The Signorini perturbation scheme is a series expansion algorithm that locates solution branches for a class of symmetry-breaking bifurcation problems in nonlinear elastostatics. The relationship of the formal steps in the algorithm to geometric aspects of the problem is brought out in work of J. E. Marsden and Y.-H. Wan, where an abstract formulation is also considered. In this paper, the abstract algorithm and its geometry are explored further: the logical structure is clarified, and it is shown how the scheme adapts to the presence of additional symmetry constraints.

The purpose of this paper is to introduce a fibrewise generalisation of category, in the sense of Lusternik–Schnirelmann. This reduces to the classical concept when the space is a point. Fibrewise category may be compared with equivariant category, which has been the subject of some recent research [1,7,8]. Many variations on the basic idea of category have been discussed in the literature, for example the concept of category of a map, but since the generalisations to the fibrewise case are fairly routine they are not considered here.

We prove that, for every non-empty open subset of a rectangular plate, there exists a positive number T such that no vibration of this plate with fixed edges can remain strictly above the rest position at all points of the subset during a period T. Moreover, the result remains valid for every non-empty segment parallel to one of the sides of the plate. Our proof is based on some results of non-harmonic analysis.

Let T be a selfadjoint uniformly elliptic partial differential operator on a bounded domain in Rn, and let S be a (possibly indefinite) L∞ multiplication operator. Estimates of the form σλ + o(λ) and σλ + β + o(1) are sought for the eigenvalues μ(λ) of λS – T as λ→ ±∞. A necessary and sufficient condition is also obtained for existence of linear eigencurves, i.e. μ(λ) = σλ + β.

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

Noncoercive variational inequalities with sublinear functionals are considered. Necessary and sufficient conditions are given for the solvability of such problems. These conditions are in the form of compatibility conditions-for the data, as well as the boundedness of the solutions to related problems. These results are used for the obstacle problems for the membrance and the elastic contact in the presence of friction.

In this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type. This class includes the solution map for functional differential equations and certain partial differential equations. In particular, we present necessary and sufficient conditions for one-to-oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator.

An asymptotic theory is developed for linear differential equations of odd order. Theory is applied with large coefficients. The forms of the asymptotic solutions are given under general conditions on the coefficients.

Our paper concerns the existence of a positive solution for the equation:

A new condition, which guarantees the existence of a solution of the above equation, has been established. It has also given some sharp information in the cases where: (1) a(x) = λ = const. and Ω is a “thin” domain; (2) Ω is a ball and a(x) is a radially symmetrical function.

Existence and uniqueness of solutions of an integro-differential equation that arises in population genetics are proved. This equation describes the evolution of type densities in a population that is subject to mutation and directional selection on a quantitative trait. It turns out that a certain Fréchet space is the natural framework to show existence and uniqueness. One of the main steps in the proof is the investigation of perturbations of generators of differentiable semigroups in Fréchet spaces.

Let ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].

Let K1, K2,…, Kd+1 be d+1 convex bodies in Ed, the interiors of which have no point in common. If mX denotes the measure of set X, we prove

and this inequality is best possible.

Let be the sum of the m-th powers of the zeros of the Bessel polynomial yn(x). It is known that for m = 0, 1, 2, …,

where cm(v) is the Hawkins polynomial. In this paper we find rational functions wm(v) such that for m = 0, 1, 2, …

Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.