The geometry of optimal control solutions on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Multiple-scattering in radar systems: a review

Although extensively studied within the lidar community, the multiple scattering phenomenon has always been considered a rare curiosity by radar meteorologists. Up to few years ago its appearance has only been associated with two- or three-body-scattering features (e.g. hail flares and mirror images) involving highly reflective surfaces. Recent atmospheric research aimed at better understanding of the water cycle and the role played by clouds and precipitation in affecting the Earth's climate has driven the deployment of high frequency radars in space. Examples are the TRMM 13.5 GHz, the CloudSat 94 GHz, the upcoming EarthCARE 94 GHz, and the GPM dual 13-35 GHz radars. These systems are able to detect the vertical distribution of hydrometeors and thus provide crucial feedbacks for radiation and climate studies. The shift towards higher frequencies increases the sensitivity to hydrometeors, improves the spatial resolution and reduces the size and weight of the radar systems. On the other hand, higher frequency radars are affected by stronger extinction, especially in the presence of large precipitating particles (e.g. raindrops or hail particles), which may eventually drive the signal below the minimum detection threshold. In such circumstances the interpretation of the radar equation via the single scattering approximation may be problematic. Errors will be large when the radiation emitted from the radar after interacting more than once with the medium still contributes substantially to the received power. This is the case if the transport mean-free-path becomes comparable with the instrument footprint (determined by the antenna beam-width and the platform altitude). This situation resembles to what has already been experienced in lidar observations, but with a predominance of wide- versus small-angle scattering events. At millimeter wavelengths, hydrometeors diffuse radiation rather isotropically compared to the visible or near infrared region where scattering is predominantly in the forward direction. A complete understanding of radiation transport modeling and data analysis methods under wide-angle multiple scattering conditions is mandatory for a correct interpretation of echoes observed by space-borne millimeter radars. This paper reviews the status of research in this field. Different numerical techniques currently implemented to account for higher order scattering are reviewed and their weaknesses and strengths highlighted. Examples of simulated radar backscattering profiles are provided with particular emphasis given to situations in which the multiple scattering contributions become comparable or overwhelm the single scattering signal. We show evidences of multiple scattering effects from air-borne and from CloudSat observations, i.e. unique signatures which cannot be explained by single scattering theory. Ideas how to identify and tackle the multiple scattering effects are discussed. Finally perspectives and suggestions for future work are outlined. This work represents a reference-guide for studies focused at modeling the radiation transport and at interpreting data from high frequency space-borne radar systems that probe highly opaque scattering media such as thick ice clouds or precipitating clouds.

Small-angle x-ray scattering study of flow alignment of a thermotropic liquid crystal in the nematic and smectic phases

WThe capillary flow alignment of the thermotropic liquid crystal 4-n-octyl-4′-cyanobiphenyl in the nematic and smectic phases is investigated using time-resolved synchrotron small-angle x-ray scattering. Samples were cooled from the isotropic phase to erase prior orientation. Upon cooling through the nematic phase under Poiseuille flow in a circular capillary, a transition from the alignment of mesogens along the flow direction to the alignment of layers along the flow direction (mesogens perpendicular to flow) appears to occur continuously at the cooling rate applied. The transition is centered on a temperature at which the Leslie viscosity coefficient α3 changes sign. The configuration with layers aligned along the flow direction is also observed in the smectic phase. The transition in the nematic phase on cooling has previously been ascribed to an aligning-nonaligning or tumbling transition. At high flow rates there is evidence for tumbling around an average alignment of layers along the flow direction. At lower flow rates this orientation is more clearly defined. The layer alignment is ascribed to surface-induced ordering propagating into the bulk of the capillary, an observation supported by the parallel alignment of layers observed for a static sample at low temperatures in the nematic phase.

The Structure of the chiral Pt{531} surface: a combined LEED and DFT study

The structure of the chiral kinked Pt{531} surface has been determined by low-energy electron diffraction intensity-versus-energy (LEED-IV) analysis and density functional theory (DFT). Large contractions and expansions of the vertical interlayer distances with respect to the bulk-terminated surface geometry were found for the first six layers (LEED: d(12) = 0.44 angstrom, d(23) = 0.69 angstrom, d(34) = 0.49 angstrom, d(45) = 0.95 angstrom, d(56) = 0.56 angstrom; DFT: d(12) = 0.51 angstrom, d(23) = 0.55 angstrom, d(34) = 0.74 angstrom, d(45) = 0.78 angstrom, d(56) = 0.63 angstrom; d(bulk) = 0.66 angstrom). Energy-dependent cancellations of LEED spots over unusually large energy ranges, up to 100 eV, can be explained by surface roughness and reproduced by applying a model involving 0.25 ML of vacancies and adatoms in the scattering calculations. The agreement between the results from LEED and DFT is not as good as in other cases, which could be due to this roughness of the real surface.

