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

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

Cellular Scaling Rules of Insectivore Brains

Insectivores represent extremes in mammalian body size and brain size, retaining various “primitive” morphological characteristics, and some species of Insectivora are thought to share similarities with small-bodied ancestral eutherians. This raises the possibility that insectivore brains differ from other taxa, including rodents and primates, in cellular scaling properties. Here we examine the cellular scaling rules for insectivore brains and demonstrate that insectivore scaling rules overlap somewhat with those for rodents and primates such that the insectivore cortex shares scaling rules with rodents (increasing faster in size than in numbers of neurons), but the insectivore cerebellum shares scaling rules with primates (increasing isometrically). Brain structures pooled as “remaining areas” appear to scale similarly across all three mammalian orders with respect to numbers of neurons, and the numbers of non-neurons appear to scale similarly across all brain structures for all three orders. Therefore, common scaling rules exist, to different extents, between insectivore, rodent, and primate brain regions, and it is hypothesized that insectivores represent the common aspects of each order. The olfactory bulbs of insectivores, however, offer a noteworthy exception in that neuronal density increases linearly with increasing structure mass. This implies that the average neuronal cell size decreases with increasing olfactory bulb mass in order to accommodate greater neuronal density, and represents the first documentation of a brain structure gaining neurons at a greater rate than mass. This might allow insectivore brains to concentrate more neurons within the olfactory bulbs without a prohibitively large and metabolically costly increase in structure mass

Continuous and discrete Clebsch variational principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics

A new class of IMP dehydrogenase with a role in self-resistance of mycophenolic acid producing fungi

<p>Abstract</p> <p>Background</p> <p>Many secondary metabolites produced by filamentous fungi have potent biological activities, to which the producer organism must be resistant. An example of pharmaceutical interest is mycophenolic acid (MPA), an immunosuppressant molecule produced by several <it>Penicillium </it>species. The target of MPA is inosine-5'-monophosphate dehydrogenase (IMPDH), which catalyses the rate limiting step in the synthesis of guanine nucleotides. The recent discovery of the MPA biosynthetic gene cluster from <it>Penicillium brevicompactum </it>revealed an extra copy of the IMPDH-encoding gene (<it>mpaF</it>) embedded within the cluster. This finding suggests that the key component of MPA self resistance is likely based on the IMPDH encoded by <it>mpaF</it>.</p> <p>Results</p> <p>In accordance with our hypothesis, heterologous expression of <it>mpaF </it>dramatically increased MPA resistance in a model fungus, <it>Aspergillus nidulans</it>, which does not produce MPA. The growth of an <it>A. nidulans </it>strain expressing <it>mpaF </it>was only marginally affected by MPA at concentrations as high as 200 μg/ml. To further substantiate the role of <it>mpaF </it>in MPA resistance, we searched for <it>mpaF </it>orthologs in six MPA producer/non-producer strains from <it>Penicillium </it>subgenus <it>Penicillium</it>. All six strains were found to hold two copies of IMPDH. A cladistic analysis based on the corresponding cDNA sequences revealed a novel group constituting <it>mpaF </it>homologs. Interestingly, a conserved tyrosine residue in the original class of IMPDHs is replaced by a phenylalanine residue in the new IMPDH class.</p> <p>Conclusions</p> <p>We identified a novel variant of the IMPDH-encoding gene in six different strains from <it>Penicillium </it>subgenus <it>Penicillium</it>. The novel IMPDH variant from MPA producer <it>P. brevicompactum </it>was shown to confer a high degree of MPA resistance when expressed in a non-producer fungus. Our study provides a basis for understanding the molecular mechanism of MPA resistance and has relevance for biotechnological and pharmaceutical applications.</p

