The theory of differential equations is largely concerned with properties of solutions of individual, or classes of, equations. This paper is given over to the converse problem - that of seeking properties of functions which require them to be, in some respect, solutions of a differential equation, and to determining all possible such differential equations.

From this point of view this paper discusses only linear ordinary quasi-differential equations of the second order. However, the methods can be extended to quasi-differential equations of general order.

In catalysis theory there is interest in the number of solutions to the equation

with the boundary conditions

the parameters λ,β, γ being all positive and p a non-negative integer. The paper answers this question when γ is large, which is the interesting situation physically. Although the treatment is somewhat different in the cases p = 0 and p ≠ 0, the final answer is the same, that is, given β, there exist two positive functions λ1(γ) and λ2(γ) such that the problem has one solution if λ<λ1(γ), or λ>λ2(γ), three solutions if λ1(γ)<λ <λ2(γ), and two solutions if λ=λ1(γ) or λ=λ2(γ).

The equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.

It is shown that, generically, scalar one-dimensional parabolic equations ut = (a2(x)ux)x + f(u), x ∈ [0, 1], with Neumann boundary conditions, have all the equilibrium solutions hyperbolic.

Moreover, the bifurcations of these equilibria are generically of the saddle-node type.

The ring of a finite projective plane, introduced by the author in a previous paper, is shown to be a Gorenstein ring. Its integral closure and conductor are also determined.

We consider boundary value problems for second order differential systems of the form (1)x” = A(t)x′ + f(t, x) and (2) x” = A(t)x′ + f(t, x) + q(t, x). By assuming the existence of a solution to (1) with a given region in (t, x) space, we derive conditions under which there exists a solution to (2) which stays in a certain neighbourhood of and satisfies given boundary conditions.

We consider linear differential equations of the form F(x, z)≡ xn + a1(z)x(n-1)+…+an(z)x = 0 with power-logarithmic coefficients or coefficients which are asymptotically similar to power-logarithmic functions in a central sector S of a complex plane for z →∞, z∈S. The main result of this paper is that in a sufficiently small central sector SE⊂S there is a fundamental system of solutions {xi(z) = exp [∫γi(z)dz)} where each function γi(z) is equivalent to a power-logarithmic function or has an estimate of the form O(z−∞). Furthermore, a precise estimate is obtained for a partial solution of a nonhomogeneous equation F(x, z) = α(z), where the function α(z) grows like a power.

We study the extreme points of two classes of polynomials of degree at most n:

It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.

The results of part I (see [5]) are applied to pairs of formally symmetric differential expressions, to Hermitian differential systems and to a reduced operator moment problem.

Motivated by results of G. K. Pedersen, showing how a simple C*-algebra must contain an abundance of projections whenever it contains a single nontrivial projection, we provide generalisations and new proofs using more algebraic methods.

This paper gives a complete analysis of the integral inequality

where the integral on the left may be only conditionally convergent. It is shown that the best possible number K is 4 and all cases of equality are identified in terms of Hankel-Bessel functions of order ⅓.

This note characterises those Banach space valued, scalar-type spectral operators T = ∫ z dP(z), where P is the resolution of the identity for T, whose indefinite spectral integral E→∫EzdP(z) as a set function of the Borel sets of the complex plane is countably additive with respect to the uniform operator topology.

We give a direct and simple proof of a central result of the above paper.

We consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E}, E↑0, holds. We shall study the validity of this formula for potentials which show oscillatory behaviour as x →∞, like e.g. q(x) = −(1 + x)−α(a + b sin x) with 0<α <2, a≧0, b≠0. We shall obtain the leading-order behaviour of both n(E) and vol n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E} as E↑0 for a certain class of q's, and we shall see that the classical formula fails in most cases, but there are some noteworthy exceptions.

It is shown that every weakly l1-stable linear and bounded operator (which represents a linear discrete-time system) on a Hilbert space is power stable. It solves (at least partially) a discrete-time version of a problem posed by A. J. Pritchard and J. Zabczyk for strongly continuous semigroups of bounded linear operators.

For a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).