Search

Necessary and sufficient conditions are presented to insure that the direct sum of two reflexive representations of a finite dimensional algebra is reflexive, and it is shown that for each such algebra, there is an integer k such that the direct sum of k copies of each of its representations is reflexive. Given a ring i\, our results are actually presented in the more general setting of A\-representations of a ring R. Over the last thirty years, a significant amount of attention has been given to the problem of determining when an algebra A of operators on a vector space V (often a Hilbert space) over a field K is reflexive in the sense that no larger algebra of operators on that space has the same lattice of invariant subspaces. (See, for example, [2], [3], [6], [7], [12], [13], [14].) Of course, A is an algebra of operators on a vector space V if and only if V is a faithful left A-module. Thus, to attack this problem from a more module theoretic point of view, based on notation of Halmos [14], the following notions were presented in [9]: For any R A-bimodule M= RMA one defines alglat(M) {a E End(MA\) I aL C L for all RL alglat(MA) denote the canonical ring homomorphism, M is called a reflexive bimodule (or A-representation of R) if A is surjective. Thus the K-algebra of operators A on V is reflexive if and only if A VK is a reflexive bimodule (and then we simply say that the A-module V is reflexive). The problems we shall deal with here have their roots in the papers [5] and [6] of Deddens and Fillmore and [2] of Azoff. Stated in module theoretic terms, in the first pair of papers the question was posed and answered (affirmatively) of whether a direct sum of two finitely generated reflexive modules over an algebra generated by a single complex matrix is again reflexive; and Azoff showed that for each finite dimensional module over a C-algebra, there is a positive integer k, depending on its dimension, such that the direct sum of k copies of that module is reflexive. Other results in this vein can be found in [4] where Brenner and Butler showed that the direct sum of two copies of the regular representation of a finite dimensional algebra is reflexive, and in Habibi and Gustafson's [12] from which the same result follows for any faithful representation of a split serial algebra. (See [9], [10] and [11] for related and more general results.) Our main objectives are to provide, in Theorem 1.2, a necessary and sufficient condition for a direct sum of two reflexive bimodules to be reflexive; and to show, in Received by the editors September 10, 1998 and, in revised form, November 10, 1998. 1991 Mathematics Subject Classification. Primary 16D20, 16G99, 16P10; Secondary 47A15. ?)2000 American Mathematical Society 2855 This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms 2856 V. P. CAMILLO AND K. R. FULLER Theorem 3.2, that whenever R is a left artinian ring with composition length c(RR), there is a positive integer k (am, fn). Thus if -y E alglat(M e N), then a = (-YIM, -YIN). A common subquotient of a pair of left modules RM and RN is a module RX that is isomorphic to a subquotient of both RM and RN. This notion together with the following lemma allows us to determine just when the direct sum of a pair of reflexive modules is reflexive. Lemma 1.1. If r, s E R, the ordered pair (r, s) E alglat(M D N) if and only if (r s)X = 0, for every (equivalently, every cyclic) common subquotient of RM and RN. Proof. Note that a E R annihilates every common subquotient of M and N if and only it annihilates each of their cyclic common subquotients. According to Goursats' Lemma, W is an R-submodule of M ED N, if and only if there are submodules M2 N,N2 such that W {(ml, ni) E Ml @ N, f(ml + M2) ni + N2}. (Given W and the orthogonal projections 7rM and 7rN for M ED N, one checks that M1 = rM(W), M2 = M n W, N1 = 7rN(W), and N2 = N n W.) (=>) If (ml,ni) E W and (rmi,sni) E W, then f(rmi + M2) = sn1 + N2 = sf(ml + M2) so, since f is an R-isomorphism, (r s)(Ml/M2)= 0. (^=) Suppose (r s)(Ml/M2) = 0, and f: M1/M2 -* N1/N2 is an isomorphism and (m1,ri) E W. Then f(rmi + M2) = rf(ml + M2) = rnm + N2 = sn, + N2 so (rm1,sn1) E W. Now we are able to provide the promised characterization in terms of annihilators of subquotients of M and N. The left annihilator of a module M is ?R(M) {r E RI rM= O}. This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms REFLEXIVITY OF DIRECT SUMS 2857 Theorem 1.2. Let M and N be reflexive, and let {Xi I i E I} represent one copy of each of the (cyclic) common subquotients of RM and RN. Then M ED N is reflexive if and only if ?R(M) + tR(N) = iR(D xi) iEl Proof. ( ) Let r E iR(GiE' Xi). Then by Lemma 1.1, (r, 0) E alglat(M ED N). Thus, assuming that M ED N is reflexive, (r,0) = (s,s) and r = (r-s) + s EE R(M) + eR(N). Since always ?R(M) + ?R(N) C iR(@iEI Xi), the proof is complete. F We note that the proof (=>) above shows that if M e N is reflexive (regardless of reflexivity of M and N), then ?R(M) + ?R(N) = ?R(@iCI Xi). From the inclusions 0 C iR(M) + eR(N) C ?R(ED Xi) C iR(Xi) C R iCI we easily obtain the following two corollaries: Corollary 1.3. If M and N are reflexive and have a common faithful subquotient, then M D N is reflexive. Corollary 1.4. If RZ is semiperfect and M and N are reflexive and have no common composition factor, then M ffl N is reflexive. Proof. Suppose that e1, ..., em, em?1, ..., en is a complete set of primitive idempotents such that Rei/Jej is not a composition factor of M for i = 1, ..., m, and Rei/Jej is not a composition factor of N for i = m+ 1, ..., n. Then ?R(M) + iR( N) R. (See [1, Section 27].) Regarding the Deddens-Fillmore result, the algebra generated by a matrix over a field K, being isomorphic to a proper factor of the polynomial ring K[x], is an example of a split commutative uniserial algebra, i.e., a direct product of local uniserial rings. Since a module over a direct product of rings is reflexive precisely when its corresponding components are reflexive, to show that direct sums of reflexive modules are reflexive over such an algebra R, we may assume that R is a local uniserial algebra, so that the ideals of R are linearly ordered and every R-module is a direct sum of factors of R (see [1, Section 32]). In this case, R/IR(M) embeds in M and every subquotient of RM is a factor of R/lR(M). In the presence of these facts Theorem 1.2 yields the following corollary almost at once. We note, however, that [10, Theorem 1] is more general. This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms 2858 V. P. CAMILLO AND K. R. FULLER Corollary 1.5. The direct sum of a finite number of reflexive modules over a split uniserial algebra is reflexive. Proof. Since we may assume that the ideals of R are linearly ordered, given modules M and N, we may also assume that ?R(M) C ?R(N) C ?R(X) for every common subquotient X of M and N. So since R/lR(N) embeds in N, we see that both sides of the desired equality are equal to ?R(N). Next, as further applications of this theorem, we shall see that it yields particularly nice proofs of some key results in [10]. In [10], in order to show that direct sums of reflexive modules may fail to be reflexive over a split K-algebra R with radical J whose quiver contains a triple arrow, a pair of reflexive modules was constructed with diagrams (as in [8]) u V w M: al b\ cl and N: a, c x y z with a, b, c linearly independent elements of J \ J2. (The diagram indicates that au = x, aw = cw = z, etc.) Here ?R(M) + ?R(N) = ?R(M) + K(a c) + Kb, but the only common subquotients of M and N are simple modules, so ?R(G X =) =R(M) + J R ?R(M) + tR(N). iEI Thus the theorem shows that M D N is not reflexive. In the positive vein we shall employ the following lemma and Theorem 1.2 to obtain a simple proof of a key part of [10, Proposition 4]. Lemma 1.6. Let R be a split local K-algebra of dim(KR) < 3. If M is a faithful left R-module, then every proper cyclic module is isomorphic to a subquotient of M. Proof. (This proof is essentially contained in the proof of [10, Proposition 4].) If R is uniserial, then RR embeds in M, so we may assume that J2 = 0 and that dim(KJ) = 2. Let u E M\SocM and suppose that WR(u) : 0. Then, since u ? SocM, there is a b C J with ?R(u) = Kb. But then there is a v E M with bv z 0, and there is an a E J with Ka =R(V). Thus it follows that J = Ka DKb.

