Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455     

b1Σ+ Emissions from group V–VII diatomic molecules. b 0+ → X1 0+, X2 1 emissions of AsI and SbI

Chemiluminescence from the b 0⁺ → X₁ 0⁺ band system of AsI and of the b 0⁺ → X₁ 0⁺, X₂ 1 systems of SbI in the near-infrared spectral region has been observed in a discharge flow system. Analysis of the spectra led to the spectroscopic constants (in cm^?1) of AsI: ω_e(X₁, X₂) = 257 ± 2, ω_ex_e(X₁, X₂) = 0.82 ± 0.2, T_e(b 0⁺) = 11738 ± 5, ω_e(b 0⁺) = 271 ± 2, ω_ex_e(b 0⁺) = 0.66 ± 0.2, and of SbI: T_e(X₂ 1) = 965 ± 10, ω_e(X₁, X₂) = 206 ± 6, T_e(b 0⁺) = 12328 ± 10, ω_e(b 0⁺) = 211 ± 6. The intensity ratio of the two sub-systems b 0⁺ → X₂ 1 and b 0⁺→ X₁ 0⁺ was found to be ≈0.013 in the case of SbI and ? 0.01 for AsI.     

Square‐Planar Carbonylchlororhodium(I) Complexes Containing trans‐Spanning Diphosphine Ligands as Catalysts for the Carbonylation of Methanol

The rhodium(I) complexes trans‐[Rh(diphos)(CO)Cl] 7 (diphos=pbpb), 8 (diphos=nbpb), and 9 (diphos=cbpb) were synthesized (Scheme 4) and used as catalysts for the carbonylation of MeOH to AcOH (Scheme 1). The trans coordination imposed by the rigid C‐spacer framework of the diphos ligands pbpb, nbpb, and cbpb, demonstrated by ³¹P‐NMR and IR spectroscopy of 7 – 9 and unambiguously confirmed by single‐crystal X‐ray structure analysis of 7 , improved the thermal stability of the rhodium(I) system under carbonylation conditions and, hence, the catalytic performance of the complexes. For the catalytic carbonylation of MeOH, the active catalyst could be prepared in situ from the mixture of [Rh(CO)₂Cl]₂ and the corresponding diphos ligand pbpb, nbpb, or cbpb, giving the same results as carbonylation in the presence of the isolated complexes 7, 8 or 9 (see Table). The highest activity was observed for complex 7 (or the mixture [Rh(CO)₂Cl]₂/pbpb, the catalytic turnover number (TON) being 950 after 15 min (170°, 22 bar).     

Dyeing uric acid crystals with methylene blue

The growth of anhydrous uric acid (UA) and uric acid dihydrate (UAD) crystals from supersaturated aqueous solutions containing methylene blue, a cationic organic dye, has been investigated. Low concentrations of dye molecules were found to be included in both types of crystal matrixes during the growth process. Incorporation of dye into UA crystals occurs with high specificity, affecting primarily [001] and [201] growth sectors, while UAD crystals grown from solutions of similar dye concentration show inclusion but little specificity. The orientation of the UA-trapped species was determined from polarization data obtained from visible light microspectrometry. To achieve charge neutrality, a second anionic species must also be included with the methylene blue into UA and UAD crystal matrices. Under high pH conditions, crystallization of 1:1 stoichiometric mixtures of methylene blue and urate yields methylene blue hexahydrate (MBU.6(H2O). The crystal structure of MBU.6(H2O) reveals continuous pi-pi stacks of planes of dye cations and urate anions mediated by water molecules. This structure provides an optimal geometry for methylene blue-urate pairs and additional support for the incorporation of these dimers in uric acid single-crystal matrices. The strikingly different inclusion patterns in UA and UAD demonstrate that subtle changes in the crystal surfaces and/or growth dynamics can greatly affect recognition events.     

