Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken?

The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum.

Let L be a positive invertible self-adjoint operator in L2(X;C). Using transference methods for locally bounded groups of operators we obtain multipliers for the group of complex powers Liu on vector-valued Lebesgue spaces. Using a Mellin inversion formula, we derive a sufficient condition for a function to be a multiplier of the semigroup e-tL on Lp(X;E) where E is a UMD Banach space and 1<p<∞.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

It is well known that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, semi-noetherian and artinian. Our main result shows that finite length is often achieved by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle.

We present some applications of monotonicity methods to the solution of certain nonscalar reaction-diffusion problems. In particular we prove existence under appropriate conditions and we introduce a convergent algorithm.

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

The space βℕ is the Stone-Čech compactification of the discrete space of positive integers. The set of elements of βℕ which are in the kernel of every continuous homomorpnism from βℕ to a topological group is a compact semigroup containing the idempotents. At first glance it would seem a good candidate for the smallest such semigroup. We produce an infinite nested sequence of smaller such semigroups all defined naturally in terms of addition on ℕ.

We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X), MAX(y)) = Γ2(X, Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of “free” unitaries.

We consider the finite-dimensional Banach spaces ℓp(n), where p>l. On these spaces there is a unique homogeneous semi-inner-product [.,.] consistent with the norm. If p≠2 this semi-inner product is not symmetric. We define a pair of vectors x and y to be biorthogonal if [x, y] = [y, x] = 0. For a given non-zero x, let τ(X) be the number of elements in a maximal linearly independent set of vectors biorthogonal to x. If p = 2 it is well-known that this number is n–1. The aim of this paper is to find τ(X) when p≠2. Our investigation shows that the situation differs from the Euclidean case in that the value of τ(X) can be either n–l or n –2. The ‘exceptional’ vectors x for which τ(x) = n –2 are characterised.

We show that every continuous nest of bounded multiplicity is unitarily equivalent to itself in a non-trivial way. Along the way, it is shown that no finite (measurable) partition of the unit interval can separate absolutely continuous homeomorphisms.

The structure of completely prime ideals in any structural matrix near-rings is determined. Partial descriptions are obtained for prime, nil, nilpotent, and locally nilpotent ideals of structural matrix near-rings. Their associated radicals are also studied in this paper.

Let be a polynomial with distinct real zeros whose separation is defined by δ(f) = min i≠j(ai-aj ). We establish upper estimates for δ(f′-kf) in terms of n, k, and δ(f). The results give sufficient conditions for the inverse operator (D – kl)−1 to preserve the reality of the zeros of a polynomial.

A semigroup is said to be completely regular if and only if each of its elements lies in a subgroup. It is shown that the algebra of a completely regular monoid (semigroup with identity) over a field of characteristic zero is directly finite.

Let S be a nontrivial monoid with zero and let F be a field. A sufficient condition, on the 0-simple principal factors of S and the characteristic of F, is given for the contracted monoid algebra of S over F to be directly finite.

We consider the distance between a fixed Hermitian operator B and the unitary orbit of another Hermitian operator A and show that in each Schatten p-class, 1<p<∞, critical points of this distance function are at operators commuting with B. As a consequence we obtain a perturbation bound for the eigenvalues of Hermitian operators in these Schatten classes.

In this paper we give a differential characterization of homogeneous Kähler submanifolds of complex projective spaces in terms of the existence of a tensor field, the homogeneous structure S. We show that for any m∈M, Sm determines a unitary representation whose orbit at m is a compact, complete Kähler submanifold which extends M. We consider the U(n) × U(N ~ n) (n = dim ℂM) module of the space of these tensors and we find its irreducible factors.

In this paper we show that away from umbilic points certain measures of the local reflectional symmetry of a surface in Euclidean 3-space are detected by the extrema of the sectional curvatures along lines of curvature. There are two types of reflectional symmetry, with one detected by the contact between the surface and spheres, and in this case the result is due to Porteous and is 20 years old. We show that an analogous result remains true for the second type of symmetry.