We prove a strong version of a theorem of Balcerzak-RoslanowskiShelah by showing, in ZFC, that there exists a simply definable Borel CT-ideal for which both the ccc and property (M) fail. The proof involves Polish group actions. Definition. A Borel ideal is an ideal I on a Polish space X with the following property: for all A E I, there exists a Borel subset B of X such that A C B and B E I. We consider three types of properties for a Borel ideal I on a Polish space X. (1) I has a H' definition if the following set is Hl: C.1 = {c c 2': c is a Borel code for a subset B, of X and B, c I}. (2) For K a cardinal, I satisfies the /c-cc if there does not exist a family of i I-almost disjoint Borel sets that are not in I. The w1-cc is usually called the ccc. (3) I satisfies property (M) if there exists a Borel-measurable function f : X -? 2W with f-1(y) ? I for all y E 2W. Property (M) was introduced in Balcerzak [1]. Obviously, an ideal satisfying property (M) violates the ccc. Both the above paper and the later paper of Balcerzak-Roslanowski-Shelah [2] are concerned with circumstances in which it is possible for both the ccc and property (M) to fail. We refer the reader to these two references for information on these properties-and several other properties of ideals. The latter paper contains the following result. Theorem 1 (Balcerzak-Roslanowski-Shelah [2, 5.6]). Assume that either CH fails or every Al set of reals has the Baire property. Then there exists a Borel a-ideal I*, containing all singletons, such that: (a) I* has a H1 definition; (b) the ccc fails for I*; (c) property (M) fails for I*; (d) I* satisfies the w2-cc. Andrzej Roslanowski presented this result in a seminar talk at Ohio State University in 1993, and asked whether or not it is provable in ZFC. I thank him for bringing this question to my attention, and I thank Ohio State for their support. Received by the editors October 23, 1998 and, in revised form, December 3, 1998. 2000 Mathematics Subject Classification. Primary 03E15.

In reply to a problem posed by Jean Leray in 1950, a nontrivial example of application of the Nielsen fixed-point theory to differential systems is given. So the existence of two entirely bounded solutions or three periodic (harmonic) solutions of a planar system of ODEs is proved by means of the Nielsen number. Subsequently, in view of T. Matsuoka's results in Invent. Math. (70 (1983), 319-340) and Japan J. Appl. Math. (1 (1984), no. 2, 417434), infinitely many subharmonics can be generically deduced for a smooth system. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal. 1. HISTORICAL REMARKS Unlike any standard fixed-point principle, Nielsen theory gives us additional information about the lower estimate of the fixed points number. It was originated by the Danish mathematician Jakob Nielsen in 1927 [N], who studied with this respect self-maps of compact surfaces. Later on, finite polyhedra were systematically treated by FRanz Wecken in 1941-42 [W], who also gave an alternative definition of the Nielsen relations which we use below. The crucial step in this development was also accomplished by Andrzej Granas who defined a strictly related fixed-point index for arbitrary (i.e. also infinite-dimensional) ANRs in 1972 [G]. Thus, a sufficiently general fixed-point principle, preserving the number of essential classes of fixed points under homotopy, could be formulated by U. K. Scholz [S], Boju Jiang [Bol], R. F. Brown [Brl], and some others (see the references in [AGJ1], [Bol], [Brl]). By sufficiency, we mean the appropriate form which was suitable for answering the question posed already by Jean Leray [L] at the first International Congress of Mathematicians held after World War II in Cambridge, Mass., in 1950. Namely, he suggested the problem of adapting the Nielsen theory to the needs of nonlinear analysis and, in particular, of its application to differential systems for obtaining multiplicity results (for more details see e.g. [Br3], [Br4]). Since that time only several papers have been devoted to this problem (see e.g. [BKM], [Br2], [Br3], [Br4], [Fl], [F2]), but either additional parameters had to be Received by the editors May 4, 1998 and, in revised form, November 6, 1998. 1991 Mathematics Subject Classification. Primary 34B15, 47H10.

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyriere computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Holder exponent of the initial measure which is the critical parameter in the Peyriere theory. We start by defining the space of Galton-Watson trees and introducing some related terminology. 1. GALTON-WATSON TREES Let Tr be the space of labelled tree graphs rooted at 4. An element r of ?7 is a set of finite sequences of positive integers (v1, v2,... , Vn) E r such that: (i) X5E r is coded as the empty sequence. (ii) If (Vl,...,Vk) ET, then (vl,...,vi) ETV 1 1. The countable dense subset To of finite labelled tree graphs rooted at 0 makes Tr a Polish space. An important class of probability distributions on the Borel sigma field B(T) of Tr for this paper is the Galton-Watson distribution with single progenitor and offspring distribution Pk, k = 0, ,.. ., for which the probability Received by the editors May 14, 1998 and, in revised form, October 8, 1998. 2000 Mathematics Subject Classification. Primary 60G57, 60G30, 60G42; Secondary 60K35. @)2000 American Mathematical Society

This paper introduces two main concepts, called a generalized Watson transform and a generalized skewWatson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

This paper presents an indecomposable finite-dimensional division algebra of p-power index that decomposes over a prime-to-p degree field extension, obtained by adjoining p-th roots of unity to the base. This shows that the theory of decomposability has an arithmetic aspect. Suppose F is a field and D is an indecomposable F-division algebra, that is, a division algebra that cannot be expressed as the tensor product of two nontrivial F-division algebras. It is easy to see that the (Schur) index of D must be a power of some prime p. In "Problem 6" of [Sa], Saltman asks if in general D remains indecomposable upon arbitrary prime-to-p extension. At issue is the nature of indecomposability, in particular whether or not it is "geometric". For example in [K], Karpenko showed a certain generic class of division algebras are indecomposable by computing the degrees of cycles on their Brauer-Severi varieties. As noted in [K], it is immediate from the geometric nature of the proof that these algebras remain indecomposable over all prime-to-p extensions. This paper presents an indecomposable division algebra that decomposes over a prime-to-p extension, namely the cyclotomic extension defined by p-th roots of unity. Thus it is proved that (in)decomposability can have an arithmetic aspect. Let p be an odd prime of Q, let k be a number field that does not contain a pth root of unity, and let k[s, t] be the polynomial ring in two variables over k. Define v :k[s, t] -+Z fDlZ, f --* (a, b) where b is smallest such that f E (tb) and a is smallest such that f E (Sa,tb+l). The map v is a valuation, with value group Z 20 Z ordered reverse lexicographically, so (a, b) < (a', b') if b < b', or if b = b' and a < a'. The field of iterated power series F =k((s))((t)) is Henselian with respect to v, with valuation ring R = k[[s]] +tk((s))[[t]] c k((s))[[t]] . R is a non-Noetherian 2-dimensional Henselian local ring, with maximal ideal (s) = (s, t) and residue field k. The ideal (s) properly contains the (infinitely generated) prime ideal tk((s))[[t]] = (t, t, I,. t Received by the editors June 10, 1998 and, in revised form, October 6, 1998. 1991 Mathematics Subject Classification. Primary 16K20; Secondary 1R37. ?)2000 American Mathematical Society

In this paper we present results on existence and regularity of integral representations of feedback operators arising from parabolic control problems. The existence of such representations is important for the design of low order compensators and in the placement of sensors. This paper extends earlier results of J. A. Burns and B. B. King to problems with N N spatial dimensions.

Kadets and Lyubich have shown that a weakly almost periodic representation of a topological group on a Banach space has the property that a vector with almost periodic coefficients is almost periodic. In this paper we obtain the analogous result for distal representations of semigroups on quasi-complete locally convex spaces, removing the condition that the representation be weakly almost periodic and requiring only that the coefficients of the representation be weakly almost periodic.

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n., Comment: 10 pages, to appear in PAMS

[EN] A group G is said to be an involutive Yang¿Baxter group, or simply an IYB-group, if it is isomorphic to the permutation group of an involutive, nondegenerate set-theoretic solution of the Yang-Baxter equation. We give new sufficient conditions for a group that can be factorised as a product of two IYB-groups to be an IYB-group. Some earlier results are direct consequences of our main theorem., The research of this paper was supported by the grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades and the Agencia Estatal de Investigacion, Spain, and FEDER, European Union, and by the grant PROMETEO/2017/057 from the Generalitat, Valencian Community, Spain. The first author was supported by the predoctoral grant 201606890006 from the China Scholarship Council. The fourth author was supported by a predoctoral grant from the "Atraccio del talent" Programme from the Universitat de Valencia.

In this paper we show that given any compact set $E \subset \hat{\mathbb{C}}$, we can always find a conformally removable subset with the same Hausdorff dimension as $E$.

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups., Comment: 13 pages, 3 figures

In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.

In the space I r I_r of invariant r r -dimensional subspaces of a null system in ( 2 r + 1 ) (2r+1) -dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections φ : I r → I r \varphi :I_r\to I_r , for which both φ \varphi and φ − 1 \varphi ^{-1} preserve adjacency. In the present paper we show that the two conditions φ : I r → I r \varphi :I_r\to I_r is a surjection and φ \varphi preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.

