Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability

We study Poisson equations and averaging for Stochastic Differential Equations (SDEs). Poisson equations are essential tools in both probability theory and partial differential equations (PDEs). Their vast range of applications includes the study of the asymptotic behaviour of solutions of parabolic PDEs, the treatment of multi-scale and homogenization problems as well as the theoretical analysis of approximations of solution of Stochastic Differential Equations (SDEs). The study of Poisson equations in non-compact state space is notoriously difficult. Results exists, but only for the case when the coefficients of the PDE are either bounded or satisfy linear growth assumptions (and even the latter case has only recently been achieved). In this paper we treat Poisson equations on non-compact state spaces for coefficients that can grow super-linearly. This is one of the two building blocks towards the second (and main) result of the paper, namely in this paper we succeed in obtaining a {\em uniform in time} (UiT) averaging result (with a rate) for SDE models with super-linearly growing coefficients. This seems to be the first UiT averaging result for slow-fast systems of SDEs (and indeed we are not aware of any other UiT multiscale theorems in the PDE or ODE literature either). Key to obtaining both our UiT averaging result and to enable dealing with the super-linear growth of the coefficients (both of the slow-fast system and of the associated Poisson equation) is conquering exponential decay in time of the space-derivatives of appropriate Markov semigroups. We refer to semigroups which enjoy this property as being Strongly Exponentially Stable.Comment: 66 pages, 2 figure

Quantifying performance of ultrasonic immersion inspection using phased arrays for curvilinear disc forgings

Use of full-matrix capture (FMC), combined with the total focusing method (TFM), has been shown to provide improvements to flaw sensitivity within components of irregular geometry. Ultrasonic immersion inspection of aerospace discs requires strict specifications to ensure full coverage – one of which is that all surfaces should be machined flat. The ability to detect defects through curved surfaces, with an equivalent sensitivity to that obtained through flat surfaces could bring many advantages. In this work, the relationship between surface curvature and sensitivity to standard defects was quantified for various front wall radii. Phased array FMC immersion inspection of curved components was simulated using finite element modelling, then visualized using surface-compensated focusing techniques. This includes the use of BRAIN software developed at the University of Bristol for production of TFM images. Modelling results were compared to experimental data from a series of test blocks with a range of curvatures, containing standard defects. The sensitivity to defects is evaluated by comparing the performance to conventional methods. Results are used to highlight the benefits and limitations of these methods relating to the application area of aerospace engine disc forgings

Parallel and Sequential Pathways of Molecular Recognition of a Tandem-Repeat Protein and Its Intrinsically Disordered Binding Partner.

The Wnt signalling pathway plays an important role in cell proliferation, differentiation, and fate decisions in embryonic development and the maintenance of adult tissues. The twelve armadillo (ARM) repeat-containing protein β-catenin acts as the signal transducer in this pathway. Here, we investigated the interaction between β-catenin and the intrinsically disordered transcription factor TCF7L2, comprising a very long nanomolar-affinity interface of approximately 4800 Å2 that spans ten of the twelve ARM repeats of β-catenin. First, a fluorescence reporter system for the interaction was engineered and used to determine the kinetic rate constants for the association and dissociation. The association kinetics of TCF7L2 and β-catenin were monophasic and rapid (7.3 ± 0.1 × 107 M-1·s-1), whereas dissociation was biphasic and slow (5.7 ± 0.4 × 10-4 s-1, 15.2 ± 2.8 × 10-4 s-1). This reporter system was then combined with site-directed mutagenesis to investigate the striking variability in the conformation adopted by TCF7L2 in the three different crystal structures of the TCF7L2-β-catenin complex. We found that the mutation had very little effect on the association kinetics, indicating that most interactions form after the rate-limiting barrier for association. Mutations of the N- and C-terminal subdomains of TCF7L2 that adopt relatively fixed conformations in the crystal structures had large effects on the dissociation kinetics, whereas the mutation of the labile sub-domain connecting them had negligible effect. These results point to a two-site avidity mechanism of binding with the linker region forming a "fuzzy" complex involving transient contacts that are not site-specific. Strikingly, the two mutations in the N-terminal subdomain that had the largest effects on the dissociation kinetics showed two additional phases, indicating partial flux through an alternative dissociation pathway that is inaccessible to the wild type. The results presented here provide insights into the kinetics of the molecular recognition of a long intrinsically disordered region with an elongated repeat-protein surface, a process found to involve parallel routes with sequential steps in each

