James C. Robinson

From finite to infinite dimensional dynamical systems

Preface. Introduction J.C. Robinson, P.A. Glendinning. Spatial correlations and local fluctuations in host-parasite models M.J. Keeling, D.A. Rand. Lattice dynamical systems L.A. Bunimovich (assisted by C. Giberti). Attractors and dynamics in partial differential equations J.K. Hale. Nonlinear dynamics of extended systems P. Collet. Three lectures on mathematical fluid mechanics P. Constantin. Low-dimensional models of turbulence P.J. Holmes, et al.

Attractors for infinite-dimensional non-autonomous dynamical systems /

The pullback attractor.- Existence results for pullback attractors.- Continuity of attractors.- Finite-dimensional attractors.- Gradient semigroups and their dynamical properties.- Semilinear Differential Equations.- Exponential dichotomies.- Hyperbolic solutions and their stable and unstable manifolds.- A non-autonomous competitive Lotka-Volterra system.- Delay differential equations.-The Navier-Stokes equations with non-autonomous forcing.- Applications to parabolic problems.- A non-autonomous Chafee-Infante equation.- Perturbation of diffusion and continuity of attractors with rate.- A non-autonomous damped wave equation.- References.- Index.-

Upper semicontinuity of attractors for small random perturbations of dynamical systems

The relationship between random attractors and global attractors for dynamical systems is studied. If a partial differential equation is perturbed by an E-small random term and certain hypotheses are satisfied, the upper semicontinuity of the random attractors is obtalned as c goes to zero. The results are applied to the Navier-Stokes equations and a problem of reaction-diffusion type, both perturbed by an additive white noise.

A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, du=( Δu+βu- u 3 ) dt+σuod W t ,x∈D⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β as the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m = 1, that as β passes through λ1 (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite–dimensional setting. Central to our approach is the existence of a random unstable manifold.

Stability, instability, and bifurcation phenomena in non-autonomous differential equations

There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.

Bayesian inverse problems for functions and applications to fluid mechanics

In this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often ill-posed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinite-dimensional version of Bayes theorem, leads to a well-posedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem, leading to the desired results on well-posedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire time-dependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the two-dimensional Navier–Stokes equation on a torus. The case of Eulerian observations—direct observations of the velocity field itself—is then a model for weather forecasting. The case of Lagrangian observations—observations of passive tracers advected by the flow—is then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinite-dimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical setting for the inverse problem of determining an unknown tensor arising in a constitutive law for a PDE, given observations of the solution. The paper is structured so that the abstract theory can be read independently of the particular problems in fluid mechanics which are subsequently studied by application of the theory.

Selective small molecule inhibitors of the potential breast cancer marker, human arylamine N-acetyltransferase 1, and its murine homologue, mouse arylamine N-acetyltransferase 2.

The identification, synthesis and evaluation of a series of rhodanine and thiazolidin-2,4-dione derivatives as selective inhibitors of human arylamine N-acetyltransferase 1 and mouse arylamine N-acetyltransferase 2 is described. The most potent inhibitors identified have submicromolar activity and inhibit both the recombinant proteins and human NAT1 in ZR-75 cell lysates in a competitive manner. (1)H NMR studies on purified mouse Nat2 demonstrate that the inhibitors bind within the putative active site of the enzyme.

Arylamine N‐acetyltransferase 1 expression in breast cancer cell lines: A potential marker in estrogen receptor‐positive tumors

The prognosis for patients with estrogen receptor (ER)‐positive breast cancer has improved significantly with the prescription of selective ER modulators (SERMs) for ER‐positive breast cancer treatment. However, only a proportion of ER‐positive tumors respond to SERMs, and resistance to hormonal therapies is still a major problem. Detailed analysis of published microarray studies revealed a positive correlation between overexpression of the drug metabolizing enzyme arylamine N‐acetyltransferase type 1 (NAT1) and ER positivity, and increasing evidence supports a biological role for NAT1 in breast cancer progression. We have tested a range of ER‐positive and ER‐negative breast cancer cell lines for NAT1 enzyme activity, and monitored promoter and polyadenylation site usage. Amongst ER‐positive lines, NAT1 activities ranged from 202 ± 28 nmol/min/mg cellular protein (ZR‐75‐1) to 1.8 ± 0.4 nmol/min/mg cellular protein (MCF‐7). The highest levels of NAT1 activity could not be attributed to increased NAT1 gene copy number; however, we did detect differences in NAT1 promoter and polyadenylation site usage amongst the breast tumor‐derived lines. Thus, whilst all cell lines tested accumulated transcripts derived from the proximal promoter, the line expressing NAT1 most highly additionally initiated transcripts initiating at a more distal, “tissue”‐specific promoter. These data pave the way for investigating NAT1 transcripts as candidate prognostic markers in ER‐positive breast cancer. This article contains Supplementary Material available at http://www.interscience.wiley.com/jpages/1045‐2257/suppmat.

A Comparison between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball (1997) to treat equations whose solutions may not be unique, and that due to Melnik and Valero (1998) tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Balls theory to differential inclusions.

Stability of random attractors under perturbation and approximation

Abstract The comparison of the long-time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals. However, one can still compare their asymptotic behaviour using the global attractor, and this is now standard in the deterministic autonomous case. For random dynamical systems there is an additional problem, since the convergence of numerical methods for such systems is usually given only on average. In this paper the deterministic approach is extended to cover stochastic differential equations, giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi-continuous with respect to a given family of perturbations or approximations.