We study in one dimension the semi-classical limit of the exact eigenfunctionΨE(N,h)h\Psi _{E(N,h)}^{h}of the HamiltonianH=−12h2Δ+V(x)H=-\frac {1}{2} h^{2} \Delta +V(x), for a potentialVVbeing analytic, bounded below andlim|x|→∞V(x)=+∞\lim _{|x|\to \infty }V(x)=+\infty. The main result of this paper is that, for any givenE>minx∈R1V(x)E>\min _{x\in R^{1}} V(x)with two turning points, the exactL2L^{2}normalized eigenfunction|ΨE(N,h)h(q)|2|\Psi ^{h}_{E(N,h)}(q)|^{2}converges to the classical probability density, and the momentum distribution|Ψ^E(N,h)h(p)|2|\hat \Psi ^{h}_{E(N,h)}(p)|^{2}converges to the classical momentum density in the sense of distribution, ash→0h\to 0andN→∞N\to \inftywith(N+12)h=1π∫V(x)>E2(E−V(x))dx(N+\frac {1}{2} )h =\frac {1}{\pi } \int _{V(x)>E} \sqrt {2(E-V(x))}dxfixed. In this paper we only consider the harmonic oscillator Hamiltonian. By studying the semi-classical limit of the Wigner’s quasi-probability density and using the generating function of the Laguerre polynomials, we give a complete mathematical proof of the Correspondence Principle.

In this paper we present two automatic surjectivity results concerning ring homomorphisms between p-classes of an H*-algebra which, in some sense, improve the main theorem in a recent paper by the author (Proc. Amer. Math. Soc. 124 (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups G, G', no quotient ring of L2(G) is isomorphic to LP(G') (2 < p < oo), a statement we conjecture to be true for every pair LP(G), Lq (G') of group rings corresponding to different exponents 1 < p, q < oo. The study of ring homomorphisms between Banach algebras has a long history. Probably the first such result is due to Eidelheit [Eid] who proved that every ring isomorphism between the algebras of all bounded linear operators acting on real Banach spaces is implemented by an invertible operator and hence it is continuous and linear. For the case of complex spaces and a recent far-reaching generalization concerning not only ring but semigroup isomorphisms of standard operator algebras we refer to [Arn] and [Sem], respectively. Moreover, in the case of general semisimple Banach algebras, the most important result on the linearity of ring isomorphisms is due to Kaplansky [Kap]. Motivated by the theory of operator ranges, in our papers [Mol2], [Mol3] we studied homomorphism ranges and considered surjective and "almost" surjective ring homomoprhisms (continuity is never assumed) between some operator algebras and function algebras, respectively. As the main corollary in [Mol3], we obtained the existence of ring theoretical differences among some important function algebras. The result of [Mol2] is in a close relation to our present investigations. It states that there is no surjective ring homomorphism between different p-classes of an infinite-dimensional H*-algebra. These classes are common generalizations of the p-classes of compact operators and Ip spaces. Our mentioned result shows that Received by the editors March 4, 1997 and, in revised form, March 10, 1998. 1991 Mathematics Subject Classification. Primary 46K15, 47D50, 47B49; Secondary 43A15, 43A22.

We consider the problem of the classification of semisimple Hopf algebras A of dimension pq where p < q are two prime numbers. First we prove that the order of the group of grouplike elements of A is not q, and that if it is p, then q = 1 (mod p). We use it to prove that if A and its dual Hopf algebra A* are of Frobenius type, then A is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension 3p, and a partial classification in dimensions 5p and 7p. In this paper we consider semisimple Hopf algebras of dimension pq over an algebraically closed field k of characteristic 0, where p and q are distinct prime numbers. Masuoka has proved that a semisimple Hopf algebra of dimension 2p over k, where p is an odd prime, is trivial (i.e. is either a group algebra or a dual of a group algebra) [Mal]. Izumi and Kasaki have proved that Kac algebras (i.e. semisimple Hopf algebras over the field of complex numbers, with an additional condition on the existence of an involution), of dimension 3p over k, where p is prime, are trivial [IK]. Thus, a natural conjecture is: Conjecture 1. Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is trivial. A well known property of A, a finite dimensional semisimple group algebra or a dual of a group algebra, is that it is of Frobenius type; that is, the dimension of any irreducible representation of A divides the dimension of A (the definition is due to Montgomery [Mo]). A special case of Kaplansky's 6th conjecture [K] is: Conjecture 2. Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is of Frobenius type. In this paper we prove among the rest that Conjecture 1 is equivalent to Conjecture 2 (see Theorem 3.5). A major role in the analysis is played by G(A) (where G(A) denotes the group of grouplike elements of A). By [NZ], IG(A)I is either l,p, q or pq. We prove in Theorem 2.1 that if p < q, then IG(A)| /: q, and if [G(A)| = p, then q = 1 (modp). Consequently, we prove in Theorem 2.2 that if IG(A)| 71 and q 1 (modp), then A is a commutative group algebra. Received by the editors August 1, 1997 and, in revised form, March 17, 1998. 1991 Mathematics Subject Classification. Primary 16W30. The second author's research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. (1999 American Mathematical Society

Let F G FG be the group ring of a group G G over a field F F , with characteristic different from 2 2 . Let ∗ \ast denote the natural involution on F G FG sending each group element to its inverse. Denote by ( F G ) + (FG)^{+} the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided G G has no 2 2 -elements, if ( F G ) + (FG)^{+} is Lie nilpotent, then so is F G FG . In this paper, we determine when ( F G ) + (FG)^{+} is Lie nilpotent, if G G does contain 2 2 -elements.

In this paper, we present some relations between generalized distributivity of quotient algebras and Mahlo operations, and show that the distributivity implies some variants of stationary relections. The notion of generalized distributivity appeared in [3] for the first time. In that paper, we began investigating the relations between the notion and other fundamental principles in set theory, especially in the area called large cardinal axioms. For example, we showed in [3] that: Let I be any non-trivial es-complete ideal on a regular uncountable cardinal re and A any cardinal with 2 es. In this paper, we shall show that some versions imply some variants of stationary reflections. We shall organize our paper as follows: In ?1, we shall introduce notations and terminologies, and several definitions of distributivity will be given. In ?2, we shall investigate distributive conditions which imply some closedness of the Mahlo operation. Our main result in this section is the following: Theorem A. Let a be any function of S into p(T) such that for a, b in S, /bta jp(c(a))I No } has positive Imeasure, {a E S: t E a(a) } has I-measure one for each t E T and if g is a function on A E I+ with g(a) E a(a), then there exists a subset B of A of positive I-measure such that g[B is constant. Then if X is a

In this paper we prove the following result: Let A A be a nonunital Banach algebra with a bounded approximate identity. Then A A cannot be both Arens regular and weakly sequentially complete. The paper also contains some applications of this result.

In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type (p, p) on LP(lRTn) if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery's result is sharp in a certain sense. We also obtain a weighted analogue of Carbery's result. Some applications of our results are also given.

We derive sharp upper bounds for eigenvalues of the Laplacian under Neumann boundary conditions on convex domains with given diameter in Euclidean space. We use the Brunn-Minkowski theorem in order to reduce the problem to a question about eigenvalues of certain classes of Sturm-Liouville problems. 1. STATEMENT OF THE RESULTS Let Q be a bounded convex domain in Euclidean space Rn. By considering a manifold obtained by glueing two copies of Q together, it is possible to show that the upper bounds obtained by S.-Y. Cheng in [3] for eigenvalues of the LaplaceBeltrami operator on a Riemannian manifold yield upper bounds for the eigenvalues of the Neumann Laplacian on a bounded convex region in Euclidean space (see [5], Section 6). Our goal in this paper is to improve those bounds. It remains an open question whether our results can be extended to Riemannian manifolds. The author was unable to replace the argument in the proof of Theorem 1 using the Brunn-Minkowski theorem by arguments related to the Bishop comparison theorem without strong additional assumptions on the cut loci of the end points of a diameter of the manifold and for other than two-dimensional manifolds (cf. section III.3 in [2] for Bishop's comparison theorem and related material). We consider in this paper the eigenvalue problem AO+ ,a = 0 in Q, aq = 0 on OQ. The spectrum of this problem is discrete and the corresponding eigenvalues can be increasingly ordered as follows: 0 = -o(Q) < /,1 (Q) ?< 22(Q) < .... We obtain our bounds by comparison with eigenvalues of balls in Euclidean space under Dirichlet boundary conditions. It is well known that the Dirichlet eigenvalues for balls can be given explicitly in terms of zeros of Bessel functions. The positive zeros of the Bessel function J, of order v are in increasing order j,,1 < j,,2 < ... (see [1], Table 9.5 for a table of numerical values). The following theorem gives an upper bound for j,am(Q) in terms of the diameter of 2. Received by the editors May 1, 1997 and, in revised form, September 2, 1997. 1991 Mathematics Subject Classification. Primary 35P15; Secondary 58G25. ?1999 American Mathematical Society

