Gaussian perturbations of circle maps: A spectral approach

In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this phenomenon a $\lambda$-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.Comment: Published in at http://dx.doi.org/10.1214/08-AAP573 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Traveling waves of selective sweeps

The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponentially growing population. To better understand this process, Beerenwinkel et al. [PLoS Comput. Biol. 3 (2007) 2239--2246] consider a Wright--Fisher model in which cells from an exponentially growing population accumulate advantageous mutations. Simulations show a traveling wave in which the time of the first $k$-fold mutant, $T_k$, is approximately linear in $k$ and heuristics are used to obtain formulas for $ET_k$. Here, we consider the analogous problem for the Moran model and prove that as the mutation rate $\mu\to0$, $T_k\sim c_k\log(1/\mu)$, where the $c_k$ can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of $X_k(t)={}$the number of cells with $k$ mutations at time $t$.Comment: Published in at http://dx.doi.org/10.1214/10-AAP721 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evolution in predator-prey systems

We study the adaptive dynamics of predator-prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see (i) the number of coexisting predators remains tight and the differences in traits from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we calculate the asymptotic rate at which the traits of the least and most "fit" predators in the population increase. This last result is obtained by comparison with a branching random walk killed to the left of a linear boundary and a finite branching-selection particle system.Comment: Major revisions to the introduction and definitions of the evolutionary processes considered, minor revisions to the proofs of Propositions 1.1 and 1.

Math 57: Applied Differential Equations I

This book is designed for the fourth semester, “capstone” course in a calculus sequence with an emphasis on modeling with linear differential equations. Students will learn to translate verbal descriptions of physical problems into differential equation models, solve and visualize solutions to differential equations using MATLAB, calculate and investigate the behavior of analytic solutions to linear differential equations, discuss how solutions to differential equations depend on parameters, and interpret solutions to differential equations in the context of applications.https://scholarlycommons.pacific.edu/open-textbooks/1025/thumbnail.jp

Doctors qualified from Chinese Universities with “English Parallel” courses registered with the General Medical Council

AbstractObjectiveTo identify the number of Chinese graduates from medical schools in China with an English Parallel course through a Freedom of Information search of the current UK medical register.MethodA Freedom of Information request was submitted to the General Medical Council to investigate the number of graduates from Chinese medical schools with MB BS courses using English as the medium of instruction.ResultsDetails of 73 graduates were obtained. Twenty one of the registrants were Chinese and 52 South Asian. During the last decade there has been a significant increase in the number of Pakistani and Indian graduates who have registered with the GMC compared to Chinese graduates.ConclusionIf the experience with Central and Eastern European “English Parallel” courses is repeated we should expect a significant growth in the numbers of British and EU citizens training and graduating in China and subsequently practicing in the UK

Does the failure to provide equitable access to treatment lead to action by NHS organisations: the case of biologics for South Asians with inflammatory bowel disease?

Aims: The purpose of this study was to identify whether NHS Trusts where discrimination in the delivery of care to patients from the South Asian community had been demonstrated had taken any actions to address the issue over the subsequent year.Methods:   Freedom of information requests were sent to three trusts which had provided evidence of disparate provision of biologic therapy to patients with Crohn’s disease, their associated Clinical Commissioning Groups and Healthwatch organisations to seek evidence they had remedied the situation. Requests were also sent to the Care Quality Commission, NHS Improvement and the Equality and Human Rights Commission seeking examples where they had responded to inequitable delivery of care related to ethnicity.Results: No organisation had any evidence of responses to the situation, many unable to accept its existence.Conclusion: Legal duties are discussed and the only remedy appears to be through the tort of negligence