] NAVASIMAYAN,S., AND ORTIZ, E.L.: 'Best approximation and the numerical solution of partial differential equations with the tau method', Portugal. Math. 41 (1985), 97-119. [23] ONUMANYI, P., AND ORTIZ, E.L.: 'Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method', Math. Comput. 43 (1984), 189-203. [24] ORTIZ, E.L.: 'The tau method', SIAM J. Numer. Anal. 6 (1969), 480--492. [25] ORTm, E.L.: 'On the numerical solution of nonlinear and functional differential equations with the tau method', in R. ANSORGEAND W. ThRmC (eds.): Numerical Treatment of Differential Equations in Applications, Berlin, 1978, pp. 127139. [26] ORTIZ, E.L., AND PHAM NGOC DINH, A.: 'Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the tau method', SIAM J. Math. Anal. 18 (1987), 452-464. [27] ORTIZ, E.L., AND SAMARA,H.: 'An operational approach to the tau method for the numerical solution of nonlinear differential equations', Computing 27 (1981), 15-25. [28] OaTm, E.L., AND SAMARA,H.: 'Numerical solution of partial differential equations with variable coefficients with an operational approach to the tau method', Computers Math. Appl. 10, no. 1 (1984), 5-13. [29] PUN, K.S., AND ORTm, E.L.: 'A bidimensional tau-elements method for the numerical solution of nonlinear partial differential equations, with an application to Burgers equation', Computers Math. Appl. 12B (1986), 1225-1240. [30] RIVLIN,T.J.: The Chebyshev polynomials, New York, 1974, 2nd. ed. 1990. [31] WRICHT, K.: 'Some relationships between implicit RungeKutta, collocation and Lanczos tau methods', BIT 10 (1970), 218-227.

Eduardo L. Ortiz

T h e c o m m u t i n g n-tuple A is said to be non-singular on X if R a n D A = K e r D A . T h e Taylor joint spectrum, or simply the Taylor spectrum, of A on X is the set aT (A, X) : = {A • C ~ : A - A is singular}. T h e decomposition A = O~=1Ak gives rise to a cochain complex K ( A , X), the so-called K o s z u l c o m p l e x associated to A on A/, as follows:

z): 0

on-1

A°(X)

4

AN(X) -+ 0,

where D k denotes the restriction of DA to the subspace Ak(X). Thus,

a T ( A , X ) = {A • C ~ : K ( A -

A , X ) is not e x a c t } .

J.L. Taylor showed in [18] t h a t if X is a B a n a c h s p a c e , then aT(A, 2() is c o m p a c t , non-empty, and contained in a t(A), the (joint) algebraic s p e c t r u m of A (cf. also S p e c t r u m o f a n o p e r a t o r ) with respect to the commutant of A, (A)' : = {B • £ ( X ) : B A = A B } . Moreover, aT carries an analytic f u n c t i o n a l c a l c u l u s with values in the double c o m m u t a n t of A, so that, in particular, aT possesses the projection property.

Example: n = 1. For n = 1, DA admits the following (2 x 2)-matrix relative to the direct sum decomposition

( z ® e0) • (x ®

00) T h e n Ker D A / R a n DA = Ker A ® ( X / R a n A). It follows at once t h a t aT agrees with ~, the s p e c t r u m of A.

Example: n = 2. For n = 2,

M S C 1991: 65Lxx

DA = Let A = A[e] = An[e] be the e x t e r i o r a l g e b r a on n generators e l , . . . , e m with identity e0 - 1. A is the algebra of forms in e l , . . . , en with complex coefficients, subject to the collapsing p r o p e r t y eiej + ejei = 0 (1 _< i, j < n). Let E~: A --+ A denote the creation operator, given by Ei~ : = ei~ (~ • A, 1 _< i < n). If one declares { e q , . . . , e i ~ : 1 < il < ... < ik < n} to be an o r t h o n o r m a l basis, the exterior algebra A becomes a H i l b e r t s p a c e , a d m i t t i n g an orthogonal decomposition A = ~Jk=lZ'~nA k, where dim A k = ( ; ) . Thus, each ~ • A adm r s a unique orthogonal decomposition ~ = e ~ t + ~tt 1 where ~1 and ~" have no ei contribution. It then readily follows t h a t E*~ = ~. Indeed, each Ei is a partial isometry, satisfying E~Ej + E j E [ = 5ij (1 _< i , j < n). TAYLOR

JOINT

SPECTRUM

-

Let X be a n o r m e d s p a c e , let A =- ( A 1 , . . . , An) be a c o m m u t i n g n-tuple of b o u n d e d operators on X" and set A(X) := X ® c A. One defines DA: A(X) --+ A(X) by DA : = E l L 1 Ai ® El. Clearly, D ~ = 0, so R a n DA C_ Ker DA.

so

KerDA/RanDA

A1

0

0

2

0

0

-A2

A1

=

(KerA1

'

N KerA2)

®

{ ( x l , x 2 ) : A2xl = A l x 2 } / { ( A l x o , A 2 x o ) : x0 e X} (9 ( X / ( R a n A1 + R a n A2)). Note t h a t since aT is defined in terms of the actions of the operators Ai on vectors of X, it is intrinsically 'spatial', as opposed to a I, a " and other algebraic joint spectra, aT contains other well-known spatial spectra, like ap (the point spectrum), a~ (the approximate point spectrum) and a5 (the defect spectrum). Moreover, if /3 is a c o m m u t a t i v e B a n a c h algebra, a -= ( a l , . . . , a , 0 , with each ai E /3, and L~ denotes the n-tuple of left multiplications by the ais, it is not hard to show t h a t aT (L~,/3) = a• (a). As a m a t t e r of fact, the same result holds w h e n / 3 is not c o m m u t a t i v e , provided all the ais come from the centre of/3.

Spectral permanence. W h e n / 3 is a C*-algebra, s a y / 3 C £(7-0, then aT(La, B) = aT(a, 7-0 [5]. This fact, known as spectral permanence for the Taylor spectrum, shows 399

TAYLOR J O I N T S P E C T R U M that for C*-algebra elements (and also for Hilbert space operators), the non-singularity of La is equivalent to the invertibility of the associated Dirac operator Da + D t . .

to be Fredholm on X if the associated Koszul complex K ( A , 2() has finite-dimensional cohomology spaces. The Taylor essential spectrum of A on A~ is then

Finite-dimensional ease. When dim A" < oc,

0-Te(A, 2() := {A C C n : A - A is not Fredholm}.

0.p = 0"1 = 0-7r -= 0-5 = G-r = 0"T = 0-1 = 0-H = ~ ,

where 0-1, 0"r and ~ denote the left, right and polynomially convex spectra, respectively. As a matter of fact, in this case the commuting n-tuple A can be simultane/ (k) ~dim W ously triangularized as Ak = ~ai, j h,j=l , and 0-T(A, X) = (~ ~"(a!~) ~ , ' " , u i-(~)' i J: l < i < d i m X

}.

Case of compact operators. If A is a commuting n-tuple of compact operators acting on a Banach space 32, then 0-T(A, 2() is countable, with ( 0 , . . . , 0 ) as the only accumulation point. Moreover, a . ( A , 2() = 0.5(A,X) =

0-T(A, X). Invariant subspaces. If 2( is a Banach space, Y is a closed subspace of X and A is a commuting n-tuple leaving y invariant, then the union of any two of the sets (7T (A, ,-32'), 0-T(A,Y) and aT(A, X / y ) contains the third [18]. This can be seen by looking at the long cohomology sequence associated to the Koszul complex and the canonical short exact sequence 0 --+ J; --~ 2( --+ 2(/32 -+ O.

Additional properties. In addition to the abovementioned properties of O-T, the following facts can be found in the survey article [6] and the references therein: i) 0-T gives rise to a compact non-empty subset M~ T (B, W) of the maximal ideal space of any commutative Banach algebra B containing A, in such a way that 0.T(A, Z ) = .4(M~T (~ , W)) [18]; ii) for n = 2, 00.T(A,7/) C c90.H(A,7/), where ~H := 0-10 0"r denotes the Harte spectrum; iii) the upper semi-continuity of separate parts holds for the Taylor spectrum; iv) every isolated point in 0-B(A) is an isolated point of 0-T(A, 7/) (and, afortiori, an isolated point of al (A, 7/) N O'r ( A , 7 / ) ) ;

v) if 0 C 0.T(A, 7-t), up to approximate unitary equivalence one can always assume that Ran DA ~ Ker DA

[7]; vi) the functional calculus introduced by Taylor in [17] admits a concrete realization in terms of the BochnerMartinelli kernel (cf. B o e h n e r - M a r t i n e l l i r e p r e s e n t a t i o n f o r m u l a ) in case A acts on a Hilbert space or on a C * - a l g e b r a [20]; vii) M. Putinar established in [13] the uniqueness of the functional calculus, provided it extends the polynomial calculus.

Fredholm n-tuples. In a way entirely similar to the development of Fredholm theory, one can define the notion of Fredholm n-tuple: a commuting n-tuple A is said 400

The Fredholm index of A is defined as the E u l e r c h a r a c t e r i s t i c of K ( A , X ) . For example, if n = 2, index(A) = d i m K e r D ° - dim(Ker D1A/RanD °) + d i m ( X / R a n D y ) . In a Hilbert space, o-we(A,7/) = 0"T(La, Q(7/)), where a := 7r(A) is the coset of A in the Calkin algebra for 7/.

Example. If 7/ = H2(S 3) and Ai := Mz, (i = 1,2), then 0-1(A) = 0-1e(A) = 0-re(A) = 0"Te(A) = S 3, 0"r(A) = 0.T(A) = B4, and index(A - A) = 1 (A ¢ B4). The Taylor spectral and Fredholm theories of multiplication operators acting on Bergman spaces over Reinhardt domains or bounded pseudo-convex domains, or acting on the Hardy spaces over the Shilov boundary of bounded symmetric domains on several complex variables, have been described in [3], [4], [8], [9], [10], [15], [16], [19], and [21]; for Toeplitz operators with H °° symbols acting on bounded pseudo-convex domains, concrete descriptions appear in [11]. Spectral inclusion. If S is a subnormal n-tuple acting on 7/ with minimal normal extension N acting on ]C (cf. also N o r m a l o p e r a t o r ) , 0.T(N,]C) _C 0-T(S, 7/) C_ ~(N, K)[14]. Left and right multiplications. For A and B two commuting n-tuples of operators on a Hilbert space 7/, and LA and RB the associated n-tuples of left and right multiplication operators [7], 0-T((LA, RB ), £(7/)) = 0.T( A, 7/) X 0.T( B, 7/), and 0-Te((LA, RB), £(7/)) = = [aTe(A,7/) X 0.T(B,7/)] U [0-T(A,7/) X 0"Te(B,7/)]. During the 1980s and 1990s, Taylor spectral theory has received considerable attention; for further details and information, see [2], [11], [20], [6], [1]. There is also a parallel 'local spectral theory', described in [11], [12]

and [20]. References [1] ALBaEeHT, E., aND VASILESCU, F.-H.: 'Semi-Fredholm complexes', Oper. Th. Adv. Appl. 11 (1983), 15-39. [2] AMBROZIE, C.-C-., AND VASILESCU, F.-H.: Banach space complexes, Kluwer Acad. Publ., 1995. [3] BERGER, C., AND COBURN, L.: 'Wiener Hopf operators on U2', Integral Eq. Oper. Th. 2 (1979), 139 173. [4] BERGER, C., COBURN, L., AND KORANYI, A.: 'Opfirateurs de W i e n e r - H o p f sur les spheres de Lie', C.R. Acad. Sci. Paris Sdr. A 290 (1980), 989-991. [5] CURTO, R.: 'Spectral p e r m a n e n c e for joint spectra', Trans. Amer. Math. Soc. 270 (1982), 659-665.

THEODORSEN [6] CURTO, R.: 'Applications of several complex variables to multiparameter spectral theory', in J.B. CONWAYAND B.B. MORREL (eds.): Surveys of Some Recent Results in Operator Theory II, Vol. 192 of Pitman Res. Notes in Math., Longman Sci. Tech., 1988, pp. 25-90. [7] CURTO, R., AND FIALKOW, L.: 'The spectral picture of ( L A , R B ) ' , J. Funct. Anal. 71 (1987), 371-392. [8] CURTO, R., AND MUHLY, P.: 'C*-algebras of multiplication operators on Bergman spaces', J. Funct. Anal. 64 (1985), 315-329. [9] CURTO, R., AND SALINAS, N.: 'Spectral properties of cyclic subnormal m-tuples', Amer. J. Math. 107 (1985), 113-138. [10] CURTO, R., AND VAN, K.: 'The spectral picture of Reinhardt measures', J. Funct. Anal. 131 (1995), 279-301. [11] ESCHMEIER, J., AND PUTINAR, M.: Spectral decompositions and analytic sheaves, London Math. Soc. Monographs. Ox-

ford Sci. Publ., 1996. [12] LAURSEN,K., AND NEUMANN,M.: Introduction to local spectral theory, London Math. Soc. Monographs. Oxford Univ. Press, 2000. [13] PUTINAR, M.: 'Uniqueness of Taylor's functional calculus', Proc. Amer. Math. Soc. 89 (1983), 647-650. [14] PUTINAR, M.: 'Spectral inclusion for subnormal n-tuples', Proc. Amer. Math. Soc. 90 (1984), 405 406. [15] SALINAS, N.: 'The cg-formalism and the C*-algebra of the Bergman n-tuple', J. Oper. Th. 22 (1989), 325 343. [16] SALINAS,N., SHEU~A., AND UPMEIER, H.: 'Toeplitz operators on pseudoconvex domains and foliation C*-algebras', Ann. of Math. 130 (1989), 531 565. [17] TAYLOR, J.L.: 'The analytic functional calculus for several commuting operators', Acta Math. 125 (1970), 1-48. [18] TAYLOR, J.L.: 'A joint spectrum for several commuting operators', g. Funct. Anal. 6 (1970), 172-191. [19] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric domains', Ann. of Math. 119 (1984), 549-576. [20] VASlLESOU,F.-H.: Analytic functional calculus and spectral decompositions, Reidel, 1982. [21] VENUGOPALKRISHNA, U.: 'Fredholm operators associated with strongly pseudoconvex domains in C '~', Y. Funct. Anal. 9 (1972), 349 373.

Ragl E. Curto

M S C 1991: 47Dxx One of several results, of which the most i m p o r t a n t is the T a y l o r f o r m u l a and its various generalizations, e.g., to wider function classes, to a stochastic setting or to multiple centres (in which case one deals with interpolation-type formulas). TAYLOR

THEOREM

-

M S C 1991: 41A05, 41A58

THEODORSEN

INTEGRAL

EQUATION

-

T h e o d o r s e n ' s integral equation [7] is a well-known tool

for computing numerically the c o n f o r m a l m a p p i n g g of the unit disc D onto a star-like region A given by the polar coordinates r, p(r) of its b o u n d a r y F. T h e m a p p i n g g is assumed to be normalized by g(0) = 0, g'(0) > 0. It is uniquely determined by its b o u n d a r y correspondence function 0, which is implicitly defined

INTEGRAL EQUATION

by

g(e it)

=

p

(0(t)) e

/o ~ O(t)

dt

(vt c R),

=

2~ 2 "

T h e o d o r s e n ' s e q u a t i o n follows from the fact t h a t the function h ( w ) := l o g ( g ( w ) / w ) is analytic in D a n d can b e extended to a h o m e o m o r p h i s m of the closure D o n t o the closure A. It simply states t h a t the 2~r-periodic function y: t ~-~ 0 - t is the conjugate periodic function of x: t ~ l o g p ( O ( t ) ) , t h a t is, y = K x , where I4 is the conjugation o p e r a t o r defined on L[0, 21r] by the principal value integral (Kx)(t)

:=

P.V.

x ( s ) cot t - s ds

(a.e.).

W h e n restricted to L2[0, 2rF], K is a skew-symmetric end o m o r p h i s m of n o r m 1 with a very simple diagonal representation in Fourier space: when x has the real Fourier coefficients a o , a l , .., b l , b 2 , . . . , t h e n y has the coemcients 0, - b l , - b 2 , .., al, a 2 , . . . . Hence, while T h e o d o r s e n ' s integral equation is normally written as

o(t)

- t = _~ P.V. /o 21r logp(0(s)) cot -t g - 8 d,, -

-

for practical purposes the conjugation is executed by t r a n s f o r m a t i o n to Fourier space: x is a p p r o x i m a t e d by a t r i g o n o m e t r i c p o l y n o m i a l of degree N , whose Fourier coefficients are quickly f o u n d by the fast Fourier transform, which then can also be applied to determine values at 2 N equi-spaced points of the trigonometric polynomial t h a t a p p r o x i m a t e s y = K x (cf. also F o u r i e r s e r i e s ) . Before the fast Fourier transform b e c a m e the s t a n d a r d tool for this discrete conjugation process, the transition from the values of z to those of y was based on multiplication by a matrix, called the Wittich m a t r i x in [1]. The fast Fourier t r a n s f o r m m e a n t a cost reduction from O ( N 2) to O ( N log N ) operations per iteration. Until the end of the 1970s the r e c o m m e n d a t i o n was to solve a so-obtained discrete version of T h e o d o r s e n ' s equation by fixed-point (Picard) iteration, an a p p r o a c h t h a t is limited to J o r d a n regions with piecewise differentiable b o u n d a r y satisfying IP'/Pl < 1, and is very slow when the b o u n d 1 is nearly attained. Other regions, like those from airfoil design, which was the s t a n d a r d application t a r g e t e d by T. Theodorsen, could be handled by using first a suitable preliminary conformal mapping, which turned the exterior of the wing cross-section into the exterior of a J o r d a n curve t h a t is close to a circle; see [6, Chapt. 10]. Moreover, for this application, the equation has to be modified slightly to map the exterior of the disc onto the exterior of a J o r d a n curve. 401

THEODORSEN

INTEGRAL EQUATION

M. G u t k n e c h t [2], [3] extended the applicability of T h e o d o r s e n ' s equation by applying more refined iterative m e t h o d s and discretizations, and O. H/ibner [5] improved the convergence order from linear to quadratic by a d a p t i n g R. W e g m a n n ' s t r e a t m e n t of a similar equation o b t a i n e d by choosing h(w) := g ( w ) / w instead. Wegm a n n ' s m e t h o d [9], [10] applies the N e w t o n m e t h o d and solves the linear equation for the corrections by interpreting it as a R i e m a n n - H i l b e r t problem that can be solved with four fast Fourier transforms. A c o m m o n framework for conformal m a p p i n g methods based on function conjugation is given in [4]; T h e o d o r s e n ' s restriction to regions given in polar coordinates can be lifted. B o t h T h e o d o r s e n ' s [8] and Wegm a n n ' s [11] equations and m e t h o d s can be extended to the d o u b l y connected case. References

[1] GAIER, D.: Konstruktive Methoden der konformen Abbildung, Springer, 1964. [2] GUTKNECHT, M.H.: 'Solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods', Numer. Math. 36 (1981), 405-429. [3] GUTKNECHT, M.H.: 'Numerical experiments on solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods', SIAM a. Sci. Statist. Comput. 4 (1983), 1 30. [4] GUTKNECHT,M.H.: 'Numerical conformal mapping methods based on function conjugation', J. Comput. Appl. Math. 14 (1986), 31-77. [5] H/iBNEa, O.: 'The Newton method for solving the Theodorsen equation', a. Comput. Appl. Math. 14 (1986), 19-30. [6] KYTHE, P.K.: Computational conformal mapping, Birkhguser, 1998. [7] THEODORSEN, T.: 'Theory of wing sections of arbitrary shape', Rept. NACA 411 (1931). [8] THEODORSEN,T., AND GARRICK,I.E.: 'General potential theory of arbitrary wing sections', Rept. NACA 452 (1933). [9] WEGMANN, R.: 'Ein Iterationsverfahren zur konformen Abbildung', Numer. Math. 30 (1978), 453-466. [10] WEGMANN,R.: 'An iterative method for conformal mapping', J. Comput. Appl. Math. 14 (1986), 7-18, English translation of [9]. (In German.) [11] WEGMANN, R.: 'An iterative method for the conformal mapping of doubly connected regions', J. Comput. Appl. Math. 14 (1986), 79-98. Martin H. Gutknecht

M S C 1991: 30C20, 30C30 THIELE DIFFERENTIAL E Q U A T I O N - Consider an n year t e r m life insurance, with sum insured S and level p r e m i u m P per time unit, issued at time 0 to an x years old person. Denote by py the force of mortality at age y and by d the force of interest. If the insured is still alive at time t E [0, n), then the insurer m u s t provide a reserve, Vt, which by s t a t u t e is the m e a n value of future discounted benefits less premiums. Splitting into 402

p a y m e n t s before and after time t + dt leads to Vt = #x+t dt S - P dt+

(1)

+(1 - #x+t dt)e -~ atvt+at + o(dt), from which one obtains t h a t Vt is the solution to dv,~ = ~/

P + ~vt - ~x+~(s

- vd,

(2)

subject to the condition V~ = 0. This is the celebrated Thiele differential equation, proclaimed 'the f u n d a m e n t of m o d e r n life insurance m a t h e m a t i c s ' in the a u t h o r i t a t i v e t e x t b o o k [1], and n a m e d after its inventor Th.N. Thiele (1838-1910). It dates back to 1875, b u t was published only in 1910 in the o b i t u a r y on Thiele by J.P. G r a m [2], and appeared in a scientific text [7] only in 1913. As is a p p a r e n t from the p r o o f sketched in [1], Thiele's differential equation is a simple example of a Kolm o g o r o v backward equation (cf. K o l m o g o r o v e q u a t i o n ) , which is a basic tool for determining conditional expected values in intensity-driven M a r k o v processes. Thus, t o d a y there exist Thiele differential equations for a variety of life insurance p r o d u c t s described by multistate Markov processes and for various aspects of the discounted payments, e.g. higher order m o m e n t s and probability distributions. T h e technique is an indispensable constructive device in theoretical and practical life insurance m a t h e m a t i c s and also forms the basis for numerical procedures, see [8]. Thiele was Professor of A s t r o n o m y at the University of C o p e n h a g e n from 1875, cofounder and Director (actuary) of the Danish life insurance c o m p a n y Hafnia from 1872, and first president of the Danish Actuarial Society founded in 1901. In 52 written works (three monographs; [11], [12], [13]) he m a d e contributions (a n u m b e r of t h e m fundamental) to astronomy, m a t h e m a t i c a l statistics, numerical analysis, and actuarial mathematics. Biographical/bibliographical accounts are given in [3], [4], [51, [6], [9], [10]. References [1] BERCER, A.: Mathematik der Lebensversicherung, Springer Wien, 1939. [2] GRAM, J.P.: 'Professor Thiele sore aktuar', Dansk Forsikringsdrbog (1910), 26-37. [3] HALD, A.: 'T.N. Thiele's contributions to statistics', Internat. Statist. Rev. 49 (1981), 1-20. [4] HALD, A.: A history of mathematical statistics from 1750 to 1930, Wiley, 1998. [5] HOEM, J.M.: 'The reticent trio: Some little-known early discoveries in insurance mathematics by L.H.F. Oppermann, T.N. Thiele, and J.P. Gram', Internat. Statist. Bey. 51 (1983), 213-221. [6] JOHNSON, N.L., AND KOTZ, S. (eds.): Leading personalities in statistical science, Wiley, 1997. [7] JORGENSEN, N.R.: Grundz@e einer Theorie der Lebensversicherung, G. Fischer, 1913.

TILTED A L G E B R A [8] NORBERG, R.: 'Reserves in life and pension insurance', Scan& Actuarial d. (1991), 1-22. [9] NORBERG, R.: Thorvald Nicolai Thiele, statisticians of the centuries, Internat. Statist. Inst., 2001. [10] SCHWEDER, W.: 'Scandinavian statistics, some early lines of development', Scan& J. Statist. 7" (1980), 113-129. [11] THIELE, T.N.: Element~er Iagttagelseslaere, Gyldendal, Copenhagen, 1897. [12] THIELE, T.N.: Theory of observations, Layton, London, 1903, Reprinted in: Ann. Statist. 2 (1931), 165-308. (Translated from the Danish edition 1897.) [13] THIELE, T.N.: Interpolationsrechnung, Teubner, 1909.

1) For every surjective stratified morphism f : M N, the restriction of f to the inverse image f - 1 (S) of a stratum S is a f i b r a t i o n . 2) If there is a sequence of stratified morphisms M N 2~ I, where f is a Thorn mapping (an 'application sans ficlatement') and I is a segment, then the mappings fa and fb over two points a and b in I have the same topological type, i.e. there are homeomorphisms h and h' such that the following diagram commutes:

Ragnar Norberg MSC 1991:62P05 THOM-MATIIER

STRATIFICATION - A stratifi-

c a t i o n of a space such that each stratum has a neigh-

bourhood which fibres over that stratum, with levels defined by a tubular function (called 'fonction tapis' in Thorn's and 'distance function' in Mather's terminoIogy), and the fibrations and tubular functions associated to the strata are compatible with each other. Thorn Mather stratifications satisfy the Thorn first and second isotopy lemmas (see below), providing results such as local topological triviality of the stratification, local topological triviality along the strata of a morphism and topological stability of generic smooth mappings ('generic' meaning transverse to the natural stratifiestion of the jet space). The word 'stratification' has been introduced by R. Thorn in [5]. He proposed regularity conditions on how the strata of a stratification should fit together and stated the isotopy lemmas. The notes [4] of J. Mather provide a detailed proof, with improvements and sireplifications (cf. [2], which contains an excellent history of stratification theory). A Thom-Mathcr stratification of a space M consists of a tube system (Tx, 7Cx, p x ) associated to the strata X of M, such that T x is a t u b u l a r n e i g h b o u r h o o d of X in M, 7rx : T x --+ X is the fibre projection associated to Tx and the tubular function Px : T x -+ R is a continuous mapping satisfying p } 1 (0) = X. These data are controlled in the following sense: If X and Y are two strata such that X is in the frontier of Y, then

M~

h

M6

N~

-~

Nb

h'

The importance of T h o m - M a t h e r stratifications is emphasized by their applications to stability and topological triviality theorems. Among other important results in singularity theory is the fact that any Whitney stratification (see S t r a t i f i c a t i o n ) is a T h o m - M a t h e r stratification. Hence, a Whitney stratification satisfies topological triviality. The converse is false [1]; in fact, being a Whitney stratification is equivalent to topological triviality for all sections by a generic flag [3]. References [1] BRIAN~ON, J., AND SPEDER, J.P.: 'La trivialit~ topologique n'implique pas les conditions de Whitney', Note C.R. Acad. Sci. Paris Ser. A 280 (1975), 365. [2] GORESKY, M., AND MACPHEHSON, R.: Stratified Morse theory, Springer, 1988. [3] Lg, D.T., AND TEISSIER, B.: 'Cycles fivanescents, sections planes et conditions de Whitney II': Proe. Syrup. Pure Math., Vol. 40, Amer. Math. Soc., 1983, pp. 65-103. [4] MATHER, J.: Notes on topological stability, Harvard Univ., 1970. [5] THOU, R.: 'La stabilit~ topologique des applications polynomiales', Enseign. Math. 8, no. 2 (1962), 24 33. [6] THOU, R.: 'Ensembles et morphismes stratifies', Bull. Amer. Math. Soc. 75 (1969), 240-284. [7] WHITNEY, H.: 'Local properties of analytic varieties', in S. CAIRNS (ed.): Differential and Combinatorial Topology, Princeton Univ. Press, 1965, pp. 205-244. [8] WHITNEY, H.: 'Tangents to an analytic variety', Ann. of Math. 81 (1965), 496-549.

Jean-Paul Brasselet MSC 1991:57N80 T I L T E D A L G E B R A - The endomorphism ring of a

i) the restriction mapping (zcx, Px) : T x n Y --+ X x ]0, ec[ is a smooth s u b m e r s i o n ; ii) for a E T x N T y such that Try(a) C T x , there are commutation relations C1) ~rx o Try(a) = rex(a), 02) p x o ~ y (a) = ~ x (a)

whenever both sides of the formulas are defined. T h o m - M a t h e r stratifications satisfy the isotopy lemmas (as proposed by Thom):

t i l t i n g m o d u l e over a finite-dimensional hereditary al-

gebra (cf. also A l g e b r a ; E n d o m o r p h i s m ) . Let H be a finite-dimensional hereditary K-algebra, K some field, for example the path-algebra of some finite q u i v e r without oriented cycles. A finite-dimensional Hmodule HT is called a tilting module if i) p. d i m T < 1, which always is satisfied in this context; ii) E x t ~ ( T , T ) = 0; and 403

TILTED ALGEBRA iii) there exists a short e x a c t s e q u e n c e 0 --+ H -+ Tz --+ T.2 -+ 0 with r l and T~ in add T, the category of finite direct sums of direct summands of T. Here, p. dim is projective dimension. The third condition also says that T is maximal with respect to the property E x t , ( T , T) = 0. Note further, that a tilting module T over a hereditary algebra is uniquely determined by its composition factors. Cf. also T i l t i n g module. The algebra B = EndH(T) is called a tilted algebra of type H, and T becomes an (H, B)-bimodule (cf. also Bimodule).

In H-mod, the c a t e g o r y of finite-dimensional Hmodules, the module T defines a torsion pair (G,$-) with torsion class G consisting of modules, generated by T and torsion-free class • = {Y: H o m H ( T , Y ) = 0}. In B-mod it defines the torsion pair (X,3;) with torsion class 2( = {Y: T ®B Y = 0} and torsion-free class ~2 = {Y: TorB(T,Y) = 0}. The Brenner-Butler theorem says that the functors H o m H ( T , - ) , respectively T ®B --, induce equivalences between G and J;, whereas E x t f / ( T , - ) , respectively T o r B ( T , - ) , induce equivalences between )c and X. In B-rood the torsion pair is splitting, that is, any indecomposable B-module is either torsion or torsion-free. In this sense, B-mod has 'less' indecomposable modules, and information on the category H - m o d can be transferred to B-mod. For example, B has global dimension at most 2 and any indecomposable B-module has projective dimension or injective dimension at most 1 (cf. also D i m e n s i o n for dimension notions). These condition characterize the more general class of quasi-tilted algebras. The indecomposable injective H-modules are in the torsion class ~ and their images under the t i l t i n g f u n c t o r HomH (T, - ) are contained in one connected component of the Auslander Reiten quiver F(B) of B-rood (cf. also Q u i v e r ; R i e d t m a n n classification), and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of F(B) characterizes tilted algebras. Moreover, the Auslander-Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components. If H is a basic hereditary algebra and He is a simple projective module, then T = H(1 - e) ® TrD He, where TrD denotes the Auslander-Reiten translation (cf. R i e d t m a n n classification), is a tilting module, sometimes called APR-tilting module. The induced torsion pair (G,¢-) in H-rood is splitting and He is the unique indecomposable module in F. The tilting functor H o m H ( T , - ) corresponds to the reflection functor introduced by I.N. BernshteYn, I.M. Gel'land and V.A. Ponomarev for their proof of the Gabriel theorem [3]. 404

If the hereditary algebra H is representation-finite (cf. also A l g e b r a o f f i n i t e r e p r e s e n t a t i o n t y p e ) , then any tilted algebra of type H also is representationfinite. If H is tame (cf. also R e p r e s e n t a t i o n o f a n a s s o c i a t i v e a l g e b r a ) , then a tilted algebra of type H either is tame and one-parametric, or representationfinite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and preinjective direct summands simultaneously. If H is wild (cf. also R e p r e s e n t a t i o n o f a n a s s o c i a t i v e a l g e b r a ) , then a tilted algebra of type H may be wild, or tame domestic, or representation-finite. See also T i l t i n g t h e o r y . References [1] ASSEM, I.: 'Tilting theory - an introduction', in N. BALCERZYK ET AL. (eds.): Topics in Algebra, Vol. 26, Banach Center Publ., 1990, pp. 127-180. [2] AUSLANDER, M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter functors without diagrams', Trans. Amer. Math. Soc. 250

(1979), 1-46. [3] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAROW, V.A.: 'Coxeter functors and Gabriel's theorem', Russian Math. Surveys 28 (1973), 17-32. [4] BONGARTZ, K.: 'Tilted algebras', in M. AUSLANDER AND E. LLUIS (eds.): Representations of Algebras. Proc. I C R A III, Vol. 903 of Lecture Notes in Mathematics, Springer, 1981, pp. 26 38. [5] BRENNER, S., AND BUTLER, M.: 'Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors', in V. DLAB AND P. GABRIEL (eds.): Representation Theory II. Proc. ICRA II, Vol. 832 of Lecture Notes in Mathematics, Springer, 1980, pp. 103 169. [6] HAPPEL, D.: Triangulated categories in the representation theory of finite dimensional algebras, Vol. 119 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1988. [7] HAPPEL, D., REITEN, I., AND SMAL0, S.O.: 'Tilting in abelian categories and quasitilted algebras', Memoirs Amer. Math. Soc. 575 (1996). [8] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras', Trans. Amer. Math. Soc. 274 (1982), 399-443. [9] KERNER, O.: 'Tilting wild algebras', J. London Math. Soc. 39, no. 2 (1989), 29-47. [10] KERNER, O.: 'Wild tilted algebras revisited', Colloq. Math. 73 (1997), 67-81. [11] LIu, S.: 'The connected components of the Auslander-Reiten quiver of a tilted algebra', J. Algebra 161 (1993), 505-523. [12] RINGEL, C.M.: Tame algebras and integral quadratic forms, Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984. [13] RINGEL, C.M.: 'The regular components of the AuslanderReiten Quiver of a tilted algebra', Chinese Ann. Math. Set. B. 9 (1988), 1-18. [14] STRAUSS, H.: 'On the perpendicular category of a partial tilting module', J. Algebra 144 (1991), 43-66. O. K e r n e r

MSC 1991: 16G10, 16G20, 16G60, 16G70 When studying an a l g e b r a A, it is sometimes convenient to consider another algebra, given for instance by the endomorphism of an TILTING F U N C T O R -

TILTING THEORY appropriate A-module, and functors between the two module categories. For instance, this is the basis of the M o r i t a e q u i v a l e n c e or the construction of the socalled Auslander algebras. An important example of this strategy is given by the t i l t i n g t h e o r y and the tilting functors, as now described. Let A be a finite-dimensional k-algebra, where k is a field, T a tilting (finitely-generated) A-module (cf. T i l t i n g m o d u l e ) and B = EndA(T). One can then assign to T the functors HomA ( T , - ) , - ® B T, Ext~ ( T , - ) , and TOrlB ( - , T), which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categories mod A and rood B, results first established by S. Brenner and M.C.R. Butler. Namely, H o m A ( T , - ) and its adjoint - ®B T give an equivalence between the subcategories

T(TA) = {MA: E x t l ( T , M) = 0} and

Y(TA) = {NB: TorB(N,T) = 0}, while Ext}4(T , - ) and T o r B ( - , T ) give an equivalence between the subcategories

Y(TA) = {NB: Tor~(N,T) = O} and

X(TA) = {NB: N ® , T = 0}. It is not difficult to see that (T(TA),5(TA)) and (X(TA), Y(TA)) are torsion pairs in rood A and rood B, respectively. Clearly, one can now transfer information from rood A to rood B. One of the most interesting cases occurs when A is a hereditary algebra and so the totsion pair (X(TA), Y(TA)) splits, giving in particular that each indecomposable B-module is the image of an indecomposable A-module either by H o m A ( T , - ) or by E x t , ( T , - ) (in this case, the algebra B is called tilted, cf. also T i l t e d algebra). This procedure has been generalized in several ways and it is worthwhile mentioning, for instance, its connection with derived categories (cf. also D e r i v e d category), or the notion of quasi-tilted algebras. It has also been considered for infinitely-generated modules over arbitrary rings. For referenes, see also T i l t i n g t h e o r y ; T i l t e d algebra.

ii) Ext (T,T) = 0; and iii) the number of non-isomorphic indecomposable summands of T equals the number of simple A-modules. The fundamental work by S. Brenner and M.C.R. Butler, and D. Happel and C.M. Ringel, on tilting theory have established the relations between the module categories rood A and rood B, where B = EndA(T), through the tilting functors HOmA (T, - ) and Ext~ (T, - ) (cf. also T i l t i n g f u n c t o r ) . The particular case where A is a hereditary algebra gives rise to the notion of a t i l t e d algebra, which nowadays (as of 2000) plays a very important role in the representation theory of algebras. One can also consider the dual notion of eotiltin9 rood-

ules. T i l t i n g t h e o r y goes back to the work of I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev on the characterization of representation-finite hereditary algebras through their ordinary quivers (cf. also Quiver). Their reflection functors on quivers has led to a module-theoretical interpretation by M. Auslander, M.I. Platzeck and I. Reiten. Next steps in this theory are the work by Brenner and Butler and Happel and Ringel, which gave the basis for all its further development. Worthwhile mentioning is the connection of tilting theory with derived categories established by Happel (cf. also D e r i v e d c a t e g o r y ) . The success of this strategy to study a bigger class of algebras through tilting theory has led to several generalizations. On one hand, one can relax the condition on the projective dimension and consider tilting modules of finite projective dimension. In this way it was possible to show the connection between tilting theory and some other homological problems in the representation theory of algebras. On the other hand, this concept can be generalized to a so-called tilting object in more general Abelian categories. For instance, this has led to the notion of a quasPtilted algebra. Recently (as of 2000), there has been much work also on exploring such notions in categories of (not necessarily finitely-generated) modules over arbitrary rings. For references, see also T i l t i n g t h e o r y ; T i l t e d algebra.

Fldvio Ulhoa Coelho MSC 1991: 16Gxx

Fldvio Ulhoa CoeIho MSC 1991: 16Gxx TILTING MODULE - A (classical) tilting module over a finite-dimensional k-algebra A (cf. also A l g e b r a ) , where k is a field, is a (finitely-generated) A-module T satisfying:

i) the projective d i m e n s i o n of T is at most one;

TILTING THEORYA r t i n algebras. A finitely-generated m o d u l e T over an Artin algebra A (cf. also A r t i n i a n m o d u l e ) is called a tilting module if p. dim A T _< 1 and E x t , ( T , T) = 0 and there is a short e x a c t s e q u e n c e 0 -+ A --+ To --> T1 ~ 0 with To,T1 C a d d T . Here, p. dimAT denotes the projective dimension of T and add T is the category

405

TILTING T H E O R Y of finite direct sums of direct summands of T (see T i l t ing m o d u l e ) . Dually, a A-module T is called a cotilting module if the A°P-module D(T) is a tilting module, where D denotes the usual duality. If T is a tilting roodule and F = E n d r ( T ) °p, then T is a tilting module over F °p. Hence D(T) is a cotilting F-module. Let T be a tilting module, and let T = F a c t be the c a t e g o r y of finitely-generated A-modules generated by T. The category T is a torsion class in the category m o d A of finitely-generated A-modules. This yields an associated torsion pair (T,~C), where • = {C: HomA(T, C) = 0}. Dually, there is associated with a cotilting module T the subcategory y = Sub T of Amodules cogenerated by T. The category 3; is a torsionfree class and there is an associated torsion pair (2(, y ) where 2( = {C: HomA(C,Y) = 0}. An important feature of tilting theory is the following connection between modA and m o d F when F = EndA(T) °p for a tilting module T: If (T, 2r) denotes the torsion pair in mod A associated with T and (2(, y ) the torsion pair associated with D(T), then there are equivalences of categories: HomA(T, .): T --+ 32 and

Ext ,(T, .): f

2(.

(Cf. also T i l t i n g f u n c t o r . ) In the special case where T is a projective generator one recovers the M o r i t a e q u i v a l e n c e HOmA(T, .) : rood A --+ mod F, where T is a projective generator of mod A. For a general module T, the Artin algebras A and F may be quite different, but they share many homological properties; in particular, one uses the tilting functors Homa(T, .) and Ext~ (T, .) in order to transfer properties between mod A and mod F. The transfer of information is especially useful when one already knows a lot about mod A and when the torsion pair (2(, y ) splits, that is, when each indecomposable F-module is in 2( or in y . This is the case when A is hereditary. In this case, F is called a tilted algebra (cf. also T i l t e d a l g e b r a ) . Tilted algebras have played an important part in representation theory, since many questions can be reduced to this class of algebras. Tilting theory goes back to the reflection functors introduced by I.N. BernshteYn, I.M. Gel'fand and V.A. Ponomarev [5] in the early 1970s. A module-theoretic interpretation of these functors was given by M. Auslander, M.I. Platzeck and I. Reiten [3]. Further generalizations where given by S. Brenner and M.C.R. Butler [6], where the equivalence Homa(T, .): T -~ 3; was established. The above definitions where given by D. Happel and C.M. Ringel [13], who developed an extensive theory of tilted algebras. A good reference for the early work in tilting theory is [2]. 406

An important theoretical development of tilting theory was the connection with derived categories established by Happel [10]. The functor HomA(T, .) : rood A --+ rood F when T is a tilting module induces an equivalence R H o m A ( T , - ) : Db(A) -+ Db(F), where Db(A) denotes the d e r i v e d c a t e g o r y whose objects are the bounded complexes of A-modules. The set of all tilting modules (up to isomorphism) over a k-algebra A, k an a l g e b r a i c a l l y c l o s e d field, has an interesting combinatorial structure: It is a countable s i m p l i c i a l c o m p l e x E. This complex has been investigated by L. Unger in [21] and [22], where it was proved that E is a shellable simplicial complex provided it is finite, and that certain representation-theoretical invariants are reflected by its structure. A n a l o g u e s a n d g e n e r a l i z a t i o n s . There is an analogous concept of a tilting sheaf T for the category coh X of coherent sheaves of a weighted projective line X (cf. also C o h e r e n t s h e a f ) as studied in [9]. The canonical algebras introduced in [19] can be realized as endomorphism algebras of certain tilting sheaves. To obtain a common treatment of both the class of tilted algebras and the canonical algebras, in [12] tilting theory was generalized to hereditary categories 7{, that is, 7{ is a connected Abelian k-category with vanishing Yoneda functor Ext2( ., .) and finite-dimensional homomorphism and extension spaces. Here, k denotes an algebraically closed field. An object T in 7{ with E x t ~ ( T , T ) = 0 such that H o m ~ ( T , X ) = 0 = Ext~t(T,X ) implies X = 0, is called a tilting object in 7{. The endomorphism algebra End~ T of a tilting object T is called a quasi-tilted algebra. Tilted algebras and canonical algebras furnish examples for quasi-tilted algebras. There are two types of hereditary categories 7-/with tilting objects: those derived equivalent to m o d H for some finite-dimensional hereditary k-algebra H and those derived equivalent to some category coh X of coherent sheaves on a weighted projective line X. Two categories are called derived equivalent if their derived categories are equivalent as triangulated categories. In 2000, Happel [11] proved that these are the only possible hereditary categories with tilting object. This proved a conjecture stated, for example, in [17].

Generalizations and applications of tilting modules. A A-module T is called a generalized tilting module if pd A T = n < ec and Ext~ (T, T) = 0 for i > 0 and there is an exact sequence 0 ~ A --+ T1 --+ ' . . --+ Tn --+ 0 with Ti C add T. Generalized tilting modules were introduced in [16]. This concept was generalized to the notion of tilting complexes by J. Rickard [18], who established some :Morita theory for derived categories'.

TITS QUADRATIC FORM Let R be a ring and let PA be the category of finitelygenerated projective A-modules. Denote by Kb(PA) the category of bounded complexes over PA modulo homotopy. A complex T E Kb(pA) is called a tilting complex if Homgb(pA)(T,T[i]) = 0 for a l l / # 0 (here, [.] denotes the shift functor) and if a d d T generates Kb(pA) as a triangulated category. Rickard proved that two rings R and R ~ are derived equivalent (i.e. their module categories are derived equivalent) if and only if R ~ is the endomorphism ring of a tilting complex T E Kb(PA). The results mentioned above uses tilting modules/objects mainly to compare modA and modF, where F = EndA T for some tilting module/object. There are other approaches, which use tilting modules to describe subcategories of mod A. Kerner [15] and W. Crawley-Boevey and Kerner [7] used tilting modules to investigate subcategories of regular modules over wild hereditary algebras. Q u a s i - h e r e d i t a r y algebras. Auslander and Reiten [4] proved that there is a one-to-one correspondence between basic generalized tilting modules and certain covariantly finite subcategories of rood A. This correspondence was further investigated [14]. The AuslanderReiten correspondence was applied to quasi-hereditary algebras by Ringel [20] and his results served as a basis for applications to Schur algebras by S. Donkin [8] and to q u a n t u m g r o u p s by H.H. Andersen [1]. In dealing with quasi-hereditary algebras and highest-weight categories, the notion of a tilting module is now (2000) used in a related but deviating way, namely for all the objects or modules that have both a A-filtration and a V-filtration. The isomorphism classes of the indecomposables that have both a A-filtration and a V-filtration correspond bijectively to the elements of the weight poset, and their direct sum is a tilting module in the sense considered above.

References [i] ANDERSEN, H.H.: 'Tensor products of quantized tilting modules', Commun. Math. Phys. 149, no. 1 (1992), 149-159. [2] ASSEM, I.: 'Tilting theory - an introduction': Topics in Algebra, Vol. 26 of Banach Center Publ., PWN, 1990, pp. 127180. [3] AUSLANDER,M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter functors without diagrams', Trans. Amer. Math. Soe. 250 (1979), 1-12. [4] AUSLANDER,M., AND REITEN, I.: 'Applications of contravariantly finite subcategories', Adv. Math. 86, no. 1 (1991), 111152. [5] BERNSTEIN, I.N., CELFAND, I.M., AND PONOMAREV, V.A.: 'Coxeter functors and Gabriel's theorem', Russian Math. Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973),

19-33.)

[6] BRENNER, S., AND BUTLER, M.C.R.: 'Generalization of Bernstein-Gelfand-Ponomarev reflection functors': Proc. Ottawa Conf. on Representation Theory, 1979, Vol. 832 of Lecture Notes in Mathematics, Springer, 1980, pp. 103-169. [7] CRAWLEY-BOEVEY, W., AND KERNER, O.: 'A functor between categories of regular modules for wild hereditary algebras', Math. Ann. 298 (1994), 481-487. [8] DONKIN, S.: 'On tilting modules for algebraic groups', Math. Z. 212, no. 1 (1993), 39-60. [9] GEIGLE, W., AND LENZING, H.: 'Perpendicular categories with applications to representations and sheaves', J. Algebra 144 (1991), 273 343. [10] HAPPEL, D.: 'Triangulated categories in the representation theory of finite dimensional algebras', London Math. Soc. Lecture Notes 119 (1988). [11] HAPPEL, D.: 'A characterization of hereditary categories with tilting object', preprint (2000). [12] HAPPEL, D., REITEN, R., AND SMALO, S.O.: 'Tilting in abelian categories and quasitilted algebras', Memoirs Amer. Math. Soc. 575 (1996). [13] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras', Trans. Amer. Math. Soc. 274 (1982), 399-443. [14] HAPPEL, D., AND UNGER, L.: 'Modules of finite projective dimension and cocovers', Math. Ann. 306 (1996), 445-457. [15] KERNER, O.: 'Tilting wild algebras', J. London Math. Soc. 39, no. 2 (1989), 29-47. [16] MIYASHITA, Y.: 'Tilting modules of finite projective dimension', Math. Z. 193 (1986), 113-146. [17] REITEN, I.: 'Tilting theory and quasitilted algebras': Proc. Internat. Congress Math. Berlin, Vol. II, 1998, pp. 109-120. [18] RICKARD, J.: 'Morita theory for derived categories', J. London Math. Soc. 39, no. 2 (1989), 436-456. [19] RINGEL, C.M.: 'The canonical algebras': Topics in Algebra, Vol. 26:1 of Banach Center Publ., PWN, 1990, pp. 407-432. [20] RINGEL, C.M.: 'The category of modules with good filtration over a quasi-hereditary algebra has alost split sequences', Math. Z. 208 (1991), 209-224. [21] UNGER, L.: 'The simplicial complex of tilting modules over quiver algebras', Proc. London Math. Soc. 73, no. 3 (1996), 27-46. [22] UNGER,L.: 'Shellability ofsimplicial complexes arising in representation theory', Adv. Math. 144 (1999), 221-246. L. Unger

MSC 1991: 16Gxx

TITS QUADRATIC F O R M - Let Q = (Qo, Q1) be a finite quiver (see [8]), that is, an oriented graph with vertex set Q0 and set Q1 of arrows (oriented edges; cf. also G r a p h , o r i e n t e d ; Quiver). Following P. Gabriel [8], [9], the Tits quadratic form qQ: Z Qo -+ Z of Q is defined by the formula 2 jCQo

i,jCQo

where x = (xi)icQo E Z Q° and diy is the number of arrows from i to j in Q1. There are important applications of the Tits form in representation theory. One easily proves that if Q is connected, then qQ is positive definite if and only if Q (viewed as a non-oriented graph) is any of the Dynkin 407

TITS QUADRATIC FORM diagrams An, D~, E6, ET, or Es (cf. also D y n k i n diag r a m ) . On the other hand, the Gabriel theorem [8] asserts that this is the case if and only if Q has only finitely many isomorphism classes of indeeomposable K-linear representations, where K is an a l g e b r a i c a l l y closed field (see also [2]). Let rePK(Q ) be the A b e l i a n c a t e g o r y of finite-dimensional K-linear representations of Q formed by the systems X = (Xi,¢9)jeQo,9~Q~ of finite-dimensional vector/(-spaces Xj, connected by Klinear mappings CZ : Xi --+ Xj corresponding to arrows /3: i --+ j of Q. By a theorem of L.A. Nazarova [12], given a connected quiver Q the category rePK(Q) is of tame representation type (see [7], [10], [19] and Quiver) if and only if qQ is positive semi-definite, or equivalently, if and only if Q (viewed as a non-oriented graph) is any of the extended Dynkin diagrams A~, Dn, E6, ET, or Es (see [1], [10], [19]; and [4] for a generalization). Let Ko(Q) = K0(reptc(Q)) be the G r o t h e n d i e c k g r o u p of the category repK(Q ). By the J o r d a n HSlder t h e o r e m , the correspondence X dimX = (dimKXj)jeQo defines a group isomorphism dim: Ko(Q) -+ Z Q°. One shows that the Tits form qQ coincides with the E u l e r c h a r a c t e r i s t i c XQ: Ko(Q) --+ Z, IX] ~ XQ([X]) = dimK EndQ(X) - dimN E x t ~ ( X , X ) , along the isomorphism dim: Ko(Q) --+ Z Q°, that is, qQ(dimX) = XQ([X]) for any X in repg(Q) (see [10], [17]). The Tits quadratic form qQ is related with an alg e b r a i c g e o m e t r y context defined as follows (see [9], [10], [19]). For any vector v = (vj)jcQo E N Q°, consider the att:ine irreducible K-variety AQ(V) = I-Ii,jEQo H(j3:j--~i)CQ1 M ~ ×~j (K)z of K-representations of Q of the dimension type v (in the Zariski topology), where M ~ ×vj (K)Z = M ~ x~j (K) is the space of (vi x vj)-matrices for any arrow fl : j -+ i of Q. Consider the algebraic g r o u p GgQ(d) = [IjcQo GI(vj,K) and the algebraic group action *: ~gQ(d) x AQ(d) --+ AQ(d) defined by the formula (hi) * (M~) = (h~-lM~hj), where fl: j ~ i is an arrow of Q, M~ C Mvj×v~(K)9, hj E GI(vj,K), and hi C Gl(vi,K). An important role in applications is played by the Tits-type equality qQ(v) = dimGfQ(V) - dimAQ(v), v C N Q°, where dim denotes the d i m e n s i o n of the a l g e b r a i c v a r i e t y (see [8]). Following the above ideas, Yu.A. Drozd [5] introduced and successfully applied a Tits quadratic form in the study of finite representation type of the Krull-Schmidt category Mats of matrix K-representations of partially ordered sets ([, -<) with a unique maximal element (see [10], [19]). In [6] and [7] he also studied bimodule matrix problems and the representation type of boxes 13 by means of an associated Tits quadratic form qB : Z ~ ~ Z 408

(see also [18]). In particular, he showed [6] that if 13 is of tame representation type, then q~ is weakly nonnegative, that is, q~(v) > 0 for all v C N ~. K. Bongartz [3] associated with any finitedimensional basic K-algebra R a Tits quadratic form as follows. Let { e l , . . . , en} be a complete set of primitive pairwise non-isomorphic orthogonal idempotents of the algebra R. Fix a finite quiver Q = (Qo,Q1) with Qo = { 1 , . . . , n } and a K-algebra isomorphism R ~- K Q / I , where KQ is the path K-algebra of the quiver O (see [1], [10], [19]) and I is an ideal of R contained in the square of the J a c o b s o n r a d i c a l rad R of R and containing a power of rad R. Assume that Q has no oriented cycles (and hence the global dimension of R is finite). The Tits quadratic form qn : Z n -+ Z of R is defined by the formula

qR(x) = Z jCQo

2

Z (~: i--+j)cQ1

xixj+

Z

r ,j i j,

(fl: i--+j)cQ1

where ri,j = IL M ejIei[, for a minimal set L of generators of I contained in ~i,j~Qo ejIei. One checks that rid = dimK Ext~(Sj,Si), where St is the simple Rmodule associated to the vertex t E Q0. Then the definition of qR depends only on R, and when R is of global dimension at most two, the form qR coincides with the Euler characteristic XR: K0(modR) -+ Z, [X] ~ Xn([X]) = Y2~=0(-1) m dimg E x t , ( X , X), under a group isomorphism dim: K0(modR) -+ Z Q°, where K0(mod R) is the G r o t h e n d i e c k g r o u p of the category mod R of finite-dimensional right R-modules (see [17]). Note that qR = qQ if R = KQ. By applying a Tits-type equality as above, Bongartz [3] proved that if R is of finite representation type, then qn is weakly positive, that is, qn(v) > 0 for all non-zero vectors v E N ~. The converse implication does not hold in general, but it has been established if the AuslanderReiten quiver of R (see R i e d t m a n n classification) has a post-projective component (see [10]), by applying an idea of Drozd [5]. J.A. de la Pefia [14] proved that if R is of tame representation type, then qR is weakly nonnegative. The converse implication does not hold in general, but it has been proved under a suitable assumption on R (see [13] and [16] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of R). Let (I, ~) be a partially ordered set with partial order relation -< and let max I be the set of all maximal elements of (I, __). Following [5] and [15], D. Simson [20] defined the Tits quadratic form q~: Z I --+ Z of (I, _-3) by the formula

iEI

i~j

jEI\max I

pCmax I

T O E P L I T Z C*-ALGEBRA and applied it in the study of prinjective KI-modules, that is, finite-dimensional right modules X over the incidence K-algebra K I = K(I, ~_) of (I,__) such that there is an exact s e q u e n c e 0 -+ P1 -+ P0 ~ X -+ 0, where P0 is a projective K I - m o d u l e and P1 is a direct sum of simple projectives. The additive Krull-Schmidt category p r i n K I of prinjective KI-modules is equivalent to the category of matrix K-representations of (I, _) [20]. Under an identification Ko(prinKI) ~- Z I, the Tits form qI is equal to the Euler characteristic XKZ: Ko(prin KI) -+ Z. A Tits-type equality is also valid for qr [15]. It has been proved in [20] that q1 is weakly positive if and only if prin K I has only a finite number of iso-classes of indecomposable modules. By [15], if p r i n K / i s of tame representation type, then qI is weakly non-negative. The converse implication does not hold in general, but it has been proved under some assumption on (I, _) (see [11]). A Tits quadratic form qA : Z n --+ Z for a class of classical D-orders A, where D is a complete discrete valuation domain, has been defined in [21]. Criteria for the finite lattice type and tame lattice type of A are given in [21] by means of qA. For a class of K-co-algebras C, a Tits quadratic form --+ Z is defined in [22], and the co-module types of C are studied by means of qc, where Ic is a complete set of pairwise non-isomorphic simple left Cco-modules and Z (Ic) is a free Abelian group of rank

q c : Z (Iv)

Ircl. References [1] AUSLANDER, V.I., REITEN, I., AND SMAL0, S.: Representation theory of Artin algebras, Vol. 36 of Studies Adv. Math., Cambridge Univ. Press, 1995. [2] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAREV, V.A.: 'Coxeter functors and Gabriel's theorem', Russian Math. Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973),

19 33.) [3] BONGARTZ, N.: 'Algebras and quadratic forms', J. London Math. Soe. 28 (1983), 461-469. [4] DLAB, V., AND RINGEL, C.M.: Indecomposable representations of graphs and algebras, Vol. 173 of Memoirs, Amer. Math. Soc., 1976. [5] DROZD, Yu.A.: 'Coxeter transformations and representations of partially ordered sets', Funkts. Anal. Prilozhen. 8 (1974), 34 42. (In Russian.) [6] DROZD, Yu.A.: 'On tame and wild matrix problems': Matrix Problems, Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev, 1977, pp. 104-114. (In Russian.) [7] DROZD, Yu.A.: 'Tame and wild matrix problems': Representations and Quadratic Forms, 1979, pp. 39-74. (In Russian.) [8] GABRIEL, P.: 'Unzerlegbare Darstellungen 1', Manuscripta Math. 6 (1972), 71-103, Also: Berichtigungen 6 (1972), 309. [9] GABRIEL, P.: 'Reprfisentations ind~composables': Sdminaire Bourbaki (1973/73), Vol. 431 of Lecture Notes in Mathematics, Springer, 1975, pp. 143-169.

[10] GABRIEL, P., AND ROITER, A.V.: 'Representations of finite dimensional algebras': Algebra VIII, Vol. 73 of Encycl. Math. Stud., Springer, 1992. [11] KASJAN, S., AND SIMSON, D.: 'Tame prinjective type and Tits form of two-peak posers II', J. Algebra 187 (1997), 71-96. [12] NAZAROVA,L.A.: 'Representations of quivers of infinite type', Izv. Akad. Nauk. SSSR 37 (1973), 752-791. (In Russian.) [13] PEI~A, J.A. DE LA: 'Algebras with hypercritical Tits form': Topics in Algebra, Vol. 26:1 of Banaeh Center Publ., PWN, 1990, pp. 353-369. [14] PEI~A, J.A. DE LA: 'On the dimension of the module-varieties of tame and wild algebras', Commun. Algebra 19 (1991), 1795-1807. [15] PE~A, J.A. DE LA, AND S~MSON, D.: 'Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences', Trans. Amer. Math. Soe. 329 (1992), 733-753. [16] PE~A, J.A. DE LA, AND SKOWROr@KI, A.: 'The Euler and Tits forms of a tame algebra', Math. Ann. 315 (2000), 37-59. [17] RINGEL, C.M.: Tame algebras and integral quadratic forms, Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984. [18] ROITER, A.V., AND I~[LEINER, M.M.: Representations of differential graded categories, Vol. 488 of Lecture Notes in Mathematics, Springer, 1975, pp. 316-339. [19] SIMSON, D.: Linear representations of partially ordered sets and vector space categories, Vol. 4 of Algebra, Logic Appl., Gordon & Breach, 1992. [20] SIMSON, D.: 'Posets of finite prinjective type and a class of orders', J. Pure Appl. Algebra 90 (1993), 77-103. [21] SIMSON, D.: 'Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders', Contemp. Math. 229 (1998), 307-342. [22] SIMSON,D.: 'Coalgebras, comodules, pseudoeompact algebras and tame comodule type', Colloq. Math. in p r e s s (2001).

Daniel Simson

MSC 1991: 16Gxx TOEPLITZ C*-ALGEBRA - A uniformly closed *algebra of operators on a Hilbert space (a uniformly closed C * - a l g e b r a ) . Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables. In the one-dimensional case one considers the Hardy space H 2(T) over the one-dimensional torus T (cf. also H a r d y spaces), and defines the T o e p l i t z o p e r a t o r Tf with 'symbol' function f E L°°(T) by Tfh := P(fh) for all h E H 2 ( T ) , where P : L2(T) --+ H 2 ( T ) is the orthogonal projection given by the C a u c h y i n t e g r a l t h e o r e m . The C * - a l g e b r a T ( T ) := C* (Tf : f E C(T)) generated by all operators Tf with continuous symbol f is not commutative, but defines a C*-algebra extension 0 --+ K:(H2(T)) ~ T ( T ) --+ C(T) --+ 0 of the C*-algebra ~ of all compact operators; in fact, this 'Toeplitz extension' is the generator of the Abelian group Ext(C(T)) ~ Z. C*-algebra extensions are the building blocks of K t h e o r y and i n d e x t h e o r y ; in our case a Toeplitz 409

T O E P L I T Z C*-ALGEBRA operator Tf is Fredholm (cf. also F r e d h o h n o p e r a t o r ) if f C C(T) has no zeros, and then the index Index(Tf) = dim Ker T / - dim Coker T / i s the (negative)

w i n d i n g n u m b e r of f. In the multi-variable case, Toeplitz C*-algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains D C C ~ [1], including the unit ball D = {z 6 Cn: Izll 2 + . . . + Iznl 2 < 1} [2], [10], for tube domains and Siegel domains over convex 'symmetric' cones [5], [8], and for general bounded symmetric domains in C n having a transitive semi-simple Lie g r o u p of holomorphic automorphisms [7]. Here, the principal new feature is the fact that Toeplitz operators Tf (say, on the Hardy space H2(S) over the Shilov boundary S of a pseudo-convex domain D C C ~) with continuous symbols f E Co(S) are not essentially commuting, i.e.

[T/1,T/=] f~ K.(H2(S)), in general. Thus, the corresponding Toeplitz C*-algebra T ( S ) is not a (one-step) extension of K:; instead one obtains a multi-step C*-filtration = ~i ~ " " ~ I~ ~ T ( S )

of C*-ideals, with essentially commutative subquotients Zk+l/27k, whose maximal ideal space (its spectrum) refleets the boundary strata of the underlying domain. The length r of the composition series is an important geometric invariant, called the rank of D. The index theory and K - t h e o r y of these multi-variable Toeplitz C*algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain D, regarded as a complex K i i h l e r m a n i f o l d . A general method for studying the structure and representations of Toeplitz C*-algebras, at least for Shilov boundaries S arising as a symmetric space (not necessarily Riemannian), is the so-called C*-duality [11], [9]. For example, if S is a Lie g r o u p with (reduced) group C*-algebra C*(S), then the so-called co-crossed product C*-algebra C*(S) ®5 Co(S) induced by a natural co-action 5 can be identified with 1~(L2(S)). Now the Cauehy-Szeg5 orthogonal projection E: L2(S) -+ H2(S) (cf. also C a u c h y o p e r a t o r ) defines a certain C*-completion C~(S) D C*(S), and the corresponding Toeplitz C*-algebra T(S) can be realized as (a corner of) C~(S) ®5 Co(S). In this way the well-developed representation theory of (co-) crossed product C*-algebras [4] can be applied to obtain Toeplitz C*-representations related to the boundary cgD. For example, the twodimensional torus S = T 2 gives rise to non-type-I C*-algebras (for cones with irrational slopes), and the underlying 'Reinhardt' domains (cf. also R e i n h a r d t d o m a i n ) have interesting complex-analytic properties, 410

such as a non-compact solution operator of the Neumann 0-problem [6]. References [i] BOUTET DE MONVEL, L.: 'On the index of Toeplitz operators of several complex variables', Invent. Math. 50 (1979), 249-272. [2] COBURN, L.: 'Singular integral operators and Toeplitz operators on odd spheres', Indiana Univ. Math. Y. 23 (1973), 433-439. [3] DOUGLAS,R., AND HOWE, R.: 'On the C*-algebra of Toeplitz operators on the quarter-plane', Trans. Amer. Math. Soe. 158 (1971), 203-217. [4] LANDSTAD,M., PHILLIPS, J., RAEBURN, [., AND SUTHERLAND, C.: 'Representations of crossed products by coactions and principal bundles', Trans. Amer. Math. Soe. 299(1987), 747784. [5] MUHLY, P., AND RENAULT, J.: 'C*-algebras of multivariable Wiener-Hopf operators', Trans. Amer. Math. Soc. 274 (1982), 1-44. [6] SALINAS, N., SHEU, A., AND UPMEIER, H.: 'Toeplitz operators on pseudoconvex domains and foliation algebras', Ann. Math. 130 (1989), 531-565. [7] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric domains', Ann. Math. 119 (1984), 549-576. [8] UPMEIER, H.: 'Toeplitz operators on symmetric Siegel domains', Math. Ann. 271 (1985), 401-414. [9] UPMmER, H.: Toeplitz operators and index theory in several complex variables, Birkh£user, 1996. [10] VENUGOPALKRISHNA, W.: 'Fredholm operators associated with strongly pseudoconvex domains in C n', J. Funct. Anal. 9 (1972), 349 373. [11] WASSERMANN, A.: 'Alg~bres d'op~rateurs de Toeplitz sur les groupes unitaires', C.R. Acad. Sci. Paris 299 (1984), 871874. H. U p m e i e r

MSC 1991: 46Lxx TOEPLITZ SYSTEM - A system of linear equations

Tx = a with T a T o e p l i t z m a t r i x . MSC1991:15A57

TRAVELLING SALESMAN PROBLEM A generic name for a number of very different problems. For instance, suppose that a facility (a 'machine') starting from an 'idle' position is assigned to process a finite set of 'jobs' (say n, n > 3 jobs). If the machine has to be 'calibrated' (or %et-up') for processing each of these jobs and if the machine's 'calibration time' (the distance metric) between processing of a pair of jobs in succession is dependent on the particular pair, then a reasonable objective is to organize this job assignment so it will minimize the total machine calibration time. One might want to assume that after the last job is processed the machine returns to its idle position. A very similar problem exists when the 'machine' corresponds to a computer centre which has n programs to

TRIANGLE CENTRE run, and each program requires resources such as a compiler, a certain portion of the main memory, and perhaps some other 'devices'. I.e., each program requires a specific configuration of devices. Conversion cost (or time) from one configuration to another, say from the configuration of program i to that of program j is denoted by cij (>_ 0). Thus, the question becomes that of determining the cost minimizing order in which all the programs ought to be run. If at the end of running all the programs by the computer centre the system returns to an 'idle' configuration, then the number of possible ways to run these programs one after the other equals n! (for n + 1 configurat!ons). This is the same problem as that in the story about the lonely salesman who has to visit n sales outlets (starting from his home) and wishes to travel the shortest total distance in the process. It is the salesman's problem to select a distance-minimizing travel order of outlet visits. Thus, the name travelling salesman prob-

lem.

possible order magnitude of the required number of computer operations. This casts these problems (the travelling salesman and the Hamiltonian circuit problems) as being 'hard' (cf. also A/P). Essentially, for this sort of problems, one does not presently (2000) know of any solution scheme which does not require some sort of enumeration of all possible 'configuration' sequences. See [2], [1] for recent overviews of the problem. References [1] FLEISHNER, H.: '~IYaversing graphs: The Eulerian and Hamiltonian theme', in M. DROR (ed.): Arc Routing: Theory, Solutions, and Applications, Kluwer Acad. Publ., 2000. [2] LAWLER, E.L., LENSTRA, J.K., RINNOY KAN, A.H.G., AND SHMOYS, D.B. (eds.): The traveling salesman problem, Wiley, 1985.

Moshe Dror MSC 1991:90C08 T R I A N G L E C E N T R E - Given a triangle A1A2A3, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:

In graph terminology terms, the problem is presented as that of a g r a p h G = (V, E), where V is a finite set of nodes ('cities') and E C_ V x V is the set of edges connecting the node pairs in V. If one associates a realvalued 'cost' matrix (cij), i , j = 1,..., IVI, with the set of edges E, the travelling salesman dilemma becomes that of constructing a cost-minimizing circuit on G that visits all the nodes in V exactly once, if such a circuit exists (eft also G r a p h c i r c u i t ) . If the requirement is that all the nodes in V are visited in a cost-minimizing fashion but without necessarily forming a circuit, then the problem is referred to as a travelling salesman path problem, or travelling salesman walk problem. Again, the question of the existence of such a path has to be addressed first. If the graph G = (V, A), A _C V x V, assigns a 'direction' to each element in A (a subset of arcs), then the corresponding travelling salesman problem is of the 'directed' variety. Clearly, there is the option of the mixed problem, where some of the node pairs are connected by arcs and some by edges. The question of whether a circuit exists in a graph G which visits each node in V exactly once is commonly referred to as that of determining the existence of a Hamiltonian circuit (or path; cf. also H a m i l t o n i a n t o u r ) . Graphs for which such a circuit (path) is guaranteed to exist are called Hamiltonian graphs. The difficulty of determining the existence of a Hamiltonian circuit for a graph G and that of constructing a cost-minimizing travelling salesman circuit on a graph G are very much the same when measured by the worst

• the centroid (i.e. the centre of mass), the common intersection point of the three medians (see M e d i a n (of a triangle));

• the incentre, the common intersection point of the three bisectrices (see B i s e c t r i x ) and hence the centre of the ineirele (see P l a n e t r i g o n o m e t r y ) ; • the circumcentre, the centre of the circumcircle (see P l a n e t r i g o n o m e t r y ) ; • the orthocentre, the common intersection point of the three altitude lines (see P l a n e t r i g o n o m e t r y ) ; • the G e r g o n n e p o i n t , the common intersection point of the lines joining the vertices with the opposite tangent points of the incircle; • the Format point (also called the Torrieelli point or first isogonic centre), the point X that minimizes the sum of the distances IAIX] + IA2XI + IA3XI; • the Grebe point (also called the Lemoine point or symmedean point), the common intersection point of the three symmedeans (the symmedean through Ai is the isogonal line of the median through Ai, see I s o g o n a l ) ; • the N a g e l p o i n t , the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see P l a n e t r i g o n o m e t r y ) . In [2], 400 different triangle centres are described. The Nagel point is the isotomic conjugate of the Gergonne point, and the symmedean point is the isogonal conjugate of the centroid (see I s o g o n a l for both notions of 'conjugacy'). References [1] JOHNSON, R.A.: Modern geometry, Houghton-Mifflin, 1929.

411

TRIANGLE CENTRE [2] KIMBERLING, C.: 'Triangle centres and central triangles', Congr. Numer. 1 2 9 (1998), 1-285. M. Hazewinkel

MSC 1991:51M04 T R I B O N A C C I NUMBER A member of the Trib o n a c c i s e q u e n c e . The formula for the nth number is given by A. Shannon in [1]: [~/2] In/a] m=0 r=0

m -I- r

r

B i n e t ' s f o r m u l a for the nth number is given by W.

Spickerman in [2]: fin+2

G-n+2

T~=

+

t-

~n+2 -

References [1] ATANASSOV, K., HLEBAROVA, J., AND MIHOV, S.: 'Recurrent formulas of the generalized Fibonacci and Tribonacci sequences', The Fibonacci Quart. 30, no. 1 (1992), 77-79. [2] BRUCE, I.: 'A modified Tribonacci sequence', The Fibonacci Quart. 22, no. 3 (1984), 244-246. [3] FEINBERG, M.: ' F i b o n a c c i - T r i b o n a c c i ' , The Fibonacci Quart. 1, no. 3 (1963), 71-74. [41 LEE, J.-Z., AND LEE, J.-S.: 'Some properties of the generalization of the Fibonacci sequence', The Fibonacci Quart. 25, no. 2 (1987), 111 117. [5] SCOTT, A., DELANEY, T., AND HOGGATT JR., V.: 'The Tribonacci sequence', The Fibonacci Quart. 15, no. 3 (1977), 193-200. [6] SHANNON, A.: 'Tribonacci n u m b e r s and Pascal's pyramid', The Fibonacci Quart. 15, no. 3 (1977), 268; 275. [71 VALAVIGI, C.: 'Properties of Tribonacci n u m b e r s ' , The Fibonacci Quart. 10, no. 3 (1972), 231-246.

er~ ! 91".; : 11B39 M~,~

-

K r a s s i m i r Atanassov

where

P=~1 ((19

-~- 3 X / ~ ) 1/3 -l- (19 -- 3V/~) 1/3 -~-

1)

~ = ~ 1 [2 - (19 + 3 v / ~ ) '/a - (19

-

3 V ~ ) 1/3] -~-

+ -vg~- i [(19 + 3X/~) 1/3 - (19 - 3 v / ~ ) 1/3] and ~ is the complex conjugate of ~. References [1] SHANNON, A.: 'Tribonacci n u m b e r s and Pascal's pyramid', The Fibonacci Quart. 15, no. 3 (1977), 268; 275. [2] SPICKERMAN, W.: ' B i n e t ' s formula for the Tribonacci sequence', The Fibonacci Quart. 15, no. 3 (1977), 268; 275.

Krassimir Atanassov

MSC 1991:11B39 T R I B O N A C C I S E Q U E N C E - An extension of the sequence of F i b o n a c c i n u m b e r s having the form (with a, b, c given constants):

to = a,

tl = b,

t2 = c,

tn+3 = tn+2 + t~+l + tn

(n ~

0).

The concept was introduced by the fourteen-yearold student M. Feinberg in 1963 in [3] for the case: a = b = 1, c = 2. The basic properties are introduced in [2], [5], [6], [7]. The Tribonacci sequence was generalized in [1], [4] to the form of two sequences: an÷3 = t t n + 2 -~- Wn+l ~- Yn, bn+3 = Vn+2 -}- Xn+l ~- Zn,

where u , v , w , x , y , z E { a , b } and each of the tuples (u, v), (w, x), (y, z) contains the two symbols a and b. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them. See also T r i b o n a c c i n u m b e r . 412

TRIGOiX,,~v.~ETRIC

PSEUDO-SPECTRAL

METH-

Trigonometric pseudo-spectral methods, and spectral methods in general, are methods for solving differential and integral equations using trigonometric functions as the basis. Suppose the boundary value problem L u = f is to be solved for u ( x ) on the interval x = [a, b], where L is a d i f f e r e n t i a l o p e r a t o r in x and f ( x ) is some given smooth function (cf. also B o u n d a r y v a l u e p r o b l e m , o r d i n a r y d i f f e r e n t i a l e q u a t i o n s ) . Also, u must satisfy given boundary conditions u(a) = u~ and u(b) = u b . As in most numerical methods, an approximate solution, UN, is sought which is the sum of N + 1 basis functions, ¢ , ( x ) , n = 0 , . . . , N , in the form UN = N ~ = 0 a , ¢ ~ ( x ) , where the coefficients an are the finite set of unknowns for the approximate solution. A 'residual equation', formed by plugging the approximate solution into the differential equation and subtracting the right-hand side, R ( x; ao, . . . , aN) =- L[u g ( x) ] -- f , is then minimized over the interval to find the coefficients. The difference between methods boils down to the choice of basis and how R is minimized. The basis functions should be easy to compute, be complete or represent the class of desired functions in a highly accurate manner, and be orthogonal (cf. also C o m p l e t e s y s t e m o f f u n c t i o n s ; O r t h o g o n a l s y s t e m ) . In spectral methods, t r i g o n o m e t r i c f u n c t i o n s and their relatives as well as other o r t h o g o n a l p o l y n o m i a l s are used. If the basis functions are trigonometric functions such as sines or cosines, the method is said to be a Fourier spectral method. If, instead, C h e b y s h e v p o l y n o m i a l s are used, the method is a Chebyshev spectral method. The method of mean weighted residuals is used to minimize R and find the unknowns coefficients a~. An inner product (.,-) and weight function p ( x ) are defined, ODS

-

TRIGONOMETRIC PSEUDO-SPECTRAL METHODS as well as N + 1 test functions wi such that (wi, R) = 0 for i = 0 , . . . , N and (u,v) = f : u ( x ) v ( x ) p ( x ) dx. This yields N + 1 equations for the N + 1 unknowns. Pseudospectral methods, including Fourier pseudo-spectral and Chebyshev pseudo-spectral methods, have Dirac deltafunctions (cf. also D i r a e d i s t r i b u t i o n ) as their test functions: wi(z) = 6(x - z i ) , where xi are interpolation or collocation points. The residual equation becomes R ( x i ; a o , . . . , a N ) = 0 for i = 0 , . . . , N . The G a l e r k i n m e t h o d uses the basis functions as the test functions. If L is linear, the following matrix equation can be formed: L~n,~a~ = fro, where L,~,~ = (4,~, L¢~) and f,~ = (f, ¢,~). An alternative Galerkin formulation can be found by transforming the residual equation into spectral space R(x; ao,..., aN) = ~-~.nrn(ao,...,aN)¢n(x) and setting r~ = 0 for n = O,..., N. In using G a u s s - J a c o b i integration to evaluate the inner products of the Galerkin method, the integrands are interpolated at the zeros of the iV + 1st basis function. By using the same set of points as collocation points for a pseudo-spectral method, the two methods are made equivalent. Problems can be cast in either gridpoint or spectral coefficient representation. For trigonometric bases, this result allows the complexity of computation to be reduced in m a n y problems through the use of fast transforms. A main difference between spectral and other methods, such as finite difference or finite element methods, is that in the latter the domain is divided into smaller subdomains in which local basis functions of low order are used. With the basis functions frozen, more accuracy is gained by decreasing the size of the subdomains. In spectral methods, the domain is not subdivided, but global basis functions of high order are used. Accuracy is gained by increasing the number and order of the basis functions. The lower-order methods produce algebraic systems which can be represented as sparse matrices. Spectral methods usually produce full matrices. The solution then involves finding the inverse. Through the use of orthogonality and fast transforms, full matrix inversion can usually be accomplished with a complexity similar to the sparse matrices. Boundary conditions are handled in a reasonably straightforward manner. Sometimes the boundary conditions are satisfied automatically, such as with periodicity and a Fourier method. With other types of conditions, an extra equation may be added to the system to satisfy it, or the basis functions may be modified to automatically satisfy the conditions. The attractiveness of spectral methods is that they have a greater than algebraic convergence rate for

smooth solutions. A simple finite difference approximation has a convergence l u - UNI = O(h~), where h = (b - a ) / N and c~ is an integer; double the number of points in the interval and the error goes down by a factor 2% The convergence rate of spectral methods is O(hh), sometimes called exponential, infinite oi" spectral, stemming from the fact that convergence of a trigonometric series is geometric. If the solution is not smooth, however, spectral methods will have an algebraic convergence linked to the continuity of the solution. Rapid convergence allows fewer unknowns to be used, but more computational processing per unknown. Hence spectral methods are particularly attractive for probl e m / c o m p u t e r matches in which m e m o r y and not computing power is the critical factor. Multi-dimensional problems are handled by tensorproduct basis functions, which are basis functions that are products of 1-dimensional basis functions. Other o r t h o g o n a l p o l y n o m i a l s can be used in pseudo-spectral methods, such as Legendre and Hermite and spherical harmonics for spherical geometries. A disadvantage of spectral methods is that only relatively simple domains and boundaries can be handled. Spectral element methods, a combination of spectral and finite element methods, have in m a n y cases overcome this difficulty. Another difficulty is that spectral methods are, in general, more complicated to code and require more analysis to be done prior to coding t h a n simpler methods.

Aliasin 9 is a phenomenon in which modes of degree higher than in the expansion are interpreted as modes t h a t are within the range of the expansion. This occurs in, say, a problem with quadratic non-linearity where twice the range is created. If the coefficients near the upper limit are sufficiently large in magnitude, there may be a significant error associated with aliased modes. For a Fourier pseudo-spectral method, the coefficient aN/2+k is interpreted as a coefficient aN/2-k. By zeroing the upper 1/3 of the coefficients, the quadratic nonlinearity will only fill a range from 2 / ( 3 N / 2 ) to 4/(3N/2). This will only produce aliasing errors for modes 2 / ( 3 N / 2 ) to N/2, but these are to be zeroed anyway. This '2/3' rule removes errors for one-dimensional problems with quadratic non-linearity. It is debatable, however, whether in a 'well-resolved' simulation there is need to address aliasing errors. References [1] BOYD, J.P.: Chebyshev and Fourier spectral methods, second ed., Dover, 2000, pdf version: http://www-

personal.engin.umich.edu/~jpboyd/book_spectral2OOO.html. ['2] CANUTO, C., HUSSAINI, M.Y., QUARTERONI,A., AND gANG,

T.A.: Spectral methods in fluid dynamics, Springer, 1987. 413

TRIGONOMETRIC PSEUDO-SPECTRAL METHODS [3] FORNBERG, B.: A practical guide to pseudospectral methods, Vol. 1 of Cambridge Monographs Appl. Comput. Math., Cambridge Univ. Press, 1996. [4] GOTTLmB, D., HUSSAINI, M.Y., AND ORSZAC, S.A.: 'Theory and application of spectral methods', in R.G. VOIGT,

414

D. GOTTLIEB, AND M.Y. HUSSAINI (eds.): Spectral Methods for Partial Differential Equations, SIAM, 1984. [5] GOTTLIEB, D., AND ORSZAG, S.A.: Numerical analysis of spectral methods: Theory and applications, SIAM, 1977. Richard B. Pelz

MSC 1991: 65M70, 65Lxx

U sequence a = (ak) over some set S satisfying the condition ULTIMATELY

PERIODIC

SEQUENCE

-

A

ak+r ~- ak

for all sufficiently large values of k and some r > 1 is called ultimately periodic with period r; if this condition actually holds for all k > 0, a is called periodic (with period r). The smallest number r0 among all periods of a is called the least period of a. The periods of a are precisely the multiples of to. Moreover, if a should be periodic for some period r, it is actually periodic with period r0. One may characterize the ultimately periodic sequences over some field F by associating an arbitrary sequence a = (ak) over F with the f o r m a l p o w e r series =

a

,x

C

Depending on the definition of the term 'concentration', one gets various concrete manifestations of this principle, one of them (see the Heisenberg uncertainty inequality below), correctly interpreted, is in fact the celebrated Heisenberg uncertainty principle of quantum of mechanics in disguise ([13]). A comprehensive discussion of various (mathematical) uncertainty principles can be found in [10]. H e i s e n b e r g u n c e r t a i n t y i n e q u a l i t y . Defining concentration in terms of s t a n d a r d d e v i a t i o n leads to the Heisenberg uncertainty inequality. If f C L 2 (R) and a ¢ R, the quantity f Ix - a[21f(z)[ 2 dx is a measure of tile concentration of f around a. Roughly speaking, the more concentrated f is around a, the smaller will this quantity be. If one normalizes f such that II/112 -- 1, then by the P l a n c h e r e l t h e o r e m II~J2 ; 1. Here, the F o u r i e r t r a n s f o r m of f , defined by

)'is

k=0

Then a is ultimately periodic with period r if and only if (1 - x~)a(x) is a p o l y n o m i a l over F. Any ultimately periodic sequence over a field is a s h i f t r e g i s t e r seq u e n c e . The converse is not true in general, as the Fibonacci sequence over the rationals shows (cf. S h i f t r e g i s t e r s e q u e n c e ) . However, the ultimately periodic sequences over a G a l o i s field are precisely the shift register sequences. Periodic sequences (in particular, binary ones) with good correlation properties are important in engineering applications (cf. C o r r e l a t i o n p r o p e r t y for s e q u e n c e s ) . References [1] JUNGNICKEL, D.: Finite fields: Structure and arithmetics, Bibliographisches Inst. M a n n h e i m , 1993. Dieter Jungnickel

MSC 1991:11B37 UNCERTAINTY

PRINCIPLE~

MATHEMATICAL

-

following meta-theorem: It is not possible for a nontrivial function and its F o u r i e r t r a n s f o r m to be simultaneously sharply localized/concentrated. The

)'(y) =

/?

f(x)

-2

xy dx,

O0

the convergence of the integral being interpreted suitably. Then, for a, b C R one has the Heisenber 9 inequal-

ity

>-- 1 -

167c2 •

Thus, the above says that if f is concentrated around a C R , then no m a t t e r what b E R is chosen, ) ' cannot be concentrated around b. Equality is attained in the above if and only if f is, modulo translation and multiplication by a phase factor, a Gaussian function (i.e. of the form Ke-cx~). B e n e d i c k s ' t h e o r e m . Concentration can also be measured in terms of the 'size' of the set on which f is supported (cf. also S u p p o r t o f a f u n c t i o n ) . If one takes 'size' to mean L e b e s g u e m e a s u r e , then M. Benedicks ([4], [1]) has proved the following result: If f C £ 2 ( R ) is a non-zero function, then it is impossible for both

UNCERTAINTY PRINCIPLE, MATHEMATICAL

A = {x: f ( x ) ¢ 0} a n d B = {y: f ( y ) ¢ 0} to have finite Lebesgue measure. (This is a significant generalization of the fact, well known to communication engineers, t h a t a function c a n n o t be b o t h time limited and band limited.) For various other uncertainty principles of this kind, see [12]. H a r d y ' s u n c e r t a i n t y p r i n c i p l e . A n o t h e r natural way of measuring c o n c e n t r a t i o n is to consider the rate of decay of the function at infinity. A result of G.H. H a r d y [11] states t h a t b o t h f and f cannot be simultaneously 'very rapidly decreasing'. More precisely: If If(x)[ < A e - ~ x ~ , If(Y)] <- Be-~bY~ for some positive constants A, a, B, b and for all x , y E R , and if ab > 1, then f =- 0. (If ab < 1, t h e n there are infinitely m a n y linearly independent functions f satisfying the inequalities, and if ab = 1, then f m u s t be necessarily a Gaussian function.) Actually, the first p a r t of H a r d y ' s result can be deduced from the following more general result of A. Beurling [14]: If f E L 2 ( R ) is such t h a t

/ff If(x)l O0

e

<

O0

then f ~_ 0. T h e r e are various refinements of H a r d y ' s theorem (see [7] for one such refinement). O t h e r d i r e c t i o n s . A p a r t from the three instances of the m a t h e m a t i c a l u n c e r t a i n t y principle described above, there are a host of u n c e r t a i n t y principles associated with different ways of measuring concentration (see, e.g., [2], [3], [5], [6], [8], [9], [15], [16], [18], [19]). If G is a locally c o m p a c t group (including the case G = R~), then it is possible to develop a Fourier transform theory for functions defined on G (cf. also H a r m o n i c a n a l y s i s , a b s t r a c t ) . There is a considerable b o d y of literature devoted to deriving various uncertainty principles in this context also. (See the bibliography in [10].) The Fourier inversion formula can be t h o u g h t of as an eigenfunction expansion with respect to the s t a n d a r d Laplacian (cf. also L a p l a c e o p e r a t o r ; E i g e n f u n c t i o n ) . So it is n a t u r a l to seek uncertainty principles associated with other eigenfunction expansions. A l t h o u g h this has not been as systematically developed as in the case of s t a n d a r d Fourier t r a n s f o r m theory, there are several results in this direction as well (see [17] and the bibliography in [10]).

416

References [1] AMREIN, W.O., AND BERTH1ER, A.M.: 'On support properties of L p functions and their Fourier transforms', J. Funct. Anal. 24 (1977), 258 267. [2] BECKNER, W.: 'Pitt's inequality and the uncertainty principle', Proc. Amer. Math. Soc. 123 (1995), 1897-1905. [3] BENEDETTO, J.J.: 'Uncertainty principle inequalities and spectrum estimation', in J.S. BYRNES AND J.L. BYRNES (eds.): Recent Advances in Fourier Analysis and Its Applications, Kluwer Acad. Publ., 1990. [4] BENEDICKS, M.: 'On Fourier transforms of functions supported on sets of finite Lebesgue measure', Y. Math. Anal. Appl. 106 (1985), 180 183. [5] BIaLYNICKI-BmULA,I.: 'Entropic uncertainty relations in quantum mechanics', in L. ACCARDIAND W. VON WALDENFELS (eds.): Quantum Probability and Applications III, Vol. 1136 of Lecture Notes in Mathematics, Springer, 1985, pp. 90-103. [6] BauIJN, N.G. DE: 'Uncertainty principles in Fourier anMysis', in ©. SmSHA(ed.): Inequalities, Acad. Press, 1967, pp. 55-71. [7] COWLING,M.G., AND PRICE, J.F.: 'Generalisations of Heisenberg's inequality', in G. MAUCERI,F. RICCI, AND G. WEISS (eds.): Harmonic Analysis, Vol. 992 of Lecture Notes in Mathematics, Springer, 1983, pp. 443-449. [8] DONOHO,D.L., AND STARK,P.B.: 'Uncertainty principles and signal recovery', SIAM J. Appl. Math. 49 (1989), 906-931. [9] FaRm, W.G.: 'Inequalities and uncertainty principles.', Y. Math. Phys. 19 (1978), 461-466. [10] FOLLAND, G.B., AND SITARAM, t . : 'The uncertainty principle: a mathematical survey.', J. Fourier Anal. Appl. 3 (1997), 207-233. [11] HARDY, G.H.: 'A theorem concerning Fourier transforms.', J. London Math. Soc. 8 (1933), 22~231. [12] HAVIN,V., AND J6RIeKE, B.: The Uncertainty Principle in Harmonic Analysis, Springer, 1994. [131 HEISENBERG,W.: @ber den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik', Z. Physik 43 (1927), 172-198. [141 H6RMANDER, L.: 'A uniqueness theorem of Beurling for Fourier transform pairs', Ark. Mat. 29 (1991), 237-240. [15] LANDAU,H.J., AND POLLAK, H.O.: 'Prolate spheroidal wave functions, Fourier anaIysis and uncertainty II', Bell Syst. Techn. J. 40 (1961), 65-84. [16] LANDAU,H.J., AND POLLAK, H.O.: 'Prolate spheroidal wave functions, Fourier analysis and uncertainty III: the dimension of the space of essentially time-and band-limited signals', Bell Syst. Techn. J. 41 (1962), 1295-1336. [17] PATI, V., SITARAM,A., SUNDARI,M., AND THANGAVELU,S.: 'An uncertainty principle for eigenfunction expansions', J. Fourier Anal. Appl. 2 (1996), 427-433. [18] PRICE, J.F.: 'Inequalities and local uncertainty principles', J. Math. Phys. 24 (1983), 1711-1714. [19] SLEPAIN,D., AND POLLAK, H.O.: 'Prolate spheroidal wave functions, Fourier analysis and uncertainty I', Bell Syst. Techn. J. 40 (1961), 43-63. A. Sitaram

MSC 1991:42A63

V VAPNIK-CHERVONENKIS

D I M E N S I O N , Vapnik-

Cervonenkis dimension, VC-dimension - Let H = ( V ( H ) , E ( H ) ) be a h y p e r g r a p h . The VapnikChervonenkis dimension of H is the largest cardinality of a subset F of V(H) that is scattered by E(H), i.e. such that for all A C F there is an E C E(H) with A = F n E. Thus, it is the same as the index of a V a p n i k - C h e r v o n e n k i s class. It is usually denoted by VC(H). Computing the Vapnik-Chervonenkis dimension is A/P-hard (cf. also A/79) for many classes of hypergraphs, [3], [7]. The Vapnik-Chervonenkis dimension plays an important role in learning theory, especially in PAC learning (probably approximately correct learning). Thus, learnability of classes of {0, 1}-valued functions is equivalent to finiteness of the Vapnik-Chervonenkis dimension, [1].

[5] NAIMAN, D.Q., AND WYNN, H.P.: 'Independence number and the complexity of families of sets', Discr. Math. 154 (1996),

203-216. [6] PACH, J., AND AGARWAL,P.K.: Combinatorial geometry, Wiley/Interscience, 1995, pp. 247-254. [7] PAPADIMITRIOU, C.H., AND YANNAKAKIS, M.: 'On limited nondeterminism and the complexity of VC-dimension', J. Comput. Syst. Sci. 53, no. 2 (1996), 161-170.

M. Hazewinkel MSC 1991: 05C65, 05D05, 68T05, 68Q15 V A R I G N O N P A R A L L E L O G R A M - Take an arbitrary quadrangle and take the midpoints of each of the four sides. Join adjoining midpoints. The result is a parallelogram, called the Varignon parallelogram. This theorem is due to P. Varignon (around 1700). The assertion that the bimedians (i.e. the lines joining opposite midpoints) bisect each other is equivalent to it.

For the role of the Vapnik-Chervonenkis dimension in neural networks, see, e.g., [2], [4]. The independence number of a hypergraph H is the maximal cardinality of a subset A of V(H) that does not contain any E E E(H) (see also G r a p h , n u m e r i cal c h a r a c t e r i s t i c s of a). This notion is closely related with VC(H), [5], [6]. References [1] BEN-DAVID, S., CEsA-BIANCHI, N., HAUSSLER, D., AND LONG, P.M.: 'Characterizations of learnability of {0,..., n}valued functions', J. Comput. Syst. Sci. 50, no. 1 (1995), 74-86. [2] HOLDEN, S.B.: 'Neural networks and the VC-dimension', in J.G. MCWHIRTER (ed.): Mathematics in Signal Processing, Vol. III, Oxford Univ. Press, 1994, pp. 73-84. [3] KRANAKIS, E., KRIZANC, D., RUF, B., URRUTIA, J., AND WOGINGER, G.: 'The VC-dimension of set systems defined by graphs', Discr. Appl. Math. 77, no. 3 (1997), 237-257. [4] MAASS, W.: 'Perspectives of current research about the complexity of learning on neural nets', in V. ROYCHOWDHURY ET AL. (eds.): Theoretical Advances in Neural Computation and Learning, Kluwer Acad. Publ., 1994, pp. 295-336.

The centre of mass of the Varignon parallelogram is the centroid of the original quadrangle (the centre of mass of four equal masses placed at the four vertices). A different V a r i g n o n t h e o r e m deals with sliding vectors. References [1] COXETER, H.S.M.: Introduction to geometry, Wiley, 1969, p. 199.

[2] COXETER,H.S.M., AND GREITZEa,S.L.: Geometry revisited, Math. Assoc. America, 1967, pp. 51-56. M. Hazewinkel MSC 1991:51M04

VAUGHAN IDENTITY VAUGHAN IDENTITY - In 1937, I.M. Vinogradov [9] proved the odd case of the Goldbach conjecture (cf. also G o l d b a c h p r o b l e m ) ; i.e., he proved that every sufficiently large odd number can be written as a sum of three prime numbers (cf. also V i n o g r a d o v m e t h o d ) . The essential new element of his proof was a non-trivial estimate for an exponential sum involving prime numbers (cf. also E x p o n e n t i a l s u m e s t i m a t e s ) . Let e(a) denote e 2 ~ and let S ( a , N) = E p < N e(pa), where p runs over the prime numbers. By simply observing that le(pa)l _< 1 for all p, a and using the prime number theorem (cf. d e la V a l l d e - P o u s s i n t h e o r e m ) , one immediately sees that S ( a , N) = O(N/log N). Vinogradov was able to improve on this estimate on the 'minor arcs'; in other words, he obtained a better estimate for those values of a t h a t could not be well approximated by a rational number with a small denominator. Vinogradov's estimate used the sieve of Eratosthenes (cf. also E r a t o s t h e n e s , s i e v e of; S i e v e m e t h o d ) to decompose the sum S(a, N) into subsums of the form

where the product is over all prime numbers p. Taking the reciprocal of the Euler product, one sees that 1

-E

C(s) where

At(n) is

n=l

the MSbius function defined by

(0--1)k

i f n = Pl " " P k for distinct prime numbers Pl, • • -, P~,

#(n)

=

if n is divisible by the square of some prime number.

By looking at the coefficients of C(s)'C(s) -1 , one obtains the useful identity

a~l~#(n)

=

otherwise.

By taking the logarithmic derivative of the Euler product formula, one sees t h a t

n=l

n~_u

where the coefficients A(n) are defined as

rn<_N/n

{~ gp

and of the form A(n) = m
The sums have become known as sums of type I and type II, respectively. Vinogradov's method is quite powerful and can be adapted to general sums of the form ~2p<_Nf(P)" However, the technical details of his method are formidable and, consequently, the method was neither widely used nor widely understood. In 1977, R.C. Vaughan [7] found a much simpler approach to sums over prime numbers. Vaughan's identity is most easily understood in the context of D i r i c h l e t series. Suppose that oo

0o

F(s) = E f ( n ) n - ~

and

G(s) = Eg(n)n-~

n=l

n=l

are both absolutely convergent in the half-plane Re s > a. Then

ifn=p~ for some prime number p, otherwise.

This is the M a n g o l d t product

f u n c t i o n . By computing the -~

T(s) in two different ways,

one sees that

Z A ( d ) = log . din

For technical reasons, it is often simpler to work with sums of the form }-2~<x A(n)f(n) than with sums of the form ~p<_Nf(P), and estimates for the latter sum can usually be easily derived from estimates for the former. Let u, v be arbitrary real numbers, both exceeding 1, and define

M(s) = E#(d)d-S' d
r(s)

= EA(e)e-S' e
Thus, M and F are partial sums of 1/~ and -C~/~ respectively. In particular, in this same half-plane. One of the simplest and most useful Dirichlet series is the Riemann zeta-function (cf. also Z e t a - f u n c t i o n ) , which is defined as ~(s) = ~ - - 1 n-~ for complex numbers s with real part exceeding 1. The Euler product formula states that ~(s) = l - I (1 - p - ~ ) - I , P

418

VAUGHT CONJECTURE Vaughan's identity. In this case, let M(s) be as before, but take

Now consider the Dirichlet series identity

<,

-(+F=~'M+FM;+

+F

(1 - ~M).

r(s) =

is( )n -s. n
Comparing coefficients of n - s on both sides of the equation, one sees that if n > v, then A(n) = E

is(d)logr-

-

a(k)- E

E

dr=n d
kr=n k
where

a(k) = E

A(d)is(e)

b(k) = Eis(d)"

and

d
dik d
If one multiplies this equation by f(n) and sums over v < n < N , one obtains the Vaughan identity:

Z

A(n)f(n) =

d
E

- E a(k) -

(logr)f(rd)+

v/d
k<_uv

~

E

f(rk)+

v/k
~

A(e)b(k)f(dk).

v<e<_N/u u
In general, the first and second sums can be treated as type-I sums, and the third sum can be treated as a typeII sum. The logarithm factor in the first sum is easily finessed with partial summation. In some applications, it is useful to divide the second sum into subsums with k < K and K <_ k < uv, where the first subsum is treated as type-I and the second subsum as type-II. For a brief and very accessible account of how Vaughan's identity is applied, see Vaughan's original article [7]. There, he proves that

n
+M

1

(1-~M)=~+F-M-FM(

one can obtain an identity for sums of the form ~-~n
v
= E Is(d)

( ~ )

A(e)b(k),

ek=n e>v k>u

-

From the equation

0 ((Nq -112 + N 415 + Nll2q'12)log 4 N)

whenever l a - alq I <_ llq 2. Another self-contained account of this can be found in [1]. There are many applications of Vaughan's identity in the literature. Vaughan [6] used it to obtain new estimates on the distribution of ap (rood 1), and he also used it to give an elegant proof of the BombieriVinogradov theorem on prime numbers in arithmetic progressions [8]. H.L. Montgomery and Vaughan [5] obtained a new estimate for the error term in the formula for the number of square-free integers up to x, conditional on the Riemann hypothesis (cf. R i e m a n n h y p o t h e s e s ) . This requires a slightly different form of

It] DAVENPORT, H.: Multiplieative number theory, second ed., Springer, 1980.

[2] HARMAN,G.: 'Eratosthenes, Legendre, Vinogradov, and beyond', in G.R.H. GREAVES,G. HARMAN,AND M.N. HUXLEY (eds.): Sieve Methods, ExponentialSums, and their Applications in Number Theory,Vol. 237 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1996. [3] HEATH-BROWN,D.R.: 'Prime numbers in short intervals and a generalized Vaughan identity', Canad. J. Math. 34 (1982), 1365-1377.

[4] HEATH-BROWN, D.R., AND PATTERSON, N.J.: 'The distribution of Kummer sums at prime arguments', J. Reine Angew. Math. 310 (1979), 110-130. [5] MONTGOMERY, H.L., AND VAUGHAN,R.C.: COn the distribution of square-freenumbers', in H. HALBERSTAM AND C. HOOLEY (eds.): Recent Progress in Analytic Number Theory,

Vol. 1, 1981, pp. 247-256. [6] VAUGHAN, R.C.: 'On the distribution of ap modulo one', Mathematika 24 (1977), 135-141. [7] VAUGHAN,R.C.: 'Sommes trigonomdtriques sur les nombres premiers', C.R. Aead. Sci. Paris Sdr. A 285 (1977), 981-983. [8] VAUGHAN, R.C.: 'An elementary method in prime number theory', Acta Arith. 37 (1980), 111-115. [9] VINOGRADOV, [.M.: 'A new estimation of a certain sum containing primes', Mat. Sb. 44 (1937), 783-791. (In Russian.) [10] VINOGRADOV, I.M.: The method of trigonometric sums in the theory of numbers, Wiley/Interscience, 1954. (Translated from the Russian.)

S. W. Graham MSC1991: 11M06, 11P32, 11L07 VAUGHT CONJECTURE - Let T be a countable complete first-order theory (cf. also L o g i c a l c a l c u l u s ) and let n(T) be the number of countable models of T, up to isomorphism (cf. also M o d e l t h e o r y ) ; n(T) < 2 s°. In 1961, R. Vaught [17] asked if one can prove, without using the c o n t i n u u m h y p o t h e s i s CH, that there is some T with n(T) = 1~1. Vaught's conjecture is the statement: If n(T) > R0, then n(T) = 2 ~°.

419

VAUGHT CONJECTURE Variants of this conjecture have been formulated for incomplete theories, and for sentences in Lwx w. In 1970, M. Morley [9] proved, using d e s c r i p t i v e s e t t h e o r y , t h a t if n ( T ) > R0, then n ( T ) = ~1 or 2 ~° (actually, he proved this for any ~ C L~I~ ). Let M o ~ ( T ) be the set of all models of T haying a~ as their universe (cf. also M o d e l t h e o r y ) . Morley equipped AJoD(T) with a Polish topology (cf. also D e s c r i p t i v e s e t t h e o r y ) . Associated with each M E M o ~ ( T ) is a countable o r d i n a l n u m b e r , S H ( M ) , called the Scott height (or Scott rank) of M. Let SH(T) = SUPMeMo~(T ) S H ( M ) and, for a < COl, let M o O ~ ( T ) = {2t4o~(T): S H ( M ) = a}. The isomorphism relation ---- is analytic (El; cf. also L u z i n set) on 340~(T); however, 3 J o ~ ( T ) is Borel (cf. also B o r e l s y s t e m o f s e t s ) and ~ restricted to A d o ~ ( T ) is a Borel equivalence relation, so [ 3 / l o ~ ( T ) / - ~ ] < R0 or = 2 ~°. Hence (if CH fails) the only possibility for T to have R1 countable models is t h a t SH(T) = HI and for each So the Vaught conjecture m a y be restated as follows: If SH(T) = wl, then for some a < c~1, ] M o ~ ( T ) / ~ - [ = 2 ~°. This formulation does not depend explicitly on CH. The above Morley analysis led to the so-called topological Vaught conjecture, which is a question regarding the n u m b e r of orbits of a Polish topological group (cf. also T o p o l o g i c a l g r o u p ) G acting in a Borel way on a Polish space X [1], [5]. Vaught's conjecture was proved for theories of trees [16], u n a r y function [6], [8], varieties [4], o-minimal theories [7], a n d theories of modules over certain rings [13]. In stable model theory, the combinatorial tools (like forking, cf. also F o r k i n g ) developed by S. Shelah in [14] enabled him to prove the Vaught conjecture for w-stable theories [15], which are at the lowest level of the stability hierarchy. Regarding superstable theories (the next level of the hierarchy), V a u g h t ' s conjecture was proved for weakly minimal theories [2], [10], and then for superstable theories of finite U - r a n k [3] and in some other cases [12]. T h e proofs in these cases use advanced geometric properties of forking [11]. References

[1] BECKER, H.: 'The topological Vaught's conjecture and minimal counterexamples', J. Symbolic Logic 59 (1994), 757 784. [2] BUECHLER, S.: 'Classification of small weakly minimal sets, I', in J.T. BALDWIN(ed.): Classification Theory, Proceedings, Chicago, 1985, Springer, 1987, pp. 32 71. [3] BUECHLER, S.: 'Vaught's conjecture for superstable theories of finite rank', Ann. Pure Appl. Logic (to appear). [4] HART, B., STARCHENKO, S., AND VALERIOTE,M.: 'Vaught's conjecture for varieties', Trans. Amer. Math. Soe. 342 (1994), I73-196. [5] HJORTH, G., AND SOLECKI,G.: 'Vaught's conjecture and the Glimm-Effros property for Polish transformation groups', Trans. Amer. Math. Soc. 351 (1999), 2623-2641. 420

[6] MARCUS,L.: 'The number of countable models of a theory of unary function', Fundam. Math. 108 (1980), 171-181. [7] MAYER, L.: 'Vaught's conjecture for o-minimal theories', Y. Symbolic Logic 53 (1988), 146-159. [8] MILLER, A.: 'Vaught's conjecture for theories of one unary operation', Fundam. Math. 111 (1981), 135-141. [9] MORLEY, M.: 'The number of countable models', J. Symbolic Logic 35 (1970), 14-18. [10] NEWELSKI,L.: 'A proof of Saffe's conjecture', Fundam. Math. 134 (1990), 143-155. [11] NEWELSKI, L.: 'Meager forking and m-independence', Documenta Math. Extra I C M (1998), 33-42. [12] NEWELSKI,L.: 'Vaught's conjecture for some meager groups', Israel J. Math. 112 (1999), 271-299. [13] PUNINSKAYA, V.: 'Vaught's conjecture for modules over a Dedekind prime ring', Bull. London Math. Soc. 31 (1999), 129-135. [14] SHELAH,S.: Classification theory, second ed., North-Holland, 1990. [15] SHELAH, S., HARRINGTON, L., AND MAKKAI,M.: 'A proof of Vaught's conjecture for R0-stable theories', Israel J. Math. 49 (1984), 259-278. [16] STEEL, J.: 'On Vaught's conjecture', in Y. MOSCHOVAKIS A. KECHRIS(ed.): Cabal Seminar '76-77, Vol. 689 of Lecture Notes in Mathematics, Springer, 1978, pp. 193-208. [17] VAUGHT, R.: 'Denumerable models of complete theories': Infinitistic Methods (Proc. Symp. Foundations Math., Warsaw, 1959), Pafistwowe Wydawnictwo Nauk. Warsaw/Pergamon Press, 1961, pp. 303 321.

Ludomir Newelski

M S C 1 9 9 1 : 03C15, 03C45, 03E15

VERTEX OPERATOR - The term 'vertex operator' in mathematics refers mainly to certain operators (in a generalized sense of the term) used in physics to describe interactions of physical states at a 'vertex' in string theory [9] and its precursor, dual resonance theory; the term refers more specifically to the closely related operators used in mathematics as a powerful tool in many applications, notably, constructing certain representations of affine Kac-Moody algebras (cf. also Kac-Moody algebra) and other infinite-dimensional Lie algebras, addressing the problems of the 'Monstrous Moonshine' phenomena for the Monster finite simple group, and studying soliton equations (cf. also Moonshine conjectures). The term 'vertex operator' also refers, more abstractly, to any operator corresponding to an element of a vertex operator algebra or a related operator.

Vertex operators arose in mathematics in the following construction of the 'basic' highest weight representation of the simplest affine Lie algebra s[(2) by means of formal differential operators in infinitely many formal variables (cf. also Representation of a Lie algebra): Consider the space C[Yl/2,Y3/2,...] of polynomials in

VERTEX OPERATOR ALGEBRA the formal variables y~, n C N + 1/2. The formal expression

ncN+I/2

ncN+I/2

constructed in [12], is a basic example of a vertex operator (for other examples, see K a c - M o o d y algebra). Here exp denotes the formal exponential series and x is another formal variable commuting with all y~. For each j E (1/2)Z, the coefficient Aj ofx j in the expansion of the vertex operator in powers of x is a well-defined linear operator on C[yl/2, Ya/2,...], and the main point is that the operators 1, Yn, O/Oy~ (n E N + 1/2) and Aj (j C (1/2)Z) span a Lie a l g e b r a of operators isomorphic to the affine Lie algebra s[(2) [12]. This vertex operator had been considered by physicists [3] for other purposes. It was interpreted [4] as the infinitesimal B~icklund t r a n s f o r m a t i o n for the K o r t e w e g - d e Vries e q u a t i o n in soliton theory. This work was generalized [11] to all the basic representations of the simply-laced (equal-root-length) affine Lie algebras and their Dynkin-diagram-induced twistings. H. Garland remarked that the differential operators reminded him of the 'vertex operators' that physicists had been using, starting in [8], in dual resonance theory. The resemblance turned into a coincidence in the construction by I. Frenkel and V. Kac [5] and G.B. Segal [13] of the untwisted vertex operator realization of the basic representations of the simply-laced affine Lie algebras. This construction had been anticipated by physicists ([10], [1]) in the case of s[(n). The untwisted vertex operator representations allowed one to look in a new way at the finite-dimensional simple Lie algebras, viewed as subalgebras of affine Lie algebras. The case of E8 was used in string theory in the construction of the heterotic string by D. Gross, J. Harvey, E. Martinec, and R. Rohm (cf. [9]). The operators constructed in [12] and [11] are understood as examples of twisted vertex operators, and give the principally-twisted vertex operator realization of the basic representations. Untwisted and twisted vertex operators entered fundamentally into the construction of the 'moonshine module' [6] for the Fischer-Griess Monster group and into the discovery of its canonical structure of v e r t e x o p e r a t o r a l g e b r a ([2], [7]). There is a great variety of interesting examples of vertex operators. The notion of vertex (operator) algebra is an abstraction of fundamental properties of vertex operators discovered by physicists and mathematicians, and provides an elegant and powerful framework for the study and application of vertex operators.

References [1] BANKS, T., HORN, D., AND NEUBERGER, H.: 'Bosonization of the S U ( N ) Thirring models', Nucl. Phys. B 1 0 8 (1976), 119. [2] BORCHERDS, R.E.: 'Vertex algebras, Kac-Moody algebras, and the monster', Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071. [3] CORmOAN, E.F., AND FAIRLIE, D.B.: 'Off-shell states in dual resonance theory', Nucl. Phys. B 9 1 (1975), 527-545. [4] DATE, E., HASHIWARA, M., AND MIWA, T.: 'Vertex operators and r functions: transformation groups for soliton equations II', Proc. Japan Acad. Ser. A ~ 7. ~qci. 57 (1981), 387-392. [5] FRENKEL~ I.B., AND ~TAC, "V." . .:~ic re~resentations of affine Lie algebras and dual resonance _m~r~*ls', Invent. Math. 62 (1980), 23-66. [6] FRENKEL, I.B., LEPOWSKY, J., AND MEURMAN, A.: 'A natural representation of the Fischer-Cr~,~zs monster with the modular function J as character', Proc. ,',:~. Acad. Sci. USA 81 (1984), 3256-3260. [7] FRENKEL, I.B., LEPOWSKY~ J.~ AND MEURMAN, A.: Vertex operator algebras and the monster, Vol. 134 of Pure Appl. Math., Acad. Press, 1988. [8] FUBINI, S., AND VENEZIANO, G.: 'Duality in operator formalism', Nuovo Cimento 6 7 A (1970), 29. [9] GREEN, M.B., SCHWARZ, J.H., AND WITTEN, E.: Superstring theory, Cambridge Univ. Press, 1987. [10] HALPERN, M.B.: ' Q u a n t u m "solitons" which are S U ( N ) fermions', Phys. Rev. D 1 2 (1975), 1684-1699. [11] KAC, V., KAZHDAN, D., LEPOWSKY, J., AND WILSON, R.L.: 'Realization of the basic representations of the Euclidean Lie algebras', Adv. Math. 42 (1981), 83 112. [12] LEPOWSKY, J., AND WILSON, R.L.: 'Construction of the affine Lie algebra A[ 1)', Comm. Math. Phys. 62 (1978), 43 53. [13] SEGAL, G.: 'Unitary representations of some infinitedimensional groups', Comm. Math. Phys. 80 (1981), 301342.

Y.-Z. Huang J. Lepowsky MSC 1991: 17B65, 81T30, 81R10

17B67,

17B10,

20D08,

11Fll,

VERTEX OPERATOR ALGEBRA - V e r t e x (operator) algebras are a fundamental class of algebraic structures that arose in mathematics and physics in the 1980s. These algebras and their representations are deeply related to many directions in mathematics and physics, in particular, the representation theory of the Fischer-Griess Monster s i m p l e finite g r o u p and the connection with the phenomena of 'Monstrous Moonshine' (cf. also M o o n s h i n e c o n j e c t u r e s ) , the representation theory of the V i r a s o r o a l g e b r a and a n n e Kac-Moody Lie algebras (cf. also K a c - M o o d y algebra), modular functions (cf. also M o d u l a r f u n c t i o n ) , the theory of Riemann surfaces (cf. also R i e m a n n surface), knot invariants and invariants of three-manifolds (cf. also K n o t t h e o r y ; Three-dimensional manifold), q u a n t u m g r o u p s , monodromy associated with differential equations, and conformal and topological field theory and string theory in physics. In fact, the 421

V E R T E X O P E R A T O R ALGEBRA theory of vertex operator algebras and their representations can be thought of as an algebraic foundation of a great number of constructions in these theories. Various equivalent definitions of the notion of vertex algebra and of the variant notion of vertex operator algebra are given below. The notion of vertex algebra was defined by R. Borcherds [2] and is a mathematically precise algebraic counterpart of the concept of 'chiral algebra' in twodimensional conformal quantum field theory (a physical theory foundational in string theory [I0] and in two-dimensional statistical mechanics), as formalized by A. Belavin, A.M. Polyakov and A. Zamolodchikov [I]. This fundamental notion reflects deep features of the traditional notions of commutative associative algebra and at the same time of Lie algebra. The theory of vertex operator algebras has a distinctly non-classical flavour, which can be thought of as analogous to the non-classical flavour of string theory. The elements of a vertex operator algebra correspond to (abstract) 'vertex operators', which in special cases include many of the 'vertex operators' introduced by physicists in the early days of string theory to describe hypothesized interactions of certain elementary particles at a 'vertex' (cf. also Vertex operator). One of the main original motivations for the introduction of the notion of vertex (operator) algebra arose from the problem of realizing the Monster M as a symmetry group of a certain infinite-dimensional graded vector space with natural additional structure. The additional structure can be expressed in terms of the axioms defining these new algebraic objects (which are not algebras, not even non-associative algebras, in the usual sense). This problem arose in the study of the remarkable subject of Monstrous Moonshine. In 1978-1979, J. McKay, J.G. Thompson, J.H. Conway and S.P. Norton [4] conjectured the existence of a natural infinite-dimensional Z-graded representation (call it V ~ = ®n>-lV~) of the then-conjectured Monster group such that the formal series ~n>-, (dim V~n)q ~ would be the modular functlon J(q) = q-l+ 196884q+. - • and such that the action of every element of the Monster on V ~ would give rise to a certain modular function with special properties. After R. Griess [ii] proved the existence of M by constructing it as an automorphism group of a remarkable new algebra of dimension 196884, I. Frenkel, J. Lepowsky and A. Meurman [7] gave a construction, incorporating a vertex operator realization of the Griess algebra, of what they called a 'moonshine module' V ~ for M having the desired relation with J(q), and they showed that the action on V ~ of certain elements of M gave rise to modular functions. This structure V ~ was interpreted by physicists as a 'toy model' physical theory of a 26-dimensional 422

string compactified on a 24-dimensional 'orbifold' associated with the L e e c h l a t t i c e , so that M turned out to be the symmetry group of an idealized physical theory. Then Borcherds introduced the axiomatic notion of vertex algebra [2] and perceived that V ~ could be endowed with an M-invariant vertex algebra structure. An M-invariant vertex operator algebra structure on V ~ was indeed constructed in [8]. The Monster is in fact the full symmetry group of this special vertex operator algebra V~, just as the Mathieu finite group 2]//24 (cf. also M a t h i e u g r o u p ) is the symmetry group of a special error-correcting code, the G o l a y c o d e , and the Conway finite group .0 is the symmetry group of a special positive definite even lattice, the L e e c h l a t t i c e . All three of these special objects possess and can be characterized by the following properties (the uniqueness being conjectural in the case of V~): a) 'self-dual', b) 'rank 24', c) 'having no small elements'. These properties have appropriate definitions for each of the three types of mathematical structures. In fact, these structures and analogies enter into the construction of V~ ([7], [8]). Using the vertex operator algebra structure on V ~ and other ideas and results, Borcherds completed in [3] the proof of the Conway-Norton Monstrous Moonshine conjecture (cf. also M o o n s h i n e c o n j e c t u r e s ) concerning the modular functions associated to elements of M, acting on V~. The notion of vertex operator algebra ([8], [6]) is a modification of the notion of vertex algebra. There are several equivalent formulations of these notions, including formulations in terms of 'minimal' and 'maximal' axioms. The canonical 'maximal' axiom is a formalvariable identity called the 'Jacobi identity' or the ' J a c o b i - e a u c h y identity' ([8], [6]), on which the fundamental principles are heavily based. This identity contains the 'full' necessary information in compact form; it is analogous to the classical Jacobi identity in the definition of the notion of Lie a l g e b r a ; and it is invariant in a natural sense under the symmetric group on three letters. The Jacobi identity and the underlying formalvariable calculus are discussed in detail in [8] and [6]. The definition of vertex algebra in [2] involved certain special cases of the Jacobi identity (see below). There are also 'minimal' axioms, stemming from the fact that the (suitably formulated) 'commutativity' of vertex operators implies 'associativity' (again suitably formulated) and hence the Jacobi identity, as explained in [8], [6] (cf. [1], [9]). The simplest 'minimal' axiom in the definition of the notion of vertex algebra (see v) below) was found in [5]. A general and systematic approach

VERTEX OPERATOR ALGEBRA and solution to the problem of efficiently constructing examples of vertex operator algebras and their modules, and related problems, was first carried out in [15]. A program to construct (geometric) conformal field theory using vertex operator algebras and the 'sewing' of Riemann spheres with punctures was initiated by Frenkel. A precise geometric formulation of the notion of vertex operator algebra in terms of partial operads of complex powers of the determinant line bundle over the moduli space of Riemann spheres with punctures and local coordinates was given in [12], [14], [13]. Complete definitions of the notions of vertex algebra and vertex operator algebra, using commuting formal variables x, x0, zl and x2 are given below. The definition of vertex algebra with either 'minimal' or 'maximal' axioms is equivalent to Borcherds' definition in [2]; see below. V e r t e x a l g e b r a s . A vertex algebra is a v e c t o r s p a c e V (over C, say) equipped with a linear mapping V (End V)[[x, x-l]] (the vector space of formal Laurent series in the formal variable x with coefficients in End V), written as v ~ Y(v, x) (the vertex operator corresponding to the element v), and a distinguished vector 1 • V, satisfying the following conditions for u, v • V: i) the formal series Y(u,x)v involves only finitely many negative powers of x; ii) Y(1, x) = 1 (the identity operator on V); iii) Y ( v , x ) l involves only non-negative powers of x and its constant term is v; iv) [D,Y(v,x)] = (d/dx)Y(v,x), where D : V -+ V is the mapping such that Dv is the constant term of (d/dx)Y(v,x)l for v • V; v) there exists a non-negative integer k (depending on u and v) such that (X 1 -- x2)k[Y(7~, Xl) , ]/-(v, x2) ] = 0.

In this definition, conditions i-v) are often called the

truncation condition, the vacuum property, the creation property, the D-bracket-derivative formula, and weak cornmutativity, respectively. The last condition was formulated and exploited in [5]. The 'main property' of a vertex algebra, the Jacobi identity, states: vi) For u,v • V,

where ~(x) is the formal Laurent series Y~'-ncZxn and, more precisely, x0

nEZ

I--i) x 1

x2x o

,

nEZ k>0 and similarly for the other two &function expressions. (All the expressions are well defined.) It can be proved that the axioms ii), iv) and v) can be replaced by the Jacobi identity in the definition. This definition in terms of the Jacobi identity is the definition with 'maximal' axioms. The Jacobi identity is actually the generating function of an infinite family of identities. The use of the three formal variables, rather than complex variables (which can also be used, but with changes in the formulas), allows the full symmetry of the Jacobi identity to reveal itself, as explained in [8] and [6]. Vertex algebras also satisfy a 'commutativity' condition, which asserts, roughly speaking, that for u, v C V, x l ) Y ( v , x2) ~ Y ( v , x2)Y(

, Xl),

where ' ~ ' denotes equality up to a suitable kind of generalized analytic continuation, and also an 'associativity' condition,

Y(u, xl)Y(v, x.2) ~ Y ( Y ( u , Xl - x2)v, x2), where the right-hand side and the generalized analytic continuation have to be understood suitably, as explained in [8] and [6] (of. [1], [9]). The associativity condition corresponds to the 'operator product expansion' for holomorphic fields in conformal field theory, together with its 'associativity'. As mentioned above, these two conditions, and even the commutativity condition alone, essentially imply the Jacobi identity. Commutativity and associativity are intimately related to the geometric interpretation of the notion of vertex operator algebra. V e r t e x o p e r a t o r a l g e b r a s . A vertex operator algebra is a vertex algebra V equipped with a Z-grading V = ®~ezV(~) and a distinguished vector ~ C V satisfying the following additional conditions: vii) dim V(~) < oc for n C Z and V(.~) = 0 for n sufficiently small; viii) for rn, n C Z, =

:

\

-xo

/

= x~16 ( x l ~ x° ) Y ( Y ( u , xo)v, x2),

-

+ n) +

-

where L(n), n E Z, are the operators on V defined by Y(w, x) = ~ e z L(n) x-~-2 and where c E C; ix) L(O)v = nv for n e Z and v • V(~); 423

VERTEX OPERATOR ALGEBRA x) Y ( L ( - 1 ) v , x) = ( d / d x ) Y ( v , x). Conditions vii-x) are often called the gradingrestriction conditions, the Virasoro algebra relations, the L(O)-grading property and the L(-1)-derivative property, respectively. In [2], Borcherds gave the following definition: A vertex algebra is a vector space V (actually, this definition works over Z or any other commutative ring) equipped with an element 1, linear operators D (i) on V for i E Z and bilinear operations (u,v) ~-+ u~(v) from V × V to V, for n E Z, satisfying the following relations a)-e) for u,v,wEVandrn, n C Z: a) us(w) = 0 for n sufficiently large (depending on u and w); b) l~(w) = 0 i f n ¢ - l , wifn=-l; c) u~(1) = D(-n-1)(u); d) un(v) = ~ i > 0 ( - 1 ) i + ~ + l D (0(v~+i(u)); e)

:

i(7)

-

(-1)mvm+n-i(ui(w))). It can be proved t h a t the other definitions above of the notion of vertex algebra are equivalent to this one, over a field of characteristic O. The relation between the vertex operator mapping and the bilinear operations is given by Y(

,x)v =

[5] DONG, C., AND LEPOWSKY, J.: Generalized vertex algebras and relative vertex operators, Vol. 112 of Progress in Math., Birkhguser, 1993. [6] FRENKEL, I.B., HUANG, Y.-Z., AND LEPOWSKY, J.: On axiomatic approaches to vertex operator algebras and modules, Vo]. 104 of Memoirs, Amer. Math. Soc., 1993, preprint: 1989. [7] FRENKEL, I.B., LEPOWSKY, J., AND MEURMAN, A.: 'A natural representation of the Fischer-Griess monster with the modular function J as character', Proc. Nat. Acad. Sci. USA 81 (1984), 3256-3260. [8] FP~ENKEL, I.B., LEPOWSKY, J., AND MEURMAN, A.: Vertex operator algebras and the monster, Vol. 134 of Pure Appl. Math., Acad. Press, 1988. [9] GODDARD, P.: 'Meromorphic conformal field theory', in V. KAC (ed.): Infinite Dimensional Lie Algebras and Groups, Vol. 7 of Adv. Ser. in Math. Phys., World Sci., 1989, pp. 556587. [10] GREEN, M.B., SCHWARZ, J.H., AND WITTEN, E.: Superstring theory, Cambridge Univ. Press, 1987. [11] GRmSS, R.: 'The friendly giant', Invent. Math. 69 (1982), 1-102. [12] HUANG, Y.-Z.: 'Geometric interpretation of vertex operator algebras', Proc. Nat. Acad. Sci. USA 88 (1991), 9964-9968. [13] HUANG, Y.-Z.: Two-dimensional conformal geometry and vertex operator algebras, Vol. 148 of Progress in Math., Birkhguser, 1997. [14] HUANG, Y.-Z., AND LEPOWSKY, J.: 'Vertex operator algebras and operads', in L. CORWIN, I. GEL'FAND, AND J. LEPOWSKY (eds.): The Gelfand Mathematical Seminar, 19901992, Birkhguser, 1993, pp. 145-161. [15] LI, H.: 'Local systems of vertex operators, vertex superalgebras and modules', J. Pure Appl. Algebra 109 (1996), 143195, preprint: 1993.

nEZ Y.-Z. H u a n 9

for u, v E V. Equating the coefficient of the formal variable x~-1 on the two sides of the Jacobi identity above recovers e), in generating-function form. Among the sources of examples of vertex (operator) algebras are conformal field theory and string theory and related mathematical structures: suitable representations of the V i r a s o r o a l g e b r a , of Heisenberg Lie algebras (cf. also C o m m u t a t i o n a n d a n t i c o m m u t a t i o n r e l a t i o n s h i p s , r e p r e s e n t a t i o n of) and of K a c - M o o d y algebras, including affine Lie algebras, and analogues, generalizations and modifications of such structures. Vertex algebras are typically infinitedimensional, although any commutative associative algebra with derivation carries the structure of a vertex algebra [2].

J. Lepowsky M S C 1991: 17B65, 17B67, 1 1 F l l , 81T30, 81T40, 81R10

17B68,

17B10,

20D08,

VISCOUS FINGERING - An interface between two fluids presents a resistance towards an increase of its surface area. This interface has a surface tension which opposes such an increase. This is the property giving a liquid drop its spherical shape: A sphere has a minimal surface for a given volume of fluid. T. Young and P.S. Laplace rationalized this scenario by postulating the Young-Laplace law. In viscous fingering, where a less viscous fluid, such as air, pushes a more viscous fluid, such as an oil, in thin rectangular cells (known as Hele-Shaw cells), surReferences face tension plays an important role in determining the [1] BELaVIN, A.A., POLYAKOV, A.M., AND ZAMOLODCHIKOV, shape and properties of the fingers. The basic experiA.B.: 'Infinite conformal symmetries in two-dimensional mental situation to obtain viscous fingers is as follows quantum field theory', Nucl. Phys. B241 (1984), 333 380. (here, the focus is on the linear channel geometry, but [2] BORCHERDS, REx 'Vertex algebras, Kac-Moody algebras, and the Monster', Proc. Nat. Acad. Sei. USA 83 (1986), such experiments can also be obtained in circular cells). 3068-3071. One usually uses two thick glass plates and a thin plastic [3] BORCHERDS, R.E.: 'Monstrous moonshine and monstrous Lie sheet to obtain a linear Hele-Shaw cell [5]. The plastic superalgebras', Invent. Math. 109 (1992), 405-444. sheet serves as a spacer between the glass plates once [4] CONWAY,J.H., AND NORTON, S.P.: 'Monstrous moonshine', they are brought into contact with each other. A long Bull. London Math. Soc. 11 (1979), 308 339. 424

VISCOUS F I N G E R I N G rectangular canal can be cut in the middle of the plastic sheet, which is then sandwiched between the glass plates to obtain a thin linear rectangular channel. The thickness of this canal is fixed by the thickness of the plastic sheet. Through holes drilled in the upper glass plates one then injects the viscous fluid to fill up the cell. Subsequently, air is injected at constant pressure or constant flux to push the oil out of this canal (see Fig. 2 for a scheme of the cell). The interface between the air and the oil is then unstable: the flat interface takes a curved shape. This instability is well known and from the pioneering work of P.G. Saffman and G.I. Taylor in the 1950s [10], it is known that the finger of air has a width which decreases with the velocity of the finger. The most mysterious aspect observed by Saffman and Taylor is that the limiting width of the fingers obtained at high velocities is very close to half the channel width. This property was not explained until the 1980s, although Saffman and Taylor did calculate the shape of the finger they obtained in their experiments.

The use of a Hele-Shaw cell permits one to use a simplified version of the equations governing the flow of the fluids. Basically, the N a v i e r - S t o k e s e q u a t i o n s are simplified to keep just the pressure gradient term and the dissipative term. This approximation is known as Darcy's law and is obeyed by flows in porous media, for example. In a sense, the linear Hele-Shaw cell is a simple porous medium, as was shown by H.J.S. Hele-Shaw [5]. This equation is written as follows:

Here, the gradient is in the plane of the cell, denoted by (x,y) (the finger propagates along the x direction with y being the transverse direction), P is the pressure, # is the fluid shear viscosity, 17 is the velocity of the fluid in the (x,y)-plane far from the fingertip and averaged over the vertical gap of thickness b (b being the spacing between the plates fixed by the thickness of the plastic sheet). One can rewrite this equation in terms of a velocity potential ¢ such that IP = V¢; due to the incompressibility of the fluid this potential obeys the L a p l a c e e q u a t i o n V2¢ = 0. Suppose the following boundary conditions on the interface apply: re)

Fig. 1: A Saffman-Taylor finger obtained with air injected in a viscous fluid. This finger has a width very close to half the channel width. The gray region delimiting the finger contains the fluid, while the white region contains air. The black dots represent the shape obtained from the Saffman Taylor solution.

=

i.

Here, ff is the normal to the interface delimiting the finger, ~ is the unit vector in the direction of propagation of the finger and U is the velocity of the tip. The finger is supposed to advance at constant velocity U and keeps a constant width AW, where W is the width of the channel and ~ is the relative width of the finger [10]. The potential at the interface is 7b2~ 12#

¢ i n t ---~ ¢ 0 "~ - -

l¸

Fig. 2: Scheme of the linear Hele-Shaw cell: two thick glass plates with two tubes attached to inject the less viscous fluid and to recover the fluid being pushed. The gray part is the plastic sheet of thickness b into which a linear channel of width W is cut.

See below for the basic equations governing this problem and some hints as to how it was resolved using techniques from complex analysis and perturbation theory in the 1980s. See [9], [7], [1] for accessible reviews of the experiment and phenomena.

Here, 7 is the surface tension of the interface and ~ is the curvature of the interface. The additional term in the potential expresses the effect of surface tension due to the Young-Laplace law. ¢o is the potential inside the less viscous fluid (which can be taken as inviscid without loss of generality). It is clear that far behind the fingertip the potential is zero while far ahead of the fingertip the viscous fluid is moving with velocity V = ~U. The problem is now to determine the relation between the coordinates of the interface x and y. In 1980, J.W. McLean and P.G. Saffman [8] suggested a way of rewriting the equations in complex representation which came in handy for later work. One writes the coordinates in the complex plane as Z = x + iy and w = ¢ + i¢ (where is the s t r e a m f u n c t i o n ) . The transformation Z ~ w is a c o n f o r m a l m a p p i n g from the flow region delimited by the finger and the walls of the channel into an infinite strip of unit width in the potential plane. The 425

VISCOUS F I N G E R I N G conformal mapping w --+ (r = s + it = e -(w-C°)" maps the flow region into the upper a-plane. This transforms the interface between the tails and the tip of the finger into the real segment 0 < s < 1. The boundary s = 0 corresponds to the tail of the finger, for which w --+ +ee, and s = 1 corresponds to the tip, for which w = ¢0. The problem now reduces to finding Z in terms of w. However, it turns out to be simpler to study the quantity d w / d Z ; it is the complex velocity V~ - iVy and may be written as qe (-iO), since the velocity is tangent to the interface (with 0 the angle between x and the tangent to the interface). The modulus q at the interface can be written as d ¢ / d S , where S measures arclength along the interface starting from the tip. By invoking the analytic properties of l n ( d w / d Z ) , the following relation can be established between q' = (1 - ),)q and 0' = 0 - 7r: :

lnq'

S p ,1 7 J00'(s')ds'

Besides this relation there is a second one, dictated by the Young-Laplace law:

d

[q,s~sJ + cos0 -

kq'Sds [

q,

= 0,

where k =

7b27r2 12#Ua2(1 - A)2"

These two equations are to be solved with the following boundary conditions: • at the tail of the finger: 0(0) = 0 and q(0) = 1; • at the tip of the finger: 0(1) = - r r / 2 and q(1) = 0. If a solution of these equations is found, the value of A (which is the quantity that needs to be determined) as well as its dependence on physical parameters, such as the velocity of the finger, the viscosity of the fluid and especially the surface tension, can be found from the relation: O(s') ln(1-A)= 1--.£1

7

- 7 - as'.

At this point the equations are complete. Of course, the dimculty is in solving these equations. For the case of zero surface tension (k = 0), one finds a continuum of solutions for the finger shape, parametrized by the value of ~ which remains unknown. The family of solutions first found by Saffman and Taylor is written as: qo(s)

1- s = [ ~ ]

1/2

'

Oo(s) = c o s < qo(s),

with a = (2~ - 1)/(1 - ~)2. In the coordinates x and y, this solution for the interface shape reads: x -

In - 1 + cos . rc 2 The problem now is to find a selection mechanism for the allowed values of k. This problem was not resolved

426

until 1986, when three different groups [3], [4], [11], [6] studied the small-k limit of these equations using techniques from p e r t u r b a t i o n theory and showed that the continuum of solutions breaks into a discrete set of solutions with the value of k decreasing to its well-known value of 1/2 as k approaches 0. This calculation is rather involved; see [3], [4], [11], [6]. The use of complex fluids such as, e.g., gels, clays, polymeric solutions, mixtures of water and surfactants can give rise to new morphologies of the interfaces in Hele-Shaw cells [7]. In the above, the viscosity of the fluid as well as the surface tension of the interface were considered to be constant. For complicated fluids these physical quantities may depend on the velocity of the fluid and therefore become variable. This complicates the m a t h e m a t ical analysis of the instability; however, some progress has been made [2]. References [i] BENSIMON, D., KADANOFF, L.P., LIANa, S., SHRAIMAN, B.I., AND TANG, C., Rev. Mad. Phys. 58 (1986), 977. [2] BONN, D., KELLAY, H., AND MEUNIER, J., Philos. Mag. B 78 (1998), 131-142. [3] COMBESCOT, R., DOMBRE, T., HAKIM, V., POMEAU, Y., AND PUMIR, A., Phys. Rev. Lett. 56 (1986), 2036.

[4] COMBESCOT,R., DOMBRE,T., HAKIM,V., POMEAU,Y., AND PUMIR, A., Phys. Rev. A 37 (1988), 1270. [5] HELE-SHAW,H.J.S., Nature 58 (1898), 34. [6] HONG, D.C., AND LANaER, J., Phys. Rev. Lett. 56 (1986), 2032. [7] McCLOUD,K.V., AND MAHER, J.V., Phys. Rept. 260 (1995), 139. [8] MCLEAN, J.W., AND SAFFMAN, P.G., g. Fluid Mech. 102 (1981), 455. [9] PELCI~, P., AND LIBCHABER, A.: Dynamics of curved fronts, Perspectives in Physics. Acad. Press, 1997. [10] SAFFMAN,P.G., AND TAYLOR, G.I., Proc. Royal Sac. London A 245 (1958), 312. [11] SHRAIMAN,B.I., Phys. Rev. Lett. 56 (1986), 2028. Hamid Kellay MSC1991: 76S05, 76Exx V M O A - S P A C E , space of analytic functions of vanishing mean oscillation - The class of analytic functions on the unit disc that are in VMO (see also B M O - s p a c e ; BMOA-space; VMO-space). Fefferman's duality theorem (see B M O - s p a c e ) gives the characterization that an a n a l y t i c f u n c t i o n in BMO is in VMOA if and only if its boundary values can be expressed as the sum of a c o n t i n u o u s f u n c t i o n and the harmonic conjugate (cf. also H a r m o n i c f u n c t i o n } of a continuous function. This suggests that functions in VMOA are close to being continuous, but one has to be careful because their behaviour can be quite wild. For example, it can be show that any c o n f o r m a l m a p p i n g onto a region of finite area is in VMOA.

VON K~g~RMftN VORTEX SHEDDING D. Sarason [4] used the fact that VMOA is the closure of the disc algebra A in B M O A to prove that H ~ + C, with C the class of continuous functions, is a closed subalgebra of L ~ and consequently the simplest example of a Douglas algebra (see V M O - s p a c e ) . The distance between a function f in BMOA and VMOA has attracted some interest, [1], [2], [5]. Let f be an analytic function on the unit disc, ~ a point on the boundary T and write K ; for the c l u s t e r set Cl(f, ~). Using an assortment of tools from f u n c t i o n a l analysis, S. Axler and J. Shapiro [1] proved that

flow fields. This is a situation where the energy subtracted from the flow field by the body drag is not dissipated directly into an irregular motion in the wake, but it is first transferred to a very regular vortex motion. The structure of the flow is then as schematically indicated in Fig. 1:

-

- @

- @ - - @ - -@ _@_ - @ _

Fig. 1.

Ill + VMOAll, <

C limsup 4cT

~ ~ .

This led to a search for the optimal geometric condition for the right-hand side above, see [5] for the answer. References [1] AXLER, S., AND SHAPIRO, J.: ' P u t n a m ' s theorem, Alexander's spectral area estimate and V M O ' , Math. Ann. 271 (1985), 161-183. [2] CARMONA, J., AND CUFI, J.: 'On the distance of an analytic function to VMO', or. London Math. Soc. (2) 34 (1986), 5266. [3] FEFFERMAN, C.: 'Characterization of bounded mean oscillation', Bull. Amer. Math. Soc. 77 (1971), 587-588. [4] SARASON, D.: 'Functions of vanishing mean oscillation', Trans. A m e r . Math. Soc. 2 0 7 (1975), 391-405. [5] STEPHENSON, K., AND STEGENGA, D.: 'Sharp geometric estimates of the distance to VMOA', Contemp. Math. 137 (1992), 421-432. D. Stegenga

MSC 1991: 30D50, 46Exx

VOLTERRA FUNCTIONAL SERIES, WienerVolterra functional series series.

The same as a V o l t e r r a

M S C 1991:41A58

VOLTERRA-STIELTJES INTEGRAL EQUATION, Stieltjes-Volterra integral equation - A (linear or nonlinear) V o l t e r r a e q u a t i o n in which the integral terms involve a S t i e l t j e s i n t e g r a l , i.e., 'ds' gets replaced by 'dg(s)' for appropriate functions g. MSC1991:45D05

VON K.~RM.~N VORTEX SHEDDING, Kdrrndn vortex shedding - A term defining the periodic detachment of pairs of alternate vortices from a bluff-body immersed in a fluid flow, generating an oscillating wake, or vortex street, behind it, and causing fluctuating forces to be experienced by the object. The phenomenon was first observed and analyzed on two-dimensional cylinders in a perpendicular uniform flow, but it is now widely documented for three-dimensional bodies and non-uniform

The alternate vortex shedding is considered as responsible for a remarkable number of collapses of civil structures and damage to industrial equipment. The vortices at either side of the body have opposite intensities (directions of rotation) and are arranged in a particular geometrical pattern which can be observed even at some distance behind the obstacle. These vortices do not mix with the outer flow and are dissipated by viscosity only after a long time. It is this feature that allows a basic explanation of the phenomenon to be carried out in terms of inviseid flow, even if its origin can only be attributed to the viscosity, as will be discussed below. This idealized explanation was first proposed by Th. von K~rm~n [7], [8] (1911, 1912), and the phenomenon is since associated to his name, even if the first experimental observations were reported by the French physicist H. Bdnard [1], [2], [3], [4], (1908, 1913, 1926) and A. Mallock [11] (1907); cf. also G i n z b u r g - L a n d a u e q u a t i o n . Early representations may be found in the drawings of Leonardo da Vinci and in Flemish paintings of the XVI century. Its idealized mathematical description, restricted to a two-dimensional flow, is based on the investigation of the stability of two parallel vortex sheets with vortices of equal but opposite intensity. Linear vortices of intensity F are located at equal distance from each other along these sheets and the motion resulting from the mutually induced velocity is investigated. (The intensity F of a vortex is based on its circulation and is defined as r :=

fO.d<)

A quick investigation (discussed in detail below) shows that only two different vortex arrangements do not produce transversal induced velocities on each other: a) one with the eddies of one row situated exactly opposite those of the other row; b) the second with the opposite eddies symmetrically staggered, as in Fig. 1.

However, a linear stability analysis with respect to small disturbances shows that the first configuration is always 427

VON KL~RMAN VORTEX SHEDDING dynamically unstable, while the second one becomes stable for one geometrical configuration specified by a definite value of the ratio b/l of the separation b between the sheets, to the longitudinal vortex separation l. To be more accurate, at specific values of this ratio, the vortex pattern is in a situation of indifferent equilibrium with respect to disturbances of wave-length 21, the important ones since they are responsible for the induced velocities between vortices. The result obtained by von K~rm£n for the above ratio is b _ 1 cosh_ 1 v ~ ~ 0.2806, h 7r a value which is considered in very good agreement with the experimental observations for uniform flow around circular cylinders. Associated to this geometry there appears a particular longitudinal velocity for the ensemble of the alternating eddies, which was determined to be

function of the velocity difference A U and separation l, F = A U . l . Under these conditions it is easy to compute, for the two-dimensional case in a fluid at rest [5], the relocity induced on each vortex by all the others, and the eventual global displacement velocity of the sheets using the complex potential formulation of the velocity, where ¢ = ¢-i¢,

with w(z)

¢(z)

-Q

I n d u c e d v e l o c i t i e s in v o r t e x s h e e t s . In inviscid fluid dynamics a vortex sheet can be defined as the ideal plane that separate two streams of uniform but different velocity. One assumes that all the vorticity associated with the velocity j u m p is concentrated on this plane. If the vorticity is carried by distinct vortices placed at equal intervals, then the individual intensity P is only a 428

=

ir

z = x + iy.

log(z-

zj).

i

iv/~,

that is, the vortices detach from the body at a lower speed than the external flow and trail behind it. This means that around each individual vortex there is a number of closed streamlines. It should be noted that the von K~rm£n approach does not give any hint on the values of b and l relative to the dimension, or shape, of the body. This justifies the continuous actual research for the numerical solution to the problem, one of the challenging problems of present computational fluid dynamics (as of 2000). This idealized formulation is based on the assumption of an inviscid fluid, while the generation of the vortices requires the fluid to be viscous. Also, as a consequence of the velocity defect of the vortex trail, the body shedding the vortices experiences a drag, leading to an apparent paradox. As stated by L. Prandtl [13], the explanation is given by the b o u n d a r y l a y e r theory. From it, it can be seen that in the limit of viscosity equal to zero (p = 0), the fluid can be considered frictionless everywhere except in a thin layer close to the body, where a different limit process (# --+ 0 and not p = 0) must be performed in tile N a v i e r - S t o k e s e q u a t i o n s . It is in this region, so small to be negligible in the context of the overall flow field, but in which friction forces cannot be neglected no matter how small the viscosity, t h a t the vorticity found in the vortex sheet is created.

de

With this convention, a single vortex of intensity F located at zj has a complex potential

F Uvortex =

= Ux - iUy -

Fig. 2. For an infinite single sheet of vortices with separation l and position z = ml, ra an integer, in a fluid at rest (cf. Fig. 2), the expression becomes ¢(z)-

iF 2~

~

log(z-(z0-ml)),

m ~ - - o o

or og

?Tt~

-

27r

-

=

-- OO

+const.

This corresponds to a velocity w(m, l) of the vortices at Z = rrt/:

w(m,1)_de~ dz

iF [ 7~z 2~ cotan l

1

z - ml

]

-0.

The velocity of each vortex is equal to zero and the single vortex row remains at rest. However [14], the sheet is not stable: Under even the slightest disturbance it becomes undulatory and experimental observation shows that it then curls up into a series of large vortices. Consider now a double sheet of vortices [10] with distance b between the sheets. All the vortices in each row have same intensity but the two rows have opposite senses of rotation. Even if the separation l between the vortices is the same, m a n y arrangements are possible, since those in opposite rows are not necessarily opposed to each other. Applying the same development as for the single sheet, it can be derived that only two different symmetrical vortex arrangements do not produce transversal induced velocities [12]: the first with the eddies of one row situated exactly opposite those of the other row, and the second with the opposite eddies

VON KL~RMAN VORTEX SHEDDING symmetrically staggered. In these cases the velocities induced by one row on the other are such as to cause only a forward motion of the facing row. F

- -@

@

@

-@- -QP

@

-@- -Q

--

3_

= 2-

F

•

/-- , Fig. 4.

A =

For the case of vortices in opposition (Fig. 3) the induced velocity at vortex positions z = ml + b/2, z = ml - b/2 is

P

7cb

U = ~ coth T '

while for the alternating case (Fig. 4) one has U =

r

tanh --/-.

In both cases, since the velocity is the same for each row, the result is a global displacement of the double vortex street. None of these arrangements is unconditionally stable under the effect of small perturbations, but it was the merit of yon KSrm4n to show that, while the first is never stable, the alternate street m a y become so in certain particular geometrical vortex arrangements. For this case the complex potential is ¢(z)=-~-~F~ [ l o g s i n ( / ( z - ~ ) ) -logsin(/

l

(z--~+~))]

+ +const.

It is based on this formulation that the small perturbation analysis is carried out. Evaluation of the stream function from the relation above shows that around each individual vortex there is a number of closed stream lines. S t a b i l i t y a n a l y s i s . In the original analysis, vortices which are at a given time at position (ml + Ut, ~b/2) are assumed to be displaced to the position (xm,j + ml + Ut, y,~,j + b/2) by a disturbance of the type Xrn,j = ozje iraO, [z = 7 j e imO,

Ym,j = f l j e imO 7 ~- o~ -~- i/3],

0<0<% index j indicating the upper or lower row. If 0 is small, this has the character of an undulation of wave-length

2~rl/O.

r

y(B

T 4T

-

where A, B and C are long functions of 0 and of the geometrical vortex arrangement. It suffices to say t h a t for stability A 2 _< C 2 and t h a t for 0 = 7r (wave-length of disturbance equal to 2l), C = 0. So, for stability A must vanish:

_@_-@--Q--@ -Q- -QQ- -QP

case

/

Fig. 3.

_

It is then possible to derive from the complex potential the equations for d a j / d t , d ~ j / d t as functions of the various parameters. The solutions are of the type e At, a positive A indicating instability. This is always the case for the symmetrical double row. For the non-symmetric

cosh~(lrb/1) - O,

0(27r - 0)

or cosh27c~---2, 7c~ ~

.8814,

b ~

.2806,

which is the result indicated in the introduction, and again F :rb F U= ~tanh l 2lye" In the original p a p e r yon K~rm~n stated, but did not prove, t h a t stability exists for all values of 0 in the range 0 to 2:r, that is, for all possible disturbance wavelengths. The proof is given by H. L a m b [10], who proved it to hold true except for 0 = 0 or 0 = 27r, a situation in which the period of the disturbance is infinite and all fluid particles will move as a whole. D r a g d u e t o v o r t e x s h e d d i n g . As stated at the beginning, the drag of a b o d y moving in a uniform flow can be computed using potential flow theory applied to the alternate shedding vortices. This result, also due to yon K~rm4n for the case of a circular cylinder [9], is an apparent paradox, since inviscid fluids cannot produce drag, and is justified by the fact that the vorticity in the sheets is generated inside the body boundary layer, a region not taken in account in the above analysis. For a cylinder moving in a fluid at rest with velocity V, the vortex street moves with a velocity V - U. The motion in the wake of the cylinder is obviously unsteady, due to the periodic vortex shedding, but this difficulty is overcome by averaging over an observation time equal to the period between the release of two vortices of the same sheet: 1 T=

V-U"

Over this time the transfer in m o m e n t u m to the vortex sheet between a cross-section far upstream of the cylinder and a corresponding one far downstream is equal to M1 = p A V l b = pFb. Per unit time this becomes

dM1 = p ? ( - U ) , 429

VON K~RMAN VORTEX SHEDDING where b is the distance between the vortex sheets. Between the two reference sections, two other sources of resistance must be accounted for: that of the amount of fluid the velocity of which is reduced from that of the free flow, V, to that of the wake, U, and equal to [13] dM~ = P l

References

(V - V),

and the momentum and pressure integral difference over the front and back sections, equal to F2 riM3 = p 27d .

The total drag is then the sum

r2 D = p

~ p~-

(V - 2U) + p2-

1.587

- 0.628

where St is the S t r o u h a l n u m b e r , defined on the vortex street width and velocity. The fact that the drag coefficient and the Strouhal number are inversely proportional [6] for a wide range of two-dimensional bodies has been observed in a number of experiments.

,

[1] BENARD, H., C.R. Acad. Sci. Paris 147 (1908), 839-842; 970-972. [2] B~NARD, H., C.R. Aead. Sci. Paris 156 (1913), 1003 1005; 1225-1228. [3] BI~NARD,H., C.R. Acad. Sci. Paris 182 (1926), 1375-1377; 1523-1525. [4] BENARD, H., C.R. Acad. Sci. Paris 183 (1926), 20 22; 184186; 379. [5] C-YUON, E., HULIN, J.P., AND PETIT, L.: Hydrodynamique physique, Intereditions/Ed. du CNRS, 1957. [6] HOERNER, S.: 'Fluid dynamic drag', in S. HOERNER (ed.): 1962.

[7] KJ~RMJ~N, T. VON, Nachr. Ges. Wissenschaft. GSttingen

if the values previously obtained for the stable sheet are used. Again, it should by noted that with the von K&m~n theory it is not possible to calculate l, b and U for a given obstacle. However, it allows one to determine the drag when they are measured, for instance from a visualisation of the vortex trail and the frequency of vortex release. Experimental results are then in good agreement with the expected theoretical results. Finally, denoting by Cd the drag coefficient, defined as

Cd-

430

l

f,.~,

1

509-517.

D

D. Olivari

MSC 1991: 76Cxx

pV2b '

and by f the frequency of the vortex release, then Cd~,

(19u),

[8] KJtRM£N, T. VON, Nachr. Ges. Wissensehaft. GSttingen (1912), 547-556. [9] KJ~RMAN, T. VON, AND RUBACH, H.L.: 'On the mechanisms of fluid resistance', Physik Z. 13 (1912), 49-59. [10] LAMS, H.: Hydrodynamics, Cambridge Univ. Press, 1932. [11] MALLOCK, A., Proe. Royal Soc. A79 (1907), 262-265. [12] MACE, A.W.: 'Zur Stabilitgt der Karmansche Wirbelstrasse', Z. Angew. Math. Mech. 20 (1940), 129-137. [13] PRANDTL, L., AND TIETJENS, O.G.: Applied hydro and aeromechanics, Dover, 1957. [14] ROBERTSON, J.M.: Hydrodynamics in theory and applications, Prentice-Hall, 1965.

fU

Cd~ ~-,

1

Cd~s~ ,

W W A L S H - P A L E Y SYSTEM, P a l e y - W a l s h s y s t e m The same as the W a l s h s y s t e m of functions. MSC 1991:33C45 W A R I N G - G O L D B A C t I PROBLEM - The same as the W a r i n g p r o b l e m and the G o l d b a c h - W a r i n g problem. MSC 1991: 11Pxx W E A K P - P O I N T - A point in a t o p o l o g i c a l s p a c e that is not an a c c u m u l a t i o n p o i n t of any countable subset of the space. Every P - p o i n t is a weak P-point. Weak P-points were introduced by K. Kunen [2] in his proof that N*, the remainder ~ N \ N of the S t o n e C e c h c o m p a c t i f i c a t i o n of the natural numbers, is not homogeneous. In fact, Kunen proved that N* contains points that are very much like P-points, so-called c-OK points: A point is c - O K if for every sequence (Un: n C N} of neighbourhoods there is a c-sequence (V~ : c~ < ¢) of neighbourhoods such that C?~cFV~ C U,~ whenever F has n elements. A c-OK point cannot be an accumulation point of any set that satisfies the countable chain condition (cf. C h a i n c o n d i t i o n ) , hence it is not an accumulation point of any countable set either. Weak P-points and similar types of points have been used to give so-called 'effective' proofs that many spaces are not homogeneous [4], [3]. These proofs are generally considered simpler than the proof by Z. Frolfk [1] of the non-homogeneity of N*, which takes a countably infinite discrete subset D of N* (whose closure is homeomorphic to/~N) and shows that a point p of N* cannot be mapped by any auto-homeomorphism of N* to its copy in the closure of D. 'Simpler' does not necessarily mean that the proof is easier, but that the properties used to distinguish the point are of a simpler nature. References [1] FROLIK, Z.: 'Non-homogeneity of ~P - P', Comment. Math. Univ. Carolinae 8 (1967), 705 709.

[2] KUNEN, K.: 'Weak P-points in N*', in A. CS~SZ£R (ed.): Topology (Proc. Fourth Colloq., Budapest, 1978), Vol. II, North-Holland, 1980, pp. 741-749. [3] MILL, J. VAN:'Sixteen topological types in f~w - w', Topoi. Appl. 13, no. 1 (1982), 43-57. [4] MILL, J. VAN:'Weak P-points in Cech-Stone compactifications', Trans. Amer. Math. Soc. 273 (1982), 657-678. K.P. Hart

MSC 1991: 54D40, 54G10 W E I E R S T R A S S REPRESENTATION OF A MINIMAL SURFACE - Let M be a R i e m a n n s u r f a c e . A harmonic eonformal mapping X : M -4 R n then defines a m i n i m a l s u r f a c e in R n, n _> 3 (cf. also H a r m o n i c f u n c t i o n ; C o n f o r m a l m a p p i n g ) . Let z = u + iv be local i s o t h e r m a l c o o r d i n a t e s ; then

5=1 \-52

)

= o.

Since X is harmonic, 035 =

2~oxj

dz

is a holomorphic 1-form on M. Hence any (branched) minimal surface in R n can be given by n meromorphic n l 03t2 = 0, and X can be ex1-forms 03j satisfying ~ j = pressed as X ( p ) = Re

/;

(c~1,..., 03n).

(1)

o

Such an X is well defined on M if and only if for any l o o p C in M,

Re

= (0,...,0).

(2)

For n = 3, one gets a m e r o m o r p h i c f u n c t i o n g and a meromorphic i-form ~/, 03-]1 Jr- i032

g-

033

__

033

031-iwe'

--1

~=g

033.

On the other hand, given a meromorphic function g and a meromorphic 1-form r/on M, define 031 :

1 - g2)r/,

w2 =

1 + g2)%

033 = gr/;

(3)

W E I E R S T R A S S R E P R E S E N T A T I O N OF A MINIMAL SURFACE s I Cdj2 = 0. Thus, (3) together with (1) defines a then E j = minimal surface in R 3 and is called the Weierstrass representation of the minimal surface via the Weierstrass

data (g, TI). The meromorphic function g has the geometric meaning that it is the composite of the spherical mapping (or unit normal vector) N : M ~ S 2 and the s t e r e o g r a p h i c p r o j e c t i o n from the north pole, where X - ~(2Reg, Igl ~ + 1

with ~ denoting the Lie algebra of K . The relative Weil algebra for (G, K ) is defined by

W(G,K) =

®

K.

With regards to the universal classifying bundle EG --+ BG (cf. also B u n d l e ; C l a s s i f y i n g space; U n i v e r s a l space), there are canonical isomorphisms in c o h o m o l ogy

H*(W(G,K))

~

H*(EG/K,R)

2 I m g , lgl ~ - 1)

and g is also called the Gauss map of the minimal surface. The first f u n d a m e n t a l f o r m and the G a u s s i a n c u r v a t u r e of the surface X ( M ) can be expressed via

I@0

R)

where I(K) denotes the AdK-invariant polynomials. For a given integer k > 0, one has the ideal

F W = FS(k+I)w(G,K) C W ( a , K ) , generated by St(g*), for ~ > k + 1. This leads to the

ds 2 =

41@1 K = Hence X ( M ) 3

truncated Weil algebra

(1+[g[2)217/12= 2 ~ [ w j [ 2 '

E j = I I jl ¢ 0

j=l

Wk = W ( a , K)k = W(G, K ) / F W .

)2

The cohomology H*(Wk) plays a prominent role in the study of secondary characteristic classes (cf. also C h a r a c t e r i s t i c class) of foliations and foliated bundles [3] (see also [2]).

(1j~)~lq

is a regular surface

if and only if

on M.

The s e c o n d f u n d a m e n t a l expressed as

f o r m of X ( M ) can be

II(W, V) = - Re(~(W) dg(V)). Moreover, W is an asymptotic direction if and only if ~(W) dg(W) C iR, and W is a principal curvature direction if and only if rl(W) dg(W) C R. The local Weierstrass representation was discovered in the 1860s by K. Enneper and K. Weierstrass. R. Osserman gave the general form on a Riemann surface in the 1960s, see [1] for more details. References [1] OSSERMAN., R.: A survey of minimal surfaces, Dover, 1986.

Yi Fang MSC1991: 53A10, 53C42 W E I L ALGEBRA OF A LIE ALGEBRA - Let G

be a connected Lie g r o u p with Lie a l g e b r a ~. The Weil algebra W(g) of 9 was first introduced in a series of seminars by H. Cartan [1], in part based on some unpublished work of A. Weil. As a differential g r a d e d a l g e b r a , it is given by the tensor product =

®

where Aft* and S9" denote the exterior and symmetric algebras, respectively (cf. also E x t e r i o r a l g e b r a ; Symmetric algebra). The Weil algebra and its generalizations have been studied extensively by F.W. Kamber and Ph. Tondeur [3] [4]. Let K C_ G be a maximal compact subgroup, 432

References [1] CARTAN, H.: 'Cohomologie r~elle d'un espace fibrd principal differentiable': Sere. H. Cartan 1949/50, Exp. 19-20, 1950. [2] DUPONT, J.L., AND KAMBER, F.W.: 'On a generalization of Cheeger Chern-Simons classes', Illinois d. Math. 34 (1990). [3] KAMBER, F.W., AND TONDEUR, PH.: Foliated bundles and characteristic classes, Vol. 493 of Lecture Notes in Mathematics, Springer, 1975. [4] KAMBER, F.W., AND TONDEUR, PH.: 'Semi-simplicial Weil algebras and characteristic classes', T6hoku Math. J. 30 (1978), 373-422.

James F. Glazebrook MSC1991: 57Rxx, 55R40 W E I L - P E T E R S S O N METRIC - A. Weil introduced a K~ihler m e t r i c for the T e i c h m i i l l e r s p a c e Tg,~ , the space of homotopy-marked Riemann surfaces (cf. R i e m a n n s u r f a c e ) of genus g with n punctures and negative E u l e r c h a r a c t e r i s t i c , [1]. The cotangent space at a marked Riemann surface {R} (the space Q(R) of holomorphic quadratic differentials on R; cf. also Q u a d r a t i c d i f f e r e n t i a l ) is considered with the Petersson Hermitian pairing. The Weil-Petersson metric calibrates the variations of the complex structure of {R}. The uniformization theorem implies that for a surface of negative Euler characteristic, the following two determinations are equivalent: a complex structure and a complete hyperbolic metric. Accordingly, the Weil-Petersson metric has been studied through q u a s i - c o n f o r m a l m a p ping, solution of the inhomogeneous R-equation (cf. also N e u m a n n R - p r o b l e m ) , the prescribed curvature equation, and global analysis, [1], [8], [12].

WEIL-PETERSSON METRIC The quotient of the Teichmiiller space T~,n by the action of the mapping class group is the moduli space of Riemann surfaces Adg,~ (cf. also M o d u l i o f a Riem a n n surface; M o d u l i t h e o r y ) ; the Weil-Petersson metric is a mapping class group invariant and descends to Adg,~. 3dg,~ (the stable-curve compactification of Adg,~) is a projective variety with ~gg,~ = Adg,n - 214~,,~ (the divisor of noded stable-curves, i.e. the Riemann surfaces 'with disjoint simple loops collapsed to points' and each component of the nodM-complement having negative Euler characteristic). Expansions for the WeilPetersson metric in a neighbourhood of ~Dg,n provide that the metric on 3dg,~ is not complete and that there is a distance completion separating points on 2tds,,~, [6]. The Weil-Petersson metric has negative sectional curvature, [11], [15]. The behaviour near 77g,n shows that the sectional curvature has as infimum negative infinity and as supremum zero. The holomorphic sectional, Ricci and scalar curvatures are each bounded above by genus-dependent negative constants. A modification of the metric introduced by C.T. McMullen [7] is K//hlerhyperbolic in the sense of M. Gromov (cf. also Grom o v h y p e r b o l i c space), has positive first eigenvalue and provides that the sign of the 3dg,n orbifold Euler characteristic is given by the parity of the dimension. The Weil-Petersson KShler form wwp appears in several contexts. L.A. Takhtayan and P.G. Zograf [10] considered the local index theorem for families of ~operators and calculated the first Chern form of the determinant line bundle det i n d 0 using Quillen's construction of a metric based on the hyperbolic metric; the Chern form is 1 127r2-wwp" The ~universal curve' is the fibration C9,~ over Tg,~ with fibre R above the class {R}. The uniformization theorem provides a metric for the vertical line bundle Fg,n of the fibration. The setup extends to the compactification: The pushdown of the square of the first Chern form of Fg,n for the hyperbolic metric is the current class of 1 27r 2 WWP,

[17]. This result is the basis for a proof of the projectivity of 2tdg,~, [13]. The Weil-Petersson volume element appears in the calculation by E. D'Hoker and D.H. Phong [4] of the partition function integrand for the string theory of A.M. Polyakov. Generating functions have also been developed for the volumes of moduli spaces, [5], [18]. J.F. Brock considers a coarse combinatorial estimate for the Weil-Petersson distance in terms of the edge path metric in the pants complex, [3].

W. Fenchel and J. Nielsen presented 'twist-length' coordinates for Tg,n, as the parameters {(~-j, ~j)} for assembling pairs of pants, three-holed spheres with hyperbolic metric and geodesic boundaries, to form hyperbolic surfaces. The Kghler form has a simple expression in terms of these coordinates: wwp = Ej d~j A d~-j, [14]. Each geodesic length function f. is convex along Weil-Petersson geodesics, [16]. Consequently, Tg,n has an exhaustion by compact Weil-Petersson convex sets, [16]. A. Verjovsky and S. Nag [9] considered the WeilPetersson geometry for the infinite-dimensional universal Teichmfiller space and found that the form wwp coincides with the Kirillov Kostant symplectic structure coming from Diff+(Sl)/Mob(S1). I. Biswas and Nag [2] showed that the analogue of the Takhtayan-Zograf result above is valid for the universal moduli space obtained from the inductive limit of Teichmiiller spaces for characteristic coverings. References [1] AHLFORS, L.V.: 'Some remarks on Teichmfiller's space of Riem a n n surfaces', Ann. of Math. 74, no. 2 (1961), 171-191. [2] BIswAs, I., AND NAG, S.: ' W e i l - P e t e r s s o n geometry and det e r m i n a n t bundles on inductive limits of moduli spaces': Lipa's legacy (New York, J995), Amer. Math. Soc., 1997, pp. 51-80. [3] BROCK, J.F.: ' T h e W e i l - P e t e r s s o n metric and volumes of 3dimensional hyperbolic convex cores', Preprint (2001). [4] D'HOKER, E., AND PHONe, D.H.: 'Multiloop amplitudes for the bosonic Polyakov string', Nucl. Phys. B 269, no. 1 (1986), 205-234. [5] I~[AUFMANN,I~., MANIN, YU., AND ZAGIER, D.: 'Higher WeilPetersson volumes of moduli spaces of stable n-pointed curves', Comrnun. Math. Phys. 181, no. 3 (1996), 763 787. [6] MASUR, H.: 'Extension of the Weil-Petersson metric to the b o u n d a r y of Teichmuller space', Duke Math. J. 43, no. 3 (1976), 623-635. [7] MCMULLEN, C.T.: 'The moduli space of R i e m a n n surfaces is Kghler hyperboiic', Ann. of Math. 151, no. 1 (2000), 327357. [8] NAG, S.: The complex analytic theory of Teichmiiller spaces, Wiley/Interscience, 1988. [9] NAC, S., AND VERJOVSKY, A.: 'Diff(S 1) and the Teichmiiller spaces', Commun. Math. Phys. 130, no. 1 (1990), 123-138. [10] TAKHTAJAN, L.A., AND ZOGRAF, P.G.: 'A local index theorem for families of O-operators on punctured R i e m a n n surfaces and a new K~ihler metric on their moduli spaces', Commun. Math. Phys. 137, no. 2 (1991), 399-426. [11] TROMBA, A.J.: ' O n a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmfiller space with respect to its Weil-Petersson metric', Manuscripta Math. 56, no. 4 (1986), 475 497. [12] TROMBA, A.J.: Teichmiiller theory in Riemannian geometry, BirkhSuser, 1992. [13] WOLPERT, S.A.: 'On obtaining a positive line bundle from the Weil-Petersson class', Amer. J. Math. 107, no. 6 (1985), 1485-1507.

433

WEIL PETERSSON METRIC [14] WOLPERT, S.A.: 'On the Weil Petersson geometry of the [15] [16] [17] [18]

moduli space of curves', Amer. J. Math. 107, no. 4 (1985), 969-997. WOLPERT, S.A.: 'Chern forms and the Riemann tensor for the modnli space of curves', Invent. Math. 85, no. 1 (1986), 119 145. WOLPERT, S.A.: 'Geodesic length functions and the Nielsen problem', J. Differential Geom. 25, no. 2 (1987), 275 296. WOLPERT,S.A.: 'The hyperbolic metric and the geometry of the universal curve', g. Differential Geom. 31, no. 2 (1990), 417-472. ZOGRAF,P.: 'The Weil Petersson volume of the moduli space of punctured spheres': Mapping Class Groups and Moduli Spaces of Riemann Surfaces (G6ttingen/Seattle, WA, 1991), Amer. Math. Soc., 1993, pp. 367-372. Scott A. Wolpert

distinct even imaginary roots in R, the fit are distinct odd imaginary roots in R, (~kl~;) = (&lfl;) = o

if k # / , (~klg;) = o

for all k, l, (&lflk) = o

ifpk > 1 , and (;q~)

for all k, 1. Set r = s = 0 if fl = 0, and define s Pj ' T h e n 171 =r+~-~-j=l

MSC1991: 14H15, 30F60 ch V(A) =

Weyl Kac formula, Kac-Weyl character formula, Kac-Weyl formula, Weyl-Kae-Borcherds character formula - A forWEYL-KAC

CHARACTER FORMULA,

mula describing the character of an irreducible highest weight module (with dominant integral highest weight) of a K a c - M o o d y algebra. The formula is a generalization of Weyl's classical formula for the character of an irreducible finite-dimensional representation of a semisimple Lie a l g e b r a (cf. C h a r a c t e r f o r m u l a ) . The formula is very robust and has been steadily applied (with increasing technical complications) to the representations of ever wider classes of algebras, see [3] for representations of Kac-Moody algebras and [2] for generalized Kac-Moody (or Borcherds) algebras. Let 1~be a Borcherds (colour) s u p e r a l g e b r a (cf. also B o r c h e r d s Lie algebra) with charge m and integral Borcherds-Cartan matrix A = (aij)~ restricted with respect to the colouring matrix C. (The charge counts the multiplicities of the simple roots.) Let 0 denote the C a r t a n s u b a l g e b r a of g and let V be a weight gmodule with all weight spaces finite-dimensional. The formal character of V is ch V ----- E

(dim Vt~)eu.

#EO*

For V(A) an irreducible highest weight module with dominant integral highest weight A, U. Ray [6] and M. Miyamoto [5] have established the following generalization of the Weyl Kac Borcherds character formula. Let W be the W e y l g r o u p , A - the negative roots and R the set of simple roots counted with multiplicities. Let p E ~* be such that 1

p(h~) = ~ a ~

for all i. Define Sx = e x+p ~ ( - 1 ) l ~ l e - ~ , where the sum runs over all elements of the weight lattice of the ?~ S form 7 = E i = I °zi ~- E j = l P J f l J such that the ak are 434

= (~19;) = o

e-O EwEW(-1)Z(W)w(S),) 1 - L c a - (1 -

O(a, a)e~)°(~, ~) dim ~o'

where 0 is the colouring map induced by C and $~ is the a root space of g. In the case of Kac-Moody algebras, there are no imaginary simple roots and 0 ( a , a ) = 1 for all a, so one recovers the Weyl-Kac formula chV(A) = ~cw(-1)z(W)eW(a+P)-P l - I a e A - (1 -- ec~) dim $~

These character formulas may also be applied to representations of associated q u a n t u m g r o u p s where quantum deformation theorems are known (see [4] and [1], for example). References

[1]

[2] [3] [4]

[5] [6]

BENKART, G., KANG,

S.-J.,

AND MELVILLE, D.J.:

'Quan-

tized enveloping algebras for Borcherds superalgebras', Trans. Amer. Math. Soc. 350 (1998), 3297-3319. BORCHERDS, R.E.: 'Generalized Kac-Moody algebras', d. Algebra 115 (1988), 501 512. KAC, V.G.: 'Infinite-dimensional Lie algebras and Dedekind's ~7 function', Funct. Anal. Appl. 8 (1974), 68-70. KANG, S.-J.: ' Q u a n t u m deformations of generalized KacMoody algebras and their modules', J. Algebra 175 (1995), 1041-1066. MIYAMOTO, M.: 'A generalization of Boreherds algebras and denominator formula', J. Algebra 180 (1996), 631-651. RAY, U.: 'A character formula for generalized Kac-Moody superalgebras', J. Algebra 177 (1995), 154-163.

Duncan J. Melville MSC1991: 17B10, 17B65 WHITHAM

E Q U A T I O N S - Perhaps the proper be-

ginning of Whitham theory is Whitham's work [74], [75], which can be viewed as a crucible of various averaging ideas subsequently developed in e.g. [1], [4], [21], [22], [20], [27], [28], [29], [43], [49], [50], [60], [73] to theories involving multi-phase averaging, Hamiltonian systems and weakly deformed soliton lattices. The term 'Whitham equations' then became associated with the moduli dynamics of Riemann surfaces and this fits naturally into work on topological field theories, Frobenius

W H I T H A M EQUATIONS manifolds, renormalization groups, coupling constants, and Seiberg Witten theory (cf. also S e i b e r g - W i t t e n e q u a t i o n s ) , along with singularity theory, isomonodromy deformations, quantum cohomology and K t h e o r y , G r o m o v - W i t t e n invariants, Witten-DijkgraafVerlinde-Verlinde equations, etc. (see the references below or the survey material in [5], [8], [10], [7], [6], [9]).

Here, the coefficients {c~j,/~j} are determined by /A

= 0.

(1)

Key early papers on averaging for this equation include [27] and [49]. The basic ideas of G. Whitham are discussed in [75], [74]. Other important papers include [3], [2], [16], [65]. The key idea is averaging out fast scales; one introduces two scales: the 'fast' scale (x,t) and 'slow' scale (X = ex, T = et), c small. One obtains a class of ('finitegap') solutions of the form u(x,t)

(2)

= u =

where fg is a m e r o m o r p h i e f u n c t i o n of g variables and Oi = ~i + wi + Oi, where the parameters U, ~i, wi depend only on the slow variables. One can then write down the evolution equations for g-phase wave trains in terms of differentials on an associated Riemann surface. The Whitham equation for the Korteweg de Vries equation is given by Odwl O~-

Odwa 0~-'

29

-

i=0

and the branch points hi are real and are assumed to satisfy ~0 < " " < 12g. Explicitly, dWl (.~) = Hig=l ('~ -- O~i) d), ~ d~ - - ÷ (holomorphic),

(4)

as A --+ oc,

4X = a ÷1 _

+

x/~d,~ + (holomorphic),

J = l, . . . , g,

t) = i=0

j=0

Note that when g = 0, the equation reduces to the dispersionless Korteweg-de Vries equation (HopfBurgers equation) with A0 = 2g and A1 . . . . . A2g = a l = ' " = c~g = 0, i.e., 0~ 0~ OT OX " The Whitham equation for the discrete Toda lattice (cf. T o d d l a t t i c e s ) is treated in [4] where shock formation is analyzed. Shocks for the Korteweg-de Vries

equation are analyzed in [34], [35]. The discrete Ablowitz-Ladik equations are analyzed in [59]. The Whitham equations are also important in the analysis of the non-linear SchrSdinger equation (cf. also B e n j a m i n - F e i r i n s t a b i l i t y ) and non-linear optics, see for example [36], [42], [64] and references therein. G e n e r a l W h i t h a m t h e o r y . More generally, for any compact R i e m a n n s u r f a c e Eg of genus g and point Q ~ oc, the Baker Akhiezer function ¢ gives rise to a KP-hierarchy (cf. also K P - e q u a t i o n ) . In particular, following [11], [27], [43], ¢ can be written as

(3)

where dC~l and dcu3 are Abelian differentials on the Riemann surface of genus 9 given by y2 = Rg(A) (cf. also Differential o n a R i e m a n n surface; A b e l l a n diff e r e n t i a l ; R i e m a n n s u r f a c e ) , where

= 1-[(x

dw3 = O, J

the vanishing of the contour integral along the canonical 2g Aj-cycle, and ch = ~ i = 0 hi. Then the averaged solution of the Korteweg-de Vries equation is given by

A v e r a g i n g . One of the most important applications of averaging theory and the Whitham equation is to the K o r t e w e g - d e Vries e q u a t i o n ut - 6uu~ + u ~

dWl = / A J

+

+

da ~

as A -+ ec.

(5)

where dam ~ d(~ ~) + . . . with fB~ d a n = V i n

near ec and fA~ dftn = 0

~ (lZn)i. Here, ¢ is periodic and

f ~ = ~n(T,~) with 17~ = 17n(Tm) for slow times Tm defined via T m = etm and 0* = ~ t~l?~ is a point in the Jacobian Jac(Eg) (cf. also J a e o b i v a r i e t y ) . Assuming, for simplicity, that the periods are incommensurable, by ergodicity one finds --2L

L ¢ d t i = (¢) = \ 2 7 r /

"'"

Cd2gO"

With ¢* corresponding to the adjoint Baker-Akhiezer function, one can think now of multi-scale analysis of ¢~b* dE with Oi --+ Oi + e(O/OTi) plus averaging over the fast times (here, dE = dA + O(~ -2) dA near ec is canonically specified). This corresponds to looking at an e expansion and setting the average first-order term to zero, leading to the Whitham equations OT,~

- -OT,,

(7) 435

W H I T H A M EQUATIONS S e i b e r g - W i t t e n t h e o r y . Given a low-energy effective action for an N = 2 susy gauge theory with partition function

Z(t, ¢) = / J¢

Z)¢exp[S(t, ¢)],

(8)

(n < 2N for technical reasons and dcoj ~ holomorphic differentials). The standard Whitham theory is now based on M

M

with ¢ ,~ fields, t ~ coupling constants and G ~ gauge group in the background, it turns out (e.g. in matrix models) that Z(t, ¢) will often be a tau-function of K P Toda type via Ward identities and Virasoro (origin of integrability). Recall that tau-functions are basic ingredients in integrable system theory (cf. also K P - e q u a t i o n ; T o d a l a t t i c e s ) and e.g. = exp ( E tn 1~) T(tj -- (1/jAJ))

1

1

Odf~A OTB

quantum

moduli

arena shifts to the as renormal-

and as deforma-

in another. The

T goes to a quasi-classical tau-function

(13)

with OdS/Oaj = dwj and OdS/OTn = dwn for (Tin aj) independent. The pre-potential F arises via

dS

(14)

J

space and the Tn appear of moduli

Odf~B - (gTA

-

ized coupling constants in one approach tion parameters

--

(9)

For Z ~ T one has an effective (classical-type) dynamics in the t variables and averaging corresponds in some sense to suppressing fast oscillations (which suggests a renormalization procedure); alternatively, it is also in some sense related to a quantization procedure in the first W K B J a p p r o x i m a t i o n , which produces slow dynamics on the action variables (Hamiltonians Casimirs from ~ = Lie(G); cf. also C a s i m i r e l e m e n t ; K a e - M o o d y a l g e b r a ) , which is equivalent in many situations to dynamics on the moduli of the underlying the quantum

(12) 1

where M < 2N and To = 0 for N , = 0. One has then Whitham equations

(tj)

spectral curves. Thus,

g

dS=ET~d~n=ET~dg~+Ec~idwj,

o

tau-function

whose

logarithm

(after e adjustment) is called the pre-potential F and this serves as a generating function for correlators and as a vehicle for expressing further renormalization effects. Consider (cf. [31], [32], [33], [37], [38], [39], [40], [41], [62]) the following example of Seiberg-Witten Toda curves for iV = 2 susy Yang-Mills with G = SU(N), N = Arc, no masses and moduli uk E Ad = quantum moduli space of inequivalent vacua: y2 = p2 _ 4A2N,

(10)

and O,~F = (1/27rin)Reso ~-~ dS, where Res0 involves o c i and the S e i b e r ~ W i t t e n differential is

Thus, for T~ = 6~,1 one has the Seiberg-Witten situation F sw = / v and one writes then also ai O~i. =

G e n e r a l f r a m e w o r k . The Whitham formulation of I. Krichever, developed in great detail with D.H. Phong (cf. [45], [46], [44], [48], [47]), involves a Riemann surface Eg with M punctures P~. One picks in an ad hoe manner two Abelian differentials dE and dQ having certain properties and sets dS = Q d E as a Seiberg-Witten-type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times TA with corresponding differentials df~A. For suitable choices of dE and dQ the formulation is adequate for Seiberg-Witten-type situations and topological field theories with Witten Dijkgraaf-Verlinde-Verlinde equations will arise as well. Soft s u s y b r e a k i n g . There is another role for Whitham times, via (of. [26], [55])

T~

=

T,~TF 1,

uk

=

Tlkuk,

(16)

and 3i = ai(Uk,T1,Tn>l = O) = Tlai(uk,A = 1) = a i ( ~ , A = T1) (note T1 ~ A in the Seiberg Witten situation). Then one defines

N

= -ilog(

)

(17)

2

Here, A is the quantum scale, ( is a local coordinate at oc+ with A~ ,~ w ~:(1/N) with w -+ oe at oo+ and w -+ 0 at oe_, and 9 = N - i. One

defines

d~n:p+/N(~)

(11)

and d~

436

and sn = - i T ~ and these are promoted to spurion superfields $~ = s~ + 02F~ and V~ = (1/2)D~02-02 in i v = 1 superfield language (0 and 0 are Grassmann variables while D~ and E~ are auxiliary fields). One has a family of non-susy theories and soft susy breaking A / : 2 --+ 3 / = 0 is achieved by fixing s~ = 0 for n > 1 and using D~, F~ (n _> 1) as susy breaking parameters (actually, the F~ alone will suffice). In any event, one can develop formulas involving I, T~ and a i derivatives

W H I T H A M EQUATIONS of the pre-potential and eventually parametrize soft susy breaking terms induced by all of the Casimirs.

there exist 39 - 3 vector fields 1)~ = n0~ + {Ha, "} which annihilate ®. With {'}0 ~ co°-structure this gives

I s o m o n o d r o m y . Various isomonodromy problems can be treated by multi-scale analysis to produce results indicating that isomonodromy deformations in W K B approximation correspond to modulation of isospectral problems (with Whitham-type equations as modulation equations). One can generate a pre-potential, period integrals, etc. as in Seiberg-Witten theory (see e.g. [68], [69], [67], [66], [72]). There are also isomonodromy connections to the Knizhnik-Zamolodchikov-Bernard equations (cf. [52], [51], [53], [63]); these equations arise in various ways in conformal field theory, geometric quantization of fiat bundles, etc. Here one takes FB(Eg, G) as fiat vector bundles over Eg with G = GL(N, C) and smooth connections A ~ (A, A). 'Flat' means zero curvature and with an arbitrary t~ this has the form

nOsH~ - nO~H8 + {Hs, Hr}0 = 0.

(nco + A)¢ = 0

(18)

and (0 + A)¢ = 0. Let # E ft-~'l(Eg) (Beltrami differentials), so # = #(z,-5)O~ ® d~ and set # = Eel t~p °, where g = 3g - 3 (g > 1) and #o is a basis in TAdg. Then (18) becomes (n0 + A)~ = 0

(19)

and (0 + # 0 + A)¢ = 0. Let 3" be a homotopically nontrivial cycle in Eg such that (Zo, g0) E 3' with ¢(Zo, go) = I and write 32(3') = ¢(z0,go)l~ = P e x p ( f A) (pathordered exponential), which yields a representation of Hl(Eg,z0) in G L ( N , C ) . The independence of monodromy y to complex structure deformation corresponds to OJ;/Ota = COa3; = 0 for a = 1 , . . . , ~ . Compatibility with (19) requires

O~A = 0

and

c0X= (1/n)A# °.

(20)

These equations are Hamiltonian when FB(ag, G) has a symplectic form w ° = f ~ (hA, 5A) with Hamiltonians

(1/2) fE~ (A,A)p°a • Consider the bundle 79 over 34g with fibre F B (using ( A , A , t ~ ta) as local coHa

=

ordinates). A gauge fixing plus flatness corresponds to reduction from F B -4 F B and one can (via WZW theory) fix the gauge to get a bundle ?5 with fibre F~-B and equations (~cO + L ) ¢ = 0

(21)

with (0 + #cO + L)~b = 0 and (ncoa + M a ) ~ ) O, where Ma comes from the gauge transformation. Putting in the canonical form via local coordinates (vi, ui) in F B , where i = 1 , . . . , M = (N 2 - 1)(9 - 1), one can write =

~o = (~v, 5u)

(22)

Using the PoincardCartan invariant form 0 = (u, bv) - ( l / n ) ~ H j t a

with w = w° - ( l / n ) ~ S H a S t a .

(23)

These equations define fiat connections in 75 and are referred to as a Whitham hierarchy of isomonodromic deformations. For a given f(u, v, t) on 75 they take the form

df = ~Osf + {Hs, f } dts

(24)

and one can introduce a pre-potential F on 75 giving Hamilton-Jacobi equations (cf. H a m i l t o n - J a c o b i theory)

~O~F+H~\bu

u,t

=0.

(25)

Thus, one has a derivation of deformation equations, properly referred to as a Whitham hierarchy, which involves no averaging or multi-scale analysis. One can also compare the Baker-Akhiezer function ~b in the Whitham hierarchy of isomonodromic deformations with elements of a certain Hitchin hierarchy (cf. also H i t e h i n syst e m ) using a WKB approximation with fast times t H and slow times Ts ~ ts. C o n t a c t t e r m s . For Af = 2 susy gauge theory on a 4-manifold with b2+ = 1 there is a u-plane integral for, say, SU(N) situations, which can be related to a Toda theory with fast and slow (Whitham) times (cf. [55], [56], [57], [58], [61], [70], [71]). Witten-Dijkgraaf-Verlinde-Verlinde. There is a beautiful and elaborate theory of B. Dubrovin and others based on Frobenius manifolds (cf. [15], [13], [12], [14], [23], [24], [25], [19], [18], [17]). This approach is especially pleasing since there is a great deal of motivation and natural structure. There are many connections to mathematics and physics and this approach has led to extensive development in Frobenius manifolds, quantum cohomology and K-theory, singularity theory, W i t t e n Dijkgraaf-Verlinde-Verlinde, etc. (see e.g. [15], [13], [12], [14], [23], [24], [25], [30], [54]). A simple Hurwitz-space Korteweg-de Vries-Landau Ginsburg model is as follows. Let 3dg,n+l be the moduli space of g gap Kortewegde Vries solutions based on L = 0 n+l qlco n - 1 . . . . . q~ with ramification based on W = p,~+l _qlp,~-i . . . . . q~. One defines Whitham times -

Ti Tn+~ =

n +1 n+l-i

Res~ W 1-[i/(~+1)] dp,

1 /A p dW, 2~ri ~

Tg+n+~ = J ;

(26)

dP,

where 1 < i _< n and 1 < a < g. These are flat times for a certain metric and determine a 437

WHITHAM

Whitham

EQUATIONS

hierarchy,

logical field theory

a Frobenius of Landau

ing the Witten-Dijkgraaf-Verlinde (associativity

equations

manifold Ginsburg

and

a topo-

type

Verlinde

satisfy-

equations

for related field correlators).

References [I] ABLOWITZ, M., AND BENNEY, D.: 'The evolution of multiphase modes of nonlinear dispersive waves', Stud. Appl. Math. 49 (1970), 225-238. [2] AVILOV, V.V., KRICHEVER, I.M., AND NOVIKOV, S.P.: 'Evolution of a Whitham zone in the Korteweg de Vries theory', Soviet Phys. Dokl. 32 (1987), 564-566. [3] AVILOV, V.V., AND NOVIKOV, S.P.: 'Evolution of the Whitham zone in KdV theory', Soviet Phys. Dokl. 32 (1987), 366-368. [4] BLOCH, A., AND KODAMA, Y.: 'Dispersive regularization of the Whitham equation for the Toda lattice', S I A M J. Appl. Math. 52 (1992), 909-928. [51 BRADEN, H., AND KRICHEVER, I. (eds.): Integrability: The Seiberg-Witten and Whitham equations, Gordon &: Breach, 2000. [6] CARROLL,R.: 'Some survey remarks on Whitham theory and EM duality', Nonlin. Anal. 30 (1997), 187-198. [7] CARROLL, R.: 'Remarks on Whitham dynamics', Applic. Anal. 70 (1998), 127-146. [81 CARROLL, R.: 'Various aspects of Whitham times', math-ph 9905010 (1999). [9] CARROLL~R.: Quantum theory, deformation and integrability, Elsevier, 2000. [10] CARROLL, l:{.: 'Various aspects of Whitham times', Acta Applic. Math. 60 (2000), 225-316. [II] CARROLL, I:~.,AND CHANG, J.: 'The Whitham equations revisited', Applic. Anal. 64 (1997), 343-378. [12] DUBROVIN, B.: 'Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models', Commun. Math. Phys. 145 (1992), 195 207. [13] DUBROVIN, B.: 'Integrable systems in topological field theory', Nucl. Phys. B 379 (1992), 627 689. [14] DUBROVIN, B.: 'Integrable systems and classification of 2dimensional topological field theories', in O. BABELON ET AL. (eds.): Integrable Systems: The Verdier Memorial Conf., Birkh/~user, 1993, pp. 313-359. [15] DUBROVIN, B.: 'Geometry of 2D topological field theories', in M. FRANCAVIGLIAET AL. (eds.): Integrable Systems and Quantum Groups, Vol. 1620 of Lecture Notes in Mathematics, Springer, 1996, pp. 120-348. [16] DUBROVIN,B.: 'Punctionals of the Peierls-Frhhlich type and the variational principle for the Whitham equations', in V.M. BUCHSTAHERET AL. (eds.): Solitons, Geometry and Topology: On the Crossroad, Vol. 179 of Amer. Math. Soc. Transl. (2), 1997, pp. 35-44. [17] DUBROVIN, B.: 'Flat pencils of metrics and Frobenius manifolds', Math. DG 9803106 (1998). [18] DUBROWN, B.: 'Geometry and analytic theory of Frobenius manifolds', Math. A G 9807034 (1998). [19] DUBROWN, B.: 'Painlev~ transcendents and two-dimensional topological field theory', Math. A G 9803107 (1998). [20] DUBROVIN,B., KRICHEVER, I., AND NOVIKOV, S.: 'Topological and algebraic geometry methods in contemporary mathematical physics II', Math. Phys. Rev. 3 (1982), 1-150. [21] DUBROVIN,B., AND NOVIKOV, S.: 'Hydrodynamics of weakly deformed soliton lattices', Russian Math. Surveys 44 (1989), 35-124.

438

[22] DUBROVIN, B., AND NOVIKOV, S.: 'Hydrodynamics of soliton lattices', Math. Phys. Rev. 9 (1991), 3-136. [23] DUBROVIN, B., AND ZHANG, Y.: 'Extended affine Weyl group and Frobenius manifolds', hep-th 9611200 (1996). [24] DUBROVIN, B., AND ZHANG, Y.: 'Bi-Hamiltonian hierarchies in 2D topologigieal field theory on one-loop approximation', hep-th 9712232 (1997). [251 DUBROVIN,B., AND ZHANG, Y.: 'Frobenius manifolds and Virasoro constraints', hep-th 9808048 (1998). [26] EDELSTEIN, J., MARI~O, M., AND MAS, J.: 'Whitham hierarchies, instanton corrctions and soft supersymmetry breaking in N = 2 S U ( N ) super Yang-Mills theory', hep-th 9805172

(1998). [27] FLASCHKA~H., FOREST, M., AND MCLAUGI-ILIN~D.: 'Multiphase averaging and the inverse spectral solution of KdV', Commun. Pure Appl. Math. 33 (1979), 739-784. [28] FLASCHKA, H., AND NEWELL, A.: 'Monodromy- and spectrum preserving deformations I', Commun. Math. Phys. 76, no. 190 (1980), 65-116. [29] FLASCHKA,H., AND NEWELL, A.: 'Multiphase similarity solutions of integrable evolution equations', Physica 3D (1981), 203-221. [30] GIVENTAL, A.: 'On the WDVV-equation in quantum Ktheory', Math. A G 0003158 (2000). [31] GORSKY, A., KRICHEVER, I., MARSHAKOV,A, MIRONOV, A., AND MOROZOV, A.: ~N = 2 supersymmetric QCD and integrable spin chains: Rotational case N f < 2Nc', Phys. Lett. B 355 (1996), 466-474. [32] GORSKY, A., MARSHAKOV,A., MIRONOV, A., AND MOHOZOV, A.: 'RG equations from Whitham hierarchy', hep-th 9802007 (1998). [33] GORSKY, A., MARSHAKOV,A., MIRONOV, A., AND MOROZOV, A.: 'RG equations from Wbitham hierarchy', Nuel. Phys. B 527 (I998), 690 716. [34] GUREVICH, A.V., AND PITAEVSKII, L.P., J E T P Letters 17 (1974), 193-195. [35] GUREVICH, A.V., AND PITAEVSKI~, L.P., Soviet Phys. J E T P 38 (1974), 291 297. [36] HASEGAWA,A., AND KODAMA, Y.: Solitons in optical communications, Oxford Univ. Press, 1999. [37] ITOYAMA,H., AND MOROZOV, A.: 'Integrability and Seiberg Witten theory: Curves and periods', hep-th 9511126 (1995). [381 ITOYAMA,H., AND MOROZOV, A.: 'Prepotential and SeibergWitten theory', hep-th 9512161 (1995). E39] ITOYAMA,H., AND MOROZOV, A.: 'Integrability and SeibergWitten theory', hep-th 9601168 (1996). [40] ITOYAMA,H., AND MOROZOV, A.: 'Integrability and SeibergWitten theory - curves and periods', Nucl. Phys. B 477 (1996), 855-877. [41] ITOYAMA, H., AND MOROZOV, A.: 'Prepotential and SeibergWitten theory', Nucl. Phys. B 491 (1997), 529-573. [42] KODAMA, V.: 'The Whitham equations for optical communication: Mathematical theory of NRZ', SIAM Y. Appl. Math. 59, no. 66 (1999), 2162 2192. [43] KRICHEVER, I.: 'The averaging method for the twodimensional "integrable" equations', Funct. Anal. Appl. 22 (1988), 200-213. [44] KRICHEVER, I.: 'The dispersionless Lax equations and topological minimal methods', Commun. Math. Phys. 143 (1992), 415-429. [45] KRICHEVER, I.: 'The 7-function of the universal Whitham hierarchy, matrix models and topological field theories', Commun. Pure Appl. Math. 47 (1994), 437-475.

WIENER-IT® DECOMPOSITION

[46] I~RICHEVER,I.: 'Algebraic-geometrical methods in the theory of integrable equations and their perturbations', Acta Applic. Math. 39 (1995), 93-125. [47] KRICHEVER,I., AND PHONG, D.: 'On the integral geometry of soliton equations and N = 2 supersymetric gauge theories', J. Diff. Geom. 45 (1997), 349-389. [48] KRICHEVER, I., AND PHONG, D.: 'Symplectic forms in the theory of solitons', hep-th 9708170 (1998). [49] LAX, P., AND LEVERMORE, D.: 'The small dispersion limit of the Korteweg de Vries equation I-III', Commun. Pure Appl. Math. 36 (1983), 253-290; 571-593; 809-829. [50] LEVERMORE, D.: 'The hyperbolic nature of the zero dispersion KdV limit', Commun. Partial Diff. Eqs. 13 (1988), 495514. [51] LEVIN, A., AND OLSHANETSKY, M.: 'Classical limit of the Kniznik-Zamolodchikov-B~nard equations as hierarchy of isomonodromic deformations. Free fields approach', hep-th 9709207 (1997). [52] LEVIN, A., AND OLSHANETSKY,M.: 'Painle~-Calogero correspondence', alg-yeom 9706010 (1997). [53] LEVIN, A., AND OLSHANETSKY,M.: 'Non-autonomous Hamiltonian systems related to highest Hitchin integrals', math-ph 9904023 (1999). [54] MANIN, V.: Frobenius manifolds, quantum cohomology and moduli spaces, Vol. 47 of Colloq. Publ., Amer. Math. Soc., 1999. [55] MARIiqO, M.: 'The uses of Whitham hierarchies', hep-th 9905053 (1999). [56] MAmNo, M., AND MOORE, G.: 'Integrating over the Coulomb branch in N = 2 gauge theory', hep-th 9712062 (1997). [57] MARI~O, M., AND MOORE, G.: 'The Donaldson Witten function for gang groups of rank larger than one', hep-th 9802185 (1998). [58] MARIIqO,M., AND MOORE, G.: 'Donaldson invariants for nonsingularly connected manifolds', hep-th 9804104 (1998). [59] MmLER, P.D., ERCOLANL N.M., KRICHEVER,I.M., AND LEVERMORE, C.D.: 'Finite genus solutions to the Ablowitz-Ladik equations', Commun. Pure Appl. Math. 48 (1996), 13691440. [60] MWRa, R., AND KRUSKAL, M.: 'Application of a nonlinear WKB method in Korteweg-de Vries equation', S I A M J. Appl. Math. 26 (1974), 376-395. [61] MOORE, G., AND WITTEN, E.: 'Integration over the u-plane in Donaldson theory', hep-th 9709193 (1997). [62] NAKATSU,T., AND TAKASAKLK.: 'Isomonodromic deformations and supersymmetric gauge theories', Int. J. Modern Phys. A 11 (1996), 5505-5518. [63] OLSHANETSKY,M.: 'Painlev6 type equations and Hitchin systems', math-ph 9901019 (1999). [64] PAVLOV, M.V.: 'Nonlinear Schrhdinger equation and the Bogolyubov-Whitham method of averaging', Theoret. Math. Phys. 71 (1987), 584 588. [65] POTEMIN,O.: 'Algebro-geometric consttruction of self-similar solutons of the Whitham equations', Russian Math. Surveys 43 (1988), 252-253. [66] TAKASAKI, K.: 'Gaudin model, KZ equation, and isomonodromic problem on torus', hcp-th 9711058 (1997). [67] TAKASAKI, K.: 'Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type', solv-int 9704004 (I997). [68] TAKASAKI, K.: 'Dual isomonodromic problems and Whitham equations', hep-th 9700516 (1998).

[69] TAKASAKI,K.: 'Dual isomonodromic problems and Whitham equations', Lett. Math. Phys. 43 (1998), 123-135. [70] TAKASAKI,K.: 'Integrable hierarchies and contact terms in uplane integrals of topologically twisted supersymmetric gauge theories', Int. J. Modern Phys. A 14 (1998), 1001-1014. [71] TAKASAKI, K.: 'Whitham deformations of Seiberg Witten curves for classical gauge groups', Int. J. Modern Phys. A 15 (2000), 3635-3666. [72] TAKASAKI,K., AND NAKATSU,T.: 'Isomonodromic deformations and supersymmetric gauge theories', Int. J. Modern Phys. A 11 (1996), 5505-5518. [73] VENAKIDES, S.: ~The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation', Commun. Pure Appl. Math. 38 (1985), 883-909. [74] WmTHAM,G.: 'Nonlinear dispersive waves', Proc. Royal Soe. A 283 (1965), 238-261. [75] WHITHAM, G.: Linear and nonlinear waves, Wiley, 1974. A. Bloch t~. Carroll

M S C 1991: 14Jxx, 57R57, 35Q53, 35A25

WIENER-IT® composition

Hilbert

DECOMPOSITION, ItS-Wiener

de-

A n o r t h o g o n a l d e c o m p o s i t i o n of the

-

space

of s q u a r e - i n t e g r a b l e f u n c t i o n s on a

G a u s s i a n space. It was first proved in 1938 by N. W i e n e r [6] in t e r m s of h o m o g e n e o u s chaos (cf. also W i e n e r c h a o s d e c o m p o s i t i o n ) . I n 1951, K. It6 [1] defined multiple W i e n e r integrals to i n t e r p r e t h o m o g e n e o u s chaos a n d gave a different p r o o f of t h e d e c o m p o s i t i o n theorein. Take a n a b s t r a c t W i e n e r space (H, B) [3] (cf. also Wiener

space,

abstract).

Gaussian measure

on

B.

Let # be the s t a n d a r d The

abstract

version of

W i e n e r - I t 5 d e c o m p o s i t i o n deals with a special orthogonal d e c o m p o s i t i o n of the real H i l b e r t space L2(#). Each h C H defines a n o r m a l r a n d o m

v a r i a b l e h on

B with m e a n 0 a n d variance Ih]~r [3]. Let F0 = R . For n > 1, let F~ be the L2(p)-closure of the linear space s p a n n e d by 1 a n d r a n d o m variables of the form h i ' " hk with k < n a n d h j E H for 1 < j < k. T h e n {Fn}n°%_0is a n i n c r e a s i n g sequence of closed subspaces of L 2 ( # ) . Let Go = R and, for n > 1, let G n be the o r t h o g o n a l comp l e m e n t of F ~ _ I in F~. T h e elements in Gn are called hom o g e n e o u s chaos of degree n. Obviously, the spaces G~ are orthogonal. Moreover, the H i l b e r t space L 2 (#) is the

direct s u m of G~ for n > 0, namely, L 2 (#) = ~,~--0 G~. Fix n > 1. To describe Gn more precisely, let P~ be the o r t h o g o n a l p r o j e c t i o n of L 2(#) onto the space G~. For hi ® " " ® hn C H ®n, define

O (hl o . . . ®

=

T h e n On(h 1 ® ".. ® hn) : O n ( h l @ . . . @ h n ) (where denotes the s y m m e t r i c t e n s o r p r o d u c t ) a n d

II0

(hl ® . . .

® h

)lJL2(.) = 439

WIENER-IT0 DECOMPOSITION Thus, On extends by continuity to a continuous l i n e a r o p e r a t o r from H ®n into Gn and is an i s o m e t r l e m a p p i n g (up to the constant v ~ . ~) from H ~n into Gn. Actually, On is surjective and so for any V) E Gn, there exists a unique f ¢ H en such that O,~(f) = ~ and II IIL (.) = Therefore, for any g) • L2(#),

where the right-hand side is a multiple Wiener integral of order n as defined by It6 in [1] and [lIn(g)tlL2(,) =

v~.V]~lL=([0,1],,)(where ~ is the symmetrization of g.) For any q# E L2(#) there exists a unique sequence {gn}~°°=0 of symmetric functions gn E L2([0, 1]n) such that oo

there exists a unique sequence {f~}n~__0 with fi~ • H ®~ such that

= £

n=O oo 2

On(/n),

II IIL2(.) =

2

2

n:O

This is the abstract version of the Wiener-It5 decomposition theorem [2], [4], [5]. Let r ( U ) ---~ xA~n=0 - ' ~ H 6~ Define a n o r m on P(H) by 1/2

=

n! IfnlH®

OG

e(fo, k , . . . ) =

Let {ek : k _> 1} be an orthonormaI basis (cf. also Ort h o g o n a l basis) for H . For any non-negative integers n , , n 2 , . . , such that n, + n 2 + . . . . n, define 1 x/nl !n2! " "

where ~tk(X) = (--1) n e x2/2 D ,ke --x2/2 is the Hermite polynomial of degree k (cf. also H e r m i t e p o l y n o m i als). The set {F~,~= .... : nj >_ O, nl + n2 + . . . . n} is an orthonormal basis for the space Gn of homogeneous chaos of degree n. Hence the set {Pnl,~2 .... : n/ > 0, nl + n2 + . . . . n, n > 0} forms an orthonormal basis for L2(p). Consider the classical Wiener space (C ~, C) [3]. The Hilbert space C ~ is isomorphic to L2([0, 1]) under the unitary operator ~(g) = J , g • C ~. The standard GaussJan measure p on C is the Wiener measure and B(t, w) = w(t), w C C, is a B r o w n i a n m o t i o n . For g • L2([0, 1]), the random variable (~-Zg) is exactly the W i e n e r integral I(g) = f~ g(t)dB(t). Let gj • L2([0,1]), 1 < j < n. The random variable @ g n ) = On(l'--l g l @ ' ' "

@ I'--l g n )

is a homogeneous chaos in the space Gn. The mapping In extends by continuity to the space L2([0, 1]~). For g • L2([0, 1]n),

In(g) = J[of,ip g(tl, . . . , tn) dB(ts ) . . . dB(tn), 440

lL2([0.1>)

•

1 ~/nx!n2!"'

"~l

(folel(t)

d B ( t ) ) JG~2 ( / o l e 2 ( t ) d B ( t ) ) "'" , nl + n2 + . . . .

n,

n > O,

where {ek: k _> 1} is an orthonormal basis for L2([0, 1]) and the integrals are Wiener integrals. References

On(f ). n~0

Zn(gl 0''"

2

This is the Wiener-It5 decomposition theorem in terms of multiple Wiener integrals. An orthonormal basis for L 2(#) is given by the set

nj > O, The Hilbert space F ( H ) is called the Fock space of H (cf. also F o c k space). The spaces F ( H ) and L2(p) are isomorphic under the u n i t a r y o p e r a t o r O defined by

~gnl'n2 .... --

n! Ig n=0

r~:O O0

II(f0, fl,...)llr(H)

= )__£ In(gn),

[1] IT6, K.: 'Multiple W i e n e r integral', d. Math. Soc. Japan 3 (I951), 157-169. [2] KALLIANPUR, G.: Stochastic filtering theory, Springer, 1980. [3] Kuo, H.-H.: Gaussian measures in B a n a c h spaces, Vol. 463 of Lecture N o t e s in M a t h e m a t i c s , Springer, 1975. [4] Kuo, H.-H.: W h i t e noise distribution theory, CRC, 1996. [5] OBATA, N.: W h i t e noise calculus and Fock space, Vol. 1577 of Lecture N o t e s in M a t h e m a t i c s , Springer, 1994. [6] WIENER, N.: ' T h e homogeneous chaos', A m e r . J. Math. 60

(1938), 897-936.

Hui-Hsiung Kuo MSC 1991: 60J65, 60Hxx, 60G15 Let fl(t), t >_ 0, be the standard B r o w n i a n m o t i o n in R d (i.e. the M a r k o v p r o cess with generator A/2) starting at 0. Let P, E denote its probability law and expectation on path space. The Wiener sausage with radius a > 0 is the process defined by WIENER

SAUSAGE

W~(t) =

U

-

B~(fl(,)),

t > O,

O<s
where Ba(X) is the open ball with radius a around

x E R d. The Wiener sausage is an important mathematical object, because it is one of the simplest examples of a non-Markovian functional of Brownian motion. It plays a key role in the study of various stochastic phenomena, such as heat conduction and trapping in random media, as well as in the analysis of spectral properties of random SchrSdinger operators (ef. also S e h r 5 d i n g e r equation).

W I E N E R SAUSAGE A lot is known

about the behaviour of the volume of

W~(t) as t -+ co. For instance,

The above analysis of the large deviation behaviour has recently been extended to cover the moderate deviation behaviour. It is proved in [2] that for d _> 3,

Sv/~, d=l, E IWa(t) l

~

<

l2~t ogt'

[,t%t,

d=2, d _> 3,

with *% = aa-22¢cd/S/r((d- 2)/2) the Newtonian capacity of Ba(0) associated with the Green's function of ( - A / 2 ) -1 (cf. also G r e e n f u n c t i o n ; C a p a c i t y ) , and t,

Var]Wa(t)l×

I

d = 1, t2

~o-p~' d = 2 , tlogt, d=3,

I,t,

d_>4

([9], [8]). Moreover, II/V~(t) l satisfies the s t r o n g law o f l a r g e n u m b e r s and the c e n t r a l l i m i t t h e o r e m for d _> 2; the limit law is Gaussian for d _> 3 and nonGaussian for d = 2 ([7]). Note that for d _> 2 the Wiener sausage is a sparse object: since the Brownian motion typically travels a distance v~ in each direction, the last two displays show that most of the space in the convex hull of W ~ (t) is not covered. The large deviation behaviour of IW ~(t) l in the downward direction has been studied in [5], [4] and [10]. For d ___2 the outcome, proved in successive stages of refinement, reads as follows: lim f(t)2/a logP(IW~(t)l < f(t)) = - ~1 Ad t-~eo

t

for any f : R + -+ R + satisfying l i m t ~

f(t) =

1ogt ' [o(t),

f(t) = oc and

d > 3,

where Ad > 0 is the smallest Dirichlet eigenvalue of - A on the ball with unit volume. The optimal strategy for the Brownian motion to realize the large deviation is to stay inside a ball with volume f(t) until time t, i.e., the Wiener sausage covers this ball entirely and nothing outside. This comes from the Faber-Krahn isoperimetric inequality (cf. also R a y l e i g h - F a b e r - K r a h n i n e q u a l ity), and the cost of staying inside the ball is [ 1A t exp [ - ~ d ~ j

]

to leading order. Note that, apparently, a large deviation below the scale of the mean 'squeezes all the empty space out of the Wiener sausage'.

lim

1

t--+co t ( d - 2 ) / d

= -r

l o g P ( I W a ( t ) l < bt) =

(1)

(b) e ( - o o , 0),

for all0 < b < ~ , and a variational representation is derived for the rate function I ~ . The optimal strategy for the Brownian motion to realize the moderate deviation is such that the Wiener sausage 'looks like a Swiss cheese': Wa(t) has random holes whose sizes are of order 1 and whose density varies on scale t 1/d. This is markedly different from the optimal strategy behind the large deviation. Note that, apparently, a moderate deviation on the scale of the mean 'does not squeeze all the empty space out of the Wiener sausage'. (I) has also been extended to d=2. It turns out that the rate function b F-~ I ~ (b) exhibits rich behaviour as a function of the dimension. In particular, for d > 5 it is non-analytic at a certain critical value inside (0, ~), which is associated with a collapse transition in the optimal strategy. Finally, the moderate and large deviations of lW a (t) l in the upward direction are a complicated issue. Here the optimal strategy is entirely different from the previous ones, because the Wiener sausage tries to expand rather than to contract. Partial results have been obtained in [3] [1], and [6]. More background can be found in [11]. References [1] BERG, M. VAN DEN, AND BOLTHAUSEN, E.: ' A s y m p t o t i c s of the generating function for the volume of the W i e n e r sausage', Probab. Th. Rel. Fields 99 (1994), 389-397. [2] BERG, M. VAN DEN, BOLTHAUSEN, E., AND HOLLANDER, F. DEN: 'Moderate deviations for the volume of the Wiener sausage', A n n . of Math. (to appear in 2001). [3] BERG, M. VAN DEN, AND TdTH, B.: 'Exponential estimates for the Wiener sausage', Probab. Th. Rel. Fields 88 (1991), 249-259. [4] BOLTHAUSEN, E.: ' O n the volume of the Wiener sausage', A n n . Probab. 18 (1990), 1576-1582. [5] DONSKER, M.D., AND VARADHAN, S.R.S.: 'Asymptotics for the Wiener sausage', C o m m u n . Pure Appl. Math. 28 (1975), 525-565. [6] HAMANA, Y., AND KESTEN, H.: 'A large deviation result for the range of r a n d o m walk and for the Wiener sausage', p r e p r i n t March (2000). [7] LE GALL, J.-F.: ' F l u c t u a t i o n results for the Wiener sausage', A n n . Probab. 16 (1988), 991-1018. [8] LE GALL, J.-F.: 'Sur une conjecture de M. Kac', Probab. Th. Rel. Fields 78 (1988), 389-402. [9] SPITZER, F.: 'Electrostatic capacity, heat flow and Brownian motion', Z. Wahrsch. Verw. Gebiete 3 (1964), 110-121.

441

W I E N E R SAUSAGE

[10] SZNITMAN,A.-S.: 'Long time asymptotics for the shrinking Wiener sausage', Commun. Pure Appl. Math. 43 (1990), 809820. [11] SZNITMAN, A.-S.: Brownian motion, obstacles and random media, Springer, 1998. F. den Hollander

MSC 1991: 60J65, 60J55, 60Gxx

WIENER-WINTNER THEOREM, Wiener Wintner ergodic theorem - A strengthening of the pointwise ergodic theorem (ef. also E r g o d i c t h e o r y ) announced in [22] and stating that if (X, 5v, #, T) is a d y n a m i c a l s y s t e m , then given f E L 1 (p) one can find a set of full m e a s u r e X I such that for x in this set the averages

can be found in [11] and [19]. Previous partial results can be found in [8]. Wiener-Wintner return-time theorem and the C o n z e - L e s i g n e a l g e b r a . A natural generalization of the return-time theorem is its Wiener-Wintner version, in which averages of the sequence f(T~x)g(Sny)e 2~i~ are considered. Such a generalization was obtained in [7] and one of the tools used to prove it was the ConzeLesigne algebra. This algebra of functions was discovered by J.P. Conze and E. Lesigne [12] in their study of the L 1 norm convergence of the averages N

1

f(T~x)e2~i~

n=l

converge for all real numbers e. In other words, the set X f 'works' for an uncountable number of e. This introduces into ergodic theory the study of general phenomena in which sampling is 'good' for an uncountable number of systems. Since [22], several proofs of the 'Wiener-Wintner theorem' have appeared (e.g., see [8] for a spectral path and [14] for a path using the notion of disjointness in [13]). Uniform Wiener-Wintner theorem and Kron e c k e r factor. For (X, F , p, T) an ergodic dynamical system (cf. also E r g o d i e i t y ) , the Kronecker factor K. of T is defined as the closed linear span in L2(#) of the eigenfunctions of T. The orthocomplement of K: can be characterized by the Wiener-Wintner theorem. More precisely, a function f is in K± if and only if for #-a.e. with respect to x,

EI-iF o

Ti n

(1)

n = l i=1

N

1 E

H

for H = 3. These averages were introduced by H. Furstenberg. (The functions f.i are in L ~ ( # ) . The L 1norm convergence of (1) for H >_ 4 is still an open problem (as of 2001).) It is shown in [7] that the orthocomplement of the Conze Lesigne factor characterizes those functions for which outside a single null set of x independent of g or S one has u-a.e. lim sup N--+cx~

l~-~f(Tnx)9(Sny)e2~ine

= O.

n=l

Several results related to the ones above can be found in [4], [1], [3], [15], [18], [2O], [171, and [21]. In [5] it was shown that many dynamical systems have a WienerWintner property, based on the speed of convergence in the uniform Wiener Wintner theorem; this allows one to derive the results in [9] and [10] for such systems in a much simpler way. References

f(T~x)e 2~i~ = O.

lim sup

N--+oc

n=l

This theorem was announced by J. Bourgain [10]. Other proofs of this result can be found in [2] and [16], for instance. A sequence of scalars an is a good universal weight (for the pointwise ergodic theorem) if the averages 1

N n~l

converge v-a.e, for all dynamical systems (Y,/3, v, S) and all functions g E LI(p). Bourgain's return-time theorem states that given a dynamical system (X, ~c, p, T) and a function f in L °~, then for #-a.e. with respect to x, the sequence f(T~z) is a good universal weight (see [9]). By applying this result to the irrational rotations on the one-dimensional torus given by S~(y) = y + ct and to the function g(y) = e 2~iv, one easily obtains the Wiener-Wintner theorem. Another proof of his result 442

[1] ASSANI, I.: 'Uniform W i e n e r - W i n t n e r theorems for weakly mixing dynamical systems', Preprint unpublished (1992). [2] AssAm, I.: 'A W i e n e r - W i n t n e r property for the helical transform', Ergod. Th. Dynam. Syst. 12 (1992), 185-194. [3] ASSANI, I.: 'Strong laws for weighted sums of independent identically distributed random variables', Duke Math. J. 88, no. 2 (1997), 217-246. [4] ASSAm, I.: 'A weighted pointwise ergodic theorem', Ann. IHP 34 (1998), 139-150. [5] ASSANL I.: ' W i e n e r - W i n t n e r dynamical systems', Preprint (1998). [6] ASSANI, I.: 'Multiple return times theorems for weakly mixing systems', Ann. IHP 36, no. 2 (2000), 153-165. [7] ASSANI, I., LESIGNE, E., AND RUDOLPH, D.: 'Wiener-Wintner return times ergodic theorem', Israel Y. Math. 92 (1995), 375-395. [8] BELLOW, n . , AND LOSERT, V.: 'The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences', Trans. Amer. Math. Soc. 288 (1995), 307-345. [9] BOURGAIN, J.: 'Return times sequences of dynamical systems', Preprint IHES (1988). [10] BOURGAIN, J.: 'Double recurrence and almost sure convergence', d. Reine Angew. Math. 404 (1990), 140-161.

W I L L M O R E FUNCTIONAL [11] BOURGAIN, J., FURSTENBERG, H., KATZNELSON, Y., AND OR.NSTEIN, D.: 'Appendix to: J. Bourgain: Pointwise ergodic theorems for arithmetic sets', IHES 69 (1989), 5-45. [12] CONZE, J.P., AND LESIGNE, E.: 'Thdor~mes ergodiques pour des mesures diagonales', Bull. Soc. Math. France 112 (1984), 143 175. [13] FURSTENBERC, H.: 'Disjointness in ergodic theory', Math. Systems Th. 1 (1967), 1-49. [14] LESIGNE, E.: 'Th~or~mes ergodiques pour une translation sur une nilvariete', Ergod. Th. Dynam. Syst. 9 (1989), 115 126. [15] LESIGNE, E.: 'Un th@or~me de disjonction de syst~mes dynamiques et une g~ndralisation du th~or~me ergodique de Wiener-Wintner', Ergod. Th. Dynam. Syst. 10 (1990), 513521. [16] LESIGNE, E.: 'Spectre quasi-discret et th~or@me ergodique de W i e n e r - W i n t n e r pour les polynSmes', Ergod. Th. Dynam. Syst. 13 (1993), 767-784. [17] ORNSTEIN, D.~ AND WEISS, B.: 'Subsequence ergodic theorems for amenable groups', Israel J. Math. 79 (1992), 113127. [18] ROBINSON, E.A.: 'On uniform convergence in the Wiener Wintner theorem', J. London Math. Soc. 49 (1994), 493 501. [19] RUDOLPH, D.: 'A joinings proof of Bourgain's return times theorem', Ergod. Th. Dynara. Syst. 14 (1994), 197-203. [20] RUDOLPH, D.: 'Fully generic sequences and a multiple-term return times theorem', Invent. Math. 131, no. 1 (1998), 199228. [21] WALTERS,P.: 'Topological Wiener-Wintner ergodic theorem and a random L 2 ergodic theorem', Ergod. Th. Dynam. Syst. 16 (1996), 179-206. [22] WIENER, N., AND WINTNER, t . : 'Harmonic analysis and ergodic theory', Amer. J. Math. 63 (1941), 415-426.

I. Assani MSC 1991: 28D05, 54H20 WIJSMAN CONVERGENCE - R. Wijsman [4] introduced a convergence for sequences of proper lower semicontinuous convex functions in R ~. Workers in topologies on hyperspaces found this convergence and the resulting topology quite useful and subsequently a vast body of literature developed on this topic (see [1], [2]).

Suppose (X, d) is a m e t r i c space and let CL(X) denote the family of all non-empty closed subsets of X. For each x C X and A E CL(X) one sets d(x,A) = inf(d(x,a): a E A}. One says that a net A~ E CL(X) (cf. also N e t (of sets in a t o p o l o g i c a l space)) is Wijsman convergent to A E CL(X) if and only if for each x C X, d(x, Ax) ~ d(x,A), i.e. the convergence is pointwise. The resulting topology Uwd on CL(X) is called the Wijsman topology induced by the metric d. The dependence of the Wijsman topology on the metric d is quite strong in as much as even two different uniformly equivalent metrics may induce different Wijsman topologies. Necessary and sufficient conditions for two metrics to induce the same Wijsman topology have been found by C. Costantini, S. Levi and J. Zieminska, among others. G. Beer showed that if (X,d) is complete and

separable (cf. also C o m p l e t e m e t r i c space; S e p a r a ble space), then TWd is a Polish space, i.e. it is separable and has a compatible complete metric. If the pointwise convergence d(x, A~) -~ d(x, A) is replaced by u n i f o r m c o n v e r g e n c e , then Hausdorff convergence is obtained, which has been known for a long time. The associated Hausdorff topology THg is derived from the H a u s d o r f f m e t r i c dH given by dH(A,B) = s u p ( I d ( x , A ) - d ( x , B ) l : x E X}). It is known that TWd = THd if and only if (X, d) is totally bounded (cf. also T o t a l l y - b o u n d e d space). A natural question arises: W h a t is the supremum of the Wijsman topologies induced by the family of all metrics that are topologically (respectively, uniformly) equivalent to d. It was shown by Beer, Levi, A. Lechicki, and S. Naimpally that the supremum of topologically (respectively, uniformly) equivalent metrics is the Vietoris topology Tv (cf. E x p o n e n t i a l t o p o l o g y ; respectively, the proximal topology Ts). These are hit-andmiss type topologies; the former has been known for a long time while the latter is a rather recent discovery (1999; cf. also H i t - o r - m i s s t o p o l o g y ) . It is known that Twd = Tv if and only if (X, d) is compact, while Twd = T5 is equivalent to (X, d) being totally bounded. G. Di Maio and Naimpally discovered a (hit-and-miss) proximal ball topology TB5 which equals TWd in almost convex metric spaces (these include normed linear spaces) [3]. L. Hol£ and R. Lucchetti have discovered necessary and sufficient conditions for the equality of TWd and TBS. The Wijsman topology Twd is always a Tikhonov topology (cf. also T i k h o n o v space) and a remarkable theorem of Levi and Lechicki shows that the separability of X is equivalent to Twa being metrizable or first countable or second countable. Wijsman's original work has been generalized by U. Mosco, Beer and others. Naimpally, Di Maio and Hol~ have studied Wijsman convergence in function spaces (see [2]). References

[1] BEER, G.: Topologies on closed and closed convex sets, Kluwer Acad. Publ., 1993. [2] BEER, G.: 'Wijsman convergence: A survey', Set-Valued Anal. 2 (1994), 77-94. [3] DI MAIO, G., AND NAIMPALLY, S.: 'Comparison of hypertopologies', Rend. Ist. Mat. Univ. Trieste 22 (1990), 140161. [4] WIJSMAN, R.: 'Convergence ofsequences of convex sets, cones, and functions II', Trans. Amer. Math. Soc. 123 (1966), 3245.

Sore NaimpaUy MSC 1991: 54Bxx WILLMORE FUNCTIONAL - The Willmore functional of an immersed surface E into the Euclidean space

443

WILLMORE FUNCTIONAL R 3 is defined by

W = / z H2 dA, where H = (hi + n.))/2 is the m e a n c u r v a t u r e of the surface. Here hi, n.~ are the two classical principal curvatures of the surface (cf. also P r i n c i p a l c u r v a t u r e ) and dA is the area element of the induced metric on E. Moreover, it is assumed that the integral W converges, which is guaranteed if E is compact, as is usually assumed. Critical points of the functional W are called Willmore surfaces and are characterized by the Euler equation A H + 2H(H 2 - K ) = 0 corresponding to the variational problem 5W = 0 (cf. also V a r i a t i o n o f a functional; Variational calculus; Variational problem). Here, K = nln2 is the G a u s s i a n c u r v a t u r e of the surface and A is its Laplace-Beltrami operator (cf. L a p l a c e B e l t r a m i e q u a t i o n ) . An alternative functional to W is the functional given by

W =

(H 2 - K ) dA.

Because of the G a u s s - B o n n e t t h e o r e m , if E is assumed to be compact and without boundary, then W = W - 27rX(E), where )/(E) denotes the E u l e r c h a r a c teristic of the surface, so that W and W have the same critical points. The functional W was first studied by W. Blaschke (1929) and G. Thomsen (1923), who established the most important property of W: The functional W is invariant under conformal changes of metric of the ambient space R 3. They considered it as a substitute for the area of surfaces in conformal geometry. For that reason, Willmore surfaces were called Konformminimalfliichen (conformally minimal surfaces; cf. also M i n i m a l s u r face). These results were forgotten for some time and were rediscovered by T.J. Willmore in 1965, reopening interest in the subject. He proved that for any compact orientable surface E immersed in R 3 one has W > 47c, equality holding if and only if E is embedded as a round sphere. In an a t t e m p t to improve this inequality for surfaces of higher gem, s, Willmore also showed that for anchor rings, obtained by rotating a circle of radius r about an axis in its plane at distance R > r from its centre, it holds that W _> 27r2, equality holding when I~/r = x/2. For various reasons, Willmore also conjectured that any torus immersed in R a satisfies the inequality W >_ 2:r 2. This inequality is known as the Will-

more conjecture. Although the general case remains an open problem (as of 2000), the Willmore conjecture has been proved for various special classes of tori. For instance, it is known to be true for a torus embedded in R a as a tube 444

of constant circular cross-section (K. Shiohama and R. Takagi, 1970, and Willmore, 1971) as well as for tori of revolution (J. Langer and D. Singer, 1984). In 1982, P. Li and S.T. Yau showed t h a t the Willmore conjecture is true for conformal structures near that of the special x/2-torus. The set of conformal structures for which the Willmore conjecture is true was enlarged by S. Montiel and A. Ros (1985). Recently (2000), B. A m m a n n proved it under the condition t h a t the LP-norm of the Gaussian curvature is su~ciently small. On the other hand, in 1978 J.L. Weiner generalized the Willmore functional by considering immersions of an orientable surface E, with or without boundary, into a Riemannian manifold of constant sectional curvature c. Instead of W he considered the integral

/

(1~1 ~ + c) dA,

where 7/ denotes the mean curvature vector field of the surface, and obtained the corresponding Euler equation. In the particular case when the ambient space is the unit sphere S 3 C R4, the Euler equation becomes A H + 2 H ( H 2 - If + 1) = 0, so that every minimal surface in S 3 is a Willmore surface. An interesting consequence of Weiner's result is the proof t h a t stereographic projections of compact minimal surfaces in S 3 produce Willmore surfaces in R 3. For instance, the special x/2torns in R 3 for which W = 27r2 corresponds to the stereographic projection of the Clifford torus, embedded as a minimal surface in S 3. Moreover, H.B. Lawson proved in 1970 that every compact, orientable surface can be minimally embedded in $3; it follows from this that there are embedded Willmore surfaces in R 3 of arbitrary genus (cf. also G e n u s o f a s u r f a c e ) . On the other hand, Weiner also considered questions of stability of Willmore surfaces by considering the second variation of the Willmore functional. In particular, he showed that the special minimizing v ~ - t o r u s is stable. Most of the known examples of embedded Willmore surfaces in R a come from compact minimal surfaces in the unit sphere S 3 C R 4. In 1985, U. Pinkall found the first examples of compact embedded Wilhnore surfaces that are not stereographic projections of compact embedded minimal surfaces in S 3. Using results of Langer and Singer on elastic curves on S ~, he exhibited an infinite series of embedded Willmore tori in R 3 that cannot be obtained by stereographic projection of minimal surfaces in S 3. All of these tori are, however, unstable critical points of W and hence are not candidates for absolute minima. I m p o r t a n t contributions to the study of Willmore surfaces are made by R.L. Bryant, who classified all Willinore immersions of a topological sphere into S 3

WODZICKI RESIDUE and determined the critical values of the Willmore functional; Li and Yau, who introduced the concept of conformal volume; M. Gromov, who introduced the concept of visual volume, closely related to the conformal volume; and others. Finally, it is worth pointing out the relation between the Willmore functional and quantum physics, including the theory of liquid membranes, two-dimensional gravity and string theory. For instance, in string theory the functional W = f H 2 dA is known as the Polyakov extrinsic action and in membrane theory it is the Helfrich

free energy. References

[1] PINKALL, U., AND STERLING, I.: 'Willmore surfaces', M a t h . Intelligencer 9, no. 2 (1987), 38 43. [2] WILLMORE, T.J.: 'A survey on Willmore immersions': Geometry and Topology of Submanifolds, IV (Leuven, 1991), World Sci., 1992, pp. 11-16. [3] WILLMORE, T.J.: Riemannian geometry, Oxford Univ. Press, 1993, p. Chap. 7. [4] WILLMORE, T.J.: 'Total mean curvature squared of surfaces': Geometry and Topology of Submanifolds, VIII (Brussels/Nordfjordeid, 1995), World Sci., 1996, pp. 383-391.

Luis J. Alias MSC 1991:53C42

a version of the sinc function (sinc(0) = 1, sinc(x) = x - l s i n x for x ¢ 0), see [1, pp. 61, 104]. In terms of the Heaviside function H(x) (g(x) = 0 for x < 0, H(0) = 1/2, H(x) = 1 for x > 0), r(x) is given by r ( x ) = H ( x + 1) - H ( x - 1).

There is also a relation with the Dirac delta-function lim

7/-->00

n

r(nx) = 5(x).

References [1] CHAMPENEY, D.C.: A handbook of Fourier transforms, Cambridge Univ. Press, 1989. [2] DAUBECHIES, I.: Ten lectures on wavelets, SIAM, 1992, p. Chap. l.

[3] SAICHEV, A.I., AND WOYCZYNSKI, W.A.: D i s t r i b u t i o n s i n t h e physical and engineering sciences, Vol. 1: Distribution and fractal calculus, integral t r a n s f o r m s and wavelets, Birkh/iuser, 1997, p. 195ff.

M. Hazewinkel

MSC 1991: 42Cxx, 94A12 WITTENBAUER T H E O R E M - Take an arbitrary quadrangle and divide each of the four sides into three equal parts. Draw the lines through adjacent dividing points. The result is a parallelogram. This theorem is due to F. Wittenbauer (around 1900).

W I N D O W F U N C T I O N - A function used to restrict consideration of an arbitrary function or signal in some way. The terms time-frequency localization, time localization or frequency localization are often used in this context. For instance, the windowed Fourier transform is given by

( F w i n f ) (02 , t )

=

/" f(s)g(s

-

t)e -i~s ds,

where g(t) is a suitable window function. Quite often, scaled and translated versions of g(t) are considered at the same time, [2], [3]. An example is the G a b o r t r a n s f o r m . (See also B a l i a n - L o w t h e o r e m ; C a l d e r S n t y p e r e p r o d u c i n g f o r m u l a . ) Such window functions are also used in numerical analysis. More specifically, the phrase window function refers to the function r(t) that equals 1 on the interval ( - 1 , 1) and zero elsewhere (at - 1 and +1 it is arbitrarily defined, usually 1/2 or 0). This function, as well as its scaled and translated versions, is also called the rectangle function or pulse function [1, pp. 30, 35, 60, 61]. However, the phrase 'pulse function' is also sometimes used for the d e l t a - f u n c t i o n , see also T r a n s f e r f u n c tion. The F o u r i e r t r a n s f o r m of the specific rectangle function r(t) (with r ( + l ) = 1/2) is the function ~ ~@sin 2zcy, y ¢ 0 , g(Y) = [ 2 ,

y = 0,

The centre of the parallelogram is the centroid (centre of mass) of the lamina (plate of uniform density) defined by the original quadrangle. References [1] BLASCHKE, W.: Projektive Geometric, Birkh~user, 1954, p. 13. [2] COXETER, H.S.M.: Introduction to geometry, Wiley, 1969, p. 216.

11/I. Hazewinkel

MSC 1991:51M04 WODZICKI

RESIDUE,

non-commutative residue -

In algebraic quantum field theory (cf. also Q u a n t u m field t h e o r y ) , in order to write down an action in operator language one needs a functional that replaces integration [1]. For the Yang-Mills theory (cf. Y a n g - M i l l s field) this is the Dixmier trace, which is the unique extension of the usual t r a c e to the ideal £(1,~) of the compact operators T such that the partial sums of its 445

WODZICKI RESIDUE spectrum diverge logarithmically as the number of terms in the sum. The Wodzicki (or non-commutative) residue [3] is the only extension of the Dixmier trace to the class of pseudo-differential operators (@DOs; cf. 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 ) which are not in £0,oo). It is the only trace one can define in the algebra of ~DOs (up to a multiplicative constant), its definition being: res A = 2 Rest=0 tr(AA-~), with A the L a p l a c e o p e r a t o r . It satisfies the trace condition: res(AB) = res(BA). A very important property is that it can be expressed as an integral (local form):

(i.e. the best linear least squares predictors converge to zero, if the forecasting horizon tends to infinity) and (zt) is linearly singular (i.e. the prediction errors for the best linear least squares predictors are zero). 2) Every linearly regular process (Yt) can be represented as

Yt = E K S t - J '

(1)

j=0

KjcR

Ko=±,

ltKjll2<

,

j=0

resA = f tr a_~(x, ~) d~ Ys *M with S*M C T * M the co-sphere bundle on M (some authors put a coefficient in front of the integral, this gives the Adler-Mania residue). If dim M = n = - ord A (M a compact R i e m a n n i a n m a n i f o l d , A an elliptic operator, n E N), it coincides with the Dixmier trace, and one has 1 Res~=l ~A(S) = -- resA -1. n

The Wodzicki residue continues to make sense for ~DOs of arbitrary order and, even if the symbols a j ( x , ~ ) , j < m, are not invariant under coordinate choice, their integral is, and defines a trace. All residues at poles of the zeta-function of a ~ D O can be easily obtained from the Wodzicki residue [2]. References [1] CONNES, A.: Noncommutative geometry, Acad. Press, 1994. [2] ELIZALDE, E.: 'Complete d e t e r m i n a t i o n of the singularity structure of zeta functions', J. Phys. A d 0 (1997), 2735. [3] WODZICKI, M.: ' N o n c o m m u t a t i v e residue I', in Yu.I. MANIN (ed.): K-Theory, Arithmetic and Geometry, Vol. 1289 of Lecture Notes in Mathematics, Springer, 1987, pp. 320-399.

E. Elizalde

where at is w h i t e n o i s e (i.e. Eat = 0, Eatals = 5stE) and ct is obtained by a causal linear transformation of

(yd. The construction behind the Wold decomposition in the H i l b e r t s p a c e H spanned by the one-dimensional process variables x~i) is as follows: If Hx(t) denotes the subspace spanned by { x ! i ) : s < t, i = 1 , . . . , n } , then z}~) is obtained from projecting x~i) on the space NtczHx(t), and c~i) is obtained as the perpendicular by projecting y}i) on the space H y ( t - 1) spanned by {y!~) : s < t, i = 1 , . . . ,n}. Thus ct is the innovation and the one-step-ahead prediction error for Yt as well as for Xt.

The implications of the above-mentioned results for (linear least squares) prediction are straightforward: Since (Yt) and (zt) are orthogonal and since Hx (t) is the direct sum of Hy(t) and Hz(t), the prediction problem can be solved for the linearly regular and the linearly singular part separately, and for a linearly regular process (Yt), Hy(t) = H~ (t) implies that the best linear least squares r-step ahead predictor for Yt+r is given by

MSC 1991: 46Lxx, 47Axx, 35Sxx

oo

t,r = W O L D DECOMPOSITION - A decomposition introduced by H. Wold in 1938 (see [7]); see also [5], [8]. Standard references include [6], [3]. The Wold decomposition of a (weakly) s t a t i o n a r y s t o c h a s t i c p r o c e s s {xt : t E Z}, xt : ft -+ R ~, provides interesting insights in the structure of such processes and, in particular, is an important tool for forecasting (from an infinite past). The main result can be summarized as:

1) Every (weakly) stationary process {xt: t C Z} can uniquely be decomposed as x t = Yt + z t ,

where the stationary processes (Yt) and (zt) are obtained by causal linear transformations of (xt) (where 'causal' means that, e.g. Yt, only depends on x~, s < t), (Yt) and (zt) are mutually uncorrelated, (Yt) is linearly regular 446

Kj

t+r-j

j=r

and thus the prediction error is r--1

Y +r --

= Z j=0

Thus, when the representation (1) is available, the prediction problem for a linearly regular process can be solved. The next problem is to obtain (1) from the second moments of (Yt) (cf. also M o m e n t ) . The problem of determining the coefficients Kj of the Wold representation (1) (or, equivalently, of determining the corresponding transfer function k(e -i~) = Y~'-o KJ e-i~j) from the spectral density

f(A) = (2~)-lk(e-i~)Ek*(e-iA),

(2)

WOLD DECOMPOSITION (where the * denotes the conjugate transpose) of a linearly regular process (Yt), is called the spectral factorization problem. The following result holds:

3) A stationary process (Yt) with a spectral density f , which is non-singular A-a.e., is linearly regular if and only if logdet f(A) dA > - ~ .

/ 7~

In this case the factorization (k, E) in (2) corresponding to the Wold representation (1) satisfies the relation det E = exp

27~)-1

/;

log det 27rf(A) d l

}

.

The most important special case is that of rational spectral densities; for such one has (see e.g. [4]): 4) Any rational and A-a.e. non-singular spectral density f can be uniquely factorized, such that k(z) (the extension of k(e -i~) to C) is rational, analytic within a circle containing the closed unit disc, det k(z) ~ 0, [z[ < 1, k(0) = I (and thus corresponds to the Wold representation (1)), and E > 0. Then (1) is the solution of a stable and miniphase ARMA or a (linear) finite-dimensional state space system. Evidently, the Wold representation (1) relates stationary processes to linear systems with white noise inputs. Actually, Wold introduced (1) as a joint representation for AR and MA systems (cf. also M i x e d a u t o r e gressive m o v i n g - a v e r a g e process). The Wold representation is used, e.g., for the construction of the state space of a linearly regular process

and the construction of state space representations, see [I], [4]. As mentioned already, the case of rational transfer functions corresponding to stable and miniphase ARMA or (finite-dimensional) state space systems is by far the most important one. In this case there is a wide class of identification procedures available, which also give estimates of the coe~cients Kj from finite data

y l , . . . , yT (see e.g. [4]). Another case is that of stationary long memory processes (see e.g. [2]). In this case, in (1), [IE~-0 K/II 2 : oo, so that f is infinity at frequency zero, which causes the long memory effect. Models of this kind, in particular so-called ARFIMA models, have attracted considerable attention in modern econometrics. References

[1] AKAIKE, H.: 'Stochastic theory of minimal realizations', I E E E Trans. Autom. Control A C - 1 9 (1974), 667-674. [2] GRANGER, C.W.J., AND JOYEUX, R.: 'An introduction to long memory time series models and fractional differencing', J. Time Set. Anal. 1 (1980), 15-39. [3] HANNAN, E.J.: Multiple time series, Wiley, 1970. [4] HANNAN, E.J., AND DEISTLER, M.: The statistical theory of linear systems, Wiley, 1988. [5] KOLMOGOROV,A.N.: 'Stationary sequences in Hilbert space', Bull. Moscow State Univ. 2, no. 6 (1941), 1-40. [6] ROZANOV, Y.A.: Stationary random processes, Holden Day, 1967. [7] WOLD, H.: Study in the analysis of stationary time series, second ed., Almqvist and Wiksell, 1954. [8] ZASUKHIN,V.N.: 'On the theory of multidimensional stationary processes', Dokl. Akad. Nauk SSSR 33 (1941), 435-437.

M. Deistler MSC 1991: 62M20, 60G25, 93E12, 93B15, 93B10

447

Z Z-TRANSFORM, Z-transformation - This transform method may be traced back to A. De Moivre [5] around the year 1730 when he introduced the concept of 'generating functions' in p r o b a b i l i t y t h e o r y . Closely related to generating functions is the Z-transform, which may be considered as the discrete analogue of the L a p l a c e t r a n s f o r m . The Z-transform is widely used in the analysis and design of digital control, and signal processing [4], [2], [3], [6]. The Z-transform of a sequence x(n), n E Z, that is identically zero for negative integers, is defined as

x(j)z-~,

for

Izl > m

(1)

j:0

k)] = z - k Z ( x ( n ) ) ,

b) Left-shifting: Z ( x ( n + k)) = zkZ(x(n)) ~ r k--1 = 0 x(r) zk-r, for Iz[ > R. iii) Initial and final value. a) Initial value theorem: limbl_+oo 2(z) = x(0); b) Final value theorem: x(oc) = limn-+oo x(n) = limz--+l(Z - 1)Z(x(n)). iv) Convolution: The convolution of two sequences x(n) and y(n) is defined by

x(n) * y(n) =

oo

~(z) = Z ( x ( n ) ) : ~

a) Right-shifting: Z [ x ( n -

x ( n - j)y(j) : E x ( n ) y ( n j:0

j)

j:0

and its Z-transform is given by

where z is a complex number. By the root test, the series (1) converges if Izl > R, where R = limsup~_~ o Ix(n)] 1/n. The number R is called the radius of convergence of the series (1). Example 1. The Z-transform of {a n} is given by oo

Z(an) = E aJz-J z for I~1 > 1. z--a j=O Example 2. The Z-transform of the Kronecker-delta sequence

Z(x(n) * y(n)) = Z(x(n)) - Z(y(n)). I n v e r s e Z - t r a n s f o r m . If 9"(z) : Z(x(n)), then the inverse Z-transform is defined as Z - l ( ~ ( z ) ) : x(n). Notice that by Laurent's theorem [1] (cf. also L a u r e n t series), the inverse Z-transform is unique [2]. Consider a circle c centred at the origin of the z-plane and enclosing all the poles of z~-15(z). Then, by the C a u c h y i n t e g r a l t h e o r e m [1], the inversion formula is given by

x(n) = --27rii~c~(Z)zn_l dz 5k(n) =

ifn#k,

is given by oo

Z(6k(n)) : Z hk(j)z-J = z j:0 P r o p e r t i e s of t h e Z - t r a n s f o r m .

-k

for allz.

i) Linearity: Let R1 and R2 be the radii of convergence of the sequences x(n) and y(n). Then for any a,3EC,

Z(~x(~) + 9y(~)) = ~Z(x(,~)) +

~z(~(n)),

for I~1 > m ~ { R , , n ~ } . ii) Shifting: Let R be the radius of convergence of Z(x(n)). Then, for k E Z +,

and by the residue theorem (cf. also R e s i d u e of a n ana l y t i c f u n c t i o n ) [1], x(n) = ~(residues of zn-l~(z)). If ~(z)z ~-1 = h(z)/g(z) in its reduced form, then the poles of 9"(z)z n-1 are the zeros of g(z). a) If g(z) has simple zeros, then the residue/4/ corresponding to the zero zi is given by Ki :

lira

Z--+Zi

, h(z) 1 g[z)]

b) If g(z) has multiple zeros, then the residue/4/ at the zero zi with multiplicity r is given by Ki-

1 d~ (r - 1 ) ! l i r a ~

[

.rh(z)] L(Z-Zi) g-~].

ZAHORSKI P R O P E R T Y The most practical method of finding the inverse Ztransform is the use of partial-fractions techniques as illustrated by the following example.

Example. See also [2]. Suppose the problem is to solve the difference equation x(n + 4) + 9x(n + 3) + 30x(n + 2) + 20x(n + 1 + 24x(n) = 0, where x(0) = 0, x(1) = 0, x(2) = 1, x(a) = 10. Taking the Z-transform yields z(~(n)) =

_

~(~-I) (~ + 2)~(~ + 3) - 4 z + - 4z - -3z + z+2 (z+2)2 (z+2) 3

=

4z z+3

Taking the inverse Z-transform of both sides yields

x(n) = ( ~ n 2 - 1 - - - ~ n - 4 ) Pairs

( - 2 ) ~ + 4 ( - 3 ) ~.

of Z-transforms. x(n)

Z(z(n))

an

z/z - a

nk

k ! z / ( z - 1) k+l

nka ~ sinnw cosnw 5k(n) sinhnw coshnw

(--1)kDk(z/(z -- 1); D = z d / d z z s i n w / ( z 2 - 2z cosw + 1) z ( z - c o s w ) / ( z 2 - 2 z c o s w + 1) z -~ z s i n h w / ( z 2 - 2 z c o s h w + l) z ( z - c o s h w ) / ( z 2 - 2 z c o s h w + l).

References

[1] CHURCHILL, R.V., AND BROWN, J.W.: Complex variables and applications, McGraw-Hill, 1990. [2] ELAYDI, S.: A n introduction to difference equations, second ed., Springer, 1999. [3] Jt~RRI, A.J.: Linear difference equations with discrete transform methods, Kluwer Acad. Publ., 1996. [4] JURY, E.: Theory and application of the z-transform method, Robert E. Krieger, 1964. [5] MOIVRE, A. DE: Miscellanew, Analytiea de Seriebus et Quatratoris, London, 1730. [6] POULARIKAS, A.D.: The transforms and applications, CRC, 1996. S. Elaydi

MSC 1991: 39A12, 93Cxx, 94A12 ZAHORSKI PROPERTY - In his fundamental paper [4], Z. Zahorski studied (among other topics) zero sets of approximately continuous functions (cf. A p p r o x i m a t e c o n t i n u i t y ; the zero set of a real-valued function f is the set of points at which the value of f is precisely 0). In modern language, Zahorski proved [4, Lemma 11] that given a subset Z of the real line R, there is an approximately continuous non-negative bounded function f on R such that Z = {x E R : f ( x ) = 0} if and only if the set Z is of type G5 (cf. also Set o f t y p e F~ (Gs)) and closed in the density topology. (Recall that the density topology on R is formed by the collection of all Lebesgue

measurable sets having each of their points as a d e n s i t y point.) Notice that the class of approximately continuous functions was introduced by A. Denjoy in [1] as a generalization of the notion of c o n t i n u i t y . It is known that a function f is approximately continuous if and only if f is continuous in the density topology. Functions that are approximately continuous have many pleasing properties. For example, they have the D a r b o u x p r o p e r t y and belong to the first Baire class (cf. B a i r e c l a s s e s ) . Moreover, any bounded approximately continuous function is a d e r i v a t i v e . Hence Zahorski's theorem can be used in constructing functions with peculiar behaviour. For example, it is easy to construct functions of Pompeiu type: A function f on R is a Pompeiu function if it has a bounded derivative f ' and if the sets on which f ' is zero or does not vanish, respectively, are both dense in R (cf. also D e n s e set). Also, Zahorski's theorem can serve as a main tool in proving a strengthened form of an old Ward's result from [3]: Given a set E C (0, 1) of L e b e s g u e m e a s u r e zero, there is an approximately continuous function f such that f+ap = - ] - ~ and f - a p ~--- --(N3 o n E (here, f - a p , respectively f+ap, denote the left-hand upper, respectively right-hand lower, approximative derivative of f). Consider now a m e t r i c s p a c e (P,p) equipped with another topology ~- which is finer than the original metric topology To. The topology T has the Luzin-Menshov property with respect to ~-p if for each pair of disjoint sets F, F~ C P with F ~-p-closed and F~ T-closed, there is a pair of disjoint sets G, G~ C P with G Tp-open and G~ r-open, such that F~ C G and F C G~. If the topology ~- has the Luzin Menshov property with respect to the metric topology, then it has the Zahorski property: Any ~--closed subset of P which is of the metric type G5 is the zero set of a bounded T-continuous and metric upper s e m i - c o n t i n u o u s f u n c t i o n on P. Note that, conversely, the Zahorski property does not imply the Luzin-Menshov property. The density topology on the real line has the Luzin-Menshov property. Therefore it has the Zahorski property. Even very general density topologies, or also fine topologies of potential theory, have the Luzin-Menshov property, hence they have the Zahorski property as well. The Zahorski property can be introduced also in a very general framework of bitopological spaces. By this one understands a set X equipped with two topologies. If such a bitopological space satisfies the so-called 'binormality condition', it has the Zahorski property. A detailed study of the Zahorski property and its applications is given in [2]. 449

ZAHORSKI P R O P E R T Y References [1] DENJOY, A.: 'Sur les fonctions d~riv~es somrnables', Bull. Soc. Math. France 43 (1915), 161-248. [2] LUKES, J., MAL'), J., AND ZAJICEK, L.: F i n e topology methods in real analysis and potential theory, Vol. 1189 of Lecture N o t e s in Mathematics, Springer, 1986. [3] WARD, A.J.: 'On the points where A D + < A D - ' , J. London Math. Sac. 8 (1933), 293-299. [4] ZAHORSKI, Z.: 'gut la premi@e d~riv~e', Trans. A m e r . Math. Soc. 69 (1950), 1-54.

J. Luke5 MSC1991: 54E55, 26A21, 26A24, 28A05 ZAK TRANSFORM, Gel'land mapping, k-q representation, Weil-Brezin mapping - The Zak transform was discovered by several people in different fields and was called by different names, depending on the field in which it was discovered. It was called the 'Gel'fand mapping' in the Russian literature because I.M. Gel'fand [3] introduced it in his work on eigenfunction expansions associated with SchrSdinger operators with periodic potentials. In 1967, almost 17 years after the publication of Gel'land's work, the transform was rediscovered independently by a solid-state physicist, J. Zak, who called it the 'k-q representation'. Zak introduced this representation to construct a quantum-mechanical representation for the motion of a Bloch electron in the presence of a magnetic or electric field [8], [9]. It has also been said [7] that some properties of another version of the Zak transform, called the 'Weil-Brezin mapping' in [1], [7], were even known to the mathematician C.F. Gauss. Nevertheless, there seems to be a general consent among experts in the field to call it the Zak transform, since Zak was indeed the first to systematically study that transform in a more general setting and recognize its usefulness. The Zak transform Z~(f) of a function f is defined by

Za[f](t, w) = (Zaf)(t, w) =

(1)

oo

= v/a ~

f(at + ak)e -2~ia~,

k=--oo

where a > 0 and t and w are real. When a = 1, one denotes Zaf by Z f. If f represents a signal, then its Zak transform can be considered as a mixed time-frequency representation of f , and it can also be considered as a generalization of the discrete Fourier transform of f in which an infinite sequence of samples in the form f(at + ak), k = 0, +1, -t-2,..., is used (cf. also F o u r i e r t r a n s f o r m ) . E x a m p l e s . If a = 1 and f(t) = 0 outside [-b,b], 0 O,

is easily shown to be

(Z f)(t, w) = (27)1/4 e-~'~t2 03(w - iTt, e - ~ ) , where 03 is the third theta-function, defined by

03(z,q) = ~

qk2e- 2 7 r i k z

k=--oo

E x i s t e n c e . If f is integrable or square integrable (cf. I n t e g r a b l e f u n c t i o n ) , its Zak transform exists almost everywhere. In particular, if f is a c o n t i n u o u s f u n c t i o n such that If(t)l < C(1 + [tl) -(1+c), for some e > 0, for all t, then its Zak transform exists and defines a continuous function. Elementary properties. 1) (linearity): for any complex numbers a and b,

Z[af(t) + bg(t)] (t, w) = aZ[f (t)] (t, w) + bZ[g(t)] (t, w). 2) (translation): for any integer rn,

z[/(t

+ m)](t, w) =

e2 m Z[f](t, w);

in particular,

(Zf)(t + 1, w) = e 2 ~ ( Z f ) ( t , w). 3) (modulation):

Z [e2"imtf] (t, w) = J'i'~t(Zf)(t, w). 4) (periodicity): The Zak transform is periodic in w with period one, that is,

(Zf)(t,w + 1) = (Zf)(t,w). 5) (translation and modulation): By combining 2) and 3) one obtains

Z [e2~imtf(t + n)] (t, w) = e2"imte 2Èin~ (Z f)(t, w).

6) (conjugation): (ZY)(t, w)

=

(zf)(t,

7) (symmetry): If f is even (cf. also E v e n f u n c t i o n ) , then

(Z f)(t, w) = (Z f ) ( - t , -w), and if f is odd, then

(z f)(t,

= - (z f)(-t,

From 6) and 7) it follows that if f is real-valued and even, then

(Z f)(t, w) = (Z f ) ( t , - w ) = (Z f ) ( - t , - w ) . Because of 2) and 4), the Zak transform is completely determined by its values on the unit square Q = [0,1] x [0, 1].

Z A R A N K I E W I C Z CROSSING N U M B E R C O N J E C T U R E The Zak transform has been used successfully to study the orthogonality and the completeness of G a b o r frames in the crucial case where ab = 1; see [2], [10].

8) (convolution): Let

//

h(t) =

R ( t - s)f(s) ds; O0

References

then =

/o1

s,w)(zf)(s,w) ds.

(ZR)(t-

A n a l y t i c p r o p e r t i e s . If f is a continuous function such that f(t) = O ((1 + ]tl) -1-~) as Itl -~ oo for some e > 0, then Z f is continuous on Q. A rather peculiar property of the Zak transform is t h a t if Z f is continuous, it must have a zero in Q. The Zak transform is a u n i t a r y t r a n s f o r m a t i o n from L2(7~) onto L2(Q); see [10, p. 481]. I n v e r s i o n f o r m u l a s . The following inversion formulas for the Zak transform follow easily from the definition, provided that the series defining the Zak transform converges uniformly (eft also U n i f o r m c o n v e r g e n c e ) :

f(t) =

(Zf)(t,w) dw,

f(-2~rw)-

fl

vl1~

-oo < t < oc,

e-2~i"t(zf)(t,w) dt,

and

[l] AUSLANDER,L., AND TOLIMIERI, Pc.: 'Radar ambiguity functions and group theory', S I A M J. Math. Anal. 16 (1985), 577-601. [2] DAUBECHIES,I.: Ten lectures on wavelets, SIAM, 1992. [3] GEL'FAND, I.: 'Eigenfunction expansions for an equation with periodic coefficients', Dokl. Akad. Nauk. SSR 76 (1950), 1117-1120. (In Russian.) [4] JANSSEN, A.J.: 'Bargmann transform, Zak transform, and coherent states', J. Math. Phys. 23 (1982), 720-731. [5] JANSSEN, A.J.: 'The Zak transform: A signal transform for sampled time-continuous signals', Philips J. Research 43 (1988), 23-69. [6] KLAUDER, J., AND SKAGERSTAM, B.S.: Coherent states, World Sci., 1985. [7] SCHEMPP, W.: 'Radar ambiguity functions, the Heisenberg group and holomorphic theta series', Proc. Amer. Math. Soc. 92 (1984), 103-110. [8] ZAK, J.: 'Finite translation in solid state physics', Phys. Bey. Lett. 19 (1967), 1385-1397. [9] ZAK, J.: 'Dynamics of electrons in solids in external fields', Phys. Rev. 168 (1968), 686-695. [10] ZAYED, A.I.: Function and generalized function transformations, CRC, 1996.

Ahmed I. Zayed _ ~1

ffoo1e_2~ixt(Zf)(x,t)

dx,

where f is the F o u r i e r t r a n s f o r m of f , given by

1

//

f(x)e iwx dx.

A p p l i c a t i o n s . The Zak transform has been used successfully in various applications in physics, such as in the study of the coherent states representation in q u a n t u m field t h e o r y [6], and in electrical engineering, such as in time-frequency representation of signals and in digital data transmission; see [5], [4]. The applications of the Zak transform are not limited to only physics and engineering. More recent applications of it in mathematics have proved to be very useful; in particular, to simplify proofs of some important results. A case in point is the Gabor representation problem. The Gabor representation problem can be stated as follows: Given g E L 2 (7~) and two real numbers, a, b, different from zero, is it possible to represent any function f E L 2 (7~) by a series of the form

f =

fi

Cm,ngmb,na,

Tn~n:--(x3

where grab,ha are the Gabor functions, defined by gm

,na(x)

=

--

and Cm,~ are constants? And under what conditions is the representation unique?

MSC1991: 44A55, 44-XX, 42Axx

ZARANKIEWICZ CROSSING NUMBER CONJECTURE, Turdn brick factory problem - P. ~hrAn [6] tells about how he posed the following problem while in a forced labour camp in World War II: 'There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected by rail with all storage yards. -.. the trouble was only at crossings. The trucks generally jumped the rails there, and the bricks fell out of them; in short this caused a lot of trouble and loss of time • • • the idea occurred to me t h a t this loss of time could have been minimized if the number of crossings of the rails had been minimized. But what is the minimum number of crossings?' Recall that a drawing of a finite g r a p h G on the plane consists of placing the vertices of G on the plane and drawing the edges of G using continuous curves of the plane, connecting corresponding vertices as endpoints of the curve such t h a t no curve has a vertex as an internal point and no point is an internal point of 3 curves. The crossing number cr(G) of a graph G is the minimum number of intersection points among the curves representing edges, over all possible drawings of the graph. It is not hard to see t h a t the crossing number can always be realized by a drawing with the following properties: i) there is no self-crossing of edges; ii) edges with the same endpoint do not cross; 451

ZARANKIEWICZ CROSSING NUMBER C O N J E C T U R E iii) intersection points among the curves representing edges are crossing points, i.e. the curves do not touch each other; and iv) any two edges intersect at most once. For variations in the definition of crossing numbers, see [4]. P u t in the technical terms above, Zarankiewicz' crossing number conjecture, or Tur~n's brick factory problem, is as follows: 'what is the crossing number cr(K~,,~) of the complete bipartite graph K~,,~?' Place [n/2J vertices to negative positions on the x-axis, [n/2] vertices to positive positions on the zaxis, [rn/2] vertices to negative positions on the y-axis, [rn/2] vertices to positive positions on the y-axis, and draw nrn edges by straight line segments to obtain a drawing of Kn,m. It is not hard to check that the following formula gives the number of crossings in this particular drawing:

K. Zarankiewicz [10] and K. Urbanfk [7] independently claimed and published that cr(Kn,m) was actually equal to (1), their argument was reprinted in a book, cited, and used in follow-up papers. However, P. Kainen and G. Ringel discovered a flaw in the argument and the flaw has withstood all attempts for correction (up till 2000). R. Guy deserves much credit for rectifying this confused state of art, see [1] and [2]. D.J. Kleitman showed that (1) holds for m _< 6 [3] and also proved that the smallest counterexample to the Zarankiewicz conjecture must occur for odd n and rn. D.R. Woodall [9] used an elaborate computer search to show that (1) holds for K7,7 and K7,9. Thus, the smallest unsettled instances of Zarankiewicz's conjecture are K7,n and K9,9. It is known that •

c =

n

cr(K

,n

2

exists; however, the value of the limit is not known (as of 2000) [5]. Woodall's result for K7,9 implies 4/21 _< c by a standard counting argument, while c < 1/4 follows from the drawing shown. If (1) always holds, then c = 1/4. References [1] GUY, R.K.: 'The decline and fall of Zarankiewicz's theorem', in F. HARARY (ed.): Proof Techniques in Graph Theory, Acad. Press, 1969, pp. 63 69. [2] GuY, R.K.: '#21749', Math. Rev. 58 (1974). [3] KLEITMAN,D.J.: 'The crossing number of Ks,n', J. Combin. Th. 9 (1970), 315-323. [4] PACH, J., AND TdTH, G.: 'Which crossing number is it anyway?': Proc. 39th Ann. Syrup. Foundation of Computer Sci., IEEE Press, 1998, pp. 617-626.

452

[5] RICHTER, R.B., AND THOMASSEN, C.: 'Relations between crossing numbers of complete and complete bipartite graphs', Amer. Math. Monthly 104 (1997), 131-137. [6] TURIN, P.: 'A note of welcome', J. Graph Th. 1 (1977), 7 9. [7] URBANfK, K.: 'Solution du probl~me pos~ par P. Tur/m', Colloq. Math. 3 (1955), 200-201. [8] WHITE, A.T., AND BEINEKE, L.W.: 'Topological graph theory', in L.W. BEINEKE AND R.J. WILSON (eds.): Selected Topics in Graph Theory, Acad. Press, 1978, pp. 15 50. [9] WOODALL, D.R.: 'Cyclic-order graphs and Zarankiewicz's crossing-number conjecture', J. Graph Th. 17 (1993), 657671. [10] ZARANKIEWIeZ, K.: 'On a problem of P. Turin concerning graphs', Fundam. Math. 41 (1954), 137-145.

Ldszld A. Szdkely MSC 1991: 05C35, 05C10 ZARISKI-LIPMAN

CONJECTURE

-

Let k be a

field of characteristic zero and let R be a finitelygenerated k-algebra, that is, a homomorphic image of a ring of polynomials R = k [ x l , . . . , x~]/I. A k-derivation of R is a k-linear mapping 5: R --~ R that satisfies the Leibniz rule

5(ab) = aS(b) + bS(a) for all pairs of elements of R. The set of all such mappings is a Lie algebra (often non-commutative; cf. also C o m m u t a t i v e a l g e b r a ) that is a finitely-generated R-module ~ = Derk (R). The algebra and module structures of ~ often code aspects of the singularities of R. A more primitive object attached to R is its module of K5hler differentials, f~k (R), of which ~ is its R-dual,

= HomR(ftk(R), R). More directly, the structure of f~k(R) reflects many properties of R. Thus, the classical Jacobian criterion asserts that R is a smooth algebra over k exactly when f~k(R) is a projective R-module (el. also P r o j e c t i v e module)• For an algebra R without non-trivial nilpotent elements, local complete intersections are also characterized by saying that the projective dimension of ftk (R) (cf. also D i m e n s i o n ) is at most one. The technical issues linking these properties are the comparison between the set of polynomials that define R, represented by the ideal I, and the syzygies of either f~k(R) or ~ (cf. also S y z y g y ) . The Zariski-Lipman conjecture makes predictions about D, similar to those properties of f~k(R). The most important of these questions is as follows. If D is R-projective, then R is a r e g u l a r r i n g (in c o m m u t a t i v e a l g e b r a ) . More precisely, it predicts that if p is a p r i m e i d e a l for which ~p is a free Rp-module, then Rp is a regular ring.

ZASSENHAUS C O N J E C T U R E In [3], the question is settled affirmatively for rings of Krull dimension 1 (cf. also D i m e n s i o n ) , and in all dimensions the rings are shown to be normal (cf. also N o r m a l r i n g ) . Subsequently, G. Scheja and U. Storch [4] established the conjecture for hypersurface rings, t h a t is, when R is defined by a single equation, I = (f). As of 2000, the last m a j o r progress on the question was the proof by M. Hochster [2] of the graded case. A related set of questions is collected in [5]: whether the finite projective dimension of either f~k (R) or 2 necessarily forces R to be a local complete intersection. It is not known (as of 2000) whether this is true if is projective, a fact which would be a consequence of the Zariski-Lipman conjecture. Several lower dimension cases are known, but the most significant progress was made by L. A v r a m o v and J. Herzog when they solved the graded case [1]. References [1] AVRAMOV, L., AND HERZOG: J.: 'Jacobian criteria for complete intersections. The graded case', Invent. Math. (1994),

75 88. [2] HOCHSTER, M.: 'The Zariski-Lipman conjecture in the graded case', J. Algebra 47 (1977), 411-424. [3] LIPMAN, J.: 'Free derivation modules', Amer. J. Math. 87

(1965), 8~4-898. [4] SCHEJA, G., AND STORCH, U.: 'Differentielle Eigenschaften der Lokalisierungen analytischer Algebren', Math. Ann. 197 (1972), 137-170. [5] VASCONCELOS,W.V.: 'On the homology of I / I 2', Commun. Algebra 6 (1978), 1801 1809.

W. Vaseoneelos

results about function fields of genus zero [1]. Using a wide range of ideas from a l g e b r a i c g e o m e t r y , [2] provides a family of counterexamples to the problem. In particular, there exist a field K and extension fields L of transcendence degree two over K t h a t are not rational and yet L ( x l , x2, x3) is a pure transcendental extension of K in five variables. Finally, in [4] it is shown t h a t the problem does have an affirmative answer most of the time, i.e., if the original varieties are of general type. Again, this result uses [6] and, in an essential way, the results from [7]. References [1] AMH~SUR,S.: 'Generic splitting fields for central simple algebras', Ann. of Math. 2, no. 62 (1955), 8-43. [2] BEAUVILLE, A., COLLIOT-THELENE, J.-L., SANSUC, J.-J., AND SWINNERTON-DYER, l~.: 'Varietes stablement rationnelles non rationnelles', Ann. of Math. 121 (1985), 283-318. [3] DEVENEY, J.: 'Ruled function fields', Proc. Amer. Math. Soc. 86 (1982), 213-215. [4] DEVENEY, J.: 'The cancellation problem for function fields', Proc. Amer. Math. Soc. 103 (1988), 363-364. [5] NAGATA, M.: 'A theorem on valuation rings and its applications', Nagoya Math. J. 29 (1967), 85-91. [6] OHM, J.: 'The ruled residue theorem for simple transcendental extensions of valued fields', Proc. Amer. Math. Soc. 89

(1983), 16-18. [7] ROQUETTE, P.: 'Isomorphisms of generic splitting fields of simple algebras', J. Reine Angew. Math. 2 1 4 / 5 (1964), 207 226. James K. Deveney

MSC1991: 14Axx

MSC1991: 13B10, 13C15, 13C40 ZARISKI

PROBLEM

ON

FIELD

EXTENSIONS

-

Zariski problem has its motivation in a geometric question. For example, one could ask the following: Given two curves C1 and C2, make two surfaces by crossing a line with each curve. If the resulting surfaces are isomorphic, must the original curves also be isomorphic? In general, one starts with two affine varieties, V1 and V2, of dimension n (cf. also Affine v a r i e t y ) and crosses each with a line. Associated to each V/ is its coordinate ring c[V/], and from an algebraic point of view, one wants to know if the polynomial rings c[V1][xl] and c[V2][xe] being isomorphic forces the coordinate rings to be isomorphic (cf. also I s o m o r p h i s m ) . For n larger than two, this is an open problem (as of 2000). However, also associated to each Vi is its function field, c(Vi), and one wants to know if isomorphism of the rational function fields in one variable over the function fields forces the function fields to be isomorphic. This is the so-called The

Zariski problem. The problem has an affirmative answer for varieties of dimension one. This result appears in [3], but uses ideas from [5] and in an essential way depends on Amitsur's

ZASSENHAUS CONJECTURE - Just as the only roots of unity in a c y c l o t o m i c field Q(¢) are of the form ± ~ i there is the classical theorem of G. Higman stating t h a t the torsion units in the integral group ring Z G of a finite A b e l i a n g r o u p are of the form ±g, g C G. Of course, if G is non-Abelian, then any conjugate of ~ g is also of finite order; however, these are not all the torsion units in ZG. The famous Zassenhaus conjecture says t h a t for a f i n i t e g r o u p G all torsion units of Z G are rationally conjugate to ±g, g C G:

ZC1) Let u C ZG, u n = 1 for some n; then u = ± x - l g x for some g C G and some unit x E QG. This conjecture was proved to be true by A. Weiss, first for p-groups [16] and then for nilpotent groups [17] (aft also N i l p o t e n t g r o u p ) . In fact, Weiss proved the following stronger Zassenhaus conjecture for nilpotent groups: ZC3) If H is a finite subgroup of units of augmentation one in ZG, then there exists a unit x E Q G such that

x - l H x C G. A special case of this is the following conjecture: 453

ZASSENHAUS C O N J E C T U R E ZC2) If H is a subgroup of ZG of augmentation one of order IGI such that ZG = ZH, then there exists a unit x E Q G with x - l H x = G. This last conjecture was earlier proved by K. Roggenkamp and L.R. Scott [12] for nilpotent groups. Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [5]. Clearly, ZC3) implies ZC1) and ZC2). Also, ZC2) implies that if two group rings ZG and Z H are isomorphic, then the groups G and H are isomorphic. This isomorphism problem was proposed in [3]: ZG _ Z H

G~_H.

(1)

Of course, then, (1) is true for nilpotent groups. Moreover, it was proved by A. Whitcomb [18] that (1) is true for metabelian groups. M. Hertweck [2] has given a counterexample to (1). Conjecture ZC1) is open in general (as of 2000). Besides nilpotent groups, it is known to be true for certain split metacyclic groups [10]: If G = (a} x (b} is the semidirect product of two cyclic groups (a) and (b) of relative prime orders, then ZC1) holds for G. This result has been strengthened to ZC3) [15]. There are several useful and interesting extensions of the above conjectures. Suppose that A is a n o r m a l s u b g r o u p of index n in G. Then ZG can be represented by (n x n)-matrices over ZA. Any torsion unit u of ZG that is mapped by the natural homomorphism G -~ G/A to 1 E Z(G/A) gives rise to a torsion matrix U E SGLn(ZA). Here, SGL~(ZA) denotes the subgroup of the g e n e r a l l i n e a r g r o u p GL~(ZA) consisting of the matrices U that are mapped by the augmentation homomorphism ZA --~ Z, when applied to each entry, to the identity matrix. Thus, ZC1) translates to the question about diagonalization of U in GL~(QA): Is a torsion matrix U E SGL~(ZG), where G is a finite group, conjugate in (QG)~xn to a matrix of the form diag(gl,... ,gn), gi E G? This was answered positively in [16] for p-groups (cf. also p - g r o u p ) . See [1] for an explicit example of a matrix U E SGL~(Z(C6 x C6)) that cannot be diagonalized but for which U 6 = I. Such a matrix U exists for a finite nilpotent group G and some n if and only if G has at least two non-cyclic Sylow p-subgroups [1] (cf. also Sylow subgroup). However, it was proved in [6] that if n = 2 and G is finite Abelian, then U is conjugate in (QG)nxn to diag(gl,g2). This has been extended to n < 5 in [9], bridging the gap between 2 and 6. The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups F. The statements remain the same and the group F is arbitrary. A counterexample to ZC1) was provided in [8]. 454

Conjecture ZC2) also does not hold for infinite groups, as shown by S.K. Sehgal and A.E. Zalesskil (see [14, p. 279]). However, one can ask if any torsion unit U E SGLn(F) can be stably diagonalized to d i a g ( 7 1 , . . . , 73), ~/i E F. This has been proved [7] to be true for p-elements U when F is nilpotent. The isomorphism problem also has a positive answer for finitely-generated nilpotent groups of class 2, cf. [11]. In general for nilpotent groups the problem remains open (as of 2000). References [1] CLIFF~ C-., AND WEISS, A.: 'Finite groups of matrices over group rings', Trans. Amer. Math. Soc. 352 (2000), 457-475. [2] HERTWECK, M.: 'A solution of the isomorphism problem for integral group rings'. [3] HIGMAN, G.: 'Units in group rings': D. Phil. Thesis Univ. Oxford, 1940. [4] HIGMAN, G.: 'The units of group rings', Proc. London Math. Soc. 46 (1940), 231-248. [5] KLINGLER, L.: 'Construction of a counterexample to a conjecture of Zassenhaus', Commun. Algebra 19 (1991), 2303-2330. [6] LUTHAR, I.S., AND PASSI, I.B.S.: 'Torsion units in matrix group rings', Commun. Algebra 20 (1992), 1223-1228. [7] )J[ARCINIAK, Z., AND SEHGAL, S.K.: 'Finite matrix groups over nilpotent group rings', J. Algebra 181 (1996), 565-583. [8] MARCINIAK, Z., AND SEHGAL, S.K.: 'Zassenhaus conjecture and infinite nilpotent groups', J. Algebra 184 (1996), 207212. [9] MARCINIAE, Z., AND SEHGAL, S.K.: 'Torsion matrices over abelian group rings', J. Group Th. 3 (2000), 67 75. [10] MIMES, C. POLCINO, PdTTER~ J., AND SEHGAL, S.K.: 'On a conjecture of Zassenhaus on torsion units in integral group rings, II', Proc. Amer. Math. Soc. 97 (1986), 201-206. [11] RITTER, J., AND SEHGAL, S.K.: 'Isomorphism of group rings', Archiv Math. 40 (1983), 32-39. [12] ROGGENKAMP, K., AND SCOTT, L.: 'Isomorphisms for p-adic group rings', Ann. Math. 126 (1987), 593-647. [13] SEHGAL, S.K.: Topics in group rings, M. Dekker, 1978. [14] SEHGAL,S.K.: Units in integral group rings, Longman, 1993. [15] VALENTI, A.: 'Torsion units in integral group rings', Proc. Amer. Math. Soc. 120 (1994), 1-4. [16] WEISS, A.: 'Rigidity of p-adic p-torsion', Ann. of Math. 127 (1988), 317-332. [17] WEISS, A.: 'Torsion units in integral group rings', J. Reine Angew. Math. 415 (1991), 175-187. [18] WHITCOMB, A.: 'The group ring problem', PhD Thesis Univ. Chicago (1968).

S.K. Sehgal MSC 1991: 20Dxx, 20C05 Polynomials (cf. also P o l y n o m i a l ) constructed by F. Zernike [5] and by Zernike and H. Brinkman [6] for the purpose of approximating certain functions, such as the aberration function of geometrical optics, on the disc D = {(x,y) E R 2 : x 2 +y2 _< 1}. The underlying premise is that errors in circular optical elements can be quantified by meansquare deviation per unit area. Given a function f on D and n E No = { 0 , 1 , 2 , . . . } , the problem of finding ZERNIKE

POLYNOMIALS

-

ZETA-FUNCTION M E T H O D F O R REGULARIZATION a polynomial p ( x , y ) of degree n which minimizes the L 2-norm fir-P]]2 =

]f(x,y)-p(x,y)]

2

dxdy)

and satisfy a Rodrigues formula:

-(-1)

(a + 1)~+~ (1 - z~) -~

is solved by means of orthogonal polynomials (cf. Ort h o g o n a l p o l y n o m i a l s ) . This means that for each n E No there is an orthogonal basis for the space ~ of polynomials of degree n, which are orthogonal to each polynomial of lower degree (orthogonality is with respect to the inner product (f, g) = f fD f ( x , y)g(x, y) dx dy). The dimension of )2~ is n + 1. In the case of the disc there are at least two useful approaches to constructing orthogonal polynomials, based on the Cartesian or on the polar coordinate system. The Zernike polynomials are associated with the polar coordinate system (x = r cos 0, y = r sin 0) and with complex coordinates (z = x + iy = re ~°, r 2 = z~). For n E No and m = n - 2j with j = 0 , . . . , n, the Zernike circle polynomial is y) =

The orthogonal polynomials of degree n (that is, Vn = span{V~~-2j : 0 < j < n}) satisfy a differential equation: ( 4 ~ (02

7)2-2(a+

0

=

l~k+ 1~r;

(a + 1)k+l 1 ) i z k _ 2 1 F ( - k , - l ; - k (ct + 1)k(a +

=

- l - a; ~-~ 1) .

The Zernike radial polynomial is Rk-1 k+l (r" \ 7~) = min(k,l)

=

(~ + 1)k+l (a+l)k(a+l)l

V" ~0=

(-k)J(-lb

r k+~-~j

(-k-l-a)jj!

'

The normalization of the polynomials comes from the equation v ,k+l k - l ( k1 , 0; O0 = 1. The orthogonality relations are /./r)

k--1 --a-b Vi+ l (x, y; c~)Va+b(X, y; a)(1 - x 2 _ y 2 ) a d x d y = (~kahlb

=

k! l! 7c (~ + 1)k(c~ + 1)~ k + / + ~ + 1"

The polynomials can be expressed in terms of J a c o b i p o l y n o m i a l s : for k > l, ~¢-~(r; ~) l! rk-~P~ (c~'k-5 (2r 2 -- 1), R~+~ (~ + 1)~

0

There is an important i n t e g r a l t r a n s f o r m used in the diffraction theory of aberrations (see [1, Chap. 9]): Let k,1 E No, k >_ l, and s > 0, then ~o 1 -~k+~, R k - l ( r., ~ ) g k - ~ ( ~ ) ( 1

2

: e

1)79)f=

= -n(n + 2 + 2a)f,

- ( - 1 ) l F ( a + 1)

where R m ( r ) is a polynomial of degree n in r, of the same parity as n. This family has been generalized to 'disc polynomials', associated with the weight function (1 - x 2 - y2)~ d x d y with arbitrary a > - 1 (see [3]). The formulas will be stated for the general case since they are no more complicated than for the Zernike polynomials (a = 0). A convenient indexing is obtained from setting n = k + l , m = k - 1 for arbitrary k,1 C No. Then (using the P o c h h a m m e r symbol (a)~ = l-Ii~=l(a + i - 1) and the h y p e r g e o m e t r i c f u n c t i o n F), define

(1 - z~) k+~+~.

- r ~ ) ~ r dr =

Jk+l+~+l(s),

where J~ denotes the Bessel function of index a (cf. B e s s e l f u n c t i o n s ) . The coefficients of the orthogonal expansion of an aberration function in terms of the Zernike polynomials are related to the so-called primary aberrations (such as astigmatism, coma, distortion), see [1, Chap. 5]. The disc polynomials for a C No appear as s p h e r i c a l f u n c t i o n s on the homogeneous spaces U(a + 2 ) / U ( a + 1) (where U denotes the unitary group, see [2, Vol. 2, Sec. 11.5, pp. 359 363]). The Zernike polynomials are key tools in two-dimensional t o m o g r a p h y ; see [4].

References [1] BORN, M., AND WOLF, E.: Principles of optics, third ed., Pergamon, 1965. [2] KLIMYK, A., AND VILENKIN, N.: Representations of Lie groups and special functions, Kluwer Acad. Publ., 1993. [3] KOORNWINDER, T.: 'Two-variable analogues of the classical orthogonal polynomials', in R. ASKEY (ed.): Theory and Applications of Special Functions, Acad. Press, 1975, pp. 435495. [4] MARR, R.: 'On the reconstruction of a function on a circular domain from a sampling of its line integrals', Y. Math. Anal. Appl. 45 (1974), 357-374. [5] ZERNmE, F.: 'Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode', Physica 1 (1934), 689-704. [6] ZERNIKE, F., AND BRINKMAN, H.: 'Hypersph~risehe Funktionen und die in sphgrischen Bereichen orthogonalen Polynome', Proc. K. Akad. Wetensch. 38 (1935), 161-170.

Charles F. Dunkl

MSC1991: 33C50, 78A05

ZETA-FUNCTION METHOD FOR REGULARIZATION, zeta-function regularization - Regularization and 455

ZETA-FUNCTION M E T H O D FOR REGULARIZATION renormalization procedures are essential issues in contemporary physics - - without which it would simply not exist, at least in the form known today (2000). They are also essential in supersymmetry calculations. Among the different methods, zeta-function regularization - - which is obtained by a n a l y t i c c o n t i n u a t i o n in the complex plane of the zeta-function of the relevant physical operator in each case - - might well be the most beautiful of all. Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system (with constraints of any kind, in principle). Assuming the corresponding Hamiltonian operator, H, has a spectral decomposition of the form (think, as simplest case, of a quantum harmonic oscillator): {Ai, ~i}i~I, with I some set of indices (which can be discrete, continuous, mixed, multiple, etc.), then the quantum vacuum energy is obtained as follows [4], [2]: E

process (it has taken over a decade already) of generalization of previous results and derivation of new expressions of this kind [4], [2]. [1].

References [1] BYTSENKO, A.A., COGNOLA, G., VANZO, L., AND ZERBINI, S.: 'Quantum fields and extended objects in space-times with constant curvature spatial section', Phys. Rept. 266 (1996), 1-126. [2] ELIZALDE,E.: Ten physical applications of spectral zeta functions, Springer, 1995. [3] ELIZALDE,E.: 'Multidimensional extension of the generalized Chowla Selberg formula', Commun. Math. Phys. 198 (1998), 83-95.

[4] ELIZALDE, E., ODINTSOV,

E. Elizalde ~,X~

EiCI ~i = ice

-s

MSC 1991: 81Qxx

=

s--1

(H(--1),

where ~H is the zeta-function corresponding to the operator H. The formal sum over the eigenvalues is usually ill-defined, and the last step involves analytic continuation, inherent to the definition of the zeta-function itself. These mathematically simple-looking relations involve very deep physical concepts (no wonder that understanding them took several decades in the recent history of q u a n t u m field t h e o r y , QFT). The zetafunction method is unchallenged at the one-loop level, where it is rigorously defined and where many calculations of Q F T reduce basically (from a mathematical point of view) to the computation of determinants of elliptic pseudo-differential operators (~DOs, cf. also P s e u d o - d l f f e r e n t i a l o p e r a t o r ) [3]. It is thus no surprise that the preferred definition of determinant for such operators is obtained through the corresponding zeta-function. When one comes to specific calculations, the zetafunction regularization method relies on the existence of simple formulas for obtaining the analytic continuation above. These consist of the reflection formula of the corresponding zeta-function in each case, together with some other fundamental expressions, as the Jacobi theta-function identity, Poisson's resummation formula and the famous Chowla-Selberg formula [3]. However, some of these formulas are restricted to very specific zeta-functions, and it often turned out that for some physically important cases the corresponding formulas did not exist in the literature. This has required a painful 456

A., BYTSENKO,

(~i, H p i ) = t r H =

iCI =

S.D., ROMEO,

A.A., AND ZERBINI, S.: Zeta regularization techniques with applications, World Sci., 1994. [5] HAWKING, S.W.: 'Zeta function regularization of path integrals in curved space time', Commun. Math. Phys. 55 (1977), 133-148. [6] NAKAHARA,M.: Geometry, topology, and physics, Inst. Phys., 1995, pp. 7-8.

ZFC, Zermelo-Fraenkel set theory with the axiom of choice - ZFC is the acronym for Zermelo-Praenkel set theory with the a x i o m o f choice, formulated in firstorder logic. ZFC is the basic axiom system for modern (2000) set t h e o r y , regarded both as a field of mathematical research and as a foundation for ongoing mathematics (cf. also A x i o m a t i c set t h e o r y ) . Set theory emerged from the researches of G. Cantor into the transfinite numbers and his c o n t i n u u m h y p o t h e s i s and of R. Dedekind in his incisive analysis of natural numbers (see [5] or [11]). E. Zermelo [20] in 1908, under the influence of D. Hilbert at GSttingen, provided the first fullfledged axiomatization of set theory, from which ZFC in large part derives. Although several axiom systems were later proposed, ZFC became generally adopted by the 1960s because of its schematic simplicity and openendedness in codifying the minimally necessary set existence principles needed and is now (as of 2000) regarded as the basic framework onto which further axioms can be adjoined and investigated. A modern presentation of ZFC follows. The language of set theory is first-order logic with a binary predicate symbol C for membership ('first-order' refers to quantification only over individuals, not e.g. properties). This language has as symbols an infinite store of variables; logical connectives (7 for 'not', V for 'or', A for 'and', -+ for 'implies', and ++ for 'is equivalent to'); quantifiers (V for 'for all' and ~ for 'there exists'); two binary predicate symbols, = and C; and parentheses. (A more parsimonious presentation is possible, e.g.

ZFC one can do with just 7, V and V, and leave out parentheses with a different syntax.) The formulas of the language are generated as follows: x = y and x • y are (the atomic) formulas whenever x and y are variables. If and ¢ are formulas, then so are ( ~ ) , (~ V ¢), (~ A ~), (~ ~ @), (~ ~ @), Vx~, and 3x~, whenever x is a variable. The various further notations can be regarded as abbreviations; for example, x C y for 'x is a subset of y' abbreviates V z ( z • x --+ z • y). The axioms of ZFC are as follows, with some historical and notational commentary. A1) A x i o m of extensionality: V x V y ( V z ( z • x ++ z • y) ~ x = y).

This is a fundamental principle of sets, that sets are to be determined solely by their members. The arrow '--+' can be replaced by ' ~ ' since the other direction is immediate. Indeed, the axiom can then be taken to be a means of introducing = itself as an abbreviation, as a symbol defined in terms of •. The term 'extensionality' stems from a traditional philosophical distinction between the intension and the extension of a term, where loosely speaking the extension of a term is the collection of things of which the t e r m is true of, and the intension is some more intrinsic sense of the term. A clear statement of the principle of extensionality had already appeared in the pioneering work of Dedekind [3], which provided a development of the natural numbers in set-theoretic terms and anticipated Zermelo's axiomatic, abstract approach to set theory. Cf. also A x i o m o f extensionality. A2) A x i o m of the e m p t y set: q x V y ( - , y • x). This axiom asserts the existence of an empty set; by A1), such a set is unique, and is denoted by the term ~. Terms are similarly introduced in connection with other axioms below, and in general such terms can be eliminated in favour of their definitions; for example, 0 • z can be regarded as an abbreviation for 3 x ( V y ( ~ y • x) A x • z). A3) A x i o m of pairs: VxVy~W,(v

• ~ ~

(~ = ~ v ,~ = y ) ) .

This axiom asserts, for any sets x and y, the existence of their (unordered) pair, the set consisting exactly of x and y. This set is denoted by {x, y}. A3) implies, taking its y to be x, that for any set x there is a set consisting solely of x, denoted by {x}. The existence of 0 and the distinction between a set x and the single-membered {x} were not clearly delineated in the early development of set theory, and equivocations in these directions can be found, e.g., in [3].

A4) A x i o m of union: Y x B z Y v ( v E z ++ 3 y ( y E x A v E y)). This axiom asserts, for any (generalized) union, the set members of members of x. U x. Note that for two sets a union a U b. A5) A x i o m of power set: Vx3zVv(v

set x, the existence of its consisting exactly of the This union is denoted by and b, U{a, b} is the usual

• z ++ V w ( w • v -~ w • x ) ) .

This axiom asserts, for any set x, the existence of its power set, the set consisting exactly of those sets v that are subsets of x. This power set is denoted by 79(x). The axioms Ag)-A5) are generative axioms, providing various means of collecting sets together to form new sets. The generative process can be started with A2), an outright existence axiom. The next axiom is another outright existence axiom, which for convenience is stated via terms defined above: A6) A x i o m of infinity: ~x(~ • x A Vy(y • x -+ y u {y} • x)).

Among various possible approaches, this axiom asserts the existence of an infinite set of a specific kind: the set contains the e m p t y set and is moreover closed in the sense t h a t whenever y is in the set, so also is y U {y}. Hence, 0, {0}, {0, {0}}, {~, {~},{~, {0}}},... are to be members; these are indeed sets by A2) and A3) and are moreover distinct from each other by A1). Zermelo himself had {y} in place of y U {y}, but the modern formulation derives from the formulation by J. von Neum a n n [15] of the ordinal numbers within set theory (cf. also O r d i n a l n u m b e r ) . Dedekind [3] had (in)famously 'proved' the existence of an infinite set; Zermelo was first to see the need to postulate the existence of an infinite set. In the presence of A6), A5) becomes a much more powerful axiom, purportly collecting together in one set all arbitrary subsets of an infinite set; Cantor famously established that no set is in bijective correspondence with its power set, and this leads to an infinite range of transfinite cardinalities (cf. also T r a n s f i n i t e n u m b e r ) . AT) A x i o m of choice: Vx : V v V w ( ( ( v • x A w • x) A 3t(t • v A t • w)) ~ v = w) $

3yVv((v • x A (~v = 0)) -+ 3sVt((t • v A t • y) ~ s = t)). This is one of the most crucial axioms of Zermelo's axiomatization [20] (cf. also A x i o m o f choice). To unravel it, the hypothesis asserts that x consists of pairwise disjoint sets, and the conclusion, that there is a set y t h a t 457

ZFC with each non-empty member of x has exactly one common member. Thus, y serves as a 'selector' of elements from members of x. AT) is usually stated in terms of functions: The theory of functions, construed as sets of ordered pairs with the univalent property on the second coordinate, is first developed with the previous axioms. Then AT) has an equivalent formulation as: Every set has a choice function, i.e. a function f whose domain is the set and such that for each non-empty member y of the set, f ( y ) E y. Zermelo [18] formulated A7) and with it, established his famous well-ordering theorem: Every set can be wellordered (cf. also Z e r m e l o t h e o r e m ) . Zermelo maintained that the axiom of choice is a 'logical principle' which 'is applied without hesitation everywhere in mathematical deduction'. However, Zermelo's axiom and result generated considerable criticism because of the positing of arbitrary functions following no particular rule governing the passage from argument to value. Since then, of course, the axiom has become deeply embedded in mathematics, assuming a central role in its equivalent formulation as Zorn's lemma (cf. also Z o r n l e m m a ) . In response to critics, Zermelo [19] published a second proof of his well-ordering theorem, and it was in large part to buttress this proof that he published [20] his axiomatization, making explicit the underlying set-existence assumptions used (see [14]). A8) Axiom (schema) of separation: For any formula ~o with unquantified variables among v, v l , . . . , v~,

VXVVl " " "Vvn3yVv(v C y ++ (V E X A ~) ). This is another crucial component of Zermelo's axiomatization [20]. Actually, it is an infinite package of axioms, one for each formula p, positing for any set x the existence of a subset y consisting of those members of x 'separated' out according to ~. Zermelo was aware of the paradoxes of logic emerging at the time, and he himself had found the famous R u s s e l l p a r a d o x independently (cf. also P a r a d o x ; A n t i n o m y ) . Russell's paradox results from 'full comprehension', the allowing of any collection of sets satisfying a property to be a set: Consider the property (-~y G y); if there were a set R consisting exactly of those y satisfying this property, one would have the contradiction (R E R ++ (~R E R)). Zermelo saw that if one only allowed collections of sets satisfying a property 'and drawn from a given set' to be a set, then there are no apparent contradictions. Thus was Zermelo able to retain, in an adequate way as it has turned out, the important capability of generating sets corresponding to properties. The first theorem in [20] applies A8) together with the Russell paradox argument to assert that the universe of sets (cf. also U n i v e r s e ) is not itself a set. 458

Zermelo's version of A8) retained an intensional aspect, with his ~ being some 'definite' property determinate for any y C x whether the property is true of y or not. However, this became unsatisfactory in the development of set theory, and eventually the suggestion of T. Skolem [17] of taking Zermelo's definite properties as those expressible in first-order logic was adopted, yielding AS). Generally speaking, logic loomed large in the formalization of mathematics at the turn into the twentieth century, at the time of G. Frege and B. Russell, but in the succeeding decades there was a steady dilution of what was considered to be logical in mathematics. Many notions came to be considered distinctly set-theoretic rather than logical, and what was retained of logic in mathematics was first-order logic. A9) Axiom (schema) of replacement: For any formula in two unquantified variables v and w,

Vv3u(Vw~ ~ u = w) 4

Vx3yVw(w E y ++ 3v(v E x A ~)). This also is an infinite package of axioms, one for each ~. To unravel it, the hypothesis asserts that qo is 'functional' in the sense that to each set v there is a unique corresponding set u satisfying qo, and the conclusion, that for any set x there is a set y serving as the 'image of x under qY. In short, for any definable function correspondence and any set, the image of that set under the correspondence is also a set. Ag) was not part of Zermelo's original axiomatization [20], and to meet its inadequacies for generating certain kinds of sets, A. Fraenkel [6] and Skolem [17] independently proposed adjoining A9). Because of historical circumstance, it was Praenkel whose initial became part of the acronym ZFC. However, it was Von Neumann's incorporation [16] of a method into set theory, t r a n s f i n i t e r e c u r s i o n , that necessitated the full exercise of Ag). In particular, he [15] defined (what are now called the yon Neumann) ordinals within set theory to correspond to Cantor's original, abstract ordinal numbers, and Ag) is needed to establish that every well-ordered set is orderisomorphic to an ordinal. By a simple argument, A9) implies A8). A10) Axiom of foundation: Vx((~x = ~) -~ 3y(y e x A V z ( z e x -~ ~ z e y))).

This asserts that every non-empty set x is well-founded, i.e. has a 'minimal' member y in terms of C. A10) also was not part of Zermelo's axiomatization [20], but appeared in his final axiomatization [21]. A10) is an elegant form of the assertion that the formal universe V of sets is stratified into a cumulative hierarchy:

ZFC T h e a x i o m is e q u i v a l e n t to t h e a s s e r t i o n t h a t V is laye r e d into sets V~ for (yon N e u m a n n ) o r d i n a l s a , where:

Vo=O; V~ = U "P(Vz+t); and v=UVo=

D. M i r i m a n o f f a n d yon N e u m a n n h a d also f o r m u l a t e d t h e c u m u l a t i v e hierarchy, b u t m o r e to specific p u r p o s e s . Zermelo s u b s t a n t i a l l y a d v a n c e d t h e s c h e m a t i c generative p i c t u r e w i t h his a d o p t i o n of A10), a n d K. GSdel u r g e d this view of t h e s e t - t h e o r e t i c universe. A10) is t h e one a x i o m u n n e c e s s a r y for t h e r e c a s t i n g of m a t h e m a t i c s in s e t - t h e o r e t i c t e r m s , b u t t h e a x i o m is also t h e salient f e a t u r e t h a t d i s t i n g u i s h e s i n v e s t i g a t i o n s specific to set t h e o r y as an a u t o n o m o u s field of m a t h e m a t i c s . Indeed, it can fairly be s a i d t h a t c u r r e n t set t h e o r y is at base t h e s t u d y of well-foundedness, t h e C a n t o r i a n well-ordering d o c t r i n e s a d a p t e d to t h e Z e r m e l i a n g e n e r a t i v e concept i o n of sets. Z F C , again, is t h e s t a n d a r d s y s t e m of a x i o m s for set theory, given b y t h e a x i o m s A 1 ) - A 1 0 ) above. 'Z' is t h e c o m m o n a c r o n y m for Z e r m e l o s e t t h e o r y , t h e a x i o m s a b o v e b u t w i t h A9), t h e a x i o m (schema) of replacement, deleted. Finally, ' Z F ' is t h e c o m m o n a c r o n y m for Z e r m e l o - F r a e n k e l s e t t h e o r y , t h e a x i o m s a b o v e b u t with A7), t h e a x i o m of choice, deleted. T h e r e has been a t r e m e n d o u s a m o u n t of w o r k done in t h e a x i o m a t i c i n v e s t i g a t i o n of set theory. T h e first s u b s t a n t i a l result was G S d e l ' s r e l a t i v e c o n s i s t e n c y res u l t [7], [8], t h a t if Z F is consistent, t h e n so also is Z F C (and this t o g e t h e r w i t h C a n t o r ' s c o n t i n u u m h y p o t h e s i s ; cf. also C o n s i s t e n c y ) . P. Cohen [1], [2], in famous work leading to t h e F i e l d s M e d a l , e s t a b l i s h e d t h e relative i n d e p e n d e n c e result, t h a t if Z F is consistent, t h e n so also is Z F t o g e t h e r w i t h t h e n e g a t i o n of t h e a x i o m o f c h o i c e (and so also is Z F C t o g e t h e r with t h e n e g a t i o n of t h e c o n t i n u u m h y p o t h e s i s ) . (Cf. also F o r c i n g m e t h o d . ) For t h e s e results, see [10], [13]. A g r e a t deal of t h e work of t h e last several decades (as of 2000) has been d e v o t e d to t h e i n v e s t i g a t i o n of large c a r d i n a l axioms a d j o i n e d to Z F C a n d t h e i r consequences a n d interactions with ongoing m a t h e m a t i c s (see [11]). References [1] COHEN, P.J.: 'The independence of the continuum hypothesis I', Proc. Nat. Acad. Sci. USA 50 (1963), 1143-1148. [2] COHEN, P.J.: Set theory and the continuum hypothesis, Benjamin, 1966. [3] DEDEKIND,R.: Was sind und was sollen die Zahlen?, Vieweg, 1888, English transl.: R. Dedekind, Essays on the theory of numbers, Dover 1963; W.B. Ewald, From Kant to Hilbert: A source book in the foundations of mathematics, Oxford Univ. Press, 1996, pp. 790-833.

[4] EWALD, W.B. (ed.): From Kant to Hilbert: A source book in the foundations of mathematics, Oxford Univ. Press, 1996. [5] FERREIRSS, J.: Labyrinth of thought: A history of set theory and its role in modern mathematics, Birkhguser, 1999. [6] FRAENKEL, A.A.: '0bet die Zermelosche Begriindung der Mengenlehre (abstract)', Jahresber. Deutsch. Math. Verein. 30, no. II (1921), 97-98. [7] G(3DEL, K.F.: 'The consistency of the axiom of choice and of the generalized continuum-hypothesis', Proc. Nat. Acad. Sci. USA 24 (1938), 556 557. [8] G(~DEL, K.F.: The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Vol. 3 of Ann. of Math. Stud., Princeton Univ. Press,

1940. [9] HEIJENOORT, J. VAN (ed.): From Frege to GSdel: A source book in mathematical logic 18"79-1931, Cambridge Univ. Press, 1967. [10] JECH, TH.J.: Set theory, second corrected ed., Acad. Press, 1997. [11] KANAMOHI,A.: 'The mathematical development of set theory from Cantor to Cohen', Bull. Symbolic Logic 2 (1996), 1-71. [12] KANAMORI,A.: The higher infinite, Springer, 1997, Corrected second printing. [13] KUNEN, K.: Set theory: A n introduction to independence proofs, North-Holland, 1980. [14] MOORE, G.H.: Zermelo's axiom of choice: Its origins, development and influence, Springer, 1982. [15] NEUMANN, J. VON: 'Zur Einfiihrung der transfiniten Zahlen', Acta Litterarum ac Scientiarum Regiae Univ. Hung. Francisco-Josephinae, Sectio Sci. Math. 1 (1923), 199-208,

English transl.: J. van Heijenoort (ed.), From Frege to GSdeh A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 346-354. [16] NEUMANN, J. VON: 'Eine Axiomatisierung der Mengenlehre', J. Reine Angew. Math. (Crelle's J.) 154 (1925), 219-240, Berichtigung, J. Reine Angew. Math. 155 (1925), 128; English transl. J. van Heijenoort (ed.), From Frege to G5deh A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 393-413. [17] SKOLEM, T.: 'Einige Bemerkungen zur axiomatischen Begriindung der Mengenlehre': Matematikerkongressen i Helsingfors den 4-7 Juli, 1922, Den femte Skand. Matematikerkongressen, Redog5relse Helsinki, Akad. Bokhandeln,

[18]

[19]

[20]

[21]

1923, pp. 217-232, English transl.: J. van Heijenoort (ed.), From Frege to G6del: A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 290-301. ZERMELO,E.: 'Beweis, dass jede Menge wohlgeordnet werden kann (Aus einem an Herrn Hilbert gerichteten Briefe)', Math. Ann. 59 (1904), 514-516, English transl.: J. van Heijenoort (ed.), From Frege to G5del: A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 139-141. ZERMELO, E.: 'Neuer Beweis fiir die M5glichkeit einer Wohlordnung', Math. Ann. 65 (1908), 107-128, English transl.: J. van Heijenoort (ed.), From Frege to G5del: A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 183-198. ZERMELO, E.: 'Untersuchungen fiber die Grundlagen der Mengenlehre I', Math. Ann. 65 (1908), 261-281, English transl.: J. van Heijenoort (ed.), From Frege to G5deh A source book in mathematical logic 1879-1931, Cambridge Univ. Press, 1967, pp. 199-215. ZERMELO,E.: @ber Grenzzahlen und Mengenbereiche: Neue Untersuchungen fiber die Grundlagen der Mengenlehre', Fun-

459

ZFC dam. Math. 16 (1930), 29-47, E n g l i s h transl.: W . B . Ewald (ed.), P r o m K a n t to Hilbert: A source b o o k in t h e f o u n d a t i o n s of m a t h e m a t i c s , Oxford Univ. Press, 1996, pp. 1219-1233.

Akihiro Kanamori MSC 1991:03E30 ZIPF L A W - In 1949 G.K. Zipf published [27]. A large, if not the main, part of it was devoted to the principles of human use of a language and the following was the main thesis of the author [27, pp. 20-21]:

From the viewpoint of the speaker, there would doubtless exist an important latent economy in a vocabulary that consisted exclusively of one single word a single word that would mean whatever the speaker wanted it to mean. Thus, if there were m different meanings to be verbalized, this word would have m different meanings. [...] But from the viewpoint of the auditor, a single word vocabulary would represent the acme of verbal labour, since he would be faced by the impossible task of determining the particular meaning to which the single word in given situation might refer. Indeed, from the viewpoint of the auditor [-..] the important internal economy of speech would be found rather in vocabulary of such size that [...] if there were m different meanings, there would be m different words, with one meaning per word. Thus, if there are an m number of different distinctive meanings to be verbalized, there will be 1) a speaker's economy in possessing a vocabulary of one word which will refer to all the distinctive meanings; and there will also be 2) an opposing auditor's economy in possessing a vocabulary of m different words with one distinctive meaning for each word. Obviously the two opposing economies are in extreme conflict. In the language of these two [economies or forces] we may say that the vocabulary of a given stream of speech is constantly subject of the opposing Forces of Unification and Diversification which will determine both the number n of actual words in the vocabulary, and also the meaning of those words. As far as this was the main thesis, the main data and the main empirical fact discussed thoroughly in the book is: James Joyce's novel Ulysses, with its 260,430 running words, represents a sizable sample of running speech that may fairly be 460

said to have served successfully in the communication of ideas. An index to the number of different words therein, together with the actual frequencies of their respective occurrences, has already been made with exemplary methods by Dr. Miles L. Hanley and associates ([5]) [...]. To the above published index has been added an appendix from the careful hands of Dr. M. Joos, in which is set forth all the qualitative information that is necessary for our present purposes. For Dr. Joos not only tells us that there are 29,899 different words in the 260,430 running words; [...] and tells us the actual frequency, f , with which the different ranks, r, occur. It is evident that the relationship between the various ranks, r, of these words and their respective frequencies, f , is potentially quite instructive about the entire matter of vocabulary balance, not only because it involves the frequencies with which the different words occur but also because the terminal rank of the list tells us the number of different words in the sample. And we remember that both the frequencies of occurrenee and the number of different words will be important factors in the counterbalancing of the Forces of Unification and Diversification in the hypothetical vocabulary balance of any sample of speech. Now, let {fi~}i=l N be the frequencies of N, N < c~, different disjoint events in a s a m p l e of size n, like occurrences of different words in a text of n running words, and let #n = ~N= 1 l{fi~_>l} be an 'empirical vocabulary', that is, the number of all different events in the sample. Furthermore, let fo,n) > "'" k f(,,,~) be the o r d e r s t a t i s t i c based on the frequencies and G~(x) = E~=I l{f,,>x}. Clearly, AGn(x) -- #n(X) = E l{A~=x} is the number of events that occurred in the sample exactly x times and G~(1) = #~. Consider the statement that

kf(k,n)~#n,

k=1,2,...,

(1)

in some sense [27, II.A, p.24]. There is a bit of a problem with such statement for high values of the rank k. Indeed, though in general G~(f(k,~)) = max{M: f(k',~) = f(k,~)} for k = 1, 2 , . . . , typically Gn(f(k,~)) = k and one can replace (1) by an(X) x ~ #n,

x = f(1,n), f(2,n),- • • •

(2)

For k close to #~ there will be plenty of frequencies of the same value which must have different ranks, and therefore the meaning of (1) becomes obscure. However,

ZIPF LAW one can use (2) for all values of x = 1 , . . . , form

#n(X) ~ A 1 _ 1 #n X X(X + 1)'

f(1,n). In the

x = 1,2,...

is frequently called the Zip/-Mandelbrot law. Among earlier work on distributions of the sort, f(k,n)

(3)

it becomes especially vivid, for it says t h a t even for very large n (of m a n y thousands) half of the empirical vocabulary should come from events which occurred in the sample only once, x = 1. The events that occurred only twice, x = 2, should constitute 1/6 of p~, and the events that occurred not more than 10 times constitute 10/11th of empirical vocabulary. So, somewhat counterintuitive even in very big samples, rare events should be presented in plenty and should constitute almost total of M1 different events that occurred. Any of statements (1), (2) or (3) are called Zipf's law. Assume for simplicity that {fin}i=1 N are drawn from a m u l t i n o m i a l d i s t r i b u t i o n with probabilities {pi~} N which form an array with respect to n. The asymptotic behaviour of the statistics {f(k,n) }k=l' tt~ {~n( k )}k=l' P~ G~('), which all are symmetric, is governed by the distribution function of the probabilities {p~}N: N

=Z

i=1

Namely, if

1Gp,,~ Y+ G,

-

(4)

depending on some p a r a m e t e r ft. H.A. Simon [21] proposed and studied a certain Markov model for {#n(X): x = 1, 2 , . . . } as a process in n and obtained (8) and a somewhat more general form of it as a limiting expression for #n(x)/#~. His core assumption was most natural: as there are x#n(x) words used x times in the first n draws, the probability t h a t a word from this group will be used on the (n + 1)st draw is proportional to xp,(z). In other words,

l i m s u p rl~zR~(dz)=O, ¢-+0

n

~--- ] ~ ( n ) [ ( X

J0

where

R~(x) = f o ( 1 _ e_Z)Gp,~(dz),

n #n

then e- ~ = - ~0 ~a /~ Xx!

(5)

(see, e.g., [9]). Therefore the limiting expression 1/x(x + 1) is by no means unique and there are as many possible limiting expressions for the spectral statistics #~(x)/#n as there are measures with f o ( 1 - e - ; ~ ) R ( d A ) = 1. The right-hand side of (5) gives Zipf's law (3) if and only if

R(x) = fo ~ 1 +1 z e -zx dz. The concepts of C. Shannon in i n f o r m a t i o n t h e o r y were relatively new when B. Mandelbrot [13] used them to make Zipf's reasoning on a compromise between speaker's and auditor's economies rigorous and derive (3) formally. Therefore the statement _#n(x) _

#,~

~

1

(a + bx) 2

--

1)#n(X

--

1)

--

(9)

X#n(X)].

His other assumption t h a t a new (unused) word would occur on each trial with the same constant probability was, perhaps, a little bit controversial. Indeed, it implies that the empirical vocabulary #~ increases with the same rate as n, while for Zipf's law in its strict sense of (3) (or for q = 1 in (7)) one obviously has

while if

£

(8)

El#,+1 (X)I#n(')] -- #~(X) =

L-

and

r(fl + 1)r(x)

r ( x + fi + 1 ) '

n

Rn2+R

(7)

is, of course, the famous P a r e t o d i s t r i b u t i o n of population over income [19]. In lexical data, b o t h Mandelbrot [14] and Zipf [27] note t h a t the so-called Zipf's law was first observed by J.B. Estoup [3]. (Note that Zipf's role was not so much the observation itself, which, by the way, he never claimed to be the first to make, as the serious a t t e m p t to explain the mechanism behind it.) G.U. Yule [26], in his analysis of data of J.G. Willis [24] on problems of biological taxonomy, introduced the following expression for a(x):

then n

" Ak-(l+q)

(6)

--

2 x = 1 xpn(x) #n

- -

1

-9

C~.

X+ 1 x:l

This and some other concerns of analytic nature (cf. the criticism in [14], [15] and the replies [22], [23]) led Simon to the third assumption: '[a word] may be dropped with the same average rate'. This is described in very clear form in [22]: We consider a sequence of k words. We add words to the end of the sequence, and drop words from the sequence at the same average rate, so t h a t the length of the sequence remains k. For the birth process we assume: the probability t h a t the next word added is one that now occurs i times is proportional to (i + c)#(i). The probability that the next word is a new word is a, a constant. For the death process we assume: the probability 461

Z I P F LAW t h a t the next word dropped is one t h a t now occurs i times is proportional to (i + d)#(i). The terms c and d are constant parameters. The steady state relation is: #(i, m + 1) - p(i, m) =

(I

-

a)

In a certain sense, this was actualized in a very elegant model of B. Hill and M. Woodroofe (see [61, [7],

[25]). According to Hill, if N individuals are to be distributed to M non-empty genera in accordance with B o s e - E i n s t e i n s t a t i s t i c s , i.e. all allocations are equiprobable, each with probability

k + crnk •[(i -

1 +

1 N--1 (M--1)

c ) p ( i - 1 , m ) - (i + e ) p ( i , m ) ] +

1 -[(i + d)p(i,m) - (i + d + 1)p(i + 1,m)] = 0.

and if P { M / N <_x} 2+ F(x), 0 < x < 1, F(0) = 0, then for the number of genera with exactly x species one has

A solution to this equation, independent of

#N(x) d U(1 - U) x-l,

k +dnk

/rt, is:

#(i, m) = d)~iB(i + c, d - c + 1),

(10)

where = (1 - a ) ( k + dnk)

(12) M where the random variable U has distribution F. Note that if U has a u n i f o r m d i s t r i b u t i o n , then, as a consequence of (12),

EPN(X) M

(k + cnk) and B is the beta function. See [14] for an analytically beneficial replacement of the difference equations (9) (for expectations E#~(x)) by certain partial differential equations. See also [41 for the exact solution of these difference equations. Simon suggested to call (8) the Yule distribution, but (8) and (10) are more appropriately and more commonly called the Yule-Simon distribution. Approximately at the same time, R.H. MacArthur [12] published a note on a model for the relative abundancy of species. His model of 'non-overlapping niches' was very simple: The environment is compared with a stick of unit length on which m - 1 points are thrown at random. The stick is broken at these points and lengths of the m resulting segments are proportional to the abundance of the m species. [-..] The expected length of r-th shortest interval is given by

1E m

1

i = 1 m -- i "}-

1

-

p(z)

(11)

so that the expected abundance of the r-th rarest species among m species and n individuals is rap(z). It appears (see [2]) that 'bird censuses' from tropical forests and many temperate regions fit this almost pet'fectly. The rate of p(z) in z is, certainly, very different from z -(l+q). Very interestingly, MacArthur suggests, then, to view divergence of a distribution from (11) as an indication and a measure of heterogeneity of a population. His idea was to 'use several sticks of very different size' to produce other curves. 462

'

÷

1 x(x + 1)'

which is exactly (3). By considering a t r i a n g u l a r a r r a y of independent (or exchangeable, as the authors have it) families of genera Mik with allocations of Nik species in each family i = 1 , . . . , k (the different 'sticks' of MacArthur [12]!), Hill and Woodroofe have shown that for combined statistics #Nk (X) : E i =kI #iNi (X) and

Mk = ~ =k 1 Mik one has

~N(X) -~ ~1 M

u(1 - u)x-lF(dx).

(13)

A. Rouault [20] considered a certain M a r k o v p r o cess as a model for formation of a text word by word and obtained the following expression for the limit of spectral statistics: # r ( x - #) a(z) = r(1 - # ) r ( x + 1)'

(14)

depending on a p a r a m e t e r / 3 . This Karlin-Rouault law appeared in a very different context in [10]: Let the unit interval be divided into two in the ratio a: 1 - a. Let each of these two be again subdivided in the same ratio, etc. At step q one obtains probabilities Pi, i = 1 , . . . , 2 q, of the form a k ( 1 - a) q-k for some k = 0 , . . . , q. One can think of filling a questionnaire with 'yes-no' questions at random with probability a for one of the answers in each question. Then the Pi are the probabilities of each particular answer to the questions or 'opinion'. It was proved t h a t if a ¢ 1/2, then #~ --+ oo but P-2-~-~ 0 (15) and that ,n(x)

- #-n

f~_~ as(x+g)e -~s dN(s) -~ F(x + 1) f~_~ aSZe - ~ dN(s)'

(16)

Z O L O T A R E V POLYNOMIALS with a = a/(1 - a ) and fl = fl(a,c) < 1 and with N(s) being a counting measure (concentrated on integers). With dN(s) replaced by the L e b e s g u e m e a s u r e ds and using the change of variables u = a s, the righthand side of (16) will coincide with (14). A possible heuristic interpretation of (15) is that if a ~ 1/2, then saying 'as many men as many minds' is incorrect -- though the number of 'minds' is infinitely large it is still asymptotically smaller than the number of 'men'. One of the present time (2000) characteristics of the study of Zipf's law is the increase of interest in the more general concept of 'distributions with a large number of rare events' (LNRE distributions), which was introduced and first systematically studied in [8] (partly reproduced in [9]). For a systematic application of LNRE distributions to the analysis of texts, see [i]. On the other hand, studies on specific laws continue. In a series of papers, Yu. Orlov (see, e.g., [16], [17], [18]) promoted the hypothesis that the presence of Zipf's law is the characteristic property of a complete, finished and well-balanced text. Though the hypothesis looks beautiful and attractive, for some readers the evidence may look thin. The problem of non-parametric estimation of the functions G and R through the expressions (3) and (5) provides another most natural example of an 'inverse problem', studied recently by C. Klaassen and R. Mnatsakanov [II].

In conclusion, the following quote from [14], whose views I share, is given: The form of Zipf's law [see (7)] is so striking and also so very different from any classical distribution of statistics that it is quite widely felt that it 'should' have a basically simple reason, possibly as 'weak' and general as the reasons which account for the role of Gaussian distribution. But, in fact, these laws turn out to be quite resistant to such an analysis. Thus, irrespective of any claim as to their practical importance, the 'explanation' of their role has long been one of the best defined and most conspicuous challenges to those interested in statistical laws of nature. References [1] BAAYEN, R.H.: Word frequency distribution, Kluwer Acad. Publ., 2000. [2] DAVIS, L.I., Aud. Field Notes 9 (1955), 425. [3] ESTOUP, J.B.: Gammes stenographiques, 4th ed., Paris, 1916. [4] GUNTHER, R., SCHAPIRO, B., AND WAGNER, P.: 'Physical complexity and Zipf's law', Internat. J. Theoret. Phys. 31, no. 3 (1999), 525-543. [5] HANLEY, L.M.: Word index to James Joyee's Ulysses, Madison, Wisc., 1937.

[6] HILL, B.M.: 'Zipf's law and prior distributions for the composition of a population', J. Amer. Statist. Assoc. 65 (1970), 1220-1232. [7] HILL, B.M., AND WOODROOFE, M.: 'Stronger forms of Zipf's law', J. Amer. Statist. Assoc. 70 (1975), 212-219. [8] I~HMALADZE, E.V.: 'The statistical analysis of large number of rare events', Techn. Rept. Dept. Math. Statist. CWI, Amsterdam M S - R 8 8 0 4 (1987). [9] KHMALADZE, E.V., AND CHITASHVILI, R.J.: 'Statistical analysis of large number of rate events and related problems', Trans. Tbilisi Math. Inst. 91 (1989), 196-245. [10] KHMALADZE, E.V., AND TSIGROSHVILI, Z.P.; 'On polynomial distributions with large number of rare events', Math. Methods Statist. 2, no. 3 (1993), 240-247. [11] KLAASSEN, C.A.J., AND MNATSAKANOV, R.M.: 'Consistent estimation of the structural distribution function', Scan& d. Statist. 27 (2000), 1-14. [12] MACARTHUR, R.H.: 'On relative abundancy of bird species', Proc. Nat. Acad. Sci. USA 43 (1957), 293-295. [13] MANDELBROT, B.: 'An information theory of the statistical structure of language', in W.E. JACKSON (ed.): Communication Theory, Acad. Press, 1953, pp. 503-512. [14] MANDELBROT, B.: 'A note on a class of skew distribution functions: Analysis and critque of a paper by H.A. Simon', Inform. ~4 Control 2 (1959), 90-99. [15] MANDELBROT, B.: 'Final note on a class of skew distribution functions: Analysis and critique of a model due to H.A. Simon', Inform. ~ Control 4 (1961), 198-216; 300-304. [16] ORLOV, Ju.K.: 'Dynamik der Haufigkeitsstrukturen', in H. GUITER AND M.V. AaAPOV (eds.): Studies on Zipf's law, Brockmeyer, Bochum, 1983, pp. 116-153. [17] ORLOV, Ju.K.: 'Ein Model der Haufigskeitstruktur des Vokabulars', in H. GUITER AND M.V. ARAPOV (eds.): Studies on Zipf's law, Brockmeyer, Bochum, 1983, pp. 154-233. [18] ORLOV, Ju.K., AND CHITASHVILI, R.Y.: 'On the statistical interpretation of Zipf's law', Bull. Acad. Sci. Georgia 109 (1983), 505-508. [19] PARETO, V.: Course d'economie politique, Lausanne and Paris, 1897. [20] ROUAULT,A.: 'Loi de Zipf et sources markoviennes', Ann. Inst. H. Poincard 14 (1978), 169-188. [21] SIMON, H.A.: 'On a class of skew distribution functions', Biometrika 42 (1955), 435-440. [22] SIMON, H.A.: 'Some further notes on a class of skew distribution functions', Inform. ~4 Control 3 (1960), 80-88. [23] SIMON, H.A.: 'Reply to 'final note' by Benoit Mandelbrot', Inform. ~ Control 4 (1961), 217-223. [24] WILLIS,J.G.: Age and area, Cambridge Univ. Press, 1922. [25] WOODROOFE, M., AND HILL, B.M.: 'On Zipf's law', J. Appl. Probab. 12, no. 3 (1975), 425-434. [26] YULE, G.U.: 'A mathematical theory of evolution, based on the conclusions of Dr.J.C. Willis, F.R.S.', Philos. Trans. Royal Soc. London Ser. B 213 (1924), 21-87. [27] ZIPF, G.K.: Human behaviour and the principles of least effort, Addison-Wesley, 1949. E. Khmaladze

MSC 1991: 92B15, 62Pxx, 60Exx, 62Exx, 92K20 ZOLOTAREV POLYNOMIALS, Zolotareffpolynomials, Solotareff polynomials - For each cr E R, the Zolotarev polynomial Z,~(x; or) is the unique solution of

463

ZOLOTAREV POLYNOMIALS the problem min m a x Ix n - o x n - 1 -t- a n - 2 x n-2 -1- " " " -t- ao [ . a0,...,an-2 - - l < x < l

That is, the Zolotarev polynomials of degree n are the polynomials whose leading two coefficients are fixed, and that deviate least from zero on the interval [-1, 1], in the uniform norm (cf. also U n i f o r m a p p r o x i m a t i o n ; P o l y n o m i a l l e a s t d e v i a t i n g f r o m zero). This problem was formulated and solved by E.I. Zolotarev. It is the first of four Zolotarev problems in a p p r o x i m a t i o n t h e o r y . Zolotarev's second problem asks for the monic polynomial p of degree n that deviates least from zero on the interval [-1, 1] while satisfying the extra condition p(~) = r] for some ~ > 1 and r] E R. The class of solutions (dependent on ~ and r]) are normalized Zolotarev polynomials. The third and fourth problems of Zolotarev deal with rational functions. In each of these problems the influence of P.L. Chebyshev can be seen, whose lectures Zolotarev attended, as these problems are all generalizations of problems of Chebyshev. T~, the monic Chebyshev polynomial of degree n, solves problem one but without any restriction on the coefficient of x n - l , cf. C h e b y s h e v p o l y n o m i als. (Prom symmetry considerations, T~(.) = Z~(.; 0).) It follows from the equi-oscillation theorem in the theory of best uniform approximation that Z~(x; ~) is uniquely determined by the fact that the coefficients of x ~ and X n - 1 a r e as given, and it attains its norm, alternately in sign, at least n times in [-1, 1]. Thus it is easily verified that for 0 < ~r < ( l / n ) tan2(Tr/2n),

2 coefficients, are fixed was considered by A. Akhiezer and by N.N. Meiman. For a generalization of the first Zolotarev problem to different (symmetric) domains and C C, see [7] and references therein. Zolotarev perfect splines and Zolotarev w-polynomials were introduced when generalizing the pointwise Markov inequalities to other spaces of functions, see e.g. [5], [81, and [2]. Due to fashions in transliteration, Zolotarev's name is sometimes written as Zolotareff or Solotareff. References [1] ACHIESER, A.: T h e o r y of a p p r o x i m a t i o n , F. Ungar, 1956. (Translated from the Russian.) [2] BAGDASAROV, S.K.: 'Zolotarev w-polynomials in WrH~[0, 1]', J. A p p r o x . Th. 90 (1997), 340-378. [3] CARLSON, B.C., AND TODD, J.: 'Zolotarev's first problem: the best approximation by polynomials of degree < n - 2 to x ~ - n r r x ~ - 1 in [-1, 1]', Aequat. M a t h . 26 (1983), 1-33. [4] ERDOS, P., AND SZEGO, G.: 'On a problem of I. Schur', A n n . M a t h . 43 (1942), 451-470. [5] I(ARLIN, S.: 'Oscillatory perfect splines and related extremal problems', in S. KARLIN, C.A. MICCHELLI, A. PINKUS, AND I.J. SCHOENBERG (eds.): S p l i n e F u n c t i o n s a n d A p p r o x i m a tion Theory, Acad. Press, 1976, pp. 371-460. [6] PASZKOWSI(I, S.: 'The theory of uniform approximation I. Non-asymptotic theoretical problems', Rozp. M a t . 26 (1962). [7] PEHERSTORFER, F., AND SCHIEFERMAYR, K.: 'Explicit generalized Zolotarev polynomials with complex coefficients II', E a s t J. A p p r o x . 3 (1997), 473-483. [8] PINKUS, A.: 'Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval', J. A p prox. Th. 23 (1978), 37-64. [9] VORONOVSKAJA,E.V.: 'The functional method and its applications', Transl. M a t h . M o n o g r a p h s 28 (1970).

Allan Pinkus MSC1991: 41-XX, 41A50

Zn(X;O" )

=

(1+ a)nT~

(which is what one gets when the Chebyshev polynomial is fixed at - 1 and pulled to the right as long as the resulting polynomial attains its norm alternately at n points in [-1, 1]). For a > (l/n)tan2(zr/2n) the Zolotarev polynomial is given in terms of elliptic integrals, see e.g. [1], [4], [6], [3] for more details. It suffices to consider cr > 0, since Zn(x;-or) = ( - 1 ) ~ Z n ( - x ; @. Zolotarev polynomials, suitably normalized, are also the class of extremal polynomials in the pointwise Markov inequalities, cf. M a r k o v i n e q u a l i t y . T h a t is, they maximize

for fixed ~ E ( - 1 , 1) and 1 < k < n - 1, over all polynomials of degree at most n which are bounded by 1 on [-1, 1]. (For I~1 > 1 the extremal polynomial is always the Chebyshev polynomial.) This problem is considered in detail in [9] (see especially the index). Various generalizations of Zolotarev polynomials can be found in the literature. The problem of finding the extremal polynomials when r coefficients, rather than just 464

ZONAL HARMONICS, zonal harmonic polynomials - Zonal harmic polynomials are spherical harmonic polynomials (cf. also S p h e r i c a l h a r m o n i c s ) that assume constant values on circles centred on an axis of symmetry. They characterize single-valued harmonic functions on simply-connected domains with rotational symmetry. To be specific, one introduces the s p h e r i c a l c o o r d i n a t e s (r, 0, 9) as xl = r sin 0 cos 9, x2 = r sin 0 sin 9, x3 = r c o s 0 , where ( x l , x 2 , x ~ ) E R 3. The zonal hatmonics Hn are the polynomial solutions of the L a p l a c e equation

o00 +

+

=0

that are axially symmetric (i.e. independent of the angle 9). They can be expressed in terms of L e g e n d r e p o l y n o m i a l s P~ of degree n, as Ha(r, 0) = rnP,~(cos 0) for n = 0, 1,..., and form a complete orthogonal set of functions in L2[D], where D: r _< r0. The H~ vanish on cones that divide a sphere centred at the origin into

ZORICH T H E O R E M n zones, hence the name zonal harmonics. The Hn are sometimes referred to as solid zonal harmonics and the P~ as surface zonal harmonics.

harmonics play an important role in axially symmetric problems in R 3 (see [1], [2], [3], [5]).

A p p l i c a t i o n s . Two types of applications arise in classical p o t e n t i a l t h e o r y (see [4], [6], [7]). In the first, one determines the potential in a sphere from its boundary values H(ro, 0). By specifying appropriate regularity conditions, the orthogonality of the Legendre polynomials is used to expand H(ro,O) as the Fourier-Legendre series Y-~=0 a~r~P~ (cos 0). The potential in the sphere is recovered as H(r, 0). The exterior boundary value problem is formulated by means of the K e l v i n t r a n s f o r m a t i o n H(r,O) --+ ( 1 / r ) H ( 1 / r 2,0). The potential between two concentric spheres is determined by combining solutions of the interior and the exterior problems. In the second, one determines the potential at points in space from its values on a segment of the symmetry axis. The solution relies on the fact that along this axis the zonal harmonics H~(r,O) = r '~, n = O, 1,.... Thus, if H(r,O) = ~°°= o a~H~(r,O), then H(r,O) = ~n=oa~H~(r,O) for r < r0, where r0 is the radius of convergence of the Taylor series.

[1] BEGHEa, H., AND GILBERT, R.P.: Transmutations, transformations and kernel functions, Vol. 58-59 of Monographs and Surveys in Pure and Applied Math., Pitman, 1992. [2] BERGMAN, S.: Integral operators in the theory of linear partim differential equations, Springer, 1963. [3] GILBERT, R.P.: Function theoretic methods in partial differential equations, Vol. 54 of Math. in Sci. and Engin., Acad. Press, 1969. [4] KELLOGG, O.D.: Foundations of potential theory, F. Ungar, 1929. [5] KRACHT, M., AND ~[REYSZIG, E.: Methods of complex analysis in partial differential equations with applications, Wiley/Interscience, 1988. [6] MACMILLAN, W.D.: The theory of the potential, Dover, 1958. [7] MORSE, P.M., AND FESHBACH, H.: Methods of theoretical physics, Vol. 1-2, McGraw-Hill, 1953. [8] SZEG6, G.: 'On the singularities of real zonal harmonic series', J. Rat. Mech. Anal. 3 (1954), 561-564.

R e l a t i o n w i t h a n a l y t i c f u n c t i o n s . There are many connections between the properties of the potentials H and those of analytic functions f of a complex variable (cf. also A n a l y t i c f u n c t i o n ; H a r m o n i c f u n c t i o n ) . One such connection, related to the previous example, concerns singularities and uses the generating function for zonal harmonics to construct reciprocal integral transforms connecting H with f . The following fact is immediate (see [3], [8]). Let {a~}~__0 be a sequence of real constants for which l i m s u p ~ _ ~ fan] 1/n = 1. Consider the associated harmonic and analytic functions H(r, O) = E~°°=oanH~(r, O) and f ( z ) = ~7~=o a~z '~, which are regular for r = Izl < 1. Then the boundary point (1, 00) is a singularity of H(r, O) if and only if the boundary point z = exp(i00) is a singularity of f(z). Thus, the singularities of solutions of a singular partial differential equation are characterized in terms of those of associated analytic functions and vice versa. From the 1950s onwards, an extensive literature has developed using integral transform methods to study solutions of large classes of multi-variable partial differential equations. The analysis is based on the theory of analytic and harmonic functions in several variables. Zonal

References

Peter A. McCoy MSC1991: 33C55, 31B05 In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space (see also Q u a s i - r e g u l a r m a p p i n g ) : A locally homeomorphic quasi-regular mapping f : R n -+ R n, n _> 3, is, in fact, a h o m e o m o r p h i s m of R n. This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for n = 2. In 1971, O. Martio, S. Rickman and J. V/iisglg proved a stronger quantitative result: For n > 3 and K > 1 there exists a number ~(n, K ) E (0, 1), the radius of injectivity, such that every locally injective K-quasi-regular mapping f: B n -+ R ~, where B n = B~(1) and B~(r) = {x C R n : Ixl < r}, for r > 0, is injective in B ~ ( ¢ ( n , K ) ) . ZORICH

THEOREM

-

References

[1] MARTIO, O., AND SEBRO, U.: 'Universal radius of injectivity for locally quasiconformal mappings', Israel J. Math. 29 (1978), 17-23. [2] RICKMAN,S.: Quasiregular mappings, Vol. 26 of Ergeb. Math. Grenzgeb., Springer, 1993. [3] ZORICH,V.A.: 'The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems', in M. VUORINEN (ed.): Quasiconformal Space Mappings, Vol. 1508 of Lecture Notes in Mathematics, 1992, pp. 132-148. M. Vuorinen MSC 1991: 30C20, 26Bxx

465

SUBJECT INDEX

.0 see: Conwaygroup-O grammar s e e : type- --

1/4-Cantor set [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) 1/2 fermion see: spin- -1 / n surgery [57M25] (see: Positive link) 1 see: isomorphism in codimension --; surjectivity in codimension --

[57M25]

[47H17]

(see: Positive link)

(see: Approximation solvability)

4-groupsee:

Klein --

A-sequence see:

(5,2) positivetoms knot [57M25] (see: Positive llnk) 6-transpositionpropertyof the monster [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) 62-knot

[57M25] (see: Positive link)

l-co-connected space

90theorem

[55Pxx, 55P15, 55U35] (see: Algebraic homotopy) 1-cycle see: relative R- --

~1 additive topological space [54610] (see: P-space)

1-cycles s e e : tive R- - -

see:

Hilbert - -

~¢/2-torus [53C421 (see: Willmore functional)

A s-abundant number primitive unitary - -

see:

primitive

[55Pxx, 55P15, 55U35]

2.4-1-dimensional Harry Dym equation

[35Q53, 58F07] (see: Harry Dym equation)

26-dimensional string [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, glT30, 81T40] (see: Vertex operator algebra) 2D software s e e : Global Manifolds -3 bifurcation s e e : codimensioe- -3-colourabihty problem

[68Q151 Average-case computational complexity) (see:

sufficiently-

(see: Baumslag-Solitar group) 3-move

[57P25] (see: Montesinos-Nakanishi conjecture) 3-sphere see: 3-string braid

homology--

[57Mxx] (see: Fibonaeci manifold) 4-ball genus of a knot

see:

[11Axx] (see: Abundant number) * - a l g e b r a see:

uniformly closed - -

*-Autonomous category (18D10, 18D15) (refers to: Category; Closed monoidal category; Functor) *-autonomous category

[18D10, 18D15] (see: *-Autonomous category) A-anti-symmetric set

[46E25, 54C35] (see: Bishop theorem) A-anti-symmetricset see: partially -A arithmeticalsemi-group s e e : axiom- -A # arithmetiealsemi-group s e e : axiom-

A k curve singularities

[14H20] (see: Tacnode) A-function

[11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) a priori-condition belief function

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) A-process see: van dot Corput --

A-proper [47H17] (see: Approximation solvability) A-proper mapping

see:

Abelian functions see: algebraicindependence of values of - - ; transcendence of values of - - ; transcendence t h e o r y of - Abelian group s e e : finite - - ; meta- - -

Abelian groups on finite - -

o~-non-deficient number

[57M25] (see: Jaeger composition product)

[62Jxx, 62Mxx] (see: Cox regression model) Abel functional equation [39B05, 39B 12] (see: Schr6der Iunetional equation)

o~-favourable topological ~Tmce

2-design s e e :

Hadamard - -

AaIen multiplicative intensity model

Abelian group invariant of links [57M25] (see: Fox n-colouring)

(x-favourable topological space w e a k l y --

2-1abelling of a graph

unique - -

c~-favourable space [26A15, 54C05] (see: Namioka space)

(see: Algebraic homotopy) 2-coeycle see: ChevalIey - -

3-design see: Hadamard - 3-manifold s e e : nice - - ; large - 3-manifold group [05C25, 2 0 F x x , 2 0 F 3 2 ]

--;

[54E52] (see: Banach-Mazur game)

2-co-connected space

A-solvability s e e :

Abelian differential [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Abelian function [11J85] (see: Gel'fond-Sehneider method)

1-median problem

1-unrectifiabie set s e e : example of a purely -15th problem s e e : Hilbert - 1D software s e e : Global Manifolds --

[47H171 (see: Approximation solvability)

Abelian B a u m s l a g - S o l i t a r group meta- - -

numerically equivalent r e i n .

[90B85] (see: Fermat-Torricelll problem) 1-rectifiable set see: example of a --

weak--

A-solvability

see:

fundamentaltheorem

Abelian integral [11J85] (see: Gel'fond-Schneider method) Abelian monopole

[81V10] (see: Dirac monopole) Abelian p-extension s e e : maximal - - ; unramified - Abelian p - g r o u p s e e : e l e m e n t a r y - -

Abelian variety [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compaetification) Abelian variety see: semi- - aberration s e e : coma - - ; distortion --

aberration functionof optics [33C50, 78A05] (see: Zernike polynomials) aberrations s e e : primary - -

diffraction theory of - - ;

Ablowitz-Kaup-Newell-Segur hierarchy

[22E65, 22E70, 35Q53, 35Q58, 5817071 (see: AKNS-hierarchy) Ablowitz-Ladik equations [14Jxx, 35A25, 35Q53, 57R57] (see: Whltham equations) absolute continuity of measures

[28-XX] (see: Absolutely continuous measures) absolute Galois group over Q [I 1R32] (see: Shafarevlch conjecture) absolute Galois group over Qab [11R32]

(see: Shafarevich conjecture) absolute retract [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) absotutevalue see:

p-adic - -

absolute value on a number field

[12J10, 12J20, 13A18, 16W60] (see: S-integer)

Absolutely continuous invariant measure (28Dxx, 54H20, 58F1 I, 58F13) (refers to: Absolutely eontinuous measures; Accumulation point; Chaos; Compactness; Dirac distribution; Dynamical system; Ergodic theorem; Haar measure; Invariant measure; Lebesgue measure; Measure; Shift dynamical system; Strange attractor; Topological group) Absolutely continuous measures (28-xx) (referred to in: Absolutely continuous invariant measure; Sobolevinner product) (refers to: Absolute continuity; Cantor set; Haar measure; Integrable function; Lebesgue measure; Measurable space; Measure; RadonNikod~m theorem; Topological group) absolutely continuous with respect to a given measure s e e : measure, --

absolutely free algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic; Algebraic logic) Abstract algebraic logic (03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35) (referred to in: Algebraic logic) (refers to: Algebraic systems, variety of; Boolean algebra; Equational logic; Gentzen formal system; Heyting formal system; Horn clauses, theory of; Intermediate logic; Manyvalued logic; Modal logic; Modus ponens; Permissible law (inference); Propositional calculus; Propositional connective; Quasi-variety; Universal algebra) abstract algebraic logic [03Gxx] (see: Algebraic logic) abstract algebraic logic semantics-based - -

see:

logistic - - ;

Abstract analytic number theory (llNxx, 11N32, 11N45, 11N80) (referred to in: Abstract prime number theory) (refers to: Abelian group; Abstract prime number theory; Algebraic number; Analytie number theory; 467

ABSTRACT ANALYTICNUMBER THEORY

Associative rings and algebras; Category; Cyclic group; de la ValldePoussin theorem; Finite field; Globally symmetric Riemannian space; Irreducible polynomial; Lie algebra; Mdbius function; p-group; Pseudometric space; Ring; Semi-group; Semi-simple ring; Topological space; Zeta-function) abstract arithmetical function [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) abstract inverse prime number theorem [llNxx, 11N32, 11N45] (see: Abstract prime number theory) abstract prime element theorem [llNxx, 11N32, 11N45, 11N801 (see: Abstract analytic number theory) abstract prime number theorem [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory; Abstract prime number theory) abstract prime number theorem [llNxx, 11N32, 11N45] (see: Abstract prime number theory) abstract prime number theorem verse additive --

see:

in-

Abstract prime number theory (11Nxx, 11N32, 11N45) (referred to in: Abstract analytic number theory) (refers to: Abstract analytic number theory; Algebraic function; Algebraic number; de la Vall~e-Poussin theorem; Finite field; Graph; Ideal; Irreducible polynomial; Mdbius function; Polyhedron; Ring; Semigroup) abstract programming [90Cxx] (see: Fritz John condition) Abundant number (11Axx) (refers to: Divisor; Number of divisors; Perfect number; Prime number; Totient function) abundant number [11Axx] (see: Abundant number) abundant number [11Axx] (see: Abundant number) abundant number see: highly - - ; primitive a- - - ; primitive - - ; primitive unitary eL--abundant numbers see: Erdds theorem on--

AC unification [06Exx, 68T15] (see: Rabbius equation) acceleration in spatial form [73Bxx, 76Axx] (see: Material derivative method) acceleration in spatial form for --

see:

formula

acceleration of a particle [73Bxx, 76Axx] (see: Material derivative method) acceleration of a particle [73Bxx, 76Axx] (see: Material derivative method) Acceptance-rejection method (62D05) (refers to: Cauchy distribution; Density era probability distribution; Distribution; Laplace distribution; Normal distribution; Sample; Student distribution) accepted input in a decision problem [03D15, 68Q15] (see: Computational complexity classes) access machine see: quantum random - - ; random - -

Accessibility for groups (20E22, 20Jxx, 57Mxx) 468

(refers to: Cohomologieal dimension; Finitely-generated group; Free product; Group; Group without torsion; HNN-extension; Hyperbolic group; Kneser theorem; Three-dimensional manifold) accessibility of finitely-generated groups see: Wallconjecture on - accessibilitytheorem see: Dunwoody --

accessible group [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) Acnode (14Hxx) (refers to: Algebraic curve)

acylindrical graph of groups see: decomposition as a k - - addition see: Nim -addition decomposable measure see: pseudo- - -

addition theoremfor the exponentialfunction [11J85] (see: Gel'fond-Schneider method) additive abstract prime number theorem see: inverse -additive additive set function see: finitely --

additive arithmetical semi-group [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) de-

see:

-

Additive basis (11Pxx) additive basis [1 lPxx] (see: Additive basis) additive basis see: asymptotic

- - ; minimal asymptotic - - ; order of an - - ; thin - -

Additive basis for the natural numbers [1 IPxx] (see: Additive basis) additive Cauchy equation [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) additive function see: additive measure see:

approximately-Non- - - ; o-- - -

additive quantum code [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) additive set function see: - - ; non- - - ; null- - additive topological space

finitely additive see:

~1"

--

additivity-excision of the Brouwer degree [55M25] (see: Brauwer degree) adble [11F03, 11F70] (see: Selberg conjecture) adelic group [1 IF25, 11F60] (see: Hecke operator) adequateGentzen system

adiabatic limit

see:

see:

adjoint Baker-Akhiezer function [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations)

active constraint [90Cxx] (see: Fritz John condition) acyclic polynomial of a graph [05Cxx, 05D15] (see: Matching polynomial of a graph) acyclic space [54H15, 55R35, 57S17] (see: Smith theory of group actions)

-

adie absolute value see: p - - adic L-function see: p - - adic Weierstrass preparation theorem p--

AKNS-equations

adjoining grammar see: tree - adjoining grammar parser s e e : tree - -

action see: fixed-point-freegroup--; fixedpoint set of a group - - ; Polyakov extrinsic -- ; spherically transitive group - - ; YangMills-Higgs - action of a semi-group see: right - action of the Steenrod algebra see: unstable - action on a rooted tree see: group - actions see: Smith theory of group --

additive arithmetical semi-group gree on an

[22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hlerarchy)

[53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) adiabatic limit [81Txx, 81T05] (see: Massless field)

fury - -

adjoint matrix see; Hermitian - adjoint operator see: essentially self- - - ; self- - - ; triangular model of a non-self- - adjoint orbit see: co- - -

Adler-Manin residue [35Sxx, 46Lxx, 47Axx] (see: Wodzicki residue) admissibility condition for a reconstruction formula for the continuous wavelet transform [42Cxx] (see: Daubechies wavelets) admissible measure [33C45, 33Exx, 46E35] (see: Sobolev inner product) admittance matrix

see:

node- - -

affine coordinate ring of an algebraic curve [12F10, 14H30, 201306, 20E22] (see: Chasles-Cayley-Brillformula) Affine design (05B30) (referred to in: Net (in finite geometry)) (refers to: Afline space; Galois field; Hadamard matrix; Net (in finite geometry); Plane; Primitive group of permutations; Tactical configuration) affine design [05B30] (see: Affine design) affine Grassmannian [52A35] (see: Geometric transversal theory) affme Kac-Moody algebra [ l l F l l , 17B 10, 17B65, 17B67, 20D08, 81R10, 81T30] (see: Vertex operator) affine Kac-Moody Lie algebra [llF11, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) affme Lie algebra [IlFll, 17B10, 17B65, 17B67, 20]308, 81R10, 81T30] (see: Vertex operator) affine Lie algebra twisted --

see:

simply-laced - - ;

affine plane [05B30] (see: Affine design) affine plane [05Bxx] (see: Net (in finite geometry)) affine plane see: classical --; finite -affine resolvable design [05BJ0] (see: Affine design) AGM method [26Dxx, 65D20] (see: Arithmetic-geometric mean process) AGM process [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Akhiezer function Baker- --

see:

adjoint Baker- - - ;

Akhiezer-Kac formula [42A16, 47B35] (see: Szeg6 limit theorems) AKNS-equations

see:

stationary --

AKNS-hierarchy (22E65, 22E70, 35Q53, 35Q58, 58F07) (refers to: Bundle; Cartan subalgebra; Connections on a manifold; Differential equation, partial, discontinuous initial (boundary) conditions; Fundamental system of solutions; Hamiltonlan system; Homogeneous space; Hyper-elliptic curve; KacMoody algebra; Korteweg-de Vries equation; KP-equatlon; Lie algebra; Poisson brackets; Regular element; Soliton) AKNS-hierarchy see: Lax equations of the - AKNS-potentialsee: algebro-geometric--

Aleksandrov problem for isometric mappings (54E35) (refers to: Homeomorphism; Metric space) Alexander-Conway polynomial (57M25) (referred to in: Conway polynomial; Jones-Conway polynomial) (refers to: Alexander invariants; Conway skein triple; Knot theory) Alexander duality theorem [55P25] (see: Spanier-Whltehead duality) Alexander module [57M25] (see: Fox n-colouring) Alexander polynomial [57M25] (see: Alexander-Conway polynomial; Positive link; Rotor) Alexander theorem on braids (57M25) (referred to in: Jones-Conway polynomial) (refers to: Braid theory; Knot theory; Link) algebra see: absolutely free - - ; affine Kac-Moody - - ; affine Kac-Moody Lie - - ; affine Lie - - ; angular momentum - - ; Artin - - ; Auslander - - ; Azumaya - - ; B a n a c h - J o r d a n - - ; B a x t e r - - ; Beurling - - ; BM- - - ; B o r c h e r d s - - ; Borcherds Kac-Moody - - ; Borcherds Lie - - ; BoseM e s n e r - - ; Calkin - - ; Cellular - - ; character of a Borcherds - - ; character on a Banach - - ; charge of a Borcherds - - ; chiral - - ; co-crossed product C * - - ; co-invariant - ; Coherent - - ; Colombeau generalized function - - ; complex function - - ; c o n v o l u t i o n - - ; C o n w a y - - ; Conze-Lesigne - - ; corona - - ; Corona O * - - - ; corona of a C * - - - ; crossed product C * - - - ; cyclotomic - - ; cylindric - - ; Dirac - - ; disc - - ; Douglas - - ; dual space of the Beurling - - ; e-perturbation of e Banach - - ; e-metric perturbation of a Banach - - ; c-perturbation of a Banach - - ; effective - - ; Egorov generalized function - - ; Ext monoid of a C * - - - ; extension of a separable ( 7 * - - - ; Eymard - - ; factor of a von Neumann - - ; factor representation of a J B . - - ; Fig&Talamanca - - ; Fig&-Talamanca-Herz - - ; finitely-generated k - - - ; flexible - - ; f o r m u l a - - ; F o u r i e r - - ; Fourier-Stie[tjes - - ; full C,'*- - - ; function - - ; generalized function - - ; generalized Kac-Moody - - ; Griess - - ; H e c k e - - ; Heisenberg - - ; Heisenberg Lie - - ; h e r e d i t a r y - - ; Heyting - - ; homogeneous Heisenberg - - ; imaginary root of a Borcherds - - ; imaginary simple root of a Borcherds - - ; i n c i d e n c e - - ; indicator - - ; infinitedimensional Grassmann - - ; E-ternary

--; d*- --; JB- --; JB *- - ; JBW-

ALMOST CONTINUITY

- - ; J C - - - ; Jordan-Banach - - ; Jordan O * - - - ; Kac-Moody - - ; K&hler differential on a k- - ; local-global principle in commutative - - ; locally spectrally associative - - ; logical - - ; logistic path from logic to - - ; Long H-dimodule Azumaya - - ; Lorentzian Kac-Moody - - ; meaning - - ; M 6 b i u s - - ; m o n o d i c - - ; Monster Lie - - ; n-ary representable cyJindric - - ; Noetherian Banach - - ; n o n - t y p e - / O * - - ; nowhere-dense generalized function - - ; oetonian - - ; oetonion - - ; Okubo - - ; partial C o n w a y - - ; path - - ; P a u l i - - ; polyadic - - ; primitive Banach-Jordan - - ; projective C * - - ; quasi-hereditary - - ; quasi-polyadic - - ; quasi-tilted - - ; quatern i o n - - ; quaternion division - - ; queer Lie super- - - ; real function - - ; real root of a Borcherds - - ; real simple root of a Borcherds -- ; Rees -- ; relation - - ; relative W e l l - - ; repetitive - - ; representable cylindric - - ; r e p r e s e n t a t i o n - f i n i t e - - ; representation-tame - - ; Robbins - - ; Rosinger nowhere-dense generalized function - - ; rule-based path from logic to - - ; o--unital - - ; Schur --; setf-injective - - ; semantical path from logic to - - ; semi-primitive Jordan - - ; semi-simple Jordan - - ; simply-laced affine Lie - - ; smooth - - ; spectral properties of the Beufling - - ; spectrum of a O * - - - ; spectrum of an element in a Banach-Jordan --; stable Banaeh - - ; standard - - ; standard basis of a cellular - - ; standard basis of a coherent - - ; standard Baxter - - ; standard embedding Lie - - ; structurable - - ; structure constants of a coherent - - ; symmetfizable Borcherds - - ; symmetrizable Kac-Moody - - ; synthesis problem for the Beurling - - ; tame - - ; tame domestic - - ; Tilted - - ; Toeplitz - - ; Toeplitz O * - - - ; topological - - ; truncated Well - - ; twisted affine Lie -- ; uniformly closed O * - - - ; uniformly closed * - - - ; universal Borcherds - - ; universal partial Conway - - ; unstable action of the Steenrod - - ; vertex - - ; Vertex o p e r a t o r - - ; Weft algebra of a Lie - - ; Wiener - - ; wild - algebra and quasi-symmetric functions see: Leibniz-Hopf - algebra associated with a vector space see: Lie - algebra closed under conjugation see: function -algebra (in algebraic eombinatorics) see: Cellular -algebrain tilting theory see: canonical - algebra of a Banach-Jordan pair see: local - algebra of a group see: Burnside -algebra of a Lie algebra see: Well -algebra of a logical matrix see: underlying -algebra of a quiver see: path- -algebraof a system see: quantum - algebra of compact operators see: O * - -algebra of Fourier series with summable majorant of coefficients see: Beurling - -

algebra of functions of compact support [22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10] (see: Fourier-Stieltjes algebra) algebra of multipliers [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) algebra of n-ary relations [03Gxx] (see: Algebraic logic) algebra of retarded distributions [46FI0] (see: Multiplication of distributions) algebraof type H see: tilted - algebraoperator vessel see: Lie -algebra relations see: V i r a s o r o - algebra separating the points of a sat see: function --

algebra with a unit see: invertible element in a Jordan - algebra with operators see: Boolean - -

algebraic combinatorics [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mfbius inversion) algebraic combinatorics) see: algebra (in - -

Cellular

algebraic convergencerate [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) algebraic curve see: affine coordinate ring of an - - ; centre on an - - ; first neighbourhood of a point on an - - ; function field of ' an - - ; local ring of a point on an -- ; plane - - ; second neighbourhood of a point on an - - ; simple point on an - - ; singular point on an - algebraic curves see: Dedekind formula for - - ; Riemann approach to - -

algebraic forest [05Exxl (see: Cellular algebra) algebraic function field [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) algebraic group see: Q-roots of a semisimple - ; R-roots of a semi-simple - -

algebraic hierarchy [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraic hierarchy [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Algebraic homolopy (55Pxx, 55P15, 55U35) (refers to: Algebraic K-theory; Classifying space; Crossed complex; Crossed module; CW-complex; Functor; Fundamental group; Groupoid; HomologieaI algebra; Homotopy) algebraic independence see: of--

measure

algebraic independence of values of AbelJan functions [llJ81] (see: Schneider method) algebraic independence of values of elliptic functions [llJ81] (see: Schneider method) algebraic independence of values of exponential functions [llJ81] (see: Schneider method) algebraic index of a link [57M25] (see: Algebraic tangles) algebraic integer see: non-reciprocal - algebraic integers see: ring of -algebraic K - t h e o r y see: excision in --

algebraic kernel method [65Lxx] (see: Tau method) algebraic link see:

n- --; ( n,h )- --

Algebraic logic (03Gxx) (refers to: Abstract algebraic logic; Boolean algebra; Free algebra; Gfdel incompleteness theorem; Mathematical logic; Ordinal number; Propositional calculus; Quasivariety; Set theory; Stone space) algebraic logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic; Algebraic logic)

algebraiclogic see: Abstract - - ; completeness theorem in - - ; concrete - - ; equivalence theorems in - - ; logistic abstract - - ; semantics-based abstract - a[gebraicm-tangle see: n - - - ; ( n , k ) - - -

algebraic models of homotopy types [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) algebraic multiplicity of an eigenvalue [47A10, 47B06] (see: Spectral theory of compact operators) algebraic number see: an--

Mahler measure of

algebraic number field [llNxx, 11N32, 11N45, llN80, 11R29] (see: Abstract analytic number theory; Odlyzko bounds) algebraic number field see: discriminanf of an - - ; L-function of an - - ; norm of a prime ideal in an - - ; signature of an - - ; units of an - algebraic number fields see: prime ideal of degree one in an extension of - - ; splitting prime ideal of an extension of -algebraic numbers see: linear independence of logarithms of - -

algebraic quantum field theory [35Sxx, 46Lxx, 47Axx] (see: Wodzicki residue) algebraic semantics [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraic semantics see:

equivalent - -

Algebraic tangles (57M25) (referred to in: Tangle) (refers to: Tangle) algebraictangles see:

n- --; (n,k)-

-

algebraic tangles in the sense of Conway [57M25] (see: Algebraic tangles) algebraic variety see: normal - - ; Qfactorial - ; singular point on an - ; terminal -

algebraic variety with terminal singularity [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) atgebraicallyclosed field see:

pseudo- - -

algebraically irreducible representation of a Gel'fond quantale [03G25, 06D99] (see: Quantale) algebraically strong homomorphism of Gel'fand quantales [03G25, 06D99] (see: Quantale) algebraizable deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraizable deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraizable deductive system see: finitely - - ; second-order finitely - - ; strongly finitely - - ; weakly - algebraizable deductive systems see: characterization theorem of --

algebraizable general semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraizable general semantical system see: finitely - -

algebraizable logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) algebraizablelogic see: second-order-algebraizable semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35]

(see: Abstract algebraic logic) atgebraizable semantical system see: finitely - algebras see: amalgamation property of the variety of Boolean - - ; associativity in vertex - - ; Automatic continuity for Banach - - ; Bishop theorem for real function - - ; Brauer theorem on splitting fields for group - - ; categorical characterization of Azumaya - - ; characterization of Borcherds - - ; Colombeau g e n e r a l i z e d function - - ; commutativity in vertex - - ; denominator identity for Borcherds - - ; ~isometry of Banach - - ; e-isomorphism of Banaeh - - ; epimorphism over a class of - - ; equational logic of a class of - - ; equational logic of Boolean - - ; extension of O * - - - ; Generalized function - - ; homomorphism Ko-extensible over a class of --; Jacobi identity for vertex - - ; Multipliers of O * - - - ; short exact sequence of C * - - ; structure theorem for Boreherds Lie - - ; variety of Boolean - - ; variety of Heyting - - ; variety of monadic - - ; variety of universal - algebras of a logic see: meaning - - ; representable - -

algebro-geometric AKNS-potential [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) algorithm see: Alternating - - ; anytime - - ; average-case behaviour of an - - ; Brent-Salamin - - ; depth-first - - ; Dijkstra - - ; Dijkstra shortest-path--; E a r l e y - - ; efficiency of an - - ; Gauss-Salamin - - ; Grover - - ; Grover search - - ; Lagrange arithmetic-geometric mean - - ; learning - - ; LLL- - - ; M a r k o v - - ; node-labeling greedy - - ; projection pursuit - - ; quantum - - ; quantum factoring - - ; Rutishauser q d - - ; Salamin-Brent - - ; satisfiability - - ; Sehur continued-fraction-like--; Shor factoring -- ; Shot quantum - - ; simplex - algorithm for'rr see: quadratic-algorithm for neural networks see: backpropagation - algorithm of linear programming see: dual - -

algorithmic geometric transversal theory [52A35] (see: Geometric transversal theory) algorithms see: Shafer - -

Hermann - - ; Shenoy-

aIiasing error [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) Allard regularity theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Ailard regularity theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) allele see:

gone frequency of an - -

Allison-Hein triple system (17A40) (referred to in: FreudenthaI-Kantor triple system) (refers to: Freudenthal-Kantor triple system; Jordan algebra; Lie algebra; Module; Non-associative rings and algebras; Vector space) Allison-Hein triple system [17A40] (see: Allison-Hein triple system) alloy see: free energy of a binary - - ; phase transition in a binary - -

Almost continuity (54C08) (refers to: Closed-graph theorem; Continuous function; Discontinuity point; Discontinuous function; Lipsebitz condition; Riemann integral; Separate and joint continuity) 469

ALMOST CONVEX GROUP PRESENTATION

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

almost convex group presentation [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) almost convexmetric space [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) almost positive link

see:

rrt- --

Almost-split sequence (16G70) (referred to in: Riedtmann classification) (refers to: Representation of an associative algebra; Riedtmann classification) almost-split sequences see: Reiten theorem on --

Auslander-

Alternating algorithm (46Cxx) (refers to: Banach space; Best approximation; Diliberto-Straus algorithm; HUbert space; Orthogonal projector; Reflexive space; Smooth space) alternating groups see: Projective representations of symmetric end - alternating links see: Tait conjectures on --

alternating Turing machine [03D15, 68Q15] (see: Computational complexity classes) alternation theorem see: ChandraKozen-Stockmeyer - alternative see: Fredholm - - ; Gordan theorem of the - -

alternative in linear inequalities [15A39, 90C05] (see:Motzkin transposition theorem) alternativesforvectorinequalitiessee:

the-

orem of --

amalgam space [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) amalgamated product [20F05, 20F06, 20F32] (see: HNN-extension) amalgamation property [03Gxx] (see: Algebraic logic) amalgamation property super- - -

see:

strong - - ;

amalgamation property of the variety of Boolean algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F351 (see: Abstract algebraic logic) ambiguity in a natural language [68S05] (see: Natural language processing) ambiguityin a natural language - - ; structural - -

see:

sense

amenable group [22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Figh-Talamanea algebra; Fourier algebra; Fourier-Stieltjes algebra) American option [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula; Option pricing) Amitsur theorem on function fields of genus zero [14Axx] (see: Zariskl problem on field extensions) Amp~recapacity Amp6re operator

see: see:

Monge--Monge---

amphicheiral knot [57M25] (see: Positive link) amplitude see: s c a t t e r i n g - analogical reasoning [68T05] (see: Machine learning) analogue of the Baker finite basis theorem 470

analogue of the Denjoy-Wolff theorem see: Fan-analogue of the E u l e r - P o i s e o n - D a r b o u x o p erator see: q-difference - analogue of the Schur inequality see: permanental - analogy see: derivatienal - - ; relevancebased - - ; similarity-based - analysis see: approximate unit in harmonic - - ; categorical variable in covariance - - ; completely crossed factors in covariance - - ; crossed factors in coveriance - - ; crossing factors in coveriance - - ; dependent variable in regression - - ; fundamental identity of sequential - - ; incompletely factors in eovariance - - ; independent variable in regression - - ; infinitesimal - - ; linear stability -- ; Mallievin theorem in harmonic - - ; multi-scale - - ; natural language -- ; nested factors in covariance -- ; nesting factors in covariance -- ; non-smooth - - ; partly crossed factors in covariance - - ; phase-space - - ; profile - - ; qualitative factors in cevariance - - ; quantitative factors in ooverianee - - ; regression - - ; s t a b i l i t y - - ; survival - - ; text - analysis for the tau method see: error - analysis in texts see: word - -

analysis of covariance [62Jxxl (see: ANOVA) analysis of variance [62Jxx] (see: ANOVA) analysis of variance see: generalized multivariate -- ; multivariate - analysis software see: statistica{ - -

analytic disc [32E201 (see: Polynomial convexity) analytic disc [32E20] (see: Polynomial convexity) analytic exponential sum [11L07] (see: Exponential sum estimates) analytic function [31B05, 33C55] (see: Zonal harmonies) analytic functions see: transcendence properties of values of - analytic functions of bounded mean oscillation see: space of - analytic functions of vanishing mean oscillation see: space of - -

analytic isomorphismrelation [03C15, 03C45, 03E15] (see: Vaught conjecture) analytic number theory

see:

Abstract - -

analytic properties of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) analytic regularization [46F10] (see: Multiplication of distributions) analytic relation [03C15, 03C45, 03E15] (see: Vaught conjecture) analytic space

see:

normal - - ; rigid - -

analytic structure [32E20] (see: Polynomial convexity) analytic structure on a polynomialhull [32E20] (see: Polynomial convexity) analytic subsets see: Lelong theorem on-analytic torsion see: R a y - S i n g e r - -

analytical learning [6gT05] (see: Machine learning) anchor ring [53C42]

(see: Willmore functional) Anderson-Wengertheorem [52A35] (see: Geometric transversal theory) angle

see:

Brocard - -

angle between Hilbert subspaces [46Cxx] (see: Alternating algorithm) angle between subspaces [46Cxx] (see: Alternating algorithm) angular derivative [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath6odory theorem) angular momentum [37J15, 53D20, 70H33] (see: Momentum mapping) angular momentum algebra [15A66, 81R05, 81R25] (see: Paull algebra) angular momentumquantumnumber [34B24, 34L40] (see: Sturm-Liouville theory) double - anomaly see: determinant - - ; multiplicative - - ; Non-commutative - - ; Wodzicki formula for multiplicetive - anomaly for zeta-function regularization s e e : non-commutative - a n n i h i l a t o r see:

ANOVA (62Jxx) (referred to in: GMANOVA;MANOVA) (refers to: Confidence interval; Least squares, method of; Maximumlikelihood method; Most-powerful test; Normal distribution; Point estimator; Statistical estimation; Statistical hypotheses, verification of; Unbiased estimator) ANOVA [62Jxx] (see: ANOVA) A N O V A see: canonical form for - A N O V A model see: mixed - -

anti-BRST transformations [81Qxx, 81Sxx, 8IT13] (see: Faddeev-Popov ghost) anti-chain in a partially ordered set [05D05, 06A07] (see: Sperner property) anti-commutative relation [15A66, 81R05, 81R25] (see: Pauli algebra) anti-ghost field [8lQxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) anti-holomorphic function [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) Anti-Lie triple system (17A40, I7B60) (refers to: Lie algebra; Lie triple system; Superalgebra; Vector space) anti-Lie triple system [17A40, 17B60] (see: Anti-Lie triple system) anti-Pasch Steiner triple system [05B07, 05B30] (see: Pasch configuration) anti-Pasch STS [05B07, 05B30] (see: Pasch configuration) anti-self-dual connection [53C15, 57R57, 58D271 (see: Atiyah-Floer conjecture) anti-symmetric set A--

see:

A- -;

partially

anytime algorithm [68S051 (see: Natural language processing) .AT9 see: class -Ap6ry numbers (llAxx, 11J72, 11M06) applications of index theorems

[46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) applications of the Zak transform [4-2Axx, 44-XX, 44A55] (see: Zak transform) applications of zonal harmonicpolynomials [31B05, 33C55] (see: Zonal harmonics) applying a quantum gate [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) approach region see: non-tangential - approach to algebraic curves

Rie-

see:

mann --

approach to D e m p s t e r - S h a f e r theory see: axiomatic - - ; marginally correct approximation - - ; naive - - ; qualitative - - ; quantative - approach to machine learning see: inductive inference - approach to portfolio optimization see: martingale - approaches to the Sturm-Liouville spectral problem see: numerical -approximate eigenfunctions see: Weyl sequence of --

approximate rn -tangent plane [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) approximate point spectrum [47Dxx] (see: Taylor joint spectrum) approximate solvability

see:

unique - -

approximate tangent [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) approximate unit [03G25, 06D99, 22D10, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Fourier algebra; Quantale) approximate unit

see:

countable - -

approximate unit in harmonic analysis [43A45, 43A46] (see: Ditkin set) approximate units see: bounded -approximately additive function [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) approximately correct learning see: probably -approximation see: best - - ; best uniform - - ; Galerkin - - ; rational - - ; tau method - - ; wavelet - - ; W K B - approximation approach to D e m p s t e r Shafertheorysee: marginallycorrect --

approximationerror of functionsin Sobolev spaces [46E35, 65N30] (see: Bramble-Itilbert lemma) approximationproper [47H17] (see: Approximation solvability) Approximation solvability (47H17) (refers to: Accretive mapping; Benach space; Basis; Biorthogonal system; Functional analysis; Galerkin method; Hilbert space; Non-linear operator; Orthogonal projector; Reflexive space) approximation solvability [47H17] (see: Approximation solvability) approximation theorem stress - -

see:

Weier-

approximative derivative [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) APR-tilting module [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra)

AUSLANDER-REITENTRANSLATION AR system [60G25, 62M20, 93B10, 93B15, 93E121 (see: Wold decomposition) arbitrage assumption s e e :

no- --

arbitrage-free assumption [90A09] (see: Option pricing) arborescence [05C12, 90C27] (see: Dijkstra algorithm) arborescent tangles [57M25] (see: Algebraic tangles) arcbody [57M25] (see: Tangle) Archimedean place of a number field [12J10, 12J20, 13A18, 16W60] (see: S-integer) Arehimedean place of a number field s e e : non-

--

area of the unit sphere in R '~ [31A05, 31A10] (see: Po~ssonformula for harmonic functions) ARFIMA model [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) arithmetic exponential sum [llL07] (see: Exponential sum estimates) arithmetic-geometric average [26Dxx, 65D20] (see: Arlthmetic-geometric mean process) arithmetic-geometric mean [26Dxx, 65D20] (see: Arithmetic-geometric mean process) arithmetic-geometric mean algorithm s e e : Lagrange - -

arithmetic-geometric mean method [26Dxx, 65D20] (see: Arithmetic-geometric mean

process) Arithmetic-geometric mean process (26Dxx, 65D20) (refers to: Arithmetic mean; Gammafunction; Geometric mean; Jacobi elliptic functions; Lemnlseates; Pi (number rr); Theta-function) arithmetic group [20F38] (see: Fibonaeci group) arithmetic manifold [57Mxx] (see: Fibonaeci manifold) arithmetic of the Baily-Borel compactification [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactitleatiou) arithmetic progression [05D10] (see: Hales-Jewett theorem) arithmetic progressions s e e : Szemerodi theorem on - - ; van der Waerden theorem OR--

arithmetic quotient [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactifieation) arithmetic quotient [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactifieation) arithmetic subgroup [llFxx, 20Gxx, 22E46] (see: Baily-Borel compaetificatlon) arithmetical category [11Nxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) arithmetical function see: Dirichlet inverse of an -arithmetical partition theory

abstract - - ;

[llNxx, 11N32, 11N45, llN80]

(see: Abstract analytic number theory) arithmetical semi-group [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) arithmetical semi-group [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory; Abstract prime number theory) arithmetical semi-group s e e : additive - - ; a x i o m - A - - ; a x i o m - A # - ; axiom-C;' - ; axiom-qb - ; a x i o m - G 1 - - ; axiom-{~l - ; axiom-{~.x - - ; classical - - ; degree on an additive - - ; norm on an - - ; primes in an - -

arithmetical variety [03Gxx] (see: Algebraic logic) arithmetically Buchsbaum scheme [13A30, 13H10, 13H30] (see: Buchsbaum ring) arity of a logical connective [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Arkhangel'skiI-Froh'kcoveting theorem [26A15, 54C05] (see: Namioka theorem) ARMA system [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) A R M A system s e e :

miniphase - -

Amol'd conjecture [58Fxx] (see: Conley index) Artin algebra [16G70] (see: Almost-split sequence) Artin algebra [16Gxx] (see: Tilting theory) Artin character [11R23] (see: Iwasawa theory) ascending Fitting chain [20F17, 20F18] (see: Fitting chain) ascending HNN-extension [20F05, 20F06, 20F32] (see: HNN-extension) ASN [62Lxx] (see: Average sample number) A S P A C E T I M E s e e : complexity class - asset of a European call option s e e : underlying -assignment s e e : subjective belief -assignment function s e e : probability--

associated orthogonal Laurent polynomials [44A60] (see: Strong Stieltjes moment prob-

lem) associated subgroups of an HNNextension [20F05, 20F06, 20F32] (see: HNN-extenslon) associated with a Dirichlet problem s e e : spectral measure -associated with a vector space s e e : Lie algebra -association s e e : measure of - -

association scheme [03Exx, 03E05] (see: Coherent algebra) associative algebra s e e : tratly --

locally spec-

associative-commutative unification [06Exx, 68T15] (see: Robbins equation) associative operation see: partially - assoeiativity s e e :

partial --

associativity equations for field correlators [14Jxx, 35A25, 35Q53, 57R57]

(see: Whitham equations) associativity in vertex algebras [11Fll, 17B10, 17B65, 17B67, 171368, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) assumption s e e : arl3itrage-free - - ; complete market - - ; no-arbitrage - -

astigmatism [33C50, 78A05] (see: Zernike polynomials) asymptotic additive basis [llPxx] (see: Additive basis) asymptotic additive basis s e e :

minimal --

asymptotic completeness [81Txx] (see: Massive feld) asymptotic completeness [81Uxx] (see: Enss method) asymptotic direction [53A10, 53C42] (see: Weierstrass representation of a minimal surface) asymptotic distribution of eigenvalues [42A16, 47B35] (see: Szeg6 limit theorems) asymptotic distributionof singular values [42A16, 47B35] (see: SzegOlimit theorems) asymptotic enumeration [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) asymptotic formula for the eigenvalue distribution of Laplacians s e e : W e y / - asymptotic quantum field s e e : free - -

asymptotically stable dynamical system [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) asymptotically stable equilibrium of a dynamical system [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) asymptoties see: Weyl -asymptotics for Dirichlet eigenvalues s e e : Weyl - -

asynchronousautomatic group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) ATIMEALT s e e :

complexity class - -

Atiyah-Bott fixed-pointformulas [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Atiyah-Floer conjecture (53C15, 57R57, 58D27) (refers to: Almost-complexstructure; Chern-Simons functional; Mapping cylinder; Stiefel-Whitney class; Symplectie manifold) Atiyah-Floer conjecture [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) Atiyah-Floer conjecture for mapping cylinders [53C15, 57R57, 58D27] (see: Atlyah-Floer conjecture) Atiyah-Hitchin manifold [35Qxx, 78A25] (see: Magnetic mouopole) Atiyah L2-index theorem [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Atiyah-Patodi-Singer index theorem [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) Atiyah-Segal indexformulas [46L80, 46L87, 55N15, 58G10, 58GI 1, 58G121 (see: Index theory) Atiyah-Singer index formulas

[46L80, 46L87, 55N15, 58G10, 58GI 1, 58G12] (see: Index theory) Atiyah-Singer index formulas [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Atiyah-Singer index theorem [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) Atiyah-Singer operator [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) atom in a lattice [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mfibins inversion) atom of a measure [28-XX] (see: Absolutely continuous measures) atomic formula [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) atomic formulas s e e : set of formulas defining a set of atomic formulas explicitly over another set of - - ; set of formulas defining a set of atomic formulas implicitly over another set of - - ; strong implicit definition of a set of atomic formulas over another set of - atomic formutas explicitly over another set of atomic formulas s e e : set of formulas defining a set of -atomic formulas implicitly over another set of atomic formulas s e e : set of formulas defining a set of - atomic formulas over another set of atomic formulas s e e : strong implicit definition of a s e t o f - -

atomic lattice [03G25, 06D99] (see: Quantale) atomic yon Neumann quantale [03G25, 06D99] (see: Quantale) atoroidal manifold [57N10] (see: Haken manifold) attraction s e e :

basin of - -

attractive fixed point [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) attractor s e e :

periodic - - ; strange - -

attribute-value representation [68T05] (see: Machine learning) Auslander algebra [16Gxx] (see: Tilting functor) Auslander-Reiten correspondence [16Gxx] (see: Tilting theory) Auslander-Reiten quiver [16G70] (see: Riedtmann classification) Auslander-Reiten quiver [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Almost-split sequence; Tilted algebra; Tits quadratic form) Auslander-Reiten sequence [16G70] (see: Almost-split sequence) Auslander-Reiten theorem [16G70] (see: Almost-split sequence) Auslander-Reiten theorem on almostsplit sequences [16G70] (see: Almost-split sequence) Auslander-Reiteu translation [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra)

471

AUTO SOFTWARE

AUTO software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) AUTO97 software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) automated theorem-proving [06Exx, 68T 15] (see: Robbins equation) Automatic continuity for Banach algebras (46H40) (refers to: Banach algebra; C*algebra; Closed-graph theorem; Continuous function; Continuum hypothesis; Derivation in a rlng; Fr6chet algebra; Homomorphism) automatic continuity theory [46H40] (see: Automatic continuity for Banaeh algebras) automatic group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) automatic group see: a s y n c h r o n o u s - automatic indexing [68S051 (see: Natural language processing) automatic theorem-proving [06Exx, 68T15] (see: Robbins equation) automaton see: probabilistic finite-state --;

quantum - -

automorphic form [11Fxx, 11F27, l 1F70, 20G05, 46L80, 46L87, 55N15, 58GI0, 58G11, 58G12, 81R05] (see: Index theory; Satake compactification; Segal-Shale-Weil representation) automorphic form of half-integral weight [11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) automorphic forms s e e : Langlands formula for the dimension of spaces of - -

automorphic product [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) automorphic representation [11F27, llF70, 20G05, 81R05] (see: Segal-Shale-Weil representation) automorphic representation s e e : cuspidal - automorphism see: biholomorphic - - ; polynomial representation of the Frobenius - -

automorphismgroup [05Bxx[ (see: Net (in finite geometry)) automorphism group see: r e g u l a r - automorphism of the infinite-dimensional torus s e e : e r g o d i e - automorphism on a curve s e e : Frobenius - automorpNsms s e e : group of biholomorphic -- ; regular group of - -

automorphisms of Baumslag-Solitar groups [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) autonomous categorysce: *- -autonomous Schr6der functional equation see; non- - -

AvDTime (T) [68Q15] (see: Average-ease computational complexity) AvDTimeDis (T, V) [68Q15[ (see: Average-case computational complexity) average s e e : arithmetic-geometric space - - ; time - -

472

--;

average-case behaviour of an algorithm [68Q15] (see: Average-case computational complexity) Average-case computational complexity (68Q15) (refers to: Complexity theory; Computational complexity classes; Density of a probability distribution; N'79; Turing machine) average-case time complexity [68Q15] (see: Average-case computational complexity) average-79 [68Q15] (see: Average-case computational complexity) average-']=' see: complexity class -Average sample number (62Lxx) (referred to in: Sequential probability ratio test) (refers to: Error; Likelihood-ratio test; Random variable; Random walk; Sequential analysis; Statistical hypotheses, verification of; Stopping time; Weld identity) average sample number [62Lxx] (see: Average sample number) average time complexity on--

see:

polynomial

averaged solution of the Korteweg~le Vries equation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) averaged Taylorpolynomial [46E35, 65N30] (see: Bramble-Hilbert lemma) averaging see: multi-phase - averaging theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Avram-Parter theorems [42A16, 47B35] (see: Szeg6 limit theorems) axially symmetric function [31B05, 33C55] (see: Zonal harmonics) axiom-A arithmetical semi-group [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) axiom-A arithmetical semi-group [llNxx, 11N32, 11N45, llNS0] (see: Abstract analytic number theory; Abstract prime number theory) ax'om-A l # arithmetical semi-group [lINxx, 11N32, 11N45, llNS0] (see: Abstract analytic number theory) axiom-A# arithmetical semi-group [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory; Abstract prime number theory) axiom-C arithmetical semi-group [11Nxx, 11N32, 11N45, tlN80] (see: Abstract analytic number theory) axiom-C arithmetical semi-group [llNxx, 11N32, 11N45, IIN80] (see: Abstract analytic number theory) axiom-a~ arithmetical semi-group [llNxx, 11N32, 11N45] (see: Abstract prime number theory) axiom-q5 arithmetical semi-group [llNxx, 11N32, 11N45] (see: Abstract prime number theory) axiom-G1 arithmetical semi-group [llNxx, 11N32, 11N45, 11N80[ (see: Abstract analytic number theory) axiem-G 1 arithmetical semi-group [llNxx, 11N32, 1IN45, I1N80[

(see: Abstract analytic number theory) axiom-G1 arithmetical semi-group [llNxx, 11N32, 11N45] (see: Abstract prime number theory) axiom-~l arithmetical semi-group [11Nxx, 11N32, 11N45] (see: Abstract prime number theory) axiom ~ see: WarlJmont-axiom-~,x arithmetical semi-group [llNxx, 11N32, 11N45] (see: Abstract prime number theory) axiom-~;~ arittwnetical semi-group [llNxx, 11N32, 11N45] (see: Abstract prime number theory) axiom of choice [03E30] (see: ZFC) axiom of choice s e e : Zermelo-Fraenkel set theory with the - axiom of constructibility s e e : G6del - -

axiom of extensionality [03E30] (see: ZFC) axiom of Jbundation [03E30] (see: ZFC) axiom of infinity [03E30] (see: ZFC) axiom of natural decomposition [06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) axiom of pairs [03E30] (see: ZFC) axiom of power set [03E30] (see: ZFC) axiom of replacement [03E30] (see: ZFC) axiom of separation [03E30] (see: ZFC) axiom of the empty set [03E30] (see: ZFC) axiom of union [03E30] (see: ZFC) axiom schema of replacement [03E30] (see: ZFC) axiom schema of separation [03E30] (see: ZFC) axiomatic approach to Dempster-Shafer theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) axiomatic characterization of the Brouwer degree [55M25] (see: Brouwer degree) axiomatizabilitysee: finite-axiomatizable s e e : finitelyschema - axiomatizabletheory s e e : finitely - -

axiomatization of set theory [03E30] (see: ZFC) axiomatized by a set of formulas ory - axioms s e e : large cardinal - -

see:

the-

axioms of Brown [55P25] (see: Spanier-Whitehead duality) axioms of ZFC [03E30[ (see: ZFC) axis s e e : half- -axis case

direct scattering problem on the see:

Inverse scattering, half- - -

Azumaya algebra [13-XX, 16-XX, 17-XX] (see: Skolem-Noether theorem)

A z u m a y a algebra s e e : dimodule -A z u m a y a algebras s e e : acterization of - -

Long H categoricalchar-

B fl-defavourablespaee sec: •- -B-process s e e : van der Corput - -

backpropagation algorithm for neural networks [68T05] (see: Machine learning) backscattering [35P25, 47A40, 81U20] (see: Inverse scattering, multidimensional case) backward reasoning [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) Bade-Curtis boundedness theorem [46H40] (see: Automatic continuity for Banach algebras) Baily-Borel compactification (llFxx, 20Gxx, 22E46) (referred to in: Satake compaetifieation) (refers to: Algebraic torus; Algebraic variety; Arithmetic group; Diagonalizable algebraic group; Intersection homology; Linear algebraic group; Moduli theory; Normal analytic space; Number field; Satake compactification; Scheme; Semi-simple algebraic group; Sheaf; Shimura variety; Stabilizer; Theta-series; Zariski topology) Baily-Borel compactification [11Fxx, 20Gxx, 22E46J (see: Baily-Borel compaetification) Baily-Borel compactificatioe see: arithmetic of the - - ; cohomology of the - - ; moduli of the - - ; S a t a k e - - Baily-Borel compactifications s e e : examples of - -

Baire category [54E52] (see: Banaeh-Mazur game) Baireclass

see:

first - -

Baker-Akhiezerfunction [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Baker-Akhiezerfunction s e e : adjoint -Baker finite basis theorem [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Baker finite basis theorem see: analogue of the - balance law s e e : e n e r g y - balanced collections of blocks in a Steiner triple system s e e : mutuallyl:- - -

balanced design for statistical experiments [62Jxx] (see: ANOVA) balanced domain [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) balanced Freudenthal-Kantor triple system [17A40] (see: Freudenthal-Kantar triple system) balanced incomplete block design s e e : symmetric - ball s e e : Poisson kernel for a - ball genus of a knot s e e : 4 - - ball in a Banaeh space s e e : extreme point of the closed unit - -

ball measure

BERNSHTEI~-GEL'FAND-GEL'FANDFORMULA [47H10] (see: Darbo fixed-point theorem) ball theorem s e e : hairy - ball topology see: proximal - Banach algebra s e e : character on a - - ; e-perturbation of a - - ; e-metric perturbation of a - - ; e-perturbation of a - - ; J o r d a n - -- ; Noetherian - - ; stable - Banacbalgebrassee: Automatic continuity f o r - - ; e-isometry of - - ; e-isomorphism of--

Banach-Jordan algebra (17C65, 46H70, 46L70) (referred to in: Banach-Jordan pair; .[B * -triple) (refers to: Banach algebra; Banaeh space; C*-algebra; Derivation in a ring; Divisionalgebra; Jacobsen radical; Jordan algebra)

base group of an HNN-extension

[20F05, 20F06, 20F32] (see: HNN-extension) base-homomorpbisms [03Gxx] (see: Algebraic logic) based abstract algebraic logic s e e : semantics- -based analogy s e e : relevance- - - ; similarity- -basedlearning see: explanation- - based on relations deforming n - m o v e s s e e : skein m o d u l e -based on the J o n e s - C o n w a y relation s e e : skein m o d u l e -based on the Kauffman polynomial s e e : skein module - based path from logic to algebra s e e : rule-

Banach-Jordanalgebra see: primitive - - ; spectrum of an element in a - -

based proof s e e : resolution--based variety s e e ; finitely --

Banach-Jordan pair (I7A40, 17C65, 46H70, 46L70) (refers to: Associative rings and algebras; Banach-Jordan algebra; Banaeh space; Chain condition; Ideal; Jacobsen radical; JB *-triple; Jordan triple system; Linear operator; Noetherian ring; Norm; Socle; Vector space)

baseline hazard

Banach-Jordan pair s e e : a--

~ocal a l g e b r a o f

Banach--Jordan pairs s e e : ditions in --

finiteness con-

basis s e e : Additive - - ; asymptotic additive --; Bernstein--; Bemstein-Bezier--; minimal asymptotic additive - - ; normal - - ; order of an additive - - ; polynomial - - ; Schauder - - ; self-dual - - ; thin additive - - ; w e a k l y self-dual polynomial -basis for the identities of a variety s e e : finite - basis for the natural numbers see: Additive -basis function s e e : linear radial - - ; Radial - - ; tensor-product -basis fuzzy topology s e e : variable- - basis generatorsee: normal - -

Banach-Jordan triple system [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Banaeh-Mazur game (54E52) (referred to in: Namioka theorem) (refers to: Baire classes; Baire space; Category of a set; Complete metric space; Completely-regular space; Non-measurable set; Topological space) generalized--; play in the - - ; stationary strategy in the generalized - - ; stationary w i n n i n g strategy in the generalized - - ; strategy in the generalized - - ; tactics in the generalized - - ; winning strategy in the generalized - Banach space s e e : b o u n d e d symmetric domain in a - - ; extreme point of the closed unit ball in a - - ; helomorphic function on a - - ; nest in a - Banaeh-Mazur

game see:

Banach-Stone property

[46Exx] (see: Banaeh-Stone theorem) Banach-Stone theorem (46Exx) (referred to in: Function vanishing at infinity) (refers to: Banaeh algebra; Banach space; Hausdorff space; Homeomorphism; Uniform algebra) Banach-Stone theorem

[46Exx] Banach theorem [46E22] (see: Reproducing-kernel Hilbert space) Banachtheorem see:

Stone---

limited function

[42A63] (see: Uncertainty principle, mathe-

matical) banks s e e :

[90Cxx] (see: Fritz John condition)

basin of attraction [28A80] (see: Sierpifiski gasket)

basis in a module

[13Pxx, 14Q20] (see: Hermann algorithms) standard-basis of a coherent algebra s e e : standard - basis of a field s e e : dual basis of an ordered - basis of an ordered basis of a field s e e : dual -basis problem for varieties s e e : finite -basis theorem s e e : analogue of the Baker finite - - ; Baker finite - - ; Normal - - ; primitive normal - b a s i s of a c e l l u l a r a l g e b r a s e e :

basis theorem for Schubert cycles

[14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) basis topology s e e : fixed- - - ; variable- -Basor-Tracy conjecture [42A16, 47B35] (see: Szeg6 limit theorems) Bass number of a module

(see: Banach-Stone theorem)

band

[62Jxx, 62Mxx] (see: Cox regression model) basic Fritz John condition

parsetree - -

Bargmarm-Segal space [46Cxx, 47B35] (see: Berezin transform) Bartlett-Nanda-Pillai test

[62Jxx] (see: ANOVA) base s e e : domain knowledge - - ; knowledge - base change s e e : quadratic --

[I6D40] (see: Flat cover) Bass n u m b e r of a module s e e :

dual - -

Bass-Serre theory [20E22, 20Jxx, 57Mxx] (see: Accessibilityfor groups) Bass-Serre theory of groups acting on trees [20F05, 201=06, 20F32] (see: HNN-extension) Bauer-Fike theorem (15A42) (refers to: Complete set; Eigen value; Eigen vector; Functional; Gershgorin theorem; Hermitian matrix; Linear algebra; Normal matrix; Symmetric matrix; Unitary matrix) Bauer-Fike theorem

[15A42] (see: Bauer-Fike theorem)

Bauer theorem s e e : B r e l o t - -Baumslag group s e e : Solitar---

Baumslag-Solitar group (05C25, 20Fxx, 20F32) (referred to in: HNN-extension) (refers to: Cayley graph; Cyclic group; Epimorphism; Formal languages and automata; Free group; Fundamental group; Grammar, regular; Group with a finiteness condition; HNN-extension; Homomorphism; Hopf group; Hyperbolic group; Identity problem; Isomorphism; Klein surface; Meta-Abelian group; Metric space; Nilpotent group; Non-Hopf group; Normal subgroup; Polynomial and exponential growth in groups and algebras; Quasi-isometric spaces; Residuallyfinite group; Solvable group; Threedimensional manifold; Torus; Word metric) Baumslag-Sofitar group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) B a u m s l a g - S o l i t a r g r o u p s e e : convex rigid - - ; meta-Abelian - - ; presentation of a - - ; rigid - Baumstag-Solitar groups s e e : automorphisms of - - ; normal forms in - - ; subgroups of - -

Baumslag-Solitar groups as examples [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Bautin point bifurcation

[34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Bautista-Brunner theorem

[16G70] (see: Almost-split sequence)

Baxter algebra (05E05, 60G50) (refers to: Algebra; Banach algebra; Characteristic function; Endomorphism; Free algebra; Identity problem; Integration by parts; Linear operator; Random variable; Symmetric function) Baxter algebra

[05E05, 60G50] (see: Baxter algebra) Baxter algebra s e e : standard - Baxter equation see: Y a n g - --

Baxter operator

[05E05, 60G50] (see: Baxter algebra) Bayesian classifier see: na'fve -Bayesian network

[68T05] (see: Machine learning)

behaviour of an algorithm s e e : averagecase - behaviour of the Z-transform s e e : shift - behaviour of the Zak transform s e e : conjugation - - ; modulation - - ; translation -belief s e e : degree of - - ; subjective - belief assignment s e e : subjective - -

belief function

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory)

belief function [28-XX]

(see: Non-additive measure) belief function s e e : a priori-condition - - ; conditional - - ; focal point of a - - ; g r a p h o i d a l properties of a - - ; independent variable sets for a - - ; vacuous - -

belieffunction theory

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) belief functions s e e : rule of combination of t w o independent - belief model s e e : t r a n s f e r a b l e - belief theory s e e : graphoidal structure in--

Beltrami differential [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Beltrami operator s e e :

Laplace- --

Benard-Schacher theorem

[11R34, 12G05, 13A20, 16S35, 20C05] (see: Sehur group) Benedicks theorem [42A63] (see: Uncertainty principle, mathematical) Benjamin-Bona-Mahony equation (35Q53, 76B15) (refers to: Cauchy problem; Fourier transform; Korteweg-de Vries equation; Laplace operator; Pseudodifferential operator; Pseudometric; Soholev space; Soliton) B e n j a m i n - B o n a - M a h o n y equation s e e : conservation taws for the - - ; generalized - - ; variable-coefficient - -

Bennequin conjecture [57M25] (see: Positive link) Berezin integral [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) Berezin transform (46Cxx, 47B35) (refers to: Analytic function; Bergman spaces; Compact operator; Harmonic function; Hilhert space; Linear functional; Toeplitz operator; Unitary operator) Berezin transform

[46Cxx, 47B35]

Bayesian network [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) Bayesian statistical inference [90Cxx] (see: Fritz John condition)

(see: Berezin transform) Berezin transformation

BBM equation

(see: Reproducing kernel) Bergman operator

[35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) BBM equation s e e :

generalized --

BCK logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) BDF theory

[i 9K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore the-

ory) Becctfi-Rouet-Stora-Tyutin transformadons [81Qxx, 81Sxx, 81T131 (see: Faddeev-Popov ghost) behaviour s e e : chaotic - - ; large deviation - - ; moderate d e v i a t i o n - -

[46Cxx, 47B35] (see: Berezin transform) Bergman kernel

[46E22] [17Cxx, 46-XX] (see: JB *.trlple)

Bcrgman space [47Dxx] (see: Taylor joint spectrum) Berkson-Portaparametric representation

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphie mappings) Bernard equations s e e : KnizhnikZ a m o l o d c h i k o v - -Bernoulli equation s e e : E u l e r - --

BemshteIn-GeI'fand-Gel'fand formula [14C15, 14M15, 14N15, 20G20, 57TI5] (see: Schubert calculus) 473

BERNSHTEi'N-GEL'FAND-PONOMAREVREFLECTION FUNCTOR

Bernshtet'n-Gel'fand-Ponomarev reflection functor [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting module; Tilting theory) Bernshteln inequality [42B05, 42B08] (see: Hyperbolic cross) Bernshtern-Szeg5 polynomials [33C45] (see: Szeg6 polynomial) Bernstein basis [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) Bernstein-Bdzier basis [4IA10, 41A15, 68U05] (see: Bernstein-B~zier form) Bernstein-B~zier form (4IA10, 41A15, 68U05) (refers to: Approximation of functions, linear methods; Bernstein polynomials; Weierstrass theorem) Bernstein-B~zierform [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) Bernsteinform [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) Bernstein operator [41A10, 41A15, 68U05] (see: Bernsteln-B~zier form) Bernstein polynomial [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) Bernstein set [54E52] (see: Banach-Maznr game) Besicovitch-Federerprojection theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Besicovitch-Federer projection theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Besov space se~ S o b o l e v - -Bessetcoefficients see:

Fourier---

Bessel function of the first kind [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) best approximation [65Lxx] (see: Tau method) best linear least squarespredictor [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) best linear unbiased estimator [62Jxx] (see: ANOVA) best uniform approximation [41-XX, 41A50] (see: Zolotarev polynomials) beta-function [11J85] (see: Gel'fond-Schneider method) Beth definabifity [03Gxx] (see: Algebraic logic) Beth definability see:

local - - ; w e a k --

Beth definability property [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Beth definability property see:

weak --

Beth definability theorem [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Beurfing algebra (42A16, 42A24, 42A28) 474

(refers to: Absolute continuity; Banach algebra; Borel measure; Continuity, modulus of; Fourier coefficients; Fourier series; Fourier transform; Function of bounded variation; Hankel operator; Lacunary sequence; Lebesgue measure; Maximal ideal; Nikol'skil space; Reflexive space; Separable space; Spectral synthesis; Synthesis problems) Beurling algebra [42A16, 42A24, 42A28] (see: Benrling algebra) Beurling algebra [43A45, 43A46] (see: Ditkin set) Beurlingalgebrasee: dualspaceofthe --; spectral properties of the - - ; synthesis

problem for the --

Beurling algebra of Fourier series with summable majorant of coefficients [42A16, 42A24, 42A28] (see: Beurling algebra) Beurling-Pollard theorem [42A16, 42A24, 42A28] (see: Beurling algebra) Bending theorem (30D55, 46115, 47A15) (refers to: Beurling-Lax theorem; Hardy classes) B6zier basis see:

Bernsteie---

Btzier curve [4tAt0, 41A15, 68U05] (see: Bernstein-B~zier form) B~zier form see:

Bernstein---

B~zier polynomial [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) Bezout domain (13Fxx) (refers to: Bezout ring) Bezout equation [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Bezout matrix [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) B ( G ) see: idempetency theorem for bidual [17Cxx, 46-XX] (see: JB * -triple) BIFTOOL software see: DDE- - -

--

bifurcation

[58Fxx] (see: Conley index) bifurcation see: Bautin point - - ; Bogdanov-Takens - - ; codJmension-3 - - ; cusp - - ; double Hopf - - ; flip - - ; generalized Hopf - - ; generalized Hopf point -- ; Ne'fmark-Saeker -- ; resonant double Hopf - - ; subcritieal - - ; s u p e r c d t i c a l - - ; swallowtail - - ; torus - - ; triple zero - - ; zero-Hopf - -

bifurcation in the Kuramoto-Sivashinsky equation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) big horosphere [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) biholomorphic automorphism [17Cxx, 46-XX] (see: JB *-triple) biholomorphic automorphisms see: of-bilinear relations see: Hirota - -

bimedian of a quadrangle [51M04] (see: Varignon parallelogram) binary alloy see: free energy of phase transition in a - -

Binet-Cauchy theorem [05C50] (see: Matrix tree theorem) Binet formula [11B39]

group

a --;

(see: Tribonacci number) binomial moment [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) binormal bitopological space [26A21, 26A24, 28A05, 54E55, 54G20] (see: Sorgenfrey topology; Zahorski property) binormality condition [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) bipolar structure on a group [20F05, 20F06, 20F32] (see: HNN-extension) bipolarstructureson groups see: Stallings characterization of - -

Birch-Swinnerton-Dyerconjecture [llFll, 11F12, 11R23] (see: Iwasawa theory; Shimura correspondence) Birkhoff decomposition [22E65, 22E70, 35Q53, 35Q58, 581=07] (see: AKNS-hierarehy) Birkhoff-Rott equation (76C05) (refers to: Cauchy integral; CauchyKovalevskaya theorem; Elliptic partial differential equation; Hyperbolic partial differential equation; Von Kiirmlln vortex shedding) birth process [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) Bishop theorem (46E25, 54C35) (refers to: Continuous function; Hahn-Banach theorem; Hausdorff space; Homeomorphism; Locally convex space; Riesz theorem; Stone~ Weierstrass theorem; Uniform convergence; Weierstrass theorem) Bishop theorem [46E25, 54C35] (see: Bishop theorem) Bishop theorem for real function algebras [46E25, 54C35] (see: Bishop theorem) bisimulation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) bisimulation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) bit [68Q05, 68Q10, 68Q15, 68Q25, 8lPxx, 81P15, 94Axx] (see: Quantum information processing, science of) bit see:

quantum --

bitopological space [26A21, 26A24, 28A05, 54E55, 54G20] (see: Sorgenfrey topology; Zahorski property) bitopologicelspace see:

binormal --

bivariate Fibonacci polynomials [33Bxx] (see: Fibonacci polynomials) bivariate Lucas polynomials [11B39] (see: Lncas polynomials) bivariate normal distribution [62H20] (see: Pearson product-moment correlation coefficient) black box representation of a multivariate polynomial [12D05] (see: Factorization of polynomials) black Listing polynomial [57M25] (see: Listing polynomials) Black-Scholes formula

(60Hxx, 90A09, 93Exx) (referred to in: Option pricing) (refers to: Brownian motion; Controlled stochastic process; Diffusion equation; Martingale; Normal distribution; Option pricing; Stochastic differential equation; Stochastic process) Black-Seholes geometric Brownian motion model [90A09] (see: Option pricing) Black-Scholes-Merton option pricing [90A09] (see: Option pricing) Blaschke-Potapov factor [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator eolligation) Bloch electron [42Axx, 44-XX, 44A55] (see: Zak transform) Bloch space [46Cxx, 47B35] (see: Berezin transform) block design see: incomplete - -

symmetric balanced

block Hankel matrix [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) block of a group [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) block of a Steiner triple system [05B07, 05B30] (see: Pasch configuration) blocks) see: trade (pair of -blocks in a Steiner triple system see: tually t - b a l a n c e d collections of - -

mu-

Blomqvist coefficient [62H20]

(see: Kendall tau metric) Blomqvist coefficient see: rameter of the --

population pa-

Blomqvist q coefficient [62H20] (see: Kendall tau metric) blow-up see:

Kirby move --

blowing-up [13A30, I3H10, 13H30] (see: Buchsbaum ring) BLUE [62Jxx] (see: ANOVA) BM-algebra [03Exx, 03E05] (see: Coherent algebra) BMO [46Cxx, 47B35] (see: Berezin transform) BMOA -space (30Axx, 46Exx) (referred to in: VMOA -space) (refers to: Analytic function; BMOspace; Brownian motion; Hardy classes; Hardy spaces; Logarithmic capacity; Riemann surface; Univalent function) B N-pair [20G05] (see: Steinberg module) Boas-TeIyakow~kE space [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) Boctmer-Martinelli kernel [47Dxx] (see: Taylor joint spectrum) Bochner-Riesz operator [47B06] (see: Riesz operator) Bochner-Riesz summability [47B06] (see: Riesz operator) Bockstein operation [20J06]

BRENT-SALAMIN ALGORITHM

(see: Serre theorem in group cohomology) body theorem see: Minkowski convex -B ogdanov-Takens bifurcation [34-04, 35-04, 58-04, 5 8 F 1 4 ]

(see: Dynamical systems software packages) Bogomolny equations [35Qxx, 78A25] (see: Magnetic monopole) Bogomolny-Prasad-S ommerfield limit [35Qxx, 78A25] (see: Magnetic monopole) Bombieri-Iwaniec method (llLxx, 11L03, llL05, llL15) (referred to in: Exponential sum estimates) (refers to: Analytic number theory; Bessel functions; Circle method; Fourier series; Gauss sum; Geometry of numbers; H61der inequality; Large sieve; Lattice of points; Number of divisors; Poisson summation method; Riemann zeta-function; Stationary phase, method of the; Taylor series; Theta-series; Weyl sam) Bombieri-Vinogradovtheorem [llL07, llM06, 11P32] (see: Vaughan identity) Bena-Mahony equation see: Benjamin- ; conservation laws for the Benjamin- ; generalized Benjamin- - - ; variablecoefficient Benjamin- - -

Bonferroni bounds [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) Book see:

Scottish - -

Boolean algebra with operators [03Gxx] (see: Algebraic logic) Boolean algebras see: amalgamation property of the variety of --; equational logic of - - ; variety of --

Boolean circuit [68Q15] (see: Average-case computational complexity) Boolean expansion lemma [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mfbins inversion) Boolean lattice [05D05, 06A07] (see: Sperner property) Borcherds algebra E11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Borcherds algebra [17B10, 17B65] (see: Weyl-Kac character formula) Borcherds algebra see: character of a - - ; charge of a - - ; imaginary root of a - - ; imaginary simple root of a - - ; real root of a - - ; real simple root of a - - ; symmetrizable - - ; universal -Borcherds algebras see: characterization of - - ; denominator identity for - -

Borcherds-Cartan matrix [17B10, 17B65] (see: Weyl-Kae character formula) Borcherds character formula see: Kac- - Borcherds colour superalgebra

Weyl-

[17B10, 17B65] (see: Weyl-Kac character formula) Borcherds Kac-Moody algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Borcherds Lie algebra (11Fxx, 17B67, 20D08) (referred to in: Moonshine conjectures; Weyl-Kae character formula) (refers to: Cartan subalgebra; Coxeter group; Kac-Moody algebra; Lie

algebra; Lie algebra, graded; Representation of a Lie algebra; Weyl group) Borcherds Lie algebras see: theorem for - -

structure

Borda count [90A28] (see: Condorcet paradox) Betel compactification see: arithmetic of the Baily- - - ; Baily- - - ; cohomo]ogy of the Bai[y- - - ; moduli of the Baily- - - ; Satake-Baily- - Borel compactifications see: examples of Baily- - -

Borel construction [54H15, 55R35, 57S17] (see: Smith theory of group actions) Borel equivalence relation [03C15, 03C45, 03E15] (see: Vaught conjecture) Borel-Serre compactificafion [llFxx, 11F67, 20Gxx, 22E46] (see: Baily-Borel compactification; Eisenstein cohomology) BoreI-Serrecompactification see: tive - -

reduc-

Borromeanrings [57Mxx] (see: Fibonacci manifold) Bose-Mesner algebra [03Exx, 03E05, 05Exx] (see: Cellular algebra; Coherent algebra) Bose-Mesner algebra [03Exx, 03E05] (see: Coherent algebra) boson see: Goldstone -Bott fixed-point formulas see: A t i y a h - - bottle see: Klein - bound see: Chernoff - - ; GilbertVarshamov - - ; Holevo -- ; log-rank lower - - ; Minkowski -bound conjecture see: u p p e r - -

bound state [35P25, 47A40, 58F07, 8IU20] (see: Inverse scattering, full-line case; Inverse scattering, half-axis case) bound-state eigenfunctions [35Q53, 58F07] (see: Harry Dym equation) boundary see: Shilov - boundary component see:

rational - -

boundary component of a symmetric space [11Fxx] (see: Satake compactification) boundary component of a symmetric space see: rational --

boundary compressible surface in a threedimensional manifold [57N10] (see: Haken manifold) boundarycondition see: tion --

Calder6n projec-

boundary incompressible surface in a three-dimensional manifold [57N10] (see: Haken manifold) boundary of a current [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) boundaryvalue problem see: Dirichlet - - ; Neumann - bounded approximate uuits [43A07, 43A15, 43A45, 43A46,

46JI0] (see: Figh-Talamanca algebra) bounded digraph see:

Iocallywalk- - -

bounded distortion [26B99, 30C62, 30C65] (see: Quasi-regular mapping) bounded distortion see: mapping with -bounded-error polynomial-time computable language [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx]

(see: Quantum computation, theory of) bounded-error quantum polynomial-time computable language [68Q05, 68QI0, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) bounded mean oscillation [30Axx, 46Exx] (see: BMOA-space) bounded mean oscillation see: function of - - ; space of analylic functions of - -

bounded probabilistic polynomial time complexity class [03D15, 68Q15] (see: Computational complexity classes) bounded symmetricdomain [llFxx, I7A40, 17C65, 20Gxx, 22E46, 46H70, 46L70] (see: Baily-Borel compactification; Banach-Jordan pair) bounded symmetricdomain in a Banach space [17Cxx, 46-XX] (see: JB * -triple) bounded variation [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) bounded variation see: function of - - ; space of functions of -boundedness theorem see: Bade-Curtis - - ; uniform -bounds see: Bonferroni - - ; Odlyzko - bounds for the Hilbert Nullstellensatz see: complexity - - ; degree - - ; height - -

Bourgain return-time theorem [28D05, 54H20] (see: Wiener-Wintner theorem) BOV-method software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Bowen-Ruelle measure see: box [ ] 6Gxx]

Sinai- - -

(see: Tits quadratic form) box dimension [28A80] (see: Sierpifiski gasket) box product [54G10] (see: P-space) box representation of a multivariate polynomial see: black - boxes see: representationtype of - -

bpa function [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) BPP [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) BPP see:

complexity class - -

BPS state [11Fxx, 17B67, 20D08] (see: Boreherds Lie algebra) BQP [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of)

braid see:

3-string - -

braid index

[57M25] (see: Alexander theorem on braids) braid theorem see: M a r k o v - braids see: Alexander theorem on --

Bramble-Hilbert lemma (46E35, 65N30) (refers to: Approximation of functions; C o n e condition; Fourier transform; Hermite interpolation formula; Imbedding theorems; Lagrange interpolation formula; Linear functional; Norm; Semi-norm; Sobolev space; Spline interpolation) Bramble-Hilbert lemma [46E35, 65N30] (see: Bramble-Hilbert lemma) Bramblelemma see:

Hilbert---

Branch group (20E08, 20E18, 20Fxx) (refers to: Automorphism; Group; Normal subgroup; p-group; Polynomial and exponential growth in groups and algebras; Profinite group; Residually-finite group; Simple group; Stabilizer; Transitive group; Tree) branch group [20E08, 20El 8, 20Fxx] .(see: Branch group) branch group see: nite - -

just infinite - - ; profi-

branch group of finite width [20E08, 20E18, 20Fxx] (see: Branch group) branch index [20E08, 20E 18, 20Fxx] (see: Branch group) branch indexfor a tree level [20E08, 20E18, 20Fxx] (see: Branch group) branch set [26B99, 30C62, 30C65] (see: Quasi-regular mapping) branch switching [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Brandt-Lickorish-Millett-Ho polynomial (57M25) (referred to in: Conway skein triple) (refers to: Conway skein triple; Kanffman polynomial; Link) Branges-Rovnyak functional model see: de--

Braner Cassini oval [15A18] (see: Gershgorin theorem) Brauer theorem on eigenvalues [15A18] (see: Gershgorin theorem) Brauer theorem on splitting fields" for group algebras [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) Braaer-Witt theorem [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) breaking see:

soft susy - - ; symmetry - sym-

b r e a k i n g in q u a n t u m fJeld t h e o r y see:

metry - -

bracket see: Peierls - - ; quantum - braeket-derivativeformula see: D- -bracket polynomial see: Kauffman - bracket skein module see: Kauffman - bracket skein relation see: Kauffman - bracket skein triple see: Kauffman - -

Brelot-Bauer theorem [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Brenner-Butler theorem [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra)

Braess paradox (60K30, 68M10, 68M20, 90B10, 90B15, 90B18, 90B20, 94C99) (refers to: Graph; Graph, oriented; Queue)

Brent-Salamin algorithm [26Dxx, 65D20] (see: Arithmetic-geometric mean process)

Brent algorithm see:

Salamin---

475

BREZIN MAPPING

Brezin mapping s e e : W e l l - -brick factory problem s e e : Tur#.n -bridge n-tangle s e e : n - - Brill formula s e e : Cayley- --; ChaslesCay[ey- - -

(see: Sierpiliski gasket) BRST transformations [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost)

Britton lemma

Bruck net

Brocard angle

Bruck net s e e :

maximal - -

Bruhat order [05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials)

[51M04] (see: Broeard point) Brocard circle

[51M041 (see: Brocard point)

Bruijn function see: D i c k m a n - D e - Brunn theorem on knots

Brocard configuration

Burnside problem

[20F05, 20F06, 20F32, 20F50] (see: Burnside group) Burnside problem see: restricted -Busby invariant

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Butler theorem

see:

Brenner---

[57M25]

[51M041

(see: Alexander theorem on braids) B r u n n e r t h e o r e m see: B a u t i s t a - - -

(see: Brocard point) Brocard point

(51M04)

bubble s e e :

(refers to: Isogonal) Brocard point

C

double --

Buchsbaum invariant

[13A30, 13H10, 13H30]

[51M04] (see: Brocard point) Brocard point s e e : first - - ; negative - - ; positive - - ; second - -

broken-circuit complex [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mtbius inversion) Brouwer degree (55M25) (referred to in: Conley index) (refers to: Algebraic topology; Brouwer theorem; Degree of a mapping; Differential topology; Homology group; Homomorphism; Jacobian; Mathematical analysis; Sard theorem; Weierstrass theorem; Winding number) Brouwer degree

[55M25] (see: Brouwer degree) Brouwer degree s e e : additivity-excision of the - - ; axiomatic characterization of the - - ; existence property of the - - ; homotopy invariance of - - ; homotepy invarianee of the - - ; local - - ; normalization property of the - - ; product theorem for the --

Brouwer fixed-point theorem [55M25] (see: Brouwer degree) Browder theorem

(see: Buchsbaum ring) Buchsbaum local ring of maximal e m b e ~ ding dimension

[13A30, 13H10, 13H30] (see: Buehsbaum ring) Buchsbaum module

[13A30, 13H10, 13H30] (see: Buchsbaum ring) Buchsbaum module s e e : maximal - - ; quasi- - Buehsbaum modules over regular local rings see: structuretheorem for maximal -Buchsbaum-representation type s e e : Noetherian local ring of finite --

Buchsbaum ring (13A30, 13H10, 13H30) (refers to: Blow-up algebra; CohenMacaulay ring; Field; Formal power series; Integral domain; Local cohomology; Local ring; Maximal ideal; Noetherian ring; Normal ring; Regular ring (in commutative algebra); Vector bundle) Buchsbaum ring

[13A30, 13H10, 13H30] (see: Buchsbaum ring) Buehsbaum ringssee: surjectivitycriterion for - Buchsbaumschemesee: arithmetically-Bukhstab function

[llAxx] (see: Dickman function) Bukhstab identity

[32E20] (see: Polynomial convexity) Brown see: axioms of - -

Brown-Douglas-Fillmore theory (19K33, 19K35, 49L80) (refers to: C*-algebra; CW-complex; Exact sequence; Fredholm operator; Generalized cohomology theories; Group; Hilbert space; Index of an operator; K-theory; Monoid; Normal operator; Nuclear space; Selfadjoint operator; Spectral measure; Spectrum of an operator; SteenrodSitnikov homology; yon Neumann

algebra) dual - -

Brown-Gitler spectra (55P42) (refers to: Algebraic topology; Category; Co-algebra; Homology; Homotopy; Hopf algebra; Immersion; Spectrum of spaces; Steenrod algebra) Brown-Gitler spectra

[55P42] (see: Browu-Gitier spectra) Brown-Gitlerspectra see: dual - Brownian motion s e e : geometric - - ; nonMarkovian functional of - - ; obliquely reflecting - - ; reflecting - Brownian motion model s e e : BlackScholes geometric --

Brownianmotionon the Sierpifiskigasket [28A80] 476

anti- - -

[05Bxx] (see: Net (in finite geometry))

[20F05, 20F06, 20F32] (see: HNN-extension)

Brown-Gitler modules s e e :

BRSTtransformations see:

Burnside group see: conjugacy problem for presentations of a free --; free - - ; word problem for presentations of a free - Burnside groups s e e : construction of free - -

[llAxx] (see: Dickman function)

Bunce-Chu structure theorem [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) bundle s e e : determinant - - ; flat --;

C ' * -algebra s e e : co-crossed product - - ; C o r o n a -- ; corona of a - - ; crossed product - - ; Ext monoid of a - - ; extension of a separable - - ; full - - ; Jordan - - ; nont y p e - [ - - ; projective - - ; spectrum of a - - ; Toeplitz - - ; uniformly closed - -

C* -algebra of compact operators [46Lxx] (see: Toeplitz C*-algebra) C * - a l g e b r a s see: extension of - - ; Multipliers of - - ; short exact sequence of - C arithmeticalsemi-group s e e : axiom- - -

C * -condition

[17Cxx, 46-XX] (see: JB *-triple) C-domain

[35P25] (see: Obstacle scattering) C*-duality

[46Lxx] (see: Toeplitz C*-algebra) C* -filtration [46Lxx] (see: Toeplitz C* -algebra) C*-filtration see: spectrum of a -C for s e e : Ordinary differential equations, property - - ; Partial differential equations, property - C + for ordinary differential equations s e e : property - C ~ for ordinary differential equations see: properly - C.¢ for ordinary differential equations s e e : property - C for partial differential equations s e e : property - C p for partial differential equations s e e : property --

foliated - - ; real vector - - ; spin - - ; spinet -bundles s e e : spectral curve of a family of line - -

C-matrix [05C50] (see: Matrix tree theorem)

Buntinas-Tanovic-Miller space [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) Bures-Uhlmannfidelity [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 8IP15, 94Axx] (see: Quantum information processing, science of) Burgers equation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Burgers equation see: H o p f - -Burnside algebra of a group [05B35, 05Exx, 05E25, 06A07, l 1A25] (see: M6bfus inversion) Burnside group (201705, 20F06, 20F32, 201750) (refers to: Burnside problem; Conjugate elements; Finitely-generated group; Free group; Hyperbolic group; Identity problem)

c-O K point

(see: Cahn-Hilliard equation) see: s t a t i o n a r y - -

Cahn-Hilliard equation

Cake-cutting problem (00A08, 90Axx) (refers to: Sperner lemma) calculus s e e : Church )~- - - ; Clarke classical prepositional - - ; functional intuitionistic propositional - - ; Kirby lambda - - ; Riesz-Dunford functional Schubert - - ; Schubert enumerative stochastic - -

--; --; --; --; --;

calculus of relations [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) calculus of subgradients [90C30] (see: Clarke generalized derivative) Calder6n projection boundary condition [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Caldertn set

[43A45, 43A46J (see: Ditkin set) Calkin algebra

[19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory)

Calkin algebra [46J10, 46L05, 46L80, 46L85, 47Dxx] (see: Multipliers of C*-algebras; Taylor joint spectrum) call option s e e : European - - ; expiration time of a European - - ; strike price of a European - - ; underlying asset of a European -call option at expiration s e e : value of a European - -

Cameron-Martin~irsanov theorem

[90A09] (see: Option pricing) CANDYS/QA software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Cant conditionals [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) canonical algebra in tilting theory [16Gxx] (see: Tilting theory) canonical form for ANOVA [62Jxx] (see: ANOVA) canonical form for GMANOVA [62Jxx] (see: ANOVA) canonical form for MANOVA

Co -semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings)

[62Jxx] (see: ANOVA) canonical measure [28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invariant measure) canonical Poisson structure [37J15, 53D20, 70H33] (see: Momentum mapping) canonical polynomialsin the tan method [65Lxx] (see: Tau method)

C -set

canonical quantization

Ca-moves see:

Habiro --

[54D40, 54G10] (see: Weak P-point)

[43A45, 43A46] (see: Ditkin set) C-set-S set problem

[43A45, 43A46] (see: Ditkin set) cadlag function

[60B 10, 60G05] (see: Skorokhod space)

Cahn-Hilliard equation (82B26, 82D35) (refers to: Lagrange multipliers; Laplace operator; Spinodal decomposition) Cahn-Hilliard equation

[82B26, 82D35]

[81Qxx] (see: Dirac quantization) canonical sequence s e e : gap in a -Cantor set s e e : l / 4- --

capacity [26B99, 30C62, 30C65] (see: Quasi-regular mapping) capacity see: Choquet - - ; Jordan pair of finite - - ; l o g a r i t h m i c - - ; Lees classification of simple Jordan pairs of finite - - ; M o n g e - A m p & e - - ; Newton - -

capacity function [31C10, 32F05] (see: Pluripotential theory) capacity of a condenser

CHARACTERIZATIONTHEOREM OF ALGEBRAIZABLEDEDUCTIVESYSTEMS [26B99, 30C62, 30C65] (see: Quasi-regular mapping) capacitytheorem see: Shannon --

Carath6odory distance [31C10, 32F05] (see: Pluripotential theory) Carath6odory function [33C45] (see: Szeg6 polynomial) Caratheodory theorem s e e : generalized Julia-Wolff---; Julia- --; Julia-Wolffcardinal axioms s e e :

large - -

Cardinal function

[41A10, 41A50, 42A10, 65Txx] (see: Chebyshev pseudo-spectral method; Fourier pseudo-spectral method) cardinal n u m b e r s e e :

regular--

Carleman-Kaplansky theorem

[43A07, 43A15, 43A45, 43A46, 46J10] (see: Figh-Talamanca algebra) Carleman momentcondition [33C45] (see: Szegi~polynomial) Carleman type s e e : of--

integral operators

Carleman-type integral operator [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) carpet see: Sierpifiski - - ; universality of the Sierpir~ski - carrying particle s e e : spin- - C a r t a n invadant form s e e : Poincar~,-.- - -

Cartan matrix

[llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) Cartan matrix see: Borcherds- - - ; generalized -- ; symmetrizable - -

Cartesian-closed category (18D15) (refers to: Category; Small category; Topos) Cartier-Dieudonn6 module [55P421 (see: Brown-Gitler spectra) Cartier divisor [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) Cassini oval

[15A18] (see: Gershgorin theorem) Cassini oval see:

Brauer--

Castelnuovo-Mumfurdregularity [13A30, 13H10, 13H30] (see: Buchsbaum ring) cat ~ -group [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) Catalan constant (11M06, 11M35, 33B15) (refers to: Riemann zeta-function) Catalan numbers

[68S05] (see: Natural language processing) categorical characterization of Azumaya algebras

[13-XX, 16-XX, 17-XX] (see: Skolem-Noether theorem) categorical variable in covariance analysis [62Jxx] (see: ANOVA) categories s e e : derived equivalent - - ; Morita theory for derived - category s e e : * - a u t o n o m o u s - - ; arithmetical - - ; Baire - - ; C a r t e s i a n - c l o s e d - - ; closed - - ; Closed monoidal - - ; exact - - ; hereditary - - ; highest-weight - - ; K r u l I - R e m a k - S c h m i d t - - ; KrulI-Schmidt - - ; mesh - - ; monoidal - - ; symmetric closed monoidal - - ; symmetric monoidal - - ; topological - - ; triangulated - -

category CQML [03G10, 06Bxx, 54A40] (see: Fuzzy topology) category L-FTOP

[03G10, 06Bxx, 54?,40]

(see: Cellular algebra) cellular matrix ring

(see: Fuzzy topology) category L - T O P

[05Exx]

[03G10, 06Bxx, 54A40] (see: Fuzzy topology)

(see: Cellular algebra) cellular ring

C a u c h y e q u a t i o n s e e : additive - Cauchyidentity see: Jacobi- --

Cauchy-RiemannO-operator [53C15, 55N351 (see: Spencer cohomology) Cauchy-Szeg6 orthogonal projection

[46Lxx] (see: Toeplitz C* -algebra) C a u c h y t h e o r e m see: causal dependency

Binet---

[68T05] causal linear transformation [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) Cayley-Brill formula

[12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) Chasles---

Cayley colour diagram

[05C501 (see: Matrix tree theorem) Cayley graph (05C25) (referred to in: Baumslag-Solitar group) (refers to: Graph; Graph, oriented; Group; Petersen graph; Regular group) edge-transitive--

Cayley graph of a group

[05C25] (see: Cayley graph) Cayley graphs s e e : characterization of - Cay[ey-Hamilton theorem s e e : generalized - -

Cayley map

[05C25] (see: Cayley graph) Cech compactification the S t o n e - - -

see:

remainder in

ceiling function

[26Axx] (see: Floor function) cell

[05Exx] (see: Cellular algebra) see: H e l e - S h a w - - ; Schubert -cell in design of statistical experiments

cell

[62Jxx] (see: ANOVA) Cellular algebra (05Exx) (referred to in: Coherent algebra) (refers to: Centre of a ring; Coherent algebra; Galois correspondence; Graph isomorphism; Permutation group) cellular algebra

[05Exx]

[41A05, 41A30, 41A63] (see: Radial basis function) centre of a triangle see: first isogonic -centre of mass of a triangle

[51M04] [12F10, 14/-I30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) [51M04] (see: Wittenbauer theorem) centroid of a quadrangle

[51M04]

standard basis of

torics)

[05Exx] (see: Cellular algebra)

[30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator eoUigation) characteristic polynomialof a graph [05Cxx, 05D15] (see: Matching polynomial of a graph) [11B37, llT71, 93C05] (see: Shift register sequence)

[51M04] (see: Triangle centre)

characteristic polynomial of a matrix Vapnik- --

[15A18] (see: Gershgorin theorem) characteristic polynomial of a rankedpartially ordered set

Cevian lines

[51M04] (see: Isogonal) chain s e e : ascending Fitting - - ; descending Fitting - - ; Fitting - - ; length of a Fitting - chain condition s e e : countable - chain in a partially ordered set s e e : anti- - chain order s e e : symmetric - -

Chandro-Kozen-Stackmeyer alternation theorem

complexity

change see: convective rate of - - ; local rate of - - ; quadratic base - change of metric s e e : conformal - channel s e e : noisy q u a n t u m - - ; quantum -chaos s e e : homogeneous - - ; spatiotemporal - chaos decomposition theorem s e e : homogeneous - -

chaotic behaviour [34-04, 35-04, 58-04, 58Fxx, 58F14] (see: Conley index; Dynamical systems software packages) chaotic dynamical system

[28Dxx, 54H20, 58Fll, 58F13] (see: Absolutely continuous invariant

measure) chaotic solution [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) character see: Artin --; Dirichlet - - ; spin - - ; Teichm011er - character formula s e e : Kac-Weyl - - ; WeyI-Kac - - ; W e y l - K a c - B o r c h e r d s - characterincyclichomologysee: Chern - -

(see: Borcherds Lie algebra) character of a symmetric group see: - - ; spin - character of a weight m o d u l e s e e : mal --

[05B35, 05Exx, 05E25, 06A07, I 1A25] (see: Mdbius inversion) characterization s e e : generalized meanvalue - - ; Mean-value - characterization for harmonic functions s e e : mean-value - characterization for holomorphie functions see: mean-value - characterization of A z u m a y a algebras s e e : categorical -characterization of bipolar structures on groups s e e : Stallings --

characterization of Borcherds algebras [llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) characterization of Cayley graphs [05C251 (see: Cayley graph) characterization of local optimality on a region of stability

[90Cxxl (see: Fritz John condition) characterization of optimality s e e : saddlepoint - characterization of pluriharmonic functions see: mean-value - characterization of separately harmonic functions s e e : mean-value - characterization of the Brouwer degree s e e : axiomatic - -

characterization property for scattering data

[35P25, 47A40, 58F07, 81U20] full-line case) (see: Inverse scattering,

characterization theorem s e e : logical - -

characterization systems linear for-

character on a Banach algebra

[46H40] (see: Automatic continuity for Ba-

nach algebras)

characteristic of a square matrix s e e : Segre - -

eharacteristic polynomial of a linear feedback shift register

(see: Varignon parallelogram) centroid of a triangle

[03D15, 68Q15] Computational classes)

characteristic initial-value problemfor the Korteweg-de Vries equation [35Q53, 58F07] (see: Harry Dym equation) characteristic operator-valued function of an operator colligation

(see: Triangle centre) centre on an algebraic curve

Cervonenkis dimension s e e :

[30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) characteristic function of an operator vessel see: joint - - ; normalized joint - -

[08Bxx, 16R10, 17B01, 20El0[ (see: Speeht property)

[11Fxx, 17B67, 20D08]

Cellular algebra (in algebraic combina-

[05Exx]

centre-by-metabelian identity

character of a Borcherds algebra

(see: Cellular algebra)

cellular closure

[53C15, 55N35] (see: Spencer eohomology) characteristic function of a colligation

centra[extension see: universal - centralizer s e e : double - - ; left - - ; right - centre s e e : Triangle - -

(see:

Cech-complete semi-topologicalgroup [54C08] (see: Almost continuity) Cech-complete space [54C05, 54C08] (see: Strongly countably complete topological space)

cellular aigebra s e e : a--

[05Exx] (see: Cellular algebra) censoring [62Jxx, 62Mxx] (see: Cox regression model)

centroid

[05C25] (see: Cayley graph) Cayley formula for the number of labelled trees

Cayleygraph see:

characteristic at an eigenvalue s e e : Segre - characteristic class s e e : secondary - characteristic covector

centre for interpolation

(see: Machine learning)

Cayley-Brillformula see:

character variety [57Mxx, 57M25] (see: Skein module)

mete-

theorem for deductive

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) characterization theorem of algebraizable deductive systems

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) 477

CHARACTERIZATIONTHEOREM OF INVERSESCATTERINGTHEORY

characterization theorem of inverse scattering theory [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis ease) characterization theorems in logic [03Gxx] (see: Algebraic logic) charge see: magnetic - charge of a Borcherds algebra [17B10, 17B65] (see: Weyl-Kac character formula) Chasles-Cayley-BrUl formula (12F10, 14H30, 20D06, 20E22) (refers to: Algebraic curve; Algebraic geometry; Algebraically closed field; Bezout theorem; Birational morphisin; Differential field; Extension of a field; Genus of a curve; Local ring; Localization in a commutative algebra; Maximal ideal; Riemann surface; Separable extension) Chasles-Cayley-Brillformula [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) Chatzidakistheorem s e e :

Mel'nikov- --

Chebotarev density theorem (11R32, 11R45) (referred to in: Factorization of polynomials) (refers to: Dirichlet density; Frobenius automorphism) Chebyshevcoefficients [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) Chebyshevcoefficients s e e :

Fourier---

Chebyshev polynomial [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) Chebyshev polynomials s e e : relation for derivatives of - -

recurrence

Chebyshev pseudo-spectral method (41A10, 41A50, 42A10) (referred t o in: Fourier pseudospectral method) (refers to: Chebyshev polynomials; Fourier pseudo-spectral method; Lagrange interpolation formula; Polynomial; Trigonometric pseudospectral methods) Chebyshevseries see: F o u r i e r - - Chebyshev spectral method [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) Chebyshev tau method [65Lxx] (see: Tau method) Chebyshev-typepair of inverse relations [11B39] (see: Lucas polynomials) Chebyshev weightfunction [33C45] (see: Szeg6 polynomial) chemical index of a tree [15A15, 20C30] (see: Immanent) Chern character in cyclic homology [46L80, 46L87, 55N15, 58G10, 58G1 i, 58G12] (see: Index theory) Chem form [14H15, 30F60] (see: Weil-Petersson metric) Chern numbersee:

first --

Chernoff bound [05C80, 60D05] (see: Jansen inequality) Chervonenkisdimension s e e :

Vapnik---

Chevalley 2-cocycle [37J15, 53D20, 70H33] (see: Momentum mapping) Chevalley formula [14C15, 14M15, 14N15, 20G20, 57T15] 478

(see: Schubert calculus) chiral algebra [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 81RI0, 81T10, 81T30, 8IT40] (see: Moonshine conjectures; Vertex operator algebra) chiral Dirac operator [46L80, 46L87, 55N15, 58G10, 58GI1, 58G12] (see: Index theory) chiral field [81Qxx] (see: Dirac quantization) choice s e e : axiom of - - ; social - - ; Zermelo-Fraenkel set theory with the axiom of - -

choice function [03E30] (see: ZFC) Chomsky transformationalgrammar [68S05] (see: Natural language processing) Choquet capacity [31C10, 32F05, 68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory; Pluripotential theory) Chequer integral (28-XX) (refers to: Lebesgue integral; Measurable function; Measurable space; Measure; Set function) Choquet integral [28-XX] (see: Choquet integral) Choquet integral see: generalized - chore-division problem [00A08, 90Axx] (see: Cake-cutting problem) Chow coordinates [11J68, 14Q20] (see: Liouville-Lojasiewicz inequality) Chow form [13Pxx, 14Q20] (see: Hermann algorithms) Chowla-Selbergformula [81Qxx] (see: Zeta-functlon method for regularization) chromatic number of a random graph [05C80, 60D05] (see: Jansen inequality) chromatic polynomial [05Cxx, 05D15] (see: Matching polynomial of a graph) chromatic polynomial of a graph [05B35, 05Exx, 05E25, 06A07, i 1A25] (see: M6bius inversion) Chu construction [18D10, 18D15] (see: *-Autonomous category) Chu structure theorem s e e : C h u r c h A-calculus

[03D15, 68Q15] (see: Computational classes) Church thesis [03D15, 68Q15] (see: Computational classes)

B u n c e - --

complexity

complexity

circle s e e : Brocard - - ; fibration over a - - ; polynomials orthogona[ on a - - ; quasi- ; Seifert -circle polynomial s e e : Z e m i k e --

circle problem [11Lxx, 11L03, llL05, llL15] (see: Bombieri-Iwaniec method) circle problem s e e : Littlewood one- -circles s e e : isogonal - circuit s e e ; Boolean - - ; quantum -circuit complex s e e : broken- -c i r c u i t c o m p l e x i t y class

[03D15, 68Q15]

