The propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.

This paper considers a natural extension of the following inequality of Hardy and Littlewood

to an inequality of the form

where μ>0 and 0<K(μ)<∞. The best possible value of k(μ) is determined, and all the cases of equality.

Measurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

The relationship between weak and sequential weak continuity for mappings between Banach spaces and semigroups on Banach space is studied.

1. In this article the following problem proposed by W. N. Everitt is considered: ‘Under what conditions is it possible to transform a differential equation

which is limit-circle at b1 into an equation of the form

which is limit-point at b2?'

A differential equation of the above type on [a, b) is said to be limit-circle at b iff b is a singular point and every solution . If b is singular and there exists a , then the equation is said to be limit-point at b. See [3] or [4] page 501, also for further details.

This paper is devoted to a study of differential equations and inequalities of the form

and

The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

As is well known, the deficiency indices of the minimal symmetric differential operator of even order 2m with real coefficients on the half-line lie within the limits m and 2m, and are equal. I. M. Glazman [6] first showed, by examples, that for such operators the deficiency indices can actually take all the intermediate values m ≦ P ≦ 2m. Subsequently fresh examples of operators with real coefficients, whose deficiency indices take all possible values, were constructed by various authors (S. A. Orlov [11], M. V. Fedoryuk [4], and others).

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.

This paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

An acoustic line source is moving parallel to the vortex sheet dividing two fluids in relative motion and the lifetime of the source is finite. The resulting phenomena are compared with those created by a fixed pulsed source. It is found that the waves produced by the source in motion are not so singular as, though qualitatively similar to, those due to a source which remains in one position but Helmholtz instability still has a dominant part to play. This statement is true whether the source be moving subsonically or supersonically so long as it has non-zero velocity.

The usual ‘closed-graph theorems’ are concerned with the question of which topological structures on sets X, Y will ensure equivalence of continuity and closed-graph conditions for certain mappings t: X → Y. In this paper some stronger closed-graph like conditions are introduced on a mapping between uniform spaces, by considering the ‘hypergraph’, or graph of the induced mapping between hyperspaces. The central result, included in Theorem 1, states that for arbitrary uniform spaces X, Y, t is continuous if its hypergraph is closed. Thus for topological vector spaces (and some others) the closed-hypergraph condition is equivalent to continuity. In section 2 some situations are found in which continuity is implied by certain intermediate conditions to closed-graph and closed-hypergraph conditions. In particular, a closed-hypergraph theorem is shown to hold for locally convex spaces, provided only that X is barrelled. Finally, in section 3, the hypergraph of a relation is studied, and the separatedness of the quotient space of an equivalence relation is shown to replace continuity with regard to closed-graph and closed-hypergraph conditions.

‘Quadratic system’ is here used in the sense introduced by W. A. Coppel for a pair of simultaneous Riccati equations with constant coefficients. The problem is to study the occurrence of movable singularities for solutions of a system

under suitable restrictions on the lines

The aim of the restrictions is to obtain systems where the trajectories have four finite and three infinite topological singularities which form specified configurations. For the movable (function theoretical) singularities the infinitary (topological) singularities are decisive. They are the endpoints of the coordinate axes and of the line F: L2–L1 = 0. A detailed qualitative discussion is given for the trajectories. The system forces various properties of monotony on the integral curves.

The qualitative discussion is completed by quantitive data in the form of series expansions for the solutions. Here the value of the parameter

is critical.

If q is not an integer, then there is one and only one solution whose trajectory is associated with the upper end of the line F which has a simple pole at t = 0 (or any other preassigned value). All other trajectories in this family belong to solutions with expansions in terms of non-logarithmic psi-series of the form

for a singularity at t = 0. Here – 1 ≦ m, 0 ≦ n and c is an arbitrary constant.

On the other hand, if q is a positive integer, then we have logarithmic psi-series of the form

where Pn(s) and Qn(s) are polynomials in s of degree ≦ [n/q].

Solutions associated with the ends of the axes have non-logarithmic psi-series. One coordinate becomes infinite like 1/t at zero, the other goes to zero like tλ where λ = –b1/a1 for the x-axis and –a2/b2 for the y-axis. The psi-series have exponents of the form j + Kλ, where j and k are integers, –1 ≦ j, 0 ≦ k.

The results are used to construct new convergence proofs for the solutions of Emden's equation at a movable singularity.

This paper is concerned with the rate of growth or decline as t → ∞ of oscillatory solutions of equations of the form

we have in mind equations which resemble in some degree the Lane-Emden-Fowler equation

where n ≧ is integral, and σ is either positive, or if negative is not too large, so that oscillatory solutions exist.

The paper discusses the nature of the singularities at any finite point t = t0 of real solutions x(t), y(t) of the system of differential equations x′ = P(x, y), y′ = Q(x, y) in which P, Q are polynomials in both x and y. Its main interest is in cases when the leading terms of P, Q are of the same degree n. Conditions are given under which the only possible singularities are algebraic poles and pseudo-poles. Conditions are also given under which the only possible singularities are of wildly oscillatory type.

In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

Conjugate points are defined in terms of solutions of a linear fourth-order differential equation satisfying two homogeneous boundary conditions at x = α and either u(β) = u′(β) = 0 or u′(γ) = u″(γ) = 0. The smallest β > α and γ > α such that these boundary conditions are satisfied by a non-trivial solution of the equation are denoted by η(α) and ῆ-(α), respectively. Upper bounds are established for min [η(α), ῆ(α)] relative to the conjugate points of a self-adjoint differential equation which is majorised by the more general equation under study.

For a certain class of operators in the direct product of two Hilbert spaces, two problems are solved: the inhomogeneous operator equation, and the eigenvalue problem. Illustrative examples are given.

In the Hilbertspace with (,) as inner product we consider the linear operator L with domain D(L) and the sesqui-linear form 〈, 〉 defined by

Let the symmetric operator L0 be the restriction of L to D(L0), where

For a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

The propagation of scalar waves in a certain two-dimensional medium is considered. The incident field, which is due to the presence of a line source, is scattered by two coupled half-planes on each of which the impedance takes a constant value. The Wiener-Hopf technique is used to find a solution which is then examined asymptotically for high frequency. It is found that there is an illuminated region in which the solution is expressed in terms of geometrical optics rays, and a shadow region in which the solution is described by creeping modes. The point of impedance discontinuity may be regarded as producing secondary radiation. The nature of this secondary radiation is quite different according as the point of impedance discontinuity lies in the illuminated or shadow region of the geometrical optics field produced by the source.