A general method to construct invariant PDEs on homogeneous manifolds

Let M = G/H be an (n + 1)-dimensional homogeneous manifold and Jk(n,M) =: Jk be the manifold of k-jets of hypersurfaces of M. The Lie group G acts naturally on each Jk. A G-invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G-invariant hypersurface E of Jk. We describe a general method for constructing such invariant partial differential equations for k>1. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup H(k-1) of the (k-1)-prolonged action of G. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space n+1 and in the conformal space n+1. Our method works under some mild assumptions on the action of G, namely: A1) the group G must have an open orbit in Jk-1, and A2) the stabilizer H(k-1) in G of the fiber Jk → Jk-1 must factorize via the group of translations of the fiber itself

Methyl group dynamics in a confined glass

We present a neutron scattering investigation on methyl group dynamics in glassy toluene confined in mesoporous silicates of different pore sizes. The experimental results have been analysed in terms of a barrier distribution model, such a distribution following from the structural disorder in the glassy state. Confinement results in a strong decreasing of the average rotational barrier in comparison to the bulk state. We have roughly separated the distribution for the confined state in a bulk-like and a surface-like contribution, corresponding to rotors at a distance from the pore wall respectively larger and smaller than the spatial range of the interactions which contribute to the rotational potential for the methyl groups. We have estimated a distance of 7 Amstrong as a lower limit of the interaction range, beyond the typical nearest-neighbour distance between centers-of-mass (4.7 Amstrong).Comment: 5 pages, 3 figures. To be published in European Physical Journal E Direct. Proceedings of the 2nd International Workshop on Dynamics in Confinemen

Traces for star products on the dual of a Lie algebra

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the \ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.Comment: 18 pages, LaTeX2e. Updated reference

The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.Comment: 21 pages, 15 figure

Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

Seabed images from Southern Ocean shelf regions off the northern Antarctic Peninsula and in the southeastern Weddell Sea

Recent advances in underwater imaging technology allow for the gathering of invaluable scientific information on seafloor ecosystems, such as direct in situ views of seabed habitats and quantitative data on the composition, diversity, abundance, and distribution of epibenthic fauna. The imaging approach has been extensively used within the research project DynAMo (Dynamics of Antarctic Marine Shelf Ecosystems) at the Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research Bremerhaven (AWI), which aimed to comparatively assess the pace and quality of the dynamics of Southern Ocean benthos. Within this framework, epibenthic spatial distribution patterns have been comparatively investigated in two regions in the Atlantic sector of the Southern Ocean: the shelf areas off the northern tip of the Antarctic Peninsula, representing a region with above-average warming of surface waters and sea-ice reduction, and the shelves of the eastern Weddell Sea as an example of a stable high-Antarctic marine environment that is not (yet) affected by climate change. The AWI Ocean Floor Observation System (OFOS) was used to collect seabed imagery during two cruises of the German research vessel Polarstern, ANT-XXIX/3 (PS81) to the Antarctic Peninsula from January to March 2013 and ANT-XXXI/2 (PS96) to the Weddell Sea from December 2015 to February 2016. Here, we report on the image and data collections gathered during these cruises. During PS81, OFOS was successfully deployed at a total of 31 stations at water depths between 29 and 784 m. At most stations, series of 500 to 530 pictures ( >  15 000 in total, each depicting a seabed area of approximately 3.45 m2 or 2.3  ×  1.5 m) were taken along transects approximately 3.7 km in length. During PS96, OFOS was used at a total of 13 stations at water depths between 200 and 754 m, yielding series of 110 to 293 photos (2670 in total) along transects 0.9 to 2.6 km in length. All seabed images taken during the two cruises, including metadata, are available from the data publisher PANGAEA via the two persistent identifiers at https://doi.org/10.1594/PANGAEA.872719 (for PS81) and https://doi.org/10.1594/PANGAEA.862097 (for PS96)

On Two Theorems About Symplectic Reflection Algebras

We give a new proof and an improvement of two Theorems of J. Alev, M.A. Farinati, T. Lambre and A.L. Solotar : the first one about Hochschild cohomology spaces of some twisted bimodules of the Weyl algebra W and the second one about Hochschild cohomology spaces of the smash product G * W (G a finite subgroup of SP(2n)), and as an application, we then give a new proof of a Theorem of P. Etingof and V. Ginzburg, which shows that the Symplectic Reflection Algebras are deformations of G * W (and, in fact, all possible ones).Comment: corrected typo

Design of Review Systems – A Strategic Instrument to shape Online Reviewing Behavior and Economic Outcomes

As online reviews play a decisive role in consumers’ purchase decisions, e-commerce platforms are using review systems strategically to obtain a competitive advantage. However, the strategic potential can only be leveraged if the review system is designed appropriately. Research on the design of review systems and the effects of design choices has not yet been summarized or synthesized in a review article. We aim to close this gap by providing a scoping review. In our synthesis we posit that the design of review systems moderates the impact of online reviews on economic outcomes and the factors that drive the formation of reviews. After reviewing current research findings, we identify gaps and provide a research agenda covering three key themes: Design features, environments, and devices