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.

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.

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.

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.

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

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.

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.

Concavity arguments have been used by Knops, Levine and Payne [1] to discuss evolutionary properties of weak solutions to an abstract non-linear differential equation in a Hilbert space. These authors demonstrate that provided the non-linearity is suitably restricted and for specified initial data, the norm of the solution becomes unbounded in a finite time. In other words, the solution possesses a finite escape time.

In this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.

Although integral equations and relations for ellipsoidal wave functions have been known for some time, it had previously been impossible to obtain asymptotic estimates for these integrals, with large wave parameter, as the available asymptotic expansions for the integrand were non-uniform. However, uniform asymptotic expansions have recently been calculated for the functions occurring in the integrand, and these expansions are now used to determine asymptotic expansions for ellipsoidal wave functions of the third kind. A different type of normalisation is proposed for ellipsoidal wave functions and this is compared with previous normalisations by evaluating asymptotically a double integral.

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).