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.

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.

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.

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.

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.

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.

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.

We consider the hyperbolic equation uxy + c(x, y, u) =f(x, y) and the wave equation

We show that, under suitable conditions, there are bounded domains in which every solution to certain problems has a zero. Characteristic initial value problems and initial boundary value problems are considered.

We prove that there exists an ᾱ > 0 such that for each 0 ≦ α ≦ ᾱ, there is at least one β = β(α) < 0 such that y“ − (x/2)y' + ey − 1 =0, y(0) = α, y'(0) = β has a solution y(x, α,β) satisfying the asymptotic property y(x, α,β)∼ −2 In x + Kα as x → ∞.

We consider the second order, linear, differential equation

where q is real-valued. In the case we calculate the Titchmarsh-Weyl m-function associated with (*).

The homogenisation of a linearly viscoelastic composite material is performed without any periodicity assumptions and with no restrictions on the initial data. A term of fading memory is evidenced in the expression of the homogenised stress tensor.

Let E be a Hausdorff locally convex space, Q a convex closed subset of E and U an open subset of Q. We develop an index theory for a class of locally compact maps f: U → E for which the usual assumption f(U) ⊂ Q is replaced by an appropriate “pushing condition”. Moreover, from this index theory, we deduce a general continuation principle and some global results for nonlinear eigenvalue problems.

We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≧ 0, and converges to a given constant state at the rate t − ¼ as t → ∞. Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t − ½ +α, α > 0, as t →∞. The proof is essentially based on the fact that for t → ∞ the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.