(see: Computational complexity classes) circuit depth [68Q151 (see: Average-case computational complexity) circuit flowsee: electrical -circuit problemsee: Hamiltonian-circuits s e e :

delay operation for - -

circulant graph [05c25] (see: Cayley graph) circular consecutive k-out-of-r~ system [60C05, 60K10] (see: Consecutive k-out-of-n: Fsystem) circular orbit [58F22, 58F25] (see: Seifert conjecture) eircumcentre of a triangle [51M04] (see: Triangle centre) clamped membrane [35J05, 35J25] (see: Diriehlet eigenvalue) clamped plate [35P15] (see: Rayleigh-Faber-Krahn inequality) clamped plate s e e : eigenvalue problem for the - - ; Rayleigh conjecture for the - -

Clarke calculus [90C30] (see: Clarke generalized derivative) Clarke generalized derivative (90C30) (refers to: Banach space; Hilbert space; Lipschitz condition; Semicontinuous function) Clarke tangent cone to a set [90C30] (see: Clarke generalized derivative) class s e e : bounded probabilistic polynomial time complexity - - ; circuit complexity - - ; complementary complexity - - ; complete problem for a complexity - - ; exponential-timecomplexity - - ; first Baire --; homotopy--; Iogspace complexity - - ; non-deterministic Iogspace complexity - - ; non-deterministic polynomial time complexity - - ; polynomial space complexity - - ; polynomial time complexity - - ; problem complete for a complexity - - ; quantum complexity -- ; quasi-equational - - ; Schubert - - ; secondary characteristic - - ; Szeg6 - - ; torsion - - ; torsion-free --

class A ? [68Q15] (see: Average-case computational complexity) class A S P A C E T I M E s e e : complexity - class ATIMEALT s e e : complexity - class average-7 9 s e e : complexity -class BPP s e e : c o m p l e x i t y - class closed under reduction s e e : complexity -class DSPACE s e e : complexity - class DTIME s e e : c o m p l e x i t y - -

class field towerproblem [11R29, 11R32] (see: Odlyzko bounds; Shafarevich conjecture) class group s e e : ideal - - ; mapping - class L s e e : complexity - class NC s e e : complexity -class NL s e e : complexity -class NP s e e : complexity -class NSPACE s e e : c o m p l e x i t y - class NTIME s e e : c o m p l e x i t y - class of algebras s e e : epimorphism over a - - ; equational logic of a - - ; homomorphism Ko-extensible over a - class of interpretations s e e : defining set of formulas for a - - ; e q u i v a l e n c e of formulas over a - - ; Fregean equivalence of formulas over a --

class" of models

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) class-one representation [11F03, 11F701 (see: Selberg conjecture) class operatorsee: trace- -class operatordeterminant s e e :

trace- - -

class 79 [68Q15] (see: Average-ease computational complexity) class P s e e : complexity - class PH s e e : complexity - class PSPACE s e e : c o m p l e x i t y - classes s e e : Computational complexity - - ; reducibility of complexity - -

classical affme plane [05B30] (see: Affine design) classical arithmetical semi-group [11Nxx, 11N32, 11N45, 11N80] (see: Abstract analytic number thedry) classical Euler product formula [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) classical information [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) classical M6bius inversion formula [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) classical Poisson formula [31A05, 31A10] (see: Poisson formula for harmonic functions) classical propositionalcalculus [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) classical state [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) classical state space [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) classical Stieltjes moment problem [44A60] (see: Strong Stieltjes moment problem) classification see: document --; Riedtmann

--

classification error [68T05] (see: Machine learning) classification of finitely generated groups with more than one end s e e : Sta]lings - classification of simple Jordan pairs of finite capacity s e e : Lops - classifier s e e : na'ive Bayesian --

classifier in a learning system [68T05] (see: Machine learning) classifying space [55Pxx, 55P15, 55P42, 55U35] (see: Algebraic homotopy; BrownGitler spectra) clause see: Horn - Clausenfunction [llM06, 11M35, 33B15] (see: Catalan constant) Clifford multiplication [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Clifford torus [53C42] (see: Winmore functional)

COMPACTSUBGROUP

climbing search see: hill- -closed see: ultraproduct--closed *-algebra see: uniformly -closed C * - a l g e b r a s e e : uniformly - -

closed category

[18D10] (see: Closed monoidal category) closed category [18D10] (see: Closed monoidal category) see: Cartesian- - closed cone of curves closed category

[14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) closed current

[32C30, 53C65, 58A251 (see: Current) closed field

see:

pseudo-algebraically--

Closed monoidal category (18D10) (referred t o in: *-Autonomous category) (refers to: Bifunctor; Category) closed monoidal category ric - -

see:

symmet-

coefficient see: Blomqvist - - ; Blomqvist q differencesign correlation - - ; drag - - ; first Lyapunov - - ; grade correlation - - ; medial correlation - - ; Pearson productmoment correlation - - ; population parameter of the BIomqvist -- ; reflection - - ; sample correlation - - ; transmission -coefficient B e n j a m i n - B o n a - M a h o n y equation see: v a r i a b l e - coefficient representation see: spectral - coefficients see: Beurling algebra of Fourier series with summable majorant of - - ; Chebyshev -- ; Fourier-Bessel -- ; F o u r i e r - C h e b y s h e v - - ; Fourier-Franklin --; Fourier-Hear--; Fouder-Jacobi--; Fourier-Laguerre--; Fourier-Legendre ; Fourier-Walsh - coefficients of a linear feedback shift register see: feedback - - - ;

- -

Cohen idempotent theorem [22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10] (see: Fourler-Stieltjes algebra) Cohen-Macaulayring

closed orbit

[58F22, 58F25] (see: Seifert conjecture) closed projections see: Kuratowski theorem on -closed under conjugation see: function algebra - closed under reduction see: complexity class - closed unit ball in a Banach space see: extreme point of the - -

closedness under consequence

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) closure see: cellular - - ; deductive - - ; Weisfeiler-Leman - closure of a group see: normal - closuretheorem see: Federer-Fleming - co-adjoint orbit [37J15, 5 3 D 2 0 , 7 0 H 3 3 ]

(see: Momentum mapping) co-connected space

(see: Dynamical systems software packages)

see:

1- - - ; 2 - - -

co-crossed product O*-algebra

see:

generalized--

Cohen-Macaulay simplicial complex

[05Cxx, 05D15] (see: Matching polynomial of a graph) co-monotone functions

[28-XX] (see: Choquet integral) co-Namioka space

[26A15, 54C05] (see: Namioka space) Cobb-Douglas function (90A11) cocycle see: Chevalley 2- - code see: additive quantum - - ; D e l s a r t e Goethals - - ; e r a s u r e - c o r r e c t i n g - - ; Kerdock - - ; linear - - ; G F 2-linear - - ; quantum - - ; quantum error-correcting - - ; Reed-Muller -- ; stabilizer quantum - code for the Dickman function see: Mathematica - codimension 1 see: isomorphism in - - ; surjectivity in - -

codimension-3 bifurcation [34-04, 35-04, 58-04, 58F14]

colligation see: characteristic function of a -- ; characteristic operator-valued function of an operator - - ; co-isometric operator - - ; isometric operator - - ; main operator of a unitary operator - - ; Operator ; regular - - ; rigged operator - - ; unitary operator - - -

47A48,

47A65,

47D40,

47N70] (see: Operator vessel)

coherent algebra see: standard basis of a - - ; structure constants of a - -

coherent configuration

[03Exx, 03E05, 05Exx] (see: Cellular algebra; Coherent al-

gebra)

coherent pair of measures [33C45, 33Exx, 46E35] (see: Sobolev inner product) k- --

(see: Zak transform) cohomological dimension one see; of-cobomologiea] dimension two [05C25, 20Fxx, 20F32]

group

(see: Baumslag-Solitar group) cohomological variety in representation theory [20J06] (see: Serre theorem in group cohomology) cohomology see: cuspidal - - ; Dolbeault - - ; Eisenstein - - ; inner - - ; intersection - - ; invariant - - ; L 2- - ; L p- - ; quantum - - ; Quillen theorem on Krull dimension of group - - ; Serre theorem in group ; Spencer - - ; stable - cohomology of a differential operator see: Spencer - c o h o m o l o g y o f flag m a n i f o l d s - -

[14C15, I4M15, 14N15, 20G20, 57T15] (see: Schubert calculus) cohomologyof the Baily-Borel compactideation [1lFxx, 20Gxx, 22E46] (see: Baily-Borel compactification) coincidence

[05A99, 11N35, 60A99, 60EI5]

[30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) collineation group [05B30] (see: Affine design) Collins conjugacy theorem for HNNextensions [20F05, 20F06, 20F32] (see: HNN-extension) collocation

see:

combinatorial Radon transform [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mgbius inversion) combinatorics see: algebraic - - ; probabilistic method in - combinatorics) see: Cellular algebra (in algebraic - -

commensurable subgroups

[11F25, 11F60] (see: Hecke operator)

commensurablesubgroups [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) commonality function

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory)

conditions

collocation method [65Lxx] (see: Tau method) collocation point [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods)

[03Exx, 03E05, 05Exx] (see: Cellular algebra; Coherent algebra)

coherent pair of measures see: coherent states [42Axx, 44-XX, 44A55]

co-matching graphs

colliding gravitational waves [35L15] (see: Euler-Poisson-Darboux equation)

[47A45,

combinatorial line (see: Hales-Jewett theorem)

collection of sets see: (k--1)separated -collections of blocks in a Steiner triple system see: mutually E-balanced - -

colligation

[20F05, 20F06, 20F32, 55Pxx, 55P15, 55U351 (see: Algebraic homotopy; HNNextension) [05D10]

collapsing property [47Dxx] (see: Taylor joint spectrum)

coherent algebra

co-isometric operator colligation

[46E22] (see: Reproduclng-kernel Hilbert space)

[llLxx, llL03, llL05, llL15] (see: Bombieri-Iwaniec method)

colligation of operators

coherent configuration see: fibre of a - - ; homogeneous - - ; intersection numbers of a -

co-isometry

coincidences in the double large sieve

[05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) Cohen-Macaulayfication [13A30, 13HI0, 13H30] (see: Buchsbaum ring) Coherent algebra (03Exx, 03E05) (referred to in: Cellular algebra) (refers to: Association scheme; Cellular algebra; Centralizer; Finite group, representation of a; Permutation group)

[46Lxx] (see: Toeplitz O*-algebra) co-invariant algebra [05E05, I3PI0,' 14C15, 14MI5, 14N15, 20G20, 57T15] (see: Schubert polynomials) [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation)

(see: I n c l u s l o n - e x c l u s i o n formula) coincidences see: problem of - -

orthogonal - -

Colombeau generalized function algebra

[46F30] (see: Colombeau generalized func-

tion algebras) Colombeau generalized function algebras (46F30) (referred to in: Generalized function algebras) (refers to: Generalized function algebras; Generalized function, derivative of a; Laplace operator; Net (directed set); Sobolevspace) Colombeau GFA

[46F30] (see: Colombeau generalized function algebras) colour diagram see: Cayley -colour superalgebra see: Borcherds - - ; colouring map for a - colourabilityproblem see: 3- - -

coloured Jones-Conway polynomial

[57M25] (see: Jones-Conway polynomial) colouredlink diagram see: colouring see: Fox n - - -

n- --

colouring group of a link [57M25] (see: Fox n-colouring) colouring map for a colour superalgebra [17B10, 17B65] (see: Weyl-Kae character formula) coma aberration [33C50, 78A05] (see: Zernike polynomials) combination see: Dempster rule of evidence - combination of two independent belief functions see: rule of - c o m b i n a t o r i a l group theory

communication see: quantum - communication complexity see: tum - -

quan-

communicationnetwork [05C25] (see: Cayley graph) see: double - commutant of an operator

eommutant

[47Dxx] (see: Taylor joint spectrum) commutative algebra see: local-global principle in - commutative anomaly see: Non- - commutative anomaly for zeta-function regularization see: non- -commutative integration see: non- -commutative linear logic see: non- - commutative logic see: foundations of n o n - - - ; non- - -

commutative operatorvessel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) commutative relation see: anti- - commutative residue see: non- - commutativetopology see: non- -commutative two-operator vessel see: quasi-Hermitian - commutative unification see: associativecommutativity

see:

weak--

commutativity in vertex algebras [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) commutatorrelation [46Ji0, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) commutingHamiltonian flows [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) commuting integrals [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) commuting operators see: essentially-compact see: Corson - - ; Eberlein - - ; Valdivia - compact derivative see: Gil de Lamadrid and Sova - compact group see: representation ring of e--

compact-open topology

[54C35] (see: Exponential law (in topology)) compact operators see: C * - a l g e b r a of ; Riesz theory of - - ; Spectral theory of -- ; Taylor spectrum for -compact space see: locally countably - - ; scattered - compact subgroup see: maximal -- -

479

COMPACT SUPPORT

compact support s e e : ot--

algebra of functions

compacfification [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) compactification see: arithmetic of the B a i l y - B o r e l - - ; B a i l y - B o r e l - - ; BorelSerre -- ; cohomology of the Baily-Borel - - ; maximal Satake - - ; moduli of the Belly-Betel - - ; non-Hermitian Satake - - ; reductive BoreI-Serre - - ; remainder in the Stone-Cech - - ; Satake - - ; SatakeBelly-Bore[ - - ; toroidal - compactifications s e e : examples of BailyBorel --

compactly generated topological space [54C35] (see: Exponential law (in topology)) compactness see: measure of non- - compactnessof a logic [03Gxx] (see: Algebraic logic) companion matrix [11B37, llT7I, 93C05] (see: Shift register sequence) competence [90A28] (see: Condorcet jury theorem) complement of a lattice element [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bins inversion) complementary complexity class [03D15, 68Q15] (see: Computational complexity classes) complementary series representation [11F03, llF70] (see: Selberg conjecture) complementation theorem s e e : Crape - complete see: Jkf79---; sequentially --; weakly sequentially - -

complete exponential sum [llL07] (see: Exponential sum estimates) complete for a complexityclass s e e : lem - -

prob-

complete function set [35P25] (see: Partial differential equations, property C for) complete holomorphic vectorfield [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) complete holomorphic vector field s e e : semi- - -

complete hyperbolicmetric [14H15, 30F60] (see: Wcil-Petersson metric) complete intersection see: local - complete market [90A09] (see: Option pricing) complete market assumption [90A09] (see: Portfolio optimization) complete net [05Bxx] (see: Net (in finite geometry)) completeorthomodu[ar[atticessee: semi-group of - -

Foulis

complete problem [68Q15] (see: Average-case computational complexity) complete problem see: .IV'7~ - - complete problem for a complexity class [03D15, 68Q15] (see: Computational complexity classes) complete projection scheme [47H17] (see: Approximation solvability) complete quasi-monoidal lattice [03G10, 06Bxx, 54A40]

480

(see: Fuzzy topology) complete sample [62Jxx] (see: ANOVA)

quantum - - ; quantum communication - - ; quantum computational - - ; worst-case - -

complexity bounds for the Hilbert Nullstellensatz [14A10, 14Q20] (see: Effective Nullstellensatz)

complete semi-topological group s e e : Cech- - complete space s e e : (~ech- - complete sup-lattice

[03G25, 06D99] (see: Quantale) complete topological space s e e : countably - -

Strongly

complete vector field [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) complete vector field s e e :

semi- --

completely crossed fitctors in covariance

analysis [62Jxx] (see: ANOVA) completely crossed layout [62Jxx] (see: ANOVA) completely free element in a field extension [12E20] (see: Galois field structure) completely free element in a Galois extension r1 IR32] (see: Normal basis theorem) completely-integrable system [35Q53, 58F071 (see: Harry Dym equation) completely normal element in a field extension [12E20] (see: Galois field structure) completely normal element in a Galois extension [11R32] (see: Normal basis theorem) completeness see: asymptotic --; .?~'~completeness theorem in algebraic logic [03Gxxl (see: Algebraic logic) complex see: broken-circuit --; CohenMacaulay simplicial - - ; dualizing - - ; f vector of a simp[icial - - ; formally exact - - ; h-vector of a simpticial - - ; pants - - ; second Spencer - - ; shellable simplicial - - ; sophisticated Spencer - - ; Spencer - - ; Stanley-Reisner ring of a simplicial - - ; tilting -- ; Tits simplicial --

complex decision problem [03D15, 68Q151 (see: Computational classes) complex dimension [28A80] (see: Sierpifiski gasket) complex function algebra [46E25, 54C35] (see: Bishop theorem)

complexity

complex of a partially ordered set s e e : der - complex structure deformation [14Jxx, 35A25, 35Q53, 57R57]

or-

(see: Whitham equations) complex structure on a manifold [14H15, 30F60] (see: Weil-Petersson metric) complex variable s e e : Quasi-symmetric function of a -complexes s e e : Sharp conjecture on dualizing - -

complexity class s e e : bounded probr abilistic polynomial time - - ; circuit - - ; complementary - - ; complete problem for a - - ; exponential-time - - ; Iogspace - - ; non-deterministic Iogspace - - ; nondeterministic polynomial time - - ; polynomial space - - ; polynomial time - - ; problem complete for a - - ; quantum - -

complexity class ASPACETIME [03D15, 68Q15] (see: Computational complexity classes) complexity class ATIMEALT [03D15, 68QI5] (see: Computational complexity classes) complexity class average-79 [68Q151 (see: Average-case computational complexity) complexity class BPP [03D15, 68Q15] (see: Computational complexity classes) complexity class closed under reduction [03D15, 68Q15] (see: Computational complexity classes) complexity class DSPACE [03D15, 68Q15] (see: Computational complexity classes) complexity class DTIME [03D15, 68Q15] (see: Computational complexity classes) complexity class L r03D15, 68Q15] (see: Computational complexity classes) complexity class NC [03D15, 68Q15] (see: Computational complexity classes) complexity class NL [03D15, 68Q15] (see: Computational complexity classes) complexity class NP [03D15, 68Q15] (see: Computational complexity elasses) complexity class NSPACE [03D15, 68Q15] (see: Computational complexity classes) complexity class NTIME [03D15, 68Q15] (see: Computational complexity classes) complexity class P [03D15, 68Q15] (see: Computational complexity classes) complexity class PH [03D15, 68Q15] (see: Computational complexity classes) complexity class PSPACE [03D15, 68Q15] (see: Computational complexity classes)

complexions-symbol [57M25] (see: Listing polynomials)

complexity classes s e e : Computational - - ; reducibility of - complexity of a sequence s e e : Linear - complexity of a shift register sequence s e e : linear --

complexity s e e : Average-case computational - - ; average-case time - - ; computational - - ; Kolmogorov - - ; polynomial on average time - - ; polynomial time - - ;

complexity of groups [203"061 (see: Serre theorem in group cohomology)

complexity of the membership problem over a module [13Pxx, 14Q20] (see: Herlnann algorithms) complexity profile of a sequence s e e : linear - component s e e : post-projective - - ; rational boundary -component KP-hierarchysee: two- - component of a symmetric space s e e : boundary - - ; rational boundary - component systems s e e : reliability of multi- --

composite event [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Sbafer theory) compositionoperator [39B05, 39B 12] (see: SchrSder functional equation) composition operators s e e : semi-group of-composition product s e e : Jaeger -eompositionalitysee: rule of - -

compressibleNavier-Stokes equation [76Axx] (see: Knudsen number) compressible surface in a threedimensional manifold [57N10] ( s e e : Haken manifold) compressible surface in a three-dimensional manifold s e e : b o u n d a r y - - ; O- - computability s e e : polynomial-time - computable language s e e : bounded-error polynomial-time - - ; bounded-error quantum polynomial-time - computation s e e : measurement in quantum - - ; model of - - ; model of quantum - - ; quantum - computation, theoryof s e e : Quantum --

computational complexity [03D15, 68Q15] (see: Computational classes)

complexity

computational complexity s e e : case - - ; quantum - -

Average-

Computational complexity classes (03D15, 68Q15) (referred to in: Average-case computational complexity; Quantum computation, theory of) (refers to: Algorithm; Boolean algebra; Church thesis; Computable function; Decision problem; Acalculus; Parallel random access machine; Turing machine) computational fluid dynamics [76Cxx] (see: Von K~rm~invortex shedding) computational intractability [68S05] (see: Natural language processing) computational learning theory [68T05] (see: Machine learning) computer s e e :

quantum--

computerized tomography [41A30, 92C55] (see: Ridge function) computing see: distributed quantum

--

concavity s e e : logarithmic -concentration s e e : metastable --

concentration field [82B26, 82D35] (see: Cahn-Hilllard equation) concentration of afunction around a point [42A63] (see: Uncertainty principle, mathematical) concentration of a function around a point see: measure of - -

concept formation system [68T05] (see: Machine learning) concept learning [68T05] (see: Machine learning)

CONSTRAINT QUALIFICATIONS

concordance [62H20] (see: Kendall tau metric; Spearman rho metric) concordant pairs of real numbers [62H20] (see: Spearman rho metric) concordant sample elements [62H20] (see: Kendall tau metric) concrete algebraic logic [03Gxx] (see: Algebraic logic) concurrency theory [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) concurrent lines in a triangle [51M04] (see: Isogonal) condenser [26B99, 30C62, 30C65] (see: Quasi-regular mapping) condenser s e e : capacityof a - condition s e e : basic Fritz John - - ; binormality - - ; C * - - ; Calder6n projection boundary - - ; Carleman moment - - ; countable chain - - ; Ditkin - - ; essential radius--; Fritz J o h n - - ; generalized mean-value - - ; Karush-Kuhn-Tucker --; Kuhn-Tucker - - ; MangasarianFromovitz--; mass-gap--; matchedends - - ; positive-energy - - ; primal optimality--; radiation--; SAW*-; Slater-; strong cone - ; Szeg5 -; Tarski finiteness - - ; tFL - - ; trace - - ; trapped-orbit - - ; truncation - - ; Winker - condition belief function s e e : a priori- - condition for a reconstruction formula for the continuous wavelet transform s e e : admissibility -condition for decay s e e : Faddeev - condition in obstacle scattering s e e : Dirichlet - - ; Neumann - - ; Robin - condition of flow invariance s e e : tangency - condition of substitution invarianee s e e : Tarski - -

condition IFL [26A21, 54E55, 54G20] (see: Sorgenfrey topology) conditional belieffunction [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) conditionally negative-definite matrix [41A05, 41A30, 41A63] (see: Radial basis function) conditionally positive-definite matrix [41A05, 41A30, 41A63] (see: Radial basis function) conditionals see: Cane - conditioning s e e : Dempster rule of -conditions s e e : colligation - - ; gradingrestriction - - ; node - - ; r e g u l a d z a t i o n - - ; Tarski -conditions for groups s e e : finiteness - conditions in Banaeh-Jordan pairs s e e : finiteness - conditions of a linear feedback shift register see: initial - -

Condorcet jury theorem (90A28) (referred to in: Condorcet paradox) (refers to: Condorcet paradox; Independence; Probability; Social choice; Statistical test) Condorcet paradox (90A28) (referred to in: Condorcet jury theorem) (refers to: Arrow impossibility theorem; Condorcet jury theorem; Probability; Social choice; Voting paradoxes) Condorcet paradox [90A28] (see: Condorcet paradox) Condorcet winner

[90A28] (see: Condorcet paradox) conduction s e e :

heat--

conductor of a local ring [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) cone s e e : convex - - ; finitely generated - - ; generator of a - - ; Kleiman-Mori - cone condition s e e : strong -cone of curves s e e : closed - -

cone theorem [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) cone to a set s e e :

Clarke tangent - -

confidence interval [62Jxx] (see: ANOVA) confidence interval see: Scheff~-type simultaneous - - ; Tukey-type simultaneous confidence intervals see: simultaneous --

-

confidence set [62Jxx] (see: ANOVA) confidence set [62Jxx] (see: ANOVA) configuration s e e : t3rocard - - ; coherent - - ; constant - - ; fibre of a coherent - - ; homogeneous coherent - - ; intersection numbers of a coherent - - ; ( p , 1 ) - - - ; Pasch - - ; variable --

L e o p o l d t - - ; M a h l e r - - ; Milnor unknottiny - - ; modified Seifert - - ; monomial - - ; Montesinos-Nakanishi - - ; M o n t e s i n o s Nakanishi three-move - - ; N a k a y a m a --; Novikov--; permanent*on-top--; permanental dominance - - ; POT - - ; Ramanujan--; Ramanujan-Petersson - - ; SchinzeI-Zassenhaus - - ; second flip - - ; Seifert - - ; Selberg - - ; Shafarevich - - ; strong Ditters -- ; topological Vaught - - ; upper bound - - ; Vandiver - - ; Vaught - - ; w e a k Ditters - - ; Willmore - - ; wrapping - - ; Zarankiewicz crossing number - - ; Zariski-Lipman - - ; Zassenhaus - - ; Zucker - conjecture at infinity s e e : RamanujanPetersson -conjecture for DiricNet eigenva/ues s e e : P61ya - conjectureformanifoldssee: immersion - conjecture for mapping cylinders s e e : Atiyah-Floer - conjecture for Neumann eigenvalues s e e : Pdlya - conjecture for plane domains s e e : nodal line - conjecture for the clamped plate s e e : Rayleigh - conjecture in inverse Galois theory s e e : Shafarevich - conjecture on accessibility of finitelygenerated groups s e e : Wall - conjecture on dualizing complexes s e e : Sharp - conjectures s e e : flip - - ; generalized moonshine - - ; Monstrous Moonshine -- ; Moonshine - - ; Tait - conjectureson alternatinglinks s e e : Tait - -

configuration in a translation quiver [16670] (see: Riedtmann classification) cnnformal change of metric [53C421 (see: Willmore functional) conjugacy problemfor Fibonacci groups conformal field theory [20F38] [1IF11, 14Jxx, 17B67, 20D08, 35A25, (see: Fibonacci group) 35Q53, 57R57, 81T10] conjugacy problem for presentationsof a (see: Moonshine conjectures; Whitham free Burnside group equations) [20F05, 20F06, 20F32, 20]750] conformal geometry (see: Burnside group) [53C42] conjugacytheorem for HNN-extensions s e e : (see: WiUmorefunctional) Collins - conformal invariant conjugate group s e e : finite - [26B99, 30C62, 30C65] conjugate point s e e : isogonnl - - ; iso(see: Quasi-regular mapping) tomic - eonformal mapping s e e :

quasi- --

conformal quantum field theory [11Fll, I7BI0, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) conformal volume [53C42] (see: Wilhnore functional) conformally minimal surface [53C42] (see: Wilhnore functional) congruence s e e : equationally definable principal relative - - ; Leibniz - - ; principal Q - - ; Q - - ; second-order Leibniz - - ; Tarski - - ; theorem on equivalence systems and the Suszko -congruence extension property s e e : relative - congruence o! a theory over $5 s e e : Suszko -congruence relation s e e : principal -congruencesubgroup s e e : principal - congruences s e e : equationally definable principal -congruentialmethod s e e : L i n e a r - conjecture s e e : Amol'd - - ; A t i y a h Fleer--; Basor-Tracy--; Bennequin - - ; Birch-Swinnerton-Dyer -- ; C o n w a y Norton Monstrous Moonshine - - ; counterexample to the Seifert - - ; E r d t s Tur~.n - - ; first f l i p - - ; Fisher-Hartwig - - ; four exponentials - - ; Fried-VOlklein - - ; Frobenius - - ; generalized Shafarevich - - ; G o l d b a c h - - ; Greenberg--; Gromov-Lnwson --; Gr0nbaum - - ; Iwasawa - - ; Iwasawa main - - ; Jones unknotting -- ; Langlands -- ; Lehmer - - ;

conjugate t r a n s f o r m a t i o n

[35L15] (see: Euler-Poisson-Darboux equation) conjugation s e e : function algebra closed under - - ; isogonal - - ; isotomie - -

conjugation behaviour of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) conjugation of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) conjunctive normal form [68Q15] (see: Average-ease computational

complexity) Conley index (58Fxx) (refers to: Brouwer degree; Contractible space; Flow (continuoustime dynamical system); Homotopy; Homotopy type; Metric space; Morse index) Conley index [58Fxx] (see: Conley index) C o n l e y i n d e x s e e : example of the - connected space s e e : l - c o - - - ; 2-co- - connection s e e : anti-self-dual -- ; flat - - ; Shirnikov - connection theorem s e e : Galois - connections s e e : moduli space of flat - - ; Quillen theory of super- - connective s e e : arity of a logical - - ; Iogieat - - ; rank of a logical - -

Connes index theoremfor foliations

[46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Connes-Moscovici higher index theorem for coverings [46L80, 46L87, 55N15, 58610, 58G1 I, 58G12] (see: Index theory) c o n o r m i n t e g r a l s e e : t- - conormmeasure see:

f;- - -

Consecutive k-out-of-n: F-system (60C05, 60K10) (referred to iu: Fibonacci polynomials; Lucas polynomials) (refers to: Fibonacci polynomials; Lucas polynomials) consecutive k-out-of-n: G-system [60C05, 60K10] (see: Consecutive k-out-of-n: Fsystem) consecutive k-out-of-n structure [60C05, 60K10] (see: Consecutive k-out-of-n: Fsystem) consecutive k - o u t - o f - n system s e e : cular - - ; linear -

consecutive system [60C05, 60K10] (see: Consecutive k-out-of-n: system) consequencesee:

cir-

F-

closedness under --

consequence relation [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) conservation law [46F10] (see: Multiplication of distributions) c o n s e r v a t i o n l a w s e e : local - c o n s e r v a t i o n l a w s s e e : infinitely many - -

conservation laws for the BenjaminBona-Mahony equation [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) conservation of energy [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) conservative discrete-time system [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) conservative linear operator [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphie

mappings) consistency of an inversion method [35P25, 47A40, 8 iU20] (see: Inverse scattering, half-axis

ease) consistencyof set theory s e e : G t d e l relative - constant s e e : Catalan - - ; coupling - - ; Euler-Mascheroni - - ; Euler theorem on the Euler-Mascheroni - - ; Gauss - - ; Gauss lemniscate - - ; Lebesgue - - ; lemniscate - - ; norming - - ; renormalized - - ; renormalized coupling --

constant configuration [05B07, 05B30] (see: Pasch configuration) constant curvature s e e : space of -constants of ncoherent algebra s e e : structure - constants of muJti-dimensional partial F o u r i e r s u m s s e e : Lebesgue -constitutive relation

[76Axx] (see: Knudsen number) constrained local minimum [90Cxx] (see: Fritz John condition) constraint s e e : constraint

active - - ; fuzzy --

qualifications

[90Cxx] (see: Fritz John condition) 481

CONSTRUCTIBILITY

constructibility s e e : G 6 d e l axiom of - c o n s t r u c t i b l e f u n c t i o n see: space- --;

c o n t i n u o u s with respect to a given m e a s u r e see: m e a s u r e , a b s o l u t e l y - -

time- - construction see: Borel - - ; C h u - - ; coupling --; Evans-Griffith --; gardener

continuum mechanics [73Bxx, 76Axx] (see: Material derivative method) continuummechanicssee: motionin --

string--; Gerfand-NaTmark-Segal--; Seifert - c o n s t r u c t i o n m e t h o d in t h e stability of functional e q u a t i o n s s e e :

direct - -

construction of free Burnside groups [20F05, 20F06, 20F32, 20F50] (see: Burnside group) constructive function theory [46E35, 65N30] (see: Bramble-Hilbert lemma)

contraction s e e :

divisorial - - ; small - -

contraction-and-deletion relation [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) contraction morphism

[14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) contraction of a function

constructive induction system

[42A16, 42A24, 42A28]

[68T05] (see: Machine learning) contact form [53C20, 53C22, 58F22, 58F25] (see: Santal6 formula; Seifert conjecture) contact term [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations)

(see: Beurling algebra)

content of a marked shifted tableau

[05E10, 05E99, 20C25]

contraction theorem [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) contractive projection [17Cxx, 46-XX] (see: JB *-triple) contrast

[62Jxx] (see: ANOYA) see: q u a n t u m - convective rate of change

control

(see: Schur Q-function)

CONTENT software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages)

[73Bxx, 76Axx] (see: Material derivative method) convergence see: linear --; p o w e r --; quadratic --; rate of --; Wijsman--

context see: discourse -context in n a t u r a l l a n g u a g e s e e :

c o n v e r g e n c e o f a n u m e r i c a l series s e e :

tional

situa-

--

continuation invariance

[58Fxx] algorithm

see:

f u z z y - - ; n e a r - - ; quasi- - ; s e p a r a t e - continuityfor Banach algebras see: Automatic - -

continuity ideal

[46H40] (see: Automatic continuity for Ba-

nach algebras) continuity of m e a s u r e s s e e :

absolute --

Abso-

continuousregularizationsee:

[90Cxx] (see: Fritz John condition)

series)

Absolutely-uppersemi-

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) continuous semi-group see: differentiabie - - ; e x p o n e n t i a l f o r m u l a r e p r e s e n t a t i o n of a - - ; exponential representation of a - - ; g e n e r a t e d - - ; g e n e r a t o r of a - - ; locally uniformly - - ; p r o d u c t f o r m u l a r e p r e s e n tation of a - differentiabil-

ity of - - ; flow-invariance for - - ; p a r a m e t ric representations of g e n e r a t o r s of - -

continuous wavelet transform

[42Cxx] (see: Daubechies wavelets) ad-

missibility condition for a reconstruction f o r m u l a for the - - ; reconstruction f o r m u l a for the - -

482

almost - -

[42A20, 42A32, 42A38]

fuzzy --; L- --;

c o n t i n u o u s wavelet t r a n s f o r m s e e :

convexgroup presentation see: almost - c o n v e x hull s e e : polynomially --

(see: Integrability of trigonometric

continuous semi-group

continuoussemi.groupssee:

convexdomain see: p s e u d o - - - ; rank of a p s e u d o - - - ; strictly pseudo- - -

convex null sequence

L - f u z z y - - ; lattice- - - ; lattice-fuzzy - continuous measures see:

Minkowski - -

convex cone [06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property)

convex model

continuous location theory [90B85] (see: Fermat-Torricelli problem) continuous mapping see:

convex bodytheorem see:

c o n v e x metric s p a c e s e e :

quasi- - s p a c e of - -

c o n t i n u o u s invariant m e a s u r e s e e : lutely - -

[31A05, 31B05, 31CI0, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization)

convex image [37J15, 53D20, 70H33] (see: Momentum mapping)

automatic - -

continuous Denjoy-Wolfftheorem [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) c o n t i n u o u s function s e e : c o n t i n u o u s functions s e e :

weak --

converse of Gauss mean-value theorem for harmonic functions

Schur -continuity s e e : Almost - - ; H61der - - ; joint - - ; L - - - ; L - f u z z y - ; lattice- - - ; lattice-

continuitytheory see:

ra-

c o n v e r g e n c e rate s e e : algebraic - - ; exponential - - ; infinite - - ; spectral - -

(see: Conley index) continued-fraction-like

dius of - c o n v e r g e n c e of m e a s u r e s s e e :

convex programming

[90Cxx] convex programming [90Cxx] (see: Fritz John condition) convex programming see:

partly - -

convex rigid Baumslag-Solitargroup [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) quasi- - -

convexset see: polynomially -convexity s e e : Polynomial - convolution s e e : Dirichlet - - ; S I - - - ; unitary --

convolution algebra [46F10, 46H40] (see: Automatic continuity for Banach algebras; Multiplication of distributions) convolutionand Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) convolution and Zak transformation

[42Axx, 44-XX, 44A55] (see: Zak transform)

Corput method see:

Conway see:

van der--

correct a p p r o x i m a t i o n a p p r o a c h to D e m p s t e r Shafertheory see: marginally --

[42Axx, 44-XX, 44A55] (see: Zak transform)

correct l e a r n i n g s e e :

algebraic t a n g l e s in t h e

s e n s e of - -

p r o b a b l y approxi-

mately -erasure---; quantum

correctingcodesee:

Conway algebra (57P25) (refers to: Conway skein triple; Jones-Conway polynomial; Link; Quasi-group; Skein module; Torus knot) C o n w a y aIgebra s e e :

partial - - ; u n i v e r s a l

partial - -

error- - correction s e e :

q u a n t u m error- - -

correlation coefficient s e e : difference sign --; g r a d e - - ; m e d i a l - - ; Pearson productmoment -- ; sample -correlatorssee: field - -

associativityequationsfor

correspondence see:

Conway group .0 [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) Conway-Norton Monstrous Moonshine conjecture [llFI 1, lVBI0, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra)

--; --;

Shimura --

correspondence theorem f o r 7~-filters

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Corson compact [26A15, 54C05] (see: Namioka space) coset diagram see:

[57M251 (see: Rational tangles) Conway polynomial (57P25) (refers to: Alexander-Conway polynomial)

coset ring of [43A45, 43A46] (see: Ditkin set)

C o n w a y polynomial s e e : Alexander- --; coloured Jones- --; Jones- -C o n w a y relation s e e : Jones- --; skein m o d u l e based on t h e J o n e s - - -

Conway skein equivalence (57P25) (refers to: Conway skein triple; Jones-Conway polynomial; Link; Signature) Conway skein relation

[57M25] Alexander-Conway polynomial) Conway skein triple (57P25) (referred to in: Alexander-Conway polynomial; Brandt-LiekorishMillett-Ho polynomial; Conway algebra; Conway skein equivalence; Homotopy polynomial; JonesConway polynomial) (refers to: Brandt-Lickorish-MillettHo polynomial; Kauffman polynomial; Skein module; Threedimensional manifold) Conze-Lesigne algebra [28D05, 54H20] (see: Wiener-Wintner theorem) Conze-Lesigne factor [28D05, 54H20] (see: Wiener-Wintner theorem) (see:

see:

region of - -

coordinate ring of an algebraic c u r v e s e e : affine - coordinates s e e : Chow --; Fenchet-

Auslander-Reiten

shifted R o b i n s o n - S c h e n s t e d - K n u t h

Conway notation for rational tangles

cooperation

(see: Fritz John condition)

convexsequence see:

see: p- - convolution under Zak transformation convolution o p e r a t o r

cosets s e e :

double

Schreier--

- -

cosine transform [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) cosine transform see:

discrete - - ; fast

discrete - cost s e e :

link - - ; route - -

cost in quantum information processing

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81PI5, 94Axx] (see: Quantum information processing, science of) cotangent space at a marked Riemann surface

[I4H15, 30F60] (see: Weil-Petersson metric) cotilting module

[16Gxx] (see: Tilting module; Tilting theory)

cotorsion theory [16D40] (see: Flat cover) count see:

Borda - -

c o u n t of involutions s e e : Schur --

countable approximateunit [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) countable chain condition [54D40, 54G10] (see: Weak P-point) eountablycompact space see:

Nielsen - - ; F e n c h e I - N i e l s e n intrinsic - - ; trilinear - -

counter function

[90D05]

corona algebra [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Corona G'*-algebra [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) corona of a C'*-algebra [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C'* -algebras) corona problem [30Axx, 46Exx] (see: BMOA-space) corona set [46J10, 46L05, 46L80, 46L85] (see: Multipliers of G'* -algebras)

(see: Sprague-Grundy function) counterexample to the Seifert conjecture

[58F22, 58F25] (see: Seifert conjecture)

counting parenthesations [68S05] (see: Natural language processing) see: counting processes c o u n t i n g problem

[62Jxx, 62Mxx] coupled Dirac operator [35Qxx, 78A25] (see: Magnetic monopole) coupling constant [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations)

van der --

Corput k t h derivative e s t i m a t e s e e : der --

real root - - ; root - -

(see: Cox regression model)

van der-van der --

C o r p u t e x p o n e n t pair s e e :

locally - -

c o u n t a b l y c o m p l e t e topological s p a c e s e e : Strongly --

countably infinite set [03E99, 04A99] (see: Hilbert infinite hotel)

C o r p u t A-process s e e : C o r p u t B-process s e e :

Frobenius-

van

coupling constant see:

couphng construction

renormalized --

DARBO FIXED-POINTTHEOREM [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) Courant nodal line theorem

cross-section

[35J05, 35J25]

[35P25]

(see: Diriehlet elgenvalne) covariance s e e : analysis of - covariance analysis s e e : categoricalvariable in - - ; completely crossed factors in - - ; crossed factors in - - ; crossing factors in - - ; incompletely factors in - - ; nested factors in - - ; nesting factors in - - ; partly crossed factors in - - ; qualitative factors in -- ; quantitative factors in --

covariant quantization

[81Qxx] (see: Dirac quantization) covariate

[62Jxx] (see: ANOVA) covector s e e : characteristic-cover s e e : ,T'- - - ; Flat - - ; flat pre- - - ; metaplectic - - ; pre- - - ; projective -cover in e graph s e e : matching - covering s e e : flat - covering domain s e e : plane- - -

covering relation in a partially ordered set

[05D05, 06A07] (see: Sperner property)

Cox regression model (62Jxx, 62Mxx) (refers to: Central limit theorem; Conditional distribution; Errors, theory of; Likelihood-ratio test; Martingale; Random variable; Regression analysis; Stochastic process) cqml

[03G10, 06Bxx, 54A40] (see: Fuzzy topology) CQML see: c a t e g o r y Craig interpolation theorem [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Crank-Nicolson method [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Crapo complementation theorem

[05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdbius inversion) creation property

[llF11, 17B10, 17B65, 17B67, 17B68, 20D08, 81RI0, 81T30, 81T40] (see: Vertex operator algebra) criterion see: Hdrmander wave front set - - ; Jacobian - - ; Schneider-Lang -criterion for Buchsbaum rings s e e : surjectivity - -

critic in a learning system

[68T05] (see: Machine learning)

[62Jxx] (see: ANOVA) crossed factors in covariance analysis s e e : completely - - ; partly - crossed layout s e e : c o m p l e t e l y - crossed p r o d u c t

[11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) crossed product C*-algebra [46Lxx] (see: Toepfitz C* -algebra) crossed product C*-algebra s e e : co- - crossing s e e :

dexiotropic - - ; l e o t r o p i c --

crossing factors in covariance analysis

[62Jxx] (see: ANOVA) crossing number

[57M25] (see: Jones-Conway polynomial) crossing n u m b e r conjecture s e e : kiewiez --

Zaran-

[05C10, 05C35] (see: Zarankiewicz crossing number conjecture) crosswise topology

[26A21, 54E55] (see: Slobodnik property) Crumtransformation see:

Darboux---

Cmmeyrolle-Pr~istaro quantization [81Qxxl (see: Dirae quantization) crunode

[14H201 (see: Tacnode) public-key --

cryptography [I2D05] (see: Factorization of polynomials) public-key - - ; quan-

cube-free superabundant number

[1 lAxx] (see: Abundant number) cubic s e e :

cuspidal - -

cumulative hierarchy of the universe of sets

[03E30] (see: ZFC)

Current (32C30, 53C65, 58A25) (referred to in: Geometric measure theory) (refers to: Analytic manifold; Analytic set; Complex manifold; Differentiable manifold; Differential form; Frdchet space; Generalized function; Geometric measure theory; Vector space) [32C30, 53C65, 58A25] (see: Current)

[11M06] (see: Riemann (-functlon)

critical point theory [55M25] (see: Brouwer degree) critical problemfor matroids [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdbius inversion) critically finite self-similarset s e e : post- -Hyperbolic - - ; Step hyper-

current [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) current s e e : boundary of a - - ; closed - - ; exact - - ; exterior differential of a - - ; integral - - ; m-rectifiable - - ; mass of a --; positive - -

current of integration

[32C30, 53C65, 58A25] (see: Current) curtailed version of a statistical test

- -

[62Lxx]

cross-cut in a lattice

05Exx,

05E25,

11A25]

06A07,

(see: Average sample number) Curtis b o u n d e d n e s s t h e o r e m s e e :

(see: Mi3bius inversion)

cross-cuttheorem

Curtis formula

(see: Index theory) see:

curvature s e e : space of constant - curvature direction s e e : principal - curvature relations see: zero- - curve s e e : affine coordinate ring of an algebraic - - ; Bdzier - - ; centre on an algebraic - - ; first neighbourhood of a point on an algebraic - - ; Flecnode on a planar -- ; Frobenius automorphism on a - - ; function field of an algebraic --; genus of a - - ; Jordan - - ; local ring of a point on an algebraic - - ; noded stable- - - ; plane a l g e b r a i c - - ; polynomial - - ; resonance - - ; second neighbourhood of a point on an algebraic - - ; Seiberg-Witten Toda - - ; simple point on an algebraic - - ; singular point on an algebraic - - ; spectral - - ; zeta-function of a -curve of a family of line bundles s e e : spectral -curve of an operator vessel s e e : discriminant - - ; input determinantal representation of the discriminant - - ; output determinantal r e p r e s e n t a t i o n of the discriminant --

Bade-

cyclic h o m o l o g y in--

curve of periodic orbits of a dynamical system

[34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) curve on the sphere s e e : elastic - curve singularities see: A k - -curves s e e : closed cone of - - ; Dedekind formula for algebraic - - ; Huxley t h e o r y of resonance - - ; modelling growth - - ; modulus of a family of -- ; Riemann approach to algebraic - -

(see: Dynamical systems software

packages) curves over finite fields s e e : hypothesis for --

Riemann

cusp

[11Fxx, 14H20, 20Gxx, 22E46] (see: Baily-Borel eompactifieation; Tacnode) cusp see: double --; ramphoid - cusp bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) cusp form [11F67, 11Lxx, 11L03, 11L05, 11L15] (see: Bombleri-Iwanlec method; Eisenstein cohomology) cuspidal automorphicrepresentation [11F03, l l F 0 ] (see: Selberg conjecture)

Chern character

cyclotomic algebra

[11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) cylinder s e e : mapping - cylinders s e e : Atiyah-F]oer c o n j e c t u r e for mapping --

cylindric algebra

[03Gxx] (see: Algebraic logic)

cylindric algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) cylindric algebra see: r~-aryrepresentable --; representable - -

cylindrifications

[03Gxx] (see: Algebraic logic)

D

curve of infinite length [28A80] (see: Sierpifiski gasket)

[34-04, 35-04, 58-04, 58FI4]

cryptographicsystem s e e :

cryptography s e e : t u m --

[20G05] (see: Steinberg module)

curves of equilibria of a dynamical system

current

critical line for the zeta-function

[05B35,

(see: Obstacle scattering) crossed factors in covariance analysis

crossing number of a graph

covering theorem see: Arkhanget'skiTFrolfk - coverings s e e : Connes-Moscovici higher index theorem for - - ; higher index theorem for - -

cross s e e : bolic

[05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdblus inversion)

A-filtration [16Gxx] (see: Tilting theory) A -move

[57M25] (see: Reldemelster theorem; Tangle

move) J-Poincar6 lemma [53C15, 55N35] (see: Spencer cohomology) O-compressible surface dimensional manifold

in a three-

[57N10] (see: Haken manifold)

O-equation [14H15, 30F60] (see: Weil-Petersson metric) O-incompressible surface in a threedimensional manifold

[57N10] (see: Haken manifold) O-operator s e e :

Cauchy-Riemann --

D-bracket-derivative formula

[11F11, 17BI0, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) "D-filters s e e : for - -

correspondence theorem

[1117671

D-graph [05Cxx, 05D15] (see: Matching polynomial of a graph) D-optimal design [05C50] (see: Matrix tree theorem)

(see: Eisenstein cohomology)

d-sequence

cuspidal cohomology

cuspidal cubic [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) cuspidal representation [11F03, 11FT0] (see: Selberg conjecture) cut in a lattice s e e : cross- - cut theorem s e e : cross- - cutting problem see: Cake- - CW-space

[55Pxx, 55P15, 55U35] (see: Algebraic homotopy) C(X) see: Ext group of - cycle s e e : heteroclinic - - ; relative R - l - ; Schubert -cycles s e e : basis t h e o r e m for Schubert - - ; duality theorem for Schubert - - ; numerically equivalent relative R - l - --

cyclic homology [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12]

[13A30, 13H10, 13H30] (see: Buchsbaum ring) d - s e q u e n c e see: unconditionedstrong d +-sequence

--

[13A30, 13H10, 13H30] (see: Buchsbaum ring)

d'Alembert equation for finite sum decompositions (26B40) Danzer theorem [52A35] (see: Geometric transversal theory) Darbo fixed-point theorem (47H10) (refers to: Banaeh space; Compact mapping; Compact operator; Completely-contlnuous operator; Continuous mapping; Contraction; Fixed point; Hausdorff measure; Lipschltz condition; Metric space; Schauder theorem) 483

DARBOUX-CRUM TRANSFORMATION

Darboux-Crum transformation [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-line case) Darboux equation see: Euler-Poissen- - ; generalized E u l e r - P o i s s o n - -Darboux operator see: q-difference analogue of the E u l e r - P o i s s o n - - -

Darboux transformation [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, fulMine case) Darcy law [76Exx, 76S05] (see: Viscous fingering) data see: characterization property for scattering - - ; extra - - ; local tomographic - - ; missing - - ; multipleinstruction multiple- - - ; nested missing - - ; noise in - - ; scattering - - ; singleinstruction multiple- - data for a minimal surface see: Weierstrass - -

data noise [68T051 (see: Machine learning) Daubechies wavelets (42Cxx) (refers to: Fourier transform; Function of compact support; Orthonorreal system; Wavelet analysis) Dbar equation [14H15, 30F60] (see: Weil-Petersson metric) DDE-BIFTOOL software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) de Branges-Rovnyakfunctionalmodel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) De Bruijn function see: D i c k m a n - - de Lamadrid and Sova compact derivative see: Gil --

de Rbam differential [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) de Rham isomorphism [11F67] (see: Eisenstein cohomology) de Rham operator [46L80, 46L87, 55N15, 58G10, 58GIl, 58G12] (see: Index theory) de Vries equation see: averaged solution of the K o r t e w e g - - - ; characteristic initialvalue problem for the K o r t e w e g - - - ; dispersionless K o r t e w e g - - - ; K o r t e w e g - - - ; shocks for the K o r t e w e g - - - ; Whitham equation for the K o r t e w e g - - de V r i e s - L a n d a u - G i n s b u r g model see: Hurwitz-space K o r t e w e g - - de Vries solution see: gap K o r t e w e g - - -

death process [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) decay see: Faddeevcondition for -decay of a function at infinity see: of-decaying function see: slowly -decidability see: semi- - -

rate

decidable equational theory [03Gxx] (see: Algebraic logic) decimation method [28A80] (see: Sierpifiski gasket) decision

see:

484

complexity

deduction-detachment theorem equivalence of E D P R C and - -

see:

decision problem see: accepted input in a - - ; complex - - ; rejected input in a - decision process see: Markov - -

deduction property of a logic [03Gxx] (see: Algebraic logic)

decision theory [28-XX, 62Lxx] (see: Average sample number; Nonadditive measure) decision tree in machine learning [68T05] (see: Machine learning) decoherence [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of)

deduction system

decomposable measure see: pseudoaddition -decomposition see: axiom of natural - - ; Birkhoff - - ; domain - - ; It6-Wiener - - ; JSJ - - ; Lebesgue - - ; optimal domain - - ; resolution in group - - ; spectral - - ; spinodal - - ; West - - ; Wiener-It6 - - ; Wold - -

deductive system see: algebraizable - - ; equivalential - - ; extensional - - ; faithful interpretation of a -- ; filter-distributive - - ; filter of a - - ; finitely algebraizable - - ; finitely e q u i v a l e n t i a l - - ; Fregean - - ; intensional - - ; interpretation of a - - ; logical equivalence of formulas with respect to a --;logically equivalent formulas with respect to a - - ; matrix model of a - - ; model of a - - ; protoalgebraie -- ; second-order finitely algebraizable -- ; self-extensional - - ; strongly finitely algebraizable - - ; theorem of a - - ; theory of a - - ; underlying - - ; weakly algebraizable - deductive systems see: characterization theorem for - - ; characterization theorem of algebraizable - defavourablespace see: o - - ~ - - -

decomposition as a k-acylindrical graph of groups [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) decomposition method see: domain - decomposition of a group see: JSJ - decomposition property see: dominated - - ; Riesz - -

decomposition property ofF. Riesz [06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) decomposition theorem see: homogeneous chaos - - ; J a e o - S h a l e n Johansson - - ; Lebesgue - - ; probabiiistic Riesz - - ; Riesz -- ; Wiener-It6 -decomposition theorem for harmonic spaces see: Riesz -decomposition theorem for operators see: Riesz - -

decomposition theorem for sets of finite Hausdorff measure [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) decomposition theorem for subharmonic functions see: Riesz - decomposition theorem for superharmonic functions see: Riesz - decompositions see: d'Alembert equation for finite sum -decreasing rearrangement see: spherical - -

decreasing rearrangement of a function [35P151 (see: Rayleigh-Faber-Krahn inequality) Dedekind definition of infinity [03E99, 04A99] (see: Hilbert infinite hotel) Dedekind domain (13F05) (refers to: Dedeklnd ring) Dedekindformula for algebraic curves [12FI0, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) Dedekind zeta-function [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) Dedekind zeta-function [llNxx, 11N32, 11N45, IINS0] (see: Abstract analytic number theory) Dedekind zeta-function factorization of the - -

multi-stage--

decision problem [03D15, 68Q15] (see: Computational classes) decision problem [68Q15]

(see: Average-case computational complexity)

see:

Hadamard

deduction-detachment system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) deduction-detachment theorem [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

see:

uninterpreted - -

deduction theorem [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) deductive closure [68T05] (see: Machine learning) deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

defeat [90A28] (see: Condorcet paradox) defect spectrum [47Dxx] (see: Taylor joint spectrum) deficient number [11Axx] (see: Abundant number) deficient number see: o~-non- - - ; non- -definability [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35]

(see: Abstract algebraic logic) definability see: Beth - - ; equational - - ; local Beth - - ; weak Beth - definability property see: Beth - - ; w e a k Beth - definabilitytheorem see: Beth - definable principal congruences s e e : equationally - definable principal relative congruence see: equationally - defining a set of atomic formulas explicitly over another set of atomic formulas see: set of formulas - defining a set of atomic formulas implicitly over another set of atomic formulas see: set of formulas - defining equations see: system of - -

defining set of formulas for a class of interpretations [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) definite Hermitian matrices see: generalization of the H a d a m a r d - F i s c h e r inequality for positive semi- -definite kernel see: non-negative- - definite matrix see: conditionally negative- ; conditionally positive- - - ; positive- -definition see: ideal of - definition of a set of atomic formulas over another set of atomic formulas see: strong implicit - definition of infinity see: Dedekind - deformation see: complex structure - - ; isomonodromy - -

deformation of a structm'e [53C15, 55N35] (see: Spencer cohomology) deformation parameters of moduli

[14Jxx, 35A25, 35Q53, 57R57] (see: WhRham equations) deformation quantization [81Qxx[ (see: Dirac quantization) deformations see: Whitham hierarchy

of isomonodromic - deformed soliton lattice see: w e a k l y - deforming n - m o v e s see: skein module based on relations - degenerate Jordan pair see: non- - degree s e e : additivity-excision of the 8 r o u w e r - - ; axiomatic characterization of the Brouwer - - ; Brouwer - - ; existence property of the Brouwer - - ; homotopy invariance of Brouwer - - ; homotepy invariance of the Brouwer - - ; L e r a y - S c h a u d e r - - ; local B r o u w e r - - ; necessity - - ; normatization property of the B r e u w e r - - ; possibility - - ; product theorem for the Brouwer - - ; topological - -

degree bounds for the HilbertNullstellensatz [14A10, 14Q20] (see: Effective Nullstellensatz) degree factorization see: distinct- - - ; equal- - degree f o r a S o b o l e v f u n c t i o n space

[55M25] (see: Brouwer degree) degree of a net [05Bxx] (see: Net (in finite geometry)) degree of a permutation group [20-XX] (see: Regular group) degree of belief [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) degree of polarization [78A40] (see: Stokes parameters) degree of symmetricmappings [55M25] (see: Brouwer degree) degree on an additive arithmetical semigroup [l 1Nxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) degree one in an extension of algebraic number fields see: prime ideal of - degrees s e e : generalized - -

degrees of freedom [62Jxx] (see: ANOVA) delay differential equation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) delay operation for circuits [68Q15] (see: Average-case computational complexity) deletion relation

see:

contraction-and---

Delsarte-Goethals code (94Bxx) (refers to: Error-correcting code; Kerdock and Preparata codes) Delsarte theorem [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) Delsarte type see: theorem of - - ; tworadius theorem of - -

Delsarte-type theorem [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) delta-net see: model - - ; strict -deltasequence see: Kronecker- - demand s e e : f l o w - -

Demazure formula [14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) Dembowski theorem

DILWORTHTHEOREM

[05B30] (see: Affine design) Dempster rule of conditioning [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) Dempster rule of evidence combination [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) Dempster-Shafer theory (68T30, 68T99, 92Jxx, 92K10) (refers to: Capacity; Hypergraph; N'79; Probability theory) Dempster-Shafer theory see: axiomatic approach to - - ; marginally correct approximation approach to - - ; naive approach to - - ; qualitative approach to - - ; quantative approach to -Denjoyintegral s e e : narrow--

Denjoy-Perron integrability [28A25] (see: Denjoy-Perron integral) Denjoy-Perron integral (28A25) (refers to: Denjoy integral; KurzweilHenstoek integral; Lebesgue integral; Perron integral) Denjoy set [58F22, 58F25] (see: Seifert conjecture) Denjoytheorem see:

Wolff---

Denjoy-Wolff point [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Denjoy-Wolff theorem (30D05, 32H15, 46G20, 47H17) (referred to in: Semi-group of holomorphic mappings) (refers to: Analytic function; Banach space; Compact mapping; Contraction operator; Dunford integral; Frdchet derivative; Functional calculus; Hilbert space; Horocycle; Hyperbolic metric; Inner product; Jordan curve; Julia-WolffCaratModory theorem; Poincar~ model; Schwarz lemma; Semi-group of holomorphic mappings; Spectrum of an operator; Uniform convergence) Denjoy-Wolff theorem [301305, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Denjoy-Wolff theorem s e e : - - ; Fan analogue of the --

continuous

Dennis-Stein symbol [19Cxx] (see: Steinberg symbol) denominator identity for Borcherds algebras [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) denotation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) denotation function [03Gxx] (see: Algebraic logic) dense generalized function algebra s e e : nowhere- -- ; Rosinger nowhere- -denseideal s e e : nowhere - -

dense representation of a multivariate polynomial [12D05] (see: Factorization of polynomials) densities s e e :

examples of Dirichlet - -

density [28-XX] (see: Absolutely continuous measures) density see: Dirichlet --; vorticity -density Hales-Jewett theorem [05DI01 (see: Hales-Jewett theorem) density Hales-Jewett theorem Fu rstenberg-Katznelson - -

density operator

see:

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) density Ramsey theory [05D10] (see: Hales-Jewett theorem) density theorem see: Chebotarev - - ; Preiss - -

(see: Abstract algebraic logic) detachment see: 17,- -detachment system see: deduction- -detachment theorem s e e : deduction- -- ; equivalence of E D P R C and deduction- - determinant s e e : Faddeev-Popov --; Fredholm operator -- ; Hanke[ - - ; infinite - - ; operator - - ; Toeplitz - - ; trace-class operator - -

density topology [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) dependencysee: causal -dependency graph [05C80] (see: Lovfisz local lemma) dependent variable in regression analysis [62Jxx] (see: ANOVA)

determinant anomaly [81T501 (see: Non-commutative anomaly) determinant bundle [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) determinant of the Dirac operator [81T50] (see: Non-commutative anomaly)

depth s e e :

determinantal representation s e e : maximal -determinantal representation of the dieerJminant curve of an operator vessel s e e : input - - ; output - determinants s e e : regularization of infinite --

circuit --

depth-first algorithm [90D05] (see: Sprague-Grundy function) derivatioo see: k- - derivational analogy [68T05] (see: Machine learning) derivative s e e : angular - - ; approximative - - ; Clarke generalized - - ; exterior - - ; formal - - ; generalized directional - - ; Gil de Lamadrid and Sova compact - - ; local - - ; local time - - ; material - - ; material time - - ; mixed - - ; mobile time - - ; particle - - ; Radon-Nikod~2m -derivative estimate s e e : van der Corput kth --

derivative following a particle [73Bxx, 76Axx] (see: Material derivative method) derivativeformula s e e : D-bracket-derivative in spatial form see: formula for the material - - ; material - derivative m e t h o d s e e : Material -derivative operator s e e : material - derivative property s e ¢ : L (-- 1)- -derivatives of Chebyshev polynomials s e e : recurrence relation for - derived categories s e e : Morita theory for - -

derived equivalent categories [16Gxx] (see: Tilting theory) derived equivalent rings [16Gxx] (see: Tilting theory) descending Fitting chain [20F17, 20F18] (see: Fitting chain) descriptionlength s e e : m i n i m u m -description of a gas s e e : kinetic - design s e e : Affine - - ; affine resolvable - - ; D - o p t i m a l - - ; Hadamard 2- - - ; Hadamard 3- - - ; line in a - - ; resolvable ~,-- ( v , k , ) ~ ) - - ; resolvable transversal - - ; symmetric balanced i n c o m p l e t e block - - ; symmetric transversal - - ; transversal - d e s i g n for statistical experiments s e e : balanced - -

design matrix [62Jxx] (see: ANOVA) design of experiments [62Jxx] (see: ANOVA) design of statistical experiments [62Jxx] (see: ANOVA) design of statistical experiments see:

cell in -- ; effect in - - ; interaction in - - ; main effect in --

design theory [05C25] (see: Cayley graph) designated set of a logical matrix [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35]

determinate strong Stieltjes moment problem [44A60] (see: Strong Stieltjes moment problem) determined operator [53C15, 55N35] (see: Spencer eohomology) deterministic dynamical system [28Dxx, 541-120, 58F11, 58F13] (see:Absolutely continuous invariant measure)

[05C25] (see: Cayley graph) difference sign correlation coefficient [6211201 (see: Kendall tau metric) different [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) different ideal [12F10, 14H30, 20I)06, 20E22] (see: Chasles-Cayley-Brill formula) different words in Ulysses [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) different words in Ulysses see: number of-differentiability s e e : Silva - differentiability of continuous

semi-

groups [321-115, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphie mappings) differentiable continuous semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings)

deterministic Iogspace complexity class s e e : non- - deterministic polynomial time complexity class see: non- - deterministic Turning machine s e e : non- ; time for a non- - deviation s e e : large - deviation behaviour s e e : large - - ; moderate - -

differentiable mappings s e e : space of infinitely Silva- - differential s e e : Abelian - - ; 8eltrami - - ; de R h a m - - ; Seiberg-Witten - differential-difference equations s e e : Toda-type - differentialequation s e e : d e l a y - - ; neutral - - ; q u a n t u m spectral measure of a partial - - ; singular partial - - ; Sturm-Liouville - - ; Thiele -differential equations s e e : elliptic partial - - ; formal Diree quantization of partial - - ; property C ' + for ordinary - - ; property C~, for ordinary - ; property C ' ~ for ordinary - - ; property C for partial - - ; property C v for partial -differential equations, property C for s e e : Ordinary - - ; Partial - differential of a current see: exterior --

dexiotropic crossing [57M25] (see: Listing polynomials)

differential of afield [12F10, I4H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula)

"DEXT'

differential on a k - a l g e b r a s e e : K#,hler -differential operator s e e : hypo-elliptic pseudo- - - ; hypo-elliptic symbol of a pseudo- - - ; index of an elliptic partial - - ; principal symbol of a - - ; pseudo- - - ; Spencer cohomology of a - - ; symbol of a pseudo- -- ; zeta-function of a pseudo- - differentials s e e : m o d u l e of K~ihler - differentiation s e e : Szeg6 fractional --

[68Q151 (see: Average-case computational complexity) diagonalizable matrix [15A42] (see: Bauer-Fike theorem) diagonafizable matrix [15A42] (see: Bauer-Fike theorem) diagonalization problem s e e : matrix - diagram s e e : Cayley eolour - - ; 'n,coloured link - - ; positive - - ; rotant of a link - - ; Schreier coeet - - ; shifted Young --

dichromatic polynomial [57M25] (see: Homotopy polynomial; Kauffman bracket polynomial) Dickman-De Bruijn function [11Axx] (see: Diekman function) Dickman function (11Axx) (refers to: Inclusion-exclusion formula; Laplace transform; Prime number; Riemann hypotheses) Diekman function s e e : Mathematiea code for the - Dieudonnd module s e e : Cartier-difference s e e : divided - difference analogue of the E u l e r - P o i s s o n Darbouxoperator s e e : q- -difference equations s e e : Toda-type differential- - difference method s e e : finite - -

difference set

diffraction theory of aberrations [33C50, 78A05] (see: Zernike polynomials) diffusion system s e e :

reaction- --

Digamma function i11M06, 11M35, 33B15] (see: Catalan constant) digraph s e e : leaf of a - - ; locally walkbounded -Dijkg raaf-Verlinde-Verlinde equations s e e : Wilt e n - - Dijkgraaf-Verlinde-Verlinde theory s e e : Witten- --

Dijkstra algorithm (05C 12, 90C27) (refers to: Graph; Greedy algorithm) Dijkstra shortest-path algorithm [05C12, 90C27] (see: Dijkstra algorithm) dilatation s e e : outer - -

inner--;

maximal - - ;

Dilworth number of a partially ordered set [05D05, 06A07] (see: Sperner property) Dilworth theorem [05B35, 05Exx, 05E25, 06A07, 11A25]

485

DILWORTHTHEOREM

(see: Mdbius inversion) dimension s e e : box - - ; Buchsbaum local ring of maximal embedding - - ; complex ; fractal - - ; Gel'fand-Kirillov - - ; global - - ; invariance of - - ; j - - - ; Kroll - - ; L y a p u n o v - - ; Minkowski - - ; projective - - ; s i m i l a r i t y - - ; s p e c t r a l - - ; Vapnikd : e r v o n e n k i s - - ; Vapnik-Chervonenkis - - ; VC- - dimension of group cohomology see: Quillen theorem on Krul[ - dimension of spaces of automorphic forms see: Langlands formula for the - dimension of the Sierpifiski gasket see: fractal - dimension one see: group of cohomological - dimension two see: cohomological - -

-

dimensional Jbrmula [17A401 (see: Freudenthal-Kantor triple system) diminutos number [11Axx] (see: Abundant number) dimodule Azumaya algebra see: Long H---

Dini subderivate [90C30] (see: Clarke generalized derivative) Dirac algebra (15A66, 81Q05, 81R25, 83C22) (refers to: Clifford algebra; Dirac matrices; Hypercomplex number; Minkowski space; Panii matrices; Schrddinger equation) Dirac distribution (46Fxx) (referred to in: Absolutely continuous invariant measure; Fig$~alamanca algebra; Pluripotential theory; Poisson formula for harmonic functions; Trigonometric pseudo-spectral methods) (refers to: Delta-function; Dirac delta-function) Dirac equation [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) Dirac gamma matrices [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) Dirac measure [43A07, 43A15, 43A45, 43A46, 46F10, 46F30, 46J10] (see: Egorov generalized function algebra; Figg-Talamanca algebra; Generalized function algebras; Multiplication of distributions) Dirac monopole (81V10) (refers to: Connection; Connections on a manifold; Four-dimensional manifold; Gauge transformation; Hopf fibration; Lens space; Magnetic field; Maxwell equations; Planck constant; Principal fibre bundle; Spherical coordinates) Dirac operator [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Dirac operator [47Dxx] (see: Taylor joint spectrum) Dirac operator see: chira[ --; coupled ; determinant of the - - ; g e n e r a l i z e d spin - - ; twisted - -

-

-- ;

Dirac quantization (81Qxx) (refers to: Algebraic topology; Boolean algebra; Commutative algebra; Hilbert space; Hopf algebra; Lie algebra; Linear operator; Locally convex space; Measure space; Quantum field theory; Quantum 486

groups; Self-adjoint operator; Topological vector space) Dirae quantization of partial equations see: formal - -

differential

Dirac representation [15A66, 81Q05, 81R25, 83C22| (see: Dirac algebra) Dirac string singularities [81V10] (see: Dirac monopole) direct construction method in the stability offunctional equations [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) direct potential scattering [35P25, 47A40, 81U20] (see: Inverse scattering, multidimensional case) direct scattering problem [35P25, 47A40, 81U20] (see: Inverse scattering, multidimensional case) direct scattering problem [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-llne case) direct scattering problem on the half-axis [35P25, 47A40, 8lU20] (see: Inverse scattering, half-axis case) directed graph of a matrix [15A18] (see: Gershgorin theorem) direction see: asymptotic --; principal curvature - -

direction for a ridge function [41A30, 92C55] (see: Ridge function) directionaIderivative

see:

generalized - -

Dirichlet boundary value problem [35J05, 35J25] (see: Diriehlet eigenvalue) Dirichlet character [llLxx, 11L03, 11L05, llL15] (see: Bombieri-Iwaniec method) Dirichlet condition in obstacle scattering [35P25] (see: Obstacle scattering) Dirichlet convolution ( 11A25) (refers to: Arithmetic function; Binary relation; Commutative ring; Dirichlet series; Mfbius function) Diriehlet densities

see:

examples of - -

Dirichlet density (1 IR44, 11R45) (referred to in: Chebotarev density theorem) (refers to: Algebraic number) Dirichlet eigenfunction [35J05, 35J25] (see: Dirichlet eigenvalue) Dirichlet eigenvalue (35J05, 35J25) (referred to in: Neumann eigenvalue; Rayleigh-Faber-Krahn inequality) (refers to: Brownian motion; Diriehlet boundary conditions; Heat equation; Laplace operator; Neumann eigenvalue; Potential theory; Rayleigh-Faber-Krahn inequality) Difichlet eigenvalue [60Gxx, 60J55, 60J65] (see: Wiener sausage) Dirichlet eigenvalues see: Pdlya eoniecture for - - ; Weyl asymptotics for - -

Dirichlet eigenvalues of the Laplacian [35P15] (see: Rayleigh-Faber-Krahn inequality) Dirichlet form [60Hxx, 60J55, 60J65] (see: Skorokhod equation) Dirichlet inverse of an arithmetical function [11A25] (see: Dirichlet convolution)

Dirichlet Laplacian [35J05, 35J25, 35P25] (see: Dirlehlet eigenvalue; Obstacle scattering) Dirichlet polynomial [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdbius inversion)

discriminant curve of an operator vessel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) discriminant curve of an operalorvessel see:

Oirichlet problem see: associated with a - -

discriminant form [11F25, 11F60] (see: Hecke operator) discriminant of an algebraic number field [11R29] (see: Odlyzko bounds)

spectral measure

Dirichlet Sturm-Liouville operator [34B24, 34L40] (see: Sturm-Liouvine theory) Dirichlet-to-Neumann mapping [35P25, 47A40, 81U20] (see: Inverse scattering, multidimensional case) Dirichlet unit theorem [llR23] (see: lwasawa theory) disc see: Hen._

analytic - - ; Gershgorin - - ;

disc algebra [30D50, 46Exx] (see: VMOA-space) disc polynomials [33C50, 78A05] (see: Zernike polynomials) discernment see: frame of - discontinuoustangential velocity [76C05] (see: Birkhoff-Rott equation) discordant pairs of real numbers [62H20] (see: Spearman rho metric) discordant sample elements [62H20] (see: Kendall tau metric) discourse context [68S05] (see: Natural language processing) discourse in natural language of-

see:

domain

discovery system [68T05] (see: Machine learning) discrepancysee:

global - - ; lattice point - -

discrete cosine transform [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) discrete cosine transform

see:

fast - -

discrete dynamical system [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) discrete Fourier transform [65Txx] (see: Fourier pseudo-spectral method) discrete Fourier transform [42Axx, 44-XX, 44A55] (see: Zak transform) discrete Gel'fand quantale [03G25, 06D99] (see: Quantale) discrete homomorphism of Gel'fand quantales [03G25, 06D99] (see: Quantale) discrete Laplace transform [39A12, 93Cxx, 94A12] (see: Z-transform) discrete logarithm [12E20] (see: Galois field structure) discrete logarithm [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) discrete-timesystem

see:

conservative--

discrete Toda lattice [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) discrete wavelet tranfform [42Cxx] (see: Daubechies wavelets)

input determinantal representation of the - - ; Output determinantal representation of the - -

disedminant polynomialsee:

irreducible--

discriminant polynomial of an operator vessel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) discriminator variety [03Gxx] (see: Algebraic logic) discs see: glueing of -dispersioniess Korteweg~le Vries equation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) displacement rank [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) displacement rank

see:

low - -

displacement vector [73Bxx, 76Axx] (see: Material derivative method) dispute resolution [00A08, 90Axx] (see: Cake-cutting problem) dissipative trapped ion mode in a plasma [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) dissipativity of the Kuramoto-Sivashinsky dynamics [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Dist N"P [68Q151 (see: Average-case computational complexity) distance see: Carath~odory--; Hausdorff - - ; integral flat - - ; Kobayashi - - ; mapping preserving a - -

distance functionfor stratifications [57N80] (see: Thom-Mather stratification) distance metric [90C08] (see: Travelling salesman problem) distance-preserving mapping [54E35] (see: Aleksandrov problem for isometric mappings) distinct-degree factorization [12D05] (see: Factorization of polynomials) distinguished polynomial [11R23] (see: Iwasawa theory) distortion see: bounded --; mapping with bounded - -

distortion aberration [33C50, 78A05] (see: Zernike polynomials) distortion of a mapping [26B99, 30C62, 30C65] (see: Quasi-regular mapping) distributed quantumcomputing [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) distribution see: bivariate normal - - ; Dirac - - ; double exponential--; Laplace - - ;

DYNAMICS SOLVERSOFTWARE LNRE - - ; log-normal - - ; malign - - ; me* meat of a probability - - ; normalized restriction of a probability - - ; positive - - ; principal value - - ; retarded - - ; uncorrelated random variables with joint normal - - ; universal -- ; Yule - - ; Y u l e - S i m o n - distribution of e i g e n v a l u e s s e e : asymptotic - distribution of Laplacians s e e : Weyl asymptotic formula for the eigenvalue - distribution of singular values s e e : asymptotic - -

distributional problem

[68Q15] (see: Average-case computational complexity) distributional product s e e : individual - distributions s e e : algebra of retarded --; localization of - - ; Multiplication of - - ; multiplier theory of -distributivedeductivesystem s e e : filter- - -

distributive lattice [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) distributivelattices s e e :

varietyof --

Ditkin condition

[43A45, 43A46] (see: Ditkin set) Ditkin problem s e e :

synthesis- - -

Ditkin set (43A45, 43A46) (refers to: Algebra of functions; Fourier algebra; Fourier transform; Harmonic analysis; Locally compact space; Metrizable space; Net (directed set); Orthogonality; Scattered space; Sequence; Spectral synthesis; Uniform boundedness) Ditkin set

[43A45, 43A46] (see: Ditkin set) Ditkin set s e e : Wiener- --

strong - - ; wide-sense - - ;

Ditkin set in the wide sense

[43A45, 43A46] (see: Ditkin set) Ditkin sets s e e : injection theorem for - Ditkin theorem s e e : W i e n e r - -bitters conjecture s e e : strong - - ; w e a k -divergence s e e : infrared --

divided difference

[05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) division see: envy-free --; fair -division algebra s e e : quaternion -division problem s e e : chore- - - ; fair - divisor s e e : C a r t i e r - -

divisor function [llLxx, llL03, llL05, llL15] (see: Bombieri-Iwaniee method) divisor problem [llLxx, llL03, llL05, llL15] (see: Bombleri-Iwanlec method) divisorial contraction

[14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) Dixmier trace [35Sxx, 46Lxx, 47Axx] (see: Wodzicki residue) document classification [68S051 (see: Natural language processing) document retrieval [68S05] (see: Natural language processing) Dolbeault cohomology [53C15, 55N35] (see: Spencer cohomology) domain see: balanced - - ; Bezout - - ; bounded symmetric - - ; C- - - ; Dedekind - - ; E- -- ; hypercenvex - - ; i nvariance of - - ; Lipschitz - - ; nodal -- ; plane-covering -- ; pseudo-convex - - ; rank of a pseudoconvex - - ; Reinhardt - - ; Riemann m a p ping function of a - - ; rough - - ; Siegel - - ;

star-shaped --; strictly pseudo-convex - - ; summation -- ; symmetric --

domain decomposition [46E35, 65N30] (see: Bramble-Hilbert lemma) domain decomposition s e e :

optimal - -

domain decompositionmethod [46Cxx] (see: Alternating algorithm) domain in a Banach space see:

double point

doubling bounded

symmetric - -

Domain (in ring theory) (13-XX, 16-XX) (refers to: Associative rings and algebras; Commutative ring) domain knowledgebase [68S05] (see: Natural language processing)

dual system

[14Hxx, 14H20] (see: Flecnode; Tacnode) double Scbubertpolynomials [05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) see:

period- - -

duality functor

Doug~as-Fillmoretheory s e e : Brown- -Douglas function s e e : Cobb-Downs-Thomson paradox

dualitytheorem see: man - -

[18D10, 18D15] (see: *-Autonomous category)

[60K30, 68M10, 68M20, 90BI0, 90B 15, 90B 18, 90B20, 94C99] (see: Braess paradox) drag coefficient (see: Von K~irm~invortex shedding)

[68Q15] (see: Average-case computational complexity) dominant s e e :

homozygous - -

dominant integral highest weight [17B10, 17B65] (see: Weyl-Kac character formula) dominated decomposition property

[06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) domination of measures

drag due to vortex shedding [76Cxx] (see: Von K~irm~invortex shedding) drag due to vortex shedding s e e :

total --

draw in a game

[90D05] (see: Sprague-Grundy function) drawing of a finite graph

[05CI0, 05C35] (see: Zarankiewicz crossing number conjecture) dressing method s e e : Shabat - drift s e e : singular - -

Zakharov-

Drinfel'd-Turaev quantization (16Wxx, 57P25) (referred to in: Skein module) (refers to: Commutative ring; Epimorphism; Free module; Knot theory; Poisson algebra; Skein module) Driiffel'd-Turaev quantization

[28-XX]

[16Wxx, 57P25]

(see: Absolutely continuous mea-

(see: Drinfel'd-Turaev quantization)

sures) domino tiling problem

[68Q15] (see: Average-case computational complexity) Donaldson invafiant [53C15, 57R57, 58D27, 81V10] (see: Atlyah-Floer conjecture; Dirac monopole) double annihilator

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) double bubble [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) double centralizer

Drinferd-Turaev quantization see: w e a k -driven phrase structure g r a m m a r s e e : head- - DSPACE s e e : complexity class - -

DsTool software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) DTIME see: eomplexityclass - dual s e e : Spanier-Whitehead - - ; topological --

dual algorithmof linear programming [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) see: self- -dual basis of an ordered basis of afield dual basis

[46J10, 46L05, 46L80, 46L85]

[12E20]

(see: Multipliers of C* -algebras)

(see: Galois field structure)

double commutant [47Dxx] (see: Taylor joint spectrum) double cosets [11F25, 1IF60] (see: Hecke operator) double cusp

[14H20] [62D05] (see: Acceptance-rejection method) double exponentialhat function [62D05] (see: Acceptance-rejection method) double Hopf bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) double Hopf bifurcation s e e :

resonant --

double large sieve

[11Lxx, 11L03, 11L05, llL15] (see: Bombieri-Iwaniec method) double large sieve the --

dual Bass number of a module [16D40] (see: Flat cover) dual Brown-Gitler modules

[55P42] (see: Brown-Gitler spectra) dual Brown-Gitler spectra

[55P42]

(see: Tacnode) double exponential distribution

see:

coincidences in

(see: Brown-Gitler spectra) dual connection s e e :

Alexander - - ; Feffer-

duality theorem for Schubert cycles

[14C15, I4M15, 14N15, 20G20, 57T15] (see: Schubert calculus) duality theory

[76Cxx]

[68S05] (see: Natural language processing) domain withrotational symmetry [31B05, 33C55] (see: Zonal harmonics)

dominance of problem reduction

dual Y a n g - M i l l s equations s e e : self- - duality s e e : C * - - - ; electric-magnetic --; Spanier-Whitehead--; Stone--; Vecten-Fasbender --; WhiteheadSpanier - -

Douglas algebra [30D50, 46Exx] (see: VMOA-space)

domain of discourse in natural language

domains see: nodal line conjecture for plane - domestic algebra s e e : tame -dominance conjecture s e e : permanentel - -

[15A39, 90C05] (see: Motzkln transposition theorem)

anti-self- - -

dual of a non-empty set

[15A39, 90C05] (see: Motzkin transposition theorem) dual polynomial basis see: w e a k l y self- -dual reductive pairs

[11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representa-

tion) dual resonancetheory [llFll, 17B10, 17B65, 17B67, 20D08, 81R10, 8iT30] (see: Vertex operator) dual space of the Beurling algebra

[42A16, 42A24, 42A28] (see: Beurling algebra)

[03Gxx] (see: Algebraic logic)

dualiziug complex [13A30, 13H10, 13H30] (see: Buchsbaum ring) dualizing complexes s e e : ture on - -

Sharp conjec-

dualizing object [18D10, 18D15] (see: *-Autonomous category) DubovitskiI-Milyutin theorem [90Cxx] (see: Fritz John condition) due to vortex shedding see: drag - - ; total drag - Dunfordfunctionalcalculus s e e : R i e s z - - Dunfordintegral s e e : Riesz---

Dunwoody accessibility theorem [20F05, 20F06, 20F32] (see: HNN-extension) Dym equation s e e : 2+l-dimensional Harry - - ; extended Harry -- ; generalized Harry - - ; Harry --

dynamic loading of flexible structures [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) dynamic portfolio optimization

[90A09] (see: Portfolio optimization) dynamical system

[34-04, 35-04, 58-04, 58F14, 58F22, 58F25] (see: Dynamical systems software packages; Seifert conjecture) dynamical system s e e : asymptotically stable - - ; asymptotically stable equilibrium of a --; chaotic --; curve of periodic orbits of a - - ; curves of equilibria of a - - ; deterministic - - ; discrete - - ; equilibrium of a - - ; minimal set of a - - ; period of a trajectory of a - - ; period of an orbit of a - - ; stochastic - - ; strange - - ; unstable equilibrium of a - - ; wild - d y n a m i c a l systems s e e : software for - -

dynamical systemssoftware [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Dynamical systems software packages (34-04, 35-04, 58-04, 58FI4) (refers to: Bifurcation; Codimensiontwo bifurcations; Differential equations, ordinary, retarded; Dynamical system; Evolution equation; Floquet exponents; Floquet theory; Hopf bifurcation; Jacobi matrix; Limit point of a trajectory; Lyapunov characteristic exponent; Neutral differential equation; Poincar~ return map; Stability theory) dynamics s e e : computational fluid - - ; dissipativity of the Kuramoto--Sivashinsky - - ; monopole - - ; phase - -

dynamics in a natural language

[68S051 (see: Natural language processing)

Dynamics Solver software [34-04, 35-04, 58-04, 58F14] 487

DYNAMICS SOLVERSOFTWARE

Dynamical systems software packages) (see:

[62Lxx] (see: Average sample number) efficient parameter

[90Ali] (see: Cobb-Douglas function) efficient portfolio s e e :

E e-entropy [42B05, 42B08] (see: Hyperbolic cross) e-isometry of Banach algebras [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) e-isomorphismof Banach algebras [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) e-metric perturbation of a Banach algebra

[46Exx] (see: Banach-Stone theorem)

e-perturbation of a Banach algebra [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) e-perturbation of a Banach algebra

[46Exx] (see: Banach-Stone theorem) [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) E-domain [35P251

(see: Obstacle scattering) E-theory [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Earley algoridma [68S05] (see: Natural language processing) Eberlein compact [26A15, 54C05] (see: Namioka space) EBL

[68T05] (see: Machine learning)

echelon matrix [14L35, 14M15, 20G20] (see: Schubert cell) economy s e e :

see:

eigenfunction s e e : Dirichlet - - ; Neumann - eigenfunctions s e e : bound-state - - ; Weyl sequence of approximate - -

eigenvalue

[70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) eigenvalue see: algebraic multiplicity of an - - ; Dirichlet - - ; Hecke -- ; index of an --; i s o p e r i m e t r i c inequality for the lowest --; Neumann - - ; Segre characteristic at an-eigenvalue distribution of Laplacians s e e : Weyl asymptotic formula for the - eigenvalue problem s e e : isospectral linear --

eigenvalue problem for the clamped plate

E k s e e : logic of - E-detachment

economical model

mean-variance - -

Egorov generalized function algebra (46F30) (referred to in: Generalized function algebras) (refers to: Generalized function algebras; Sheaf)

macro-

--; micro-

mathematical - -

edge Laplacian matrix of a graph

[05C50] (see: Matrix tree theorem) edge-transitive Cayley graph [05C251 (see: Cayley graph) E D P R C and d e d u c t i o n - d e t a c h m e n t t h e o r e m see: equivalence of --

[35P15] (see: Rayleigh-Faber-Krahn equality)

eigenvalues and geometry [35J05, 35J25] (see: Dirichlet eigenvalue) eigenvalues of the Laplace operator s e e : Neumann -eigenvalues of the Laplacian s e e : Dirichlet -eight knot s e e : figure --

Eisenstein cohomology (11F67) (refers to: Arithmetic group; Cohomology; de Rham cohomology;Exact sequence; Holomorphic function; Lfunction; Lie algebra; Parabolic subgroup; Reductive group; Representation of a group; Riemannian manifold; Sheaf; Symmetric space) Eisenstein series [11Fxx, 11F67, 20Gxx, 22E46] (see: Baily-Borel compaetification; Eisenstein cohomology) Eiseostein series

see:

Poincard- - -

[62Jxx] (see: ANOVA)

election voting

effect in design of statistical experiments

effect in design of statistical experiments see: main --

effective algebra [13Pxx, 14Q20] (see: Hermann algorithms) effective Hilbert Nullstellensatz s e e : eralized --

gen-

Effective Nullstellensatz (14A10, 14Q20) (referred to in: Hermann algorithms) (refers to: Field; Masser-Phifippon/ Lazard-Mora example; .M7~) effects model s e e :

fixed - - ; random --

efficiency of a representation for machine learning

[68T05] (see: Machine learning)

efficiency of an algorithin [68Q15] (see: Average-case computational complexity) efficiency of statistical tests 488

[90A28] (see: Condorcet paradox)

electric-magnetic duality [SIV10] (see: Dirac monopole) electrical circuit flow [60K30, 68M10, 68M20, 90B10, 90B 15, 90B 18, 90B20, 94C99] (see: Braess paradox) electrical network s e e :

element of a Steiner triple system

[05B07, 05B30] (see: Pasch configuration)

passive --

electromagnetic radiation [78A40] (see: Stokes parameters) electromagnetic wave scattering s e e : Smatrix for - electron s e e : Bloch - electron equation s e e : relativistic - element s e e : complement of a lattice - - ; FC- - - ; idempotent - - ; Matrix - - ; normai - - ; primitive - - ; right-sided - element in a Banach-Jordan algebra s e e : spectrum of an - -

(refers to: Schr6dinger equation) entailment logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) entangled quantum state

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) entanglement s e e :

quantum - -

entier function

see: abstract prime - elementarily equivalent models element theorem

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Elementarteilerform [13Pxx, 14Q20] (see: Hermann algorithms)

[26Axx] (see: Floor function) entropic fight quasi-group [57P25] (see: Conway algebra)

[20J06]

entropy s e e : e- - - ; strong subadditivity inequality for von Neumann - - ; yon Neumann - entry s e e : matrix - enumerabletheory see: recursively--

(see: Serre theorem in group coho-

enumeration

elementary Abelian p-group

[llNxx, 11N32, 11N45, 1IN80]

mology) elementary equivalence

[03Gxx, 03G05, 03G10, 03GI5, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) elementary event

[68T30, 68T99, 92Jxx, 92K10] in-

eigenvalues s e e : asymptotic distribution of - - ; Brauer theorem on - - ; localization of - - ; localization theorem for matrix - - ; max-min principle for - - ; Pdlya conjecture for Dirichlet - - ; Pdlya conjecture for Neumann - - ; Weyl asymptotics for Dirichlet - -

elastic curve on the sphere [53C42] (see: Willmore functional)

element in a field extension s e e : completely free - - ; completely normal -- ; free - - ; normal -element in a Galois extension s e e : completely free - - ; completely n o r m N - - ; free - - ; norm of an - - ; trace of an - element in a Jordan algebra with a unit s e e : invertible - element method s e e : finite - - ; spectral --

(see: Dempster-Shafer theory) elements s e e : concordant sample - - ; discordant sample - elimination s e e : Gaussian - ellipse s e e : multifocal - - ; polarization - - ; poly- --

elliptic function (see: Gel'fond-Schneider method) elliptic functions see: algebraic independence of values of - - ; transcendence of values of -- ; transcendence theory of - -

elliptic integral [11J85, 41-XX, 41A50] (see: Gel'fond-Schneider method; Zolotarev polynomials) elliptic partial differential equations [46Cxx] (see: Alternating algorithm) elliptic partial differential operator s e e : index of an - elliptic pseudo-differential operator s e e : hypo- - elliptic symbol of a pseudo-differential operator see: hypo- - embedding s e e : regular - - ; toroidal - embedding dimension s e e : Buchsbaum local ring of maximal -embedding homomorphism of Gel'land quantales s e e : right - embedding Lie algebra s e e : standard - embedding Lie superalgebra s e e : standard - embedding problem s e e : Galois - empty set s e e : axiom of the - - ; dual of a non- - - ; polar of a non- - encryption s e e : p u b l i c - k e y - end s e e : Stallings classification of finitely generated groups with more than one - -

end of a group [20F05, 20F06, 20F32] (see: HNN-extension) endomorphisms see: quantale of -ends condition see: matched- -energy s e e : conservation of - - ; Helfrich free - - ; quantum v a c u u m - -

energy balance law

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) see: positive- - energy-momentum operator energy condition

[81Txx, 81T05] (see: Massless field)

Enss method (81 Uxx)

enumeration [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) enumeration see: a s y m p t o t i c - enumerativecalculus s e e : Schubert - -

enumerative geometry

[14C15, 14M15, 14NI5, 20G20, 57T15] (see: Schubert calculus) envelope s e e : pro- - envelope of a module s e e :

l11J85]

energy of a binary altoy s e e :

(see: Abstract analytic number theory)

free

--

injective - -

envy-free division

[00A08, 90Axx] (see: Cake-cutting problem) epimorphism over a class of algebras

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) EQP [06Exx, 68T15] (see: Robbins equation) EQP proof [06Exx, 68T15] (see: Robbins equation) equal-degree factorization

[12D05] (see: Factorization of polynomials) equality s e e : Horn logic with - - ; infinitary universal Horn logic without - - ; Titstype -equation s e e : 2 + l - d i m e n s i o n a l Harry Dym - - ; Abel functional - - ; additive Cauchy - - ; averaged solution of the K o r t e w e g - d e Vries - - ; B B M - - ; Benjamin-Bona-Mahony--; Bezout - - ; bifurcation in the Kuramoto-Sivashinsky - - ; Birkhoff-Rott - - ; Burgers - - ; C a h n Hilliard - - ; characteristic initial-value problem for the K o r t e w e g - d e Vries - - ; compressible Navier-Stokes - - ; conservation laws for the B e n j a m i n - B o n a Mahony--; O- - - ; D b a r - - ; delay d i f f e r e n t i a l - - ; Dirac - - ; dispersionless K o r t e w e g - d e Vries - - ; Euler-Bernoulli --; Euler-Poisson-Darboux --; extended Harry Dym - - ; generalized BBM -- ; g e n e r a l i z e d B e n j a m i n - B o n a - M a h o n y - - ; generalized E u l e r - P o i s s o n - D a r b o u x - - ; generalized Harry Dym - - ; generalized Lax - - ; Hardy-Weinberg - - ; Hardy-Weinberg equilibrium - - ; Harry Dym - - ; H o p f - B u r g e r s - - ; Huntington - - ; Kardar-Parisi-Zhang - - ; Kolmogorov backward - - ; K o r t e w e g - d e Vries - - ; KS --; Kuramoto-Sivashinsky--; Lie--; Massless Klein-Gordon - - ; neutral differential - - ; non-autonomous Schrdder functional - - ; non-linear evolution - - ; non-linear Sch rddinger - - ; normally solvable - - ; optimality in the Fritz John - - ;

EUROPEAN CALL OPTION

quantum spectral measure of a partial differential --; regularized long wave - - ; relativistic electron - - ; residual - - ; Robbins - - ; Schreder functional - - ; shocks for the Korteweg-de Vriee - - ; singular partial differential - - ; singularity manifold--; Sivashinsky-Kuramoto--; S k e r o h o d - - ; S k o r o k h o d - - ; stability in the Fritz John - - ; stationary CahnHilliard - - ; Stieltjes-Volterra integral - - ; Sturm-Liouville differential - - ; superstable f u n c t i o n a l - - ; Theodorsen integral - - ; Thiele differential - - ; Toda molecule - - ; translation functional - - ; variablecoefficient 8enjamin-Bona-Mahony --; VoRerra-Stieltjes integral - - ; Whitham equation for the Korteweg-de Vries - - ; Yang-Baxter -equation for finite sum decompositions s e e : d'Alembert -equation for the Korteweg-de Vries equation see: Whitham -equation for the Riemann ~-function s e e : functional -equation of e variational problem s e e : E u ler -equationalclass s e e : quasi- -equational definability [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35]

(see: Abstract algebraic logic) see: faithfuLinterpretation

equationaltogic of an --

equational logic of a class of algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equational logic of Boolean algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equational theory [03Gxx] (see: Algebraic logic) equational theory decidable - -

see:

decidable - - ; un-

equationally definable principal congruences [03Gxx] (see: Algebraic logic) equationally definable principal relative congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equations s e e : Ablowitz-Ladik - - ; AKNS- - - ; Bogomolny--; direct construction method in the stability of functional - - ; elliptic partial differential - - ; formal Dirac quantization of partial differential--; Hamilton-Jacobi--; KnizhnikZamolodehikov-Bernard--; L o r e n z - - ; Nahm - - ; property C + for ordinary differential - - ; property (7,a for ordinary differential - - ; property C~h for ordinary differential - - ; property C for partial differential - - ; property C p for partiar differential - - ; Seiberg-Witten - - ; self-dual Yang-Mills - - ; soliton - - ; stability problem of functional - - ; stationary AKNS- - ; system of defining - - ; Toda-type differential-difference--; W h i t h a m - - ; Witt en-Dijkg raaf-Ve rlinde-Verlinde -equations for field correlators s e e : associativity -equations for least-squares estimation s e e : normal -equations of the AKNS-hierarchy s e e : Lax -equations, property (7 for s e e : Ordinary differential - - ; Partial differential --

equi-measurablefunctions [35P15] (see: Rayleigb-Faber-Krahn equality) equi-oscillation theorem [41-XX, 4IA50]

in-

(see: Zolotarev polynomials) equilibria of a dynamical system see: curves el -equilibrium equation Weinberg -equilibrium flow s e e :

Hardy-

see:

user --

equilibrium link flow [60K30, 68M10, 68M20, 90B10, 90B15, 901318, 90B20, 94C99] (see: Braess paradox) equilibrium of a dynamical system [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) equilibrium of a dynamical system s e e : asymptotically stable - - ; unstable --

equilibrium traffic flow [60K30, 68M10, 68M20, 90BI0, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) equivalence s e e : Conway skein - - ; elementary - - ; Frege T - - ; logical --

equivalence of EDPRC and deductiondetachment theorem [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equivalence of formulas over a class of interpretations [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equivalence of formulas over a class of interpretations see: Fregean -equivalence of formulas with respect to a deductive system s e e : logical -equivalence of operator vessels s e e : unitary --

equivalence of tangles [57M25] (see: Tangle) equivalence relation

see:

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equivalential general semantical see: finitely --

equivalential semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equivalential finitely --

semantical

Bore1 - -

equivalence system s e e : proto- -equivalence systems end the Suszko congruence s e e : theorem on --

equivalence theoremsin algebraic logic [03Gxx] (see: Algebraic logic) equivalent algebraic semantics [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) equivalent categories s e e : derived -equivalent formulas s e e : Frege T - -equivalent formulas with respect to a deductive system s e e : logically --

equivalent measures [28-XX] (see: Absolutely continuous measures)

system

see:

equivariant index [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) equivariant index theorem [46L80, 46L87, 55N15, 58GI0, 58Gll, 58G12] (see: Index theory) erasure-correctingcode [05B07, 05B30] (see: Pasch configuration) Erdes theorem on abundantnumbers [1 IAxx] (see: Abundant number) ErdSs-Turdn conjecture [llPxx] (see: Additive basis) ergodic automorphism of the infinitedimensional toms [11C08, 11R04] (see: Lehmer conjecture) ergodic invariant measure [28Dxx, 541-I20, 58F11, 58F13] (see: Absolutely continuous invariant measure) ergodic theorem [28Dxx, 541120, 58F11, 58F13] (see: Absolutely continuous invariant measure) ergodic theorem s e e : Wiener-Wintner --

equivalence system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

system

pointwise

error

[62Jxx] (see: ANOVA) error s e e : aliasing - - ; classification prediction - - ; typeq - - ; type-ll -error analysis for the tan method

error-correctingcode s e e : quantum -error-correction s e e : quantum -error of functions in Sobolev spaces s e e : approximation --

error of the Lagrange interpolation formula [46E35, 65N30] (see: Bramble-Hilbert lemma) error polynomial-time computable language see: bounded- --

error probability [62Lxx] (see: Average sample number) error quantum polynomial-time computable language s e e : bounded- --

espace tamisable [54E52] (see: Banach-Mazur game) essential ideal [46J10, 46L05, 46L80, 46L85] (see: Multipliers of O* -algebras) essential radius condition [26A21, 54E55, 54G20] (see: Sorgenfrey topology) essential spectrum [34B24, 34L40] (see: Sturm-Liouville theory)

equivalent tangles [57M25] (see: Tangle) equivalential deductive system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

essentialspectrum see:

deductive system

see:

equivalentia[ general semantical system

--;

[65Lxx] (see: Tau method)

equivalent models s e e : elementarily -equivalent operator vessels s e c : unitarily -equivalent over a theory s e e : formulas Frege -equivalent quasi-variety see: secondorder -equivalent relative R-t-cycles s e e : numerically -equivalent rings s e e : derived -equivalent sentences in logic s e e : logically --

equivalential finitely --

--;

Taylor --

essential spectrum of an operator [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) essential submodule [16D40] (see: Flat cover) essentially commuting operators [46Lxx]

(see: Toeplitz C'* -algebra) essentially normal operator [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) essentially self-adjoint operator [11F03, 11F70] (see: Selberg conjecture) estimable parametricfunction in statistics [62Jxx] (see: ANOVA) estimate s e e : van der Corput kth derivative - - ; Yudin -estimates s e e : Exponentialsum - -

estimation [62Jxx] (see: ANOVA) estimation s e e : exponent pair in exponential sum - - ; exponent pairs in exponential sum - - ; normal equations for leastsquares - - ; point - - ; statistical - estimator s e e : best linear unbiased - - ; least-squares - - ; minimum variance unbiased - -

Euclidean Taub--NUTmetric [35Qxx, 7gA25] (see: Magnetic monopole) Euler-Bernoulli equation [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) Euler equation of a vadafinnal problem [53C42] (see: Willmore functional) Euler-Mascheroni constant [11M06, 11M35, 331315] (see: Catalan constant) Euler-Mascheroni constant [11M06, 11M35, 33B15] (see: Catalan constant) Euler-Maseheroni constant see: theorem on the -Euler method s e e : implicit --

Euler

Euler-Poincar~ theorem [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brifi formula) Euler-Poisson-Darboux equation (35L15) (refers to: Fractional integration and differentiation; Gamma-function; Hyperbolic partial differential equation; Imbedding theorems) Euler-Poisson-Darboux equation see: generalized -operator difference analogue of the --

Euler-Poieson-Darboux

see:

q-

Euler product formula [llL07, llM06, llNxx, 11N32, 11N45, 11N80, 11P32] (see: Abstract analytic number theory; Vaughan identity) Euler product formula modified - -

see:

classical - - ;

Euler quadratic form [16Gxx] (see: Tits quadratic form) Euler system [11R23] (see: Iwasawa theory) Euler theorem on the Euler-Mascheroni constant [11M06, 11M35, 33B15] (see: Catalan constant) Euler totient function [051335, 05Exx, 05E25, 06A07, 11A25[ (see: Mebius inversion) Euler zeta-function [11M06] (see: Riemann ~-function) Eulerian function of a finite group [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) Eulerian identity [05E05, 60G50] (see: Baxter algebra) European call option 489

EUROPEAN CALL OPTION

[60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) European call option see: expiration time of a - - ; strike price of a - - ; underlying asset of a - European call option at expiration see: value of a - -

European option

[60Hxx, 90A09, 93Exx] (see: Black-Scholes formula; Option pricing) Evans-Griffithconstruction [13A30, 13HI0, 13H30] (see: Buehsbaum ring) event see: eventually

composite - - ; elementary -sequencevanishing --

see:

evidence

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) evidence see: mathematical theory of - evidence combination see: Dempster rule of--

evolution

[92D101 (see: Hardy-Weinherg law) evolution equation

see:

non-linear--

exact category [16670] (see: Almost-split sequence) exact complex

see:

formally - -

exact current

[32C30, 53C65, 58A25] (see: Current) exact genus formula

[12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) exact sequence see: minimal non-split short - - ; non-split short - - ; R o s e n b e r g Zelinsky - - ; short - exact sequence of C * - a l g e b r a s see: short - -

exact test of a hypothesis

[62Jxx] (see: ANOVA) example see: Gamelin-Sibony - - ; Lazard-Mora--; Lazard-Mora/MasserPhilippon - - ; Masser-Philippon --; Masser-Philippon/Lazard-Mora - -

example of a 1-rectifiable set [28A78, 49Qxx, 49Q15, 53C65, 58A251 (see: Geometric measure theory) example of a purely 1-uurectifiable set [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) example of a Taylor spectrum [47Dxx] (see: Taylor joint spectrum) example of a Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) example of the Conley index [58Fxx] (see: Conley index) example of the use of the Jansen inequality [05C80, 60D05] (see: Jansen inequality) example of using the Lovfiszlocal lemma [05C80] (see: Lov~iszlocal lemma) examples as--

see:

Baumslag-Solitar groups

examples of Baily-Borel compactificadons [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compaefification) examples of Dirichlet densities [11R44, 11R45] (see: Diriehlet density) examples of Jordan triple systems [17A40] (see: Jordan triple system) examples of reproducingkernels [46E22] (see: Reproducing kernel) 490

examples of Z-transforms [39A12, 93Cxx, 94AI2] (see: Z-transform) examples of Zak transforms [42Axx, 44-XX, 44A55] (see: Zak transform) excessive function [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) excessive measure [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) excision in algebraic K-theory [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) excision of the Brouwer degree see: additivity- - exclusion see: principle of inclusion- - exclusion formula see: Inclusion- - exclusion method see: inclusion- - exclusion principle see: inclusion- --

exclusive or [90D05] (see: Sprague-Grundy function) exhaustion function [31C10, 32F05] (see: Pluripotential theory) existence of Zak transforms [42Axx, 44-XX, 44A55] (see: Zak transform) existence property of the Brouwer degree

[55M25] (see: Brouwer degree) exit set

[58Fxx] (see: Conley index)

expander [05C25] (see: Cayley graph) expansion see: partial-fraction - - ; uniform - expansion lemma see: Boolean -expansion remainder see: T a y l o r - expansivesemi-group see: non- --

expected utility in portfolio optimization

[90A09] (see: Portfolio optimization) experiencein a learning system ing - -

see:

train-

experiment generator in a learning system

[68T05] (see: Machine learning) experiments see: balanced design for statistical - - ; cell in design of statistical - - ; design of - - ; design of statistical - - ; effect in design of statistical - - ; interaction in design of statistical - - ; main effect in design of statistical -expiration see: value of a European call option at - -

expiration time of a European call option

[60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) explanation-based learning

[68T05] (see: Machine learning) explicitly over another set of atomic formulas see: set of formulas defining a set of atomic formulas - exponent n see: group of - -

exponent of a group

[20F05, 20F06, 20F32, 20F50] (see: Burnside group) exponent pair

see:

van der Corput - -

exponent pair in exponential sum estimation

(see: Trigonometric pseudo-spectral methods) exponentialdistribution

see:

double - -

exponential formula representation for a linear semi-group

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) exponential formula representation of a continuous semi-group

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) exponentialfunction see: additiontheorem for the - exponential functions see: algebraic independence of values of - - ; transcendence of values of - - ; transcendence theory of -exponential growth see: group of - exponential hat function see: double - exponentiallaw see: fibred =

exponential law jbr sets"

[54C35] (see: Exponential law (in topology))

Exponential law (in topology) (54C35) (refers to: Algebraic topology; Compact-open topology; Compact space; Hausdorff space; Locally compact space; Metric space; Separation axiom; Space of mappings, topological; Topologicalspace; Topos) exponential representation of a continuous semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Seml-group of holomorphic mappings) exponential sum

[11L07] (see: Exponential sum estimates) exponential sum [I1Lxx, llL03, 11L05, llL15] (see: Bombieri-lwaniec method) exponential s u m see: analytic - - ; arithmetic - - ; complete - - ; monomial - -

Exponential sum estimates (11L07) (referred to in: Selberg conjecture; Vaughan identity) (refers to: Analytic number theory; Bombieri-Iwaniec method; Lindelrf hypothesis; Poisson summation formula; Riemann zeta-function; Trigonometric sum; Vinogradov method; Waring problem; Zetafunction) exponential sum estimation see: exponent pair in - - ; exponent pairs in - -

exponential sum of type I

[11L07, 11M06, 11P32] (see: Vaughan identity) exponential sum of type H

[llL07, llM06, 11P32] (see: Vaughan identity)

exponential time [68Q15] (see: Average-case computational complexity) exponential-time complexity class [68Q15] (see: Average-case computational complexity)

[11L07] (see: Exponential smn estimates) exponentpairs in exponential sum estimation [11L07] (see: Exponential sum estimates)

exponentials conjecture see: f o u r - exponentials theorem see: Roy strong six - - ; six - - ; strong six - -

exponentialsee:

(see: Machine learning) Ext group of C ( X )

path-ordered--

exponential convergencerate [65Lxx, 65M70]

expressiveness of a representation for machine learning

[68T05] [19K33, 19K35, 49L80]

(see: Brown-Douglas-Fillmore theory) Ext monoid of a C* -algebra [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) extended GMANOVA

[62Jxx] (see: ANOVA)

extended Harry Dym equation [35Q53, 58F07] (see: Harry Dym equation) extensible over a class of algebras see: homomorphism Ko- -extension see: ascending HNN- - - ; associated subgroups of an HNN- - - ; base group of an H N N - - - ; completely free element in a field - - ; completely free element in a Galois - - ; completely normal element in a field - - ; completely normal element in a Galois - - ; free element in a field - - ; free element in a Galois - - ; HNN- - - ; maximal Abelian p - - - ; norm of an element in a Galois - - ; normal element in a field - - ; stable letter of an HNN- - - ; trace of an element in a Galois - - ; universal central - - ; unramified Abelian p- - - ; unramified field - extension of a number field see: Z v - - -

extension era separable C* algebra

[19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore the-

ory) extension of a set of vectors

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) extension of a set of vectors ous

vacu-

see:

- -

extension of a term

[03E30] (see: ZFC) extension of a vector

[68T30, 68T99, 92Jxx, 92KI0] (see: Dempster-Shafer theory) extension of algebraic number fields see: prime ideal of degree one in an - - ; splitting prime ideal of an - -

extension of C* algebras

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) extension operator extension property ence extension theorem

see:

vacuous relative congru-

see:

Tietze - -

see:

- -

- -

extensional deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) extensionaldeductivesystem see: self- - extensionalitysee: axiom of -extensions see: Collins conjugacy theorem for HNN- -- ; normal form theorem for HNN- - - ; reduced sequence in the theory of HNN- - - ; torsion theorem for HNN- - - ; Zariski problem on field - -

exterior derivative [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) exterior differential era current

[32C30, 53C65, 58A25] (see: Current) exterior face ring

[05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) external space of an operator vessel

[47A45, 47A4g, 47A65, 47D40, 47N70] (see: Operator vessel) extra data

[62Jxx] (see: ANOVA) extra-grammatical usage of a natural language

[68S05] (see: Natural language processing) extraordinary homologytheory

FIBRED EXPONENTIALLAW

[55P42] (see: Brown-Gitier spectra) extremal graph [05C25] (see: Cayley graph) extremal polynomial [41-XX, 41A50] (see: Zolotarcv polynomials) extremal ray [14Exx, 14E30, 14Jxx] (see: Mort theory of extremal rays) extremal rays s e e :

Mort theory of - -

extreme point of the closed unit ball in a Banach space [17Cxx, 46-XX] (see: JB * -triple) extremum problem s e e : Szeg6 - extrinsic action s e e : Polyakov - -

Eymard algebra [221310, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Fourier algebra)

F (b arithmetical semi-group

see:

axiom- - -

.U-cover [16D40] (see: Fiat cover)

f'-nef

[14Exx, 14E30, 14Jxx] (see: Mort theory of extremal rays) .U-pc [16D40] (see: Flat cover) F-polynomial [05Cxx, 05D15] (see: Matching polynomial of a graph) f-smoothing vertices of a graph [57M25] (see: Jaeger composition product) F - s y s t e m see: Consecutive k-out-ofn:

--

F-test

[62Jxx] (see: ANOYA) f-vector of a simplicial complex [05Exx, 13C14, 55U10] (see: Staniey-Reisner ring) Faber-Krahn inequality see:

Rayleigh---

Faber-Krahn isopefimetricinequafity [60Gxx, 60J55, 60J65] (see: Wiener sausage) face lattice of a polytope [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Miibius inversion) face-monomials [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) face ring [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) face ring s e e : exterior - - ; StanleyReisner -facilitylocation problem s e e : single - factor s e e : Blaschke-Potapov - - ; C o n z e Lesigne - - ; Fitting - - ; I I do - - ; J B W - ; Kronecker - - ; level of a statistical - -

factor large numbers [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) factor of avou Neumann algebra [03G25, 06D99] (see: Quantale) factor of a weight function s e e : spectral -factorquantaM s e e : von Neumann - -

factor representation of a JB -algebra [17C65, 46H70, 46L70] (see: Banach-Jordan algebra)

factorial algebraic variety s e e :

Q- -

factorial layout [62Jxx] (see: ANOVA) factoring algorithm s e e : Shor - -

quantum - - ;

factoring polynomials [12D05] (see: Factorization of polynomials) factorization s e e : distinct-degree - - ; equal-degree - - ; W i e n e r - H o p f - factorizationmethod s e e : Kaltofen-Trager random polynomial-time - -

Factorization of polynomials (12D05) (refers to: Chebotarcv density theorem; Cryptalogy; Factorial ring; Field; Finite field; Frobenius automorphism; Hilbert theorem; Legendre symbol; LLL basis reduction method; Polynomial; Turing machine; Undecidability) factorization of the Dedekind zeta-function see: Hadamard -factorization problem s e e : spectral - factorization theorem s e e : Gauss - - ; Hadamard - factors s e e : statistical -factors in covarianco analysis s e e : completely crossed - - ; crossed - - ; crossing - - ; incompletely - - ; nested - - ; nesting - - ; partly crossed - - ; qualitative - - ; quantitative - factory problem s e e : Turin brick - -

Faddeev condition for decay [35Q53, 58F07] (see: Harry Dym equation) Faddeev-Popov determinant [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) Faddeev-Popov ghost (81Qxx, 81Sxx, 81T13) (refers to: Connection; Exterior algebra; Lie algebra; Principal fibre bundle; Super-manifold; Yang-Mills field) Faddeev-Popov method [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) failures s e e :

run of - -

fair division [00A08, 90Axx] (see: Cake-cutting problem) fair division problem [00A08, 90Axx] (see: Cake-cuttlng problem) faithful interpretation of a deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) faithful interpretation of an equational logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) fake Monster [11Fxx, 17B67, 20D08] (see: Boreherds Lie algebra) families of operators s e e : a--

index theory for

family s e e : S p e m e r - family in a partially ordered set s e e : k- -family of curves see: modulus of a - family of line bundles s e e : spectral curve of o - family of subsets s e e : lower shadow of a--

family of subsets of a set [05D05, 06A07] (see: Kruskal-Katona theorem) Fan analogue of the Denjoy-Wolff theorem [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Fano-Mori fibre space [14Exx, 14E30, 14Jxx]

(see: Mort theory of extremal rays) Farkas-M inkowski-Weyl theorem [15A39, 90C05] (see: Motzkin transposition theorem) Farkas theorem [15A39, 90C05] (see: Motzkin transposition theorem) Fasbenderduality s e e :

Vecten- --

fast discrete cosine transform [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) fast Fourier transform [30C20, 30C30, 65Txx] (see:Fourier pseudo-spectral method; Theodorsen integral equation) fault tolerant quantumprocessing [68Q05, 68QI0, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) favourablospace s e e : o<- - favourable topological space s e e : weakly ~ - - -

ci- --;

FC-element [20F24] (see: FC-group) FC-group (20F24) (refers to: CC-group; Group; Group with a finiteness condition) Federer-Fleming closure theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Federer projection theorem s e e : vitch- --

Besieo-

Fedosov trace formula (81Q05) (refers to: Hamiltonian system; Pseudo-differential operator; Symbol of an operator) feedback coefficients of a linear feedback shift register [11B37, llT71, 93C05] (see: Shift register sequence) feedback matrix of a linear feedback shift register [11B37, 11T71, 93C05] (see: Shift register sequence) feedback polynomial of a linear feedback shift register [11B37, llT71, 93C05] (see: Shift register sequence) feedback shift register s e e : characteristic polynomial of a linear - - ; feedback coefficients of a linear - - ; feedback matrix of a linear - - ; feedback polynomial of a linear - - ; impulse-response sequence of a linear - - ; initial conditions of a linear - - ; length of a linear - - ; linear - - ; reciprocal polynomial of a linear - - ; state vector of a linear - -

feedforward neural net [41A30, 92C55] (see: Ridge function) feedforward neural net s e e :

Fermat-Torrice[li problem s e e : ized - - ; unweighted - -

general-

Format-Weber problem (90B85) (refers to: Weber problem) Fermi-Pasta-Ulam problem [35Q53, 58F07] (see: Harry Dym equation) fermion s e e :

spin-l/2

--

Feynman-Kac formula [60Hxx, 90A09, 93Exx] (see: Blaek-Scholes formula) Feynman path method [81Qxx] (see: Dirac quantization) Fibonacei group (20F38) (referred to in: Fibonacci manifold) (refers to: Arithmetic group; Cyclic group; Fibonacci manifold; Fibonaeci numbers; Ftaitely-presented group; Group calculus; Hyperbolic group; Identity problem; Lens space; Noetherian group; Presentation; Riemannian manifold) Fibonaeci group s e e : fractional - - ; generalized - Fibonaeci groupssee: conjugacyproblem for - - ; word problem for - -

Fibonacci manifold (57Mxx) (referred to in: Fibonacci group) (refers to: Arithmetic group; Dehn surgery; Fibonacci group; Hyperbolic metric; Lens space; Listing knot; Orientation; Threedimensional manifold) Fibonacci manifold [20F38] (see: Fibonacei group) Fibonacci numbers of order k [33Bxx] (see: Fibonacci polynomials) Fibonacci polynomials (33Bxx) (referred to in: Consecutive k-out-ofn: F-system; Lucas polynomials) (refers to: Binomial distribution; Consecutive k-out-of-n: F-system; Fibonacci numbers; Generating

function; Irreducible polynomial; Multinomial coefficient) Fibonacci polynomials s e e :

bivariate - -

Fibonacci polynomials of order k [33Bxx] (see: Fibonaeci polynomials) multi-Myer--

Fefferman duality theorem [30Axx, 46Exx] (see: BMOA -space) Fefferman duafity theorem [30D50, 46Exx] (see: VMOA-space) Felt-Solomon theorem [1IR34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) Fenchel-Nielsen coordinates [14H15, 30F60] (see: Weil-Petersson metric) Fenchel-Nielsen intrinsic coordinates [14H15, 30F60] (see: Wcil-Petersson metric) Fermat point [51M04] (see: Triangle centre) Format problem s e e :

Fcrmat-Torricelli problem (90B85) (refers to: Approximation theory; Computational geometry; Ellipse; Galois theory; General position; Gradient method; Inner product; Non-linear programming; Operations research; Robust statistics; Steiner point; Steiner tree problem; Weber problem)

Yorriee[li- - -

polynomials of order h s e e : tivariate - -

Fibonacei

mul-

Fibonacci sequence [11B37, llT71, 93C05] (see: Shift register sequence) Fibonacci-type polynomials [11B39] (see: Lncas polynomials) Fibonacci-type polynomials of order k [33Bxx] (see: Fibonacci polynomials) fibration over a circle [32E20] (see: Polynomial convexity) fibre of a coherent configuration [03Exx, 03E05] (see: Coherent algebra) fibrespace s e e :

Fend--Mort - -

fibred exponential law [54C35] (see: Exponential law (in topology))

491

FIDELITY

,fidelity [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) fidelity see: B u r e s - U h l m a n n -field s e e : absolute value on a number - - ; algebraic function - - ; algebraic n u m b e r - - ; anti-ghost - - ; Archimedean place of a n u m b e r -- ; chiral - - ; complete holomorphic vector - - ; complete vector -- ; concentration - - ; differential of a - - ; discriminant of an algebraic n u m b e r - - ; dual basis of an ordered basis of a - - ; free asymptotic q u a n t u m - - ; function - - ; ghost - - ; Hamiltonian vector - - ; Higgs - - ; Hilbertian - - ; L-function of an algebraic n u m b e r - - ; locally Hamiltonian vect o r - - ; Massive - - ; Massless - - ; metric on a n u m b e r - - ; menlo polynomial over a finite - - ; non-Archimedean place of a n u m b e r - - ; norm of a prime ideal in an algebraic n u m b e r - - ; PAC - - ; pseudoalgebraically closed -- ; Reeb vector - - ; residue - - ; semi-complete holomorphie vector -- ; semi-complete vector -- ; signature of an algebraic n u m b e r -- ; units of an algebraic n u m b e r - - ; Z p - e x t e n s i o n of a number -field correlators s e e : associativity equations for - field extension s e e : completely free element in a - - ; completely normal element in a - - ; free element in a - - ; normal element in a - - ; unramified - field extensions s e e : Zariski problem on -field isomorphism see: fixed place of a - - ; valence of a - field of an algebraic curve s e e : function --

field of modular functions [11Fll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) fieldof modularfunctionssee: Hauptmodul for a - field structure s e e : Galois - field theories s e e : topological -field theory s e e : algebraic q u a n t u m - - ; conformal - - ; eonformal q u a n t u m - - ; massive q u a n t u m - - ; symmetry breaking in q u a n t u m - - ; topological - - ; topological quantum -field theory of Landau-Ginsburg type s e e : topological -field tower problem s e e : class -field (update) s e e : Galois - fields s e e : prime ideal of degree one in an extension of algebraic n u m b e r - - ; Riemann hypothesis for curves over finite - - ; splitting prime ideal of an extension of algebraic n u m b e r -fields for group algebras s e e : Brauer theorem on splitting - fields of genus zero s e e : Amitsur theorem on function --

Figh-Talamanca algebra (43A07, 43A15, 43A45, 43A46, 46J10) (referred to in: Fourier algebra; Fourier-Stieltjes algebra) (refers to: Banach algebra; Dirac distribution; Fourier algebra; Haar measure; Harmonic analysis, abstract; Linear operator; Measure; Weak topology) Fig~t-Talamanca algebra [43A07, 43A15, 43A45, 43A46, 46J10] (see: Figh-Talamanca algebra) Figd- Talamanca-Herz algebra [43A07, 43A15, 43A45, 43A46, 46J10] (see: Figh-Talamanca algebra) figure eight knot [57M25] (see: Jones-Conway polynomial; Positive link) figure-eight knot 492

[57M25] (see: Listing polynomials) Fike theorem s e e : Bauer-Fillmore theory s e e : B r o w n - D o u g l a s - -film s e e : singularity in a soap - films on inclines s e e : fluid - filter s e e : full second-order - - ; truth - -

filter-distributive deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) filter of a deductive system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) filtering s e e : Kalman - filtering problem s e e : non-linear-filters s e e : correspondence theorem for "D-filtrationsee: C*---; A---; ~---;spect r u m of a C * - - - ; VassiIiev-Gusarov --

final value theoremfor the Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) financing portfolio see: self- -financing portfolio strategy s e e : self- - finding the inverse Z-transform s e e : partial-fractions technique for -fine l i m i t [26A21, 54E55, 54G20]

(see: Sorgenfrey topology) fine topologyin potential theory [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) fingersee: Saffman-Yaylor-fingering s e e : Viscous -finite A b e l i a n g r o u p

[llNxx, 11N32, 11N45, llNS0] (see: Abstract analytic number theory) finite Abelian groups see: fundamental theorem on - -

finite affine plane [05B30[ (see: Affine design) finite algebra see: r e p r e s e n t a t i o n - finite axiomatizability [03Gxx] (see: Algebraic logic) finite basis for the identities era variety [08Bxx, 16R10, 17B01, 20El0[ (see: Specht property) finite basis problem for varieties [08Bxx, 16R10, 17B01, 20El0[ (see: Specht property) finite basis theorem see: analogue of the Baker--; Bakerfinite Buchsbaum-representation type s e e : Noetherian local ring of - finite capacity s e e : Jordan pair of - - ; Lees classification of simple Jordan pairs of - -

finite conjugate group [20F24] (see: FC-group) finite difference method [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) finite-dimensional case s e e : trum in t h e - -

Taylor spec-

finite-dimensional state space system [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) finite element method [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) finite field s e e : monic polynomial over a - finite fields s e e : Riemann hypothesis for curves over - -

finite-gap solution [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) finite geometry) see: Net (in -finite geometry) (update) s e e : Net (in -finite graph s e e : drawing of a - -

finite group see: Eulerian function of a - - ; residually- - finite Hausdedf measure see: decomposition theorem l o t sets of - finite partially ordered set s e e : locally - finite seif-similarset s e e : post-critically--

finite spectrum [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) finite-stateautomaton see: probabilistic-finitesum decompositionssee: equation for - -

d'Alembert

finite-type knot invariant [57Mxx, 57M25] (see: Skein module) finite width s e e :

branch group of - -

finite Wiener-Hopf operator [42A16, 47B35] (see: Szegii limit theorems) finitely additive additive set function [28-XX] (see: Non-additive measure) finitely algebra&able deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely algebraizab~e deductive system s e e : second-order - - ; strongly - -

finitely algebraizable general semantical system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely algebraizable semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely axiomatizable theory [03Gxx] (see: Algebraic logic) finitely based variety [08Bxx, 16R10, 17B0I, 20El0[ (see: Specht property) finitely equivalential deductive system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely equivalential general semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely equivalential semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) finitely generated cone [15A39, 90C05[ (see: Motzkin transposition theorem) finitely-generated groups s e e : Wall conjecture on accessibility of - finitely generated groups with more than one end s e e : Stallings classification of - -

finitely-generated k-algebra [13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) finitely-presented non-Hopfiangroup [20F05, 20F06, 20F32] (see: HNN-extension) fiuitely ramified fractal [28A80] (see: Sierpifiski gasket) finitely schemaaxiomatizable [03Gxx] (see: Algebraic logic) finiteness condition s e e :

Tarski - -

finiteness" conditions for groups [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) finiteness conditions in Banach-Jordan pairs [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) finitistic space [54H15, 55R35, 57S17] (see: Smith theory of group actions) first algorithm s e e :

depth- --

first Baire class [26A21, 54E55] (see: Slobodnik property) first Baire class [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) first Brocard point [51M04] (see: Brocard point) first Chern number [81Vm] (see: 13irac monopole) first flip conjecture [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) first isogonic centre era triangle [51M04] (see: Triangle centre) first kind s e e :

Bessel function of the --

first Listing polynomial [57M25] (see: Listing polynomials) first Lyapunovcoefficient [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) first neighbourhood era point on an algebraic curve [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) first-order language [03Gxx] (see: Algebraic logic) first-orderlanguage see: full restricted - first-order l o g i c [03E30, 0 3 G x x ]

(see: Algebraic logic; ZFC) first-order logic s e e : formula in - - ; n vadable fragment of - - ; symbol in - -

first-order predicate logic [68S05] (see: Natural language processing) first-order predicate logic in machine learning [68T05] (see: Machine learning) first-order trace formula [42A16, 47B35] (see: Szeg6 limit theorems) first spacing problem [llLxx, llL03, 11L05, llL15] (see: Bombieri-Iwaniec method) first Szeg5 limit theorem [42A16, 47B35] (see: Szeg6 limit theorems) first variation of an m-varifold [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) first Zolotarev problem [41-XX, 41A50] (see: Zolotarev polynomials) Fischer-GriessMonster [llFll, 17B67, 201)08, 81T10] (see: Moonshine conjectures) Fischer-Griess monster group [llFII, 17B10, 17B65, 17B67, 17B68, 201308, 81R10, 81T30, 81T40] (see: Vertex operator; Vertex operator algebra) Fischer inequality s e e : Hadamard--Fischer inequality for positive semi-definite Hermitian matrices s e e : generalization of the H a d a m a r d - --

Fisher-Hartwig conjecture [42A16, 47B35] (see: Szeg5 limit theorems) fit problem s e e : linear - fitting s e e : over- - -

Fitting chain (20F17, 20F18) (referred to in: Fitting length) (refers to: Characteristic subgroup; Finite group; Fitting length; Fitting subgroup; Nilpotent group; Normal subgroup; Solvable group)

FORMULA REPRESENTATIONOF A CONTINUOUSSEMI-GROUP

Fitting chain s e e : ascending - - ; descending - - ; length of a - -

Fitting factor [20FI 7, 20F18] (see: Fitting chain) Fitting height [20F18] (see: Fitting length) Fitting length (20F18) (referred to in: Fitting chain; Regular group) (refers to: Automorphism; Carter subgroup; Finite group; Fitting chain; Nilpotent group; Solvable group) Fitting length of a group [20F181 (see: Fitting length) Fitting series [20F17, 20F18] (see: Fitting chain) FiRing series see: lower --; upper -fixed-basis topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) fixed effects model [62Jxx] (see: ANOVA) fixed place of a field isomorphism [12F10, 14H30, 20D06, 20E22] (see: Chaslcs-Cayley-BriU formula) fixed point see: attractive--; hyperbolic - fixed-point formula [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) fixed-point formulas s e e :

Atiyah-Bott

fixed-point-free group action [20-XX]

(see: Regular group) fixed-point iteration [30C20, 30C30] (see: Theodorsen integral equation) fixed-point operator see:

least- - -

fixed-point set of a group action [54H15, 55R35, 57S17] (see: Smith theory of group actions) fixed-point theorem s e e : Brouwer --; Darbo - - ; Knaster-KuratowskiMazurkiewicz - - ; Schauder --

fixed-point theorems [55M25] (see: Brouwer degree) fixed prime ideal [54G101 (see: P-point; P-space) flag manifolds s e e : cohomologyof - flame front s e e : laminar - -

fiat bundle [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) flat connection [14Jxx, 35A25, 35Q53, 53C15, 57R57, 58D27[ (see: Atiyah-Floer conjecture; Whltham equations) flat connections see: moduli space of -Flat cover (16D40) (refers to: Automorphism; Category; Dimension; Injective module; Isomorphism; Module; Morphism; Noetherian ring; Projective module; Ring) fiat cover [16D40] (see: Flat cover) .flat covering [16D40] (see: Flat cover) flat distance see: integral - flat in a geometric lattice [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) flat module

[16D40] (see: Flat cover) flat pre-cover [16D40] (see: Flat cover) flats of a matroid see: lattice of -FLC ring [13A30, 13H10, 13H30] (see: Buchsbaum ring) Flecnode (I4Hxx) (refers to: Node; Point of inflection) Flecnode on a planar curve [14Hxx] (see: Flecnode) Fleming closuretheorem s e e :

Federer---

flexible algebra [17A35, 17D25, 83C20] (see: Okubo algebra) flexible programming [90C70] (see: Fuzzy programming) flexible structure [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) flexible structures s e e : of-

dynamic loading

.flip

[14Exx, 14E30, 14Jxx] (see: Morl theory of extremal rays) flip see: log - flip bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) flip conjecture s e e :

first - - ; second --

flip conjectures [14Exx, 14E30, 14Jxx] (see: Mort theory of extremal rays) Floor conjecture s e e : Atiyah- - Fleer c o n i e c t u r e for mapping cylinders s e e : Atiyah- - Fleer h o m o l o g y

[53C15, 57R57, 58D27, 58Fxx] (see: Atiyah-Floer conjecture; Conley index) Fleer homology s e e : eymplectic -Fleer homology for a symplectic mapping see: symplectic - Fleer homology for Lagrangian intersections see: symplectic - Fleer homology for three-dimensional manifolds s e e : instanton - -

Floor function (26Axx) (referred to in: Linear congruential method) Floquet multipliers [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) flow [73Bxx, 76Axx] (see: Material derivative method) flow see: electrical circuit - - ; equilibrium link - - ; equilibrium traffic -- ; inviscid --; link - - ; LiouviJle theorem on invariance under geodesic - - ; route -- ; traffic -- ; user equilibrium - incompressible

,flow demand [60K30, 68M10, 68M20, 90BI0, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) flow invariance s e e : of--

tangency condition

flow-invariance for continuous semigroups [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) flowtheory see: potential - flows s e e :

commuting Harniltonian --

fluctuation theory [05E05, 60G50] (see: Baxter algebra) fluctuations in thermodynamics

[82B35, 82C35] (see: Onsager-Machlup function) fluid s e e : incompressible - fluid dynamics s e e : computational - -

fluid films on inclines [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) fluid shear viscosity [76Exx, 76S05] (see: Viscous fingering) flUX tensor [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) jocal point of a belieffunction [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) Fock space [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Foias functional model s e e : fold groupoid s e e : ~ - - -

Sz.-Nagy- --

fold point [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) foliated bundle [55R40, 57Rxx[ (see: Well algebra of a Lie algebra) foliadon [46L80, 46L87, 55N15, 55R40, 57Rxx, 58G10, 58G11, 58G12] (see: Index theory; Well algebra of a Lie algebra) foliations s e e : Cennes index theorem for - - ; index theorem for - following a particle s e e : derivative--

Fomin space [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) fonction tapis [57N80] (see: Thom-Mather stratification) footrule s e e : Spearman -force s e e : long-range--

forecasting [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) forest s e e : algebraic - form s e e : acceleration in spatial - - ; aut o m o r p h i c - - ; Bernstein --; BernsteinB d z i e r - - ; Chern - - ; C h o w - - ; conjunctive normal - - ; contact - - ; cusp - - ; Dlrichlet--; diseriminant--; Eufer quadratic - - ; formula for acceleration in spatial - - ; formula for the material derivative in spatial - - ; function in material - - ; function in spatial - - ; inertial - - ; Jac o n - - ; Jordan normal - - ; i 2 harmonic - - ; Maase - - ; Maase wave - - ; material derivative in spatial - - ; P o i n c a r d - C a r t a n invariant -- ; Siegel modular - - ; Tits quadratic - - ; weakly non-negative quadratic - - ; weakly positive quadratic - - ; WeilPetersson KAhler - form for ANOVA s e e : canonical - form for GMANOVA s e e : canonical - form for MANOVA s e e : canonical -form of a JB *-triple s e e : real - form of half-integralweight s e e : automorphic - form theorem for HNN-extensions s e e : normal - -

formal character of a weight module [17B10, 17B65] (see: Weyl-Kac character formula) formal derivative [26B99, 30C62, 30C65[ (see: Quasi-regular mapping) formal Dirac quantization of partial differential equations [81Qxx] (see: Dirae quantization) formal scheme

[11R32] (see: Shafarevich conjecture) formally exact complex [53C15, 55N35] (see: Spencer cohomology) formation see: shock - - ; spatio-temporal pattern - formation system s e e : concept - forms s e e : Hida theory of modular - - ; Langlands formula for the dimension of spaces of automorphic -forms in Baumslag-Solitar groups s e e : normal - formula s e e : Akhiezer-Kac - - ; atomic --; Bernshte~n-GeFfand-GePfand - - ; Binet - - ; Black-Scholes - - ; Cayley-Brill --; Chasles-Cayley-Brill--; Chevalley--; C h o w l a - S e l b e r g - - ; classical Euler product - - ; classical Mdbius inversion - - ; classical Poieson - - ; Curtis - - ; D-bracket-derivative--; Demazure - - ; dimensional - - ; error of the Lagrange interpolation - - ; Euler product - - ; exact genus - - ; Fedoeov trace - - ; FeynmanKac - - ; first-order trace - - ; fixed-point --; Giambelli - - ; Inclusion-exclusion - - ; K a c - W e y l - - ; Kac-Wey[ character - - ; McKean-Singer i n d e x - - ; Mdbius inversion - - ; modified Euler product - - ; M u m a g h a n - N a k a y a m a - - ; numbertheoretic Mdbius inversion - - ; Pieri - - ; Poisson resummation - - ; RiemannH u r w i t z - - ; Santal6 - - ; second-order trace - - ; sieve - - ; spectral radius - - ; Stark - - ; Szeg5 quadrature - - ; transition - - ; Voronof summation - - ; WeyI-Kac - - ; WeyI-Kac-Borcherds character - - ; Weyt-Kae character - - ; Wilton summation - - ; Zeuthen --

formula algebra [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) formula algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) formula for a zeta-function s e e : tion --

reflec-

formula for acceleration in spatial form [73Bxx, 76Axx] (see: Material derivative method) formula for algebraic curves s e e : Dedekind -formula for harmonic functions s e e : Potsson - formula for multiplicative anomaly s e e : Wodzicki - formula for the continuous wavelet transform see: admissibility condition for a reconstruction - - ; reconstruction - formula for the dimension of spaces of automorphic forms s e e : Langlands - formula for the eigenvalue distribution of Lapla6ans s e e : Weyl asymptotic - -

formula for the material derivative in spatial form [73Bxx, 76Axx] (see: Material derivative method) formula for the number of labelled trees s e e : Cayley - formula for the pseudo-local tomography function s e e : inversion -formula for Zernike polynomials s e e : ROdrigues - -

formula in a logical language [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) formula in first-order logic [03E30] (see: ZFC) formula representation for a linear semigroup s e e : exponential -formula representation of a continuous semigroup s e e : exponential - - ; product --

493

FORMULAS

formulas s e e : Atiyah-Bott fixed-point - - ; A t i y a h - S e g a l index - - ; A t i y a h - S i n g e r i n d e x - - ; Frege T - e q u i v a l e n t - - ; LSZreduction - - ; model of a set of - - ; replication - - ; set of formulas defining a set of atomic formulas explicitly over another set of atomic - - ; s e t of formulas defining a set of atomic formulas implicitly over another set of atomic - - ; strong implicit definition of a set of atomic formulas over another set of atomic - - ; theory axiomatized by a set of - formulas defining a set of atomic formulas explicitly over another set of atomic formulas see: set of -formu[as defining a set of atomic formulas implicitly over another set of atomic formulas s e e : set of - formulas explicitly over another set of atomic formulas s e e : set of formulas defining a set of atomic - formulas for a class of interpretations see: defining set of - -

formulas Frege equivalent over a theory

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) formulas implicitly over another set of atomic formulas see: set of formulas defining a set of atomic -formulas over a class of interpretations see: equivalence of - - ; Fregean equivalence of-formulas over another set of atomic formulas see: strong implicit definition of a set of atomic - formulas with respect to a deductive system see: logical equivalence of - - ; logically equivalent -formulation see: Krichever-Phong Whitham - formulation of the tau method see: operational - -

Forster-Swan theorem (13B30, 13C15, 16Lxx, 16P60) (refers to: Commutative ring; CWcomplex; Dimension; Jacobsen radical; Local-global principles for large rings of algebraic integers; Localglobal principles for the ring of algebraic integers; Module; Noetherian ring; Prime ideal; Spectrum of a ring; Vector bundle; Vector space) Forster-Swan theorem

[13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) Forstertheorem see:

Swan---

Forstnerig theorem

[32E201 (see: Polynomial convexity)

forward reasoning [68T30, 68T99, 92Jxx, 92KI0] (see: Dempster-Shafer theory) Foulis quantale [03G25, 06D99] (see: Quantale) Foulis semi-groupof complete orthomodular lattices [03G25, 06D99] (see: Quantale) foundation see:

axiom of - -

foundations of non-commutativelogic [03G25, 06D991 (see: Quantale) foundations of quantummechanics [03G25, 06D99] (see: Quantale) four exponentials conjecture

ill J81] (see: Schneider method) Fourier algebra (22D10, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10) (referred to in: Ditkin set; Fig~Talamanca algebra; FourierStieltjes algebra) 494

(refers to: Banach algebra; Compact group; Figh-Talamanea algebra; Fourier-Stieltjes algebra; von Neumann algebra) Fourier algebra

[22DI0, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Fourier algebra) Fourier-Bessel coefficients

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Chebyshev coefficients

[42C10, 42C15] (see: Fonrier-Haar series) Fourier-Chebyshev series

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Franklin coefficients

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Franklin series

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Haar coefficients

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Haar series (42C10, 42C15) (refers to: Bessel functions; Chebyshev polynomials; Fourier-Bessel integral; Fourier-Bessel series; Fourier series; Fourier series in orthogonal polynomials; Franklin system; Haar system; Jacobi polynomials; Laguerre polynomials; Laguerre transform; Legendre polynomials; Measure; Orthonormal system; Walsh system) Fourier-Haar series

42C10, 42C15] see: Fourier-Haar series) Fourier-Jacobi coefficients

[42C10, 42C15] see: Fourier-Haar series) Fourier-Jacobi series

[42C10, 42C15] see: Fourier-Haar series) Fourier-Laguerre coefficients

[42C10, 42C15] see: Fourier-Haar series) Fourier-Laguerre series

[42C10, 42C15] see: Fourier-Haar series) Fourier-Legendre coefficients

[42C10, 42C15] see: Fourier-Haar series) Fourier-Legendre series

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Legendre series [31B05, 33C55] (see: Zonal harmonics) Fourier multiplier operator [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) Fourier pseudo-spectral method (65Txx) (referred to in: Chebyshev pseudospectral method) (refers to: Chebyshevpseudo-spectral method; Differential operator; Lagrange interpolation formula; Trigonometric polynomial; Trigonometric pseudo-spectral methods) Fourier representation (42Axx) (refers to: Fourier series; Function; Trigonometric series) Fourier series s e e : H e a r - - - ; Hadamardlacunary - - ; linear means of a - - ; summability of - -

Fourier series representation

[42Axx] (see: Fourier representation) Fourier series with s u m m a b l e majorant of coefficients see: Beurling algebraof - -

Fourier spectral method

Fredholm index

[65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) Fourier-Stieltjes algebra (221)10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10) (referred to in: Fourier algebra) (refers to: Banach algebra; Compact group; Fig~-Talamanea algebra; Fourier algebra; Unitary representation) Fourier-Stieltjes algebra

[22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10] (see: Fourier-Stieltjes algebra) Fourier sum see: multi-dimensional partial - F o u r i e r sums see: hyperbolic partial - - ; Lebesgue constants of muRi-dimensional partial - - ; spherical partial - Fouriertransform see: discrete - - ; fast - - ; windowed - -

Fourier-Walsh coefficients

[42C10, 42C15] (see: Fourier-Haar series) Fourier-Walsh series

Fox n-eolouring (57M25) (referred to in: Rotor; Tangle move) (refers to: Abefian group; Kauff-

man polynomial; Knot and link diagrams; Knot and link groups; Link; Montesinos-Nakanishi conjecture; Symplectic structure) finitely ramified - - ; soft-

fractai dimension [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) fractal dimension of the Sierpi~ski gasket

[28A80] (see: Sierpifiski gasket) fraction see: T- - fraction expansion see: partial- - fraction-like algorithm see: Schur continued- - fractionaldifferentiation see: Szeg6 - fractional

Fibonacci

group

[20F38] (see: Fibonacci group) fractionalintegration see:

Fredholm index [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) Fredholm n - t u p l e s see: for - Fredholm operator see:

Taylor spectrum Semi- - -

Fredholm operator determinant [81T50] (see: Non-commutative anomaly) Fredholm solvability (47A53) (refers to: Adjoint operator; Banach space; Duality; Linear operator; Normal solvability) Fredholm tuple of operators

[47Dxx] (see: Taylor joint spectrum) free algebra see: absolutely-free assumption see: arbitrage- --

flee asymptoticquantum field [81Txx, 81T05] (see: Massless field) free Burnside group

[42C10, 42C15] (see: Fourier-Haar series)

fractal see: similar - -

[47Dxx] (see: Taylor joint spectrum)

Szeg6 - -

fractional-linear transformation [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) fractional part function

[26Axx] (see: Floor function) fractions technique for finding the inverse Ztransform see: partial- -Fraenkel set theory see: Zermelo- -Fraenkel set theory with the axiom of choice see: Z e r m e l o - - fragment see: guarded - fragment of first-order logic see: nvariable -fragmentablespace see: o-- - frame see: G a b o r - - ; Varignon - -

frame of discernment

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) framed link

[57M27] (see: Kirby calculus) Franke theorem

[11F67] (see: Eisenstein cohomology) Franklin coefficients see: Fourier-Franklin series see: Fourier---

Fredholm alternative

[47A53] (see: Fredholm solvability)

[20F05, 20F06, 20F32, 20F50] (see: Burnside group) free Burnside group s e e : eonjugacy problem for presentations of a - - ; word problem for presentations of a - free Burnside groups see: construction of-freeclass see: torsion- - free division see: envy- - -

free element in afield extension

[12E20] (see: Galois field structure) free element in a field extension see: pletely --

com-

free element in a Galois extension

[11R32] (see: Normal basis theorem) free element in completely -free energy see:

a

Galois extension see: Helfrich --

free energy of a binary alloy [82B26, 82D35] (see: Cahn-Hilliard equation) free group action see: fixed-point- -free net see: t r a n s v e r s a l - free path see: mean - free Steiner triple system see: eral - -

quadrilat-

free subgroupof rank two [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) free superabundant n u m b e r see:

cube- - -

free ullyafilter [54G10] (see: P-point) freedom see: degrees of - Frege equivalent over a theory see: mulas --

for-

Frege principle

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Frege relation

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Frege relation of a theory

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Frege T-equivalence

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Frege T-equivalent formulas

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Fregean deductive system

FUNCTIONS

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Fregean equivalence of formulas over a class of interpretations [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) frequencies s e e :

Natural - -

frequencies" of occurrence [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) frequency s e e :

word - -

time- - gene -natural - -

Freudenthal-Kantor triple system (17A40) (referred to in: Allison-Hein triple system) (refers to: Allison-Heintriple system; Field; Jordan triple system; Lie algebra; Lie triple system; Steiner system; Vector space) Freudenthal-Kantor triple system [17A40] (see: Freudenthal-Kantor triple system) FreudenthaI-Kantortriplesystemsee: anced --

bal-

Fried-VOlklein conjecture [11R32] (see: Shafarevich conjecture) Fritz John condition (90Cxx) (refers to: Borel measure; Convex function (of a real variable); Convex programming; Differentiable function; Implicit function; KarushKuhn-Tucker conditions; Lagrange method; Mathematical programming; Polar set; Saddle point) Fritz John condition see: basic -Fritz John equation s e e : optimality in the - - ; stability in the -Frobenius automorphism s e e : polynomial representation of the --

Frobenius automorphism on a curve [11R231

(see: Iwasawa theory) Frobenius conjecture [20F181 (see: Fitting length) Frobenius manifold [14Jxx, 35A25, 35Q53, 57R57] (see: Wbitham equations) Frobenius matrix norm (15A60) (refers to: Norm) Frobenius norm [15A60] (see: Frobenins matrix norm) Frobenius-Schur count of involutions" [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) Frobenius symbol [l 1R32, 11R45] (see: Chebotarev density theorem) Frolikcoveringtheoremsec:

full second-order model s e e : reduced - -

Leibniz-

fully adequate Gentzen system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) function see: ,At- --; a priori-condition be-

frequency localization [42Cxx, 94A12] (see: Window function) frequencylocalization s e e : frequency of an allele s e e : frequency resonance s e e :

[03Gxx] (see: Algebraic logic) full second-order filter [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) full second-order model [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

Arkhangel'skiT-

Fromovitzcondition s e e : M a n g a s a r i a n - - front s e e : laminar flame -front set criterion s e e : Hermander wave - -

frontier of a stratum [57N80] (see: Thom-Mather stratification) FTOP see: categoryL- full C*-algebra [22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10] (see: Fourier-Stieltjes algebra) full-linecase see: Inverse scattering, - full restricted first-order language

lief - - ; Abelian - - ; abstract arithmetical - - ; addition theorem for the exponential - - ; adjoint Baker-Akhiezer - - ; analytic - - ; anti-holomorphic--; approximately additive -- ; axially symmetric -- ; BakerAkhiezer - - ; band limited - - ; belief - - ; beta- - - ; bpa - - ; Bukhstab - - ; cadlag - - ; capacity - - ; Carath~odory - - ; Cardinal - - ; ceiling - - ; Chebyshev weight - - ; choice - - ; Clausen - - ; Cobb-Douglas - - ; commonality - - ; conditional belief - - ; contraction of a - - ; counter - - ; critical line for the zeta- --; decreasing rearrangement of a - - ; Dedekind zeta- - - ; denotation - - ; Dickman - - ; Dickman-De Bruijn - - ; Digamma - - ; direction for a ridge - - ; Dirichlet inverse of an arithmetical - - ; divisor - - ; double exponential hat - - ; elliptic - - ; entier - - ; Euler totient - - ; Euler zeta- - - ; excessive -- ; exhaustion - - ; finitely additive additive set - - ; Floor - - ; focal point of a belief - - ; fractional part - - ; functional equation for the Riemann c . _ ; ,7. _ ; 9. _ ; Gaussian - - ; Gegenbauer weight - - ; generalized matrix - - ; generalized Sprague-Grundy - - ; g e n e r a t i n g - - ; Gddel r e c u r s i v e - - ; graphoidal properties of a belief - - ; greatest i n t e g e r - - ; Green - - ; G r u n d y - - ; Grundy-Sprague - - ; Hadamard factorization of the Dedekind zeta- - - ; HardyKrause variation of a - - ; harmonic - - ; hat - - ; hazard - - ; Heaviside - - ; Herglotz - - ; Hurwitz zeta- - - ; I - - - ; impedance - - ; independent variable sets for a belief - - ; inner - - ; integral part - - ; inversion formula for the pseudo-local tomography - - ; isotone - - ; j - - - ; Jacobi weight - - ; Jacobi zeta- - - ; Jest - - ; K - - ; Klimek Green - - ; L a g u e r r e weight - - ; linear radial basis - - ; M-quasi-symmetric - - ; Machlup-Onsager - - ; mass - - ; Mathemalice code for the Dickman - - ; meaning - - ; M6bius - - ; modified zeta- - - ; modular j - - - ; monotone set - - ; multiquadric - - ; nearest i n t e g e r - - ; Nevanlinna - - ; non-additive set -- ; null-additive set - - ; Onsager-Machlup - - ; optimal hat - - ; outer - - ; p-adic L - - ; plausibility - - ; pluricomplex Green - - ; pluriharmonic - - ; plurisubharmonic--; Pompeiu - - ; preference - - ; probability assignment - - ; production - - ; pseudo-local tomography - - ; Psi - - ; pulse - - ; quasi-continuous - - ; quasi-symmetric -- ; Radial basis - - ; Ramanujan - - ; rectangle - - ; reflection formula for a zeta- - - ; resolvent - - ; reversible transition -- ; Ridge - - ; Riemann ( - - - ; Riemann zeta- - - ; Robin - - ; sawtooth - - ; Schur Q - - ; separately harmonic - - ; simple measurable - - ; sinc - - ; slowly decaying - - ; Smarandache - - ; space-constructible--; sparse - - ; spectral factor of a weight - - ; SpragueGrundy - - ; squeeze - - ; standard local tomography - - ; Student-~ hat - - ; subadditive set - - ; superadditive set - - ; superparabolic - - ; symmetric Green - - ; symmetrization of a - - ; Szeg5 - - ; "r- - - ; table

mountain- - - ; tau- -- ; tensor-product basis -- ; third theta- - - ; time-constructible - - ; time limited - - ; trace - - ; transfer - - ; triangular hat - - ; triangular set - - ; tubular -- ; utility -- ; vacuous belief - - ; Weyl - - ; Weyl m - - - ; Wightman - - ; Window - - ; ( - - - ; Zak transform of the Gaussian - - ; zero set of a - - ; zeta- - -

function algebra [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C'* -algebras) function algebrasee: Colombeau generalized - - ; complex - - ; Egorov generalized - - ; generalized - - ; nowhere-dense generalized - - ; real - - ; Rosinger nowheredense generalized --

function algebra closed under conjugation [46E25, 54C35] (see: Bishop theorem) function algebra separating the points of a set

[46E25, 54C35] (see: Bishop theorem) function algebras s e e : 13ishop theorem for real - - ; Colombeau generalized - - ; Generalized - function around a point s e e : concentration of a - - ; measure of concentration of a - function at infinity s e e : rate of decay of a--

function field [14Axx] (see: Zariski problem on field extensions) function field s e e :

algebraic - -

function field of a n algebraic curve [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brillformula) function fields of genus zero s e e : Amitsur theorem on - function for stratifications s e e : distance - function identity s e e : Jacobi theta- - function in a learning system s e e : target - -

function in material form [73Bxx, 76Axx] (see: Material derivative method) function in spatial form [73Bxx, 76Axx] (see: Material derivative method) function in statistics s e e : estimable parametric -- ; parametric - function method for regularization s e e : Zeta- -function of a colligation s e e : characteristic -function of a complexvariable s e e : Quasisymmetric - function of a curve s e e : zeta- - function of a domain s e e : Riemann mapping -function of a finite group s e e : Eulerian - function of a group s e e : growth - function of a pseudo-differentia/ operator see: zeta- - function of a state model of a physical system s e e : partition - function of a statistical test s e e : power - function of an algebraic number field s e e :

L-function of an operator

see: zeta- - function of an operator colligation s e e : characteristic operator-valued - function of an operator vessel s e e : joint characteristic - - ; normalized joint characteristic - -

function of bounded mean oscillation [30Axx, 46Cxx, 46Exx, 471335] (see: Berezin transform; BMOAspace) function of bounded variation [34B24, 34L40] (see: Sturm-Liouville theory) function of KP-Toda type s e e : tau- - function of optics s e e : aberration - function of the first kind s e e : Bessel - -

function on a Banach space s e e : holomorphic - function on a part{aLly ordered set s e e : level of a rank - - ; rank - function on T s e e : M-quasi-symmetric - - ; quasi-symmetric - function regularization s e e : noncommutative anomaly for zeta- - - ; zetafunction set s e e : complete - - ; total - function space s e e : degree for a Sobolev -function theorem s e e : implicit - function theory s e e : belief - - ; constructive - -

Function vanishing at infinity (54C35) (refers to: Banach-Stone theorem; Continuous function) function vanishing at infinity [22D10, 22D25, 43A07, 43A15, 43A25, 43A30, 43A35, 46Exx, 46J10] (see: Banach-Stone theorem; FourierStieltjes algebra) function version of the matrix tree theorem see: generating - function with pole at infinity s e e : Green - functional s e e : integral representation of a - - ; Lempert - - ; Willmore - -

functional calculus [17C65, 46H70, 46L70] (see: Banach-Jordau algebra) functionalcalculus s e e : Riesz-Dunford-functional equation s e e : Abel - - ; nonautonomous Schreder - - ; Schreder - - ; superstable - - ; translation --

functional equation for the Riemann ~function [llM06] (see: Riemann ~-function) functional equations s e e : direct construction method in the stability of - - ; stability problem of - functional grammar see: lexical- --

functional integration [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) functional model see: de BrangesRovnyak - - ; Sz.-Nagy-Foias --

functional model of a pair of operators [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) functional of Brownian motion s e e : nonMarkovian - functionalseries s e e : Volterra--; WienerVolterra -functions s e e : algebraic independence of values of Abelian - - ; algebraic independence of values of elliptic - - ; algebraic independence of values of exponential - - ; c o - m o n o t o n e - - ; converse of Gauss mean-value theorem for harmonic - - ; equi-measurable--; field of m o d u l a r - - ; G a b o r - - ; HalI-Littlewood - - ; Hauptmodul for a field of modular - - ; Jaoobi theta- - - ; Leibniz-Hopf algebra and quasi-symmetric - - ; local ring of - - ; mean-value characterization for harmonic - - ; mean-value characterization for holomorphic -- ; mean-value characterization of pluriharmonic - - ; meanvalue characterization of separately harmonic - - ; mean-value theorem for harmonic - - ; modular - - ; multiplication of generalized - - ; natural order on a space of real-valued - - ; Poisson formula for harmonic - - ; 13iesz decomposition theorem for subharmonic - - ; Riesz decomposition theorem for superharmonic - - ; rule of combination of two independent belief - - ; space of continuous - - ; transcendence of values of Abelian - - ; transcendence of values of elliptic - - ; transcendence of values of exponential - - ; transcendence properties of values of ana-

495

FUNCTIONS

lytic - - ; transcendence theory of Abelian - - ; transcendence theory of elliptic - - ; transcendence theory of exponential - functions in machine learning see: probabilistic - functions in S o b o l e v s p a c e s see: approximation error of - functions of bounded mean oscillation see: space of analytic - functions of bounded variation see: space of-functions of compact support see: algebra of-functions of vanishing mean oscillation see: space of analytic - functor see: Bernshtein-GeI'fandPonomarev reflection - - ; duality - - ; k,- ; reflection - - ; simple - ; Tilting - - ; Yoneda - functoriality see: L a n g l a n d s - -

fundamental identity of sequential analysis [62L10] (see: Sequential probability ratio test) fundamental theorem on finite Abelian groups [llNxx, 11N32, 11N45, I1N80] (see: Abstract analytic number theory) Furstenberg-Katznelson density HalesJewett theorem [05D10] (see: Hales--Jewett theorem) fuzzy constraint [90C70] (see: Fuzzy programming) fuzzy continuity see:

L- -;

lattice- - -

fuzzy continuous mapping [03G10, 06Bxx, 54A40] (see: Fuzzy topology) fuzzy continuous mapping see:

L-

-;

lattice- - -

fuzzy goal [90C70] (see: Fuzzy programming) fuzzy measure [28-xx] (see: Non-additive measure) Fuzzy programming (90C70) (refers to: Linear programming; Mathematical programming; Nonprecise data) fuzzy topological space [03G10, 06Bxx, 54A40] (see: Fuzzy topology) fuzzy topological space see:

L - - - ; lattice-

Fuzzy topology (03G10, 06Bxx, 54A40) (refers to: Category; Complete lattice; Continuous function; Sets, category of; Tensor product; Topological space) fuzzy topology see: variable-basis - -

L - - - ; lattice- - - ;

(see: Sprague-Grundy function) G6 -insertion property [26A21, 54E55, 54G20] (see: Sorgenfrey topology) .g-module see: highest-weight -G - s y s t e m see: consecutive k-out-ofn: G a b o r frame [42Axx, 44-XX, 44A55]

(see: Zak transform) Gabor functions [42Axx, 44-XX, 44A55] (see: Zak transform) Gabor representation problem [42Axx, 44-XX, 44A55] (see: Zak transform) Gabriel theorem [16Gxx] (see: Tits quadratic form) Gabriel theorem [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) GAIO software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Galerkin approximation [35L15] (see: Euler-Poisson-Darbnux equation) Galerkin method [47H17, 65Lxx] (see: Approximation solvability; Tau method) Galerkin method see: G a l l a i theorem

Petrov- --

[05D10] (see: Hales-Jewett theorem) Gallai-type theorem [52A35] (see: Geometric transversal theory) Galois connection theorem [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mtbius inversion) Galois embedding problem [11R32] (see: Shafarevlch conjecture) Galois extension see: completely free element in a - - ; completely normal element in a - - ; free element in a - - ; norm of an element in a - - ; trace of an element in a--

Galois field structure

(12E20) (referred to in: Shift register sequence) (refers to: Block design; Coding and decoding; Cryptography; Cryptology; Cyclic group; Difference set; Elliptic curve; Frobenlus automorphism; Galois extension; Galois field; Galois geometry; Galois theory; Irreducible polynomial; Normal basis theorem; Primitive polynomial; Symmetric design; Vector space) Galois field (update) [12E20] (see: Galois field structure) Galois group over Q see:

G 7-function [90D05] (see: Sprague-Grundy function) (~ see: coset ring of -Warlimont axiom - ~}~ arithmetical semi-group see:

axiom-

G 1 arithmetical semi-group see:

axiom-

g.~ see:

G1 arithmetiealsemi-groupsee:

9-function [901)05]

496

absolute --

Galois g r o u p o v e r Q a b see:

axiom---

absolute --

Galois problem over Q a b see; inverse Galois theory see: Shafarevichconjecture in inverse -game see: B a n a c h - M a z u r - - ; draw in e - - ; generalized B a n a c h - M a z u r -- ; impartial - - ; normal play of a - - ; octal - - ; play in the 8 a n a c h - M a z u r - - ; Sierpifiski - - ; stationary strategy in the generalized B a n a e h - M a z u r - - ; stationary winning strategy in the generalized B a n a c h Mazur - - ; strategy in the generalized B a n a c h - M a z u r - - ; succinct -- ; sum- - - ; tactics in the generalized B a n a c h - M a z u r - - ; topological - - ; winning strategy in the generalized B a n a c h - M a z u r - -

game graph [90D051 (see: Spragne-Grundy function) game-graphof a sum [90D05] (see: Sprague-Grundy function) game of Nim [90D05] (see: Sprague-Grundy function) Gamelin-Sibonyexample [31C10, 32F05] (see: Pluripotentlal theory) games see: sum of - g a m m a matrices see: Dirac -gap see: mass- - gap condition see: mass- - -

gap in a canonical sequence [65Lxx] (see: Tau method) gap Korteweg-de Vries solution [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) gap solution see: finite- - GAR-realizability of a group [11R32] (see: Shafarevich conjecture) gardener string construction [90B85] (see: Fermat-Torricelli problem) gas see: kinetic description of a - - ; Knurlsen - - ; rarefied -gasket see: Brownian motion on the Sierpifiski - - ; fractal dimension of the Sierpifiski - - ; Laplacian on the Sierpifiski - - ; Sierpifiski - gate see: applying a quantum - - ; quantum -gauge theory see: S U ( 2 ) - - ; susy - -

gauge transformation [14Jxx, 35A25, 35Qxx, 35Q53, 57R57, 78A25] (see: Magnetic monopole; Whitham equations) Gauss* constant [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Gauss factorization theorem [12D05] (see: Factorization of polynomials) Gauss-Jacobiintegration [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) Gauss lemniscate constant [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Gauss-Lobatto points [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method) Gauss map [53A10, 53C42] (see: Weierstrass representation of a minimal surface) Gauss-Marker theorem [62Jxx] (see: ANOVA) Gauss mean-value theorem for harmonic functions see: converse of - -

Gauss quadratic reciprocity law [19Cxx] (see: Steinberg symbol) Gauss-Salamin algorithm [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Gauss sum [11R23] (see: Iwasawa theory) Gauss theoremon products of monic irreducible polynomials [12D051 (see: Factorizatiou of polynomials) Gaussian elimination [14L35, 14M15, 20G20]

(see: Schubert cell) Gaussian function [41A05, 41A30, 41A63, 42A63] (see: Radial basis function; Uncertainty principle, mathematical) Gaussian function see: Zak transform of the - -

Ganssian measure [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Gaussian space [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Gegenbaner-Sobolev orthogonalpolynomials [33C45, 33Exx, 46E35] (see: Sobolev inner product) Gegenbauer weight function [33C45, 33Exx, 46E35] (see: Sobolev inner product) Gel'fand formula Gel'land- -Gel'fand-Gerfand shteib- --

see:

Bernshtein-

formula see:

Bern-

Gel'fand-Kirillov dimension [11F27, llF70, 20G05, 81R05] (see: Segal-Shale-Well representation) Gel'fand mapping [42Axx, 44-XX, 44A55] (see: Zak transform) Gel' fand-NaYmark-Segalconstruction [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Gel'fand-Narmark theorem [17Cxx, 46-XX] (see: J13 *-triple) G e l ' f a n d - P o n o m a r e v reflection functor see: Bernshte'fn- --

Gel'fand problem [32E20] (see: Polynomial convexity) Gel'fand quantale [03G25, 06D99] (see: Qnantale) Gel'fand quantale see: algebraically irreducible representation of a - - ; discrete - - ; irreducible representation of a - - ; point of a - - ; representation of a - - ; spatial - GeJ'fand quantales see: algebraically strong homomorphism of - - ; discrete homomorphism of - - ; right embedding homomorphism of --

Gel'fond-Schneider method (11J85) (referred to in: Mahler method) (refers to: Algebraic independence; Algebraic number; Diophantine approximations; Dirichlet principle; Gel'fond-Baker method; Hilbert problems; Meromorphic function; Modular function; Number field; Schneider method; Transcendental extension; Transcendental number) gene frequency of an allele [92D10] (see: Hardy-Weinberg law) gene pool [92D10] (see: Hardy-Weinberg law) general case of the Lov~szlocal lemma [05C80] (see: Lov~iszlocal lemma) general logic [03Gxx] (see: Algebraic logic) general mean [62Jxx] (see: ANOVA) general semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

GERSHGORIN THEOREM

general semantical system see: algebraizable - - ; equivalential - - ; finitely algebraizable - - ; finitely equivalential - - ; pretoalgebraic - general will s e e : Rousseau theory of - -

generalization of the Hadamard-Fischer inequality for positive semi-definite Hermitian matrices [15A 25, 20C30] (see: Immanant) generalized Banach-Mazur game [54E52] (see: Banach-Mazur game) generalized B a n a c h - M a z u r game see: stationary strategy in the - - ; stationary winning strategy in the - - ; strategy in the - - ; tactics in the - - ; winning strategy in the - -

generalized BBM equation [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) generalized Benjamin-Bona-Mahony equation [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) generalized Benjamin-Bona-Mahony equation [35Q53, 76B 15] (see: Benjamin-Bona-Mahony equation) generalized Cartan matrix [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) generalized Cayley-Hamilton theorem [47A45, 47A48, 47A65, 47D40, 47N701 (see: Operator vessel) generalized Choquctintegral [28-XX/ (see: Choquet integral) generalized Cohen-Macaulay ring [13A30, 13H10, 13H30] (see: Buchsbaum ring) generalized degrees [55M25] (see: Brouwer degree) generalized derivative see: Clarke - generalized Dirac operator [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) generalized directional derivative [90C30] (see: Clarke generalized derivative) generalized effective Hilbert Nullstellensatz [14A10, 14Q20] (see: Effective Nullstellensatz) generalized Euler-Poisson-Darboux equation [35L15] (see: Euler-Poisson-Darboux equation) generalized Fermat-Torricelli problem [90B85] (see: Fermat-Torricelli problem) generalized Fibonacci group [20F38] (see: Fibonacci group) generalized function algebra [46F30] (see: Generalized function algebras) generalized function algebra s e e : Colombeau - - ; Egorov - - ; nowhere-dense - - ; Rosinger nowhere-dense --

Generalized function algebras (46F30) (referred to in: Colombeau generalized function algebras; Egorov generalized function algebra; Multiplication of distributions; Rosinger nowhere-dense generalized function algebra)

(refers to: Colombeau generalized function algebras; Differentiable function; Differential algebra; Egorov generalized function algebra; Generalized function; Generalized functions, space of; Multiplication of distributions; Net (directed set); Non-standard analysis; Rosinger nowhere-dense generalized function algebra; Sheaf; Ultrafilter) generalizedfunctionalgebrassee: Colembeau -generalized functions s e e : multiplication of--

generahzed GMANOVA [62Jxx] (see: ANOVA) generalized gradients s e e :

set of - -

generalized Hadamard matrix [05Bxx] (see: Net (in finite geometry)) generalized Harry Dym equation [35Q53, 58F07] (see: Harry Dym equation) generalized homologytheory [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) generalized Hopf bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) generalized Hopf point bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) generalized index theorems [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) generalized Jacobian [90C30] (see: Clarke generalized derivative) generalized Julia-Wolff-Carathdodory theorem [30C45, 47HI0, 47H20] (see: Julia-Wolff-Carath~odery theorem) generalized Kac-Moody algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) generalized Lax equation [35Q53, 58F07] (see: Harry Dym equation) generalized Mal'cev theorem [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) generalized matrix function [15A15, 20C30] (see: Immanant) generafized mean-valuecharacterization [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) generalized mean-value condition [31A05, 31B05, 31CI0, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) generalized moments [44A60, 47A57] (see: Moment matrix) generalized moonshineconjectures [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) generalized Morse index [58Fxx] (see: Conley index) generalized multivariate analysis of variance [62Jxx] (see: ANOVA;GMANOVA) generalized Nim-sum [90D05] (see: Sprague-Grundy function) generalized Shafarevichconjecture [11R32]

(see: Shafarevich conjecture) generalized Skorokhod space [60B10, 60G05] (see: Skorokhod space) generalized Sprague-Grundyfunction [90D051 (see: Sprague-Grundy function) generalized tilting module [16Gxx] (see: Tilting theory) generalized dlting module [16Gxx] (see: Tilting theory) generafized topology [18D10, 18D15] (see: *-Autonomous category) generalizer in a learning system [68T05] (see: Machine learning) generated by a hotomerphic mapping s e e : semi-group -generated cone s e e :

accessibility of finitely- -generated groups with more than one end see: Stallings classification of finitely - generated k-algebra s e e : finitely- - generated topological space s e e : compactly - -

generating function [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) generating function version of the matrix tree theorem [05C50] (see: Matrix tree theorem) naturallanguage - -

generative grammar [68S051 (see: Natural language processing) generator s e e : Lie - - ; normal basis - - ; projective - - ; Tausworth -generator in a learning system s e e : experiment - -

generator of a cone [15A39, 90C05] (see: Motzkin transposition theorem) generator of a continuous semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) generators of a module s e e : local number of-generators of continuous semi-groups s e e : parametric representations of - -

generic smoothmapping [57N80] (see: Thom-Mather stratification) genetics s e e :

geometric average s e e :

algebro-

arithmetic- - -

geometric Brownian motion [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) geometric Brownian motion model s e e : Black-Scholes - -

geometric group theory [05C25, 20Fxx, 20F05, 20F06, 20F32] (see: Baumslag-Solitar group; HNNextension) geometric lattice [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) geometric lattice geometric mean geometric mean arithmetic- - geometric mean

see:

flat in a - arithmetic- - algorithm s e e : Lagrange see:

method s e e :

arithmetic-

geometric mean process s e e :

Arithmetic-

finitely --

generated continuous semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Seml-group of holomorphle mappings) g e n e r a t e d groups see: Wall conjecture on

generation s e e :

geometric AKNS-potential s e e :

population - -

Gentzen system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logle) Gentzen system s e e : fully adequate - - ; model of a -genus s e e : Witten -genus formula s e e : exact - -

genus of a curve [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-BrUl formula) genus of a knot [57M25] (see: Positive link) genus of a knot s e e : 4-ball - - ; planar -genuszero s e e : Amitsur theorem on function fields of -geodesic flow s e e : Liouville theorem on invariance under -geodesic submanifold s e e : totally - -

Geometric measure theory (28A78, 49Qxx, 49Q15, 53C65, 58A25) (referred to in: Current) (refers to: Absolute continuity; Cantor set; Current; Differential form; Exterior algebra; Grassmann manifold; Hausdorff measure; Lebesgue measure; Mutually-singular measures; Plateau problem; Radon measure; Rectifiable curve; Variational calculus; Weak topology)

geometric measure theory [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) geometric permutation [52A35] (see: Geometric transversal theory) geometric quantization [81Qxx] (see: Dirac quantization) geometric quantization [14Jxx, 35A25, 35Q53, 46Lxx, 57R57] (see: Toeplltz G'*-algebra; Whitham equations) geometric Ramsey theorem [05D101 (see: Hales-Jewett theorem) Geometric transversal theory (52A35) (refers to: Computational geometry; General position; Higherdimensional geometry; Locally convex space; Matroid) geometric transversaltheory [52A35] (see: Geometric transversal theory) geometric transversal theory s e e : algorithmic - geometry s e e : conformal - - ; eigenvalues and - - ; enumerative - - ; m e t a s y m p l e c t i c - - ; quantum - - ; Schwarzschild - - ; uniform - geometry) s e e : Net (in finite - geometry) (update) s e e : Net (in finite -geophysicalscattering s e e : inverse -geophysical scattering problem s e e : inverse - -

Gerschgorin theorem [15A18] (see: Gershgorin theorem) GerYgorin theorem [15A18] (see: Gershgorin theorem) Gershgorin disc [15A18] (see: Gershgorin theorem) Gershgorin theorem (15A18) (referred to in: Bauer-Fike theorem) (refers to: Cassini oval; Eigen value; Graph, oriented; Matrix)

497

GERSHGORIN-TYPE THEOREM

Gershgorin-type theorem [15A42] (see: Bauer-Fike theorem) Gevrey norm [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) GF 2-linear code [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) GFAsee: Colombeau -ghost see: Faddeev-Popov-ghost field [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) ghost field s e e :

anti- - -

Giambelli formula [14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) Gil de Lamadrid and Sova compact derivative [90Cxx[ (see: Fritz John condition) Gilbert-Varshamovbound [68Q05, 68Q10, 68Q15, 68Q25, 81pxx, 81P15, 94Axx] (see: Quantum information processing, science of) Gillman-Henriksen P-space 154G10] (see: P-space) Ginsburg model s e e : Hurwitz-space Korteweg-de Vries-Landau- -Ginsburgtype s e e : topological field theory of Landau- --

Girard quantale [03G25, 06D99] (see: Quantale)

Delsarte---

Massless K l e i n - -

correlation coefficient [62H20] (see: Spearman rho metric) gradients see: set of generalized grade

Cameron-Martin-

Girsanov ~'ansformation [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula; Option pricing) dual B r o w n - - Brown- - - ; dual

Gleason-Kahane-Zelazko theorem (46Hxx) (refers to: Banach algebra; Entire function; Functional; Linear functional; Spectrum of an element) global dimension [16GI0, 16G20, 16G60, 16G70] (see: Tilted algebra) global discrepancy [65C10] (see: Linear congruential method) Global Manifolds 1D software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Global Manifolds 2D software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) global principle in commutative algebra s e e : local- --

glueing of discs [32E20] (see: Polynomial convexity) GMANOVA (62Jxx) (refers to: ANOVA) GMANOVA [62Ixx] (see: ANOVA) GMANOVA [62Jxx] (see: ANOVA) GMANOVA see: canonical form for - - ; extended - - ; generalized - goal s e e : fuzzy --

Gtdel axiom of constructibility 498

Goethals code s e e :

Goldbach conjecture [11L07, IlM06, 11P32] (see: Vaughan identity) Goldbach problem see: W a r i n g - - Goldie theorem [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) Goldstone boson [8lTxx, 81T05] (see: Massless field) good universal weight [28D05, 54H20] (see: Wiener-Wintner theorem) Goodman-Pollack theorem [52A351 (see: Geometric transversal theory) Gordan theorem [15A39, 90C05] (see: Motzkin transposition theorem) Gordan theorem of the alternative [90Cxx] (see: Fritz John condition) Gordian number [57M25] (see: Positive link) Gordon equation s e e :

Girsanovtheorem s e e :

Gitler modules s e e : Gitler spectra s e e : Brown- - -

[54G10] (see: P-space) GiJdel recursive function [03D15, 68Q151 (see: Computational complexity classes) GOdel relative consistency of set theory [03E30] (see: ZFC) Goeritz matrix [57M251 (see: Listing polynomials)

grading property s e e : grading-restriction

--

L ( O)- --

conditions

[llFI1, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex uperator algebra) grammar s e e : Chomsky transformational - - ; g e n e r a t i v e - - ; head-driven phrase structure - - ; lexical-functional--; probabilistic - - ; tree adjoining - - ; type-0 - - ; unification - grammar parser s e e : tree adioining -grammatical usage of a natural language see: extra- -graph s e e : 2-labelling of a - - ; acyclic polynomial of a -- ; Cayley - - ; characteristic polynomial of a - - ; chromatic number of a random - - ; chromatic polynomial of a - - ; circulant -- ; crossing number of a - - ; D- - - ; dependency - - ; drawing of a finite - - ; edge Laplacian matrix of a - - ; edge-transitive Cayley - - ; extremal - - ; f-smoothing vertices of a - - ; game - - ; Hamiltonian - - ; incidence matrix of a -- ; interval - - ; k-matching in a - - ; Laplacian matrix of a - - ; Laplacian of a - - ; matching cover in a - - ; matching in a - - ; Matching polynomial of a --; matching unique ; mixed Laplacian matrix of a - - ; Ramanujan - - ; reduced i n t e r v a l - - ; rotational number of a - - ; simple matching polynomial of a - - ; sum- - - ; unit-interval - - ; unlabelled -- ; vertex-transitive - - ; weight of a matching in a - -

graph isomorphism problem [05Exx] (see: Cellular algebra) graph manifold s e e : Waldhausen -graph model s e e : random - graph of a group s e e : Cayley - graph of a matrix s e e : directed - graph of a sum s e e : game- - graph of groups s e e : decomposition as a k-acylindrical -graph theory s e e : n - r o t o r in -- ; rotor in --

graphoidal properties of a belief function [68T30, 68T99, 92Jxx, 92K10[ (see: Dempster-Shafer theory) graphoidal structure in belief theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) graphs s e e : characterization of Cayley -- ; co-matching -- ; Laplace operator on - Grassmann algebra s e e : infinitedimensional - -

Grassmann variable [I4Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Grassmannian s e e : affine --; isotrepic -gravitationalwaves s e e : colliding - gravitysee: two-dimensional - -

Gray image [94Bxx] (see: Delsarte-Goethals code) greatest integer function [26Axx] (see: Floor function) Grebe point [51M04] (see: Triangle centre) greedyalgorithm s e e :

node-labeling --

Green function [31C10, 32F05] (see: Phiripotential theory) Green function s e e : Klimek - - ; pluricomplex - - ; symmetric - -

Green function with pole at infinity [31C10, 32F05] (see: Phiripotential theory) Greenberg conjecture [llR23] (see: Iwasawa theory) grid scheme s e e :

multi- - -

Griess algebra [ i l F l l , 17B67, 20D08, 81T10] (see: Moonshine conjectures) Griess Monster s e e : F i s c h e r - - Griess monster group s e e : Fischer- -Griffith construction s e e : Evans- - -

Gromov hyperbolic group [20F05, 20F06, 20F32, 20F50] (see: Burnside group) Gromov-Lawson conjecture [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) Gromov-Witten invariants [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Grothendieck topos [18D15] (see: Cartesian-closed category) group see: 3-manifold --; accessible - - ; adelic - - ; amenable - - ; arithmetic - - ; asynchronous automatic - - ; automatic - - ; automorphism - - ; Baumslag-Solitar - - ; bipolar structure on a - - ; block of a--; Branch - - ; Burnside - - ; Burnside algebra of a - - ; Cayley graph of a --; (~ech-complete semi-topological - - ; collineation - - ; conjugacy problem for presentations of a free Burnside - - ; convex rigid Baumslag-So]itar - - ; degree of a permutation - - ; elementary Abelian p- - - ; end of a - - ; entropic right quasi- - ; Eulerian function of a finite - - ; exponent of a - - ; FC- - - ; Fibonacci - - ; finite Abeiian --; finite c o n j u g a t e - - ; finitelypresented non-Hopfian - - ; FischerGriess monster - - ; Fitting length of a - - ; fractional Fibonacci - - ; free Burnside - - ; GAR-realizability of a - - ; generalized Fibonacci - - ; Gromov hyperbolic - - ; growth function of a - - ; Heisenberg - - ; hereditarily just infinite - - ; Hilbert-Siegel m o d u l a r - - ; Hopf - - ; Hopfian - - ; ideal class - - ; Jacobi - - ; JSJ decomposition of a - - ; just infinite - - ; just infinite branch - - ; Kac-Moody - - ; Klein 4- - - ; linear character of a symmetric - - ; little - - ; loop - - ; mapping class - - ; meta-Abelian - - ; meta-Abelian Baumslag-Solitar - - ;

metabetian - - ; metaplectic - - ; monster - - ; nilpotent length of a - - ; Noetherian - - ; non-Hopfian - - ; normal closure of a - - ; cat s - - - ; orthochronous proper P o i n c a r 6 - - ; p e r i o d i c - - ; permutation - - ; Polish t o p o l o g i c a l - - ; presentation of a - - ; presentation of a BaumslagS o l i t a r - - ; p r o - p - - - ; profinite branch - - ; projective profinite - - ; Q-roots of a semi-simple algebraic - - , R-roots of a semi-simple algebraic - - ; Regular - - ; regular automorphism - - ; regular p- - - ; regular permutation - - ; renormalization - - ; representation - - ; representation ring of a compact - - ; residually-finite - - ; rigid Baumslag-Solitar - - ; Schur - - ; semi-topological - - ; Siegel modular - - ; simplicial - - ; Solitar-Baumslag - - ; spin character of a symmetric - - ; transitive - - ; transitive regular permutation - - ; weak order of a symmetric - - ; Weyl - - ; word problem for presentations of a free Burnside - group .0 s e e : Conway - -

group acting on a tree [20F05, 20F06, 20F32] (see: HNN-extension) groupaction s e e : fixed-point-free--; fixedpoint set of a - - ; spherically transitive - -

group action on a rooted tree [20E08, 20E18, 20Fxx] (see: Branch group) group actions s e e : Smith theory of -group algebras s e e : Brauer theorem on splitting fields for -group cohomology s e e : Quillen theorem on Krull dimension of - - ; Serre theorem in-group decomposition s e e : resolution in -group invariant of links s e e : Abelian -group of a link s e e : colouring - group of an HNN-extension s e e : base -group of automorphisms s e e : regular--

group of biholomorphic automorphisms [17Cxx, 46-XX] (see: JB * -triple) grouper C ( X ) s e e : E x t - group of cohomological dimension one [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) group of exponent n [20F05, 20F06, 20F32, 20F50] (see: Burnside group) group of exponential growth [20E08, 20E 18, 20Fxx] (see: Branch group) group of finite width s e e :

branch - -

group of intermediate growth [20E08, 20El 8, 20Fxx] (see: Branch group) group of permutations [20-XX] (see: Regular group) groupof permutations s e e : regular-group o f polynomial growth

[20E08, 20E18, 20Fxx] (see: Branch group) group of units [12J10, 12J20, 13A18, 16W60] (see: S-integer) group over Q s e e :

absolute Galois - -

groupover Qab s e e : absolute Galois - group presentation s e e : almost c o n v e x - group rings s e e : isomorphism problem for - group theory s e e : combinatorial -- ; geometric - groupoid s e e : n-fold - - ; simpflcial - groups s e e : Accessibility for - - ; automorphisms of Baumslag-Solitar - - ; complexity of - - ; conjugacy problem for Fib o n a c c i - - ; construction of free Burnside - - ; decomposition as a k-acylindrical graph of - - ; finiteness conditions for - - ; fundamentattheorem on finite Abelian - - ; growth of - - ; isoperimetric inequality in

HEAD-DRIVENPHRASE STRUCTUREGRAMMAR

; normal forms in Baumslag-Solitar - - ; Projective representations of symmetric and alternating - - ; Stallings characterization of bipolar structures on - - ; subgroups of Baumslag-Solitar - - ; Wall conjecture on accessibility of finitely-generated - - ; word problem for Fibonacci - groups acting on trees s e e : Sass-Serre theory of -groups as examples s e e : BaumslagSolitar - groups with more than one end s e e : Stallings classification of finitely generated --

-

Grover algorithm [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

o13 Grover search algoritinn [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

o13 growth s e e : group of exponential - - ; group of intermediate - - ; group of polynomial - growth curves s e e : modelling - -

growth function of a group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) growth of groups [05C251 (see: Cayley graph) Grtinbaumconjecture [52A35] (see: Geometric transversal theory) Gdinbaum theorem [52A35] (see: Geometric transversal theory) Grundy function [90D05] (see: Sprague-Grundy function) Grundy function s e e : S p r a g u e - - - ; Sprague- - -

generalized

Grundy-Sprague function [90D05] (see: Sprague-Grundy function) Grushko theorem [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) guarded fragment [03Gxx] (see: Algebraic logic) Gusarov filtration s e e : V a s s i l i e v - - Gusarov invariants s e e : Vassiliev- - GusarovskeJn module s e e : Vassiliev- - -

Hear series see: Fourier- -Haar wavelet [42Cxx] (see: Daubechies wavelets) Habiro Cn-moves [57M25] (see: Tangle move) Habiro moves [57M25] (see: Tangle move) Hadamard 2-design [05B30] (see: Affine design) Hadamard 3-design [05B30] (see: Affine design) Hadamard factofization of the Dedekind zeta-function [11R29] (see: Odlyzko bounds) Hadamard factorization theorem [11M06] (see: Riemann ~-functlon) Hadamard-Fischer inequality [15A15, 20C30] (see: Immanant) Hadamard-Fischer inequality for positive semi-definite Hermitian matrices s e e : generalization of the - -

Hadamard inequality for Hermitian matrices [15A15, 20C30] (see: Immanant) Hadamard-lacunary Fourier series [42A16, 42A24, 42A28] (see: Beurling algebra) Hadamard matrix s e e : generalized-Hadamard multiplication s e e : S c h u r - - -

Hadwiger theorem [52A35] (see: Geometric transversal theory) Hadwiger transversal theorem [52A35] (see: Geometric transversal theory) Hadwiger-type transversaltheorem [52A35] (see: Geometric transversal theory) hairy ball theorem [58F22, 58F25] (see: Seifert conjecture) Haken manifold (57N10) (refers to: Connected sum; Covering surface; Dehn lemma; Fundamental group; Homology group; Piecewiselinear topology; Projective plane; Three-dimensional manifold) Haken manifold s e e :

.H *-homomorphism [46J10, 46L05, 46L80, 46L85] (see: Multipliers of G'* -algebras) H see: tilted algebraof type - H-dimodule Long --

Azumaya algebra

see:

H ~ -disc [32E20] (see: Polynomial convexity) n °° symbol s e e : with - -

Toeplitz operators

H*-triple theory [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) h-vector of a simplicial complex [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) Haag-Ruelle scatteringtheory [81Txx] (see: Massive field) Haarcoefficients s e e :

Fourier--

series [42C10, 42C15] (see: Fourier-Haar series)

Hear-Fourier

non- --

Hales-Jewett theorem (05D10) (refers to: Alphabet; Games, theory of; Topological dynamics; van der Waerden theorem) Hales-Jewett theorem [05D10] (see: Hales-Jewett theorem) Hales-dewett theorem see: density - - ; Furstenberg-Katznetson density - - ; infinitary - - ; polynomial - Hales theorem s e e : J e w e t t - - half-axis s e e : direct scattering problem on the half-axiscase see: Inverse scattering, - half-integralweight s e e : automorphicform of-half-open interval topology s e e : right -half-open square topology s e e : Sorgenfrey - half-plane s e e : u p p e r - half-space s e e : Siegel u p p e r - Hallidentity s e e : E - H a l l - L i t t l e w o o d functions -

-

[05El0, 05E99, 20C25] (see: Schur Q-function) Hamburger moment problem [15A57, 44A60, 47A57, 47B35, 65F05, 93B15]

(see: Hankel matrix; Moment matrix) Hamilton-Jacobi equations [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Hamilton theorem s e e : Cayley- - -

generalized

harmonic function s e e : separately - harmonic functions s e e : converse of Gauss mean-value theorem for - - ; meanvalue characterization for - - ; mean-value characterization of separately - - ; meanvalue theorem for - - ; Poisson formula for - -

Hamiltonian circuit problem [68Q15] (see: Average-case computational complexity)

harmonic measure [31A05, 31A10] (see: Poisson formula for harmonic functions)

Hamiltonian flows s e e :

harmonic minorant s e e : hypo- - harmonic polynomial s e e : spherical - harmonic polynomials s e e : applications of zonal - - ; zonal - -

commuting - -

Hamiltonian graph [90C08] (see: Travelling salesman problem) Hamiltonian system [14Jxx, 35A25, 35Q53, 57R57] (see: Whltham equations) Hamiltonian systems s e e :

reduction of - -

Hamiltonian vector field [37J15, 53D20, 70H33] (see: Momentum mapping) Hamiltonianvector field s e e : locally - handle slide s e e : Kirby move - -

Hankel determinant [15A57, 44A60, 47B35, 65F05, 93B15] (see: Hankel matrix; Strong Stieltjes moment problem) Hankel matrices s e e :

Krvnecker theorem

on--

Hankel matrix (15A57, 47B35, 65F05, 93B15) (referred to in: Moment matrix; Strong Stieltjes moment problem) (refers to: Algebraic geometry; Berlekamp-Massey algorithm; Borel measure; Carleman operator; Euclidean algorithm; Hilbert space; Matrix; Moment problem; Orthogonal polynomials; Pad~ approximation; Sturm theorem) Hankel matrix s e e :

block - -

Hankel moment matrix [44A60, 47A57] (see: Moment matrix) Hankel operator [15A57, 47B35, 65F05, 93B15] (see: Hankei matrix) Hanke/operator s e e : Hi[bert- - - ; symbol of a - -

Hantzche-Wendt manifold [20F38, 57Mxx] (see: Fibonacci group; Fibonacci manifold) hard s e e : ./k['79 - -hard problem [90C08] (see: Travelling salesman problem) Hardy-Kranse variation of a function [65C101 (see: Linear congruential method) Hardy space [46Cxx, 47B35, 47Dxx] (see: Berezin transform; Taylor joint spectrum) Hardy uncertainty principle [42A63] (see: Uncertainty principle, mathematical) Hardy-Weinberg equation [92D10] (see: Hardy-Weinberg law) Hardy-Weinberg equilibrium equation [92D10] (see: Hardy-Weinberg law) Hardy-Wcinberg law (92D10) Harish-Cbandra realization [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactifieation) harmonic analysis s e e : approximate unit in - - ; Malliavin theorem in - harmonic form s e e : L 2 - -

harmonic function [31B05, 33C55] (see: Zonal harmonics)

harmonic regularization [46F10, 46F30] (see: Generalized function algebras; Multiplication of distributions) harmonic spaces see: Riesz decomposition theorem for - harmonics s e e : solid zonal - - ; surface zonal - - ; Zonal - -

Harry Dym equation (35Q53, 58F07) (referred to in: Inverse scattering, fullline case) (refers to: B~icklund transformation; Completely-integrable differential equation; Darboux transformation; Evolution equation; Fredholm equation; Hamiltonian system; Inverse scattering, full-line case; Korteweg-de Vries equation; KPequation; Moutard transformation; Non-linear partial differential equation; Painlev~ test; Painlev~-type equations; Schrddinger equation; Sofiton; Spectral theory of differential operators) Harry Dym equation s e e : 2+ldimensional--; alized - -

extended--;

gener-

Harte spectrum [47Dxx] (see: Taylor joint spectrum) Hartwigconjecture s e e :

Fisher--

hat function [62D05] (see: Acceptance-rejection method) hat funcdon [62D05] (see: Acceptance-rejection method) hat function see: double exponential - - ; optimal - - ; Student-t - - ; triangular - -

Hauptmodul for a field of modular functions [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) Hauptmodul property [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) Hausdorff distance [35P25] (see: Obstacle scattering) Hausdorff k-space [54C35] (see: Exponential law (in topology)) Hausdorff measure [47H10] (see: Darbo fixed-point theorem) Hausdorffmeasure [28A80] (see: Sierpifiski gasket) Hausdorff measure s e e : decomposition theorem for sets of finite - -

Hausdorff topology [54Bxx] (see: Wijsman convergence) hazardsee:

baseline--

hazard function [62Jxx, 62Mxx] (see: Cox regression model) hazards see:

proportional-

head-driven phrase structuregrammar [68S051 (see: Natural language processing) 499

HEAD KNOT

h e a d k n o t see: T u r k - heat conduction [60Gxx, 60J55, 60J65] (see: Wiener sausage) heat kernel [35J05, 35J25, 46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Diriehlet eigenvalue; Index theory) heat-kernel method [46L80, 46L87, 55N15, 58G10, 5gG11, 58G12] (see: Index theory) Heaviside function [42Cxx, 94A 12] (see: Window function) Hecke algebra it 1F25, 11F60] (see: Hecke operator) Hecke algebra [1 IF67, 57M25] (see: Eisenstein cohomology; JonesConway polynomial) Hecke eigenvalne [llFll, 11F12] (see: Shimura correspondence) Hecke operator (11F25, 11F60) (referred to in: Shimura correspondence) (refers to: Group; Modular form; Modular group) Hecke operator [11F25, 11F60] (see: Hecke operator) Hecke operator [llF03, llFll, llF70, 17B67, 20D08, 81T10] (see: Moonshine conjectures; Selberg conjecture) hedge strategy [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) Heegaard splitting [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) Heegner point [llFll, 11F12] (see: Shimura correspondence) height

see:

Fitting - - ; Scott - -

height bounds for the Hilbert Nullstellensatz [t4A10, 14Q201 (see: Effective Nullstellensatz) Hein triple system

see:

Allison- --

Heisenberg algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Heisenberg algebra see:

homogeneous--

Heisenberg group [11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) Heisenberg inequality [42A63] (see: Uncertainty principle, mathematical) Heisenberg Lie algebra [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) Heisenberg uncertainty inequality [42A63] (see: Uncertainty principle, mathematical) Hele-Shaw cell [76Exx, 76S05] (see: Viscousfingering) Helfrich free energy [53C42] (see: Willmore functional) helicity [81Txx, 81T05] (see: Massless field) Helly theorem [52A35] (see: Geometric transversal theory) 500

Helly-type transversaltheorem [52A35] (see: Geometric transversal theory) Helmholtz instability see: Kelvin- -Henriksen P - s p a c e

see:

Gillman---

Hensel tiffing [12D05] (see: Factorization of polynomials) Henstock integrabitity [28A25] (see: Denjey-Perron integral) HenstoekintegrabiLity see: Kurzweit--hereditarily just infinite group [20E08, 20El 8, 20Fxx] (see: Branch group) hereditarily submetacompactspace [26A15, 54C05] (see: Namieka space) hereditary algebra [I6G10, 16G20, 16G60, 16G70] (see: Tilted algebra) hereditary algebra see: quasi- -hereditary category [16Gxx] (see: Tilting theory) Herglotz function [34B24, 34L40] (see: Sturm-Liouville theory) Herglotz kernel see: R i e s z - - Herglotztransform

see:

R i e s z - --

Hermann algorithms (13Pxx, 14Q20) (refers to: Commutative algebra; Cramer rule; Effective Nullstellensatz; Grdbner basis) Hermann algorithms [13Pxx, 14Q20] (see: Hermann algorithms) hermit point [14Hxx] (see: Acnode) Hermitian adjoint matrix [03Exx, 03E05] (see: Coherent algebra) Hermitian commutative two-operator vessel see: quasi- - Hermitian matrices see: generalization of the H a d a m a r d - F i s c h e r inequality for positive semi-definite - - ; Hadamard inequality for - Hermitian Satake compactification see: non- -

Hermitian symmetric space [11Fxx] (see: Satake compactification) Herz algebra

see:

Fig&-Talamanca---

Herz theorem [42A16, 42A24, 42A28] (see: Beurling algebra) heteroctiuic cycle [35Q35, 58F13, 76Exx] (see: Kuramoto--Sivashinsky equation) heteroctiuic orbit [58Fxx] (see: Conley index) heterotic string [IIFll, 17B10, 17B65, 17B67, 20D08, 81R10, 81T30] (see: Vertex operator) heterozygous [92D10] (see: Hardy-Weinberg law) heuristic

see:

incomplete --

Heyting algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Heyling algebra [18D151 (see: Cartesian-closed category) Heyting algebras

see:

varietyof --

Hida theory of modular forms [11R23] (see: Iwasawa theory) hidden Markov process [68S051

(see: Natural language processing) hierarchy see: Ablowitz-Kaup-NewellSegur - - ; A K N S - - - ; algebraic - - ; Hitchin - - ; L a x equations of the AKNS- - ; polynomial-time - - ; stability - - ; twocomponent KP- - - ; Whitham - -

hierarchy for a three-dimensional manifold [57N10] (see: Haken manifold) hierarchy for a three-dimensional manifold see: length of a - - ; partial - hierarchy of isomonodromic deformations see: Whitham -hierarchy of the universe of sets see: cumulative --

hierarchy theorem [03D15, 68Q15] (see: Computational classes) hierarchy theorem [03D15, 68Q15] (see: Computational classes) Higgs action

see:

complexity

complexity

Yang-Mills---

Higgs field [35Qxx, 78A25] (see: Magnetic monopole) higher index theorem for coverings [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) higher index theorem for coverings see: Connes-Moscovici -higher-orderlogic see: typed - highest weight see: dominant integral - -

highest-weightcategory [16Gxx] (see: Tilting theory) highest-weight D-module [llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) highest weight module

see:

irreducible --

highest weightrepresentation [llFll, 17B10, 17B65, 17B67, 20D08, 81R10, 81T30] (see: Vertex operator) highly abundant number [llAxxl (see: Abundant number) Higman theorem [20C05, 20Dxx] (see: Zassenfiaus conjecture) Hilbert 15th problem [14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) Hilbert 90 theorem [13-XX, 16-XX, 17-XX] (see: Skolem-Noether theorem) Hilbert-Bramble lemma [46E35, 65N30] (see: Bramble-Hilbert lemma) Hilbert-Hankel operator [15A57, 47B35, 65F05, 93B 15] (see: Hankel matrix) Hilbert hotel [03E99, 04A99] (see: Hilbert infinite hotel) Hilbert infinite hotel (03E99, 04A99) (refers to: Infinity) Hilbert irreducibility theorem [12D05] (see: Factorization of polynomials) Hitbert lemma

see:

Bramble---

Hilbert matrix [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Hilbert Nullstellensatz [14A10, 14Q20] (see: Effective Nulistellensatz) Hilbert Nullstellensatz [13Pxx, 14Axx, 14Q20] (see: Hermann algorithms; MasserPhilippon/Lazard-Mora example; Rabinowitsch trick)

Hilbert Nullstellensatz s e e : complexity bounds for the - - ; degree bounds for the - - ; generalized effective - - ; height bounds for the - -

Hilbert paradox [03E99, 04A99] (see: Hilbert infinite hotel) Hilbert quantale [03G25, 06D99] (see: Quantale) Hilbert series [05Exx, 13C14, 55U10] (see: Stauley-Reisner ring) Hilbert seventh problem ill J81, 11J85] (see: Gel'fond-Schneider method; Schneider method) Hilbert-Siegel modular group [llFxx, 20Gxx, 22E46] (see: Baily-Berel eompaetifieation) HiIbert space see: Reproducing-kernel - - ; rigged - - ; universal - Hilbert subspaces s e e : an~ I~ between --

Hilbert triple [17A40, 17C65,46I-V ¢6L70] (see: Banach-Jorda. ~,,air) Hilbertian field [11R32] (see: Shafarevich conjecture) Hildebrand identity [11Axx] (see: Dickman function) hill-climbing search [68T05] (see: Machine learning) Milliard equation Cahn- --

see:

Cahn---;

stationary

Hirota bilinear relations [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-bierarchy) Hirzebruch prize question [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) Hirzebruch signature theorem [46L80, 46L87, 55N15, 58G10, 58Gli, 58G12] (see: Index theory) Hitchin hierarchy [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Hitchin manifold see: A t i y a h - - HNN-extension (20F05, 20F06, 20F32) (referred to in: Accessibility for groups; Baumslag-Solitar group) (refers to: Amalgam of groups; Baumslag-Solitar group; Conjugate elements; Free group; Group; Homomorphism; Isomorphism; Monomorphism; Non-Hopf group; Normal subgroup; Presentation; Quotient group) HNN-extension [20F05, 20F06, 20F32] (see: HNN-extension) I-INN-extension [20F05, 20F06, 20F32] (see: HNN-extenslon) HNN-extension see: ascending - - ; associated subgroups of an - - ; base group of an - - ; stable letter of an - HNN-extensions see: Collins conjugacy theorem for - - ; normal form theorem for ; reduced sequence in the theory of - - ; torsion theorem for -He polynomial see: Brandt-LickorishMillett- - - -

H61der continuity [26B99, 30C62, 30C65, 30C99] (see: Quasi-regular mapping; Quasisymmetric function of a complex variable) Holevo bound [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx]

IDENTITY (see: Quantum information processing, science of) holomorphic function s e e :

anti- - -

holomorphic function on a Banach space [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) holomorphic functions s e e : mean-value characterization for - holomorphic mapping s e e : matrix-valued - - ; operator-valued - - ; semi-group generated by a -holomorphic mappings s e e : Semi-group of-holomorphic v e c t o r field s e e : c o m p l e t e - - ; semi-complete - -

homeomorphism problem for threedimensional manifolds [57N10] (see: Haken manifold) Homily polynomial (57M25) (refers to: Jones-Conway polynomial) Homily polynomial [57M25] (see: Jones-Conway polynomial) Homflypt [57Mxx, 57M25] (see: Skein module) Homflypt polynomial [57M25] (see: Jones-Conway polynomial) homoclinic orbit [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) homogeneous chaos [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) homogeneouschaos [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) homogeneouschaos decompositiontheorem [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) homogeneous coherent configuration [03Exx, 03E05] (see: Coherent algebra) homogeneous Heisenberg algebra [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) homogeneousmanifold [53C15, 55N35] (see: Spencer cohomology) homogeneous tree s e e :

spherically--

homology 3-sphere [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) homology s e e : Chern character in cyclic - - ; cyclic - - ; Fleer - - ; K - - - ; Steenrod K - - - ; symplectic Fleer - homology for a symplectic mapping s e e : symplectic Fleer - homology for Lagrangian intersections s e e : symplectic Fleer - homology for three-dimensional manifolds see: instanton F l e e r - homology theory s e e : extraordinary - - ; generalized

--

homomorphism s e e : * - - - ; meaning - - ; strong - - ; superstable - -

homomorphism Ko-extensible over a class of algebras [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) homomorphism of Gerfand quantales s e e : algebraically strong - - ; discrete - - ; right embedding - homomorphisms s e e : base- - - ; stabiIity of - - ; Ulam question on stability of - homotepe s e e : ! I " - homotopy s e e : Algebraic - -

homotopyclass [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) homotopyinvariance of Brouwerdegree [58Fxx] (see: Conley index) homotopy invariance of the Brouwer degree [55M25] (see: Brouwer degree) homotopy of spheres see: stable - Homotopy polynomial (57M25) (refers to: Conway skein triple; Graph culturing; Knot theory; Link; Skein module) homotopy skein module [57Mxx, 57M25] (see: Skein module) homotopyskein module s e e : q - -homotopy theory [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) homotopy theory see: rational --; simple - -

homotopytype [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) hemotopy types s e e : of--

algebraic models

homozygousdominant [92D10] (see: Hardy-Weinherg law) homozygousrecessive [92D10] (see: Hardy-Weinberg law) hook immanant [15A15, 20C30] (see: Immanant) Hopf algebra and quasi-symmetric functions see: Leibniz- - Hopf bifurcation s e e : double - - ; generalized -- ; resonant double -- ; zero- - -

Hopf-Burgers equation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Hopf factorization s e e :

W i e n e r - --

Hopf group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Hopf link [57M25] (see: Positive link) Hopf operator s e e : finite W i e n e r - - - ; truncated W i e n e r - -- ; W i e n e r - - -

Hopf point [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Hopf point bifurcation see: generalized -Hopfian group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Hopfian groupsee: - ; non- - -

finitely-presentednon-

strict --

Hotelling T 2 [62Jxx] (see: ANOVA)

Huntington equation [06Exx, 68TI5] (see: Robbins equation) Hurwitz formula see: R i e m a n n - - Hurwitz-space Korteweg-de VriesLandan-Ginsburgmodel [14Jxx, 35A25, 35Q53, 57R57] (see: Whltham equations) Hurwitz zeta-function [11M06, 11M35, 33B15] (see: Catalan constant) Huxley theory of resonance curves [llLxx, 11L03, 11L05, 11L15] (see: Bombieri-Iwaniec method) Hyers theorem [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) Hyers-Ulam-Rassias stability (39B72, 46B99, 46Hxx) (refers to: Banach space; Group; Metric) Hyers-Ulam-Rassias stability [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) Hyers-Ulam stability [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) hyper-K~hlermetric [35Qxx, 78A25] (see: Magnetic monopole) Hyperbolic cross (42B05, 42B08) (referred to in: Lebesgue constants of multi-dimensional partial Fourier sums; Step hyperbolic cross) (refers to: Approximation theory; Differential operator; Eigen value; Fourier series; Partial Fourier sum) hyperbolic cross [42B05, 42B08] (see: Hyperbolic cross) hyperbolic cross see: Step -hyperbolic fixed point [58Fxx] (see: Conley index) hyperbolic group s e e :

Gromov --

hyperbolic metric [141115, 30F60] (see: Weil-Petersson metric) hyperbolic metric see: complete

--;

hyperbolic partial Fourier sums [42B05, 42B08] (see: Hyperbolic cross) hyperbolic partial sum s e e :

infinitary

Hem theory [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) horocycle [30D05, 32H15, 46G20, 47H17]

step - -

hyperconvex domain [31C10, 32F05] (see: Pluripotential theory) hypergraphsee: a--

hypo-elliptic symbol of a pseudodifferential operator [44A12, 65R10, 92C55] (see: Local tomography) hypo-harmonicminorant [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) hypotheses s e e :

Hotelling test s e e : L a w l e y - -hull s e e : analytic structure on a polynomial -- ; polynomially convex - -

independencenumberof

hyperptane problem s e e :

Horn logic with equality [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Horn logic without equality s e e : universal - -

horosphere s e e : big - - ; small -hotel s e ¢ : Hilbert - - ; Hilbert infinite - hotel paradox s e e : infinite - -

Poincar~ - -

HOrmander wave front set criterion [46F10] (see: Multiplication of distributions) HOrmander-Wermer theorem [32E20] (see: Polynomial convexity) Horn clause [68T05] (see: Machine learning) Horn logic s e e :

(see: Denjoy-Wolff theorem) horodisc [30C45, 47H10, 47H20] (see: Julia-Wolff-Carathtodory theorem)

median - -

hyperplane transversal [52A351 (see: Geometric transversal theory) hypersurface ring [13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) hypertree [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) hypo-elliptic pseudo-differential operator [44A12, 65R10, 92C55] (see: Local tomography)

testing of - -

hypotheses testing [62H20, 62Jxx, 62Mxx] (see: ANOVA; Cox regression model; Kendall tau metric) hypothesis s e e : exact test of a -- ; linear - - ; m a x i m u m a posteriori - - ; Riemann - - ; Well Riemann -hypothesis for curves over finite fields s e e : Riemann - -

hypothesis testing [62Lxx] (see: Average sample number)

I I see: exponentialsum of type - I C * - a l g e b r a s e e : non-type- - I error s e e : type- - -

[-.function [34A55, 34L25, 35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case; Ordinary differential equations, property C' for) ideal s e e : c o n t i n u i t y - - ; different - - ; essential - - ; fixed prime - - ; integral - - ; M- - - ; nowhere dense - - ; parameter - - ; principal inner - - ; triple - - ; unramified prime - ideal at x s e e : prime - ideal class group

[11R23] (see: Iwasawa theory) ideal in norm ideal of fields

an algebraic number field s e e : of a prime -an extension of algebraic number see: splitting prime --

ideal of definition [13Hxx] (see: System of parameters of a module over a local ring) ideal of degree one in an extension of algebraic number fields s e e : prime - ideal space s e e : maximal - i d e a l t h e o r e m s e e : Landau prime - -

idealizer [46310, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) ideals s e e : regular set of prime - i d e m p o t e n c y theorem f o r B ( G )

[22D10, 221)25, 43A07, 43A15, 43A25, 43A30, 43A35, 46J10] (see: Fourier-Stieltjes algebra) idempotent s e e :

Steinberg --

idempotent element [03G25, 06D99] (see: Quantale) idempotent element [46J10, 46L05, 46Lg0, 46L85] (see: Multipliers of C* -algebras) idempotent theorem s e e :

Cohen - -

identical in law [60Hxx, 60J55, 60J65] (see: Skorokhod equation) identification of systems [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) identities see: Macdonald -identities of a variety s e e : finite basis for the - identity s e e : Bukhstab - - ; centre-bymetabelian - - ; Eulerian - - ; Hildebrand

501

IDENTITY

--; Jacobi-Cauchy--; Jacobi thetafunction - - ; Jordan - - ; Jordan triple -- ; E Hall - - ; polarization - - ; R a m a n u j a n Petersson - - ; Spitzer - - ; Vaughan - - ; Ward - - ; Waring -identity for Borcherde algebras s e e : denominator - identity for vertex algebras s e e : Jacobi - identityof sequentialanalysis s e e : fundamental - II s e e : e x p o n e n t i a l s u m of type - II error s e e : type- --

I1 ~ factor [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) ILP [68T05] (see: Machine learning) image see: convex - - ; Gray -image operator [03G10, 06Bxx, 54A40] (see: Fuzzy topology) image operator s e e :

pro- - -

imaginary root of a Borcherds algebra [llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) imaginary simple root of a Boreherds algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) imbedding theorem s e e :

Sobolev--

Immanent

(I5A15, 20C30) (refers to: Character of a group; Determinant; Graph; Hermitian matrix; Partial order; Permanent; Permutation; Schur functions in algebraic combinatorics; Symmetric group) immanent [15A15, 20C30] (see: Immanent) immanent s e e : hook --; normalized -immanantal polynomial [15A15, 20C30] (see: Immanent) immanants s e e :

Sehur inequalityfor --

Immerman-Szelepcs~nyi theorem [03D15, 68Q15] (see: Computational complexity classes) immersed surface [53C42] (see: Willmore functional) immersion s e e :

Willmore --

immersion conjecture for manifolds [55P42] (see: Brown-Gitler spectra) immersion of a surface into a Riemannian manifold [53C421 (see: Willmore functional) impartial game [90D05] (see: Sprague-Grundy function) impedance function [34A55, 34L25] (see: Ordinary differential equations, property C for)

in-tree [o5c5o] (see: Matrix tree theorem) incentre of a triangle [51M04] (see: Triangle centre) incidence algebra [16Gxx} (see: Tits quadratic form) incidence matrix [05B35, 05Exx, 05E25, 06A07, 11A25] (see: MObius inversion) incidence matrix of a graph [o5c5o] (see: Matrix tree theorem) incidence structure [05Bxx] (see: Net (in finite geometry)) incident plane wave in scattering [35P25] (see: Obstacle scattering) incircle [51M04] (see: Triangle centre) inclines see: fluid films on - inclusion-exclusion s e e :

inclusion for Taylor spectrum s e e : spectral - incomplete block design s e e : symmetric balanced - -

incomplete heuristic [06Exx, 68T15] (see: Robbins equation) incomplete sample [62Jxx] (see: ANOVA) incompletely factors in covariance analysis [62Jxx] (see: ANOVA) incompressible flow see: inviseid - incompressible fluid [76Exx, 76S05] (see: Viscous fingering) incompressible limit [76Axx] (see: Knudsen number) incompressible surface [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) incompressible surface in a threedimensional manifold [57N10] (see: Haken manifold) incompressible surface in a th ree-dimensional manifold s e e : b o u n d a r y - - ; O- - -

implicit definition of a set of atomic formulas over another set of atomic formulas s e e : strong --

incompressible torus [57Mxx, 57M25] (see: Skein module)

implicit Euler method [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) implicit function theorem [90Cxx] (see: Fritz John condition)

independence s e e : braic --

implicitly over another set of atomic formulas see: set of formulas defining a set of atomic formulas - -

irnpulse-response sequence of a linear feedback shift register [11B37, llT71, 93C05] (see: Shift register sequence)

502

principle of --

Inclusion-exclusion formula (05A99, 11N35, 60A99, 60E15) (referred to in: Dickman function) (refers to: Bonferroni inequalities; Brnn sieve; Montmort matching problem; Probability space; Probability theory; Sieve method) inclusion-exclusion method [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) inclusion-exclusion principle [05A99, 11N35, 60A99, 60E15] (see: Inclusion-excluslon formula)

measure of alge-

independence number of a hypergraph [05C65, 05D05, 68Q15, 68T05] (see: Vapnik-Chervonenkis dimension) independence of logarithms of algebraic numbers s e e : linear - independence of values of Abelian functions see: atgebraic - independence of values of elliptic functions see: algebraic - independence of values of exponential functions s e e : algebraic--

independent belief functions combination of two - -

see:

rule of

independent variable in regression analysis [62Jxx] (see: ANOVA) independent variable sets for a belief function [68T30, 68T99, 92Jxx, 92KI0] (see: Dempster-Shafer theory) indeterminate strong Stieltjes moment problem [44A60] (see: Strong Stieltjes moment problem) index see: braid -- ; branch -- ; Conley - - ; equivadant - - ; example of the Conley - - ; Fredholm - - ; generalized Morse - - ; ramification - - ; yon N e u m a n n -- ; Wiener - index for a tree level s e e : branch - indexformula see: McKean-Singer-index formulas s e e : Atiyah-Segal --; Atiyah-Singer - index of a link s e e : algebraic - -

index of a net [05Bxx] (see: Net (in finite geometry)) index of a tree s e e :

chemical --

index of a n eigenvalue [47A 10, 47B06] (see: Spectral theory of compact operators) index of an elliptic partial differential operator [53C15, 55N35] (see: Spencer cohomology) index of an operator [47A53] (see: Semi-Fredholm operator) index pair for an isolated invariant set [58Fxx] (see: Conley index) index theorem see: Atiyah L 2- --; Atiyah-Patodi-Singer--; Atiyah-Singer ; equivariant -- ; local - indextheorem for coverings s e e : C o n n e s Moscovici higher - - ; higher - - -

index theorem for fofiations [46L80, 46L87, 55N15, 58Gll, 58G12] (see: Index theory) index theorem for Connes -index theorems s e e : generalized - -

foliations

58G10,

see:

applications of - - ;

Index theory (46L80, 46L87, 55N15, 58G10, 58Gll, 58G12) (referred to in: Spencer cohomology; Toeplitz G'* -algebra) (refers to: Bott periodicity theorem; C*-algebra; Chern character; Clifford algebra; Cohomology; Connection; Covering; Cyclic cohomology; de Rham eohomology; Elliptic operator; Eta-invariant; Fibration; Fredholm operator; Fundamental class; Heat content asymptotics; Index formulas; Index of an operator; Irreducible representation; K-theory; Lie group; Malliavin calculus; Manifold; Metric; Monodromy transformation; Orientation; Orthogonal basis; Principal part of a differential operator; Probability theory; Pseudo-differential operator; Selfadjoint operator; Signature; Symbol of an operator; Them isomorphism; Todd class) index theory see: K-theoryin -index theory for a families of operators [46L80, 46L87, 55N15, 58Gl0, 58Gll, 58G12] (see: Index theory) index theory for a single operator [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12]

(see: Index theory) indexing see: automatic -indicator algebra [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) indices see: Wedderburn theorem

on

Schur - -

individual distributional product [46F10] (see: Multiplication of distributions) individual distributional product [46F10] (see: Multiplication of distributions) induced by a metric s e e :

Wijsman topol-

ogy - -

induced velocities in vortex sheets [76Cxx] (see: Von Kdrmdn vortex shedding) induction system s e e : constructive -inductive inference [68T05] (see: Machine learning) inductive inference approach to machine learning [68T05] (see: Machine learning) inductive learning [68T051 (see: Machine learning) inductive logic programming [68T05] (see: Machine learning) inductive reasoning [68T05] (see: Machine learning) inequalities see: alternative in linear --; theorem of alternatives for vector - inequality s e e : Bernshtefn - - ; example of the use of the Jansen - - ; Faber-Krahn isoperimetric--; Hadamard-Fischer--; Heieenberg - - ; Heieenberg uncertainty - - ; ieoperimetric - - ; Jansen - - ; linear ; Liouville - - ; Liouville-Lojasiewicz - - ; Lojasiewiez--; LYM--; permanental analogue of the Schur - - ; pointwise Markey - - ; R a m a n u j a n - - ; R a y l e i g h - F a b e r Krahn - - ; Sidon-type - - ; Suen - - ; vector - - ; Yff - inequality for a matrix norm s e e : submultiplicative - inequalityforavectornormsee: triangle-inequalityfor Hermitian matrices s e e : Hadamard - inequality for immanants s e e : Schur - inequalityfor permanents s e e : Marcus - inequality for positive semi-definite Hermitian matrices s e e : generalization of the Hadamard-Fischer -inequality for the lowest eigenvalue s e e : isoperimetric - inequality for yon Neumann entropy s e e : strong subadditivity - inequality in groups s e e : isoperimetric -- -

inertial form [35Q35, 58F13, 76Exx] (see: Kuramoto--Sivashinsky equation) inertial manifold [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) inference s e e : Bayesian statistical - - ; inductive - inference approach to machine learning s e e : inductive --

inference rule [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) infinitary Hales-Jewett theorem [05D101 (see: Hales-Jewett theorem) infmitary universal Horn logic without equality [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

INVERSE GEOPHYSICALSCATTERINGPROBLEM

infinite branch group s e e : i n f m i t e convergence rate

just - -

[65Lxx, 65M701 (see: Trigonometric pseudo-spectral methods) infinite determinant [81T50] (see: Non-commutative anomaly) infinite determinants s e e : of--

regu/arization

inlinite-dimensional Grassmannalgebra [81Qxx, 8lSxx, 81T13] (see: Faddeev-Popov ghost) infinite-dimensional t o m s see: ergodic automorphism of the - infinite group s e e : hereditaNy just - - ; just - infinite hotel s e e : Hilbert - -

infinite hotel paradox

[03E99, 04A99] (see: Hilbert infinite hotel) infinitelength s e e :

curve of --

infinite set [03E99, 04A99] (see: Hilbert infinite hotel)

[26B99, 30C62, 30C65] (see: Quasi-regular mapping) inner function [30D55, 46J15, 47A15] (see: Beurling theorem) innerideal s e e : principal -inner product s e e : Legendre-Sobolev--; Sobolev - -

input determinantal representation of the discriminant curve of an operator vessel

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) input in a decision problem s e e : - - ; rejected - input multiple-outputsystem s e e :

accepted multiple-

input single-output system s e e : single- - input tape s e e : r e a d - o n l y - insertion propertysee: G ¢,- - instability s e e : intrinsic - - ; KelvinH e f m h o t t z - - ; Saffman-Taylor - -

instantaneous velocity at a point

infinitely many conservationlaws [35Q53, 581=07] (see: Harry Dym equation)

[73Bxx, 76Axx] (see: Material derivative method) instanton [35Qxx, 78A25] (see: Magnetic monopole)

infinitely Silva-differentiable mappings s e e : space of - -

instanton Fleer homology for threedimensional manifolds

infinite set

see:

countably

-

-

infinitesimal analysis [46F301 (see: Generalized function algebras) infinity

[03E99, 04A99] (see: Hilbert infinite hotel) infinity s e e : axiom of - - ; Dedekind definition of - - ; Function vanishing at - - ; Green function with pole at - - ; Ramanujan-Petersson conjecture at - - ; rate of decay of a function at - - ; vanish at-inflection point

[14Hxx] (see: Flecnode) information see: classical - information processing s e e : cost in quantum - - ; resource in quantum - information processing, science of s e e : Quantum --

[53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) instruction multiple-data s e e : single- -insurance s e e : life --

multiple- - - ;

insurance theory [90A11] (see: Cobb-Douglas function) integer s e e : k-representation of an - - ; non-reciprocal algebraic - - ; norm of an --;S-integer function s e e : greatest - - ; nearest - integers s e e : ring of algebraic - integrability s e e : Denjoy-Perron - - ; Henstock--; KurzweiI-Henstock--; Virasore - -

informationretrieval [68S05] (see: Natural language processing)

Integrability of trigonometric series (42A20, 42A32, 42A38) (refers to: Absolute continuity; Dirichlet kernel; Fourier series; Lebesgue constants; Trigonometric series)

information theory s e e :

integrable module

quantum - -

inffaparticle problem [81Txx, 81T05] (see: Massless field) infrared divergence [81Txx, 81T05] (see: Massless field) initial conditions eta linear feedback shift register

[11B37, 11TYl, 93C05] (see: Shift register sequence) initial-value problem for the K o r t e w e g - d e Vfies equation s e e : c h a r a c t e r i s t i c - -

integrablesystem s e e : completely- - integral see: Abelian - - ; Berezin - - ; Choquet - - ; Denjoy-Perron - - ; elliptic - - ; generalized Choquet - - ; Kronecker - - ; multiple Wiener - - ; narrow Denjoy - - ; q- - - ; Riesz-Dunford - - ; Sugeno - - ; ~conorm --

integral current

[28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) integral equation see: Stieltjes-VoIterra - - ; Theodorsen - - ; Volterra-Stieltjes - -

injection theorem for Ditkin sets"

integral flat distance

injective algebra see:

self- - -

injective envelope e t a module

[16D40] (see: Flat cover) injective linear operator

[46E22] (see: Reproducing-kernel Hilbert space) iniective module s e e : pre- - injectivity s e e : radius of - inner c o h o m o l o g y

[11F67] (see: Eisnnstein cohomology) inner dilatation

integraltransforms s e e : reciprocal - integral weight s e e : automorphic form of half- - integrals s e e : commuting -integration s e e : current of - - ; functional - - ; G a u s s - J a c o b i - - ; non-commutative - - ; Szeg6 fractional --

intension of a term

[03E30] (see: ZFC) intensional deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) intensitymodel s e e :

Aalen m u l t i N i c a t i v e - -

intensity e t a vortex

[76Cxx] (see: Von Kfirmfin vortex shedding) intensity process

[62Jxx, 62Mxx] (see: Cox regression model) interaction in design of statistical experiments"

[62Jxx] (see: ANOVA) interactive proof

[03D15, 68Q15] (see: Computational classes)

complexity

interface s e e : naturallanguage - intermediate growth s e e : group of --

internal space of an operator vessel

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) interpolation - - ; quasiinterpolation Lagrange

centre f o r - - ; polynomial -formula s e e : error of the --

see:

interpolation point [65Lxx, 65M70[ (see: Trigonometric pseudo-spectral methods) interpolation polynomial see: L a g r a n g e - interpolation scheme s e e : rational -interpolation theorem s e e : Craig -interpretation s e e : Ne'eman-ThierryMieg - -

interpretation e t a deductive system

[11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra)

initial value theoremfor the Z-trnnsform [39A12, 93Cxx, 94A12] (see: Z-transform) [43A45, 43A46[ (see: Ditkin set)

integral representation of a functional [44A60, 47A57] (see: Moment matrix) integral transformmethod [31B05, 33C55] (see: Zonal harmonies)

[28A78, 49Qxx, 49Q15, 53C65, 58A25[ (see: Geometric measure theory) integral highest weight s e e :

dominant --

[03Gxx, 03G05, 03G10, 03GI5, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) interpretation of a deductive system s e e : faithful - interpretation of an equational logic s e e : faithful - interpretations s e e : defining set of formulas for a class of - - ; equivalence of formulas over a class of - - ; Fregean equivalence of formulas over a class of - - ; mutually inverse - intersection s e e : Iocalcomplete - -

intersection cohomology [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) intersection numbers of a coherent configuration

[03Exx, 03E05] (see: Coherent algebra)

integral ideal [llNxx, 11N32, 11N45, IlNS0] (see: Abstract analytic number theory)

intersection pairing

integral operator s e e :

intersections see: sympIectic Floerhomology for Lagrangian - interval s e e : confidence - - ; Scheffd-type simultaneous confidence -- ; Tukey-type simultaneous confidence - -

Carleman-type --

integral operators of Carleman type [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) integral part function

[26Axx] (see: Floor function) integral representation s e e :

[14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus)

interval graph

[llNxx, 11N32, 11N45] Riesz - -

(see: Abstract prime number theory)

interval graph s e e :

reduced - - ; unit- - -

interval in a partially ordered vector space

[06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) intervaltopology s e e : right half-open - intervals s e e : simultaneous confidence - intractability s e e : computational - -

intractable problem [68Q15] (see: Average-case computational complexity) intrinsic coordinates Nielsen - -

Fenchel-

see:

intrinsic instability [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) intuifionistic propositionalcalculus [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) invariance s e e : continuation - - ; positive - - ; tangency condition of flow - - ; Tarski condition of substitution -invariance for continuous semi-groups s e e : flow- - invariance of B r o u w e r d e g r e e s e e : homotopy - -

invariance of dimension [55M25] (see: Brouwer degree) invariance of domain [55M25] (see: Brouwer degree) invariance of the Brouwer degree s e e : homotopy - invariance under geodesic flow s e e : Liouville theorem on - invariant s e e : Buchsbaum - - ; Busby - - ; conformal - - ; Donaldson - - ; finite-type knot - - ; Iwasawa - - ; Vassiliev - invariant algebra s e e : co- --

invariant cohomology [53C15, 55N35] (see: Spencer cohomology) invariantform s e e :

PoincarC::,-Cartan - -

invariant Laplacian [3IA05, 31B05, 3IC10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) invariant measure s e e : Absolutelycontinuous - - ; ergodic - invariant of a matroid s e e : MObius - -

invariant of links [57M25] (see: Fox n-colouring) invariant of links see: Abelian group -invariantset s e e : i n d e x p a i r f o r a n isolated - - ; isolated - invariant subspace s e e : shift- - - ; Taylor spectrum for an - -

invarinnt toms [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) invariants s e e : Gromov-Witten - - ; o p e r a tor--; Reshetikhin-Turaev--; SeibergWitten - - ; Vassiliev-Gusarov - -

inverse additive abstract prime number theorem [11Nxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) inverse Galois problem over Qab i11R32]

(see: Shafarevich conjecture) inverse Galois theory s e e : conjecture in --

Shafarevich

inverse geophysicalscattering [35P25, 47A40, 81U20] (see: Inverse scattering, dimensional case)

multi-

inverse geophysical scattering problem

[35P25, 47A40, 81U20] 503

INVERSE GEOPHYSICALSCATTERINGPROBLEM (see: Inverse scattering, dimensional case)

inverse interpretations s e e :

multi-

mutually - -

inverse obstacle scattering [35P25] (see: Obstacle scattering) inverse of an arithmetical function s e e : Dirichlet - -

inverse potential scattering [35P25, 47A40, 81U20] (see: Inverse scattering, nmltidimensional case; Obstacle scattering) inverse potential scattering problem [35P25, 47A40, 81U20] (see: Inverse scattering, multidimensional case) inverse prime n u m b e r theorem s e e : stract --

ab-

inverse problem [34A55, 34L25, 35P25] (see: Ordinary differential equations, property C for; Partial differential equations, property C for) inverse relationssee: of--

Chebyshev-typepair

Inverse scattering, full-line case (35P25, 47A40, 58F07, 81U20) (referred to in: Harry Dym equation) (refers t o : Evolution equation; Harry Dym equation; Korteweg-de Vries equation; Ordinary differential equations, property C for; Scattering matrix; SchrSdinger equation) Inverse scattering, half-axis case (35P25, 47A40, 81U20) (refers to: Dirichlet boundary conditions; Entire function; Homeomorphism; Meromorphic function; Scattering matrix; Self-adjoint operator; Spectral function) Inverse scattering, multi-dimensional case (35P25, 47A40, 81U20) (referred to in: Partial differential equations, property C for) (refers to: Analytic function; Obstacle scattering) inverse scattering problem [35P25, 47A40, 58F07, 81U201 (see: Inverse scattering, full-line case; Inverse scattering, half-axis case) inverse scattering theory s e e : characterization theorem of - - ; reconstruction theorem of - - ; uniqueness theorem of --

inverse scattering transform [35Q53, 58F07] (see: Harry Dym equation) inverse spectral methodsfor Jacobi matrices [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) inverse spectral transform [35Q53, 58F07] (see: Harry Dym equation) inverse Sturm-Liouville problem [34B24, 34L40] (see: Sturm-Liouville theory) inverse Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) inverse Z-transform s e e : partial-fractions technique for finding the --

inverse Z-transformation [39A12, 93Cxx, 94A12] (see: Z-transform) inversion s e e :

M~Sbius - classical M6bius -- ; MObius - - ; number-theoretic Mdbius - -

inversionformulasee:

inversion formula for the pseudo-local tomography function [44A12, 65R10, 92C55] (see: Pseudo-local tomography) inversionmethodsee:

consistencyofan --

inversionof the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) 504

invertible element in a Jordan algebra with a unit [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) inviscid incompressibleflow [76C05] (see: Birkhoff-Rott equation) involutions s e e :

Frobenius-Schur count

of--

involutive quantale [03G25, 06D99] (see: Quantale) ion mode in a p l a s m a s e e : trapped - -

[14Hxx] (see: Acnode) isolated point in the spectrum of an operator [47B06] (see: Riesz operator) isolating neighbourhood [58Fxx] (see: Conley index) isometric mappings s e e : problem for - isometric metric spaces s e e :

dissipative

irrationality of ~ (3) [llAxx, 11J72, llM06] (see: Ap~ry numbers) irreducibilitytheorem see: Hilbert -irreducible discriminant polynomial [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) irreducible highest weight module [17B10, 17B65] (see: Weyl-Kac character formula) irreducible matrix [15A18] (see: Gershgorin theorem) irreducible polynomials s e e : Gauss theorem on products of monic - -

irreducible representation of a Gel'fund quantale [03G25, 06D99] (see: Quantale) irreducible representation of a Gel'fund quantale s e e : algebraically--

irreducible three-dimensional manifold [57N101 (see: Haken manifold) irreducible three-dimensional manifold see: p2._

irreversibility [76Axx] (see: Knudsen number) Isogonal (51M04) (referred to in: Brocard point; Triangle centre) (refers to: Anti-conformal mapping; Baryeentric coordinates; Bisectrix; Ceva theorem; Conformal mapping; Gergonne point; Isogonal trajectory; Nagel point; Stereographic projection) isogonal circles [51M04] (see: Isogonal) isogonal circles [51M04] (see: Isogonal) isogonal conjugate point [51M04] (see: Isogonal) isogonal conjugation [51M04] (see: Isogonal) isogonal line [51M04] (see: Isogonal) isogonal mapping [51M04] (see: Isogonal) isogonal trajectory [51M04] (see: Isogonal) isogonic centre of a triangle s e e : first -isolated invariant set [58Fxx] (see: Conley index) isolated invariant set see: index pair for an-

isolated part of the spectrum of a linear operator [31A10, 31D05, 47AI0, 47A15, 47A60] (see: Riesz decomposition theorem) isolated point

Aleksandrov quasi- - -

isometric operator colligation [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) isometricoperatorcolligation s e e :

co- - -

isometry [54E35] (see: Aleksandrov problem for isometric mappings) isometry see: co- --; quasi- -isometry of Banach algebras s e e : ~- - isomonodromic deformations see: Whitham hierarchy of - isomonodromy

[14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) isomonodromydeformation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) isomorphism s e e : de Rham - - ; fixed place of a field - - ; valence of a field - -

isomorphism in codimension 1 [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) isomorphism of Banach algebras see: eisomorphism problem s e e :

graph - -

isomorphism problem for group rings [20C05, 20Dxx] (see: Zassenhaus conjecture) isomorphism relation see: analytic -isoperimetric inequality [53C20, 53C22] (see: Santal6 formula) isoperimetrie inequality s e e : Krahn - -

Faber-

isoperimetric inequality for the lowest eigenvalue [35P15] (see: Rayleigh-Faber-Krahn inequality) isoperimetric inequality in groups [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) isospectral linear eigenvalue problem [35Q53, 58F07] (see: Harry Dym equation) isospectral problem [35Q53, 58F07] (see: Harry Dym equation) isospectral problem [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) isospin [15A66, 81R05, 81R25] (see: Paufi algebra) isotomic conjugate point [51M04] (see: Isogonal) isotomic conjugation [51M04] (see: Isogonal) isotomic line [51M04] (see: Isogonal) isotone function [03G10, 06Bxx, 54A40] (see: Fuzzy topology) isotopy lemma [57N80] (see: Thom-Mather stratification) i s o t o p y l e m m a s see: T h e m - isotropic Grassmannian [05E10, 05E99, 20C25] (see: Schur Q-function)

1SP [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-line

case) item at risk [62Jxx, 62Mxx] (see: Cox regression model) iteration s e e : f i x e d - p o i n t - - ; Pieard - It6 decomposition s e e : Wiener-It6 decomposition theorem s e e : Wiener-

ltg~-Wiener decomposition [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Iwahori subgroup [20G05] (see: Steinberg module) Iwaniee method s e e :

Bombieri---

Iwaniec theorem [llLxx, llL03, llL05, 11L15] (see: Bombieri-Iwanlee method) Iwasawa conjecture [11R23] (see: Iwasawa theory) Iwasawainvariant [11R23] (see: lwasawa theory) Iwasawamain conjecture [11R23] (see: Iwasawa theory) lwasawa module [11R23] (see: Iwasawa theory) Iwasawamodule [11R23] (see: Iwasawa theory) lwasawa theorem [11R32] (see: Shafarevich conjecture) Iwasawatheorem [11R32] (see: Shafarevich conjecture) Iwasawa theory (11R23) (refers to: Class field theory; Fermat last theorem; Irregular prime number; Weierstrass theorem)

J J* -algebra [17Cxx, 32H15, 34G20, 46-XX, 46G20, 47D06, 47H20] (see: JB *-triple; Semi-group of holo-

morphlc mappings) J* -algebra [30C45, 30D05, 32H15, 46G20, 47H10, 47H17, 47H20] (see: Denjoy-Wolff theorem; JufiaWolff-Carath~odory theorem) j-dimension [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) j-function [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) j-function see:

modular--

j-prime [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) j-Spec (R) [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) J-ternary algebra [17A40] (see: AIIison-Hein triple system) Jaco-Shalen-Johansson decomposition theorem [20E22, 20Jxx, 57Mxx] (see: Accessibilityfor groups) Jaccr-Shalen-Johansson splitting theorem

JULIA-WOLFF-CARATHI~ODORYTHEOREM

[20E22, 20Jxx, 57Mxx]

3]3 *-algebra

(see: Accessibility for groups)

[17Cxx, 17C65, 46-XX, 46H70, 46L70] (see: Banaeh-Jordan algebra; JB *triple) JB *-triple (17Cxx, 46-XX) (referred to in: Banach-Jordan pair) (refers to: C*-algebra; BanachJordan algebra; Banach space; Bergman spaces; Harmonic analysis; Hermitian operator; Hilbert space; Lie group, Banach; Linear operator; Spectrum of an operator; Symmetric space; Topological space; Unitary representation; yon Neumann algebra)

Jacobi-Cauchy identity [11F11, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) Jaeobi coefficients s e e : Fourier--Jacobi equations s e e : Hamilton- --

Jacobi form [llFll, 11F12] (see: Shimura correspondence) Jacobi group [11Fll, 11F12] (see: Shimera correspondence) Jacobi identity for vertex algebras

[llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) Jacobi identity for vertex algebras [llF11, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) daeobiintegration s e e : Jacobi matrices s e e : methods for - -

G a u s s - -inverse spectral

Jacobi matrix [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Jacobi polynomials [33C45, 33Exx, 46E35] (see: Sobolev inner product) dacobi series s e e :

generalized - -

[13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) Jacobsen representation theory [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Jacobsen-typeradical [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Jaeger composition product (57M25) (referred to in: Jones-Conway polynomial) (refers to: Hopf algebra; JonesConway polynomial; Skein module) Janson inequality (05C80, 60D05) (refers to: Chebyshev inequality; FKG inequality; Graph colouring; Graph, random; Laplace transform; Lov~isz local lemma; Martingale; Probability; Probability space; Random variable) Janson inequality [1 IPxx] (see: Additive basis) example of the use

prime -- ; real - - ; real

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) J B W -factor

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) J B W * -triple

[17Cxx, 46-XX] (see: JB * -triple) JBW *-triple [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) JC -algebra [I7C65, 46H70, 46L70] (see: Banach-Jordan algebra) JC * -triple [17Cxx, 46-XX] (see: JB *-triple) Jensen measure

[32E20] (see: Polynomial convexity) jet space [57Ng0] (see: Thom-Mather stratification) Jewett-HaIes theorem

[05D101 (see: Hales-Jewett theorem) Jewett theorem s e e : density H a l e s - - - ; Furstenberg-Katznelson density H a l e s - ; H a l e s - - - ; infinitary H a l e s - - - ; polynomial H a l e s - - Johansson decomposition theorem s e e : Jaco-Shalen- -Johansson splitting theorem s e e : JacoShalen- -John condition s e e : basic Fritz - - ; Fritz - John equation s e e : optimality in the Fritz - - ; stability in the Fritz --

Johnson-Sinclair theorem

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra)

Johnson-Sinclair theorem [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Johnson uniqueness-of-norm theorem

[46H40] (see: Automatic continuity for Ba-

nach algebras) Johnsonuniqueness-of-normtheorem [17C65, 46H70, 46L70] (see: Banach-Jordan algebra)

JB -algebra

a--

JB *-triple s e e : form of a - -

J B W -algebra

Jacobian criterion

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) JB -algebra [17Cxx, 46-XX] (see: JB * -triple) JB -algebra s e e : factor representation

JB * -triple [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair)

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holoInorphie mappings)

Fourier- - -

Jansen inequalitysee: of the - -

[17Cxx, 46-XX] (see: JB *-triple)

JB * triple system

Jacobi theta-functionidentity [81Qxx] (see: Zeta-function method for regularization) Jacobi theta-functions [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Jacohi-Trudi matrix [15A15, 20C30] (see: Immanent) Jacobi weightfunction [33C45, 33Exx, 46E35] (see: Sobolev inner product) Iacobi zeta-function [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Jaeobian s e e :

JB * -triple

joint characteristic function of an operator vessel of

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel)

joint characteristic function of an operator vessel s e e : normalized - -

joint continuity [54C05, 5 4 C 0 8 ] (see: Strongly countably complete topological space) joint normal distribution see: random variables with - joint spectrum s e e :

uncorrelated

Jordan pair

[17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Jordan pair s e e : B a n a c h - - - ; local algebra of a B a n a c h - - - ; non-degenerate -- ; socle of a - -

Jordan pair of finite capacity

Taylor --

joint spectrum of operators [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) Jones-Conway polynomial (57M25) (referred to in: Conway algebra; Conway skein equivalence; Homily polynomial; Jaeger composition product; Kauffman bracket polynomial; Markov braid theorem; Rotor) (refers to: Alexander-Conway polynomial; Alexander theorem on braids; Arf-invariant; Braid theory; Connected sum; Conway skein triple; Jaeger composition product; Jones unknotting conjecture; Kauffman polynomial; Marker braid theorem; Reidemeister theorem; Skein module; Statistical mechanics, mathematical problems in; Yang-Baxter equation) Iones-Conway polynomial [57M25, 57P25] (see: Conway skein equivalence; Jones-Conway polynomial) Oones-Conwaypolynomialsee:

coloured--

[17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Jordan pairs s e e : finiteness conditions in B a n a c h - -Jordan pairs of finite capacity s e e : Lees classification of simple - -

Jordan product

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Jordan separation theorem

[55M25] (see: Brouwer degree) Jordan triple identity

[17Cxx, 46-XX] (see: J13 * -triple)

Jordan triple system (17A40) (referred to in: Banach-Jordan pair; Freudenthal-Kantor triple system) (refers to: Algebraically closed field; Associative rings and algebras; Jordan algebra; Lie algebra; Lie triple system; Steiner system; Vector space) Jordan triple system

[17A40] (see: Freudenthal-Kantor triple system; Jordan triple system) Jordan triple system s e e : Jordan triple systems s e e :

Jones-Conway relation

[57Mxx, 57M25] (see: Skein module) Oones-Conwayrelation s e e : based on the - -

(see: Segre characteristic of a square matrix)

Banaeh- -examples of - -

Josefson theorem skein m o d u l e

Jones-Ocneanu trace [57M251 (see: Jones-Conway polynomial) Jones polynomial

[57M25] Jones-Conway polynomial; Kauffman bracket polynomial) Jones polynomial [20F36, 57Mxx, 57M25] (see: Fax n-eolouring; JonesConway polynomial; Jones unknotting conjecture; Marker braid theorem; Positive link; Rotor; Skein module) Jones-type skein module [57Mxx, 57M25] (see: Skein module) Jones unknotting conjecture (57M25) ( r e f e r r e d to in: Jones-Conway polynomial) (refers to: Knot theory) (see:

Jordan algebra see: B a n a c h - - - ; primitive B a n a e h - -- ; semi-primitive - - ; semisimple - - ; spectrum of an element in a B a n a c h - -Jordan algebra with a unit s e e : invertible element in a - -

Jordan-Banach algebra

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra)

Jordan-Banach algebra [17Cxx, 46-XX] (see: JB * -triple) Jordan C*-algebra

[17Cxx, 46-XX] (see: JB *-triple) Jordan curve

[28A803 (see: Sierpifiski gasket) Jordan identity

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Jordan normal form [15A18, 15A21]

[31C10, 32F05] (see: Pluripotential theory) Jest function

[34A55, 34L25, 35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case; Ordinary differential equations, property C for) Jest solution

[35P25, 47A40, 81U20] (see: Inverse scattering, half-axis

case) JSJ decomposition

[201322, 20Jxx, 57Mxx] (see: Accessibility for groups)

JSJ decompositionof a group [20F05, 20F06, 20F32] (see: HNN-extension) JSJ theorem

[20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) Julia-Carath~odory theorem

[30C45, 47H10, 47H20] (see: Julia-Woiff-Carath~odory theorem)

Julia-Carath~odory theorem [30C45, 47H10, 47H20] (see: Jnlia-Wolff-Carath~odory theorem) Julia lemma [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath~odory theorem) Julia theorem [30C45, 47H10, 47H20] (see: Julia-Wolff-Carathtodory theorem) Julia-Wolff-Carath~odor y theorem (30C45, 47H10, 47H20) (referred to in: Denjoy-Wolff theorem) (refers to: Analytic function; Hero-

sphere; Hyperbolic metric; Julia theorem; Kobayashi hyperbolicity; Poincar~ model; Schwarz lemma; Semi-group of holomorphic mappings) Julia-Wolff-Carathdodory theorem

505

JULIA-WOLFF-CARATHI~ODORYTHEOREM

[30C45, 47H10, 47H20] (see: Julia-Wolff-Carath~odory the-

orem) dulia-Wolff-Carath~'odory theorem s e e : generalized - -

Julia-Wolff theorem

[30C45, 47H10, 47H20] (see: Julia-Wolff-Carathtodory theorem) jury theorem s e e :

Condorcet --

just infinite branch group [20E08, 20El 8, 20Fxx] (see: Branch group) just infinite group

[20E08, 20El 8, 20Fxx] (see: Branch group) just infinite group s e e :

hereditarily--

(see: Kruskal-Katona theorem) k-set problem [90B85] (see: Fermat-Torricelli problem) k-space

see:

Katonatheorem see: Kruskal--Katznelson density Hales-Jewett t h e o r e m see: Furstenberg- - -

Hausdo#f--

k-tangle [57P25] (see: Conway skein triple) K - t h e o r y see: Kasparov --

excision in algebraic - - ;

K-theory in index theory [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) k-transversal

[52A35] (see: Geometric transversal theory) ( k-- 1)-separated collection of sets

Jutila method

[llLxx, llL03, llL05, llL15] (see: Bombieri-Iwaniec method) Jutila method [llLxx, llL03, 11L05, llL15] (see: Bombieri-lwaniec method)

[52A351 (see: Geometric transversal theory) Kac-Borcherds character formula s e e : W e y l - -Kaccharacterformula see: W e y l - -Kacformulasee: Akhiezer---; Feynman--; Weyl- --

Kac-Moody algebra

K see: Fibonaeci numbers of order - - ; Fibonacci polynomials of order - - ; Fibonacoi-type polynomials of order - - ; Lucas-type polynomials of order - - ; multivariate Fibonecci polynomials of order - k-acylindrical graph of groups s e e : decomposition as a - k-algebra see: finitely-generated - - ; Ktihler differential on a - k

k-coherent pair of measures

[33C45, 33Exx, 46E35] (see: Sobolev inner product) [I3B10, 13C15, 13C40] (see: Zariski-Lipman conjecture)

-

-

Kac-Moody group [22E65, 22E70, 35Q53, 35Q58, 58F071 (see: AKNS-hierarchy) K a c - M o o d y Lie algebra s e e :

affine - -

Kac problem [35J05, 35J25] (see: Dirlchlet eigenvalue)

[16G70] (see: Almost-spfit sequence) K-homology [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) K-homologysee:

Steenrod --

k-matching in a graph

[05Cxx, 05D15] (see: Matching polynomial of a graph) K-move

[57M27] (see: Kirby calculus) k-order type

[52A35] (see: Geometric transversal theory)

[17B10, 17B651 (see: Weyl-Kae character formula) [17B10, 17B65] (see: Weyl-Kac character formula) see:

[13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) K~ihler differentials s e e : module of -K&hler form s e e : WeiI-Petersson -KAhler metric s e e : hyper- --

K'dhler potential [35Qxx, 78A25] (see: Magnetic monopole) Kalman filtering [15A57, 47B35, 65F05, 93B 15] (see: Hankel matrix) Kaltofen-Trager random polynomialtime factorization method [12D05] (see: Factorization of polynomials) K a m p e n theorem s e e : Van - Kantor triple system s e e : balanced Freudenthal- - - ; Freudenthaf- -Kaplanskytheorem see: Carleman---

Kardar-Parisi-Zhang equation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Karlin-Rouault law

k-q representation

Karush-Kuhn-Tucker condition

[42Axx, 44-XX, 44A55] [26B99, 30C62, 30C65] (see: Quasi-regular mapping) k-representation of an integer

[05D05, 06A07] 506

Gleason-

[60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) K~trman relation see:

yon - -

Kdrmdn vortex shedding

[76Cxx] (see: Von K~irm~invortex shedding) Kfirm~n v o r t e x s h e d d i n g see: Von -[90Cxx] (see: Fritz John condition) Kasparov K-theory [19K33, 19K35, 46J10, 46L05, 46L80, 46L85, 49L80] (see: Brown-Douglas-Fillmore theory; Multipliers of C'* -algebras)

Kirby moves [57M25] (see: Tangle move) Kirillov dimension s e e :

Gel'fand---

K K-theory

[19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) Kleiman-Mori cone

[14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays)

skein m o d u l e

[57P25] hierarchy

[20-XX] (see: Regular group) Klein bottle [05C25, 20Fxx, 20]?32] (see: Baumslag-Solitar group) Massless - -

Klimek Green function

(see: Conway skein triple) see:

Kelvin-Helmholtz instability [76C05] (see: Birkhoff-Rott equation) Kemer theorem [08Bxx, 16R10, 17B01, 20El0] (see: Specht property)

[31C10, 32F05] (see: Pluripotential theory) Kloosterman sum [11F03, 11F70] (see: Selberg conjecture) Knapsack problem (90C27) (refers to: Greedy algorithm; N'79) Knaster-Kuratowski-Mazurkiewicz fixed-point theorem

[55M25] (see: Brouwer degree) Kneser theorem

[20E22, 20Jxx, 57Mxx] (see: Accessibility far groups)

(see: Kendall tau metric) Kendall tau s e e : the --

[57M27] (see: Kirby calculus)

K l e i n - G o r d o n equation s e e :

[57P25] (see: Conway skein triple)

[62H20]

Kac-Weyl character formula

k-order type s e e : simple -k-out-of-m: F - s y s t e m s e e : Consecutive - k-out-of-'n,: G - s y s t e m s e e : consecutive - k - o u t - o f - n structure s e e : consecutive - k-out-of-r~ system s e e : circularconsecut[ve - - ; linear consecutive - -

(see: Zak transform) K-quasi-regular mapping

Skein module) Kauffman bracket skein triple

Kendall tau

Kiihler differential on a k-algebra

k-functor

[57Mxx, 57M25] (see: Kauffman bracket polynomial;

Kaup-Newell-Segur Ablowitz- --

[57M271 (see: Kirby calculus) Kirby moves

Klein 4-group

Kauffman skein quadruple

k-family in a partially ordered set

[68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory)

[57Mxx, 57M25] (see: Skein module) Kauffman bracket skein relation

[11Fxx, 17B67, 20D08]

Kahane-Zelazko theorem

K function

Kauffman bracket skein module

(see: Boreherds Lie algebra) K a c - M o o d y algebra s e e : affine - - ; Borcherds - - ; generalized - - ; Lorentzian ; symmetrizable - -

K o - e x t e n s i b l e over a class of algebras s e e : homomorphism --

[05D05, 06A07] (see: Sperner property)

Kauffman bracket polynomial (57M25) (referred to in: Listing polynomials; Rotor) (refers to: Jones-Couway polynomial; Kauffman polynomial; Kirby calculus; Reldemeister theorem; Skein module)

Kauffman polynomial s e e : based on the - -

Kac-Weyl formula

k-derivation

Kasparov theory [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory)

population version of

Kendall tau metric (62H20) (referred to in: Kendall tau metric; Spearman rho metric) (refers to: Copula; Correlation coefficient; Kendall tau metric; Median (in statistics); Pearson product-moment correlation coefficient; Random variable; Sample; Spearman rho metric) Kerdock code [94Bxx] (see: Delsarte-Goethals code) kernel s e e : Bergman - - ; B o c h n e r Martinelli--; heat--; non-negativedefinite - - ; normalized reproducing - - ; Poisson--; Reproducing--; RieszHerglotz - kernel for a ball s e e : Poisson - kernel Hilbert space s e e : Reproducing- - kernel method s e e : algebraic - - ; heat- - kerneltheorem see: Peano - kernels s e e : e x a m p l e s o f reproducing - keycryptographiosystem see: public- - keycryptographysee: public- -keyencryption s e e : public- - KI-module see: prinjective --

Kiefer-Weissproblem [62L10] (see: Sequential probability ratio test) kind s e e : Bessel function of the first - - ; Szeg5 polynomial of the second - -

kinetic description of a gas

[76Axx] (see: Knudsen number)

Kirby calculus (57M27) (referred to in: Kauffman bracket polynomial; Tangle move) (refers to: Surgery) Kirby move blow-up

[57M27] (see: Kirby calculus) Kirby move handle slide

Knizhnik-Zamolodchikov-Bemard equations [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) knot see: 4-ballgenus of a --; (5,2) positive torus - - ; 6 2 - - - ; amphicheiral ; figure eight - - ; figure-eight -- ; genus of a - - ; planar genus of a - - ; positive torus - - ; pretzel - - ; slice - - ; Stevedore - - ; torus - - ; trefoil - - ; Turk head - - ; virtual - knot invariant s e e : finite-type -knot theory see: 'n,-rotor in - - ; rotor in - knots s e e : Brunn theorem on - -

-

knowledge base [68T05] (see: Machine learning) k n o w l e d g e base s e e :

domain - -

Knudsen gas

[76Axx] (see: Knudsen number)

Knudsen number (76Axx) (refers to: Boltzmann equation; Brownlan motion; ChapmanEnskog method; Euler equation; Mach number; Navier-Stokes equations; Newton laws of mechanics; Prandtl number; Reynolds number) Knudsen number

[76Axx] (see: Knudsen number) Knuth correspondence s e e : R o b i n s o n - S c h e n s t e d - --

shifted

Kobayashi distance [31C10, 32F05] (see: Pluripotential theory) Kobayashi metric [30C45, 30D05, 32H15, 46G20, 47H10, 47H17, 47H20] (see: Denjoy-Wolff theorem; JuliaWolff-CaratModory theorem) Koenig method [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Kolmogorovbackward equation [62P05] (see: Thiele differential equation)

LAPLACE DISTRIBUTION

Kolmogorov complexity [68T05] (see: Machine learning) Kolmogorovwidth [42B05, 42B08] (see: Hyperbolic cross) Konformminimalflache [53C42] (see: Willmore functional) Korobovmethod s e e :

Vinogradov---

Korteweg-de Vries equation [35Q53, 58F07] (see: Harry Dym equation) Korteweg-de Vries equation s e e : averaged solution of the - - ; characteristic initial-value problem for the - - ; dispersionless - - ; shocks for the - - ; Whitham equation for the - Korteweg-de Vries-Landau-Ginsbu rg model s e e : Hurwitz-space -Korteweg-de Vries solution s e e : gap - Kozen-Stoekmeyer alternation theorem s e e : Chandra- -KP-hierarchy s e e : two-component-KP-Todatype s e e ; tau-funetion of - Krahninequality s e e : Rayleigh-Faber--Krahnisopedmetrieinequalitysee: Faber-

Hardy-

Kre~n method [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) KreYnmethod [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) Kre~n-Mil'man theorem [46E25, 52A35, 54C35] (see: Bishop theorem; Geometric transversal theory) Krichever-Phoug Whithamformulation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Kronecker-delta sequence [39A12, 93Cxx, 94A12] (see: Z-transform) Kroneckerfactor [28D05, 54H20] (see: Wiener-Wintrier theorem) Kronecker factor [28D05, 54H20] (see: Wiener-Wintrier theorem) Kronecker integral [55M25] (see: Brouwer degree) Kronecker theorem [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Kronecker theorem [11C08, 11R04] (see: Lehmer conjecture) Kronecker theorem on Hankel matrices [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Kronecker-Weber theorem [ l 1R321 (see: Shafarevieh conjecture) Kronheimer-Mr6wka theorem [57P25] (see: Milnor unknotdng conjecture) Krull dimension [05Exx, 13B10, 13B30, 13C14, 13C15, 13C40, 16Lxx, 16P60, 55U10] (see:Forster-Swan theorem; StanleyReisner ring; Zariski-Lipman conjecture) Krull dimension of group cohomology s e e : Quillen theorem on - -

Krull-Remak-Schmidt category [16G70] (see: Almost-split sequence) Krull-Schmidt category

kth derivative estimate s e e : put --

van der Cor-

Kuhn-Tucker condition [90Cxx] (see: Fritz John condition) Kuhn-Tuckercondition s e e : K a r u s h - - Kuramotoequation s e e : Sivashinsky--Kuramoto--Sivashinskydynamics s e e : dissipativity of the - -

Krasny method [76C05] (see: Birkhoff-Rott equation) Krausevariation of a function s e e :

[16Gxx, 16G70] (see: Almost-split sequence; Tits quadratie form) Krull-Schmidt category [16Gxx] (see: Tits quadratic form) Krull-Schmidt-type theorem [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) Kruskal-Katona theorem (05D05, 06A07) (referred to in: Sperner theorem) (refers to: Sperner property; Sperner theorem; Totally ordered set) Kruskal-Katona theorem [05D05, 06A07] (see: KruskaI-Katona theorem) KS equation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation)

Kuramoto-Sivashinsky equation (35Q35, 58F13, 76Exx) (refers to: Bifurcation; Dissipative system; Dynamical system; Gevrey class; Hausdorff dimension; Lyapunov characteristic exponent; Navier-Stokes equations; Painlcv6 test; Sobolev space) Kuramoto-Sivashinskyequation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Kuramoto-Sivashinsky equation s e e : bifurcation in the - Kuratowski-Mazurkiewicz fixed-point theorem s e e ; K n a s t e r - - Kuratowski theorem on closed projections

[26A15, 54C05] (see: Namioka theorem) Kurzweil-Heustockintegrability [28A25] (see: Denjoy-Perron integral) Kuttamethod s e e :

Runge---

L A-calculus s e e : Church - L see: complexity class -L~I~ see: sentencein-

L ~-cohomology [11Fxx, 20Gxx, 22E46] (see: Baily-Borel eompactification; Satake compactification) LV-cohomology [11Fxx] (see: Satake compactification) L -continuity [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L-continuous mapping [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L - F T O P see: L-function s e e :

category-p-adic --

L-function of an algebraic number field [11R23] (see: Iwasawa theory) L-fuzzy continuity [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L-fuzzy continuous mapping

[33C45, 33Exx, 46E35] (see: Sobolev inner product)

[03G10, 06Bxx, 54A40] (see: Fuzzy topology) L-fuzzy topological space [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L-fuzzy topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L 2 harmonic form [35Qxx, 78A25] (see: Magnetic monopole) L e - i n d e x t h e o r e m see:

Lamadrid and Sova compact derivative s e e : Gil de - -

Atiyah - -

L-powerset [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L1 regression problem [90B85] (see: Fermat-Torricelli problem) L-subset [03G10, 06Bxx, 54A40] (see: Fuzzy topology) l~-sum [17Cxx, 46-XX] (see: JB *-triple) L-TOPsee: category-L-topological space [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L -topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) L (0) -grading property [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) L ( -- 1) -derivative property [llFll, 17B10, 17B65, 171367, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) labeling s e e : S p e r n e r - labeling greedy algorithm s e e : node- - labelled trees s e e : Cayley formula for the number of -labelling of a graph s e e : 2 - - laced affine Lie algebra s e e : simply- - lacunary Fourier series s e e : HadamardLadik equations s e e :

Ablowitz---

Lagrange arithmetic-geometric mean algorithm [26Dxx, 65D20] (see: Arithmetic-geometric mean process) Lagrangeinterpolation formula s e e : of the - -

error

Lagrange interpolation polynomial [65Txx] (see: Fourier pseudo-spectral method) Lagrange theorem on squares [I 1Pxx] (see: Additive basis) Lagrungian [90Cxx] (see: Fritz John condition) kagrangJan s e e : quantum - Lagrangian intersections s e e : Fleer homology for --

symplectie

Lagrangian subgroup [11F27, I 1F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) Lagrangian submanifold [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) Lagrangian subspace [57M25] (see: Fox n-colouring) kaguerrecoefficients s e e :

Fourier---

Laguerre polynomials [33C45, 33Exx, 46E35] (see: Sobolevinner product) Laguerreseries s e e :

Fourier---

Laguerre-Sobolev orthogonal polynomials [33C45, 33Exx, 46E35] (see: Sobolev inner product) Laguerre weightfunction

lambda calculus [03D15, 68Q15] (see: Computational complexity classes) lamina [51M04] (see: Wittenbauer theorem) laminar flame front [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Lanczos tau method [65Lxx] (see: Tau method) Lanczos tridiagonalization process [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) Landau-Ginsburg model s e e : Hurwitzspace Korteweg-de Vries- -Landau-Ginsburg type s e e : topological field theory of --

Landau prime ideal theorem [llNxx, 11N32, 11N45] (see: Ahstract prime number theory) Landau theorem [llNxx, 11N32, I1N45] (see: Abstract prime number theory) Landan-Weber theorem [11Nxx, 11N32, 11N45] (see: Abstract prime number theory) Langcriterion s e e :

Schneider---

Langlands conjecture [11F03, 11F70] (see: Selberg conjecture) Langlands formula for the dimension of spaces of automorphicforms [46L80, 46L87, 55N15, 58G10, 58GI 1, 58G12] (see: Index theory) Langlands functoriality [11F27, llF70, 20G05, 81R05] (see: Segal-Shale-Weil representation) language s e e : ambiguity in a natural - - ; bounded-error polynomial-time computable - - ; bounded-error quantum polynomial-time computable - - ; domain of discourse in natural - - ; dynamics in a natural - - ; extra-grammatical usage of a natural - - ; first-order -- ; formula in a logical - - ; full restricted first-order - - ; pragmatics of natural - - ; regular - - ; semantics of a natural - - ; sense ambiguity in a natural - - ; situational context in natural - - ; structural ambiguity in a natural - - ; syntax of a natural - - ; term in a logical - - ; u n d e r e o n s t r a i n e d n e s s in natural - - ; ungrammatical usage of a natural - language analysis s e e : natural -language generation s e e : natural - language interface s e e : natural - -

language of set theory [03E30] (see: ZFC) language parsing s e e : natural - language processing s e e : Natural - languagesynthesis s e e : natural - language system s e e : natural - -

language type [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) pragmatics of natural - - ; robustness of natural - -

languageusagesee:

Laplace-Beltrami operator [35J05, 35J25, 53C42, 82B35, 82C35] (see: Dirichlet eigenvalue; OnsagerMachlup function; Willmore functional) Laplace distribution [62D05] (see: Acceptance-rejection method) Laplacelaw s e e :

Young---

507

LAPLACE OPERATOR

Laplaceoperatorsee: ues of the - -

Neumann eigenval-

Laplace operator on graphs [20E08, 20E18, 20Fxx] (see: Branch group) Laplace sequence [35L15] (see: Euler-Poisson-Darboux equation) Laplace transform see: discrete -Laplacian see: Dirichlet - - ; Dirichlet eigenvaluee of the -- ; invariant -- ; Neumann - -

Laplacian matrix of a graph [05C50, 15A15, 20C30] (see: Immanent; Matrix tree theorem) kaplacian matrix of a graph mixed - -

see:

edge - - ;

Laplacian of a graph [05C501 (see: Matrix tree theorem) Laplacian on the Sierpifiski gasket [28A80] (see: Sierpifiski gasket) kapLacians see: Weyl asymptotic formula for the eigenvalue distribution of -large 3-manifold see: sufficiently- --

large cardinal axioms [03E30] (see: ZFC) large deviation [60Gxx, 60J55, 60J65] (see: Wiener sausage) large deviation behaviour [60Gxx, 60J55, 60J65] (see: Wiener sausage) large numbers

see:

factor--

large-sample theory [62H20] (see: Kendall tau metric) large sieve see: coincidencesin the double - - ; double - large three-dimensional manifold see: sufficiently- - lattice see: atom in a - - ; atomic - - ; Boolean - - ; complete quasi-monoidal - - ; complete sup- -- ; cross-cut in a -- ; discrete Toda - - ; distributive - - ; flat in a geometric --; geometric --; Iocalic quasimonoidal - - ; Lorentzian - - ; orthocomplemented sup- - - ; orthomodular -- ; partition - - ; spanning set in a - - ; topological molecular -- ; uniform -- ; weakly deformed soliton - -

lattice-continuity [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-continuous mapping [03G10, 06/3xx, 54A40] (see: Fuzzy topology) lattice element

see:

complement of a - -

lattice-fuzzy continuity [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-fuzzy continuous mapping [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-fuzzy topological space [03G10, 06/3xx, 54A40] (see: Fuzzy topology) lattice-fuzzy topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice of a polytope

see:

face - -

lattice of fiats of a matroid [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mebius inversion) lattice point discrepancy [llLxx, llL03, llL05, 11L15] (see: Bombieri-lwaniec method) lattice-powerset [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-subset [03G10, 06Bxx, 54A40] 508

(see: Fuzzy topology) tattice-theoretictopology

scc:

point-set --

lattice-topological space [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) lattice-valued topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) Foulis semi-group of complete o r t h o m o d u l a r - - ; varietyof distributive - - ; variety of orthomodular --

latticessee:

Laurent polynomial [44A60] (see: Strong Stieltjes moment problem) Laurent polynomial [65D32] (see: Szeg~ quadrature) kaurent polynomials see: associated orthogonal -- ; orthogonal --

Laurent theorem [39A12, 93Cxx, 94A12] (see: Z-transform) law s e e : conservation - - ; D a m y - - ; energy balance - - ; fibred exponential - - ; Gauss quadratic reciprocity - - ; H a r d y Weinberg--; identical i n - - ; KarlinRouault - - ; local conservation - - ; van der Weals - - ; Young-Laplace - - ; Zipf --; Zipf-Mandelbrot - law (in topology) see: Exponential - l a w f o r sets see: exponential - -

Lawley-Hotelling test [62Jxx] (see: ANOVA) laws see: infinitely many

conservation - laws for the B e n j a m i n - B o n a - M a h o n y equation s e e ; conservation - Lawson conjecture see: G r o m o v - - Lax equation see: generalized --

Lax equations of the AKNS-hierarchy [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hlerarchy) Lax pair [35Q53, 58F07] (see: Harry Dym equation) layer feedforward neural net see: multi- -layer neural net see: multi- -layout see: completely crossed - - ; factorial - - ; two-way - -

Lazard-Mora example [14Axx, 14Q20] (see: Masser-Philippon/LazardMore example) Lazard-Mora/Masser-Philippon example [14Axx, 14Q20] (see: Masser-Philippon/LazardMora example) leading principal submatrix [15A57, 47B35, 651=05, 93B15] (see: Hankel matrix) leaf of a digraph [90D05] (see: Sprague-Grundy function) learning see: analytical - - ; concept - - ; decision tree in machine - - ; efficiency of a representation for machine --; explanation-based--; expressiveness of a representation for machine - - ; first-order predicate logic in machine - - ; inductive - - ; inductive inference approach to machine - - ; Machine - - ; neural network in machine - - ; PAC - - ; policy - - ; probabilistic functions in machine - - ; probably approximately correct - - ; reinforcement - - ; rule - - ; speedup - - ; supervised -- ; unsupervised - -

learning algorithm [68T05] (see: Machine learning) learning neural network [68T05]

(see: Machine learning) learning program [68T05] (see: Machine learning) learning system [68T05] (see: Machine learning) learning system see: classifier

Leibnizcongruence

in a --; critic in a - - ; experiment generator in a - - ; generalizer in a - - ; performance system in a - - ; representation structure in a - - ; target function in a - - ; training experience in a - learning theory see: computational - -

least-fixed-point operator [03D15, 68Q15] (see: Computational complexity classes) least period of an ultimately periodic sequence [11B37] (see: Ultimately periodic sequence) least-squares estimation equations for - -

see:

normal

least-squares estimator [62Jxx] (see: ANOVA) least squaresprediction [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) least squares prediction see: linear - least squares predictor see: best linear --

least-squaresregression [90B85] (see: Fermat-Torricelli problem) Lebesgue constant [42B05, 42B08] (see: Hyperbolic cross) Lebesgue constants of multi-dimensional partial Fourier sums (42B05, 42B08) (refers to: Continuous function; Dilatation; Fourier series; Fourier transform; Hyperbolic cross; Integrable function; Lebesgue constants; Partial Fourier sum; Principal curvature; Step hyperbolic cross) Lebesgue decomposition [28-XXl (see: Absolutely continuous measures) Lebesgue decompositiontheorem [34B24, 34L40] (see: Sturm-Liouville theory) kebesguelemma

see:

Riemann---

Lebesgue point [42A16, 42A24, 42A28] (see: Beurling algebra) Lee variety see:

Vaughan- - -

left centralizer [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Legendrecoefficients Legendreseries see:

see:

$econd-order--

Leibniz-Hopf algebra and quasisymmetric functions (05E05, 16W30) (refers to: Leibuiz-Hopf algebra; Quasi-symmetric function)

Fourier--Fourier---

see:

Legendre-Sobolev inner product [33C45, 33Exx, 46E35] (see: Sobolevinner product) Legendre tau method [65Lxx] (see: Tau method)

Lehmer conjecture (11C08, 11R04) (refers to: Algebraic number; Entropy; Jensen formula; Mahler measure) Lehmer conjecture [11C08, 11R04] (see: Lchmer conjecture) Leibniz congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Leibniz congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

Leibniz-reducedfull second-order model [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Leibniz-reduction [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Leibniz rule [13B10, 13C15, 13C40] (see: Zariski-Lipmau conjecture) Lelong theorem on analytic subsets [32C30, 53C65, 5gA25] (see: Current) Leman closure s e e : W e i s f e i l e r - - lemma see: Boolean expansion - - ; Bramble-Hilbert - - ; Britton - - ; 5Poincar6 - - ; example of using the Lovgtsz local - - ; general case of the Lovasz local - - ; H i l b e r t - B r a m b l e - - ; i s o t o p y - - ; Julia--; Lov~isz l o c a l - - ; Nakayama --; Poincar6 - - ; R i e m a n n - L e b e s g u e - - ; S c h w a r z - P i c k - - ; stability - - ; symmetric case of the Lovasz local - - ; Thue-Siegel - - ; Weld - - ; Wolff-Schwarz - lemmas see: T h e m i s o t o p y - -

lemniscate constant [26Dxx, 65D20] (see: Arlthmetic-geometric

mean

process) lemniscate constant

see:

Gauss - -

Lemoine point [51M04] (see: Brocard point; Triangle centre) Lempert functional [31C10, 32F05] (see: Pluripotential theory) length see: curve of infinite - - ; Fitting - - ; minimum description - - ; nilpotent - - ; vector of shortest - -

length of a Fitting chain [20F18] (see: Fitting length) length of a group see:

Fitting

--;

nilpo-

tent - -

length of a hierarchy for a threedimensional manifold [57N10] (see: Haken manifold) length of a linearfeedback shift register [11/337, llT71, 93C05] (see: Shift register sequence) Leopoldt conjecture [11R23] (see: Iwasawa theory) Leopoldt conjecture [llJB1, 11R23] (see: Iwasawa theory; Schneider

method) leotropic crossing [57M25] (see: Listing polynomials) Leray-Schauder degree [55M25] (see: Brouwer degree) Leray-Serre spectral sequence [53Ci5, 55N35] (see: Spencer cohomology) kesigne algebra see: C o n z e - - Lesigne factor see: C o n z e - - letter of an HNN-extension see: stable -level see: branch index for a tree - - ; rigid stabilizer of a tree - -

level of a rank function on a partially ordered set [05D05, 06A07] (see: Sperner property) level of a statistical factor [62Jxx] (see: ANOVA) Levinesignature

see:

Tristram----

LITTLEWOOD-RICHARDSONRULE

Levine signatureof a link s e e :

Tristram---

Lrvy local time [60Hxx, 60J55, 60J65] (see: Skorokhod equation) lexical-functional grammar [68S05] (see: Natural language processing) lexical system [68S05] (see: Natural language processing) LFSR [11B37, 11T71, 93C05] (see: Shift register sequence) Liekorish-Mitlett-Ho polynomial s e e : B r a n d t - -Lie algebra see: affine - - ; affine K a c Moody - - ; Borcherds - - ; Heisenberg - - ; Monster - - ; simply-laced affine - - ; standard embedding - - ; twisted affine - - ; Well algebra of a - -

Lie algebra associated with a vector space

[17A40] (see: FreudenthaI-Kantor triple system) Lie algebra operator vessel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) Lie algebras s e e : Borcherds - -

structure theorem for

Lie equation [53C15, 55N35] (see: Spencer eohomology) [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) queer-standard embed-

Lie triple system (17A40) (referred to in: Anti-Lie triple system; Frendenthal-Kantor triple system; Jordan triple system) (refers to: Derivation in a ring; Homogeneous space; Lie algebra; Steiner system; Symmetric space; Vector space) Lie triple system [17A40] (see: Lie triple system) Lie triple system s e e :

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Lindenbaum-Tarski process [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) line s e e : combinatorial - - ; isogonal - - ; isotomic - - ; Sorgenfrey - - ; weighted projective - line bundles s e e : spectral curve of a family of-line case s e e : Inverse scattering, full- - line conjecture for plane domains s e e : nodal -line for the zeta-function s e e : critical - -

line in a design

[05B30] (see: Affine design) line in a net

[05Bxx] (see: Net (in finite geometry)) line of a Steiner triple system

line program s e e : line theorem s e e : linear character o f [05El0, 05E99,

Anti- - -

fife insurance [62P05] (see: Thiele differential equation) [16G70] (see: Almost-split sequence)

liftable property [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) see: Hensel -lifting problem lifting

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) likelihood s e e : partial -likelihood ratio

[62L101 (see: Sequential probability ratio

test) likelihood-ratio test [62Jxx] (see: ANOVA)

straight- -Courant nodal - a s y m m e t r i c group 20C25]

limit see: adiabatic - - ; B o g o m o l n y Prasad-Sommerfield -- ; fine -- ; incompressible - - ; non-tangential - - ; thermodynamic - limit theorem s e e : first Szeg6 - - ; strong Szeg6 -limit theorems s e e : Szeg6 - limited function s e e : band - - ; time - -

Lin singulartoil technique [57Mxx, 57M25] (see: Skein module) Lindemann-Weierstrass theorem

[90B85] (see: Fermat-Torricelli problem) linear hypothesis

[62Jxx] (see: ANOVA)

linear independence of logarithms of algebraic numbers ill J81] (see: Schneider method) linear inequalities s e e :

alternative in - -

linear inequality [15A39, 90C05] (see: Motzkin transposition theorem) linear least squares prediction [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) best - -

linear logic s e e : non-commutative - linear mapping s e e : separating space of

G F 2- - -

Linear complexity of a sequence (65C10, 68Q15, 93B99, 94A60) (refers to: Berlekamp-Massey algorithm; Cryptography; Galois field; Mersenne number; Shift register sequence; Ultimately periodic sequence) linear complexity of a sequence

[65C10, 68Q15, 93B99, 94A60] (see: Linear complexity of a sequence) linear complexity of a shift register sequence

[65C10, 68Q15, 93B99, 94A60] (see: Linear complexity of a sequence) linear complexity profile of a sequence

[65C10, 68Q15, 93B99, 94A60] (see: Linear complexity of a sequence) Linear congruential method (65C10) (refers to: Euclidean algorithm; Floor function; LLL basis reduction method; Minkowski theorem; Primitive root; Pseudo-random numbers; Random and pseudo-random numbers; Uniform distribution; Variation of a function) linear consecutive k-out-of-n system

a--

linear means of a Fourier series [42A16, 42A24, 42A28] (see: Beurfing algebra) linear model s e e : log- - - ; univariate - linear operator s e e : conservative - - ; injective - - ; isolated part of the spectrum of a - - ; positive - linear potentialtheory s e e : non- - linear programming s e e : dual algorithm of--

linear programming problem

F-

system) linear convergence [46Cxx] (see: Alternating algorithm)

linear radial basis function

[41A05, 41A30, 41A63] (see: Radial basis function) linear recurrence relation

[11B37, 11T71, 93C05] (see: Shift register sequence) linear regression

[62Jxx] (see: ANOVA) linear regression [62Jxx] (see: ANOVA)

Linear skein (57M25) (referred to in: Skein module) (refers to: Skein module) linear skein

linear stability analysis [76Cxx] (see: Von K~irmhnvortex shedding) causal - - ; best - -

linearity of the Yak transform

linear feedback shift register

linearly regular stochastic process

[11B37, 11T71, 93C05] linear feedback shift register [65C10, 68Q15, 93B99, 94A601 (see: Linear complexity of a sequence)

[57M25] (see: Tangle) link s e e : algebraic index of a - - ; eolouring group of a - - ; framed - - ; Hopf - - ; ~ almost positive - - ; Murasugi signature of a--; n-algebraic--; (n,k)-algebraic --; Positive--; relative--; TristramLevine signature of a - - ; wrapping number of a - -

link cost

[60K30, 68M10, 68M20, 901310, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) link diagram s e e : n-coloured --; rotant of a -

link flow [60K30, 68M10, 68M20, 90B10, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) link flow s e e : equilibrium - links s e e : Abelian group invariant of - - ; invariant of - - ; mutation of - - ; Taft conjectures on alternating - Liouville differential equation s e e : Sturm--

Liouville inequality

[11J68, 14Q20] (see: Liouville-Lojasiewicz inequal-

ity) Liouville--Lojasiewicz inequality (11J68, 14Q20) (referred to in: Masser-Philippon/ Lazard-Mora example) (refers to: Liouville theorems) Liouville-:Lojasiewicz inequality

[11168, 14Q20] (see: Liouville-Lojasiewicz inequality) Liouville measure

[53C20, 53C22] Liouville operator s e e : Dirichlet S t u r m - - ; Stu r m - - Liouville problem s e e : inverse S t u r m - - Liouville spectral problem s e e : numerical approaches to the S t u r m - -- ; S t u r m - --

Liouville theorem on invariance under geodesic flow

[53C20, 53C22] (see: Santal6 formula)

(see: Obstacle scattering) Lipscbitz mapping [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Lipschitz retract [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) Listing polynomial s e e : second - - ; white - -

[57Mxx, 57M25]

linear eigenvalue problem s e e : isospectral -linearevolution equation s e e : non- --

(see: Shift register sequence)

link

kiouvilletheory s e e : S t u r m - -Lipman conjecture s e e : Z a r i s k i - - Lipschitz domain [35P25]

linear resolvent s e e : non- - linear Schr6dinger equation s e e : non- - linearsemi-groupsee: exponentialformula representation for a - -

linear transformation s e e : fractional- - linear unbiased estimator s e e :

(see: Wold decomposition) lines s e e : Cevian - lines in n triangle s e e : c o n c u r r e n t - linguistics) s e e : stem (in - -

(see: Santal6 formula)

[90C70] (see: Fuzzy programming)

(see: Skein module)

[60C05, 60K10] (see: Consecutive k-out-of-n:

linear fit problem

linear logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35, 18D10, 18D15] (see: *-Autonomous category; Abstract algebraic logic)

linear code [68Q05, 68Q10, 68QI5, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) linear code s e e :

linear feedback shift register s e e : characteristic polynomial of a - - ; feedback coefficients of a - - ; feedback matrix of a - - ; feedback polynomial of a - - ; impulseresponse sequence of a --; initial conditions of a - - ; length of a - - ; reciprocal polynomial of a - - ; state vector of a - linear filtering problem s e e : non- - -

linear least squares predictor s e e :

[05B07, 05B30] (see: Pasch configuration)

(see: Schur Q-function)

Lie generator

Lie super-algebrasee: Lie superalgebra s e e : ding - -

[11385] (see: Gel'fond-Schneider method) Lindenbaum-Tarski model

[42Axx, 44-XX, 44A55] (see: Yak transform) [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) linearly singular stochastic process

[60G25, 62M20, 93B10, 93B15, 93E12]

black - - ; first - - ;

Listing polynomials (57M25) (refers to: Kauffman bracket polynomial; Knot and link diagrams; Listing knot; Statistical mechanics, mathematical problems in; Torus knot) little group

[81Txx, 81T05] (see: Massless field) Littlewood functions s e e :

Hall- - -

Littlewood one-circle problem

[3IA05, 31B05, 31CI0, 31C35, 32A10, 46FI0, 60Y65] (see: Mean-value characterization) Littlewood-Richardson rule 509

LITTLEWOOD-RICHARDSON RULE

[05El0, 05E99, 14C15, 14M15, 14N15, 20C25, 20G20, 57T15] (see: Schubert calculus; Schur Qfunction) LLL [05C801 (see: Lovfisz local lemma) LLL-algorithm [65C10] (see: Linear congruential method) LNRE distribution [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) loading of flexible structures s e e : namic - Lobatto points s e e : G a u s s - --

dy-

local algebra of a Banach-Jordan pair [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) local Beth definability [03Gxx] (see: Algebraic logic) local Brouwer degree [55M25] (see: Brouwer degree) local complete intersection [13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) local conservationlaw [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) local derivative [73Bxx, 76Axx] (see: Material derivative method) local-global principle in commutative algebra [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) local index theorem [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) local lemma see: example of using the Lov~.sz - - ; general case of the Lov~tsz - - ; Lovasz - - ; symmetric case of the Lov~,sz - local minimum s e e : constrained --

local number of generators of a module [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) local optimality on a region of stability s e e : characterization of --

local rate [73Bxx, 76Axx] (see: Material derivative method) local rate of change [73Bxx, 76Axx] (see: Material derivative method) local representationtheoremsee: Riesz-local ring s e e : conductor of a - - ; regular - - ; System of parameters of a module over a - -

local ring of a point on an algebraic curve [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) local ring of finite Buehsbaum-representation type see: Noetherian - -

local ring of functions [13Hxx] (see: System of parameters of a module over a local ring) local ring of maximal embedding dimension see: Buchsbaum -local rings see: s t r u c t u r e t h e o r e m f o r m a x imal Buchsbaum modules over regular --

local spectral theory [47Dxx] (see: Taylor joint spectrum) local time [60Hxx, 60J55, 60J65] (see: Skorokhod equation) local time see: L~vy - local time derivative [73Bxx, 76Axx] (see: Material derivative method) local lime derivative 510

[73Bxx, 76Axx] (see: Material derivative method) local tomographic data [44A12, 65R10, 92C55] (see: Local tomography; Pseudolocal tomography) Local tomography (44A12, 65R10, 92C55) (referred to in: Pseudo-local tomography) (refers to: Entire function; Fourier transform; Pseudo-differential operator; Radon transform; Tomography; Wave front) local tomographysee: Pseudo- - local tomography function s e e : inversion formula for the pseudo- - - ; pseudo- - - ; standard --

local tomography operator [44A12, 65R10, 92C55] (see: Local tomography) local topological triviality [57N80] (see: Thom-Mather stratification) locale [03G10, 06Bxx, 54A40] (see: Fuzzy topology) localic quasi-monoidal lattice [03G10, 06Bxx, 54A40] (see: Fuzzy topology) localization see:

frequency - - ; time - - ;

time-frequency --

localization of distributions [46F10] (see: Multiplication of distributions) localization of eigenvalues [15A42] (see: Bauer-Fike theorem) localization problem see: real root --; root -localization theorem for matrix eigenvalues [15A42] (see: Bauer-Fike tbeorem) locally countably compact space [54C05, 54C08] (see: Strongly eountably complete topological space) locally finite partially ordered set [05B35, 05Exx, 05E25, 06A07, 11A25] (see: MSbius inversion) locally Hamiltonian vector field [37J15, 53D20, 70H33] (see: Momentum mapping) locally spectrally associative algebra [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) locally uniformlycontinuoussemi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) locally walk-bounded digraph [90D05] (see: Sprague-Grundy function) location problem see: single facility -location science [90B85] (see: Fermat-Torricelli problem) location theory [90B85] (see: Fermat-Torricelli problem) location theory s e e : continuous -log flip [14Exx, 14E30, 14Jxx] (see: Mort theory of extremal rays) log-linear model [62Jxx] (see: ANOVA) log-normal distribution [44A60] (see: Strong Stieltjes moment problem) log-rank lowerbound [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx]

(see: Quantum computation, theory

of) log terminal s e e : weakly - logarithm s e e : discrete - - ; Zech - -

of)

logarithmic capacity [31C10, 32F05] (see: Pluripotential theory) logarithmic concavity [05D05, 06A07] (see: Sperner property) logarithms of algebraic numbers ear independence of - -

see:

lin-

logic [03Gxx] (see: Algebraic logic) logic s e e : Abstract algebraic - - ; Algebraic--; algebraizable--; BCK --; characterization theorems in - - ; compactness of a - - ; completeness theorem in algebraic - - ; concrete algebraic - - ; deduction property of a - - ; entailment - - ; equivalence theorems in algebraic - - ; faithful interpretation of an equational - - ; f i r s t - o r d e r - - ; first-order predicate - - ; formula in first-order - - ; foundations of non-commutative - - ; general - - ; linear - - ; logically equivalent sentences in - - ; logistic abstract algebraic - - ; meaning algebras of a - - ; modal - - ; monadic predicate - - ; n-variable fragment of first-order - - ; non-commutative - - ; non-commutative l i n e a r - - ; orthom o d u l a r - - ; propositional - - ; protoalgebraic - - i quantum - - ; relevance - - ; repr e s e n t a b l e aJgebras of a - - ; second-order algebraizable - - ; semantics-based abstract algebraic - - ; sentential - - ; strict Horn -- ; symbol in first-order -- ; syntactical theory of a - - ; typed higher-order - logic) s e e : matrix (in - logic in machine learning s e e : first-order predicate -logic of a class of algebras s e e : equational - logic of Boolean algebras s e e : equational - -

logic of E k [81Qxx] (see: Dirae quantization)

