If E is a Hausdorff barrelled space, which does not already have its finest locally convex topology, then the continuous dual E′ may be enlarged within the algebraic dual E*. Robertson and Yeomans [10] have recently investigated whether E can retain the barrelled property under such enlargements. Whereas finite-dimensional enlargements of the dual preserve barrelledness, they have shown that this is not always so for countable-dimensional enlargements E′+M. In fact, if E contains an infinitedimensional bounded set, there always exists a countable-dimensional M for which the Mackey topology τ(E, E′+M) is not barrelled [10, Theorem 2].

Let K be a topological field. On introducing the vector-space topology on the n × n matrices over K, it becomes clear that the determinant map φ enjoys the following properties:

(A) φ is a continuous surjective homomorphism from GLn(K) to K*,

(B) φ(μa) = μn φ(a) for each non-zero φ in K, and all a in GLn(K).

A completely regular semigroup is a semigroup that is a union of groups. The aim here is to provide an alternative characterization of the free completely regular semigroup Fcrx on a set X to that given by J. A. Gerhard in [3, 4].

Although the structure theory for completely regular semigroups was initiated in 1941 [1] by A. H. Clifford it was not until 1968 that it was shown by D. B. McAlister [5] that Fcrx exists. More recently, in [7], M. Petrich demonstrated the existence of Fcrx by showing that completely regular semigroups form a variety of unary semigroups (that is, semigroups with the additional operation of inversion).

This note extends classical results on certain Galois groups attached to onedimensional algebraic groups. We prove that the fields arising from the division of a fixed set of rational points on the product of an elliptic curve by the multiplicative group are as “large” as possible.

Let S be a prime Noetherian ring and G a finite group acting on 5 such that Gis x-outer on S. We give sufficient conditions for the skew group ring S * Gto be a prime maximal order. If we impose the further hypothesis that the order of Gbe a unit of S, then these conditions are also necessary. Moreover, if S is a commutative Noetherian domain, then there are necessary and sufficient conditions for S*Gto be a prime maximal order, without requiring that the order of G be a unit in S.

The question “Does a Banach space with a symmetric basis and weak cotype 2 (or Orlicz) property have cotype 2?” is being seriously considered but is still open though the similar question for the r.i. function space on [0, 1] has an affirmative answer. (If X is a r.i. function space on [0, 1] and has weak cotype 2 (or Orlicz) property then it must have cotype 2.) In this note we prove that for Lorentz sequence spaces d(a, 1) they both hold.

Let Q(X) denote and let BTr denote the classifying space of the r-torus. In [8], Segal showed that Q(BT1) is homotopy equivalent to a product BU × F where BU denotes the classifying space for stable complex vector bundles and F a space with finite homotopy groups. This result has been a very useful one. For example, in [5] it was used to show that up to a stable homotopy equivalence there is only one loop structure on the 3-sphere at each odd prime p. (The subsequent work of Dwyer, Miller, and Wilkerson shows this result is even true unstably, at every prime p.) In [6] it was used to classify, up to homology, the stable self maps of the projective spaces ℂPn and ℍPn. In [5] I asked if a splitting similar to Segal's might exist for Q(BTr) when r≥2. In particular, since the homotopy and homology groups of BU are torsion free it seemed natural to ask if Q(BTr), when r>, could likewise contain a retract with torsion free homology and homotopy groups and whose complement is rationally trivial. The purpose of this note is to show that the answer is no.

An exact solution of triple trigonometrical equations is obtained by using the finiteHilbert transform. The solution of these equations is used to solve a two-dimensional electrostatic problem. The problem of determining the electrostatic potential due to two parallel coplanar strips of equal length, charged to equal and opposite potentials, each parallel to and equidistant from an earthed strip, is considered. Both the charged strips lie along the x-axis and they are equally spaced with respect to the y-axis. Finally the expression for the surface charge density (per unit depth) of the strip is derived

Let M be a von Neumann algebra and a strongly continuous one-parameter group of *-automorphisms of M. Recently, considerable interest has been shown in the associated mappings (see for instance [2], [3] and [4]). The paper of Thaheem, Van Daele and Vanheeswijk [4] exclusively deals with these mappings. The main result of [4] is formulated as follows. Let and be strongly continuous one-parameter groups of *-automorphisms on a von Neumann algebra M satisfying the operator equation

