Heat Kernel on Homogeneous Bundles over Symmetric Spaces

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.Comment: 55 page

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease

Molecular phylogenetic evidence for the reorganization of the Hyperiid amphipods, a diverse group of pelagic crustaceans

[Display omitted] ► We present the first multi-locus phylogeny of the hyperiid amphipods. ► We sampled 16 of the 23 recognized hyperiidian families for 18S, 28S, H3, and COI. ► Three reciprocally monophyletic clades were supported within Physocephalata. ► Our analyses place Cystisomatidae and Paraphronimidae at the base of Physosomata. ► Results are discussed within the context of pelagic hyperiid amphipod life history. Within the crustaceans, the Amphipoda rank as one of the most speciose extant orders. Amphipods have successfully invaded and become major constituents of a variety of ecosystems. The hyperiid amphipods are classically defined as an exclusively pelagic group broadly inhabiting oceanic midwater environments and often having close associations with gelatinous zooplankton. As with other amphipod groups they have largely been classified based on appendage structures, however evidence suggests that at least some of these characters are the product of convergent evolution. Here we present the first multi-locus molecular phylogenetic assessment of relationships among the hyperiid amphipods. We sampled 51 species belonging to 16 of the 23 recognized hyperiidian families for three nuclear loci (18S, 28S, and H3) and mitochondrial COI. We performed both Bayesian Inference and Maximum Likelihood analyses of concatenated sequences. In addition, we also explored the utility of species-tree methods for reconstructing deep evolutionary histories using the Minimize Deep Coalescence (MDC) approach. Our results are compared with previous molecular analyses and traditional systematic groupings. We discuss these results within the context of adaptations correlated with the pelagic life history of hyperiid amphipods. Within the infraorder Physocephalata (Bowman and Gruner, 1973) we inferred support for three reciprocally monophyletic clades; the Platysceloidea, Vibilioidea, and Phronimoidea. Our results also place the enigmatic Cystisomatidae and Paraphronimidae at the base of the infraorder Physosomata (Bowman and Gruner, 1973) suggesting that Physosomata as traditionally recognized is paraphyletic. Based on our multilocus phylogeny, major rearrangements to existing taxonomic groupings of hyperiid amphipods are warranted