There are 432 strongly squarefree symmetric bilinear forms of signature (2, 1) defined over $\mathbb{Z}[\sqrt{2}]$ whose integral isometry groups are generated up to finite index by finitely many reflections. We adapted Allcock's method (based on Nikulin's) of analysis for the 2-dimensional Weyl chamber to the real quadratic setting, and used it to produce a finite list of quadratic forms which contains all of the ones of interest to us as a sub-list. The standard method for determining whether a hyperbolic reflection group is generated up to finite index by reflections is an algorithm of Vinberg. However, for a large number of our quadratic forms the computation time required by Vinberg's algorithm was too long. We invented some alternatives, which we present here.

We study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

In this note we show that Propositions 5·2 and 5·3 of [1] are incorrect as stated. Thus statement (5) of Theorem 1·3 must be considered as a conjecture and certainly does not follow from our constructions. Our error does not affect any other results in the paper, and, indeed, § 5 was included mainly to provide some conceptual motivation for our constructions.

Let X be a real Banach space which is partially ordered by a closed cone C. Let X* denote the Banach dual space, and let

X is called a simplex space (2) if (X*, C*,‖·‖) is an AL-space. We proved in (3) that X is a simplex space if and only if X is an approximate order-unit-normed space and has the Riesz decomposition property. When the given norm on X is an order-unit norm, some representation theorems are obtained by Effros ((2), theorem 4·8) and Sterrmer ((6), theorem 4·3). In this note, we shall study the representation problem for more general simplex spaces. One of our aims is to prove that X is equivalent to C0(ω) if and only if X has the Urysohn property (for definitions, see below).

We investigate the operation of torus surgery on tori embedded in S4. Key questions include which 4–manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3–manifolds into 4–manifolds by viewing Dehn surgery as a cross section of a surgery on a surface. In particular, we give new embeddings of homology spheres into S4.

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.

Let Bn denote the Artin braid group on ‘n-strings[ and PBn its normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representation G: PBn → Mn × n(λ), into the group of n-by-n matrices over , is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn → Mn × n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

For a hyperelliptic curve ${\hbox{\ax C}}$ of genus $g$ with a divisor class of order $n = g + 1$, we shall consider an associated covering collection of curves ${\hbox{\ax D}}_\delta$, each of genus $g^2$. We describe, up to isogeny, the Jacobian of each ${\hbox{\ax D}}_\delta$ via a map from ${\hbox{\ax D}}_\delta$ to ${\hbox{\ax C}}$, and two independent maps from ${\hbox{\ax D}}_\delta$ to a curve of genus $g(g-1)/2$. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all ${\Bbb Q}$-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

The well-known Steinhaus theorem (2) with respect to the set of distances of linear sets of positive Lebesgue measure has been generalized to the case of linearly measurable subsets of rectifiable curves in the Euclidean plane by Besicovitch and Miller (1). An extension of the theorem to rectifiable curves in Euclidean n-space is immediate. Prof. A. P. Morse has suggested the problem as to whether or not the theorem still remains true in a general metric space. By defining a particular curve ℒ in a metric space we prove that the answer to this question is in the negative.

We shall work with rational Mn(K)-modules, where K is an infinite field of prime characteristic p > 0, and Mn(K) is the full matrix semigroup of n × n matrices over K. Recall that equivalence classes of simple rational Mn(K)-modules are parametrized by the set of all partitions λ = (λ1, λ2, …, λl) of length l = l(λ) ≤ n, and that the socle L(λ) of the Schur module (or dual Weyl module) H0(λ) is a simple Mn(K)-module whose highest weight corresponds to the partition λ.

In 1962 Kodaira[11] proved that ifX ↪ Y is a compact complex submanifold with normal bundle N such that H1(X, N) = 0, then X belongs to a locally complete family {Xt: t ∈ M} of complex submanifolds Xt of Y with the moduli space M being a (dimcH0(X, N))-dimensional complex manifold, and there exists a canonical isomorphism

between the tangent space of M at a point t ∈ M and the space of all global sections of the normal bundle Nt of the embedding Xt ↪ Y.

Certain interesting distributions are characterized by two or more symmetry properties, and it is a challenge to link these properties in an effective way. Dynkin's advice Discuss the simplest case first is always the best, and here we follow it (and avoid technical jargon) by discussing in detail just one concrete example, which concerns a discounted version of the arc-sine law. Theorem 1 B gives the symmetry properties for this case, and Theorem 1 A shows how they may be exploited to obtain an asymptotic result which we find rather surprising. Section 5 points to one of the generalizations which we shall examine in a sequel, and describes the natural formulation of the strange symmetry property which we first meet at (16b). (The sequel will be in many ways very different from this paper, but will share with it the fact that its results are not those which we originally conjectured.) In Sections 6 and 7 we consider some of the bizarre numerical analysis associated with the current problem.

Explicit forms are given for the Wigner for values of J from J = 0 to J = 7/2. The tedious labour involved in evaluating available formulas is reduced by use of a computer.

We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.

In 1935 Baer[1] introduced the concept of kern of a group as the subgroup of elements normalizing every subgroup of the group. It is of interest from three points of view: that of its structure, the nature of its embedding in the group, and the influence of its internal structure on that of the whole group. The kern is a Dedekind group because all its subgroups are normal. Its structure is therefore known exactly (Dedekind [7]): if not abelian it is a direct product of a copy of the quaternion group of order 8 and an abelian periodic group with no elements of order 4. As for the embedding of the kern, Schenkman[13] shows that it is always in the second centre of the group: see also Cooper [5], theorem 6·5·1. As an example of the influence of the structure of the kern on its parent group we cite Baer's result from [2], p. 246: among 2-groups, only Hamiltonian groups (i.e. non-abelian Dedekind groups) have nonabelian kern.

The object of this note is to direct attention to an entirely false inference which was drawn from the analysis given in my previous paper with the same title.

The success of Stoner's extensions of the theory of collective electron ferromagnetism(1) opens up new possibilities of considerable importance. The background of the theory can be found in standard works on the theory of metals(2) and statistical mechanics(3) and will not be given here.

A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be finitary if it consists of elements of finite rank. We study the situation when L acts irreducibly on the infinite-dimensional vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I. Moreover I is simple and L/I is solvable.