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.     

Possible application of X-ray optical elements for reducing the spectral bandwidth of an X-ray SASE FEL

A new design for a single pass X-ray Self-Amplified Spontaneous Emission (SASE) FEL is proposed. The scheme consists of two undulators and an X-ray monochromator located between them. The first stage of the FEL amplifier operates in the SASE linear regime. After the exit of the first undulator the electron bunch is guided through a non-isochronous bypass and the X-ray beam enters the monochromator. The main function of the bypass is to suppress the modulation of the electron beam induced in the first undulator. This is possible because of the finite value of the natural energy spread in the beam. At the entrance to the second undulator the radiation power from the monochromator dominates significantly over the shot noise and the residual electron bunching. As a result the second stage of the FEL amplifier operates in the steady-state regime when the input signal bandwidth is small with respect to that of the FEL amplifier. Integral losses of the radiation power in the monochromator are relatively small because grazing incidence optics can be used. The proposed scheme is illustrated for the example of the 6 nm option SASE FEL at the TESLA Test Facility under construction at DESY. As shown in this paper the spectral bandwidth of such a two-stage SASE FEL (Δλ/λ 5 × 10⁻⁵) is close to the limit defined by the finite duration of the radiation pulse. The average brilliance is equal to 7 × 10²⁴ photons/(s × mrad² × mm² × 0.1% bandw.) which is by two orders of magnitude higher than the value which could be reached by the conventional SASE FEL. The monochromatization of the radiation is performed at a low level of radiation power (about 500 times less than the saturation level) which allows one to use conventional X-ray optical elements (grazing incidence grating and mirrors) for the monochromator design.     

Stereochemie von Metallocenen, 40. Mitt.

Starting from optically active methylferrocene-- and--carboxylic acids (1 a and1 b) of known absolute configuration and enantiomeric purity about 15 chiral ferrocenes (of each isomeric series— and ) were obtained by suitable ligand transformations. Thereby (almost) all possible chiral combinations of the ligands CH₃, COOH (COO^–), COOCH₃, CN and NH₂ (NH₃ ⁺) were accessible which are necessary for a potential test of approximations of chirality functions for compounds with basic symmetry C_5v. The chiroptical properties of these disubstituted ferrocenes are recorded.Preliminary tests using a shortened Ansatz revealed large discrepancies between calculated () and found [M]_D-values. Possible reasons for this failure are discussed.

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61. Mitt. über Ferrocenderivate

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39. Mitt.:A. Meyer, H. Neudeck undK. Schlögl, Chem. Ber.110, 1403 (1977).

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60. Mitt.:V. Rapi, K. Schlögl undB. Steinitz, J. Organometal. Chem.94, 87 (1975).     

Wirkungsgrade in der Welt des Lebens und der Technik

Zusammenfassung Um der oft gehörten Meinung entgegenzutreten, die energetischen Wirkungsgrade lebender Organismen seien wesentlich höher als die technischer Verfahren, werden entsprechende Werte miteinander verglichen. Der sehr niedrige Wirkungsgrad der Photosynthese zusammen mit der kleinen Energiedichte der Sonnenstrahlung erzwang ab einer kritischen Größe die Aufspaltung in Energiesammler, die Pflanzen, und Verbraucher chemischer Energie, den Tieren. Dank den 10 mal höheren Wirkungsgraden in der Technik, lassen sich Solarmobile bauen, die von der auf sie fallenden Sonnenstrahlung angetrieben werden. Wegen des Multiplikationsgesetzes der Teilwirkungsgrade ergeben sich sowohl für die Speicherung der Sonnenenergie in den fossilen Brennstoffen wie in der Nahrungskette extrem niedrige Wirkungsgrade.

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Efficiencies in the world of life and in engineering

One often hears the opinion, that the efficiencies in the world of life are higher than in engineering. To disprove this, many values of efficiency of corresponding processes are compared. The very low efficiency of photosynthesis in living plants together with the low density of solar radiation has led the world of life to split into collectors of energy, the plants, and consumers of chemical energy, the animals. Thanks to the ten times higher efficiencies of solar cells, it is possible, to produce solar mobiles, vehicles that move driven by solar radiation.

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Herrn Prof. Dr.-Ing. U. Grigull zum 75. Geburtstag gewidmet     

Analysis of the 3-dimensional electron distribution in silicon using directional Compton profile measurements

A method of reconstructing the electron momentum density (p), and its Fourier transform,B(t), from a series of directional Compton profiles is described. It is based on a double Fourier inversion technique and an expansion in lattice harmonic functions. The effect of random errors has been analysed, and the implications for the data collection discussed. We have used the reconstruction technique to obtain (p) andB(t) for silicon from six directional Compton profiles measured with 412 KeV gamma-radiation. The experimental result is in good agreement with earlier measurements and with available solid state theories. A recent Wannier function calculation for silicon using orthogonalised bond orbitals provides a useful tool for identifying the physical origins of the observed anisotropies. A comparison between the information presented in position and momentum space shows that the ease of interpretation depends upon the degree to which the various interactions give rise to localised features in each representation.     

Umsetzung von hydrobromiertem 1,4-Polybutadien mit lithiumorganischen Verbindungen

Ohne ZusammenfassungMit 1 Abbildung und 1 Tabelle     

Die Schmelzdiagramme der Systeme CsN3/Zn(N3)2 und KN3/Zn(N3)2

The phase diagrams of the systems CsN₃/Zn(N₃)₂ and KN₃/Zn(N₃)₂ have been obtained employing the microscopic technique ofL. Kofler andA. Kofler. Within the system CsN₃/Zn(N₃)₂ three eutectics at 148°C, 142°C, and 210°C were found. Besides Cs₂Zn(N₃)₄, melting incongruently in the interval 153°C to 170°C, there exist two further compounds of the most probable composition Cs₃Zn₂(N₃)₇ and CsZn₂(N₃)₅, melting congruently at 170°C and 210°C, resp. In the system KN₃/Zn(N₃)₂ there exist two eutectics at 203°C and 172°C and two compounds, one of them, i.e. K₂Zn(N₃)₄, melting congruently at 206°C, the other one, with composition KZn₃(N₃)₇ or KZn₄(N₃)₉, melting incongruently at 210°C.

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Mit 8 Abbildungen

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Herrn Professor Dr.Heribert Grubitsch zum 70. Geburtstag gewidmet.     

η-Allylpentacarbonylvanadium

The preparation of η-C₃H₅V(CO)₅ from allyl chloride and sodium hexacarbonylvanadate (–I) is reported. Composition, IR and NMR spectra are in accordance with a π-bonded C₃H₅-moiety; the preparation has been extended to give complexes with substituted allyl groups. Furthermore (1-3-η-2-butenyl)pentacarbonylvanadium can be obtained by the addition of butadiene to HV (CO)₆.