The conference Groups–St Andrews 1985 was held at the University of St Andrews from 27 July to 10 August 1985. The conference received financial support from the Edinburgh Mathematical Society, the London Mathematical Society and the British Council. There were 366 participants from 43 countries registered for the conference. Although the conference did not specialize in a particular area of group theory, a glance at Mathematical Reviews shows that the work of the participants is mainly under classifications 20D, 20E and 20F. In part this is because the conference followed an earlier conference [6] which was primarily based on topics falling under 20F.

Recently Collins (2) has studied triple series equations involving series of Legendre polynomials. These equations arise in the study of mixed boundary value problems and can be regarded as extensions of the dual series equations considered by Collins in (1).

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**.

Simple and subsimple objects were introduced in [6]. It was shown that if there are enough simple objects in a category , then there is no room for injectives in . This idea was exploited in [6] and [2] to show that several classes of groups, rings and classes belonging to other categories do not possess non-trivial injectives or retracts. In this note, the above results will be strengthened by introducing a weaker condition than subsimple of [6]. As a consequence, and by employing some embedding theorems, we show that some important classes do not possess non-trivial retracts.

All the categories are assumed to have a zero object.

Let R = ⊕n∈zRn be a ℤ-graded commutative Noetherian ring and let M be a ℤ-graded R-module. S. Goto and K. Watanabe introduced the graded Cousin complex *C(M)* for M, a complex of graded R-modules. Also one can ignore the grading on M and construct the Cousin complex C(M)* for M, discussed in earlier papers by the second author. The main results in this paper are that *C(M)* can be considered as a subcomplex of C(M)* and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)* is exact if and only if *C(M)* is (and this is the case precisely when M is Cohen-Macaulay).

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.

In the first paper of this series (L.Q.I)1 we have shown that the logarithmetic LQ of a finite quasigroup Q is a quasigroup with respect to addition and that it is a subdirect union of the logarithmetics of the elements of Q.

In this second part we shall discuss further the structure of LQ in its additive aspect, and obtain results concerning the order N of LQ. For plain quasigroups (§3) the structure of LQ(-\-) is studied in more detail and it is shown that N is a power of n, the order of Q.

The integral evaluated in this note was suggested by the famous one connected with the Poincaré polynomials of the classical groups (see (1)).

Let X be an n × n matrix whose elements depend on k parameters. Denote by a manifold in Euclidean space of dimension n2, with the property that if X ∈ , then so does XI−i for 1≦i≦n, where I−i is the unit matrix I altered by a minus sign in the (i, i)th place. Suppose further that there exists on a measure which is invariant under the transformation X→XI−i. Such manifolds and measures exist. For example (see (2), § 5), the set of all proper and improper n×n orthogonal matrices H is such a manifold, the H depending on ½n(n−1) parameters because of the orthogonality and normality of the columns of H. Since the set of all H is a compact topological group, an invariant measure exists.

The standard method of establishing rigidly the tests for a maximum or minimum value of a function of two independent variables, depending as it does on the use of Taylor's Theorem and on a very critical consideration of the Remainder in that theorem, presents difficulties so considerable that it is not surprising that most text-books on the Calculus frankly decline to enter on the discussion, and assume the necessity and sufficiency of the well known Lagrange's Condition. It is the object of this paper to show that a satisfactory proof of the tests may be given, from the purely geometrical standpoint, without recourse to Taylor's Theorem. The method requires only an elementary knowledge of the process of changing the independent variables in partial derivatives, and may therefore be introduced comparatively early in the Calculus course.

Let R be an arbitrary near-ring and define the multiplicative centre Z(R) by

In previous papers (2,3,5) we have established additive or multiplicative commutativity for various near-rings R in which selected elements were restricted to lie in Z(R); the near-rings involved were usually distributively-generated (d-g) and were frequently assumed to have a multiplicative identity element as well.

The class of non-associative algebras over a field of characteristic zero named in the title is studied using a result of Ouattara [9]. As an application, the differential equation for overlapping generations in the time-continuous model is solved.

Given amalgams of semigroups [U; Ti] and [U;Si] with U⊆Ti⊆Si, it was proved by Howie in [6] that need not be embedded in . We use the homological techniques developed by Renshaw in [7, 9] and study three new conditions each of which imply the embeddability of the above free products.

We study the profile near quenching time for the solutions of the first and second initial boundary value problems (IBVP) for a semilinear heat equation. Under certain conditions, one-point quenching occurs for both first and second IBVPs. Furthermore, we derive the asymptotic self-similar quenching rate for both problems.

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