Reduction of Time-Dependent Systems Admitting a Superposition Principle

The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.     

Spin phase

The existence of a complementary quantity in the Weyl sense for the third component of spin is discussed. This is called the spin phase and the possibilities of measuring this quantity are considered. Connections between the spin phase and the indeterminacy of the direction for the spin are shown. On leave from the Institute of Physics, Nicolas Copernicus University, Torun, Poland.     

Three-dimensional pictures of industrial scenes applying an optical radar

For small scale ranging, a light beam, modulated with a wavelength typical for radio waves, is a suitable means of scanning. The ranging beam can be focussed to illuminate only a small spot. The diffuse reflection on the spot ensures that an echo appears. By applying phase comparison techniques to the ranging and echo beam the optical path length and thus the height of the illuminated surface element can be determined. By systematically deflecting the beam and thereby scanning the scene, a discrete height map is obtained. The realisation of such a small scale ‘optical radar’ is described, the main elements of which are a semiconductor diode laser for emission, an avalanche photo diode for detection, and a two-dimensional beam deflection mechanics for scanning. The typical performance is taking a picture of a volume of about 0·5 m × 0·5 m × 0·5 m within 1s, consisting of about 64 000 pixels.     

Synthesis of (±)-cartorimine

Heating pyranulose 4 and cinnamate 2 in the presence of 2,6-di-t-butylpyridine in CH₃CN afforded the [5+2] cycloadduct, which was hydrolyzed to give 13% of cartorimine (1).     

Repeatable measurements in quantum theory: Their role and feasibility

Recent advantages in experimental quantum physics call for a careful reconsideration of the measurement process in quantum mechanics. In this paper we describe the structure of the ideal measurements and their status among the repeatable measurements. Then we provide an exhaustive account of the interrelations between repeatability and the apparently weaker notions of value reproducible or first- kind measurements. We demonstrate the close link between repeatable measurements and discrete observables and show how the ensuing measurement limitations for continuous observables can be lifted in a way that is in full accordance with actual experimental practice. We present examples of almost repeatable measurements of continuous observables and some realistic models of weakly disturbing measurements.Dedicated to Peter Mittelstaedt on the occasion of his 65th birthday.Leaving the Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany.     

Intramolecular carbonyl?carbonyl interactions in W, Mo and Fe complexes containing the η-N-maleimidato ligand: X-ray, DFT and AIM studies

The crystal structures of (η⁵-C₅H₅)W(CO)₃(η¹-N-maleimidato) and (η⁵-C₅H₅)Fe(CO)₂(η¹-N-maleimidato) complexes were determined by single crystal X-ray diffraction. The molecular geometries of both structures are compared with those of the Mo analog of the W complex and ethyl-N-maleimide in order to find a relation between the geometrical features and the rate constants of the addition reaction of the sulfhydryl group of biomolecules to the ethylenic bond of the maleimidato fragment. For a deeper insight into the problem DFT calculation were performed. An analysis of atomic charges, using the CHELPG scheme, and of theoretical electron density function, using the AIM theory, was performed. In the (η⁵-C₅H₅)W(CO)₃(η¹-N-maleimidato), likewise in its Mo analog, the carbonyl?carbonyl interaction was found both for experimental and calculated structures. It is probably the first approach to explain this type of intramolecular interactions acting in organometallic compounds. This interaction can play the essential role in the reaction mechanism of nucleophilic addition to the maleimidato moiety. The AIM investigations indicate also the differences in the character of bonding between the η-N-maleimidato ligand and the central metal atom.