The inequality, for suitable sets of non-negative numbers x1, x2, …, xn,

is undecided in the cases n = 11, 12, 13, 15, 17, 19, 21 and 23. In this note we present the results of numerical analysis supporting the conjecture that the inequality is valid in these cases. A new counter–example for n = 25 and a new elementary proof when n = 7 are given.

Consider the differential expression

It is shown here that given pn > 0 with pn ∈ C2n(0, ∞) there exist coefficients

such that all solutions of the equation Ry = 0 are in L2(0,∞). The pi for i<n can be explicitly obtained in terms of pn and a parameter function.

Let S be a compact subset of the open unit disc in C. Associate to S the set

Let R(X) be the uniform algebra on X generated by the rational functions which are holomorphic near X. It is shown that the spectrum of R(X) is determined in a simple wayby the potential-theoretic properties of S. In particular, the spectrum of R(X) is X if and only if the functions harmonic near S are uniformly dense in the continuous functions on S. Similar results can be obtained for other subsets of C2 constructed from compact subsets of C.

An integral operator is constructed which maps solutions of the reduced wave equation defined in exterior domains onto solutions of ∆n u+λ2(l+B(r))u = 0 (*) defined in exterior domains, where B(r) is a continuously differentiable function of compact support. This operator is then used to construct a solution to the exterior Neumann problem for (*) satisfying the Sommerfeld radiation condition at infinity. Such problems arise in connection with the scattering of acoustic waves in a non-homogeneous medium, and this paper gives a method for solving these problems which is suitable for analytic and numerical approximations.

We study the asymptotic behaviour for ɛ→+0 of the solution Φ of the elliptic boundary value problem

is a bounded domain in ℝ2, 2 is asecond-order uniformly elliptic operator, 1 is a first-order operator, which has critical points in the interior of , i.e. points at which the coefficients of the first derivatives vanish, ɛ and μ are real parameters and h is a smooth function on . We construct firstorder approximations to Φ for all types of nondegenerate critical points of 1 and prove their validity under some restriction on the range of μ.

In a number of cases we get internal layers of nonuniformity (which extend to the boundary in the saddle-point case) near the critical points; this depends on the position of the characteristics of 1 and their direction. At special values of the parameter μ outside the range in which we could prove validity we observe ‘resonance’, a sudden displacement of boundary layers; these points are connected with the spectrum of the operator ɛ2 + 1 subject to boundary conditions of Dirichlet type.

This paper, extending the work of Stratford [6] considers a boundary layer with uniform pressure when x < x0, and with the pressure in x > x0 so chosen that the layer is just on the point of separation for all x >x0. The required pressure distribution is shown to be

The displacement and momentum thicknesses are also derived as series in powers of ξ (and log ξ), and the shape parameter H then obtained as a similar series. The continuous change in H from the Blasius value (when ξ = 0) towards the Falkner-Skan [3] separation value is convincingly demonstrated, with the aid of the leading terms of an asymptomatic expansion for large ξ.

Sufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain of En. Such results apply in particular to the n-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

An inequality of C. R. Putnam involving a Dirichlet functional in the singular theory of the Sturm-Liouville differential equation is generalised. The corresponding result for the Sturm-Liouville theory over a compact interval is established and this is extended to the singular case by means of a Tauberian argument.

In a certain factor analysis model sampling formulae for maximum likelihood estimators of the parameters are found by inverting an augmented information matrix. The formulae were derived previously by other methods; but the techniques here employed may be of some general interest and applicable to other problems.

This paper extends some recent results of V. Barbu and H. Brézis. It is concerned with bounded solutions of the problem pu″+qu′ ∈ Au, u(0) = a, where A is a maximal monotone operator in a real Hilbert space H and p and q are real functions. Existence and uniqueness theorems are proved, with results on integrability of solutions in various measure spaces on R+. T(t) denotes the family of contractions of D(A) generated by the equation and we obtain a regularising effect on the initial data. Some properties of this family of contractions are studied.

It is shown that three independent axioms uniquely determine the topological degree of set-valued maps of the form I – G, where G is a convex-valued, limit compact map. This extends earlier work of Amann and Weiss, Nussbaum, and others, in that, apart from dealing with set-valued maps, a larger class of maps is considered even in the single-valued case.

The theorems in this paper deal with the analogues for integrals of Hardy's Series Inequality and its generalisations.

A proof is given of a formula connecting Weyl's function m(λ) with a spectral integral, in the setting of the singular theory of the Sturm-Liouville differential equation. In addition, a relatively short treatment is given of the argument introducing the function. The theory applies to both the limitpoint and limit-circle cases.

Let d(L) denote the deficiency indices (which are equal) of a formally symmetric differential expression L with real coefficients. Then it is shown that

(a)if k>1, d(Lk)−d(Lk−1) ≤ order L, and

(b)if j ≥ k > 1, d(Lj)−d(Lj−1) ≥ d(Lk)−d(Lk−1).

In a previous publication [1] we determined the structure of some new types of ordered inverse semigroup in which the ordering need not be the natural ordering. At the moment, very little is known about ordered regular semigroups and the purpose of this paper is to generalise the structure theorems in [1] to corresponding theorems on ordered regular semigroups. The results obtained provide an interesting bridge between the theory of semigroups and that of ordered semigroups, these theories currently having very little in common.

A fundamental solution for equations of pseudoparabolic type is constructed by a method analogous to Hadamard's construction for elliptic equations. By the use of this fundamental solution we obtain a regularity theorem and integral representations for the solution to the more common initial boundary value problems that are associated with these equations.

Several criteria are given for some or all polynomials in an expression M(y) = –(py′)′+qy which is non-oscillatory on (0, ∞) to be limit-point. One of these states that if M is non-oscillatory and , then every polynomial in M is limit-point.