The Burau estimate for the entropy of a braid

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for $t=-1$. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved.Comment: 28 pages, 8 figure

Human candidate gene polymorphisms and risk of severe malaria in children in Kilifi, Kenya: a case-control association study

Background: Human genetic factors are important determinants of malaria risk. We investigated associations between multiple candidate polymorphisms—many related to the structure or function of red blood cells—and risk for severe Plasmodium falciparum malaria and its specific phenotypes, including cerebral malaria, severe malaria anaemia, and respiratory distress. Methods: We did a case-control study in Kilifi County, Kenya. We recruited as cases children presenting with severe malaria to the high-dependency ward of Kilifi County Hospital. We included as controls infants born in the local community between Aug 1, 2006, and Sept 30, 2010, who were part of a genetics study. We tested for associations between a range of candidate malaria-protective genes and risk for severe malaria and its specific phenotypes. We used a permutation approach to account for multiple comparisons between polymorphisms and severe malaria. We judged p values less than 0·005 significant for the primary analysis of the association between candidate genes and severe malaria. Findings: Between June 11, 1995, and June 12, 2008, 2244 children with severe malaria were recruited to the study, and 3949 infants were included as controls. Overall, 263 (12%) of 2244 children with severe malaria died in hospital, including 196 (16%) of 1233 with cerebral malaria. We investigated 121 polymorphisms in 70 candidate severe malaria-associated genes. We found significant associations between risk for severe malaria overall and polymorphisms in 15 genes or locations, of which most were related to red blood cells: ABO, ATP2B4, ARL14, CD40LG, FREM3, INPP4B, G6PD, HBA (both HBA1 and HBA2), HBB, IL10, LPHN2 (also known as ADGRL2), LOC727982, RPS6KL1, CAND1, and GNAS. Combined, these genetic associations accounted for 5·2% of the variance in risk for developing severe malaria among individuals in the general population. We confirmed established associations between severe malaria and sickle-cell trait (odds ratio [OR] 0·15, 95% CI 0·11–0·20; p=2·61 × 10−58), blood group O (0·74, 0·66–0·82; p=6·26 × 10−8), and –α3·7-thalassaemia (0·83, 0·76–0·90; p=2·06 × 10−6). We also found strong associations between overall risk of severe malaria and polymorphisms in both ATP2B4 (OR 0·76, 95% CI 0·63–0·92; p=0·001) and FREM3 (0·64, 0·53–0·79; p=3·18 × 10−14). The association with FREM3 could be accounted for by linkage disequilibrium with a complex structural mutation within the glycophorin gene region (comprising GYPA, GYPB, and GYPE) that encodes for the rare Dantu blood group antigen. Heterozygosity for Dantu was associated with risk for severe malaria (OR 0·57, 95% CI 0·49–0·68; p=3·22 × 10−11), as was homozygosity (0·26, 0·11–0·62; p=0·002). Interpretation: Both ATP2B4 and the Dantu blood group antigen are associated with the structure and function of red blood cells. ATP2B4 codes for plasma membrane calcium-transporting ATPase 4 (the major calcium pump on red blood cells) and the glycophorins are ligands for parasites to invade red blood cells. Future work should aim at uncovering the mechanisms by which these polymorphisms can result in severe malaria protection and investigate the implications of these associations for wider health. Funding: Wellcome Trust, UK Medical Research Council, European Union, and Foundation for the National Institutes of Health as part of the Bill & Melinda Gates Grand Challenges in Global Health Initiative

LOW-Γ JETS from COMPACT STELLAR MERGERS: CANDIDATE ELECTROMAGNETIC COUNTERPARTS to GRAVITATIONAL WAVE SOURCES

The American Astronomical Society. All rights reserved.Short gamma-ray bursts (GRBs) are believed to be produced by relativistic jets from mergers of neutron stars (NSs) or NSs and black-holes (BHs). If the Lorentz-factors Γ of jets from compact stellar mergers follow a similar power-law distribution to those observed for other high-energy astrophysical phenomena (e.g., blazars, active galactic nuclei), the population of jets should be dominated by low-Γ outflows. These jets will not produce prompt gamma-rays, but jet energy will be released as X-ray/optical/radio transients when they collide with the ambient medium. Using Monte Carlo simulations, we study the properties of such transients. Approximately 78% of merger jets Mpc result in failed GRBs if the jet Γ follows a power-law distribution of index -1.75. X-ray/optical transients from failed GRBs will have broad distributions of their characteristics: light-curves peak tp ∼ 0.1-10 days after a merger; flux peaks for X-ray 10-6 mJy ≲ Fx ≲ 10-2 mJy; and optical flux peaks at 14 ≲ mg ≲ 22. X-ray transients are detectable by Swift XRT, and ∼85% of optical transients will be detectable by telescopes with limiting magnitude mg ≳ 21, for well localized sources on the sky. X-ray/optical transients are followed by radio transients with peak times narrowly clustered around tp ∼ 10 days, and peak flux of ∼10-100 mJy at 10 GHz and ∼0.1 mJy at 150 MHz. By considering the all-sky rate of short GRBs within the LIGO/Virgo range, the rate of on-axis orphan afterglows from failed GRBs should be 2.6(26) per year for NS-NS(NS-BH) mergers, respectively. Since merger jets from gravitational-wave (GW) trigger events tend to be directed to us, a significant fraction of GW events could be associated with the on-axis orphan afterglow

MiR-137-derived polygenic risk: effects on cognitive performance in patients with schizophrenia and controls

