For a dynamics on the whole line, for both discrete and continuous time, we extend a result of Pliss that gives a characterization of the notion of a trichotomy in various directions. More precisely, the result gives a characterization in terms of an admissibility property in the whole line (namely, the existence of bounded solutions of a linear dynamics under any nonlinear bounded perturbation) of the existence of a trichotomy, i.e. of exponential dichotomies in the future and in the past, together with a certain transversality condition at time zero. In particular, we consider arbitrary linear operators acting on a Banach space as well as sequences of norms instead of a single norm, which allows us to consider the general case of non-uniform exponential behaviour.

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

We answer a question of Takesaki by showing that the following can be derived from the thesis of Shen: if A and B are σ-unital hereditary C*-subalgebras of C such that ‖p – q‖ < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the σ-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between σ-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the σ-unital case. In particular, every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between σ-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersen's non-commutative Tietze extension theorem.) We elaborate the relations of the above with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady. We discuss the relationship of our results to the theory of perturbations of C*-algebras.

We are interested in non-negative non-trivial solutions of

where 1 < p and Ω is a bounded smooth domain in ℝN with 3 ⩽ N ⩽ 9. We show that given a non-negative integer M there is some large p(M,Ω) such that the only non-negative solution u, of Morse index at most M, is u = 0.

In this paper we study the problem

where Ω = ℝ N or Ω = B 1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.

We introduce a new affinely invariant structure on smooth surfaces in ℝ3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ‘ points’, which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.

We derive a stage-structured model for an insect population in which a larva matures on reaching a certain size, and in which there is intra-specific competition among larvae that hinders their development, thereby prolonging the larval phase. The model, a system of delay differential equations for the total numbers of adults and larvae, assumes two forms. One of these is a system with a variable state-dependent time delay determined by a threshold condition, the other has constant and distributed delays, a size-like independent variable replacing time t, and no threshold condition. We prove theorems on boundedness and on the linear stability of equilibria.

We study the following polyharmonic Hénon equation:

where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

In 1992 Ermentrout and McLeod published in this journal a landmark study of travelling wavefronts for a differential–integral equation model of a neural network. Since then a number of authors have extended the model by adding an additional equation for a ‘recovery variable’, thus allowing the possibility of travelling-pulse-type solutions. In a recent paper, Faye gave perhaps the first rigorous proof of the existence (and stability) of a travelling-pulse solution for a model of this type, treating a simplified version of equations originally developed by Kilpatrick and Bressloff. The excitatory weight function J used in this work allowed the system to be reduced to a set of four coupled ordinary differential equations (ODEs), and a specific firing-rate function S, with parameters, was considered. The method of geometric singular perturbation was employed, together with blow-ups. In this paper we extend Faye's results on existence by dropping one of his key hypotheses, proving the existence of pulses at least two different speeds, and, in a sense, allowing a wider range of the small parameter in the problem. The proofs are classical and self-contained aside from standard ODE material.

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.