Stability theory and hamiltonian dynamics in the Euler ideal fluid equations

The study of shear flow steady states has led to a wealth of research in the field of fluid dynamics. By studying shear flows, we can understand how a fluid behaves and how coherent structures arise. We primarily study the stability of shear flows in the Euler equations. The Euler equations describe the dynamics of an ideal fluid which is incompressible, inviscid, and experiences no external forces. We study the family of shear flows of the Euler equations with vorticity of the form Ω(x,y)=cos(κxpxx+κypyy) on a two-dimensional periodic domain of size [0,2π/κx)×[0,2π/κy), and formulate this as a Poisson system. We prove that if py=0 and κx|px|(3+2×31/2 )/2, we prove the flow is nonlinearly unstable. We discuss the full spectrum of the linearisation of shear flows and a related Jacobi problem. We prove analogous stability results in a known Poisson structure preserving truncation and discuss the qualitative differences. We extend a previously known Poisson integrator for this truncation of the Euler equations to a general two-dimensional periodic domain. The Euler equations on a three-dimensional periodic domain are less well-understood. In this domain we formulate the dynamics in terms of the vorticity Fourier modes. This is then used to study shear flows and prove similar stability results as for the two-dimensional case. The linearised equations split into subsystems which have equivalent dynamics to those of the two-dimensional subsystems. We prove the existence of a family of linearly stable shear flows, and another of linearly unstable shear flows. For a dense set of parameter values, the linearised system has a nilpotent part. This is linked to nonnormality and indicates a transition to turbulence. We formulate the Euler equations on a three-dimensional periodic domain as a Poisson system. We finally present some numerical results demonstrating and exploring the results of this thesis

Impact of the COVID-19 pandemic on faecal immunochemical test-based colorectal cancer screening programmes in Australia, Canada, and the Netherlands: a comparative modelling study

Background: Colorectal cancer screening programmes worldwide have been disrupted during the COVID-19 pandemic. We aimed to estimate the impact of hypothetical disruptions to organised faecal immunochemical test-based colorectal cancer screening programmes on short-term and long-term colorectal cancer incidence and morta

Informed Conditioning on Clinical Covariates Increases Power in Case-Control Association Studies

Genetic case-control association studies often include data on clinical covariates, such as body mass index (BMI), smoking status, or age, that may modify the underlying genetic risk of case or control samples. For example, in type 2 diabetes, odds ratios for established variants estimated from low–BMI cases are larger than those estimated from high–BMI cases. An unanswered question is how to use this information to maximize statistical power in case-control studies that ascertain individuals on the basis of phenotype (case-control ascertainment) or phenotype and clinical covariates (case-control-covariate ascertainment). While current approaches improve power in studies with random ascertainment, they often lose power under case-control ascertainment and fail to capture available power increases under case-control-covariate ascertainment. We show that an informed conditioning approach, based on the liability threshold model with parameters informed by external epidemiological information, fully accounts for disease prevalence and non-random ascertainment of phenotype as well as covariates and provides a substantial increase in power while maintaining a properly controlled false-positive rate. Our method outperforms standard case-control association tests with or without covariates, tests of gene x covariate interaction, and previously proposed tests for dealing with covariates in ascertained data, with especially large improvements in the case of case-control-covariate ascertainment. We investigate empirical case-control studies of type 2 diabetes, prostate cancer, lung cancer, breast cancer, rheumatoid arthritis, age-related macular degeneration, and end-stage kidney disease over a total of 89,726 samples. In these datasets, informed conditioning outperforms logistic regression for 115 of the 157 known associated variants investigated (P-value = 1×10−9). The improvement varied across diseases with a 16% median increase in χ2 test statistics and a commensurate increase in power. This suggests that applying our method to existing and future association studies of these diseases may identify novel disease loci