Units of compatible nearrings

We study singly-generated wavelet systems on ${\mathbb {R}^2}$ that are naturally associated with rank-one wavelet systems on the Heisenberg group N. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N, we give an explicit construction for Parseval frame wavelets that are associated with I. We say that ${g\in L^2(I\times \mathbb {R})}$ is Gabor field over I if, for a.e. ${\lambda \in I}$ , |??|^1/2 g(??, ·) is the Gabor generator of a Parseval frame for ${L^2(\mathbb {R})}$ , and that I is a Heisenberg wavelet set if every Gabor field over I is a Parseval frame (mother-)wavelet for ${L^2(\mathbb {R}^2)}$ . We then show that I is a Heisenberg wavelet set if and only if I is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.     

Quantification of the electrophilic reactivities of aldehydes, imines, and enones

The rates of the epoxidation reactions of aldehydes, of the aziridination reactions of aldimines, and of the cyclopropanation reactions of α,β-unsaturated ketones with aryl-stabilized dimethylsulfonium ylides have been determined photometrically in dimethyl sulfoxide (DMSO). All of these sulfur ylide-mediated cyclization reactions as well as the addition reactions of stabilized carbanions to N-tosyl-activated aldimines have been shown to follow a second-order rate law, where the rate constants reflect the (initial) CC bond formation between nucleophile and electrophile. The derived second-order rate constants (log k(2)) have been combined with the known nucleophilicity parameters (N, s(N)) of the aryl-stabilized sulfur ylides 4a,b and of the acceptor-substituted carbanions 4c-h to calculate the electrophilicity parameters E of aromatic and aliphatic aldehydes (1a-i), N-acceptor-substituted aromatic aldimines (2a-e), and α,β-unsaturated ketones (3a-f) according to the linear free-energy relationship log k(2) = s(N)(N + E) as defined in J. Am. Chem. Soc.2001, 123, 9500-9512. The data reported in this work provide the first quantitative comparison of the electrophilic reactivities of aldehydes, imines, and simple Michael acceptors in DMSO with carbocations and cationic metal-π complexes within our comprehensive electrophilicity scale.     

Ein numerisches Verfahren zur Lösung des axialsymmetrischen Torsionsproblems

Übersicht Es wird ein numerisches Berechnungsverfahren für den axialsymmetrisch tordierten linearelastischen Drehkörper entwickelt. Die wesentlichen Gleichungen werden angeschrieben und Hinweise für die Behandlung mit dem Computer gegeben. Das Verfahren ist in erster Linie für Probleme geeignet, bei denen die Verschiebungen und Spannungen auf dem Körperrand interessieren; durch einen nachgeschalteten Rechenschritt kann jedoch auch jeder beliebige Innenpunkt untersucht werden. Die Brauchbarkeit des Verfahrens wird anhand von Testrechnungen aufgezeigt.

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Summary A numerical procedure for the solution of the equations of an axially symmetric twisted linear elastic body is given. The basic equations are derived and hints for computer programming are given. The procedure is suited primarily for problems involving displacements and tractions on the boundary of the body; however, a subsequent computational routine also allows the investigation of arbitrary interior points. Some numerical examples are given to illustrate the usefulness of the procedure.

     

Internationales Biometrisches Seminar über Planungsforschung und Symposium über Wachstums- und Ertragsgesetze,Beurteilung der Wirkung von Heilmitteln bei chronischen Erkrankungen,die Gültigkeit des t-Testes. Linz a. d. Donau, 24. September bis 3. Oktober 1956

Ohne Zusammenfassung     

The DMAP-catalyzed acetylation of alcohols--a mechanistic study (DMAP = 4-(dimethylamino)pyridine)

The acetylation of tert-butanol with acetic anhydride catalyzed by 4-(dimethylamino)pyridine (DMAP) has been studied at the Becke3 LYP/6-311 + G(d,p)//Becke3 LYP/6-31G(d) level of theory. Solvent effects have been estimated through single-point calculations with the PCM/UAHF solvation model. The energetically most favorable pathway proceeds through nucleophilic attack of DMAP at the anhydride carbonyl group and subsequent formation of the corresponding acetylpyridinium/acetate ion pair. Reaction of this ion pair with the alcohol substrate yields the final product, tert-butylacetate. The competing base-catalyzed reaction pathway can either proceed in a concerted or in a stepwise manner. In both cases the reaction barrier far exceeds that of the nucleophilic catalysis mechanism. The reaction mechanism has also been studied experimentally in dichloromethane through analysis of the reaction kinetics for the acetylation of cyclohexanol with acetic anhydride, in the presence of DMAP as catalyst and triethylamine as the auxiliary base. The reaction is found to be first-order with respect to acetic anhydride, cyclohexanol, and DMAP, and zero-order with respect to triethyl amine. Both the theoretical as well as the experimental studies strongly support the nucleophilic catalysis pathway.