A claim of H. H. Teh on Archimedean full orders ((2), ‘only if’ of Theorem 2, p. 480) fails. His statement (p. 478, line 5) that the direct product of n copies of the additive rational numbers admits no full order of Archimedean rank more than n is proved without Hahn's embedding theorem. A few other remarks are made.

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

Throughout X will denote a connected, finite C.W. complex.

Let G be a subgroup of ∑n, the symmetric group, which acts transitively on the Cartesian product, Xn, of the space X. A map f:Xn→X is G-symmetric if it commutes with the action of G. If x0εX is a base point let i:X→Xn denote the inclusion, i(x) = (x,x0, …,x0). In ((6); (7); (11)–(13)) the following problem is posed: if X is an orientable topological n-manifold, what is the set of integers which may be obtained as the degree of(f.i) where f is a G-symmetric map? The degree of(f.i) is called the James number of f. If G = ∑n(G = Zn) a G-symmetric map will be called a symmetric map (a cyclic map). If X = Sn, the n-sphere, this problem has been studied in ((6)–(8), (11)–(13))

Atiyah(2) defined the geometrical dimension of an element to be less than k + 1 (g dim (x) ≤ k) if there is a k-dimensional bundle over X whose stable equivalence class is x. If ξ is an n-plane complex bundle over X, we say that it has r sections if there is an (n − r)-plane bundle η such that ξ is isomorphic to η ⊕ εr where εr is the trivial r-plane bundle over X. If X has dimension 2n or less, then ξ has r sections if and only if g dim (ξ − n) ≤ n − r.

An ordered topological space is a set X endowed with a topology τ and a partial order ≤. We shall denote such a space by (X, τ), it being understood that (unless otherwise stated) the symbol ≤ is used to denote the partial order on X. An account of some of the properties of these spaces can be found in (5), (6) and (7).

We consider some special operators on a locally convex Hausdorff space to itself, which have neat spectral theories and prove some perturbation results. This leads us to define and study a few special classes of locally convex spaces in which various subsets of the algebra of continuous linear operators either coincide or are closely related with each other. These are then compared to the classes of barrelled, infrabarrelled and DF-spaces and examples are given to distinguish them from one another.

Let p = (pk) and q = (qk) be any sequences of strictly positive numbers. Denote by c0(p) the set of all complex sequences x = (xk) such that . The Köthe–Toeplitz dual of c0(p) was determined in Theorem 6(4). In the case in which p∈l∞ (the space of all bounded sequences), c0(p) becomes a locally convex FK space under the paranorm where M = max (1, sup pk). It has been recently proved by Lascarides(3) that the following three properties are equivalent:

(a) p∈c0(i.e.pk → 0),

(b) c0(p) is perfect (in the sense of Köthe-Toeplitz duality),

(c) c0(p) has the Schur property (i.e. weak and strong sequential convergence are equivalent).

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:

Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac

Let 0 ≤ λ0 < λ1 < … < λn → ∞, and let

If as ω → ∞, we write

Suppose that

converges for σ > 0,

where τp(h) is finite for |h| < δ. If

we write

A general method is given and illustrated by application to particular cases for obtaining subdominant solutions of stiff difference and differential equations, i.e. when rapidly varying solutions – transients or otherwise – are possible but are in fact excluded by the initial conditions.

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that

Several years ago, Fisher and Fuller(1) proved that if P is a real square matrix with all members of its ‘nested set’ of principal minors non-zero, then there exists a real diagonal matrix, D, such that the characteristic roots of DP are all real, negative and distinct. This interesting and powerful result, used by the authors to derive further results concerning convergence of linear iterative processes, has since also proved of interest to economists studying the stability of economic general equilibrium.

The problem is governed by a Jones integral equation and the solution is shown to depend upon a single function which occurs naturally after a contour deformation has produced extensive cancellation in the work. The far-scattered field off the axis of symmetry is found in detail, yielding terms which are believed to be new.

Initially a general incident field is considered and the equations are split into Fourier components. Each Fourier component gives rise to an integral equation similar to that obtained when investigating diffraction of a normally incident plane wave. After the oblique wave has been specified an analysis similar to that used for the normal wave leads to a solution of the problem containing an infinite number of constants. It is shown, however, that these constants do not affect the leading terms of the high frequency scattering coefficient.

The propagation of electromagnetic waves through isotropic media containing a continuous energy-gain mechanism is considered and the existence of perfect resonance properties is investigated for both homogeneous and inhomogeneous layers.

A many-body variational calculation is carried out using a trial wave function which is a product of pair functions. An exact integro-differential equation for the pair function is derived, valid for any system of more than three Bose particles.