(see: Abstract algebraic logic) theorem

see:

logical connective [03Gxx] (see: Algebraic logic) logical connective s e e : of a - -

arity of a - - ; rank

logical equivalence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logical equivalence of formulas with respect to a deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logical language s e e : in a - -

formula in a - - ; term

logical matrix [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logical matrix s e e : designated set of a - - ; underlying algebra of a - logical property s e e : recta- - -

logical state

logical truth [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logically equivalentformulas with respect to a deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logically equivalent sentences in logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logistic abstract algebraic logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logistic path from logic to algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logistic system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) logspace complexity class [03D15, 68Q15] (see: Computational complexity classes) Iogspaee complexity class s e e : deterministic - -

non-

Lojasiewicz inequality [11J68, 14Q20] (see: Liouville-Lojasiewicz inequality) {_ojasiewicz inequality see: Liouville- - Long H-dimodule Azumayaalgebra [13-XX, 16-XX, 17-XX] (see: Skolem-Noether theorem) long memory process [60G25, 62M20, 93BI0, 93BI5, 93E12] (see: Wold decomposition) long m e m o r y process s e e :

logic programming s e e : inductive - logic to algebra s e e : logistic path from - - ; rule-based path from - - ; semantical path from -logic with equatity s e e : Horn - logic without equality s e e : infinitary universal Horn - logical algebra [03Gxx, 03G05, 03G10, 03GI5, 03G25, 06Exx, 06F35] logical characterization meta- --

[68Q05, 68Q10, 68Q15, 68Q25, 81pxx] (see: Quantum computation, theory

stationary --

long-range force [81Txx, 81T05] (see: Massless field) long wave equation s e e :

regularized --

longest path problem [05C12, 90C27] (see: Dijkstra algorithm) loop group [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) loop network [05C25] (see: Cayley graph) Lops classificationof simple Jordan pairs of finite capacity [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Lorch theorem [46Cxx] (see: Alternating algorithm) Lorentz manifold [81Q05, 81Txx, 81T20] (see: Massless Klein-Gordon equation) Lorentz surface [53C20, 53C22] (see: Santal6 formula) Lorentzian Kac-Moody algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Lorentzian lattice [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Lorenz equations [58Fxx] (see: Conley index) Lovfisz local lemma

(05C80)

MAPPING (referred to in: Jansen inequality) (refers to: Graph; Independence; LLL basis reduction method; Probability theory) Lov~,sz local lemma see: example of using the - - ; general case of the - - ; symmetric case of the -low displacement rank

[15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) lower bound see:

log-rank--

lower Fitting series [20F17, 20F18] (see: Fitting chain) lower nilpotent series [20F17, 20F18] (see: Fitting chain) lower shadow of a family of subsets [05D05, 06A07] (see: Kruskal-Katona theorem) lowest eigenvalue see: equality for the --

isoperimetric in-

LR test [62Jxx] (see: ANOVA) LR test see:

Wilks --

LSE

[62Jxx] (see: ANOYA) LSZ-reduction formulas [81Txx] (see: Massive field) Lucas numbers [11B39] (see: Lncas polynomials) Lucas polynomials (11B39) (referred to in: Consecutive k-out-ofn: F-system) (refers to: Chebyshev polynomials; Consecutive k-out-of-n: F-system; Fibonacci polynomials; Generating function; Irreducible polynomial) Lucas polynomials see: bivariate Lucas-type polynomials [11B39] (see: Lucas polynomials) Lucas-type polynomials [60C05, 60K10] (see: Consecutive k-out-of-n: Fsystem) Lucas-type polynomials of order k [11B39] (see: Lucas polynomials) Luzin-Menshov property [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) - -

Lyapunov coefficient see:

first --

Lyaptmovdimension [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) LYM inequality [05D05, 06A07] (see: Sperner property; Sperner theorem) LYM partially ordered set [05D05, 06A07] (see: Sperner property) LYM partially ordered set [05D05, 06A07] (see: Sperner theorem) LYM poset [05D05, 06A07] (see: Sperner property) LYM poset [05D05, 06A07] (see: Sperner theorem) LYM posers see:

quotient theorem for --

Lyndon word [05E05, 16W30] (see: Leibniz-Hopf algebra and quasi-symmetric functions)

Logic programming; Markov process; Neural network; Predicate calculus; Probability theory; Search algorithm; Turing machine; Undecidability)

M

machine learning see: decision tree in - - ; efficiency of a representation for - - ; expressiveness of a representation for -- ; first-order predicate logic in - - ; inductive inference approach to - - ; neural network in - - ; probabilistic functions in --

m-almost positive link [57M25] (see: Positive link) m-function see: Weyl M-ideal [17Cxx, 4 6 - X X ]

- -

(see: JB *-triple) M-quasi-symmetric function [30C62, 30C99] (see: Quasi-symmetric function of a complex variable) M-quasi-symmetric function on T [30C62, 30C99] (see: Quasi-symmetric function of a complex variable) m-rectifiable current [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) m-rectifiable Radon measure [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) m-rectifiable set [28A78, 49Qxx, d9Q15, 53C65, 58A25] (see: Geometric measure theory) m-rectifiable varifold [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) m-sequence [11B37, 11T71, 93C05] (see: Shift register sequence) m-tangent plane see: approximate m-tangle see: r~-algebraie - - ; (n,k)- -

algebraic -m,-unreotifiable Radon measure see: purely -m-unrectifiable set see: purely - -

m -varifold [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) m-varifold see:

first variation of an --

MA system [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) Maass form [11Fll, llF12] (see: Shlmura correspondence) Maass wave form [11F03, 11F70] (see: Selberg conjecture) Macaulay ring see: generafized C o h e n - Macaulay simplicial complex see: Macaulayfication see:

Cohen-

Cohen- --

Macdonald identities [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Macdonald polynomials [05El0, 05E99, 20C25] (see: Schur Q-function) Mach number [76Axx] (see: Knudsen number) machine see: alternating Turing - - ; non-deterministic Turning - - ; probabilistic Turing - - ; quantum random access - - ; quantum luring - - ; random access - - ; time for a non-deterministic Turning - - ; universal Turing - -

Machine learning (dgT05) (referred to in: Natural language processing) (refers to: Bayes formula; Boolean function; Conditional distribution;

machine translation [68S051 (see: Natural language processing) Machlup function see:

Onsager---

Machlup~Onsager function [82B35, 82C35] (see: Onsager-Machlup function) macro-economical model [90A11] (see: Cobb-Douglas function) MacWilhams transform [9aBxx] (see: Delsarte-Goethals code) magnetic charge [35Qxx, 78A25, 81V10] (see: Dirac monopole; Magnetic monopole) magnetic duality see:

electric- - -

Magnetic monopole (35Qxx, 78A25) (refers to: Algebraic curve; Connections on a manifold; Covariant derivative; Curvature; Dynkin diagram; IGqhler-Einstein manifold; Lie group; Moduli theory; Principal G-object; Riemann theta-function; S-duality; Yang-Mills field) magnetic monopole [81VlO] (see: Dirac monopole) Mahler conjecture [1 lj85] (see: Gel'fond-Schneider method) Mahler measure of a minimal polynomial [11C08, 11R04] (see: Lehmer conjecture) Mahler measure of an algebraic number [11C08, 11R04] (see: Lehmer conjecture) Mahler method (11Fll, 11J82, 11J85, 11J91) (refers to: Algebraic number; Analytic function; Diophantine approximations; Dirichlet principle; Formal languages and automata; Gel'fondSchneider method; Modular function; Polynomial; Schneider method; Transcendental number) MabJer vanishingtheorem [llFll, 11J82, 11J85, 11391] (see: Mahler method) Mahony equation see: Benjamin-Bona- ; conservation laws for the BenjaminBona- -- ; generalized Benjamin-Bona- ; variable-coefficient Benjamin-Bonamain conjecture see:

Iwasawa --

main effect in design of statistical experiments [62Jxx] (see: ANOVA) main operator of a unitary operator colligation [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator eolligation) Majorana representation [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) majorant see; monotone - majorant of coefficients see: Beurling algebra of Fou tier series with summable --

majorization [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx]

(see: Quantum information processing, science of) Mal'cevtheorem see:

generalized--

malign distribution [68Q15] (see: Average-case computational complexity) Malliavin theorem in harmonic analysis [43A45, 43A46] (see: Ditkin set) malnormal subgroup [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Mandelbrot law see:

Zipf- --

Mangasarian~Fromovitz condition [90Cxx] (see: Fritz John condition) manifold see: arithmetic - - ; AtiyahHitchin - - ; atoroidal--; boundary compressible surface in a three-dimensional - - ; boundary incompressible surface in a three-dimensionaJ - - ; complex structure on a - - ; c o m p r e s s i b l e surface in a three-dimensional - - ; O-compressible surface in a three-dimensional - - ; O-incompressible surface in a threedimensional - - ; Fibonacci - - ; Frobenius - - , Haken - - ; Hantzche.-Wendt - - ; hierarchy for a three-dimensional - - ; homogeneous - - ; immersion of a surface into a Riemannian -- ; incompressible surface in a three-dimensional - - , inertial - - ; irreducible three-dimensional - - ; length of a hierarchy for a three-dimensional - - ; L o r e n t z - - ; nice 3- - - ; non-Hakee _; pZ.irreducible three-dimensional - - ; partial hierarchy for a three-dimensional - - ; Poisson - - ; quantum - - ; reducible three-dimensional - - ; spin e. _ ; stable - - ; sufficiently-large 3- - - ; sufficientlylarge three-dimensional - - ; two-sided surface in a - - ; unstable - - ; Waldhausen graph -manifold at a surface see: splitting a threedimensional - manifold equation see: singularity -manifold group see: 3- - Manifolds 1D s o f t w a r e see: GlobaJ -Manifolds2D software see: Global -manifolds see: eohomoJogy of flag - - ; homeomorphism problem for threedimensional -- ; immersion conjecture for - - ; instanton Fleer homology for threedimensional - - ; uniformization for threedimensional - - ; word problem for threedimensional -Mania residue see: A d l e r - - -

MANOVA (62Jxx) (refers to: ANOVA) MANOVA [62Jxx] (see: ANOVA) MANOVA [62Jxx] (see: ANOVA) MANOVA see: canonical form for -Manton programme

[35Qxx, 78A25] (see: Magnetic monopole) many conservation laws see:

infinitely --

many-one problem reduction [68Q15] (see: Average-case computational complexity) many-one problem reduction see: polynomial-time - map see: C a y t e y - - ; Gauss - - ; Poinear6 -mapforacoloursuperalgebrasec: colouring -mapping see: A-proper - - ; Dirichletto-Neumann - - ; distance-preserving - - ; distortion of a - - ; fuzzy continuous --; Gel'fand--; generic s m o o t h - - ; isogonal--; K-quasi-regular--; Lcontinuous - - ; ,L-fuzzy continuous - - ;

511

MAPPING

lattice-continuous - - ; lattice-fuzzy continuous--; L i p s c h i t z - - ; matrix-valued holomorphic - - ; Momentum - - ; operatorvalued holomorphic - - ; Poincar6 - - ; positive - - ; q u a s i - c o n f o r m a l - - ; Quasiregular - - ; p-monotone - - ; semi-group generated by a holomorphic - - ; separating s p a c e of a linear - - ; sesquilinear - - ; spherical - - ; symplectic Fleer homology for a symplectic - - ; T h e m - - ; WeiI-Brezin - -

Markovian functional of Brownian motion

see:

non-

-

-

Martin-Girsanovtheorem see:

Cameron-

Martin theorem on semi-groups [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) Martinelli kernel s e e : B o c h n e r - -martingale s e e : super- - -

matching graphs s e e :

mapping class group [14H15, 30F60] (see: Weil-Petersson metric) mapping cyhnder [53C15, 57R57, 58D27] (see: Atlyah-Floer conjecture)

martingale approach to portfolio optimization [90A09] (see: Portfolio optimization)

mapping cylinders s e e : Atiyah-Floer conjecture for - mapping function of a domain s e e ; Riemann --

mass function [68T30, 68'I"99, 92Jxx, 92K10] (see: Dempster-Shafer theory) mass-gap [8lYxx] (see: Massive held) mass-gap condition [81Txx] (see: Massive field) mass era current [28A78, 49Qxx, 49QI5, 53C65, 58A25] (see: Geometric measure theory)

mapping, monotone with respect to the Poincar( metric [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) mapping preserving a distance [54E351 (see: Aleksandrov problem for isometric mappings) mapping with bounded distortion [26B99, 30C62, 30C65] (see: Quasi-regular mapping) mappings see: Aleksandrov problem for isometric - - ; degree of symmetric -- ; Semi-group of holomorphic - - ; space of infinitely Silva-differentiable - -

Marchenko method [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-line ease; Inverse scattering, half-axis ease) Marcinkiewicz multiplier theorem [42B05, 42B08] (see: Step hyperbolic cross) Marcus inequality for permanents [15A15, 20C30] (see: Immanant) marginalization operator [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) marginally correct approximation approach to Dempster-Shafer theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) marked Riemann surface [14HI5, 30F60] (see: Weil-Petersson metric) marked Riemann surface s e e : space at a --

cotangent

marked shifted tableau [05El0, 05E99, 20C25] (see: Schur Q-function) marked shifted tableau s e e : a--

content of

market s e e : complete - market assumption s e e : complete - -

Markov algorithm [03D15, 68Q15] (see: Computational complexity classes) Marker braid theorem (20F36, 57M25) (referred to in: Jones-Conway polynomial) (refers to: Braid theory; JonesConway polynomial) Markov decision process [68T05] (see: Machine learning) Markovinequality s e e :

peintwise - -

Markov move [20F36, 57M25] (see: Markov braid theorem) Markov process s e e : Markovtheorem see:

512

hidden -Gauss---

Euler---;

Mascheroniconstantsee:

Euler

theorem on the Euler- --

mass of atriangle s e e :

massless particle see: spinless and - matched-ends condition [58F22, 58F25] (see: Seifert conjecture) matching see: perfect -matching cover in a graph [05Cxx, 05D15] (see: Matching polynomial of a graph)

centre of --

mass operator [81Txx] (see: Massive field) mass renormalization [81Txx, 81T05] (see: Massless field) Masser-Philippon example [14Axx, 14Q20] (see: Masser-Philippon/LazardMora example) Masser-Philippon/Lazard-Mora example (14Axx, 14Q20) (referred to in: Effective Nullstellensatz) (refers to: Hilbert theorem; LiouviIleLojasiewicz inequality) Massive field (81Txx) (referred to in: Massless field; Massless Klein-Gordon equation) (refers to: rock space; Fourier transform; Generalized functions, space of; Hilbert space; Hyperbolic partial differential equation; Massless field; Poincar6 group; Quantum field theory; Scattering matrix; Unitary representation) massive quantum field theory [81Txx] (see: Massive field) Massless field (81Txx, 81T05) (referred to in: Massive field; Massless Klein-Gordon equation) (refers to: rock space; Generalized functions, space of; Hilbert space; Huygens principle; Massive field; Poincar6 group; Quantum field theory; Scattering matrix; Unitary representation) massless field [81Txx, 81T05] (see: Massless field) Massless Klein-Gordon equation (81Q05, 81Txx, 81T20) (refers to: Hyperbolic partial differential equation; Klein-Gordon equation; Massive field; Massless held; Planck constant; Quantum field theory; Schrfdinger equation; Wave equation) massless particle [81Q05, 81Txx, 81T20] (see: Massless Klein-Gordon equation)

co- - -

matching in a graph [05Cxx, 05D15] (see: Matching polynomial of a graph) matching in a graph s e e : a-

k- -;

weight of

matching polynomial [05Cxx, 05D15] (see: Matching polynomial of a graph) Matching polynomial of a graph (05Cxx, 05D15) (refers to: Chebyshev polynomials; Graph; Hermite polynomials; Laguerre polynomials) matching polynomial of a graph s e e : simple -matching property see: normalized - matching property of a partially ordered set see: normalized - -

matching theorem for operator vessels [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) matching unique graph [05Cxx, 05D15] (see: Matching polynomial of a graph) material derivative [73Bxx, 76Axx] (see: Material derivative method) material derivativein spatial form [73Bxx, 76Axx] (see: Material derivative method) materia[ derivative in spatial form s e e : mula for the --

for-

Material derivative method (73Bxx, 76Axx) (refers to: Derivative; Motion; Partial derivative; Vector) material derivative operator [73Bxx, 76Axx] (see: Material derivative method) material form see: function in - material rate [73Bxx, 76Axx] (see: Material derivative method) material time derivative [73Bxx, 76Axx] (see: Material derivative method) material time derivative [73Bxx, 76Axx] (see: Material derivative method) Mathcmadca code for the Dickman function [1 IAxx] (see: Dickman function) mathematical ple, --

see:

Uncertainty princi-

mathematical economy [28-XX] (see: Non-additive measure) mathematical theory of evidence [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) mathemahcal uncertainty principle [42A63] (see: Uncertainty principle, mathematical) Matherstratification s e e : T h e m - - matrices s e e : Dirac g a m m a - - ; generalization of the H a d a m a r d - F i s c h e r inequality for positive semi-definite Hermitian - - ; Hadamard inequality for Hermitian - - ; inverse spectral methods for Jac o n -- ; Kronecker theorem on Hankel -- ; Pauli spin - - ; perturbation of - - ; Schur

stability of polynomials and - - ; Schur theorem on - - ; stability theorem for polynomials and - matrix s e e : Bezout - - ; block Hankel - - ; B o r c h e r d s - C a r t a n - - ; O - - - ; Cartan - - ; characteristic polynomial of a - - ; companion - - ; conditionally negative-definite -- ; conditionally positive-definite - - ; design - - ; designated set of a logical - - ; diagonalizable - - ; directed graph of a - - ; echelon - - ; generalized Cartan - - ; generalized Hadamard - - ; Goeritz - - ; Hankel - - ; Hankel moment - - ; Hermitian a t joint - - ; H i l b e r t - - ; incidence - - ; irreducible - - ; Jacobi - - ; J a c o b i - T r u d i - - ; logical - - ; meaning - - ; Moment - - ; monodromy - - ; Mueller - - ; node-admittance - - ; null space of a - - ; positive-definite - - ; S - - - ; Schur - - ; Segre characteristic of a square - - ; stable - - ; support of a zero-one - - ; symmetrizable Cartan - - ; Toeplitz - - ; Toeplitz moment - - ; totally positive - - ; underlying algebra of a logical - - ; Wittich --

matrix diagonalization problem [20C05, 20Dxx] (see: Zassenhaus conjecture) matrix eigenvalues s e e : rem for - -

localization theo-

Matrix element (15-XX) matrix entry [ ] 5-XX] (see: Matrix element) matrix for electromagnetic wave scattering see: S- -matrix function s e e : generalized -matrix

(in logic)

[03Gxx, 03G05, 03G10, 03G25, 06Exx, 06F35]

03GI5,

(see: Abstract algebraic logic) matrix model of a deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) matrix norm [15A42] (see: Bauer-Fike theorem) matrix norm see: Frobenius - - ; submultiplicative inequality for a - matrix of a graph s e e : edge Laplacian - - ; incidence - - ; Laplacian - - ; mixed Laplaclan - matrix of a linear feedback shift register s e e : feedback - -

matrix problem [16Gxx] (see: Tits quadratic form) matrix ring s e e :

ceIlular--

Matrix tree theorem (05C50) (refers to: Cofaetor; Design of experiments; Graph; Marker chain; Minor; Perron-Frobenius theorem) matrix tree theorem

[05C50] (see: Matrix tree theorem) matrix tree theorem s e e : tion version of the - -

generating func-

matrix-valuedholomorphic mapping [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath~odory theorem) matroid s e e : lattice of flats of a - - ; M~bius invariant of a - - ; oriented -matroids s e e : critical problem for - -

Matsumoto theorem [19Cxx] (see: Steinberg symbol) max-min principle [35J05, 35J25] (see: Dirichlet eigenvaine) max-min principle f o r eigenvalues [35J05] (see: Neumann eigenvalue) maximal Abelian p-extension [IIR23]

MICHAEL PROBLEM

(see: Iwasawa theory) maximal Bruck net

[05Bxx] (see: Net (in finite geometry)) maximal Buchsbaum module

[13A30, 13H10, 13H30] (see: Buchsbaum ring) maximal Buchsbaum modules over regular local rings s e e : structuretheorem for --

maximal compact subgroup [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) maximal determinantal representation

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) maximal dilatation

[26B99, 30C62, 30C65] (see: Quasi-regular mapping) maximal embedding dimension Buchsbaum local ring of - -

see:

mean-valuecondition see: generalized - mean value theorem s e e : Vinogradov-mean-value theorem s e e : TemlyakovOpiaI-Siciak-type --

mean-value theorem for harmonic functions

[31A05, 31AI0] (see: Poisson formula for harmonic functions)

maximal ideal space

[32E20] (see: Polynomial convexity) maximal period sequence

mean-value theorem for harmonic functions see: converseof Gauss

[11B37, llTT1, 93C05]

- -

mean-variance efficient portfolio

(see: Shift register sequence)

[90A09]

maximal Satake compactification [llFxx] (see: Satake compactification)

(see: Portfolio optimization)

mean weightedresiduals [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods)

maximal ( s , r )-net

[05Bxx] (see: Net (in finite geometry)) maxtmum a posteriori hypothesis

meaning s e e :

semantical - -

meaning algebra

[68T05] (see: Machine learning) maximumroot test [62Jxx] (see: ANOVA)

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) meaning algebras of a logic

maximum root test s e ¢ : ,Roy - Mazur game s e e : Banach- - - ; generalized Banach- - - ; play in the Banach- - ; stationary strategy in the generalized Banach- - - ; stationary winning strategy in the generalized Banach- - - ; strategy in the generalized Banach- - - ; tactics in the generalized Banach- - - ; winning strategy in the generalized Banach- -Mazurkiewicz fixed-point theorem s e e : Knaster-Kuratowski- -

McKean-Singer index formula [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) mean s e e : arithmetic-geometric - - ; general -m e a n algorithm see: Lagrangearithmeticgeometric - -

mean free path

[76Axx] (see: Knudsen number) mean method s e e : arithmeticgeometric - mean oscillation s e e : bounded--; function of bounded - - ; space of analytic functions of bounded - - ; space of analytic functions of vanishing - mean process s e e : Arithmeticgeometric --

mean square

[62Jxx] (see: ANOVA) Mean-value characterization (31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65) (refers to: Besselfunctions; Harmonic function; Lebesgue measure; Pluriharmonic function; Reinhardt domain) mean-value" characterization s e e : alized - -

mean-value characterization for holomorphic functions [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) mean-value characterization of pluriharmonic functions [31A05, 31B05, 3IC10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) mean-value characterization of separately harmonic functions [31A05, 3IB05, 31C10, 3IC35, 32A10, 46F10, 60Y65] (see: Mean-value characterization)

gener-

mean-valuecharacterization for harmonic functions [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization)

[03Gxx] (see: Algebraic logic) meaning function

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) meaning homomorphism [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) meaning matrix

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) means of a Fourier series see: linear - measurablefunction see: simple -measurablefunctions s e e : equi- - measure s e e : Absolutely continuous invariant - - ; admissible - - ; atom of a - - ; ball - - ; canonical - - ; decomposition theorem for sets of finite Hausdorff - - ; Dirac - - ; ergodic invariant - - ; excessive - - ; fuzzy - - ; Gaussian - - ; harmonic - - ; Hausdorff - - ; Jensen - - ; Liouville - - ; m-rectifiable Radon - - ; measure, absolutely continuous with respect to a given - - ; Non-additive - - ; outer - - ; pseudo-addition decomposable - - ; purely m-unrectifiaNe Radon - - ; purely unrectifiable - - ; quasi- - - ; rectifiable - - ; representing - - ; Riemannian volume - - ; o--additive - - ; SBR - - ; self-similar - - ; Sinai-Bowen-Ruelle - - ; spectral - - ; tconorm - - ; tangent - - ; transcendence - - ; uncertainty--; upper Minkowski - - ; vector - - ; Wiener - -

[20C05, 20Dxx]

[llFII, 11J82, I1J85, llJ91] (see: Mahler method) measure of an algebraic number s e e : Mahler - -

measure of association [62H20] (see: Kendall tau metric; Spearman rho metric) measure of concentration of a function around a point

[42A63] (see: Uncertainty principle, mathe-

metaplectic cover [1IFll, 11P12] (see: Shimura correspondence) metaplectic group

[llFll, liP12, 11F27, llFV0, 20G05, 81R05] (see: Segal-Shale-Weil representation; Shimura correspondence) metastable concentration

[82B26, 82D35]

matical)

(see: Cahn-HilUard equation)

measure of non-compactness

[47H10] (see: Darbo fixed-point theorem)

measure-theoretic smootlqness [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) measuretheory s e e : Geometric - measurement s e e : von Neumann - -

measurement in quantum computation

[68Q05, 68Q10, 68Q15, 68Q25, 8lPxx] (see: Quantum computation, theory of) measures s e e : absolute continuity of - - ; Absolutely continuous - - ; coherent pair of - - ; domination of - - ; e q u i v a l e n t - - ; k-coherent pair of - - ; weak convergence of-mechanics s e e : continuum - - ; foundations of quantum - - ; motion in continuum - - ; quantum - media s e e : trapping in random - -

medial correlation coefficient

[62H20] (see: Kendall tau metric) median hyperplane problem

[90B85] (see: Fermat-Torricelli problem) median problem s e e :

(see: Zassenhaus conjecture) metabelian identity see: centre-by---

1- - -

Mel' nikov-Chatzidakis theorem

[l 1R321 (see: Shafarevieh conjecture)

memberstfipproblem [68S051 (see: Natural language processing) membership problem over a module

[13Pxx, 14Q20] (see: Hermann algorithms) membership problem over a module s e e : complexity of t h e - membrane s e e : clamped - - ; vibration of e-

membrane theory [53C42] (see: Willmore functional) '

metasymplectic geometry [17A40] (see: Freudenthal-Kantor triple system) method s e e : Acceptance-rejection - - ; AGM - - ; algebraic kernel - - ; arithmeticgeometric mean - - ; Bombieri-lwaniec - - ; canonical polynomials in the tau - - ; Chebyshev pseudo-spectral - - ; Chebyshev spectral - - ; Chebyshev tau - - ; collocation - - ; consistency of an inversion - - ; Crank-Nicolson - - ; decimation - - ; domain decomposition - - ; Ense - - ; error analysis for the tau - - ; FaddeevPopov - - ; Feynman path - - ; finite difference--; finite element - - ; Fourier pseudo-spectral - - ; Fourier spectral - - ; Galerkin - - ; G e l ' f o n d - S c h n e i d e r - - ; h e a t - k e r n e l - - ; implicit E u l e r - - ; inclusion-exclusion - - ; integral transform - - ; Jutila - - ; Kaltofen-Trager random polynomial-time factorization - - ; Koenig - - ; K r a s n y - - ; Kre'~n - - ; Lanczos tau - - ; Legendre tau - - ; Linear congruential - - ; M a h l e r - - ; Marchenko - - ; Material d e r i v a t i v e - - ; M o n t e - C a r l o - - ; Nahm - - ; operational formulation of the tau - - ; Petrov-Galerkin - - ; p s e u d o - s p e c t r a l - - ; Rankin-Selberg - - ; ratio-of-uniforms --; r e c u r s i v e tau - - ; R u n g e - K u t t a - - ; sampling - - ; Schneider--; ShidlevskFSiegel - - ; SiegeI-Shidlovskii"--; sieve - - ; SOR - - ; spectral element - - ; spectral tau - - ; successive overrelaxation - - ; "1" - - ; Tau - - ; twistor - - ; van der Corput - - ; V i n o g r a d o v - K o r o b o v - - ; Wegmann - - ; weighted residual - - ; Zakharov-Sbabat dressing - method approximation s e e : tau -method for r e g u l a r i z a t i o n s e e : Zetafunction - method in c o m b i n a t o r i c s s e e : probabilistic - method in the stability of functional equations s e e : direct construction - -

memory see; quantum - memory process s e e : long - - ; stationary long

method of selected points

Menger sponge [28A80] (see: Sierpifiski gasket)

method software s e e : BOV- - methods s e e : Trigonometric pseudospectral - methods for Jacobi matrices s e e : inverse spectral - metric s e e : c o m p l e t e hyperbolic - - ; conformal change of - - ; distance - - ; Euclidean Taub-NUT - - ; hyper-Ktihler - - ; hyperbolic - - ; Kendall tau - - ; Kobayashi - - ; mapping, monotone with respect to the Poincar@ - - ; w ~ - - - ; Poincar@ - - ; Poincar@ hyperbolic - - ; Quillen - - ; Schwarz-Pick pseudo- - - ; Spearman rho--; W e i I - P e t e r s s o n - - ; Wijsman topology induced by a - -

- -

Menshov property see: k u z i n - - Morton option pricing s e e : Scholes- - -

Black-

mesh category

[16G70] (see: Riedtmann classification) Mesner algebra see:

B o s e - --

[65Lxx] (see: Tau method)

[28-XX]

meta-Abelian Baumslag-Solitargroup [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) meta-Abelian group [20C05, 20Dxx] (see: Zassenhaus conjecture)

(see: Absolutely continuous mea-

meta-logical characterization theorem

[12J10, 12J20, 13A18, 16W60]

safes)

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) recta-logical property [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) metabelian group

(see: S-integer)

measure, absolutely continuous with respect to a given measure

measure associated with a Dirichlet problem see: spectra/-measure of a minimal polynomial s e e : Mahler -measure of a partial differential equation see: quantum spectral --

measme of algebraic independence

metric on a number field

metric perturbation of a Banach algebra s e e : E-metric space s e e : almost convex - - ; strongly zero-dimensional -metric spaces s e e : quasiqsometric-metrizable space s e e : w t, - - -

Michael problem

513

MICHAEL PROBLEM

[46H40] (see: Automatic continuity for Banaeh algebras) micro-economical model [90AlI] (see: Cobb-Douglas function) Mieg interpretation s e e : Ne'emanThierry- -Miller space s e e : Buntinas-Tanovie---

Miller symmetry [35L151 (see: Euler-Poisson-Darboux equation) Millett-Ho polynomial s e e : BrandtLiekorish- - Mills equations s e e : self-dual Y a n g - - Mills-Higgs action s e e : Y a n g - - Mills potential s e e : Y a n g - - Mills theory s e e : Y a n g - - Mil'man theorem s e e : KreTn---

Milnor unknotting conjecture (57P25) (referred to in: Positive link) (refers to: Link; Positive link; Torus knot) Milyutin theorem

see:

Dubovitskfi'-

--

MIMD (68Mxx) MIMe system (93A25) (refers to: Automatic control theory) min principle s e e : max- - min principle for eigenvalues s e e :

max- - -

minimal asymptotic additive basis [11Pxx] (see: Additive basis) minimal model program [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) minimal non-split short exact sequence [16G70] (see: Almost-split sequence) minimal polynomial s e e : of a --

Mahler measure

minimal polynomial of a shift register sequence [11B37, 11T71, 93C05] (see: Shift register sequence) minimal representation [11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) minimal set era dynamical system [58F22, 58F25] (see: Seifert conjecture) minimal surface [53C42] (see: Willmore functional) minimal surface see: conformally --; Weierstrass data for a - - ; Weierstrass representation of a -m i n i m a t t h e o r y s e e : weakly --

minimal variety [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) m i n i m u m see: constrainedlocal - minimum description length [68T05] (see: Machine learning) minimum variance unbiased estimator [62Jxx] (see: ANOVA) miniphase ARMA system [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) Minkowskibound [11R29] (see: Odlyzko bounds) Minkowskiconvex body theorem [65C10] (see: Linear eongruential method) Minkowskidimension [28A80] (see: Sierpifiski gasket) Minkowski measure s ¢ ¢ :

Minkowskispace-time 514

upper--

[15A66, 81Q05, 81R25, 83C22] (see: Dirae algebra) Minkowski-Steinitz-Weyl theorem [15A39, 90C05] (see: Motzkin transposition theorem) Minkowski-Weyltheorem see: Farkas--minorant s e e : hypo-harmonic - - ; subharmonic --

missing data [62Jxx] (see: ANOVA) missing data s e e :

nested - -

mixed ANOVA model [62Jxx] (see: ANOVA) mixed derivative [42B05, 42B08] (see: Hyperbolic cross) mixed Laplacian matrix era graph [05C50] (see: Matrix tree theorem) mixed motive [11F67] (see: Eisenstein cohomology) mixed reasoning [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) ML [68T05] (see: Machine learning) M/M/1 queue [60K30, 68M10, 68M20, 90B10, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) MMP [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) mobile time derivative [73Bxx, 76Axx] (see: Material derivative method) M6bius algebra [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) MSbius function [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) Mbbius function JILL07, 11M06, 11P32] (see: Vaughan identity) M6bius invariant of a matroid [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) M6bius inversion (05B35, 05Exx, 05E25, 06A07, 11A25) (refers to: Arrangement of hyperplanes; Atom; Boolean algebra; Chain; Closure space; Combinatorics; Commutative ring; Distributive lattice; Euler characteristic; Euler function; Exact sequence; Graph colouring; Homology group; Inclusion-and-excluslon principle; Lattice; Matroid; Modular lattice; Nerve of a family of sets; Partially ordered set; Simplieial complex) MObius inversion formula [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Dirichlet convolution; M6bius inversion) Mdbius inversion formula s e e : -- ; number-theoretic - -

classical

modal logic [68S05] (see: Natural language processing) mode in a plasma s e e : dissipativetrapped ion -model s e e : Aalen multiplicative int e n s i t y - - ; A R F I M A - - ; Black-Scholee geometric 8rownian motion - - ; convex - - ; Cox regression - - ; de B r a n g e s Rovnyak functional - - ; fixed effects - - ; full second-order - - ; Hurwitz-space

Korteweg-de V r i e s - L a n d a u - G i n s b u rg - - ; Leibniz-reduced full second-order --; Lindenbaum-Tarski - - ; log-linear - - ; macro-economical --', micro-economical --; mixed ANOVA -- ; partially specified s t a t i s t i c a l - - ; Potts - - ; r a n d o m effects - - ; random g r a p h - - ; randomization - - ; Solow - - ; Suszko-reduced - - ; Sz.Nagy-Foias functional - - ; theory of a - - ; transferable belief - - ; univariate linear - -

model delta-net [46F10] (see: Multiplication of distributions) model of a deductive system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) model of a deductive system s e e :

matrix --

model of a Gentzen system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) model of a non-self-adjeint operator s e e : triangular - m o d e l of a p a i r o f o p e r a t o r s s e e : functional - - ; triangular - model of a physical system s e e : partition function of a state - -

model of a set of formulas [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) model of computation [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

o0 model of quantum computation [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of) model program s e e : minimal - m o d e l l i n g g r o w t h curves

[62Jxx] (see: ANOVA) inodelling uncertainty [28-XX] (see: Non-additive measure) models see: class of --; elementarily equivalent - models of homotopy types s e e : braic - -

alge-

moderate deviation behaviour [60Gxx, 60J55, 60J65] (see: Wiener sausage) modified Euler product formula [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) modified moments [44A60, 47A57] (see: Moment matrix) modified Seifert conjecture [58F22, 58F25] (see: Seifert conjecture) modified Seifert conjecture [58F22, 58F25] (see: Seifert conjecture) modified zeta-function [11Nxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) modular form see: Siegel - modular forms s e e :

Hida theory of - -

modular functions [ l l F l l , 17B67, 20D08, 81T10] (see: Moonshine conjectures) modular functions s e e : field of - - ; Hauptmodul for a field of - modular group s e e : Hilbert-Siegel - - ; Siegel --

modular j-function [ l l F l l , 17B67, 20D08, 81T10] (see: Moonshine conjectures) modular moonshine [l l F l l , 17B67, 20D08, 8IT10]

(see: Moonshine conjectures) modulation behaviour of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) modulation of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) module see: A l e x a n d e r - - ; APR-tilting - - ; basis in a - - ; Bass n u m b e r of a - - ; B u c h s b a u m - - ; Cartier-Dieudonne - - ; complexity of the membership problem over a - - ; cotilting - - ; dual Bass n u m b e r of a - - ; flat - - ; formal character of a w e i g h t - - ; generalized tilting - - ; highest-weight fl- - - ; homotopy skein --; injective envelope of a - - ; integrable - - ; irreducible highest weight - - ; Iwasawa - - ; Jones-type skein - - ; Kauffman bracket skein - - ; local n u m b e r of generators of a - - ; maximal Buchsbaum - - ; m e m b e r s h i p problem over a - - ; moonshine - - ; non-projective - - ; pre-injeetive - - ; pre-projeetive - - ; prinjective K I - - - ; q - h o m o t o p y skein - - ; quasi-Buchsbaum - - ; Skein - - ; Steinberg - - ; Tilting - - ; Vassiliev-Gusarov skein - - ; Verma -m o d u l e based on relations deforming n moves s e e : skein - m o d u l e based on the J o n e s - C o n w a y relation s e e : skein - m o d u l e based on the Kauffman polynomial see: skein - -

module of K?ihler differentials [13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture) m o d u l e over a local ring s e e : System of parameters of a - modules s e e : dual B r o w n - G i t l e r - modules over regular local rings s e e : structure theorem for maximal Buchsbaum -moduli s e e : deformation parameters of - -

moduli of the Baily-Borel compactificalion [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) moduli space [11Fxx, 20Gxx, 22E46, 35Qxx, 53C15, 57R57, 58D27, 78A25] (see: Atiyah-Floerconjecture; BailyBorel compactification; Magnetic monopole) moduli space s e e :

q u a n t u m --

moduli space of fiat connections [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) modulus of a family of curves [26B99, 30C62, 30C65] (see: Quasi-regular mapping) molecularlattice s e e : topological -molecule equation s e e : Toda --

mollifier [46F30] (see: Colombeau generalized function algebras; Egorov generalized function algebra) moment see: binomial - m o m e n t condition s e e : Carleman - m o m e n t correlation coefficient s e e : Pearson product- --

Moment matrix (44A60, 47A57) (refers to: Complex moment problem, truncated; Gram matrix; Hankel matrix; Inner product; Laurent series; Linear functional; Matrix; Moment; Moment problem; Moments, method of (in probability theory); Probability distribution; Toeplitz matrix) m o m e n t matrix s e e :

Hankel - - ; Toeplitz --

moment of a probability distribution [44A60, 47A57] (see: Moment matrix) m o m e n t of random variables s e e : uct - -

prod-

MURNAGHAN-NAKAYAMAFORMULA

moment problem [44A60, 47A57] (see: Moment matrix) m o m e n t problem see: classical Stieltjes - - ; determinate strong Stieltjes - - ; Hamburger - - ; indeterminate strong Stieltjes - - ; strong - - ; Strong Stieltjes - - ; strong symmetric Stieltjes - - ; trigonometric - m o m e n t s see: g e n e r a l i z e d - - ; modified - m o m e n t u m see: angular - m o m e n t u m algebra see:

angular--

Momentum mapping (37J15, 53D20, 70H33) (refers to: de Rham cohomology; Lie algebra; Lie derivative; Lie group; Polsson algebra; Poisson brackets; Symplectic manifold; Symplectic structure) momentum mapping [37J15, 53D20, 70H33] (see: Momentum mapping) m o m e n t u m operator see: energy- - m o m e n t u m q u a n t u m n u m b e r see: angular - -

monadic algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) monadic algebras see:

varietyof --

monadic predicate logic [68S051 (see: Natural language processing) Monge-Ampkre capacity [31C10, 32F051 (see: Pluripotential theory) Monge-Amp~recapacity [31C10, 32F05] (see: Pluripotential theory) Monge-Amp~re operator [31C10, 321705] (see: Pluripotentlal theory) monic irreducible polynomials see: Gauss theorem on products of - -

monic polynomial over a finite field [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory) monodromy matrix [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) monoid of a C * - a l g e b r a see:

Ext - -

monoidal category [18D10] (see: Closed monoidal category) monoida[ category see: Closed - - ; symmetric -- ; symmetric closed - monoidal lattice see: complete quasi- - - ; Iocalic quasi- - -

monomialconjecture [13A30, 13H10, 13H30] (see: Buchsbaum ring) monomial exponential sum [llLxx, llL03, llL05, 11L15] (see: Bombleri-Iwaniec method) monomialpropertyfor parameter systems [13A30, 13H10, 13H30] (see: Buchsbaum ring) monomiala see: face- - monopole see: Abelian - - ; Dirac - - ; Magnetic - - ; S U ( 2 ) - --

monopole dynamics [35Qxx, 78A25] (see: Magnetic monopole) monopole theory [35Qxx, 78A25J (see: Magnetic monopole) monopoles see: t h e o r y o f - monotone functions see: co- - monotone majorant [42A16, 42A24, 42A28]

(see: Beurling algebra) pmonotone set function [28-XX] (see: Choquet integral) monotone mapping see:

--

monotone with respect to the Poincare metric see: mapping, -Monster see: fake - - ; Fiseher-Griess -monster see: 6-transposition property of the - -

morpheme [68S05] (see: Natural language processing)

monster group [llF11, 17B10, 17B65, 17B67, 20D08, 81R10, 81T10, 81T30] (see: Moonshine conjectures; Vertex operator)

morphology [68S051 (see: Natural language processing)

monster group see:

Fischer-Gdess --

Monster Lie algebra [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) monstrous moonshine [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) Monstrous Moonshine conjecture see: Conway-Norton --

Monstrous Moonshineconjectures [11Fxx, 17B67, 20D081 (see: Borcherds Lie algebra) Monte-Carlo method [65C10] (see: Linear congrnential method) Montesinos-Nakanishi conjecture (57P25) (referred toin: Fox n-colouring; Skein module; Tangle move) (refers to: Link) Montesinos-Nakanishi three-move conjecture [57Mxx, 57M25] (see: Skein module) Moody algebra see: affine K a c - - - ; Borcherds K a c - - - ; generalized K a c - - - ; K a c - - - ; Lorentzian K a e - - - ; symmetrizable K a c - -M o o d y g r o u p see: K a c - -Moody Lie algebra see: affine K a c - - moonshine see: m o d u l a r - - ; monstrous - Moonshine conjecture see: ConwayNorton Monstrous --

Moonshine conjectures (IlFll, 17B67, 20D08, 81T10) (referred to in: Vertex operator; Vertex operator algebra) (refers to: Borcherds Lie algebra; Group; Kac-Moody algebra; Modular function; Normal subgroup; Sporadic simple group; ThompsonMcKay series; Vector space; Vertex operator algebra) Moonshineconjecturessee: Monstrous-moonshine conjectures see: generalized --

moonshine module [llFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) moonshinemodule [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator; Vertex operator algebra) Moore singularity [76C05] (see: Birkhoff-Rott equation) More example see: L a z a r d - - - ; MasserPhilippon/Lazard- - Mora/Masser-PhiLippon example see: Lazard- -Mori cone see: Kle]man- -Mori fibrespace see: F a n o - - -

Mori theory of extremal rays (14Exx, 14E30, 14Jxx) (refers to: Algebraic variety; Birational mapping; Category; Divisor; Intersection index (in algebraic geometry); Kawamata rationality theorem; Kawamata-Viehweg vanishing theorem; Vector space) Morita P-space [54G101 (see: P-space) Morita theory for derived categories [16Gxx] (see: Tilting theory)

morphism see: contraction - - ; Poisson - - ; proper - - ; quotient - - ; stratum of a - -

Morse i n d e x s e e : generalized - Morse n u m b e r see: T h u e - - Moscoviei higher index t h e o r e m for coverings see: Connes--most p o w e r f u l t e s t see: uniformly -motion see: geometric Brownian - - ; non-Markovian functional of Brownian - - ; obliquely reflecting Brownian - - ; reflecting Brownian - -

motion in continuummechanics [73Bxx, 76Axx] (see: Material derivative method) motion mode~ see: B l a c k - S c h o l e s geometric Brownian -motion on the Sierpifiski gasket see: Brownian - motive see: mixed --

Motzkin transposition theorem (15A39, 90C05) Motzkin ~xanspositiontheorem [15A39, 90C05] (see: Motzkin transposition theorem) mountain-function see:

table - -

move see: 3- --; A- --; K - --; (re,q)- ; Markov --; p/q-rational--; Tangle -move blow-up see: Kirby - move conjecture see: MontesinosNakanishi three- -move handle slide see: Kirby - moves s e e : Habiro - - ; Habiro C ~ - - ; K i r b y - - ; n - - - ; R e i d e m e i s t e r - - ; skein m o d u l e based on relations deforming n -

(m,q)-move [57M25] (see: Tangle move) Mr6wka theorem see:

Kronheimer---

Mueller matrix [78A40] (see: Stokes parameters) Muller code see: R e e d - -multi-component systems s e e : reliability of_ multi-dimensional case see: Inverse scattering, - -

multi-dimensional partial Fourier sum [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) multi-dimensional partial Fourier sums see: Lebesgue constants of - -

multi-grid scheme [46Cxx] (see: Alternating algorithm) multi-layer feedforwardneural net [41A30, 92C55] (see: Ridge function) multi-layer neural net [41A30, 92C55] (see: Ridge function) multi-phase averaging [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) multi-quadric function [41A05, 4IA30, 41A63] (see: Radial basis function) multi-scale analysis [14Jxx, 35A25, 35Q53, 57R57] (see: Whltham equations) multi-stage decision [28-xx] (see: Non-additive measure) multifoeal ellipse [90B85] (see: Fermat-Torricelll problem) multiple-data see: multiple-instruction - - ; single-instruction - -

multiple-input multiple-output system

[93A251 (see: MIMO system) multiple-instruction multiple-data [68Mxx] (see: MIMD) multiple-output system see: multipleinput - -

multiple regres'sion [62Jxx] (see: ANOVA) multiple Wiener integral [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) multiplication see: Clifford - - ; S c h u r Hadamard -- ; Taylor spectrum under - -

Multiplication of distributions (46F10) (referred to in: Generalized function algebras) (refers to: Differential algebra; Fourier transform; Generalized function; Generalized function algebras; Generalized functions, product of; Harmonic function; Linear operator; Net (directed set); Poisson integral; Sobolev classes (of functions); Wave front) multiplication of generalizedfunctions [46F101 (see: Multiplication of distributions) multiplicative anomaly [glT50] (see: Non-commutative anomaly) multiplicative anomaly see: Wodzicki form u l a for - multiplieativeintensitymodelsee: A a l e n - multiplicity of an eigenvalue see: algebraic - multiplieroperatorsee: Fourier--

multiplier operators [47B06] (see: Riesz operator) multiplier problems [42A20, 42A32, d2A38] (see: Integrability of trigonometric series) multiplier theorem see:

Maminkiewiez - -

multiplier theory of distributions [46F10] (see: Multiplication of distributions) multiplier theoryof distributions [46F101 (see: Multiplication of distributions) multipliers see:

algebra of - - ; Floquet

--

Multipliers of C*-algebras (46J10, 46L05, 46L80, 46L85) (refers to: C*-algebra; Exact sequence; Extension theorems; Hausdorff space; Hilbert space; Net (directed set); Self-adjoint operator; Separable algebra; Stone-Cech compactification; Strong topology; Uniform convergence; von Neumann algebra) multivariate analysis of variance [62Jxx] (see: ANOVA;MANOVA) multivariate analysis of variance s e e : eralized - -

gen-

multivariate Fibonaccipolynomials of order k [33Bxx] (see: Fibonacei polynomials) multivariate polynomial see: black box representation of a - - ; dense representation of a - - ; sparse representation of a - Mumford regularity see: C a s t e l n u o v o - - -

Murasugi signature [57P251 (see: Conway skein equivalence) Murasugi signature of a link [57P25] (see: Conway algebra) Murnaghan-Nakayamaformula [20C25] (see: Projective representations of symmetric and alternating groups) 515

MURNAGHAN-NAKAYAMARULE Murnaghan-Nakayamarule [05El0, 05E99, 20C251 (see: Schur Q-function) mutation of links [57M25] (see: Rotor) mutually inverse interpretations [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) mutually t-balanced collections of blocks in a Steiner triple system [05B07, 05B30] (see: Pasch configuration)

Nakayamarule see:

n-algebraic link [57M25] (see: Algebraic tangles) n-algebraic m-tangle [57M25] (see: Algebraic tangles) n-algebraic tangles [57M25] (see: Algebraic tangles) algebraof --

n-ary representable cylindric algebra [03Gxx] (see: Algebraic logic) n-bridge n-tangle [57M25] (see: Rational tangles) n-coloured link diagram [57M25] (see: Fox n-colouring) n - c o l o u r i n g see: Fox - Consecutive k-out-of-

n-fold groupoid [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) n: G-system see:

consecutive k-out-of-

n-moves [57M25] (see: Tangle move) n-moves see: skein module based on relations deforming --

N-position [90D05] (see: Sprague-Grundy function) n-rotor in graph theory [57M25] (see: Rotor) n-rotor in knot theory [57M25] (see: Rotor) n structure s e e : consecutive k-out-of-

-n system s e e : circular consecutive k-outof- - - ; linear consecutive k-out-of- -

n-tangle [57M25] (see: Tangle) n-tangle [57M25, 57P25] (see: Fox n-colouring; MontesinosNakanishi conjecture) n-tangle see: n-bridge-n-tuples see: Taylor spectrum for Fredholm - n - v a r i a b l e f r a g m e n t o f first-order l o g i c [03Gxx]

(see: Algebraic logic) Nagy-Foias functionalmodel s e e :

Nahm equations" [35Qxx, 78A25] (see: Magnetic monopole)

516

Nakanishiconjecturesee: Montesinos--Nakanishi three-move coniecture s e e : Montesinos- - -

Murnaghan---

Nakayama lemma [13B30, 13C15, 16Lxx, I6P60] (see: Forster-Swan theorem)

P - p o i n t ultrafilteron - -

n: F-system see:

naive approach to Dempster-Shafer theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) na~)e Bayesian classifier [68T051 (see: Machine learning)

Nakayamaformula see:

V-filtration [16Gxx] (see: Tilting theory) n see: grouper exponent--

n - a r y relations s e e :

NaTmark-Segal construction see: Gel'fand- -Na'fmarktheorem s e e : Gel'fand---

Nakayama conjectnre [20C25] (see: Projective representations of symmetric and alternating groups)

N

N see:

Nahm method [35Qxx, 78A25] (see: Magnetic monopole)

Sz.- - -

Murnaghan---

Namioka property [26A15, 54C05] (see: Namioka space) Namioka space (26A15, 54C05) (referred to in: Namioka theorem; Strongly countably complete topological space) (refers to: Baire space; Blumberg theorem; Namioka theorem; Pseudometric space; Separate and joint continuity; Set of type F~ (G~); Sorgenfrey topology; Strongly eountably complete topological space; Topological space) Namioka space [26A15, 54C05] (see: Namioka space) Namioka space s e e :

co- --

Namioka theorem (26A15, 54C05) (referred to in: Namioka space; Strongly countably complete topological space) (refers to: Baire space; BanachMazur game; Compact space; Namioka space; Pseudo-metric space; Separate and joint continuity; Set of type F~ (G~); Strongly countably complete topological space; Topological space) Nanda-Pillaitest see:

Bartlett- --

Napoleon theorem [90B85] (see: Fermat-Torricelli problem) narrowDenjoy integral [28A25] (see: Denjoy-Perron integral) naturaldecomposition s e e :

axiom of --

Natural frequencies (70Jxx, 70Kxx, 73Dxx, 73Kxx) (referred to in: Neumann eigenvalue; Rayleigh-Faber-Krahn inequality) (refers to: Compact operator; Eigen value; Euler formula; Laplace operator; Neumann eigenvalue; RayleighFaber-Krahn inequality; Resolvent; Schr6dinger equation; Self-adjoint operator; Spectrum of an operator) natural frequencies [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) natural frequency resonance [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) natural language s e e : ambiguity in a - - ; domain of discourse in - - ; dynamics in a - - ; extra-grammatical usage of a - - ; pragmatics of - - ; semantics of a - - ; sense ambiguity in a - - ; situational context in -- ; structural ambiguity in a - - ;

syntax of a - - ; underconstrainedness in - - ; ungrammatical usage of a - -

natural language analysis [68S05] (see: Natural language processing) natural language generation [68S05] (see: Natural language processing) natural languageinterface [68S05] (see: Natural language processing) natural language parsing [68S05] (see: Natural language processing) Natural language processing (68S05) (refers to: Automatic translation; Automaton, finite; Automaton, probabilistic; Complexity theory; Formal languages and automata; Formalized language; Grammar, contextfree; Grammar, context-sensitive; Grammar, generative; Grammar, transformational; Logic programming; Machine learning; Markov process; Predicate calculus; Syntactic structure; Types, theory of) natural language synthesis [68S05] (see: Natural language processing) natural language system [68S05] (see: Natural language processing) natural l a n g u a g e usage s e e : pragmatics of -- ; robustness of - natural numbers s e e : Additive basis for the - -

natural order on a space of real-valued functions

[31AI0, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Navier-Stokes equation s e e : ible - -

compress-

Nazarova theorem [16Gxx] (see: Tits quadratic form) NC s e e :

complexity class - -

near continuity [54C08] (see: Almost continuity) nearest integer function [26Axx] (see: Floor function) necessitation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) necessitation s e e :

rule of - -

necessity degree [90C70] (see: Fuzzy programming) Ne'eman-Thierry-Mieginterpretation [8iQxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) nef see: f ' - negative Brocard point [51M04] (see: Brocard point) negative-definitekernel s e e : negative-definite matrix s e e : ally - negative quadratic form s e e :

non- - conditionw e a k l y non-

neighbourhood s e e : isolating -neighbourhood of a point on an algebraic curve see: first - - ; second - -

NeYmark-Sackerbifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) nest in a Banach space [46H40] (see: Automatic continuity for Banach algebras) nestedfactors in covariance analysis [62Jxx]

(see: ANOVA) nested missing data [62Jxx] (see: ANOVA) nested projection scheme [47H17] (see: Approximation solvability) nesting factors in covariance analysis [62Jxx] (see: ANOVA) net [05Bxx] (see: Net (in finite geometry)) net s e e : Bruck - - ; complete - - ; degree of a - - ; feedforward neural - - ; index of a - - ; line in a - - ; maximal ( s , r ) - - - ; maximal Bruck - - ; model delta- - - ; multi-layer feedforward neural - - ; multi-layer neural - - ; order of a - - ; ( s , r ) --; (s,r;,u)- - ; strict delta- - - ; symmetric - - ; translation - - ; transversal-free - - ; transversal i n a --

Net (in finite geometry) (05Bxx) (referred to in: Affine design) (refers to: Affine design; Affine space; Block design; Galois field; Hadamard matrix; Orthogonal array; Orthogonai Latin squares; Plane; Projective plane; Tactical configuration; Transversal system) Net (infinite geometry) (update) [05Bxx] (see: Net (in finite geometry)) networksee: Bayesian - - ; communication - - ; learning neural - - ; loop - - ; neural - - ; passive electrical - - ; q u a n t u m - - ; social --; stochastic - networkin machine learning s e e : neural - networks s e e : backpropagation algorithm for neural - Neumann algebra s e e : factor of a yon - -

Neumann boundary value problem [35J05] (see: Neumann eigenvalue) Neumann condition in obstacle scattering [35P25] (see: Obstacle scattering) Neumann eigenfunction [35J05] (see: Neumann eigenvalue) Neumann eigenvalue (35J05) (referred to in: Dirichlet eigenvalue; Natural frequencies; RayleighFaber-Krahn inequality) (refers to: Dirichlet eigenvalue; Laplace operator; Natural frequencies; Neumann boundary conditions; Rayleigh-Faber-Krahn inequality) Neumann eigenvalue [35J05, 35J25] (see: Dirichlet eigenvalue) Neumann eigenvalues s e e : ture for - -

Pdlya conjec-

Neumann eigenvalues of the Laplace operator [35P15] (see: Rayleigh-Faber-Krahn inequality) N e u m a n n entropy see: strong subadditivity inequality for von -- ; yon - N e u m a n n factor quantale s e e : von - N e u m a n n i n d e x s e e : von - -

NeumannLaplacian [35J05, 35P25] (see: Neumann eigenvalue; Obstacle scattering) Neumann mapping see: Dirichlet-to- - N e u m a n n measurement s e e : yon - N e u m a n n ordinal s e e : von - Neumann quantale s e e : atomic von - - ; von - Neumann regularity see: von - N e u m a n n theorem s e e : Stone-von --; Weyl-von - -

NORMAL FORMS IN BAUMSLAG-SOLITARGROUPS

neural net see: f e e d f o r w a r d - - ; multi-layer - - ; multi-layer feedforward - -

neural network [05C65, 05D05, 68Q15, 68T05] (see: Vapnik-Chervnnenkis dimension) neural network see:

learning - -

neural network in machine learning [68T05] (see: Machine learning) neural networks see: algorithm for - -

baekpropagation

neutral differential equation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Nevanlinna function [34B24, 3aL40] (see: Sturm-Liouville theory) NewelI-Segur hierarchy see: K a u p - --

Ablowitz-

Newton capacity [60Gxx, 60J55, 60J65] (see: Wiener sausage) N E2#1o

[68Q15] (see: Average-case computational complexity) Neyman-Pearson probability ratio test [62L10] (see: Sequential probability ratio test) nice 3-manifold [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Nieolson method see: Crank-Nielsen coordinates s e e : Fenchel- -Nielsen intrinsic coordinates see: Fenchel- -Nikodgm derivative s e e :

Radon---

nilpotency relation [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) nilpotent length [20F18] (see: Fitting length) nilpotent length of a group [20F18] (see: Fitting length) nilpotent operator see:

quasi- - -

nilpotent series [20F17, 20F18] (see: Fitting chain) nilpotent series see: lower - - ; upper -Nim see: game of - -

Nim addition [901:)05] (see: Sprague-Grundy function) Nim-sum see:

generalized - -

( r~,k )-algebraic link [57M25] (see: Algebraic tangles) ( n,k )-algebraic m-tangle [57M25] (see: Algebraic tangles) ( n , k )-algebraic tangles [57M25] (see: Algebraic tangles) NL see;

complexity class --

NLP

[68S05] (see: Natural language processing) no-arbitrage assumption [90A09] (see: Option pricing) nodal domain [35J05, 35J25] (see: Dirichlet eigenvalue) nodal line conjecture for plane domains [35J05, 35J25] (see: Dirichlet eigenvalue) nodal line theorem see:

Courant - -

node [14Hxx, 14H20] (see: Flecnode; Taenode) node see:

operator--

node-admJttance matrix

[05C50] (see: Matrix tree theorem) node conditions [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) node-labeling greedy algorithm [05C12, 90C27] (see: Dijkstra algorithm) node of operators [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) noded Riemann surface [14H15, 30F60] (see: Weil-Petersson metric) noded stable-curve [14H15, 30F60] (see: Weil-Peterssnn metric) Noether theorem see:

Skolem- --

Noetherian Banach algebra [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Noetherian group [20F38] (see: Fibonacci group) Noetherian local ring of finite Buchsbaumrepresentation type [13A30, 13H10, 13H30] (see: Buchsbaum ring) noise see: data --; q u a n t u m - noise in data [68T05] (see: Machine learning) noise sequence see:

pseudo- - -

noisy quantumchannel [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) Non-additive measure (28-XX) (refers to: Capacity; Idempotent analysis; Measure; Non-precise data; Semi-group; Set function) non-additive set function [28-XX] (see: Non-additive measure) non-Archimedean place of a number field [12J10, 12J20, 13A18, 16W60] (see: S-integer) non-autonomous Schr6der functional equation [39B05, 39B12] (see: Schr6der functional equation) Non-commutative anomaly (81T50) (refers to: Determinant; Pseudodifferential operator; Quantum field theory; Wodzickl residue; Zetafunction method for regularlzation) non-commutative anomaly [81T50] (see: Non-commutative anomaly) non-commutative anomaly for zetafunction regularization [glT50] (see: Non-commutative anomaly) non-commutativeintegration [22D10, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Fourier algebra) non-commutativelinear logic [03G25, 06D99] (see: Quantale) non-commutativelogic [03G25, 06D99] (see: Quantale) non-commutative logic see: foundations of--

non-commutative residue [35Sxx, 46Lxx, 47Axx] (see: Wodzlcki residue) non-commutativetopology [03G25, 06D99] (see: Quantale) non-compactness see: measure

of - -

non-deficient number [llAxx] (see: Abundant number) non-deficientnumbersee: o~- - non-degenerate Jordan pair [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) non-deterministic logspace complexity class [03D15, 68Q15] (see: Computational complexity classes) non-deterministic polynomial time complexity class [03D15, 68Q15] (see: Computational complexity classes) non-deterministic Turning machine [03D15, 68Q15] (see: Computational complexity classes)

non-split short exact sequence [16G70] (see: Almost-split sequence)

non-deterministic Turning machine see: time for a - n o n - e m p t y set see: dual of a - - ; polar of a-

norm of a primeideal in an algebraic number field [11R44, 11R45] (see: Dirichlet density) norm of an element in a Galois extension [12E20] (see: Galois field structure) norm of an integer [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory; Abstract prime number theory) norm on an arithmetical semi-group [llNxx, 11N32, 11N45, llNS0] (see: Abstract analytic number theory)

non-expansive senti-group

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) non-Haken manifold [57N10] (see: Haken manifold) non-Hermitian Satake compactification [11Fxx] (see: Satake compactilieation) non-Hopfiangroup [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) non-Hopfian group presented --

finitely-

see:

non-linear evolution equation [35Q53, 58F07] (see: Harry Dym equation) non-linear filtering problem [82B35, 82C35] (see: Onsager-Machlup function) non-linear potential theory [26B99, 30C62, 30C65] (see: Quasi-regular mapping) non-linear resolvent [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) non-linear Schr6dingerequation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) non-Markovian functional of Brownian motion [60Gxx, 60J55, 60165] (see: Wiener sausage) non-negative-definite kernel [46E22] (see: Reproducing kernel) non-negative weakly --

quadratic

form

see:

non-probabilistic uncertainty [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) non-projective module [16G70] (see: Almost-split sequence) non-reciprocal algebraic integer [1 IC08, 11R04] (see: Lehmer conjecture) nen-self-adjoint operator see: triangular model of a - -

non-separatingsurface [57Mxx, 57M25] (see: Skein module) non-singular tuple of operators [47Dxx] (see: Taylor joint spectrum) non-smooth analysis [90C30] (see: Clarke generalized derivative)

non-split short exact sequence s e e : mal --

mini-

non-tangential approach region [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath6odory theorem) non-tangential limit [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath6odory theorem) non-type-I C* -algebra [46Lxx] (see: Toeplitz C*-algebra) n o r m see: Frobenius - - ; Frobenius matrix - - ; G e v r e y - - ; matrix - - ; operator - - ; eubmuJtiplicative inequality for a matrix - - ; / : - - - ; triangle inequality for a vector - - ; vector - -

n o r m problem see: u n i q u e n e s s - o f - n o r m theorem see: Johnson uniquenessof- - - ; uniqueness-of- - normal see: proximal - -

normal algebraic variety [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) normal analytic space [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) normal basis [11R32, 12E20] (see: Galois field structure; Normal basis theorem) normal basis generator [11R32] (see: Normal basis theorem) Normal basis theorem (11R32) (referred to in: Galois field structure) (refers to: Galois extension; Galois field; Galois group) normal basis theorem see:

primitive - -

normal closure of a group [20F05, 20F06, 20F32] (see: HNN-extension) normal distribution see: bivariate - - ; log- ; uneorrelated random variables with joint - -

normal element [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C*-algebras) normal element in a field extension [12E20] (see: Galois feld structure) normal element in a field extension see: completely - normal element in a Galois extension see: completely - -

normal equations for least-squares estimation [62Jxx] (see: ANOVA) normal form see: dan - -

conjunctive - - ; Jor-

normal f o r m theorem for HNNextensions [20F05, 20F06, 20F32] (see: HNN-extcnsion) normalformsin Baumslag-Solitargroups [05C25, 20Fxx, 20F32] 517

NORMAL FORMS IN BAUMSLAG-SOLITARGROUPS (see: Baumslag-Solitar group) normal operator s e e :

essentially - -

normal play era game

[90D05] (see: Sprague-Grundy function) normalization property of the Brouwer degree

[55M251 (see: Brouwer degree) normalized immanant

[15A15, 20C30] (see: Immanant) normalized joint characteristic function of an operator vessel

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) normalized matching property

[05D05, 06A07] (see: Sperner tileorem) normalized matching property of a partially ordered set

[05D05, 06A07] (see: Sperner property) normalized reproducing kernel

[46Cxx, 47B35] (see: Berezin transform) normalized restriction of a probability distribution

[68Q15] Average-case computational complexity) normalized Zolotarev polynomials [4I-XX, 41A50] (see: Zolotarev polynomials) (see:

normally solvable equation

[47A53] (see: Fredholm solvability)

normally solvable operator [47A53] (see: Fredholm solvability) norming constant

[35P25, 47A40, 81U20] (see: Inverse scattering, half-axis

case) Norton Monstrous Moonshine conjecture scc: Conway- -notation for rational tangles s e e : Conway --

Novikov conjecture [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) nowhere-dense generalized function algebra

[46F30] (see: Rosinger nowhere-dense generalized function algebra) nowhere-dense generalized function algebra s e e : R o s i n g e r - -

nowhere dense ideal

[46F30] (see: Generalized function algebras) NP see: complexity class -J~'P-complete

[90D05] (see: Sprague-Grnndy function)

.MT>-complete problem [68Q15] (see: Average-case computational complexity) AfT%completeness [68S051 (see: Natural language processing) JV'79-hard [05C65, 05D05, 57M25, 68Q15, 68T05, 68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory; Jones-Conway polynomial; VapnikChervonenkis dimension) NSPACE see: complexity class -NTIME s e e : complexity class --

nucleation

[82B26, 82D35] (see: Cahn-Hilliard equation) null-additive set function

[28-XX] 518

(see: Non-additive measure) null-additive set function [28-XX] (see: Non-additlve measure)

numerically cycles

null sequence s e e :

N U T metric s e e :

convex--

equivalent

relative

R-I-

[14Exx, 14E30, 14Jxx] (see: Marl theory of extremal rays) Euclidean T a u b - - -

null space era matrix

[15A39, 90C05] (see: Motzkin transposition theorem) Nullstellensatz s e e : complexity bounds for the Hilbert - - ; degree bounds for the Hilbert - - ; Effective - - ; generalized effective Hilbert - - ; height bounds for the Hilbert - - ; Hilbert - number s e e : (x-non-deficient - - ; Abundant - - ; angular momentum quantum -- ; Average sample - - ; crossing - - ; cubefree superabundant - - ; deficient - - ; diminutos - - ; first Chern - - ; Gordian - - ; highly abundant - - ; Knudsen - - ; Mach - - ; Mahler measure of an algebraic - - ; non-deficient - - ; perfect - - ; Prandlt - - ; primitive el-abundant - - ; primitive abundant - - ; primitive unitary el-abundant -- ; regular cardinal - - ; Reynolds - - ; smooth - - ; s u p e r a b u n d a n t - - ; superfluous - - ; Taft - - ; T h u e - M o r s e - - ; Tribonacci - - ; unknotting--; weird--; winding - - ; writhe - number conjecture s e e : Zarankiewicz crossing -number field s e e : absolute value on a - - ; algebraic - - ; Archimedean place of a - - ; discriminant of an algebraic - - ; L function of an algebraic - - ; metric on a - - ; non-Arehimedean place of a - - ; norm of a prime ideal in an algebraic - - ; signature of an algebraic - - ; units of an algebraic - - ; Z v - e x t e n s i o n of a -number fields s e e : prime ideal of degree one in an extension of algebraic - - ; splitting prime ideat of an extension of algebraic -number of a graph s e e : crossing - - ; rotational - number of a hypergraph s e e : independence - number of a link s e e : wrapping -number of a module s e e : Bass - - ; dual Bass - number of a partially ordered set s e e : Dilworth - - ; S p e m e r -number of a random graph s e e : chromatic - number of a t r e e s e e : Wiener - -

number of different words in Ulysses

[60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) number of generators of a module s e e : local - number of labelled trees s e e : Cayley formula for the -number set s e e : place of the rational - number theorem s e e : abstract inverse prime - - ; abstract prime - - ; inverse additive abstract prime - - ; prime - -

number-theoretic MSbius inversion formula

[05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) number theory s e e : Abstract analytic -- ; Abstract prime -numbers s e e : Additive basis for the natural -- ; Ap~ry -- ; Catalan -- ; concordant pairs of real - - ; discordant pairs of real - - ; Erd6s theorem on abundant -- ; factor large -- ; linear independence of logarithms of algebraic - - ; Lucas -numbers of a coherent configuration s e e : intersection -numbers of order k s e e : Fibonacci - -

numerical approaches to the SturmLiouville spectral problem [34B24, 34L40] (see: Sturm-Liouville theory) numerical series s e e : gence of a - -

radius of conver-

O w~ -metric

[54Gi0] (see: P-space) w ~-metrizable space

[54G10] (see: P-space) w-polynomials see:

Zolotarev

--

w-stable theory [03C15, 03C45, 03E15] (see: Vaught conjecture) see: soft -object s e e : dualizing - - ; tilting - 0

obliquely reflecting Brownian motion

[60Hxx, 60J55, 60J65] (see: Skorokhod equation) Obstacle scattering (35P25) (referred to in: Inverse scattering, multi-dimensional case) (refers to: Analytic function; Continuous function; Dirichlet boundary conditions; Fourier transform; Hausdorff measure; Limitabsorption principle; Logarithmic branch point; Meromorphic function; Neumann boundary conditions; Scattering matrix; Sobolev space; Unitary operator) obstaclescatteringsee: Dirichletcondition in - - ; inverse - - ; Neumann condition in - - ; Robin condition in --

obstacle scattering problem

I35P25] (see: Obstacle scattering) Occurrence s e e : frequencies of - Ocneanutrace see: Jones- --

octal game [90D05] (see: Sprague-Grundy function) octonian algebra [17Cxx, 46-XX] (see: JB * -triple) actonion algebra [17A35, 17D25, 83C20] (see: Okubo algebra) Odlyzko bounds (11R29) (refers to: Algebraic number; Class field theory; Dedekind zeta-function; Discriminant; Geometry of numbers; Number field; Riemann hypotheses; Tower of fields; Zetafunction) see: Oka theorem

O K point

e- --

[32E20] (see: Polynomial convexity) Okubo algebra (17A35, 17D25, 83C20) (refers to: Algebra with associative powers; Division algebra; Lieadmissible algebra; Lie algebra; Trace of a square matrix; Vector space) one s e e : group of cohomological dimension - one-circleproblem s e e : Littlewood - one end s e e : Stallings classification of finitely generated groups with more than - one in an extension of algebraic number fields s e e : prime ideal of degree - one matrix see: support of a zero- - one problem reduction s e e : many- - - ; polynomial-time many-

-

one representation s e e : class- - Onsagerfunction s e e : M a c h l u p - --

Onsager-Machlup function (82B35, 82C35) (refers to: Diffusion process; Lagrange function; Lagrangian; Laplace-Beltrami equation; Maximum-likelihood method; Random field; Riemannian manifold; Vector field) ontology [68S05] (see: Natural language processing) ontology server [68S05] (see: Natural language processing) open interval topology s e e : right half- - open set s c c : semi- - open square topology s e e : Sorgenfrey half- - open topology s e e : compact- - operation s e e : Bockstein - - ; partially associative--; squaring--; Steenrod squaring - operation for circuits s e e : delay - -

operational formulationof the tau method [65Lxx] (see: Tau method) operator s e e : Atiyah-Singer - - ; Baxter - - ; Bergman - - ; Bernstein - - ; B o c h n e r Riesz - - ; Carleman-type integral - - ; C a u c h y - R i e m a n n O- - - ; chiral Dirac - - ; commutant of an - - ; composition - - ; conservative linear - - ; coupled Dirac - - ; de Rham - - ; density - - ; determinant of the Dirac - - ; determined - - ; Dirac - - ; Dirichlet Sturm-Liouville - - ; energymomentum - - ; essential spectrum of an ; essentially normal -- ; essentially selfadjoint - - ; finite W i e n e r - H o p f - - ; Fourier multiplier - - ; generalized Dirac - - ; Hankel--; Hecke--; Hilbert-Hankel--; hypo-elliptic pseudo-differential - - ; hypoelliptic symbol of a pseudo-differential - - ; image - - ; index of an - - ; index of an elliptic partial differential - - ; index theory for a single - - ; injective linear - - ; isolated part of the spectrum of a linear - - ; isolated point in the spectrum of an - - ; Laplace-Beltrami - - ; least-fixed-point - - ; local tomography --; marginalization--; mass--; material derivative - - ; M o n g e - A m p ~ r e - - ; Neumann eigenvalues of the Laplace - - ; normally solvable - - ; p-convolution - - ; pole of an - - ; positive linear - - ; powerset--; pre-image--; principal symbol of a differential - - ; projection - - ; pseudo-differential - - ; q-difference analogue of the E u l e r - P o i s s o n - D a r b o u x - - ; quasi-nilpotent - - ; random SchrSdinger - - ; resoIvent - - ; Riesz - - ; SchrOdinger - - ; s e l f - a d j o i n t - - ; Semi-Fredholm - - ; signature - - ; spectrum of an - - ; Spencer cohomology of a differential - - ; spin Dirac - - ; Sturm-Liouville - - ; substitution - - ; symbol of a Hankel - - ; symbol of a pseudo-differential - - ; Toeplitz - - ; traceclass -- ; transfer -- ; triangular model of a non-self-adjoint - - ; truncated W i e n e r Hopf - - ; twisted Dirac -- ; twisted vertex - - ; vacuous extension - - ; V e r t e x - - ; wave - - ; W i e n e r - H o p f - - ; zeta-function of a pseudo-differential - - ; zeta-function of an - operatoralgebra s e e : V e r t e x - -

-

Operator colligation (30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70) (referred to in: Operator vessel) (refers to: Contraction; Hilbert space; Krein space; Linear operator; Nonself-adjoint operator; Pontryagin space; Rigged Hilbert space; Schur functions in complex function theory; Self-adjoint operator; Spectrum

P-ADIC ABSOLUTEVALUE

of an operator; Transfer function; Unitary operator) operator colligation

[30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) operator colligation s e e : characteristic operator-valued function of an - - ; coisometric - - ; isometric - - ; main operator of a unitary - - ; rigged - - ; unitary - -

operator determinant [42A16, 47B35] (see: Szeg6 limit theorems) operator determinant s e e : trace-class

Fredholm - - ;

- -

operator invariants [46L80, 46L87, 55N15, 58G10, 58GI 1, 58G12] (see: Index theory) operator node

[30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator eolligation) operator norm

[15A42] (see: Bauer-Fike theorem) operator norm [15A60] (see: Frobenius matrix norm) operatorof a unitary operatorcolligation s e e : main - operatoron graphs s e e : Laplace - operatortopologysee: ultraweak - operator-valued function of an operator colligation s e e : characteristic--

operator-valued holomorphic mapping [30C45, 47H10, 47H20] (see: Jufia-Wolff-Carathtodory theorem) Operator vessel (47A45, 47A48, 47A65, 47D40, 47N70) (refers to: Algebraic geometry; Hilbert space; Irreducible polynomial; Isometric mapping; Jacobi variety; Lie algebra; Lie group; Non-self-adjoint operator; Operator colligation; Overdetermined system; Plane real algebraic curve; Resolution of singularities; Riemann surface; Self-adjoint operator; Sheaf; Spectral analysis; Taylor joint spectrum; Transfer function) operator vessel s e e : commutative - - ; discriminant c u r v e of an - - ; discriminant polynomial of an - - ; external space of an - - ; input determinantal representation of the discriminant curve of an - - ; internal space of an - - ; joint characteristic function of an - - ; Lie algebra - - ; normalized joint characteristic function of an - - ; output determinantal representation of the discriminant curve of an - - ; quasiHermitian commutative two- - operator vessels s e e : matching theorem for - - ; unitarily equivalent - - ; unitary equivalence of - operators s e e : Boolean algebra with - - ; CJ* -algebra of compact -- ; colligation of - - ; essentially commuting - - ; Fredholm tuple of - - ; functional model of a pair of --; index theory for a families of - - ; joint spectrum of - - ; multiplier - - ; node of - - ; non-singular tuple of - - ; Riesz decomposition theorem for --; Riesz theory of compact - - ; semi-group of composition - - ; spectral theory of - - ; Spectral theory of compact -- ; Taylor spectrum for compact - - ; triangular model of a pair of - - ; Zerlegungssatz for -operators of Carlemen type s e e : integral - operators with H °° symbol s e e : Toeplitz - OpiaI-Siciak-type mean-value theorem s e e : Temlyakov- - -

optical theorem

[35P25] (see: Obstacle scattering) optics see: aberration function of - optimal design s e e :

D-

--

optimal domain decomposition [46E35, 65N30] (see: Bramble-Hilbert lemma) optimal hat function [62D05] (see: Acceptance-rejection method) optimal stopping problem [90A09] (see: Option pricing) optimality s e e : saddle-point characterization of - - ; Wald-Wolfowitz - optimality condition s e e : primal - -

optimality in the Fritz John equation [90Cxx] (see: Fritz John condition) optimality on a region of stabiJity s e e : characterization of local -optimality property for the sequential probability ratio test s e e : strong - optimization s e e : dynamic portfolio - - ; expected utility in portfolio - - ; martingale approach to portfolio - - ; Portfolio - - ; static portfolio --

option

[90A09] (see: Option pricing) option s e e : American - - ; European - - ; European call - - ; expiration time of a European call - - ; r e p l i c a t e d - - ; strike price of a European call - - ; underlying asset of a European call -option at expiration s e e : value of a European call - -

Option pricing (90A09) (referred to in: Black-Scholes for-

mula) (refers to: Black-Scholes formula;

Brownian motion; Differential equation, partial, free boundaries; Diffusion equation; Martingale; Portfolio optimization; Probability measure; Random variable; Stochastic differential equation; Stopping time) option pricing Merton - -

see:

Black-Scholes-

option replication

[90A09] (see: Option pricing) option right

[90A09] (see: Option pricing) oracle

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of) oracle see: quantum - - ; querying an - oracle problem

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) orbifold [11Fll, 14H15, 17BI0, 17B65, 17B67, 17B68, 20D08, 30F60, 81R10, 81T30, 81T40] (see: Vertex operator algebra; WeilPetersson metric) orbit

[58F22, 58F25] (see: Seifert conjecture) orbit s e e : circular --; closed --; co-adjoiet - - ; heteroclinic - - ; homodinic - - ; periodic - - ; stationary -orbit condition s e e : trapped- - orbit of a dynamical system s e e : period of an--

orbits of a dynamical system s e e : curve of periodic - order s e e : Bruhat - - ; symmetric chain - -

order complex of a partially ordered set

[05B35, 05Exx, 05E25, 06A07, 11A25] (see: M/ibins inversion) order k s e e : Fibonacci numbers of - - ; Fibonacci polynomials of - - ; Fibonaccitype polynomials of - - ; Lucas-type polynomials of -- ; multivariate Fibonacci polynomials of - -

orthomodular lattices s e e : Foulis semigroup of complete - - ; variety of - -

o1~homodularlogic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) orthonormal Szeg6 polynomial

[33C45] (see: Szeg6 polynomial)

order of a net

[05Bxx] (see: Net (in finite geometry)) order of a symmetric group s e e :

weak - -

order of an additive basis

[llPxx] (see: Additive basis) order on a space of real-valued functions see: natural - order type s e e : k- - ; simple k - - ordered basis of a field s e e : dual basis of an--

ordered exponential s e e : path- - ordered set s e e : anti-chain in a partially - - ; characteristic polynomial of a ranked partially - - ; covering relation in a partially - - ; Dilworth number of a partially - - ; kfamily in a partially - - ; level of a rank function on a partially - - ; locally finite partially - - ; LYM partially - - ; normalized matching property of a partially - - ; order complex of a partially - - ; Peck partially - - ; rank function on a partially - - ; rank of a partially - - ; Sperner number of a partially - - ; width of a partially - ordered vector space s e e : interval in a partially - - ; partially - ordering theorem s e e : well- - ordinal s e e : von Neumann - ordinary differential equations s e e : property C + for - - ; property C~, for - - ; property C ¢ for - -

Ordinary differential equations, property C for (34A55, 34L25) (referred to in: Inverse scattering, fullline case)

oscillation s e e : bounded mean - - ; function of bounded mean - - ; space of analytic functions of bounded mean - - ; space of analytic functions of vanishing mean - oscillation theorem s e e : equi- - osculation s e e : point of - -

osculation point

[14H20] (see: Taenode) Ostwald ripening

[82B26, 82D35] (see: Cahn-Hilliard equation) OTTER [06Exx, 68T15] (see: Robbins equation) Otter theorem [llNxx, 11N32, 11N45] (see: Abstract prime number theory) out-tree

[05C50] (see: Matrix tree theorem) outer dilatation

[26B99, 30C62, 30C65] (see: Quasi-regular mapping)

outer function [30D55, 33C45, 46M5, 47A15] (see: Beurling theorem; Szeg6 polynomial) outer measure [28-XX] (see: Non-additive measure) output determinantal representation of the discriminant curve of an operator vessel

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel)

oriented matroid

[52A35] (see: Geometric transversal theory) orthocentre of a triangle

[51M04] (see: Triangle centre) erthochronousproper Poincar6 group [81Txx, 81T05] (see: Massive field; Massless field) orthocomplement [03G25, 06D99] (see: Quantale) orthocomplement

see:

output system s e e : multiple-input multiple- - ; single-input single- -oval s e e : Brauer Cassini - - ; Cassini - -

over-fitting

[68T051 (see: Machine learning) overrelaxation method s e e :

successive--

Oxtoby theorem

[54E52] (see: Banach-Mazur game)

pseudo- - -

orthocomplemented sup-lattice [03G25, 06D99] (see: Quantale)

P

orthogonal collocation

[65Lxx] (see: Tau method) orthogonai Laurent polynomials [44A60] (see: Strong Stieltjes moment problem) orthogonal Laurent polynomials s e e : associated - orthogonaIon a circle s e e : polynomials - orthogonalpolynomialssee: GegenbauerSobolev - - ; Laguerre-Sobolev - - ; paraorthogonal projection s e e : Szeg6 - -

Cauchy-

orthogonality relation [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) orthogonality relations for Zernike polynomials

[33C50, 78A05] (see: Zernike polynomials) orthomodular lattice [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

rr see:

quadraticalgorithm f o r - -

',#DO

[81T501 (see: Non-commutative anomaly)

P [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) see: a v e r a g e - - - ; class - - ; complexity class average- -P see: complexity class --

[68Q05, 68Q10, 68Q15, 6gQ25, 81Pxx, 81P15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) [14A10, 14Q20, 68Q15] Average-case computational complexity; Effective Nullstellensatz) (see:

p-adic absolute value

519

P-ADIC ABSOLUTEVALUE

[12J10, 12J20, 13A18, 16W60] (see: 5'-integer) p-adic L-function [11R23] (see: Iwasawa theory) p-adic Weierstrasspreparation theorem [11R23] (see: Iwasawa theory) p-convolution operator [43A07, 43A15, 43A45, 43A46, 46J10] (see: Fig&-Talamanca algebra) p-extension s e e : maximal Abelian - - ; unramified Abelian -p-group see: elementary Abelian - - ; pro- - ; regular - -

t~ Hall identity [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) PZ-irreducible three-dimensional manifold [57N10] (see: Haken manifold) P-polnt (54G10) (referred to in: P-space; Weak Ppoint) (refers to: Completely-regular space; Continuous function; Continuum hypothesis; Homogeneous space; Maximal ideal; Prime ideal; StoneCech compactification; Ultrafilter) P-point see: W e a k - P-point ultrafilter on N [54G10] (see: P-point) P-position [90D051 (see: Sprague-Grundy function) P-space (54G10) (refers to: Cardinal number; Completely-regular space; G6del constructive set; Maximal ideal; Metric; Metric space; Metrizable space; Normal space; P-point; Paracompact space; Prime ideal; Set of type F~ (G~); Topological product; Topological space; Zero-dimenslonal space) P-space see: Morita - -

Gillman-Henriksen - - ;

PAC field [11R32] (see: Shafarevich conjecture) PAC learning [68T05] (see: Machine learning) PAC learning [05C65, 05D05, 68Q15, 68T05] (see: Vapnik-Chervoncnkis dimension) packages s e e :

Dynamical systems soft-

ware --

Pairdev6 property [35Q53, 58F07] (see: Harry Dym equation) pair s e e : BN--; Banach-Jordan --; Jordan - - ; Lax - - ; local algebra of a Banach-Jordan -- ; non-degenerate Jordan - - ; socle of a Jordan - - ; splitting torsion -- ; torsion - - ; van der Corput exponent -pair for an isolated invariant set s e e : index -pair in exponential sum estimation s e e : exponent - (pair of blocks) s e e : trade -pair of finite capacity see: Jordan - pair of inverse relations s e e : Chebyshevtype -pair of measures s e e : coherent - - ; kcoherent -pair of operators s e e : functional model of a - - ; triangular model of a - -

pair of pants [14H15, 30F60]

520

(see: Weil-Petersson metric) pairing s e e : intersection - - ; Petersson - pairs s e e : axiom of - - ; dual reduclive - - ; finiteness conditions in B a n a e h Jordan -pairs in exponentiN s u m estimation s e e : exponent - pairs of finite capacity s e e : Lads classification of simple Jordan - pairs of real numbers s e e : concordant - - ; discordant - Paley system s e e : Walsh- --

Paley-Walsh system [33C45] (see: Walsh-Paley system) Palmer theorem s e e : V i d a v - -pants s e e : pair of - -

pants complex [14H15, 30F60] (see: Weil-Petersson metric) para-orthogonal polynomials [65D32] (see: Szeg6 quadrature) parabolic subgroup [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) paradox s e e : Braess - - ; Condorcet - - ; D o w n s - T h o m s o n - - ; Hilbert - - ; infinite hotel -parallelogram see: Varignon - parameter s e e : efficient - - ; population - - ; regression - - ; Schur - - ; Szeg5 - - ; uniformizing - -

parameter ideal [13Hxx] (see: System of parameters of a module over a local ring) parameter ideal [13A30, 13HI0, 13H30] (see: Buchsbaum ring) parameter of the Blomqvist coefficient s e e : population -parameter systems s e e : monomial property for - parameters s e e : regular system of - - ; Stokes - - ; system of -parameters of a m o d u l e over a local ring s e e : System of - parameters of moduli s e e : deformation - -

parametric function in statistics [62Jxx] (see: ANOVA) parametric function in statistics s e e : estimable -parametric representation s e e : BerksonPorta - -

parametric representations of generators of continuous semi-groups [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphie mappings)

paramodulatiou [06Exx, 68T15] (see: Robbins equation) paramultiplication [46F10] (see: Multiplication of distributions) paraproduct [46F10] (see: Multiplication of distributions) parenthesationssee: counting-Parisi-Zhang equation s e e : Kardar---

parse tree banks [68S051 (see: Natural language processing) parser s e e : TAG - - ; tree adjoining grammar - parsing s e e : natural language -part function s e e : fractional - - ; integral -part of the spectrum of a linear operator s e e : isolated -Partertheorems s e e : Avram---

partial associativity [46F10] (see: Multiplication of distributions) partial Coaway algebra

[57P25] (see: Conway algebra)

p a r t i a l C o n w a y algebra s e e : universal - partial differential equation s e e : quantum spectral measure of a - - ; singular - partial differential equations s e e : elliptic - - ; formal Dirac quantization of - - ; property C for - - ; property (70 for - -

Partial differential equations, property (7 for (35P25) (refers to: Algebraic variety; Differential operator; Harmonic function; Inverse scattering, multidimensional case; Schr6dinger equation; Sobolev space; Transversality) partial differential operator s e e : index of an elliptic - partial Fourier s u m s e e : multidimensional - partial Fourier sums s e e : hyperbolic - - ; Lebesgue constants of multi-dimensional --; spherical - -

partial-fraction expansion [39A12, 93Cxx, 94A12] (see: Z-transform) partial-fractions technique for finding the inverse Z-transform [39A12, 93Cxx, 94AI2] (see: Z-transform) partial hierarchy for a three-dimensional manifold [57N10] (see: Hakcn manifold) partial likelihood [62Jxx, 62Mxx] (see: Cox regression model) partial reahzation problem of system theory [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) partial sum see: step hyperbolic - partially A-anti-symmetric set [46E25, 54C35] (see: Bishop theorem) partially associative operation [46F10] (see: Multiplication of distributions) partiallyordered set s e e : anti-chain in a - - ; characteristic polynomial of a ranked - - ; covering relation in a - - ; Dilworth n u m b e r of a - - ; k-family in a - - ; level of a rank function on a - - ; locally finite - - ; LYM - - ; normalized matching property of a - - ; order complex of a - - ; Peck - - ; rank function on a - - ; rank of a -- ; S p e r n e r n u m b e r of a - - ; width of a - -

partially ordered vector space [06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) partially ordered vector space see:

interval

i n a --

partially specified statistical model [62Jxx, 62Mxx] (see: Cox regression model) particle s e e : acceleration of a - - ; derivative following a - - ; massless -- ; relativistic -- ; spin-carrying - - ; spinless - - ; spinless and massless - - ; velocity of a --

particle derivative [73Bxx, 76Axx] (see: Material derivative method) particle physics s e e :

su ( 3 ) - -

partition function of a state model of a physical system [57M25] (see: Listing polynomials) partition lattice [05D05, 06A071 (see: Sperner property) partition theory s e e : arithmetical - partitioning s e e : rent - -

partly convex programming [90Cxx] (see: Fritz John condition) partly crossed factors in covariance analysis

[62Jxx] (see: ANOVA) Pasch configuration (05B07, 05B30) (refers to: Steiner system) Pasch configuration [05B07, 05B30] (see: Pasch configuration) Paseh Steiner triple system s e e : Pasch STS s e e : anti- - -

anti- - -

Pasch switch [05B07, 05B30] (see: Pasch configuration) passive electrical network [05C50] (see: Matrix tree theorem) Pasta-Ulam problem s e e : path s e e : mean free - -

F e r m i - --

path algebra [16Gxx] (see: Tits quadratic form) path-algebra of a quiver [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) path algorithm s e e : Dijkstra shortest- - path from logic to algebra s e e : logistic - - ; rule-based - - ; semantical - path method s e e : Feynman - -

path-ordered exponential [14Jxx, 35A25, 35Q53, 57R57[ (see: Whitham equations) path problem s e e : longest - - ; shortest- ; travelling salesman - Patodi-Singerindextheorem see: Atiyahpattern formation s e e :

spatio-temporal - -

Pauli algebra (15A66, 81R05, 81R25) (refers to: Clifford algebra; Pauli matrices; Quaternion) Pauli algebra [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) Pauli spin matrices [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) Pauli-Villars regularization [81T501 (see: Non-commutative anomaly) pc s e e :

.~- --

p,c.f, self-similarset [28A80] (see: Sierpifiski gasket) PCT theorem [81Txx] (see: Massive field) PDECONT software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Peano kernel theorem [46E35, 65N30] (see: Bramble-Hilbert lemma) Pearson probability Neyman- --

ratio

test

see:

Pearson product-moment correlation coefficient (62H20) (referred to in: Kendall tau metric; Spearman rho metric) (refers to: Correlation; Correlation coefficient; Covariance; Independence; Normal distribution; Random variable; Regression; Regression analysis) Peck partially ordered set [05D05, 06A07] (see: Sperner property) Peck partially ordered set [05D05, 06A07] (see: Sperner theorem) Peckposet [05D05, 06A07] (see: Sperner property) Peck poset [05D05, 06A07] (see: Sperner theorem)

POINT ON AN ALGEBRAIC CURVE Peierls bracket [81Qxxl (see: Dirac quantlzatlon) pencil [12F10, 14H30, 20I)06, 20E22] (see: Chasles-Cayley-Brill formula) perceptrons [68T05] (see: Machine learning) perfect matching [05Cxx, 05D15] (see: Matching polynomial of a graph) perfect number [11Axx] (see: Abundant number) perfect splines see: Z o l o t a r e v - performance ratio [90C27] (see: Knapsack problem) performance system in a learning system [68T05] (see: Machine learning) period-doubling [34-04, 35-04, 58-04, 58F14[ (see: Dynamical systems software packages) period of a periodic sequence [11B37] (see: Ultimately periodic sequence) period of a trajectory of a dynamical system [34-04, 35-04, 58-04, 58F14[ (see: Dynamical systems software packages) period of an orbit of a dynamical system [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) period of an ultimately periodic sequence [11B37] (see: Ultimately periodic sequence) period of an ultimately periodic sequence see: least -period sequence s e e : maximal - -

periodic attractor [28Dxx, 54H20, 58F11, 58F13] (see: Absolutelycontinuous invariant measure) periodic group [20F05, 20F06, 20F32, 20F50] (see: Burnside group) periodic orbit [34-04, 35-04, 58-04, 58F14, 58F22, 58F25] (see: Dynamical systems software packages; Seifert conjecture) periodic orbit [58Fxx] (see: Conley index) periodic orbits of a dynamical system see: curve of --

periodic sequence [11B37]

(see: Ultimately periodic sequence) periodic sequence s e e : least period of an ultimately - - ; period of a - - ; period of an ultimately - - ; Ultimately - -

periodicity of the Zak transform [42Axx, 44-XX, 44A55[ (see: Zak transform) permanence for the Taylor spectrum s e e : spectral --

permanent-on-top conjecture [15A15, 20C30[ (see: Immanant) permanental analogue of the Schur inequality [15A15, 20C30] (see: Immanant) permanental dominance conjecture [15A15, 20C30] (see: Immanant) permanents s e e : permutation s e e :

Marcus inequalityfor -geometric - - ; r e g u l a r - -

permutation group [2o-xx]

(see: Regular group) permutation group s e e : degree of a - - ; regular - - ; transitive regular - permutation representationsee: regular-permutations s e e : group of - - ; regular group of -Perron integrabilitysee: Denjoy--Perron integral s e e : D e n j o y - - -

Perron set [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) perturbation of a Banach algebra s e e : - - ; z- - - ; z-metric --

e-

perturbation of matrices [15A42] (see: Bauer-Fike theorem) Ramanujan- -Petersson conjecture at infinity s e e : Ramanujan- -Petersson identity s e e : R a m a n u j a n - -Petersson K~.hler form s e e : W e l l - - Petersson metric s e e : W e i l - - Petersson pairing Peterssonoonjeeturesee:

[14H15, 30F60] (see: Weil-Petersson metric) Petrov-Galerkin method [47H17] (see: Approximation solvability) PH [03D15, 68Q15] (see: Computational complexity classes) PH see: complexity class -phase averaging s e e :

multi- --

phase dynamics [35Q35, 58F13, 76Exx] (see: Kuramoto--Sivashinsky equation) phase shift [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) phase space [81Uxx] (see: Enss method) phase-space analysis [81Uxx] (see: Enss method) phase transition [68Q15] (see: Average-case computational complexity) phasetransition s e e :

SAT - -

phase transitionin a binary alloy [82B26, 82D35] (see: Cahn-Hilliard equation) phase turbulence [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Philippon example s e e : Mora/Masser- --; Masser- -Philippon/Lazard-Mora example Masser- --

Lazardsee:

phoneme [68S051 (see: Natural language processing) phonetics [68S05] (see: Natural language processing) Phong Whitham formulation see: Krichever- - -

phonology [68S05] (see: Natural language processing) phrase structure grammar s e e : headdriven -physical system s e e : partition function of a state model of a -physics s e e : su ( 3 ) particle - -

Picard iteration [30C20, 30C30] (see: Theodorsen integral equation) Pick lemma see: S c h w a r z - -Pick pseudo-metric see:

Pieriformula

Schwarz- --

[14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) Pillaitest s e e :

Bartlett-Nanda---

(p,1)-configuration [05B07, 05B30] (see: Pasch configuration) place of a field isomorphism s e e : fixed -place of a number field s e e : Arehimedean - - ; non-Archimedean - -

place of the rational number set [12J10, 12J20, 13A18, 16W60] (see: S-integer) planar curve see: Fleenode on a - planar genus of a knot [57M25] (see: Positive link) planar vortex sheet [76C051 (see: Birkhoff-Rott equation) plane s e e : affine - - ; approximate t';,~tangent - - ; classical affine - - ; finite affine - - ; translation - - ; upper half- - -

plane algebraic curve [14Hxx] (see: Acnode) plane-covering domain [35J05, 35J25] (see: Dirichlet eigenvaine; Neumann eigenvalue) plane domains see: nodal line conjecture for - -

plane wave [41A30, 92C55] (see: Ridge function) plane wave s e e : scattering by a - plane wave in scattering s e e : incident -plasma s e e : dissipative trapped ion mode ina -plate s e e : clamped - - ; eigenvalue problem for the clamped - - ; Rayleigh conjecture for the clamped - plate spline s e e : thin- - -

plausibilityfunction [68T30, 68T99, 92Jxx, 92KI0] (see: Dempster-Shafer theory) play [03E50, 54-XX, 90D80] (see: Sierpifiski game) play in the Banach-Mazur game [54E52] (see: Banaeh-Mazur game) play of a game s e e :

normal - -

PLTMG software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) plug see: Wilson - -

p~uralityvoting s e e :

straight - -

pluricomplex Green function [31C10, 32F05] (see: Pluripotenfial theory) pluriharmonic function [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) pluriharmonic functions s e e : characterization of - -

mean-value

pluripolar set [31C10, 32F05] (see: Pluripotential theory) Pinripotential theory (31C10, 32F05) (refers to: Analytic function; Analytic manifold; Bergman spaces; Capacity; Capacity potential; Dirac distribution; Dirichlet problem; Fundamental solution; Green function; Hermitian matrix; Hyperbolic metric; Laplace operator; Logarithmic capacity; Monge-Amp~re equation; Pinrisubharmonic function; Potential theory; Pseudo-convex and pseudo-concave; Riesz decomposition theorem; Riesz theorem; Subharmonic function) plurisubharmonic function

[31C10, 32F05] (see: Pluripotential theory) PN-sequence [11B37, 11T71, 93C05] (see: Shift register sequence) Pochhammer symbol [33C50, 78A05] (see: Zernike polynomials) Poincar#-Cartan invariant form [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Poincard-Eisenstein series [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) Poinear@ group s e e : proper - -

orthochronous

Poincar6 hyperbolic metric [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Woiff theorem) Poincar6 lemma [53C15, 55N35] (see: Spencer cohomology) Poincar@lemma s e e :

~- --

Poincar# map [58Fxx] (see: Poinear~ mapping) Poincar6 mapping (58Fxx) (refers to: Poincar6 return map; Poincar~ return theorem) Poincar6 metric [30C45, 47H10, 47H20] (see: Julia-Wolff-Carath~odory theorem) Poincar@ metric s e e : mapping, monotone with respect to the - -

Poincar6 polynomial [05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bins inversion) Poincar6 series [1iFxx, 20Gxx, 22FA6J (see: Baily-Borel compactification) Poincar@theorem s e e : E u l e r - - point s e e : attractive fixed - - ; Broeard - - ; c-OK - - ; collocation - - ; concentration of a function around a - - ; Denjoy-Welff - - ; double - - ; Fermat - - ; first Broeard - - ; fold - - ; Grebe - - ; Heegner - - ; hermit - - ; Hopf - - ; hyperbolic fixed - - ; inflection - - ; instantaneous velocity at a - - ; interpolation - - ; isogonal conjugate - - ; isolated - - ; isotomic conjugate - - ; Lebesgue - - ; L e m o i n e - - ; m e a s u r e of concentration of a function around a - - ; negative Broeard - - ; osculation - - ; P - - - ; positive Broeard - - ; Riesz - - ; second Brocard - - ; sink - - ; s y m m e d e a n - - ; Torrieelli - - ; W e a k 19. _ point bifurcation s e e : Bautin - - ; generalized Hopf - point characterization of optimality s e e : saddle- - point discrepancy s e e : lattice - -

point estimation [62Jxx] (see: ANOVA) point formula s e e : fixed- - point formulas s e e : Atiyah-Bott fixed- -point-freegroup action s e e : fixed- - point in the spectrum of an operator s e e : isolated - point iteration s e e : fixed- - point of a belief function s e e : focal --

point of a Gel'fand quantale [03G25, 06D99] (see: Quantale) point of a Steiner triple system [05B07, 05B30] (see: Pasch configuration) point of osculation [14H20] (see: Tacnode) point of the closed unit ball in a Banaeh space s e e : extreme - point on an algebraic curve s e e : first neighbourhood of a - - ; local ring of a - - ;

521

POINT ON AN ALGEBRAIC CURVE

second neighbourhood of a - - ;simpfe - - ; singular - point on an algebraic variety see: singular -point operator see: least-fixed- - -

point-set lattice-theoretic topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) point set of a group action

see:

fixed- --

point spectrum [47Dxx] (see: Taylor joint spectrum) point spectrum see: approximate -point theorem see: Brouwer fixed- - - ; Darbo fixed- - - ; K n a s t e r - K u r a t o w s k i Mazurkiewiez fixed- - - ; Schauder fixedpoint theorems see: fixed- -point theory see: critical - point ultrafilteron N see: P - points see: G a u s s - L o b a t t o - - ; method of selected - points in space see: potential at - points of a set see: function algebra separating the - -

pointwiseergodic theorem [28D05, 54H20] (see: Wiener-Wintrier theorem) pointwise Markov inequality [41-XX, 41A50] (see: Zolotarev polynomials) Poisson-Darboux equation see: Euler- - ; generalized Euler- -Poisson-Darboux operator see: qdifference analogue of the Euler- - Poisson formula see: classical - -

Poisson formula for harmonic functions (31A05, 31A10) (refers to: Dirac distribution; Diriehlet problem; Green function; Hardy spaces; Harmonic function; Weak convergence of probability measures) Peisson kernel [46F101 (see: Multiplication of distributions) Poisson kernel for a ball [31A05, 31A10] (see: Poisson formula for harmonic functions) Poisson manifold [37J15, 53D20, 70H33] (see: Momentum mapping) Poisson morphism [37Ji5, 53D20, 70H33] (see: Momentum mapping) Poisson resummationformula [81Qxx] (see: Zeta-function method for regularization) Poisson structure

see:

canonical - -

Poisson tensor [37J15, 53D20, 70H33] (see: Momentum mapping) polar of a non-empty set [15A39, 90C05] (see: Motzkin transposition theorem) polarization see: degree of - polarization ellipse [78A40] (see: Stokes parameters) polarization identity [46E22] (see: Reproducing-kernel Hilbert space) pole at infinity

see:

Green function with - -

pole of an operator [47A10, 47B06] (see: Spectral theory of compact operators) policy learning [68T05] (see: Machine learning) Polish space [03E50, 54-XX, 54Bxx, 60B10, 60G05, 90D80] 522

(see: Sierpifiski game; Skorokhod space; Wijsman convergence) Polish topological group [03C15, 03C45, 03E15] (see: *Caughtconjecture) Polish topology [03C15, 03C45, 03E15] (see: Vaught conjecture) Pollack theorem Pollard theorem

see: see:

Goodman--Beurling---

poly-ellipse [90B85] (see: Fermat-Torrieelli problem) Pdlya conjecturefor Dirichlet eigenvalues [35J05, 35J25] (see: Diriehlet eigenvalue) PSlya conjecture for Neumann eigenvalues [35J05] (see: Neumann eigenvalue) polyadic algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) polyadic algebra see:

quasi- --

Polyakov extrinsic action [53C42] (see: Willmore functional) polyhedral set [15A39, 90C05] (see: Motzkin transposition theorem) polynomial see: A l e x a n d e r - - ; A l e x a n d e r C o n w a y - - ; averaged Taylor - - ; Bernstein - - ; Bdzier - - ; black box representation of a multivariate - - ; black Listing - - ; Brandt-Lickorish-Millett-Ho--; Chebyshev - - ; chromatic - - ; coloured J o n e s Conway - - ; Conway - - ; dense representation of a multivariate - - ; dichromatic -- ; Dirichlet - - ; distinguished - - ; extremal - - ; F- - - ; first Listing - - ; Homily - - ; Homflypt - - ; Homotopy - - ; immanantel ; irreducible discriminant -- ; Jones - - ; J o n e s - C o n w a y -- ; Kauffman bracket - - ; Lagrange interpolation - - ; Laurent - - ; Mahler measure of a minimal - - ; matching - - ; orthonormal Szegd - - ; Poincare - - ; primitive - - ; rook - - ; Sehur - - ; second Listing - - ; skein - - ; skein module based on the Kauffman - - ; sparse representation of a multivariate - - ; spherical harmonic -- ; stable - - ; Swinnerton-Dyer ; Szeg6 - - ; ultimately - - ; white Listing ; Zernike circle - - ; Zernike radial - - -

- -

- -

polynomial basis [12E20] (see: Galois field structure) polynomial basis

see:

weaklyself-dual --

Polynomial convexity (32E20) (refers to: Algebra of functions;vAnalytic function; Analytic set; Cech eohomology; CR-submanifold; Foliation; Hardy spaces; Hausdorff measure; Homology group; Lebesgue measure; Maximum principle; Oka theorems; Probability measure; Riesz theorem; Spectrum of an operator; Uniform algebra; Weak topology) polynomial curve [41A10, 41A15, 68U05] (see: Bernstein-B~zier form) polynomial growth see: group of - polynomial Hales-Jewett theorem [05D10] (see: Hales-Jewett theorem) polynomial hull a--

see:

analytic structure on

polynomial interpolation [41A05, 41A30, 4IA63] (see: Radial basis function) polynomial of a graph see: acyclic - - ; characteristic - - ; chromatic - - ; Matching simple matching - - - ;

polynomial of a linear feedback shift register see: characteristic - - ; feedback - - ; reciprocal - polynomial of a matrix see: characteristic - polynomial of a ranked partially ordered set see: characteristic - polynomial of a shift register sequence s e e : minimal -polynomial of an operator vessel see: diecriminant - polynomial of the second kind see: Szeg6 --

polynomial on average time complexity [68Q15] (see: Average-case computational complexity) polynomial over a finite field

see:

monic - -

polynomial representation of the Frobenius automorphism [12D05] (see: Faetorization of polynomials) polynomial space complexity class [03D15, 68QI5] (see: Computational complexity classes) polynomial time complexity [68Q15] (see: Average-ease computational complexity) polynomial time complexity class [03D15, 68Q15] (see: Computational complexity classes) polynomial time complexity class see: bounded probabi]istic - - ; non-deterministic - -

polynomial-time computability [68Q15] (see: Average-case computational complexity) polynomial-time computable language see: b o u n d e d - e r r o r - - ; bounded-error quantum - polynomial-time factorization method see: Kaltofen-Trager random - -

polynomial-time hierarchy [03D15, 68Q15] (see: Computational complexity classes) polynomial-time many-one problem re duction [68Q15] (see: Average-case computational complexity) polynomially convex hull [32E20] (see: Polynomial convexity) polynomially convex set [32E20] (see: Polynomial convexity) polynomials see: applications of zonal harmonic - - ; associated orthogonal Laurent--; Bernshtefn-Szeg6--; bivariate Fibonacci - - ; bivariate Lucas - - ; disc - - ; double Schubert - - ; factoring - - ; Factorization of - - ; Fibonacci - - ; F i b o n a c c i - t y p e - - ; Gauss theorem on products of monic irreducible - - ; G e g e n b a u e r - S o b o l e v orthogonal - - ; Jacobi - - ; Laguerre - - ; L a g u e r r e - S o b o l e v orthogonal--; Listing--; Lucas--; L u c a s - t y p e - - ; Macdonald - - ; normalized Zolotarev - - ; orthogonal Laurent ; orthogonality relations for Zernike - - ; para-orthogonal - - ; quantum Schubert - - ; recurrence relation for derivatives of Chebyshev - - ; Rodrigues formula for Zernike - - ; S c h u b e r t - - ; Solotareff - - ; universal S c h u b e r t - - ; Zernike - - ; Zoiotareff - - ; Zolotarev - - ; Zolotarev w - - ; zonal harmonic - polynomials and matrices see: Schur stability of -- ; stability theorem for - polynomials in the tau method see: canonical - - -

polynomials of order k s¢¢: Fibonacci - - ; Fibonacci-type - - ; Lucas-type - - ; multivariate Fibonaeci - -

polynomials orthogonal on a circle [33C45] (see: Szeg6 polynomial) polytope

see:

face lattice of a - -

Pompeiu function [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) Ponomarev reflection functor see: Bernshtei'n-G el'f a n d - - pool see: g e n e - P o p o v d e t e r m i n a n t see: F a d d e e v - - P o p o v g h o s t see: F a d d e e v - - Popov method s e e : Faddeev---

population [92D10] (see: Hardy-Weinberg law) population genetics [92DI0] (see: Hardy-Weinberg law) population parameter [62I-I20] (see: Spearman rho metric) population parameter of the Blomqvist coefficient [62H20] (see: Kendall tau metric) population version of the Kendall tau [62H20] (see: Kendall tau metric) Porta parametric representation see: Berkson- -portfolio see: mean-variance efficient - - ; self-financing - -

Portfolio optimization (90A09) (referred to in: Option pricing) (refers to: Average; Dispersion; Dynamic programming; Quadratic programming; Viscosity solutions) portfolio optimization see: dynamic - - ; expected utility in - - ; martingale approach to - - ; static - portfolio strategy sec: self-financing - -

portfolio value process [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) portfolio weight process [60Hxx, 90A09, 93Exx] (see: Black-Seholes formula) poset [05D05, 06A07] (see: Sperner property) poset see: LYM - - ; Peck -posets see: quotient theorem for LYM - position see: N - - - ; P - -

positive Brocard point [5IM04] (see: Brocard point) positive current [32C30, 53C65, 58A25] (see: Current) positive-definite matrix [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) positive-definite matrix ally - -

see:

condition-

positive diagram [57M25] (see: Jones-Conway polynomial) positive distribution [44A60, 47A57] (see: Moment matrix) positive-energy condition [81Txx] (see: Massive field) positive invariance [58Fxx] (see: Conley index) positive linear operator [44A60] (see: Strong Stieltjes moment problem) Positive link

PROBABLY APPROXIMATELYCORRECT LEARNING

(57M25) (referred to in: Miinor unknotting conjecture) (refers to: Knot theory; Link; Miinor unknotting conjecture; Surgery) positive link s e e :

rn,-almost - -

positive mapping [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, glPI5, 94Axx] (see: Quantum information processing, science of) positive matrix see: t o t a l l y - positive quadratic form s e e : weakly - positive semi-definite Hermitian matrices see: generalization of the H a d a m a r d Fischer inequality for - -

positive toms knot [57M25] (see: Positive link) positivetorus knot s e e :

(5,2)

--

maximum a --

POT conjecture [15A15, 20C30] (see: Immanent) Potapov factor s e e :

Blasehke- - -

potential [35P25, 47A40, 81U20[ (see: Inverse scattering, dimensional ease)

multi-

potential s e e : aIgebro-geometric AKNS--; K~hler--; pre- - - ; scattering - - ; Yang-Mills --

potential at points in space [31B05, 33C55] (see: Zonal harmonics) potential flow theory [76Cxx] (see: Von Kfirm~invortex shedding) potential in a sphere [31B05, 33C55] (see: Zonal harmonics) potential scattering see: direct - - ; inverse -potential scattering problem s e e : inverse - potential theory s e e : fine topology in - - ; non-linear -Potts m o d e ]

[57M25] (see: Kauffman bracket polynomial) power convergence [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) power function of a statistical test [62L10] (see: Sequential probability ratio test) power of a statistical test [62Jxx] (see: ANOVA) power set s e e : axiom of -power spectrum [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation)

pricingsee: Black-Scholes-Mertonoption - - ; Option - -

rational -(p,q) type s e e : theorem of -pragmatics of natural language [6gs05] (see: Natural language processing) pragmatics of natural language usage [68S05] (see: Natural language processing) PRAM [03DI5, 68Q15] (see: Computational complexity classes) Prandlt number [76Axx] (see: Knudsen number) p/q-tanglesee:

Prasad-Sommerfieldlimitsee:

poslat topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) possibilistic programming [90C70] (see: Fuzzy programming) possibility degree [90C70] (see: Fuzzy programming) possibility theory [90C70] (see: Fuzzy programming) post-critically finite self-similarset [28A801 (see: Sierpifiski gasket) post-projective component [16Gxx] (see: Tits quadratic form) pesteriori hypothesis s e e :

(see: Fuzzy topology) p/q-rational move [57M25] (see: Tangle move)

Pr#.staro quantization s e e :

BogomolnyCrumeyrolle-

pre-cover [16D401 (see: Flat cover) pre-coversee:

flat --

pre-envelope [16D40] (see: Fiat cover) pre-image operator [03G10, 06Bxx, 54A40] (see: Fuzzy topology) pre-injective module [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) pre-potential [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) pre-projective module [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) pre-specification of a program s e e : weakest -predicate logic s e e : first-order - - ; monadic -predicate logic in machine learning s e e : first-order -prediction s e e : least squares - - ; linear least squares - p r e d i c d o n error

[60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) predictor s e e :

best linear least squares --

preference function [90All1 (see: Cobb-Douglas function) Preiss density theorem [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) preparation theorem s e e : p-adic Weierstress - presentation s e e : almost convex group --

presentation of a Baumslag-Solitar group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) presentation of a group [20F05, 20F06, 20F32] (see: HNN-extension) presentation of a group [20F05, 20F06, 20F32] (see: HNN-extension) presentations of a free Burnside group s e e : conjugacy problem for - - ; word problem for -presented non-Hopfian group see: finitelypreserving a distance s e e : mapping -preserving mapping s e e : distance- - -

powerfultest s e e : uniformly most - powerset s e e : L - - ; l a t t i c e - - -

pretzel knot [57M25] (see: Positive link)

powerset operator [03G10, 06Bxx, 54A40]

price of a European strike - -

primal optimality condition [90Cxx] (see: Fritz John condition) primal system [15A39, 90C05] (see: Motzkln transposition theorem) primality problem [03D15, 68Q15] (see: Computational complexity classes) primary aberrations [33C50, 78A05] (see: Zernlke polynomials) prime see: j- -prime element theorem s e e : abstract - prime ideal s e e : fixed - - ; unramified - -

prime ideal at x [54610[ (see: P-point) prime ideal in an algebraic number field s e e : norm of a - prime ideal of an extension of algebraic number fields s e e : splitting - -

prime ideal of degree one in an extension of algebraic number fields [11R44, 11R45] (see: Dirlchlet density) primeideal theorem s e e : Landau -prime ideals s e e :

regular set of --

prime JB *-triple [17Cxx, 46-XX] (see: JB *-triple) prime number theorem [llNxx, 11N32, 11N45, 11N80] (see: Abstract analytic number theory; Abstract prime number theory) prime number theorem [11L07, 11M06, 11P32] (see: Vaughan identity) prime number theorem s e e : abstract - - ; abstract inverse - - ; inverse additive abstract - prime numbertheory s e e : Abstract - prime theorem s e e : Zel'manov --

primes in an arithmetical semi-group [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) primitive o:-abundant number [llAxx] (see: Abundant number) primitive abundant number [11Axx] (see: Abundant number) primitive Banach-Jordan algebra [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) primitive element [12E20] (see: Galols field structure) primitive Jordan algebra s e e :

principal congruences s e e : definable - call option

see:

semi- - -

primitive normal basis theorem [12E20] (see: Galois field structure) primitive polynomial [12E20] (see: Galois field structure) primitive root [12E201 (see: Galois field structure) primitive unitary o~-abundant number [11Axx] (see: Abundant number) principal congruencerelation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) principal congruence subgroup [11F25, 11F60] (see: Heeke operator) principal curvature direction [53A10, 53C42]

equationally

(see: Weierstrass representation of a minimal surface) principal inner ideal [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) principal Q-congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) principal relative congruence see: equationally definable - -

principal series representation [11F03, 11F70] (see: Selberg conjecture) principal submatrix see: leading -principal symbol of a differential operator [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) principal value distribution [46F10] (see: Multiplication of distributions) principle s e e : Frege - - ; Hardy uncertainty - - ; inclusion-exclusion - - ; mathematical uncertainty - - ; max-min - - ; Rayleigh - principle for eigenvalues s e e : max-min -principle in commutative algebra s e e : local-global - principle, mathematical s e e : Uncertainty - -

principle of inclusion-exclusion [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdblus inversion) prinjective Kl-module [16Gxx] (see: Tits quadratic form) priori-condition belief function s e e : prize question s e e : Hirzebrueh - -

a --

pro-p-group [20J06] (see: Serre theorem in group cohomology) probabilistic finite-state automaton [68S05] (see: Natural language processing) probabilistic functions in machine learning [68T05] (see: Machine learning) probabilistic grammar [68S05] (see: Natural language processing) probabilistic methodin combinatorics [05C80] (see: Lov~iszlocal lemma) probabilistic polynomial time class s e e : bounded --

complexity

probabifistic Riesz decomposition theorem [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) probabilistic Turing machine [03D15, 68Q15] (see: Computational complexity classes) probabilistic uncertainty see: non- -probability s e e : error - - ; runs in - -

probability assignment function [68T30, 68T99, 92Jxx, 92KI0] (see: Dempster-Shafer theory) probability distribution s e e : moment of a - - ; normalized restriction of a - -

probability ratio [62L10] (see: Sequential probability ratio

test) probability ratio test s e e : NeymanPearson - - ; Sequentia[ - - ; strong optimality property for the sequential - -

probably approximately correct learning [68T05] (see: Machine learning) probably approximatelycorrect learning [05C65, 05D05, 68Q15, 68T05]

523

PROBABLY APPROXIMATELYCORRECT LEARNING

(see: Vapnik-Chervonenkis dimension) problem see: 1-median --; 3-colouraNlity accepted input in a decision - - ; Burnside - - ; C - s e t - S - s e t - ; Cake-cutting - - ; chore-division - - ; circle - - ; class field tower - - ; classical Stieltjes moment - - ; complete - - ; complex decision - - ; corona - - ; decision - - ; determinate strong Stieltjes moment - - ; direct scattering - - ; Dirichlet boundary value - - ; distributional - - ; divisor - - ; domino tiling - - ; Euler equation of a variational - - ; fair division -- ; Fermat-TorriceIli -- ; Format-Weber --; Fermi-Pasta-Ulam - - ; first spacing - - ; first Zolotarev - - ; Gabor representation - - ; Galois embedding--; Gel'land--; generalized Fermat-TorricelIi - - ; graph isomorphism - - ; Hamburger moment - - ; Hamiltonian circuit - - ; h a r d - - ; Hilbert 15th - - ; Hilbert seventh - - ; indeterminate strong Stieltjes moment - - ; infraparticle - - ; intractable - - ; inverse - - ; inverse geophysical scattering - - ; inverse potential scattering - - ; inverse scattering - - ; inverse Sturm-Liouville -- ; isospectral - - ; isospectral linear eigenvalue - - ; k-set - - ; Kac - - ; Kiefer-Weiss - - ; Knapsack - - ; L 1 regression - ; lifting - - ; linear fit - - ; linear programming - - ; Littlewood one-circle - - ; longest path - - ; matrix - - ; matrix diagonalization - - ; median hyperplane - - ; membership - - ; Michael - - ; moment - - ; Neumann boundary value --; non-linear filtering - - ; iV'tO-complete - - ; numerical approaches to the S t u r m Liouvil[e spectral - - ; obstacle scattering --; optimal stopping --; oracle --; prireality - - ; real root counting - - ; real root localization - - ; rejected input in a decision - - ; restricted Burnside - - ; Riemann - - ; Robbins - - ; root counting - - ; root localization - - ; Saffman-Taylor - - ; sarisfiability -- ; scattering - - ; second spacing - - ; second Zolotarev - - ; shortest-path - - ; single facility location - - ; Specht - - ; spectral f a c t o r i z a t i o n - - ; spectral measure associated with a Dirichlet - - ; Steiner--; Steiner-Weber--; strong moment - - ; Strong Stieltjes moment - - ; strong symmetric Stieltjes moment - - ; Sturrn-Liouville spectral - - ; submartingale - - ; synthesis-Ditkin - - ; Szegb extremum - - ; tau - - ; Torricelli-Fermat - - ; Travelling salesman - - ; travelling salesman path - - ; travelling saIesman walk -- ; trigonometric moment - - ; Turfin brick factory - - ; uniqueness-of-norm - - ; unweighted Fermat-Torrioelli - - ; Vincensini - - ; Waring-Gotdbach - - ; word - - ; Zariski - --;

problem complete for a complexity class [03D15, 68Q15] (see: Computational complexity classes) problem for a complexity class see: complete - problem for Fibonacci groups see: conjugacy - - ; word - problem for group rings see: isomorphism -problem for isometric mappings see: Aleksandrov -problem for matroids see: critical -problem for presentations of a free Burnside group see: e o n j u g a c y - - ; word - problem for the Beurling algebra see: synthesis - problem for the clamped plate see: eigenvalue -problem for the Korteweg-de Vries equation see: characteristicinitial-value - problem for three-dimensional manifolds see: homeomorphism - - ; word - problem for varieties see: finite basis - -

524

problem in union - -

spectral

synthesis

see:

problem of coincidences [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) problem of functional equations see: stability - problem of system theory s e e : partial realization -problemon field extensions see: Zariski -problem on the half-axis see: direct scattering -problem over a module see: complexity of the membership - - ; membership - problem over Q a b see: inverse Galois - problem reduction see: dominance of - - ; many-one - - ; polynomial-time manyone

--

problems s e e : multiplier --; reduction between -- ; Zolotarev - procedure see: quantization - - ; renormalization - process see: A G M - - ; Arithmeticgeometric mean - - ; birth - - ; death - - ; hidden M a r k e r -- ; intensity - - ; Lanczos tridiagonalization - - ; Lindenbaum-Tarski --; linearly regular stochastic - - ; linearly singular stochastic - - ; long m e m o r y -- ; M a r k e r decision - - ; portfolio value - - ; portfolio weight -- ; state space representation of a - - ; stationary long memory - - ; van der Corput A- - - ; van der Corput B processes see: counting - processing see: cost in quantum information -- ; fault tolerant quantum -- ; Natural language - - ; resource in quantum information --; signal -processing, science of see: Quantum information - product see: amalgamated - - ; automorphic - - ; box - - ; crossed - - ; individual distributional - - ; Jaeger composition -- ; Jordan - - ; L e g e n d r e - S o b o l e v inner - - ; smash - - ; Sobolev inner - - ; triple - product basis function see: tensor- -product C * - a l g e b r a see: co-crossed - - ; crossed - product formula see: classical Euler - - ; Euler - - ; modified Euler - -

product formula representation of a continuous semi-group [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holemorphic mappings) product-moment correlation coefficient Pearson --

see:

product moment of random variables" [62H20] (see: Pearson product-moment correlation coefficient) product theorem for the Brouwer degree [55M25] (see: Brouwer degree) product theoremfor the Brouwerdegree [55M25] (see: Brouwer degree) product topology [26A21, 54E55, 54G20] (see: Sorgenfrey topology) production function [90A11] (see: Cobb-Douglas function) products of monic irreducible polynomials see: Gauss theorem on - -

profile analysis [62Jxx] (see: ANOVA) profile of a sequence ity - -

see:

projection [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) projection see: C a u c h y - S z e g d o r t h o g o n a l - - ; contractive - - ; Riesz - - ; spectral - projection boundary condition see: Calderdn --

projection operator [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) projection pursuit algorithm [41A30, 92C55] (see: Ridge function) projection scheme s e e : complete

--; nested - projection theorem see: BesicovitchFederer -projections see: Kuratowski theorem on closed --

projective C* -algebra [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) projective component

see:

post- - -

projective cover [16D40] (see: Flat cover) projective dimension [13B10, 13C15, 13C40, 16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting module; Tilting theory; Zariski-Lipman conjeeture) projective generator [16Gxx] (see: Tilting theory) projectiveline see: weighted - projective module see: non- - - ; pro- - -

projective profinite group [11R32]

(see: Shafarevich conjecture) Projective representations of symmetric and alternating groups (20C25) (referred to in: Schur Q-function) (refers to: Alternating group; Finite group; Projective representation; Representation of the symmetric groups; Schur functions in algebraic combinatorics; Schur multiplicator; Schur Q-function; Specht module; Symmetric group; Young diagram) projective resolution [16G?0] (see: Almost-split sequence) projectorsee:

Riesz--

Prolla theorem [46E25, 54C35] (see: Bishop theorem) proof see: EQP --;

interactive

--;

resolution-based -propagation see: sound - propagator

see:

linear complex-

riskless --

--

orsee:

see:

properties of the Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) properties of the Z a k transform see: analytic - properties of values of analytic functions see: transcendence - properties o f Z a k transforms [42Axx, 44-XX, 44A55]

(see: Zak transform) property see: amalgamation - - ; B a n a c h Stone - - ; Beth definability - - ; collapsing - - ; creation - - ; dominated decomposition - - ; G,~-insertion - - ; Hauptmodul - - ; L ( 0)-grading - - ; L ( - - 1 )-derivative - - ; liffable -- ; L u z i n - M e n s h o v - - ; metalogical - - ; Namioka - - ; normalized matching - - ; Painlev~ - - ; relative congruence extension - - ; reproducing - - ; Riesz decomposition - - ; Slobodnik - - ; Specht - - ; Spurner - - ; strong amalgamation -- ; strong Spurner - - ; super-amalgamation - - ; u n i m o d a l i t y - - ; vacuum - - ; w e a k Beth definability - - ; Zahorski -property C for see: Ordinary differential equations, - - ; Partial differential equations, - -

property C+ for ordinary differential equations [34A55, 34L25] (see: Ordinary differential equations, property C for) property C~ for ordinary differential equations [34A55, 34L25] (see: Ordinary differential equations, property C for) property C¢ for ordinary differential equations [34A55, 34L25] (see: Ordinary differential equations, property C for) property C for partial differential equations [35P25] (see: Partial differential equations, property C for) property C for partial differential equations [35P25] (see: Partial differential equations, property C for) property Cp for partial differential equations [35P25] (see: Partial differential equations, property C for) property for parameter systems see: monomial - property for scattering data see: characterization - property for the sequential probability ratio test see: strong o p t i m a l i t y - propertyof a logic see: deduction - propertyof a partially ordered set see: normalized matching - propertyof E Riesz see: decomposition -property of the Brouwer degree see: existence - - ; normalization - property of the monster see: 6transposition - property of the variety of Boolean algebras see: amalgamation - -

propositional calculus intuitionistic - -

proper morphism [46J10, 46L05, 46L80, 4.6L85] (see: Multipliers of C* -algebras) proper Poincar6 group see: thochronous - properties of a belief function graphoidal - -

properties of the Beurling algebra spectral - -

proportional hazards [62Jxx, 62Mxx] (see: Cox regression model)

[81Qxx] (see: Dirac quantization) proper see: A- - - ; approximation - proper mapping see: A- - -

profmite branch group [20E08, 20El 8, 20Fxx] (see: Branch group) profinite group see: projective profit

program s e e : learning - - ; minimal model - - ; straight-line - - ; weakest prospecification of a -programme see: Manton - programming see: abstract - - ; c o n v e x - - ; dual algorithm of linear - - ; flexible - - ; Fuzzy - - ; inductive logic - - ; partly c o n v e x --; possibilistic - - ; robust - programming problem see: l i n e a r - progression see: arithmetic - progressions see: Szemergdi theorem on arithmetic -- ; van der Waerden theorem on arithmetic - -

see:

classical - - ;

propositional logic [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic; Algebraic logic) proto-equivalence system

QUANTUM COMPUTATION

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) protoalgebraic deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) protoalgebraic general semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) protoalgebraic logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) protoalgebraic semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) protocolsee: transmission - provabilityrelation see: syntactical - prover see: theorem -proving see: automated theorem- - - ; automatic theorem- - -

proximal ball topology [54Bxx] (see: Wijsman convergence) proximal normal [90C30] (see: Clarke generalized derivative) proximal subgradient [90C30] (see: Clarke generalized derivative) proximal topology [54Bxx] (see: Wijsman convergence) pseudo-addition decomposablemeasure [28-XX] (see: Non-additive measure) pseudo-algebraically closed field [11R32] (see: Shafarevich conjecture) pseudo-convexdomain [47Dxx] (see: Taylor joint spectrum) pseudo-convexdomain see: rank of a - - ;

[68Q05, 68Q10, 68Q15, 68Q25, 81PxxJ (see: Quantum computation, theory ol3 PSPACE see: complexity class - public-key cryptographic system [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) public-key cryptography [12E201 (see: Galois field structure) public-key encryption [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) pull-back [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) pulse function [42Cxx, 94A 12] (see: Window function) pure submodule [16D401 (see: Flat cover) pure unrectifiability [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) purely 1-unreetifiable set see: example of [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) purely ra-unrectifiable set [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) purely unrectifiable measure [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) pursuit algorithm see:

pseudo-differential operator [42A16, 47B35] (see: Szeg6 limit theorems) pseudo-differential operator see:

projection - -

Q

hypoelliptic - - ; hypo-elliptic symbol of a - - ; symbol of a - - ; zeta-function of a - -

pseudo-localtomographyfunetion see: version formula for the - pseudo-metric see: Schwarz-Pick - -

in-

pseudo-noise sequence [11B37, llT71, 93C05] (see: Shift register sequence) pseudo-orthocomplement [03G25, 06D99] (see: Quantale) pseudo-randomsequence [11B37, 11T71, 93C05] (see: Shift register sequence) pseudo-Schurring [05Exx] (see: Cellular algebra) pseudo-spectral method [65Lxx] (see: Tau method)

Q

see:

quadratic algorithmfor 7r [26Dxx, 65D20] (see: Arithmetic-geometric mean process) quadratic base change [11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) quadratic convergence [26Dxx, 65D20] (see: Arithmetic-geometric mean process)

absolute Galois group over - -

Q a b see: absoluteGalois group o v e r - - ; inverse Galois problem over -q coefficient see: BIomqvist --

Q-congruence [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Q-congruencesee:

principal -

q-difference analogue of the EulerPoisson-Darboux operator [35L15] (see: Euler-Poisson-Darboux equation) Q-factorial algebraic variety [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) Q-function see:

Schur --

pseudo-spectralmethod see: Chebyshev - - ; Fourier - pseudo-spectral methods see: Trigonometric - -

q-homotopy skein module [57Mxx, 57M25] (see: Skein module) q-integral [05E05, 60G50] (see: Baxter algebra) q representation see: k- - Q-roots of a semi-simplealgebraic group [11Fxx, 20Gxx, 22E46] (see: BaUy-Borel compacfification) qd algorithm see: R u t i s h a u s e r - -

Psi function [llM06, 11M35, 33B15] (see: Catalan constant) PSPACE

Qvr [81Qxx] (see: Zeta-function method for regularization)

[81Qxx] (see: Dirac quantization) quantum algorithm [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81PI5, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) q u a n t u m algorithm see:

Shor--

quadratic regression [62Jxx] (see: ANOVA) quadrature see: Szag6

quantum automaton [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum bit [68Q05, 6gQ10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum bit [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

quadrilateral [05B07, 05B30] (see: Pasch configuration) quadrilateral free Steiner triple system [05B07, 05B30] (see: Pasch configuration) quadruple see: Kauffman skein - -

quantum bracket [81Qxx] (see: Dirae quantization) quantum channel [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of)

quadratic form see: Euler - - ; Tits - - ; w e a k l y non-negative - - ; w e a k l y positive - quadratic reciprocitylaw see: Gauss - -

-quadrature formula see: Szeg6 -quadric function see: multi- --

qualifications see:

a - -

purely rn-unrectifiable Radon measure

strictly - -

Pseudo-local tomography (44A12, 65R10, 92C55) (refers to: Cauchy integral; Continuous function; Local tamography; Radon transform; Tomography) pseudo-local tomography function [44A12, 65R10, 92C55] (see: Pseudo-local tomography)

quadranglesee: bimedian of a - - ; centroid of a - quadrangles see: Varignon theorem on - -

constraint - -

qualitative approach to Dempster-Shafer theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) qualitative factors in covariance analysis [62Jxx] (see: ANOVA) quantal set [03G25, 06D99] (see: Quantale) Quantale (03G25, 06D99) (refers to: Assoeiativity; Atomic lattice; Boolean algebra; C*-algebra; Distrihutivity; Hilbert space; Lattice; Locale; Spectrum of a C*algebra; yon Neumann algebra) quantalesee: algebraica/lyirreduciblerepresentation of a Gei'fand - - ; atomic von Neumann - - ; discrete Gel'fend - - ; Foulis - - ; Gel'fend - - ; Girard - - ; Hilbert - - ; involutive - - ; irreducible representation of a Gel'land - - ; point of a Gel'fend - - ; representation of a Gel'land - - ; sheaf on a - - ; spatial Gel'fend - - ; yon Neumann - - ; von N e u m a n n factor --

quantale of endomorphisms [03G25, 06D99] (see: Quantale) quantale of relations [03G25, 06D99] (see: Quantale) quantales see: algebraically

strong hom o m o r p h i s m of Gel'fend - - ; discrete hom o m o r p h i s m of Gel'fend - - ; right embedding h o m o m o r p h i s m of Gel'fend --

quantative approach to Dempster-Shafer theory [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) quantitative factors in covariance analysis [62Jxx] (see: ANOVA) quantization see: canonical - - ; eovariant - - ; Crumeyrolle-Prfistaro - - ; defermation - - ; Dirac - - ; Drinfel'd-Turaev - - ; geometric - - ; w e a k Ddnfel'd-Tu raev - quantization of partial differential equations see: formal Dirac --

quantization procedure [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) quantum algebra of a system

of)

q u a n t u m channel see:

noisy - -

quantum circuit [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of) quantum code [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) q u a n t u m code see: additive --; stabilizer - -

quantum cohomology [14Jxx, 35A25, 35Q53, 53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture; Whitham equations) quantum communication [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum communication [68Q05, 68Q10, 68Q15, 68Q25, glPxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum communication complexity [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) quantum complexity [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) quantum complexity class [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory ot3 quantum computation [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum computation [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) 525

QUANTUMCOMPUTATION

quantum computation in - - ; model of -

see:

measurement

-

Quantum computation, theory of (68Q05, 68Q10, 68Q15, 68Q25, 81Pxx) (referred to in: Quantum information processing, science of) (refers to: Complexity theory; Computational complexity classes; Entropy; Error-correcting code; Hilbert space; N'79; Quantum field theory; Quantum information processing, science of; Turing machine; Unitary operator) quantum computational complexity [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) quantum computer [68Q05, 68Q10, 68Ql5, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum computing

see:

distributed - -

quantum control [68Q05, 68QI0, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum control [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum cryptography [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum entanglement [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum error-correcting code [68Q05, 6gQ10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum error-correctingcode [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of)

of) quantum field s e e : free asymptotic - quantum field theory s e e : algebraic - - ; conformal - - ; massive - - ; symmetry breaking in - - ; topological --

quantum gate [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 8lP15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) see:

applying a - -

see:

foundationsof--

quantum memory [68Q05, 68Q10, 68Q15, 68Q25, 8lPxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum modnii space [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) quantum network [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of)

quantum information processing in - - ; resource in - -

tum - -

quantum oracle [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory quantum polynomial-time computable language s e e : b o u n d e d - e r r o r - quantum processing s e e : fault tolerant - -

quantum RAM [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of) quantum random access machine [68Q05, 68Q10, 68QI5, 68Q25, 81Pxx] (see: Quantum computation, theory

of) quantum Schubertpolynomials [05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) quantum search [68Q05, 68QI0, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory quantum search

see:

cost

Quantum information processing, science of (68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx) (referred to in: Quantum computation, theory of)

see:

unstructured--

quantum situs [81Qxx] (see: Dirac quantization) quantum spectral measure of a partial differential equation [81Qxx] (see: Dirac quantization) quantum state

[68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum Turingmachine [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) quantum vacnnmenergy [81Qxx] (see: Zeta-function method for regularlzation) quark [17A40] (see: Frcudenthal-Kantor triple system) quasi-Buchsbaum module [13A30, 13H10, 13H30] (see: Buchsbaum ring) quasi-circle [30C62, 30C99] (see: Quasl-symmetric function of a complex variable) quasi-conformaI mapping [26B99, 30C62, 30C65] (see: Quasi-regular mapping) quasi-continuity [54C081 (see: Almost continuity) quasi-continuous function [31C10, 32F05] (see: Pinripotential theory) quasi-convex sequence [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) quasi-equational class [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) quasi-groupsee:

quantum noise [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum number see: angular momen-

of)

quantum geometry [81Qxx] (see: Dirac quantlzation)

526

quantum mechanics

of)

quantum error-correction [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) quantum factoring algorithm [68Q05, 68Q10, 68Q15, 68Q25, 81pxx] (see: Quantum computation, theory

quantum gate

(refers to: Complexity theory; Cryptography; Entropy; Error-correcting code; Finite field; Hilbert space; Information, amount of; Information, source of; Majorization ordering; N'79; Quantum communication channel; Quantum computation, theory of; Quantum field theory; Shannon theorem; "luring machine; Unitary operator) quantum informationtheory [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81p15, 94Axx] (see: Quantum information processing, science of) quantum Lagrangian [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) quantum logic [81Qxx] (see: Dirac quantization) quantum manifold [81Qxx] (see: Dirac quantization) quantum mechanics [68Q05, 68Q10, 68Q15, 68Q25, 78A40, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of; Stokes parameters)

see:

quantum topology

entangled - -

entropic right --

quasi-hereditary algebra [16Gxx] (see: Tilting theory) quasi-Hermitian commutative twooperator vessel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) quasi-interpolation [41A05, 41A30, 41A63] (see: Radial basis function) quasi-isometric metric spaces [05C25, 20Fxx, 20F32] (see: Baumslag-Sofitar group) quasi-isometry [05C25, 20Fxx, 20F32] (see: Banmslag-Solitar group) quasi-measure [28-XXl (see: Non-additive measure) quasi-monoidal lattice Iocalic --

see:

complete - - ;

quasi-nilpotent operator [47B06] (see: Riesz operator) quasi-polyadic algebra [03Gxx] (see: Algebraic logic) Quasi-regular mapping (26B99, 30C62, 30C65) (referred to in: Zurich theorem) (refers to: Analytic function; Conformal mapping; Elliptic partial differential equation; Extremal length; Hdlder condition; Homeomorphlsm; Jacobian; Laplace equation; Lebesgue measure; Pieard theorem; Potential theory; Quasl-conformal mapping; Sobolev space; Valuedistribution theory; Zorich theorem) quasi-regular mapping [26B99, 30C62, 30C65] (see: Quasi-regular mapping) quasi-regularmapping

[26B99, 30C62, 30C65] (see: Quasi-regular mapping) quasi-regularmapping

see:

K.

--

quasi-symmetric function [30C62, 30C99] (see: Quasl-symmetrlc function of a complex variable) quasi-symmetricfunction

see:

M.

