Prädikatives und funktionales Denken in der Wahrscheinlichkeitsrechnung

In order to emphasize functional thinking in mathematical education, arguments are brought forward for an early and frequent use of tree diagrams in teaching stochastics. Since set and tree diagrams can be regarded as two sides of the same matter, Bayes’ theorem, which is strongly associated with predicative set diagrams, need not be dealt with explicitly any longer. Set diagrams are replaced by tree diagrams that are labelled in detail and by tree inversion as a functional instrument of dealing with conditioned probabilities. This technique is demonstrated by three typical problems in the context of conditioned probability. Finally, two letters by Pascal to Fermat are analysed in order to illustrate that predicative and functional approaches were already pursued in the early stages of probability calculus in the 17^th century.     

Narrow linewidth 2 μm optical parametric oscillation in periodically poled LiNbO<Subscript>3</Subscript> with volume Bragg grating outcoupler

An optical parametric oscillator using a periodically poled LiNbO₃ crystal and a volume Bragg grating output coupler is presented. Signal and idler wavelengths of 2008 and 2264 nm were generated from the 1064 nm Nd:YVO₄ pump laser. The Bragg grating was reflecting in a narrow band around 2008 nm, creating a purely singly resonant cavity. Signal and idler linewidths of approximately 0.44 and 0.72 nm, respectively, were measured. This is a reduction of 80 and 60 times compared to when using a mirror as output coupler. A total output energy of 156 μJ was measured with 47% slope efficiency. PACS 42.65.Yj; 42.72.Ai; 42.40.Eq     

Deep trench etching in macroporous silicon

We present a method to create at the same time trenches and ordered macropore arrays during photo-electrochemical etching of n-type silicon. This novel method allows in situ separation of single devices with a submicrometer precision. It also enables new device structures in macroporous silicon in the areas of photonics, sensing and electronics. The limits of this new process are simulated using electrostatic models and are verified experimentally. PACS 82.45.Yz; 81.16.-c     

Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs

A probabilistic result of Bollobás and Catlin concerning the largest integer p so that a subdivision of K_p is contained in a random graph is generalized to a result concerning the largest integer p so that a subdivision of A_p is contained in a random graph for some sequence A₁, A₂,… of graphs such that A_i+1 contains a subdivision of A_i. A similar result is proved for subdivisions with odd paths or cycles. The result is applied to disprove a conjecture of Chartrand, Geller, and Hedetniemi. The maximum number of edges in a graph without a subdivision of K_p, p = 4, 5, with odd paths or cycles is determined.     

QCD corrections to the forward-backward asymmetries of c and b quarks at the Z pole

Measurements of the forward-backward production asymmetry of heavy quarks in Z decays provide a precise determination of . The asymmetries are sensitive to QCD effects, in particular hard gluon radiation. In this paper QCD corrections for and are discussed. The interplay between the experimental techniques used to measure the asymmetries and the QCD effects is investigated using simulated events. A procedure to estimate the correction needed for experimental measurements is proposed, and some specific examples are given. Received: 26 February 1998 / Published online: 2 June 1998     

High dose iron implantation into silicon and metals

High dose implantations of Fe into metals and semiconductors have been performed with beam energies up to 1 MeV at the UNILAC-injector at GSI. Unusual high concentrations of 70 atomic % for Si and 20 atomic % for Cu have been obtained, with doses of 10¹⁸ Fe/cm² in the case of Si and several 10¹⁷ Fe/cm² in the case of Cu. For Si the thickness of the layers were determined by Rutherford backscattering to be 4500 Å. These results are consistent with calculations, which show that these high concentrations are due to the reduction of the sputter yield at the relative high particle energies. Samples have been characterized using several complementary methods (Mössbauer Spectroscopy (MS), Rutherford Backscattering Spectroscopy (RBS), Auger electron Spectroscopy (AES). Scanning Electron Microscopy (SEM), X-ray diffraction (XRD)).