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

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

Protostellar clusters in intermediate-mass (IM) star forming regions

The transition between the low density groups of T Tauri stars and the high density clusters around massive stars occurs in the intermediate-mass (IM) range (M$_*$$\sim$2--8 M$_\odot$). High spatial resolution studies of IM young stellar objects (YSO) can provide important clues to understand the clustering in massive star forming regions. Aims: Our aim is to search for clustering in IM Class 0 protostars. The high spatial resolution and sensitivity provided by the new A configuration of the Plateau de Bure Interferometer (PdBI) allow us to study the clustering in these nearby objects. Methods: We have imaged three IM Class 0 protostars (Serpens-FIRS 1, IC 1396 N, CB 3) in the continuum at 3.3 and 1.3mm using the PdBI. The sources have been selected with different luminosity to investigate the dependence of the clustering process on the luminosity of the source. Results: Only one millimeter (mm) source is detected towards the low luminosity source Serpens--FIRS 1. Towards CB 3 and IC1396 N, we detect two compact sources separated by $\sim$0.05 pc. The 1.3mm image of IC 1396 N, which provides the highest spatial resolution, reveal that one of these cores is splitted in, at least, three individual sources.Comment: 4 pages, 3 figures, accepted for publication in Astronomy and Astrophysics Letters (Special Feature IRAM/PdB

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

Spin and Statistics and First Principles

It was shown in the early Seventies that, in Local Quantum Theory (that is the most general formulation of Quantum Field Theory, if we leave out only the unknown scenario of Quantum Gravity) the notion of Statistics can be grounded solely on the local observable quantities (without assuming neither the commutation relations nor even the existence of unobservable charged field operators); one finds that only the well known (para)statistics of Bose/Fermi type are allowed by the key principle of local commutativity of observables. In this frame it was possible to formulate and prove the Spin and Statistics Theorem purely on the basis of First Principles. In a subsequent stage it has been possible to prove the existence of a unique, canonical algebra of local field operators obeying ordinary Bose/Fermi commutation relations at spacelike separations. In this general guise the Spin - Statistics Theorem applies to Theories (on the four dimensional Minkowski space) where only massive particles with finite mass degeneracy can occur. Here we describe the underlying simple basic ideas, and briefly mention the subsequent generalisations; eventually we comment on the possible validity of the Spin - Statistics Theorem in presence of massless particles, or of violations of locality as expected in Quantum Gravity.Comment: Survey based on a talk given at the Meeting on "Theoretical and experimental aspects of the spin - statistics connection and related symmetries", Trieste, Italy - October 21-25, 200

Overview of (pro-)Lie group structures on Hopf algebra character groups

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai

Looking at the bright side of the ρ Ophiuchi dark cloud. Far infrared spectrophotometric observations of the ρ Oph cloud with the ISO-LWS

We present far infrared (45-195 μm) spectrophotometric observations with the ISO-LWS of the active star forming ρOph main cloud (L 1688). The [CII] 158 μm and [OI] 63 μm lines were detected at each of the 33 positions observed, whereas the [OI] 145 μm line was clearly seen toward twelve. The principal observational result is that the [CII] 158 μm line fluxes exhibit a clear correlation with projected distance from the dominant stellar source in the field (HD 147889). We interpret this in terms of PDR-type emission from the surface layers of the ρOph cloud. The observed [CII] 158 μm/[OI] 63 μm flux ratios are larger than unity everywhere. A comparison of the [CII] 158 μm line emission and the FIR dust continuum fluxes yields estimates of the efficiency at which the gas in the cloud converts stellar to [CII] 158 μm photons (χCII≳0.5%). We first develop an empirical model, which provides us with a three dimensional view of the far and bright side of the dark ρOph cloud, showing that the cloud surface towards the putative energy source is concave. This model also yields quantitative estimates of the incident flux of ultraviolet radiation (G0 ~ 101 - 102) and of the degree of clumpiness/texture of the cloud surface (filling of the 80" beam ~0.2). Subsequently, we use theoretical models of PDRS to derive the particle density, n(H), and the temperature structures, for Tgas and Tdust, in the surface layers of the ρOph cloud. Tgas is relatively low, ~60 K, but higher than Tdust ( ~30 K), and densities are generally found within the interval (1-3) 104 cm-3 . These PDR models are moderately successful in explaining the LWS observations. They correctly predict the [OI] 63 μm and [CII] 158 μm line intensities and the observed absence of any molecular line emission. The models do fail, however, to reproduce the observed small [OI] 63 μm/[OI] 145 μm ratios. We examine several possible explanations, but are unable to uniquely identify (or to disentangle) the cause(s) of this discrepancy. From pressure equilibrium arguments we infer that the total mass of the ρOph main cloud (2pc2) is ~2 500 M⊙, which implies that the star formation efficiency to date is ≲4%, significantly lower than previous estimates