Geodesics on Lie groups: Euler equations and totally geodesic subgroup

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler-Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given. The setting works both in the classical nite dimensional case, and in the category of in nite dimensional Fr echet Lie groups, in which di eomorphism groups are included. Using the framework we give new examples of both nite and in nite dimensional totally geodesic subgroups. In particular, based on the cross helicity, we construct right invariant metrics such that a given subgroup of exact volume preserving di eomorphisms is totally geodesic. The paper also gives a general framework for the representation of Euler-Arnold equations in arbitrary choice of dual pairing

Semi-invariant Riemannian metrics in hydrodynamics

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev Hk-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid’s internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted

Symplectic integrators for index one constraints

We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.Comment: 13 pages, accepted in SIAM J Sci Compu

Kvalitet i nya deponiers lakvatten - resultat från Renovas deponi Fläskebo

The European landfill directive increases demands on landfills. The amount of organic matter being landfilled is restricted and landfills should be as dry as possible. The consequences of these changes are not yet fully understood. In this study the first landfill in Sweden to be constructed according to the directive, Fläskebo, has served as an example of future landfills. Its leachate has been studied with multivariate techniques (principal component analysis and canonical correlation analysis) and with the geochemical model PHREEQC. Several interesting correlations were found and the results clearly show that multivariate statistics can be of great use when studying landfill leachate. Their ability to extract the most important information is one thing that can be very useful. Combined with geochemical models they can provide an increased understanding of processes governing leachate quality. The next part of the study will be to use multivariate statistics to compare Fläskebo’s leachate with that from older landfills to find interesting differences and similarities

Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the symplectic structure in the case $\epsilon=0$. In the case $0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order

Diffeomorphic random sampling using optimal information transport

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.Comment: 8 pages, 3 figure

Modeling and characterization of the morphology of multiphase polymeric nanoparticles

Multiphase polymeric nanoparticles that synergistically combine the properties of their constituents present enhanced properties and display new functionalities. Therefore, they are used in a wide range of applications including anticorrosive, superhydrophobic and anti-molding coatings; switchable adhesives; photoswitchable fluorescent particles; energy storage; gene and drug delivery; anticounterfeiting and LEDs. Although it is recognized that application properties strongly depend on the morphology of the nanoparticles, there is a surprising lack of progress towards the knowledge-based synthesis of these materials with well controlled morphologies. There are two main reasons for this. Firstly, the difficulties associated to the accurate characterization of the morphology of the polymeric nanoparticles, and secondly, the lack of quantitative understanding of the processes controlling the morphology. Please click Additional Files below to see the full abstrac

Geometric Generalisations of SHAKE and RATTLE

A geometric analysis of the Shake and Rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises Shake and Rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for Shake and Rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given

Hierarchical organization and molecular diffusion in gold nanorod/silica supercrystal nanocomposites

Hierarchical organization of gold nanorods was previously obtained on a substrate, allowing precise control over the morphology of the assemblies and macroscale spatial arrangement. Herein, a thorough description of these gold nanorod assemblies and their orientation within supercrystals is presented together with a sol?gel technique to protect the supercrystals with mesoporous silica films. The internal organization of the nanorods in the supercrystals was characterized by combining focused ion beam ablation and scanning electron microscopy. A mesoporous silica layer is grown both over the supercrystals and between the individual lamellae of gold nanorods inside the structure. This not only prevented the detachment of the supercrystal from the substrate in water, but also allowed small molecule analytes to infiltrate the structure. These nanocomposite substrates show superior Raman enhancement in comparison with gold supercrystals without silica owing to improved accessibility of the plasmonic hot spots to analytes. The patterned supercrystal arrays with enhanced optical and mechanical properties obtained in this work show potential for the practical implementation of nanostructured devices in spatially resolved ultradetection of biomarkers and other analytes.This work was supported by the European Research Council (ERC Advanced Grant #267867 Plasmaquo) and the European Union's Seventh Framework Programme (FP7/2007–2013 under grant agreement no. 312184, SACS). A. C. and E. M. acknowledge financial support from the FP7-PEOPLE-2011-IRSES N295180 (MagNonMag) project and the "International Projects for Scientific Cooperation" Program of FEFU. The authors are grateful to Prof. Jan Vermant (ETH Zurich) for useful discussions