On M–multisplittings of singular M–matrices with application to Markov chains

Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.     

类锂离子体系高角动量态1s2ng(n=5～9)的精细结构的理论计算

利用全实加关联方法，以类锂等电子序列为研究对象，通过自旋－轨道相互作用和自旋－其它轨道相互作用算符的期待值计算了类锂离子等电子序列（Z=9～20）的高角动量1s2ng(n=5～9)激发态的精细结构劈裂.为了获得更精确的理论结果，在类氢近似下估算了高阶相对论修正和量子电动力学（QED）效应对精细结构的贡献，得到的结果与等电子序列规律符合的很好。     

Improved Gram–Schmidt Type Downdating Methods

The problem of deleting a row from a Q–R factorization (called downdating) using Gram–Schmidt orthogonalization is intimately connected to using classical iterative methods to solve a least squares problem with the orthogonal factor as the coefficient matrix. Past approaches to downdating have focused upon accurate computation of the residual of that least squares problem, then finding a unit vector in the direction of the residual that becomes a new column for the orthogonal factor. It is also important to compute the solution vector of the related least squares problem accurately, as that vector must be used in the downdating process to maintain good backward error in the new factorization. Using this observation, new algorithms are proposed. One of the new algorithms proposed is a modification of one due to Yoo and Park [BIT, 36:161–181, 1996]. That algorithm is shown to be a Gram–Schmidt procedure. Also presented are new results that bound the loss of orthogonality after downdating. An error analysis shows that the proposed algorithms’ behavior in floating point arithmetic is close to their behavior in exact arithmetic. Experiments show that the changes proposed in this paper can have a dramatic impact upon the accuracy of the downdated Q–R decomposition. AMS subject classification (2000) 65F20, 65F25     

Redox-active ligands as a challenge for electronic structure methods

To improve the catalytic activity of 3d transition metal catalysts, redox-active ligands are a promising tool. These ligands influence the oxidation state of the metal center as well as the ground spin-state and make the experimental determination of both properties challenging. Therefore, first-principles calculations, in particular employing density functional theory with a proper choice of exchange-correlation (xc) functional, are crucial. Common xc functionals were tested on a simple class of metal complexes: homoleptic, octahedral tris(diimine) iron(II) complexes. The spin-state energy splittings for most of these complexes showed the expected linear dependence on the amount of exact exchange included in the xc functionals. Even though varying redox-activity affects the electronic structure of the complexes considerably, the sensitivity of the spin-state energetics to the exact exchange admixture is surprisingly small. For iron(II) complexes with highly redox-active ligands and for a broad range of ligands in the reduced tris(diimine) iron(I) complexes, self-consistent field convergence to local minima was observed, which differ from the global minimum in the redox state of the ligand. This may also result in convergence to a molecular structure that corresponds to an energetically higher-lying local minimum. One criterion to detect such behavior is a change in the sign of the slope for the dependence of the spin-state energy splittings on the amount of exact exchange. We discuss possible protocols for dealing with such artifacts in cases in which a large number of calculations makes checking by hand unfeasible.     

Indirect detection of nitrogen-14 in solid-state NMR spectroscopy.

NMR spectra of (14)N (spin I=1) are obtained by indirect detection in powders spinning at the magic angle. The method relies on the transfer of coherence from a neighboring "spy" nucleus with S=1/2, such as (13)C or (1)H, to single- or double-quantum transitions of (14)N nuclei. The transfer of coherence can occur through a combination of scalar and residual dipolar splittings (RDS); the latter are also known as second-order quadrupole-dipole cross terms. The two-dimensional NMR spectra reveal powder patterns determined by second- and third-order quadrupolar couplings. These spectra depend on the quadrupolar coupling constant C(Q) (typically a few megahertz), on the asymmetry parameter eta(Q) of the (14)N nucleus, and on the orientation of the internuclear vector r(IS) between the I ((14)N) and S (spy) nuclei with respect to the quadrupolar tensor. These parameters, which can be subject to motional averaging, can reveal valuable information about the structure and dynamics of nitrogen-containing solids. Application of this technique to various amino acids, either enriched in (13)C or with natural carbon isotope abundance, with spectra recorded at various magnetic fields, illustrates the scope of the method.     

Local Multigrid in H（curl）

We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.     

A phenomenological bag model with variable bag pressure

We examine the consequences of a variable (density-dependent) bag pressure term and a fixed hadronic size in the phenomenological MIT bag model for hadron spectroscopy. Mass spectrum of the low-lying baryons and mesons, baryon magnetic moments and the hadron mass splittings are estimated. These are found to be in closer agreement with experiment than the MIT results.     

119Sn electric field gradients in model clusters of chalcogenide glasses

Calculations of ¹¹⁹Sn electric field gradients (EFG) have been performed using the Extended Hückel approximation on characteristic molecular clusters simulating possible types of sites in chalcogenide glasses. The motivation for these calculations derives from theoretical concepts on varying near neighbor relationships in these types of glasses, and from recent ¹¹⁹Sn Mössbauer experiments on Sn-doped Ge_x(Se or S)_1–x bulk glasses which reveal three types (A, B and C) of chemically inequivalent sites, with distinct values and composition dependences for their isomer shifts and quadrupole splittings. The model clusters chosen for the calculations were the ethane-like (Ge₂Se₃)_n quasi-one-dimensional chains of varying lengths which have been proposed as possible sources of the B site. In addition, calculations were also carried out on several additional types of clusters, in order to help in interpreting the results for the chains. We find that the magnitude of the quadrupole splitting in isolated linear ethane-like chains is very small, and almost independent of the particular site along the chain at which Sn replaces Ge. It therefore seems unlikely that such isolated linear clusters would be the source of the B sites. These sites are more likely to be related to distortions of the ethane-like clusters into non-linear configurations, as well as interactions with neighboring clusters, as forced by the constraints of the packing in the structure of the glass.