Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

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.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

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 E³, the spheres S³ and the hyperboloids H³ 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 is illustrated.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

The diabatic contour advective semi-Lagrangian model

This article describes a novel algorithmic development extending the contour advective semi-Lagrangian model to include nonconservative effects. The Lagrangian contour representation of finescale tracer fields, such as potential vorticity, allows for conservative, nondiffusive treatment of sharp gradients allowing very high numerical Reynolds numbers. It has been widely employed in accurate geostrophic turbulence and tracer advection simulations. In the present, diabatic version of the model the constraint of conservative dynamics is overcome by including a parallel Eulerian field that absorbs the nonconservative ( diabatic) tendencies. The diabatic buildup in this Eulerian field is limited through regular, controlled transfers of this field to the contour representation. This transfer is done with a fast newly developed contouring algorithm. This model has been implemented for several idealized geometries. In this paper a single-layer doubly periodic geometry is used to demonstrate the validity of the model. The present model converges faster than the analogous semi-Lagrangian models at increased resolutions. At the same nominal spatial resolution the new model is 40 times faster than the analogous semi-Lagrangian model. Results of an orographically forced idealized storm track show nontrivial dependency of storm-track statistics on resolution and on the numerical model employed. If this result is more generally applicable, this may have important consequences for future high-resolution climate modeling.

The timing of Quaternary calcrete development in semi-arid southeast Spain: Investigating the role of climate on calcrete genesis

Quaternary-aged calcrete horizons are common weathering products in arid and semi-arid regions. It is, however, unclear how calcrete forming processes respond to the major oscillations in climate that occur over the Quaternary period. This paper presents a U-series-based calcrete age database from the Sorbas basin, southeast Spain. The study constructs an age frequency distribution of these ages which is consequently compared to a range of palaeoenvironmental records from the Mediterranean. The age distribution presented here suggests that the formation of pedogenic calcrete horizons in the Sorbas basin primarily occurs during 'warm' isotope stages (MIS 1 and 5), with very few calcrete ages occurring during cold glacial/stadial stages (MIS 2, 3 and 4). It is suggested that this is a function of the environments that existed during 'warm' isotope stages being more conducive to calcrete development than those that existed during cold climate episodes. In a semi-arid region such as the Sorbas basin it is likely that increased aridity during glacial stages, coupled with reduced vegetation and accelerated landscape instability, was crucial in reducing rates of calcrete formation. In a semi-arid region such as southeast Spain, calcrete formation during the Quaternary, therefore, oscillates with climate change but is primarily a "warm" episode phenomenon. It is suggested that further studies are required to see how calcrete genesis responds to environmental change in more humid parts of the Mediterranean. (C) 2009 Elsevier B.V. All rights reserved.

A semi-quantitative approach to deriving a model structure for the uptake of organic chemicals by vegetation

An expert elicitation exercise was undertaken to determine those components and processes that are most important for modeling plant uptake of organic chemicals. The state of our knowledge of these processes was also assessed. This semi-quantitative analysis allowed the construction of an idealized model with seven compartments; soil bulk, soil water, roots, stem, leaves, fruit, and air. Three main areas were identified further research: 1) the uptake of organic chemicals by fruit; 2) the internal transfer of organic chemicals between plant structures (e.g., stem and leaves); and 3) the transfer via the soil-air-plant pathway. Until new data becomes available to quantify these processes, it is proposed that an equilibrium partitioning approach is used between plant components other than fruit or that models consist of both an edible and inedible compartment.

Effects of manure and fertilizer on grain yield, soil carbon and phosphorus in a 13-year field trial in semi-arid Kenya

Long-term indicators of soil fertility were assessed by measuring grain yield, soil organic carbon (SOC) and soil Olsen phosphorous for a P-deficient soil. In one set of treatments, goat manure was applied annually for 13 years at 0, 5 and 10 t ha(-1), and intercrops of sorghum/cowpea, millet/green gram and maize/pigeonpea were grown. Yield depended on rainfall and trends with time were not identifiable. Manure caused an upward trend in SOC, but 10 t ha(-1) manure did not give significantly more SOC than 5 t ha(-1). Only 10 t ha(-1) manure increased Olsen P. Measurements of both SOC and Olsen P are recommended. In another set of treatments, manure was applied for four years; the residual effect lasted another seven to eight years when assessed by yield, SOC and Olsen P Treatment with mineral fertilizers provided the same rates of N and P as 5 t hat manure and yields from manure and fertilizer were similar. Fertilizer increased Olsen P but not SOC. Management systems with occasional manure application and intermediate fertilizer applications should be assessed. Inputs and offtakes of C, N and P were measured for three years. Approximately 16, 25 and 11% of C, N and P respectively were stabilized into soil organic matter from 5 t ha(-1) a(-1) manure. The majority of organic P was fixed as soil inorganic P.

Effect of manure application on crop yield and soil chemical properties in a long-term field trial of semi-arid Kenya

The sustainability of cereal/legume intercropping was assessed by monitoring trends in grain yield, soil organic C (SOC) and soil extractable P (Olsen method) measured over 13 years at a long-term field trial on a P-deficient soil in semi-arid Kenya. Goat manure was applied annually for 13 years at 0, 5 and 10 t ha(-1) and trends in grain yield were not identifiable because of season-to-season variations. SOC and Olsen P increased for the first seven years of manure application and then remained constant. The residual effect of manure applied for four years only lasted another seven to eight years when assessed by yield, SOC and Olsen P. Mineral fertilizers provided the same annual rates of N and P as in 5 t ha(-1) manure and initially ,gave the same yield as manure, declining after nine years to about 80%. Therefore, manure applications could be made intermittently and nutrient requirements topped-up with fertilizers. Grain yields for sorghum with continuous manure were described well by correlations with rainfall and manure input only, if data were excluded for seasons with over 500 mm rainfall. A comprehensive simulation model should correctly describe crop losses caused by excess water.

A contour-advective semi-lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields

This paper describes a novel numerical algorithm for simulating the evolution of fine-scale conservative fields in layer-wise two-dimensional flows, the most important examples of which are the earth's atmosphere and oceans. the algorithm combines two radically different algorithms, one Lagrangian and the other Eulerian, to achieve an unexpected gain in computational efficiency. The algorithm is demonstrated for multi-layer quasi-geostrophic flow, and results are presented for a simulation of a tilted stratospheric polar vortex and of nearly-inviscid quasi-geostrophic turbulence. the turbulence results contradict previous arguments and simulation results that have suggested an ultimate two-dimensional, vertically-coherent character of the flow. Ongoing extensions of the algorithm to the generally ageostrophic flows characteristic of planetary fluid dynamics are outlined.