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.     

High-Resolution,Higher-Order UV/VIS Derivative Spectrophotometry

Derivative spectrophotometry has gained increasing importance in the past two years and is currently experiencing vigorous development. Following and introduction the present article provides a review of this extremely effective method. In particular, the advantages of higher-order derivative spectrophotometry (HODS method, n > 2) are discussed on the basis of practical examples from a wide range of analytical fields. The results are achieved with the aid of a newly developed analog computer unit, whereby for the first time readily reproducible, low-noise, on-line spectra can be obtained up to the 7th order and even, in favorable cases, up to the 9th order. In practice it has proved valuable to work with spectra of the 3rd to 5th order; but even higher derivations could be profitable for the separation of strongly superposed signals or for “fingerprinting”.     

Asymmetric synthesis of unsaturated, fused bicyclic proline analogues through amino alkylation of cyclic bis(allylsulfoximine)titanium complexes and migratory cyclization of delta-amino alkenyl aminosulfoxonium salts

Described is an asymmetric synthesis of new Delta(3a,4)-unsaturated, fused bicyclic proline analogues from cyclic bis(allylsulfoximine)titanium complexes and N-tert-butylsulfonyl imino ethyl ester. Treatment of the enantiomerically pure five-, six-, seven-, and eight-membered cyclic bis(allylsulfoximine)titanium complexes with the imino ester gave mixtures of the corresponding (E,syn)- and (Z,syn)-configured, delta-sulfoximine substituted, cyclic gamma,delta-unsaturated alpha-amino acid esters with high regio- and diastereoselectivities in good yields. Activation of the N-methyl sulfoximine group of these amino acid derivatives through methylation with Me(3)OBF(4) afforded in nearly quantitative yields the corresponding (dimethylamino)sulfoxonium salts. A novel migratory cyclization of these salts with DBU gave via an isomerization to the corresponding allylic (dimethylamino)sulfoxonium salts and an intramolecular substitution of the (dimethylamino)sulfoxonium group the enantio- and diastereomerically pure, bicyclic, N-tert-butylsulfonyl protected proline analogues having a six- and eight-membered unsaturated carbocyclic ring. Cyclization of the alkenyl (dimethylamino)sulfoxonium salts was independent of the configuration of the double bond. N,N-Dimethylphenylsulfinamide of > or =99% ee was obtained in good yield as a further reaction product. Conversion of the sulfinamide to N,S-dimethyl-S-phenylsulfoximine of > or =99% ee, the starting material for the synthesis of the allylic sulfoximines, had been accomplished previously. Finally, cleavage of the tert-butylsulfonyl protecting group with anhydrous acid furnished the fused bicyclic proline analogue containing an unsaturated six-membered ring in high yield.     

Synthesis of pyrroles from 1-dialkylamino-3-phosphoryl(or phosphanyl)allenes through 1,5-cyclization of conjugated azomethine ylide intermediates

1-Dialkylamino-1,3-diaryl-3-diphenylphosphanylallenes 3a-e are thermally converted into a-annulated 3,5-diarylpyrroles 6a-f and [a]-annulated benzo[c]azepines 7a,b,d. These transformations are likely to include conjugated azomethine ylide intermediates that can undergo either a 1,5- or a 1,7-electrocyclization. The periselectivity is markedly shifted toward 1,5-cyclization when the diphenylphosphanyl substituent is replaced by the diphenylphosphoryl group. Thus, 1-dialkylamino-3-(diphenylphosphoryl)allenes 4a-f yield pyrroles 6 exclusively and with improved yields, unless the 3-aryl substituent in the allene is too electron-rich (e.g., benzodioxol-5-yl, 4f --> 7f). The preparation and thermal transformation of aminoallenes 4 over three or four steps can be conducted as a one-pot procedure, thus providing a convenient synthesis of [a]-annulated 3,5-diarylpyrroles from enaminoketones.     

Structure and reactivity of xanthocorrinoids. Part V. Formation of trans-diol derivatives of 5,6-dihydrocobyrinic acid from xanthocorrinoids under acidic conditions

On treatment with H₂SO₄/MeOH, epimerization of hexamethyl cis-5,6-dihydroxycobyrinate c,8-lactam ( 3 ) takes place quantitatively at C(6), yielding the corresponding trans-diol 4 . The corresponding lactone 7 , whose structure has been established by X-ray analysis, is obtained from xanthocorrinoids 5 and 6 under similar conditions.     

Was ist „Molybdänsäure”︁ oder „Polymolybdänsäure”︁?

What is “Molybdic Acid” or “Polymolybdic Acid”? According to a comparative study of the literature, supplemented by well-aimed experimental investigations and equilibrium calculations, the terms “molybdic acid” or “polymolybdic acid”, used for many substances, species, or solutions in the literature, are applicable to a species, a solution, and two solids:

• a) The monomeric molybdic acid, most probably having the formula MoO₂(OH)₂(H₂O)₂(? H₂MoO₄, aq), exists in (aqueous) solution only and never exceeds a concentration of ≈ 10^?3 M since at higher concentrations it reacts with other monomemeric molybdenum (VI) species to give anionic or cationic polymers.
• b) A concentrated (>0.1 M Mo^VI) aqueous molybdate solution of degree of acidification P = 2 (realized, e. g., by a solution of one of the Mo^VI oxides; by any molybdate solutions whose cations have been exchanged by H₃O⁺ on a cation exchanger; by suitable acidification of a molybdate solution) contains 8 H₃O⁺ and the well-known polyanion Mo₃₆O₁₁₂(H₂O)₁₆^8? exactly in the stoichiometric proportions.
• c) A glassy substance, obtained from an alkali metal salt-free solution prepared according to (b), refers to the compound (H₃O)₈[Mo₃₆O₁₁₂(H₂O)₁₆]·xH₂O, x = 25—29.
• d) A solid having the ideal composition [(H₃O)Mo₅O₁₅(OH)H₂O·H₂O]∞ consists of a polymolybdate skeleton (the well-known ?decamolybdate”? structure), in the tunnels of which H₃O⁺ and H₂O are intercalate. The structure is very unstable if only H₃O⁺ cations are present, but it is enormously stabilized by a partial exchange of H₃O⁺ by certain alkali or alkaline earth metal cations.

For the compounds MoO₃, MoO₃·H₂O, and MoO₃·2H₂O the term ?molybdic acid”? is unjustified. The commercial product ?molybdic acid, ≈85% MoO₃”? is the well-known polymolybdate (NH₄)₂O·4 MoO₃ with a layer structure of the polyanion.