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We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.

A subset A A of a topological space is said to be semi-open if there exists an open set U U such that U ⊆ A ⊆ Cl ⁡ ( U ) U \subseteq A \subseteq \operatorname {Cl} (U) where Cl ⁡ ( U ) \operatorname {Cl} (U) denotes the closure of U U . The class of semi-open sets of a given topological space ( X , T ) (X,\mathcal {T}) is denoted S .O . ( X , T ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) . In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if T \mathcal {T} and T ∗ {\mathcal {T}^ \ast } are two topologies for a set X X such that S .O . ( X , T ) ⊆ S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T ⊆ T ∗ \mathcal {T} \subseteq {\mathcal {T}^ \ast } . In a corollary to this theorem, Levine states that if S .O . ( X , T ) = S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T = T ∗ \mathcal {T} = {\mathcal {T}^ \ast } . An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.

where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.

The incorrectness of Birkhoff's result, with which the paper [2] is concerned, was noted already by F. R. Gantmacher in 1954 in his book Theory of matrices (Russian); cf. the recent English translation of the second part of this, [l, pp. 175-176]. Example C in [2] is, in fact, the analogue of Gantmacher's counter-example when z= a> is taken instead of 2 = 0 as the singular point. Gantmacher does not, however, discuss the cases in which the result is correct; cf. [2, D]. Unfortunately, the writer became aware of Gantmacher's work only when it was too late to have paper [2] withdrawn or even to have a note added to it.

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas., Comment: 13 pages, no figures

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887-898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval’s identity.

In this paper we consider continuous, SL ( 2 , R ) \text {SL}(2,\mathbb {R}) -valued, Schrödinger cocycles over irrational rotations. We prove two generic results on the Lyapunov exponents which improve the corresponding ones contained in a paper by Bjerklöv, Damanik and Johnson.

In this paper, all spectra are localized at an odd prime. Let T ( 1 ) T(1) be the Ravenel spectrum characterized by B P ∗ BP_{\ast } -homology as B P ∗ [ t 1 ] BP_{\ast }[t_1] , T ( 1 ) / ( v 1 ) T(1)/(v_1) be the cofiber of the self-map v 1 : Σ 2 p − 2 T ( 1 ) → T ( 1 ) v_1: \Sigma ^{2p-2}T(1)\rightarrow T(1) and L 2 L_2 denote the Bousfield localization functor with respect to v 2 − 1 B P ∗ v_2^{-1}BP_{\ast } . In this paper, we determine the homotopy groups of L 2 T ( 1 ) / ( v 1 ) L_2T(1)/(v_1) .

Refinable functions have been widely investigated because of their importance in wavelet theory and multiresolution analysis, as well as because of intrinsic interest. Problems involving refinability can be challenging and interesting problems in mathematics. Several papers have investigated refinability of splines and other classes of functions. The purpose of this paper is to develop necessary and sufficient conditions for the refinability of the class of step functions on the real line taking complex values.

We prove that ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. Hopkins [2] proved that ΩSp(2) and ΩSp(3) are stably indecomposable. Later Hubbuck [3] added ΩG2 and ΩF4 to the list of such spaces. We [4] also proved that ΩE6 and ΩE7 are stably indecomposable. In this paper we will show that ΩSp(n) are stably indecomposable at the prime 2 for n ≥ 2, which was conjectured by Hubbuck. Theorem. ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. In contrast to our result ΩSU(n) is stably decomposable [1]. From now on until the end of this paper all spaces are assumed to be localized at the prime 2 and H∗(X) stands for H∗(X;F2). First we recall the ring structure of H∗(ΩSp(n)) and the dual action of the Steenrod algebra on them [5]: H∗(ΩSp(n)) ∼= F2[z1, z3, . . . , z2n−1], where z2i−1 ∈ H4i−2(ΩSp(n)). To state the action of the Steenrod algebra on z2i−1, for a positive integer i we define zi = (zb(i)) 2a(i) , where a(i) and b(i) are unique non-negative integers such that i = 2b(i) and b(i) is odd. Then we have Sqz2i−1 = ( 2i− 2− j j ) z2i−1−j. In particular, Sqz2i−1 = z2i−2. We also have Sqzi = ( i− 1− j j ) zi−j for any positive integer i. 2000 Mathematics Subject Classification. Primary 55P35, 55P42, 55S10.

