In this note we shall study random labelled graphs. Denote by

the set of all graphs with n given labelled vertices and M(n) edges. As usual, we turn (M(n)) into a probability space by giving all graphs the same probability. The question we address ourselves to is the following. Given a graph H and a constant p, 0 < p < 1, for what functions M(n) is it true that the probability PM(n)(H ⊂ G) that a graph G∈ (M(n)) contains H tends to p as n∞→? This question was posed by Erdös and Rényi (3), (4), who also proved several beautiful and surprising theorems. In order to state the main general result of Erdös and Rényi in this direction, and for our use later, we introduce some definitions.

This paper relates the concept of n-connection for graphs to Tutte's theory of n-connection for matroids (12). In particular, we show how Tutte's definition may be modified to give a matroid concept directly generalizing the graph-theoretic notion of n-connection.

Let P(x) denote a polynomial with degree n and integer coefficients. By the height h of P we mean the maximum of the absolute values of the coefficients.

Let A be a (commutative, Noetherian) local ring (with identity) and let a1,…, an be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay.A-module with respect to a1,…, an if a1,…, an is an M-sequence, that is if M ‡ = (a1,…, an) M and, for each i = 1,…, n,

One of the main open problems in commutative algebra at the present time is that of establishing the existence of a big Cohen–Macaulay module with respect to a specified s.o.p. in an arbitrary local ring. The work and writings of Hochster, such as (5), show that, if the existence of such modules could be established, then several conjectures in commutative algebra, some of which are quite long-standing, would be settled. Moreover, Hochster has established the existence of such big Cohen–Macaulay modules whenever the local ring A concerned contains a field as a subring, or has dimension not exceeding 2: see ((5), chapters 4, 5) and (4).

1. The present paper continues the investigation begun in (5). This earlier paper concluded with the description of a pencil P of canonical curves of genus 6 on a del Pezzo quintic surface F, but had first to describe the relevant features of the mapping of prime sections of F on a plane p by cubic curves circumscribing a quadrangle in p. It remains now to consider F and P in their ambient [5] in relation to G, the group, isomorphic to the symmetric group of degree 5, of self-projectivities of F.

Let G be a finite group and let g be the augmentation ideal of the integral group ring G. Following Gruenberg(5) we let (g̱) denote the category whose objects are short exact sequences of zG-modules of the form and in which the morphisms are commutative diagrams

In this paper we describe the projective objects in this category. These are the objects which satisfy the usual categorical definition of projectivity, but they may also be characterized as the short exact sequences

in which P is a projective module.

The inequality

for fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extension

for T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney class

in the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,

(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and

(b) stable, i.e. w(V) = 1 when F is a trivial representation.

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

Let A be the Steenrod algebra over the field F2. For any graded A-module M we construct here a small chain complex of A-modules EM. This complex computes the homology of the Steenrod algebra with coefficients in M, in the sense that:

The complexes F2 ⊗AEM are comparable in size but slightly larger than those that arise from the Λ-algebra(4). We are motivated to introduce them because they can be used to prove an embedding theorem for the homology of the Steenrod algebra. In fact, we construct a functor K on the category of graded. A-modules, and using the complexes EM, prove that there is a natural, one-one map:

Suppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.

Theorem 2. With probability one we have

for almost all x.

Theorem 3. With probability one we have

In a previous paper (4), the author showed that the martingale approach to point processes may be used to derive Poisson distributional limit theorems. Here this approach is extended to Cox convergence, general increasing processes are considered, and a regularity condition needed in (4) is removed (that of calculability). In Section 3 we give examples to show the need for the various hypotheses used in the main theorem of Section 2. This theorem is applied to various (dependent) thinnings and compound-ings of point processes in Section 4.

In this paper the major concern is in investigating various divisibility properties of the random variable W, the limit of the Heyde martingale in supercritical branching processes. Throughout the paper ø(s) will be used to denote the Laplace-Stieltjes transform of W.

Let I be a countable set, with discrete topology, and let X = (Xn), Y = (Yn) be stationary stochastic processes taking values in I. To a probabilist, the natural topology on processes (strictly speaking, on distributions of processes) is weak convergence:

It was first pointed out by Sommerfeld, around the turn of the century, that certain multi-valued solutions of the wave equation in ℝ3 can be used to deal with the problem of scattering by a wedge, or reflection in a corner. The older literature on this subject is extensive; see ((4), chapter 5) for references up to 1958. The object of this Note is to give an explicit and elementary construction of a forward fundamental solution of the wave equation, of this type, in ℝn+1, where n ≥ 2; for n = 2 this includes Sommer-feld's original result.

Bona and Smith (6) have suggested that the coupled system of equations

has the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariant

they show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

The solution set for the (sourceless) Yang-Mills equations on a spacetime with compact Cauchy surface is a smooth manifold (i.e. the equations are linearization stable) except at solutions that are symmetric. At such symmetric solutions, the structure is described by a homogeneous quadratic form. The degeneracy space for this form is tangent to a manifold of symmetric solutions. Symmetry breaking occurs for perturbations in the nondegenerate directions of the quadratic form. The terms ‘symmetry’ and ‘stability’ in the present work are compared to these terms as used elsewhere in the literature.

In the spirit of generalized coordinates in Lagrangian mechanics, the finite deformation of a material element is considered to be specified by arbitrary geometric variables, and the associated generalized stresses are generated by work conjugacy. The comparative quantitative influences of such coordinate choices are investigated for the main state variables in anisotropic thermoelasticity, namely the moduli, compliances, thermal coefficients of expansion, pressure coefficients, and specific heats. Transformation formulae are obtained to connect the respective values of the state variables for different choices. Special attention is given to the identification of coordinate-invariant magnitudes and relationships. The entire analysis is carried farther for coordinates in a class of tensor measures of finite strain; such measures are used widely in current theories of the mechanical response of solid materials, but apparently not yet in regard to thermal phenomena.