We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

The problem of evaluating the non-homogeneous critical determinant of the plane region |y − x2| ≤ 1 is solved as a particular case of a result concerning the non-homogeneous properties of convex regions proved in the author's thesis (not yet published). The search for an alternative method of proof led to a proof of the existence of an infinite sequence of minima analogous to the Markoff chain. This is the subject of the present paper.

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)

We consider the equation:

where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,

f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) is

so that

Using this relation, Churchhouse has shown that

and

where

with

Moreover

and b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,

and

The object of this note is to prove these conjectures.

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:

THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequencies

We shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

The drag on a fragment moving through the air at supersonic velocity tends to be proportional to the projected area of the fragment perpendicular to the line of flight and to be independent of the Reynolds number, depending only on the Mach number. Also the minimum velocity at which the fragment perforates a target is a function of the projected area at the moment of strike. Hence the probability that a fragment, with a given initial velocity, perforates the target at a given distance, depends on the mode of rotation and the distribution of projected areas of the fragment. It is customary to assume that the fragment rotates randomly so that we require to know the distribution of the projected areas when all orientations are equally likely. The distributions for fragments in the form of cylinders and rectangular parallelepipeds are derived in this paper.

Let F be a free group. We explain the classification of finitely presented subgroups of F×F in geometric terms. The classification emerges as a special case of results concerning the structure of 2-complexes which are built out of squares and have the property that the link of each vertex has no reduced circuits whose length is odd or less than four. We obtain these results using tower arguments and elements of the theory of non-positively curved spaces.

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we have

and

as x → ∞.

The term ‘critical graph’ has been used most frequently to mean a graph such that the removal of any line decreases the chromatic number. This can be generalized to define a graph G to be ‘critical (μ)’ for an arbitrary invariant, μ if, for each line e, μ(G – e) ≠ μ(G). Analogously we may consider any line e which joins two points not already adjacent in G and define G as ‘anticritical (μ)’ if μ(G + e) ≠ μ(G). Anticritical graphs are investigated here with respect to the following invariants: chromatic number, line-chromatic number, point-connectivity, line-connectivity, point-independence number, line-independence number, diameter, radius, and cyclability number.

Let ℕ be the natural numbers object in , the effective topos. It was shown in [1] that the maps ℕ → ℕ are (internally or externally) just the total recursive functions. Now a subset A ⊆ ω of natural numbers corresponds to a ¬¬-closed subobject A ↣ ℕ; let kA be the least topology forcing A to be decidable, and let ℕA be the sheafification of ℕ with respect to this topology. Then one would expect the maps ℕ → ℕA to be the total functions recursive in A.

We show that if $f$ is a transcendental meromorphic function with a finite number of poles and $f$ has a cycle of Baker domains of period $p$, then there exist $C > 1$ and $r_0>0$ such that $\bigg\{z:\frac1C r\lt |z|\lt Cr\bigg\}\cap \mbox{sing} (f^{-p})\ne\varnothing,{\for}r\ge r_0.$ We also give examples to show that this result fails for transcendental meromorphic functions with infinitely many poles.

In this paper we investigate a time harmonic plane strain problem in a homogeneous isotropic semi-infinite elastic solid with mixed boundary conditions on its surface. The boundary is free from applied stress apart from on an infinite strip where the tangential components of stress are zero, and the normal component of displacement is a prescribed function. By reducing the mixed boundary value problem to the solution of a Fredholm integral equation of the second kind, an existence theorem is proved and a simple low frequency asymptotic formula relating the normal stress to the prescribed displacement is derived rigorously. The results are applied to the forced heaving and rocking motion of a rigid cylindrical body in smooth contact with an elastic half space. Some further applications to scattering problems are suggested.

A unital C*-algebra A is said to have unitary 1-stable range (8) if for all pairs a, b of elements of A satisfying aA + bA = A there exists a unitary u in A such that a + bu is invertible. This concept is somewhat stronger than the usual stable range condition of algebraic K-theory ((3), chapter V). Handelman(8) shows among other things that finite AW*-algebras have unitary 1-stable range and uses this fact to study the algebraic K1 of a finite AW*-algebra. We prove below that a unital C*-algebra has unitary 1-stable range if and only if its group of invertible elements is dense. In addition we give some consequences of this fact and consider the related question of (unitary) polar decomposition in C*-algebras.

Let G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

Introductory. In the following pages a miscellany of information about M12, the Mathieu group of degree 12, is presented, mainly concerning its subgroups. The development is inspired by the work of Coxeter and Todd on the representation of M12 as a group of collineations in PG (5,3)—i.e. five-dimensional projective space over the field of 3 elements.

An account is given of attempts to detect radiation in thunder clouds by means of photographic plates attached to balloons. Eleven balloons were sent up and the plates attached to ten were recovered and examined.

The result was negative in every case.

A small ionisation chamber with quartz condenser intended for detecting radiation in thunder clouds is described.

These observations were made at the Solar Physics Observatory, Cambridge, and I have to thank the Director for giving me every facility.

I am also greatly indebted to Professor C. T. R. Wilson for his continued advice and interest.

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectra

inducing a morphism

of homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequence

where Z is viewed as a Z[t] module via the augmentation

and, is induced by η*in the natural way.