For elastic or plastic media the linkages between basic properties at two levels of description are rigorously analysed. At one level the material is structurally heterogeneous, while at the other it behaves macroscopically as if homogeneous. The deformation is unrestricted as to magnitude, but is treated incrementally. Full account is taken of coupling between stresses and rotations within a representative volume. Theorems on spatial averages of tensor variables and their products are reviewed and extended. Relations between constitutively significant quantities at both levels are developed by means of tensor influence functions associated with auxiliary elastic fields. The analysis is conducted primarily in terms of non-objective variables and a Lagrangian reference configuration. A final transformation to intrinsic variables is effected with the help of some novel operators.

A random submatroid ωr of the projective geometry PG(r − 1, q) is obtained from PG(r − 1, q) by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability p of being retained. The properties of such random structures were studied in [5] and [6]. In the first of these papers, p was kept fixed, while in the second, motivated by Erdös and Rényi's work ([3), [4]) on random graphs, p was taken to be a function of r. A recent paper of Bollobás[2] strengthens and extends a number of the results of Erdös and Rényi. In this paper we prove matroid analogues of several of Bollobàs's results thereby extending some of the results of [6].

Two, essentially different, criteria for truncating a potential function of a conservative structural system are reviewed, one based on deflection considerations and the other on the mathematical concept of determinacy. Formulations in which they give different results are discussed; most importantly, these include Koiter's 1976 form for mode interaction in stiffened structure [11], which exhibits Thorn's parabolic umbilic catastrophe [15]. For this case, two alternative schemes of perturbation analysis are presented, each describing asymptotically the post-buckling response. The more obvious approach turns out to be the less successful, being linked to an indeterminate, too severely truncated, form of the potential function.

The second, determinacy linked, approach generates as a first-order response a distinctive looping pattern of equilibrium paths, which has recently been identified in a simple model due to Budiansky and Hutchinson[1], by exact, closed-form, solution [8]. The convergence of each asymptotic scheme to the exact solution is briefly reviewed for this simple model, by plotting first and second order results.

In [3], Barth, Brannan and Hayman proved that if u(z) is any non-constant harmonic function in ℝ2, ø(r) is a positive increasing function of r for r ≥ 1 and

then there exists a path going from a finite point to ∞, such that u(z) > ø(|z|) on Γ. Moreover, they showed by example that the integral condition above cannot be relaxed.

This paper presents an alternative analysis of the behaviour of the Moran model. Using some of the techniques developed in the study of interactive particle systems, a process which is dual to the Moran model is constructed. Exact analysis of this dual process is considerably easier than for the original model. Because of the relationship between the two processes it is then possible to find the distribution of the Moran model.

Lemma 4·1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4·2, the only result dependent upon it, is true. The following supplies the necessary modification to the proof.

We consider second order differential equations of Liénard type:

Such equations have been very widely studied and arise frequently in applications. There is an extensive literature relating to the existence and uniqueness of periodic solutions: the paper of Staude[6] is a comprehensive survey. Our interest is in the number of periodic solutions of such equations.

Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Then

is a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvalues

of T satisfy as n → ∞. We prove a related result. Let W12[0, 1]2 denote the space of all K(x, t) ε L2[0, 1]2 which are absolutely continuous in x for each t and absolutely continuous in t for each x, and the partial derivatives ∂K/∂x(x, t), ∂K/∂t(x, t) are both in L2[0, 1]2. We slow that the eigenvalues of any satisfy .

Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.

Volume 92 (1982), 115–119

K. J. Falconer

School of Mathematics, University of Bristol

‘Growth conditions on powers of Hermitian elements’

The above paper aimed to obtain Banach algebra equivalents of the elegant theorems of Roe [2] and Burkill [1] which gave characterizations of the sine function. Professor John Duncan has kindly pointed out that the paper contains two oversights, with the result that the theorems stated are not the strict Banach algebra equivalents as was claimed. (The assertion on p. 117 that T is Hermitian is false if μ > 0, and at the bottom of p. 117 it follows from Theorem 2 that F0(t) = c1eit + c2eit so that a2 – e = 0.) Whilst Theorems 1–3 are correct as stated, the conclusions of Theorems 1 and 3 must be strengthened to provide equivalent versions when μ > 0. In the basic case when μ = 0, the analysis remains as given.

We characterize absolutely continuous and continuous measures by means of the g-function and distribution function, respectively, of the Poisson integral in a half space. Some other ways of measuring the Poisson integral are found to make such measures indistinguishable. A variant of the Poisson integral is also studied.

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 < EX ≡ μ < ∞ and finite positive variance 0 < var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .

Consider the second order nonlinear equation

where, r, p, f ε C ([a, + ∞), R), R = (–∞, +∞) with r > 0 and λ > 0 is the quotient of odd integers. One of the prototypes of this class of equations is the generalized Thomas-Fermi of Emden-Fowler equation

This paper describes a criterion which, if satisfied, guarantees uniqueness of solution for the problem involving the radiation or diffraction of small amplitude water waves by a totally submerged body. This criterion was originally stated by a Russian mathematician, V. G. Maz'ja, although his work does not appear to be widely known. For this reason a brief outline of Maz'ja's proof is given here, together with a discussion of the condition to be satisfied by the submerged body.

Maz'ja's criterion is not satisfied by all submerged bodies and so his result does not provide a general proof of uniqueness. However, the criterion is satisfied in some important cases and examples are presented here. The partial nature of this result is due entirely to the method of proof and not to any intrinsic difficulty with the underlying hydrodynamics.

Amongst the various modifications of the general theory of relativity (GRT), the scalar-tensor theory of Brans and Dicke (BDT) is treated most seriously ([2], [17], [13]). The BDT is consistent with observations as long as the coupling parameter ω between the scalar and tensor components of gravitation is about equal to or greater than 500 [18]. However, there are no a priori theoretical reasons for excluding other values of ω. In the limit ω → ∞, the BDT reduces to the GRT for a constant BDT-scalar field Φ

Born partial wave integrals are considered for electron-neutral atom collisions. It is shown that for relatively general atomic wave functions these may be evaluated analytically. These form the Born reactance matrix RB and can be used to calculate the collision strengths in the approximations ωI, ωII.