Multiple sclerosis risk sharing scheme: two year results of clinical cohort study with historical comparator

Objective To generate evidence on the longer term cost effectiveness of disease modifying treatments in patients with relapsing-remitting multiple sclerosis

Trauma care in the tropics: addressing gaps in treating injury in rural and remote Australia

In Australia, over half a million people are admitted to hospital every year as a result of injury, and where you live matters. Rural populations have disproportionately higher injury hospitalisation rates (1.5 to 2.5-fold), higher rates of preventable secondary complications, higher mortality rates (up to 5-fold), and higher costs (3-fold) than patients injured in major cities. These disparities scale up rapidly with increased remoteness, and shift the service needle from ‘scoop and run’ to ‘continuum-of-care’ . Poorer outcomes, however, are not solely due to longer retrieval distances or delays; but arise from inefficiencies in one or more potentially modifiable factors in the chain-of-survival. After discussing the burden of injury in Australia, we present a brief history of retrieval services in Queensland and discuss how remoteness requires a different kind of service delivery with many moving parts from point-of-injury to definitive care. We next address the ongoing challenges for the Australian Trauma Registry, and how centralisation of data from the metropolitan cities masks the inequities in rural and remote trauma. There is an urgent need for accurate data from all service providers around Australia to inform state and federal governments, and we highlight the paucity of trauma data analysis in North Queensland. Lastly, we identify some major gaps in treating rural and remote polytrauma and en-route patient stabilisation, and discuss the relevance of combat casualty care research and practices. We conclude that a greater emphasis should be placed on collecting more robust trauma patient records, as only accurate data will drive change

Test statistics for comparing two proportions with partially overlapping samples

Standard tests for comparing two sample proportions of a dichotomous dependent variable where there is a combination of paired and unpaired samples are considered. Four new tests are introduced and compared against standard tests and an alternative proposal by Choi and Stablein (1982). The Type I error robustness is considered for each of the test statistics. The results show that Type I error robust tests that make use of all the available data are more powerful than Type I error robust tests that do not. The Type I error robustness and the power among tests introduced in this paper using the phi correlation coefficient is comparable to that of Choi and Stablein (1982). The use of the test statistics to form confidence intervals is considered. A general recommendation of the best test statistic for practical use is made

Critical behaviour and phase structure of 3d Scalar+Gauge Field Theories in the adjoint representation

In a class of holographic models for cosmology, the dual theory is given by a massless super-renormalisable QFT in 3 dimensions. In order to obtain cosmological observables, correlators of this QFT may be obtained via lattice field theory. Previous work has focused on scalar $\phi^4$ matrix theories in the adjoint representation of SU(N). In this work we present preliminary results in the critical behaviour and phase structure of the theory with an SU(N) scalar field coupled to gauge fields by utilising the Heatbath-Overrelaxation (HBOR) algorithm in lattice field theory

Statistical thermodynamics for choice models on graphs

Formalism based on equilibrium statistical thermodynamics is applied to communication networks of decision making individuals. It is shown that in statistical ensembles for choice models, properly defined disutility can play the same role as energy in statistical mechanics. We demonstrate additivity and extensivity of disutility and build three types of equilibrium statistical ensembles: the canonical, the grand canonical and the super-canonical. Using Boltzmann-like probability measure one reproduce the logit choice model. We also propose using q-distributions for temperature evolution of moments of stochastic variables. The formalism is applied to three network topologies of different degrees of symmetry, for which in many cases analytic results are obtained and numerical simulations are performed for all of them. Possible applications of the model to airline networks and its usefulness for practical support of economic decisions is pointed out.Comment: 17 pages, 13 figure