Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.

Relations between coal petrology and gas content in the Upper Newlands Seam, central Queensland, Australia

The Upper Newlands Seam in the northern Bowen Basin, Queensland, Australia consists of six benches (A-F) that have different petrographic assemblages. Benches C and E contain relatively abundant inertodetrinite and mineral matter, as well as anomalously high reflectance values; these characteristics support a largely allochthonous, detrital origin for the C and E benches. Fractures and cleats in the seam show a consistent orientation of northeast- southwest for face cleats, and a wide range of orientations for fractures. Cleat systems are well developed in bright bands, with poor continuity in the dull coal. Both maceral content and cleat character are suggested to influence gas drainage in the Upper Newlands Seam. A pronounced positive correlation between vitrinite abundance and gas desorption data suggests more efficient drainage from benches with abundant vitrinite. Conversely, inertinite-rich benches are suggested to have less efficient drainage, and possibly retain gas within pore spaces, which could increase the outburst potential of the coal. (C) 2001 Elsevier Science BN. All rights reserved.

Patch dynamics and metapopulation theory: the case of successional species

We present a mathematical framework that combines extinction-colonization dynamics with the dynamics of patch succession. We draw an analogy between the epidemiological categorization of individuals (infected, susceptible, latent and resistant) and the patch structure of a spatially heterogeneous landscape (occupied-suitable, empty-suitable, occupied-unsuitable and empty-unsuitable). This approach allows one to consider life-history attributes that influence persistence in patchy environments (e.g., longevity, colonization ability) in concert with extrinsic processes (e.g., disturbances, succession) that lead to spatial heterogeneity in patch suitability. It also allows the incorporation of seed banks and other dormant life forms, thus broadening patch occupancy dynamics to include sink habitats. We use the model to investigate how equilibrium patch occupancy is influenced by four critical parameters: colonization rate? extinction rate, disturbance frequency and the rate of habitat succession. This analysis leads to general predictions about how the temporal scaling of patch succession and extinction-colonization dynamics influences long-term persistence. We apply the model to herbaceous, early-successional species that inhabit open patches created by periodic disturbances. We predict the minimum disturbance frequency required far viable management of such species in the Florida scrub ecosystem. (C) 2001 Academic Press.

Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations

In this paper we discuss implicit methods based on stiffly accurate Runge-Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge-Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs. (C) 2001 Elsevier Science B.V. All rights reserved.

NMR structure and backbone dynamics of a concatemer of epidermal growth factor homology modules of the human low-density lipoprotein receptor

The ligand-binding region of the low-density lipoprotein (LDL) receptor is formed by seven N-terminal, imperfect, cysteine-rich (LB) modules. This segment is followed by an epidermal growth factor precursor homology domain with two N-terminal, tandem, EGF-like modules that are thought to participate in LDL binding and recycling of the endocytosed receptor to the cell surface. EGF-A and the concatemer, EGF-AB, of these modules were expressed in Escherichia coli. Correct protein folding of EGF-A and the concatemer EGF-AB was achieved in the presence or absence of calcium ions, in contrast to the LB modules, which require them for correct folding. Homonuclear and heteronuclear H-1-N-15 NMR spectroscopy at 17.6 T was used to determine the three-dimensional structure of the concatemer. Both modules are formed by two pairs of short, anti-parallel beta -strands. In the concatemer, these modules have a fixed relative orientation, stabilized by calcium ion-binding and hydrophobic interactions at the interface. N-15 longitudinal and transverse relaxation rates, and {H-1}-N-15 heteronuclear NOEs were used to derive a model-free description of the backbone dynamics of the molecule. The concatemer appears relatively rigid, particularly near the calcium ion-binding site at the module interface, with an average generalized order parameter of 0.85 +/- 0.11. Some mutations causing familial hypercholesterolemia may now be rationalized. Mutations of D41, D43 and E44 in the EGF-B calcium ion-binding region may affect the stability of the linker and thus the orientation of the tandem modules. The diminutive core also provides little structural stabilization, necessitating the presence of disulfide bonds. The structure and dynamics of EGF-AB contrast with the N-terminal LB modules, which require calcium ions both for folding to form the correct disulfide connectivities and for maintenance of the folded structure, and are connected by highly mobile linking peptides. (C) 2001 Academic Press.

Enzymology, structure, and dynamics of acetohydroxy acid isomeroreductase

Acetohydroxy acid isomeroreductase is a key enzyme involved in the biosynthetic pathway of the amino acids isoleucine, valine, and leucine. This enzyme is of great interest in agrochemical research because it is present only in plants and microorganisms, making it a potential target for specific herbicides and fungicides. Moreover, it catalyzes an unusual two-step reaction that is of great fundamental interest. With a view to characterizing both the mechanism of inhibition by potential herbicides and the complex reaction mechanism, various techniques of enzymology, molecular biology, mass spectrometry, X-ray crystallography, and theoretical simulation have been used. The results and conclusions of these studies are described briefly in this paper.

