1H and 13C NMR of platinum porphyrins—III. Platinum deuteroporphyrin IX dimethyl ester

¹H and ¹³C NMR spectra of the title compound are reported and peaks assigned by comparison with the free-base porphyrin. Long-range spin—spin coupling is observed between platinum and the macrocyclic protons and carbon atoms. Meso-protons are assigned from line-broadening observed for three of the peaks, possibly caused by residual coupling of these protons to the pyrrole β-protons. Allylic coupling of the pyrrole β-protons to adjacent methyl groups is also noted, and supports the tautomeric model of an 18-atom delocalisation pathway around the macrocycle. The ¹H NMR spectrum of the title compound is more concentration-dependent than that of the free-base porphyrin and probably reflects additional macrocyclic interactions via the platinum atoms.     

Limiting distributions of homogeneous functions of sample spacings for distributions approaching the uniform distribution

Summary X ₁,...,X _n are independent random variables, identically distributed over the unit interval, with common probability density function 1 + r(x)/n for all sufficiently large n, where is a positive constant, and |r(x)| <D. V ₁, ..., V _n+1 are the sample spacings generated by X ₁,..., X _n . It is shown that in many cases, the asymptotic joint distribution of homogeneous functions of V ₁,..., V _n+1 can be found directly from the asymptotic joint distribution of homogeneous functions of independent exponential random variables.Research supported by NSF Grant GP 3783.     

Conforming finite element computations applied to a spatially periodic,harmonic Navier-Stokes problem

In this paper computations in the two dimensional case of a harmonic Navier-Stokes problem with periodic boundary conditions are presented. This study of an incompressible viscous fluid leads to a non-symmetric linear problem (very low Reynolds number). Moreover unknown functions have complex values (monochromatic dynamic behaviour). Numerical treatment of the incompressibility condition is a generalization of the classical treatment of Stokes problem. A mixed formulation, where discrete pressure plays the role of Lagrange multipliers is used (Uzawa algorithm). Two conforming finite element methods are tested on different meshes. The second one uses a classical refinement in the shape function: the so-called bulb function. All computational tests show that the use of a bulb function on each element gives better results than refinement in the mesh without introducing too many degrees of freedom. Finally numerical results are compared to experimental data.     

Rotating ring-disk electrode studies of the reaction of stable radical cations with styrene and isobutyl vinyl ether

An electroanalytical technique has been utilized in a new method for the study of reactive intermediates in polymerization reactions. A ring-disk electrode system generated persistent carbocation radicals whose stability decreased in the order: 1,3,6,8-tetraphenylpyrene (TPP), rubrene (Ru), 9,10-diphenylanthracene (DPA), and 9,10-dimethylanthracene (DMA). Radical cations from these parent compounds flowed to a collecting ring which was controlled potentiostatically to reduce unreacted cations. When styrene or isobytyl vinyl ether was added to the solution, the concentration of carbocation radicals reaching the electrode was reduced. Current collection efficiencies N were determined as a function of rotation speed ω for each monomer concentration. Plots of N^?1 as ω^?1 in the absence of monomer show no dependence on ω (indicative of stable intermediates), but a linear dependence is found with each concentration of monomer. This indicates a first-order dependence on radical cation concentration. The rate constants show a trend in cation reactivities which is in agreement with that obtained by other methods. This technique, however, extends the range of investigation to a much shorter time scale.     

Well-posedness of a degenernate parabolic equation issuing from two-dimensional perfect fluid dynamics

A degenerate parabolic equation of convection-diffusion type has been proposed by Robert and Sommeria in [12] to describe the relaxation towards statistical equilibrium states in 2D incompressible perfect fluid dynamics. The paper is concerned with the Cauchy problem for this equation. The local existence of a variational soluation is obtained in using the decrease of the (negative) mixing entropy and Schauder theorem. A smoothing effect is used when proving the uniqueness of the variational solutions by youdovitch's method. Finally, global existence of solutions and their asymptotic convergence towards Gibbs states are shown for a large class of initial data.     

Structure and properties of potassium niobato-borophosphate glasses

Bulk glasses of the series (1 ? x)[0.5K₂O–0.1B₂O₃–0.4P₂O₅]–xNb₂O₅ with x = 0–45.7 mol% Nb₂O₅ were prepared by slow cooling in air and investigated by Raman, ³¹P, and ¹¹B MAS NMR spectroscopy. The incorporation of Nb₂O₅ into the parent borophosphate glass results in a substantial increase in the glass transition temperature and chemical durability of glasses. Raman spectra showed that Nb atoms form distorted NbO₆ octahedra, which are isolated at low Nb₂O₅ content, whereas at higher Nb₂O₅ content they form clusters. ¹¹B NMR spectra of the glasses revealed the interaction between Nb₂O₅ and BO₄ tetrahedral units, which results in a partial transformation of tetrahedral BO₄ units to trigonal BO₃ units and the formation of mixed B(OP)_4?n(ONb)_n units.     

ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

The (mixed finite element) discretization of the linearized Navier–Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ‘grad‐div’ (GD) stabilization, as well as on the discrete side where it is known as an ‘augmented Lagrangian’ (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of ??‐LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of ??‐LU factorization in the preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.