Imaginary cones and limit roots of infinite Coxeter groups

Let (W,S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropy cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots (see arXiv:1112.5415). In this article we study the close relations of the imaginary cone (see arXiv:1210.5206) with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.Comment: v1: 63 pages, 14 figures. v2: Title changed; abstract and introduction expanded and a few typos corrected. v3: 71 pages; some further corrections after referee report, and many additions (most notably, relations with geometric group theory (7.4) and Appendix on links with Benoist's limit sets). To appear in Mathematische Zeitschrif

Asymptotical behaviour of roots of infinite Coxeter groups

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain that this subset is built from the intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.Comment: 19 pages, 11 figures. Version 2: 29 pages, 11 figures. Reorganisation of the paper, addition of many details (section 5 in particular). Version 3 : revised edition accepted in Journal of the CMS. The number "I" was removed from the title since number "II" paper was named differently, see http://arxiv.org/abs/1303.671

Asymptotical behaviour of roots of infinite Coxeter groups I

Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$

Trisomy for Synaptojanin1 in Down syndrome is functionally linked to the enlargement of early endosomes

Enlarged early endosomes have been observed in neurons and fibroblasts in Down syndrome (DS). These endosome abnormalities have been implicated in the early development of Alzheimer's disease (AD) pathology in these subjects. Here, we show the presence of enlarged endosomes in blood mononuclear cells and lymphoblastoid cell lines (LCLs) from individuals with DS using immunofluorescence and confocal microscopy. Genotype-phenotype correlations in LCLs carrying partial trisomies 21 revealed that triplication of a 2.56 Mb locus in 21q22.11 is associated with the endosomal abnormalities. This locus contains the gene encoding the phosphoinositide phosphatase synaptojanin 1 (SYNJ1), a key regulator of the signalling phospholipid phosphatidylinositol-4,5-biphosphate that has been shown to regulate clathrin-mediated endocytosis. We found that SYNJ1 transcripts are increased in LCLs from individuals with DS and that overexpression of SYNJ1 in a neuroblastoma cell line as well as in transgenic mice leads to enlarged endosomes. Moreover, the proportion of enlarged endosomes in fibroblasts from an individual with DS was reduced after silencing SYNJ1 expression with RNA interference. In LCLs carrying amyloid precursor protein (APP) microduplications causing autosomal dominant early-onset AD, enlarged endosomes were absent, suggesting that APP overexpression alone is not involved in the modification of early endosomes in this cell type. These findings provide new insights into the contribution of SYNJ1 overexpression to the endosomal changes observed in DS and suggest an attractive new target for rescuing endocytic dysfunction and lipid metabolism in DS and in A

DYRK1A, a Novel Determinant of the Methionine-Homocysteine Cycle in Different Mouse Models Overexpressing this Down-Syndrome-Associated Kinase

BACKGROUND:Hyperhomocysteinemia, characterized by increased plasma homocysteine level, is associated with an increased risk of atherosclerosis. On the contrary, patients with Down syndrome appear to be protected from the development of atherosclerosis. We previously found a deleterious effect of hyperhomocysteinemia on expression of DYRK1A, a Down-syndrome-associated kinase. As increased expression of DYRK1A and low plasma homocysteine level have been associated with Down syndrome, we aimed to analyze the effect of its over-expression on homocysteine metabolism in mice. METHODOLOGY/PRINCIPAL FINDINGS:Effects of DYRK1A over-expression were examined by biochemical analysis of methionine metabolites, real-time quantitative reverse-transcription polymerase chain reaction, and enzyme activities. We found that over-expression of Dyrk1a increased the hepatic NAD(P)H:quinone oxidoreductase and S-adenosylhomocysteine hydrolase activities, concomitant with decreased level of plasma homocysteine in three mice models overexpressing Dyrk1a. Moreover, these effects were abolished by treatment with harmine, the most potent and specific inhibitor of Dyrk1a. The increased NAD(P)H:quinone oxidoreductase and S-adenosylhomocysteine hydrolase activities were also found in lymphoblastoid cell lines from patients with Down syndrome. CONCLUSIONS/SIGNIFICANCE:Our results might give clues to understand the protective effect of Down syndrome against vascular defect through a decrease of homocysteine level by DYRK1A over-expression. They reveal a link between the Dyrk1a signaling pathway and the homocysteine cycle

The E-ELT first light spectrograph HARMONI: capabilities and modes

Trabajo presentado en SPIE Astronomical Telescopes, celebrado en San Diego (California), del 26 de junio al 1 de julio de 2016HARMONI is the E-ELT's first light visible and near-infrared integral field spectrograph. It will provide four different spatial scales, ranging from coarse spaxels of 60 × 30 mas best suited for seeing limited observations, to 4 mas spaxels that Nyquist sample the diffraction limited point spread function of the E-ELT at near-infrared wavelengths. Each spaxel scale may be combined with eleven spectral settings, that provide a range of spectral resolving powers (R 3500, 7500 and 20000) and instantaneous wavelength coverage spanning the 0.5 - 2.4 ¿m wavelength range of the instrument. In autumn 2015, the HARMONI project started the Preliminary Design Phase, following signature of the contract to design, build, test and commission the instrument, signed between the European Southern Observatory and the UK Science and Technology Facilities Council. Crucially, the contract also includes the preliminary design of the HARMONI Laser Tomographic Adaptive Optics system. The instrument's technical specifications were finalized in the period leading up to contract signature. In this paper, we report on the first activity carried out during preliminary design, defining the baseline architecture for the system, and the trade-off studies leading up to the choice of baseline

A pan-European epidemiological study reveals honey bee colony survival depends on beekeeper education and disease control

Reports of honey bee population decline has spurred many national efforts to understand the extent of the problem and to identify causative or associated factors. However, our collective understanding of the factors has been hampered by a lack of joined up trans-national effort. Moreover, the impacts of beekeeper knowledge and beekeeping management practices have often been overlooked, despite honey bees being a managed pollinator. Here, we established a standardised active monitoring network for 5 798 apiaries over two consecutive years to quantify honey bee colony mortality across 17 European countries. Our data demonstrate that overwinter losses ranged between 2% and 32%, and that high summer losses were likely to follow high winter losses. Multivariate Poisson regression models revealed that hobbyist beekeepers with small apiaries and little experience in beekeeping had double the winter mortality rate when compared to professional beekeepers. Furthermore, honey bees kept by professional beekeepers never showed signs of disease, unlike apiaries from hobbyist beekeepers that had symptoms of bacterial infection and heavy Varroa infestation. Our data highlight beekeeper background and apicultural practices as major drivers of honey bee colony losses. The benefits of conducting trans-national monitoring schemes and improving beekeeper training are discussed