On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

Incorporating Legacy Soil pH Databases into Digital Soil Maps

Soil data and reliable soil maps are imperative for environmental management. conservation and policy. Data from historical point surveys, e.g. experiment site data and farmers fields can serve this purpose. However, legacy soil information is not necessarily collected for spatial analysis and mapping such that the data may not have immediately useful geo-references. Methods are required to utilise these historical soil databases so that we can produce quantitative maps of soil propel-ties to assess spatial and temporal trends but also to assess where future sampling is required. This paper discusses two such databases: the Representative Soil Sampling Scheme which has monitored the agricultural soil in England and Wales from 1969 to 2003 (between 400 and 900 bulked soil samples were taken annually from different agricultural fields); and the former State Chemistry Laboratory, Victoria, Australia where between 1973 and 1994 approximately 80,000 soil samples were submitted for analysis by farmers. Previous statistical analyses have been performed using administrative regions (with sharp boundaries) for both databases, which are largely unrelated to natural features. For a more detailed spatial analysis that call be linked to climate and terrain attributes, gradual variation of these soil properties should be described. Geostatistical techniques such as ordinary kriging are suited to this. This paper describes the format of the databases and initial approaches as to how they can be used for digital soil mapping. For this paper we have selected soil pH to illustrate the analyses for both databases.

Conservation strategy maps: a tool to facilitate biodiversity action planning illustrated using the heath fritillary butterfly

1. The UK Biodiversity Action Plan (UKBAP) identifies invertebrate species in danger of national extinction. For many of these species, targets for recovery specify the number of populations that should exist by a specific future date but offer no procedure to plan strategically to achieve the target for any species. 2. Here we describe techniques based upon geographic information systems (GIS) that produce conservation strategy maps (CSM) to assist with achieving recovery targets based on all available and relevant information. 3. The heath fritillary Mellicta athalia is a UKBAP species used here to illustrate the use of CSM. A phase 1 habitat survey was used to identify habitat polygons across the county of Kent, UK. These were systematically filtered using relevant habitat, botanical and autecological data to identify seven types of polygon, including those with extant colonies or in the vicinity of extant colonies, areas managed for conservation but without colonies, and polygons that had the appropriate habitat structure and may therefore be suitable for reintroduction. 4. Five clusters of polygons of interest were found across the study area. The CSM of two of them are illustrated here: the Blean Wood complex, which contains the existing colonies of heath fritillary in Kent, and the Orlestone Forest complex, which offers opportunities for reintroduction. 5. Synthesis and applications. Although the CSM concept is illustrated here for the UK, we suggest that CSM could be part of species conservation programmes throughout the world. CSM are dynamic and should be stored in electronic format, preferably on the world-wide web, so that they can be easily viewed and updated. CSM can be used to illustrate opportunities and to develop strategies with scientists and non-scientists, enabling the engagement of all communities in a conservation programme. CSM for different years can be presented to illustrate the progress of a plan or to provide continuous feedback on how a field scenario develops.

Fast training of self organizing maps for the visual exploration of molecular compounds

Visual exploration of scientific data in life science area is a growing research field due to the large amount of available data. The Kohonen’s Self Organizing Map (SOM) is a widely used tool for visualization of multidimensional data. In this paper we present a fast learning algorithm for SOMs that uses a simulated annealing method to adapt the learning parameters. The algorithm has been adopted in a data analysis framework for the generation of similarity maps. Such maps provide an effective tool for the visual exploration of large and multi-dimensional input spaces. The approach has been applied to data generated during the High Throughput Screening of molecular compounds; the generated maps allow a visual exploration of molecules with similar topological properties. The experimental analysis on real world data from the National Cancer Institute shows the speed up of the proposed SOM training process in comparison to a traditional approach. The resulting visual landscape groups molecules with similar chemical properties in densely connected regions.

Spin spaces and positive decomposition of linear maps on ordered Banach spaces

Spin factors and generalizations are used to revisit positive generation of B(E, F), where E and F are ordered Banach spaces. Interior points of B(E, F)+ are discussed and in many cases it is seen that positive generation of B(E, F) is controlled by spin structure in F when F is a JBW-algebra.

Harmonic force fields from scaled SCF calculations: Program ASYM40

We report an extended version of our normal coordinate program ASYM40, which may be used to transform Cartesian force constants from ab initio calculations to a force field in nonredundant internal (symmetry) coordinates. When experimental data are available, scale factors for the theoretical force field may then be optimized by least-squares refinement. The alternative of refining an empirical force field to fit a wide variety of data, as with the previous version ASYM20, has been retained. We compare the results of least-squares refinement of the full harmonic force field with least-squares refinement of only the scale factors for an SCF calculated force field and conclude that the latter approach may be useful for large molecules where more sophisticated calculations are impractical. The refinement of scale factors for a theoretical force field is also useful when there are only limited spectroscopic data. The program will accept ab initio calculated force fields from any program that presents Cartesian force constants as output. The program is available through Quantum Chemistry Program Exchange.

Analytical potentials for triatomic molecules IX. The prediction of anharmonic force constants from potential energy surfaces based on harmonic force fields and dissociation energies for SO2 and O3

Analytical potential energy functions which are valid at all dissociation limits have been derived for the ground states of SO2 and O3. The procedure involves minimizing the errors between the observed vibrational spectra and spectra calculated by a variational procedure. Good agreement is obtained between the observed and calculated spectra for both molecules. Comparisons are made between anharmonic force fields, previously determined from the spectral data, and the force fields obtained by differentiating the derived analytical functions at the equilibrium configurations.

Microwave spectrum of CHD2CN and CHD2NC, and the harmonic force field and rz structure of methyl cyanide and isocyanide

The microwave spectra of CHD2CN and CHD2NC have been measured from 18 to 40 GHz; about 20 type A and 30 type C transitions have been observed for each molecule. These have been fitted to a Hamiltonian using 3 rotational constants, and 5 quartic and 4 sextic distortion constants, in the IrS reduction of Watson [in “Vibrational spectra and structure” Vol. 6 (1977)]; the standard error of the fit is 26 kHz. For methyl cyanide the 5 quartic distortion constants have been used to further refine the recent harmonic force field of Duncan et al. [J. Mol. Spectrosc. 69, 123 (1978)], but the changes are small. Finally, for both molecules, the harmonic force field has been used to determine zero point average moments of inertia Iz from the ground state rotational constants for many isotopic species, and these have been used to determine an rz structure. The results are compared with rs structure calculations.

Redundant coordinates in harmonic force-field calculations

Redundancy relations between vibrational coordinates may be linear (as for rectilinear coordinates used in deriving a G matrix), or non-linear (as for curvilinear coordinates used in formulating model force fields). It is shown that geometrically defined internal coordinates are necessarily curvilinear. Hence it is shown that linear force constants can occur in model force field calculations involving redundant coordinates, in disagreement with the recent proposal of Gussoni and Zerbi.