A permutation group is quasiregular if it acts regularly on each of its orbits (i.e. the stabiliser of an element fixes every other element in its orbit). So, in particular, any permutation representation of an abelian or hamiltonian group must be quasiregular.

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

The problem of Josephus is the following. We are given two positive integers n, q. There are n places arranged around a circle, and numbered clockwise 1, 2, …, n. Each of n people takes one of the places, then (please excuse this, but we didn't invent the problem!) every qth one is executed, until just one remains. More precisely, the occupant of place q is ‘removed’ first, and in general, if some place j has just been vacated, then the qth one of the places clockwise around from j that are still occupied will be vacated next. One question is this: if you would like to be the last survivor, then into what place should you go initially? We denote the answer to this question by Jq(n). For example, if n = 5 and q = 2, the order of execution is 2, 4, 1, 5, 3, and J2(5) = 3. Other questions have been raised about the problem, and it has an extensive literature (see [1]–[10]). In this paper we deal with the Jq(n)'s.

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

In the theory of Hilbert space operators an important question is whether an operator is subnormal [3], [4], [7], [8]. A densely defined linear operator S in a complex Hilbert space H is subnormal if there exists a normal operator N in a complex Hilbert space K ⊃ H such that S ⊂ N.

In this paper we obtain the general solution of the equation

whose kernel is

where 0<α, β<1, μ, η, σ > 0, δ>0, are real parameters, z = max(x, y), φ(t) and g(y) are prescribed functions and ƒ(x) is to be determined.

Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), …, c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer.

Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

If (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.

There are two well-known methods to build up algebras from given algebras, the direct and inverse limits, and a systematic account of these constructions may be found in [2]. It is known that every algebra can be represented as a direct limit of finitely generated algebras although in some cases the representation is trivial. Furthermore, Haimo [3] has established a certain inverse limit representation for the class of all infinite Boolean algebras which generalises, in actual fact, to the class of all infinite lattices with 1. The purpose of this note is to exhibit a certain nontrivial inverse limit representation which is peculiar to the class of infinite, complete Boolean algebras.

Let A = (aij) be an n × n complex matrix. The permanent of this matrix is

where the sum is taken over all permutations p of the set {1, …, n}.

In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.

The space of Schwartz distributions on the unit circle Г in the plane is topologically a considerable generalization of the space of regular, finite Borel measures on Г. However, the order structure of is usually taken to be the same as that of : there are no “positive” distributions which are not measures. This perhaps warrants consideration, since the order structure of generates its topology. In this paper we construct a system of order structures for which is a more natural complement in the intermediate stages to the topology of and which provides an interpretation of with its Schwartz topology as a quotient of a generalized base norm space V′. Where denotes the space of continuous functions on Г with its supremum norm topology, V′ is the dual of . The space contains the infinitely differentiable functions on Г with their usual topology, and (via the pointwise ordering on ) in its product ordering is realized as a generalized order unit space. Some consequences for harmonic functions are discussed.

For any group G, denote by φf(G respectively L(G)) the intersection of all maximal subgroups of finite index (respectively finite nonprime index) in G, with the usual provision that the subgroup concerned equals Gif no such maximals exist. The subgroup φf(G) was discussed in [1] in connection with a property v possessed by certain groups: a group G has v if and only if every nonnilpotent, normal subgroup of G has a finite, nonnilpotent G-image. It was shown there, for instance, that G/φf(G) has v for all groups G. The subgroup L(G), in the case where G is finite, was investigated at some length in [3], one of the main results being that L(G) is supersoluble. (A published proof of this result appears as Theorem 3 of [4]). The present paper is concerned with the role of L(G) in groups G having property v or a related property a, the definition of which is obtained by replacing “nonnilpotent” by “nonsupersoluble” in the definition of v. We also present a result (namely Theorem 4) which displays a close relationship between the subgroups L(G) and φf(G) in an arbitrary group G. Some of the results for finite groups in [3] are found to hold with rather weaker hypotheses and, in fact, remain true for groups with v or a. We recall that if a group has a it also has v ([2]Theorem 2) but not conversely. For example, G = (x, y: y-1xy = x2)has v but not a. It is a well-known result of Gaschütz ([8], 5.2.15) that, in a finite group G, if His a normal subgroup containing φ(G) such that H/φ(G) is nilpotent than His nilpotent. This remains true in the case where G is any group with v [1, Proposition 1]. Our first result is in a similar vein and is a generalization of Theorem 9 of [7] and Theorem 1.2.9 of [3], the latter of which states that, for a finite group G, if G/L(G) is supersoluble, then so is G.

R. A. Rankin [3] considered the problem of finding, for each integer n ≧ 3, a sequence of positive integers containing no n−term geometric progression. He constructed such sets Bn having asymptotic density

For example A3 ≑ 0·71975, A4 ≑ 0·8626, and An→1 as n → ∞.

The existence of r-regular graphs such that each edge lies in exactly t triangles, for given integers t < r, is studied. If t is sufficiently close to r then each such connected graph has to be the complete multipartite graph. Relations to graphs with isomorphic neighborhoods are also considered.

In this note homomorphisms of (2, 3, n) = 〈x, y: x2 = y3 = (xy)n = 1) into PSL3(q) are considered. Of particular interest is (2, 3, 7) whose finite factors are referred to as Hurwitz groups. It is known (see [3]) that for certain q, PSL2(q) is a Hurwitz group, so that one might suppose that PSL3(q) is a natural place to search for new Hurwitz groups. This intuition turns out to be ill-founded, for as we shall see all Hurwitz subgroups of PSL3(q) have already been discovered in [3].