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.

The diffraction of a line source of sound by an absorbing semi-infinite half plane in the presence of a fluid flow is examined. It is found that the radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter. For subsonic flow the system exhibits no acoustic instabilities.

Given a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1

where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

New proofs are given for some main results in the theory of linear-quadratic optimal control.

A generalisation is considered of the potential barrier problem beyond the familiar case in which the barrier is bounded by transition points of order one. Here, the two transition points involved are of arbitrary odd order. The approximate method employed, though formal in character, avoids certain pitfalls often made in the past whereby certain exponentially small terms within the barrier are confounded with inherent error terms. This confusion is avoided in the treatment given here by tracing uniformly approximate solutions round the transition points in the complex plane by means of the Stokes phenomenon, the method not requiring the dubious concept of a subdominant term existing in the presence of a dominant term on a Stokes line. At the same time, the solutions to which the reflection and transmission coefficients may be attached are carefully discussed, so that the appearance of a small exponential term may be seen to be genuine when taken in conjunction with inherent error terms. The resulting formula for the modulus of the reflection coefficient generalizes the more elementary formula.

The principal results of this paper concern the spectral properties of the maximal realisation Pp in Lp(Rn) of a formally self adjoint constant coefficient strongly elliptic partial differential operator P(D), assumed to be homogeneous of order 2m, for 1 ≦ p ≦ ∞ and n ≧ 2. If we assume that , for 2n/n + l ≦ p ≦ 2n/n−1, together with certain assumptions on the associated real zero surfaces P(ξ) = λ, λ > 0, then σ(Pp) = σc(Pp) = [0, ∞). We obtain an estimate on the norm of the resolvent of Pp for points near the real axis, which allows us to establish the existence of a generalised resolution of the identity in the sense of Kocan.

This paper studies Sobolev spaces under perturbation of domains of definition. It establishes the basic concepts, methods and results for the convergence of sequences of sets in ℝn, the strong and weak convergence of sequences of Sobolev spaces, the discrete compactness of natural embeddings and the continuous convergence of continuous linear functionals, boundary integrals and trace operators for sequences of Sobolev spaces.

The operator Ik(η, α) and its adjoint Kk(η, α) have obvious boundedness properties for α ≧½ because of their resemblance to fractional integrals. By expressing Ik(0, α) as the product of two Hankel transformations and a translation Heywood and Rooney [1] have shown that if 0<α<½ then Ik(η, α) and Kk(η, α) can be extended to bounded operators from a weighted Lp space to itself provided that 2/(1+α) ≦ p ≦ 2/(1−α) and the weight is suitably restricted. Heywood and Rooney conjectured that this p range could be improved, andin the present paper it is extended to

which may be best possible.

An analogue of the Riemannian theory of uniqueness of Fourier coefficients is developed for Sturm Liouville series, using asymptotic formulae for the eigenfunctions and other quantities. This theory generalises earlier work by Haar in that the coefficient function in the differential equation is only assumed to be integrable.

Let X be the boundary of a compact set which does not separate the plane, C. Let Φ and Ψ be homeomorphisms of C to C with opposite orientations. Then every continuous complex-valued function on X is the uniform limit on X of sums p(Φ)+q(ψ), where p and q are analytic polynomials.

This paper examines the mathematical problem of the propagation of a smooth fronted wavein the context of shallow water theory. Here, a smooth fronted wave will be taken to be one in which the surface slope is continuous across a line in the free-surface, while the second derivative of the surface slope is discontinuous across that same line. This discontinuity line in the surface then plays the role of the wavefront. After establishing that such wavefronts propagate along the characteristics, and deriving the appropriate transport equations, the explicit form is found for the acceleration with respect to distance of the horizontal component of the water velocity of the surface immediately behind the wavefront as a function of position and seabed profile when the wave propagates into still water. The result is then used to prove that in this approximation such a wave cannever break immediately behind the wavefront before the shore line is reached.

This paper is concerned with the L2 classification of ordinary symmetrical differential expressions defined on a half-line [0, ∞) and obtained from taking formal polynomials of symmetric differential expression. The work generalises results in this area previously obtained by Chaudhuri, Everitt, Giertz and the author.

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.

It is well known that the existence of transcendental meromorphic solutions of non-linear ordinary differential equations puts severe restrictions on the equation, the most striking example being the theorem of Malmquist [3]. The value distribution theory of R. Nevanlinna was applied to such questions by K. Yosida [8] who used it to prove Malmquist's theorem as well as important generalisations. An alternate approach was given by H. Wittich [4,5,6] and in his argument the finiteness of the order played an essential role. Wittich estimated the corresponding enumerative and proximity functions via the calculus of residues. In this note a geometric argument is proposed instead (closest packing of small discs in a bigger circle or on its rim). This method seems to generalise more readily to Yosida's extensions.

1. The generalisation of von Kármán's equations of swirling flow studied by Serrin [7] and Hartman [3, 4] is the system

with the boundary conditions

When α = β = ½ this system reduces to von Kármán's equations studied by several authors recently (see [1, 6] for many references).

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.

Using the constitutive relationship for a Maxwell body, as an example of a viscoelastic fluid, the equations of motion are derived in the Fourier wavenumber-time domain, and specialised to the case of isotropic turbulence. It is shown that, for grid-generated turbulence, the model predicts increased spectral intensity levels, reduced decay rates and steepening of the spectrum in wavenumber, relative to the Newtonian case. These forms of behaviour have been observed in dilute solutions of drag reducing polymers. The key factor in this is found to be the presence of an ‘elastic’ non-linear term in the equations of motion: this term reverses the normal direction of turbulent energy transfer in wavenumber.

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.