Many problems involving the solution of partial differential equations require the solution over a finite region with fixed boundaries on which conditions are prescribed. It is a well known fact that the numerical solution of many such problems requires additional conditions on these boundaries and these conditions must be chosen to ensure stability. This problem has been considered by, amongst others, Kreiss [11, 12, 13], Osher [16, 17], Gustafsson et al. [9] Gottlieb and Tarkel [7] and Burns [1]

Let R be a ring with identity and let Ω be a totally ordered set. Let Ω′ be a totally ordered set which is disjoint from and equipotent to Ω′ with ′:Ω→Ω′ an order preserving bijection. Define Ω1=Ω∪Ω′ and let Ω1, be totally ordered by inheriting the order from Ω and Ω′ and with ω<λ for all ω∈Ω and λ′∈Ω′. Let M be the free R-module R(Ω1).(We define the alternate bilinear form (*, *) on M by

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.

We are concerned here with question: to what extent can the structure of a group G be recaptured from information about the structure of its group of automorphismsAut G? For example, one might try to find all groups which have some specific group astheir (full) automorphism group, a point of view adopted by Iyer in a recent paper [5]. Nothing is known about this question in general except the result of Nagrebeckü [7] that there are only finitely many finite groups with a given group as automorphismgroup.

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:

Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

There are a number of classes of distributive lattices whose members can be characterised as the coproduct A * L of suitable distributive lattices A and L. For example, Post algebras [1], pseudo-Post algebras [4], Post Lalgebras ([6], [8[9]) and the lattices [D]n of [4]. Moreover, the α-completeness and α-representability of some(though not all) of these algebras have been investigated in [7], [2], [6], and [10].

Recently A. D. Sands [7] solved a problem posed in [6], and characterised the semisimple classes of associative rings as classes being regular, coinductive and closed under extensions. It is the purpose of this note to prove the same assertion for alternative rings. This result is perhaps not surprising, nevertheless its proof cannot be considered an easy one, and it requires a technique of dealing with ideals of ideals. In addition, semisimple classes of hereditary radicals and those of supernilpotent radicals will be characterised as easy consequences of our theorem.

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).

A near-ring N is a set N with binary operations + and · satisfying the conditions (1) (N, +) is a group, (2) (N, ·) is a semigroup, and (3) · satisfies one of the distributive laws over +. (N, +) need not be an abelian group and if the left distributive law holds, i.e. a · (b + c) = a · b + a · c for all a, b, c ∈ N, then N is called a left near-ring. Similarly, the notion of a right near-ring may be defined.

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

Let K be a field of characteristic p>0, G a finite p-solvable group, P a p-Sylow subgroup of G of order pa, KG the group algebra of G over K, and J(KG) the Jacobson radical of KG. In the present paper we study the nilpotency index t(G) of J(KG), which is the least positive integer t with J(KG)t= 0. Since J(EG) = E⊗KJ(KG) for any extension field E of K (cf. [7, Proposition 12.11]), we may assume that K is algebraically closed.

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then

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.

Let T: A → B be a linear operator between two Banach algebras A and B. The basic problem in the theory of automatic continuity is to find algebraic conditions on T, A, and B which ensure that T is continuous. As a means to study continuity properties of T the separating space of T has played a crucial role. It is defined as

In studying the cohomology of the symmetric groups and its applications in topology one is led to certain questions concerning the representation rings of special subgroups of . In this note we calculate the Chern classes of the regular representation of (Z/p)n where p is a fixed odd prime in terms of certain modular invariants first described by L. E. Dickson in 1911. In a later paper [9] we apply these results to study the odd primary torsion in the PL cobordism ring. Some indications of this application are given in Sections 10–12 where we apply the result above to obtain information about the cohomology of . After circulation of this note in preprint form we learned that H. Mui [10], has also proved Theorem 6.2.

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.

It has been proved, see [1], that a closed infinite dimensional subspace of c0 is isomorphic to c0 if and only if it is the range of a bounded linear projection. In [6] we proved half of the order-theoretic analogue of this result. In fact we showed that an infinite dimensional subspace of c0 which is the range of a positive projection is order-isomorphic to c0. We left open the question whether the converse holds also true. In this paper we answer this question negatively by providing an example in Section 4. In Section 3 we give necessary and sufficient conditions in order that an ordered-subspace of c0 be the range of a positive projection.