--

Quasi-symmetric function of a complex variable (30C62, 30C99) (refers to: Automorphism; Conformal mapping; Fourier series; Homeomorphism; Jordan curve; Modulus of an annulus; Quasi-conformal mapping) quasi-symmetric function on T [30C62, 30C99] (see: Quasi-symmetrlc function of a complex variable) quasi-symmetricfunetion on T s e e : M - - quasi-symmetric functions s e e : LeibnizHopf algebra and - -

quasi-tilted algebra [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting theory) quasi-tilted algebra [16Gxx] (see: Tilting functor; Tilting module) quasi-variety lent - -

see:

second-order equiva-

quaternion algebra [15A66, 8IR05, 81R25] (see: Pauli algebra) qnnternion division algebra [15A66, 81R05, 81R25] (see: Pauli algebra) qubit [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) qubit [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) queer Lie super-algebra [05El0, 05E99, 20C25] (see: Schur Q-function) querying an oracle [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory

of) question s e e : Hirzebruch prize -question on stability of homomorphisms Ulam -queue s e e : M/M/1 - -

see:

quick-sort [68Q15] (see: Average-case computational complexity) Quillen metric [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) Quillen theorem on Krull dimension of group cohomology [20J06] (see: Serre theorem in group cohomology) Quillen theory of super-connections [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) quiver [i6Gxx] (see: Tits quadratic form) quiver s e e : Auslander-Reiten - - ; configuration in a translation - - ; path-algebra of a - - ; slice in a - - ; translation -quotient s e e : arithmetic - -

quotient morphism [46JI0, 46L05, 46L80, 46L85] (see: Multipliers of C'* -algebras)

RECURSIVE TAU METHOD

(see: Selberg conjecture) ramification index [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group)

quotient theoremfor LYM posets [05D05, 06A07] (see: Spurner property)

ramified fraetal see:

(see: Acceptance-rejection method)

[14H20] (see: Taenode)

p-monotone mapping

[321-I15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) R n see:

area of the unit sphere in - R - l - c y c l e see; relative - R - l - c y c l e s see: numerically equivalent relative - -

R-roots of a semi-simplealgebraic group [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) Rabinowitsch trick (14Axx) (refers to: Field; Hilbert theorem) Radial basis function (41A05, 41A30, 41A63) (refers to: Algebraic polynomial of best approximation; Approximation of functions; Continuous function; Interpolation; Uniform convergence) radial basis function

[41A05, 41A30, 41A63] (see: Radial basis function) radial basis function see: linear

Ramsey theorem [05D10] (see: Hales-Jewett theorem) Ramseytheorem

see:

geometric--

Ramsay theory [05910] (see: Hales-Jewett theorem) R a m s a y t h e o r y see: density - r a n d o m access m a c h i n e

[68Q151 (see: Average-case computational complexity) random access machine see:

quantum --

random effects model

[62Jxx] (see: ANOVA)

random effects model [62Jxx] (see: ANOVA) random graph see:

chromatic n u m b e r of

a--

random graph model [68Q15] (see: Average-case computational complexity)

-radial polynomial see: Zernike - radiation see: e l e c t r o m a g n e t i c - -

random media see: trapping in - random polynomial-time factorization method see: Kaltofen-Trager--

radiation condition

random Schr6dingeroperator [60Gxx, 60J55, 60J65] (see: Wiener sausage)

[35P25, 47A40, 81U20] Inverse scattering, multidimensional case; Obstacle seattaring) (see:

radical see: Jacobsen-type - radius condition see: essential - radius formula see: spectral - -

radius of convergence of a numerical series

[39A12, 93Cxx, 94A12] (see: Z-transform) radius of injectivity

random sequence see: pseudo- - randomvariabtes see: product m o m e n t of - - ; uncorrelated -random variables with joint normal distribution see: uncorrelated - randomization model

[62Jxx] (see: ANOVA) rangeforce see: long- - rank s e e : d i s p l a c e m e n t - - ; low displace-

[26Bxx, 30C20]

ment - - ; Scott - - ; U - -

(see: Zorich theorem) radius theorem of Delsarte type s e e :

rank function on a partially ordered set two-

[05D05, 06A07] (see: Sperner property)

Radon measure see: ~n-rectifiable - - ; purely rr~-unrectifiable - -

Radon-Nikod~m derivative

[28-xx] sures) Radon transform RAM

rank function on a partially ordered set see: level of a - rank lower b o u n d see: log- --

rank of a logical connective

(see: Absolutely continuous measee:

combinatorial - -

[68Qt5] (see: Average-case computational complexity) RAM see: q u a n t u m -Ramanujan conjecture

[11F03, 11F70] (see: Selberg conjecture) Ramanajan function [11F25, 11F60] (see: Heeke operator) Ramanujan graph [05C25] (see: Cayley graph) Ramanujan inequality

[11F03, 11FT0] (see: Selberg conjecture) Ramanujan-Petersson conjecture

[11F03, 11F70] (see: Selberg conjecture)

Ramanujan-Peterssonconjecture at infinity [11F03, 11F70] (see: Selberg conjecture) Ramanujan-Peterssonidentity [11F03, 11F70]

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) rank of a partially ordered set

[05D05, 06A07] (see: Spurner property) rank of a pseudo-convex domain

[46Lxx] (see: Tueplitz C*-algebra) r a n k t w o see: free subgroup of -ranked partially ordered set see: characteristic polynomial of a - -

Ranldn-Selberg method [llFll, 11F12] (see: Shimura correspondence) rarefied gas

[76Axx] (see: Knudsen number) Rassias stability see: Hyers-UPam- - rate s e e : algebraic convergence -- ; exponential convergence -- ; infinite convergence - - ; local -- ; material - - ; spectral convergence -rate of c h a n g e see: convective - - ; local - -

rate of convergence [46Cxx] (see: Alternating algorithm) rate of decay of a function at infinity

[42A63]

probability - -

ratio-of-uniforms method

[62D05]

finitely --

ramphoid cusp

R

(see: Uncertainty principle, mathematical) ratio see: likelihood --; performance --;

ratio test see: likelihood- - - ; N e y m a n Pearson probability - - ; Sequential probability - - ; strong optimality property for the sequential probability - -

[11Fxx, 20Gxx, 22E46] (see: Baily-Borel eompactifieation) rational boundary component of a symmetric space

[llFxx] (see: Satake compactification) rational homotopy theory [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) rational interpolation scheme [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) rational move see: p / q rational n u m b e r set see:

--

place of the - -

rational p/q-tangle

[57M25] (see: Rational tangles) rational tangle

[57M25] (see: Rational tangles) slope of a - -

Rational tangles (57M25)

(see: Borcherds Lie algebra) r e a l s u b m a n i f o l d see: t o t a l l y - real-valued functions see: natural order on a space of - -

real vector bundle [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) realizability of a group see: GAR- - realization s e e : Harish-Chandra-realization problem of system t h e o r y s e e : partial - rearrangement see: spherical d e c r e a s ing - rearrangementof a function see: decreasing - reasoning see: analogical - - ; backward - - ; forward - - ; inductive - - ; mixed - recessive s e e : homozygous - reciprocal algebraic integer see: non- - -

reciprocal integral transforms [31B05, 33C55] (see: Zonal harmonics) reciprocal polynomial of a linear feedback shift register

(referred to in: Tangle) (refers to: Tangle)

[11B37, 11T71, 93C05] (see: Shift register sequence)

C o n w a y notation

reciprocity l a w see:

Gauss quadratic - -

reconstruction formula for the continuous wavelet transform

Ray-Singer analytic torsion [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Rayleigh conjecture for the clamped plate

[35P15] Rayleigh-Faber-Krahn inequality) Rayleigh-Faber-Krahn inequality (35P15) (referred to in: Dirichlet eigeuvalue; Natural frequencies; Neumann eigenvalue; Wiener sausage) (refers to: Bessel functions; Dirichlet boundary conditions; Dirichlet eigenvalue; lsoperimetrie inequality; Laplace operator; Natural frequencies; Neumann eigenvalue) Rayleigh principle [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) (see:

rays s e e :

[11Fxx, 17B67, 20D08] [llFxx, 17B67, 20D08]

rational boundary component

rational tangles see: for -raysee: extramal --

real root of a Borcherds algebra (see: Borcherds Lie algebra) real simple root of a Borcherds algebra

rational approximation [65Lxx] (see: Tau method)

rationaltangle see:

[47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) real root counting problem [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) real root localization problem [15A57, 47B35, 65F05, 93B15] (see: Hankei matrix)

Mori theory of extremal - -

[42Cxx] (see: Daubechies wavelets) reconstruction formula for the continuous wavelet transform see: admissibility condition for a - -

reconstruction theorem of inverse scattering theory

[35P25, 47A40, 81U20] (see: Inverse scattering, half-axis

case) rectangle function

[42Cxx, 94A12] (see: Window function) rectifiability

[28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) rectifiability [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) rectifiable current see:

m-

--

reaction-diffusion system [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation)

rectifiable measure [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory)

read-only input tape

rectifiable Radon measure see: m - - rectifiable set see: example of a 1- - - ; m---

[03D15, 68Q15} (see: Computational classes) real form of a JB *-triple [17Cxx, 46-XX] (see: JB * -triple)

complexity

rectifiablevarifold see: m- -recurrence relation see: l i n e a r - -

recurrence relation for derivatives of Chebyshevpolynomials [41A10, 41A50, 42A10] (see: Chebyshev pseudo-spectral method)

real function algebra

[46E25, 54C35] (see: Bishop theorem) real function algebras see: rem for --

Bishop theo-

real JB *-triple [17Cxx, 46-XX] (see: JB *-triple) real numbers see: concordant pairs of - - ; discordant pairs of - -

real Riemann surface

recurrent sequence

[11B37, llT71, 93C05] (see: Shift register sequence) reeursivefunction see:

GeOel - -

recursive sequence

[11B37, 11T71, 93C05] (see: Shift register sequence)

recursive tan method 527

RECURSIVE TAU METHOD [65Lxx] (see: Tau method) recursively enumerabletheory [03Gxx] (see: Algebraic logic) reduced full second-order model s e e : Leibniz- - -

reduced interval graph [llNxx, 11N32, 11N45] (see: Abstract prime number theory) reduced model s e e :

Suszko- - -

reduced sequence in the theory of HNNextensions [20F05, 20F06, 20F32] (see: HNN-extension) reducibility of complexity classes [03D15, 68Q15] (see: Computational complexity classes) reducible three-dimensional manifold [57NI0] (see: Haken mantibld) reduction s e e : complexity class closed under - - ; dominance of problem - - ; Leibniz- - - ; many-one problem - - ; polynomiaFtime many-one problem - - ; Suszko- - -

reduction between problems [68Q15] (see: Average-ease computational complexity) reduction formulas s e e :

LSZ- - -

reduction of Hamiltonian systems [37J15, 53D20, 70H33] (see: Momentum mapping) reduction theory [11Fxx] (see: Satake eompactification) reduction theory. [11F67] (see: Eisenstein cohomology) reductive Borel-Serre compactification [11Fxx] (see: Satake compactification) reductive pairs s e e :

dual - -

Reeb vector field [58F22, 58F25] (see: Seifert conjecture) Reed-Muller code [94Bxx] (see: Delsarte-Goethals code) Reeh-Schfieder theorem [81Txx, 81T05] (see: Massless field) Rees algebra [I3A30, 13H10, 13H30] (see: Buchsbaum ring) reflecting Brownian motion [60Hxx, 60J55, 60J65] (see: Skorokhod equation) reflecting Brownian motion s e e :

obliquely--

reflection coefficient [33C45, 35P25, 35Q53, 47A40, 58F07, 81U20] (see: Harry Dym equation; Inverse scattering, full-llne ease; Szeg6 polynomial) reflection formulafor a zeta-function [81Qxx] (see: Zeta-function method for regularizalion) reflection functor [16Gxx, I6GI0, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting module; Tilting theory) reflection functor s e e : BernshtefnGel'fand-Ponomarev - region s e e ; non-tangentialapproach --

region of cooperation [90Cxx] (see: Fritz John condition) region of stability s e e : characterization of local optimality on a - -

528

register s e e : characteristic polynomial of a linear feedback shift - - ; feedback coefficients of a linear feedback shift - - ; feedback matrix of a linear feedback shift -- ; feedback polynomial of a linear feedback shift - - ; impulse-response sequence of a linear feedback shift - - ; initial conditions of a linear feedback shift - - ; length of a linear feedback shift - - ; linear feedback shift - - ; reciprocal polynomial of a linear feedback shift - - ; state vector of a linear feedback shift - register sequence s e e : linear complexity of a shift - - ; minimal polynomial of a shift - - ; Shift --

regression [62Jxx] (see: ANOVA) regression s e e : least-squares - - ; linear - - ; multiple - - ; quadratic - -

regression analysis [62Jxx] (see: ANOVA) regression analysis see: dependent variable in -- ; independent variable in -regression model s e e : Cox - -

regression parameter [62Jxx, 62Mxx] (see: Cox regression model) regression problem s e e : L1 -regressor variable [62Jxx] (see: ANOVA) regular automorphismgroup [20-XX] (see: Regular group) regular cardinal number [54G10] (see: P-space) regular colligation [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) regular embedding [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Regular group (20-XX) (referred to in: Cayley graph) (refers to: Fitting length; p-group) regular groupof automorphisms [20-XX] (see: Regular group) regular group of permutations [20-XX] (see: Regular group) regular group of permutations [20-XX] (see: Regular group) regular language [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) regular local ring [13Hxx] (see: System of parameters of a module over a local ring) regularlocal rings s e e : structure theorem for maximal Buohsbaum modules over - regular mapping s e e : K - q u a s i - - - ; Quasi-

regular p-group [20-XX] (see: Regular group) regular p-group [2o-xx] (see: Regular group) regular permutation [20-XX] (see: Regular group) regular permutation group [20-XX] (see: Regular group) regular permutation group s e e : tive --

regular permutation representation [20-XX] (see: Regular group)

transi-

regular set of prime ideals [l 1R44, 11R45] (see: Dirichlet density) regular set of prime ideals [11R32, 11R45] (see: Chebotarev density theorem) regularstochastic process s e e :

linearly - -

regular surface [53A10, 53C42] (see: Weierstrass representation of a minimal surface) regular system of parameters [13Hxx] (see: System of parameters of a module over a local ring) regularity see: Castelnuovo-Mumford - - ; von Neumann - regularitytheorem s e e : Allard - regularization s e e : analytic - - ; harmonic - - ; non-commutative a n o m a l y for zetafunction -- ; Pauli-Villars - - ; upper semic o n t i n u o u s - - ; z e t a - f u n c t i o n - - ; Zetafunction method for - -

regularization conditions [90Cxx] (see: Fritz John condition) regularization of infinite determinants [81T50] (see: Non-commutatlve anomaly) regularized long wave equation [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) Reidemeister moves [57M25] (see: Reidemeister theorem) Reidemeister moves [57M25] (see: Tangle) Reidemeister theorem (57M25) (referred to in: Jones-Conway polynomial; Kauffman bracket polynomial; Tangle) reinforcement learning [68T05] (see: Machine learning) Reinhardt domain [47Dxx] (see: Taylor joint spectrum) Reisner face ring s e e : Stanley--Reisner ring s e e : Stanley--Reisner ring of a simplicial complex s e e : Stanley- -Reiten correspondence s e e : AuslanderReiten quiver s e e : A u s l a n d e r - - Reiten sequence s e e : A u s l a n d e r - - Reitentheorem s e e : A u s l a n d e r - - Reiten theorem on almost-split sequences see: Auslander- -Reiten translation s c c : Auslander---

rejected input in a decision problem [03D15, 68Q15] (see: Computational complexity classes) rejection method s e e : Acceptance- - related topologies s e e : S- -relation see: analytic - ; analytic isomorphism - - ; anti-commutative - - ; Borel e q u i v a l e n c e - - ; c o m m u t a t o r - - ; consequence - - ; constitutive - - ; contractionand-deletion -- ; C o n w a y skein - - ; Frege - - ; J o n e s - C o n w a y - - ; Kauffman bracket skein - - ; l i n e a r recurrence - - ; nilpotency - - ; o r t h o g o n a l i t y - - ; principal congruence - - ; skein - - ; skein m o d u l e based on the J o n e s - C o n w a y - - ; syntactical preyability -- ; validity - - ; v o n K~,rmAn - -

relation in a partially ordered set s e e : covering -relation of a theory s e e : Frege - relations s e e : algebra of r~-ary - - ; calculus of - - ; Chebyshev-type pair of inverse - - ; Hi rota b i l i n e a r - - ; quantale of - - ; Virasoro algebra - - ; zero-curvature - relations deforming r~-moves s e e : skein m o d u l e based on - relations for Zernike polynomials s e e : orthogonality - relative congruence s e e : equationallydefinable principal - -

relative congruence extension property [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) relative consistency of set theory s e e : G6del --

relative link [57M25] (see: Tangle) relative R-l-cycle [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) relative R-l -cycles see:

numerieallyequiv-

alent --

relative Well algebra [55R40, 57Rxx] (see: Well algebra of a Lie algebra) relativistic electron equation [15A66, 81Q05, 81R25, 83C22] (see: Dirac algebra) relativistic particle [81Txx, 81T05] (see: Massless field) relevance-based analogy [68T05] (see: Machine learning) relevance logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) reliability of multi-component systems [60C05, 60KI0] (see: Consecutive k-out-of-n: F system) reliability theory [05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusion formula) remainder s e e :

Taylor expansion --

remainder in the Stone-Cech compactification [54G10] (see: P-point) R e m a k - S c h m i d t category s e e :

Krull---

renormalization [81Qxx] (see: Zeta-function method for regularization) renormalization s e e :

mass --

renormalization group [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) renormalization procedure [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) renormalized constant [81T50] (see: Non-commutative anomaly) renormalized couplingconstant [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) rent partitioning [00A08, 90Axx] (see: Cake-cutting problem) repetitive algebra [16G70] (see: Riedtmann classification) replacement s e e : axiom of - - ; axiom schema of - - ; rule of - -

relation algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic; Algebraic logic)

replication see:

relation for derivatives of Chebyshev polynomials s e e : recurrence --

replication formulas [llFI1, 17B67, 20D08, 8IT10]

replicated option [90A09] (see: Option pricing) option - -

RIESZ DECOMPOSITION THEOREM (see: Moonshine conjectures) representable algebras e t a logic [03Gxx] (see: Algebraic logic) representable cylindric algebra [03Gxx] (see: Algebraic logic) representable cylindric algebra s e e : ary --

representations of generators of continuous semi-groups s e e : parametric -representations of symmetric and alternating groups s e e : Projective --

r~.

attribute-value--; automorphic - - ; B e r k s o n - P o r t a parametric - - ; c l a s s - o n e - - ; complementary series - - ; cuspidal -- ; euspidal automorphic -- ; Dirac - - ; F o u r i e r - - ; Fourier series - - ; highest weight - - ; k - q - - ; Majorana - - ; maximal determinantal - - ; minimal - - ; principal series - - ; regular permutation - - ; Riesz - - ; Riesz integral - - ; S e g a l Shale-Well - - ; special - - ; spectral coefficient - - ; spin - - ; Steinberg - - ; Well - - ; Wold - -

representation see:

representation-finite algebra [16G70] (see: Riedtmann classification) representation-finitealgebra [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting module) representation for a linear semi-group s e e : exponential formula - representation for machine learning s e e : efficiency of a - - ; expressiveness of a - -

representation group [20C251 (see: Projective representations of symmetric and alternating groups) representation of a continuous semi-group see: exponential - - ; exponential formula - - ; product formula - representation of a functional s e e : gral - -

inte-

representation e t a Gel'land quantale [03G25, 06D99] (see: Quantale) representation of a Gel'fund quantale s e e : algebraically irreducible - - ; irreducible - representation of a J B -algebra s e e ; factor -representation of a minimal surface s e e : Weierstrass -representation of a multivariate polynomial see: black box - - ; dense - - ; sparse - representation of a process s e e : state space - representation of an integer s e e : k- representation of the discriminant curve of an operator vessel s e e : input determinantal - - ; output determinantal - representation of the Frobenius automorphism s e e : polynomial - -

representation of uncertainty [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) representation problem s e e :

Gabor--

representation ring e t a compact group [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) representation structure in a learning systern [68T05] (see: Machine learning) representation-tamealgebra [16G70] (see: Riedtmann classification) representation theorem s e e : Riesz local - -

Riesz - - ;

representation theory [37J15, 53D20, 70H33] (see: Momentum mapping) representationtheory see: cohomological variety in -- ; Jacobsen - representation type s e e : Noethorian local ring of finite Buchsbaum- - - ; tame - -

representation type of boxes [16Gxx] (see: Tits quadratic form)

representing measure [32E20] (see: Polynomial convexity) Reproducing kernel (46E22) (referred to in: Reproducing-kernel Hilbert space) (refers to: Analytic function; Basis; Bergman kernel function; Cauchy inequality; Compact operator; Conformal mapping; Fourier coefficients; Hilbert space; Injection; Inner product; Linear independence; Measure; Orthogonal system; Parseval equality; Reproducing-kernel Hilbert space; Rigged Hilbert space; Simply-connected domain; Surjection) reproducing kernel [46Cxx, 46E22, 47B35] (see: Berezin transform; Reproducing kernel) reproducing kernel s e e :

normalized - -

Reproducing-kernel Hilbert space (46E22) (referred to in: Reproducing kernel) (refers to: Adjoint operator; Deltafunction; Fourier transform; Hilbert space; Inner product; Isometric operator; Linear functional; Linear operator; Measure; Pre-Hilbert space; Reproducing kernel; Riesz theorem; Rigged Hilbert space; Self-adjoint operator; Total set) reproducing-kernel Hilbert space [46E22] (see: Reproducing-kernel Hiibert space) reproducing kernels s e e :

examples of - -

reproducing property [46E22] (see: Reproducing-kernel Hilbert space) Reshetikhin-Turaev invariants [57M25] (see: Kanffman bracket polynomial; Tangle) Reshetnyak theory [26B99, 30C62, 30C65] (see: Quasi-regular mapping) residual equation [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) residual method see: weighted -residual set [54E52] (see: Banach-Mazur game) residually-finite group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) residually-finite group [20F24] (see: FC-group) residuals s e e : mean weighted -residue s e e : Adler-Manin commutative - - ; Wodzicki --

--;

non-

residue field [11R44, 11R45] (see: Diriehlet density) residue theorem [39A12, 93Cxx, 94A12] (see: Z-transform) resolution s e e : dispute - - ; projective -resolution-based p r o o f

[68T05] (see: Machine learning) resolution in group decomposition [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) resolvable design see: affine -resolvable t-- ( v , k , X )-design [05B30]

(see: Affine design) resolvable transversal design [05Bxx] (see: Net (in finite geometry)) r e s o l v e n t see: non-linear-resolvent function [47A10, 47B06] (see: Spectral theory of compact operators) resolvent of a sheaf [53C15, 55N35] (see: Spencer cohomology) resolvent operator [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) resonance [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) resonance see: natural f r e q u e n c y - resonance curve [11Lxx, 11L03, 11L05, 11L15] (see: Bombieri-Iwaniec method) resonancecurvessee: H u x l e y t h e o r y o f -resonance in scattering theory [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) r e s o n a n c e t h e o r y see: dual - resonant double Hopf bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) resource s e e :

space - - ; time - -

resource in quantum information processing [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) response sequence of a linear feedback shift register s e e : impulse- - -

restricted Burnside problem [20]=05, 20]?06, 20F32, 201750] (see: Burnside group) restricted first-orderlanguage see: full -conditions s e e : grading- - restriction of a probability distribution s e e : normalized - result s e e : Ward - resummation formula s e e : Poisson - restriction

retarded distribution [46F10] (see: Multiplication of distributions) retarded distributions see: algebraof - retract s e e : absolute - - ; Lipschitz -retraction

[16G701 (see: Almost-split sequence) retrieval s e e :

document - - ; information --

return-time theorem [28D05, 54H20] (see: Wiener-Wintner theorem) return-time

theorem s e e :

Wiener-Wintner

Bourgain - - ;

--

reversible transition function [68Q05, 68Ql0, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) Reynolds number [76Axx] (see: Knudsen number) Rham differential see: de -Rham isomorphism s e e : de - R h a m operator s e e : de - the s e e : Spearman - the metric s e e : Spearman - Richardson rule s e e : L i t t l e w o o d - - -

Ridge function (41A30, 92C55) (refers to: Curse of dimension; Neural network; Tomography) ridge function s e e : direction for a -Riedtmann classification (16G70)

(referred to in: Almost-split sequence; Tilted algebra; Tits quadratic form) (refers to: Algebra; Algebraically closed field; Almost-split sequence; Cyclic group; Dynkin diagram; Group algebra; Injective module; Jacobsen radical; Morita equivalence; Quiver; Representation of an associative algebra; Tilted algebra; Tilting theory) Riemann approach to algebraic curves [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brillformula) Riemann O-operator s e e :

Cauchy---

Riemann-Hurwitz formula [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brillformula) Riemann hypothesis [11M06] (see: Riemann ~-tuncfion) Riemann hypothesis [11Axx] (see: Dickman function) Riemann hypothesis see: Well -Riemann hypothesis for curves over finite fields [11F03, l 1FT0] (see: Selberg conjecture) Riemann-Lebesgue lemma [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) Riemann mapping function of a domain [30Axx, 46Exx] (see: BMOA -space) Riemann problem [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) Riemann surface s e e : cotangent space at a marked - - ; marked - - ; noded - - ; real - -

Riemann ~-function (11M06) (refers to: de ia Vallte-Poussin theorem; Dirichlet series; Entire function; Fourier transform; Gammafunction; Hadamard theorem; Rie-

mann hypotheses; Riemann zetafunction; Zeta-function) Riemann E-function [11M06] (see: Riemann E-function) Riemann E-function s e e : tion for the --

functionalequa-

Riemann zeta-function [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) Riemann zeta-function [llL07, 11M06, llNxx, 11N32, 11N45, llN80, 11P32] (see: Abstract analytic number theory; Vaughan identity) Riemannian manifold s e e : immersion of a s u r f a c e into a -R i e m a n n i a n s y m m e t r i c space

[17A40] (see: Lie triple system) Riemannian volume measure [28A80] (see: Sierpifiski gasket) Riesz s e e :

decomposition propertyof E - -

Riesz decomposition property (06F20, 31D05, 46A40, 46L05) (referred to in: Riesz decomposition theorem) (refers to: C*-algebra; Partial order; Potential theory, abstract; Riesz decomposition theorem; Riesz space;

Vector lattice; Vector space) Riesz decomposition property [06F20, 31D05, 46A40, 46L05] (see: Riesz decomposition property) Riesz decomposition theorem 529

R1ESZ DECOMPOSITIONTHEOREM (31AI0, 31D05, 47A10, 47A15, 47A60) (referred to in: Pluripotential theory; Riesz decomposition property; Spectral theory of compact operators) (refers to: Bauaeh space; Functional calculus; Krein space; Linear operator; Potential theory, abstract; Riesz decomposition property; Riesz space; Riesz theorem; Spectral synthesis; Subharmonic function; Vector lattice) Riesz decomposition theorem see: abilistic --

prob-

Riesz decomposition theorem for harmonic spaces [31AI0, 31D05, 47A10, 47A15, 47A601 (see: Riesz decomposition theorem) Riesz decomposition theorem for operators [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decompos:.tion theorem) Riesz decompositAontheoremfor subharmonic functions [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz decomposition theorem for superharmonic functions [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz-Dunford functional calculus [3IAI0, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz-Dunford integral [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Riesz-Herglotz kernel [33C45] (see: Szeg6 polynomial) Riesz-Herglotz transform [33C45] (see: Szeg6 polynomial) Riesz integral representation [31A10, 31D05, 47AI0, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz local representation theorem [31A10, 31D05, 47A10, 47A15, 47A60] (see: Ricsz decomposition theorem) Riesz operator (47B06) (refers to: Banach space; Compact operator; Hilbert space; Riesz summation method; Spectral theory of compact operators) Riesz operator [47B06] (see: Riesz operator) Riesz operator see:

Bochner---

Riesz point [47B06] (see: Riesz operator) Riesz projection [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz projector [31A10, 31D05, 47A10, 47A15, 47A60, 47B06] (see: Riesz decomposition theorem; Spectral theory of compact operators) Riesz representation [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz representationtheorem [32E20, 46E25, 54C35] (see: Bishop theorem; Polynomial convexity) Riesz spectral theory

530

[47B06] (see: Riesz operator) Riesz splitting theorem [31AI0, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz summability see: B o c h n e r - - Riesz theorem [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Riesz theory of compact operators [47A10, 47B06] (see: Riesz operator; Spectral theory of compact operators) rigged Hilbert space [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator co~!igation) rigged operator colligation [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator coUigation) right see: option - right action of a semi-group [39B05, 39B12] (see: Schrfder functional equation) right centralizer [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) right embedding homomorphism of Gel'fand quantales [03G25, 06D99] (see: Quantale) right halfCopen interval topology [26A21, 54E55, 54G20] (see: Sorgenfrey topology) right quasi-group see:

entropic - -

right-sided element [03G25, 06D99] (see: Quantale) rigid analytic space [11R32] (see: Shafarevich conjecture) rigid Baumslag-Solitargroup [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) rigid Baumslag-Solitar group see: vex --

con-

rigid stabilizer of a tree level [20E08, 20El 8, 20Fxx] (see: Branch group) rigidity [53C20, 53C22] (see: Santal6 formula) ring see: anchor --; Buchsbaum --;

cellular - - ; cellular matrix - - ; conductor of a local - - ; exterior face - - ; face - - ; FLC --; generalized C o h e n - M a c a u l a y - - ; hypersurface -- ; pseudo-Sehur - - ; regular local - - ; Sehur - - ; Stanley-Reisner - - ; Stanley-Reisner face - - ; System of parameters of a m o d u l e over a local - ring of a compact group see: representation -ring of a point on an algebraic curve see: local -ring of a simplicial complex see: StanleyReisner - -

ring of algebraic integers [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) ring of an algebraic curve see: affine coordinate - ring of finite Buchsbaum-representation type see: Noetherian local -ring of functions see: local -ring of (~ see: coset - ring of maximal e m b e d d i n g dimension see: Buchsbaum local -ring theory) see: Domain (in -rings see: Borromean - - ; derived equivalent -- ; isomorphism problem for group -- ; structure theorem for maximal Buchsb a u m modules over regular local -- ; suriectivity criterion for Buchsbaum --

(see: Jaeger composition product)

ripening see: Ostwald -risk see: item at - risk theory

rotationalsymmetry see:

[90A11] (see: Cobb-Douglas function) riskless profit [90A09] (see: Option pricing) Robbins algebra [06Exx, 68T15] (see: Robbins equation) Robbins equation (06Exx, 68T15) (refers to: Boolean algebra) Robbins problem [06Exx, 68T15] (see: Robbins equation) Robin condition in obstacle scattering [35P25] (see: Obstacle scattering) Robin function [31C10, 32F05] (see: Pluripotential theory) Robinson-Sehensted-Knuth dence see: shifted - -

correspon-

robust programming [90C70] (see: Fuzzy programming) robustness of natural language usage [68S051 (see: Natural language processing) Rodrigues formula for Zernike polynomials [33C50, 78A05] (see: Zernike polynomials) roll solution [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) rolled-up vortex sheet [76C05] (see: Birkhoff-Rott equation) rook polynomial [05Cxx, 05D15] (see: Matching polynomial of a graph) rook theory [05Cxx, 05D15] (see: Matching polynomial of a graph) root see:

primitive - -

root countingproblem [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) root counting problem see:

real - -

root localization problem [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) root localization problem see: real - root of a Borcherds algebra see: imaginary -- ; imaginary simple - - ; real - - ; real simple - root test see: m a x i m u m - - ; Roy maxim u m -rooted tree see: group action on a - roots of a semi-simple algebraic group see:

Q- --;R- -Rosenberg-Zelinsky exact sequence [13-XX, 16-XX, 17-XX] (see: Skolem-Noether theorem) Rosinger nowhere-dense generalized function algebra (46F30) (referred to in: Generalized function algebras) (refers to: Flabby sheaf; Generalized function algebras; Generalized functions, space of) Rota sign theorem [05B35, 05Exx, 05E25, 06A07, 1IA25] (see: MSbius inversion) rotant of a link diagram [57M25] (see: Rotor) rotational number of a graph [57M251

domain with

--

Rotor (57M25) (referred to in: Tangle) (refers to: Cyclic group; Fox ncolouring; Graph; Joncs-Conway polynomial; Kauffman bracket polynomial; Knot and link diagrams; Statistical mechanics, mathematical problems in; Yang-Baxter equation) rotor in graph theory [57M25] (see: Rotor) rotor in graph theory see:

n- --

rotor in knot theory [57M25] (see: Rotor) rotor in knot theory see: n- -Rott equation see: Birkhoff- -Rouault law see: Karlin--R o u e t - S t o r a - T y u t i n transformations see: Becch[- - -

rough domain [35P25] (see: Obstacle scattering) round voting s e e :

two- - -

Rousseau theory of general will [90A28] (see: Condoreet jury theorem) route [60K30, 68M10, 68M20, 90BI0, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) route cost [60K30, 68M10, 68M20, 90B10, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) route flow [60K30, 68M10, 68M20, 90B10, 90BI5, 90B18, 90B20, 94C99] (see: Braess paradox) Rovnyak functional model see: Branges- --

de

Roy maximum root test [62Jxx] (see: ANOVA) Roy strong six exponentials theorem [llJ811 (see: Schneider method) Ruelle measure see: Sinai-Bowen--Ruelle scatteringtheory see: Haag---

Ruelle theorem [81Uxx] (see: Enss method) rule see: inference - - ; Leibniz - - ; Littlewood-Richardson--; MurnaghanNakayama - -

rule-based path from logic to algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) rule learning [68T05] (see: Machine learning) rule of combination of two independent belief functions [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) rule of compositionality [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) r u l e o f conditioning see: Dempster-rule of evidence combination see: Dempster - -

mle of necessitation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) rule of replacement [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) mn of failures [60C05, 60K10]

SCHUBERT POLYNOMIALS

(see: Consecutive k-out-of-n:

F-

system) mn of successes [60C05, 60KI0] (see: Consecutive k-out-of-m system) Runge-Kutta method [65Lxx] (see: Tau method) runs in probability [60C05, 60K10] (see: Consecutive k-out-of-n: system) Rutishauser qd algorithm [15A57, 47B35, 651=05, 93B15] (see: Hankel matrix)

F-

F-

discordant -sample n u m b e r s e e : Average - sample theory s e e : large- - -

Santald formula

[53C20, 53C22] (see: Santal6 formula)

~r-additive measure [28-XX]

(see: Choquet integral)

cr-~-defavourable space [26A15, 54C05] (see: Namioka space; Namioka theorem) ~r-fragmentable space [26A15, 54C05] (see: Namioka space) o'-unital algebra

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) S' -convoludon [46F10] (see: Multiplication of distributions) S-integer (12J10, 12J20, 13A18, 16W60) (refers to: Diophantine equations; Mutually-prime numbers; Number field; Place of a field; Valuation) S-integer

[12J10, 12J20, 13A18, 16W60] (see: S-integer) S-matrb:

[35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-line case; Inverse scattering, half-axis case) S-matrix [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) S-matrix for electromagnetic wave scattering

[35P25] (see: Obstacle scattering) S-related topologies

[26A21, 54E55] (see: Slobodnik property) S-set problem see: C - s e t - -S unit

[12J10, 12J20, 13A18, 16W60] (see: S-integer) $5 see: Suszko congruence of a

theory

Santal6 theorem [52A35] (see: Geometric transversal theory) SAS (62-04) SAT [6gQ15] (see: Average-case computational complexity) SAT phase transition [68Q15] (see: Average-case computational complexity) Satake-Baily-Borel compactification

[11Fxx, 20Gxx, 22E46] (see: Baily-Borel eompactification)

Satake compactification (11Fxx) (referred to in: Baily-Borel compactlfication) (refers to: Abelian variety; Algebraic curve; Mgebraic geometry; Algebraic gr,-n;. Arithmetic group; Baily-Bnrel-~mp.~ctification; Compactificat~. :, ]~¢.~mitian symmetric space; K d,.e~;y; Lie group; Lie group, semi-simple; Modular curve; Moduli theory; Parabolic subgroup; Representation of a Lie algebra; Resolution of singularities; Riemann surface; Symmetric space) Satakecompactification s e e : non-Hermitian --

[90Cxx] (see: Fritz John condition) Saffman-Taylor finger [76Exx, 76S05] (see: Viscous fingering) Saffman-Taylorinstability [76Exx, 76S05] (see: Viscous fingering) Saffman-Taylorproblem [35Q53, 58F07] (see: Harry Dym equation) Brenf---; Gauss-

Salamin-Brent algorithm

[26Dxx, 65D20] (see: Arithmetic-geometric mean process)

maximal - - ;

[11Fxx, 20Gxx, 22E46] (see: Baily-Borel eompactification) satisfiability algorithm [68Q15] (see: Average-case computational complexity) satisfiability problem [68Q15] (see: Average-case computational complexity) Wiener--

Savitch theorem

[03D15, 68Q15] (see: Computational classes) SAW

complexity

* -condition

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) sawtooth function

[llLxx, llL03, 11L05, 11L15] (see: Bombieri-Iwaniec method)

SBIBD (05B05) (refers to: Block design) SBR measure

[28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invariant measure) scale analysis s e e :

multi- - -

scattered compact space [26A15, 54C05]

(see: Namioka space) scattered set

S c h e n s t e d - K n u t h correspondence shifted R o b i n s o n - - -

[05C65, 05D05, 68Q15, 6gT05] (see: Vapnik-Chervonenkis dimension) scattered set [43A45, 43A46] (see: Ditkin set)

Schiuzel-Zassenhausconjecture [11C08, 11R04] (see: Lehmer conjecture)

scatterer s e e : spherically symmetric - scattering s e e : direct potential - - ; Dirichlet condition in obstacle - - ; incident plane wave in - - ; inverse geophysical - - ; inverse obstacle - - ; inverse potential - - ; N e u m a n n condition in obstacle - - ; Obstacle -- ; Robin condition in obstacle - - ; S - m a t r i x for electromagnetic wave - -

scattering amplitude

[35P25, 47A40, 81U20] Inverse scattering, multidimensional ease; Obstacle scattering) (see:

scattering by a plane wave

scattering data

[35P25, 47A40, glU20] (see: Inverse scattering, half-axis case) scattering data [35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-fine case) scattering data s e e : characterization property for - scattering, full-line case s e e : Inverse -scattering, half-axis c a s e s e e : Inverse - scattering, multi-dimensionalcase s e e : Inverse - -

scattering potential

[35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) scattering problem

[glUxx] (see: Enss method)

scattering problem [35P25] (see: Obstacle scattering) scattering problem s e e : direct - - ; inverse - - ; inverse geophysical - - ; inverse potential - - ; obstacle -scattering problem on the half-axis s e e : direct --

[35P25] (see: Obstacle scattering) scattering solution [35P25, 47A40, 81U20] (see: Inverse scattering, dimensional ease)

multi-

scattering theory s e e : characterization theorem of inverse - - ; H a a g - R u e l l e - - ; r e c o n s t r u c t i o n theorem of inverse - - ; resonance in - - ; uniqueness t h e o r e m of inverse - scatteringtransform s e e : inverse - Schachertheorem see: Benard--Schauder basis

[47H171 (see: Approximation solvability) Schauder degree see: L e r a y - - -

Schauder fixed-point theorem [47H10] (see: Darbo fixed-point theorem) Schef/E4ype simultaneous confidence intervaI

[62Jxx] (see: ANOVA) s c h e m a a x i o m a t i z a b l e see: f i n i t e l y - schema of replacement s e e : axiom - schema of separation s e e : axiom - scheme s e e : arithmetically B u c h s b a u m - - ; association - - ; complete projection - - ; formal - - ; multi-grid - - , nested projection -- ; rational interpolation - -

s¢¢:

Schliedertheorem see: Reeh--Schmidt category s e e : Krull- --; KrullRemak- -Schmidt-typetheorem see: Krull- --

Schneider-Lang criterion

[11J851 (see: Gel'fond-Schneider method)

Schneider method (11Jg1) (referred to in: Gel'fond-Schneider method; Mahler method) (refers to: Algebraic number; Dirichlet principle; Hilbcrt problems; Linear independence; Number field; Transcendency, measure of; Transcendental number) Schneider m e t h o d s e e :

[35P25, 47A40, 58F07, 81U20] (see: Inverse scattering, full-line case)

scattering solution

Satake topology

sausage s e e : Ne)'mark---

saddle-point characterization of optimality

Salamin algorithm s e e :

sample correlation coefficient [62H20] (see: Pearson product-moment correlation coefficient) sample elements see: concordant --;

sampling method [62D05] (see: Acceptance-rejection method) Santal6 formula (53C20, 53C22) (refers to: Contact structure; Geodesic flow; Riemannian manifold)

S

over -Sacker bifurcation s e e :

salesman path problem s e e : travelling - salesman problem s e e : Travelling -salesman w a l k problem s e e : travelling - s a m p l e s e e : complete - - ; i n c o m p l e t e - -

Gel'fond---

Scholes formula s e e : Slack- -Scholes geometric 8rownian motion model see: Black- -S c h o l e s - M a r t o n option pdcing s e e : Slack- -Schreier coset d i a g r a m

[05C25] (see: Cayley graph)

Schrrder functional equation (39B05, 39B12) (referred to in: Semi-group of holomorphic mappings) (refers to: Functional equation; Semigroup) SchrSder functional equation s e e : nonautonomous -Schrbdingar equation see: n o n - l i n e a r -

Schrrdinger operator [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) Schrbdinger operator s e e :

random --

Schubert calculus (14C15, 14MI5, 14N15, 20G20, 57T15) (referred to in: Schubert cycle) (refers to: Algebraic geometry; Algebraic topology; Borel subgroup; Chow ring; Flag; Grassmann manifold; Hilbert problems; Homology; Intersection theory; Schubert cycle; Schubert polynomials; Schur functions in algebraic combinatorics) Schubert cell (14L35, 14M15, 20G20) (refers to: Affine space; Algebraically closed field; Borel subgroup; CWcomplex; Elimination theory; Field; Grassmann manifold; Linear algebraic group; Maximal torus; Parabolic subgroup; Weyl group) Schubert class

[14C15, 14C17, 14M15, 20G20, 57T15] (see: Schubert cycle) Schubert cycle (14C15, 14C17, 14M15, 20G20, 57T15) (referred to i~: Schubert calculus; Schubert polynomials) (refers to: Chern class; Cohomology group; Cohomologyring; Flag structure; Linear algebraic group; Parabolic subgroup; Schubert calculus; Schubert variety) Schubert cycle [05El0, 05E99, 20C25] (see: Schur Q-function) Schubert cycles see: basis theorem for - - ; duality theorem for - -

Schubert enumerative calculus

[14C15, 14M15, 14N15, 20G20, 57Tt5] (see: Schubert ealculus) Schubert polynomials 531

SCHUBERT POLYNOMIALS (05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15) (referred to in: Schubert calculus) (refers to: Borel subgroup; Cohomology ring; Lexicographic order; Reduetive group; Schubert cycle; Symmetric group; Symplectic group) Schubert polynomials see: quantum - - ; universal - -

double - - ;

Sehwarz lemma

Schubert variety [14CI5, 14M15, 14N15, 20G20, 57T15] (see: Schubert calculus) Schur algebra (20Bxx, 20B 15, 20C30) (refers to: Schur ring) Scbur algebra [16Gxx] (see: Tilting theory) Scliur continued-fraction-like algorithm [33C45] (see: Szeg/i polynomial) Schurcount of involutions

see:

Frobenius-

Schur group (11R34, 12G05, 13A20, 16S35, 20C05) (refers to: Algebraic number; Brauer group; Character of a group; Clifford theory; Field; Finite group; Galois extension; Galois group; Group algebra; Number field; Schur index) Schur group [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) Schur-Hadamard multiplication [03Exx, 03E05] (see: Coherent algebra) Schur-Hadamard multiplication [05Exx] (see: Cellular algebra) Schur indices see: Wedderburn theorem on-Schur inequality see: logue of the - -

permanental ana-

Schur inequality for immanants [15A15, 20C30] (see: Immanant) Schur matrix [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) Schur parameter [33C45] (see: Szeg6 polynomial) Schur polynomial [05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) Schur Q-function (05El0, 05E99, 20C25) (referred to in: Projective representations of symmetric and alternating groups) (refers to: Pfaffian; Projective representations of symmetric and alternating groups; Representation of the symmetric groups; RobinsonSchensted correspondence; Sehur functions in algebraic combinatorics; Young diagram) Schur Q function [20C25] (see: Projective representations of symmetric and alternating groups) Schur ring [05Exx] (see: Cellular algebra) Schur ring

see:

pseudo-

-

-

Sehur stability of polynomials and ma-

trices (15A18, 93C05, 93D15) (refers to: Characteristic polynomial; Dynamical system; Eigen value; Matrix; Normal matrix; Pole assignment problem)

532

Schur theorem on matrices [15AI8, 93C05, 93D15] (see: Schur stability of polynomials and matrices) Schwartz space [11F27, I1F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) Wolff---

see:

Schwarz-Pick lemma [30C45, 47H10, 47H20] (see: Jniia-Wolff-Carath~odory theorem) Schwarz-Pick pseudo-metric [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) Schwarzscbild geometry (53B30, 53B50, 83C20, 83F05) (refers to: Schwarzschild metric) Schwarzschild solution (53B30, 53B50, 83C20, 83F05) (refers to: Schwarzsehild metric) science see: location - science of see: Quantum information processing, --

Scott height [03C15, 03C45, 03E15] (see: Vanght conjecture) Scott rank [03C15, 03C45, 03E15] (see: "Caughtconjecture) Scottish Book [54E52] (see: Banach-Mazur game) search [68T05] (see: Machine learning) search see: hill-climbing - - ; quantum

--;

second Brocard point [51M04] (see: Brocard point) second flip conjecture [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) second kind see: Szeg6 polynomial of the - -

second Listing polynomial [57M25] (see: Listing polynomials) second neighbourhood of a point on an algebraic curve [12Fl0, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) second-order algebraizable logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) second-order equivalent quasi-variety [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) see:

full - -

second-order finitely algebraizable deductive system [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) second-order Leibniz congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) socond-ordermodel reduced full - -

see:

full - - ; Leibniz-

second-order trace formula [42A16, 47B35] (see: Szegti limit theorems) second spacing problem [llLxx, llL03, 11L05, 11L15] (see: Bombieri-Iwaniee method) second Spencer complex [53C15, 55N35] (see: Spencer cohomology) second Zolotarev problem

Segat construction NaYmark- - Segal indexformulas

Gerfand-

see:

see:

Atiyah- --

Segal-Shale-Weil representation (11F27, 11F70, 20G05, 81R05) (refers to: Addle; Automorphism; Base change; Centralizer; Character of a group; Global field; Group; Intertwining operator; Irreducible representation; Lie algebra, exceptional; Local field; Poisson summation formula; Projective representation; Quadratic reciprocity law;

Symplectic group; Theta-function; Vector space) Segal-Shale-Weil representation [11F27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) Sega[space

unstructured quantum -search algorithm see: G r o v e r - -

second-orderfilter

[41-XX, 41A50] (see: Zolotarev polynomials) secondary characteristic class [55R40, 57Rxx] (see: Well algebra of a Lie algebra) section [16G70] (see: Almost-split sequence) section see: cross- - -

Bargmann---

see:

Segre characteristic at an eigenvalue [15A18, 15A21] (see: Segre characteristic of a square matrix) Segre characteristic of a square matrix (15A18, 15A21) (refers to: Eigen value; Field; Jordan matrix; Matrix) Segre characteristic of a square matrix [15A18, 15A21] (see: Segre characteristic of a square matrix) Segur hierarchy Newell- - -

see:

Ablowitz-Kaup-

Seiberg-Witten differential [i4Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Seiberg-Witten equations [81V10] (see: Dirac monopole) Seiberg-Witten invariants [81V10] (see: Dirac monopole) Seiberg-Witteu theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Seiberg-Witten Toda curve [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Seifert circle [57M25] (see: Positive link) Seifert conjecture (58F22, 58F25) (refers to: Contact structure; Dynamical system; Essential mapping; Flow (contlnuous-time dynamical system); Hamiltonian system; Hopf fibration; Vector field) Seifert conjecture see: the - - ; modified --

eounterexampleto

Seifert construction [57M251 (see: Positive link) Selberg conjecture (11F03, I1F70) (refers to: Addle; Exponential sum estimates; Irreducible representation; Laplace operator; Modular form; Principal series; Riemann hypotheses; Self-adjoint operator; Trigonometric sum; Zeta-function) Selberg formula Selberg method

see: see:

Chowla--Rankin---

Selberg theorem [I1F03, 11F70] (see: Selberg conjecture) selected points

see:

method of - -

selector [03E30] (see: ZFC) self-adjoint operator [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) self-adjoint operator s e e : essentially - - ; triangular model of a non- - -

self-dual basis [12E20] (see: Galois field structure) self-dualconnection s e e : self-dual polynomial basis

anti- - weakly - -

see:

self-dual Yang-Mills equations [35Qxx, 78A25] (see: Magnetic monopole) self-extensional deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) self-financing portfolio [90A09] (see: Option pricing) self-financing portfolio strategy [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) self-injective algebra [16G70] (see: Riedtmann classification) self-injective algebra [16G70] (see: Almost-split sequence) self-similar fractal [28A80] (see: Sierpifiski gasket) self-similar measure [28A80] (see: Sierpirlski gasket) self-similarset see: p.e.f. --;post-critically finite - self-similaritysee: strict - -

semantical meaning [03Gxx] (see: Algebraic logic) semantical path from logic to algebra [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) semantical system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) semantical system see: algebraizable ; algebraizable general - - ; equivalential - - ; equivalential general - - ; finitely algebraizable - - ; finitely algebraizable general - - ; finitely equivalential - - ; finitely equivalential general - - ; general - - ; protoalgebraic - - ; protoalgebraic general - semantics see: algebraic - - ; equivalent algebraic - -

-

semantics-basedabstract algebraic logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) semantics of a natural language [68S05] (see: Natural language processing) semi-Abelian variety [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactification) semi-complete holomorphic vector field [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) semi-complete vector field [32H15, 34G20, 46G20, 47D06, 47H20] (see: Seml-group of holomorphic mappings) semi-continuous regularization per - semi-decidability

see:

up-

[68S05] (see: Natural language processing)

SEVENTHPROBLEM

semi-definiteHermitian matrices see: generalization of the H a d a m a r d - F i s c h e r inequality for positive - -

Semi-Fredhelm operator (47A53) (refers to: Banach space; Compact operator; Continuous operator; Fredholm operator; Index of an operator; Normally-solvable operator) semi-Fredholm operator

[47A53] (see: Semi-Fredholm operator) semi-group s e e : additive arithmetical - - ; arithmetical - - ; a x i o m - A arithmetical - - ; a x i o m - A # arithmetical - - ; axiomC arithmetical - - ; a x i o m - ~ arithmetical - - ; a x i o m - O 1 arithmetical - - ; a x i o m - O l arithmetical--; axiom-O;~ arithmetical - - ; C o - - - ; classical arithmetical - - ; continuous - - ; degree on an additive arithmetical - - ; differentiable continuous - - ; exponential formula representation for a linear - - ; exponential formula representation of a continuous - - ; exponential representation of a continuous - - ; generated continuous -- ; generator of a continuous - - ; locally uniformly continuous - - ; nonexpansive - - ; n o r m on an arithmetical - - ; primes in an arithmetical - - ; product formula representation of a continuous - - ; right action of a - -

semi-group generated by a holomorphic mapping [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) semi-group of complete orthomodular lattices s e e : Foulis - -

semi-group of composition operators

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphie mappings) semi-groupof compositionoperators [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) Semi-group of holomorphic mappings (32H15, 34G20, 46G20, 47D06, 47H20) (referred to in: Denjoy-Wolff theorem; Julia-Wolff-Carathdodory theorem) (refers to.: Accretive mapping; Adjoint space; Banach space; Banach space of analytic functions with infinite-dimensional domains; Cauchy problem; Contraction semigroup; Denjoy-Wolff theorem; Duality; Evolution equation; Fr~chet derivative; Functional equation; Hyperbolic metric; Lie algebra; Maximum principle; Poinear~ model; Schrdder functional equation; Semigroup of operators; Strong topology; Transvection; Uniform convergence; Vector field) semi-groups see: differentiability of continuous - - ; flow-invadance for continuous - - ; Martin theorem on -- ; parametric representations of generators of continuous - -

semi-open set

[54C08] (see: Almost continuity) semi-primitive Jordan algebra

(see: Abstract algebraic logic)

semi-topological group s e e : Cechcomplete - s e n s e s e e : Ditkin set in the wide --

sense ambiguity in a natural language

[68S05] (see: Natural language processing) Ditkin set s e e : wide- -s e n s e of Conway s e e : algebraic tangles in the - sense

sentence in Lw i w [03C15, 03C45, 03E15] (see: Vaugbt conjecture) sentences in logic s e e : lent --

Q-roots

semi-simple Jordan algebra

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) semi-topological group

fundamental

Sequential probability ratio test (62L10) (refers to: Average sample number; Distribution function; Likelihoodratio test; Random variable; Sequential analysis; Statistical hypotheses, verification of; Stochastic process) sequential probability ratio test

[62Lxx, 62LI0] quential probability ratio test)

logically equiva-

sentential logic [03Gxx, 03G05, 03G10, 03GI5, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) separable C * - a l g e b r a s e e : a--

sequential analysis s e e : identity of - -

(see: Average sample number; Se-

extension of

separate continuity [54C05, 54C08J (see: Strongly countably complete topological space) separated collection of sets s e e :

(k--1)-

separately harmonic function

[31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) separately harmonic functions s e e : value characterization of - -

mean-

separating space of a linear mopping

[46H40] (see: Automatic continuity for Ba-

nacb algebras) separating surface s e e : non- - separatingthe points of a set s e e : function algebra -separation s e e : axiom of - - ; axiom s c h e m a of --

separation theorem

[15A39, 90C05] (see: Motzkin transposition theorem)

separation theorem [15A39, 90C05] (see: Motzkin transposition theorem) separation theorem s e e : Jordan - sequence s e e : Almost-split - - ; Auslander-Reiten - - ; convex null - - ; d- - ; d + - - ; Fibonacci - - ; gap in a canonical--; Kronecker-delta--; Laplace - - ; least period of an ultimately periodic - - ; Leray-Serre spectral - - ; Linear complexity of a - - ; linear complexity of a shift register - - ; linear complexity profile of a - - ; m - -- ; maximal period -- ; minimal non-split short exact -- ; minimal polynomial of a shift register - - ; non-split short exact -- ; period of a periodic - - ; period of an ultimately periodic - - ; periodic - - ; PN- - - ; p s e u d o - n o i s e - - ; pseudorandom - - ; q u a s i - c o n v e x - - ; recurrent --; recursive--; Rosenberg-Zelinsky exact - - ; Shift register - - ; short exact - - ; Tribonacci -- ; Ultimately periodic -- ; unconditioned strong d- - - ; USD- - - ; weak A - sequence in the theory of HNN-extensions see: reduced - sequence of a linear f e e d b a c k shift register see: impulse-response - sequence of approximate eigenfunctions see: Weyl - sequence of C * - a t g e b r a s s e e : short exact - -

sequence vanishing eventually

[17C65, 46H70, 46L70] (see: Banach-Jordan algebra) semi-simple algebraic group s e e : of a - - ; R-roots of a --

[54C08] (see: Almost continuity)

[46F30] (see: Rosinger nowhere-dense gener-

alized function algebra) sequences s e e : Auslander-Reiten theorem on almost-split --

sequent

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35]

sequential probability ratio test s e e : optimality property for the - -

strong

sequential test [62L10] (see: Sequential probability ratio test) sequentially complete [22D10, 43A07, 43A30, 43A35, 43A45, 43A46, 46J10] (see: Fourier algebra) sequentiallycomplete s e e : weakly -series see: Eisenstein --; Fitting --; F o u r i e r - C h e b y s h e v - - ; Fourier-Franklin --; Fourier-Haar--; Fourier--Jacobi--; Feurier-Laguerre--; Fourier-Legendre - - ; Fourier-Walsh - - ; H a a r - F o u r i e r - - ; Hadamard-lacunary Fourier - - ; Hilbert - - ; Integrability of t r i g o n o m e t r i c - - ; linear means of a Fourier - - ; lower Fitting - - ; lower nilpotent - - ; nilpotent - - ; Peincare - - ; Poincard-Eisenstein - - ; radius of c o n v e r g e n c e of a numerical - - ; summability of Fourier - - ; upper Fitting - - ; upper nilpotent -- ; Volterra functional - - ; Wiener-Volterra functional - - ; Young theorem on trigonometric -series representation s e e : complementary - - ; Fourier - - ; principal - series with s u m m a N e majorant of coefficients see: Beading algebra of Fourier - Serre compactification s e e : BorN- --; reductive B e t e l - -Serre spectral sequence s e e : L e r a y - - -

Serrc theorem in group cohomology (20J06) (refers to: Cohomological dimension; Cohomology; Cohomology of groups; Cohomology operation; Dimension; Finite group; p-group; Pro-p-group; Profinite group) Serretheory see: Bass- -Serre theory of groups acting on trees s e e : B a s s - -server see: ontology - -

sesquilinear mapping [17Cxx, 46-XX] (see: JB *-triple) set s e e : 1/4-Cantor--; A-antisymmetric - ; anti-chain in a partially ordered - - ; axiom of power - - ; axiom of the empty - - ; Bernstein - - ; branch - - ; C - - ; Calderdn - - ; characteristic polynomial of a ranked partially ordered - - ;CIarke tangent cone to a - - ; complete function - - ; confidence - - ; corona - - ; countably infinite - - ; covering relation in a partially ordered - - ; Denjoy - - ; difference - - ; Dilworth number of a partially ordered - - ; Ditkin - - ; dual of a nonempty - - ; example of a 1-rectifiable - - ; example of a purely 1-unrectifiable - - ; exit - - ; family of subsets of a - - ; function algebra separating the points of a - - ; index pair for an isolated invariant - - ; infinite - - ; isolated invariant - - ; k-family in a partially ordered - - ; level of a rank function on a partially ordered - - ; locally finite p a r t i a l l y ordered - - ; LYM partially ordered - - ; m-rectifiable - - ; normalized matching property of a partially ordered - - ; order complex of a partially ordered

- - ; p.c.f, self-similar - - ; partially A - a n t i symmetric - - ; Peck partially ordered - - ; Perron - - ; place of the rational number - - ; pluripolar - - ; polar of a non-empty - - ; p o l y h e d r a l - - ; polynomially convex - - ; post-critically finite self-similar - - ; purely m-unrectifiable - - ; quantal - - ; rank function on a partially ordered - - ; rank of a partially ordered - - ; residual - - ; scattered - - ; semi-open - - ; skein - - ; Sperner number of a partially ordered - - ; strong Ditkin - - ; test - - ; total function - - ; training - - ; well-founded - - ; wide-sense Ditkin - - ; width of a partially ordered - - ; Wiener-Ditkin - set criterion s e e : Hdrmander wave front -set function s e e : finitely additive additive - - ; monotone - - ; non-additive - - ; nulladditive - - ; subadditJve - - ; superadditive - - ; triangular - set in a lattice s e e : spanning - set in the wide sense s e e : Ditkin - setlattice-theoretictopologysee: point- -set of a dynamical system s e e : minimal - set of a function s e e : zero - set of a group action s e e : fixed-point - set of a logical matrix s e e : designated - set of atomic formulas s e e : set of formulas defining a set of atomic formulas explicitly over another - - ; set of formulas defining a set of atomic formulas implicitly over another - - ; strong implicit definition of a set of atomic formulas over another - set of atomic formulas explicitly over another s e t of atomic formulas s e e : set of formulas defining a -set of atomic formulas implicitly over another set of atomic formulas s e e : set of formulas defining a - set of atomic formulas over another set of atomic formulas s e e : strong implicit definition of a - set of formulas s e e : model of a - - ; theory axiomatized by a --

set of formulas defining a set of atomic formulas explicitly over another set of atomic formulas

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) set of formulas defining a set of atomic" formulas implicitly over another set of atomic formulas

[03Gxx, 03G05, 03G10, 03GI5, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) set of formulas for a class of i n t e r p r e t a t i o n s see: defining --

set of generalized gradients

[90C301 (see: Clarke generalized derivative) set

of prime ideals s e e :

regular--

set of spectral synthesis

[43A45, 43A46] (see: Ditkin set)

set of synthesis [42A16, 42A24, 42A28] (see: Beurling algebra) set of vectors s e e : extension of a - - ; vacuous extension of a - set problem s e e : C - s e t - S - - ; k - s e t - S - s e t problem s e e : C . - set theory s e e : axiomatization of - - ; Gddel relative consistency of - - ; language of -- ; Zermelo -- ; Zermelo-Fraenkel -set theory) s e e : Z (in - - ; ZF (in - set theory with the axiom of choice s e e : Zermelo-Fraenkel - sets s e e : cumulative hierarchy of the universe of - - ; exponential law for - - ; injection theorem for Ditkin - - ; ( k - - ] ) separated collection of - - ; universe of -sets for a belief function s e e : independent variable --

sets of finite Hausdorff measure s e e : composition theorem for - seventh problem s e e : Hilbert - -

de-

533

SHABATDRESSINGMETHOD

Shabat dressing method s e e :

Zakharov-

shadow s e e : u p p e r - shadow of afamily of subsets s e e :

lower --

Shafarevich conjecture ( l 1R32) (refers to: Algebraically closed field; Class field theory; Cohomologiealdimension; Finite field; Finite group; Galois group; Galois theory, inverse problem of; Global field; Profinite group; Projective group; Tower of fields) Shafarevich conjecture [1IR32] (see: Shafarevich conjecture) Shafarevich conjecture ized - -

see:

general-

Shafarevich conjecture in inverse Galois theory

111R32] (see: Shafarevich conjecture) Shafer algorithms s e e : S h e n o y - - Sharer theory s e e : axiomatic approach to D e m p s t e r - - - ; D e m p s t e r - - ; marginally correct approximation approach to D e m p s t e r - - - ; naive approach to D e m p s t e r - - - ; qualitative approach to D e m p s t e r - - - ; quantative approach to D e m p s t e r - -Shale-Weil representation see: S e g a l - - Shalen-Johansson decomposition theorem see: Jaco--Shaten-Johansson splitting theorem s e e : Jade- --

Shannon capacity theorem [68Q05, 68QI0, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) shaped domain s e e :

star- - -

Sharp conjecture on dualizing complexes

[13A30, 13H10, 13H30] (see: Buchsbaum ring) Shawcell see: Hole- -sheaf s e e :

resolvent of a - - ; tilting --

sheaf on a quantale [03G25, 06D99] (see: Quantale) shearviscosity s e e : fluid - shedding see: drag due to vortex - - ; Kfirmfin vortex - - ; total drag due to vortex -- ; Von K & m f i n vortex - sheet s e e : planar vortex - - ; rolled-up vortex - - ; vortex - sheets s e e : induced velocities in vortex - -

shellable simplicial complex [16Gxx] (see: Tilting theory) Shenoy-Shaferalgorithms [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) see: S i e g e l - -Shidlovskff-Siegel method Shidlovskiimethod

[11R99]

Shift register sequence (11B37, llT71, 93C05) (referred to in: Linear complexity of a sequence; Ultimately periodic sequence) (refers to: Cryptography'; Fibonacci numbers; Field; Formal power series; Galois field; Galois field structure; Pseudo-random numbers; Recursive sequence; Ultimately periodic sequence) shift register sequence s e e : linear complexity of a - - ; minimal polynomial of a - -

shifted Robinson-Schensted-Knuth correspondence [05El0, 05E99, 20C25] (see: Schur Q-functlon) shifted tableau s e e : - - ; marked - -

content of a marked

shifted Youngdiagram [05El0, 05E99, 20C25] (see: Schur Q-function) shifting technique s e e :

Spurner --

short exact sequence s e e : split - - ; non-split --

minimal non-

short exact sequence of C'* -algebras [46110, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras)

(see: Siegel-Shidlovskii method) shift see: phase - -

shortest length see: vector of - shortest-pathalgorithm s e e : D i j k s t r a - -

shift behaviourof the Z-transform [39A12, 93Cxx, 94A12] (see: Z-transform) shift-invariant subspace [30D55, 46J15, 47A15] (see: Beurling theorem)

shortest-path problem

shift of the Zak transform

Sidon-TelyakovskE space

[05C12, 90C27] Sieiak-type mean-value Temlyakov-Opial- --

theorem

see:

[42Axx, 44-XX, 44A55]

[42A20, 42A32, 42A38] (see: Integrability of trigonometric

534

series) Sidon-type inequality [42A20, 42A32, 42A38] (see: Integrability of trigonometric series) Siegel domain [46Lxx] (see: Tooplitz C* -algebra) Siegellemma see: Siegel method s e e :

[1iFxx, 20Gxx, 22E46] (see: Baily-Borel compactification) Siegel modular group s e e :

Thue--Shidlovskii- - -

Siegel modularform

Hilbert---

Siegel-Shidlovskil method (11R99) (refers to: Siegel method) Siegel upper half-space [I 1Fxx] (see: Satake compactification)

simple Jordan algebra s e e : semi- - simple Jordan pairs of finite capacity s e e : Lops classification of - -

simple k-order type

[52A35] (see: Geometric transversal theory)

[28A80] (see: Sierpifiski gasket) Sierpifiski carpet [28A80] (see: Sierpifiski gasket)

simple matching polynomial of a graph

universalityof the - -

Sierpifiski game (03E50, 54-XX, 90D80) (refers to: Borel set; Cantor discontinhum; Complete metric space; Continuum hypothesis; Descriptive set theory; Separable space; Topological space; Vague topology) Sierpifiski gasket (28A80) (refers to: Algebraic number; Cantor set; Connected space; Dimension; Dynamical system; Fractals; Hausdorff dimension; Ising model; Peano curve; Rectifiable curve; Totally-disconnected space; Zerodimensional space) Sierpifiski gasket s e e : Brownian motion on the - - ; fractal dimension of the - - ; Laplaeian on the - sieve s e e : c o i n c i d e n c e s in the double large - - ; double large - -

sieve formula

[05A99, 11N35, 60A99, 60E15] (see: Inclusion-exclusionformula) sieve method

[05A99, 11N35, 60A99, 60El 5] (see: Inclusion-exclusionformula) sign correlation coefficient s e e : ence --

differ-

sign theorem

[05B35, 05Exx, 05E25, 06A07, 11A25] (see: M6bius inversion) Rote - -

signal processing [12E201 (see: Galois field structure)

siguamre of an algebraic number field [llR29] (see: Odlyzko bounds) signature operator [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) Hirzebruch - -

Silva differentiability [46F30] (see: Colombeau generalized function algebras) Silva-differentiable mappings of infinitely - -

see:

[05Cxx, 05D15] (see: Matching polynomial of a graph) simple measurablefunction [28-XX] (see: Chequer integral) simple point on an algebraic curve

[12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) simple root of a Borcherds algebra s e e : imaginary - - ; real - -

simplex algorithm [68Q15] (see: Average-case computational complexity) simplicial complex s e e : Cohen-Macaulay - - ; f - v e c t o r of a - - ; h-vector of a - - ; shellable - - ; S t a n l e y - R e i s n e r ring of a - - ; Tits - -

simplicial group [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) simplicial groupoid [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) simply-laced affine Lie algebra [11FI 1, 17B10, 17B65, 17B67, 20D08, 81R10, 81T30] (see: Vertex operator) simuRaneous confidence interval Scheffe-type - - ; Tukey-type - -

see:

simultaneousconfidence intervals [62Jxx] (see: ANOVA) Sinai-Bowen-Ruelle measure

[28Dxx, 541120, 58F11, 58F13] (see: Absolutely continuous invariant measure) sinc function

signature see: Murasugi - - ; TristramLevine -signature of a link s e e : Murasugi - - ; Tristram-Levine --

see:

[16G70] (see: Almost-split sequence) [55Pxx, 55P15, 55U35] (see: Algebraic homotopy)

see: tamis de - Sierpir~ski carpet

sign theorem s e e :

space

SIMD (68Q10)

[42Cxx, 94A12] (see: Window function) Sinclair theorem s e e : J o h n s o n - Singer analytic torsion s e e : R a y - Singer index formula s e e : M c K e a n - -Singer index formulas s e e : A t i y a h - - Singer index theorem s e e : Atiyah- --; Atiyah-Patodi- -Singer operator s e e : A t i y a h - --

single facility location problem

[90B851 (see: Fermat-Torricelli problem) single-input single-output system

[73Axx] (see: SISO system) single-instruction multiple-data

[68Q10] (see: SIMD) single operator s e e : index theory for a - single-output system s e e : single-input--

singular drift

similarfractal see: self- -similar measure s e e : self- - similar set s e e : p.c.f, self- - - ; postcritically finite self- -similarity s e e : strict self- --

[60Hxx, 60J55, 60J65] (see: Skorokhod equation) singular partial differential equation [31B05, 33C55] (see: Zonal harmonics)

similarity-based analogy

singular point on an algebraic curve

[68T05] (see: Machine learning) similarity dimension [28A80]

of a

semi- - ; R-roots of a semi- - simple functor

simple homotopy theory

Sierpifiski

signature theorem

(see: Dijkstra algorithm) S i b o n y e x a m p l e see: G a m e l i n - - -

(see: Zak transform) shift register s e e : characteristic polynomial of a linear feedback - - ; feedback coefficients of a linear feedback - - ; feedback matrix of a linear feedback - - ; feedback polynomial of a linear feedback - - ; impulse-response sequence of a linear feedback - - ; initial conditions of a linear feedback - - ; length of a linear feedback - - ; linear feedback - - ; reciprocal polynomial of a linear feedback - - ; state vector of a linear feedback --

Siegel modular group

Sierpifiskiearpetsee:

Shilov boundary [46Lxx, 47Dxx] (see: Taylor joint spectrum; Toeplitz C* -algebra) Shil'nikov connection [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Shimura correspondence (llFll, 11FI2) (refers to: Automorphic form; Dirichlet L-function; Elliptic curve; Euler product; Fourier coefficients; Hecke operator; Modular form; Modular function; Thole-series) shock formation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) shocks for the Korteweg-de Vries equation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Shor factoring algorithm [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum computation, theory of; Quantum information processing, science of) Shor quantum algorithm [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory of) short exact sequence [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mtbius inversion)

(see: Sierpifiski gasket) Simon distributionsee: Yule-simple algebraic group s e e : Q-roots

[llFll, 11F12] (see: Shimura correspondence)

[12F10, 14I-I30,20D06, 20E22] (see: Chasles-Cayley-Brill formula)

singular point on an algebraic variety [13Hxx]

SPACE

(see: System of parameters of a module over a local ring) singularstochasticprocesssee: linearly-singulartoritechnique see: Lin -singular tuple of operators see: non- -singular values s e e : asymptotic distribution of - singularities see: Ak-curve - - ; Dirac string - singularity see: algebraic variety with terminal - - ; Moore - -

singularityin a soap film [28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) singularity manifold equation [35Q53, 58F07] (see: Harry Dym equation) singularity theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) sinkpoint [30D05, 321115, 46G20, 47H17] (see: Denjoy-Wolff theorem) SISO system (73Axx) (refers to: Automatic control theory) situational context in natural language [68S051 (see: Natural language processing) situs see: quantum Sivashinskydynamics see: dissipativityof the Kuramoto- - Sivashinsky equation see: bifurcation in the Kuramoto- - - ; Kuramoto- - - -

Sivashinsky-Kuramoto equation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) six exponentials theorem i l l J81] (see: Schneider method) six exponentials theorem see: Roy strong - - ; strong - skein see: Linear -skein equivalence see:

Conway --

Skein module (57Mxx, 57M25) (referred to in: Conway algebra; Conway skein triple; Drinfel'd-Turaev quantization; Homotopy polynomial; Jaeger composition product; Jones-Conway polynomial; Kauffman bracket polynomial; Linear skein) (refers to: Algebra; Algebraic topology based on knots; Drinfel'dTuraev quantization; Fundamental group; Homology group; Hopf algebra; Linear skein; Link; Manifold; Module; Montesinos-Nakanishi conjecture; Three-dimensional manifold) skein module [57Mxx, 57M25] (see: Skein module) skein module see: homotopy --; Jones-type - - ; Kauffman bracket - - ; q-homotopy - - ; Vassiliev-Gusarov - -

skein module based on relations deforming n-moves [57Mxx, 57M25] (see: Skein module) skein module based on the Jones-Conway relation [57Mxx, 57M25] (see: Skein module) skein module based on the Kauff}nan polynomial [57Mxx, 57M25] (see: Skein module) skein polynomial [57M25] (see: Jones-Conway polynomial) skein quadruple see:

skein relation [57M251

Kauffman --

(see: Brandt-Lickorish-MiUett-Ho polynomial) skein relation [57Mxx, 57M25] (see: Skein module) skein relation see: C o n w a y - - ; Kauffman bracket --

skein set [57P25] (see: Conway skein triple) skein triple see: bracket --

Conway - - ; Kauffman

Skolem-Noether theorem (13-XX, 16-XX, 17-XX) (refers to: Brauer group; Central algebra; Cross product; Field; Hilbert theorem; Inner automorphism; Picard group; Separable algebra; Simple algebra) Skorohod equation [60Hxx, 60J55, 60J65] (see: Skorokhod equation) Skorokhod equation (60Hxx, 60J55, 60J65) (refers to: Brownian motion; Markov process; Stochastic process) Skorokhod space (60B 10, 60G05) (refers to: Banach space; Central limit theorem; Metric; Metric space; Norm; Separable space; Skorokhod topology; Stochastic process; Weak convergence of probability measures) Skorokhod space [60B10, 60G05] (see: Skorokhod space) Skorokhodspace see:

generalized --

Skorokhod topology [60B10, 60G05] (see: Skorokhod space) Slater condition [90Cxx] (see: Fritz John condition) slice in a quiver [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) slice knot [57M25] (see: Positive link) slide see: Kirby move handle Slobodnik property (26A21, 54E55) (refers to: Baire classes; Baire space; Category of a set; Dense set; Topological space) Slobodnik property [26A21, 54E55] (see: Slobodnik property) Slobodnik theorem [26A21, 54E55] (see: Slobodnik property) slope of a rational tangle [57M25] (see: Rational tangles) slowly decaying function [34B24, 34L40] (see: Sturm-Liouville theory) small contraction [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) small horosphere [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Smarandache function (11A05) smash product [55P25] (see: Spanier-Whitehead duality) Smith-Solman-Waguer theorem [46Cxx] (see: Alternating algorithm) Smith theory of group actions (54H15, 55R35, 57S17) - -

(refers to: Action of a group on a manifold; Classifying space; Cohomology; Covering (of a set); EilenbergMacLane space; Finite group; General topology; Homology; p-group; Spectral sequence; Transformation group) smooth algebra [13B10, 13C15, 13C40] (see: Zariski-Lipman conjecture)

Solitar groups see: automorphisms of 8aumsrag- - - ; normal forms in Baumslag- - - ; subgroups of Baumslag-

smooth analysis see: smooth mapping see:

soliton lattice see:

non- - generic - -

smooth number [llAxx] (see: Dickman function) smoothing vertices of a graph see: f- -smoothness see: measure-theoretic-soap fiEm see: singularity in a - -

Sobolev-Besov space [46F10] (see: Multiplication of distributions) Sobolev function space see:

degree for

a--

Sobolev imbedding theorem

[46E35, 65N30] (see: Bramble-Hilbert lemma) Sobolev inner product (33C45, 33Exx, 46E35) (refers to: Absolutely continuous measures; Borel measure; Inner product; Lebesgue measure; Orthogonal polynomials; Polynomial) Sobolev inner product [33C45, 33Exx, 46E35] (see: Sobolev inner product) Sobolevinner product see: kegendre--Sobolev orthogonal polynomials see: Gegenbauer- -- ; Laguerre--

Sobolev space [35P25] (see: Obstacle scattering) Sobolev spaces see: of functions in -Sobolov spaces

approximation error

[46F10] (see: Multiplication of distributions) social choice [28-XX] (see: Non-additive measure) social network [05C501 (see: Matrix tree theorem) socle [l 7A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) socle of a Jordan pair [I 7A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) softO [12D05] (see: Factorization of polynomials) soft susy breaking [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations)

Solitar groups as BaumsIag- - -

examples

see:

soliton equations [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) weakly deformed - -

soliton solution [35Q53, 58F07] (see: Harry Dym equation) Solman-Wagnertheorem see: Smith- -Solomon theorem see: Feit---

Solotareff polynomials [41-XX, 41A50] (see: Zolotarev polynomials) Solow model [90A I 1] (see: Cobb-Douglas function) solution see: chaotic - - ; finite-gap - - ; gap Korteweg-de Vries - - ; Jest - - ; toil - - ; scattering - - ; Schwarzsehild - - ; so/iton - - ; subordinate - solution of the Korteweg-de Vries equation see: averaged -solvability

[47H17] (see: Approximation solvability) solvability see: A- - - ; Approximation - - ; Fredholm - - ; unique A- - - ; unique approximate - solvable equation see: normally - soIvableoperatorsee: normally - Solversoftware see: Dynamics - S o m m e d i e l d limit see: BogomolnyPrasad- - -

sophisticated Spencer complex [53C15, 55N35] (see: Spencer cohomology) SOR method (65F10) (refers to: Acceleration methods; Relaxation method) Sorgenfrey half-open square topology [26A21, 54E55, 54G20] (see: Sorgenfrey topology) Sorgenfrey line [26A21, 54E55, 54G20] (see: Sorgenfrey topology) Sorgenfrey line [26A15, 54C05] (see: Namioka space) Sorgenfrey topology (26A21, 54E55, 54G20) (referred to in: Namioka space) (refers to: Baire classes; Completelyregular space; Density topology; Fine topology; First axiom of countability; Hausdorff space; Lindel6f space; Locally compact space; Locally connected space; Metrizable space; Normal space; Nowhere-dense set; Paracompact space; Perfectly-normal space; Second axiom of countability; Separable space; Topological product; Topological structure (topology); Zero-dimensional space)

software see: AUTO - - ; AUTO97 - - ; BOV-method - - ; CANDYS/QA - - ; CONTENT - - ; DDE-BtFTOOL - - ; DsTool - - ; dynamical systems - - ; Dynamics Solver - - ; GAIO - - ; Global Manifolds 1D - - ; Global Manifolds 2D - - ; PDECONT - - ; PLTMG - - ; statistical analysis - - ; XPP - -

sort see:

software for dynamical systems [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages)

sound propagation [35L15] (see: Euler-Poisson-Darboux equation)

software packages see: tems

Sova compact derivative see: Gil de Lamadrid and space see: 1-co-connected - - ; 2-coconnected - - ; i'll-additive topological - - ; c~-favourable - - ; o~-favourable topol o g i c a l - - ; a c y c l i c - - ; almost convex metric - - ; amalgam - - ; BargmannSegal - - ; Bergman - - ; binormal bitopol o g i c a l - - ; bitopologica[--; Bloch - - ; BMOA - --; Boas-Telyakovskfi - - ; boundary component of a symmetric - - ; bounded symmetric domain in a

Dynamical sys-

- -

solid zonal harmonics" [31B05, 33C551 (see: Zonal harmonics) Solitar-BaumsIag group [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) Solitar group see: Baums[ag- - - ; convex rigid Baumslag- - - ; metaAbelian Baumslag- - - ; presentation of a Baumslag- - - ; rigid Baumslag- --

quick- - -

535

SPACE Banach --; BUntin as-Tan eric-Mille r --; (~ech-complete --; Oassical state --; classifying - - ; co-Namioka - - ; compactly generated topological - - ; CW- - - ; degree for a Sobolev function - - ; extreme point of the closed unit ball in a Banach - - ; Fano-Mori fibre - - ; finitistic - - ; Fock - - ; Fomin - - ; fuzzy topological - - ; Gaussian - - ; generalized Skorokhod - - ; GillmanHenriksen /9- - - ; Hardy - - ; Hausdorff k - - - ; hereditarily submetacompact - - ; Hermitian symmetric - - ; holomorphic function on a Banaeh - - ; interval in a partially ordered vector - - ; jet - - ; L-fuzzy topological--; L-topological - - ; latticefuzzy topological - - ; lattice-topological - - ; Lie algebra associated with a vector - - ; locally countably compact - - ; maximal ideal - - ; moduli - - ; Morita /9_ _ ; Namioka - - ; nest in a Banach - - ; normal analytic - - ; w o - m e t r i z a b l e - - ; /9- - - ; partially ordered vector - - ; phase - - ; Polish - - ; potential at points in - - ; quantum moduli - - ; rational boundary component of a symmetric - - ; Reproducing-kernel Hilbert - - ; Riemannian symmetric - - ; rigged Hilbert - - ; rigid analytic - - ; o-,G-defavourable--; o--fragmentable--; scattered compact - - ; Schwartz - - ; Sidon-TelyakovskiT - - ; Siegel upper half- ; Skorokhod - - ; Sobolev - - ; SobolevBesov - - ; Strongly countably complete topological - - ; strongly zero-dimensional metric - - ; sub-Stonean topological - - ; submetacompact - - ; universal Hilbert - - ; V M O A - - - ; weakly cx-favourable topological -space analysis see: phase- -space at a marked Riemann surface see: cotangent - -

space average [28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invarlant measure) space complexity class

see:

polynomial --

space-constructible function [03D15, 68Q15] (see: Computational complexity classes) space Korteweg-de Vries-kandauGinsburg model see: Hurwitz- -space of a linear mapping see: separating -space of a matrix see: null -space of an operator vessel see: external - - ; internal --

space of analytic functions of bounded mean oscillation [30Axx, 46Exx] (see: BMOA-space) space of analytic functions of vanishing mean oscillation [30D50, 46Exx] (see: VMOA-space) space of constant curvature [35P15] (see: Rayleigh-Faber-Krahn inequality) space of continuous functions [54E52] (see: Banach-Mazur game) space of flat connections

see:

moduli --

space of functions of bounded variation [42A20, 42A32, 42A3g] (see: Integrability of trigonometric series) space of infinitely Silva-differentiable mappings [46F30] (see: Colombeau generalized function algebras) space of real-valuedfunctions see: natural order on a -space of the Beurling algebra see: dual -space representation of a process see: state - -

space resource

536

[68Q15] (see: Average-case computational complexity) space system see: finite-dimensional state -space-time see: Minkowski - spaces see: approximation error of functions in Sobolev - - ; quasi-isometric metric - - ; Riesz decomposition theorem for harmonic - - ; Sobolov - spaces of automorphic forms see: Langlands formula for the dimension of -spacing problem see: first - - ; second - Spanierduality see: Whitehead- --

Spanier-Whitehead dual [55P25] (see: Spanier-Whitehead duality) Spanier-Whitehead duality (55P25) (refers to: Spectrum of spaces) Spanier-Whitehead duality [55P42] (see: Brown-Gitier spectra) spanning set in a lattice [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mtbius inversion) spanning subtree [05C12, 90C27] (see: Dijkstra algorithm) sparse function [34B24, 34L40] (see: Sturm-Liouville theory) sparse representation of a multivariate polynomial [12D05] (see: Factorization of polynomials) spatial form see: acceleration in - - ; formula for acceleration in - - ; formula for the material derivative in - - ; function in - - ; material derivative in - -

spatial Gel'Jhnd quantale [03G25, 06D99] (see: Quantale) spatio-temporal chaos [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) spafio-temporalpattern formation [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Spearman footrule [62H20] (see: Spearman rho metric) Spearman rho [62H20] (see: Spearman rho metric) Spearman rho metric (62H20) (referred to in: Kendall tau metric) (refers to: Copula; Correlation coefficient; Kendall tau metric; Pearson product-moment correlation coefficient; Random variable; Rank statistic) Specht problem [08Bxx, 16R10, 17B01, 20E10] (see: Specht property) Specht property (08Bxx, 16RI0, 17B01, 20El0) (refers to: Lie algebra; Universal algebra; Variety of universal algebras) Specht property [08Bxx, 16R10, 17B01, 20El0] (see: Specht property) special representation [20G05] (see: Steinberg module) specification of a program see: weakest pre- - specified statistical model see: partially -spectra see: Brown-Gitler - - ; dual Brown-Gitler - -

spectral coefficient representation [65Txx] (see: Fourier pseudo-spectral method)

spectral convergencerate [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) spectral curve [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) spectral curve of a fomily of line bundles [35Qxx, 78A25] (see: Magnetic monopole) spectral decomposition [34B24, 34L40] (see: Sturm-Liouville theory) spectral dimension [28A80] (see: Sierpifiski gasket) spectra1 element method [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) spectral factor of a weight function [33C45] (see: Szeg6 polynomial) spectral factorization problem [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) spectral inclusionfor Taylor spectrum [47Dxx] (see: Taylor joint spectrum) spectral measure [34B24, 34L40] (see: Sturm-Liouville theory) spectral measure associated with a Dirichlet problem [34B24, 34L40] (see: Sturm-Liouville theory) spectral measure of a partial differential equation see: quantum - spectral method see: Chebyshev - - ; Chebyshev pseudo- - - ; Fourier - - ; Fourier pseudo- -- ; pseudo- -spectral methods see: Trigonometric pseudo- -spectral methods for Jacobi matrices see: inverse --

spectral permanence for the Taylor spectrum [47Dxx] (see: Taylor joint spectrum) spectral permanence for the Taylor spectrum [47Dxx] (see: Taylor joint spectrum) spectral problem see: numerical approaches to the Sturm-Liouville - -

Sturm-Liouville

--;

spectral projection [47A10, 47B06] (see: Spectral theory of compact operators) spectral propertiesof the Beurling algebra [42A16, 42A24, 42A28] (see: Beurling algebra) spectral radius formula [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) spectralsequence see: k e r a y - S e r r e - spectral statistics [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) spectra[ synthesis problem in --

see:

set of - - ; union

spectral tau method [65Lxx] (see: Tau method) spectral theory

see:

local - - ; Riesz - -

Spectral theory of compact operators (47A10, 47B06) (referred to in: Riesz operator) (refers to: Banach space; Compact operator; Riesz decomposition theorem) spectral theory of operators [34B24, 34L40] (see: Sturm-Liouville theory) spectraltransform see: inverse --

spectrally associative algebra s e e : locally -spectrum see: approximate point - - ; defect - - ; essential - - ; example of a Taylor - - ; example of the Taylor - - ; finite - - ; Harts - - ; point - - ; power - - ; spectral inclusion for Taylor - - ; spectral permanence for the Taylor - - ; sphere - - ; Taylor - - ; Taylor essential - - ; Taylor joint - - ; weak - spectrum for an invariant subspace see: Taylor - spectrum for compact operators see: Taylor - spectrum for Fredholm r~-tuples see: Taylor -spectrum in the finite-dimensional case see: Taylor - -

spectrum of a C*-algebra [03G25, 06D99] (see: Qnantale) spectrum of a C*-filtration [46Lxx] (see: Toeplitz C*-algebra) spectrum of a linear operator part of the - -

see:

isolated

spectrum of an element in a BanachJordan algebra [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) spectrum of an operator [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) spectrum of an operator s e e : essential - - ; isolated point in the - spectrum of operators see: joint -spectrum under multiplication see: Taylor - -

speedup learning [68T05] (see: Machine learning) Spencer cohomology (53C15, 55N35) (refers to: Cohomology;de Rham cohomology; Duality in complex analysis; Elliptic partial differential equation; Euler characteristic; Index formulas; Index theory; Lie group; Linear differential operator; Manifold; Prolongation of solutions of differential equations; Resolvent; Sheaf; Spectral sequence; Symbol of an operator; Vector bundle) Spencer cohomology of a differential operator [53C15, 55N35] (see: Spencer cohomology) Spencer complex [53C15, 55N35] (see: Spencer cohomology) Spencer complex ticated - -

see:

second - - ; sophis-

Sperner family [05D05, 06A07] (see: Sperner property; Sperner theorem) Spemer labeling [00A08, 90Axx] (see: Cake-cutting problem) Sperner number of a partially ordered set [05D05, 06A07] (see: Sperner property) Sperner property (05D05, 06A07) (referred to in: Kruskal-Katona theorem; Sperner theorem) (refers to: Partially ordered set; Sperner theorem) Sperner property [05Dfi5, 06A07] (see: Sperner property) Spernerproperty

see:

strong - -

Sperner shifting technique [05D05, 06A07] (see: Sperner theorem) Sperner system [05D05, 06A07]

STATIONARYSTRATEGYIN THE GENERALIZEDBANACH-MAZUR GAME

(see: Sperner theorem) Sperner theorem (05D05, 06A07) (referred to in: Kruskal-Katona theorem; Sperner property) (refers to: Kruskal-Katona theorem; Sperner property) Sperner theorem [05D05, 06A07] (see: Sperner theorem) Sperner theory [05D05, 06A07] (see: Sperner theorem) sphere see: elastic curve

on the --; homology 3- - - ; potential in a -sphere in R ~" see: area of the unit - -

sphere spectrum [55P25] (see: Spanier-Whitehead duality) spheres see: stable h o m o t o p y o f spherical decreasing rearrangement [35P15] (see: Rayleigh-Faber-Krahn inequality) spherical harmonicpolynomial [31B05, 33C55] (see: Zonal harmonics) spherical mapping [53A10, 53C42J (see: Weierstrass representation of a minimal surface) spherical partial Fourier sums [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) spherically homogeneoustree [20E08, 20E18, 20Fxx] (see: Branch group) spherically symmetric scatterer [35P25] (see: Obstacle scattering) spherically transitive group action [20E08, 20E18, 20Fxx] (see: Branch group) spin-1/2 fermion [15A66, 81R05, 81R25] (see: Panli algebra) spin bundle [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) spin-carryingparticle [81Q05, 81Txx, 81T20] (see: Massless Klein-Gordon equation) spin character [20C25] (see: Projective representations of symmetric and alternating groups) spin character of a symmetric group [05El0, 05E99, 20C25] (see: Schur Q-function) spin Dirac operator [46L80, 46L87, 55N15, 58GI0, 58G11, 58G12] (see: Index theory) spin C_manifold [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) --

spin matrices see:

Pauli - -

spin representation [20C25] (see: Projective representations of symmetric and alternating groups) spinless and masslessparticle [81Q05, 81Txx, 81T20] (see: Massless Klein-gordon equation) spinlessparticle [81Q05, 81Txx, 81T20] (see: Massless Klein-Gordon equation) spinodal decomposition [82B26, 82D35] (see: Cahn-Hilliard equation) spinode

[14H20] (see: Tacnode) spinet bundle [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) Spitzer identity [05E05, 60G50] (see: Baxter algebra) spline see: t h i n - p t a t e - splines see: Zolotarev perfect - split sequence see: Almost- -split sequences see: Auslander-Reiten theorem on almost- -split short exact sequence see: minimal n o n -

--;

n o n -

splitting see:

--

Heegaard - -

splitting a three-dimensional manifold at a surface [57N10] (see: Haken manifold) splitting fields f o r group algebras see: Brauer theorem on --

splitting prime ideal of an extension of algebraic number fields [11R44, 11R45] (see: Dirichlet density) splitting theorem see: Jaco-ShalenJohansson - - ; Riesz - -

splitting torsion pair [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting theory) splitting torsion pair [16Gxx] (see: Tilting fuuctor) s p o n g e see: Monger - Spraguefunction see: G r u n d y - - -

Sprague-Grundy function (90D05) (refers to: Arithmetic progression; Complexity theory; Two-person zero-sum game) Sprague~;rundy function [90D05] (see: Sprague-Grundy function) S p r a g u e - G r u n d y function see: ized - -

general-

SPRT [62L10] (see: Sequential probability ratio test) spurion superfield [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) square see: mean --

square matrix see: a--

Segre characteristic of

square topology see: Sorgenfrey halfopen - squares see: Lagrange theorem on - - ; s u m of -squares estimation see: normal equations for least- - squares estimator see: least- - squares prediction see: least - - ; linear least - squares predictor see: best linear least -squares regression see: least- --

squaring operation [20J06] (see: Serre theorem in group cohomology) squaring operation see:

Steanrod - -

squeeze function [62D05] (see: Acceptance-rejection method) (*,,')-net [05Bxx] (see: Net (in finite geometry)) (s,r)-net see: maximal-(s,r;l*)-net [05Bxx] (see: Net (in finite geometry)) stability see: characterization of local optimality on a region of -- ; Hyers-Ulam - - ; Hyers-Ulam-Rassias - - ; topological - -

stability analysis [76Cxx] (see: Yon K~irmhnvortex shedding) stability analysis s e e :

linear - -

stability hierarchy [03C15, 03C45, 03E15] (see: Vaught conjecture) stability in the Fritz John equation [90Cxx] (see: Fritz John condition) stability lemma [46H40] (see: Automatic continuity for Banach algebras) stabilityof functional equations see: c o n s t r u c t i o n m e t h o d in the - -

direct

stability of homomorphisms [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) stability of h o m o m o r p h i s m s see:

Ulam

question on - stability of polynomials and matrices see: Schur - -

stability of Willmore surfaces [53C42] (see: Willmore functional) stability problem or functional equations [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) s t a b i l i t y t h e o r e m see:

topological - -

(see: Accessibility for groups) standard algebra [16G70] (see: Riedtmann classification) standard basis of a cellular algebra [05Exx] (see: Cellular algebra) standard basis of a coherent algebra [03Exx, 03E05] (see: Coherent algebra) standard Baxter algebra [05E05, 60G50] (see: Baxter algebra) standard embedding Lie algebra [17A40] (see: Freudenthal-Kantor triple system; Lie triple system) standard embedding Lie superalgebra [17A40, 17B60] (see: Anti-Lie triple system) standard local tomography function [44A12, 65R10, 92C55] (see: Local tomography) standing wave [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Stanley-Reisner face ring [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring)

stability theorem for polynomials and matrices [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) stability theory [26B99, 30C62, 30C65] (see: Quasi-regular mapping)

Stauley-Reisner ring (05Exx, 13C14, 55U10) (refers to: Cohen-Macaulay ring; Dimension; Exterior algebra; Field; Ideal; Simplicial complex) Stanley-Reisner ring [13A30, 13H10, 13H30] (see: Buchsbaum ring)

stabilizer of a tree level see:

Stanley-Reisner ring of a simplicial complex [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) star-shaped domain [46E35, 65N30] (see: Bramble-Hilbert lamina) Stark formula [11R29] (see: Odlyzko bounds)

rigid - -

stabilizer quantum code [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) stabilizer subgroup [20-XX] (see: Regular group) stable Banach algebra [46Exx] (see: Banach-Stone theorem) stable cohomology [11Fxx] (see: Satake compactification) stable-curve see: noded -stable dynamical system see: asymptotically - stable equilibrium of a dynamical system see: asymptotically - -

stable homotopyof spheres [55P421 (see: Brown-Giller spectra) stable letter of an HNN-extension [20F05, 20F06, 20F32] (see: HNN-extension) stable manifold [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) stable matrix [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) stable polynomial [15A18, 93C05, 93D15] (see: Schur stability of polynomials and matrices) stable theory see:

co- - -

Stallings characterization of bipolar structures on groups [20F05, 20F06, 20F32] (see: HNN-extension) Stallings classification of finitely generated groups with more than one end [20F05, 20F06, 20F32] (see: HNN-extension) Stallings theorem [20E22, 20Jxx, 57Mxx]

state see: b o u n d - - ; BPS - - ; classical - - ; entangled q u a n t u m - - ; JogJcal - state automaton see: probabilistic finitestate eigenfunetions see: bound- - state model of a physical system see: tition function of a - statespace see: classical --

par-

state space representationof a process [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) state space system dimensional - -

see:

finite-

state vector of a linearfeedback shift register [11B37, 11T71, 93C05] (see: Shift register sequence) states see:

coherent--

static portfolio optimization [90A09] (see: Portfolio optimization) stationary AKNS-equations [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) stationary Cahn-Hilliard equation [82B26, 82D35] (see: Cahn-Hilliard equation) stationary long memoryprocess [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) stationary orbit [58FxxJ (see: Conley index) stationary strategy in the generalized Banach-Mazur game [54E52]

537

STATIONARYSTRATEGYIN THE GENERALIZEDBANACH-MAZURGAME

(see:

Banach-Mazur game)

stationary varifold

[28ATg, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) stationary winning strategy in the generalized Banach-Mnzur game

[54E52] (see: Banach-Mazur game) statistical analysis software

[62-04] (see: SAS) statistical estimation [62Lxx] (see: Average sample number) statistical experiments s e e : balanced design for - - ; cell in design of - - ; design of - - ; effect in design of - - ; interaction in design of - - ; main effect in design of - statistical factor s e e : level of a - -

statistical factors

[62Jxx] (see: ANOVA) statisticalinference see: Bayesian - statistical model s e e : partiallyspecified -statistical test s e e : curtailed version of a ; power function of a -- ; power of a - statisticaltests s e e : efficiency of -statistics s e e : estimable parametric function in - - ; parametric function in - - ; spectral - - ; truncation in - - ; Wald test - Steenrod algebra s e e : unstable action of the --

-

Steenrod K-homology [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) Steenrod squaringoperation [20J06] (see: Serre theorem in group cohomology) Stein symbol s e e : Steinberg

Dennis--

idempotent

(see: Segal-Shale-Weil representa-

Steiner-Weber problem

tion)

[90B85] (see: Fermat-TorriceUi problem) Steinitz-Weyl theorem s e e :

Minkowski-

Steinness (32E10) space) stem (in linguistics) [68S05] (see: Natural language processing) step-downtest [62Jxx] (see: ANOVA) Step hyperbolic cross (42B05, 42B08) (referred to in: Lebesgue constants of multi-dimensional partial Fourier sums) (refers to: Approximation theory; Fourier series; Hyperbolic cross; Interpolation of operators; Lebesgue constants) step hyperbolic cross

[42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) step hyperbolic partial sum

[42B05, 42B08] (see: Step hyperbolic cross) Stevedore knot [57M25] (see: Positive link) Stickelberger theorem [11R23] (see: Iwasawa theory) Stieltjes algebra see: F o u r i e r - - Stieltjes integralequation s e e : V o l t e r r a - - Stieltjes moment problem s e e : classical ; determinate strong - - ; indeterminate strong -- ; Strong - - ; strong symmetric - -

Stieltjes-Volterra integral equation

[45D05]

[20G05] (see: Steinberg module) Steinberg module (20G05) (refers to: Algebraic group; Borel subgroup; Field; Finite group; Fundamental domain; Group algebra; Mumford hypothesis; p-group; Sylow subgroup; Tits building; Unipotent group; Weyl group)

tion) Stiemke theorem [15A39, 90C05] (see: Motzkin transposition theorem) stochastic calculus [46L80, 46L87, 55N15, 58GI0, 58G11, 58G12] (see: Index theory)

Steinberg module

stochastic dynamical system

[20G05] (see: Steinberg module) Steinberg representation

[20G05] (see: Steinberg module) Steinberg symbol (19Cxx) (refers to: Abelian group; Algebraic K-theory; Differential field; Extension of a group; Field; Fundamental group; Group; Homomorphism; Linear representation; Projective representation; Quadratic reciprocity law; Sehur multiplieator; Topological field) Steinberg symbol

[19Cxx] (see: Steinberg symbol) Steiner problem (05C35, 51M16) (refers to: Steiner tree problem) Steiner triple system

[05B05, 05B07, 05B30, 51El0] (see: Pasch configuration; STS) Steiuer triple system [17A401 (see: Jordan triple system; Lie triple system) Steiner triple system s e e : anti-Pasch - - ; block of a -- ; element of a -- ; line of a - - ; mutually t-balanced collections of blocks in a - - ; point of a - - ; quadrilateral free - - ; trade in a - - ; triple of a --

538

(see: Volterra-Stieltjes integral equa-

[28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invariant measure) stochastic network [60Hxx, 60J55, 60J65] (see: Skorokhod equation) stochastic process s e e : linearly regular ; linearly singular - Stockmeyer alternation theorem s e e : Chandra-Kozen- --

-

Stot'Iow theorem

[26B99, 30C62, 30C65] (see: Quasi-regular mapping) Stokes equation s e e : Navier- - -

compressible

Stokes parameters (78A40) (refers to: Electromagnetism) Stokes vectors

[78A40]

strange attractor [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashlnsky equation) strange dynamical system

[28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invariant measure) strategy s e e : hedge - - ; self-financing portfolio -- ; winning - -

strategy in the generalized BanachMazur game

[54E52] (see: Banach-Mazur game) strategy in the generalized B a n a c h - M a z u r game see: stationary - - ; stationary winning - - ; winning - stratification s e e : T h o m - M a t h e r - - ; Whitney -stratifications s e e : distance function for -stratum s e e ; frontier of a - -

stratum of a morphism [57N80] (see: Thom-Mather stratification) vortex - -

strict delta-net

[46F10] (see: Multiplication of distributions) strict Horn logic [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) strict self-similarity

[28A80] (see: Sierpifiski gasket) strict topology

[46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras)

strictly pseudo-convexdomain [46Lxx] (see: Toeplitz C* -algebra) strike price of a European call option

[60Hxx, 90A09, 93Exx] (see: Blaek-Seholes formula) string --; string string string

see:

see: 26-dimensional - - ; heterotic vibration of a - braid s e e : 3- - construction s e e : gardener-singularities s e e : Dirac - -

string theory [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 53C42, 81R10, 81T30, 81T40] (see: Vertex operator; Vertex operator algebra; Willmore functional) strong amalgamationproperty [03Gxx] (see: Algebraic logic) strong cone condition [46E35, 65N30] (see: Bramble-Hilbert lemma) see:

unconditioned--

[43A45, 43A46] (see: Ditkin set)

Stone duality [03Gxx] (see: Algebraic logic) B a n a c h - -B a n a c h - --

Stone-yon Neumann theorem

[11F27, 11F70, 20G05, 81R05]

re-

[44A60, 47A57] (see: Moment matrix) strong optimality property for the sequential probability ratio test

[62L10]

[90A28] (see: Condorcet paradox)

strong Ditkin set

[46Exx] (see: Banach-Stone theorem)

Stone property s e e : Stone theorem s e e :

straight plurality voting

strong d-sequence

(see: Stokes parameters) Stone-Banach theorem

S t o n e - ~ e c h compactification mainder in the - -

[68Q15] (see: Average-case computational complexity)

street s e e :

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) strong moment problem

straight line program

(refers to: Stein manifold; Stein

-

Stonean topological space see: sub- - stopping problem s e e : optimal - Stora-Tyutintransformationssee: BecchiR o u e t - --

strong implicit definition of a set of atomic formulas over another set of atomic formulas

strong Ditters conjecture

[05E05, 16W30] Leibnlz-Hopf algebra and quasi-symmetric functions) (see:

strong homomorphism

[03G25, 06D99] (see: Quantale) strong homomorphism of Gel'fend quantales s e e : a l g e b r a i c a l l y - -

(see: Sequential probability ratio

test) strong six exponentials theorem

[llJ81] (see: Schneider method) strong six exponentials theorem Roy - -

see:

strong Sperner property

[05D05, 06A07] (see: Sperner property)

Strong Stieltjes moment problem (44A60) (refers to: Continued fraction; Hankel matrix; Inner product; Krein condition; Linear operator; Moment problem; Normal distribution) strong Stieltjes m o m e n t problem s e e : terminate - - ; indeterminate - -

de-

strong subadditivity [68Q05, 68Qi0, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) strong subadditivity inequality for von Neumann entropy [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) strong symmetric Stieltjes moment problem

[44A60] (see: Strong Stieltjes moment prob-

lem) strong Szeg6 limit theorem

[42A16, 47B35] (see: Szeg6 limit theorems)

strong turbulence [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) Strongly eountably complete topological space (54C05, 54C08) (referred to in: Namioka space; Namioka theorem) (refers to: Baire space; Namioka space; Namioka theorem; Separate and joint continuity) strongly finitely algebraizable deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) strongly zero-dimensionalmetric space [54G10] (see: P-space) structurable algebra [17A40] (see: Freudenthal-Kantor triple system) structural ambiguity in a natural fan guage

[68S05] (see: Natural language processing) structure s e e : analytic - - ; canonical Potsson - - ; consecutive k - o u t - o f - n -- ; deformation of a - - ; flexible - - ; Galois field - - ; incidence - -

structure constants of a coherent algebra

[03Exx, 03E05] (see: Coherent algebra) structure deformation s e e : structure grammar s e e : phrase --

complex-head-driven

SWINNERTON-DYERPOLYNOMIAL

structure in a learning system see; representation -structurein belief theory see: graphoidal -structure on a group see: bipolar - structure on a manifold see: c o m p l e x - structure on a polynomial hull s e e : analytic -structure theorem see: B u n c e - O h u -- ; Z e l ' m a n o v --

structure theoremfor Borcherds Lie algebras [11Fxx, 17B67, 20D0g] (see: Borcherds Lie algebra) structure theorem for maximal Buchsbaum modules over regular local rings [13A30, 13H10, 13H30] (see: Buehsbaum ring) structures see: dynamic loading of flexible - structures on groups see: acterization of bipolar -STS

Stallings char-

(05B05, 05B07, 51El0) (refers to: Steiner system) STS [05B07, 05B30] (see: Pasch configuration) STS s e e :

antJ-Paseh --

Dirichlet -inverse - -

Sturm-Liouville spectralproblem [341324, 34L40] (see: Sturm-Liouville theory) Sturm-Liouvillespectral problem see: merical approaches to the - -

nu-

Sturm-Liouville theory (34B24, 34L40) (refers to: Analytic function; Eigen value; Hausdorff dimension; Herglotz formula; Lebesgue measure; Lebesgue theorem; Linear ordinary differential equation of the second order; Radon-Nikod#m theorem; Schr6dinger equation; Selfadjoint operator; Spectral analysis; Spectral function; Spectral theory; Spectrum of an operator; SturmLionvilie problem; Stnrm-Liouville problem, inverse; Titchmarsh-Weyl m-function; WKB method) SU (2) gauge theory [81V101 (see: Dirae monopole) SU (2) -monopole [35Qxx, 78A25] (see: Magnetic monopole) su (3) particle physics [17A35, 17D25, 83C20] (see: Okubo algebra) sub- Stonean topological space [46J10, 46L05, 46L80, 46Lg5] (see: Multipliers of G'* -algebras) subadditive set function [28-XX] (see: Non-additive measure) subadditivitysee: strong - subadditivity inequality for von Neu m a n n entropysee: strong --

subcritical bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) subderivate see: Dini -subdifferentialsee:

proximal --

sobgradientssee: calculus of - subgroup see: adthmetic - - ; Iwahofi - - ; Lagrangian - - ; malnormal - - ; maximal compact - - ; p a r a b o l i c - - ; principal congruence - - ; stabilizer -subgroup of rank two see: free -subgroups see: commensurable - subgroups of an HNN-extension see: sociated - -

as-

subgroupsof Baumslag-Solitar groups [05C25, 20Fxx, 20F32] (see: Baumslag-Solitar group) subharmonicfunctions see: position theorem for --

Riesz decom-

subharmonicminorant [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) subjective belief [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) subjective behef assignment [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) submanifold see: Lagrangian - - ; totally geodesic - - ; totally real --

Student-t hat function [62D05] (see: Acceptance-rejection method) Sturm-Liouville differential equation [34B24, 34L40] (see: Sturm-Liouville theory) Sturm-Liouville operator [34B24, 34L40] (see: Sturm-Liouville theory) Sturrn-Liouville operator see: Sturm-Lieuville problem see:

subgradient see:

viscosity--

subgradient [90Cxx, 90C30] (see: Clarke generalized derivative; Fritz John condition)

submartingaleproblem [60Hxx, 60J55, 60J65] (see: Skorokhod equation) submatdxsee:

s'um-game [90D05] (see: Sprague-Grundy function) sum-graph [90D051 (see: Sprague-Grundy function) sum of games [90D05] (see: Sprague-Grundy function) sum of squares [62Jxx] (see: ANOVA) sum of t y p e I see: exponential - sum of t y p e II see: exponential - summability see: Bochner-Riesz --

summability of Fourier series [42A16, 42A24, 42A28] (see: Beurling algebra)

leading principal --

s u m m a b l e majorant of coefficients see: Beurling algebra of Fourier series with --

submeasure [28-xx] (see: Non-addltive measure) submetacompactspace [26A15, 54C05] (see: Namioka space)

summation domain [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums)

submetacompact space see: hereditarily - s u b m o d u l e see: essential - - ; pure --

submultiplicative inequality for a matrix norm [15A42] (see: Bauer-Fike theorem) subordinate solution [34B24, 34L40] (see: Sturm-LiouviUe theory) subresnltanttheory [15A57, 47B35, 65F05, 93B15] (see: Hankel matrix) subset see: L- --; lattice- --; Suslin -subsets see: Lelong theorem on analytic -; lower s h a d o w of a family of - subsets of a set see: family of - subspace see: Lagrangian - - ; invariant - - ; Taylor spectrum for variant - subspaces see: angle between - - ; between Hilbert -substitution invariance see: Tarski tion of - -

s u m see: analytic exponential - - ; arithmetic exponential - - ; complete exponential - - ; exponentiaf - - ; game-graph of a - - ; Gauss - - ; generalized Nim- - ; K l o o s t e r m a n - - ; I c e - - - ; monomial exponential - - ; multi-dimensional partial Fourier - - ; step hyperbolic partial - s u m decompositions s e e : d'Alembert equation for finite - s u m estimates see: Exponential - s u m estimation see: exponent pair in exponential - - ; exponent pairs in exponential - -

shiftan inangle condi-

substitution operator [39B05, 39B12] (see: Schrfder functional equation) subtree see: spanning - successes see: run of --

successive overrelaxation method [65F10] (see: SOR method) succinct game [90D05] (see: Sprague-Grundy function) Suen inequality [05C80, 60D05] (see: Jansen inequality) sufficiently-large 3-manifold [57N10] (see: Haken manifold) sufficiently-large three-dimensional manifold [57N10] (see: Haken manifold) Sugeno integral [28-XX] (see: Choquet integral)

summation formula see: Voronof - - ; Wilton -sums see: hyperbolic partial Fourier - - ; Lebesgue constants of multi-dimensional partial Fourier - - ; spherical partial Fourier - super-algebrasee: queer Lie - -

[03C15, 03C45, 03E15] (see: Vaught conjecture) supersymmetry [46L80, 46L87, 55N15, 58G10, 58G1 l, 5gG12] (see: Index theory) supertriple system [17Ad0] (see: Lie triple system) supervised learning [683705] (see: Machine learning) support see: pact - -

algebra of functions of com-

support of a zero-one matrix [03Exx, 03E05] (see: Coherent algebra) surface s e e : eonformally minimal - - ; cotangent space at a marked Riemann --; immersed--; incompressible--; Lorentz - - ; marked Riemann - - ; minimal - - ; noded Riemann - - ; non-separating - - ; real Riemann - - ; regular - - ; splitting a three-dimensional manifold at a - - ; Weierstrass data for a minimal - - ; Weierstrass representation of a minimal --; Willmore -surface in a manifold s e e : two-sided - surface in a three-dimensional manifold see: b o u n d a r y compressible - - ; boundary incompressible - - ; compressible - - ; O-compressible - - ; O-incompressible - - ; incompressible - surface into a Riemannian manifold see: immersion of a - -

surface theory [35L15] (see: Euler-Poisson-Darboux equation) surface zonal harmonics [31B05, 33C55] (see: Zonal harmonics) surfaces see: surgery s e e :

stability of Willmore - -

super-martingale [31AI0, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) superabundant number [llAxx] (see: Abundant number)

1/ r~ - surjectivity criterion for Buchsbaum rings [13A30, 13H10, 13H30] (see: Buchsbaum ring) surjectivity in codimension 1 [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) survival analysis [62Jxx, 62Mxx] (see: Cox regression model) Suslin subset [03E50, 54-XX, 90D80] (see: Sierphiski game)

superabundant n u m b e r s e e :

susy breaking see:

super-amalgamationproperty [03Gxx] (see: Algebraic logic) super-connections see: of--

Quillen theory

cube-free --

soft --

superadditive set function [28-XX] (see: Non-additive measure)

susy gauge theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations)

superalgebra see: Borcherds colour - - ; colouring map for a colour - - ; standard e m b e d d i n g Lie --

Suszko congruence see: theorem on equivalence systems and the - -

supercritical bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) superfield see: spurion -superfluousnumber [llAxx] (see: Abundant number) superharmonic functions see: composition theorem for - -

Riesz de-

superparaboficfunction [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) superstability [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) superstable functional equation [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) superstable homomorphism [39B72, 46B99, 46Hxx] (see: Hyers-Ulam-Rassias stability) superstable theory

Suszko congruence of a theory over $5 [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Suszko-reduced model [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Suszko-reduction [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) swallowtail bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Swan-Forster theorem [13B30, 13C15, 16Lxx, 16P60] (see: Forster-Swan theorem) S w a n theorem see: Forster--Swinnerton-Dyer conjectu re see:

Birch-

Swinnerton-Dyerpolynomial [12D05] (see: Factorization of polynomials) 539

SWITCH

switch s e e : Pasch -switching s e e : branch -symbol s e e : complexions- - - ; DennisS t e i n - - ; F r o b e n i u s - - ; Pochhammer - - ; Steinberg - - ; Toeplitz operators with H ~ --; Weyl --

(see: Borcherds Lie algebra) symmetrization of a function [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition)

symbol in first-order logic [03E30] (see: ZFC)

symmetrybreaking [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) symmetrybreaking in quantum field theory [81Txx, 81T05] (see: Massless field) symmetry for the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) symplectic Fleer homology [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) symplectic Fleer homology for a symplectic mapping [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture) symplectic Fleer homology for Lagrangian intersections [53C15, 57R57, 58D27] (see: Atiyah-Floer conjecture)

symbol of a differential operator cipal --

prin-

see:

symbol of a Hankel operator [15A57, 47B35, 65F05, 93B151 (see: Hankel matrix). symbol of a pseudo-differential operator [44A12, 65R10, 92C55] (see: Local tomography) symbol of a pseudo-differential operator s e e : hypo-elliptic --

symmedean [51M041 (see: Triangle centre) symmedean point [51M04] (see: Triangle centre) symmetric and alternating groups Projective representations of --

see:

symmetric balanced incomplete block design [05B051 (see: SBIBD) symmetriccase of the Lov~iszlocal lemma [05C801 (see: Lov~iszlocal lemma) symmetric chain order [05D05, 06A07] (see: Spurner property) symmetric closed monoidal category [18D10, 18D15] (see: *-Autonomous category) symmetric domain [11Fxx, 20Gxx, 22E46, 46Lxx] (see: Baily-Borel compactification; Toeplitz C*-algebra) symmetric domain s e e : bounded -symmetric domain in a Banaeh space s e e : bounded -symmetric function s e e : axially - - ; M quasi- -- ; quasi- -symmetric function of a complex variable see: Quasi- -symmetric function on T s e e : M-quasi- ; quasi- -symmetric functions s e e : Leibniz-Hopf algebra and quasi- --

symmetric Greenfunction [31C10, 32F05] (see: Pluripotential theory) symmetric group s e e : linear character of a - - ; spin character of a - - ; weak order of a -symmetric mappings s e e : degree of --

symmetric monoidal category [18D10] (see: Closed monoidal category) symmetric net [05Bxx] (see: Net (in finite geometry)) symmetric scatterer s e e : spherically-symmetric set s e e : A-anti- - ; partially A-anti- -symmetric space s e e : boundary component of a - - ; Hermitian - - ; rational boundary component of a - - ; Riemannian -symmetric Stieltjes moment problem s e e : strong --

symmetric transversal design [05Bxx] (see: Net (in finite geometry)) symmetrizable Borcherds algebra [1lFxx, 17B67, 20D08] (see: Borcherds Lie algebra) symmetrizable Cartan matrix [llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) symmetrizable Kac-Moody algebra [llFxx, 17B67, 20D08] 540

symmetry s e e : Miller --

domain with rotational - - ;

sympleetic mapping s e e : homology for a --

symplectic Fleer

syntactical provabilityrelation [03Gxx] (see: Algebraic logic) syntactical theory of a logic [03Gxx] (see: Algebraic logic) syntax of a natural language [68S05] (see: Natural language processing) synthesis s e e : natural language - - ; set of -- ; set of spectral -- ; union problem in spectral --

synthesis-Ditkin problem [43A45, 43A46] (see: Ditkin set) synthesis problem for the Beurling algebra [42A16, 42A24, 42A28] (see: Beurling algebra) system s e e : algebraizable deductive - - ; algebraizable general semantical -- ; algebraizable semantical - - ; AllisonHein triple - - ; Anti-Lie triple - - ; antiPasch Steiner triple - - ; AR - - ; ARMA - - ; asymptotically stable dynamical - - ; asymptotically stable equilibrium of a dynamical - - ; balanced FreudenthalKantor triple - - ; Banach-Jordan triple - - ; block of a Steiner triple - - ; chaotic dynamical - - ; circular consecutive hout-of-n - ; classifier in a learning - ; completely-integrable - - ; concept formation - - ; consecutive - - ; Consecutive k-out-of-n: F - - ; consecutive k-outof-n: G - - ; conservative discrete-time - - ; constructive induction - - ; critic in a learning - - ; curve of periodic orbits of a dynamical - - ; curves of equilibria of a dynamical - - ; deduction-detachment - - ; deductive - - ; deterministic dynamical - - ; discovery--; discrete dynamical - - ; dual - - ; dynamical - - ; element of a Steiner triple - - ; equilibrium of a dynamical -- ; equivalence -- ; equivalential deductive - - ; equivalential general semantical - - ; equivalential semantical - - ; Euler - - ; experiment generator in a learning - - ; extensional deductive - - ; faithful interpretation of a deductive - - ; filter-distributive deductive - - ; filter of a deductive - - ; finite-dimensional state space -- ; finitely algebraizable deductive - - ; finitely algebraizable general semantical - - ; finitely algebraizaNe semantical - - ; finitely equivalential deductive - - ; finitely equivalential general semantical - - ; finitely equivalentia/ semantical

- - ; Fregean deductive - - ; FreudentbalKantor triple --; fully adequate Gentzen - - ; general semantical - - ; generalizer in a learning - - ; Gentzen - - ; Hamittonian - - ; intensional deductive - - ; interpretation of a deductive - - ; Jordan triple - - ; learning - - ; lexical - - ; Lie triple - - ; line of a Steiner triple - - ; linear consecutive k-out-of-n - - ; logical equivalence of formulas with respect to a deductive - - ; logically equivalent formulas with respect to a deductive - - ; logistic - - ; MA - - ; matrix model of a deductive - - ; MIMO - - ; minimal set of a dynamical - - ; miniphase ARMA - - ; model of a deductive - - ; model of a Gentzen - - ; multiple-input multiple-output - - ; mutually /,-balanced collections of blocks in a Steiner triple - - ; natural language - - ; JB * triple - - ; Paley-Walsh - - ; partition function of a state model of a physical -- ; performance system in a learning -- ; period of a trajectory of a dynamical - - ; period of an orbit of a dynamical -- ; point of a Steiner triple - - ; primal - - ; pretoequivalence -- ; protoalgebraic deductive - - ; protoa]gebraic general semantical - - ; protoalgebraie semantical - - ; publickey cryptographic - - ; quadrilateral free Steieer triple - - ; quantum algebra of a - - ; reaction-diffusion--; representation structure in a learning - - ; second-order finitely algebraizable deductive - - ; selfextensional deductive -- ; semantical -- ; single-input single-output - - ; SISO - - ; Sperner - - ; Steiner triple - - ; stochastic dynamical - - ; strange dynamical - - ; strongly finitely algebraizable deductive - - ; supertdple - - ; target function in a learning - - ; theorem of a deductive - - ; theory of a deductive - - ; Toeplitz - - ; trade in a Steiner triple - - ; training experience in a learning - - ; triple - - ; triple of a Steiner triple - - ; tube - - ; underlying deductive -- ; uninterpreted deduction -- ; unstable equilibrium of a dynamical - - ; Walsh-Paley - - ; weakly algebraizable deductive - - ; wild dynamical -system in a tearning system s e e : performance --

system of defining equations [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) system of parameters [13Hxxl (see: System of parameters of a module over a local ring) system of parameters

see:

regular--

System of parameters of a module over a local ring (13Hxx) (refers to: Algebraic variety; Dimension) systemtheorysee:

partial realization prob-

lem of -systems s e e : characterization theorem for deductive - - ; characterization theorem of algebraizable deductive - - ; examples of Jordan triple - - ; identification of - - ; monomial property for parameter - - ; reduction of Hamiltonian - - ; reliability of multi-component -- ; software for dynamical -systems and the Suszko congruence s e e : theorem on equivalence - systems software s e e : dynamical -systems software packages s e e : Dynamical --

Sz.-Nagy-Foias functionalmodel [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) SzegO class [33C45, 33Exx, 46E35] (see: Sobolev inner product) SzegO condition

[33C45] (see: Szeg6 polynomial) Szeg6 extremum problem [33C45] (see: Szeg6 polynomial) Szegti fractional differentiation [35L15] (see: Euler-Polsson-Darboux equation) Szeg~ fractional integration [35L15] (see: Euler-Poisson-Darboux equation) SzegOfunction [33C45] (see: Szeg6 polynomial) Szeg6 limit theorem

first -- ; strong --

see:

Szeg5 limit theorems (42A16, 47B35) (refers to: Analytic function; Asymptotic expansion; Continuous function; Fourier coefficients; Fourier transform; Hankel operator; Hilbert space; Integral operator; Nuclear operator; Spectrum of an operator; Toeplitz matrix; Toeplitz operator; Wiener-Hopf method; Wiener-Hopf operator; Winding number) Szegdorthogonalprojection

see:

Cauchy-

Szeg5 parameter [33C45] (see: Szeg6 polynomial) Szeg5 polynomial (33C45) (referred to in: Szeg5 quadrature) (refers to: Carathdodory class; Cayley transform; Complete system; Geometric mean; Hardy classes; Inner product; Measure; Orthogonal polynomials on a complex domain; Schur functions in complex function theory) Szeg6 polynomial

orthonormal --

see:

SzegOpolynomial of the second kind [33C45] (see: Szeg6 polynomial) Szeg6 polynomials

see:

Bemshtein---

Szeg5 quadrature (65D32) (refers to: Christoffel numbers; Gauss quadrature formula; Measure; Quadrature formula; Szeg6 polynomial) Szeg6 quadrature formula [65D32] (see: Szeg6 quadrature) Szeg6 theory [33C45] (see: Szeg6 polynomial) Szelepes~nyi theorem

see:

Immerman-

Szemer~di theorem [05DI01 (see: Hales-Jewett theorem) Szemerddi theoremon arithmeticprogressions [05D10] (see: Hales-Jewett theorem)

T 7--function [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) 7"method [65Lxx] (see: Tau method) T see: M-quasi-symmetric function on --;

quasi-symmetric function on Hotelling --

T 2 see:

THEOREM

C-balanced collections of blocks in a Steiner

triple system see:

mutually --

t-conorm integral [28-XX] (see: Choquet integral) t-conorm measure [28-XX] (see: Non-additive measure) T-equivalence see: Frege -T-equivalent formulas see: Frege -T-fraction

[44A60]

(see: Strong Stieltjes moment prob-

lem) t hat function see:

Student- --

t-norm [03G10, 06Bxx, 54A40] (see: Fuzzy topology) t--(v,k,A)-design see: resolvable-table mountain-function

[62D05] (see: Acceptance-rejection method) tableau see: content of a marked shifted --; marked shifted --

Taenode (14H20) (refers to: Cusp; Node) tactics in the generalized Banach-Mazur game

[54E521 (see: Banach-Mazur game) TAG

[68S051 (see: Natural language processing)

TAG parser [68S05] (see: Natural language processing) Tait conjectures [57M25] (see: Listing polynomials) Tait conjectures on alternating links [57M25] (see: Kauffman bracket polynomial) Tait number

[57M25] (see: Kauffman bracket polynomial) Takens bifurcation see: B o g d a n o v - - Talamanca algebra see: Fig&- -Talamanca-Herz algebra see: Fig&- --

tame algebra [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) tame algebra see: representation-tame domestic algebra

[16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) tame representationtype [16Gxx] (see: Tits quadratic form tamis de Sierpihski

[28A801 (see: Sierpifiski gasket) tamisable see:

espace --

tangency condition of flow invariance

[32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic mappings) tangent see: approximate -tangent cone to a set see: Clarke --

tangent measure

[28A78, 49Qxx, 49Q15, 53C65, 58A25] (see: Geometric measure theory) tangent plane see: approximatem- -tangential approach region see: non- -tangentiallimit see: non- -tangentialvelocity see: discontinuous --

Tangle (57M25) (referred to in: Algebraic tangles; Rational tangles; Tangle move) (refers to: Algebraic tangles; Braided group; Monoid; Rational tangles; Reidemeister theorem; Rotor) tangle see: k- - ; n- - ; n-algebraic m- - ; n-bridge n- - - ; (n,k)-algebraic m -

- - ; rational -- ; rational p / q - -- ; slope of a rational --

Tangle move (57M25) (refers to: Fox n-colouring; Kirby calculus; Montesinos-Nakanishi conjecture; Tangle) tangles see: Algebraic -- ; arborescent -- ; Conway notation for rational - - ; equivalence of - - ; equivalent - - ; n-algebraic - - ; (n,k)-algebraic - - ; Rational -tangles in the sense of Conway see: algebraic -Tanovic-Miller space see: Buntinas--tape see: read-onlyinput -tapis see: fonction --

target function in a learning system

[68T05] (see: Machine learning) Tarski condition of substitution invariance

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Tarski conditions [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Tarski congruence

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Tarski congruence [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Tarski finiteness condition

[03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) Tarskimodel see: kindenbaum--Tarski process see: Lindenbaurn--tau see: Kendall - - ; population version of the Kendall --

tan-function [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) tau-function of KP-Toda type [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Tan method (65Lxx) (refers to: Chebyshev polynomials; Linear differential operator; Special functions; Uniform approximation) tan method [65Lxx] (see: Tau method) tau method see: canonical polynomials in the -- ; Chebyshev -- ; error analysis for the - - ; Lanczos - - ; Legendre - - ; operational formulation of the -- ; recursive -- ; spectral --

tau method approximation

[65Lxx] (see: Tau method) tau metric see:

KendaJI --

tau problem

Taylor problem see:

Saffman---

Taylor spectrum

[47Dxx] (see: Taylor joint spectrum) Taylor spectrum [47Dxx] (see: Taylor joint spectrum) Taylor spectrum see: example of a - - ; example of the - - ; spectral inclusion for - - ; spectral permanence for the --

Taylor spectrumfor an invariant subspace [47Dxx] (see: Taylor joint spectrum) Taylor spectrum for compact operators [47Dxx] (see: Taylor joint spectrum) Taylor spectrumfor Fredholm n-tuples [47Dxx] (see: Taylor joint spectrum) Taylor spectrumin the finite-dimensional case [47Dxx] (see: Taylor joint spectrum) Taylor spectrumunder multipfication [47Dxx] (see: Taylor joint spectrum) Taylor theorem (41A05, 41A58) (refers to: Taylor formula) technique see: Lin singular tori - - ; Sperner shifting -technique for finding the inverse Z-transform see: partial-fractions -techniques see: Zel'manov --

Teichmfiller character [1 lR23] (see: Iwasawa theory) Telyakovski'f space see:

Boas- - - ; Sidon-

Temlyakov-Opial-Siciak-type meanvalue theorem [31A05, 31B05, 31CI0, 31C35, 32A10, 46F10, 60Y65] (see: Mean-vaine characterization) Temperley theorem [o5c5o1 (see: Matrix tree theorem) tempera[chaos see: spatio- -temporal pattern formation see: spatio- -tensor see: f l u x - - ; Poisson --

tensor-productbasis function [65Lxx, 65M70] (see: Trigonometric pseudo-spectral methods) term see: contact --; extension of a --; intension of a --

[65Lxx] (see: Tau method) Taub-NUTmetric see: Taussky theorem

(refers to: Banach space; BochnerMartinelli representation formula; C*-algebra; Euler characteristic; Exterior algebra; Functional calculus; Hilbert space; Koszul complex; Normal operator; Normed space; Spectrum of an operator) Taylor joint spectrum [47Dxx] (see: Taylor joint spectrum) Taylor polynomial see: averaged --

term in a logical language Euclidean --

[15A18] (see: Gershgorin theorem) Tausworth generator

[65CI0] (see: Linear congruential method) Taylor essential spectrum

[47Dxx] (see: Taylor joint spectrum) Taylor expansionremainder [46E35, 65N30] (see: Bramble--Hilbert lemma) Taylor finger see: Saffman--Taylor instability see: Saffman- --

Taylor joint spectrum (47Dxx) (referred to in: Operator vessel)

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) terminal see:

weaklylog --

terminal algebraic variety

[14Exx, 14E30, 14Jxx] (see: Mort theory of extremal rays) terminal singularity see: algebraic variety with -ternaryalgebra see: J - - test see: Bartlett-Nanda-Pillai - - ; curtailed version of a statistical - - ; F - - - ; Lawley-Hoteliing--; likelihood-ratio--; LR - - ; maximum root - - ; NeymanPearson probability ratio - - ; power function of a statistical - - ; power of a statistical - - ; Roy maximum root - - ; sequential - - ; Sequential probability ratio

- - ; step-down - - ; strong optimality property for the sequential probability ratio - - ; UMP - - ; uniformly most powerful - - ; Wilks LR -test of a hypothesis see: exact --

test set

[68T05] (see: Machine learning) test statistics see: Wald -testing seer hypotheses - - ; hypothesis - -

testing of hypotheses [62Jxx] (see: ANOVA) tests see:

efficiency of statistical - -

text analysis [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) texts see: word analysis in -tFL see: condition -tFL condition

[26A21, 54E55, 54G20] Sorgenfrey topology) Theodorsen integral equation (30C20, 30C30) (refers to: Conformal mapping; Fourier series; Homeomorphism; Newton method; Riemann-Hilbert problem; Trigonometric polynomial) (see:

theorem see: abstract inverse prime number - - ; abstract prime element - - ; abstract prime number - - ; Alexander duality - - ; Allard regularity - - ; analogue of the Baker finite basis - - ; AndersonWenger - - ; Arkhangei'skif-Frolfk covering - - ; Atiyah L2-index - - ; AtiyahPatodi-Singer index - - ; Atiyah-Singer i n d e x - - ; Auslander-Reiten - - ; Bade-Curtis boundedness - - ; Baker finite b a s i s - - ; B a n a c h - - ; Banach-Stone - - ; Bauer-Fike - - ; Bautista-Brunner - - ; Benard-Schacher - - ; Benedicks - - ; Besicovitch-Federer projection - - ; Beth definability--; B e u r l i n g - - ; BeuNngP o l l a r d - - ; B i n a t - C a u c h y - - ; Bishop - - ; Bombieri-Vinogradov--; Bourgain return-time--; Brauer-Witt--; BrelotB a u e r - - ; Brenner-Butler--; Brouwer fixed-point--; Browder--; Bunce---Chu structure - - ; Cameron-Martin-Girsanov - - ; Carleman-Kaplansky--; ChandraKozen-Stockmeyer alternation - - ; Chebotarev density - - ; Cohen idempotent - - ; Condorcet jury - - ; cone - - ; continuous Denjoy-Wolff - - ; contraction - - ; Courant nodal line - - ; Craig interpolation - - ; Crape complementation - - ; c r o s s - c u t - - ; D a n z e r - - ; Darbo fixed-point - - ; deduction - - ; deductiondetachment--; Delsarte--; Belsartet y p e - - ; Dembowski--; Denjoy-Wolft - - ; density Hales-Jewatt--; Dilworth - - ; Dirichlet unit - - ; Dubovitskii-Milyutin - - ; Dunwoody accessibility--; equioscillation - - ; equivalence of EDPRC and deduction-detachment - - ; equivariant index - - ; ergodic - - ; Euler-Poincar6 - - ; Fan analogue of the Denjoy-WoN - - ; Farkas - - ; Farkas-Minkowski-Weyl--; Federer-Fleming closure - - ; Fefferman duality - - ; Felt-Solomon - - ; first Szeg6 limit - - ; Forster-Swan - - ; Forstneri6 - - ; Franke -- ; Furstenberg-Katznelson density Hales-Jewett--; Gabriel--; Galiai - - ; Gallai-type - - ; Galois connection - - ; Gauss factorization - - ; Gauss-Marker - - ; Gel'fand-Naimark--; generalized Cayley-Hamilton - - ; generalized JuliaWolff--Carathdodory - - ; generalized Mercer - - ; generating function version of the matrix tree - - ; geometric Ramsey - - ; Gerschgorin - - ; Gerggorin - - ; Gershgorin - - ; Gershgorin-type - - ; Gleason-Kahane-2elazko - - ; Goldie - - ; Goodman-Pollack--; Gordan - - ; Grdnbaum - - ; Grushko - - ; Hadamard factorization - - ; Hadwiger - - ; Hadwiger

541

THEOREM

transversal - - ; Hadwiger-type transversal - - ; hairy ball - - ; H a l e s - J e w e t t - - ; H e l l y - - ; Helly-type transversal--; Herz - - ; hierarchy - - ; Higman - - ; Hilbert 90 - - ; Hilbert irreducibility--; Hirzebrueh signature - - ; homogeneous chaos decomposition - - ; HGrmander-Wermer - - ; H y e r s - - ; Immerman-Szelepcs~nyi - - ; implicit function - - ; infinitary HalesJewett - - ; inverse additive abstract prime number--; Iwaniec--; Iwasawa--; Jaco-Shalen-Johansson decomposition - - ; Jaco-Shalen-Johansson splitting - - ; Jewett-Hales--; Johnson-Sinclair--; Johnson uniqueness-of-norm -- ; Jordan separation - - ; Josefson - - ; JSJ - - ; Julia - - ; Julia-CarathGodory--; Julia-Wolff - - ; Julia-Wolff-Carath~odory - - ; Kemer --; Knaster-Ku ratowski-Mazu rkiewicz fixed-point - - ; Kneser -- ; Kre'(n-Mirman - - ; Kronecker--; Kronecker-Weber--; K r o n h e i m e r - M r o w k a - - ; KrulI-Schmidttype -- ; KruskaI-Katona - - ; Landau -- ; Landau prime ideal - - ; Landau-Weber - - ; Laurent - - ; Lebesgue decomposition - - ; Lindemann-Weierstrass - - ; local i n d e x - - ; L o r c h - - ; Mahler vanishing - - ; Marcinkiewicz multiplier - - ; Markov braid - - ; Matrix tree - - ; Matsumoto - - ; M e r n i k o v - C h a t z i d a k i s - - ; mete-logical characterization - - ; Minkowski convex body--; Minkowski-Steinitz-Weyl--; Motzkin transposition - - ; Namioka - - ; Napoleon - - ; Nazarova - - ; Normal basis - - ; Oka - - ; optical - - ; Otter - - ; Oxtoby - - ; p-adic Weierstrass preparation - - ; PCT - - ; Peano kernel - - ; pointwise ergodic - - ; polynomial Hales-Jewett - - ; Preiss density - - ; prime number - - ; primitive normal basis - - ; probabilistic Riesz d e c o m p o s i t i o n - - ; P r o l l a - - ; Ramsey - - ; R e e h - S c h l i e d e r - - ; Reidemeister - - ; r e s i d u e - - ; r e t u r n - t i m e - - ; Riesz - - ; Riesz decomposition - - ; Riesz local representation - - ; Riesz representation - - ; Riesz splitting - - ; Rote sign - - ; Roy strong six exponentials - - ; Ruelle - - ; S a n t a l 6 - - ; Savitch - - ; Schauder f i x e d - p o i n t - - ; Selberg - - ; separation - - ; Shannon capacity - - ; sign - - ; six exponentials - - ; Skolern-Noether - - ; Slobodnik - - ; Smith-Solman-Wagner - - ; Sebolev imbedding - - ; Sperner - - ; Stallings - - ; Stickelberger--; Stiemke - - ; StoTIow - - ; Stone-Banach - - ; Stonevon Neumann -- ; strong six exponentials - - ; strong Szeg6 limit - - ; Swan-Forster - - ; SzemerGdi - - ; Taussky - - ; Taylor - - ; Temlyakov-OpiaI-Siciak-type meanvalue - - ; Temperley -- ; Tietze extension - - ; topological stability - - ; topological t r i v i a l i t y - - ; t r a n s v e r s a l - - ; Tverberg - - ; uniform b o u n d e d n e s s - - ; uniform W i e n e r - W i n t n e r - - ; uniformization--; uniqueness-of-norm - - ; Van Kampen - - ; Vidav-Palmer - - ; Vinogradov mean value--; Waldspurger--; Weber--; Weierstrass approximation - - ; Weisner - - ; w e l l - o r d e r i n g - - ; Weyl-von Neumann - - ; Wiener - - ; Wiener-Ditkin - - ; Wiener-R6 decomposition - - ; WienerWintrier - - ; Wiener-Wintner ergodic - - ; Wiener-Wintrier return-time - - ; Wittenb a u e r - - ; Wolff --; W o t f f - D e n j o y - - ; Zermanov prime - - ; Zel'manov structure -- ; Zorich -theorem for B ( G ) see: idempotency -theorem for Borcherds Lie algebras s e e : structure -theorem for coverings s e e : ConnesMoscovici higher index - - ; higher index - theorem for "D-filters see: correspondence - theorem for deductive systems s e e : characterization - theorem for Ditkin sets s e e : injection --

542

theorem for foliations s e e : Connes index - - ; index -theorem for harmonic functions s e e : converse of Gauss mean-value - - ; meanvalue - theorem for harmonic spaces s e e : Riesz decomposition - theorem for HNN-extensions s e e : Collins conjugacy - - ; normal form - - ; torsion - theorem for LYM posets s e e : quotient - theorem for matrix eigenvalues s e e : localization - theorem for maximal Buchsbaum modules overregularlocalringssee: structure-theorem for operator vessels s e e : matching - theorem for operators s e e : Riesz decomposition - theorem for polynomials and matrices s e e : stability -theorem for real function algebras s e e : Bishop -theorem for Schubert cycles s e e : basis - - ; duality -theorem for sets of finite Hausdorff measure see: decomposition - theorem for subharmonic functions s e e : Riesz decomposition -theorem for superharmonic functions s e e : Riesz decomposition -theoremforthe Brouwerdegree s e e : product - theorem for the exponential function s e e : addition - theorem for the Z-transform s e e : final value - - ; initial value - theorem in algebraic logic s e e : completeness -theorem

in

group

cohomology

Serre -theorem in harmonic analysis s e e : avin - -

see:

Malli-

theorem of a deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) theorem of atgebraizable deductive systems see: characterization - -

theoremof alternatives for vector inequalities [15A39, 90C05] (see: Motzkin transposition theorem) theorem of Delsarte type [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) theorem of Delsarte type s e e : tworadius -theorem of inverse scattering theory s e e : characterization - - ; reconstruction - - ; uniqueness --

theorem of (p,q) type [52A35] (see: Geometric transversal theory) theorem of the alternative s e e : Gordan - theorem on abundant numbers s e e : ErdGs - theorem on almost-split sequences s e e : Auslander-Reiten - theorem on analytic subsets s e e : Lelong - theorem on arithmetic progressions s e e : Szemer~di - - , van der Waerden - theorem on braids s e e : Alexander-theorem on closed projections s e e : Kuratowski -theorem on eigenvalues s e e : Brauer --

theorem on equivalence systems and the Suszko congruence

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) theorem on finite Abelian groups s e e : fundamental - theorem on function fields of genus zero s e e : Amitsur - -

theorem on Hankel matrices s e e : Kronecker - theorem on invariance under geodesic flow see: Liouville -theorem on knots s e e : Brunn - theorem on Krull dimension of group cohomology s e e : Quillen - theorem on matrices s e e : S c h u r - theorem on products of monic irreducible polynomials s e e : Gauss - theorem on quadrangles s e e : Varignon - theorem on Schur indices s e e : Wedderburn - theorem on semi-groups s e e : Martin - theorem on splitting fields for group algebras see: Brauer-theorem on squares s e e : Lagrange-theorem on the Euler-Mascheroni constant see: Euler -theorem on trigonometric series s e e : Young - -

inverse scattering - - ; recursively enumerable - - ; reduction - - ; reliability--; representation--; Reshetnyak - - ; resonance in scattering - - ; Riesz spectral - - ; risk - - ; rook - - ; rotor in graph - - ; rotor in knot - - ; Seiberg-Witten - - ; Shafarevich conjecture in inverse Galois - - ; simple homotopy - - ; singularity - - ; Sperner - - ; stability - - ; string - - ; Sturm-Liouville -- ; subresultant - - ; superstable - - ; surface - - ; susy gauge - - ; symmetry breaking in quantum field - - ; Szeg6 - - ; Tilting - - ; topological field - - ; topological quantum field - - ; undecidable equational - - ; uniqueness theorem of inverse scattering - - ; utility - - ; weakly minimal - - ; Whitham -- ; Witten-Dijkgraaf-Verlinde-Verlinde - - ; Yang-Mills - - ; Zermelo-Fraenkel set - - ; Zermelo set - theory) s e e : Domain (in ring - - ; Z (in set - - ; ZF (in set --

theorem prover [06Exx, 68T15] (see: Robbins equation)

theory axiomatized by a set of formulas

theorem-proving see: automated - - ; automatic -theorems s e e : applications of index - - ; Avram-Parter -- ; fixed-point - - ; generalized index - - ; Szeg6 limit - theorems in algebraic logic s e e : equivalence -theorems in logic s e e : characterization - theoretic MGbius inversion formula s e e : number- - theoretic smoothness s e e : measure- - theoretictopologysee: point-setlattice- - theories s e e : topological field - theory s e e : Abstract analytic number - - ; Abstract prime number - - ; algebraic quantum field - - ; algorithmic geometric transversal - - ; arithmetical partition - - ; automatic continuity - - ; averaging - - ; axiomatic approach to Dempster-Shafer - - ; axiomatization of set - - ; Bass-Serre - - ; BDF - - ; belief function - - ; BrownDouglas-Fillmore - - ; canonical algebra in tilting - - ; characterization theorem of inverse scattering - - ; cohomological variety in representation - - ; combinatorial group - - ; computational learning - - ; concurrency - - ; conformal field - - ; conformal quantum field - - ; constructive function - - ; continuous location - - ; cotorsion - - ; critical point - - ; decidable equational--; decision - - ; DempsterS h a f e r - - ; density R a m s e y - - ; design - - ; dual resonance - - ; duality - - ; E- - ; equational - - ; excision in algebraic K - - ; extraordinary homology - - ; fine topology in potential - - ; finitely axiomatizable - - ; fluctuation - - ; formulas Frege equivalent over a - - ; Frege relation of a - - ; generalized h o m o l o g y - - ; geometric group - - ; Geometric measure - - ; Geometric transversal - - ; GGdel relative consistency of set - - ; graphoidal structure in belief - - ; H * - t r i p l e - - ; Haag-Rueile scattering - - ; homotopy - - ; Horn - - ; I n d e x - - ; i n s u r a n c e - - ; Iwasawa - - ; Jacobson representation - - ; K - t h e o r y in index - - ; K K --; Kasparov - - ; Kasparov K - - - ; language of set - ; large-sample - - ; local spect r a l - - ; location - - ; marginally correct approximation approach to DempsterShafer - - ; massive quantum field - - ; membrane - - ; monopole - - ; n-rotor in graph - - ; n-rotor in knot - - ; naive approach to Dempster-Shafer - - ; nonlinear potential - - ; w-stable - - ; SU ( 2 ) gauge - - ; partial realization problem of system - - , Pluripotential--; possibility - - ; potential flow - - ; qualitative approach to Dempster-Shafer - - ; quantative approach to Dempster-Shafer - - ; quantum information - - ; Ramsey - - ; rational homotopy - - ; reconstruction theorem of

