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.

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.

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.

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.

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.

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.

An ordered topological space (E, τ, ≥) is a set E endowed with a topology τ and a partial order ≥. For such a space order separation axioms have been studied by Nachbin (6) and McCartan (3). In this paper we discuss the consequences for these axioms of the imposition, in turn, of four conditions on (E, τ, ≥), namely convexity (6), continuity, anticontinuity and bicontinuity (5).

The object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

Locally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.

Let Kτ be a framed knot in S3 representing the 4-manifold which is obtained by attaching a 2-handle onto B4 along K with the framing τ (see (1)). Tristram (2) proved the following non-imbedding theorem for the generator of ).

Let A = (ank) be an infinite matrix of complex numbers; suppose I is the unit matrix and c is the space of convergent sequences. By (A) we denote the summability field of A, i.e. (A) = {x = (xk): Ax ∈ c}.

Let

and K′ = K(k′), where k2 + k′2 = 1

with

In a recent paper, Fenn (3) has established the following theorem: if ø is a piecewise linear involution on Sn without fixed points, if f: Sn → Sn and g: are continuous and the degree of f is odd, then there are points x and y on Sn such that f(x) = ø(f(y)) and g(x) = g(y). The Borsuk–Ulam theorem is the special case of this in which f is the identity and ø corresponds to reflexion in the origin. Since an infinite-dimensional version of the Borsuk–Ulam theorem is known, involving compact maps (see, for example, page 72 of (2)), it is natural to ask whether Fenn's result can also be extended to general Banach spaces, and in this paper we give such an extension when ø is reflexion in the origin. More precisely, we prove that if B is the closed unit ball in a Banach space X and f, g: B → X are compact, with deg (I − f, B, 0) odd (this is the Leray–Schauder degree and I is the identity map) and (I − g) (B) contained in a proper, closed subspace of X, then there exist x, y on the boundary ∂B of B and apositive numberα such that α(I − f) (x) = − (I − f) (y) and (I − g) (x) = (I – g) (y).

Atkinson and Reuter(1) consider travelling wave solutions for the deterministic epidemic, with or without removals, spreading along the line. In the case where there are no removals, they reformulate the problem in terms of the solutions X(·) to the integral equation

which satisfy X(− ∞) = − ∞, X( + ∞) = 0, X(u) < 0 for u ∈ (− ∞, ∞), where

is the left hand tail of the contact distribution, and where ʗ > 0 is the velocity of the wave corresponding to X. They show that no solution is possible unless

converges for λ > 0 sufficiently small, and that any solution X must satisfy

for some C > 0. They then prove the following existence theorems.

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form

The purpose of these remarks is to use elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and certain elementary catastrophes. Multi-valued Legendre transforms appear in subjects such as non-linear elasticity and non-convex optimization.

Elastic work functions may be restricted by material symmetry requirements and/ or by the Cauchy symmetry of the first order moduli with respect to the Green strain. This investigation enables the construction of any work function so constrained. The crystallographic point groups are considered in detail, and examples of such work functions are given.