Section 1 contains a list of the facts about well-quasi-orderings (wqo's) which were established in (1), (4), (7), (10), and (13), with proofs given except for the last three theorems. The theory of well-quasi-orderings is a precursor to Nash–Williams' theory of better-quasi-orderings (10, 11, 12, 6, 7). Section 1 of this paper may be viewed as a beginning to Section 1 of a forthcoming paper, in which an explication of bqo theory is given.

The term ‘critical graph’ has been used most frequently to mean a graph such that the removal of any line decreases the chromatic number. This can be generalized to define a graph G to be ‘critical (μ)’ for an arbitrary invariant, μ if, for each line e, μ(G – e) ≠ μ(G). Analogously we may consider any line e which joins two points not already adjacent in G and define G as ‘anticritical (μ)’ if μ(G + e) ≠ μ(G). Anticritical graphs are investigated here with respect to the following invariants: chromatic number, line-chromatic number, point-connectivity, line-connectivity, point-independence number, line-independence number, diameter, radius, and cyclability number.

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.

1. Let f(n) be a real-valued additive arithmetic function. Let α(x) and β(x) > 0 be real valued functions, defined for x ≥ 2. Define the frequencies

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we defined

where λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

C. Ehresmann (2) has shown that if a leaf L of a smooth foliation has a foliated neighbourhood, then there exists a fibre bundle over L, normal to the leaves, with discrete structure group. Using the concept of a microbundle and the n-isotopy extension theorem, we find a similar result for both PL and TOP categories, and, in addition, show that the structure group can be chosen to be the holonomy group of L. As for applications we show that holonomy characterizes the foliated neighbourhood of a leaf (proved by Haefliger in the differentiable case (3)). In particular, if the holonomy group of a compact leaf L is trivial then the leaf has a trivial foliated neighbourhood, and if it is finite it has a neighbourhood of compact leaves which are covering spaces of L. Another corollary is the known result that a proper submersion with compact fibre is a fibration. Finally we use the fact that the constructed normal microbundle can be chosen to have its fibres contained in the leaves of a transverse foliation to demonstrate isotopy uniqueness of normal microbundles.

The results given here are a natural continuation of those given by G. P. Shannon (4). The extension of various operators to the completions of the respective spaces are considered and the adjoint operator is more closely examined. A perturbation result for strictly cosingular operators is also given.

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

This note contains a partial generalization of the results of (3). We extend to arbitrary n and k the complete determination of the cases in which cup length is strictly less than the maximum possible value dim Gn, k(R) = nk. We also show that the Lusternik–Schnirelmann category cat Gn, k(R) < nk in the same cases.

Given a set consisting of n1, objects of type 1, n2 of type 2, …, nk of type k, we denote by the number of possible derangements of the set i.e. permutations in which no object occupies a site originally occupied by an object of the same type. A formula is found for in terms of Laguerre polynomials, and some of its implications are considered.

1. The singularities of solutions of linear partial differential equations are most appropriately described in terms of wave front sets of distributions. (For the definition and properties of wave front sets, see (4) and (5).) In particular, let Ω ⊂ RN be an open set and let P be a linear partial differential operator defined on Ω, with smooth coefficients, real principal symbol p(x, ξ), and simple characteristics. Then if u ∈ ′ (Ω) and Pu = 0, the wave front set of u, WF(u), is a set in the cotangent bundle T*Ω (minus the zero section) that is contained in p−1(0) and is invariant under the flow generated in p−1(0) by the Hamiltonian vector field (gradξp, − gradxp). In other words, WF(u) is made up of bicharacteristic strips. This is a special case of an important theorem due to Hörmander (5), and Duistermaat and Hörmander (1), which generalizes classical results on the relation between discontinuities of solutions of a partial differential equation and the characteristics of the equation.

A class of approximate stationary vacuum solutions are obtained by expanding the metric in powers of a certain parameter, and by solving explicitly the first few orders in terms of two harmonic functions ø and ψ. These solutions, to the order considered, reduce to the Weyl solutions when ψ = 0 and yield the Papapetrou solutions when ø = 0, and contain a subclass that is asymptotically flat and has non-zero mass. The physical interpretation and the possibility of proceeding further are briefly discussed.

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equations

This system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

An incompressible magnetohydrodynamic flow, bounded below by a semi-infinite solid wall insulator with a plane surface, is weakly perturbed by two-dimensional (i.e. cylindrical) sources immersed within the fluid. Certain results of Chee-Seng(1) are employed to establish an exact analytic solution for arbitrary source functions. Both direct emission and reflected disturbance fields are hyperliptic, i.e. partly hyperbolic and partly elliptic. The reflected hyperbolic mode is propagated as a Riemann invariant of a modified incident field and its Hilbert transform. This propagation occurs along reflected characteristics whose (acute) inclination with the interface is generally unequal to that of the incident characteristics, but which are, instead, parallel to those non-incident characteristic rays emanating away from the interface.