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) theory for a families of operators s e e : index - theory for a single operator s e e : index - theory for derived categories s e e : Morita - theoryin indextheory s e e : K - - theoryof s e e : Quantum computation, -

theory of a deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) theory of a logic s e e :

syntactical - -

theory of a model

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) theory of Abelian functions s e e : transcendence - theory of aberrations s e e : diffraction - theory of compact operators s e e : Riesz - - ; Spectral - theory of distributions s e e : multiplier - theory of elliptic functions s e e : transcendence -theoryof evidence s e e : mathematical - theory of exponential functions s e e : transcendence - theory of extremal rays s e e : Mori -theory of generalwill s e e : Rousseau - theory of group actions s e e : Smith - theory of groups acting on trees s e e : Bass-Serre - theory of HNN-extensions s e e : reduced sequence in the - theory of Landau--.Ginsburg type s e e : topological field -theory of modular forms s e e : Hide - -

theory of monopoles [35Qxx, 78A25] (see: Magnetic monopole) theoryof operators s e e : spectral - theoryof resonance curves s e e : Huxley - theory of super-connections s e e : Quillen - theory over $5 s e e : Suszko congruence of a - theory with the axiom of choice s e e : Zermelo-Fraenkel set - -

thermodynamic fimit [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) thermodynamics s e e : fluctuations in - thesis s e e : Church - theta-function s e e : third -theta-functionidentity s e e : Jacobi -theta-functions s e e : Jacobi --

Thiele differential equation (62P05) (refers to: Kolmogorovequation) Thiele differential equation

[62P05]

TORRICELLI POINT

(see: Thiele differential equation) Thierry-Mieginterpretationsee:

Ne'eman-

thin additive basis [11Pxx] (see: Additive basis) thin-plate spline [41A05, 41A30, 41A63] (see: Radial basis function) third theta-function [42Axx, 44-XX, 44A55] (see: Zak transform) Them isotopy lemmas [57N80] (see: Thom-Mather stratification) Thorn mapping [57N80] (see: Thom-Mather stratification) Thom-Mather stratification (57Ng0) (refers to: Fibration; Stratification; Submersion; Tubular neighbourhood) Thom-Mather stratification [57N801 (see: Thom-Matber stratification) Thomson paradox see: D o w n s - three-dimensional manifold s e e : boundary compressible surface in a -- ; boundary incompressible surface in a - - ; compressible surface in a -- ; O-compressible surface in a - - ; O-incompressible surface in a - - ; hierarchy for a -- ; incompressible surface in a - - ; irreducible - - ; length of a hierarchy for a - - ; p2.irreducible - - ; partial hierarchy for a - - ; reducible - - ; sufficiently-large - three-dimensional manifold at a s u r f a c e s e e : splitting a -three-dimensional manifolds s e e : homeomorphism problem for - - ; instanton Finer homology for - - ; uniformization for - - ; word problem for - three-move conjecture s e e : MontesinosNakanishi --

Thue-Morse number [11Fll, 11J82, 11J85, llJ9l] (see: Mahler method) Thue-Siegel lemma [llFll, llJ81, 11J82, 11J85, llJ91] (see: Gel'fond-Schneider method; Mahler method; Schneider method) Tietze extensiontheorem [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) Tietze transformations [55Pxx, 55P15, 55U35] (see: Algebraic homotopy) Tikhonov topology [54Bxx] (see: Wijsman convergence) tiling problem see: domino -Tilted algebra (16G10, 16G20, 16G60, 16G70) (referred to in: Riedtmann classification; Tilting functor; Tilting module; Tilting theory) (refers to: Algebra; Algebra of finite representation type; Bimodule; Category; Dimension; Endomorphism; Exact sequence; Quiver; Representation of an associative algebra; Riedtmann classification; Tilting functor; Tilting module; Tilting theory) tilted algebra [16Gxx] (see: Tilting functor; Tilting theory) tilted algebra s e e :

quasi- - -

tilted algebra of type H [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) tilting complex [16Gxx] (see: Tilting theory) Tilting functor (16Gxx)

(referred to in: Tilted algebra; Tilting module; Tilting theory) (refers to: Algebra; Derived category; Field; Morita equivalence; Tilted algebra; Tilting module; Tilting theory) tilting functor [16Gxx] (see: Tilting functor) Tilting module (16Gxx) (referred to in: Tilted algebra; Tilting functor; Tilting theory) (refers to: Algebra; Derived category; Dimension; Field; Quiver; Tilted algebra; Tilting functor; Tilting theory) tilting module [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting module; Tilting theory) tilting module s e e : ized - -

A P R - - - ; general-

lilting object [16Gxx] (see: Tilting theory) tilting object [16Gxx] (see: Tilting module) tilting sheaf [16Gxx] (see: Tilting theory) Tilting theory (16Gxx) (referred to in: Riedtmann classification; Tilted algebra; Tilting fnnctor; Tilting module) (refers to: Algebraically closed field; Artinian module; Category; Coherent sheaf; Derived category; Exact sequence; Module; Morita equivalence; Quantum groups; Simplicial complex; Tilted algebra; Tilting functor; Tilting module) tilting theory [16Oxx] (see: Tilting theory) tilting theory see: canonical algebrain - time s e e : exponential - - ; Levy local -- ; local - - ; Minkowski space- - - ; Whitham --

time average [28Dxx, 54H20, 58F11, 58F13] (see: Absolutelycontinuous invariant measure) time complexity see: average-case - - ; polynomial - - ; polynomial on average -time complexity class s e e : bounded probabilistic polynomial - - ; exponential- - - ; non-deterministic polynomial - - ; polynomial - time computability s e e : polynomial- - time computable language s e e : boundederror polynomial- - - ; bounded-error quantum polynomial- - -

time-constructible function [03915, 68Q15] (see: Computational classes)

complexity

time derivative s e e : Inca[ - - ; material - - ; mobile -time factorization method s e e : KaltofenTrager random polynomial- --

time for a non-deterministic Turning machine [03D15, 68Q15] (see: Computational complexity classes) time-frequency localization [42Cxx, 94A 12] (see: Window function) time hierarchy see:

polynomial- --

time limited function [42A63] (see: Uncertainty principle, mathematical) time localization

[42Cxx, 94A12] (see: Window function) time many-one problem reduction s e e : polynomial- - time of a European call option s e e : expiration - -

time resource [68Q15] (see: Average-case computational complexity) time system see: conservative d i s c r e t e - - time theorem s e e : Bourgain return- - - ; return- - - ; W i e n e r - W i n t n e r return- - -

Tits quadratic form (16Gxx) (refers to: Abelian category; Algebraic geometry; Algebraic group; Algebraic variety; Algebraically closed field; Dimension; Dynkin diagram; Euler characteristic; Exact sequence; Graph, oriented; Grothendieck group; Jacobsen radical; Jordan-H61der theorem; Quiver; Riedtmann classification; Zariski topology) Tits simplicial complex [20G051 (see: Steinberg module) Tits-type equality [16Gxx] (see: Tits quadratic form) Todacurve see: Toda lattice s e e :

Seiberg-Witten - discrete - -

Toda molecule equation [35L15] (see: Euler-Poisson-Darboux equation) Yoda type s e e :

tau-function of K P - --

Toda-type differential-difference equations [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarehy) Toeplitz algebra [46Cxx, 47B35] (see: Berezin transform) Toeplitz C*-algebra (46Lxx) (refers to: C*-algebra; Cauchy integral theorem; Cauchy operator; Fredholm operator; Hardy spaces; Index theory; K-theory; K~ihier manifold; Lie group; Neumann O-problem; Reinhardt domain; Toeplitz operator; Winding number) Toepiitz determinant [42A16, 47B35] (see: Szeg6 limit theorems) Toeplitz matrix [42A16, 47B35] (see: Szeg6 limit theorems) Toepfitz momentmatrix [44A60, 47A57] (see: Moment matrix) Toeplitz operator [46Cxx, 47B35] (see: Berezin transform) Toeplitz operator [42A16, 47B35] (see: Szeg6 limit theorems) Toeplitz operators with H ~ symbol [479xx] (see: Taylor joint spectrum) Toeplitz system (15A57) (refers to: Toeplltz matrix) tolerant quantum processing s e e : fault -tomographicdata s e e : local -tomography s e e : computerized - - ; Local --; Pseudo-Inca[ -tomography function s e e : inversion formula for the pseudo-local - - ; pseudolocal - - ; standard local - tomographyoperator s e e : local - T O P s e e : category L - - -

top conjecture s e e :

permanent-on- - -

topological algebra [54C08] (see: Almost continuity) topological category [03G10, 06Bxx, 54A40] (see: Fuzzy topology) topological degree [55M25] (see: Brouwer degree) topological dual [47A53] (see: Fredhohn solvability) topological field theories [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) topological field theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) topological field theory of LandauGinsburg type [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) topological game [54G10] (see: P-space) topological group s e e : (~ech-complete semi- - - ; Polish - - ; semi- - -

topological molecularlattice [03G10, 06Bxx, 54A40] (see: Fuzzy topology) topological quantumfield theory [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) topological space s e e : l,~l-additive - - ; a favourable - - ; compactly generated - - ; fuzzy--; L---; L-fuzzy-; lattice- ; l a t t i c e - f u z z y - - ; Strongly countably complete -- ; sub-Stonean -- ; weakly a favourable - -

topological stability [57N80] (see: Thom-Mather stratification) topological stability theorem [57N80] (see: Thom-Mather stratification) topologicaltriviality s e e :

local - -

topological triviality theorem [57N80] (see: Thom-Mather stratification) topological Vaught conjecture [03C15, 03C45, 03E15] (see: Vaught conjecture) topologies s e e : S-related - topology see: compact-open - - ; crosswise - - ; density - - ; fixed-basis - - ; Fuzzy --; generalized--; Hausdorff--; L- - ; L - f u z z y - - ; lattice- - - ; lattice-fuzzy - ; l a t t i c e - v a l u e d - - ; non-commutative - - ; point-set lattice-theoretic--; Polish - - ; posIat - - ; product - - ; proximal - - ; proximal ball - - ; quantum - - ; right halfopen interval - - ; Satake - - ; Skorokhod - - ; Sorgenfrey - - ; Sorgenfrey half-open square - - ; strict - - ; Tikhonov - - ; ultraweak operator - - ; variable-basis - - ; variable-basis fuzzy - - ; Vietoris - topology) s e e : Exponentiallaw (in - topologyin potentialtheory s e e : fine - topology induced by a metric s e e : Wijsman - topos see: G r o t h e n d i e c k - toritechnique s e e : Lin singular - -

toroidal compactification [llFxx] (see: Satake compactification) toroidal embedding [11Fxx, 20Gxx, 22E46] (see: Bully-Betel compactification) TorriceUi-Fermat problem [90B85] (see: Fermat-Torricelli problem) Torrieelli point [51M04] (see: Triangle centre) 543

TORRICELLI PROBLEM

Torricelti problem s e e : F e r m a t - - - ; generalized F e r m a t - --; u n w e i g h t e d F e r m a t torsion s e e :

R a y - S i n g e r analytic --

torsion class [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting theory) torsiou-free class [16Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting theory) torsion pair [I6Gxx, 16G10, 16G20, 16G60, 16G70] (see: Tilted algebra; Tilting functor; Tilting theory) torsion pair see: splitting -torsion theorem for HNN-extensions [20F05, 20F06, 20F32] (see: HNN-extension) torus s e e : Clifford - - ; ergodic automorphism of the infinite-dimensional - - ; incompressible - - ; invariant - - ; x / 2 - - -

toms bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) torus knot [57M25] (see: Positive link) torus knot s e e : tive - -

( 5 , 2 ) positive - - ; posi-

total drag due to vortex shedding [76Cxx] (see: Van Kfirmfin vortex shedding) total function set [35P25] (see: Partial differential equations, property C for) totally geodesic submanifold [17A40] (see: Lie triple system) totally positive matrix [15A15, 20C30] (see: Immanant) totally real submanifold [32E203 (see: Polynomial convexity) totient function see: E u l e r - tower problem s e e : class field -trace s e e : Dixmier - - ; J o n e s - O c n e a n u --

trace-class operator [42A16, 47B35] (see: Szeg6 limit theorems) trace-class operator determinant [81T50] (see: Non-commutative anomaly) trace condition [35Sxx, 46Lxx, 47Axx] (see: Wodzieki residue) traceformula s e e : Fedosov - - ; first-order - - ; second-order --

trace function [11B37, 11T71, 93C05] (see: Shift register sequence) trace of an element in a Galois extension [ 12E20] (see: Galois field structure) Tracyconjecture see: B a s o r - -trade in a Steiner triple system [05B07, 05B30] (see: Pasch configuration) trade (pair of blocks) [05B07, 05B30] (see: Pasch configuration) traffic flow [60K30, 68M10, 68M20, 90BI0, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) traffic flow see: equilibrium - Trager random polynomial-time factorization method see: K a i t o f e n - - -

training experience in a learning system [68T05] (see: Machine learning) training set

544

[68T05] (see: Machine learning) trajectory [58F22, 58F25] (see: Seifert conjecture) trajectory see: isogonal - trajectory of a dynamical system s e e : riod of a --

pe-

transcendence measure [I1Fll, 11J82, 11J85, llJ91] (see: Mahler method) transcendence of values of Abelian functions [llJ811 (see: Schneider method) transcendence of values of elliptic functions [llJ81] (see: Schneider method) transcendence of values of exponential functions [llJSi] (see: Schneider method) transcendence properties of values of analytic functions [llFI1, 11J82, 11J85, llJ91] (see: Mahler method) transcendence theory of Abelian functions [11J81] (see: Schneider method) transcendence theory of elliptic functions [11J81] (see: Schneider method) transcendencetheory of exponentialfuncdons [llJ81] (see: Schneider method) transfer function [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) transfer operator [28Dxx, 54H20, 58F11, 58F13] (see: Absolutely continuous invariant measure) transferable belief model [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) transform s e e : admissibility condition for a reconstruction formula for the continuous wavelet - - ; analytic properties of the Zak -; applications of the Zak -- ; Berezin -- ; combinatorial Radon - - ; conjugation behaviour of the Zak - - ; conjugation of the Zak - - ; continuous wavelet - - ; convolution and Z- -- ; cosine -- ; discrete cosine - - ; discrete Fourier - - ; discrete Laplace -; discrete wavelet - - ; example of a Z- - ; fast discrete cosine - - ; fast Fourier - - ; final value theorem for the Z- - - ; initial value theorem for the Z- - - ; inverse scattering - - ; inverse spectral - - ; inverse Z- - - ; inversion of the Zak - - ; linearity of the Zak - - ; MacWilliams - - ; modulation behaviour of the Zak - - ; modulation of the Zak - - ; partial-fractions technique for finding the inverse Z- - - ; periodicity of the Zak - - ; properties of the Z- -- ; reconstruction formula for the continuous wavelet - - ; Riesz-Herglotz - - ; shift behaviour of the Z- - - ; shift of the Zak - - ; symmetry for the Zak - - ; translation behaviour of the Zak - - ; translation of the Zak - - ; w i n d o w e d Fourier - - ; Z--; Zak -transform method s e e : integral -transform of the Gaussian function s e e : Zak - transformation s e e : Berezin - - ; causal linear - - ; conjugate -- ; convolution end Zak - - ; convolution under Zak - - ; Darbeux - - ; D a r b o u x - C r u m - - ; fractionallinear -- ; gauge -- ; Girsanov -- ; inverse Z- - ; Z - transformational grammar s e e : Chomsky - -

transformations s e e : anti-BRST - - ; Becchi-Rouet-Stora-Tyutin --; BRST --; Tietze -transforms s e e : examples of Z- - - ; examples of Zak - - ; existence of Zak - - ; properties of Zak - - ; reciprocal integral - transition s e e : phase - - ; SAT phase - -

transition formula [05E05, 13P10, I4C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) transition function see: r e v e r s i b l e - transition in a binary alloy s e e : phase - transitive Cayley graph s e e : edge- - transitive graph s e e : vertex- - -

transitive group [20-XX] (see: Regular group) transitive group action see: s p h e r i c a l l y - transitive regular permutation group [20-XX] (see: Regular group) translation s e e : machine - -

Auslander-Reiten --;

translation behaviour of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) translation functional equation [39B05, 39B12] (see: Schrfder functional equation) translation net [05Bxx] (see: Net (in finite geometry)) translation of the Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform) translation plane [05Bxx, 05B30] (see: Affine design; Net (in finite geometry)) translation quiver [16G70] (see: Riedtmann classification) translation quiver s e e : a--

configuration in

transmission coefficient [35P25, 35Q53, 47A40, 58F07, 81U20] (see: Harry Dym equation; Inverse scattering, full-line case) transmission protocol [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) transposition property of the monster s e e : 6--transposition theorem s e e : Motzkin - -

transvection [32H15, 34G20, 46G20, 47D06, 47H20] (see: Semi-group of holomorphic

mappings) hyperplane - - ; ktransversal design [05Bxx] (see: Net (in finite geometry))

transversal s e e :

transversal design s e e : symmetric - -

--

resolvable - - ;

transversal free net [05Bxx] (see: Net (in finite geometry)) transversal in a net [05Bxx] (see: Net (in finite geometry)) transversal theorem [52A35] (see: Geometric transversal theory) transversal theorem s e e : Hadwiger - - ; H a d w i g e r - t y p e - - ; HaPly-type - transversal theory s e e : algorithmic geometric - - ; Geometric - -

transversality [35P25] (see: Partial differential equations, property C for)

trapped ion m o d e in a plasma s e e : pative - trapped-orbit

dissi-

condition

[581722, 58F25] (see: Seifert conjecture) trapping in random media [60Gxx, 60J55, 60J65] (see: Wiener sausage) travelling salesman path problem [90C08] (see: Travelling salesman problem) Travelling salesman problem (90C08) (refers to: Graph; Graph circuit; Hamiltonian tour; .N'7~) travelling salesman problem [90C08] (see: Travelling salesman problem) travelling salesman walk problem [90C08] (see: Travelling salesman problem) travelling wave [35Q35, 58F13, 76Exx] (see: Kuramote-Sivashinsky equa-

tion) tree s e e : chemical index of a - - ; group acting on a - - ; group action on a rooted - - ; in- - - ; out- - - ; spherically homogeneous - - ; unlabelled - - ; Wiener n u m b e r of a - -

tree adjoining grammar [68S05] (see: Natural language processing) tree adjoining grammar parser [68S05] (see: Natural language processing) tree banks s e e : parse -tree in machine learning s e e : decision -tree level s e e : branch index for a - - ; rigid stabilizer of a - tree theorem s e e : generating function version of the matrix - - ; Matrix - trees s e e : Bass-Serre theory of groups acting on - - ; Cayley formula for the number of labelled - -

trefoil knot [57M25] (see: Jenes-Conway Positive link)

polynomial;

triangle s e e : centre of mass of a - - ; centroid of a - - ; circumcentre of a - - ; concurrent lines in a - - ; first isogonic centre of a - - ; incentre of a - - ; orthocentre of a--

Triangle centre (51M04) (refers to: Bisectrix; Gergonne point;

Isogonal; Median (of a triangle); Nagel point; Plane trigonometry) triangle inequality for a vector norm [15A42] (see: Bauer-Fike theorem) triangular hat function [62D05] (see: Acceptance-rejection method) triangular model of a non-self-adjointoperator [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) triangular model of a pair of operators [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) triangular set function [28-XX] (see: Non-additive measure) triangulated category [16Gxx] (see: Tilting theory) Tribonacci number (11B39) (referred to in: Tribonacci sequence) (refers to: Tribonacci sequence) Tribonacci sequence (11B39) (referred to in: Tribonacci number)

UNIFORM GEOMETRY

(refers to: Fibonacci numbers; Tribonacci number) trick s e e : Rabinowitsch -tridiagonalizationprocesssee:

truth s e e : Lanezos--

trigonometric moment problem

[44A60, 47A57] (see: Moment matrix) Trigonometric pseudo-spectral methods (65Lxx, 65M70) (referred to in: Chebyshev pseudospectral method; Fourier pseudospectral method) (refers to: Boundary value problem, ordinary differential equations; Chebysbev polynomials; Complete system of functions; Differential operator; Dirac distribution; Galerkin method; Ortbogonal polynomials; Orthogonal system; Trigonometric functions) trigonometric series s e e : - - ; Young theorem on - -

Integrability of

trilinear coordinates

logical - -

truth filter

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) tube system [57N80] (see: Tbom-Mather stratification) tubular function [57N80] (see: Thom-Mather stratification) Tucker condition

see:

Karush-Kuhn- --;

Kuhn- -Tukey-type simultaneous confidence interval

[62Jxx] (see: ANOVA) t u n e of operators s e e : Fredholm - - ; nonsingular -tuples s e e : Taylor spectrum for Fredholm n---

Turaevinvariants s e e : R e s h e t i k h i n - - Turner quantization s e e : Drinfel'd- - - ; w e a k Drinfel'd- - -

[51M04] (see: Isogonal) triple see: Conway JB*- --; JBW*man bracket skein real form of a JB * -

[62Jxx, 62Mxx] (see: Cox regression model)

skein --; Hilbert --; - - ; J C * - - - ; Kauff- - ; prime ,113* - - - ; -- ; real JB * - - -

triple ideal [17Cxx, 46-XX] (see: JB * -triple) see: Jordan - triple of a Steiner triple system

triple identity

[05B07, 05B30] (see: Pasch configuration) triple product

[17A40, 17Cxx, 46-XX] (see: JB *-triple; Jordan triple system) triple product [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair; Lie triple system) triple system

[17A40, 17B60] (see: Allison-Hein triple system; Anti-Lie triple system; FreudenthalKantor triple system; Jordan triple system; Lie triple system) triple system s e e : Allison-Hein - - ; AntiLie - - ; anti-Pasch Steiner - - ; balanced FreudenthaI-Kantor--; Banach-Jordan - - ; block of a Steinar - - ; element of a Steiner - - ; FreudenthaI-Kantor -- ; J B * Jordan - - ; Lie - - ; line of a Steiner - - ; mutually t-balanced collections of blocks in a Steiner - - ; point el a Steiner - - ; quadrilateral free Steiner - - ; Stainer - - ; trade in a Steiner - - ; triple of a Steiner - triple systems s e e : examples of Jordan - tripletheory s e e : H * - - - ;

triple zero bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) Tristram-Levinesignature [57P25] (see: Conway skein equivalence) Tristram-Levine signature of a link [57P25] (see: Conway algebra) triviality s e e : Iocaltopological - trivialitytheorem s e e : topological - Trudi matrix s e e : J a c o b i - - -

truncated Well algebra

[55R40, 57Rxx] (see: Well algebra of a Lie algebra) truncated Wiener-Hopf operator

[42A16, 47B35] (see: Szeg5 limit theorems) truncation condition

[llF11, 17B10,17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) truncation in statistics

Turdn brick factory problem

[05C10, 05C35] (see: Zarankiewicz crossing number conjecture) Tur~in conjecture s e e : E r d d s - - turbulence s e e : phase - - ; strong - - ; weak -Turing machine s e e : alternating - - ; probabilistic - - ; quantum --; universal --

Tufing undecidability [12D05] (see: Factorization of polynomials) Turk head knot [57Mxx] (see: Fibonacci manifold) Turning machine s e e : non-deterministic ; time for a n o n - d e t e r m i n i s t i c - --

Tverberg theorem [52A35] (see: Geometric transversal theory) twisted affine Lie algebra [llFll, 17B10, 17B65, 17B67, 20D08, 8IR10, 81T30] (see: Vertex operator) twisted Dirac operator [46L80, 46L87, 55N15, 58G10, 58G11, 58G12] (see: Index theory) twisted vertex operator [llFll, 17B10, 17B65, 17B67, 20D08, 81R10, 81T30] (see: Vertex operator) twister method [35Qxx, 78A25] (see: Magnetic monopole) two s e e : cohomological dimension - - ; free subgroup of rank - -

two-componentKP-hierarchy [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) two-dimensionalgravity [53C42] (see: WUlmorefunctional)

type s e e : homotopy --; integral operators of Carleman --; k - o r d e r --; language - - ; Noetherian local ring of finite Buchsbaum-representation - - ; simple k - o r d e r - - ; tame representation - - ; tan-function of K P - T o d a - - ; theorem of (p,q) - - ; theorem of Delsarte - - ; topological field theory of Landau-Ginsburg ; two-radius theorem of Delsarte - --

type-0 grammar [68S05] (see: Natural language processing) type differential-difference equations s e e : Toda- - type equality s e e : Tits- - type H s e e : tilted algebra of - type I s e e : exponential sum of - type-/C*-algebrasee: non- - -

type-I error [62Lxx] (see: Average sample number) type II s e e :

exponentialsum of - -

type-II error [62Lxx] (see: Average sample number) type inequality s e e : Sidon- - type integral operator s e e : Cademan- -type knot invariant s e e : finite- - typemean-valuetheoremsee: TemlyakovOpiaI-Siciak- - type of boxes s e e : representation -type pair of inverse relations s e e : Chebyshev- -type polynomials s e e : Fibonacci- - - ; Lucas- - type polynomials of order k s e e : Fibonacci- - - ; Lueas- - type radical s e e : Jacobsen- - type simultaneous confidence interval s e e : Scheff#- - - ; Tukey- - type skein module s e e : Jones- - type theorem s e e : Delsarte- - - ; Gallai- - ; Gershgorin- -- ; KrulI-Schmidt- - type transversal theorem s e e : Hadwiger- - ; Helly- --

typed higher-orderlogic [68S05] (see: Natural language processing) types see: algebraic models of homotopy - Tyutin transformations s e e : BecehiRouet-Stora- --

uncertainty measure [68T30, 68T99, 92Jxx, 92K101 (see: Dempster-Shafer theory) uncertainty principle ematical - -

see:

Uncertainty principle, mathematical (42A63) (refers to: Eigen function; Fourier transform; Harmonic analysis, abstract; Laplace operator; Lebesgue measure; Plancherel theorem; Stan. dard deviation; Support of a function) unconditioned strong d-sequence

[13A30, 13H10, 13H30] (see: Buchsbaum ring) uncorrelated random variables

[62H20] (see: Pearson product-moment cor-

relation coefficient) uncorrelated random variables with joint normal distribution [62H20] (see: Pearson product-moment correlation coefficient) undeeidabilitysee:

Tudng--

undecidable equational theory [03Gxx] (see: Algebraic logic)

[60Hxx, 90A09, 93Exx]

Ulam question on stability of homomorphisms

[39/372, 46B99, 46Hxx] (see: Hyers-ulam-Rassias stability)

[11B37] (see: Ultimately periodic sequence)

[62Jxx]

ultrafiltersee:

free

--

(see: Black-Scholes formula) underlying deductive system

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) ungrammatical usage of a natural language

Olam-Rassias stability see: H y e r s - - Ulam stability s e e : H y e r s - - -

(see: ANOVA)

in natural lan-

underlying asset of a European call option

Ohlmann fidelity see: B u r e s - - Ulam problem s e e : F e r m i - P a s t a - - -

ultimately periodic sequence see: period of an - - ; period of an --

Hardy - - ; math-

[03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic)

U-rank [03C15, 03C45, 03EI5] (see: Vaught conjecture)

ultimately polynomial [90D05] (see: Sprague-Grundy function)

two-way layout

UMP test [62Jxx] (see: ANOVA)

(see: Natural language processing) underlying algebra of a logical matrix

ultimately periodic sequence

[57N10] (see: Haken manifold)

Ulysses s e e : different w o r d s i n - - ; number of different words in - -

[68S051

U

two-round voting

two-sided surface in a manifold

[60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law)

underconstrainedness guage

two-radius theoremof Delsarte type [31A05, 31B05, 31C10, 31C35, 32A10, 46F10, 60Y65] (see: Mean-value characterization) [90A28] (see: Condorcet paradox)

Ulysses

unbiased estimator s e e : best linear - - ; minimum variance - uncertainty s e e : modelling - - ; nonprobabilistic - - ; representation of - uncertainty inequality see: Heisenberg - -

Ultimately periodic sequence (11B37) (referred to in: Linear complexity of a sequence; Shift register sequence) (refers to: Correlation property for sequences; Formal power series; Galois field; Polynomial; Shift register sequence)

two independent belief functions s e e : rule of combination of - two-operatorvessel s e e : quasi-Hermitian commutative --

ultrafilteron N s e e : P - p o i n t - ultrapower [17Cxx, 46-XX] (see: JB *-triple) ultraproduct-elosed [03Gxx] (see: Algebraic logic) ultraweak operatortopology [43A07, 43A15, 43A45, 43A46, 46J10] (see: Fig~-Talamanca algebra)

least

[68S05] (see: Natural language processing) unification s e e : commutative - -

AC --;

associative-

unification grammar

[68S05] (see: Natural language processing) uniform approximation see: best - -

uniform boundednesstheorem [43A45, 43A46] (see: Ditkin set) uniform expansion [28Dxx, 541120, 58F11, 58F13] (see: Absolutely continuous invariant measure) uniform geomeUy [05C25, 20Fxx, 20F32] 545

UNIFORM GEOMETRY

(see: Baumslag-Solitar group) uniform lattice [03G10, 06Bxx, 54A40] (see: Fuzzy topology) uniformWiener-Wintriertheorem [28D05, 54H20] (see: Wiener-Wintner theorem) uniformization for three-dimensional manifolds [57N10] (see: Haken manifold) uniformizadon theorem [14H15, 30F60] (see: Weil-Petersson metric) uniformizing parameter [12FI0, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) uniformly closed *-algebra [46Lxx] (see: Toeplitz C*-algebra) uniformly closed C*-algebra [46Lxx] (see: Toeplitz C* -algebra) uniformly continuous semi-group see: cally - -

lo-

uniformly most powerful test [62Jxx] (see: ANOVA) uniforms method see:

ratio-of---

unimodality property [05D05, 06A07] (see: Sperner property) uninterpreted deduction system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35[ (see: Abstract algebraic logic) union see:

axiom of - -

union problem in spectral synthesis [43A45, 43A46] (see: Ditkin set) unique A-solvability [47H17] (see: Approximation solvability) unique approximate solvability [47H17] (see: Approximation solvability) unique graph see:

matching - -

uniqueness-of-norm problem [46H40] (see: Automatic continuity for Banach algebras) uniqueness-of-norm theorem [46H40] (see: Automatic continuity for Banach algebras) uniqueness-of-norm theorem see: son -

John-

-

uniqueness theorem of inverse scattering theory [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) unit see: approximate - - ; countable approximate - - ; invertible element in a Jordan algebra with a - - ; S - - unit ball in a Banach space see: extreme point of the closed -unit in harmonic analysis see: approximate - -

unit-interval graph [llNxx, 11N32, 11N45] (see: Abstract prime number theory) unit s p h e r e i n Rn see: area of the -unit theorem see: unital algebra see:

Dirichlet - ~r. - -

unitarily equivalent operator vessels [47A45, 47A48, 47A65, 47D40, 47N70] (see: Operator vessel) unitary c~-abundant n u m b e r see: tive - -

primi-

unitary convolution [11A25] (see: Dirichlet convolution) unitary equivalence of operator vessels" [47A45, 47A48, 47A65, 47D40, 47N70]

546

(see: Operator vessel) unitary operator colligation [30E05, 47A48, 47A57, 47A65, 47Bxx, 47N50, 47N70] (see: Operator colligation) unitary operatorcolligation see: erator of a --

main op-

unitization [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) units s e e : of--

bounded approximate - - ; group

units of an algebraic number field [llJ81] (see: Schneider method univariate linear model [62Jxxl (see: ANOVA) universal algebras see:

variety of --

universal Borcherds algebra [1 IFxx, 17B67, 20D08] (see: Borcherds Lie algebra) universal central extension [19Cxx] (see: Steinberg symbol) universal distribution [68Q15] (see: Average-case computational complexity) universal Hilbert space [46J10, 46L05, 46L80, 46L85] (see: Multipliers of C* -algebras) universal Horn logic without equality see: infinitary --

universal partial Conwayalgebra [57P25] (see: Conway algebra) universal Schubertpolynomials [05E05, 13P10, 14C15, 14M15, 14N15, 20G20, 57T15] (see: Schubert polynomials) universal Taring machine [68T05] (see: Machine learning) u n i v e r s a l w e i g h t see:

good --

universality of the Sierpuiski carpet [28A80] (see: Sierpifiski gasket) universe of sets [03E30] (see: ZFC) universe of sets see: cumulative hierarchy of the - unknotting conjecture see: nor - -

Jones - - ; Mil-

unknottin g number [57M25] (see: Positive link) unlabelled graph [llNxx, 11N32, 1IN45] (see: Abstract prime number theory) unlabelled tree [llNxx, 11N32, 11N45] (see: Abstract prime number theory) unramified Abelian p-extension [11R23] (see: Iwasawa theory) unramified field extension [11F67] (see: Eisenstein cohomology) unramified prime ideal [11R32, 11R45] (see: Chebotarev density theorem) unrectifiabilitysee: pure -unrectifiable measure see: purely - unrectifiable Radon measure see: purely unrectifiable set see: 1- - - ; purely m - - -

example of a purely

unstable action of the Steenrod algebra [55P42] (see: Brown-Gitler spectra) unstable equilibrium of a dynamical system [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages)

unstable manifold [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) unstructured quantum search [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx] (see: Quantum computation, theory or) unsupervised learning [68T05] (see: Machine learning) unweighted Fermat-Torricelli problem [90B85] (see: Fermat-Torrieelli problem) (update) see: GaIois field - - ; Net (in finite geometry) --

upper bound conjecture [05Exx, 13C14, 55U10] (see: Stanley-Reisner ring) upper Fitting series [20F17, 20F18] (see: Fitting chain) upper half-plane [11Fxx, 20Gxx, 22E46] (see: Baily-Borel compactilication) upper half-space see:

Siegel - -

upper Minkowski measure [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) upper nilpotent series [20F17, 20F18] (see: Fitting chain) upper semi-continuous regularization [31C10, 32F05] (see: Pluripotential theory) upper shadow [05D05, 06A07] (see: Sperner theorem) pragmaties of natu ral l a n g u a g e - - ; robustness of natural language - usage of a natural language see: extragrammatical - - ; ungrammatical - usagesee:

USD-sequence [13A30, 13H10, 13H30] (see: Buchsbaum ring) use of the Jansen inequality see: of the - -

example

user equilibrium flow [60K30, 68M10, 68M20, 90B10, 90B15, 90B18, 90B20, 94C99] (see: Braess paradox) using the Lov~.sz local l e m m a see: ple of - -

exam-

utility function [90All[ (see: Cobb-Douglas function) utility in portfolio optimization see: pected - -

ex-

utility theory [28-XX] (see: Non-additive measure)

V vacuous belieffunction [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) vacuous extension of a set of vectors [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) vacuous extension operator [68T30, 68T99, 92Jxx, 92K10] (see: Dempster-Shafer theory) vacuum energysee:

q u a n t u m --

vacuum property [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 8IRI0, 81T30, 81T40] (see: Vertex operator algebra) Valdivia compact [26A15, 54C05] (see: Namioka space)

valence of afield isomorphism [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brill formula) validity relation [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) valuation [12J10, 12J20, 13A18, 16W60] (see: S-integer) value see: p - a d i c absolute - value distribution see: principal - -

value of a European call option at expiration [60Hxx, 90A09, 93Exx] (see: Black-Scholes formula) value on a n u m b e r field see: absolute - value problem s e e : Dirichlet b o u n d a r y - - ; Neumann b o u n d a r y - value problem for the Korteweg--de Vries equation see: characteristic initial- - value process see: portfolio-value representation see: attribute- - v a l u e t h e o r e m see: Vinogradov mean - value theorem for the Z-transform see: final - - ; initial - values see: asymptotic distribution of singular - values of Abelian functions see: algebraic independence of - - ; transcendence of - values of analytic functions see: transcendence properties of - values of elliptic functions see: algebraic independence of - - ; transcendence of - values of exponential functions see: algebraic independence of - - ; transcendence of--

van der Corput A-process [llL07] (see: Exponential sum estimates) van der Corput B-process [11L07] (see: Exponential sum estimates) van der Corputexponent pair [11Lxx, 1IL03, 11L05, 11L15] (see: Bombieri-Iwaniec method) van der Corput kth derivative estimate [11L07] (see: Exponential sum estimates) van der Corput method [11L07] (see: Exponential sum estimates) van der Waals law [76Axx] (see: Knudsen number) van der Waerden theorem on arithmetic progressions [05D10] (see: Hales-Jewett theorem) Van Kampen theorem [20F05, 20F06, 20F32] (see: HNN-extension) Vandiver conjecture [11R23] (see: Iwasawa theory) vanish at infinity [54C35] (see: Function vanishing at infinity) vanishing at infinity see: Function - vanishing eventually see: sequence - vanishing mean oscillation see: space of analytic functions of -v a n i s h i n g t h e o r e m see: Mahler--

Vapnik-Cervonenkis dimension [05C65, 05D05, 68Q15, 68T05] (see: Vapnik-Chervonenkis dimension) Vapnik-Chervonenkis dimension (05C65, 05D05, 68Q15, 68T05) (refers to: Graph, numerical characteristics of a; Hypergraph; AfT~; Vapnik-Chervonenkis class) variable see: Grassmann - - ; Quasisymmetric function of a complex - - ; regressor - -

variable-basisfuzzy topology [03G10, 06Bxx, 54A40]

VON KARMANRELATION (see: Fuzzy topology) variable-basis topology [03G10, 06Bxx, 54A40] (see: Fuzzy topology) variable-coefficient Benjamin-BonaMahony equation [35Q53, 76B15] (see: Benjamin-Bona-Mahony equation) variable configuration [05B07, 05B30] (see: Pasch configuration)

Varshamov bound see:

variable fragment of first-order logic s e e : /2-v a r i a b l e in covarianee analysis s e e : categorical - variable in regression analysis s e e : dependent - - ; independent - variable sets for a belief function s e e : independent -variables s e e : product moment of random - - ; uncorrelated random - variables with joint normal distribution s e e : uncorrelated random -variance s e e : analysis of - - ; generalized multivariate analysis of - - ; multivariate analysis of - variance efficient portfolio see: mean- - variance unbiased estimator s e e ; minimum variation s e e : bounded - - ; function of bounded - - ; space of functions of bounded - variation of a function s e e : HardyKrause - variation of an m - v a r i f o l d s e e : first -variational problem s e e : Euler equation of a--

finite basis problem for -variety s e e ; Abelian - - ; arithmetical - - ; character - - ; discriminator - - ; finite basis for the identities of a - - ; finitely based - - ; minimal - - ; normal algebraic - - ; Q-factorial algebraic - - ; Schubert - - ; second-order equivalent quasi- - - ; semiAbelian - - ; singular point on an algebraic - - ; terminal algebraic - - ; VaughanLee - variety in representation theory s e e : cohomological -varieties s e e :

variety of Boolean algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) variety of Boolean algebras s e e : mation property of the - -

amalga-

variety of distributive lattices [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) variety of Heyting algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) variety of monadic algebras [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) variety of orthomodularlattices [03Gxx, 03G05, 03GI0, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) variety of universal algebras [08Bxx, 1dR10, 17B01, 20El0] (see: Specht property) varietywith terminalsingularity s e e : braic - -

(51M04) (refers to: Varignon theorem) Varignon parallelogram [51M04] (see: Varignon parallelogram) Varignon theorem on quadrangles [51M04] (see: Varignon parallelogram)

alge-

v:Lrifold [28A78, 49Qxx, 49Q15, 53c65, 58A25] (see: Geometric measure theory) varifold s e e : first variation of an m - -- ; m - - - ; m-rectifiable - - ; stationary - -

Varignonframe [90B85] (see: Fermat-Terricefii problem) Varignon parallelogram

Gilbert---

Vassiliev-Gusarov filtration [57Mxx, 57M25] (see: Skein module) Vassiliev-Gusarovinvariants [57M251 (see: Jones-Conway polynomial) Vassiliev-Gusarov skein module [57Mxx, 57M25] (see: Skein module) Vassiliev invariant [57Mxx, 57M25] (see: Skein module) Vaughan identity (11L07, llM06, 11P32) (refers to: de la Vall~e-Poussin theorem; Dirichlet series; Eratosthenes, sieve of; Exponential sum estimates; Gauss sum; Goldbach problem; Knmmer hypothesis; Mangoldt function; Riemann hypotheses; Sieve method; Vinogradov method; Zetafunction) Vaughan identity [11L07, 11M06, 11P32] (see: Vaughan identity) Vanghan-Lee variety [08Bxx, 16R10, 17B01, 20El0] (see: Specht property) Vaught conjecture (03C15, 03C45, 03E15) (refers to: Borel system of sets; Continuum hypothesis; Descriptive set theory; Forking; Logical calculus; Luzin set; Model theory; Ordinal number; Topological group) Vaught conjecture [03C15, 03C45, 03E15] (see: Vaught conjecture) Vaught conjecture s e e :

topological --

VC-dimension [05C65, 05D05, 68Q15, 68T05] (see: Vapnik-Chervonenkis dimension) Vecten-Fasbender duality [90B85] (see: Fermat-Torricelli problem) vector s e e :

displacement - - ; extension of

a-vector bundle s e e :

real - vector field s e e : complete - - ; complete holomorphic - - ; Hamiltonian - - ; locally Hamiltonian - - ; Reeb -- ; semi-complete - - ; semi-complete holomorphic - vector inequalities s e e : theorem of alternatives for --

vector inequality [15A39, 90C05] (see: Motzkin transposition theorem) vector measure [28-XX] (see: Non-additive measure) vector norm [15A42] (see: Bauer-Fike theorem) vector norm s e e : a--

triangle inequality for

vector of a linear feedback shift register s e e : state - vector of a simplicial compTex s e e : f- --; h-v e c t o r o f shortest length

[65C10] (see: Linear congruential method) vector space s e e : interval in a partially ordered -- ; Lie algebra associated with a - - ; partially ordered --

vectors s e e : extension of a set of - - ; Stokes - - ; vacuous extension of a set of-v e l o c i t i e s in v o r t e x s h e e t s s e e : induced - velocity s e e : discontinuous tangential - velocity at a point s e e : instantaneous - -

velocity of a particle [73Bxx, 76Axx] (see: Material derivative method) velocity of a particle [73Bxx, 76Axx] (see: Material derivative method) Verlindeequations s e e : W i t t e n - D i j k g r a a f Verlinde-- -Verlinde theory s e e : Witten-DijkgraafVerlinde- -Verlinde-Verlindeequations s e e : WittenDijkgraaf- - Verlinde--Ver/inde theory s e e : WittenDijkgraaf- - -

Verma module [llFxx, 17B67, 20D08] (see: Borcherds Lie algebra) version of a statisticaltest s e e : curtailed -version of the Kendall tau s e e : population -version of the matrix tree theorem s e e : generating function - -

vertex algebra [11Fll, 17B10, I7B65, 17B67, 17B68, 20D08, glR10, 81T30, 81T40] (see: Vertex operator algebra) vertex algebra [llFll, 17B10, 17B65, 17B67, 17B68, 20D08, 81RI0, 81T30, 8IT40] (see: Vertex operator; Vertex operator algebra) vertex algebras s e e : associativity in - - ; commutativity in - - ; Jacobi identity for - -

Vertex operator (11Fll, 17B10, 17B65, 17B67, 20D08, 81R10, 81T30) (referred to in: Vertex operator algebra) (refers to: B~icklund transformation; Kac-Moody algebra; Korteweg--de Vries equation; Lie algebra; Moonshine conjectures; Representation of

a Lie algebra; Vertex operator algebra) vertex operator [11Fll, 17B10, 17B65, 17/367, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) vertex operator s e e :

twisted - -

Vertex operator algebra (11F11, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40) (referred to in: Moonshine conjectures; Vertex operator) (refers to: Commutation and anticommutation relationships, representation of; Golay code; KacMoody algebra; Knot theory; Leech lattice; Lie algebra; Mathieu group; Modular function; Moonshine conjectures; Quantum field theory; Quantum groups; Riemann surface; Simple finite group; Threedimensional manifold; Vector space;

Vertex operator; Virasoro algebra) vertex operator algebra [11Fll, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) vertex operator algebra [llF11, 17B10, 17B65, 17B67, 17B68, 20D08, 81RI0, 81T30, 81T40] (see: Vertex operator algebra) vertex-transitive graph [05C25] (see: Cayley graph) vertices of a graph s e e : f - s m o o t h i n g - vessel s e e : commutative operator - - ; discriminant curve of an operator - - ; discriminant polynomial of an operator - - ; external space of an operator - - ; input

determinantal representation of the discriminant curve of an operator - - ; internal space of an operator - - ; joint characteristic function of an operator - - ; Lie algebra operator - - ; normalized joint characteristic function of an operator - - ; Operator - - ; output determinantal representation of the discriminant curve of an operator - - ; quasi-Hermitian commutative two-operator - vessels s e e ; matching theorem for operator - - ; unitarily equivalent operator - - ; unitary equivalence of operator - -

vibration [35J05, 35J25] (see: Dirichlet eigenvalue) vibration of a membrane [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) vibration of a string [70Jxx, 70Kxx, 73Dxx, 73Kxx] (see: Natural frequencies) Vidav-Palmer theorem [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Vietoris topology [54Bxx] (see: Wijsman convergence) Vil[ars regularization s e e :

Pauli- - -

Viucensini problem [52A35] (see: Geometric transversal theory) Vinogradov-Korobovmethod [11L07] (see: Exponential sum estimates) Vinogradov mean value theorem [11L07] (see: Exponential sum estimates) V i n o g r a d o v t h e o r e m see:

Bombieri---

Virasoro algebra relations [llF11, 171310, 17B65, 17B67, 171368, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) Virasoro integrability [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) virtual knot [57M25] (see: Jones unknotting conjecture) viscosity s e e :

fluid s h e a r - -

viscosity subdifferential [90C30] (see: Clarke generalized derivative) Viscous lingering (76Exx, 76S05) (refers to: Conformai mapping; Laplace equation; Navier-Stokes equations; Stream function) visual volume [53C421 (see: Wifimore functional) VMOA -space (30D50, 46Exx) (refers to: Analytic function; BMOspace; BMOA-space; Cluster set; Conformal mapping; Continuous function; Functional analysis; Harmonic function; VMO -space) VOA [IIFll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) volatility [60Hxx, 90A09, 93Exx] (see: Blaek-Scholes formula) VOlklein conjecture s e e :

Fried---

Volterra functional series (41A58) (refers to: Volterra series) Volterra functionalseries s e e : Volterraintegralequation s e e :

Wiener- -Stieltjes---

Volterra-Stieltjes integral equation (45D05) (refers to: Stieltjes integral; Volterra equation) volume s e e : eonformal - - ; visual - volume measure s e e : Riemannian --

yon Kdrmdn relation

547

VON KARMAN RELATION

[76Axx] (see: Knudsen number) Von K~rmfin vortex shedding (76Cxx) (referred to in: Birkhoff-Rott equation) (refers to: Boundary layer; GinzburgLandau equation; Navier-Stokes equations; Strouhal number) von Neumann algebra s e e :

factor of a --

yon Neumann entropy [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81P15, 94Axx] (see: Quantum information processing, science of) von Neumann entropy s e e : ditivity inequality for - -

strong subad-

von Neumann factor quantale [03G25, 06D99] (see: Quantale) von Neumann index [46L80, 46L87, 55N15, 58G10, 58Gll, 58G12] (see: Index theory) yon Neumann measurement [68Q05, 68Q10, 68Q15, 68Q25, 81Pxx, 81PI5, 94Axx] (see: Quantum information processing, science of) yon Neumarmordinal [03E30] (see: ZFC) von Neumann quantale [03G25, 06D99] (see: Quantale) von N e u m a n n quantale s e e :

atomic --

von Neumannregularity [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) von Neumann theorem see: S t o n e - --; W e y l - -Voronol summationformula [11Lxx, llL03, llL05, llL15] (see: Bombieri-Iwaniec method) vortex s e e : intensity of a -vortex shedding s e e : drag due to - - ; K~.rm~.n - - ; total drag due to - - ; Von K&rman - -

vortex sheet [76Cxx, 76C05] (see: Birkhoff-Rott equation; Von Kfirmfin vortex shedding) vortex sheet s e e : vortex sheets s e e :

planar - - ; rolled-up - induced velocities in --

vortex street [76Cxx] (see: Von Khrmfin vortex shedding) vorticity density [76C05] (see: Birkhoff-Rott equation) voting s e e : election - - ; straight plurality --; two-round - - ; weighted - Vries equation s e e : averaged solution of the Korteweg-de - - ; characteristic initial-value problem for the K o r t e w e g de - - ; dispersionless K o r t e w e g - d e - - ; Korteweg-de - - ; shocks for the Korteweg--de - - ; Whitham equation for the Korteweg-de -V r i e s - L a n d a u - G i n s b u r g model s e e : Hurwitz-space K o r t e w e g - d e - Vries solution s e e : gap K o r t e w e g - d e - -

W Weals law s e e : van der -Waerden theorem on arithmetic progressions s e e : van der - Wagner theorem s e e : S m i t h - S o l m a n - - -

Wald lemma [62Lxx] (see: Average sample number) Weld lemma

548

[62L10] (see: Sequential probability ratio test) Weld test statistics [62Jxx, 62Mxx] (see: Cox regression model) Wald-WolJbwitz optimality [62L101 (see: Sequential probability ratio test) Waldhausen graph manifold [57M25] (see: Algebraic tangles) Waldspurgertheorem [llFll, 11F12] (see: Shimura correspondence)

wavelet [42Cxx] (see: Daubechies wavelets)

[05E05, 16W30] (see: Leibniz-Hopf algebra and quasi-symmetric functions) weak Drinfel'd-Turaev quantization [16Wxx, 57P25] (see: Drinfel'd-Turaev quanfization) weak order of a symmetricgroup [05E05, 13P10, 14C15, 14M15, 14NI5, 20G20, 57T15] (see: Schubert polynomials) Weak P-point (54D40, 54G10) (refers to: Accumulation point; Chain condition; P-point; Stone-Cech compactification; Topological space) weak spectrum [03G25, 06D99] (see: Quantale) weak turbulence [35Q35, 58F13, 76Exx] (see: Kuramoto-Sivashinsky equation) weakest pre-specification of a program [03G25, 06D99] (see: Quantale) weakly o~-favouruble topological space [54E52] (see: Banach-Mazur game) weakly algebraizable deductive system [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) weakly deformed soliton lattice [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) weakly log terminal [14Exx, 14E30, 14Jxx] (see: Mori theory of extremal rays) weakly minimal theory [03C15, 03C45, 03E15] (see: Vaught conjecture) weakly non-negative quadratic form [16Gxx] (see: Tits quadratic form) weakly positive quadratic form [16Gxx] (see: Tits quadratic form) weakly se!f-dual polynomial basis [12E20] (see: Galois field structure) weakly sequentiallycomplete [22D10, 43A07, 43A30, 43A35, 43A45, 43A46, 46110] (see: Fourier algebra)

wavelet s e e :

Weber problem s e e :

walk-bounded digraph s e e : locally -walk problem s e e : travelling salesman --

Wall conjecture on accessibility offiniteIygenerated groups [20E22, 20Jxx, 57Mxx] (see: Accessibility for groups) Walsh coefficients s e e :

Fourier---

Walsh-Paley system (33C45) (refers to: Welsh system) Walsh series s e e : Welsh system s e e :

Fourier-P a l e y - --

Ward identity [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Ward result [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) Waring-Goldbach problem (11Pxx) (refers to: Goldbach-Waring problem; Waring problem) Waring identity [05E05, 60G50] (see: Baxter algebra) Warlimont axiom Gn [11Nxx, 1IN32, 11N45] (see: Abstract prime number theory) wave s e e : plane --; scattering by a plane --; standing - - ; travelling -wave equation s e e : regularizedlong -wave form s e e : Maass -wave front set criterion s e e : Hdrmander -wave in scattering s e e : incident plane --

wave operator [81Uxx] (see: Enss method) wave scattering see: S-matrixfor

electro-

magnetic - -

Hear - -

wavelet approximation [41A05, 41A30, 41A63] (see: Radial basis function) wavelet transform s e e : admissibility condition for a reconstruction formula for the c o n t i n u o u s - - ; c o n t i n u o u s - - ; discrete -; reconstruction formula for the continuous -wavelets s e e : Daubechies - waves s e e : colliding gravitational - -

weak A-sequence [13A30, 13H10, 13H30] (see: Buchshaum ring) weak Beth definability [03Gxx] (see: Algebraic logic) weak Beth definability property [03Gxx, 03G05, 03G10, 03G15, 03G25, 06Exx, 06F35] (see: Abstract algebraic logic) weak commutativity [llFII, 17B10, 17B65, 17B67, 17B68, 20D08, 81R10, 81T30, 81T40] (see: Vertex operator algebra) weak convergence of measures [60B10, 60G05] (see: Skorokhod space) weak Ditters conjecture

Fermat- --; Steiner-

Weber theorem [llNxx, 11N32, 1IN45] (see: Abstract prime number theory) Weber theorem s e e : L a n d a u - --

Kronecker- - - ;

Wedderburn theorem on Schur indices [11R34, 12G05, 13A20, 16S35, 20C05] (see: Schur group) Wegmannmethod [30C20, 30C30] (see: Theodorsen integral equation) Weierstrass approximationtheorem [41A10, 41A15, 55M25, 68U05] (see: Bernstein-B~zier form; Brouwer degree) Weierstrass data for a minimal surface [53A10, 53C42] (see: Weierstrass representation of a minimal surface) Weierstrass preparation theorem s e e : adic - -

p-

Weierstrass representation of a minimal surface (53A10, 53C42)

(refers to: Conformal mapping; First fundamental form; Gaussian curvature; Harmonic function; Isothermal coordinates; Loop; Meromorphic function; Minimal surface; Riemann surface; Second fundamental form; Stereographic projection) Weierstrass representation of a minimal surface

[53A10, 53C42] (see: Weierstrass representation of a minimal surface) Weierstrasstheorem s e e :

Lindemann---

weight see: automorphie form of halfintegral - - ; dominant integral highest - - ; good universal - w e i g h t category see: highest- - w e i g h t function s e e : Chebyshev - - ; G e g e n b a u e r - - ; Jacobi - - ; Laguerre - - ; spectral factor of a - weight g-module see: highest- - weight module see: formal character of a -; irreducible highest - -

weight of a matching in a graph [05Cxx, 05D15] (see: Matching polynomial of a graph) w e i g h t process s e e : portfolio - weight representation s e e : highest - -

weighted projective line [16Gxx] (see: Tilting theory) weighted residual method [65Lxx] (see: Tan method) weighted residuals s e e :

mean --

weighted voting [90A28] (see: Condorcet jury theorem) Weil algebra s e e :

relative - - ; truncated

--

Weil algebra of a Lie algebra

(55R40, 57Rxx) (refers to: Bundle; Characteristic class; Classifying space; Cohomology; Exterior algebra; Graded algebra; Lie algebra; Lie group; Symmetric algebra; Universal space) Weil-Brezin mapping [42Axx, 44-XX, 44A55] (see: Zak transform) Weil-Petersson Kdhlerform [14H15, 30F60] (see: Weil-Petersson metric) Weii-Petersson metric (14H15, 30F60) (refers to: Euler characteristic; Gromov hyperbolic space; I£dihler metric; Moduli of a Riemann surface; Moduli theory; Neumann

O-problem; Quadratic differential; Quasi-conformal mapping; Riemann

surface; Teichmiiller space) Well representation [1 IF27, 11F70, 20G05, 81R05] (see: Segal-Shale-Weil representation) Weil representation see:

SegaI-Shale---

Weil Riemannhypothesis [llL07] (see: Exponential sum estimates) Weinbergequation see: Hardy--Weinberg equilibrium equation s e e : Hardy- -Weinberglaw see: Hardy--

number [llAxx] (see: Abundant number) Weisfeiler-Leman closure [05Exx] (see: Cellular algebra) Weisner theorem [05B35, 05Exx, 05E25, 06A07, 11A25] (see: Mdbius inversion) weird

Weiss problem s e e :

well-founded set

Kiefer---

WORD FREQUENCY

[03E30] (see: ZFC) well-ordering theorem [03E30] (see: ZFC) Wendt manifold see: H a n t z c h e - - Wengertheorem s e e : Wermer theorem s e e :

Anderson--Hfrmander-

-West decomposition [47B06] (see: Riesz operator) Weyl asymptotic formula for the eigenvalue distributionof Laplacians [28A80] (see: Sierpifiski gasket) Weyl asymptotics [35J05] (see: Neumann eigenvalue) Weyl asymptoticsfor Dirichlet eigenvalnes [35J05, 35J25] (see: Dirichlet eigenvalue) Weyl characterformula s e e : Weyl formura s e e : Kac- --

Kac- - -

Weyl function [35P25, 47A40, 81U20] (see: Inverse scattering, half-axis case) Weyl group [11Fxx, 17B67, 20D08] (see: Borcherds Lie algebra) Weyl-Kac-Boreherds character formula [llFxx, 17B10, 17B65, 17B67, 20D08] (see: Borcherds Lie algebra; WeylKac character formula) Weyl-Kac character formula (17BI0, 17B65) (refers to: Borcherds Lie algebra; Cartan subalgebra; Character formula; Kac-Moody algebra; Lie algebra; Quantum groups; Superalgebra; Weyl group) Weyl-Kacformula [17B10, 17B65] (see: Weyl-Kac character formula) Weyl m-function [34B24, 34L40] (see: Sturm-Liouville theory) Weyl sequenceof approximateeigenfuncdons [34B24, 34L40] (see: Sturm-Liouville theory) Weyl symbol [81Q05] (see: Fedosov trace formula) Weyl theorem s e e : Farkas-Minkowski-- ; Minkowski-Steinitz- - -

Weyl-von Neumann theorem [19K33, 19K35, 49L80] (see: Brown-Douglas-Fillmore theory) white Listing polynomial [57M25] (see: Listing polynomials) Whitehead dual s e e : S p a n i e r - - Whitehead duality see: S p a n i e r - - -

Whitehead-Spanier duality [55P25] (see: Spanier-Whitehead duality) Whitham equation for the Korteweg-de Vries equation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Whitham equations (14Jxx, 35A25, 35Q53, 57R57) (refers to: Abclian differential; Benjamin-Feir instability; Casimir element; Differential on a Riemann surface; HamUton-Jacobi theory; Hitchin system; Jacohi variety; K-theory; Kac-Moody algebra; Korteweg-de Vries equation; KPequation; Meromorphic function; Riemann surface; Seiberg-Witten equations; Toda lattices; WKBJ approximation)

Whitham formulation Phong - -

see:

Krichever-

Whitham hierarchy [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Whitham hierarchy of isomonodromic deformations [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Whitham theory [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Whitham time [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Whitney stratification [57N80] (see: Thom-Mather stratification) wide sense s e e :

Ditkin set in the - -

wide-sense Ditkin set [43A45, 43A46] (see: Ditkin set) width s e e : mogorov

branch group of finite - - ; Kol-

-

width of a partially ordered set [05D05, 06A07] (see: Sperner property) Wiener algebra [42A16, 42A24, 42A28] (see: Beurling algebra) Wiener decomposition

Itt----

see:

Wiener-Ditkin set [43A45, 43A46] (see: Ditkin set) Wiener-Ditkin theorem [42A16, 42A24, 42A28] (see: Beurling algebra) Wiener-Hopffactorization [42A16, 47B35] (see: Szeg6 limit theorems) Wiener-Hopfoperator [42A16, 47B35] (see: Szeg6 limit theorems) Wiener-Hopf operator s e e : cated - -

finite - - ; trun-

Wiener index [05C50] (see: Matrix tree theorem) Wiener integral s e e :

multiple

- -

Wiener-It~ decomposition (60G15, 60Hxx, 60]65) (refers to: Brownian motion; Fock space; Hermite polynomials; Hilbert space; Isometric mapping; Linear operator; Norm; Orthogonal basis; Random variable; Tensor product; Unitary operator; Wiener chaos decomposition; Wiener integral; Wiener space, abstract) Wiener-It6 decomposition [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Wiener-ItO decomposition theorem [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Wiener measure [60G15, 60Hxx, 60J65] (see: Wiener-It6 decomposition) Wiener number of a tree [15A15, 20C30] (see: Immanant) Wiener sausage (60Gxx, 60J55, 60J65) (refers to: Brownian motion; Capacity; Central limit theorem; Green function; Markov process; Rayleigh-Faber-Krahn inequality; Schrtdinger equation; Strong law of large numbers) Wiener theorem [43A07, 43A15, 43A45, 43A46, 46Ji0] (see: Figh-Talamanca algebra) Wiener-Volterra functional series [41A58] (see: Volterra functional series) Wiener-Wintner ergodic theorem

[28D05, 54H20] (see: Wiener-Wintrier theorem) Wiener-Wintrier return-time theorem [28D05, 54H20] (see: Wiener-Wintrier theorem) Wiener-Wintrier theorem (28D05, 54H20) (refers to: Dynamical system; Ergodic theory; Ergodicity; Measure) Wiener-Wintnertheorem s e e :

uniform - -

Wightman function [81Txx] (see: Massive field) Wijsman convergence (54Bxx) (refers to: Complete metric space; Exponential topology; Hausdorff metric; Hit-or-miss topology; Metric space; Net (of sets in a topological space); Separable space; Tikhonov space; Totally-bounded space; Uniform convergence) Wijsman convergence [54Bxx] (see: Wijsman convergence) Wijsman topology induced by a metric [54Bxx] (see: Wijsman convergence) wild algebra [16G10, 16G20, 16G60, 16G70] (see: Tilted algebra) wild dynamical system [28Dxx, 54H20, 58Fll, 58F13] (see: Absolutely continuous invariant measure) Wilks LR test [62Jxx] (see: ANOVA) will s e e :

Rousseau theory of general - -

Willmore conjecture [53C42] (see: Willmore functional) WiUmore functional (53C42) (refers to: Euler characteristic; Gauss-Bonnet theorem; Gaussian curvature; Genus of a surface; Laplace-Beltrami equation; Mean curvature; Minimal surface; Principal curvature; Riemannian manifold; Variation of a functional; Variational calculus; Variational problem) Willmore immersion [53C42] (see: Willmore functional) Willmore surface [53C42] (see: Willmore functional) Willmore surfaces s e e : stability of -Wilson plug [58F22, 58F25] (see: Seifert conjecture) Wilton summationformula [llLxx, llL03, llL05, llL15] (see: Bombieri-lwaniec method) winding number [81V10] (see: Dirac monopole) Window function (42Cxx, 94A 12) (refers to: Balian-Low theorem; Calder6n-type reproducing formula; Delta-function; Fourier transform; Gabor transform; Transfer function) window function [42Cxx, 94A12] (see: Window function) windowed Fourier transform [42Cxx, 94A12] (see: Window function) Winker condition [06Exx, 68T15] (see: Robbins equation) winner s e e :

Condorcet

--

winning strategy [03E50, 54-XX, 90D80] (see: Sierpifiski game)

winning strategy in the generalized Banach-Mazur game [54E521 (see: Banach-Mazur game) winning strategy in the generalized BanachMazur game s e e : stationary-Wintner ergodic theorem s e e : Wiener- - Wintner return-time theorem see: WienerWintner theorem s e e : uniform Wiener- ; Wiener- - without equality s e e : infinitary universal Horn logic - Witt theorem s e e : Brauer--Witten differential s e e : S e i b e r g - - -

Witten-Dijkgraaf-Verlinde-Verlinde equations [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Witteu-Dijkgraaf-Verlinde-Verfinde thetry [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Witten equations s e e :

Seiberg- - -

Witten genus [11Fll, 17B67, 20D08, 81T10] (see: Moonshine conjectures) Witten invariants see: Gromov-

--;

Seiberg- - Witten theory s e e : S e i b e r g - - Witten Toda curve s e e : Seiberg- - -

Wittenbauer theorem (51M04) Wittich matrix [30C20, 30C30] (see: Theodorsen integral equation) WKB approximation [14Jxx, 35A25, 35Q53, 57R57] (see: Whitham equations) Wodzicki formula for multiplicative anomaly [81T50] (see: Non-commutative anomaly) Wodzicki residue (35Sxx, 46Lxx, 47Axx) (referred to in: Non-commutative anomaly) (refers to: Laplace operator; Pseudodifferential operator; Quantum field theory; Riemannian manifold; Trace; Yang-Mills field) Wold decomposition (60G25, 62M20, 93B10, 93B15, 93E12) (refers to: Hilbert space; Mixed autoregressive moving-average process; Moment; Spectral density; Stationary stochastic process; White noise) Wold representation [60G25, 62M20, 93B10, 93B15, 93E12] (see: Wold decomposition) Wolff-Carathtodorytheorem s e e : alized Julia- - - ; Julia- - -

gener-

Wolff-Denjoy theorem [301)05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Wolff point s e e :

Denjoy---

Wolff-Schwarz lemma [30D05, 32H15, 46G20, 47H17] (see: Denjoy-Wolff theorem) Wolff theorem [30C45, 47H10, 47H20] (see: Julia-Wolff-Carathtodory theorem) Wolff theorem s e e : continuous Denjoy- - ; Denjoy- - - ; Fan analogue of the Denjoy- - - ; Julia- - Wolfowitz optimality s e e : Wald- -word s e e : Lyndon - -

word analysis in texts [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) word frequency [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) 549

WORD PROBLEM

word problem [05C25, 05E05, 20Fxx, 20F32, 60G50, 68S051 (see: Baumslag-Solitar group; Baxter algebra; Natural language processing) word problem for Fibonacei groups [20F38] (see: Fibonacci group) word problem for presentations of a free Burnside group [20F05, 20F06, 20F32, 20F50] (see: Burnside group) word problem.tor three-dimensional manifolds [57N10] (see: Haken manifold) words in Ulysses see: of different --

[42A20, 42A32, 42A38] (see: Integrability of trigonometric series) Yudin estimate [42B05, 42B08] (see: Lebesgue constants of multidimensional partial Fourier sums) Yule distribution [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) Yule-Simon distribution [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law)

Z

different - - ; n u m b e r

worst-case complexity [68Q15] (see: Average-case computational complexity) wrapping conjecture [57M25] (see: Kauffman bracket polynomial) wrapping number of a link [57M25] (see: Kauffman bracket polynomial) writhe number [57M25] (see: Kauffman bracket polynomial)

X

5(3) see: irrationalityof-Zp-extension of a number field [11R23] (see: Iwasawa theory) Zp-extension of a number field [11R23] (see: Iwasawa theory) Z (in set theory) [03E30] (see: ZFC) Z-transform (39A12, 93Cxx, 94A12) (refers to: Cauehy integral theorem; Laplace transform; Laurent series; Probability theory; Residue of an analytic function) Z-transform see: convolution and - - ; example of a - - ; final value theorem for the ; initial value theorem for the - - ; inverse - - ; partial-fractions technique for finding the inverse - - ; properties of the - - ; shift behaviour of the - -

Z a m o l o d c N k o v - B e r n a r d equations s e e : Knizhnik- --

Zarankiewicz crossing number conjecture (05C10, 05C35) (refers to: Graph) Zariski-Lipman conjecture (13B10, 13C15, 13C40) (refers to: Commutative algebra; Dimension; Field; Normal ring; Prime ideal; Projective module; Regular ring (in commutative algebra); Syzygy) Zariski-Lipman conjecture [13BI0, 13C15, 13C40] (see: Zariski-Lipman conjecture) Zariski problem [14Axx] (see: Zariski problem on field extensions) Zarisld problem on field extensions (14Axx) (refers to: Affine variety; Algebraic geometry; Isomorphism) Zassenhaus conjecture (20C05, 20Dxx) (refers to: Abelian group; Cyclotomic field; Finite group; General linear group; Nilpotent group; Normal subgroup; p-group; Sylow subgroup) Zassenhaus conjecture [20C05, 20Dxx] (see: Zassenhaus conjecture) Zassenhausconjecture see:

Schinzel---

Zech logarithm [12E20] (see: Galois field structure) Zelazko theorem see:

Gleason-Kahane-

- -

E-function [11M06] (see: Riemann (-function) E-function see: functional equation for the Riemann -- ; Riemann - z see: prime ideal at - Xor [90D05]

Z-transformation [39A12, 93Cxx, 94A12] (see: Z-transform) Z-transformation see: inverse

(see: Sprague-Grundy function) XPP software [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages)

Zahorski property (26A21, 26A24, 28A05, 54E55) (refers to: Approximate continuity; Baire classes; Continuity; Darboux property; Dense set; Density point; Derivative; Lebesgue measure; Metric space; Semi-continuous function; Set of type F,, (G~)) Zahorski property [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) Zak transform (42Axx, 44-XX, 44A55) (refers to: Continuous function; Even function; Fourier transform; Integrable function; Quantum field theory; Uniform convergence; Unitary transformation) Zak transform [42Axx, 44-XX, 44A55] (see: Zak transform)

Y !J-homotope [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Yang-Baxter equation [17A40] (see: Freudenthal-Kantor triple system) Yang-Mills equations see:

self-dual - -

Yang-Mills-Higgs action [35Qxx, 78A25] (see: Magnetic monopole) Yang-Mills potential [35Qxx, 78A25] (see: Magnetic monopole) Yang-Mills theory [81Qxx, 81Sxx, 81T13] (see: Faddeev-Popov ghost) Yff inequality [51M04] (see: Brocard point) Yoneda functor [16Gxx] (see: Tilting theory) Young diagram see:

shifted - -

Young-Laplacelaw [76Exx, 76S05] (see: Viscous fingering) Young theorem on trigonometric series 550

Z-transforms see:

-examples of - -

Zak transform see: analytic properties of the - - ; applications of the - - ; conjugation behaviour of the - - ; conjugation of the - - ; inversion of the - - ; linearity of the - - ; modulation behaviour of the - - ; modulation of the - - ; periodicity of the -- ; shift of the -- ; symmetry for the - - ; translation behaviour of the -- ; translation of the --

Zak transform of the Gaussian function [42Axx, 44-XX, 44A55] (see: Zak transform) Zak transformation see: convolution and - - ; convolution under - Zak transforms see: examples of - - ; existence of - - ; properties of - -

Zakharov-Shabat dressing method [22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy)

Zelinskyexactsequencesee:

Rosenberg-

Zel'manov prime theorem [17C65, 46H70, 46L70] (see: Banach-Jordan algebra) Zel'manov structure theorem [17A40, 17C65, 46H70, 46L70] (see: Banach-Jordan pair) Zel'manov techniques [17Cxx, 46-XX] (see: JB * -triple) Zerlegungssatz for operators" [31A10, 31D05, 47A10, 47A15, 47A60] (see: Riesz decomposition theorem) Zermelo-Fraenkel set theory [03E30] (see: ZFC) Zermelo-Fraenkel set theory [03E30] (see: ZFC) Zermelo-Fraenkel set theory with the axiom of choice [03E30] (see: ZFC) Zermelo set theory [03E30] (see: ZFC) Zernike circle polynomial [33C50, 78A05] (see: Zernike polynomials) Zernike polynomials (33C50, 78A05) (refers to: Bessel functions; Hypergeometric function; Integral transform; Jacobi polynomials; Orthogonal polynomials; Polynomial; Spherical functions; Tomography) Zernike polynomials see: orthogonality relations for - - ; Rodrigues formula for --

Zernike radial polynomial [33C50, 78A05] (see: Zernike polynomials) zero see: Amitsur theorem on function fields of genus - zero bifurcation see: triple - -

zero-curvaturerelations

[22E65, 22E70, 35Q53, 35Q58, 58F07] (see: AKNS-hierarchy) zero-dimensional strongly - -

metric

space

see:

zero-Hopf bifurcation [34-04, 35-04, 58-04, 58F14] (see: Dynamical systems software packages) zero-one matrix see;

support of a --

zero set of a function [26A21, 26A24, 28A05, 54E55] (see: Zahorski property) zeta-function [llNxx, I1N32, 11N45, llN80] (see: Abstract analytic number theory) zeta-function [llNxx, 11N32, 11N45, llN80] (see: Abstract analytic number theory) zeta-function see: critical line for the - - ; Dedekind - - ; Euler - - ; Hadamard factorization of the Dedekind - - ; Hurwitz - - ; Jacobi - - ; modified - - ; reflection formula for a -- ; Riemann - -

Zeta-function method for regularizalion (81Qxx) (referred to in: Non-commutative anomaly) (refers to: Analytic continuation; Pseudo-differential operator; Quantum field theory) zeta-function of a curve [11R23] (see: Iwasawa theory) zeta-function of a pseudo-differential operator [35Sxx, 46Lxx, 47Axx] (see: Wodzicki residue) zeta-function of an operator [81Qxx] (see: Zeta-function method for regularization) zeta-function regularization [81Qxx] (see: Zeta-function method for regularizafion) zeta-function regularizafion [81T50] (see: Non-commutative anomaly) zeta-function regularization see: commutative anomaly for --

non-

Zeuthen formula [12F10, 14H30, 20D06, 20E22] (see: Chasles-Cayley-Brillformula) ZF (in set theory) [03E30] (see: ZFC) ZFC (03E30) (refers to: Antinomy; Axiom of choice; Axiom of extensionality; Axiomatic set theory; Consistency; Continuum hypothesis; Forcing method; Ordinal number; Paradox; Russell paradox; Set theory; Transfinite number; Transfinite recursion; Universe; Zermelo theorem; Zorn lemma) ZFC see: axioms of - Z h a n g equation see: Kardar-Parisi---

Zipf law (60Exx, 62Exx, 62Pxx, 92B 15, 92K20) (refers to: Bose-Einstein statistics; Information theory; Lebesgue measure; Markov process; Multinomial distribution; Order statistic; Pareto distribution; Sample; Triangular array; Uniform distribution) Zipf l a w [60Exx, 62Exx, 62Pxx, 92B 15, 92K20] (see: Zipf law) Zipf-Mandelbrot law [60Exx, 62Exx, 62Pxx, 92B15, 92K20] (see: Zipf law)

ZUCKERCONJECTURE Zolotareff polynomials

[41-XX, 41A50] (see: Zolotarev polynomials) Zolotarev w-polynomials [41-XX, 41A50] (see: Zolotarev polynomials) Zolotarev perfect splines [41-XX, 41A50] (see: Zolotarev polynomials) Zolotarev polynomials (41-XX, 41A50) (refers to: Approximation theory;

Chebyshev polynomials; Markov inequality; Polynomial least deviating from zero; Uniform approximation)

zonal harmonic polynomials

harmonics)

[31B05, 33C55] (see: Zonal harmonics)

zonal harmonics s e e :

Zototarev polynomials s e e : normalized - Zolotarev problem s e e : first - - ; second - -

zonal harmonic polynomials s e e : tions of - -

Zolotarev problems

Zonal harmonics (31B05, 33C55) (refers to: Analytic function; Harmonic function; Kelvin transformation; Laplace equation; Legendre polynomials; Potential theory; Spherical coordinates; Spherical

[41-XX, 41A50] (see: Zolotarcv polynomials) Zolotarev problems [4I-XX, 41A50] (see: Zololarev polynomials)

applica-

solid - - ; surface --

Zorich theorem (26Bxx, 30C20) (referred to in: Quasi-regular map-

ping) (refers to: Homeomorphism; Quasi-

regular mapping) Zucker conjecture

[11Fxx, 20Gxx, 22FA6] (see: Baily-Borel compactiflcation)

551

AUTHOR INDEX

A Abate, M. s e e : Deujoy-Wolff theorem; Julia-Wolff-Carath6odory theorem; Semi-group of holomorphic mappings Abel, N.H. s e e : Semi-group ofholomorphic mappings Abi-Khuzam, E s e e : Brocard point Ablowitz, M.J. s e e : AKNS-hierarchy Adyan, S.1. s e e : Burnside group Afflerbach, L. s e e : Linear congruential method Ahem, R s e e : Berezin transform Ahlfors, L.V. s e e : Quasi-symmetric function of a complex variable Akhiezer, A. s e e : Zolotarev polynomials Alaoglu, L. s e e : Abundant number Aleksandrov, A.D. s e e : Aleksandrov problem for isometric mappings Aleksandrov, ES. s e e : Sierpifiski game Aleksyuk, V.N. s e e : Non-additive measure Alexander, H. s e e : Polynomial convexity Alexander, J.W. s e e : Alexander theorem on braids; Reidemeister theorem; Skein module Alford, W.A. s e e : Burnside group Alfsen, E.M. s e e i . B a n a c h - J o r d a n algebra Alikakos, N. s e e : Cahn-Hilliard equation Alladi, K. s e e : Diekman function Allard, W.K. s e e : Geometric measure theory Almgren, EJ. s e e : Geometric measure theory Althammer, P. s e e : Sobolev inner product A m m a n n , B . s e e : Willmore functional Amou, M. s e e : Mahler method Andersen, H.H. s e e : Tilting theory Aodo, T. s e e : Julia-Wolff-Carath6odory theorem Antzoulakos, D.L. s e e : Fibonacci polynomials Apdry, R. s e e : Apdry numbers Arens, R. s e e : Exponential law (in topology) Aronszaln, N. s e e : Alternating algorithm Ashbaugh, M.S. s e e : Rayleigh-FaberKrahn inequality Atiyah, M.E s e e : Atiyah-Floer conjecture; Index theory; Magnetic monopole Aumann, R.J. s e e : Non-additive measure Aupefit, B. s e e : B a n a c h - J o r d a n a l g e b r a Auslander, M. s e e : Almost-split sequence; Tilting module; Tilting theory

552

Avidon, M.R. s e e : Abundant number Avogadro, A. s e e : Knudsen number Avramov, L. s e e : Zariski-Lipman conjecture Axler, S. s e e : Berezin transform; VMOA -space

B Babenko, K. s e e : Hyperbolic cross Bachelier, L. s e e : Black-Scholes formula Bachet de Mdziriac s e e : Abundant number Bade, W.G. s e e : Automatic continuity for Banach algebras Baer, R. s e e : Flat cover Baily, W.L. s e e : Baily-Borel compactification Bah'e, R. s e e : Namioka theorem Baker, K. s e e : Abstract algebraic logic Banach, S . s e e : Bauach-Mazur game; Banach-Stone theorem Bannai, E. s e e : Coherent algebra Bafiuelos, R. s e e : Dirichlet eigenvalue Bass, H. s e e : Flat cover Bauer, EL. s e e : Bauer-Fike theorem Baues, H.-J. s e e : Algebraic homotopy Bayen, E s e e : Dirac quantization Bayes, A.J. s e e : Burnside group Beck, J. s e e : Lov~isz local lemma Becket, P.G. s e e : Mahler method Bedford, E. s e e : Pluripotential theory; Polynomial convexity Beer, G. s e e : Wijsman convergence Behrend, E s e e : Abundant number Belavin, A. s e e : Vertex operator algebra Belinsky, E. s e e : Lebesgue constants of multl-dimensional partial Fourier sums Bell, E.T. s e e : Dirichlet convolution Bellman, R.E. s e e : Fuzzy programming Belov, A.Ya. s e e : Specht property Bdnard, H. s e e : Yon K~irm~in vortex shedding Bender, E.A. s e e : Abstract prime number theory Benedicks, M. s e e : Uncertainty principle, mathematical Benguria, R.D. s e e : Rayleigh-FaberK r a h n inequality Benioff, P. s e e : Quantum computation, theory of; Quantum information processing, science of Benkovski, S.J. s e e : Abundant number Bennett, C. s e e : Quantum information processing, science of Benslimane, M. s e e : Banach-Jordan pair Berezin, E s e e : Berezin transform

Berg, I.D. see: Brown-DouglasFillmore theory Bergum, G.E. s e e : Lucas polynomials Bergvelt, M.J. s e e : AKNS-hierarchy Berkson, E. s e e : Denjoy-Wolff theorem; Semi-group of holomorphic mappings Berlekamp, E.R. s e e : Factorization of polynomials Bernshtetu, I.N. s e e : Schubert polynomials; Tilted algebra; Tilting module; Tilting theory Bernshte~n, S.N. s e e : Bernstein-Bdzier form Bernstein, E. s e e : Quantum computation, theory of Berustein, S.N. s e e : Bernstein-B6zier form Bertrand, J. s e e : Fermat-Torrieelli problem Besicovitch, A.S. s e e : Geometric measure theory Bestvina, M. s e e : Accessibility for groups Beukers, E s e e : Apdry numbers Beurling, A . s e e : Beurling algebra; Quasi-symmetric function of a complex variable; Uncertainty principle, mathematical Bdzier, R s e e : Bernstein-Bdzier form Bican, L. s e e : Flat cover Bicknell, M. s e e : Fibonacci polynomials; Lucas polynomials Birldaoff, G. s e e : Abstract algebraic logic; Bramble-Hilbert lemma Bishop, E. s e e : Bishop theorem; Polynomial convexity Bismut, J.-M. s e e : Index theory Biswas, I. s e e : Weil-Petersson metric Black, E s e e : Black-Scholes formula; Option pricing Blanksby, EE. s e e : Lehmer conjecture Blaschke, W. s e e : Willmore functional Blessenohl, D. s e e : Normal basis theorem Btocki, Z. s e e : Pluripotential theory Boas, R.P. s e e : Integrability of trigonometric series Boethius, A.M.S. s e e : Abundant number Bogomolny, E.B. see: Magnetic monopole Bohl, P.G. s e e : Brouwer degree Bollob~is, B. s e e : Janson inequality; Sperner theorem Boltzmann, L. s e e : Knudsen number Bombieri, E. s e e : Bombieri-Iwaniec method; Exponential sum estimates; Gel'fond-Schneider method Bongartz, K. s e e : Tits quadratic form Bonora, L. s e e : Faddeev-Popov ghost Bony, J.M. see:Multiplication of distributions Boppana, R. s e e : Janson inequality

Borcherds, R. s e e : Moonshine conjectures; Vertex operator algebra Borda, J.-Ch. s e e : Condorcet paradox Borel, A. s e e : Smith theory of group actions Borsuk, K. s e e : Brouwer degree Botti, Ph. s e e : Immanant Boudi, N. s e e : B a n a c h - J o r d a n pair Bourgain, J. s e e : Wiener-Wintner theorem Bouziad, A. see: Almost continuity; Namioka space Bovillus,C. see: Abundant number Braess, D. see: Braess paradox Bramble, J.H. see: Bramble-Hilbert lemma Brascamp, H. s e e : Dirichlet eigenvalue Brauer, A. s e e : Gershgorin theorem Brenner, S. s e e : Tilting functor; Tilting module; Tilting theory Br6zis, H. s e e : Brouwer degree Bdggs, G.B. s e e : R e i d e m e i s t e r t h e o r e m Blinkman, H. s e e : Zernike polynomials Britton, J.L. s e e : HNN-extension Brocaxd, H. s e e : Brocard point Brock, J.E s e e : Weil-Petersson metric Brodskit, M.S. s e e : Operator colligation Broscius, J. s e e : Abundant number Brouwer, A.E. s e e : Pasch configuration Brouwer, L.EJ. s e e : Brouwer degree Brown, E.H. s e e : Spanier-Whitehead duality Brown Jr., E.H. s e e : Brown-Gitler spectra Brown, J.B. s e e : Namioka theorem Brown, L.G. s e e : Brown-DouglasFillmore theory Brown, R. s e e : Algebraic homotopy; Exponential law (in topology) Bmmer, A. s e e : Iwasawa theory Brun, V. s e e : Inclusion-exclusion formula Brundan, J. s e e : Projective representations of symmetric and alternating groups Bryan, A.C. s e e : Benjamin-BonaMahony equation Bryant, R.L. s e e : Willmore functional Bryant, R.M. s e e : Specht property Buchsbaum, D.A. s e e : Buchsbaum ring Burnside, W. s e e : Burnside group Busby; R.C. s e e : Multipliers of C * algebras Butler, M.C.R. s e e : Tilting functor; Tilting module; Tilting theory

C Cahn-Hilliard equation Calder6n, A.P. s e e : Ditkin set Cab-n, J. s e e :

CANTOR, D.G.

Cantor, D.G. s e e : Factorization of polynomials Cantor, G. s e e : ZFC Carath6odory, C. s e e : Julia-WolffCarathdodory theorem Carroll, T. s e e : Dirichlct eigenvalue C a f t a n , H . s e e : Baily-Borel compactification; Well algebra of a Lie algebra Cartier, E s e e : Baxter algebra Catalan, E.Ch. s e e : Catalan constant Cauchy, A. s e e : Almost continuity; Brouwer degree Cavalieri, B. s e e : Fermat-Torricelli problem Cayley, A. s e e : Cayley graph Chandra, A. s e e : Computational complexity classes Chapman, S. s e e : Knudsen number Charalambides, Ch.A. s e e : Lueas polynomials Cbarney, R. s e e : Satake compactification Chasles, M. s e e : Schubert calculus Chebyshev, EL. s e e : Zolotarev polynomials Cheeger, J. s e e : Index theory Chiba, K. s e e : P-space Choi, M.D. s e e : Brown-DouglasFillmore theory Chomsky, N. s e e : Natural language processing Choquet, G. s e e : Banach-Mazur game; Non-additive measure Chowla, S. s e e : Abundant number Chu, C.-H. s e e : Denjoy-Wolff theorem Chudnovskii, G.V. s e e : Gel'fondSchneider method Church, A. s e e : Quantum computation, theory of; Quantum information processing, science of Cipolla, M. s e e : Dirichlet convolution Clausius, J.R. s e e : Knudsen number Cobb, Ch.W. s e e : Cobb-Douglas function Cohen, E. s e e : Diriehlet convolution Cohen, P. s e e : ZFC Cohen, S.D. s e e : A b s t r a c t a n a l y t i c n u m bet theory; Galois field structure Colombeau, J.E s e e : Colombeau generalized function algebras; Generalized function algebras Common, A.K. s e e : Strong Stieltjes moment problem Conley, C. s e e : Conley index Connell, LG. s e e : Abstract analytic number theory Cormes, A. s e e : Index theory Conway, J.H. s e e : Algebraic tangles; Conway algebra; Fibonacci group; Rational tangles; Skein module; Vertex operator algebra Conze, J.E s e e : Wiener-Wintrier theorem Costaatini, C. s e e : Wijsman convergence Courant, R. s e e : Rayleigh-FaberKrahn inequality Coveyou, R.R. s e e : Linear congruential method Cowling, M. s e e : Fourier-Stieltjes algebra Cox, D.R. s e e : Cox regression model Crawley-Boevey, W. s e e : Tilting theory Crelle, A.L. s e e : Brocard point Curtis, C.W. s e e : Steinberg module Curds Jr., E C. s e e : Automatic continuity for Banach algebras Czelakowski, J. s e e : Algebraic logic

D Dade, E.C.

see:

Fitting length

Danzer, L. s e e : Geometric transversal theory Darbo, G. s e e : Darbo fixed-point theorem Daubechies, I. s e e : Daubechies wavelets Dangavet, I. s e e : Lebesgue constants of multi-dimensional partial Fourier sums Davenport, H. s e e : Abundant number de Bmijn, N.G. s e e : Diekman function de Caritat, M.J.A.N. s e e : Condorcet jury theorem; Condorcct paradox de Casteljau, E s e e : Bernstein-B~zier form de Fermat, E s e e : Fermat-Torricelli problem de Finetti, B. s e e : Dempster-Shafer theory de la Pefia, J.A. s e e : Tits quadratic form De Moivre, A . s e e : Inclusion-exclusion formula; Z-transform de Neuveglise, Ch. s e e : A b u n d a n t n u m bet de Rham, G. s e e : Current; Geometric measure theory de Vries, G. s e e : Harry Dym equation Debs, G. s e e : Banach-Mazur game; Namioka space Dedekind, R. s e e : ZFC Delta, M. s e e : Reidemeister theorem Deligne, P. s e e : Iwasawa theory; Selberg conjecture Delsarte, J. s e e : Mean-value characterization Delsarte, Ph. s e e : Delsarte-Goethals code Demazure, M. s e e : Schubert polynomials Dembo, A. s e e : Onsager-Machlup function Dempster, A.P. s e e : Dempster-Shafer theory Denjoy, A. s e e : Denjoy-Wolff theorem; Zahorski property Dennis, R.K. s e e : Steinberg symbol Descartes, R. s e e : Fcrmat-Torrieelli problem Deutsch, D. s e e : Quantum computation, theory of; Quantum information processing, science of Deville, R. s e e : Namioka space D'Hoker, E. s e e : W e i l - P e t c r s s o n m e t r i c Di Maid, G. s e e : Wijsman convergence Diaz, G. s e e : Gel'fond-Schneider method Dickman, K. s e e : Dickman function Dickson, L.E. s e e : Abundant number Dieter, U. s e e : Linear congruential method Dirac, EA.M. s e e : Dirac monopole; Dirac quantization Ditldn, V.A. s e e : Ditkin set Dobrakov, I. s e e : Non-additive measure Dobrowolski, E. s e e : Lehmer conjecture Donaldson, S.K. s e e : A t i y a h - F l o e r c o n jecturc; Dirac monopole Donkin, S. s e e : Tilting theory Doob, J.L. s e e : Riesz decomposition theorem Dostoglou, S. s e e : A t i y a h - F l o e r c o n j e c tare Douglas, EH. s e e : Cobb-Douglas function Douglas, R.G. s e e : Brown-DouglasFillmore theory Downs, A. s e e : Braess paradox Drensky, V. s e e : Speeht property Drozd, Yu.A. s e e : Tits quadratic form Dubickas, A. s e e : Lehmer conjecture Dubois, D. s e e : Fuzzy programming Dabrovin, B. s e e : Whitham equations DubrovskiL V.G. s e e : Harry Dym equation Duchon, J. s e e : Radial basis function Dugundji, J. s e e : Exponential law (in topology)

Dunwoody, M.J. s e e : Accessibility for groups Dupont, T. s e e : Bramble-Hilbert lemma Dym, H. s e e : Harry Dym equation

E Eckmatm, B. s e e : Flat cover Effros, E.G. s e e : Brown-DouglasFillmore theory Egorov, Yu.V. s e e : Egorov generalized function algebra; Generalized function algebras Ehresmarm, C. s e e : Schubert cycle Einstein, A. s e e : Knudsen number Eklof, EC. s e e : Flat cover E1Bashir, R. s e e : Flat cover Englig, M. s e e : Berezin transform Eimeper, K. s e e : Weierstrass representation of a minimal surface Enochs, E. s e e : Flat cover Enskog, D. s e e : Knudsen number Enss, V. s e e : Enss method Erd6s, E s e e : Abstract analytic number theory; Abundant number; Additive basis; Pasch configuration Esterle, J.R. s e e : Automatic continuity for Banach algebras Estoup, J.B. s e e : Zipf law Euler, L. s e e : Catalan constant; Knudsen number; Material derivative method Eymard, E s e e : Fourier algebra

F Faber, G. s e e : Rayleigh-Faber-Krahn inequality Faddeev, L.D. s e e : Enss method; Faddeev-Popov ghost Fagin, R. s e e : Computational complexity classes Fagnano, G. s e e : Fermat-Torricelli problem Fan, K. s e e : Denjoy-Wolff theorem; Julia-Wolff-Carath6odory theorem Fasbender, E. s e e : Fermat-Torricelli problem Fechner, G.T. s e e : Kendall tau metric Federer, H. s e e : Geometric measure theory Fefferrnan, C. s e e : B M O A -space Feighn, M. s e e : Accessibilityfor groups Feinberg, M. s e e : Tribonacci sequence Fenchel, W. s e e : Weil-Petersson metric Fenn, R.E s e e : Kirby calculus Ferrero, B. s e e : Iwasawa theory Feynman, R. s e e : Faddeev-Popov ghost; Quantum computation, theory of; Quantum information processing, science of Fig,~t-Talamanca,A. s e e : Fig~-Talamanca algebra Fike, C.T. s e e : Bauer-Fike theorem Fillmore, EA. s e e : Brown-DouglasFillmore theory Flake, U. s e e : Linear congruential method Fisher, R.A. s e e : ANOVA Flaschka, H. s e e : AKNS-hierarchy Flato, M. s e e : Dirac quantization Fleming, W.H. s e e : Geometric measure theory F/oer, A. s e e : Atiyah-Floer conjecture Flores, M. s e e : Berezin transform Foia~, C. s e e : Operator colligation Fomenko, A.T. s e e : Index theory

Forster, R.O. s e e : Forster-Swan theorem Fouvry, E. s e e : Bomhieri-Iwaniec method Fox, R.H. s e e : Exponential law (in topology); Fox n-colouring Fraenkel, A. s e e : ZFC Franchetti, C. see: Alternating algorithm Franke, J. s e e : Eisenstein cohomology Frege, G. s e e : ZFC Frenkel, I. s e e : Vertex operator; Vertex operator algebra Freudeuthal, H. s e e : FreudenthalKantor triple system Frey, G. s e e : Shafarevich conjecture Fried, M. s e e : Shafarevicb conjecture Friedlander, L. s e e : Neumann eigenvalue Friedman, Y. s e e : JB * -triple FrShlich, A. s e e : Hermann algorithms Frol~, Z. s e e : Strongly countably complete topological space; Weak P point Fronsdal, C. s e e : Dirac quantization Furstenberg, H. s e e : Satake compactification; Wiener-Wintrier theorem Fused, G. s e e : Cahn-Hilliard equation

G Gabriel, E s e e : Tits quadratic form Galton, E s e e : Pearson productmoment correlation coefficient Gamelin, T. s e e : Pinripotential theory Gardner, C.S. s e e : Harry Dym equation Garland, H. s e e : Vertex operator Gauss, C.E s e e : Factorization of polynomials; Fermat-Torricelli problem; Zak transform Gelbart, S. s e e : Selberg conjecture Gell-Mann, M. s e e : Okubo algebra Gel'land, I.M. s e e : Automatic continuity for Banach algebras; Harry Dym equation; Schubert polynomials; Tilted algebra; Tilting module; Tilting theory; Zak transform Gel'fand, S.I. s e e : Schubert polynomials Gel'fond, A.O. s e e : Gel'fond-Schneider method Georghiou, C. s e e : Fibonacci polynomials GershgorJn, S. s e e : Gershgorin theorem Getzler, E. s e e : Index theory Giles, R. s e e : Quantale Gilkey, P. s e e : Index theory Ginzburg, V.L. s e e : Seifert conjecture Gitler, S. s e e : Brown-Gitler spectra Gleasoa, A.M. s e e : Gleason-KahaneZelazko theorem G/Jdel, K . s e e : ZFC Goebel, K. see: Denjoy-Wolff theorem Goethals, J.-M. s e e : Delsarte-Goethals code Gohberg, I.C. s e e : Semi-Fredholm operator Golomb, S.W. s e e : Shift register sequence

Goodman, J.E. s e e : Geometric transversal theory Gordon, C. s e e : Dirichlet eigcnvalue Goto, S. s e e : Buchsbaum ring Goulden, I.P. s e e : Immanant Graham, R. s e e : Hales-Jewett theorem Gram, J.P. s e e : Thiele differential equation Greene, J.M. s e e : Harry Dym equation Griess, R. s e e : Vertex operator algebra Gromov, M. s e e : Weil-Petersson metric; Willmore functional Gross, B.H. s e e : Shimura correspon. dence

553

LEMPERT, L. Gross, D. s e e : Vertex operator Grothe, H. s e e : Linear congrnential method Grothendieck, A. s e e : Index theory Grovel L. s e e : Quantum computation, theory of Griin, 0. s e e : Abundant number Griinbanm, B. s e e : Geometric transversal theory Gutknecht, M. s e e : Theodorsen integral equation Guy, R. s e e : Zarankiewicz crossing number conjecture

H Haboush, W. s e e : Steinberg module Hadamard, J. s e e : Brouwer degree Hadwiger, H. s e e : Geometric transversal theory Hahn, H. s e e : Namioka theorem Hall, M. s e e : Burnside group Hankel, H. s e e : Almost continuity Hanlon, P. s e e : Abstract prime number theory Hansen, W. s e e : Mean-value characterization Happel, D. s e e : Tilting module; Tilting theory Harbater, D. s e e : Shafarevich conjecture Hardy, G.H. s e e : A b s t r a c t a n a l y t i c n u m ber theory; Hardy-Weinberg law; Uncertainty principle, mathematical Harman, G. s e e : Vaughan identity Harris, L.A. s e e : J u l i a - W o l f f - C a r a t h 6 o . dory theorem; JB *-triple Harrison, J.M. s e e : Black-Scholes formula; Seifert conjecture Harvey, ER. s e e : Polynomial convexity Harvey, J. s e e : Vertex operator Hausdorff, E s e e : Fredholm solvability; Sierpifiskl game Havas, G. s e e : Burnside group Hayman, W. s e e : BMOA -space He Jifeng s e e : Quantale Heath-Brown, D.R. s e e : Vaughan identity Hecke, E. s e e : Shimura correspondence Heegaard, P. s e e : Reidemeister theorem Heins, M.H. s e e : Denjoy-Wolff theorem Hele-Shaw, H.J.S. s e e : V i s c o u s lingering Helling, H. s e e : Fibonacci group; Fibonacci manifold Helly, E. s e e : Geometric transversal theory Helstrom, C. s e e : Quantum information processing, science of Henzelt, K. s e e : Hermann algorithms Hermann, G. s e e : Effective Nullstellensatz; Hermann algorithms Hertweck, M. s e e : Zassenhaus conjecture Herz, C.S. s e e : Ditkin set; FighTalamanca algebra; FourierStieltjes algebra Herzog, J. s e e : Zariski-Lipman conjecture Heyfron, P. s e e : Immanant Hickel, M. s e e : Liouville-Lojasiewicz inequality Higgins, P.J. s e e : Algebraic homotopy Higman, D.G. s e e : Cellular algebra; Coherent algebra Higman, G. s e e : HNN-extension; Zassenhaus conjecture Hilbert, D. s e e : Knudsen number; ZFC Hilbert, S.R. s e e : Bramble-Hilbert lemma Hildebrand, A. s e e : Dickman function Hildeu, H.M. s e e : Fibonacci manifold Hill, B. s e e : Zipflaw Hiiliard, J. s e e : Cahn-Hilliard equation 554

Hiramine, Y. s e e : Afline design Hirzebruch, E s e e : Index theory Hitchin, N.J. s e e : Index theory; Magnetic monopole Hoare, C.A. s e e : Quantale Hochster, M. s e e : Buchsbaum ring; Zariski-Lipman conjecture Hofer, H. s e e : Seifert conjecture Hoggatt Jr., V.E. s e e : Fibonacci polynomials; Lucas polynomials Hol~, L. s e e : Wijsman convergence Holevo, A . s e e : Quantum information processing, science of Hopf, H. s e e : Baumslag-Solitar group HSrmander, L. s e e : Multiplication of distributions Houghton, C.J. s e e : Magnetic monopole Howe, R. s e e : Segal-Shale-Weil representation Hfibuer, O. s e e : Theodorsen integral equation Huneke, C. s e e : Buchsbaum ring Hunt, G.A. s e e : Riesz decomposition theorem Huntington, E.V. s e e : Robbins equation Hurtubise, J. s e e : Magnetic monopole Huxley, M.N. s e e : Bombieri-Iwaniec method; Exponential sum estimates Hyers, D.H. s e e : Hyers-Ulam-Rassias stability

I_ Immerman, N. s e e : Computational complexity classes Indlekofer, K.-H. s e e : Abstract analytic number theory Ito, T. s e e : Coherent algebra It6, K. s e e : Wiener-It6 decomposition Ivanov, S.V. s e e : B u r n s l d e group Ivanov, V.K. s e e : Generalized function algebras Ivi6, A. s e e : Abundant number Iwaniec, H. s e e : Bombieri-Iwaniec method; Exponential sum estimates; Selberg conjecture Iwasawa, K. s e e : Iwasawa theory; Shafarevich conjecture

J Jackson, D.M. s e e : Immanant Jacobi, C.G.J. s e e : Chasles-CayleyBrill formula; Galois field structure; Segal-Shale-Weil representation Jacobson, N. s e e : Banach-Jordan algebra Jacquet, H. s e e : Selberg conjecture Janson, S. s e e : Janson inequality John, E s e e : BMOA -space Johnsen, K. s e e : Normal basis theorem Johnson, B.E. s e e : Multipliers of C*algebras Johnson, D.L. s e e : Fibonacei group Jones, V.ER. s e e : Jaeger composition product; Jones-Conway polynomial Jones, W.B. s e e : Strong Slieltjes moment problem Jordan, C. s e e : Inclusion-excluslon formula Jordanus, N. s e e : Abundant number Jozsa, R. s e e : Quantum computation, theory of Julia, G. s e e : J u l i a - W o l f f - C a r a t h ~ o d o r y theorem Jutila, M . see: Bombieri-lwaniec method

K

Kostant, B. s e e : Dirac quantization; Momentum mapping Kostrikin, A.I. s e e : Burnside group Kozen, D.C. s e e : Computational complexity classes K r a h n , E . s e e : Rayleigh-Faber-Krahn inequality Krasny, R. s e e : Birkhoff-Rott equation Krautstengl, A. s e e : Gershgorin theorem Kre~n, M.G. s e e : Inverse scattering, half-axis case; Semi-Fredholm operator Kreps, D. s e e : Black-Scholes formula Krichever, I. s e e : Whitham equations KrSger, P. s e e : Neumann eigenvalue Kronecker, L. s e e : Brouwer degree; Lehmer conjecture KronheimeL P.B. s e e : Milnor unknotting conjecture Klull, W. s e e : Hermann algorithms Kruskal, J.B. s e e : Kruskal-Katona theorem Kruskal, M.D. s e e : H a r r y Dym equation Kubota, K.K. s e e : Mahler method Kubota, T. s e e : Iwasawa theory; Shimura correspondence Kuhn, H.W. s e e : Fermat-Torricelli problem Kummer, E. s e e : Vaughan identity Kummer, H. s e e : Quantale Kunen, K. s e e : Weak P-point Kuperberg, G. s e e : Seifert conjecture Kuperberg, K. s e e : Seifert conjecture Kuznetsov, Yu.A. s e e : Dynamical systems software packages Kuzuetsova, O. see:Lebesgue constants of multi-dimensional partial Fourier sums

Kac, M. s e e : Dirichlet eigenvalue; Sierpifiski gasket Kac, V. s e e : Vertex operator Kahane, J.P. s e e : Gleason-KahaneZelazko theorem Kainen, P. s e e : Zarankiewicz crossing number conjecture Kaiser, U. s e e : Skein module Kaltofen, E. s e e : Factorization of polynomials Kamber, EW. s e e : Well algebra of a Lie algebra Kanenobu, T. s e e : Jones-Conway polynomial Kantor, I.L. s e e : Freudenthal-Kantor triple system Karasev, M.V. s e e : Dirac quantization Karoubi, M. s e e : Index theory Kasparov, G . s e e : Multipliers of C*algebras Katchalski, M. s e e : Geometric transversal theory Kato, T. s e e : Semi-Fredholm operator Katok, S. s e e : Shimura correspondence Katona, G.O.H. s e e : Kruskal-Katona theorem Kanffman, L.H. s e e : B r a n d t - L i c k o r i s h Millett-Ho polynomial; Jones unknotting conjecture; Kauffman bracket polynomial Kanp, D.J. s e e : AKNS-hierarehy Kaup, W. s e e : Banach-Jordan pair; JB *-triple Kautsky, J. s e e : Burnside group Kawasaki, T. s e e : Buchsbaum ring Kay, I. s e e : Harry Dym equation Kemer, A.R. s e e : Specht property KendaI1, M.G. s e e : Kendall tan metric Kiefer, J. s e e : Sequential probability ratio test K i e r n a n , P. s e e : Baily-Borel eompaetiLagrange, J. s e e : Material derivative lication method Kim, A.C. s e e : Fibonacei group; FiLamb, H. s e e : Von K~rmlln vortex shedbonacci manifold ding K i m , H . s e e : Selberg conjecture Lanczos, C. s e e : Tan method Kiuderlnan, A.J. s e e : AcceptanceLandau, E. s e e : Abstract analytic numrejection method ber theory Kirby, R. s e e : Kirby calculus Landauer, R. s e e : Quantum information Kirchhoff, G. s e e : Matrix tree theorem processing, science of Kirillov, A.A. s e e : M o m e n t u m mapping Lang, S. s e e : Gel'fond-Schneider Klaassen, C. s e e : Zipflaw method; Schneider method Kle~man, Yu.G. s e e : Specht property Langer, J. s e e : Willmore functional Kleitman, D.J. s e e : Z a r a n k i e w i c z crossLanglands, R.P. s e e : Selberg conjecture ing number conjecture Laplace, P.S. s e e : Viscous lingering Kleshchev, A. s e e : Projective represenLascoux, A. s e e : Schubert polynomials tations of symmetric and alternating Laugwitz, D. s e e : Generalized function groups algebras Klingenberg, W. s e e : Polynomial conLaunhardt, W. s e e : Fermat-Torricelli vexity problem Knopfmacher, J. s e e : Abstract analytic Lavrent'ev, M.A. s e e : Zorich theorem number theory Lawson, H.B. s e e : Polynomial convexKnudsen, M.H.Ch. s e e : Knudsen numity; Willmore functional ber Lazard, D. s e e : Hermann algorithms; Knuth, D.E. s e e : Linear congruential Masser-Philippon/Lazard-Mora exmethod ample Kohnen, W. s e e : Shimura corresponLebesgue, H. s e e : Fermat-Torrieelli dence problem Kohnert, A . s e e : Schubert polynomials Lechicki, A. s e e : Wijsman convergence Kolesnik, G. s e e : Bombieri-Iwaniec Lee, R. s e e : Satake eompactification method Leeb, K. s e e : Hales-Jewett theorem Kolmogorov, A.N. s e e : Integrability of Lefranc, M. s e e : Fourier-Stieltjes algetrigonometric series bra Kolyvagin, V. s e e : lwasawa theory Legendre, A.M. s e e : Factorization of Ktinig, H. s e e : Generalized function alpolynomials gebras Lehmer, D.H. s e e : Lehmer conjecture Konopelchenko, B.G. s e e : Harry Dym Lelong, P. s e e : Current; Pluripotential equation theory Korobov, N.M. s e e : Exponential sum Leman, A.A. s e e : Cellular algebra; Coestimates herent algebra Korteweg, D.J. s e e : Harry Dym equaLemoine, E. s e e : Brocard point tion Lempert, L. s e e : Pluripotential theory

L

LENSTRA, A.K.

Lenstra, A.K. s e e : Factorization of polynomials; Galois field structure Lenstra, J.K. s e e : Linear congrnential method Lenstra, Jr., H.W. s e e : Factorization of polynomials; Linear congrnential method Leonardo da Vinci s e e : Von K,-irm~in vortex shedding Leopoldt, H.W. s e e : Iwasawa theory Lepowsky, J. s e e : Vertex operator algebra Leray, J. s e e : Brouwer degree Lesigne, E. s e e : Wiener-Wintner theorem Levi, E s e e : Burnside group Levi, S. see: Wijsman convergence Levitan, B.M. s e e : H a r r y Dymequation Ldvy, P. s e e : Skorokhod equation Li, P. s e e : Willmore functional Li Yi-Shen s e e : Harry Dym equation Lichnerowicz, A. s e e : Dirac quantization Lickorish, W.B.R. s e e : Kirby calculus Lie, S. s e e : Momentum mapping Lieb, E.H. s e e : Diriehlet eigenvalue; Immanant Light, W.A. s e e : Alternating algorithm LindelOf, L.L. s e e : Fermat-Torricelli problem Lip.nell, P.A. s e e : Accessibility for groups LiouviUe, J. s e e : Dirichlet convolution; Quasi-regular mapping Listing, J.B. s e e : Listing polynomials Littlewood, D.E. s e e : Immanant; Schur Q-function Livgic, M.S. s e e : Operator colligation; Operator vessel Loday, J.-L. s e e : Algebraic homotopy Long, C.T. s e e : Fibonacci polynomials Looijenga, E. s e e : Satake compactification Lord Rayleigh s e e : Rayleigh-FaberKrahn inequality Lorden, G. s e e : Sequential probability ratio test Loring, T.A. s e e : Multipliers of C*algebras Losert, V. s e e : Fourier-Stieltjes algebra Lov~isz, L. s e e : Factorization of polynomials; Krnskal-Katena theorem; Linear congruential method; Lov~isz local lemma Loxton, J.H. s e e : Mahler method Lozano, M.T. s e e : Fibonacci manifold Lubell, D. s e e : Sperner theorem Lucas, E. s e e : Abundant number Lucchetti, R. s e e : W i j s m a n convergence Luczak, T. s e e : Janson inequality Lukasiewicz, J. s e e : Abstract algebraic logic Lumiste, U. s e e : Faddeev-Popov ghost Luo, W. s e e : Selberg conjecture Lychagin, V. s e e : Dirac quantization Lysenok, I.G. s e e : Burnside group

M Maass, H. s e e : Selberg conjecture MacArthur, R.H. s e e : Zipf law Machado, S. s e e : Bishop theorem Machlup, S. s e e : Onsager-Machlup function Maclachlan, C. s e e : Fibonacci group MacLane, S. s e e : A l g e b r a i c homotopy MacPherson, R.D. s e e : Linear congrnential method Magnus, W. s e e : Burnside group Mahler, K. s e e : Mahler method Maksimova, L. s e e : Algebraic logic Mallock, A. s e e : Von K,qrm~in vortex shedding

Mal'cev, A. s e e : Abstract algebraic logic Mandelbrot, B. s e e : Zipf law Manstavi~ius, E. s e e : Abstract analytic number theory Maroon, N.S. s e e : Magnetic monopole Marchenko, V.A. s e e : Harry Dym equation Marcus, M. s e e : Immanant Markowitz, H. s e e : Portfolio optimization Marquis de Condorcet s e e : Condorcet jury theorem; Condorcet paradox Marsaglia, G. s e e : Linear congruential method Martin, R.H. see: Semi-group of holomorphic mappings Martinec, E. s e e : Vertex operator Martinet, J. s e e : Odlyzko bounds Martio, O. s e e : Quasi-regular mapping; Zorich theorem Mascheroni, L. s e e : Catalan constant Maslov, V.P. s e e : Dirac quantization; Generalized function algebras Masser, D.W. s e e : Mahler method; Masser-Philippon/Lazard-Mora example Mather, J. s e e : Thom-Mather stratification Matsumoto, H. s e e : Steinberg symbol Matsumoto, K. s e e : Shimura correspondence Matzat, B.H. s e e : Shafarevich conjecture Maxwell, J.C. s e e : Knudsen number Mazur, S. s e e : Banach-Mazur game McCabe, J. s e e : S t r o n g S t i e l t j e s m o m e n t problem McCrimmon, K. s e e : Banach-Jordan algebra McDonald, G. s e e : Berezin transform McKay, B. s e e : Cayley graph McKay, J. s e e : Moonshine conjectures; Vertex operator algebra McKean, H. s e e : Index theory McLean, J.W. s e e : Viscous fingering McMutlen, C.T. s e e : Weil-Petersson metric Meiman, N.N. s e e : Zolotarev polynomials Melas, A.D. s e e : Dirichlet eigenvalue; Rayleigh-Faber-Krahn inequality Mellon, P. s e e : Denjoy-Wolff theorem; Julia-Wolff-Carathdodory theorem Menger, K. s e e : Sierpifiski gasket Mennicke, J. s e e : Fibonacci group; Fibonacci manifold Mercer, P.R. see: Julia-WolffCarathtodory theorem Merris, R. s e e : Immanant Merton, R.C. s e e : Option pricing; Portfolio optimization Meshalkin, L. s e e : Sperner theorem Meskin, S. s e e : Baumslag-Solitar group Meurman, A. s e e : Vertex operator algebra Mihmr, J. s e e : Steinberg symbol Minkowsld, H. s e e : L i n e a r congruential method; Odlyzko bounds Minoiu, S. s e e : Fuzzy programming Mirimanoff, D. s e e : ZFC Mishchenko, A. s e e : Index theory Mityagin, B. s e e : Lebesgue constants of multi-dimensional partial Fourier sums; Step hyperbolic cross Miura, R.M. s e e : Harry Dym equation Miyamoto, M. s e e : Weyl-Kac character formula Mnatsakanov, R. s e e : Zipf law Monahan, LF. see: Acceptancerejection method Montague, R. s e e : Natural language processing Montejano, L. s e e : G e o m e t r i c t r a n s v e r sal theory

Montesinos, J.M. s e e : Fibonacci manifold Montgomery, H.L. s e e : Lehmer conjecture; Vaughan identity Montiel, S. s e e : Willmore functional Montmort, P.R. s e e : Inclusion-exclusion formula Moon, LW. s e e : Matrix tree theorem Moore, C. s e e : Steinberg symbol Moore, D. s e e : Birkhoff-Rott equation Mora, T. s e e : M a s s e r - P b i l i p p o n l L a z a r d Morn example Morita, K. s e e : P-space Morley, M. s e e : Vaught conjecture Morris, A.O. s e e : Projective representations of symmetric and alternating groups Mosco, U. s e e : Wijsman convergence Moscovici, H. s e e : Index theory Moser, L. s e e : Abundant number Moses, H.E. s e e : Harry Dym equation Motzldn, T.S. s e e : Motzkin transposition theorem Mourre, E. s e e : Enss method Mozzochi, C.J. s e e : Bombieri-Iwaniec method Mr6wka, T.S. s e e : Milnor unknotting conjecture Mulvey, C.J. s e e : Quantale Mufioz, V. s e e : Atiyah-Floer conjecture Murray, M.K. s e e : Magnetic monopole

N Nadirashvili, N. s e e : Mean-value characterization; Rayleigh-FaberKrahn inequality Nag, S. s e e : Weil-Petersson metric Nagata, M. s e e : Gel'fond-Schneider method Nagumo, M. s e e : Brouwer degree N a h m , W . s e e : Magnetic monopole Naimpally, S. s e e : Wijsman convergence Namioka, I. s e e : Namioka space; Namioka theorem Narldewicz, W. s e e : Dirichlet convolution Navier, L. s e e : Knudsen number Nazarov, M.L. s e e : Projective representations of symmetric and alternating groups Nazarova, L.A. s e e : Tits quadratic form Ne'eman, Y. s e e : Faddeev-Popov ghost Negoita, C.V. s e e : Fuzzy programming Nesterenko, Yu.V. s e e : Gel'fondSchneider method Neumann, B.H. s e e : 1-INN-extension; Specht property Neumann, H. s e e : HNN-extension Newell, A.C. s e e : A K N S . h i e r a r c h y Newman, M.E s e e : Burnside group Nicolas, J.-L. s e e : Abundant number Nicomachus, G. s e e : Abundant number Niederreiter, H. s e e : Linear congrnential method Nielsen, J. s e e : Weil-Petersson metric Nirenberg, L. s e e : Brouwer degree; B M O A -space Nishida, K. see: Bucbsbaum ring Nishioka, K. s e e : Mahler method Noether, E. s e e : Hermann algorithms; Schur group Norton, S.P. s e e : Moonshine conjectures; Vertex operator algebra Novikov, P.S. s e e : Burnside group

O Odlyzko, A.M.

see:

Odlyzko bounds

Oka, K. s e e : Pluripotential theory Okubo, S. s e e : Okubo algebra Ol'shanskiL A.Yu. s e e : B u r n s i d e g r o u p ; Specht property Onsager, L. s e e : Onsager-Machlup function Orlov, Yu. s e e : Z i p f l a w Orsag, S.A. s e e : Tau method Ortiz, E.L. s e e : Tau method Osserman, R. s e e : Weierstrass representation of a minimal surface Ostresh, L.M. s e e : Fermat-TorriceUi problem !Otter, R. s e e : Abstract prime number theory Oxtoby, J. s e e : Banach-Mazur game

P Padoa, A. s e e : Abstract algebraic logic Palmer, E.M. s e e : Abstract prime number theory Parben'y, E.A. s e e : Fibonacci polynomials Pate, Th.H. s e e : Immanant Patodi, V. s e e : Index theory Patterson, S.J. s e e : Vaughan identity Pearse, E.P.J. s e e : Sierpifiski gasket Pearson, K. s e e : Pearson productmoment correlation coefficient Petfis, B.J. s e e : Almost continuity Pflug, P. s e e : Pluripotentiai theory Philippon, P. s e e : Gel'fond-Schneider method; Masser-Pbilippon/LazardMorn example Philippou, A.N. s e e : Fibonacci polynomials Phong, D.H. s e e : Weil-Pctersson metric; Whitham equations Piateckii-Shapiro, I.I. s e e : Baily-Borel compactification Pick, G. s e e : Fermat-Torricelli problem Pinkall, U. s e e : Willmore functional Plateau, J. s e e : Geometric measure theory Platzeck, M.I. s e e : Tilting module; Tilting theory Pliska, S. s e e : Black-Scholes formula Podkorytov, A. s e e : Lebesgue constants of multi-dimensional partial Fourier sums

Pohst, M. s e e : Linear congruential method Poincar6, H. s e e : Brouwer degree; Incinsion-exclusion formula Pollack, R. s e e : Geometric transversal theory P61ya, G. s e e : Dirichlet eigenvalue; Neumann eigenvalue Polyakov, A.M. s e e : Vertex operator algebra; Weii-Petersson metric Pommerenke, Ch. s e e : BMOA -space Ponomarev, V.A. s e e : Tilted algebra; Tilting module; Tilting theory Pontryagin, L.S. s e e : D i r a c quantization Pop, E s e e : Shafarevlch conjecture Popov, V.N. s e e : Faddeev-Popov ghost Porta, H. s e e : Denjoy-Wolff theorem; Semi-group of holomorphic mappings Post, E. s e e : Abstract algebraic logic Potapov, V.P. see: Julia-WolffCarath~odory theorem Praeger, C.E. s e e : Cayley graph Prandtl, L. s e e : Yon K,'irm~in vortex shedding Prasad, M.K. s e e : Magnetic monopole PrAstaro, A. s e e : Dirac quantization Prtkopa, A. s e e : Inclusion-exclusion formula Prolla, J.B. s e e : B i s h o p theorem 555

TORRICELLI, E.

Przymusifiski, T.C. s e e : P-space Putinar, M. s e e : Taylor joint spectrum

Q Quillen, D. see:Flat cover

R Ramachandra, K. s e e : Schneider method Ramanujan, G. s e e : Abstract analytic number theory Ramm, A.G. s e e : Inverse scattering, multi-dimensional case; Obstacle scattering Ransford, T.J. s e e : Bishop theorem Rassias, Th.M. s e e : Hyers-UlamRassias stability Ratiu, T. s e e : AKNS-bierarchy Ray, U. s e e : WeyI-Kac character formula Razmyslov, Yu.P. s e e : Burnside group Reddy, D.R. s e e : Abundant number Reidemeister, K. s e e : Reidemeister theorem Reiten, I. s e e : Almost-split sequence; Tilting module; Tilting theory Reshemyak, Yu.G. s e e : Quasi-regular mapping Reufel, M. s e e : Hermann algorithms Ribet, K. s e e : lwasawa theory Richmond, L.B. s e e : Abstract prime number theory Rickard, J. s e e : Tilting theory Rickman, S. s e e : Quasi-regular mapping; Zorich theorem Pdedtmann, C. s e e : Riedtmann classification Riemana, B.G. s e e : Riemann ~-function Riesz, E s e e : Riesz decomposition theorem Ringel, C.M. s e e : Tilting module; Tilting theory Ringel, G. s e e : Zarankiewicz crossing number conjecture Riordan, J. s e e : Lucas polynomials Rival, T. s e e : Ap6ry numbers Robbins, H. s e e : Robbins equation Robinson, A. s e e : Generalized function algebras Roelcke, W. s e e : Selberg conjecture Rogers, C. s e e : Harry Dym equation Roggenkamp, K. s e e : Zassenhaus conjecture Rohm, R. s e e : Vertex operator Ros, A. s e e : Willmore functional Rosick3~,J. s e e : Quantale Rosinger, E.E. s e e : Generalized function algebras; Rosinger nowheredense generalized function algebra Rota, G.-C. s e e : Baxter algebra; M6bins inversion Rothschild, B. s e e : Hales-Jewett theorem Rouault, A. s e e : Zipf law Rourke, C.P. see: Kirby calculus Rowley, D.R. s e e : Schur Q-function Royle, G.E s e e : Cayley graph Rudin, M.E. s e e : P-space Rudin, W. s e e : Berezin transform; P point Rudnick, Z. s e e : Selberg conjecture Ruelle, D. s e e : Enss method Russell, B. s e e : ZFC Russo, B. s e e : J B * . t r i p l e 556

S Sabatier, EC. see: Harry Dym equation Saffman, EG. s e e : Viscous fingering Sagan, B.E. s e e : Scbur Q-function Saint-Raymond, J. s e e : Namioka space Salamon, D . s e e : Atiyah-Floer conjecture Sali6, H. s e e : Abundant number Samara, H. s e e : q?au method Samuelson, PA. s e e : Black-Scholes formula Sanov, I.N. s e e : Burnside group Saper, L. s e e : Satake compactifieation Sarason, D. s e e : J u l i a - W o l f f - C a r a t h 6 o dory theorem; VMOA -space Sargos, P. s e e : Bombieri-Iwaniee method Samak, P. s e e : Selberg conjecture; Shimnra correspondence Satake, I. s e e : Baily-Borel compactification; Satake compactification Scheja, G. s e e : Zariski-Lipman conjecture Schief, W.K. s e e : Harry Dym equation Schifter, M. s e e : Obstacle scattering Schinzel, A. s e e : Lehmer conjecture Schmid, W. s e e : Index theory Schmieden, C. s e e : Generalized function algebras Schneider, Th. s e e : G e l ' f o n d - S c h n e i d e r method; Schneider method Scholes, M. s e e : Black-Scholes formula; Option pricing School R.J. see: Galois field structure Sch6pf, A. s e e : Flat cover Schreier, O. s e e : Cayley graph Schr6der, E. s e e : Schr6der functional equation; Seml-group of holomorphic mappings Schubert, H. s e e : Schubert calculus Schultz, M. s e e : Bramble-Hilbert lemma Schur, I. s e e : Immanant; Projective representations of symmetric and alternating groups; Schnr Q-function; Szeg6 polynomial Schtitzenberger, M.-P. s e e : Schubert polynomials Schwartz, L. s e e : Current; Generalized function algebras; Multiplication of distributions Schweitzer, P.A. s e e : Seifert conjecture Schwenk, A.J. see: Abstract prime number theory Scott, L.R. s e e : Bramble-Hilbert lemma; Zassenhaus conjecture Segal, G.B. s e e : Magnetic monopole; Vertex operator Segal, I. s e e : Segal-Sbale-Weil representation Segur, H. s e e : AKNS-hierarchy Sehgal, S.K. s e e : Zassenhaus conjecture Seidenberg, A. s e e : Hermann algorithms Seifert, H. s e e : Seifert conjecture Sela, Z. s e e : Accessibilityfor groups Selberg, A. s e e : Selberg conjecture Selby, A. s e e : Magnetic monopole Serre, J.-P. s e e : F o r s t e r - S w a n t h e o r e m ; Iwasawa theory; Serre theorem in group cohomology Shafarevich, I.R. s e e : Shafarevich conjecture Shafer, G. s e e : D e m p s t e r - S h a f e r t h e o r y Shahidi, E s e e : Selberg conjecture Shale, D. s e e : Segal-Shale-Weil representation Shannon, A. s e e : Tribonacci number Shannon, C. s e e : Quantum information processing, science of; Zipf law Shapiro, H.N. s e e : Abundant number Shapiro, J. s e e : VMOA -space

Shapley, L.S. s e e : N o n - a d d i t i v e measure Shelah, S. s e e : P-point; Vaught conjecture Sheperdson, J.C. s e e : Hermann algorithms Shimoda, Y. s e e : Bnehsbaum ring Stfimura, G. s e e : Shimura correspondence Shiohama, K. s e e : Willmore functional Sinnulyan, Yu.L. s e e : Operator colligation Shot, P. s e e : Quantum computation, theory of; Quantum information processing, science of Shoup, V. s e e : Factorization of polynomials Shultz, EW. see: Banach-Jordan algebra Sibony, N. s e e : Pinripotential theory Sidon, S. s e e : Integrability of trigonometric series Siegel, C.L. s e e : Segal-Sbale-Weil representation Sierpifiski, W. s e e : Sierpifiski game; Sierpifiski gasket Sigal, I.M. s e e : Enss method Sikonia, W. s e e : Brown-DouglasFillmore theory Simon, H.A. s e e : Zipf law Simpson, Th. s e e : Fermat-Torricelli problem Simson, D. s e e : Tits quadratic form Sinclair, A.M. s e e : Banach-Jordan pair Singer, D. s e e : Willmore functional Singer, I. s e e : Index theory Singer, M.A. s e e : Magnetic mouopole Sinnott, W. s e e : Iwasawa theory Siva Rama Prasad, V. s e e : Abundant number Skolem, T. s e e : ZFC Skorokhod, A.V. s e e : Skorokhod equation; Skorokbod space Slodkowski, Z. s e e : Polynomial convexity Smith, K.T. s e e : Alternating algorithm Smith, EA. s e e : Smith theory of group actions Sobolev, S.L. s e e : Bramble-Hilbert lemma Softer, A . s e e : Enss method Solman, D.C. s e e : Alternating algorithm Solomon, L. s e e : Steinberg module Sommerfield, C.M. s e e : Magnetic monopole Soules, G.W. see: Immanant Souriau, J.M. s e e : Dirac quantizatlon; Momentum mapping Spaniel E. s e e : Exponential law (in topology) Spearman, C. s e e : Spearman rho metric Specht, W. s e e : Specbt property Spencer, D. s e e : Spencer cohomology Spencer, J. s e e : Additive basis; Janson inequality Spemer, E. s e e : Rayleigh-FaberKrahn inequality; Sperner property; Sperner theorem Spickerman, W. see: Tribonacci number Stachura, A. s e e : D e n j o y - W o l f f t h e o r e m Stafford, LT. s e e : Forster-Swan theorem Stallings, J.R. s e e : Accessibility for groups Stan, E. s e e : Fuzzy programming Stark, H.M. s e e : Odlyzko bounds Steane, A. s e e : Quantum computation, theory of Stegenga, D. s e e : BMOA -space Stein, M.R. s e e : Steinberg symbol Steinberg, R. s e e : Steinberg symbol Stembfidge, J.R. s e e : Schur Q-function Stephenson, K. s e e : BMOA -space Stern, M. s e e : Satake compactification Sternheimer, D. s e e : Dirac quantization Stiles, W.J. s e e : Alternating algorithm Stinespring, W.E s e e : Fourier algebra

Stockmeyer, L. s e e : Computational complexity classes Stokes, G.C. s e e : Knudsen number; Stokes parameters Stolz, S. s e e : Index theory Stolzenberg, G. s e e : Polynomial convexity Stone, M.H. s e e : Banach-Stone theorem Storch, U. s e e : Zariski-Lipman conjecture Stormer, E. s e e : Banach-Jordan algebra Slyatonovich, R.L. s e e : OnsagerMachinp function Struve, J. s e e : Abundant number Smart, A.E.G. s e e : Benjamin-BonaMahony equation StiJckrad, J. s e e : Buchsbaum ring S t u r m , R . s e e : Fermat-Torricelli problem Suen, W.C.S. s e e : J a n s o n i n e q u a l i t y Sugeno, M. s e e : Non-additive measure Sundberg, C. s e e : Berezin transform Suszko, R. s e e : Abstract algebraic logic Sutcliffe, P.M. s e e : Magnetic monopole Swan, R.G. s e e : Forster-Swan theorem Swinnerton-Dyer, H. s e e : BombieriIwaniee method Sz.-Nagy, B. s e e : Operator culligation Szeg6, G. s e e : Rayleigh-Faber-Kralm inequality; Szegii limit theorems Szekeres, G. s e e : Abstract analytic number theory

T Takagi, R. s e e : Willmore functional Takhtayan, L.A. s e e : Weil-Petersson metric Tanaka, H. s e e : Fuzzy programming Tarski, A. s e e : Algebraic logic; Robbins equation Tate, J. s e e : Steinberg symbol Tanbes, C.H. s e e : Index theory; Magnetic monopole Tauraso, R. s e e : J u l i a - W o l f f - C a r a t h ~ o . dory theorem Taussky, O. s e e : Gershgorin theorem Taylor, B.A. s e e : Pluripotential theory Taylor, G.I. s e e : Viscous fingering Taylor, J.L. see:Taylor joint spectrum Telg~rsky, R. s e e : Sierpifiski game ten Kroode, A.P.E. s e e : AKNShierarchy Thaine, E s e e : Iwasawa theory Theodorsen, T. s e e : T h e o d o r s e n i n t e g r a l equation Thiele, Th.N. s e e : Tbiele differential equation Thierry-Mieg, J. s e e : Faddeev-Popov ghost Thistlethwaite, M.B. s e e : Jones unknotting conjecture Thorn, R. s e e : Dirac quantization; Tb0m-Mather stratification Thompson, J,G. s e e : Fitting length; Vertex operator algebra Thomsen, G. s e e : Willmore functional Thomson, J.M. s e e : Braess paradox Thron, W.J. s e e : Strong Stieltjes moment problem Timoshenko, S. s e e : Natural frequencies Tits, J. s e e : Steinberg module Tobin, S.J. s e e : Burnside group Todd, A.R. s e e : Slobodnik property Tondeur, Ph. see: Weil algebra of a Lie algebra Tonin, M. s e e : Faddeev-Popov ghost T6pfer, T. s e e : Mabler method Torricelli, E. s e e : Fermat-Torricelli problem

TRACZYK, E

Traczyk, E s e e : Rotor Trlifaj, J. s e e : Flat cover Tsekanovskil, E.R. s e e : Operator colligation Tukia, P. s e e : Quasl-symmetric function of a complex variable Tullo, A.W. s e e : Banach-Jordan pair Turaev, V.G. s e e : Skein module Tur~in, P. s e e : Zarankiewicz crossing number conjecture Turing, A.M. s e e : F a c t o r i z a t i o n o f p o l y nomials; Quantum computation, theory of; Quantum information processing, science of Tutte, W.T. s e e : Matrix tree theorem Tverberg, H. s e e : Geometric transversal theory

U Ulam, S.M. s e e : Brouwer degree; Hyers-Ulam-Rassias stability Unger, L. s e e : Tilting theory Urbam'k, K. s e e : Zarankiewicz crossing number conjecture

V V/iis~il~,J. s e e : Quasi-regular mapping; Quasi-symmetric function of a complex variable; Zorich theorem van der Corput, J. s e e : Exponential sum estimates van der Poorten, A.J. s e e : Mahler method van der Waals, J.D. s e e : Cahn-HiUiard equation van der Waerden, B.L. s e e : Burnside group; Hermann algorithms Vaxdi, M.Y. s e e : Computational complexity classes Varga, R.S. s e e : Bramble-Hilbert lemma; Gershgorin theorem Varignon, P. s e e : Varignon parallelogram Vaughan-Lee, M.R. s e e : Speeht property Vanghan, R.C. s e e : Vaughan identity Vaught, R. s e e : Vaught conjecture Vazirani, U. s e e : Quantum computation, theory of Veltzke, C. s e e : Hermann algorithms Verjovsky, A. s e e : Weil-Petersson metric Vertigan, D.L. s e e : Jones-Conway polynomial Vesentini, E. s e e : Denjoy-Wolff theorem; Semi-group of holomorphic mappings Vesnin, A. s e e : Fibonacci group

Villena, A.R. s e e : Banach-Jordan pair Vincensini, E s e e : Geometric transversal theory Vinogradov, I.M. s e e : Exponential sum estimates; Vaughan identity Viviani, V. s e e : Fermat-Torricelli problem Vogel, W. s e e : Buchsbaum ring V61klein, H. s e e : Shafarevich conjecture Volterra, V. s e e : Almost continuity von K~-m,Sn, Th. s e e : K n u d s e n n u m b e r ; Von K~rm~in vortex shedding yon Neumann, J. s e e : Alternating algorithm; ZFC vun Tschirnhaus, E.W. s e e : FermatTorricelli problem von zur Gathen, J. s e e : Factorization of polynomials

W Waadeland, H. s e e : Strong Stieltjes moment problem Wagner, S.L. s e e : Alternating algorithm Wald, A . s e e : Average sample number; Sequential probability ratio test Waldspurger, J.-L. s e e : SegaI-ShaleWell representation; Shimura correspondence Wall, C.T.C. s e e : Accessibility for groups Wall, Ch.R. s e e : Abundant number Wallace, A.D. s e e : Kirby calculus Walter, M.E. s e e : Fourier-Stieitjes algebra Wamsley, J.W. s e e : Burnside group Ward, R.S. s e e : Magnetic monopole Warfield, R.B. s e e : Forster-Swan theorem Warlimont, R. s e e : Abstract analytic number theory; Abstract prime number theory Washington, L. s e e : Iwasawa theory Watt, N. s e e : B o m b i e r i - I w a n i c c m e t h o d Web, D. s e e : Dirichlet eigenvalue Webb, W.A. s e e : Fibonacci polynomials Weber, H. s e e : Abstract analytic number theory Wegmann, R. s e e : Theodorsen integral equation Weierstrass, K. s e e : Almost continuity; Non-commutative anomaly; Weierstrass representation of a minimal surface Weil, A. s e e : Segal-Shale-Wcil representation; Selberg conjecture; Shimura correspondence; Weft algebra of a Lie algebra; Weil-Petersson metric Weinberg, W. s e e : Hardy-Weinberg law Weinberger, H.E s e e : Rayleigh-FaberKrahn inequality Weiner, J.L. s e e : Willmore functional Weinstock, B.M. s e e : Polynomial con-

vexity Weisfeiler, B.Yu. s e e : Cellular algebra; Coherent algebra Weiss, A. s e e : Zassenhaus conjecture Weiss, J. s e e : Harry Dym equation Weiss, L. s e e : Sequential probability ratio test Weissauer, R. s e e : Shafarevich conjecture Weiszfeld, E. s e e : Fermat-Torricelli problem Wenger, R. s e e : Geometric transversal theory Wermer, J. s e e : Polynomial convexity West, T.T. s e e : Riesz operator Weyl, H. s e e : Dirichlet eigenvalue; Exponential sum estimates; Neumann eigenvalue Whitcomb, A. s e e : Zassenhaus conjectare White, B. s e e : Geometric measure theory Whitehead, J.H.C. s e e : A l g e b r a i c h o m o topy W h i t h a m , G . s e e : Whitham equations Whitney, H. s e e : M6bius inversion Widom, H. s e e : Szeg6 limit theorems Wiener, N. s e e : Wiener-It6 decomposition Wiesner, S. s e e : Quantum information processing, science of Wigner, E. s e e : Massive field; Massless field Wijsman, R. s e e : Wijsman convergence Wik, I. see: Ditkin set Wiles, A. s e e : Iwasawa theory Willis, J.G. see: Zipf law Willmore, T.J. s e e : Willmore functional Wilson, EW. s e e : Seifert conjecture Winker, S. s e e : Robbins equation Witten, E. s e e : Dirac monopole; Index theory Wittenbauer, E s e e : Wittenbauer theorem Witzgall, C. s e e : Fermat-Torricelll problem Wlodarczyk, K. s e e : Julia-WolffCarath4odory theorem Wold, H. s e e : Wold decomposition Wolff, J. s e e : Denjoy-Wolff theorem; Julla-Wolff-Carath4odory theorem Wolfowitz, J. s e e : Sequential probability ratio test Wolpert, S. s e e : Dirichlet eigenvaine Woodall, D.R. s e e : Zarankiewicz crossing number conjecture Woodroofe, M. s e e : Zipf law Wormald, N.C. s e e : Abstract prime number theory Wu, T.T. s e e : Dirac monopole Wiistholz, G. s e e : Gel'fond-Schneider method Wyatt, J.L. s e e : Braess paradox

X

Xu, J.

see:

Flat cover

Y Yamada, S. s e e : Jones unknotting conjecture Yamamoto, S. s e e : Sperner theorem Yang, C.N. s e e : Dirac monopole Yang, P. s e e : Denjoy-Wolff theorem Yao, A. s e e : Quantum computation, theory of Yau, S.T. s e e : Willmore functional Yff, P. s e e : Brocard point Yoshino, Y. s e e : Buchsbaum ring Young, A. s e e : Projective representations of symmetric and alternating groups Young, T. s e e : Viscous fingering Young, W.H. s e e : Integrability of trigonometric series Yule, G.U. s e e : Zipf law

Z Zabuskff, N.J. s e e : H a r r y Dym equation Zadeh, L.A. s e e : Fuzzy programming; Non-additive measure Zagier, D. s e e : Shimura correspondence Zahorski, Z. s e e : Zahorski property Z a k , J. s e e : Zak transform ZalesskiI, A.E. s e e : Zassenhaus conjecture Zamolodchikov, A. s e e : V e r t e x o p e r a t o r algebra Zarankiewicz, K. s e e : Zarankiewicz crossing number conjecture Zassenhaus, H. s e e : Factorizatioil of polynomials; Lehmer conjecture Zeitouni, D. s e e : Onsager-Machinp function Zelazko, W. s e e : Gleason-KahaneZelazko theorem Zel'manov, E.L s e e : B a n a c h - J o r d a n algebra; Burnside group Zermelo, E. s e e : ZFC ZernJke. E s e e : Zernike polynomials Zhang, W.-B. s e e : Abstract analytic number theory Zheag, D. s e e : Berezin transform Zieminska, J. s e e : Wijsman convergence Zimmermann, H.-J. s e e : Fuzzy programming Zipf, G.K. s e e : Zipf law Zograf, P.G. s e e : Weil-Petersson metric Zolotarev, E.I. s e e : Zolotarev polynomials Zorich, V.A. s e e : Zorich theorem Zukowski, C.A. s e e : Braess paradox

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