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context. 1. Introduction and definitions The problem of expanding a function with respect to a given polynomial basis has many implications in analysis. The simplest example of this kind is the Taylor's expansion theorem. In this paper, we replace the classical derivative by a difference operator of Askey-Wilson type. Our results complement the paper (3) of Ismail and Stanton and are a natural continuation of the point of view presented in (5), where a new approach to the theory of classical hypergeometric polynomials is given. In contrast with (3), our aim is to find sufficient conditions for t he Taylor series to converge, but not necessarily to the original function. In this more general setting, we may consider non-necessarily entire functions and we give an explicit expression for the limit of the remainders in terms of a contour integral. Using this and a new estimate for the q-shifted factorials, which might be of independent interest, we obtain a summation formula which is new as far as we know. As we explain below, it can be regarded as a non-symmetrized version of the non-terminating q-Saalschutz sum. As applications, we also provide a new proof of the q-Gauss summation formula and a list of binomial type summation formulas in the same line than Ismail's paper (2). Now we give some definitions which will be used in what follows. The notions we are presenting were already introduced in (5) with the aim of studying some aspects of the theory of hypergeometric polynomials. The relevance of this approach is justified in (5), where a more detailed exposition is given.

Let F \mathbb {F} be a finite field and T T a transcendental element over F \mathbb {F} . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over F ( T ) \mathbb {F}(T) in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers m m and n n , there are infinitely many function fields K K of fixed degree m m such that the class group of K K contains a subgroup isomorphic to ( Z / n Z ) m − 1 (\mathbb {Z}/n\mathbb {Z})^{m-1} and the prime at infinity is inert.

In his famous paper The image of a derivation is contained in the radical, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra A to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on A must lie in the radical of A. In this paper, we exhibit discontinuous derivations on a commutative unital Frechet algebra A such that the separating subspace is the whole of A. Thus, the situation on Frechet algebras is markedly different from that on Banach algebras.

Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ { 2 B 2 (q), 2 G 2 (q), G 2 (q), 2 F 4 (q), F 4 (q), 3 D 4 (q)} provided that p < 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over F q with q large enough.

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.

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.

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.

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 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.

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.

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.

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

Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the Parseval formula. In a special case this relation dates back to work by Bateman in 1933 and we follow a part of the historical development for these polynomials. Some applications of this relation are given. 1. Introduction and history In Askey's scheme of hypergeometric orthogonal polynomials we find the Jacobi polyno- mials and the continuous Hahn polynomials; see Askey and Wilson (5, Appendix) with the correction in (2), Koekoek and Swarttouw (20) or Koornwinder (23, §5) for information on Askey's scheme. In the hierarchy of Askey's scheme of hypergeometric orthogonal polyno- mials the continuous Hahn polynomials are above the Jacobi polynomials since they have one extra degree of freedom. In this paper we consider a way of going up in the Askey scheme from the Jacobi polynomials to the continuous Hahn polynomials by use of the Fourier transform. This method is a simple extension of some special cases introduced by Bateman in the 1930's. In his 1933 paper (9) Bateman introduces the polynomial Fn satisfying

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc. Amer. Math. Soc. 108 (1990), 587-600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that.

In this paper, we show that a reduced ring R is weakly regular (i.e., 12 = I for each one-sided ideal I of R ) if and only if every prime ideal is maximal. This result extends several well-known results. Moreover, we provide examples which indicate that further generalization of this result is limited. Throughout this paper R denotes an associative ring with identity. All prime ideals are assumed to be proper. The prime radical of R and the set of nilpotent elements of R are denoted by P(R) and N(R), respectively. The connection between various generalizations of von Neumann regularity and the condition that every prime ideal is maximal will be investigated. This connection has been investigated by many authors [2, 3, 5, 7, 12, 14]. The earliest result of this type seems to be by Cohen [3, Theorem 1]. Storrer [12] was able to provide the following result: If R is a commutative ring then the following are equivalent: (1) R is 7r-regular; (2) R/P(R) is regular; and (3) all prime ideals of R are maximal ideals. Fisher and Snider extended this result to P.I. rings [5, Theorem 2.3]. On the other hand, Chandran generalized Storrer's result to duo rings [2, Theorem 3]. Next Hirano generalized Chandran's result to right duo rings [7, Corollary 1]. More recently the result was generalized to bounded weakly right duo rings by Yao [14, Theorem 3]. As a corollary of our main result, we show that if R/P(R) is reduced (i.e., N(R) = P(R)) then the following are equivalent: (1) R/P(R) is weakly regular; (2) R/P(R) is right weakly 7r-regular; and (3) every prime ideal of R is maximal. This result generalizes Hirano's result for right duo rings. A further consequence of our main result is that if R is reduced then R is weakly regular if and only if every prime factor ring of R is a simple domain. This result can be compared to the well-known fact that when R is reduced, then R is von Neumann regular if and only if every prime factor ring of R is a division ring. We conclude our paper with some examples which illustrate and delimit our results. Received by the editors December 8, 1992. 1991 Mathematics Subject Classification. Primary 16D30, 1 6E50; Secondary 16N60.