The surface geometry of carbonmonoxide and hydrogen co-adsorbed on Ni 111

A combination of photoelectron spectroscopy, temperature programmed desorption and low energy electron diffraction structure determinations have been applied to study the p(2 x 2) structures of pure hydrogen and co-adsorbed hydrogen and CO on Ni {111}. In agreement with earlier work atomic hydrogen is found to adsorb on fcc and hcp sites in the pure layer with H-Ni bond lengths of 1.74Angstrom. The substrate interlayer distances, d(12) = 2.05Angstrom and d(23) = 2.06Angstrom, are expanded with respect to clean Ni {111} with buckling of 0.04Angstrom in the first layer. In the co-adsorbed phase Co occupies hcp sites and only the hydrogen atoms on fcc sites remain on the surface. d(12) is even further expanded to 2.08Angstrom with buckling in the first and second layer of 0.06 and 0.02Angstrom, respectively. The C-O, C-Ni, and H-Ni bond lengths are within the range of values also found for the pure adsorbates.

The interplay between geometry, electronic structure, and reactivity of Cu-Ni bimetallic (111) surfaces

The mutual influence of surface geometry (e.g. lattice parameters, morphology) and electronic structure is discussed for Cu-Ni bimetallic (111) surfaces. It is found that on flat surfaces the electronic d-states of the adlayer experience very little influence from the substrate electronic structure which is due to their large separation in binding energies and the close match of Cu and Ni lattice constants. Using carbon monoxide and benzene as probe molecules, it is found that in most cases the reactivity of Cu or Ni adlayers is very similar to the corresponding (111) single crystal surfaces. Exceptions are the adsorption of CO on submonolayers of Cu on Ni(111) and the dissociation of benzene on Ni/Cu(111) which is very different from Ni(111). These differences are related to geometric factors influencing the adsorption on these surfaces.

Adsorption and dissociation of benzene on bimetallic surfaces - the influence of surface geometry and electronic structure

This topical review discusses the influence of the surface geometry (e.g. lattice parameters and termination) and electronic structure of well-defined bimetallic surfaces on the adsorption and dissociation of benzene. The available data can be divided into two categories with combinations of non-transition metals and transition metals on the one side and combinations of two transition metals on the other. The main effect of non-transition metals in surface alloys is site blocking which can suppress chemisorption and dissociation of the molecules completely. When two transition metals are combined, the effects are less dramatic. They mainly affect the strength of the chemisorption bond and the degree of dissociation due to electronic and template effects.

The surface geometry of carbon monoxide and oxygen co-adsorbed on Ni{111}

Low energy electron diffraction (LEED) structure determinations have been performed for the p(2 x 2) structures of pure oxygen and oxygen co-adsorbed with CO on Ni{111}. Optimisation of the non-geometric parameters led to very good agreement between experimental and theoretical IV-curves and hence to a high accuracy in the structural parameters. In agreement with earlier work atomic oxygen is found to adsorb on fee sites in both structures. In the co-adsorbed phase CO occupies atop sites. The positions of the substrate atoms are almost identical, within 0.02 Angstrom, in both structures, implying that the interaction with oxygen dominates the arrangement of Ni atoms at the surface.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

High frequency scattering by convex curvilinear polygons

We consider the scattering of a time-harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.

The adsorption geometry and chemical state of lysine on Cu{110}

Chemisorbed layers of lysine adsorbed on Cu{110} have been studied using X-ray photoelectron spectroscopy (XPS) and near-edge X-ray absorption fine structure (NEXAFS) spectroscopy. XPS indicates that the majority (70%) of the molecules in the saturated layer at room temperature (coverage 0.27 ML) are in their zwitterionic state with no preferential molecular orientation. After annealing to 420 K a less densely packed layer is formed (0.14 ML), which shows a strong angular dependence in the characteristic π-resonance of oxygen K edge NEXAFS and no indication of zwitterions in XPS. These experimental results are best compatible with molecules bound to the substrate through the oxygen atoms of the (deprotonated) carboxylate group and the two amino groups involving Cu atoms in three different close packed rows. This μ4 bonding arrangement with an additional bond through the !-amino group is different from geometries previously suggested for lysine on Cu{110}.