Asymptotic behaviour of total generalised variation

The recently introduced second order total generalised variation functional $\mathrm{TGV}_{\beta,\alpha}^{2}$ has been a successful regulariser for image processing purposes. Its definition involves two positive parameters $\alpha$ and $\beta$ whose values determine the amount and the quality of the regularisation. In this paper we report on the behaviour of $\mathrm{TGV}_{\beta,\alpha}^{2}$ in the cases where the parameters $\alpha, \beta$ as well as their ratio $\beta/\alpha$ becomes very large or very small. Among others, we prove that for sufficiently symmetric two dimensional data and large ratio $\beta/\alpha$, $\mathrm{TGV}_{\beta,\alpha}^{2}$ regularisation coincides with total variation ($\mathrm{TV}$) regularisation

The effects of intrinsic noise on the behaviour of bistable cell regulatory systems under quasi-steady state conditions

We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under1 quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady state approximation based on the semi-classical approximation of the partial differential equation for the generating function associated with the Chemical Master Equation. Such approximation proceeds by optimising an action functional whose associated set of Euler-Lagrange (Hamilton) equations provide the most likely fluctuation path. We show that, under appropriate conditions granting time scale separation, the Hamiltonian can be re-scaled so that the set of Hamilton equations splits up into slow and fast variables, whereby the quasi-steady state approximation can be applied. We analyse two particular examples of systems whose mean-field limit has been shown to exhibit bi-stability: an enzyme-catalysed system of two mutually-inhibitory proteins and a gene regulatory circuit with self-activation. Our theory establishes that the number of molecules of the conserved species are order parameters whose variation regulates bistable behaviour in the associated systems beyond the predictions of the mean-field theory. This prediction is fully confirmed by direct numerical simulations using the stochastic simulation algorithm. This result allows us to propose strategies whereby, by varying the number of molecules of the three conserved chemical species, cell properties associated to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.Comment: 33 pages, 9 figures, accepted for publication in the Journal of Chemical Physic

Review of recent advances in local approaches applied to pre-stressed components under fatigue loading

Fatigue strength of mechanical components in the high cycle regime depends on both the applied loading and the intensity of any residual stress field induced by either non-homogeneous plastic deformation or the solidification of a local portion of material due to welding operations. In presence of geometric variations that are amenable to being modelled as a sharp V-notch, the residual stress distribution near the notch tip is singular and follows the same form as the solution obtained by Williams in 1952 where the intensity of the asymptotic stress field is quantified by the notch stress intensity factor (NSIF). However, the residual stress varies during fatigue loading and a stable value may be reached. Numerical models have been developed for the calculation of the residual NSIFs and their variation under fatigue loading. Taking advantage of these models, new local approaches have also been recently developed which are able to predict the fatigue strength of pre-stressed notched components. The present paper provides a brief review of such recent advances

Local generalised method of moments: an application to point process-based rainfall models

Long series of simulated rainfall are required at point locations for a range of applications, including hydrological studies. Clustered point process-based rainfall models have been used for generating such simulations for many decades. These models suffer from a major limitation, however, their stationarity. Although seasonality can be allowed by fitting separate models for each calendar month or season, the models are unsuitable in their basic form for climate impact studies. In this paper, we develop new methodology to address this limitation. We extend the current fitting approach by allowing the discrete covariate, calendar month, to be replaced or supplemented with continuous covariates that are more directly related to the incidence and nature of rainfall. The covariate-dependent model parameters are estimated for each time interval using a kernel-based nonparametric approach within a generalised method-of-moments framework. An empirical study demonstrates the new methodology using a time series of 5-min rainfall data. The study considers both local mean and local linear approaches. While asymptotic results are included, the focus is on developing useable methodology for a complex model that can only be solved numerically. Issues including the choice of weighting matrix, estimation of parameter uncertainty and bandwidth and model selection are considered from this perspective

On the asymptotic distribution of certain bivariate reinsurance treaties

Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common distribution function F and let N(t), t \ge 0 be a stochastic process which counts the number of claims that occur in the time interval [0,t], t\ge 0. In this paper we derive the joint asymptotic distribution of randomly indexed order statistics of the random sample (X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain asymptotic representations for the joint distribution of two generalised largest claims reinsurance treaties available under specific insurance settings. As a by-product we obtain a stochastic representation of a m-dimensional Lambda-extremal variate in terms of iid unit exponential random variables.Comment: 11 page

Drip Paintings and Fractal Analysis

It has been claimed [1-6] that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has also led to two new results in fractal analysis of more general scientific significance. First, the composite of two fractals is not generally scale invariant and exhibits complex multifractal scaling in the small distance asymptotic limit. Second the statistics of box-counting and related staircases provide a new way to characterize geometry and distinguish fractals from Euclidean objects

On the moving contact line singularity: Asymptotics of a diffuse-interface model

The behaviour of a solid-liquid-gas system near the three-phase contact line is considered using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. Relaxation of the classical approach of a sharp liquid-gas interface and careful examination of the asymptotic behaviour as the contact line is approached is shown to resolve the stress and pressure singularities associated with the moving contact line problem. Various features of the model are scrutinised, alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall.Comment: 14 pages, 3 figure

Limit theorems for bipower variation in financial econometrics

In this paper we provide an asymptotic analysis of generalised bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation and bipower variations which have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects