The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the form

is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.

It is obvious that the amount of the optical activity of a given compound containing an asymmetric atom of carbon depends upon the amount of difference in character among the four radicals united to the asymmetric carbon atom, so that if two of them are very nearly equal we come very near to a compound of a symmetric carbon atom, in which the optical activity is zero. The question suggests itself, How are we to measure this difference of character ? We shall assume that there is a function, capable of numerical representation, derivable from the composition and constitution of the radical and the temperature of the substance, and that it is the difference between the values of this function in the case of two radicals which gives us the difference of character referred to. For the sake of brevity, we may call this function the K of the radical. The object of this paper is to show that, if there be such a function, there are methods by means of which we may hope to ascertain its value in each case.

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

We give a complete characterization of the analytic and Darbouxian first integrals of the Rikitake system, which serves as a model for the reversal of polarity of the Earth's electromagnetic field. Our approach uses the Darboux theory of integrability.

Though it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

We study a boundary integral equation method for transmission problems for strongly elliptic differential operators, which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with Galerkin's procedure. The method is shown to work for the vector Helmholtz equation in ℝ3 with electromagnetic transmission conditions. We propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over H1. We analyse the boundary integral equations using the calculus of pseudodifferential operators. Here the concept of the principal symbol is used to derive existence and regularity results for the solution.

9. A formula is adopted which gives the probability of an individual of given ability passing a test item in terms of two quantities constant for that item. A method of estimating these two constants is given. Making certain assumptions concerning the items composing a test, formula? are then derived giving the expected value and the standard error of the test score of any person. It is shown that there is a certain reciprocity between persons and items, and corresponding formula? are given for the item scores. Finally, the results are applied to actual data obtained on a Moray House intelligence test, an estimate being made of the reliability of the test.

In conclusion I should like to thank Professor Godfrey H. Thomson for his help and valuable criticism in connection with this paper. I should also like to take this opportunity of thanking the Carnegie Trust for the Universities of Scotland for grants to cover the cost of the setting and printing of mathematical formula? in two papers previously published in the Society's Proceedings.

The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like $k^{-7/6}$ as $k\to0$. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function $k^{-7/6}$ that we obtained in an earlier paper.

The steady states of the Cahn–Hilliard equation are studied as a function of interval length, L, and average mass, m. We count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a.e. there is an even number of such solutions and for a.e. there is an odd number of such solutions. If m lies within the metastable region, then for a.e. L > 0 there is an even number of solutions. Furthermore, we prove that for all values of m, there are no secondary bifurcations.

Professor A. Landé has kindly sent me a manuscript of a paper in which he develops some most important ideas on the meaning and the value of the number 137. His work is founded on the principle of reciprocity which I recently suggested (Born, 1938). Landé adds to this principle a new assumption about the wave functions, representing an electron, and he finds a linear integral equation for these containing a parameter the value of which may be expected to be 137. It is Landé's great merit to have shown that the reciprocity principle leads to a condition for the Planck constant ћ (the value of which in natural units, c = 1, e = 1, is 137), and to have discovered the root of the peculiar order of magnitude of the number. But some difficulties which I encountered led me to reconsider the problem.

Several energies measuring jump discontinuities of a unit length gradient field are considered and are called defect energies. The main example is a total variation I(φ) of the hessian of a function φ in a domain. It is shown that the distance function is the unique minimiser of I(φ) among all non-negative Lipschitz solutions of the eikonal equation |grad φ| = 1 with zero boundary data, provided that the domain is a two-dimensional convex domain. An example shows that the distance function is not a minimiser of I if the domain is noncovex. This suggests that the selection mechanism by I is different from that in the theory of viscosity solutions in general. It is often conjectured that the minimiser of a defect energy is a distance function if the energy is formally obtained as a singular limit of some variational problem. Our result suggests that this conjecture is very subtle even if it is true.

We establish the existence of isolated local minimisers to the problem of partitioning certain two-dimensional domains into three subdomains having least interfacial area. The solution we exhibit has the special property that the three boundaries of the minimising partition meet at a common point or “triple junction”. The configuration represents a likely candidate for a stable equilibrium in the dynamical problem of two-dimensional motion by curvature and also leads to the existence of local minimisers possessing triple junction structure to the energy associated with the vector Ginzburg–Landau and Cahn–Hilliard evolutions.

We prove the existence of radial solutions of concentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov–Schmidt reduction.

We present a generalisation of the continuous Gronwall inequality and show its use in bounding solutions of discrete inequalities of a form that arise when analysing the convergence of product integration methods for Volterra integral equations. We then use these ideas to prove convergence of a numerical method which is effective in approximating Volterra integral equations of the second kind with weakly singular kernels.

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.