It is shown that if the Banach space operator T has regular normsequence, its vector orbits are asymptotically non-vanishing and there exists a complete vector orbit satisfying the growth condition of non-quasianalycity, then T has infinitely many disjoint hyperinvariant subspaces. In the recent paper [7] we initiated the study of operators possessing regular norm-sequences. The definition was based on the concept of almost convergence. We recall that a bounded, complex sequence S E '??(N,C) almost converges to the complex number c if limk,oo supneN Ic k-l Z+k 1 (i)I = 0. (The reader is referred to [9] for fundamental properties of this type of convergence.) We say that the sequence J almost converges to c in the strong sense if | cll almost converges to zero, where 1 stands for the constant 1 sequence. A gauge function is a mapping p : N -* (0, oo) with the property that {p(n + 1 )/p(n)}nEN almost converges in the strong sense to a positive number c. The set of all gauge functions is denoted by 7P. Let us consider now a complex Banach space X and let L(X) denote the set of bounded, linear operators acting on X. We say that the norm-sequence of an operator T E L(X) is compatible with the gauge function p E P, if ITnll < p(n) holds for every n E N and the sequence {lTnll/p(n)},EN does not almost converge to zero. The set of all such operators is denoted by L(p, X), and T E X is said to have a regular norm-sequence if T belongs to L(p, X) for some p E P. It was shown in [7] that {p(n + l)/p(n)}nEN almost converges to the spectral radius r(T) for every T E ?(p, X), and that a large class of operators with positive spectral radius possesses regular norm-sequence, including power bounded operators of spectral radius one. With regard to the asymptotic behaviour of vector orbits the following subclasses can be introduced. The operator T E ?(p, X) belongs to the class Ci.(p, X) (C*.(p, X)) if {llTnsll/p(n)}nEN does not almost converge to zero for all non-zero vectors x E X (for some non-zero vector x E X, respectively). In the present paper we are concerned with the hyperinvariant subspace lattice Hlat(T) of operators T of class C1. (p, X) and C*. (p, X). We remind the reader that a (closed) subspace M is hyperinvariant for T, and so it belongs to Hlat(T), if CM C M holds for every operator C commuting with T. In [7] we proved among other things the existence of a proper hyperinvariant subspace under different conditions, Received by the editors August 6, 1997. 1991 Mathematics Subject Classification. Primary 47A15, 47A60. Research partially supported by Hungarian NFS Research grant no. T 022920. (1999 American Mathematical Society

We investigate the relationships between the notions of a continuous function being monotone on no interval, monotone at no point, of monotonic type on no interval, and of monotonic type at no point. In particular, we characterize the set of all points at which a function that has one of the weaker properties fails to have one of the stronger properties. A theorem of Garg about level sets of continuous, nowhere monotone functions is strengthened by placing control on the location in the domain where the level sets are large. It is shown that every continuous function that is of monotonic type on no interval has large intersection with every function in some second category set in each of the spaces pn, Cn, and Lip1. 1. NONMONOTONICITY PROPERTIES In a series of interesting papers [4], [5], [6], [7], [8], Garg investigated level set structures and derivate structures of continuous functions. This investigation was continued in a paper by Bruckner and Garg [2]. These articles considered several notions that measure degrees of pathology in the class of continuous, nowhere monotone functions. In this section we further study the relationships among these properties. We use C and BV to denote the collections of functions from [0, 1] into R, the reals, that are continuous and of bounded variation, respectively. Df (x) and Df (x) denote the lower and upper (two-sided) Dini derivates, respectively, of a function f at a number x (see [1]). We use standard terms such as perfect sets, first category sets, sets with the Baire property, etc., whose definitions may be found in [9]. Following Bruckner and Garg [2], we say that a function f is nondecreasing at x if f(t)-f(x) > 0 for all t $& x in some neighborhood of x. That f is nonincreasing t-xat x is defined with the obvious modification. If f is either nonincreasing at x or nondecreasing at x, then we say that f is monotone at x. That f is nonmonotone at x means that f is not monotone at x. If f is a function and m E R then, following Garg [8], denote by f+m and f-r the functions defined by f+r(x) = f (x) + mx and f-m(x) = f(x) mx. Inclusion of the "+" and "-" avoids confusion with Received by the editors August 20, 1996 and, in revised form, May 7, 1997. 1991 Mathematics Subject Classification. Primary 26A48; Secondary 26A24.

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.

We give conditions on a couple of ideal Banach spaces with weights which are both necessary and sufficient for the Hardy-Littlewood maximal function to satisfy the two-weighted estimations of weak type, and we consider a modification of the Hardy-Littlewood maximal function. We also give some conditions on weights in order for the Hardy-Littlewood maximal function and the modification under consideration to fulfil the two-weighted estimations of strong type. 0. INTRODUCTION The general problem of two-weighted estimations of weak type in LP-spaces for the Hardy-Littlewood maximal function has been solved in [9]. In the case of Lorentz spaces Lpq a corresponding result has been obtained in [2, 7] (see also [3, 4]). In order to obtain a positive answer the AP-condition for weights is required in both cases mentioned. The problem of two-weighted estimations of strong type for the Hardy-Littlewood maximal function in LP-spaces has been considered in [12]. The solution is boundedness of the operator of the Hardy-Littlewood maximal function on the set of the characteristic functions of cubes. Simpler conditions, necessary and sufficient on a certain class of weights, have been considered by many authors (see for example [3] or [4]). Among recent papers we should mention [11], where, for a large class of weights, there are given necessary and sufficient conditions for boundedness of the weighted fractional maximal operator. In the present paper we give necessary and sufficient conditions for two-weighed estimations of weak type for the Hardy-Littlewood maximal function and a modification in ideal Banach spaces. We also give some conditions on weights assuring two-weighted estimations of strong type for the Hardy-Littlewood maximal function and a modification. The obtained results are new even in the case of Orlicz spaces. 1. PRELIMINARIES Let S = S(R', ,u) be the space of all Lebesgue measurable real-valued functions on R' endowed with the topology of convergence in measure on each set of finite measure. Recall that a Banach subspace X in S is said to be ideal [8] if x E X, Received by the editors December 11, 1991 and, in revised form, September 19, 1996. 1991 Mathematics Subject Classification. Primary 42B20, 42B25. ?)1999 American Mathematical Society 79 This content downloaded from 157.55.39.144 on Mon, 25 Jul 2016 03:48:01 UTC All use subject to http://about.jstor.org/terms

In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209–212) for the proof of the conjecture on the maximum number of invariant straight lines of cubic and quartic polynomial differential systems; and also prove the conjecture in a previous paper of the second author (Qualitative theory of polynomial differential systems, Shanghai Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain special case of the n n degree polynomial systems. Furthermore, we will prove that cubic and quartic differential systems have invariant straight lines along at most six and nine different directions, respectively, and also show that the maximum number of the directions can be obtained.

The work in this paper is a continuation of an earlier paper of the second author (Acta Math. 34 (1991), 191–202). We discuss the properties of finite-dimensional complete Lie algebras with abelian nilpotent radical over the complex field C \mathbf {C} . We solve the problems of isomorphism, classification and realization of complete Lie algebras with commutative nilpotent radical.

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces. Let Tm be a multiplier operator defined in terms of Fourier transforms by Tmf Q) = M(W for suitable functions f. It is well-known that there is a multiplier theorem for H1 (Rn) (see [FS]): if a > n/2 and (1) J ID3m(e)12d d 0, let us denote m6QW) = M(6077(0) It is easy to check that (1) is equivalent to (2) sup |m6||Ka 2(Rn) < 0, 6 2 where K2'2 2(R) is a non-homogeneous Herz space (see [BS]). By using some embedding relations on Herz spaces, A. Baernstein II and E. T. Sawyer [BS] weakened (2) into (3) sup iM6i||Kf1l (Rn) < 0, 6 where 0 < E < an2. In fact, this is just a special case of their theorem. In [BS], Baernstein and Sawyer showed that m is a bounded multiplier of H1 (Rn) under an even weaker condition than (3); see Theorem 3b in [BS, page 21]. By using the technique of Herz type Hardy spaces developed by the authors in [LY1]-[LY3] and [Y], in this paper, we shall first establish a multiplier theorem for the homogeneous Herz type Hardy space Hk n(1-1/q), 1(IRn) which is introduced by y ~~~q the authors of this paper in [LY1]. Then as simple consequences of this theorem, a multiplier theorem for the corresponding non-homogeneous version of the space Received by the editors April 13, 1995 and, in revised form, April 5, 1997. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 42B30.

