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

One-site density matrix renormalization group and alternating minimum energy algorithm

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT) / matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201

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

Precipitable Water Comparisons Over Ghana using PPP Techniques and Reanalysis Data

Atmospheric Water vapor is an important greenhouse gas and contributes greatly in maintaining the Earth’s energy balance. This critical meteorological parameter is not being sensed by any of the 22 synoptic weather stations in Ghana. This study presents a highly precise tool for water vapor sensing based on the concept Global Navigation Satellite Systems (GNSS) meteorology and tests the computed results against global reanalysis data. Conventional approaches used to sense the atmospheric water vapor or Precipitable Water (PW) such as radiosondes, hygrometers, microwave radiometers or sun photometers are expensive and have coverage and temporal limitations. Whereas GNSS meteorological concept offers an easier, inexpensive and all-weather technique to retrieve PW or Integrated Water Vapor (IWV) from zenith tropospheric delays (ZTD) over a reference station. This study employed precise point positioning (PPP) techniques to quantify the extend of delays on the signal due to the troposphere and stratosphere where atmospheric water vapor resides. Stringent processing criteria were set using an elevation cut-off of 5 degrees, precise orbital and clock products were used as well as nominal tropospheric corrections and mapping functions implemented. The delays which are originally slanted are mapped unto the zenith direction and integrated with surface meteorological parameters to retrieve PW or IWV. The gLAB software, Canadian Spatial Reference System (CSRS) and Automatic Precise Positioning Service (APPS) online PPP services were the approaches used to compute ZTD. PW values obtained were compared with Japanese Metro Agency Reanalysis (JRA), European Centre for Medium-Range Weather Forecasts Reanalysis (ERA-interim) and National Center for Environmental Prediction (NCEP) global reanalysis data. Correlation analysis were run on the logged station data using the three approaches and global reanalysis data. The obtained results show stronger correlation between the retrieved PW values and those provided by the ERA-interim. Finally, the study results indicate that with a more densified network of GNSS base stations the retrieved PW or IWV will greatly improve numerical weather predictions in Ghana.Keywords: GNSS Signals, PPP, Integrated Water vapour, Precipitable Water, Reanalysis Model

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

Selective cancer cell toxicity and radiosensitization using different high atomic number nanoparticles

Radiotherapy is currently used in around 50% of cancer treatments. Although it is generally effective, it is damaging to surrounding healthy tissues. This damage can be reduced by by better targeting to cancer cells. Improved radiotherapy outcomes can be achieved by targeting heavy elements to cancer cells, since these produce energetic electrons and free radicals upon irradiation that further increase the effectiveness of radiotherapy. Because of their biocompatibility and amenability to surface modification, gold nanoparticles (AuNPs) show significant promise in this area.We observe that acute 3hr exposure of cells to 2nm gold nanoparticles, bearing a 50:50 ratio of alpha-galactose and PEGamine ligands, results in selective chemotoxicity toward cancer cells at nanomolar concentrations, without affecting normal cells (Figure 1). Chemotoxicity is prevented by antioxidant co-administration and is partially prevented by a pan-caspase inhibitor (ZVAD- fmk). Our initial results suggest that AuNP toxicity is additive with using X-ray radiotherapy. Cerium oxide NPs are one of the most studied metal oxides NPs due to their high redox and oxygen transport properties [1]. They are interesting for radiotherapy applications because they adsorb oxygen and may therefore favour the radiolytic production of oxygen-containing free radicals, especially under low-oxygen conditions, such as the environment of most solid tumours. The surface oxygen vacancy of ceria NPs therefore makes them an interesting tool for prooxidant properties [2,3]. Nevertheless, the potential of ceria as a radiosensitiser has not been investigated extensively as it has a relatively low atomic number. It will therefore produce a low flux of secondary electrons upon irradiation which could potentially displace the adsorbed oxygen. We have been working on the fabrication of mixed ceria NPs, incorporating gold or bismuth, to increase the flux of secondary electrons (Figure 2). It is hoped these effects combined will further facilate the localized generation of ROS and increase effectiveness of targetted radiotherapy We predict that these new mixed-oxide ceria NPs will have better radiosensitisation potential than pure ceria NPs, and could be used to effectively target hypoxic cancer cells

Incorporation of pollen data in source maps is vital for pollen dispersion models

Information about distribution of pollen sources, i.e. their presence and abundance in a specific region, is important especially when atmospheric transport models are applied to forecast pollen concentrations. The goal of this study is to evaluate three pollen source maps using an atmospheric transport model and study the effect on the model results by combining these source maps with pollen data. Here we evaluate three maps for the birch taxon: (1) a map derived by combining land cover data and forest inventory; (2) a map obtained from land cover data and calibrated using model simulations and pollen observations; (3) a statistical map resulting from analysis of forest inventory and forest plot data. The maps were introduced to the Enviro-HIRLAM (Environment – High Resolution Limited Area Model) as input data to simulate birch pollen concentrations over Europe for the birch pollen season 2006. 18 model runs were performed using each of the selected maps in turn with and without calibration with observed pollen data from 2006. The model results were compared with the pollen observation data at 12 measurement sites located in Finland, Denmark and Russia.We show that calibration of the maps using pollen observations significantly improved the model performance for all three maps. The findings also indicate the large sensitivity of the model results to the source maps and agree well with other studies on birch showing that pollen or hybrid-based source maps provide the best model performance. This study highlights the importance of including pollen data in the production of source maps for pollen dispersion modelling and for exposure studies