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.

In some biological problems, a parasite-host system is immersed in a toxic solution in order to kill off the parasite while leaving the host as little affected as possible. A problem of this type was considered by Clements and Edelstein (2), who treated both the host and the parasite as cylindrical in shape. In a separate paper Clements (1) considered the corresponding problem where the parasite is spherical and the host cylindrical. In both cases, the concentration of the toxic solution at the boundary is taken as having a constant value, c, and the penetration of the poison into the host and parasite is treated as a linear diffusion problem with an appropriate diffusion coefficient. It is assumed also that the host and the parasite are free of the toxic substance initially. The process is terminated when the average concentration in the parasite reaches a lethal level, τ, and the problem is to see how M, the average concentration in the host, is affected by the choice of c (for a given value of τ).

If an oriented manifold M immerses in codimension k, then the normal bundle has dimension k such that its Euler class χ є Hk(M; Z) and χ2 є H2k(M; Z). (Cf. (3)).

If M is the complex Grassmann manifold G2(Cn) of 2-planes in Cn (n = 4, 5,…, 15, 17), then dim M = 4n – 8 ≡ d and we shall show that although M immerses in R2d–1 by classical results (3), M does not immerse in Rd+d/2.

The same result was obtained for n = 4 and 5 by Connell (2) and for n = 6 and 7 by the author (6). The nonimmersion results of this paper are new for n = 8, 9, …, 15, 17 and they are an improvement over the result for the general G2(Cn) obtained in (5). In this paper, we use generators of the cohomology ring of G2(Cn) different from those used in (2) and (6) and this simplifies the calculations considerably.

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).

Throughout this paper we shall work in the class of associative rings. In (4) it was shown that a class of rings is a semisimple class if and only if it is closed under extensions and ideals and is coinductive. This establishes a duality between radical classes and semisimple classes. This result has been proved also for classes of alternative rings in (2). In the original work by Kuros (1) on this subject two conditions were used for semisimple classes, one of which was weaker than the assumption that the class is closed under ideals. This condition is that every non-zero ideal of a ring in the class should have a non-zero homomorphic image in the class. It is natural to ask whether in the above set of conditions the condition of being closed under ideals can be replaced by this weaker condition. This question is raised in (3) and in (5) but it is suggested there that, in order to compensate, the coinductive condition be replaced by the stronger condition that the class is closed under subdirect sums. In fact we shall show that the weaker condition may be used without needing to replace the coinductive condition. We also give examples to show independence relations among these conditions.

On a semigroup S the relation ℒ* is defined by the rule that (a, b) ∈ ℒ* if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. It is well known that for a monoid S, every principal right ideal is projective if and only if each ℒ*-class of S contains an idempotent. Following (6) we say that a semigroup with or without an identity in which each ℒ*-class contains an idempotent and the idempotents commute is right adequate. A right adequate semigroup S in which eS ∩ aS = eaS for any e2 = e, a ∈ S is called right type A. This class of semigroups is studied in (5).

The study of the cohomological dimensions of algebraic varieties has produced some interesting results and problems in local algebra: the general local problem is that posed by Hartshorne and Speiser in (4, p. 57). We consider a (commutative, Noetherian) local ring A (with identity), a proper ideal a of A, and ask the following question.

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).

One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined by

Ramanujan (9) showed that when n is large, θn possesses the asymptotic expansion

The first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined by

then πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansion

A generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by

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.

The classification of the normal subgroups of the infinite general linear group GL(Ω, R) has received much attention and has been studied in, for example, (6), (4) and (2). The main theorem of (6) gives a complete classification of the normal subgroups of GL(Ω, R) when R is a division ring, while the results of (2) require that R satisfies certain finiteness conditions. The object of this paper is to produce a classification, along the lines of that given by Wilson in (7) or by Bass in (3) in the finite dimensional case, that does not require any finiteness assumptions. However, when R is Noetherian, the classification given here reduces to that given in (2).

We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.

Let A be a matrix over a field Φ partitioned as follows

where A11 is n×n and A22 is m×m. The objective of the present paper is to give further results on the problems mentioned in Section 1 of (3). Concretely we shall consider the following question: “we prescribe the characteristic polynomial f(λ) = λn+m−c1λn+m−1 + … of A and the principal blocks A11, A22. Find a necessary and sufficient condition for the existence of A satisfying these prescribed conditions”.

The definition of a suitable Jordan analogue of C*-algebras (which we call JB*-algebras in this paper) was recently suggested by Kaplansky (see (26)). The theory of unital JB*-algebras is now comparatively well understood due to the work of Alfsen, Shultz and Størmer (1) from which a Gelfand-Neumark theorem for unital JB*-algebras can be obtained (26). Independently, from work on simply connected symmetric complex Banach manifolds with base point, Kaup introduced the definition of C*-triple systems in (14) and subsequently in (7) it was shown that every unital JB*-algebra is a C*-triple system. In this paper, we wish to extend this result to show that every JB*-algebra is a C*-triple system.

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.

It is a fundamental fact in the theory of radicals of associative rings that if S is a radical and I is a two-sided ideal of R then S(I)⊆S(R). In view of this result it seems to be interesting to investigate radicals satisfying such or similar connections for other type of subrings. There are many works devoted to similar problems (2, 8, 8, 10). In this paper we try to get a uniform description of some facts in this area.

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identities

hold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.

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.)

Let denote the class of finite-dimensional Lie algebras L (over a fixed, but arbitrary, field F) all of whose maximal subalgebras have codimension 1 in L. In (2) Barnes proved that the solvable algebras in are precisely the supersolvable ones. The purpose of this paper is to extend this result and to give a characterisation of all of the algebras in . Throughout we shall place no restrictions on the underlying field of the Lie algebra.

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then

(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that

(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that

(iii) every prespectral operator on X is spectral.

(See also (1, Theorem 6.11, p. 165) for (iii).)