On algebraic structures of numerical integration on vector spaces and manifolds

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.Comment: 42 pages, final versio

On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

Hopf algebras in dynamical systems theory

The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota-Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus we seek an algebraic theory of integration, with the Rota-Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson-Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota-Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations.Comment: International Journal of Geometric Methods in Modern Physics, in pres

Using Qualitative Evidence in Decision Making for Health and Social Interventions: An Approach to Assess Confidence in Findings from Qualitative Evidence Syntheses (GRADE-CERQual)

Published onlineJournal ArticleResearch Support, Non-U.S. Gov'tThis is the final version of the article. Available from Public Library of Science via the DOI in this record.Simon Lewin and colleagues present a methodology for increasing transparency and confidence in qualitative research synthesis.This work was supported by funding from the Department of Reproductive Health and Research, WHO (www.who.int/reproductivehealth/about_us/en/) and Norad (Norwegian Agency for Development Cooperation: www.norad.no) to the Norwegian Knowledge Centre for the Health Services. Additional funding for several of the pilot reviews was provided by the Alliance for Health Policy and Systems Research (www.who.int/alliance-hpsr/en/). We also received funding for elements of this work through the Cochrane supported "Methodological Investigation of Cochrane reviews of Complex Interventions" (MICCI) project (www.cochrane.org). SL is supported by funding from the South African Medical Research Council (www.mrc.ac.za). The funders had no role in study design, data collection and analysis, preparation of the manuscript or the decision to publish

Cohort Profile: Pregnancy And Childhood Epigenetics (PACE) Consortium.

Development Psychopathology in context: famil

Epigenome-wide meta-analysis of blood DNA methylation in newborns and children identifies numerous loci related to gestational age

Background Preterm birth and shorter duration of pregnancy are associated with increased morbidity in neonatal and later life. As the epigenome is known to have an important role during fetal development, we investigated associations between gestational age and blood DNA methylation in children. Methods We performed meta-analysis of Illumina's HumanMethylation450-array associations between gestational age and cord blood DNA methylation in 3648 newborns from 17 cohorts without common pregnancy complications, induced delivery or caesarean section. We also explored associations of gestational age with DNA methylation measured at 4-18 years in additional pediatric cohorts. Follow-up analyses of DNA methylation and gene expression correlations were performed in cord blood. DNA methylation profiles were also explored in tissues relevant for gestational age health effects: fetal brain and lung. Results We identified 8899 CpGs in cord blood that were associated with gestational age (range 27-42 weeks), at Bonferroni significance, P < 1.06 x 10(- 7), of which 3343 were novel. These were annotated to 4966 genes. After restricting findings to at least three significant adjacent CpGs, we identified 1276 CpGs annotated to 325 genes. Results were generally consistent when analyses were restricted to term births. Cord blood findings tended not to persist into childhood and adolescence. Pathway analyses identified enrichment for biological processes critical to embryonic development. Follow-up of identified genes showed correlations between gestational age and DNA methylation levels in fetal brain and lung tissue, as well as correlation with expression levels. Conclusions We identified numerous CpGs differentially methylated in relation to gestational age at birth that appear to reflect fetal developmental processes across tissues. These findings may contribute to understanding mechanisms linking gestational age to health effects