A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.

D. Grant gave a set of defining equations and the addition formulae on the Jacobian varieties of hyperelliptic curves of the form $y^2=x^{5}+\cdots$, using the $\sigma$-function and hyperelliptic functions. In this paper, using these functions, we give the n-multiplication formulae on the Jacobian varieties of hyperelliptic curves of the form $y^2=x^{5}+\cdots$.

Using the decomposition of a certain vector space under the action of the structure group of Riemannian almost product manifolds, A. M. Naveira (9) has found thirty-six distinguished classes of these manifolds. In this article, we prove that this decomposition is irreducible by computing a basis of the space of invariant quadratic forms on such a space.

We study homomorphisms from Kähler groups to Coxeter groups. As an application, we prove that a cocompact complex hyperbolic lattice (in complex dimension at least 2) does not embed into a Coxeter group or a right-angled Artin group. This is in contrast with the case of real hyperbolic lattices.

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.

For Γ = ℤp, Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large μ-invariants. In this work, we investigate other uniform pro-p groups which are realisable as Galois groups of towers of number fields with arbitrarily large μ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p−1. Both in Iwasawa's work, and in the present one, the size of the μ-invariant appears to be intimately related to the existence of primes that split completely in the tower.

The following derivation of the equations of viscous motion, and, by analogy, the equations of motion of an isotropic elastic solid, appears to have an advantage in brevity over those in general use.

The aim of this paper is to establish a result on a family of congruences arising from the Fermat quotient: this result has an interesting application to the Fermat problem. For over 150 years the Fermat problem has been divided into two cases; the First Case being the assertion that for each odd prime p,

has no solution in non-zero integers x, y, z. Kummer's work on ideal numbers led, in 1847, to the complete solution of the Fermat problem for regular primes.

There are many results which discuss ergodicity in terms of approximate product properties. Here we work throughout with stochastically independent σ-algebras (ℰ, ℱ are independent of μ if μ (E ∩ F) =μ (E)μ(F) for all E ∊ ℰ,F ∊ ℱ), to obtain an exact product property characteristic of (a large class of) ergodic measures. The ideas are based on work of A. V. Skorokhod on admissible translates of probabilities on Hilbert space.

We prove a theorem which facilitates homotopy classification of maps into a topological group G. Some information about homotopy groups of G is obtained, including the following two results. Consider the Samelson product, as defined in (7), which constitutes a bilinear pairing of πp(G) with πq(G) to πp+q(G). The product of a α ∈ πp(G) with β ∈ πq(G) is written in the form 〈α, β〉. There exist groups having Samelson products of infinite order. Homotopy-commutative groups have zero Samelson products. We shall prove

Theorem (1·1). If G is a connected Lie group then there exists a positive integer n such that n〈α, β〉 = 0 for every pair α, β of elements in the homotopy groups of G.

Let Ωn denote the convex set of all n × n doubly stochastic matrices: chat is, the set of all n × n matrices with non-negative entries and row and column sums 1. If A = (aij) is an arbitrary n × n matrix, then the permanent of A is the scalar valued function of A defined by

where the subscripts i1, i2, …, in run over all permutations of 1, 2, …, n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that if A ∈ Ωn then 0 < cn ≤ per (A) ≤ 1, where the constant cn depends only on n. It is natural to inquire if per (A) is a convex function of A for A ∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I + ½A) ≤ ½ + ½ per (A) for all A ∈ Ωn. Here I = In is the identity matrix of order n.

1. Introduction

A class [Xscr] of groups is said to be countably recognizable, if every group all of whose countable subgroups are contained in countable [Xscr]-subgroups is itself an [Xscr]-group. Many examples of such classes are discussed in section 8·3 of [20]. In the present work we are concerned with the question of how far countable recognizability can be obtained for classes of finitary linear groups. Recall that a group is said to be finitary []-linear if it is isomorphic to a subgroup of FGL[](V), the group of all invertible []-linear transformations α of the []-vector space V with the property that the image of the endomorphism α−idV has finite []-dimension. This generalizes the notion of linearity. A survey about features of finitary linear groups is given in [18].

A method for studying the adsorption of hydrogen on a clean tungsten surface has been developed which depends on the fact that the accommodation coefficient of neon is different for a bare surface and for a surface with an adsorbed film on it.

It has been shown that on a bare tungsten surface, saturation occurs at a partial pressure of hydrogen of 4.10−4 mm. of mercury and probably much lower, so that the adsorption is of the nature of chemi-sorption.

Important deductions can be drawn concerning the process of activation if any for this chemi-sorption of hydrogen on a bare tungsten surface.

The accommodation coefficient of neon with a tungsten surface covered with a hydrogen film of the type investigated is 0·17 at 295°K, and 0·32 at 79°K.

The experiments described were carried out during my tenure of a Moseley Studentship at the Cavendish Laboratory, and I should like to thank Lord Rutherford for his interest in the work and the Council of the Royal Society for their continued support. I should also like to thank Professor Rideal for his interest in the results.

Springer's theory of regular elements in complex reflection groups is generalized to arbitrary fields. Consequences for the cyclic sieving phenomenon in combinatorics are discussed.

For a general state space Markov chain on a space (X, [Bscr ](X)), the existence of a Doeblin decomposition, implying the state space can be written as a countable union of absorbing ‘recurrent’ sets and a transient set, is known to be a consequence of several different conditions all implying in some way that there is not an uncountable collection of absorbing sets. These include

([Mscr ]) there exists a finite measure which gives positive mass to each absorbing subset of X;

([Gscr ]) there exists no uncountable collection of points (xα) such that the measures Kθ(xα, ·)[colone ](1−θ)ΣPn(xα, ·)θn are mutually singular;

([Cscr ]) there is no uncountable disjoint class of absorbing subsets of X.

We prove that if [Bscr ](X) is countably generated and separated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the structure of absorbing sets are also developed.

1. Introduction. By a local system for a group G we shall mean a collection Σ of subgroups of G such that for every finite subset of G there is a member of Σ containing it. If is a class of groups G is locally if G has a local system of subgroups.

We study blocks of the Iwahori–Hecke algebra ${\cal H}_q (\frak{S}_n)$ of weight two over a field of characteristic two. Using techniques and notation developed by Scopes, Richards, Chuang and Tan for the case of odd characteristic, we find the decomposition numbers and classify extensions between simple modules for these blocks.

It is shown in this paper that the conservation laws of non-local continuum mechanics are equivalent to, and can be derived from, the conservation of energy and the invariance conditions under superposed rigid body motions. Also, the theory of non-local thermoelasticity is reconsidered in the light of recent developments in thermodynamics, taking invariance conditions fully into account.

We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.

A wide class of scalar first order linear autonomous neutral delay differential equations with distributed type delays is considered. By the use of two distinct real roots of the corresponding characteristic equation, a new result on the behavior of the solutions is established.