Platinum(II) 1,10-phenanthroline complexes of acetylides containing redox-active groups

The new diimine ligand 3,8-di-n-pentyl-4,7-di(phenylethynyl)-1,10-phenanthroline (1) was used for the synthesis of a range of Pt(II) complexes, viz.[Pt(1)Cl2], [Pt(1)(C triple bond C-Ph)2], [Pt(1)(C triple bond C-Fc)2] and [Pt(1)(C triple bond C-p-C6H4-C triple bond C-Fc)2](Fc = ferrocenyl). Crystal structure analyses were performed for [Pt(1)Cl2] and [Pt(1)(C triple bond C-Ph)2] and revealed that the di(acetylide)pi-tweezer of the latter binds a molecule of chloroform through C-H...pi hydrogen bonds. The redox and optical properties of 1 and its complexes were investigated by (spectro-)electrochemistry, UV-Vis and luminescence spectroscopy, and an energy level diagram was derived for [Pt(1)(C triple bond C-Fc)2] and related compounds on the basis of the data collected. The ferrocenyl-substituted Pt(II) complexes are donor-sensitiser assemblies. Intramolecular quenching of the photoexcited Pt(II) diimine unit leads to very short luminescence lifetimes for [Pt(1)(C triple bond C-p-C(6)H(4)-C triple bond C-Fc)2](2 ns) and [Pt(1)(C triple bond C-Fc)2](0.3 ns), as opposed to [Pt(1)(C triple bond C-Ph)2](0.7 micros). Excimer formation has been observed for [Pt(1)(C triple bond C-Ph)(2)] at room temperature in dichloromethane and at low temperatures in frozen glassy dichloromethane and 2-methyltetrahydrofuran solution, but not in the solid state.     

Growth and melting behaviour of thin in films on Ge(100)

Growth and melting behaviour of thin indium films on Ge(100) have been investigated by Auger-electron spectroscopy (AES), atomic force microscopy (AFM) and perturbed angular correlation (PAC) spectroscopy, respectively. At room temperature inidium is found to grow in three-dimensional islands even at submonolayer coverages. A very rough film surface is observed for thicknesses up to 230 ML. The melting behaviour of such films has been studied by PAC. A reduction of the melting temperature T _m as well as a strong supercooling of the films is observed. The electric field gradient for ¹¹¹In(¹¹¹Cd) in the indium islands is determined as a function of temperature and is used to monitor the local crystalline order of the films up to temperatures just below the melting point.     

Stratospheric chemical ionization mass spectrometry: nitric acid detection by different ion molecule reaction schemes

Detailed height profiles of stratospheric nitric acid mixing ratios have been derived with a baloon borne chemical ionization mass spectrometer by applying several ion molecule reaction schemes, each associated to a specific and selective ion source. These ions (CO₃⁻, Cl_n⁻, CF₃O⁻, and CF₃O⁻H₂O) give rise to specific product ions (mainly CO₃⁻HNO₃, NO₃⁻HCl, NO₃⁻HF, and CF₃O⁻HNO₃) upon reaction with ambient nitric acid molecules. This paper reports on the instrumental details as well as on the results obtained during two balloon flights with the instrument. Within the accuracy of the measurements, the nitric acid height profiles obtained with the three different ion sources are in good agreement with one another as well as with literature data.     

Some aspects of tailoring flame retardancy in HET-acid-based flame retardants

HET acid was condensed with either ethylene glycol or 1,4-butanediol to yield the respective oligoesters. These oligoesters were characterized by molecular weight, measured by vapor-pressure osmometry. The degradation mechanism of oligoesters of HET acid with either ethylene glycol or 1,4-butanediol was elucidated. The thermal properties of these polyesters were studied using differential thermal analysis and thermal volatilization analysis. The mechanism of degradation was proposed on the basis of pyrolysis–gas chromatography studies, the degradation products being subsequently identified by mass spectrometry. It turned out that in the course of decomposition, hexachlorocyclopentadiene was formed, which is believed to be the active substance for flame retardance in these systems. The aspect of tailoring flame retardancy by changing the chemical environment, i.e., by using different diols, is discussed.