Let Aij, l≦j≦k, be bounded Hermitean operators on Hilbert spaces Hi, 1≦i≦k, and let be the induced operators on . An important operator for multiparameter theory is δ: H →H denned by δ = det the determinant being expanded formally. Various definiteness properties of δ are critical for multiparameter spectral theory.

We use the operators Aij to construct a numerical matrix δ(δ) upon which we use Geršgorin theory to investigate the non-singularity and definiteness of δ. Diagonal dominance properties of the array [Aij] are also discussed.

We consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

By an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.

Similar results are obtained for 0-simple semigroups without nonzero idempotents.

Exact solutions recently discovered for non-conservative three-wave resonance are here related to the ‘one-lump’ solutions obtained by Backlund transformation in conservative cases.

Comparison principles for systems of reaction-diffusion equations coupled via both the reaction and diffusion terms are considered. Applications to the FitzHugh–Nagumo equations and models of coupled nerve fibres are included.

Suppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.

We consider eigenvalues λ =(λ1, λ2) ∈R2 for the problem W(λ)x = 0, x ≠ 0, x ∈ H, where W(λ) = R + λ1V1 + λ2V2), and R, V1, V2 are self-adjoint operators on a separable Hilbert space H, R being bounded below with compact resolvent and V1, V2 being bounded. The i-th eigencurve Z1 is the set of eigenvalues λ, for which the i-th eigenvalue (counted according to multiplicity and in increasing order) of W(λ) vanishes. We study monotonic and asymptotic properties of Zi, and we give formulae for any asymptotes that exist. Additional results are given in the finite dimensional case.

The symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].

Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.

We show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

It is shown that even in the fourth order case there exist real symmetric ordinary differential expressions with nonempty essential spectrum which are not in the limit-point case.

J. M. Ball has introduced the notion of polyconvexity to study nonlinear problems in elasticity and he has shown that polyconvexity implies rank 1 convexity. In this paper we prove by a counterexample that the converse of this implication is false in two dimensions.

Suslin and Cohn have proved that the polynomial ring Z[t1, …,td], where Z is the ring of rational integers and d >0, is a GEn-ring if and only if n ≧ 3. (A commutative ring R with identity is called a GEn-ring if and only if SLn(R) is generated by elementary matrices, where n ≧ 2.) In this paper we consider the following question:

Given algebraic numbers α1, …,αd, for which n (if any) is the ring A =Z [α1, …,αd a GEn-ring?

By standard results from algebraic K-theory it follows that (a) A is GEn for all n ≧ 2, (b) A is not GEn for any n ≧ 2, or (c) A is GEn if and only if n ≧ 3. Examples of each type are provided. In particular, it is shown that if each αi is real or at least one αi is not an algebraic integer, then A is of type (a).

The Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.

We consider the weak Neumann problem for logarithmic potentials in plane domains. We prove that this problem can be treated by the Fredholm–Radon method if and only if the boundary of the corresponding domain is formed by finitely many curves fulfilling specified regularity conditions.

We consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].

Let S be an E-unitary regular semigroup and V a variety of bands. We prove that S can be embedded into a semidirect product of a band from V by a group if and only if S can be embedded in a canonical way into the semidirect product of the free band in V over a well-determined partial semigroup by the greatest group homomorphic image of S. Moreover, we show that every E-unitary regular semigroup with regular band of idempotents E can be embedded into a semidirect product of a band B by a group, where B belongs to the variety of bands generated by E.

We extend the notion of order of singularity for complex-valued functions, as originally set forth by Hadamard, to Banach algebra-valued functions. Restricting our attention to linear operators whose spectral radius is one, we obtain a connection between the rate of growth of the norm of iterates of a linear operator and the rate of growth of the norm of the resolvent of the operator near the spectrum of the operator. In the finite dimensional case, we obtain an upper bound on the size of the Jordan block corresponding to an eigenvalue of maximum modulus.

In this paper we study the solutions of the boundary value problem

where t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).

Coexistent steady-state solutions to a Lotka–Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka–Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander–Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.

Given an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.