Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Traceability issues in measurement

This paper reviews the current state of play of the Mutual Recognition Arrangement created by the International Committee for Weights and Measures in 1999. The aim of the MRA is to provide a framework within which National Metrology Institutes can demonstrate the equivalence of their realisations of the units and quantities of the SI system to which accredited laboratories are traceable. The article offers some views on the need for traceable measurements, their relevance to technical barriers to trade, and the use that is being made of the MRA framework by national and international bodies.     

Europium-151 Mössbauer spectroscopic and XANES investigation of europium-exchanged Y-zeolite

Eu³⁺ in ca. 10 wt% europium-exchanged Y-zeolite is partially reduced by treatment in hydrogen at 600°C to Eu²⁺. The reduction of Eu³⁺ is more readily achieved in Y-zeolite than in europium(III) oxide. The discrepancy in the extent of reduction as revealed by¹⁵¹Eu Mössbauer spectroscopy and near edge X-ray absorption fine structure (XANES) is associated with any difference in the recoil free fractions of Eu²⁺ and Eu³⁺ which may exist at 298 K and the enhanced sensitivity of the XANES to changes in the europium oxidation state.     

3,4‐[2,2‐Bis(methoxyethoxymethoxymethyl)propylenedithio]‐3′,4′‐(ethylenedithio)tetrathiafulvalene: a spiro‐substituted BEDT–TTF analogue

The crystal structure of the title compound [systematic name: 2‐(1,3‐dithiolo[4,5‐b][1,4]dithiin‐2‐ylidene)‐6,6‐bis(methoxyethoxymethoxymethyl)‐1,3‐dithiolo[4,5‐b][1,4]dithiepine], C₂₁H₃₀O₆S₈, a spiro‐substituted BEDT–TTF analogue [BEDT–TTF is bis(ethylenedithio)tetrathiafulvalene], has a strongly bent heterocyclic framework. The seven‐membered ring adopts a pseudo‐chair conformation with notably widened ring bond angles, especially at the methylene C atoms [119.49 (11) and 117.60 (11)°]. The axial side chain adopts an extended conformation, but the equatorial side chain curls back on itself and the O atom nearest the ring system is involved in three short contacts to H atoms (2.45–2.53 Å). The molecules pack in centrosymmetrically related pairs, which are isolated from each other by columns of the polyether side chains. This study emphasizes the ease of distortion of the neutral bis(propylenedithio)tetrathiafulvalene ring structure, and how the need to accommodate side chains can easily override the tendency of these donor systems to form stacks in the crystalline state.     

Structures and syntheses of layered and framework amine-bearing uranyl phosphate and uranyl arsenates

Two hydrated uranyl arsenates and a uranyl phosphate were synthesized by hydrothermal methods in the presence of amine structure-directing agents and their structures determined: (N₂C₆H₁₄)[(UO₂)(AsO₄)]₂(H₂O)₃, DabcoUAs, {NH(C₂H₅)₃}[(UO₂)₂(AsO₄)(AsO₃OH)], TriethUAs, and (N₂C₄H₁₂)(UO₂)[(UO₂)(PO₄)]₄(H₂O)₂, PiperUP. Intensity data were collected at room temperature using MoKα X-radiation and a CCD-based area detector. The crystal structures were refined by full-matrix least-squares techniques on the basis of F² to agreement indices (DabcoUAs, TriethUAs, PiperUP) wR₂=5.6%, 8.3%, 7.2% for all data, and R₁=2.9%, 3.3%, 4.0%, calculated for 1777, 5822, 9119 unique observed reflections (|F_o|?4σ_F), respectively. DabcoUAs is monoclinic, space group C2/m, Z=2, a=18.581(1), b=7.1897(4), c=7.1909(4) Å, β=102.886(1)°, V=936.43(9) Å³, D_calc=3.50 g/cm³. TriethUAs is monoclinic, space group P2₁/n, Z=4, a=9.6359(4), b=18.4678(7), c=10.0708(4) Å, β=92.282(1)°, V=1790.7(1) Å³, D_calc=3.41 g/cm³. PiperUP is monoclinic, space group Pn, Z=2, a=9.3278(4), b=15.5529(7), c=9.6474(5) Å, β=93.266(1)°, V=1397.3(1) Å³, D_calc=4.41 g/cm³. The structure of DabcoUAs contains the autunite-type sheet formed by the sharing of vertices between uranyl square bipyramids and arsenate tetrahedra. The triethylenediammonium cations are located in the interlayer along with two H₂O groups and are disordered. Both TriethUAs and PiperUP contain sheets formed of uranyl pentagonal bipyramids and tetrahedra (arsenate and phosphate, respectively) with the uranophane sheet-anion topology. In TriethUAs, triethlyammonium cations are located in the interlayer. In PiperUP, the sheets are connected by a uranyl pentagonal bipyramid that shares corners with phosphate tetrahedra of adjacent sheets, resulting in a framework with piperazinium cations and H₂O groups in the cavities of the structure.     

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n ² m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.     

Practical syntheses of [1,3,5]‐triazine dendritic molecules on solid supports

A practical and divergent synthesis of supported [1,3,5]‐triazine dendritic molecules on Wang resin, PEGA resin, SynPhase? Lanterns, and silica gel is described. The alkylamine linkers used allow derivatization with functionality for both synthetic (e.g., supported reagent and scavenger activity) and chemical biology applications. The use of supported intermediates allows differentiation of symmetric linkers without the need for protecting group chemistry. The synthetic route uses inexpensive, readily available starting materials in a straightforward and scaleable strategy. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2248–2259, 2006