We prove the existence of the thermodynamic limit of the free energy per particle for a twocomponent plasma in one space dimension and with a logarithmic pair interaction.

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.

By assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

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.

If X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

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 existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation,

are established. The limiting behaviour of these waves, when ε tends to zero and when δ tends to zero is examined together with a singular limit wherein both ε and δ tend to zero.

The deficiency index of each power of the differential expression M[y] = w−1(−(py′)′ + qy), defined on [a, ∞), is calculated exactly in terms of the behaviour of a simple function of p and w for a large class of expressions satisfying a hypothesis which requires that p be large compared with w and q. In general, not all powers of M are limit-point.

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 obtain estimates for the exponential growth of the solutions to u″(t) = (A + ζ2I)u(t) in terms of the exponential growth of the solutions to u″(t) = Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weighted L2 norms are also considered in Hilbert space.

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 consider a system of functional differential equations

where G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

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.

For a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.

Second order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.

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.

In this paper some eigenvalue problems for elliptic as well as hyperbolic equations are solved. The main tool used is an abstract critical point theorem on an unbounded manifold of the form {u | (Lu, u) = constant} (where L is a nonpositive selfadjoint operator), which makes use of a linking type argument on a manifold.

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.