The existence,uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system $\text{?x}\text{?t}\text{= Δx + F(x,y),}\text{?y}\text{?t}\text{= G(x,y)}$ is investigated, where x and y are scalar functions of time (t ? 0), and n space variables $\text{(ξ}{}_1\text{,…, ξ}{}_{\mathrm{n}}\text{), Δx ≡ ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{x}\text{?ξ}{}_{\mathrm{i}}{}^2$, and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution $\text{(x(r),y(r)),}\text{where}\text{r = (ξ}{}_1{}^2\text{+ … + ξ}{}_{\mathrm{n}}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which is bounded for r ? 0 and satisfies x(0) >x₀, y(0) > y₀, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.     

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

     

Equations of fluctuating nonlinear hydrodynamics for normal fluids

The full set of fluctuating nonlinear hydrodynamic equations for normal fluids is derived from the conventional Langevin equations extended to include multiplicative noise. The equations describing the set of conserved variables (the mass density, the momentum densityg, the energy density) agree with those found by Morozov for a case of a driving free energy which is a local function of the hydrodynamic variables. We show here that if the standard form of the hydrodynamic equations is to hold in the absence of noise, then the driving free energy must be a local function ofg and, but it may have to be a nonlocal function of the mass density.     

The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems

Summary The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an inner iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.This work was supported in part by National Science Foundation Grants DCR-8412314 and DCR-8502014The work of this author was completed while he was on sabbatical leave at the Centre for Mathematical Analysis and Mathematical Sciences Research Institute at the Australian National University, Canberra, Australia     

Jacobi matrices for sums of weight functions

Orthogonal polynomials are conveniently represented by the tridiagonal Jacobi matrix of coefficients of the recurrence relation which they satisfy. LetJ ₁ andJ ₂ be finite Jacobi matrices for the weight functionsw ₁ andw ₂, resp. Is it possible to determine a Jacobi matrix \(\tilde J\) , corresponding to the weight functions \(\tilde w\) =w ₁+w ₂ using onlyJ ₁ andJ ₂ and if so, what can be said about its dimension? Thus, it is important to clarify the connection between a finite Jacobi matrix and its corresponding weight function(s). This leads to the need for stable numerical processes that evaluate such matrices. Three newO(n ²) methods are derived that “merge” Jacobi matrices directly without using any information about the corresponding weight functions. The first can be implemented using any of the updating techniques developed earlier by the authors. The second new method, based on rotations, is the most stable. The third new method is closely related to the modified Chebyshev algorithm and, although it is the most economical of the three, suffers from instability for certain kinds of data. The concepts and the methods are illustrated by small numerical examples, the algorithms are outlined and the results of numerical tests are reported.     

Functional Pyrimidinyl Pyrazolate Pt(II) Complexes: Role of Nitrogen Atom in Tuning the Solid‐State Stacking and Photophysics

Pt(II) metal complexes are known to exhibit strong solid‐state aggregation and are promising for realization of efficient emission in fabrication of organic light emitting diodes (OLED) with nondoped emitter layer. Four pyrimidine–pyrazolate based chelates, together with four isomeric Pt(II) metal complexes, namely: [Pt(pm2z)₂], [Pt(tpm2z)₂], [Pt(pm4z)₂], and [Pt(tpm4z)₂], are isolated and systematically investigated for their structure–property relationships for practical OLED applications. Detailed single molecular and aggregated structures are revealed by photophysical and mechanochromic measurements, grazing‐incidence X‐ray diffraction, and theoretical approaches. These results suggest that these Pt(II) emitters pack like a deck of playing cards under vacuum deposition, and their emission energy is not only affected by the single molecular designs, but notably influenced by their intermolecular packing interaction, i.e., Pt···Pt separations that are arranged in the order: [Pt(tpm4z)₂] > [Pt(pm4z)₂] > [Pt(tpm2z)₂] > [Pt(pm2z)₂]. Nondoped OLED with emission ranging from green to red are prepared, to which the best performances are recorded for [Pt(tpm2z)₂], giving maximum external quantum efficiency (EQE) of 27.5% at 10³ cd m^?2, maximum luminance of 2.5 × 10⁵ cd m^?2 at 17 V, and with stable CIE_x,y of (0.56, 0.44).     

The First Metal‐Carbon Bond β‐Isomer Paddlewheel Pd24+ Compounds

The first metal‐carbon bond β‐form paddlewheel complexes containing a Pd₂⁴⁺ core, [Pd(η²‐dithio)]₂(μ‐dppa)( μ‐SCNMe₂) (dithio = S₂P(OEt)₂, 2 ; S₂COEt, 3 ; S₂CNC₄H₈, 4 ), were prepared by the reactions of the α‐form paddlewheel‐type Pd₂⁺⁴ dipalladium complex [Pd₂ (μ‐Hdppa)₂(μ‐SCNMe₂)₂][Cl]₂, 1 with various dithio‐ligands, [NH₄][S₂P(OEt)₂], [K][S₂COEt] and [NH₄][S₂CNC₄H₈], in methanol at ambient temperature (Hdppa = bis(diphenylphosphino)amine). Electronic spectra and two X‐ray structures of the Pd₂⁺⁴ species have been determined.     

Wheat digalactosyldiacylglycerol molecular species profiling using porous graphitic carbon stationary phase

The potential of porous graphitic carbon stationary phase (PGC) was assessed for the separation of molecular species of digalactosyldiacylglycerol (DGDG). Detection was by an evaporative light scattering detector (ELSD). A conventional optimization strategy allowed definition of a quaternary non-aqueous mobile phase and separation of 9 wheat DGDG molecular species with isocratic elution: methanol/toluene/tetrahydrofuran/chloroform 64.3/21.5/13.7/0.5 v/v with 0.1% of triethylamine and a stoichiometric amount of formic acid. The molecular species were identified by LC/MS. The chromatographic behavior of DGDG on PGC was then compared to previous studies. The addition of a carbon double bond on the alkyl chain decreased the retention. This contribution was less important when the number of unsaturations increased in the alkyl chain. The consequence of this retention behavior with PGC was an elution order of molecular species which did not agree with the partition number as observed with C18 grafted stationary phases.     

Bismuth nanowires for potential applications in nanoscale electronics technology.

Nanowires of bismuth with diameters ranging from 10 to 200 nm and lengths of 50 microm have been synthesized by a pressure injection method. Nanostructural and chemical compositional studies using environmental and high resolution transmission electron microscopy with electron stimulated energy dispersive X-ray spectroscopy have revealed essentially single crystal nanowires. The high resolution studies have shown that the nanowires contain amorphous Bi-oxide layers of a few nanometers on the surface. In situ environmental high resolution transmission electron microscopy (environmental-HRTEM) studies at the atomic level, in controlled hydrogen and other reducing gas environments at high temperatures demonstrate that gas reduction can be successfully applied to remove th oxide nanolayers and to maintain the dimensional and structural uniformity of the nanowires, which is key to attaining low electrical contact resistance.     

Reaction of methyl hypochlorite with certain olefins in the presence of boron trifluoride

Fluoro chlorides are major products in the reaction of methyl hypochlorite with certain olefins in the presence of boron trifluoride.