We extend a result of J. F. Toland concerning bifurcation from infinity and we made some applications to variational inequalities.

The diffraction of a line source by an absorbing semi-infinite plane in the presence of a subsonic fluid flow is examined. Expressions for the total far field for the leading edge (no wake present) and the trailing edge (wake present) situation are given.

It is found that the presence of fluid flow attenuates the sound level in the shadow region of the absorbent screen. There is greater attenuation of the sound level in the trailing edge situation than there is for the leading edge situation.

Sound level attenuation on the source side of the absorbent screen is found to depend more critically on the absorptive properties of the screen than on the fluid flow. 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.

The aim of this paper is to give a functional analytic treatment of the homogeneous and inhomogeneous linear transport equation in the case that the parameter c occurring in that equation equals 1. The larger part of the paper is devoted to the study of a certain operator T−1 A in the space L2(– 1, 1). A peculiarity not arising in the case c < 1 (treated amongst others by Hangelbroek) is that, for c = 1, the operator T−1A has a double eigenvalue 0 and that it is no longer hermitian. The Spectral Theorem is used to diagonalise the operator as far as possible, and full-range and half-range formulae are derived. The results are applied inter alia to give a new treatment of the Milne problem concerning the propagation of light in a stellar atmosphere.

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.

We use the notations and results of Part I of this paper [see 7]. Sections and formulae of that paper will be referred to as section I.4, formula (1.5), etc. In particular, T, A, TN0, K, σ, F and Λ(λ) will have the same meaning as in [7].

Let

be Hermite's function of order λ and let h = h(λ) be the largest real zero of Hλ(t). Set

In this paper we establish the inequality

where

Equality holds for S = ½. The result is also fairly accurate as S→0 and S→1. The proof is analytical except in the ranges −1·1 ≦ h ≦ −0·1 and where the argument is concluded by means of a computer.

The following deduction is made elsewhere [2, Theorem A]. If u(x) is subharmonic in Rm(m ≧ 2) and the set E where u(x) > 0 has at least k components, where k ≧ 2, then the order ρ of u(x) is at least ϕ(1/k). In particular, if ρ < 1, E is connected. This result fails for ρ = 1.

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.

The purpose of this work is to consider some analytic aspects of the matrix functional differential equation, y′(x) = Ay(λx) + By(x), (0 ≦ x < ∞), where λ is a constant with 0 < λ < 1 and A and B are constant matrices.

Let A be a closed linear transformation from a real Hilbert space ℋ, with symmetric inner product 〈, 〉, into itself; and let f ∈ ℋ be given such that the problem Aø = f has a solution ø ∈ D(A), the domain of A. Then bivariational upper and lower bounds on 〈g, ø〉 for any g ∈ ℋ are exhibited when there exists a positive constant a such that 〈AΦ, AΦ⊖ ≧ a2〈Φ, Φ〉 for all Φ ∈ D(A). The applicability of the theory both to Fredholm integral equations and also to time-dependent diffusion equations is demonstrated.

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.

For various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

The series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.

We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

A differential operator (1) Ly = y(n) + an-1(t)y(n-1) +…+ao(t)y is said to be disconjugate on an interval I if every non-trivial solution of Ly = 0 has less than n zeros on I, multiple zeros being counted according to their multiplicity. An algorithm for the construction of disconjugate operators of type (1) is given here. All such operators are obtained at least in the case when the interval I is open or compact.

This paper investigates the one-dimensional propagation of weak discontinuities, that is acceleration waves, in a homogeneous and isotropic half-space composed of an arbitrary number of non-linearly hyperelastic layers. The transmission and reflection coefficients are evaluated in terms of the initial condition at the boundary, and the steepening of the waves to form a shock is discussed. The results are specialised to the case of periodic layering.

The direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.

Some characterisations of commutatitivity for C*-algebras are given in terms of inequalities involving sums and products of self-adjoint elements, and optimal constants are obtained for the corresponding inequalities for non-commutative C*-algebras.

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.

In the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integral

associated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.

The abstract non-linear non-autonomous functional differential equation

is considered. An evolution operator is associated with the solutions of this equation and existence and stability results are obtained.

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.