In [10] Segal shows that the groups of units in certain ordinary cohomology rings are the zeroth terms of generalised cohomology theories. Geometric methods then give a multiplicative transfer on these groups of units for fibrations with finite fibres; see Kahn and Priddy [6] and Adams ([1], 4). On the other hand Evens [5] by manipulations with cochains has constructed a multiplicative transfer in the cohomology of a group G and a subgroup H of finite index. Now it is well known that the algebraic cohomology of G and H can be identified with the topological cohomology of their classifying spaces BG and BH, and that there is a fibration BH→BG with finite fibres. This suggests that Evens' algebraic transfer and the geometric transfer derived from Segal's work may be related. In the present paper I confirm this by constructing a common generalisation; I also describe some of its properties.

If M is a mathematical system and End M is the set of singular endomorphisms of M, then End M forms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroup SM of EndM generated by the idempotents EM of End M for different systems M. The first of these was by J. M. Howie [4]; here the case of M being an unstructured set X was considered. Howie showed that if X is finite, then End X = Sx.

In [3], [8], and [2], it was shown that if is an essentially Hermitian operator on l P, 1≦ p<∞, or on Lp[0,1], 1< p<∞, then T is a compact perturbation of a Hermitian operator. In [1], this result was established for operators on Orlicz sequence space l M, where 2∉[α M,β M] (the associated interval for M). In that same paper, it was conjectured that this result does not in general hold if 2∈[α M,β M]. In this paper, we show that this conjecture is correct by exhibiting an Orlicz sequence space l M and an essentially Hermitian operator on l M which is not a compact perturbation of a Hermitian operator.

Let V be a group, written additively but not necessarily abelian, and let S be a semigroup of endomorphisms of V. The set C( S; V)={ f: V→ V| fσ=σ f for all σ∈ S and f(0)=0} forms a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centraliser near-ring determined by S and V. Centraliser near-rings are very general, for if N is any zero-symmetric near-ring with 1 then there exists a group V and a semigroup S of endomorphisms of V such that N≃ C( S; V).

Rings and distributive lattices can both be considered as semirings with commutative regular addition. Within this framework we can consider subdirect products of rings and distributive lattices. We may also require that the semirings with these restrictions are regarded as algebras with two binary operations and the unary operation of additive inversion (within the additive subgroup of the semiring). We can also consider distributive lattices with the two binary operations and the identity mapping as the unary operation. This makes it possible to speak of the join of ring varieties and distributive lattices. We restrict the ring varieties in order that their join with distributive lattices consist only of subdirect products. In certain cases these subdirect products can be obtained via a general construction of semirings by means of rings and distributive lattices.

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae

Let a meromorphic differential equation

be given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expression

where F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, and

is a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.