In [8] and [9] we initiated a study of lattice theory by means of Baer semigroups. Basically, a Baer semigroup is a multiplicative semigroup with 0 in which the left annihilator L(x) of each element x is a principal left ideal generated by an idempotent, while its right annihilator R(x) is a principal right ideal generated by an idempotent. By [8, Lemma 2, p. 86], L(0) has a unique idempotent generator 1 which is effective as a two-sided multiplicative identity for S. For any Baer semigroup S, if we use set inclusion to partially order both ℒ = ℒ (S) = {L(x) | x ∈ S} and ℛ = ℛ (S) = {R(x) | x ∈ S}, we have by [8, Theorem 5, p. 86], that ℒ and ℛ form dual isomorphic lattices with 0 and 1. The Baer semigroup S is said to coordinatize the lattice L in case ℒ(S) is isomorphic to L. In connection with this, it is important to note that by [9, Theorem 2.3, p. 1214], a poset P with 0 and 1 is a lattice if and only if it can be coordinatized by a Baer semigroup.

Let ℒ V denote the algebra of all linear transformations on an n-dimensional vector space V over a field Φ. A subsemigroup S of the multiplicative semigroup of ℒ V will be said to be an affine semigroup over Φ if S is a linear variety, i.e., a translate of a linear subspace of ℒ V.

This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.

In § 2 a product of two modified Bessel Functions of the Second Kind is expressed as an integral with a function of the same type as a factor of the integrand. In § 3 an integral involving a product of these functions, regarded as functions of their orders, is evaluated in terms of another function of this kind. These results were suggested by a study of Mellin's inversion formula.

The equations. We shall solve the equation

giving the solution in two forms, and give a new solution of

originally solved by Carlemann.

The latter will be extended to the case where the limits are a and b with a < x <b.

A compact bordered Klein surface of genus g ≥ 2 has maximal symmetry [4] if its automorphism group is of order 12(g − 1), the largest possible. An M*-group [8] acts on a bordered surface with maximal symmetry. The first important result about these groups was that they must have a certain partial presentation [8, p. 5]. However, research has tended to focus more on the surfaces with maximal symmetry than on the M*-groups, and results about these groups typically deal with existence.

Hurwitz's theorem says that the limit of schlicht functions, in the topology of compact convergence, is again a schlicht function or a constant function. We generalize this to mappings between Riemann surfaces and get more precise information on the relation between the distribution of values of analytic functions and the topology of compact convergence on the space of all analytic maps.

W. A. Bogley and M. A. Gutierrez [2] have recently obtained an eight-term exact homology sequence that relates the integral homology of a quotient group Г/MN, where M and N are normal subgroups of the group Г, to the integral homology of the free product Г/M * Г/N in dimensions ≤3 by means of connecting terms constructed from commutator subgroups of Г, M, N and M ∩ N. In this paper we use the methods of [4] to recover this exact sequence under weaker hypotheses and for coefficients in /q for any non-negative integer q. Further, for q = 0 we extend the sequence by three terms in order to capture the relation between the fourth homology groups.

All graphs considered in this article are finite connected, without loops and multiple edges. Let G be a graph and x be a vertex. The vertex neighbourhood graph (or υ-neighbourhood) of x in G (denoted by is the subgraph of G induced by the set of all vertices of G adjacent to x Analogously if f = xy is any edge of G, the edge neighbourhood graph (or e-neighbourhood) of f in G is the subgraph of G (denoted or induced by the set of all vertices of G which are adjacent to at least one vertex of the pair x, y and are different from x, y.

In this paper I construct a reduction formula for the integral

the formula connects any three consecutive members of a set I0, I1, I2, …, Im. We regard m as given, and, in order to avoid wasting time over trivialities, we postulate that the constants a, α, β, γ δ (which are not restricted to be real) have real parts large enough to ensure (i) that the integrals In under consideration and the integrals related to them which will be introduced subsequently are all absolutely convergent, and (ii) that, in all the partial integrations which will be effected, the integrated parts vanish at both limits.

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.