For each integer m, Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than nm, provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namely

where L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

The number of ions produced by a neutral particle of small mass with a magnetic moment equal to n Bohr magnetons has been calculated (I–III) and found to be 103n2 per km. path in air at N.T.P., being practically independent of the mass and energy of the particle (IV). A comparatively large fraction of the secondary electrons produced have high energy and may therefore be counted if they are produced in the wall of the Geiger counter instead of in the counter itself. This fact should facilitate the experimental detection of the ionization considerably (V).

We introduce and study two new notions of amenability for Banach algebras. In particular we compare these notions with some of those studied earlier. We show that several classes of Banach algebras, including certain Banach algebras related to locally compact groups, are responsive to these notions.

Let E and F be Banach spaces. The semigroup Φ+(E,F) of semi-Fredholm operators consists of the bounded linear mappings E→F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T∈Φ+(E,F) if and only if for any bounded set B⊂E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ+(E,F) if and only if there is c ≥ 0 such that

for all bounded B⊂E. Here γ is the well known Hausdorff measure of non-compactness

with BE the closed unit ball of E.

The molecule of a dissociating gas is treated as a dynamical system with n normal modes of vibration with incommensurable frequencies. The assumption that the molecule dissociates when one coordinate attains a critical high value leads to a unimolecular velocity constant expressed as an integral; analytical expressions are derived for bounds to its value. Calculations in a simple case indicate that the velocity constant is approximately of the form

This is in general agreement with experiment if the constant W is identified with the corresponding empirical energy constant.

In this paper we shall be mainly concerned with the following three apparently widely differing questions.

(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?

In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.

(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)

Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.

(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decomposition

one of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

Let f: X → X be a mapping of the metric space X into itself. We call f non-expansive if

We show that the self-affine sets considered by McMullen in [11] and by Bedford in [1] have infinite Hausdorff measure in their dimension, except in the (rare) cases where the Hausdorff dimension coincides with the Minkowski (≡ box) dimension. More precisely, the Hausdorff measure of such a self-affine set K is infinite in the gauge

(where γ is the Hausdorff dimension of K, and c > 0 is small). The Hausdorff measure of K becomes zero if 2 is replaced by any smaller number in the formula for the gauge ø.

This paper will show that after localization at any given prime p, the infinite loop space structure on the space BSO is essentially unique. If the word ‘localization’ is replaced by ‘completion’, the result continues to hold; and both results continue to hold if the space BSO is replaced by the space BSU.

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.

Given a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

In this paper we prove some results about the stable homotopy and cohomotopy of spaces related to the infinite real protective space RP∞. These include M. E. Mahowald's conjecture on the limit of stable homotopy of stunted real projective spaces RP2N+m/RP2N−m as N, m → ∞, G. Segal's Burnside ring conjecture for

and the stable analogue of a conjecture of D. Sullivan on RP∞.

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

This paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.

In this paper we consider the equilibrium configurations of a homogeneous, incompressible, isotropic elastic body subjected to a uniform dead load surface traction of magnitude T whose direction is normal to the surface of the body in the reference configuration, and to no other forces. We concentrate on homogeneous equilibrium solutions, that is those for which the deformation gradient F is constant, and we study their bifurcations and stability (with respect to an appropriate static criterion) as T varies. Since it turns out that the equations for homogeneous equilibrium solutions, and the stability properties that we consider of these solutions, are independent of the shape of the body in the reference configuration, we can suppose if desired that this shape is a cube. (See Fig. 1.1.)

An exact solution in terms of analytic functions is given for the Stokes stream function and forces acting on two spheres in contact when the spheres move uniformly with the same velocity along their line of centres. For the case of equal sized spheres, the values of the forces agree with the Faxén limit of the Stimson–Jeffery forces for separated spheres when the minimum clearance between the spheres tends to zero. Numerical values of the forces for arbitrary sized spheres are tabulated and compared with corresponding values when the spheres are separated over a range of values of the minimum clearance. Comparison of the theoretical results with those obtained experimentally for the case of equal sized spheres is also given.

The momentum distribution for the electron in the hydrogen molecular ion has been calculated for various wave functions, including the one used by James with which he obtained such a good value for the binding energy. The method adopted for this particular wave function is outlined and the results show appreciable change with improvement in the wave function. In conclusion there are discussed the implications of the present calculations on similar work on the H2 molecule.