We work in set theory without the axiom of choice: ZF. We show that the axiom BC: Compact Hausdorff spaces are Baire, is equivalent to the following axiom: Every tree has a subtree whose levels are finite, which was introduced by Blass (cf. [4]). This settles a question raised by Brunner (cf. [9, p. 438]). We also show that the axiom of Dependent Choices is equivalent to the axiom: In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire. Here convex-compact is the notion which was introduced by Luxemburg (cf. [16]).

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed.

By an algebra Λ we mean an associative k-algebra with identity, where k is an algebraically closed field. All algebras are assumed to be finite dimensional over k (except the path algebra kQ). An algebra is said to be biserial if every indecomposable projective left or right Λ-module P contains uniserial submodules U and V such that U+V=Rad(P) and U∩V is either zero or simple. (Recall that a module is uniserial if it has a unique composition series, and the radical Rad(M) of a module M is the intersection of its maximal submodules.) Biserial algebras arose as a natural generalization of Nakayama's generalized uniserial algebras [2]. The condition first appeared in the work of Tachikawa [6, Proposition 2.7], and it was formalized by Fuller [1]. Examples include blocks of group algebras with cyclic defect group; finite dimensional quotients of the algebras (1)–(4) and (7)–(9) in Ringel's list of tame local algebras [4]; the special biserial algebras of [5, 8] and the regularly biserial algebras of [3]. An algebra Λ is basic if Λ/Rad(Λ) is a product of copies of k. This paper contains a natural alternative characterization of basic biserial algebras, the concept of a bisected presentation. Using this characterization we can prove a number of results about biserial algebras which were inaccessible before. In particular we can describe basic biserial algebras by means of quivers with relations.

A primitive permutation group is said to be of twisted wreath type if its socle is both non-abelian and regular. We reduce the study of such primitive permutation groups to the study of ‘maximal non-abelian simple sections’ of non-abelian simple groups.

The 2-modular and 3-modular decomposition numbers for the sporadic simple O'Nan group O'N and its triple cover are determined. The results are obtained with the help of the computer algebra packages MeatAxe and GAP. Permutation modules and tensor products of modules have been condensed. The result relies on the construction of a 153-dimensional representation for 3·O'N over GF(4) and a 154-dimensional representation for O'N over GF(3).

The main result is that a finitely generated group that is isomorphic to all of its finite index subgroups has free Abelian first homology, and that its commutator subgroup is a perfect group. A number of corollaries on the structure of such groups are obtained, including a method of constructing all such groups for which the commutator subgroup has a trivial centralizer. As an application, conditions are presented for the covering spaces of compact manifolds that determine when the fundamental groups of the base spaces are free Abelian.

The quantum group Uq(SLn) introduced by Drinfel'd [2] and Jimbo [5] is a Hopf algebra which is naturally paired with Oq(sln), the coordinate ring of quantum SLn. When q is not a root of unity, the finite dimensional representation theory of Uq(sln) is essentially the same as that of U(sln). Furthermore, it is known that Uq(sln) is essentially a quasitriangular Hopf algebra [2]. When q is a root of unity the situation changes dramatically, and the representation theory of U(sln) is no longer effective. Moreover, Uq(sln) is not quasitriangular. In this case one can consider the quotient Hopf algebra Uq(sln)′, introduced by Lusztig [7], which is a finite dimensional Hopf algebra with a nice representation theory. It is well known that Uq(sl2)′ is quasitriangular. Finite dimensional quasitriangular Hopf algebras are important for the study of knot invariants [11, 12]. Thus, a natural question is: when is Uq(sln)′ quasitriangular? The somewhat unexpected answer is given in Theorem 3.7: it depends sharply on the greatest common divisor of n and the order of q1/2. For these Hopf algebras we classify all the possible R-matrices and give necessary and sufficient conditions for them to be minimal quasitriangular. These conditions depend again on n and the order of q1/2. In the process we describe the groups of Hopf automorphisms of Uq(sln)′ and Oq(SLn)′, where Oq(SLn)′ is a finite dimensional quotient Hopf algebra of Qq(SLn) proved to be the dual of Uq(sln)′ in [15].

Throughout this paper, r will denote a prime. Our object in this paper and another is to describe the finite irreducible subgroups of L(r):=SL(r, [Copf ]), where [Copf ] denotes the field of complex numbers. The cases for small degrees are known (see [14] for r=2 and r=3; [1] and [23–25] for r=5 and r=7; [13, 18]). The imprimitive irreducible subgroups of L(r) are necessarily monomial because r is a prime, and we describe these in another paper [4]. In this paper, we show how the classification of finite simple groups can be used to classify the finite primitive groups of prime degree. Much of the necessary work has already been done by V. Landazuri and G. M. Seitz (see Lemma 1.3 below).

