Undecidability in number theory

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342

Tracking Alzheimer's disease

Population-based brain mapping provides great insight into the trajectory of aging and dementia, as well as brain changes that normally occur over the human life span. We describe three novel brain mapping techniques, cortical thickness mapping, tensor-based morphometry (TBM), and hippocampal surface modeling, which offer enormous power for measuring disease progression in drug trials, and shed light on the neuroscience of brain degeneration in Alzheimer's disease (AD) and mild cognitive impairment (MCI). We report the first time-lapse maps of cortical atrophy spreading dynamically in the living brain, based on averaging data from populations of subjects with Alzheimer's disease and normal subjects imaged longitudinally with MRI. These dynamic sequences show a rapidly advancing wave of cortical atrophy sweeping from limbic and temporal cortices into higher-order association and ultimately primary sensorimotor areas, in a pattern that correlates with cognitive decline. A complementary technique, TBM, reveals the 3D profile of atrophic rates, at each point in the brain. A third technique, hippocampal surface modeling, plots the profile of shape alterations across the hippocampal surface. The three techniques provide moderate to highly automated analyses of images, have been validated on hundreds of scans, and are sensitive to clinically relevant changes in individual patients and groups undergoing different drug treatments. We compare time-lapse maps of AD, MCI, and other dementias, correlate these changes with cognition, and relate them to similar time-lapse maps of childhood development, schizophrenia, and HIV-associated brain degeneration. Strengths and weaknesses of these different imaging measures for basic neuroscience and drug trials are discussed

Chronic obstructive pulmonary disease and related phenotypes: Polygenic risk scores in population-based and case-control cohorts.

Background: Genetic factors influence chronic obstructive pulmonary disease (COPD) risk, but the individual variants that have been identified have small effects. We hypothesised that a polygenic risk score using additional variants would predict COPD and associated phenotypes. Methods: We constructed a polygenic risk score using a genome-wide association study of lung function (FEV1 and FEV1/forced vital capacity [FVC]) from the UK Biobank and SpiroMeta. We tested this polygenic risk score in nine cohorts of multiple ethnicities for an association with moderate-to-severe COPD (defined as FEV1/FVC <0·7 and FEV1 <80% of predicted). Associations were tested using logistic regression models, adjusting for age, sex, height, smoking pack-years, and principal components of genetic ancestry. We assessed predictive performance of models by area under the curve. In a subset of studies, we also studied quantitative and qualitative CT imaging phenotypes that reflect parenchymal and airway pathology, and patterns of reduced lung growth. Findings: The polygenic risk score was associated with COPD in European (odds ratio [OR] per SD 1·81 [95% CI 1·74–1·88] and non-European (1·42 [1·34–1·51]) populations. Compared with the first decile, the tenth decile of the polygenic risk score was associated with COPD, with an OR of 7·99 (6·56–9·72) in European ancestry and 4·83 (3·45–6·77) in non-European ancestry cohorts. The polygenic risk score was superior to previously described genetic risk scores and, when combined with clinical risk factors (ie, age, sex, and smoking pack-years), showed improved prediction for COPD compared with a model comprising clinical risk factors alone (AUC 0·80 [0·79–0·81] vs 0·76 [0·75–0·76]). The polygenic risk score was associated with CT imaging phenotypes, including wall area percent, quantitative and qualitative measures of emphysema, local histogram emphysema patterns, and destructive emphysema subtypes. The polygenic risk score was associated with a reduced lung growth pattern. Interpretation: A risk score comprised of genetic variants can identify a small subset of individuals at markedly increased risk for moderate-to-severe COPD, emphysema subtypes associated with cigarette smoking, and patterns of reduced lung growth. Funding: US National Institutes of Health, Wellcome Trust