Mathematik f"urs Leben am Beispiel der Computertomographie

Zusammenfassung. Computertomographie ist heutzutage ein fast ebenso bekanntes Hilfsmittel des Mediziners wie das klassische R?ntgen. Weit weniger bekannt ist jedoch, dass es insbesondere die Mathematik ist, die dieses Verfahren erst erm?glicht. Es erscheint immer wichtiger, Schülern und Studierenden neben der Mathematik selbst auch deren Bedeutung für Probleme aus dem Alltag nahe zu bringen. In dieser Arbeit vermitteln wir ein Grundverst?ndnis der Computertomographie. Hierzu geh?ren Nutzen und Einsatzm?glichkeiten in der Medizin, physikalischer Hintergrund und als zentrales Thema ein einfaches mathematisches Modell und dessen numerische L?sung. Die Arbeit kann als Grundlage für Projekte in Schule und Hochschule verwendet werden. Eingegangen am 17. September 2001 / Angenommen am 15. November 2001     

Optimized look-ahead recurrences for adjacent rows in the Padé table

A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms.The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N ²) Levinson-type form, anO(N ²) Schur-type form, and a superfastO(N log² N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT ^–1 orT, respectively, can be found.     

Purification of pro- and eukaryotic superoxide dismutases by charge-controlled hydrophobic chromatography

The process of purifying superoxide dismutases was simplified using charge-controlled hydrophobic chromatography on 10-carboxydecyl Sepharose. In only one chromatographic step following ammonium sulphate precipitation, Fe-containing superoxide dismutase from Pseudomonas putida and Cu,Zn-containing superoxide dismutase from bovine erythrocytes were purified with an overall yield of about 70% to electrophoretic homogeneity. The specific activities of the crystalline enzyme preparations were expressed in McCord and Fridovich units and were 3000 and 3200 U/mg, respectively.     

On the mechanism of the chromium carbonyl photocatalyzed watergas shift reaction

The chromium hexacarbonyl catalyzed watergas shift reaction is accelerated by UV irradiation and inhibited by increased CO pressure. An activation energy of 30 kJ mol^?1 has been determined for the photochemical and one of 145 kJ mol^?1 for the thermal reaction. Light accelerates the conversion of Cr(CO)₆ into [Cr(CO)₅ formate]^?, which is thermally activated, as evidenced by in situ IR and UV spectroscopy.     

Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems

Summary. We present generalizations of the nonsymmetric Levinson and Schur algorithms for non-Hermitian Toeplitz matrices with some singular or ill-conditioned leading principal submatrices. The underlying recurrences allow us to go from any pair of successive well-conditioned leading principal submatrices to any such pair of larger order. If the look-ahead step size between these pairs is bounded, our generalized Levinson and Schur recurrences require $ operations, and the Schur recurrences can be combined with recursive doubling so that an $ algorithm results. The overhead (in operations and storage) of look-ahead steps is very small. There are various options for applying these algorithms to solving linear systems with Toeplitz matrix. Received July 26, 1993     

On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems

In this note, we study the convergence of the Levenberg–Marquardt regularization scheme for nonlinear ill-posed problems. We consider the case that the initial error satisfies a source condition. Our main result shows that if the regularization parameter does not grow too fast (not faster than a geometric sequence), then the scheme converges with optimal convergence rates. Our analysis is based on our recent work on the convergence of the exponential Euler regularization scheme (Hochbruck et al. in Inverse Probl 25(7):075009, 2009).     

Solvent isotope effect on the fluorescence of azoalkanes

A study of fluorescence quantum yields and fluorescence lifetimes of the cyclic azoalkanes I and II reveals a striking dependence of Φ_F and τ_F on solvent and on isotopic substitution (OH å OD). A mechanism involving specific deactivation of the fluorescent state from a hydrogen bonded complex is proposed to rationalize the data. The observation that the quantum yield for decomposition of I does not correlate with the variation of fluorescence parameters with solvent and isotopic change leads to the conclusion that the state responsible for photoreaction proceeds the fluorescent state.