This note is concerned with a result about spines of manifolds embedded in a sphere, which proved useful at one point in (2).

1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.

The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.

The author provides a new and very simple proof of the following theorem of Magnus (2):

Theorem. Let the group G have both a presentation with n + r generators and r relations also a presentation with n generators. Then G is the free group on n generators.

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)

An explicit formula is derived for exp (iβJz) as a finite sum of irreducible tensor components. With this formula, a technique is developed to obtain the matrix elements of exp (iβJy).

We refer the reader to the IHES notes of Zeeman (14) for basic facts about PL (or piecewise-linear) manifolds. If Mm is a locally flat PL-submanifold of Qm+2, our object will be to study the normal structure of M in Q: one of our main results is:

There exists a PL-bundle over M, with fibre a 2-simplex, which is PL-homeomorphic to a neighbourhood of M in Q; moreover, the bundle and homeomorphism are unique up to equivalence. We also make an application to smoothing theory.

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditions

for all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have

.

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying

A non-enumerable set of real numbers, Z, is concentrated provided that there exists an enumerable set K such that if G is open and G ⊂ K then Z − G is at most enumerable. Z is said to be universally concentrated if we can take for K any enumerable set H for which

In this note we prove the following result due to Milnor ((5), Theorem 4·1).

A well-known theorem of Minkowski on the product of two linear forms states that if

are two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

We extend certain results of the theory of closed operators in Banach spaces to general linear operators in normed spaces. A ‘state diagram’ for linear operators is drawn up. We prove some perturbation theorems, improving or correcting certain results of Goldberg.

In (1), Theorem 4·1·4, and hence Theorem 4·3·1, are incorrect as stated, unless h* denotes ordinary cohomology theory. However, if h* denotes complex K-theory, corresponding results still hold, provided everything is graded over Z2 instead of Z.

Let Ωn be the set of all n-square doubly stochastic matrices, i.e. matrices with non-negative entries all of whose row sums and column sums are 1. If A = (aij) is an n-square matrix let per (A) denote the permanent of A, that is.

The Peter-Weyl theorem states that for a compact topological group G, the set of finite dimensional (almost invariant) functions on G is uniformly dense in the set of continuous functions on G. In this paper, we consider the question: which subalgebras of C(G) inherit this property?

This theorem proves the equivalence of two conjectures on perfect Singer difference sets, one of which is concerned with the actual computation of the set, and the other being of interest in the theory of multipliers and fixed sets.