Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations

Understanding the limits to generalizability of experimental evolutionary models.

Post print version of article deposited in accordance with SHERPA RoMEO guidelines. The final definitive version is available online at: http://www.nature.com/nature/journal/v455/n7210/abs/nature07152.htmlGiven the difficulty of testing evolutionary and ecological theory in situ, in vitro model systems are attractive alternatives; however, can we appraise whether an experimental result is particular to the in vitro model, and, if so, characterize the systems likely to behave differently and understand why? Here we examine these issues using the relationship between phenotypic diversity and resource input in the T7-Escherichia coli co-evolving system as a case history. We establish a mathematical model of this interaction, framed as one instance of a super-class of host-parasite co-evolutionary models, and show that it captures experimental results. By tuning this model, we then ask how diversity as a function of resource input could behave for alternative co-evolving partners (for example, E. coli with lambda bacteriophages). In contrast to populations lacking bacteriophages, variation in diversity with differences in resources is always found for co-evolving populations, supporting the geographic mosaic theory of co-evolution. The form of this variation is not, however, universal. Details of infectivity are pivotal: in T7-E. coli with a modified gene-for-gene interaction, diversity is low at high resource input, whereas, for matching-allele interactions, maximal diversity is found at high resource input. A combination of in vitro systems and appropriately configured mathematical models is an effective means to isolate results particular to the in vitro system, to characterize systems likely to behave differently and to understand the biology underpinning those alternatives

Modelling Cognitive Decline in the Hypertension in the Very Elderly Trial [HYVET] and Proposed Risk Tables for Population Use

Although, on average, cognition declines with age, cognition in older adults is a dynamic process. Hypertension is associated with greater decline in cognition with age, but whether treatment of hypertension affects this is uncertain. Here, we modelled dynamics of cognition in relation to the treatment of hypertension, to see if treatment effects might better be discerned by a model that included baseline measures of cognition and consequent mortalityThis is a secondary analysis of the Hypertension in the Very Elderly Trial (HYVET), a double blind, placebo controlled trial of indapamide, with or without perindopril, in people aged 80+ years at enrollment. Cognitive states were defined in relation to errors on the Mini-Mental State Examination, with more errors signifying worse cognition. Change in cognitive state was evaluated using a dynamic model of cognitive transition. In the model, the probabilities of transitions between cognitive states is represented by a Poisson distribution, with the Poisson mean dependent on the baseline cognitive state. The dynamic model of cognitive transition was good (R(2) = 0.74) both for those on placebo and (0.86) for those on active treatment. The probability of maintaining cognitive function, based on baseline function, was slightly higher in the actively treated group (e.g., for those with the fewest baseline errors, the chance of staying in that state was 63% for those on treatment, compared with 60% for those on placebo). Outcomes at two and four years could be predicted based on the initial state and treatment.A dynamic model of cognition that allows all outcomes (cognitive worsening, stability improvement or death) to be categorized simultaneously detected small but consistent differences between treatment and control groups (in favour of treatment) amongst very elderly people treated for hypertension. The model showed good fit, and suggests that most change in cognition in very elderly people is small, and depends on their baseline state and on treatment. Additional work is needed to understand whether this modelling approach is well suited to the valuation of small effects, especially in the face of mortality differences between treatment groups.ClinicalTrials.gov NCT0012281

Socio-economic drivers of specialist anglers targeting the non-native European catfish (Silurus glanis) in the UK.

Information about the socioeconomic drivers of Silurus glanis anglers in the UK were collected using questionnaires from a cross section of mixed cyprinid fisheries to elucidate human dimensions in angling and non-native fisheries management. Respondents were predominantly male (95%), 30-40 years of age with £500 per annum. The proportion of time spent angling for S. glanis was significantly related to angler motivations; fish size, challenge in catch, tranquil natural surroundings, escape from daily stress and to be alone were considered important drivers of increased time spent angling. Overall, poor awareness of: the risks and adverse ecological impacts associated with introduced S. glanis, non-native fisheries legislation, problems in use of unlimited ground bait and high fish stocking rates in angling lakes were evident, possibly related to inadequate training and information provided by angling organisations to anglers, as many stated that they were insufficiently informed

X-Ray Spectroscopic Diagnosis of a Wind-Collimated Blast Wave and Metal-Rich Ejecta from the 2006 Explosion of RS Ophiuchi

Chandra HETG observations of RS Ophiuchi at day 13.9 of the 2006 outburst reveal a rich spectrum of emission lines from abundant ions formed over a wide temperature range (∼ 3 × 10 6 to 60 × 10 6 K) indicative of shock heating of the circumstellar medium by the expanding blast wave. Lines are asymmetric and strongly broadened (v ∼ 2400 km s −1 at zero intensity). Using simple analytical model profiles, we show how the lines are shaped by differential absorption in the red giant wind and explosion ejecta, and that shock heating to multimillion degree temperatures appears to have occurred preferentially in the direction perpendicular to the line of sight. We conclude that the asymmetric nature of the offset 1/r 2 density profile and likely equatorial circumstellar density enhancement in which the explosion occurred are responsible for both the shock collimation and broad range in plasma temperature observed. The ejecta mass deduced from X-ray absorption is more easily reconciled with the expected mass accretion rate for material enhanced in metals by up to an order of magnitude

The flow of a DAE near a singular equilibrium

We extend the differential-algebraic equation (DAE) taxonomy by assuming that the linearization of a DAE about a singular equilibrium has a particular index-2 Kronecker normal form. A Lyapunov-Schmidt procedure is used to reduce the DAE to a quasilinear normal form which is shown to posses quasi-invariant manifolds which intersect the singularity. In turn, this provides solutions of the DAE which pass through the singularity

Transversality and separation of zeros in second order differential equations

Sufficient conditions on the non-linearity f are given which ensure that non-trivial solutions of second order differential equations of the form Lu = f(t, u) have a finite number of transverse zeros in a given finite time interval. We also obtain a priori lower bounds on the separation of zeros of solutions. In particular our results apply to non-Lipschitz non-linearities. Applications to non-linear porous medium equations are considered, yielding information on the existence and strict positivity of equilibrium solutions in some important classes of equations

Trajectories of a DAE near a pseudo-equilibrium

We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity. From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general. Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE