Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems

It is shown that, if a parametrized fämily of extremals F can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity.     

Least-Squares Fitting of Algebraic Spline Surfaces

We present an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. By simultaneously approximating points and associated normal vectors, we obtain a method which is computationally simple, as the result is obtained by solving a system of linear equations. In addition, the result is geometrically invariant, as no artificial normalization is introduced. The potential applications of the algorithm include the reconstruction of free-form surfaces in reverse engineering. The paper also addresses the generation of exact error bounds, directly from the coefficients of the implicit representation.     

Computational experience with general equilibrium problems

We report on computational experience with an implementation of three algorithms for the general economic equilibrium problem. As a result we get that the projection algorithm for variational inequalities increases the size of solvable models by a factor of 5–10 in comparison with the classical homotopy method. As a third approach we implemented a simulated annealing heuristic which might be suitable to estimate equilibria for very large models.Supported by the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

UV-holographic gratings in TS-diacetylene single crystals

Holographic gratings with grating periods between 0.37 m and 4.4 m, respectively, were recorded on the surface of TS-diacetylene single crystals by two interfering 257 nm laser beams. Diffraction efficiencies of up to 35 % were obtained for readout with red light of = 633 nm. The main modulation mechanism was phase modulation. The spatial resolution was better than 1600 lines/mm. The upper limit of the polymer chain length was 0.6 m. The holographic sensitivity was 4.5 cm²/J for immediate read-out at 633 nm without processing after exposure. — A weakly exposed latent hologram may be developed simply by gentle annealing the crystal which increases the sensitivity by one order of magnitude.     

Eine variante der simplexmethode mit gleichzeitigem austausch mehrerer basisvariablen pro iterationsschritt

Zusammenfassung Das Iterationsverfahren der Simplexmethode besteht bekanntlich darin, von einer (gegebenen, aber noch nicht optimalen) Basislösung eine Variable auszuwählen und diese gegen eine Nichtbasisvariable auszutauschen, so daß hierbei eine im Sinne des Extremalwertes der Zielfunktion günstigere Basislösung entsteht. Die hier dargestellte Variante gestattet es, gegebenenfalls mehrere Basisvariable gegen eine gleiche Anzahl von Nichtbasisvariablen bei einem Iterationsschritt auszutauschen. Dieses Verfahren kürzt beispielsweise den Lösungsprozeß des Transportproblems im allgemeinen beträchtlich ab.

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Summary Solving a linear programming problem by the Simplexmethod the usual procedure is as follows: from a (given not yet optimal) basic solution one basic variable is chosen for being changed with a non-basic variable to get a better approximation to the extremum of the given linear equation. The described method sometimes allows exchanging several basic variables in each Simplex-step. Using this method usually the solution process of the transportation problem for example may be abbreviated efficiently.

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Vorgel. v.:H. P. Künzi     

Isotope effect for transport and magnetic properties of La 0.35 Pr 0.35 Ca 0.3 MnO 3 thin films

The effect of ¹⁶ O → ¹⁸ O isotope substitution on electrical resistivity, magnetoresistance, and ac magnetic susceptibility was studied for La_0.35Pr_0.35Ca_0.3MnO₃ epitaxial thin films deposited onto LaAlO₃ and SrTiO₃ substrates. For the films on LaAlO₃, the isotope substitution resulted in the reversible transition from a metal-like to insulating state. The applied magnetic field ( H ≥ 2 T) transformed the sample with ¹⁸O back to the metallic state. The films on SrTiO₃ remained metallic at low temperatures for both ¹⁶O and ¹⁸O, but the shift of the resistivity peak corresponding to onset of metallic state exceeded 63 K after ¹⁶ O → ¹⁸ O substitution. The temperature dependence of both resistivity and magnetic susceptibility was characterized by hysteresis, especially pronounced in the case of the films on LaAlO₃. Such a behavior gives certain indications of the phase separation characteristic of interplay between ferromagnetism and charge ordering. Received 11 February 2000 and Received in final form 13 September 2000