ENCYCLOPAEDIA OF MATHEMATICS Supplement Volume III
ENCYCLOPAEDIA OF MATHEMATICS Supplement Volume III
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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 , 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 _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 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) 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-
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 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 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 __ 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)
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), 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 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>
+
-
= = ( 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 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 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 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) 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 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 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 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 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}0oIn/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 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 _ 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 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 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 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:
] ~ 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 < u _< 2, and (uw(u))' = w(u - 1), u > 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 ~ 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(~) _ ~
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 _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 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 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 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, - 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
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) 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 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) 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 < i < n, such that A ¢ Gi(A), where
the last inequality following from the definition of ri (A) in (1) and the fact t h a t Ixjl 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_ O, and there is an i, with 1 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 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 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]
>_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 -- 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 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 x.
I f t h e data { r + ( k ) , i k j , ( m + ) 2 : 1 < j --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 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-
--oo.
j=l
In [5] a different approach to solving the inverse scattering problem is described for
References
/ff l 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)
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 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
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
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 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] 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[
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 )
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_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 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¢) -- : 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
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 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 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_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 _< b has a solution x; a2) ATy=O,y_>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 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+~ 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 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)) 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_ 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
2. Then Hk(C n \ K ; G ) = O ,
l 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!
/ 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 ( ~ ) 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
_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
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 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 :=
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 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~ = [_]~_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
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 0.22837, i.e., 5
IRe(sp,i)l _
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 - 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 > 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 and sequentially computes the likelihood ratio after each observation. If after n observations An _< A, one stops taking observations and decides to accept H0. If 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_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 such that the 'cross'
K(z,5):=
(tl,t2):
Izj-tjl 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 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
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}.
= 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 _ 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 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 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)[ 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 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 ([, - 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)] 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
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 3,
Sv/~, d=l, E IWa(t) l
~
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
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 < b _< 1/2, then (Zf)(t,w) = f(t), Itl _< 1/2. The Zak transform of the Gaussian function
f(t) 450
= (27) 1/4 exp (--Tr3't2),
3' > 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 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
