A multi-edge graph is said to be a restricted Euler graph when it can be decomposed into cycles of length greater than 2. Necessary and sufficient conditions are obtained in relation to vertex structure for a graph to be restricted Euler.

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

1. A tiling is a collection = {Ti|i = 1, 2, …} of closed topological discs which covers the Euclidean plane E2, and of which the individual tiles Ti have disjoint interiors. We shall assume throughout that the intersection of any two tiles is a connected set. If each tile is congruent (directly or reflectively isometric) to a given set T, then the tiling is called monohedral and T is called the prototile of . Clearly every monohedral tiling is locally finite.

Suppose that

is regular in the unit disk D = {|z| < 1}, and assumes there values ω lying in a domain Δ. It is natural to ask what effect this restriction has on the coefficients aν. The strongest order result we can hope to prove is that

It is not known if a projective plane of order 10 can contain ovals, nor if there are any structural restrictions on the ovals it might contain. In this note we take a small step toward understanding the possible embeddings by showing that no plane can contain ovals of a certain type.

Throughout this paper the word ‘ring’ will mean ‘commutative Noetherian ring with non-zero identity’, and A will always denote such a ring. It will only be assumed that A is local when this is explicitly stated, but when this is the case m (resp. k) will always be used to denote the maximal ideal (resp. residue field) of A. It is to be understood that ring homomorphisms respect identity elements.

Let G be a group and A a right G-module. If the additive group A+ of A is a Černikov group, that is, a direct sum of finitely many cyclic and quasi-cyclic groups, we shall call A a Černikov module over G or over the integral group ring . Suppose that A+ is, furthermore, a divisible p-group, where p is a prime. Since the endomorphism ring of a quasi-cyclic p-group is isomorphic to the ring of p-adic integers, we find that is a free -module of finite rank. We can make A* into a right G-module in the usual way, and since A* is actually just the Pontrjagin dual of A, Pontrjagin duality shows that A → A* gives rise to a contravariant equivalence between the categories of divisible Černikov p-torsion modules over and G-modules which are -free of finite rank. Since the latter category is to some extent familiar, at least when G is finite – for its objects determine representations of G over the field of p-adic numbers, a field of characteristic zero – we may hope to exploit this correspondence systematically to study divisible Černikov p-modules. This is our main theme.

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

A Banach space X is said to have the Banach–Saks property (BS) if every bounded sequence (xn) in X has a subsequence (), which is (C, 1) convergent in norm to a point x in X; that is,

Kakutani (7) showed that all uniformly convex spaces are (BS); moreover, all (BS) spaces are reflexive. It is further known that both these implicationsare strict: see, for example, Baernstein (1) and Diestel (4).

Here we answer a question raised by Zaleskii (5) and reiterated by Segal (3) of whether a finitely generated nilpotent group can have primitive irreducible representations.

A unital JB-algebra is a Jordan algebra A with identity together with a complete norm satisfying, for all a, b ∈ A,

(i) (a2b) a = a2(ba),

(ii) ∥a2∥ = ∥a∥2,

(iii) ∥ab∥ ≤ ∥a∥ ∥b∥,

(iv) ∥a2 + b2∥ ≥ ∥a2∥, ∥b2∥.

(It should be noted that axiom (iii) is a consequence of (ii) and (iv).) Such spaces have been studied by several authors (3, 6, 11), and as a consequence their structure is now quite well understood. Many of the results of these papers, while relying on the existence of an identity for their proofs, can be formulated for algebras which lack this property. C*-algebra theory and operator theory abound in examples of spaces which fail to be unital JB-algebras only in this one respect, and this motivates the study of the general case undertaken in this note.

Let G be an Abelian group, the symmetric algebra of G and the associated graded ring of the integral group ring ZG, where (AG denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4)

which is given on the nth component by

In general θ is not an isomorphism. In fact Bachmann and Grünenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.

The orbit space of a manifold under the smooth action of a compact Lie group is typically a ‘manifold with singularities’. The orbits of a given type each form a mani-fold, but these ‘strata’ generally do not give a manifold when pieced together. For example, the orbit space of S3 under the action of given by the matrix.

is the suspension of P(2), which fails to be a manifold at the suspension points.

In this paper I shall give examples where, in spite of having many orbit types, the orbit space is a manifold, with or without boundary.

In this paper every space will be 2-local, for example BO will mean the 2-localization of the space usually denoted BO.

The purpose of this paper is to exhibit two exact sequences involving the sets of conjugacy classes in various groups. One of them is applied to prove the following result.

The crucial step in the proof of the Topological Stability Theorem is the construction of a certain contact-invariant Whitney stratification of the sufficient jets in the jet space Jk(n, p). There is therefore a certain intrinsic interest in the construction of natural contact-invariant Whitney stratifications of the jet space. It has long been known that if there are only finitely many contact orbits then these provide a Whitney stratification, but in general there is the double problem of finding a candidate for a Whitney stratification, and then establishing that it has the desired properties. Virtually nothing is known about this problem, so it seems sensible to start by attempting to understand the first non-trivial examples which arise. Since functions are rather better understood than mappings it seems natural to try the case p = 1 first. The complexity of the problem now increases with k. When k = 1, 2 there are only finitely many contact orbits, so we start with k = 3. Having fixed p = 1, k = 3 we turn our attention to n. When n = 1, 2 we still obtain only finitely many contact orbits, as easy computations verify, so we start with n = 3. Thus our first example is J3(3, 1). It is a relatively easy matter to list the contact orbits in this case, and in so doing one is presented with an obvious candidate for a Whitney stratification. Our objective is to show that this candidate is indeed a Whitney stratification.

Let M be a W*-algebra. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, M is said to be injective [(8), § 5] if there is a projection of norm one from L(H) onto π(M). Tomiyama [(13), theorem 7.2] has shown that this definition is independent of the choice of representation π, and, moreover, that M is injective if and only if the commutant π(M)′ is injective for some such π. M is said to be semidiscrete [(8), § 3] if the identity map on M can be approximated in the topology of simple-weak* convergence by completely positive normal linear maps in L(M) of finite rank which preserve the identity. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, a *-homomorphism η of the algebraic tensor product M ⊙ π (M)′ into L(H) is defined by

If η has a bounded extension to M ⊗ π(M)′ (the completion relative to the spatial C*-cross norm) we shall say that M is amenable.

An important role in the classical theory of symmetric functions of a real n-tuple x = (x1, x2, …, xn) is played by the complete symmetric functions or homogeneous product sums hr defined by the generating function

(see Littlewood (5), p. 82). In an earlier paper (4) I conjectured that h2r is positive definite. The main object of the present paper is to prove this conjecture in a rather sharper form.

This paper follows (4) in a quantitative study of derivations. The object is to investigate the properties of the spectrum of an inner derivation. In particular, we shall be interested in obtaining a general description of the spectrum of a derivation on a W*-algebra.

Let S be both a topological space and a semigroup. For s ∈ S, define the maps λs and ρs of S to S by

We shall say that S is a right-topological (resp. left-topological) semigroup if ρs (resp. λs) is continuous for each s in S. We denote by Λ(S) the set

this is a subsemigroup of S.