Let X0, X1…Xn,… be a stationary Gaussian process. We give sufficient conditions for the expected number of real zeros of the polynomial Qn (z) = Σnj =o X jzj to be (2/ π)log n as n tends to infinity.

Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).

Let G be a countable torsion free finitely generated nilpotent group. Then the Fourier transform can be considered as a map from the space of bounded degree 1 random operators to the Fourier algebra A(G). In this paper, we recover the matrix elements of a positive random variable from the corresponding positive definite function in A(G) for such a group.

Let Λ be a lattice in R3 of determinant 1. Define the homogeneous minium of Λ as mn (Λ) = inf |u1, u2, u3| extended over all points (u1, u2, u3) of Λ other than the origin. It is shown that for any given (c1, c2, c3) in R3 there exists a point (u1, u2, u3) of Λ for which provided that ρσ > 1/64 if mn (Λ) = 0, and ρσ ≥1/16.81 if mn (A) > 0.

In this paper, the rings which have a torsion theory τ with associated torsion radical τ such that R/t(R) has a minimal τ-torsionfree cogenerator are studied. When τ is the trivial torsion theory these are precisely the left QF-3 rings. For τ = τL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on τL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators.

Certain central products of the binary polyhedral groups with finite cyclic groups are here shown to have presentations with two generators and two defining relations; this disproves a conjecture of the second author, stated in J. Austral. Math. Soc. Ser. A 38 (1985), 230–240.

In this article we establish an estimate for a sum over primes that is the analogue of an estimate for a sum over consecutive integers which has proved to be very useful in applications of exponential sums to problems in number theory.

This paper is concerned with non-trivial solvability in p–adic integers, for relatively large primes p, of a pair of additive equations of degree k > 1: where the coefficients a1,…, an, b1,…, bn are rational integers.

Our first theorem shows that the above equations have a non-trivial solution in p–adic integers if n > 4k and p > k6. The condition on n is best possible.

The later part of the paper obtains further information for the particular case k = 5. specifically we show that when k = 5 the above equations have a non-trivial solution in p–adic integers (a) for all p > 3061 if n ≥ 21; (b) for all p execpt p = 5, 11 if n ≥ 26.

It is well known that for a ring with identity the Brown-McCoy radical is the maximal small ideal. However, in certain subrings of complete matrix rings, which we call structural matrix rings, the maximal small and minimal essential ideals coincide.

In this paper we characterize a class of commutative and a class of non-commutative rings for which this coincidence occurs, namely quotients of Prüfer domains and structural matrix rings over Brown-McCoy semisimple rings. A similarity between these two classes is obtained.

In this paper a proper class of barrelled spaces which strictly contains the suprabarrelled spaces is considered. A closed graph theorem and some permanence properties are given. This allows us to prove the necessity of a condition of a theorem of S. A. Saxon and P. P. Narayanaswami by constructing an example of a non-suprabarrelled Baire-like space which is a dense subspace of a Fréchet space and is not an (LF)-space under any strong locally convex topology.

Individuals in a population which grows according to the rules defining the Markov branching process can mutate into novel allelic forms. We obtain some results about the time of the last mutation and the limiting frequency spectrum. In the present context these results refine certain results obtained in the discrete time case and they answer some conjectures still unresolved for the discrete time case.

Let λ be a simple eigenvalue of a bounded linear operator T on a Banach space X, and let (Tn) be a resolvent operator approximation of T. For large n, let Sn denote the reduced resolvent associated with Tn and λn, the simple eigenvalue of Tn near λ. It is shown that under the assumption that all the spectral points of T which are nearest to λ belong to the discrete spectrum of T. This is used to find error estimates for the Rayleigh-Schrödinger series for λ and ϕ with initial terms λn and ϕn, where P (respectively, ϕn) is an eigenvector of T (respectively, Tn) corresponding to λ (respectively, λn), and for the Kato-Rellich perturbation series for PPn, where P (respectively, Pn) is the spectral projection for T (respectively, Tn) associated with λ (respectively, λn).

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.

Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

This paper studies quintic residuacity of primes p of the form for which the expression for 4f modulo p given in the first volume of this journal becomes indeterminate, and replaces it by a much simpler expression.

We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (1/p)|G| elements to their cubes, where p is the smallest prime dividing |G|. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

The BMO norm of f is equivalent to where Pt is the Poisson kernel. In this note, we show that Pt can be replaced by a nonnegative radial function h, which is positive in a neighbourhood of 0, with and , where h is the least decreasing radial majorant of h.

We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, HαHβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).