A variational method for analyzing stochastic limit cycle oscillators

We introduce a variational method for analyzing limit cycle oscillators in $\mathbb{R}^d$ driven by Gaussian noise. This allows us to derive exact stochastic differential equations (SDEs) for the amplitude and phase of the solution, which are accurate over times over order $\exp\big(Cb\epsilon^{-1}\big)$, where $\epsilon$ is the amplitude of the noise and $b$ the magnitude of decay of transverse fluctuations. Within the variational framework, different choices of the amplitude-phase decomposition correspond to different choices of the inner product space $\mathbb{R}^d$. For concreteness, we take a weighted Euclidean norm, so that the minimization scheme determines the phase by projecting the full solution on to the limit cycle using Floquet vectors. Since there is coupling between the amplitude and phase equations, even in the weak noise limit, there is a small but non-zero probability of a rare event in which the stochastic trajectory makes a large excursion away from a neighborhood of the limit cycle. We use the amplitude and phase equations to bound the probability of it doing this: finding that the typical time the system takes to leave a neighborhood of the oscillator scales as $\exp\big(Cb\epsilon^{-1}\big)$

Synchronization of stochastic hybrid oscillators driven by a common switching environment

Many systems in biology, physics and chemistry can be modeled through ordinary differential equations, which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However the discrete nature of the Markov jump process raises some difficulties: in fact we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapinov exponent is more accurate

FLOW INJECTION AND MULTIVARIATE CALIBRATION TECHNIQUES FOR PROCESS ANALYSIS

The role of process analytical chemistry is summarised in chapter one with particular emphasis on a multidisciplinary approach and the instrumental requirements for on-plant analysis. These concepts are extended to process FIA, highlighting its potential for simultaneous multicomponent determinations. The development of an automated FIA monitor for the on-line determination of sulphite in potassium chloride brine is covered in the second chapter. Reaction stability is demonstrated and the results of on-plant validation and on-line trials are presented. The next chapter deals with the concepts of multivariate calibration. Direct multicomponent analysis, principal components regression and partial least squares regression are critically examined in practical spectroscopic terms and statistical terms. The relative predictive abilities of these techniques are compared in chapter four for the resolution of a multicomponent UV-visible spectrophotometric data set. Chapter five describes the development of an automated FIA-diode array system for the simultaneous determination of phosphate and chlorine. The implications of combining reaction chemistries and the influence of a number of calibration parameters are considered in detail. Finally, the jackknife is presented as a means of dimensionality estimation' and bias correction in PLS modelling. Data sets from the literature are analysed and the results compared with those obtaining using commercial software.ICI Chemicals & Polymer

Accuracy of a video odometry system for trains

Reliable Data Systems is developing a video-based odometry system that enables trains to measure velocities and distances travelled without the need for trackside infrastructure. A camera is fixed in the cab, taking images of the track immediately ahead, at rates in the range 25–50 frames per second. The images in successive frames are ‘unwarped’ to provide a plan view of the track and then matched, to produce an ‘optical flow’ that measures the distance travelled. The Study Group was asked to investigate ways of putting bounds on the accuracy of such a system, and to suggest any improvements that might be made. The work performed in the week followed three strands: (a) an understanding of how deviations from the camera’s calibrated position lead to errors in the train’s calculated position and velocity; (b) development of models for the train suspension, designed to place bounds on these deviations; and (c) the performance of the associated image processing algorithms

APAF1 is a key transcriptional target for p53 in the regulation of neuronal cell death

p53 is a transcriptional activator which has been implicated as a key regulator of neuronal cell death after acute injury. We have shown previously that p53-mediated neuronal cell death involves a Bax-dependent activation of caspase 3; however, the transcriptional targets involved in the regulation of this process have not been identified. In the present study, we demonstrate that p53 directly upregulates Apaf1 transcription as a critical step in the induction of neuronal cell death. Using DNA microarray analysis of total RNA isolated from neurons undergoing p53-induced apoptosis a 5–6-fold upregulation of Apaf1 mRNA was detected. Induction of neuronal cell death by camptothecin, a DNA-damaging agent that functions through a p53-dependent mechanism, resulted in increased Apaf1 mRNA in p53-positive, but not p53-deficient neurons. In both in vitro and in vivo neuronal cell death processes of p53-induced cell death, Apaf1 protein levels were increased. We addressed whether p53 directly regulates Apaf1 transcription via the two p53 consensus binding sites in the Apaf1 promoter. Electrophoretic mobility shift assays demonstrated p53–DNA binding activity at both p53 consensus binding sequences in extracts obtained from neurons undergoing p53-induced cell death, but not in healthy control cultures or when p53 or the p53 binding sites were inactivated by mutation. In transient transfections in a neuronal cell line with p53 and Apaf1 promoter–luciferase constructs, p53 directly activated the Apaf1 promoter via both p53 sites. The importance of Apaf1 as a p53 target gene in neuronal cell death was evaluated by examining p53-induced apoptotic pathways in primary cultures of Apaf1-deficient neurons. Neurons treated with camptothecin were significantly protected in the absence of Apaf1 relative to those derived from wild-type littermates. Together, these results demonstrate that Apaf1 is a key transcriptional target for p53 that plays a pivotal role in the regulation of apoptosis after neuronal injury

On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim