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

The cleavage surface of the BaFe_(2-x)Co_(x)As_(2) and Fe_(y)Se_(1-x)Te_(x) superconductors: from diversity to simplicity

We elucidate the termination surface of cleaved single crystals of the BaFe_(2-x)Co_(x)As_(2) and Fe_(y)Se_(1-x)Te_(x) families of the high temperature iron based superconductors. By combining scanning tunneling microscopic data with low energy electron diffraction we prove that the termination layer of the Ba122 systems is a remnant of the Ba layer, which exhibits a complex diversity of ordered and disordered structures. The observed surface topographies and their accompanying superstructure reflections in electron diffraction depend on the cleavage temperature. In stark contrast, Fe_(y)Se_(1-x)Te_(x) possesses only a single termination structure - that of the tetragonally ordered Se_(1-x)Te_(x) layer.Comment: 4 pages, 4 figure

Semi-Lagrangian methods in air pollution models

Various semi-Lagrangian methods are tested with respect to advection in air pollution modeling. The aim is to find a method fulfilling as many of the desirable properties by Rasch andWilliamson (1990) and Machenhauer et al. (2008) as possible. The focus in this study is on accuracy and local mass conservation. <br><br> The methods tested are, first, classical semi-Lagrangian cubic interpolation, see e.g. Durran (1999), second, semi-Lagrangian cubic cascade interpolation, by Nair et al. (2002), third, semi-Lagrangian cubic interpolation with the modified interpolation weights, Locally Mass Conserving Semi-Lagrangian (LMCSL), by Kaas (2008), and last, semi-Lagrangian cubic interpolation with a locally mass conserving monotonic filter by Kaas and Nielsen (2010). <br><br> Semi-Lagrangian (SL) interpolation is a classical method for atmospheric modeling, cascade interpolation is more efficient computationally, modified interpolation weights assure mass conservation and the locally mass conserving monotonic filter imposes monotonicity. <br><br> All schemes are tested with advection alone or with advection and chemistry together under both typical rural and urban conditions using different temporal and spatial resolution. The methods are compared with a current state-of-the-art scheme, Accurate Space Derivatives (ASD), see Frohn et al. (2002), presently used at the National Environmental Research Institute (NERI) in Denmark. To enable a consistent comparison only non-divergent flow configurations are tested. <br><br> The test cases are based either on the traditional slotted cylinder or the rotating cone, where the schemes' ability to model both steep gradients and slopes are challenged. <br><br> The tests showed that the locally mass conserving monotonic filter improved the results significantly for some of the test cases, however, not for all. It was found that the semi-Lagrangian schemes, in almost every case, were not able to outperform the current ASD scheme used in DEHM with respect to accuracy

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

Fingerprint-enhanced capacitive-piezoelectric flexible sensing skin to discriminate static and dynamic tactile stimuli

nspired by the structure and functions of the human skin, a highly sensitive capacitive‐piezoelectric flexible sensing skin with fingerprint‐like patterns to detect and discriminate between spatiotemporal tactile stimuli including static and dynamic pressures and textures is presented. The capacitive‐piezoelectric tandem sensing structure is embedded in the phalange of a 3D‐printed robotic hand, and a tempotron classifier system is used for tactile exploration. The dynamic tactile sensor, interfaced with an extended gate configuration to a common source metal oxide semiconductor field effect transistor (MOSFET), exhibits a sensitivity of 2.28 kPa−1. The capacitive sensing structure has nonlinear characteristics with sensitivity varying from 0.25 kPa−1 in the low‐pressure range (<100 Pa) to 0.002 kPa−1 in high pressure (≈2.5 kPa). The output from the presented sensor under a closed‐loop tactile scan, carried out with an industrial robotic arm, is used as latency‐coded spike trains in a spiking neural network (SNN) tempotron classifier system. With the capability of performing a real‐time binary naturalistic texture classification with a maximum accuracy of 99.45%, the presented bioinspired skin finds applications in robotics, prosthesis, wearable sensors, and medical devices

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&

A deep optical/near-infrared catalog of Serpens

We present a deep optical/near-infrared imaging survey of the Serpens molecular cloud. This survey constitutes the complementary optical data to the Spitzer "Core To Disk" (c2d) Legacy survey in this cloud. The survey was conducted using the Wide Field Camera at the Isaac Newton Telescope. About 0.96 square degrees were imaged in the R and Z filters, covering the entire region where most of the young stellar objects identified by the c2d survey are located. 26524 point-like sources were detected in both R and Z bands down to R=24.5 mag and Z=23 mag with a signal-to-noise ratio better than 3. The 95% completeness limit of our catalog corresponds to 0.04 solar masses for members of the Serpens star forming region (age 2 Myr and distance 260 pc) in the absence of extinction. Adopting the typical extinction of the observed area (Av=7 mag), we estimate a 95% completeness level down to 0.1 solar masses. The astrometric accuracy of our catalog is 0.4 arcsec with respect to the 2MASS catalog. Our final catalog contains J2000 celestial coordinates, magnitudes in the R and Z bands calibrated to the SDSS photometric system and, where possible, JHK magnitudes from 2MASS for sources in 0.96 square degrees in the direction of Serpens. This data product has been already used within the frame of the c2d Spitzer Legacy Project analysis in Serpens to study the star/disk formation and evolution in this cloud; here we use it to obtain new indications of the disk-less population in Serpens.Comment: 7 page, 5 figure

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