Rational design of selective anion receptors

The new family of the anion receptors based on oligoureas with varied flexibility was developed and studied. The preparation of the urea chains containing two different units in various sequences was elaborated. The complete sets of four cyclic trimers and six tetramers based on the two units were prepared. Their conformational and complexation properties were studied with NMR spectroscopy and X-ray structure determinations, their behaviour towards various anions was evaluated and compared. The synthesis and the same studies were performed also with four different cyclic hexamers. During these studies the remarkable templation by two halide anions was observed.

Topologically correct intersection curves of tori and natural quadrics

This thesis provides efficient and robust algorithms for the computation of the intersection curve between a torus and a simple surface (e.g. a plane, a natural quadric or another torus), based on algebraic and numeric methods. The algebraic part includes the classification of the topological type of the intersection curve and the detection of degenerate situations like embedded conic sections and singularities. Moreover, reference points for each connected intersection curve component are determined. The required computations are realised efficiently by solving quartic polynomials at most and exactly by using exact arithmetic. The numeric part includes algorithms for the tracing of each intersection curve component, starting from the previously computed reference points. Using interval arithmetic, accidental incorrectness like jumping between branches or the skipping of parts are prevented. Furthermore, the environments of singularities are correctly treated. Our algorithms are complete in the sense that any kind of input can be handled including degenerate and singular configurations. They are verified, since the results are topologically correct and approximate the real intersection curve up to any arbitrary given error bound. The algorithms are robust, since no human intervention is required and they are efficient in the way that the treatment of algebraic equations of high degree is avoided.