Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen

Summary We consider two-point boundary value problems for linear differential equations (of ordern). The main part consists in proving stronger apriori estimates for (mn–1) using discreteL ₁-norms of the involved right-hand sides; thus extending known results for discreteL _p -norms with 2p. These estimates are the main tool in proving sharp estimates for the error \y–y _h \ _h, as well asfor \P _y –P _h y _h \ _h, , whereP _h denotes a consistent approximation of the differential operatorP of ordermn–1, in terms of the discretL ₁-norm of the truncation error. By means of an interpolation technique these estimates yield a correct estimate of the order of convergence, also in case that the solution satisfies locally some Lipschitz-conditions.     

Modulated differential scanning calorimetry in the glass transition region

Temperature-modulated calorimetry (TMC) allows the experimental evaluation of the kinetic parameters of the glass transition from quasi-isothermal experiments. In this paper, model calculations based on experimental data are presented for the total and reversing apparent heat capacities on heating and cooling through the glass transition region as a function of heating rate and modulation frequency for the modulated differential scanning calorimeter (MDSC). Amorphous poly(ethylene terephthalate) (PET) is used as the example polymer and a simple first-order kinetics is fitted to the data. The total heat flow carries the hysteresis information (enthalpy relaxation, thermal history) and indications of changes in modulation frequency due to the glass transition. The reversing heat flow permits the assessment of the first and higher harmonics of the apparent heat capacities. The computations are carried out by numerical integrations with up to 5000 steps. Comparisons of the calculations with experiments are possible. As one moves further from equilibrium, i.e. the liquid state, cooperative kinetics must be used to match model and experiment.On leave from Toray Industries, Inc., Otsu, Shiga 520, Japan.This work was supported by the Division of Materials Research, National Science Foundation, Polymers Program, Grant # DMR 90-00520 and the Division of Materials Sciences, Office of Basic Energy Sciences, U. S. Department of Energy at Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. for the U. S. Department of Energy, under contract number DE-AC05-96OR22464. Support for instrumentation came from TA Instruments, Inc. Research support was also given by ICI Paints, and Toray Industries, Inc.     

The first enantiomerically pure [n]triangulanes and analogues: sigma-[n]helicenes with remarkable features

(M)-(-)- and (P)-(+)-Trispiro[2.0.0.2.1.1]nonanes [(M)- and (P)-3] as well as (M)-(-)- and (P)-(+)-tetraspiro[2.0.0.0.2.1.1.1]undecanes [(M)- and (P)-4]-enantiomerically pure unbranched [4]- and [5]triangulanes-have been prepared starting from racemic bicyclopropylidenecarboxylic [(1RS)-12] and exo-dispiro[2.0.2.1]heptane-1-carboxylic [(1RS,3SR)-13] acids. The optical resolutions of rac-12 and rac-13 furnished enantiomerically pure acids (S)-(+)-12, (R)-(-)-12, (1R,3S)-(-)-13, and (1S,3R)-(+)-13. The ethyl ester (R)-25 of the acid (R)-(-)-12 was cyclopropanated to give carboxylates (1R,3R)-26 and (1R,3S)-26. The ester (1R,3S)-26 and acids (1R,3S)-13 and (1S,3R)-13 were converted into enantiomerically pure methylene[3]triangulanes (S)-(-)- and (R)-(+)-28. An alternative approach consisted of an enzymatic deracemization of endo-[(1SR,3SR)-dispiro[2.0.2.1]heptyl]methanol (rac-20) or anti-[(1SR,3RS)-4-methylenespiropentyl]methanol (rac-18). This afforded (S)-(-)- and (R)-(+)-28 (starting from rac-20), as well as enantiomerically pure (M)-(-)- and (P)-(+)-1,4-dimethylenespiropentanes [(M)- and (P)-23] starting from rac-18. The methylenetriangulanes (S)-(-)- and (R)-(+)-28 were cyclopropanated furnishing (M)- and (P)-3. The rhodium-catalyzed cycloaddition of ethyl diazoacetate onto (S)-(-)- and (R)-(+)-28 yielded four diastereomeric ethyl trispiro[2.0.0.2.1.1]nonane-1-carboxylates in approximately equal proportions. The enantiomerically pure esters (1R,3S,4S)- and (1S,3R,4R)-30 were isolated by careful distillation and then transformed into [5]triangulanes (M)- and (P)-4 using the same sequence of reactions as applied for (M)- and (P)-3. The structures of the key intermediates (R)-12 and rac-31 were confirmed by X-ray analyses. Although [4]- and [5]triangulanes have no chromophore which would lead to any significant absorption above 200 nm, they have remarkably high specific rotations even at 589 nm with [alpha](20)D=-192.7 [(M)-3, c=1.18, CHCl(3))] or +373.0 [(P)-4, c=1.18, CHCl(3))]. This remarkable optical rotatation is in line with their helical arrangement of sigma bonds, as confirmed by a full valence space single excitation configuration interaction treatment (SCI) in conjunction with DFT computations at the B3LYP/TZVP//B3LYP/6-31+G(d,p) level of theory which reproduce the ORD very well. Thus, it is appropriate to call the helically shaped unbranched [n]triangulanes the "sigma-[n]helicenes", representing the sigma-bond analogues of the aromatic [n]helicenes.     