In this paper we shall consider the factorization of positive cones of order n of a von Neumann algebra. Namely, we shall show the existence of a *-subalgebra inducing the positive cone of order n of a von Neumann algebra. Let M be a von Neumann algebra on a Hilbert space H. It is known that a positive cone M,(M)+ (= (M 0 Mn)+), where Mn denotes an algebra of all n x n matrices, coincides with the convex hull of all elements [xlx;] for xi E M. We shall find a *-subalgebra N of M such that a positive cone of order n is generated by all elements [alc*caj] for ai E N, c E M. We may then say that the cone is factorized by N. The purpose of this paper is to consider when a positive cone of order n can be factorized or not. The factorization of a self-dual cone in the Hilbert space associated with a standard form of a von Neumann algebra was already shown in [4]. We shall use the books of Pedersen [5] and Takesaki [8] as references of concepts and results of operator algebras. We begin with the following definition, in which we consider two convex cones of order n . Definition 1. Let M be a von Neumann algebra on a Hilbert space H and N a subspace of M. For a natural number n we put Cn(N) = cos{[a c*caj] E Mn(M)jai E N, c E M}

In this paper, we study certain properties of the group ring of a nilpotent group which are related to commutativity and conjugation. We establish some relations involving conjugates of the elements of the group ring; these relations are then used to get a better understanding of torsion in abelian-bynilpotent groups; we shall see notably that given any action of a nilpotent group N on an abelian group A, then the set of torsion elements of A with respect to the action of N is actually a subgroup of A . In nilpotent (or even locally nilpotent) groups, torsion elements always form a subgroup; moreover, a nilpotent group can only have multiple roots (xl = y' with x /& y ) if it has torsion elements. In both cases, the situation is quite different if we drop the nilpotency assumption; one can find in [7] an appropriate setting for the study of these questions. Groups with unique roots as well as groups in which roots exist for each element have been extensively studied in the literature. Baumslag [ 1, 2] studied extensively those concepts in relation notably with wreath products, hence with group extensions. On the other hand, for any fixed set of prime numbers (0, Kuz'min showed in [6] that any metabelian U,,-group (xn = yn with n e w( implies x = y) could be embedded in a metabelian D,,-group (x I, xn is a bijection whenever n e w(). Let us denote by P = P(w) the set of prime numbers complementary to w(. We were able to obtain a new proof of Kuz'min's result by showing that given any extension A > G B with A and B abelian and G a U,,-group, it is possible to embed G in the quotient of Ap I Bp (the wreath product of the P-localizations of A and B) by its w(-torsion subgroup (the minimal normal subgroup with a D,,-quotient). We used the idea beautifully developed by Bludov and Medvedev [3] in their completion of ordered metabelian groups as well as results to be found in [1, 2]. By trying (unsuccessfully) to adapt the same proof to the abelian-by-nilpotent situation, we obtained the results contained in this paper which, we believe, are interesting by themselves. Let us first introduce some notation and terminology. * We denote as usual by R[B] the group ring, where R is any commutative ring with a unity element and B is an arbitrary group. Received by the editors March 10, 1992. 1991 Mathematics Subject Classification. Primary 20C07, 20F18. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

In this paper we point out an error in paper [2] and study the asymptotic behavior of the differential equation \[ L n x + f ( t , x ) = r ( t ) . {L_n}x + f\left ( {t,x} \right ) = r\left ( t \right ). \] The results obtained are extensions of some of the results of [2].

In this paper, we analyze the compatibility of Fong's reduction and the Glauberman correspondence of characters and then clarify that the p-solvable hypothesis in a paper of Harris and Linckelmann is not necessary.

In Chermak and Delgado's paper "A measuring argument for finite groups", a certain "measuring lemma" was shown to hold. This lemma has been successfully applied in many recent papers. We generalize this lemma by expanding the discussion from groups acting on groups to groups acting on sets. As applications, we obtain the main results of several earlier papers.