Wide field CO J = 3->2 mapping of the Serpens Cloud Core

Context. Outflows provide indirect means to get an insight on diverse star formation associated phenomena. On scales of individual protostellar cores, outflows combined with intrinsic core properties can be used to study the mass accretion/ejection process of heavily embedded protostellar sources. Methods. An area comprising 460"x230" of the Serpens cloud core has been mapped in 12 CO J = 3\to 2 with the HARP-B heterodyne array at the James Clerk Maxwell Telescope; J = 3\to 2 observations are more sensitive tracers of hot outflow gas than lower J CO transitions; combined with the high sensitivity of the HARP-B receptors outflows are sharply outlined, enabling their association with individual protostellar cores. Results. Most of ~20 observed outflows are found to be associated with known protostellar sources in bipolar or unipolar configurations. All but two outflow/core pairs in our sample tend to have a projected orientation spanning roughly NW-SE. The overall momentum driven by outflows in Serpens lies between 3.2 and 5.1 x 10^(-1) M\odot km s^(-1), the kinetic energy from 4.3 to 6.7 x 10^(43) erg and momentum flux is between 2.8 and 4.4 x 10^(-4) M\odot km s^(-1) yr^(-1). Bolometric luminosities of protostellar cores based on Spitzer photometry are found up to an order of magnitude lower than previous estimations derived with IRAS/ISO data. Conclusions. We confirm the validity of the existing correlations between the momentum flux and bolometric luminosity of Class I sources for the homogenous sample of Serpens, though we suggest that they should be revised by a shift to lower luminosities. All protostars classified as Class 0 sources stand well above the known Class I correlations, indicating a decline in momentum flux between the two classes.Comment: 15 pages, 10 figures, accepted for publication in A&

Evolution of the polarization of the optical afterglow of the gamma-ray burst GRB 030329

We report 31 polarimetric observations of the afterglow of GRB 030329 with high signal-to-noise and high sampling frequency. We establish the polarization light curve, detect sustained polarization at the percent level, and find significant variability of polarization degree and angle. The data imply that the afterglow magnetic field has small coherence length and is mostly random, probably generated by turbulence.Comment: Nature 426 (13. Nov. 2003), 2 figure

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

Applying GRADE-CERQual to qualitative evidence synthesis findings-paper 7: Understanding the potential impacts of dissemination bias

This is the final version. Available from BMC via the DOI in this record. Additional materials are available on the GRADE-CERQual website (www.cerqual.org).Background: The GRADE-CERQual (Confidence in Evidence from Reviews of Qualitative research) approach has been developed by the GRADE (Grading of Recommendations Assessment, Development and Evaluation) Working Group. The approach has been developed to support the use of findings from qualitative evidence syntheses in decision-making, including guideline development and policy formulation. CERQual includes four components for assessing how much confidence to place in findings from reviews of qualitative research (also referred to as qualitative evidence syntheses): (1) methodological limitations, (2) coherence, (3) adequacy of data and (4) relevance. This paper is part of a series providing guidance on how to apply CERQual and focuses on a probable fifth component, dissemination bias. Given its exploratory nature, we are not yet able to provide guidance on applying this potential component of the CERQual approach. Instead, we focus on how dissemination bias might be conceptualised in the context of qualitative research and the potential impact dissemination bias might have on an overall assessment of confidence in a review finding. We also set out a proposed research agenda in this area. Methods: We developed this paper by gathering feedback from relevant research communities, searching MEDLINE and Web of Science to identify and characterise the existing literature discussing or assessing dissemination bias in qualitative research and its wider implications, developing consensus through project group meetings, and conducting an online survey of the extent, awareness and perceptions of dissemination bias in qualitative research. Results: We have defined dissemination bias in qualitative research as a systematic distortion of the phenomenon of interest due to selective dissemination of studies or individual study findings. Dissemination bias is important for qualitative evidence syntheses as the selective dissemination of qualitative studies and/or study findings may distort our understanding of the phenomena that these syntheses aim to explore and thereby undermine our confidence in these findings. Dissemination bias has been extensively examined in the context of randomised controlled trials and systematic reviews of such studies. The effects of potential dissemination bias are formally considered, as publication bias, within the GRADE approach. However, the issue has received almost no attention in the context of qualitative research. Because of very limited understanding of dissemination bias and its potential impact on review findings in the context of qualitative evidence syntheses, this component is currently not included in the GRADE-CERQual approach. Conclusions: Further research is needed to establish the extent and impacts of dissemination bias in qualitative research and the extent to which dissemination bias needs to be taken into account when we assess how much confidence we have in findings from qualitative evidence syntheses.WHONorad (Norwegian Agency for Development Cooperation)Research Council of NorwayCochrane Methods Innovation FundSouth African Medical Research Counci