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]

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).

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.

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.

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.

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

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.

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

Throughout this paper all near-rings will be left distributive. We shall denote the zero-symmetric part of a near-ring N by N0. The fact that the near-rings under consideration may not be zero-symmetric has important consequences for what follows, particularly the results of the last section.

A (2, 3, 7)-group is a group generated by two elements, one an involution and the other of order 3, whose product has order 7. Known finite simple examples of such groups are PSL(2, 7), PSL(2, p) where p is prime and p ≡ ±1 (mod 7), PSL(2, p3) where p is prime and p≢0, ±1 (mod 7), groups of Ree type of order q3(q3 + 1)(q − 1) where q = 32n+1 and n > 0, the sporadic group of order 23 · 3 · 5 · 7 · 11 · 19 discovered by Janko, and the Hall–Janko–Wales group of order 27 · 33 · 52 · 7 [4, 2]. G. Higman in an unpublished paper has shown that every sufficiently large alternating group is a (2, 3, 7)-group. Here we show that the sporadic group Co3 discovered by Conway [1] is a (2, 3, 7)-group.

This paper contains a new proof of a theorem of D. A. R. Wallace [5] in case of a p-solvable group. An alternative proof has been given by K. Motose [4].

Let G be a finite group, F a field with prime characteristic p, P a Sylow p-subgroup of G. JFG will denote the Jacobson radical of FG.

In this paper we will show that large classes of near-rings are embeddable into (or even isomorphic to) near-rings Gν[x] of polynomials over G in a suitably chosen variety ν of Ω-groups.

For near-rings see [2], for polynomials [1]; for near-rings of polynomials over Ω-groups see [3] [4], [5].

Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.

Let X be a (real or complex) Banach space, and let K be a linear subspace of its dual X*. Denote by K1 the unit ball in K. If K is not weak-star closed, then the Krein-Šmul'yan theorem says that K1 is not weak-star closed. What, however, is its weak-star closure? Inner and outer extimates were obtained in [3] for the special case where K is a hyperplane. In the present paper we generalise these estimates to arbitrary linear subspaces. For f to belong to w*(K1) it is sufficient to have |φ(f)| ≦ 1 −∥f∥ for all φ in K0 (the annihilator of K in X**) with dist (φ, X)≦1. It is necessary to have |φ(f)| ≦ 1 + ∥f∥ for all such φ. These estimates depend on the action of each φ in K0 separately, which will often make them hard to apply in practice; in both cases, we derive a second estimate, expressed only in terms of certain constants that describe the position of X and K0 in X**.

In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the following

Theorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if

(i) , the ring of integers modulo a prime power pk,

(ii) R ≅ N2, the null ring on a cyclic group of order 2, or

(iii) R is a rigid field of characteristic zero.

Using results obtained by J. W. L. Glaisher [1, 2] for the number of representations Rr,s(n) of n as a sum of r odd and s even squares, formulae are derived for the number of Cayley integers of given norm n in certain orders ℴ. When computer generating order elements of given norm, the formulae can be used to verify that all the required elements have been obtained.