Volumes of polytopes in spaces of constant curvature

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a solution for Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find a few versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference

DNA methylation mediates the effect of maternal smoking during pregnancy on birthweight of the offspring

Background: We examined whether the effect of maternal smoking during pregnancy on birthweight of the offspring was mediated by smoking-induced changes to DNA methylation in cord blood. Methods: First, we used cord blood of 129 Dutch children exposed to maternal smoking vs 126 unexposed to maternal and paternal smoking (53% male) participating in the GECKO Drenthe birth cohort. DNA methylation was measured using the Illumina HumanMethylation450 Beadchip. We performed an epigenome-wide association study for the association between maternal smoking and methylation followed by a mediation analysis of the top signals [false-discovery rate (FDR)<0.05]. We adjusted both analyses for maternal age, education, pre-pregnancy BMI, offspring's sex, gestational age and white blood cell composition. Secondly, in 175 exposed and 1248 unexposed newborns from two independent birth cohorts, we replicated and meta-analysed results of eight cytosine-phosphate-guanine (CpG) sites in the GFI1 gene, which showed the most robust mediation. Finally, we performed functional network and enrichment analysis. Results: We found 35 differentially methylated CpGs (FDR<0.05) in newborns exposed vs unexposed to smoking, of which 23 survived Bonferroni correction (P<1×10-7). These 23 CpGs mapped to eight genes: AHRR, GFI1, MYO1G, CYP1A1, NEUROG1, CNTNAP2, FRMD4A and LRP5. We observed partial confirmation as three of the eight CpGs in GFI1 replicated. These CpGs partly mediated the effect of maternal smoking on birthweight (Sobel P<0.05) in meta-analysis of GECKO and the two replication cohorts. Differential methylation of these three GFI1 CpGs explained 12-19% of the 202 g lower birthweight in smoking mothers. Functional enrichment analysis pointed towards activation of cell-mediated immunity. Conclusions: Maternal smoking during pregnancy was associated with cord blood methylation differences. We observed a potentially mediating role of methylation in the association between maternal smoking during pregnancy and birthweight of the offspring. Functional network analysis suggested a role in activating the immune system

Glycosylation of immunoglobulin G is regulated by a large network of genes pleiotropic with inflammatory diseases

Effector functions of immunoglobulin G (IgG) are regulated by the composition of a glycan moiety, thus affecting activity of the immune system. Aberrant glycosylation of IgG has been observed in many diseases, but little is understood about the underlying mechanisms. We performed a genome-wide association study of IgG N-glycosylation (N = 8090) and, using a data-driven network approach, suggested how associated loci form a functional network. We confirmed in vitro that knockdown of IKZF1 decreases the expression of fucosyltransferase FUT8, resulting in increased levels of fucosylated glycans, and suggest that RUNX1 and RUNX3, together with SMARCB1, regulate expression of glycosyltransferase MGAT3. We also show that variants affecting the expression of genes involved in the regulation of glycoenzymes colocalize with variants affecting risk for inflammatory diseases. This study provides new evidence that variation in key transcription factors coupled with regulatory variation in glycogenes modifies IgG glycosylation and has influence on inflammatory diseases.Molecular Epidemiolog

Naval And Military Series

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