Kirby and Siebenmann (in (7)) proved the annulus conjecture in dimensions five and above, by proving the stable homeomorphism conjecture in those dimensions, that is, by proving that every orientation-preserving homeomorphism of Rn is stable, if n ≥ 5. Although this result apparently lessens the importance of stable homeomorphisms, the concept of stability should still be of use in settling the fate of the annulus conjecture in dimension 4, or for uncovering a simple (i.e. non-surgical) proof in other dimensions.

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

Let 〈a1, . . ., am ∣ wn〉 be a presentation of a group G, where n ≥ 2. We define a system of codimension-1 subspaces in the universal cover, and invoke Sageev's construction to produce an action of G on a CAT(0) cube complex. We show that the action is proper and cocompact when n ≥ 4. A fundamental tool is a geometric generalization of Pride's C(2n) small-cancellation result. We prove similar results for staggered groups with torsion.

An account is given of the seiche motion in a curved lake. It is shown that the periods of the first two modes of oscillation are effectively unchanged in comparison with those for a straight lake having a length equal to that of the median line along the curved lake. Near to the nodal positions, the surface elevation is uniform across the lake. In the neighbourhood of the anti-nodes, which includes the ends of the lake, the wave amplitude is shown to be a maximum at the outside of the bend.

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.

In the present note we discuss some properties of a ‘measure of asymmetry’ of convex bodies in n-dimensional Euclidean space. Various measures of asymmetry have been treated in the literature (see, for example, (1), (6); references to most of the relevant results may be found in (4)). The measure introduced here has the somewhat surprising property that for n ≥ 3 the n-simplex is not the most asymmetric convex body in En. It seems to be the only measure of asymmetry for which this fact is known.

An error in Dieudonné's paper, Sur les espaces de Köthe, published in Journal d' Analyse Mathématique, 1 (1951), 81–115, is pointed out and a correction put forward.

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.

Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:

[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.

F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S.

Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

Problems involving the addition of heat to a flowing gas have received considerable attention in recent years. The practical interest lies in the application to ram jets and similar engines, and the presence of entropy gradients makes the problem of theoretical interest.

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

For bond percolation on the two-dimensional triangular lattice with arbitrary retention parameter p ∈ [0, 1], we show that the number of infinite rigid components is a.s. at most 1. This proves a conjecture by Holroyd. Further results, concerning simultaneous uniqueness, and continuity (in p) of the probability that a given edge is in an infinite rigid component, are also obtained.

The methods of papers I and II (1), (2) are applied to a problem in classical elastodynamics.

The effect of heat transfer on boundary-layer growth is considered for the problem of a main stream of small Mach number which is instantaneously released past a heated cylinder, or alternatively the cylinder is accelerated impulsively into a stream at rest. In particular the effect of different values of the given wall temperature on the interval to separation and the thickness of the boundary layer is investigated. It is found that the former is decreased and the latter increased by the presence of heat transfer.

Axially symmetric static fields produced by one extended body are considered. The metric outside the body is the Weyl metric which can be written in so-called Weyl coordinates in the form ds2 = a2(dr2 + dz2) + b2dϕ2 – c2dt2 (a, b, c functions of (r, z) and r > 0) with b2 = r2c−2. Criteria are found under which these Weyl coordinates can be extended through the whole of the body so that ds2 keeps the form above (although in general b2 ǂ r2c−2 inside the body). In the spherically symmetric (⇒ axially symmetric) case it is shown that the Weyl coordinates cover the whole of spacetime only if one allows r < 0 in a certain part of the body.

In his paper 10 the author investigated the structure of m-full ideals by analysing their syzygies and, as one special case, showed how the Betti numbers of Borel stable ideals over polynomial rings can be computed. The same result, among other things, was also obtained by Eliahou and Kervaire1 by a different method.

Let ℛ be a von Neumann algebra, with predual ℛ*, acting on a Hilbert space ℋ; G a locally compact group with left Haar measure m, and α a representation of G on aut (ℛ), the group of all *-automorphisms of ℛ, i.e. α is a group homomorphism from G to aut (ℛ). We shall show that if ℋ is separable, then very weak measurability assumptions on the representation α produce strong continuity properties. This will be used to obtain results on the extension of representations from a C*-algebra to its weak closure, giving a much simpler proof of a result of Aarnes ((1), theorem 8, p. 31), and on continuity of tensor products of representations. The main result was suggested by the analogous theory concerning unitary representations of locally compact groups, and its proof employs ideas frequently used in that context. (See, for example, (5), theorem 22.20 (b), p. 347.)

In two classical papers (1, 2) J. M. Whittaker introduced the study of integral functions bounded at the lattice points m + in(m, n = 0, ± 1, …,). He succeeded in showing (cf. also G. Polya(3)) that an integral function of at most the minimum type of order 2 uniformly bounded at the lattice points was necessarily constant. This result was improved almost simultaneously by A. Pflüger(5) and V. Ganapathy Iyer(11), who showed that the result was true also for functions of type K<½12π of order 2. The example of Weierstrass's σ(z) function shows that theirs is a best possible result in this direction.