The purpose of this paper is to find the exact norm of the Rademacher projection onto {r 1 , r 2 , r 3 }. Namely, we prove formula math. The same techniques also give the relative projection constant of ker{1, ..., 1} in l n p , that is, formula math. for n = 2,3,4. We discuss the relation of the above inequalities to the famous Khintchine and Clarkson inequalities. We conclude the paper by stating some conjectures that involve the geometry of the unit ball of l n p .

In a previous paper, the author gave a combinatorial proof and refinement of Siladic's theorem, a Rogers-Ramanujan type partition identity arising from the study of Lie algebras. Here we use the basic idea of the method of weighted words introduced by Alladi and Gordon to give a non-dilated version, further refinement and companion of Siladic's theorem. However, while in the work of Alladi and Gordon, identities were proved by doing transformations on generating functions, we use recurrences and $q$-difference equations as the original method seems difficult to apply in our case. As the non-dilated version features the same infinite product as Schur's theorem, another dilation allows us to find a new interesting companion of Schur's theorem, with difference conditions very different from the original ones.

In this paper, we give two new characterizations of separable inner quasidiagonal C*-algebras. Based on these characterizations, we show that a unital full free product of two inner quasidiagonal C*-algebras is itself inner quasidiagonal. As an application, we show that a unital full free product of two inner quasidiagonal C*-algebras with amalgamation over a full matrix algebra is inner quasidiagonal. Meanwhile, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is inner quasidiagonal if there are faithful tracial states on each of these two AF algebras such that the restrictions of these states to the common subalgebra coincide.

In this paper, we use p-adic Gamma function and certain formulas on hypergeometric series to establish several new supercongruences. In particular, we give a generalization of a p-adic supercongruence conjecture due to van Hamme and Swisher.

In this paper we study the harmonic elements of (convolution) Markov maps associated to (ergodic) actions of locally compact quantum groups on (�-finite) von Neumann algebras. We give several equivalent conditions under which the harmonic elements are trivial.

The purpose of this paper is to prove that hermitian algebras are the natural framework for the results of Ky Fan on analytic functions of a paper contraction

Measure differential equations can model non-classical problems like the quantum effects. In this paper we will solve extremal problems for eigenvalues of measure differential equations by exploiting the approximation of general measures by smooth measures and the continuity results of eigenvalues in weak ∗ ^* topology of measures.

This paper investigates the Poisson (co)homology of affine Poisson algebras. It is shown that there is a twisted Poincaré duality between their Poisson homology and cohomology. The relation between the Poisson (co)homology of an affine Poisson algebra and the Hochschild (co)homology of its deformation quantization is also discussed, which is similar to Kassel’s result (1988) for homology and is a special case of Kontsevich’s theorem (2003) for cohomology.

It is well known that every homology functor on the stable homotopy category is representable, so of the form E ∗ ( X ) = π ∗ ( E ∧ X ) E_{*} (X)=\pi _{*} (E\wedge X) for some spectrum E E . However, Christensen, Keller, and Neeman (2001) have exhibited simple triangulated categories, such as the derived category of k [ x , y ] k[x,y] for sufficiently large fields k k , for which not every homology functor is representable. In this paper, we show that this failure of Brown representability does not happen on the model category level. That is, we show that a homology theory is representable if and only if it lifts to a well-behaved functor on the model category level. We also show that, for a reasonable model category M \mathcal {M} , every functor that has the same formal properties as a functor of the form X ↦ X ⊗ E X\mapsto X\otimes E for some cofibrant E E is naturally weakly equivalent to a functor of that form. This is closely related to the Eilenberg-Watts theorem in algebra, which proves that every functor with the same formal properties as the tensor product with a fixed object is isomorphic to such a functor.

Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials r(z, z) on Cn×Cn for which r(z, z) ‖z‖ = ‖h(z)‖ for some natural number d and a holomorphic polynomial mapping h = (h1, . . . , hK) from Cn to CK . When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in C[z1, . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $\deg(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a homogeneous ideal $J$ containing $x_1^{a_1},...,x_n^{a_n}$ with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris conjecture when $f_i$ splits into linear factors for all $i$.