We compute the periodic cyclic cohomology Chern character of an admissible pseudomanifold Xf with one singular stratum. As a corollary, we obtain the index theorem and spectral flow for signature operators. 0. INTRODUCTION This paper is the sequel to [Chan]. In that paper, by considering the "straight" Chern character, we gave a de-Rham type realization of the Goresky-MacPhersonSiegel L-classes of admissible pseudomanifolds with one singular stratum Xt M U (ct (L) x N) such that 2L has zero oriented cobordism, and we also obtained the index theorem for the twisted signature operators on Xt. In this paper, we will continue to study Xt by using non-commutative geometry. We will compute (Theorem 3.1 and Theorem 3.3) the corresponding periodic cyclic cohomology Chern character by using the infinite temperature limit formula in [CoM]. As in [Chan], we will choose a scaling in the conical direction such that the signature operator is essentially self-adjoint. Also, by using a singular elliptic estimate, we handle the calculation on the singular part by that on model space c0,0, (L) x N. We finish the computation by Getzler's calculus. As a consequence, we recover the index theorem for twisted signature operators on Xt [Chan, Theorem 4.1] in the even case. In the odd case, we obtain (Corollary 3.4) the spectral flow for signature operators on admissible spaces with conical singularity. 1. PRELIMINARIES To fix the notation, let us recall some definitions in [Chan]. Let M be a smooth, oriented, compact and connected m-dimensional manifold with boundary O9M = L x N where L and N are smooth, oriented, closed and connected manifolds of dimensions ? and n respectively. Let c(L) -= (0, 1) x L and ct (L) = [0, 1) x L/{O} x L be the cone and completed cone with link L respectively. Then Xt = MU (ct (L) x N) is called a pseudomanifold with one singular stratum [Chan]. We define a metric g on Xt such that (i) g is a measurable metric on Xt; (ii) glM is a smooth metric on M and is a product near OM; Received by the editors August 30, 1996. 1991 Mathematics Subject Classification. Primary 19D55; Secondary 58G12. ?1998 American Mathematical Society 669 This content downloaded from 157.55.39.127 on Wed, 29 Jun 2016 04:18:19 UTC All use subject to http://about.jstor.org/terms

The purpose of the present paper is to prove that the topological degree of the subdifferential of a coercive lower semi-continuous function on a sufficiently large ball in a reflexive Banach space is equal to one. A well known result of Krasnoselskii [7] says that the degree of the gradient of a coercive potential mapping on a sufficiently large ball in RN is equal to one. As Nirenberg pointed out in [8], this result seems intuitively clear, but the proof is not. Many authors had new views of this theorem. See [1], [9]-[11]. The aim of this note is to give a generalization of the degree of a maximal monotone mapping and prove that the topological degree of the subdifferential of a coercive convex function is equal to one on a sufficiently large ball. The main result of this paper is Theorem 5. Its proof is very simple and elementary. Let us recall the properties of the degree of mappings of class (S+), as introduced by Browder [4]. See also [3], [6]. Let E be a real reflexive Banach space of dual E* . We can always assume that both E and E* are locally uniformly convex [12]. The duality pairing between E* and E is denoted by (.,.). Let 0* be the zero element of E*, J : E -E* the duality mapping, Q C E an open bounded subset, and T: Q -E* a demi-continuous operator of class (S+) (see [4] for a definition); if 0* 0 T(Q9), then there is a topological degree deg(T, Q, 0*) satisfying Theorem 1. (a) deg(J, Q, 0*) = { 0 ifo* E J(Q). (b) If deg(T, Q, 0*) $7 0, then Tx = 0* has a solution in Q. (c) If Q1, Q2 are two disjoint open subsets of Q, then deg(T, Q, 0*) + deg(T,Q2,0*) = deg(T, Q, U Q2, 0*). (d) If {Tt} o,1] is a homotopy of class (S+), and 0* f TtQ9Q), then deg(T, Q, 0*) doesn't depend on t E [0, 1]. Now, suppose that A: D(A) C E -2 is a maximal monotone operator. Let AA = (AJ-1+A-')-', RA = I-AJ-1AA be its Yosida approximation and resolvent Received by the editors February 25, 1997. 1991 Mathematics Subject Classification. Primary 47H10, 47H15; Secondary 55M25.

The paper gives an upper bound for the valence of harmonic polynomials. An example is given to show that this bound is sharp. Interest in harmonic mappings in the complex plane has increased due to the publication of [1] in 1984. Most of this interest has centered on univalent harmonic mappings. It is, however, the purpose of this paper to present the fundamental valence results of complex valued polynomials which are harmonic in the plane. Specifically a harmonic polynomial is a function P(z) = S(z) + T(z) where S(z) and T(z) are analytic polynomials. The degree of P is defined as the larger of the degrees of S and T. Lyzzaik studied the local topology of harmonic mappings in [4]; in particular he put a bound on the local valence of P but did not obtain a global valence bound. In this paper it will be shown that, apart from certain degenerate examples, P(z) is at most n2-valent, where n is the degree of P. An example is exhibited to show this to be the sharp bound. To prove the n2 valence bound we need the following classical result from algebraic geometry; a proof may be found in [2]. Theorem 1 (Bezout's Theorem). Let A and B be relatively prime polynomials in the real variables x and y with real coefficients, and let deg A = n and deg B = m. Then the two algebraic curves A(x, y) = 0 and B(x, y) = 0 have at most mn points in common. To apply Bezout's theorem to the valence problem we restate it in an equivalent form. Theorem 2 (Alternate form of Bezout's Theorem). Let A and B be polynomials in the real variables x and y with real coefficients. If deg A = n and deg B m, then either A and B have at most mn common zeros or have infinitely many common zeros. Proof. If A and B are relatively prime, then, by Bezout's theorem, the polynomial equations A(x, y) = 0 and B(x, y) = 0 have at most mn common zeros. Otherwise A and B are not relatively prime and they have a greatest common factor, say C. Let deg C = q. If C(x, y) = 0 has no solutions, then A and B have at most (n q)(m q) common zeros. But if there exists a point on C(x, y) = 0 with either Oc 74 0 or 4c 04 o, then, by the implicit function theorem, C(x, y) = 0 Received by the editors June 26, 1995 and, in revised form, September 20, 1996 and January 2, 1997. 1991 Mathematics Subject Classification. Primary 30C55. (?)1998 American Mathematical Society

Let I ⊂ R I\subset \mathbb {R} be an interval and let f f , φ \varphi be arbitrary elements of H 1 ( I ) H^1(I) and B M O ( I ) BMO(I) , respectively, with ∫ I φ = 0 \int _I\varphi =0 . The paper contains the proof of the estimate ∫ I f φ ≤ 2 ‖ f ‖ H 1 ( I ) ‖ φ ‖ B M O ( I ) , \begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*} and it is shown that 2 \sqrt {2} cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.

The semi-dynamical system of a continuous self-map is generated by iteration of the map, however, the iteration itself, being an operator on the space of continuous self-maps, may generate interesting dynamical behaviors. In this paper we prove that the iteration operator is continuous on the space of all continuous self-maps of a compact metric space and therefore induces a semi-dynamical system on the space. Furthermore, we characterize its fixed points and periodic points in the case that the compact metric space is a compact interval by discussing the Babbage equation. We prove that all orbits of the iteration operator are bounded but most fixed points are not stable. On the other hand, we prove that the iteration operator is not chaotic.

Exceptional Laguerre and Jacobi polynomials p n ( x ) p_n(x) are bispectral, in the sense that as functions of the continuous variable x x , they are eigenfunctions of a second order differential operator and as functions of the discrete variable n n , they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.

By a finite quantum group, we will mean in this paper a finitedimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a *-algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

The least cardinal for which the Baire category theorem fails is equal to the least cardinal for which a Ramseyan theorem fails. The Baire category theorem states that the real line is not the union of countably many meager (also known as first category) sets. Let cov(M) denote the least cardinal number such that there are that many first category subsets of the real line whose union is the entire real line. Then cov(M) is the least cardinal number for which the Baire category theorem fails. This cardinal number, defined in terms of topological notions, appears in many different guises in combinatorial set theory. A long list of diverse guises of this cardinal number is already general knowledge for set theorists; Galvin gave a game-theoretic version (part of which is published in [4], and part of which is unpublished—however, see [8]), A. W. Miller gave a characterization in terms of sequences of positive integers [6], which was later given an elegant improvement by Bartoszynski [1]. It is also known to be the least cardinal number such that there is a set of real numbers of that cardinality which does not have Rothberger’s property C′′. A set X of real numbers has property C′′ if, for every sequence (Un : n = 1, 2, 3, . . . ) of open covers of X , there is a sequence (Un : n = 1, 2, 3, . . . ) such that, for each n, Un ∈ Un and {Un : n = 1, 2, 3, . . .} is a cover for X . It seems that for the purposes of applications of set theory to other areas of mathematics, it would be useful to have as many non-trivial characterizations of this cardinal number as possible. In this paper we give a few more equivalent forms of this cardinal number. To explain some of our results, we need some terminology which is well-known in other contexts. Let κ be an infinite cardinal number which will be fixed for the duration of the paper. A collection of subsets of κ is said to be a cover of κ if its union is equal to κ. We shall be interested in countable covers of κ. A cover of κ is said to be an ω-cover if it is countably infinite, κ itself is not a member of the cover, and if there is for every finite subset of κ an element of this cover which contains it. We shall let the symbol Ω denote the collection of ω-covers of κ. Borrowing from Ramsey theory (see Section 8 of [3]), we shall use the symbol

