Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems

Summary. The convergence rate of Krylov subspace methods for the solution of nonsymmetric systems of linear equations, such as GMRES or FOM, is studied. Bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse. Estimates for these quantities are available during the iteration from the underlying Arnoldi process. It is shown how these bounds can be used to study the convergence properties, in particular, the dependence on the mesh-size and on the size of the skew-symmetric part, for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems. This is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems. Received May 3, 1996     

Two modifications of a five‐coordinate silicon complex

TBPY‐5‐34‐(Butane‐1,4‐diyl)(2‐{[1‐(2‐oxidophenyl)ethylidene‐κO]amino‐κN}ethanolato‐κO)silicon, C₁₄H₁₉NO₂Si, crystallizes in two modifications. The monoclinic form, (IIm), was obtained by crystallization over a period of 2 d at room temperature; the orthorhombic form, (IIo), crystallized overnight at 248 K. The main difference between the two molecular structures involves the angles in the equatorial plane of the trigonal bipyramid around silicon. Form (IIm) has an O—Si—O angle of ca 121° and O—Si—C angles of ca 121 and 116°. In form (IIo), the corresponding angles are ∼123, 124 and 111°. There are also significant differences in the packing: (IIm) shows π stacking, whereas (IIo) does not.     

Conductance in homomorphous solvents: Tetrabutylammonium salts and alkali picrates in butyrolactone-sulfolane mixtures

Conductance measurements are reported for LiPi, NaPi, KPi, RbPi, CsPi, Bu₄NPi, Bu₄NBr, Bu₄NClO₄, Bu₄NNO₃, and Bu₄NBBu₄ at 25°C in -butyrolactone-sulfolane mixtures. In these mixtures of solvents that are practically homomorphous, isodielectic and with comparable dipole moments, the ion pair association and ionic mobilities of large ions conform to the expectations of the primitive model. Electrolytes containing lithium or sodium ions show anomalies indicating that other factors besides shape, dipole moment, and polarizability of the solvent molecules are involved in the association and transport processes of these ions.     

Quecksilber(II)‐chlorid‐ und ‐iodid‐Komplexe von Dithia‐ und Tetrathiakronenethern

Mercury(II) Chloride and Iodide Complexes of Dithia‐ and Tetrathiacrown Ethers The complexes [(HgCl₂)₂((ch)₂30S₄O₆)] ( 1 ), [HgCl₂(mn21S₂O₅)] ( 2 ), [HgCl₂(ch18S₂O₄)] ( 3 ) and [HgI(meb12S₂O₂)]₂[Hg₂I₆] ( 4 ) have been synthesized, characterized and their crystal structures were determined. In [(HgCl₂)₂((ch)₂30S₄O₆)] two HgCl₂ units are discretely bonded within the ligand cavity of the 30‐membered dichinoxaline‐tetrathia‐30‐crown‐10 ((ch)₂30S₄O₆) forming a binuclear complex. HgCl₂ forms 1 : 1 “in‐cavity” complexes with the 21‐membered maleonitrile‐dithia‐21‐crown‐7 (mn21S₂O₅) ligand and the 18‐membered chinoxaline‐dithia‐18‐crown‐6 (ch18S₂O₄) ligand, respectively. The 12‐membered 4‐methyl‐benzo‐dithia‐12‐crown‐4 (meb12S₂O₂) ligand gave with two equivalents HgI₂ the compound [HgI(meb12S₂O₂)]₂[Hg₂I₆]. In the cation [HgI(meb12S₂O₂)]⁺ meb12S₂O₂ forms with the cation HgI⁺ a half‐sandwich complex.     

In situ observations of water production and distribution in an operating H2/O2 PEM fuel cell assembly using 1H NMR microscopy

Proton NMR imaging was used to investigate in situ the distribution of water in a polymer electrolyte membrane fuel cell operating on H2 and O2. In a single experiment, water was monitored in the gas flow channels, the membrane electrode assembly, and in the membrane surrounding the catalysts. Radial gradient diffusion removes water from the catalysts into the surrounding membrane. This research demonstrates the strength of 1H NMR microscopy as an aid for designing fuel cells to optimize water management.     

