In a recent paper (1) Singh and Dhaliwal present an analysis which purports to solve a pair of dual integral equations which occur in crack theory and the object of this note is to demonstrate that their analysis is erroneous.

Recently, the first author and, independently, A. V. Jategaonkar have shown that every factor ring of U(g), the universal enveloping algebra of a finite dimensional complex Lie algebra, has a primary decomposition if g is solvable and almost algebraic. On the other hand, a suitable factor ring of U(SL(2, ℂ) fails to have a primary decomposition (1).

Let P be a semilattice. In (5), a ring T is called a supplementary semilattice sum of subringsTα (α∈P) if the following conditions hold: TαTβ⊆Tαβ for all α,β∈P, and for each α∈P. Thus, as an abelian group, T is a direct sum of the additive subgroups Tα (α∈P), and the multiplicative structure of T is strongly influenced by the semilattice P. Properties of these rings have been studied extensively in (2), (3), (5), and (6).

In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.

A graph in which each line is designated as either positive or negative is called a signed graph S. The sign of a cycle in S is a product of the signs of its lines. A signed graph in which every cycle is positive is called balanced. This concept was introduced by Harary in (3) and the following characterisation of balanced signed graphs was given.

A formally self-adjoint operator L is said to be of limit circle type at infinity if its highest order coefficients are zero-free and all solutions x of L(x) = 0 are square-integrable on [c, ∞) for some c. (We will drop “at infinity” in what follows.)

A 2-complex K is called almost-acyclic if H2(K) = 0 and H1(K) is torsion-free. This class of complexes was introduced in a previous paper (2), and applied to a problem of J. H. C. Whitehead concerning aspherical 2-complexes. In this note, the methods developed in (2) are used to study the finitely-generated subgroups of the fundamental group of an almost-acyclic 2-complex.

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

Consider the equation

where is the generalised derivative of x defined as follows: for every t ≧ T

.

Let R ⊊ T be an extension of commutative rings having the same identity. A. Wadsworth (10) studies the situation when R and T are integral domains, and all rings between R and T are Noetherian. In this case (R, T) is called a Noetherian pair. In a similar vein, E. Davis (4) studies normal pairs and I. Papick (8) shows when coherent pairs are Noetherian pairs.

A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.