We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert C∗-modules. In this context we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces. The notion of Morita equivalence of twisted coactions was introduced in [B]. In [B, Theorem 3.3] we established conditions on twisted coactions which are sufficient to ensure Morita equivalence of the corresponding crossed product C∗-algebras. Later [ER] gave a shorter proof for this result using their results on multipliers of imprimitivity bimodules. However in the proofs of both [B] and [ER], all C∗algebras and Hilbert C∗-modules need to be represented on Hilbert spaces. In this paper we present a new proof for [B, Theorem 3.3] based on the notion of crossed products of Hilbert C∗-modules introduced in [B2]. Crossed products of Hilbert C∗-modules in [B2] were defined as subspaces of adjointable operators between Hilbert C∗-modules. In this abstract context, we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces as in [B] and [ER]. As a consequence, the proof here is shorter and more elegant than that of [B]. Our approach is close to the spirit of [BS], and different from [ER]. Throughout this paper G is a locally compact group and N is a closed normal amenable subgroup of G. Recall from [M, Lemma 3] that there is a surjective homomorphism Ψ from C∗ r (G) into C ∗ r (G/N) such that Ψ(λ (r)) = λ (qN (r)), where q N : G → G/N is the quotient map, λ and λ are the left regular representations ofG andG/N . We denote by WG the unitary operator on L 2(G×G) defined by [WGξ](r, s) = ξ(r, r −1s). If f is an element of the Fourier algebra A(G), then Sf (WG) = Mf . Here Sf denotes the slice map, see [LPRS, §1]. To apply [B2, Theorem 1.6] to this paper, we need to show that WG is a regular multiplicative unitary. For any ξ, η ∈ L(G), we define ωη,ξ = 〈Tξ|η〉, ∀T ∈ B(L(G)). Then for any ω = ωη,ξ, we have 〈(id⊗ ω)(WG)ξ′|η′〉 = 〈Mω◦λGξ′|η′〉, ∀ξ′, η′ ∈ L(G). It then follows that ŜWG = C0(G), and the crossed product of [B2, Proposition 1.5] is just the crossed product of [LPRS, Definition 2.4]. The unitary operator Received by the editors October 23, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 22D25. c ©1997 American Mathematical Society 2109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

In [14] a program was initiated for using the topological theory of 3-manifolds to obtain lower bounds for volumes of hyperbolic 3-manifolds. In [1], by a combination of new geometric ideas with relatively standard (but specifically 3-dimensional) topological techniques, we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is at least 3 has volume exceeding 0.92. By contrast, the best known lower bound [10,5] for the volume of an arbitrary closed hyperbolic 3-manifold is approximately 0.0012. In [3] we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is 2 has volume exceeding 0.34. The proof depended on supplementing the results and techniques of [1] with ingenious elementary arguments due to Zagier [11] and numerical computations. In the present paper we shall show that if one excludes certain special manifolds, such as fiber bundles over S 1, then the lower bound of 0.34 also holds for hyperbolic 3-manifolds with Betti number 1. The proof depends heavily on the results of [1] and [3], but it involves much deeper topological ideas than these papers. The new topological results needed for the proof occupy most of the present paper. To some extent these results have the flavor of general topology, but the proofs make use of such specifically low-dimensional techniques as the characteristic submanifold theory [9, 8], the interaction between trees and incompressible surfaces, and Scott's theorem [12] that surface groups are locally extended residually finite. Before giving a precise statement of our main result we must review a few elementary notions from 3-manifold theory.

In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union. The space of pairs of convex bodies has been investigated in a number of papers [3], [8], [9], and [12]. This space has found an application in quasidifferential calculus (cf. [1], [5], [7], [10]). A quasidifferential is represented as a pair of convex bodies and it is essential to find the minimal representation of this pair. The notion of minimal pairs was introduced in [5] and investigated in [2], [6], [7] and [11]. Some criteria of minimality are given in [6]. In this paper we investigate pairs of convex bodies with convex union. We introduced a quotient class of pairs of convex compact sets in which every member has convex union. Moreover some criteria for the convex class are given. In this paper X = (X, τ) stands for a real locally convex vector space, and X∗ denotes the dual space of X . Denote by K(X) the family of all convex bodies in X, i.e., of all nonempty compact convex subsets of X . If A,B are nonempty subsets of X , then A+B is the usual algebraic Minkowski sum of A and B. It may be showed that K(X) satisfies the order cancellation law; i.e. for every A,B,C ∈ K(X) the inclusion A + B ⊂ B + C implies A ⊂ C (cf. [12]).Hence it follows that K(X) endowed with the Minkowski sum is a commutative semigroup satisfying the law of cancellation. Now let K2(X) = K(X)×K(X); the equivalence relation between pairs of convex bodies is given by: (A,B) ∼ (C,D) if and only if A+D = B+C. For A,B ∈ K(X) we will use the notation A∨B := conv (A∪B), where the operation ”conv” denotes the convex hull. If A,B,C ∈ K(X), and b ∈ X , then A∨B+C = (A∨B)+C and A+ b = A+ {b}. We have [a, b] = {a} ∨ {b}. Let f ∈ X∗, A ∈ K(X) and c ∈ R. We denote by pA(f) := maxx∈Af(x) the support function of the set A. Moreover, H f := {x ∈ X | f(x) = c} and HfA := {x ∈ A | f(x) = pA(f)}, where H f is the hyperplane generated by the functional f and the number c, and HfA is the face of A with respect to f. For the sum of the faces of two convex bodies A,B ⊂ X with respect to f ∈ X∗ the identity Hf (A + B) = Hf + HfB holds true. For A ⊂ X we denote by ∂A the boundary Ā \Ao of the set A, where Ā := cl A and A := intA. Received by the editors June 12, 1996. 1991 Mathematics Subject Classification. Primary 52A07, 90C30, 26A27.

Let G be a discrete group and denote by λG its left regular representation on `2(G). Denote further by Fn the free group on n generators {g1, g2, . . . , gn} and λ its left regular representation. In this paper we show that a subset S = {t1, t2, . . . , tn} of G has the Leinert property if and only if for some real positive coefficients α1, α2, . . . , αn the identity ∥∥∥∥ n ∑ i=1 αi λG(ti) ∥∥∥∥ C∗ λ(G) = ∥∥∥∥ n ∑ i=1 αi λ(gi) ∥∥∥∥ C∗ λ(Fn) holds. Using the same method we obtain some metric estimates about abstract unitaries U1, U2, . . . , Un satisfying the similar identity ∥∥∥∥∑ni=1 Ui ⊗ Ui∥∥∥∥ min = 2 √ n− 1. Notation. Throughout in this paper G will denote a discrete group with unit element e and C∗ λ(G) the sub-C ∗-algebra of B(`2(G)) generated by its left regular representation λG. This algebra is equipped with the trace state τG(X) = 〈Xδe, δe〉. A subset {t1, t2, . . . , tn} of G is called free if it generates a copy of Fn, the free group on n generators. We shall denote the canonical generators of the free group by {g1, g2, . . . , gn}. When considering the free group we shall omit the subscript in λFn . We shall almost exclusively deal with finitely generated groups and repeatedly use the fact that given a generating set {h1, h2, . . . , hn} for the group G there is a (unique) quotient mapping q : Fn → G with q(gi) = hi ∀i ∈ {1, 2, . . . , n}. The following proposition collects some results from [K, Lemma 3.1 and Theorem 3]: Proposition 1. Let {h1, h2, . . . , hn} be a generating set of the group G and denote by α1, α2, . . . , αn positive real numbers. Then ∥∥∥∥ n ∑ i=1 αi (λG(hi) + λG(hi) ∗) ∥∥∥∥ C∗ λ(G) ≥ ∥∥∥∥ n ∑ i=1 αi (λ(gi) + λ(gi) ∗) ∥∥∥∥ C∗ λ(Fn) . Moreover in the case of equal coefficients, ∥∥∥∥ n ∑ i=1 (λG(hi) + λG(hi) ∗) ∥∥∥∥ C∗ λ(G) = ∥∥∥∥ n ∑ i=1 (λ(gi) + λ(gi) ∗) ∥∥∥∥ C∗ λ(Fn) = 2 √ 2n− 1 if and only if {h1, h2, . . . , hn} is a free set. Received by the editors February 22, 1996 and, in revised form, May 21, 1996. 1991 Mathematics Subject Classification. Primary 22D25; Secondary 43A05, 43A15, 60J15.

If a smooth manifold has a Riemannian metric with positive scalar curvature, it follows immediately that the universal covering also has such a metric. The paper establishes a converse if the manifold in question is closed of dimension at least 5 and the fundamental group is an elementary abelian p-group of rank 2, where p is an odd prime. A conjecture of J. Rosenberg [Rs3] states that a closed smooth manifold with odd order fundamental group and dimension ≥ 5 admits a Riemannian metric with positive scalar curvature if and only if its universal covering admits such a metric. Standard considerations involving transfers (cf. [KS1]) reduce the conjecture to the special case of p-groups (where p is an odd prime). Results of Rosenberg [Rs1]– [Rs3] and S. Kwasik and the author [KS1] prove the conjecture if the fundamental group G is a cyclic p-group. In this work we study the conjecture when G is a finite abelian p-group. The following interim conclusion reflects many of the basic ideas and disposes of the first examples not covered by [KS1], [Rs1]–[Rs3]. Theorem. Let p be an odd prime, and let M be a closed smooth manifold of dimension n ≥ 5 with fundamental group Zp × Zp. Then M admits a Riemannian metric with positive scalar curvature if and only if its universal covering M does. In [RsS] J. Rosenberg and S. Stolz prove a stable result that yields a weaker conclusion for G ∼= Zp × Zp but applies to all finite groups. In particular, if G is a finite group of odd order and M is a closed smooth manifold with fundamental group G, their result states that the universal covering M has a Riemannian metric with positive scalar curvature if and only if some product M ×X × · · · ×X does, where X is an 8-dimensional Bott manifold; i.e., it is a spin manifold whose Â-genus is equal to 1. Although the methods should yield quantitative information on the number of factors of X that are needed, it is not clear how precise such estimates would be. Our result implies that if G ≈ Zp×Zp, then no copies of X are needed if n ≥ 5 and at most one copy of X is needed if n < 5. A more geometric proof of this result in dimensions ≤ 2p− 2 was obtained independently by Stolz (unpublished). It seems likely that the methods of this paper can be combined with the multiple Kunneth formula decomposition of [Hn] and finite induction to prove at least a semistable version of Rosenberg’s conjecture for all elementary abelian p-groups Received by the editors February 13, 1995 and, in revised form, September 13, 1995. 1991 Mathematics Subject Classification. Primary 53C21, 55N15, 57R75; Secondary 53C20, 57R85. c ©1997 American Mathematical Society

In a recent paper an author has suggested a series of dimensions which include as first terms dimension of a vector space, Gelfand-Kirillov dimenision and superdimension. In terms of these dimensions the growth of free polynilpotent finitely generated Lie algebras has been specified. All these dimensions are integers. In this paper we study for all levels q = 2, 3 . . . what numbers α > 0 can be a q-dimension of some Lie (associative) algebra. Introduction: Series of dimensions Let A be a Lie (associative) algebra over a field K, generated by a finite set X ; in this case we write A = alg(X) (moreover, all algebras in this paper are finitely generated). Denote by A(X,n) the subspace spanned by all monomials in X of length not exceeding n (including identity if A is associative). Denote γA(n) = γA(X,n) = dimK A(X,n), λA(n) = γA(n)− γA(n− 1), where dimK stands for dimension of a vector space over K. Growth of a function γA(n) is an important characteristic of A. On functions f : N → R, where R={α ∈ R |α > 0}, we consider the following partial order: f(n) g(n) iff there exist N > 0, C > 0, such that f(n) ≤ g(Cn), n ≥ N . An equivalence f(n) ∼ g(n) means that f(n) g(n), f(n) g(n). It is easy to verify that for another generating set X ′ we have an equivalent function γA(X,n) ∼ γA(X ′, n). Sometimes it is better to use another partial order: f(n) g(n) iff there exists N > 0, such that f(n) ≤ g(n), n ≥ N . The growth less then exponential (in sense ) is called subexponential. If it is also greater than any polynomial growth, then it is called intermediate. To study the intermediate growth of Lie and associative algebras the following series of dimensions has been suggested [1], [2]. Denote by iteration lnn = lnn; lnn = ln(lnn), q = 1, 2, . . . . Received by the editors July 13, 1995. 1991 Mathematics Subject Classification. Primary 16P90, 17B30; Secondary 17B35. The author was partially supported by ISF grant M22000(1994). c ©1997 American Mathematical Society 649 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 650 V. M. PETROGRADSKY Consider a series of functions Φα(n), q = 1, 2, 3, . . . , of a natural argument with the parameter α ∈ R: Φα(n) = α, Φα(n) = n , Φα(n) = exp(n ), Φα(n) = exp ( n (ln(q−3)n)1/α ) ; q = 4, 5, . . . . Supppose that f(n) is any positive valued function of a natural argument. We define the (upper) dimension of level q, q = 1, 2, 3, . . . , and lower dimension of level q by Dimf(n) = inf{α ∈ R | f(n) Φα(n)} = inf{α ∈ R | f(n) Φα(n)}, Dimf(n) = sup{α ∈ R | f(n) Φα(n)} = sup{α ∈ R | f(n) Φα(n)} (infimums as well as supremums do coincide [2]). We also call these dimensions q-dimensions. Suppose that A is a finitely generated algebra with γA(n) as above. We define the q-dimension (lower q-dimension), q = 1, 2, 3, . . . , of A by DimA = DimγA(n), Dim A = DimγA(n). By the remark above these dimensions do not depend on generating set X . If 0 0, q = 1, 2, . . . . Also for q ≥ 3 suppose that DimL = α and for q = 2 suppose that DimλL(n) = α− 1, α ≥ 1. Then DimU(L) = DimU(L) = α. The proof was outlined in [1] as well as the proof of a theorem on growth of solvable Lie algebras. In [2] we give detailed proofs and also we prove a more general result on the growth of a free polynilpotent finitely generated Lie algebra. IfM is a variety of Lie algebras then by F (M, k) we denote its free algebra of rank k. (For theory of varieties see the monograph [7].) Then the second main result of [2] is read as follows. Theorem 2 ([2]). Let L = F (Nsq . . .Ns2Ns1 , k), q ≥ 2, be a free polynilpotent Lie algebra of rank k, k ≥ 2. Then DimL = DimL = s2 dimK F (Ns1 , k). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON LIE ALGEBRAS WITH NONINTEGRAL q-DIMENSIONS 651 1. Main results We see that all q-dimensions for free polynilpotent Lie algebras are integers. For the Gelfand-Kirillov dimension of associative algebras (recall that it coincides with dimension of level 2) the following facts are known: Theorem 3 ([4]). For any σ ∈ [2,+∞) there exists a two-generated associative algebra A with Gelfand-Kirillov dimension as follows: DimA = DimA = σ. Also Gelfand-Kirillov dimension may equal 1 and 0 (latter are finite-dimensional algebras over the field). There is a gap: DimA / ∈ (0, 1) for associative (Lie) algebra A because either γA(n+ 1) ≥ γA(n) + 1, ∀n, hence γA(n) ≥ n, or ∃n γA(n+ 1) = γA(n) which easily implies that A(X,N) = A(X,n) ∀N ≥ n. It is more interesting that there is another gap: Theorem 4 (Bergman [5]). The Gelfand-Kirillov dimension of an associative algebra does not belong to the interval (1, 2). As for dimension of level 3 (which coincides with superdimension up to normalization) only the following fact is known: Theorem 5 ([4]). For any σ ∈ (0, 1] there exists a two-generated associative algebra A with DimA = σ. In this paper we study nonintegral dimensions for different levels q = 2, 3, . . . for Lie and associative algebras. For α ∈ R let {α} be the least integer greater or equal than α. As the first result we prove Proposition 1. For any σ ∈ (b − 1, b), b ≥ 2, there exists a two-generated Lie algebra H ∈ NbA with DimH = DimH = σ. Remark. By Theorem 2 for L = F (NbA, 2) one has DimL = DimL = 2b, but Proposition fills only segment [1, b]. Thus we suggest the question for which numbers σ ∈ [b, 2b] does there exist a two-generated Lie algebra H ∈ NbA with DimH = DimH = σ? In view of Proposition 1 the following fact is interesting. Lemma 3. Let L ∈ A be a finitely generated Lie algebra. Then DimL is an integer. The main result of this paper is as follows. Theorem 6. For levels q = 2, 3 . . . and any σ ∈ [1,+∞) there exists a twogenerated Lie algebra H ∈ ANsA, s = {σ} with DimL = DimL = σ. Remarks. For level q = 2 the gap σ ∈ (0, 1) is natural (see above). In case q = 2 it is interesting that unlike associative algebras Lie algebras have no gap (1, 2) (compare Theorem 4). As for higher levels q = 3, 4, . . . we have an open problem whether for all σ ∈ (0, 1) there exist Lie algebras with DimL = DimL = σ. By applying Theorem 1, and adding results of Theorems 3,4,5 one has License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 652 V. M. PETROGRADSKY Corollary. For levels q = 1, 2, 3, . . . there exist finitely generated associative algebras A with the following nonzero finite q-dimensions: DimA = DimA ∈  N, q = 1, 1, [2,+∞) q = 2, (0,+∞) q = 3, (1,+∞) q = 4, 5, . . . . Remark. For high levels q = 4, 5, . . . we also have intervals σ ∈ (0, 1) for further study. 2. Nonintegral growth of level 2 (Gelfand-Kirillov dimension) Let L = L(x, y) be a free Lie algebra with free generators x, y where y > x. Now we need some facts [8]. The basis of L is formed by “regular monomials” which uniquely correspond to “regular words” in free associative algebra A = A(x, y). Suppose that words in A are ordered lexicographically from the left. Then regular words a ∈ A are exactly those that are greater than all cyclic conjugates (i.e. a = a1a2 implies that a > a2a1). For a regular word a ∈ A there is a unique arrangement of brackets [a] ∈ L that makes it into a regular monomial. Linear order is naturally extended to monomials of L. Now we fix an integer b ≥ 2 and set I = IdL(y), L = L/I, ρ : L→ L. Denote a = a(α1, α2 . . . , αs) = yx 1yx2 . . . yxs , s = 1, 2, 3, . . . , deg a = s+ α1 · · ·+ αs, degy a = s, αi(a) = αi, i = 1, . . . , s. In case s = 0 we have only one regular word a = x. Then one has a basis L = ∞ ⊕ n=1 Ln; Ln = 〈 ρ([a]) | a = a(α1, α2, . . . , αs) is a regular word with deg a = n, 0 ≤ s ≤ b〉 (first letter of a being y by regularity). Suppose that Ψs(n) denotes the number of regular words a ∈ A with deg a = n, degy a = s. Then by the Witt formula [8, 2.2.8]: Ψs(n) = 1 n ∑ m|n,m|s μ(m) (n/m)! (s/m)! ((n− s)/m)! ≈ ns−1 s!

