Über einen Abbildungssatz von B. Sz.-Nagy

Ohne Zusammenfassung     

“Das letzte Ziel ist immer die Darstellung einer Funktion”: Grundlagen der analysis bei Weierstraß 1886, historische Wurzeln und Parallelen

Weierstrass regarded his famous approximation theorem of 1885 primarily as a result on representations. In his lectures of 1886 he aimed at showing that even some classes of discontinuous functions can be represented by “arithmetical expressions,” i.e., series of polynomials. Also, and with some surprise, he had learned from the work of Fourier that fairly arbitrary functions could appear as boundary or initial distributions of analytic solutions to differential equations. In his eyes, all this justified the Eulerian conception of a function as represented by an expression. Weierstrass was apparently unaware of related methods and results by Cauchy and his contemporaries. In particular, Weierstrass' use of “delta functions” and convergence factors looks clumsier than earlier work done at Paris around 1820. The limited knowledge and mention of early 19th-century real analysis in the lectures of 1886 and its contrast to verbal praise of the significance of history, may explain a strange fact in historiography: the Weierstrassians show a lack of understanding, or even of knowledge, of earlier French real analysis.Weierstraß hat seinen berühmten Approximationssatz von 1885 in erster Linie als einen Darstellungssatz angesehen. In seinen Vorlesungen von 1886 wollte er sogar Klassen von unstetigen Funktionen durch “arithmetische Ausdrücke” darstellen, d.h. bei ihm durch Reihen von Polynomen. Mit einiger Überraschung entnimmt er dem Werk von Fourier, daß weitgehend willkürliche Funktionen als Rand- oder Anfangswerte von analytischen Lösungen von Differentialgleichungen auftreten können. Er sieht in alledem eine Rechtfertigung des Eulerschen Funktionsbegriffs. Weierstraß scheint verwandte Methoden und Resultate von Cauchy und dessen Zeitgenossen nicht gekannt zu haben. Insbesondere ist Weierstraß' Verwendung von Methoden wie Deltafunktionen und konvergenzerzeugenden Faktoren weniger geschickt als in den Pariser Arbeiten aus der Zeit um 1820. Die Vorlesungen von 1886 zeigen, daß Weierstraß nur eine begrenzte Kenntnis der reellen Analysis des frühen 19. Jahrhunderts hatte, was im Gegensatz zu seinen Lobworten über die Geschichte steht. Damit kann eine Merkwürdigkeit der Geschichtsschreibung erklärt werden: In der Nachfolge von Weierstraß beobachtet man fehlendes Verständnis und sogar mangelnde Kenntnis der früheren französischen reellen Analysis.     

Umlagerung intermediaerer 1,4-diradikale bei der photosensibilisierten addition von maleinsaeureanhydrid an propargylhalogenide

Photosensitized addition of maleic anhydride to propargyl bromides and tertiary propargyl chlorides gives rise to formation of vinylcyclopropane derivatives. Results are discussed in terms of a mechanism involving rearrangement of intermediate 1,4-alkyl-vinyldiradicals.     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

Determination of 2-alkylcyclobutanone using fluorescent labelling

2-Alkylcyclobutanones are formed under the influence of ionising radiation from triglycerides. The determination of cyclobutaneones could be a powerful means of identifying irradiated foods, not only from a legal viewpoint but also in food quality control. In this paper 2-dodecylcyclobutaneone was used as a model substance, synthesised by means of the key substance 1-bromo-1-ethoxycyclopropane. The determination was carried out by labelling the cyclic ketone with fluorescent dyes and subsequent HPLC separation. 1-Pyrenebutyrylhydrazide and 7-diethylamino-3-carbonylazide were used as labelling reagents. Dodecylcyclobutanone was sensitively determined in fat matrices. Labelling with 1-pyrenebutyrylhydrazide was unsuccessful because of its low reactivity. Medium- and short-chain aldehydes and ketones were easily labelled. After reduction to the corresponding alcohol, dodecylcyclobutanone, as well as aldehydes and ketones, were successfully labelled with 7-diethylamino-3-carbonylazide. Separation was carried out with good results on RP-18 reversed phases using methanol water mixtures as mobile phases. The detection limit of dodecylcyclobutanone is about 5 ng. Present address: Institut für Lebensmittelchemie, Technische Universität Berlin, Gustav-Meyer-Allee 25, D-13355 Berlin, Germany