Reaction of 5,6,7,8-Tetrahydropterin with Iron(III) Acetylacetonate. Detection of Radical Cations by Electrospray Ionization Mass Spectrometry

The experimental conditions developed for the detection of rather stable radical cations in solution by electrospray-ionization mass spectrometry (ESI-MS) of a Fe^II complex of 2-amino-5,6,7,8-tetrahydro-5-methylpteridin-4 (3H)-one ( 1c ) are used to observe the formation of the more unstable radical cations formed from 2-amino-5,6,7,8-tetrahydropteridin-4(3H)-one ( 1a ) and tris(pentane-2,4-dionato)iron(III) ([Fe^III(acac)₃]; 4 ) and to monitor their oxidation to the corresponding p-quinonoid dihydropterin complexes. These results contribute to the understanding of the important role played by 6β-5,6,7,8-tetrahydro-L -biopterin ( 1b ; a homologue of 1a ) together with iron as constituent of some cofactors. The complexes obtained from 1a and iron may be considered, e.g. as a model of the cofactor of the phenylalanine hydroxylase. Moreover, we describe an improved synthesis of 1c .     

AES investigations on starting powders for high performance ceramics

AES depth profiles on ceramic powders (untreated/hydrolyzed/oxidized/ (Al, Y)₂O₃ coated Si₃N₄, [BaO, SiO₂] coated Al₂O₃) are feasible on thin, homogeneous layers or m sized agglomerations prepared on an Au foil. By means of the depth profiles one can qualitatively characterize the coating around the particles. Factor analysis of the depth profiles on the differently treated Si₃N₄ powders suggests the existence of an Si₂N₂O phase on the oxidized sample.Dedicated to Professor Günther Tölg on the occasion of his 60th birthday     

C[triple‐bond]C—H systems as hydrogen‐bond donors and acceptors: trans‐1,2‐diethynylcyclo­hexane‐1,2‐diol and trans‐1,4‐diprop‐2‐ynylcyclo­hexane‐1,4‐diol monohydrate

The title 1,2‐diol derivative, C₁₀H₁₂O₂, crystallizes with two independent but closely similar mol­ecules in the asymmetric unit. Only two of the four OH groups are involved in classical hydrogen bonding; the mol­ecules thereby associate to form chains parallel to the short c axis. The other two OH groups are involved in O—H⋯(C[triple‐bond]C) systems. Additionally, three of the four C[triple‐bond]C—H groups act as donors in C—H⋯O inter­actions. The 1,4‐diol derivative crystallizes with two independent half‐mol­ecules of the diol (each associated with an inversion centre) and one water mol­ecule in the asymmetric unit, C₁₂H₁₆O₂·H₂O. Both OH groups and one water H atom act as classical hydrogen‐bond donors, leading to layers parallel to the ac plane. The second water H atom is involved in a three‐centre contact to two C[triple‐bond]C bonds. One acetyl­enic H atom makes a very short `weak' hydrogen bond to a hydr­oxy O atom, and the other is part of a three‐centre system in which the acceptors are a hydroxy O atom and a C[triple‐bond]C bond.     

