An n-link of multiplicity is a smooth embedding of the disjoint union of μ copies of Sn in Sn+2; is said to be trivial if it extends to a smooth embedding of the disjoint union of μ copies of Dn+1. Let , and Cnμ denote the wedge product of μ copies of S1 and (μ – 1) copies of Sn+1. Then clearly, if is trivial, then X ≃ Cn,μ, where ≃ denotes homotopy equivalence.

Notation and results. The main results of this paper are (a) a new proof of the canonical Schoenflies theorem and (b) a non-compact version of the canonical Schoen-flies theorem.

The result (a) is already known(5); the existing proof is based on the proof by Brown of the ordinary Schoenflies theorem, whereas ours is based on that of Mazur and Morse.

Let Mm be a closed connected smooth manifold of dimension m, and set Pm = Mm – int Dm where Dm is a disc in M. In (4), Wall has the following exact sequence

where (M), (resp. (P)) is the Δ-set of diffeomorphisms of M (resp. P) as in(l), and [M/P] is the set of diffeomorphism classes of smooth manifolds obtained by glueing a disc to the boundary of P. In this paper we obtain some results on π0((M)) for particular M, and in the following sense: Using the techniques of (2), we can determine π0((P)), so the main portion of the paper is concerned with a discussion of the kernel of α. There is a map ω: π1((P)) → Γm+1, given by the obstruction to extending h∈π1((P)) to a concordance of the identity of M to itself, and it is clear that if this map is attached at the beginning of the sequence (*) we get exactness. We will obtain (for certain M) an alternative description of the image of α in terms of those homotopy spheres which can appear as the boundaries of thickenings.

An ideal of an integral group ring is divisible by a given integer if all of its elements share this common factor; the ideals most often encountered are rarely divisible in this sense. Only in the case of finite p-groups are powers of the augmentation ideal of the integral group ring ever divisible. For every e, there is some n for which the nth power of the augmentation ideal is divisible by pe. The smallest such integer n arises in many contexts; this paper describes its properties and interpretations.

Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p-groups. The subgroups are calculated for Abelian p-groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p-subgroups of the symmetric group arises.

We show that if E is a V-Sidon set and K a disjoint compact set then there exists a function of low tensor norm separating them.

A function analytic approach to the generalized inversion problem, for arbitrary operators between Hilbert spaces, is investigated in the present paper.

The generalized inverse is defined as the inverse of the largest invertible restriction of the given operator and it is extended by zero to the orthogonal complement of the range of the invertible restriction. This domain-dense operator satisfies, on some restricted manifolds, the defining properties of the generalized inverse in some special cases, and it provides the least extremal solution of possibly inconsistent linear equations.

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

Kwapień(2) and Schwartz (3) have shown that the inclusion mapping I: ι1→ι2 is 0-radonifying. We shall give another proof of this, using an inequality due to Khint-chine(1).

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

1. Let A be a summability method given by the sequence-to-sequence transformation

We suppose throughout that, for each n

converges; this is a much weaker assumption than the regularity of A. Then we define

We also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say that

Let A' denote the summability method given by the series-to-sequence transformation

Following Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,

Let f(x) and g(x) be rational functions of x with integer coefficients, p an odd prime, e(x) = e(2πix/p), e(u/v) = e(w), where u = vw (mod p)† and χ(x) is a character mod p. It is well known that very abstruse methods are required to find an estimate for sums such as

where in the summation, ‡ the values of x are omitted for which eitherf(x) or g(x) is not defined.

1. Professor J. W. S. Cassels (in (l)) recently enunciated a conjecture concerning Kummer sums. His conjecture suggests that there is a close connexion between the Kummer sum and a certain product of Weierstrass elliptic functions.

In this paper we consider the biquadratic Gauss sum. A theory similar to that used in (1) and in (4) suggests that there is a corresponding connexion between the biquadratic sum and a product of Jacobian elliptic functions. The enunciation of this conjecture appears in section 6. But first of all we require some preliminary results.

If f(t) belongs to L(0, R) for every positive R and is such that the integral

converges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis and

for any positive ξ at which f(ξ+) and f(ξ−) both exist.

We define an operator Lk, t[F(x)]by

Under the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),

Let Ln, x denote the differentiation operator

Suppose that

converges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,

This paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

1. It often happens that the greater parts of two variables x, y may be connected linearly but there may be other disturbances of both. A treatment given by one of us ((1), section 4·42, pp. 216–17) assumes that the other disturbances are independent, as for observational errors. This is not altogether satisfactory because unless the variations of xr, yr are much greater than the standard errors there is an asymmetry in the treatment corresponding to the well-known distinction between the two regression lines.

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

In the following a distribution or a generalized function f is denned, as by Gelfand and Shilov(2) or by Temple (3), as a continuous linear functional on the space K of infinitely differentiable test functions ø having compact support. The value of f at a test function ø will be denoted by (f, ø).

A sequence of test functions {øn} is said to be a null sequence if

(1) the support of each øn is contained in some fixed domain D independent of n,

(2) the sequence converges uniformly to zero in D, as n tends to infinity, for all m.

The algebraic invariants of the Riemann tensor for a static axially symmetric gravitational field are examined. Contrary to common expectation they are shown to be well defined in a region of the field known to be physically singular.

Two extensions of the Rikitake model are examined; the first includes viscous friction and the second includes time delay. By use of Liapounov's direct method the viscous system is shown to have an unstable limit cycle associated with two of the three singular points. By numerical integration true ‘reversals of field’ are shown to occur for both the viscous and time delay cases.