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

A serological survey of ruminant livestock in Kazakhstan during post-Soviet transitions in farming and disease control

The results of a serological survey of livestock in Kazakhstan, carried out in 1997–1998, are reported. Serum samples from 958 animals (cattle, sheep and goats) were tested for antibodies to foot and mouth disease (FMD), bluetongue (BT), epizootic haemorrhagic disease (EHD), rinderpest (RP) and peste des petits ruminants (PPR) viruses, and to Brucella spp. We also investigated the vaccination status of livestock and related this to changes in veterinary provision since independence in 1991. For the 2 diseases under official surveillance (FMD and brucellosis) our results were similar to official data, although we found significantly higher brucellosis levels in 2 districts and widespread ignorance about FMD vaccination status. The seroprevalence for BT virus was 23%, and seropositive animals were widespread suggesting endemicity, despite the disease not having being previously reported. We found a few seropositives for EHDV and PPRV, which may suggest that these diseases are also present in Kazakhstan. An hierarchical model showed that seroprevalence to FMD and BT viruses were clustered at the farm/village level, rather than at a larger spatial scale. This was unexpected for FMD, which is subject to vaccination policies which vary at the raion (county) level

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. 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

Bio-psycho-social factors’ associations with brain age: a large-scale UK Biobank diffusion study of 35,749 participants

Brain age refers to age predicted by brain features. Brain age has previously been associated with various health and disease outcomes and suggested as a potential biomarker of general health. Few previous studies have systematically assessed brain age variability derived from single and multi-shell diffusion magnetic resonance imaging data. Here, we present multivariate models of brain age derived from various diffusion approaches and how they relate to bio-psycho-social variables within the domains of sociodemographic, cognitive, life-satisfaction, as well as health and lifestyle factors in midlife to old age (N = 35,749, 44.6–82.8 years of age). Bio-psycho-social factors could uniquely explain a small proportion of the brain age variance, in a similar pattern across diffusion approaches: cognitive scores, life satisfaction, health and lifestyle factors adding to the variance explained, but not socio-demographics. Consistent brain age associations across models were found for waist-to-hip ratio, diabetes, hypertension, smoking, matrix puzzles solving, and job and health satisfaction and perception. Furthermore, we found large variability in sex and ethnicity group differences in brain age. Our results show that brain age cannot be sufficiently explained by bio-psycho-social variables alone. However, the observed associations suggest to adjust for sex, ethnicity, cognitive factors, as well as health and lifestyle factors, and to observe bio-psycho-social factor interactions’ influence on brain age in future studies.publishedVersio

Dose response of the 16p11.2 distal copy number variant on intracranial volume and basal ganglia.

Carriers of large recurrent copy number variants (CNVs) have a higher risk of developing neurodevelopmental disorders. The 16p11.2 distal CNV predisposes carriers to e.g., autism spectrum disorder and schizophrenia. We compared subcortical brain volumes of 12 16p11.2 distal deletion and 12 duplication carriers to 6882 non-carriers from the large-scale brain Magnetic Resonance Imaging collaboration, ENIGMA-CNV. After stringent CNV calling procedures, and standardized FreeSurfer image analysis, we found negative dose-response associations with copy number on intracranial volume and on regional caudate, pallidum and putamen volumes (β = -0.71 to -1.37; P < 0.0005). In an independent sample, consistent results were obtained, with significant effects in the pallidum (β = -0.95, P = 0.0042). The two data sets combined showed significant negative dose-response for the accumbens, caudate, pallidum, putamen and ICV (P = 0.0032, 8.9 × 10-6, 1.7 × 10-9, 3.5 × 10-12 and 1.0 × 10-4, respectively). Full scale IQ was lower in both deletion and duplication carriers compared to non-carriers. This is the first brain MRI study of the impact of the 16p11.2 distal CNV, and we demonstrate a specific effect on subcortical brain structures, suggesting a neuropathological pattern underlying the neurodevelopmental syndromes

Seroprevalence of infectious diseases in saiga antelope (Saiga tataricatatarica) in Kazakhstan 2012–2014

286 serum samples were collected from three sub-populations of saiga in Kazakhstan (Betpakdala, Ustyurt and Volga-Ural) between 2012 and 2014, and were tested for the presence of antibodies to Brucella spp., bluetongue virus, peste des petits ruminants (PPR) virus, Akabane virus, Schmallenberg virus, Chlamydophila, Toxoplasma, Mycobacterium avium subspecies paratuberculosis and Coxiella burnetii (Q Fever). Seropositives to Coxiella burnetii of saiga were detected and the adjusted seroprevalence of Q Fever antibodies was 0.07 (95% confidence interval (CI): 0.03-0.10). Seropositives to Akabane virus were detected in all three populations and the adjusted seroprevalence values for this virus were very high (all were>0.13). Lower adjusted seroprevalence values were estimated for PPR Virus and Mycobacterium avium subsp. paratuberculosis (0.005 and 0.006). No seropositives for bluetongue, Toxoplasma, Brucella or Schmallenberg were detected