All the Fibonacci groups in the family F(2, n) have been either fully identified or determined to be infinite, bar one, namely F(2, 9). Using computer-aided techniques it is shown that F(2, 9) has a quotient of order 152.5741, and an explicit matrix representation for a quotient of order 152.518 is given. This strongly suggests that F(2, 9) is infinite, but no proof of such a claim is available.

In the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:

A is assumed to be a maximal monotone operator and F(t) is a periodic function.

When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:

with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.

In this paper we study the wave equation, in particular the propagation of discontinuities. Two problems are considered: diffraction of a normally incident plane pulse by a plane screen and diffraction of a spherical wave by the same screen. It is shown that when an incident wave front strikes the edge of the screen a diffracted wave front is produced. The discontinuities are precisely computed in a neighbourhood of the edge for a small time interval after the arrival of the incident wave front and a theorem of Hörmander on the propagation of singularities is used to obtain a globalresult.

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

The following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

We seek non-trivial solutions (u,λ)∈C1([0,1])×[0,∞ with u(x)≧0 for all x ∈[0,1], of the nonlinear eigenvalue problem –u″(x)=λf(u(x)) for x ∈ (0,1) and u(0)=u(1)=0,where f:[0,∞)→[0,∞) is such that f(p) = 0, for p ∈ [0,1), and f(p) = K(p), for p ∈ (1,∞), and K: [1, ∞)→(0, ∞) is assumed to be twice continuously differentiable. (The value ƒ(1) is only required to be positive.)

Existence and multiplicity theorems are given in the cases where ƒ is asymptotically sub-linear and ƒ is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term ƒ we obtain the precise number of non-trivial solutions for given values of λ ∈ [0, ∞).

Let F be any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complement cF, that approximates the distance from F of any point x ∈ cF in the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

If the diffusion matrix coefficient of an Itô stochastic differential equation is everywhere non-singular, then the corresponding Chapman-Kolmogorov semi-group may be defined on L∼(Rn), the space of Lebesgue equivalence classes of essentially bounded Borei measurable functions. However, if the diffusion matrix is singular at some points of Rn, it is not clear that this can always be done. We show that in certain situations it is possible to do so.

Conditions for the existence of anti-isomorphisms and anti-automorphisms of completely 0-simple semigroups are derived and a method is given for the enumeration of finite 0-simple and simple semigroups up to isomorphism and anti-isomorphism.

Consider mild solutions on the real line of non-homogeneous differential equations in a Banach space: u′(t) = Au(t) + f(t), where A is the infinitesimal generator of a C0-semigroup.

We prove an existence result for optimal solutions (as defined in the text) in reflexive spaces and an uniqueness fact in uniformly convex B-spaces.

The construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

This paper deals with a class of interpolation series of the form

called R-series. It is equiconvergent with the Dirichlet series

If the nth Legendre polynomial for the interval (0,1) is denoted by (−1)nLn(t), then the bilinear formula

serves as generating function for the Rn(z). It also leads in easy steps to R-series expansions for rational functions.

Lagrange [7] has shown that a function holomorphic and of finite rate of growth in a right half-plane can be expanded in an R-series whose abscissa of convergence is limited by the rate of growth of f(z). The converse problem is attacked in Theorem 2 below where it is shown that

The cubic surfaces in, save for the elliptic cone, are, whatever their singularities, projections of del Pezzo's non-singular surface F, of order 9 in. It is explained how, merely by specifying the geometrical relation of the vertex of projection to F, each cubic surface is obtainable “at a stroke”, without using spaces of intermediate dimensions.

Let D be a bounded simply connected domain in the plane and Ω the unit disk. Let F(Θ;k) be the far field pattern arising from the scattering of an incoming plane wave by the obstacle D and let an(k) denote the nth Fourier coefficient of F. Then if f conformally maps ℝ2\D onto ℝ2\Ω, a “moment” problem is derived which expresses an(k) in terms of f−1 for small values of the wave number k. The solution of this moment problem then gives the Laurent coefficients of f−1 and hence ∂D.

A stability condition is derived for solutions of the Von Kárman-Batchelor equations for the flow of a viscous, incompressible fluid between rotating, coaxial (infinite) disks. The rigid motion solution which arises when the angular velocities of the disks are equal is stable with respect to perturbations which go to zero sufficiently rapidly at infinity, for all values of the Reynolds number. If the angular velocities are sufficiently close the stability condition derived applies to perturbations whose “deformation energy” is sufficiently confined to a “core” region.