We give a survey on the use of Eva Kallin's lemma. This lemma gives a condition on two polynomially convex sets in [Copf ]n under which their union is polynomially convex. This result has proved to be a useful tool in different areas of complex function theory of several variables, for instance in the study of polynomial convexity of the union of totally real surfaces, and in approximation problems in function algebras.

Any doubly stochastic finite matrix with integer entries is the sum of permutation matrices. A lower bound on how many of the permutations are equal is obtained.

The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq.

We derive an identity connecting a theta function and a sum of Lambert series. As a consequence of this identity, we deduce a number of results of Jacobi, Dirichlet, Lorenz, Ramanujan and Rademacher.

Let X be a nonsingular real algebraic variety, and let X([Copf ]) be its nonsingular projective complexification. We study the following question: when does the homology class represented by X in the homology group of X([Copf ]) vanish? An application concerning the structure of the set of regular mappings is given.

We describe a general construction of a module A from a given module B such that Ext(B, A) = 0, and we apply it to answer several questions on splitters, cotorsion theories and saturated rings.

We characterize the extremal functions for Heisenberg type inequalities of the form

(formula here)

The method also yields an alternative and elementary derivation of the sharp constant in Nash's inequality, originally found by Carlen and Loss.

We prove that for an arbitrary measurable set A ⊂ ℝ2 and a σ-finite Borel measure μ on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L containing the point is a residual set, and, moreover, μ(A) = μ({∪[lscr ] : [lscr ] ∈ L}). We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C1, C2 for which there is a set which is visible from C1 and invisible from C2.

It is shown that if a C*-algebra A contains a semi-scattered C*-algebra B such that the pure states of B and the zero functional extend uniquely to A, and the canonical mapping Bˆ → Â is injective, then there exists a (unique) projection of norm one R : A → B. In certain circumstances, the conditional expectation R can be effected by a unitary averaging process using unitary elements in the centre of the multiplier algebra M(B).

Let (G, X) be a locally compact transformation group in which G acts freely on X. We show that the associated transformation-group C*-algebra C0(X) [rtimes ] G is a Fell algebra if and only if X is a Cartan G-space.

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum σ(a) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies

(formula here)

We prove that the converse is also true, provided that A is semisimple.

We show that any gauge-equivalence class of solutions (∇, Φ) of the monopole equation F∇ = *d∇Φ, defined on an open set Ω of S3 (equipped with its standard metric), induces canonically a holomorphic function on the complex 2-dimensional space of all geodesics on S3 entirely contained in Ω.

Let G be a compact connected Lie group and K a maximal rank subgroup of G. The homogeneous space G/K has the S1-action defined by left translations induced from a homomorphism from S1 to G. In this paper, we study a problem on the realization of some deformation of the cohomology algebra H*(G/K; [ ]p) by the S1-equivariant cohomology of G/K. In consequence, for the case where G is a classical Lie group, it follows that there exists at most one essentially different homomorphism from S1 to G which realizes a given deformation, and that the homomorphism is controlled by an appropriate equation in one indeterminate.

The purpose of this note is to establish a uniform estimate for the mass function ℙ(Sm = y) of an integer-valued random walk when y → ∞ and (y − mμ)/√m → ∞, where μ is the mean of the step distribution. (The local central limit theorem provides such an estimate when (y − mμ)/√m is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that [sum ]∞n=0nκ′p(−n) < ∞ for some κ′ > 2. From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space–time Martin boundary of certain random walks.

The local Caratheodory conjecture on the index of an isolated singularity of the principal foliations in surface theory is equivalent to a conjecture of Loewner on the index of the isolated singularities of the Hessian operator of a smooth function in the plane. Here we prove the latter conjecture for a special class of functions.

Dr Jack Howlett, who died on 5 May 1999 at the age of 86, a Founder Fellow of the Institute of Mathematics and its Applications, was a mathematician with a special interest in numerical analysis who early recognised the power of computing methods and who strongly influenced the development of some mechanical computing machines and electronic computers as we know them today. The widespread advance of the use of computer models in all scientific disciplines was made possible thanks to the efforts of a small number of mathematicians who laid the foundations of modern numerical analysis during the late 1930s to early 1960s, a period which covered the greater part of Jack's working life and in which he was a star player.