We discuss when a unital homomorphism {\phi} : C(X) \rightarrow A can be approximated by finite-dimensional homomorphisms, where X is a compact metric space and A is unital simple C*-algebra with tracial rank one. In this paper, we will give a necessary and sufficient condition., Comment: 14

In this paper, we discuss the finite generation problem for the invariant subalgebras of finite abelian groups which act linearly on the 2 x 2 generic matrix algebra, and we obtain some conditions on the groups to ensure that their invariant subalgebras are finitely generated. Introduction. Let R(m) = K{X, Y} be a generic matrix algebra generated by two m x m generic matrices over a field K. A theorem of Fisher and Montgomery [2] shows that if G = (g) is a finite cyclic group such that char K t ICl, then RG the subalgebra of invariants, is not finitely generated whenever g is not scalar and mI CGI-[VI[G]+1. When m = 2 and G is a finite subgroup of SL(2, K), Formanek and Schofield [3] have proved that if char K t IGI, then RG is always finitely generated. In this paper we consider the case when m = 2 and G is a finite abelian subgroup of GL(2, K) such that char K t IGl. In this case we may assume that K is algebraically closed. So G acts diagonally on R = R(2) and RG is generated by some monomials in the generators of R. We obtain necessary and sufficient conditions on G to ensure that RG is finitely generated. They are as follows: RG is finitely generated if and only if G contains no pseudoreflections or, equivalently, G = (g) is cyclic and the eigenvalues of g in some extension field of K have the same order. In some sense, this provides a converse to Theorem 1 of Fisher and Montgomery [2]. 1. For cyclic groups. LEMMA 1.1 (FORMANEK AND SCHOFIELD). Let G be any finite subgroup of GL(2, K) such that char K t IGI and let [R, R] = J be the commutator ideal of R. Then (i) for each m > 1, Jm = J(XY YX)m-1, and (ii) G acts linearly on R/J2 and (R/J2)G is a finitely generated K-algebra. PROOF. See [3, Lemmas 6 and 7]. LEMMA 1.2. Let u E R be homogeneous and Z = XY YX. Then u = vZ + aYrXs + E bi,j,,,lY'XjZYKXl Received by the editors February 25, 1987 and, in revised form, August 19, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A38.

In this paper we will prove that if X is any space with a finite fundamental group, then Waldhausen's algebraic K-groups of X are finitely generated. We will use Dwyer's machinery developed in Twisted homokgical stabity for general linear groups (Ann. of Math. 111). Introduction. In [D] Dwyer proved the following theorem: THEOREM (DWYER). If X is simply-connected andxi(X) is finitely generated for i > 2, then rj (A(X)) is finitely generated for all j, where A(X) is Waldhausen's algebraic K-theory of a space X. In this paper we will use Dwyer's approach in order to generalize his theorem to the case when ri (X) is a finite group. We will prove the following theorem. THEOREM I. If 7ri (X) is a finite group and 7ri (X) is finitely generated for i > 2, then xrj(A(X)) is finitely generated for all j. REMARK. Theorem I is, in general, the best possible result in this direction. Consider X = S1. Then using a spectral sequence Ep2q =: irp+q(A(X)) (for details see [W, Proposition 2.6]) where (i) E.pq = rp+q(fiber(Aq-1(X) -? Aq-2(X))) if q > 2, (ii) Ep2 = p+ 1K (Z [G(X)] ) if q = 1 and where *. -+ Ai (X) . is a tower derived from the Postnikow tower for Q??S??, it is easy to see that 7r2(A(S1)) contains an infinite number of copies of Z/2Z so ir2(A(S')) is not finitely generated. I. Let R be a ring and let A be an R-bimodule which is finitely generated as an abelian group. Then MnA-the abelian group of (n x n)-matrices with entries in A-is an MnR-bimodule and MnA becomes a left GlnR-module by defining g o m = gmg-1 for m E MnA, g eGlnR where the multiplication (1.1) is taken with respect to the R-bimodule structure on A. Now let An be a left GlnR-module with the obvious action. Consider R -the left GlnR-module which is isomorphic to Rn as an abelian group and where the action of GlnR is given by (1.2) gx(ri, ...,rn) = (r1,...,rn)g-1 for all g eGlnR. Received by the editors July 19, 1985. 1980 Mathemadics Su1Wject C1a&4ifwation. Primary 55D99, 55B99, 18H10. Key wourd and phrases. Algebraic K-theory of a space, Postnikow tower. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

The paper deals with a number of generalizations of an identity first introduced by Brock, and subsequently investigated in some detail by Carlitz. The theorem presented in this paper not only extends the formulas of Carlitz, but also shows that a multi-dimensional function satisfies the identity. A result is also deduced regarding the positive nature of the recurrence relation.