In this paper we examine when a twisted group ring, Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we define Rγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.

Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form:

Lx = N(λ, x)

with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions.

While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.

Let R be a ring and let Rn be the complete ring of n × n matrices with coefficients from R.

Let s be a natural number, s ≥ 2. We seek a positive number λ(s) with the following property:

Let ε > 0. Let Q1(x1, …, xs), Q2(x1, …, xs) be real quadratic forms, then for N > C1(s, ε) we have

for some integers n1, …, ns,

Here we extend an arithmetical inequality about multiplicative functions obtained by K. Alladi, P. Erdős and J. D. Vaaler, to include also the case of submultiplicative functions. Also an alternative proof of an extension of a result used for this purpose is given.

Let Uk, for integral k, denote the set {1,2,…, k}, and Vk denote the collection of all subsets of Uk. In the following, all unspecified sets like A,…, are assumed to be subsets of Uk. Let σ = {Si} and τ = {Tj} be two given collections of subsets of Uk. Set

and

Let ′ denote complementation in Uk (but for in the proof of (3) where it denotes complementation in C). For any collection p of subsets of Uk, let p′ denote the collection of the complements of members of p.

Let S be a regular semigroup. Given x ∈ S, we shall say that a ∈ S is an associate of x if xax = x. The set of associates of x ∈ S will be denoted by A(x). Now suppose that S has an identity element 1. Let H1 denote the group of units of S. Then we say that u ∈ S is a unit associate of x whenever u ∈ A(x)∩Hl. In what follows we shall write U(x) = A(x)∩=H1, and we shall say that S is unit regular [1, 3] if (∀ x∈S)U(x)≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x∈S, the set U(x) is a singleton). It is worthy of mention that neither of the examples cited above is of this type.

This paper continues our study of non-linear multiparameter eigenvalue problems. In recent work [3], [4], [5] we have discussed bifurcation from eigenvalues in both abstract and concrete multiparameter problems. In this note we present further simple conditions which will ensure bifurcation from eigenvalues of a multiparameter Sturm-Liouville problem. We consider the regular case and the asymptotic case of bifurcation from infinity.

Let Q(x) = Q(x1, …, xn)∈ℤ[x1, …, xn] be a quadratic form. We investigate the size of the smallest non-zero solution of the congruence Q(x)≡0 (mod q). We seek a bound Bn(q), independent of Q, such that there is always a non-zero solution satisfying

The form gives the trivial lower bound Bn(q)≥(q/n)½ for all q and n, since if x≠0 and q∣ Q(x), then Q(x)≥q.

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that

∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

The definition of a power-free group will be found in [1]. It is a partial algebraic system which, roughly speaking, may be thought of as a group in which (with the exception of the identity) squares and higher powers of an element are not defined.

It has been shown [1, Theorem 3.3] that the usual cancellation laws need not hold in a power-free group. When these laws do hold, the power-free group is called cancellative. In this paper we shall be solely concerned with cancellative power-free groups and the term ‘power-free group’ is to be understood to mean ‘cancellative power-free group’.

Mumford and Suominen in [8] and Newstead in [11] have considered the moduli problem of classifying the endomorphisms of finite-dimensional vector spaces. Using similar ideas we consider the moduli problem for endomorphisms of indecomposable semistable vector bundles over a compact connected Riemann surface of genus g ≥ 2.

MV-algebras were introduced by C. C. Chang [3] in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory [1].

Let G be a group and

a free presentation of G, i.e. a short exact sequence of groups with F free. Conjugation in F induces on = R/R', the abelianized normal subgroup R, the structure of a right G-module (if r∈ R, x∈ F then (r)(xπ) = x-1rxR'). The G-module is called the relation module determined by the presentation (1). For a detailed discussion of this subject we refer to Gruenberg [3].

A Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]

and the mehler formula [1, p. 377]

This suggests the following problem. Consider the equation

where fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,

A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).

1. We use Cassels's notation and define h (m, n), Q (m, n), Zh (s), Zh (1) – ZQ (1) and G (x, y) as in [1]. Rankin [5] proved that the Epstein zeta-function Zh (s) satisfies, for s ≧ 1·035, the

THEOREM. For s > 0, Zh (s) — ZQ (s) ≧ 0 with equality if and only ifh is equivalent to Q. Rankin then asked whether the theorem is true for all s > 1. Cassels [1] answered this question in the affirmative and proved further that the theorem is true for all s > 0.

In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory.