The following paper aims at a more general treatment than has hitherto been given, of the integral expansions of arbitrary functions, from the point of view of integral equation theory.

This paper is devoted to 3-manifolds which admit two distinct Dehn fillings producing a Klein bottle.

Let M be a compact, connected and orientable 3-manifold whose boundary contains a 2-torus T. If M is hyperbolic then only finitely many Dehn fillings along T yield non-hyperbolic manifolds. We consider the situation where two distinct slopes γ1, γ2 produce a Klein bottle. We give an upper bound for the distance Δ(γ1, γ2), between γ1 and γ2. We show that there are exactly four hyperbolic manifolds for which Δ(γ1, γ2) > 4.

Introduction. Infinite systems

are discussed in which k(θ) and b(n) are differentiable. The aim is to determine the behaviour of a(n) for large n and to reduce (1) to a finite system. This involves solving

for the high order a(n), n ≥ N, in terms of the preceding a(n), with a large enough N to facilitate the iteration procedure employed. The desired N − 1, N − 1 system follows on substitution for the high order a(n) in the first N − 1 equations (1).

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) if

exists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there exist

such that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

The notation of the introduction of the first paper (1) will be used without further explanation. All page references, and also references to Theorems 1, 2, 3, 4 or to numbered equations, will be understood to refer to the pages, theorems or equations* so numbered in (1). The theorems of the present paper are denoted by primed numbers.

We solve a problem of Steinhaus (New Scottish Book, Problem 328; also Colloq. Math. 5 (1958), Problem 225); if p is a prime, does there exist a regular p-gon such that three of its diagonals cut in a point? The answer is in the negative, except in the case where the said diagonals are all parallel, which can clearly occur for any p ≥ 7.

Let H be a complex Hilbert space. Recall that a bounded linear operator A, on H, is positive if (Ax, x) ≥ 0 (x ∈ H) (so that A = A* necessarily) and positive definite if A is positive and invertible.

Introduction. For applications in differential topology it is desirable to be able to find restrictions on the possible real and complex vector bundles over a given manifold. These restrictions usually state that the top component of some specific rational multinomial in the various characteristic classes is an integral multiple of the fundamental cocycle. For example, in proving non-embedding or non-immersion theorems (compare (1); (10)) one could test whether the normal bundle is consistent with these integrality conditions.

We analyse the Gottlieb groups of function spaces. Our results lead to explicit decompositions of the Gottlieb groups of many function spaces map(X, Y)—including the (iterated) free loop space of Y—directly in terms of the Gottlieb groups of Y. More generally, we give explicit decompositions of the generalised Gottlieb groups of map(X, Y) directly in terms of generalised Gottlieb groups of Y. Particular cases of our results relate to the torus homotopy groups of Fox. We draw some consequences for the classification of T-spaces and G-spaces. For X, Y finite and Y simply connected, we give a formula for the ranks of the Gottlieb groups of map(X, Y) in terms of the Betti numbers of X and the ranks of the Gottlieb groups of Y. Under these hypotheses, the Gottlieb groups of map(X, Y) are finite groups in all but finitely many degrees.

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

There is a theorem in geometrical optics due to Helmholtz which states that if a ray of light (1) after any number of refractions and reflections at plane or nearly plane surfaces gives rise (among others) to a ray (2) whose intensity is a certain fraction f12 of the intensity of the ray (1), then on reversing the path of the light an incident ray (2)′ will give rise, among others, to a ray (1)′ whose intensity is a fraction f21 of the ray (2)′, such that

The surfaces if not plane must be such that their radius of curvature is large compared with the lateral extent of the ray and this lateral extent must itself be large compared with the wave-lengths in the ray. In view of the many recent studies of reflection and transmission (refraction) of beams of electrons at potential walls, it seems worth while to ask whether there exists an analogy of the optical theorem for beams of electrons (or other particles). It is easy to see that the analogy must exist by using the principle of detailed balancing required by an assembly in statistical equilibrium. But a direct “dynamical” proof is also easy and may prove of sufficient practical interest to put on record here.

Let P be a fixed set of primes, the category of all groups and group-homomorphisms, and the full subcategory of nilpotent groups. In [9], an idempotent functor called P-localization, was defined so as to extend the ℤ-module-theoretic localization of abelian groups. There are two well-known extensions of e to , namely, Bousfield's P-localization [2], [4], denoted by EZP, and Ribenboim's P-localization [13], usually denoted by ( )P. Ribenboim's P-localization is the maximal extension among localizations extending e to in that it maximizes the number of groups in its image [7]. The localized groups obtained after applying Ribenboim's P-localization are precisely the P-local groups, that is, the groups having unique nth-root for every n whichis co-prime to P, [13]. Being maximal is equivalent to this class of P-localized groups being the saturated class of groups generated by e–equivalences, that is, group homomorphisms between nilpotent groups which become isomorphisms after applying e.

In this paper we present two versions in several variables of the following result:

Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

We propose a smooth correction for skewness in an asymptotically normal statistic. Unlike the case of approximation by Edgeworth expansion, our correction applies uniformly in all values of the level. The correction is based on a ' comparison statistic', and, in the case of the Studentized mean, it enables removal of all effects of skewness up to terms of order .

We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.

In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

This paper determines which Stiefel–Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F2-vector space of Stiefel–Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP3 bundles. These Stiefel–Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of 3 bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

Bollobás and Erdös[1] have posed the problem:

If a is irrational, show that for 1 ≤ i < j ≤ p the number of integers t with 1 ≤ t ≤ p such that {(t–i)2a} < d and {(t–j)2a} < d, 0 < d < 1, is d2p + o(p) uniformly in i, j.