Let X be a Banach space and L(X) the Banach algebra of bounded linear operators on X. An operator T in L(X) is hermitian if ∥eitT∥ = 1 (t ∈ R), and is normal if T = R + iJ where R and J are commuting normal operators; R and J are then determined uniquely by T, and we may write T* = R–iJ. These definitions extend those for operators on Hilbert spaces. More details may be found in [1].

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

The purpose of this note is to generalize to fields of characteristic two the results obtained in [4]. We obtain necessary and sufficient conditions involving quadratic forms for certain tensor products of quaternion algebras to be division algebras.

Several writers (4), (6), (7), (9) have used orthogonal expansions in discussing properties of Fourier transformations, and Kober (3) has used such expansions to derive fractional Fourier and Hankel transformations. In 1950 Barrucand (1) noted a reciprocity holding between the coefficients in the expansions in Laguerre polynomials of pairs of functions which are transforms with respect to the kernel J0(2x½).

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

Let G be a compact abelian group and let Γ be its (discrete) dual group. Denote by M(G) the space of complex regular Borel measures on G.

Let E be a subset of Γ. Then:

(i) E is called a Rajchman set if, for all μ ∈M(G) implies

(ii) E is called a set of continuity if given ε > 0 there exists δ > 0 such that if and

(iii) E is called a parallelepiped of dimension N if |E| = 2N and there are two-element sets . (The multiplication indicated here is the group operation.)

Bullett and Macdonald [1] have used power series to simplify the statement and proof of the Adem relations for Steenrod cohomology operations. In this paper I give a similar treatment of May's generalized Adem relations [4, §4] and of the Nishida relations ([6], [2, 1.1.1(9)], [5, 3.1(7)]). Both sets of relations apply to Dyer-Lashof operations in E∞, spaces such as infinite loop spaces ([3], [2, I.I]) and in H^ ring spectra ([5, §3]).

We are here concerned with following result of Trench:

Theorem. (Trench [5]). Let v1 and v2 be two linearly independent solutions of the differential equation

where a(t) is continuous on [0, ∞), and let b(t) be a continuous function of t for t ≧ 0 satisfying

where m(t) = max {|v1(t)|2, |v2(t)|2}. Then, if α1and α2are two arbitrary constants, there exists a solution u of

which can be written in the form

,

with

fori = 1, 2.

In this paper we consider differential equations of the form

where the coefficients Ai are holomorphic functions in a domain G1 × G2 ⊂ C × C. We restrict our attention to those equations for which it is possible to represent the solutions in the form

where g1(z1) and g2(z2) are arbitrary holomorphic functions in G1 and G2 respectively. The coefficients a1, k and a2, k depend on the given differential equation. Within the last ten years a number of publications have been devoted to this kind of representation of solutions.

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functional

where Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

Cooke [1] modified a technique used by Erdelyi and Sneddon [2] to solve triple integral equations of a certain type. In this paper, we extend this method to solve the quadruple integral equations

where F1, G2, F3 and G4 are prescribed functions of p and ψ(ξ) is to be determined. With no loss of generality we shall assume that G2(p)≡0, G4(p)≡0.

Let {a, b} designate the Pythagorean ratio (a2 - b2)/2ab between the sides of a rational right angled triangle. This paper studies the circumstances in which Pythagorean ratios can occupy consecutive places in an arithmetic progression. Part I deals with sets of three such ratios, while Part II discusses sets of four ratios.

The groups G of the title are those in which EG(x) = {y | y ∈ G, [y, nx] = 1 for some n} is a subgroup for every x in G. We show that the quotient group G/F(G) is rather restricted for finite E-groups; in particular, soluble finite E-groups are of Fitting length 4 at most. Some criteria for infinite groups are given.

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].