Sharp Hypercontractivity for Global Functions

For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality for the noise operator allows bounding norms of $T_\rho f=\sum_S\rho^{|S|}\hat f(S)\chi_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$ inequality allows bounding the norm of $f^{=d}=\sum_{|S|=d}\hat f(S)\chi_S$ in terms of $E[f]$. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube is endowed with the uniform measure, they do not hold for more general discrete product spaces. Finding a `natural' generalization was a long-standing open problem. In [P. Keevash et al., Global hypercontractivity and its applications, J. Amer. Math. Soc., to appear], a hypercontractive inequality for this setting was presented, that holds for functions which are `global' -- namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group $S_n$ and to other settings where sharpness of the bound is crucial. Also, no level-$d$ inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-$d$ inequality for this setting. Our inequalities open the way for diverse applications to extremal set theory and to group theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications to the study of functions on $S_n$ -- including hypercontractivity and level-$d$ inequalities, character bounds, variants of Roth's theorem and of Bogolyubov's lemma, and diameter bounds, that were obtained using our techniques

Association of the PHACTR1/EDN1 genetic locus with spontaneous coronary artery dissection

Background: Spontaneous coronary artery dissection (SCAD) is an increasingly recognized cause of acute coronary syndromes (ACS) afflicting predominantly younger to middle-aged women. Observational studies have reported a high prevalence of extracoronary vascular anomalies, especially fibromuscular dysplasia (FMD) and a low prevalence of coincidental cases of atherosclerosis. PHACTR1/EDN1 is a genetic risk locus for several vascular diseases, including FMD and coronary artery disease, with the putative causal noncoding variant at the rs9349379 locus acting as a potential enhancer for the endothelin-1 (EDN1) gene. Objectives: This study sought to test the association between the rs9349379 genotype and SCAD. Methods: Results from case control studies from France, United Kingdom, United States, and Australia were analyzed to test the association with SCAD risk, including age at first event, pregnancy-associated SCAD (P-SCAD), and recurrent SCAD. Results: The previously reported risk allele for FMD (rs9349379-A) was associated with a higher risk of SCAD in all studies. In a meta-analysis of 1,055 SCAD patients and 7,190 controls, the odds ratio (OR) was 1.67 (95% confidence interval [CI]: 1.50 to 1.86) per copy of rs9349379-A. In a subset of 491 SCAD patients, the OR estimate was found to be higher for the association with SCAD in patients without FMD (OR: 1.89; 95% CI: 1.53 to 2.33) than in SCAD cases with FMD (OR: 1.60; 95% CI: 1.28 to 1.99). There was no effect of genotype on age at first event, P-SCAD, or recurrence. Conclusions: The first genetic risk factor for SCAD was identified in the largest study conducted to date for this condition. This genetic link may contribute to the clinical overlap between SCAD and FMD

Формирование эмоциональной культуры как компонента инновационной культуры студентов

Homozygosity has long been associated with rare, often devastating, Mendelian disorders1 and Darwin was one of the first to recognise that inbreeding reduces evolutionary fitness2. However, the effect of the more distant parental relatedness common in modern human populations is less well understood. Genomic data now allow us to investigate the effects of homozygosity on traits of public health importance by observing contiguous homozygous segments (runs of homozygosity, ROH), which are inferred to be homozygous along their complete length. Given the low levels of genome-wide homozygosity prevalent in most human populations, information is required on very large numbers of people to provide sufficient power3,4. Here we use ROH to study 16 health-related quantitative traits in 354,224 individuals from 102 cohorts and find statistically significant associations between summed runs of homozygosity (SROH) and four complex traits: height, forced expiratory lung volume in 1 second (FEV1), general cognitive ability (g) and educational attainment (nominal p<1 × 10−300, 2.1 × 10−6, 2.5 × 10−10, 1.8 × 10−10). In each case increased homozygosity was associated with decreased trait value, equivalent to the offspring of first cousins being 1.2 cm shorter and having 10 months less education. Similar effect sizes were found across four continental groups and populations with different degrees of genome-wide homozygosity, providing convincing evidence for the first time that homozygosity, rather than confounding, directly contributes to phenotypic variance. Contrary to earlier reports in substantially smaller samples5,6, no evidence was seen of an influence of genome-wide homozygosity on blood pressure and low density lipoprotein (LDL) cholesterol, or ten other cardio-metabolic traits. Since directional dominance is predicted for traits under directional evolutionary selection7, this study provides evidence that increased stature and cognitive function have been positively selected in human evolution, whereas many important risk factors for late-onset complex diseases may not have been

The Influence of Age and Sex on Genetic Associations with Adult Body Size and Shape: A Large-Scale Genome-Wide Interaction Study.

Genome-wide association studies (GWAS) have identified more than 100 genetic variants contributing to BMI, a measure of body size, or waist-to-hip ratio (adjusted for BMI, WHRadjBMI), a measure of body shape. Body size and shape change as people grow older and these changes differ substantially between men and women. To systematically screen for age- and/or sex-specific effects of genetic variants on BMI and WHRadjBMI, we performed meta-analyses of 114 studies (up to 320,485 individuals of European descent) with genome-wide chip and/or Metabochip data by the Genetic Investigation of Anthropometric Traits (GIANT) Consortium. Each study tested the association of up to ~2.8M SNPs with BMI and WHRadjBMI in four strata (men ≤50y, men >50y, women ≤50y, women >50y) and summary statistics were combined in stratum-specific meta-analyses. We then screened for variants that showed age-specific effects (G x AGE), sex-specific effects (G x SEX) or age-specific effects that differed between men and women (G x AGE x SEX). For BMI, we identified 15 loci (11 previously established for main effects, four novel) that showed significant (FDR<5%) age-specific effects, of which 11 had larger effects in younger (<50y) than in older adults (≥50y). No sex-dependent effects were identified for BMI. For WHRadjBMI, we identified 44 loci (27 previously established for main effects, 17 novel) with sex-specific effects, of which 28 showed larger effects in women than in men, five showed larger effects in men than in women, and 11 showed opposite effects between sexes. No age-dependent effects were identified for WHRadjBMI. This is the first genome-wide interaction meta-analysis to report convincing evidence of age-dependent genetic effects on BMI. In addition, we confirm the sex-specificity of genetic effects on WHRadjBMI. These results may provide further insights into the biology that underlies weight change with age or the sexually dimorphism of body shape