Dynamic thermodynamic resolution of lithiated N-Boc-N′-alkylpiperazines

Deprotonation of N-Boc-N′-alkylpiperazines with sec-BuLi in Et₂O-TMEDA gave the 2-lithio derivatives which were resolved in the presence of a chiral ligand. The best ligands for the asymmetric substitution were diamino-alkoxides that promoted a dynamic thermodynamic resolution (DTR) of the organolithium at −30 °C. Electrophilic quench gave enantiomerically enriched 2-substituted piperazines. Of a selection of piperazines, the N′-t-butyl derivative gave the best results, with the product N-Boc-N′-t-butyl-2-substituted piperazines being formed with enantiomer ratios up to 81:19.     

In situ measurements of the spectral emittance of coal ash deposits

The spectral emittance of deposits left by bituminous and sub-bituminous coals under oxidizing conditions have been measured in situ. Pulverized coal is injected into a down-fired entrained-flow reactor. Ash accumulates on a probe in the reactor effluent and radiation emitted by the ash layer is recorded using a Fourier transform infrared (FTIR) spectrometer. Values for the spectral emissive power emitted by the ash and the surface temperature of the ash are extracted from these data. These results are then used to calculate the spectral emittance of the deposit. The spectral emittances of ash deposits formed by burning Illinois #6 (bituminous) coal and Powder River Basin (sub-bituminous) coal were measured between 3000 and 500 wavenumbers. The spectral emittance of the deposit left by the bituminous coal has a constant value of approximately 0.46 between 3000 and 2400 wavenumbers. Between 2200 and 1200 wavenumbers, the spectral emittance of the deposit increases from approximately 0.47 to approximately 0.61. Between 1200 and 500 wavenumbers, the spectral emittance is relatively constant at 0.61. The spectral emittance of the deposit left by the sub-bituminous coal is also relatively constant between 3000 and 2400 wavenumbers at a value of 0.29. Between 2200 and 500 wavenumbers, the spectral emittance of deposits from the sub-bituminous coal increases from approximately 0.29 to 0.55. Differences between these spectral emittance measurements and those measured ex situ illustrate the importance of making in situ measurements. Band emittances were calculated using the measured spectral emittances, and band emittances of the deposits are reported as functions of temperature.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L ²-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.     

Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

Norm estimates for thel 2-inverses of multivariate Toeplitz matrices

This paper bounds thel ²-norms of inverses of certain Toeplitz matrices arising from Pólya frequency functions.     

Das Atomgewicht des Kaliums

Ohne Zusammenfassung