A lower bound for nodal count on discrete and metric graphs

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference

olaparib maintenance therapy in patients pts with platinum sensitive relapsed psr ovarian cancer oc and stable disease sd following platinum based chemotherapy

n/

"Tomography" of the cluster structure of light nuclei via relativistic dissociation

These lecture notes present the capabilities of relativistic nuclear physics for the development of the physics of nuclear clusters. Nuclear track emulsion continues to be an effective technique for pilot studies that allows one, in particular, to study the cluster dissociation of a wide variety of light relativistic nuclei within a common approach. Despite the fact that the capabilities of the relativistic fragmentation for the study of nuclear clustering were recognized quite a long time ago, electronic experiments have not been able to come closer to an integrated analysis of ensembles of relativistic fragments. The continued pause in the investigation of the "fine" structure of relativistic fragmentation has led to resumption of regular exposures of nuclear emulsions in beams of light nuclei produced for the first time at the Nuclotron of the Joint Institute for Nuclear Research (JINR, Dubna). To date, an analysis of the peripheral interactions of relativistic isotopes of beryllium, boron, carbon and nitrogen, including radioactive ones, with nuclei of the emulsion composition, has been performed, which allows the clustering pattern to be presented for a whole family of light nuclei.Comment: ISBN 978-3-319-01076-2. 55 pages, 28 figure

Identification of independent association signals and putative functional variants for breast cancer risk through fine-scale mapping of the 12p11 locus.

BACKGROUND: Multiple recent genome-wide association studies (GWAS) have identified a single nucleotide polymorphism (SNP), rs10771399, at 12p11 that is associated with breast cancer risk. METHOD: We performed a fine-scale mapping study of a 700 kb region including 441 genotyped and more than 1300 imputed genetic variants in 48,155 cases and 43,612 controls of European descent, 6269 cases and 6624 controls of East Asian descent and 1116 cases and 932 controls of African descent in the Breast Cancer Association Consortium (BCAC; http://bcac.ccge.medschl.cam.ac.uk/ ), and in 15,252 BRCA1 mutation carriers in the Consortium of Investigators of Modifiers of BRCA1/2 (CIMBA). Stepwise regression analyses were performed to identify independent association signals. Data from the Encyclopedia of DNA Elements project (ENCODE) and the Cancer Genome Atlas (TCGA) were used for functional annotation. RESULTS: Analysis of data from European descendants found evidence for four independent association signals at 12p11, represented by rs7297051 (odds ratio (OR) = 1.09, 95 % confidence interval (CI) = 1.06-1.12; P = 3 × 10(-9)), rs805510 (OR = 1.08, 95 % CI = 1.04-1.12, P = 2 × 10(-5)), and rs1871152 (OR = 1.04, 95 % CI = 1.02-1.06; P = 2 × 10(-4)) identified in the general populations, and rs113824616 (P = 7 × 10(-5)) identified in the meta-analysis of BCAC ER-negative cases and BRCA1 mutation carriers. SNPs rs7297051, rs805510 and rs113824616 were also associated with breast cancer risk at P < 0.05 in East Asians, but none of the associations were statistically significant in African descendants. Multiple candidate functional variants are located in putative enhancer sequences. Chromatin interaction data suggested that PTHLH was the likely target gene of these enhancers. Of the six variants with the strongest evidence of potential functionality, rs11049453 was statistically significantly associated with the expression of PTHLH and its nearby gene CCDC91 at P < 0.05. CONCLUSION: This study identified four independent association signals at 12p11 and revealed potentially functional variants, providing additional insights into the underlying biological mechanism(s) for the association observed between variants at 12p11 and breast cancer risk.UK funding includes Cancer Research UK and NIH.This is the final version of the article. It first appeared from BioMed Central via http://dx.doi.org/10.1186/s13058-016-0718-

AD51B in Familial Breast Cancer

Common variation on 14q24.1, close to RAD51B, has been associated with breast cancer: rs999737 and rs2588809 with the risk of female breast cancer and rs1314913 with the risk of male breast cancer. The aim of this study was to investigate the role of RAD51B variants in breast cancer predisposition, particularly in the context of familial breast cancer in Finland. We sequenced the coding region of RAD51B in 168 Finnish breast cancer patients from the Helsinki region for identification of possible recurrent founder mutations. In addition, we studied the known rs999737, rs2588809, and rs1314913 SNPs and RAD51B haplotypes in 44,791 breast cancer cases and 43,583 controls from 40 studies participating in the Breast Cancer Association Consortium (BCAC) that were genotyped on a custom chip (iCOGS). We identified one putatively pathogenic missense mutation c.541C&gt;T among the Finnish cancer patients and subsequently genotyped the mutation in additional breast cancer cases (n = 5259) and population controls (n = 3586) from Finland and Belarus. No significant association with breast cancer risk was seen in the meta-analysis of the Finnish datasets or in the large BCAC dataset. The association with previously identified risk variants rs999737, rs2588809, and rs1314913 was replicated among all breast cancer cases and also among familial cases in the BCAC dataset. The most significant association was observed for the haplotype carrying the risk-alleles of all the three SNPs both among all cases (odds ratio (OR): 1.15, 95% confidence interval (CI): 1.11–1.19, P = 8.88 x 10−16) and among familial cases (OR: 1.24, 95% CI: 1.16–1.32, P = 6.19 x 10−11), compared to the haplotype with the respective protective alleles. Our results suggest that loss-of-function mutations in RAD51B are rare, but common variation at the RAD51B region is significantly associated with familial breast cancer risk

