In this note the differential expression M[y] ≡ − y” + qy, q∈Lp(ℝ+) for some p ≧ l, is considered on [0,∞) together with the boundary condition either y(0) = 0 or y'(0) = 0. Lower bounds are given for the spectrum of the self-adjoint operators T generated by M[·] and these boundary conditions. The bounds depend on the Lp-norm of the coefficient q and they improve results of Everitt and Eastham. The bounds are optimal.

In this paper we obtain information on the parity of the number of singularities associated with generic mappings and families of functions. More generally we obtain results relating homology cycles (with ℤ2 coefficients) associated with certain singularity types. The methods employed are elementary, and rely on computations of local incidence numbers associated with certain manifold partitions or stratifications. These computations are carried out for some stratifications arising naturally within singularity theory.

An (n, q) graph is a graph on n labelled points and q lines without loops or multiple lines. We write ν(n, q) for the number of smooth (n, q) graphs, i.e. connected graphs without end points, and ν = V(Z, Y) = ∑n,q ν(n,q)ZnYq /n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form for V (not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied by V. We interpret this equation in purely combinatorial terms. We write Vk = ∑ n ν(n, n + k)Xn/n! and find a recurrence formula for Vk for successive k. We use these and other results to find an asymptotic expansion for ν(n,q) as n→∞ when (q/n) − log n − log log n→ + ∞ and an asymptotic approximation to ν(n,n + k) when 0 < k = o and to log ν(n, n + k) when k < (1−ε).

Some attributes of a new method of defining the convolution and multiplication of generalised functions, too singular to be encompassed by earlier theories, are developed and its extension to the case of several variables is described.

In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 of

for any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

Our basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.

In this article the Brauer characters of the irreducible p-modular representations of the Higman-Sims simple group of order 44352000 are determined when p is an odd prime.

We obtain inequalities for where Wn = anlX1 + … + annXn, the Xr being independent random variables and the Mn being certain truncated means. We then use these inequalities to study the rate at which this probability tends to zero as N→ ∞, noting that in the special case Wn = (X1 + … + Xn)/n, we obtain the estimate given by L. E. Baum and M. Katz which they show is, in a sense, best possible.

A desire to find an inequality which would lead to the result of Baum and Katz was, indeed, the impetus behind this paper.

Let G: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ and T(t) be the Co semigroup generated by G. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {x ε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions for G are satisfied. Proofs are given in terms of the Cayley transformation T = (G + I)(G − I)−1 of G. The results are applied to the damped wave equation utt + γutx + uxxxx + ßuxx = 0, 0 ≦ t < ∞ 0 < x < 1, β, γ ≧ 0, with boundary conditions u(0, t) = ux(0, t) = uxx (1, t) = uxxx(1, t) = 0.

An explicit solution is derived for the dual integral equations

when A(ξ) takes the form

It is then shown how the basic technique can be adapted to derive explicit solutions for more general forms of A(ξ) such as

and

and linear combinations of such sums.

In this paper the author investigates a system of simultaneous dual trigonometric series equations. A closed form solution is obtained by reducing the dual series to singular integral equations of Carleman type. The use of these equations is then illustrated by their application to a crack problem in the theory of elasticity.

If G is a discontinuous group of homeomorphisms of a connected, locally path connected space X, which acts freely on X, then the projection π: X → X/G is a covering map and has the homotopy lifting property. Here we allow the elements of G to have fixed points and use work of Rhodes to investigate how two loops in X are related if their projections are homotopic in X/G. This enables us to establish a formula for the fundamental group of the orbit space of a discontinuous group under very general conditions. Finally we show by means of an example that some restriction on the action near fixed points is needed for the formula to be valid.

For n ≧ 3 the independence of the associative laws

for n-ary operations (…) is proved.

The classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.

The semilinear parabolic system ut + A(x, D)u = g(u) in (0, ∞) × Ω, Ω⊂ℝn bounded, u ∈ ℝN, with homogeneous boundary conditions B(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearity g is assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solution u is relatively compact in .

Let Mm (r, f) denote the mean-value of a real-valued integrable function f over a geodesic sphere with centre m and radius r in an n-dimensional Riemannian manifold M. We obtain an expansion of Mm (r, f) in powers of r, thereby generalizing Pizzetti's formula valid in euclidean space. From this expansion we prove that the property

for every harmonic function near m, characterizes Einstein spaces. We define super-Einstein spaces and prove that they are characterized by the property

Throughout this paper the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. That is, x · 0 = 0 and (x + y)z = xz + yz for all x, y, z ∈ N. In what follows we generalise the notion of s-primitivity first introduced in an earlier paper by the author (1968), where only distributively generated (d.g.) near-rings with identity were considered. We define a Jacobson type radical Js (N) and show that J1(N)⊇Js(N) ⊇ Q(N), where Q(N) is the intersection of all 0-modular left ideals of N (Pilz). In addition we settle some of the problems remaining from Hartney (1968).