On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces

Fine-Scale Mapping of the 5q11.2 Breast Cancer Locus Reveals at Least Three Independent Risk Variants Regulating MAP3K1

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Biogeochemical status of the Paleo-Pacific Ocean: clues from the early Cambrian of South Australia

Published online: 28 Mar 2021The Ediacaran–Cambrian transition was a time of profound reorganisation of the biosphere, coinciding with the assembly of Gondwana and increasing atmospheric and oceanic oxygenation. The lower Cambrian marine sediments of the Stansbury Basin, South Australia, were deposited at low northern paleolatitudes on the western margin of the emerging Pacific Ocean. Here we report results of a multi-pronged investigation employing trace and rare earth element (REE) abundances, total organic carbon (TOC) contents and pyritic sulfur isotopic compositions (δ34Spy) in carbonaceous shales from three formations within the Normanville and Kanmantoo groups (sequences Є1.2 to Є2.2). TOC ranges from ≤0.5 wt% in the Emu Bay Shale, through 0.4–1.9 wt% in the Talisker Formation, to 0.2–2.6 wt% in the Heatherdale Shale. Covariance is demonstrated between trace elements and organic matter, with comparative uranium enrichment in the Heatherdale Shale likely linked to increased primary productivity. Heavy REEs and yttrium are typically more depleted than light REEs. Provenance appears to evolve through the upper Normanville sediments into the Kanmantoo Group, corroborating published detrital zircon interpretations that suggest an increasing dominance of southerly derived sources. The prevailing paleoredox regime for the Heatherdale Shale and the Talisker Formation was dysoxic, evolving into progressively more reducing conditions up section, the latter exhibiting a secular decline in δ34Spy (from +10 to −11‰ VCDT). Conversely, redox proxies concur in demonstrating that the basal fossiliferous Emu Bay Shale accumulated beneath an oxic water column. Comparison of trace element and REE distributions with those of slightly older sequences on the Yangtze Platform, South China, reveals striking similarities, implying that the trace element chemistry of the Paleo-Pacific and Asian oceans was homogeneous.P. A. Hall, D. M. McKirdy, G. P. Halverson, J. B. Jago and A. S. Collin

Prediction of Breast Cancer Risk Based on Profiling With Common Genetic Variants

Background: Data for multiple common susceptibility alleles for breast cancer may be combined to identify women at different levels of breast cancer risk. Such stratification could guide preventive and screening strategies. However, empirical evidence for genetic risk stratification is lacking. Methods: We investigated the value of using 77 breast cancer-associated single nucleotide polymorphisms (SNPs) for risk stratification, in a study of 33 673 breast cancer cases and 33 381 control women of European origin. We tested all possible pair-wise multiplicative interactions and constructed a 77-SNP polygenic risk score (PRS) for breast cancer overall and by estrogen receptor (ER) status. Absolute risks of breast cancer by PRS were derived from relative risk estimates and UK incidence and mortality rates. Results: There was no strong evidence for departure from a multiplicative model for any SNP pair. Women in the highest 1% of the PRS had a three-fold increased risk of developing breast cancer compared with women in the middle quintile (odds ratio [OR] = 3.36, 95% confidence interval [CI] = 2.95 to 3.83). The ORs for ER-positive and ER-negative disease were 3.73 (95% CI = 3.24 to 4.30) and 2.80 (95% CI = 2.26 to 3.46), respectively. Lifetime risk of breast cancer for women in the lowest and highest quintiles of the PRS were 5.2% and 16.6% for a woman without family history, and 8.6% and 24.4% for a woman with a first-degree family history of breast cancer. Conclusions: The PRS stratifies breast cancer risk in women both with and without a family history of breast cancer. The observed level of risk discrimination could inform targeted screening and prevention strategies. Further discrimination may be achievable through combining the PRS with lifestyle/environmental factors, although these were not considered in this report.Peer reviewe