Implicit Solution of Hyperbolic Equations with Space-Time Adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.     

Rate optimal estimation with the integration method in the presence of many covariates

For multivariate regressors, integrating the Nadaraya–Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.     

Stereoselective Ring Opening of Electronically Excited Cyclohexa-2, 4-dienones: Cause and effect

The two conformers of a cyclohexa-2, 4-dienone with different substituents at C(6) on irradiation are believed to undergo ring opening stereospecifically affording a mixture of two configurationally isomeric diene-ketenes (and descendents thereof)- Exceptions are generally found for those dienones with one C and one O substituent or even with two C substituents, if one of them carries a polar group at a site able to interact through space with the ring C?O group. In these cases, only one of the two anticipated diene-ketenes (and descendents thereof) is produced. A thorough investigation of the photochemistry of a series of structurally different cyclohexa-2, 4-dienones on analytical as well as on preparative scale extends our mechanistic knowledge of the various routes from diene-ketenes into a variety of compound classes. Novel compound classes accessible to diene-ketenes are seven-membered carbocycles (by intramolecular aldolization of the zwitterion of appropriately substituted, transiently formed diene-(N, O)-ketene acetals) and β-lactams (by Staudinger reaction).