Exponential sum approximations for t−βt^{-\beta}

Given $\beta>0$ and $\delta>0$, the function $t^{-\beta}$ may be approximated for $t$ in a compact interval $[\delta,T]$ by a sum of terms of the form $we^{-at}$, with parameters $w>0$ and $a>0$. One such an approximation, studied by Beylkin and Monz\'on, is obtained by applying the trapezoidal rule to an integral representation of $t^{-\beta}$, after which Prony's method is applied to reduce the number of terms in the sum with essentially no loss of accuracy. We review this method, and then describe a similar approach based on an alternative integral representation. The main difference is that the new approach achieves much better results before the application of Prony's method; after applying Prony's method the performance of both is much the same.Comment: 18 pages, 5 figures. I have completely rewritten this paper because after uploading the previous version I realised that there is a much better approach. Note the change to the title. Have included minor corrections following revie

Localization of adaptive variants in human genomes using averaged one-dependence estimation.

Statistical methods for identifying adaptive mutations from population genetic data face several obstacles: assessing the significance of genomic outliers, integrating correlated measures of selection into one analytic framework, and distinguishing adaptive variants from hitchhiking neutral variants. Here, we introduce SWIF(r), a probabilistic method that detects selective sweeps by learning the distributions of multiple selection statistics under different evolutionary scenarios and calculating the posterior probability of a sweep at each genomic site. SWIF(r) is trained using simulations from a user-specified demographic model and explicitly models the joint distributions of selection statistics, thereby increasing its power to both identify regions undergoing sweeps and localize adaptive mutations. Using array and exome data from 45 ‡Khomani San hunter-gatherers of southern Africa, we identify an enrichment of adaptive signals in genes associated with metabolism and obesity. SWIF(r) provides a transparent probabilistic framework for localizing beneficial mutations that is extensible to a variety of evolutionary scenarios

African-American patients with cancer Talking About Clinical Trials (TACT) with oncologists during consultations: evaluating the efficacy of tailored health messages in a randomised controlled trial—the TACT study protocol

Introduction Low rates of accrual of African-American (AA) patients with cancer to therapeutic clinical trials (CTs) represent a serious and modifiable racial disparity in healthcare that impedes the development of promising cancer therapies. Suboptimal physician–patient consultation communication is a barrier to the accrual of patients with cancer of any race, but communication difficulties are compounded with AA patients. Providing tailored health messages (THM) to AA patients and their physician about CTs has the potential to improve communication, lower barriers to accrual and ameliorate health disparities. Objective (1) Demonstrate the efficacy of THM to increase patient activation as measured by direct observation. (2) Demonstrate the efficacy of THM to improve patient outcomes associated with barriers to AA participation. (3) Explore associations among preconsultation levels of: (A) trust in medical researchers, (B) knowledge and attitudes towards CTs, (C) patient-family member congruence in decision-making, and (D) involvement/information preferences, and group assignment. Methods and analysis First, using established methods, we will develop THM materials. Second, the efficacy of the intervention is determined in a 2 by 2 factorial randomised controlled trial to test the effectiveness of (1) providing 357 AA patients with cancer with THM with 2 different ‘depths’ of tailoring and (2) either providing feedback to oncologists about the patients\u27 trial THM or not. The primary analysis compares patient engaged communication in 4 groups preconsultation and postconsultation. Ethics and dissemination This study was approved by the Virginia Commonwealth University Institutional Review Board. To facilitate use of the THM intervention in diverse settings, we will convene ‘user groups’ at 3 major US cancer centres. To facilitate dissemination, we will post all materials and the implementation guide in publicly available locations