Der Beweis des Weierstraßschen approximationssatzes 1885 vor dem hintergrund der entwicklung der fourieranalysis

Eine bisher unveröffentlichte Ausarbeitung der Weierstraßschen Vorlesung “Ausgewählte Kapitel aus der Funktionenlehre” vom Sommersemester 1886 sowie verschiedene Briefwechsel aus jener Zeit gestatten es, Weierstraß' Verhältnis zur Theorie der Funktionen reeller Variablen näher zu beluechten. Es zeigt sich, daß Weierstraß den Beweis seines Approximationssatzes (1885) bewußt in die Traditionslinie der Fourieranalysis einordnete. Dies führte ihn zur Verallgemeinerung seines Satzes auf unstetige Funktionen mittels des Cantorschen Begriffs des (äußeren) Inhalts einer beschränkten Punktmenge.An unpublished autograph version of Weierstrass' lecture-course “Selected Chapters on Function Theory” (summer semester 1886) as well as several letters shed some light upon Weierstrass' position concerning the theory of functions of real variables. I show that Weierstrass considered his Approximation Theorem (1885) a part of the tradition of Fourier analysis. This led him to an extension of the theorem to the case of discontinuous functions by means of Cantor's concept of the outer content of a bounded point set.     

Traffic of Molecular Motors Through Tube-Like Compartments

The traffic of molecular motors through open tube-like compartments is studied using lattice models. These models exhibit boundary-induced phase transitions related to those of the asymmetric simple exclusion process (ASEP) in one dimension. The location of the transition lines depends on the boundary conditions at the two ends of the tubes. Three types of boundary conditions are studied: (A) Periodic boundary conditions which correspond to a closed torus-like tube. (B) Fixed motor densities at the two tube ends where radial equilibrium holds locally; and (C) Diffusive motor injection at one end and diffusive motor extraction at the other end. In addition to the phase diagrams, we also determine the profiles for the bound and unbound motor densities using mean field approximations and Monte Carlo simulations. Our theoretical predictions are accessible to experiments.     

Investigation of multi-layer coatings on cemented carbide cutting tools by analytical electron microscopy

Multi-layer coatings on cemented carbide substrates have been investigated by analytical electron microscopy (AEM). The samples were taken from cross-sections through the coating thus enabling a study of individual layers and interfaces. Various phenomena are shown: (1) tungsten diffusion and graphite inclusions in TiC layers, (2) oxygen impurities in Ti(C, N) coatings and (3) the occurrence of titanium oxinitrides in the interface between the Al₂O₃ and TiN-layers.Dedicated to Professor Günther Tölg on the occasion of his 60th birthday     

Opticon: An algorithm for the optimal control of nonlinear stochastic models

In this paper we describe the algorithm OPTCON which has been developed for the optimal control of nonlinear stochastic models. It can be applied to obtain approximate numerical solutions of control problems where the objective function is quadratic and the dynamic system is nonlinear. In addition to the usual additive uncertainty, some or all of the parameters of the model may be stochastic variables. The optimal values of the control variables are computed in an iterative fashion: First, the time-invariant nonlinear system is linearized around a reference path and approximated by a time-varying linear system. Second, this new problem is solved by applying Bellman's principle of optimality. The resulting feedback equations are used to project expected optimal state and control variables. These projections then serve as a new reference path, and the two steps are repeated until convergence is reached. The algorithm has been implemented in the statistical programming system GAUSS. We derive some mathematical results needed for the algorithm and give an overview of the structure of OPTCON. Moreover, we report on some tentative applications of OPTCON to two small macroeconometric models for Austria.     

Ein neuer Zugang zu 2,6,6-Trimethylcyclohexa-2,4-dienon aus 4-Oxoisophoron

A New Access to 2,6,6-Trimethylcyclohexa-2,4-dienone from 4-Oxoisophorone 2,6,6-Trimethylcyclohexa-2,4-dienone ( 1 ), a versatile starting material for the preparation of some carotenoids and several natural products, was efficiently (73% yield) prepared from oxoisophorone 3 . After conversion of 3 to the alcohol 4 or the acetate 5 , H₂O was eliminated ( 4→1 ) under acidic distillative conditions, whereas AcOH could be eliminated ( 5→l ) under Pd(O) catalysis.     

A separation property of planar triangulations

Every planar triangulation G has the property that each induced cycle C of length at least 4 in G separates G, but no proper subgraph of C does. This property is trivially shared by all chordal graphs since these contain no such cycles at all. We ask to what extent maximally planar graphs and chordal graphs are unique with this property — or how much larger the class of graphs is that it determines. The answer is given in the form of a characterization of this class in terms of the simplicial decompositions of its elements. The theory of simplicial decompositions appears to be a very interesting, but still largely unexploited, method of characterization in graph theory, which seems tailor-made for problems like the one discussed.