Charge-transfer complexes of indenophanes with π-acceptors

The charge-transfer (CT) spectra of the -complexes formed by a number of -acceptors with several indenophanes as well as indene as a model compound have been measured in methylene chloride at 20 °C. Association constants and transition energies of these complexes as well as ionization potentials of the -donors have been determined. The data obtained indicate the existence of transannular electronic interactions in the indenophane nucleus. Furthermore, the pseudo-para- andmeta[2.2]indenophane isomers (3 and4) show a large difference in their -base strength. A good linear relationship has been observed between the association constants and _max of the long wavelength CT bands for the -complexes of these -donors with both tetracyanoethylene (TCNE) and 2,3-dichloro-5,6-dicyanobenzoquinone (DDQ). All CT complexes studied have a 1:1 stoichiometry.

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Charge-transfer-Komplexe aus Indenophanen und -Akzeptoren

Zusammenfassung Die Charge-transfer-Spektren (CT-Spektren) von -Komplexen aus Indenophanen bzw. der Modellverbindung Inden und verschiedenen -Akzeptoren wurden in Methylenchlorid bei 20 °C bestimmt. Die Assoziationskonstanten und Übergangsenergien dieser Komplexe sowie die Ionisationspotentiale der -Donatoren wurden ermittelt. Die Daten sprechen für das Vorliegen transannularer elektronischer Wechselwirkungen im Indenophan-System. Die isomeren pseudo-para- und -meta[2.2]indenophane3 und4 unterscheiden sich in ihrer -Basizität deutlich. Es besteht eine gute lineare Korrelation zwischen den Assoziationskonstanten und _max der langwelligen CT-Banden der verschiedenen -Donatoren mit Tetracyanoethylen (TCNE) und 2,3-Dichlor-5,6-dicyano-p-benzochinon (DDQ). Alle untersuchten CT-Komplexe besitzen 1:1-Stöchiometrie.

     

Comparison of total folate concentrations in foods determined by microbiological assay at several experienced U.S. commercial laboratories

Analysis of total folate concentration measured by microbiological assay in a variety of foods submitted in a routine manner to experienced laboratories that regularly perform folate analysis on fee-for-service basis was evaluated. Homogenates of fresh strawberries, frozen spinach, orange juice, frozen meat and vegetable pizza, dry macaroni, and dried pinto beans were prepared and stored under conditions previously determined to maintain stability of folate content. An aliquot of each composite and of 3 certified reference materials were sent on each of 4 occasions to 4 laboratories. Results for macaroni and pizza, the only folic acid-fortified foods, had considerably lower between-laboratory variation (CV(B)) with CV(B) of 9-11% versus >45% for other foods. Mean total folate ranged from 14 to 279 microg/100 g for a mixed vegetable reference material, from 5 to 70 microg/100 g for strawberries, and from 28 to 81 microg/100 g for wholemeal flour. Only 1 laboratory reported using a tri-enzyme extraction, and all laboratories used folic acid fortified foods as internal control materials. Users of commercial total folate analysis should understand the uncertainty in values determined by microbiological assay, particularly for foods containing primarily naturally occurring folate, which may not be apparent when replicate samples are not submitted for analysis.     

Anion radicals and dianions as electron transfer reagents

The catalytic regeneration of depolarizers by certain substrates has been demonstrated by means of cyclic voltammetry. As depolarizers we have employed anion radicals of aromatic hydrocarbons, heteroaromatic compounds, ketones, esters, nitriles, and olefins, and dianions of aromatic hydrocarbons, heteroaromatic compounds, ketones, and esters. As substrates we have investigated simple aromatic and aliphatic halides, 1,2- and 1,3-dihalides, carbon dioxide, and activated olefins. A requirement for the observation of a catalytic wave is that the reduced substrate undergoes a practically irreversible reaction after the reaction with the reduced depolarizer; for simple halides it is a cleavage reaction, for 1,2- and 1,3-dihalides an elimination, for carbon dioxide an addition, and for activated olefins a coupling. Electrochemical investigations of these kinds of reactions may be of interest in connection with the mechanism of several types of reactions in organic chemistry.     

The `cyclophene' [2.2.2](1,2,4)cyclophan‐9‐ene

In the title compound, C₁₈H₁₆, the [2.2]paracyclophane geometry is restrained to a considerable extent despite the introduction of the extra C=C bridge; typical paracyclophane features, such as the elongated C—C bridges, are still observed. However, the bridgehead atoms of the C=C bridge are forced into unusually close proximity [2.657 (3) Å], which in turn causes the rings to be rotated to an interplanar angle of 13.7 (2)°. The packing involves hexagonally close‐packed layers of molecules parallel to the xy plane, corresponding to the known `7,11' pattern of paracyclophanes, but without significant short intermolecular contacts.