Variants at microRNA-137 (MIR137), one of the most strongly associated schizophrenia risk loci identified to date, have been associated with poorer cognitive performance. As microRNA-137 is known to regulate the expression of ~1900 other genes, including several that are independently associated with schizophrenia, we tested whether this gene set was also associated with variation in cognitive performance. Our analysis was based on an empirically derived list of genes whose expression was altered by manipulation of MIR137 expression. This list was cross-referenced with genome-wide schizophrenia association data to construct individual polygenic scores. We then tested, in a sample of 808 patients and 192 controls, whether these risk scores were associated with altered performance on cognitive functions known to be affected in schizophrenia. A subgroup of healthy participants also underwent functional imaging during memory (n=108) and face processing tasks (n=83). Increased polygenic risk within the empirically derived miR-137 regulated gene score was associated with significantly lower performance on intelligence quotient, working memory and episodic memory. These effects were observed most clearly at a polygenic threshold of P=0.05, although significant results were observed at all three thresholds analyzed. This association was found independently for the gene set as a whole, excluding the schizophrenia-associated MIR137 SNP itself. Analysis of the spatial working memory fMRI task further suggested that increased risk score (thresholded at P=10−5) was significantly associated with increased activation of the right inferior occipital gyrus. In conclusion, these data are consistent with emerging evidence that MIR137 associated risk for schizophrenia may relate to its broader downstream genetic effects

Measurements of the pp → ZZ production cross section and the Z → 4ℓ branching fraction, and constraints on anomalous triple gauge couplings at √s = 13 TeV

Four-lepton production in proton-proton collisions, pp -> (Z/gamma*)(Z/gamma*) -> 4l, where l = e or mu, is studied at a center-of-mass energy of 13 TeV with the CMS detector at the LHC. The data sample corresponds to an integrated luminosity of 35.9 fb(-1). The ZZ production cross section, sigma(pp -> ZZ) = 17.2 +/- 0.5 (stat) +/- 0.7 (syst) +/- 0.4 (theo) +/- 0.4 (lumi) pb, measured using events with two opposite-sign, same-flavor lepton pairs produced in the mass region 60 4l) = 4.83(-0.22)(+0.23) (stat)(-0.29)(+0.32) (syst) +/- 0.08 (theo) +/- 0.12(lumi) x 10(-6) for events with a four-lepton invariant mass in the range 80 4GeV for all opposite-sign, same-flavor lepton pairs. The results agree with standard model predictions. The invariant mass distribution of the four-lepton system is used to set limits on anomalous ZZZ and ZZ. couplings at 95% confidence level: -0.0012 < f(4)(Z) < 0.0010, -0.0010 < f(5)(Z) < 0.0013, -0.0012 < f(4)(gamma) < 0.0013, -0.0012 < f(5)(gamma) < 0.0013

Cross-ancestry genome-wide association analysis of corneal thickness strengthens link between complex and Mendelian eye diseases

Central corneal thickness (CCT) is a highly heritable trait associated with complex eye diseases such as keratoconus and glaucoma. We perform a genome-wide association meta-analysis of CCT and identify 19 novel regions. In addition to adding support for known connective tissue-related pathways, pathway analyses uncover previously unreported gene sets. Remarkably, >20% of the CCT-loci are near or within Mendelian disorder genes. These included FBN1, ADAMTS2 and TGFB2 which associate with connective tissue disorders (Marfan, Ehlers-Danlos and Loeys-Dietz syndromes), and the LUM-DCN-KERA gene complex involved in myopia, corneal dystrophies and cornea plana. Using index CCT-increasing variants, we find a significant inverse correlation in effect sizes between CCT and keratoconus (r =-0.62, P = 5.30 × 10-5) but not between CCT and primary open-angle glaucoma (r =-0.17, P = 0.2). Our findings provide evidence for shared genetic influences between CCT and keratoconus, and implicate candidate genes acting in collagen and extracellular matrix regulation

Using de novo assembly to identify structural variation of eight complex immune system gene regions

Driven by the necessity to survive environmental pathogens, the human immune system has evolved exceptional diversity and plasticity, to which several factors contribute including inheritable structural polymorphism of the underlying genes. Characterizing this variation is challenging due to the complexity of these loci, which contain extensive regions of paralogy, segmental duplication and high copy-number repeats, but recent progress in long-read sequencing and optical mapping techniques suggests this problem may now be tractable. Here we assess this by using long-read sequencing platforms from PacBio and Oxford Nanopore, supplemented with short-read sequencing and Bionano optical mapping, to sequence DNA extracted from CD14+ monocytes and peripheral blood mononuclear cells from a single European individual identified as HV31. We use this data to build a de novo assembly of eight genomic regions encoding four key components of the immune system, namely the human leukocyte antigen, immunoglobulins, T cell receptors, and killer-cell immunoglobulin-like receptors. Validation of our assembly using k-mer based and alignment approaches suggests that it has high accuracy, with estimated base-level error rates below 1 in 10 kb, although we identify a small number of remaining structural errors. We use the assembly to identify heterozygous and homozygous structural variation in comparison to GRCh38. Despite analyzing only a single individual, we find multiple large structural variants affecting core genes at all three immunoglobulin regions and at two of the three T cell receptor regions. Several of these variants are not accurately callable using current algorithms, implying that further methodological improvements are needed. Our results demonstrate that assessing haplotype variation in these regions is possible given sufficiently accurate long-read and associated data. Continued reductions in the cost of these technologies will enable application of these methods to larger samples and provide a broader catalogue of germline structural variation at these loci, an important step toward making these regions accessible to large-scale genetic association studies