Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.

We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

Let$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form

$$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$

$\mu \in L^{\infty}$

$\mathbb {D}_s^2$

$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$

This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L∈(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∈(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

We consider the existence of stationary or pinned waves of reaction–diffusion equations in heterogeneous media. By combining averaging, homogenization and dynamical-systems techniques we prove under mild non-degeneracy conditions that if the heterogeneity is periodic with period ε, pinned solutions persist at most for intervals in parameter space whose length is O(e−c/√ε).

1. Fourier's Integral Theorem.—In a former paper it was shown that certain formulæ, usually obtained by means of Fourier's Integral Theorem, could also be proved by contour integration: and, indeed, Fourier's Integral Theorem can be established by this method. The functions involved must be holomorphic, so that the proof does not justify the application of the theorem to such wide classes of functions as the usual proof by means of Dirichlet Integrals. In what follows it will be assumed that all the integrals are convergent, and the changes of order of integration justified. Fourier's Integral Theorem will be proved in the following form.

In my Presidential Address of October 1920 I cited the stems of certain Zygopterideæ in illustration of size as a factor in Stelar Morphology. But only a few salient facts were adduced from the protosteles of their stems. The relation of increasing size to the elaboration of the vascular tracts of primitive plants deserves further examination: and the results would appear the more cogent if the evidence were collected not only from series of plants which are related to one another, or even from different parts of the same plant, but also from series not so nearly related. A re-examination of the published figures of axial steles and petiolar traces in the fossil Cœnopterideæ has provided interesting results.

The planet Saturn is surrounded by several concentric flattened rings, which appear to be quite free from any connection with each other, or with the planet, except that due to gravitation.

The exterior diameter of the whole system of rings is estimated at about 176,000 miles, the breadth from outer to inner edge of the entire system, 36,000 miles, and the thickness not more than 100 miles.

It is evident that a system of this kind, so broad and so thin, must depend for its stability upon the dynamical equilibrium between the motions of each part of the system, and the attractions which act on it, and that the cohesion of the parts of so large a body can have no effect whatever on its motions, though it were made of the most rigid material known on earth.

The first definite olfactory rudiments, arising as two solid ingrowths of the deep layer of ectoderm on the under-side of the head, are to be seen in sections of Stage XXVII.

A section through the left olfactory rudiment of Stage XXVIII (fig. 1) shows the early structure of the organ clearly, with its ingrowing thickening of the deep ectodermal layer covered by the thin superficial ectoderm. The cells in both layers are heavily laden with yolk particles, as illustrated in the text-figure by black dots.

In the later stages, XXIX and XXX, the rudiment is similar; the cells near their outer ends are becoming progressively less yolky.

The problem in question is—To find the number of possible arrangements of a set of n things, subject to the conditions that the first is not to be in the last or first place, the second not in the first or second place, the third not in the second or third place, and so on.

In an examination of 11,797 eggs which had failed to hatch, 433 specimens exhibiting teratological abnormalities were obtained.

The various types of monsters observed are briefly described and the frequency of each given.

The most common types were characterised by various degrees of abnormality either in the brain, cranium, or eyes, or in two of these structures, or in all three. Hyperencephaly, exencephaly, and microphthalmia constituted 93 per cent. of all the monsters observed.

Other teratological abnormalities encountered included prognathia, duplicity, ectopia, cyclopia, otocephaly, malformed limbs, and absence of premaxillæ.

Monsters accounted for at least 3·6 per cent, of all mortality in the material examined.Since this material included many embryos too decomposed for detection of teratological abnormality, the actual loss is probably considerably higher.

No significant deviations from the normal sex-ratio were evident in any class of monsters.

A decline in the frequency of such abnormalities from February to April was evident.

Evidence is adduced from Stockard's experimental teratology in fish eggs, to show that a very probable cause of the production of monsters in the chick is an arrest of development of the embryo at a critical stage.Presumably such a condition results from the chilling of those eggs laid in the early stages of gastrulation.

Some evidence to indicate the possibility of an hereditary tendency to produce monsters is briefly discussed.

For the Euclidean squared-distance function f(·) = dist2(·, K), with K ⊂ MN×n, we show that K is convex if and only if f(·) equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i) K is compact and contractible or (ii) dim C(K) = k < Nn, K is convex if and only if f equals one of the semiconvex relaxations when dist2(P, K) is sufficiently large, and for case (i), P ∈MNxn; for case (ii), P ∈ Ek—a k-dimensional plane containing C(K). We also give some estimates of the difference between dist2(P, K) and its semiconvex relaxations. Some possible extensions to more general p-distance functions are also considered.

Sufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

Let T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the cone

related to solubility of the multiparameter eigenvalue problem

with W(λ)−I nonpositive definite. The main result is as follows.

Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.

We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

We define the rational de Rham cohomology associated with the generalised confluent hypergeometric functions. Purity of the cohomology is proved and an explicit ℂ-basis of the nontrivial cohomology is computed.

For piston problems for a system of isentropic gas dynamics, convergence theorems of a difference scheme are obtained by compensated compactness theory and by analysis of the difference scheme.

Differential operators generated by the differential expression My(x) = —y″(x)+q(x)y(x) in L2(0, ∞) are considered. It is assumed that

is bounded for all x in [0, ∞) and some fixed ω > 0. The operators are shown to be bounded below and an estimate for the lower bound is obtained in terms of q(x). In the case where q(x) is LP (0, ∞) for some p ≧ 1, the results are compared with recent ones of W. N. Everitt. Some comments are made on the best-possible nature of the results.

We investigate the large-time behaviour of the solutions u = u(x, t) to the one-dimensional nonlinear heat equation with reaction

with exponents m > 1,p < 1. The initial function u(x, 0) is assumed to be measurable and nonnegative. In the case m + p ≧ 2 where the initial value does not uniquely determine the solution, we also fix the positivity set of the solution u(x, t) if the support of u(x, 0) is not the whole line ℝ, i.e. u(x, t) > 0 if and only if −s1(t) <x<s2(t), t ≧ 0, where 0≦si(t)≦∞ for t ≧ 0, i = 1, 2 are lower semicontinuous given functions. We prove that u converges to a self-similar function which depends only on the behaviour of u(x, 0) for |x| large or si(t) for t large. We classify the set of self-similar solutions and study the equation satisfied by their interfaces.

The main result of this article is the calculation of the first derivative of Teichmüller's metric from an inequality of Reich and Strebel. Furthermore, from the same inequality one is able to calculate information about the difference quotient for the second derivative. From the techniques used here it does not seem possible to determine whether the metric is C2.

Let s and t be normal elements of the Calkin algebra, and let (s) denote the unitary orbit of s. A formula ρ(s,t) is defined to measure the distance between unitary orbits, and satisfies