SNAr reactions of β-substituted porphyrins and the synthesis of meso substituted tetrabenzoporphyrins

Reaction of 2,3,7,8,12,13,17,18-octaethylporphyrin with LiR reagents containing functional groups readily yields meso substituted derivatives suitable for further transformations with residues such as -p-C₆H₅Br, -p-C₆H₅-CCH, -p-C₆H₅-NH₂ or -(CH₂)₃-CHCH₂. Similar reactions of tetrabenzoporphyrin with alkyllithium reagents afforded the first entry into meso mono- and dialkylsubstituted tetrabenzoporphyrins while reaction of bicyclo[2.2.2]oct-type masked isoindole precursors with LiR followed by in situ retro-Diels-Alder reaction also afforded the 5-phenyl and 5,10-diphenyltetrabenzoporphyrins in high purity.     

New class of potent catalysts of O2.-dismutation. Mn(III) ortho-methoxyethylpyridyl- and di-ortho-methoxyethylimidazolylporphyrins

Three new Mn(III) porphyrin catalysts of O2.-dismutation (superoxide dismutase mimics), bearing ether oxygen atoms within their side chains, were synthesized and characterized: Mn(III) 5,10,15,20-tetrakis[N-(2-methoxyethyl)pyridinium-2-yl]porphyrin (MnTMOE-2-PyP(5+)), Mn(III)5,10,15,20-tetrakis[N-methyl-N'-(2-methoxyethyl)imidazolium-2-yl]porphyrin (MnTM,MOE-2-ImP(5+)) and Mn(III) 5,10,15,20-tetrakis[N,N'-di(2-methoxyethyl)imidazolium-2-yl]porphyrin (MnTDMOE-2-ImP(5+)). Their catalytic rate constants for O2.-dismutation (disproportionation) and the related metal-centered redox potentials vs. NHE are: log k(cat)= 8.04 (E(1/2)=+251 mV) for MnTMOE-2-PyP(5+), log k(cat)= 7.98 (E(1/2)=+356 mV) for MnTM,MOE-2-ImP(5+) and log k(cat)= 7.59 (E(1/2)=+365 mV) for MnTDMOE-2-ImP(5+). The new porphyrins were compared to the previously described SOD mimics Mn(III) 5,10,15,20-tetrakis(N-ethylpyridinium-2-yl)porphyrin (MnTE-2-PyP(5+)), Mn(III) 5,10,15,20-tetrakis(N-n-butylpyridinium-2-yl)porphyrin (MnTnBu-2-PyP(5+)) and Mn(III) 5,10,15,20-tetrakis(N,N'-diethylimidazolium-2-yl)porphyrin (MnTDE-2-ImP(5+)). MnTMOE-2-PyP(5+) has side chains of the same length and the same E(1/2), as MnTnBu-2-PyP(5+)(k(cat)= 7.25, E(1/2)=+ 254 mV), yet it is 6-fold more potent a catalyst of O2.-dismutation , presumably due to the presence of the ether oxygen. The log k(cat)vs. E(1/2) relationship for all Mn porphyrin-based SOD mimics thus far studied is discussed. None of the new compounds were toxic to Escherichia coli in the concentration range studied (up to 30 microM), and protected SOD-deficient E. coli in a concentration-dependent manner. At 3 microM levels, the MnTDMOE-2-ImP(5+), bearing an oxygen atom within each of the eight side chains, was the most effective and offered much higher protection than MnTE-2-PyP(5+), while MnTDE-2-ImP(5+) was of very low efficacy.     

Hair color changes and protein damage caused by ultraviolet radiation

Ultraviolet and visible radiations are known to damage hair. However, quantitative data relating damage to hair type, proteins and color to the radiation wavelength are missing. We studied the effect of UV plus visible, UVA plus visible, visible mercury-vapor lamp radiation and sunlight on (blended) virgin dark-brown, blond and red hair and (one head) virgin black and curly dark-brown hair. All hair types showed a substantial increase in protein loss in water after lamp and sun irradiation. The damaging effect of UVB was about 2-5 times higher than that of UVA plus visible radiation, depending on the hair type. Significant color changes were also observed in every hair type, after lamp and sun irradiation, being more pronounced for the light colored hairs. The luminosity difference parameter was the major contributor to the hair color changes, but significant changes in the red-green and yellow-blue parameters of every hair were observed. In this case, the damaging effect is ascribable mainly to UVA radiation. No significant changes in the mechanical properties or topography were observed in any case. We discuss these results in terms of hair type and composition and melanin types.     

A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations

Summary We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ^{N, N} , withA nonsingular, andb ^N are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.