Let f, g be two functions of two Besov spaces (or Sobolev spaces), we look for the Besov spaces to which the product f × g belongs so that the multiplication is a continuous mapping.

So far as we are aware, the only measurements that have been made of the linear extension of a colloidal medium due to gaseous sorption are those of Dr F. T. Meehan, published in 1927. Meehan studied the response of charcoal to carbon dioxide at atmospheric pressure and different temperatures, and found it to expand almost equally along and across the grain when it took in the gas. No tests of this kind appear to have been carried out with other rigid gels or with other gases, and none with any gas under pressure.

It is shown that spectral properties of Sturm–Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

We study the issues of the reconstruction and stability of the inverse nodal problem for the one-dimensional p-Laplacian eigenvalue problem. A key step is the application of a modified Prüfer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths. Two associated Ambarzumyan problems are also solved.

A quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.

Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:

First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.

1. An examination has been made of 11,797 eggs which had failed to hatch, among which were 5050 embryos which had died after the eighteenth day.

2. Four major malpositions of the chick embryo are described and the frequency of each given.

3. It is concluded that one of these, in which the head is buried between the legs, definitely prevents hatching. In the material examined, this abnormality was responsible for 9·25 per cent. of the mortality among embryos of eighteen days or over.

4. It is suggested that the other three malpositions usually result fatally, by reason of their preventing pulmonary respiration in the embryo as well as by mechanical hindrance.

5. Of the embryos over eighteen days, nearly 56 per cent. were in one or another of the four major malpositions.

6. Possible causes are discussed, and the suggestion made that some of the abnormal positions result from an incorrect orientation of the embryo established early in cleavage.

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation

From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

Eigenvalues of both regular and singular Sturm–Liouville (S–L) problems with general coupled self-adjoint boundary conditions are characterised. This characterisation, although elementary, appears to be new even in the regular case. The singular characterisation is an exact parallel of the regular one and reduces to it. One application yields inequalities among the eigenvalues of different coupled boundary conditions. This is a far-reaching extension, even in the regular case, of the well-known relationship among the periodic and semiperiodic eigenvalues.

Karyological studies on spermatogenesis in species of aquatic bugs of the family Corixidæ have brought to light several examples of an unusual type of chromosome behaviour during the course of meiosis. It is considered to be an association of non‐homologues in a manner unexampled in other animal species.

We study some asymptotic behaviour of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a control of the total energy and chemical potential. We show that the limit interface is an integral varifold with generalized Lp mean curvature. The convergence of interfaces as 0 is in the Hausdorff distance sense.

The object of the present communication is to call attention to the remarkable consequences which follow from Carnot's proposition, established as it is on a new foundation, in the dynamical theory of heat; that there is an absolute waste of mechanical energy available to man, when heat is allowed to pass from one body to another at a lower temperature, by any means not fulfilling his criterion of a “perfect thermo-dynamic engine.” As it is most certain that Creative Power alone can either call into existence or annihilate mechanical energy, the “waste” referred to cannot be annihilation, but must be some transformation of energy. To explain the nature of this transformation, it is convenient, in the first place, to divide stores of mechanical energy into two classes—statical and dynamical.

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

We prove that for all x > 0, we have

with the best possible constants α = 0 and .

We show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

A weighted, formally self-adjoint ordinary differential operator l of order 2n is considered, and conditions are given on the coefficients of l which ensure that all self-adjoint operators associated with l have a spectrum which is discrete and bounded below. Both finite and infinite singularities are considered. The results are obtained by the establishment of certain conditions which imply that l is non-oscillatory.

The purpose of this paper is to give a compactness-continuity result for the solution of a nonlinear Dirichlet problem in terms of its domain variation. The topology in the family of domains is given by the Hausdorff metric and continuity is obtained under capacity conditions. A generalisation of Sverak's result in iV-dimensions is deduced as a particular case.

We study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Here

is bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.

This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.