Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

We study the distributions Fθ,p of the random sums [sum ]∞1εnθn, where ε1, ε2, … are i.i.d. Bernoulli-p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p=.5, Fθ,p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain β of small degree, simulation gives the Hausdorff dimension to several decimal places.

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).

Throughout this paper, D denotes a division ring (possibly commutative) and V a left vector space over D, usually, but not exclusively, infinite-dimensional. We consider irreducible subgroups G of GL(V) and are particularly interested in such G that contain an element g the fixed-point set CV(g) of which is non-zero but finite-dimensional (over D). We then use this to derive conclusions about cofinitary groups, an element g of GL(V) being cofinitary if dimDCV(g) is finite, and a subgroup of GL(V) being cofinitary if all its non-identity elements are cofinitary.

Suppose that G is a cofinitary subgroup of GL(V). There are two extreme cases. If dimDV is finite the cofinitary condition is vacuous. At the other extreme, if G acts fixed-point freely on V then the fixed-point sets CV(g) for g∈G[setmn ]〈1〉 are as small as possible, namely {0}. Work of Blichfeldt and his successors shows that certain irreducible linear groups G of dimension at least 2 over, for example, the complexes are always imprimitive. This is the case if G is nilpotent, or supersoluble, or metabelian. Apart from the two extreme cases, the same is frequently true for irreducible cofinitary subgroups G of GL(V). For example, this is the case if G is finitely generated nilpotent [9, 1.2] or more generally if G is supersoluble [10, 1.1], but not in general if G is metabelian [10, 7.1] or parasoluble (a group G is parasoluble if it has a normal series of finite length such that every subgroup of each of its factors is Abelian and normalised by G) (see [10, 7.2]). Further, it is also the case if G is Abelian-by-finite [10, 3.4], and every supersoluble group is finitely generated and nilpotent-by-finite. Collectively, these results suggest that one should consider nilpotent-by-finite groups.

The paper provides necessary and sufficient conditions for the existence of weak solutions for the Dirichlet problem within Bishop's constructive mathematics. It also considers the continuity of those weak solutions relative to the boundary data and the domain.

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….

The purpose of this paper is to give characterizations of weights for which reverse inequalities of Hölder type for quasi-monotone functions are satisfied. Our inequalities with general weights and with sharp constants complement the results of [2, 8, 9] and [16, 17] for the values of parameters p and q with 1[les ]p[les ]q<∞.

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.

Lower bounds are established for the number of zeros of the logarithmic derivative of a meromorphic function of small growth with few poles.

This paper shows that a subharmonic function in the half-space which does not grow too rapidly near the boundary and which does not have asymptotic value +∞ at too many points must have finite minimal fine limits at a boundary set of positive measure. For harmonic functions, the conclusion may be expressed in terms of non-tangential limits. A related Phragmén–Lindelöf theorem is also established.

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.

Asymptotic results are given for the heat content of planar regions with cusps.

A theorem of Hall extends the sylow theory to π-subgroups of a finite solvable group for any set of primes π. This theorem has been generalized to solvable periodic linear groups [21] and to other similar settings [19]. We consider the same problem in a model-theoretic context.

In model theory, the study of ω-stable groups generalizes the classical theory of algebraic groups in the case of a algebraically closed base field, and is closely connected with certain aspects of the general structure theory developed by Shelah, particularly in connection with the more geometric theory of Zilber, Buechler, Hrushovski, Pillay, and others. In this connection, the primary concern is with the structure of simple ω-stable groups of finite rank. One way to approach the study of ω-stable groups is by generalizing some of the methods of finite group theory to this setting. One anticipates that character theory and counting arguments will not be available, though one would hope to recover much of the ‘local analysis’. Unfortunately there is as yet no very satisfactory sylow theory for ω-stable groups, except for the prime 2 [4]. One of us (Nesin) noticed initially that there is a sylow theorem for connected solvable ω-stable groups of finite Morley rank; this prompted us to look for a generalization along the following lines.

Let G be a locally compact group not necessarily unimodular. Let μ be a regular and bounded measure on G. We study, in this paper, the following integral equation,

formula here

This equation generalizes the functional equation for spherical functions on a Gel'fand pair. We seek solutions ϕ in the space of continuous and bounded functions on G. If π is a continuous unitary representation of G such that π(μ) is of rank one, then tr(π(μ)π(x)) is a solution of [Escr ](μ). (Here, tr means trace). We give some conditions under which all solutions are of that form. We show that [Escr ](μ) has (bounded and) integrable solutions if and only if G admits integrable, irreducible and continuous unitary representations. We solve completely the problem when G is compact. This paper contains also a list of results dealing with general aspects of [Escr ](μ) and properties of its solutions. We treat examples and give some applications.