The following question of Lanski is answered positively in the case when a ring R with involution * is Noetherian with respect to two-sided *ideals. Let R be a ring with * and invertible 2, and let S be the subring of R generated by the symmetric elements in R. Does any left R-module have the same Krull dimension when considered as an R-module and S-module? Our starting point is a question implicit in a paper of Lanski [7], namely given a ring R with involution *, with 1/2 E R, and with S the subring generated by the symmetric elements in R, and given any left R-module M, does M have the same Krull dimension when considered as an R-module and as an S-module? The question is a very natural one, given developments in the theory of rings with involution in the 1970s, but is surprisingly hard to settle. Indeed Lanski [7, p. 398] frankly admits that he did not know the answer even in the case of an Artinian R-module M. To the best of our knowledge, this is still the case. Let us review some of the positive results related to this question. In an earlier paper, Lanski [5] proved that if R is an Artinian ring then so too is S and R is a finitely generated S-module. In [6], Lanski continued his investigations and proved that if R is a Noetherian ring then R is a Noetherian S-module and the Krull dimensions of R as an R-module and as an S-module are the same. The first part of this theorem of Lanski was proved independently by Chuang and Lee [1]. In this note we shall prove that if R satisfies the ascending chain condition on two-sided *-ideals then any left R-module has the same (dual) Krull dimension when viewed as an R-module and as an S-module. This theorem contains all the above results as special cases. As one might expect, the arguments in the cases of Krull dimension and dual Krull dimension are dual to each other in many places, but Lemma 7 and its "dual" Lemma 7* do differ in some important regards. To be careful and for the convenience of the reader, we give both arguments, the dual argument being given in parentheses. Naturally we are disappointed not to be able to settle the general question one way or the other, but we present our results in the belief that the methods are interesting. Basically, in the spirit of Lanski's work, we make certain reductions Received by the editors July 29, 1992 and, in revised form, September 29, 1992. 1991 Mathematics Subject Classification. Primary 16W10, 16P60. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

Let R R be a commutative Noetherian ring, and let M M be an R R -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers μ i ( p , M ) \mu _i(p,M) were defined for all primes p p and all integers i ≥ 0 i\ge 0 by use of the minimal injective resolution of M M . It is well known that μ i ( p , M ) = dim k ( p ) ⁡ Ext R p i ⁡ ( k ( p ) , M p ) \mu _i(p,M)=\dim _{k(p)}\operatorname {Ext} _{R_p}^i(k(p),M_p) . On the other hand, if M M is finitely generated, the Betti numbers β i ( p , M ) \beta _i(p,M) are defined by the minimal free resolution of M p M_p over the local ring R p R_p . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants π i ( p , M ) \pi _i(p,M) were defined by the minimal flat resolution of M M over Gorenstein rings. The invariants π i ( p , M ) \pi _i(p,M) were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that \[ π i ( p , M ) = dim k ( p ) ⁡ Tor i R p ⁡ ( k ( p ) , Hom R ⁡ ( R p , M ) ) \pi _i(p,M)= \dim _{k(p)}\operatorname {Tor}_i^{R_p}(k(p), \operatorname {Hom}_R(R_p,M)) \] for any cotorsion module M M . Comparing this with the computation of the Bass numbers, we see that Ext \operatorname {Ext} is replaced by Tor \operatorname {Tor} and the localization M p M_p is replaced by Hom R ⁡ ( R p , M ) \operatorname {Hom}_R(R_p,M) (which was called the colocalization of M M at the prime ideal p p by Melkersson and Schenzel).

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in H3(−1), and prove that it cannot omit more than four points unless the surface is a horosphere. 0. Introduction In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate “constant mean curvature one” by CMC-1. These surfaces share many properties with minimal surfaces in R. They possess the “Weierstrass representation” in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant’s representation formula we are able to answer this question as follows. Theorem. The hyperbolic Gauss map of nonflat complete CMC-1 surfaces in H3(−1) can omit at most four points. Received by the editors November 1, 1995 and, in revised form, April 2, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.

Let p p be a rational prime and D D a positive rational integer coprime with p p . Denote by N ( D , 1 , p ) N(D, 1,p) the number of solutions ( x , n ) (x, n) of the equation D x 2 + 1 = p n D x^2 + 1 = p^n in rational integers x ≥ 1 x \geq 1 and n ≥ 1 n \geq 1 . In a paper of Le, he claimed that N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 without giving a proof. Furthermore, the statement N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 is incorrect by proving that N ( 2 , 1 , 3 ) = 3 N(2, 1, 3)=3 .

We present some results associated with the existence of central configurations (c.c.’s) in the classical gravitational N N -body problem of Newton. We call a central configuration of five bodies, four of which are coplanar, a p y r a m i d a l pyramidal central configuration (p.c.c). It can be shown that there are only three types of p.c.c.’s, admitting one or more planes of symmetry, viz. (i) the case where the planar bodies lie at the vertices of a regular trapezoid, (ii) the case where the bodies lie at the vertices of a kite-shaped quadrilateral, and (iii) the case where the bodies lie at the vertices of a rectangle. In this paper we classify all p.c.c.’s with a rectangular base and, in fact, prove that there is only one such c.c., namely, the square-based pyramid with equal masses at the corners of the square. The classification of all p.c.c.’s satisfying either (i) or (ii) will be discussed in subsequent papers.

In a recent paper by the author the normset and its multiplicative structure was studied. In that paper it was shown that under certain conditions (including Galois) that a normset has unique factorization if and only if its corresponding ring of integers has unique factorization. In this paper we shall examine some of the properties of a normset and describe what it says about the class group of the corresponding ring of integers.

We give an example as indicated in the title where the weight (i.e., the minimum cardinality of a base for the topology) of the product is the smallest uncountable cardinal. 0. INTRODUCTION AND THE MAIN THEOREM History. Throughout this paper the term "space" abbreviates "Hausdorff regular topological space". Is there a normal space and a metric space with a nonnormal product? The answer is yes in ZFC and is due to E. A. Michael ([6], 1963). (ZFC denotes the standard set of axioms for set theory.) Let IP and R denote respectively the irrationals and the reals with their usual topologies; and let M denote the Michael line which is the refinement of R obtained by isolating each irrational point. In Michael's example, M is the normal space (and moreover hereditarily paracompact) and P is the metric space. Note that the weight of M, and therefore the product, is c (c denotes the cardinality of the continuum of real numbers). Is there a ZFC example of a normal space with weight RI (the smallest uncountable cardinal) whose product with IP is nonnormal? (This question is due to Ronnie Levy.) It is not possible in ZFC to construct a Lindelbf space with both R, weight and a nonnormal product with IP (Lawrence, [5], 1990). Define a Michael space to be a Lindeldf space whose product with IP is nonnormal. The existence of a Michael space follows from both the Continuum Hypothesis (Michael, [7], 1971; first mentioned in a footnote of [6]) and Martin's Axiom (K. Alster, [1], 1990). Is there a Michael space in LZFC? This is Michael's problem and is equivalent to the existence in ZFC of a Lindeldf space and a separable completely metrizable space with a nonnormal product (see [5]). Suppose we drop the requirement for completeness on the metrizable factor. In the 1963 paper, Michael also gave a ZFC example of a Lindeldf space and a separable metric space with a nonnormal product. The Lindel6f space is the refinement of R obtained by isolating each point in a Bernstein set, and the metric space is the Bernstein set as a subspace of R. A subset of R is Bernstein if both it and its Received by the editors November 24, 1992 and, in revised form, March 14, 1994. 1991 Mathematics Subject Classification. Primary 54B10; Secondary 54D20, 54E50.