We are interested in how small a root of multiplicity k can be for a power series of the form f(z:= 1+[sum ]∞n=1aizi with coefficients ai in [−1, 1]. Let r(k) denote the size of the smallest root of multiplicity k possible for such a power series. We show that

formula here

We describe the form that the extremal power series must take and develop an algorithm that lets us compute the optimal root (which proves to be an algebraic number). The computations, for k[les ]27, suggest that the upper bound is close to optimal and that r(k)∼1−c/(k+1), where c=1.230….

Let [Escr ] denote the space of all entire functions, equipped with the topology of local uniform convergence (the compact-open topology). MacLane [15] constructed an entire function f whose sequence of derivatives (f, f′, f'', …) is dense in [Escr ]; his construction is succinctly presented in a much later note by Blair and Rubel [2], who unwittingly rederived it (see also [3]). We shall call such a function f a universal entire function. In this note we show that analogous universal functions exist in the space [Hscr ]N of functions harmonic on ℝN, where N[ges ]2. We also study the permissible growth rates of universal functions in [Hscr ]N and show that the set of all such functions is very large.

For purposes of comparison, we first review relevant facts about universal entire functions. The function constructed by MacLane is of exponential type 1. Duyos Ruiz [7] observed that a universal entire function cannot be of exponential type less than 1. G. Herzog [11] refined MacLane's growth estimate by proving the existence of a universal entire function f such that [mid ]f(z)[mid ]=O(rer) as [mid ]z[mid ]=r→∞. Finally, Grosse–Erdmann [10] proved the following sharp result.

The classical transmission problem for the scattering of time-harmonic electromagnetic waves by a dielectric object is solved using layer potential techniques. The surface of the object is only assumed to be a Lipschitz manifold. Unlike in the case of smoother surfaces, neither the calculus of pseudodifferential operators nor the compactness of certain boundary operators are available tools. Nevertheless, the particular representation of the solutions and the choice of the space of boundary data used in the paper make the problem tractable via harmonic analysis techniques for elliptical partial differential equations in nonsmooth domains.

The object of this paper is to extend a result of Agnew in [1] for single series to double series, and so enable Tauberian constants between matrix transformations of double sequences to be calculated using the technique that many authors have used previously for single sequences (see, for instance, [2, 3, 8]). This method (see below) gives the best possible Tauberian constants in the form of certain upper limits, and we attempt to calculate these for Cesàro summability with various known Tauberian conditions. In Section 2, we introduce our notation and review earlier work. In Section 3, we prove our main result, and we discuss applications to Cesàro summability in Section 4.

The selection problem for strong M-bases is shown to have a negative solution in Hilbert space. The constructions are used to demonstrate that spectral synthesis for compact operators is not preserved by inflation. The results show that a number of statements in the literature are incorrect.

We study boundedness and compactness properties of the Hardy integral operator Tf(x)=∫xAf from a weighted Banach function space X(v) into L∞ and BMO. We give a new simple characterization of compactness of T from X(v) into BMO. We construct examples of spaces X(v) such that T(X(v)) is (a) bounded in L∞ but not compact in BMO; (b) compact in BMO but not bounded in L∞; (c) bounded in BMO but neither bounded in L∞ nor compact in BMO; (d) bounded in L∞, compact in BMO and yet not compact in L∞. In order to obtain the last of the counterexamples we construct a new weighted Banach function space.

It is well known that there are bounded domains Ω⊂ℝn whose boundaries ∂Ω are not smooth enough for there to exist a bounded linear extension for the Sobolev space W1p(Ω) into W1p(ℝn), but the embedding W1p(Ω)⊂ Lp(Ω) is nevertheless compact. For the Lipγ boundaries (0<γ<1) studied in [3, 4], there does not exist in general an extension operator of W1p(Ω) into W1p(ℝn) but there is a bounded linear extension of W1p(Ω) into Wγp(ℝn) and the smoothness retained by this extension is enough to ensure that the embedding W1p(Ω)⊂ Lp(Ω) is compact. It is natural to ask if this is typical for bounded domains which are such that W1p(Ω)⊂ Lp(Ω) is compact, that is, that there exists a bounded extension into a space of functions in ℝn which enjoy adequate smoothness. This is the question which originally motivated this paper. Specifically we study the ‘extension by zero’ operator on a space of functions with given ‘generalized’ smoothness defined on a domain with an irregular boundary, and determine the target space with respect to which it is bounded.

The following result is well known and easy to prove (see [14, Theorem 2.2.6]).

Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A[rarrhk ]BL(X) is continuous.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A[rarrhk ]BL(X) is contractive.

Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.

For a wide class of infinitely divisible laws μ numbers R(μ) are evaluated satisfying the property: if R>R(μ) then μ is uniquely determined by its values on the half-line (−∞, R); if R<R(μ) then μ is not determined by its values on (−∞, R). Necessary and sufficient conditions are given for ‘half-line’ convergence to μ on (−∞, R) to be equivalent to weak convergence to μ.