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.

Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Björner, Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.

A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face poset, graded in the obvious way by chain lengths, is a Koszul ring.

The main result in this paper, Theorem 1.2, generalizes a theorem of Zerner [26] concerning sufficient conditions for the holomorphic continuability of a solution of a linear holomorphic partial differential equation across a point of a hypersurface, on one side of which it is holomorphic. The point of the new theorem is, roughly speaking, that it applies also to regular solutions of partial differential equations whose coefficients may have certain kinds of singularities. This enables us to deduce some new results (see §2) on elliptic partial differential equations in ℝ2[ratio ]Theorem 2.1 extends a result of Vekua on the size of the domain of holomorphy of solutions to elliptic equations, in the case where singularities are permitted in the coefficients; Theorem 2.2 is of an apparently novel type, showing (roughly) that under certain conditions the solution to Cauchy's problem is real-analytic in a domain whose size depends only on the principal part of the operator, which is assumed to be the Laplacian, and the Cauchy data on the real axis. (Results of this kind are very delicate, as we shall illustrate in §4 with a simple counterexample.) Theorem 2.2 is new and non-trivial even for equations with analytic coefficients, in which case though, Theorem 1.2 is not needed for the proof.

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.

Weakly almost periodic compactifications have been seriously studied for over 30 years. In the pioneering papers of de Leeuw and Glicksberg [4] and [5], the approach adopted was operator-theoretic. The current definition is more likely to be created from the perspective of universal algebra (see [1, Chapter 3]). For a discrete group or semigroup S, the weakly almost periodic compactification wS is the largest compact semigroup which (i) contains S as a dense subsemigroup, and (ii) has multiplication continuous in each variable separately (where largest means that any other compact semigroup with the properties (i) and (ii) is a quotient of wS). A third viewpoint is to envisage wS as the Gelfand space of the C*-algebra of bounded weakly almost periodic functions on S (for the definition of such functions, see below).

In this paper, we are concerned only with the simplest semigroup (ℕ, +). The three approaches described above give three methods of obtaining information about wℕ. An early striking result about wℕ, that it contains more than one idempotent, was obtained by T. T. West using operator theory [13]. He considered the weak operator closure of the semigroup {T, T2, T3, …} of iterates of a single operator T on the Hilbert space L2(μ) for a particular measure μ on [0, 1]. Brown and Moran, in a series of papers culminating in [2], used sophisticated techniques from harmonic analysis to produce measures μ that permitted the detection of further structure in wℕ; in particular, they found 2[cfr ] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in wℕ was found.

The breakthrough came in 1991, and it was made by Ruppert [11]. In his paper, he created a direct construction of a family of weakly almost periodic functions which could detect 2[cfr ] different idempotents in wℕ. His method was very ingenious (he used a unique variant of the p-adic expansion of integers) and rather complicated. Our main aim in this paper is to construct weakly almost periodic functions which are easy to describe and so appear more ‘natural’ than Ruppert's. We also show that there are enough functions of our type to distinguish 2[cfr ] idempotents in wℕ.

In [17, 18, 19], we began to investigate the continuity properties of homomorphisms from (non-abelian) group algebras. Already in [19], we worked with general intertwining maps [3, 12]. These maps not only provide a unified approach to both homomorphisms and derivations, but also have some significance in their own right in connection with the cohomology comparison problem [4].

The present paper is a continuation of [17, 18, 19]; this time we focus on groups which are connected or factorizable in the sense of [26]. In [26], G. A. Willis showed that if G is a connected or factorizable, locally compact group, then every derivation from L1(G) into a Banach L1(G)-module is automatically continuous. For general intertwining maps from L1(G), this conclusion is false: if G is connected and, for some n∈ℕ, has an infinite number of inequivalent, n-dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L1(G into a Banach algebra by [18, Theorem 2.2] (provided that the continuum hypothesis is assumed). Hence, for an arbitrary intertwining map θ from L1(G), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L1(G). Even if the target space of θ is a Banach module (which implies that the continuity ideal [Iscr ](θ) of θ is closed), it is not a priori evident that θ is automatically continuous: the proofs of the automatic continuity theorems in [26] rely on the fact that we can always confine ourselves to restrictions to L1(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4]. It is not clear if this strategy still works for an arbitrary intertwining map from L1(G) into a Banach LL1(G)-module.