The grey squirrel (Sciurus carolinensis leucotus Gapper), which is a native of the North American continent, was introduced into England and Scotland quite recently and soon became of great economic importance in these countries. In his very excellent monograph, Middleton (1930) states that both sexes of the grey squirrel show diminished activity during the winter and while they do not hibernate in the true sense of the word, they probably sleep if the weather is severe. The breeding habits of the female were studied by Deanesly and Parkes (1933), who ascertained that the female definitely has a period of anœstrus during which the reproductive organs are atrophic. In the male, however, fully functional testes were found throughout the year (Allanson, 1933), and there were no histological changes which would indicate a regular period of quiescence.

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku) dx on the space BVk(Ω:ℝm) of the integrable functions, whose the f-th derivative in the sense of distributions is a Radon measure with bounded total variation.

A new method is developed for finding radially symmetric solutions of semilinear elliptic problems by phase space methods. The basic idea is to formulate a shooting argument from initial conditions at r = 0 which involves encoding the oscillation information about the trajectories under consideration. The main theorem is applied to a particular nonlinearity and produces a cascade of solutions wherein the multiplicity increases with the number of zeros.

1. Among 7135 chick embryos of nine days or over, obtained from unhatched eggs from three poultry farms, 112 cases of chondrodystrophy were observed. From 1900 eggs set at this Department, 124 chondrodystrophic embryos were obtained.

2. The mortality caused by this condition varied in different flocks from 0.80 to 7.96 per cent. of the embryos dying after the eighth day; at this Department it was responsible for loss of 6.5 per cent. of the eggs set.

3 The incidence of chondrodystrophic embryos was found to be greatest in January and February, and to decline steadily thereafter to an almost complete absence in June.

4. The occurrence of chondrodystrophy appears to be independent of breed of fowl, sex of embryo, and age of dam.

5. The number of these abnormal embryos produced by individual hens ranged from 1 to 28.

6. Evidence is advanced in support of the theory that the causal agency is an hereditary physiological abnormality in the dam which, under certain unfavourable conditions, results in the production of chondrodystrophy in the embryo.

7. It is shown that the incidence of chondrodystrophy appears to be inversely proportional to the amount of sunshine, and the suggestion is made that lack of direct sunlight is a factor in the etiology of the abnormality.

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.

This paper summarises the results of an attempt to extend the theory upon which the relationship between linear differential equations and integral equations is based. The case in which the nucleus K(x, s) of the integral equation arises as a Green's function is well known; the nucleus is there characterised by its having discontinuous derivates when x = s. The method here dealt with is virtually an extension of Laplace's and analogous methods for solving linear differential equations by definite integrals, and leads to nuclei which are continuous and have continuous derivates for x = s.

The main purpose of this paper is to study the mean value properties of certain Hardy sums over a short interval by using the mean-value theorems of the Dirichlet $L$-functions. Our main result is a mean-value formula for these sums.

Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and β be real numbers. The sequence n ↦ λ (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α; ≥ 1. And n ↦ λ(n) − βn (n = 0, 1, 2, …) is strictly increasing if and only if β < λ(1) − log 2 = 0.98519….

The main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles and such that certain combinations of derivatives of f have few non-real zeros.

In this paper we prove existence of multiple positive solutions for a Neumann problem in ℝN/(0, R), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.

We study the large-time behaviour of solutions to the Korteweg-de Vries-Burgers equation with bore-like initial data. This work relies on the methods of Amick, Bona and Schonbeck to obtain sharp rates of temporal decay of certain norms of the solution, thus obtaining an improvement over results of Naumkin and Shishmarev.

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

Decay estimates for the energy are derived for the initial boundary value problem of the wave equation with a degenerate dissipative term:

where Ω is a bounded domain in Rn, a(×) is a nonnegative function such that a1 ∊Lp(Ω) for some p > 0 and f is a function tending to 0 rapidly as t → ∞

This paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

It is shown that, for integrals of the type with Ω RN open, bounded and f: Ω × Rm × Rd → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences un → u in measure, υn → υ in Lp, Aυn → 0 in W−1, p is the Ax-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the form with A(i) ∈ C∞ (Ω; Ml × d) ∩ W1, ∞, i = 1, …, N, satisfying the condition and Ax denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υn} in Lp satisfying the condition Aυn → 0 in W−1,p, is obtained.

We prove some results on the global existence of smooth solutions for certain nonlinear parabolic systems of the form Ut + A(U)Ux = DUxx. Here U is a vector and A(U), D are matrices with D a constant, positive matrix. We show how to use our results to study the global continuous (or generalised) solutions to the corresponding nonlinear hyperbolic conservation laws and a conjecture is given.

In the context of reliability theory, two definitions are given for coherent functions of n variables, where both function and variables can take any of l possible levels. The enumeration problem for such functions is discussed and several recursive bounds are derived. In the case of l = 2 (the Dedekind problem) a recursive upper bound is derived which is better than the previous best explicit upper bound forn < 15, and also provides a systematic improvement on this bound for larger values of n.

Using some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundedness

is proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.

We consider the parabolic one-dimensional Allen–Cahn equation

The steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we find

where the functions ξj (t) satisfy a first-order Toda-type system. They are given by

for certain explicit constants γjk.

We consider periodic homogenization problems for Lévy operators with asymmetric Lévy densities. The formal asymptotic expansion used for the α-stable (symmetric) Lévy operators (α ∈ (0, 2)) is not directly applicable to such asymmetric cases. We rescale the asymmetric densities and extract the most singular parts of the measures, which average out the microscopic dependencies in the homogenization procedures. We give two conditions, (A) and (B), that characterize such a class of asymmetric densities under which the above ‘rescaled’ homogenization is available.