Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

A million and more trees for science

TreeDivNet is the largest network of biodiversity experiments worldwide, but needs to expand. We encourage colleagues to establish new experiments on the relation between tree species diversity and forest ecosystem functioning, and to make use of the platform for collaborative research

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 similar to 2.8M SNPs with BMI and WHRadjBMI in four strata (men 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= 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 providefurther insights into the biology that underlies weight change with age or the sexually dimorphism of body shape.Peer reviewe

Citalopram 20 mg, 40 mg and 60 mg are all effective and well tolerated compared with placebo in obsessive-compulsive disorder

Serotonin reuptake inhibitors appear to be uniquely effective treatments for obsessive-compulsive disorder (OCD). This double-blind, placebo-controlled study was the first trial to assess the efficacy of the most selective of the serotonin reuptake inhibitors, citalopram, in OCD. A total of 401 patients were randomized to receive citalopram 20, 40 or 60 mg/day or placebo for 12 weeks. All three doses of citalopram were significantly more effective than placebo measured on the Yale-Brown Obsessive Compulsive Scale (Y-BOCS) change score (P < 0.01). The highest response rate, defined as 25% improvement in Y-BOCS entry score, was observed in the 60 mg group (65%). This compared with 52% and 57.4% in the 40 mg and 20 mg groups. Response rate on placebo was 36.6% (P < 0.05 for all three doses of citalopram compared to placebo). There was no significant difference between the individual doses of citalopram. An advantage was seen for citalopram on the Sheehan Disability Scale compared with placebo (P < 0.05 on all three citalopram groups versus placebo for both the work situation and the family life and home responsibilities and P < 0.05 on citalopram 60 mg and 20 mg versus placebo for the social life and home activities). Citalopram was well tolerated; only 4 to 6 patients in each dose group discontinued the study prematurely due to adverse events. © 2001 Lippincott Williams & Wilkins.Articl

Group representations with empty residual spectrum

Let X be a Banach space on which a discrete group Γ acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. We show, for instance, that this occurs if X is ℓ 2(Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair, and develop some of the basic properties of this construction. The cases X = ℓ p (Γ) for 1 ≤ p < 2 or 2 < p < ∞ are more difficult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary