The problem of dependent evidence

It is appropriate to use Dempster's rule for combining belief functions only if the belief functions combined are based on independent items of evidence. What can be done in the case of dependent evidence? Often the answer is to reframe the problem. Three examples are given: one from everyday experience, one from probabilistic relaxation, and one from expert systems.     

Molecular dynamics and quantum chemical approaches in the study of the hydration of protonated cyclohexyldiamines

Explicit hydration of the neutral and charged cyclohexylamine and of the cyclohexyldiamine isomers in their mono- or diprotonated forms is investigated through classical molecular dynamics (MD) simulations in aqueous solutions combined with DFT calculations in amine–water complexes. The MD studies performed in the monoamines reveal that the structure of the hydration shell around the neutral amino group (NH₂) is quite distinct from that around the charged one (NH₃⁺). On average, the number of water molecules surrounding the two groups is calculated to be ～2 and 3–4, respectively. The variation of the hydration structure prompted by the groups’ proximity is discussed based on the data found for the mono- and diprotonated diamines. To have a more detailed picture of the water molecules’ arrangement around the amino groups and of the amine–water hydrogen bonds, geometry optimisations in hydrates with up to six water molecules are carried out at the B3LYP/aug-cc-pVDZ level. Complexation energies are also computed. The main findings emerging from these calculations are found to be very helpful to rationalise the mutual influence of the amino groups and therefore to better elucidate the MD findings. The complementary nature of the two research methods is emphasised as an excellent tool in order to closely examine the hydration of polyamines, as exemplified for the cyclohexyldiamines.     

Provably stable and time‐accurate extensions of Runge–Kutta schemes for CFD computations on moving grids

Extending fixed‐grid time integration schemes for unsteady CFD applications to moving grids, while formally preserving their numerical stability and time accuracy properties, is a nontrivial task. A general computational framework for constructing stability‐preserving ALE extensions of Eulerian multistep time integration schemes can be found in the literature. A complementary framework for designing accuracy‐preserving ALE extensions of such schemes is also available. However, the application of neither of these two computational frameworks to a multistage method such as a Runge–Kutta (RK) scheme is straightforward. Yet, the RK methods are an important family of explicit and implicit schemes for the approximation of solutions of ordinary differential equations in general and a popular one in CFD applications. This paper presents a methodology for filling this gap. It also applies it to the design of ALE extensions of fixed‐grid explicit and implicit second‐order time‐accurate RK (RK2) methods. To this end, it presents the discrete geometric conservation law associated with ALE RK2 schemes and a method for enforcing it. It also proves, in the context of the nonlinear scalar conservation law, that satisfying this discrete geometric conservation law is a necessary and sufficient condition for a proposed ALE extension of an RK2 scheme to preserve on moving grids the nonlinear stability properties of its fixed‐grid counterpart. All theoretical findings reported in this paper are illustrated with the ALE solution of inviscid and viscous unsteady, nonlinear flow problems associated with vibrations of the AGARD Wing 445.6. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of a line implicit method to fully coupled system of equations for turbulent flow problems

For unstructured finite volume methods, we present a line implicit Runge–Kutta method applied as smoother in an agglomerated multigrid algorithm to significantly improve the reliability and convergence rate to approximate steady-state solutions of the Reynolds-averaged Navier–Stokes equations. To describe turbulence, we consider a one-equation Spalart–Allmaras turbulence model. The line implicit Runge–Kutta method extends a basic explicit Runge–Kutta method by a preconditioner given by an approximate derivative of the residual function. The approximate derivative is only constructed along predetermined lines which resolve anisotropies in the given grid. Therefore, the method is a canonical generalisation of point implicit methods. Numerical examples demonstrate the improvements of the line implicit Runge–Kutta when compared with explicit Runge–Kutta methods accelerated with local time stepping.     

Influence of the surface morphology on the melting of polymer crystals. I. Loops of random length and adjacent reentry

A polymer crystal with a noncrystalline surface layer formed by chain loops of different lengths is considered. It is assumed that the length of each loop can be changed by longitudinal diffusion of the molecule through the crystal lattice. From the condition that the free energy of the system is minimum, the loop length distribution and the average loop length as function of temperature are calculated. In contrast to the results for loops of equal length, for the present model, a substantial thickness of the noncrystalline surface layer and a broad melting range is obtained also for the case of adjacent reentry. In order to get this result one has to take into account that even an “ideal fold” consists of at least four rigidly arranged CH2 groups in energetically unfavored conformation.     

The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations

In this paper, the a posteriori error estimates of Chebyshev–Petrov–Galerkin approximations are investigated. For simplicity, we choose the Poisson equation with Dirichlet boundary conditions to discuss the a posteriori error estimators, and deduce their efficient and reliable properties. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error estimators.     

Site-selective C–H nitration of N-aryl-7-azaindoles under palladium(II) catalysis

The site-selective C–H nitration reaction of 7-azaindoles with t-butyl nitrite under palladium catalysis is described. This protocol provides an efficient method for the construction of ortho-nitrated N-aryl-7-azaindoles with excellent site-selectivity and functional group compatibility. The formed 7-azaindole derivatives can be readily transformed into 7-azaindoles containing an aniline functional group under palladium-catalyzed hydrogenation conditions.     

Phragmen–lindelöf principle for weak subsolutions and cooperative reaction–diffusion systems

One of the principal techniques for treating sustems of reaction–diffusion equations is based on a comparison method using sub and super–solutions. In practice this method is much more effective if non–smooth subsolutions are allowed. In this note we extend the analysis in [2,3] for cooperative systems and prove a comparison principle for a natural and rather general class of weak subsolutions satisfying a Phragmen–Lindelöf condition. An application is then given to a biological model in involving a pair of mutualists.     

Designed Construction of SrTiO3/SrSO4/Pt Heterojunctions with Boosted Photocatalytic H2 Evolution Activity

Efficient separation of photogenerated electron–hole pairs is a crucial factor for high-performance photocatalysts. Effective electron–hole separation and migration could be achieved by heterojunctions with suitable band structures. Herein, a porous SrTiO₃/SrSO₄ heterojunction is prepared by a sol-gel method at room temperature followed by an annealing process. XRD characterization suggests high crystallinity of the heterostructure. A well-defined interface between the two phases is confirmed by high-resolution (HR)TEM. The photocatalytic H₂ evolution productivity of the SrTiO₃/SrSO₄ heterojunction with Pt as co-catalyst reaches 396.82 μmol g⁻¹ h⁻¹, which is 16 times higher than that of SrTiO₃/Pt. The boosted photocatalytic activity of SrTiO₃/SrSO₄/Pt can be ascribed to the presence of SrSO₄, which promotes the transfer and migration of photogenerated carriers by forming the heterojunction and porous structure, which provides a large amount of active sites. This novel porous heterostructure brings new ideas for the development of high-efficiency photocatalysts for H₂ release.