In recent years interest in the mixed boundary value problems of mathematical physics has increased appreciably because of various applications. The mixed boundary value problems for simply-connected regions have been investigated widely and it can reasonably be hoped that within a short time the theory will reach a satisfactory stage. It appears, however, that very few problems for multiply-connected domains have been solved. Recently Srivastav [2] has considered the problem of rinding an axisymmetric potential function for a half space with a cylindrical cavity subject to mixed type boundary conditions. In a subsequent paper [1], Srivastav extends the analysis to the asymmetric problem and formulates the problem in terms of dual integral equations involving Bessel functions of the first and second kinds whose solution leads to the solution of the potential problem. The latter paper, however, involves heavy manipulations and complicated contour integrals.

Let M(c) denote a 4n-dimensional quaternion space form of quaternion sectional curvature c, and let P(H) denote the 4n-dimensional quaternion projective space of constant quaternion sectional curvature 4. Let N be an n-dimensional Riemannian manifold isometrically immersed in M(c). We call N a totally real submanifold of M(c) if each tangent 2-plane of N is mapped into a totally real plane in M (c). B. Y. Chen and C. S. Houh proved in [1].

Let A be a closed subset of a topological space X and f a continuous mapping of A into X with the following two properties:

1.1. f {Fr (A)} & f {Int (A)} are disjoint.

1.2. The mapping f* = f | Fr (A) is 1–1.

It is proved in [5], that if X is the euclidean n–sphere Sn = {x; x e Rn+1 and ― x ― = 1}, then

1.3. f {Fr(A)} = Fr {f(A)}.

[ Hence f{Int (A)} = Int {f(A)}].

Let μ be an isomorphism which maps a subgroup A of the group G onto a second subgroup B (not necessarily distinct from A) of G; then μ is called a partial automorphism of G. If A coincides with G, that is if the isomorphism is defined on the whole of G, we speak of a total automorphism; this is what is usually called an automorphism of G. A partial (or total) automorphism μ,* extends or continues a partial automorphism μ if μ* is defined for, at least, all those elements for which μ is defined, and moreover μ* coincides with μ where μ is defined.

The present paper is closely related to a paper with the same title by A. M. Macbeath [3]. We use many notions which are defined there for a measure-space; nevertheless we define them once more because we consider the slightly different case of a measure-ring.

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.

Among such forms, let . The Epstein zeta function of f is denned to be

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,

We prove a corresponding result for theta functions. For real α > 0, let

This function satisfies the functional equation

(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

The Dedekind eta-function is defined for any τ in the upper half-plane by

where x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a function

where N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ by

when a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier series

then we define a Hecke operator Tp by

where

and

In this paper we consider mappings induced by matrix multiplication which are defined on lattices of matrices whose coordinates come from a fixed orthomodular lattice L (i.e. a lattice with an orthocomplementation denoted by ′ in which a ≦ b ⇒ a ∨ (a′ ∧ b) = b). will denote the set of all m × n matrices over L with partial order and lattice operations defined coordinatewise. For conformal matrices A and B the (i,j)th coordinate of the matrix product AB is defined to be (AB)ij = Vk(Aik ∧ BkJ). We assume familiarity with the notation and results of [1]. is an orthomodular lattice and the (lattice) centre of is defined as , where we say that A commutes with B and write . In § 1 it is shown that mappings from into characterized by right multiplication X → XP (P ∈ ) are residuated if and only if p ∈ ℘ (). (Similarly for left multiplication.) This result is used to show the existence of residuated pairs. Hence, in § 2 we are able to extend a result of Blyth [3] which relates invertible and cancellable matrices (see Theorem 3 and its corollaries). Finally, for right (left) multiplication mappings, characterizations are given in § 3 for closure operators, quantifiers, range closed mappings, and Sasaki projections.

[7] Sands raised the following questions:

(1) Must a hereditary radical which is right strong be left strong?

(2) Must a right hereditary radical be left hereditary?

(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?

Negative answers to questions (1) and (2) were given by Beidar [1].

In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of primecharacteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).

(4) There are faithful fully reducible representations of every Lie-algebra of primecharacteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w1*w2 of two distributions w1 and w2 was defined as the limit ofithe sequence {wln*w2n}, provided the limit w exists in the sense that

for all fine functions ф in the terminology of Jones [2], where

w1n(x) = wl(x)τ(x/n), W2n(x) = w2(x)τ(x/n)

and τ is an infinitely differentiable function satisfying the following conditions:

(i) τ(x) = τ(—x),

(ii)0 ≤ τ (x) ≤ l,

(iii)τ (x) = l for |x| ≤ ½,

(iv) τ (x) = 0 for |x| ≥ 1.

The definition of Ωf,a on p. 57 should read a

In 1856 Hermite showed how to determine by purely rational operations the number of zeros of a given polynomial lying in a specified half plane [1]: one inspects the signature of a certain Hermitian form. This type of result is still of interest for practical applications, and several authors have provided alternatives to Hermite's original, highly computational proof (for example [2, 3]). Recently V. Pták and the author gave a simple matrix-theoretic proof and generalization of a class of Hermite-type theorems [4]. This class included the Schur-Cohn test for zeros in a circle, but not, to our regret, the original theorem of Hermite. The purpose of this note is to show that a slight modification of our method does indeed provide a simple proof of Hermite's theorem.

In his recent study of free inverse semigroups, Munn [2] introduced and used extensively the concept of a word-tree. In this note the number of such trees is found.

In this paper we use the Bourbaki conventions for rings and modules: all rings are associative but not necessarily commutative and have an identity element; all modules are unital.

§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the form

The purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.