Vector structures and solutions of linear matrix equations

A variety of systems problems give rise to special cases of the linear matrix equation

$\text{∑}\text{i}\text{A}{}_{\mathrm{i}}\text{p×s}\text{X}\text{s×t}\text{B}{}_{\mathrm{i}}\text{t×q}\text{+}\text{∑}\text{j}\text{C}{}_{\mathrm{j}}\text{p×t}\text{X}{}^{\mathrm{T}}\text{t×s}\text{D}{}_{\mathrm{j}}\text{s×q}\text{=}\text{F}\text{p×q}\text{(Z)}$

. F(Z) may be a matrix-valued function of a matrix argument Z. It is the purpose here to summarize and extend some of the applicable solution procedures through a systematic use of operators which convert the matrix equation to vector and dimension-reduced vector forms. The format and details of the results are convenient for machine computation.     

Chemistry inside a Porphyrin Skeleton: Platinacyclopentadiene from Tellurophene

Platinum(II) binds to 21,23-ditelluraporphyrin forming a side-on complex, which can be easily transformed into an aromatic metallaporphyrin, that is, 21-platina-23-telluraporphyrin, with a platinacyclopentadiene unit built in the porphyrin skeleton in place of one pyrrole ring. The central platinum(II) ion with a CCNTe square-planar coordination sphere can be oxidized to platinum(IV) by chlorine, bromine, methyl iodide or allyl chloride to yield octahedral complexes. All platinatelluraporphyrins show dynamic behavior involving the platinum ion coordination sphere fluxionality and the porphyrin skeleton deformation, both in-plane and out-of-plane, as demonstrated by ¹H NMR spectroscopy.     

Learning from the acquired optimized pulse shapes about the isotope selective ionization of potassium dimers

Selective optimization of the 39,39K2 and 39,41K2 isotopomers in a three-photon ionization process is presented by applying evolution strategies on shaped fs pulses in a feedback loop. The optimizations at different center wavelengths show considerably large enhancements of one isotope compared to the other and reversed. We compare the acquired optimized pulse shapes for combined phase and amplitude with pure amplitude modulation. Particularly from their spectra we are able to extract information about the optimally chosen differing ionization paths via the involved vibrational states. Furthermore, a comparison of the temporal shape of the optimized pulse forms for combined phase and amplitude with pure phase optimization is given. The presented pulse form analysis demonstrates the potential of restricted optimization to gain insight into the underlying dynamical processes. Our approach reveals how the optimization algorithm precisely addresses the vibrational wave functions both spectrally and temporally.     

Äussere Potenzen von Differentialmoduln Reduzierter Vollständiger Durchschnitte

Let R be an analytic algebra over the perfect field k. Assume that R is a reduced complete intersection. Then the i-th exterior powers of the KäHLER differential module of R over k are torsionfree (resp. reflexive) for i=1,...,p if and only if the codimension of the singular locus of R is greater than inf{p,dim R} (resp. inf{p+1,dim R}).

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Bei der vorliegenden Arbeit handelt es sich um einen Teil der Habilitationsschrift des Verfassers.