A formal representation of assumptions in process modelling

In this work, we present a systematic approach to the representation of modelling assumptions. Modelling assumptions form the fundamental basis for the mathematical description of a process system. These assumptions can be translated into either additional mathematical relationships or constraints between model variables, equations, balance volumes or parameters. In order to analyse the effect of modelling assumptions in a formal, rigorous way, a syntax of modelling assumptions has been defined. The smallest indivisible syntactical element, the so called assumption atom has been identified as a triplet. With this syntax a modelling assumption can be described as an elementary assumption, i.e. an assumption consisting of only an assumption atom or a composite assumption consisting of a conjunction of elementary assumptions. The above syntax of modelling assumptions enables us to represent modelling assumptions as transformations acting on the set of model equations. The notion of syntactical correctness and semantical consistency of sets of modelling assumptions is defined and necessary conditions for checking them are given. These transformations can be used in several ways and their implications can be analysed by formal methods. The modelling assumptions define model hierarchies. That is, a series of model families each belonging to a particular equivalence class. These model equivalence classes can be related to primal assumptions regarding the definition of mass, energy and momentum balance volumes and to secondary and tiertinary assumptions regarding the presence or absence and the form of mechanisms within the system. Within equivalence classes, there are many model members, these being related to algebraic model transformations for the particular model. We show how these model hierarchies are driven by the underlying assumption structure and indicate some implications on system dynamics and complexity issues. (C) 2001 Elsevier Science Ltd. All rights reserved.

Structural Dynamics in (ND4)2[Cu(D2O)6](SO4)2

The extended X-ray absorption fine structure spectroscopy (EXAFS) of (ND4)(2)[CU(D2O)(6)](SO4)(2) at 5, 14,100, 200, and 298 K is reported. This indicates that the Cu-O bond lengths of the Cu(D2O)(6)(2+) ion do not change significantly within this temperature range, which contrasts with EPR results and X-ray and neutron diffraction experiments, which imply that two of the Cu-(D2O) bonds converge in length as the temperature is raised. The EXAFS measurements thus confirm that the bond distances yielded by the diffraction experiments refer to the average positions of ligands involved in a dynamic equilibrium in which the directions of the long and intermediate bonds of the Jahn-Teller distorted Cu(D2O)(6)(2+) ion are interchanged in the crystal lattice. Analysis of the displacement parameters is consistent with this interpretation, as are the wave functions calculated using a model involving Jahn-Teller vibronic coupling and the influence of lattice strain interactions.

Three-dimensional orthotropic viscoelastic finite element model of a human ligament

Ligaments undergo finite strain displaying hyperelastic behaviour as the initially tangled fibrils present straighten out, combined with viscoelastic behaviour (strain rate sensitivity). In the present study the anterior cruciate ligament of the human knee joint is modelled in three dimensions to gain an understanding of the stress distribution over the ligament due to motion imposed on the ends, determined from experimental studies. A three dimensional, finite strain material model of ligaments has recently been proposed by Pioletti in Ref. [2]. It is attractive as it separates out elastic stress from that due to the present strain rate and that due to the past history of deformation. However, it treats the ligament as isotropic and incompressible. While the second assumption is reasonable, the first is clearly untrue. In the present study an alternative model of the elastic behaviour due to Bonet and Burton (Ref. [4]) is generalized. Bonet and Burton consider finite strain with constant modulii for the fibres and for the matrix of a transversely isotropic composite. In the present work, the fibre modulus is first made to increase exponentially from zero with an invariant that provides a measure of the stretch in the fibre direction. At 12% strain in the fibre direction, a new reference state is then adopted, after which the material modulus is made constant, as in Bonet and Burton's model. The strain rate dependence can be added, either using Pioletti's isotropic approximation, or by making the effect depend on the strain rate in the fibre direction only. A solid model of a ligament is constructed, based on experimentally measured sections, and the deformation predicted using explicit integration in time. This approach simplifies the coding of the material model, but has a limitation due to the detrimental effect on stability of integration of the substantial damping implied by the nonlinear dependence of stress on strain rate. At present, an artificially high density is being used to provide stability, while the dynamics are being removed from the solution using artificial viscosity. The result is a quasi-static solution incorporating the effect of strain rate. Alternate approaches to material modelling and integration are discussed, that may result in a better model.

Quantum criticality

We investigate the theory of quantum fluctuations in non-equilibrium systems having large critical fluctuations. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction, and also to envisage future tests of quantum theory in regions of macroscopic quantum fluctuations. A long-term objective of this research is to identify suitable physical systems in which macroscopic 'Schrodinger cat'-like behaviour may be observed. We investigate two systems in particular of much current experimental interest, namely the degenerate parametric oscillator near threshold, and the evaporatively cooled (BEC). We compare the results obtained in the positive-P representation, as a fully quantum mechanical calculation, with the truncated Wigner phase space equation, also known as semi-classical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. In the region where the largest quantum fluctuations and Schrodinger cat-like behaviour might be expected, we find that the quantum predictions correspond very closely to the semi-classical theory. Nature abhors observing a Schrodinger car.