Associations of autozygosity with a broad range of human phenotypes

In many species, the offspring of related parents suffer reduced reproductive success, a phenomenon known as inbreeding depression. In humans, the importance of this effect has remained unclear, partly because reproduction between close relatives is both rare and frequently associated with confounding social factors. Here, using genomic inbreeding coefficients (F-ROH) for >1.4 million individuals, we show that F-ROH is significantly associated (p <0.0005) with apparently deleterious changes in 32 out of 100 traits analysed. These changes are associated with runs of homozygosity (ROH), but not with common variant homozygosity, suggesting that genetic variants associated with inbreeding depression are predominantly rare. The effect on fertility is striking: F-ROH equivalent to the offspring of first cousins is associated with a 55% decrease [95% CI 44-66%] in the odds of having children. Finally, the effects of F-ROH are confirmed within full-sibling pairs, where the variation in F-ROH is independent of all environmental confounding.Peer reviewe

Genome-Wide Association Study in BRCA1 Mutation Carriers Identifies Novel Loci Associated with Breast and Ovarian Cancer Risk

BRCA1-associated breast and ovarian cancer risks can be modified by common genetic variants. To identify further cancer risk-modifying loci, we performed a multi-stage GWAS of 11,705 BRCA1 carriers (of whom 5,920 were diagnosed with breast and 1,839 were diagnosed with ovarian cancer), with a further replication in an additional sample of 2,646 BRCA1 carriers. We identified a novel breast cancer risk modifier locus at 1q32 for BRCA1 carriers (rs2290854, P = 2.7×10-8, HR = 1.14, 95% CI: 1.09-1.20). In addition, we identified two novel ovarian cancer risk modifier loci: 17q21.31 (rs17631303, P = 1.4×10-8, HR = 1.27, 95% CI: 1.17-1.38) and 4q32.3 (rs4691139, P = 3.4×10-8, HR = 1.20, 95% CI: 1.17-1.38). The 4q32.3 locus was not associated with ovarian cancer risk in the general population or BRCA2 carriers, suggesting a BRCA1-specific associat

Variability in coral reef fish baseline and benchmark biomass in the central and western Indian Ocean provinces

Reef fish biomass is increasingly recognized as a key indicator of fishery and biodiversity status linked to ecosystem integrity on coral reefs, and yet the evaluation of appropriate baselines for biomass, and what drives variation in potential baselines, is sparse. Variability in reef fishable biomass was assessed to test for the existence of baselines or benchmarks (B&Bs), based on field studies of 223 reef sites in remote uninhabited reefs, in high-compliance closures of >5 km2, and among the increasing number of small and recent closures. The purpose of the study was to examine the effects of human habitation, travel time and distance to cities, habitat, depth, benthic cover, and environmental variables on fish B&Bs. There were large differences in the three categories of ‘no fishing’, with human habitation being the single best predictor of biomass. In remote areas without people (>9 hours of travel time), fish biomass had a mean of 2,450 kg ha–1 (95% confidence interval, 95% CI, 2,130–2,770 kg ha–1; median = 1,885 kg ha–1). In these remote areas, biomass was weakly associated with human travel time to the site and, to a lesser extent, wave energy. In high-compliance closures, fish biomass peaked at 20 years and 5–10 km2, and levelled at 910 kg ha–1 (95% CI 823–989 kg ha–1) for both closure age and size. There was little evidence that human travel time and environmental factors influenced biomass greatly in these established closures. In small and recent closures (<15 years), habitat, depth and closure age were the best predictors of fish biomass. Based on the weakness of environmental factors, country or site-specific B&Bs are not required in these two provinces. However, human habitation in the seascape as well as the size and age of closures set limits to the maximum achievable biomass. The importance of environmental factors increases as the no-fishing areas and closure times decline. Reef wilderness is not widespread in these provinces, but provides key services and therefore needs to be included in conservation and fisheries policy and management goals