Rank one transformations with singular spectral type

We show that a certain class of measures arising from generalized Riesz products is singular. In particular, cutting and stacking (i.e. rank one) transformations whose cuts do not grow too rapidly, have singular maximal spectral type. The precise condition is , wherew _h is the number of cuts at stagen. Supported by a Canada NSERC grant. Supported by an NSF grant.     

Anwendungen des Kationenaustausches in der Analyse

Zusammenfassung Die unterschiedlichen Affinitäten der Kationen zu dem Austauscherharz Wofatit KS werden untersucht und relative Zahlen dafür angegeben. Die dabei festgestellten größeren Unterschiede zwischen Lithium, Natrium und Kalium sowie zwischen Cadmium und Zink werden benutzt, um durch Eluieren derartiger gemischter Belegungen aus einer Austauschersäule mit verdünnter Säure diese Kationen voneinander zu trennen. Eine hierfür entwickelte Leitfähigkeitsmeßapparatur gestattet es, derartige Trenndiagramme registrierend aufzunehmen, die nach voraufgegangener Eichung einer quantitativen Auswertung zugänglich sind.I. M. Mitteilung diese Z. 132, 241 (1951); II. Mitteilung diese Z. 132, 321 (1951).Herrn Dr. Krüger, dem Leiter des Untersuchungslaboratoriums der Chemischen Werke Hüls, G. m. b. H., bin ich für seine wertvollen Anregungen und Ratschläge, durch die die vorliegende Arbeit stark gefördert wurde, zu großem Dank verpflichtet.     

Adaptive eigenvalue computation: complexity estimates

This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ ₂, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity. This research was supported in part by the Leibniz Programme of the DFG, by the SFB 401 funded by DFG, the DFG Priority Program SPP1145 and by the EU NEST project BigDFT.     

A Germylene Stabilized by Homoconjugation

The synthesis of a bicyclic germylene from the reaction of a germole dianion with hafnocene dichloride is reported. This germylene is stabilized by a homoconjugative interaction of the dicoordinated germanium atom with a remote C=C double bond. First reactivity studies revealed its nucleophilic character and resulted in the synthesis of bimetallic hafnium/iron and hafnium/tungsten complexes with a germylene group linker.     

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.     

Genetic microheterogeneity of human transthyretin detected by IEF

Mutations of the human transthyretin (TTR) gene have attracted medical interest as a cause of amyloidosis. Recently, we have described in detail an electrophoretic procedure with PAGE followed by IEF in urea gradients for the study of the microheterogeneity of TTR monomers (Altland, K., Winter, P., Sauerborn, M. K., Electrophoresis 1999, 20, 1349-1364). In this paper, we present a study on 49 different mutations of TTR including 33 that result in electrically neutral amino acid substitutions. The aims of the investigation were to test the sensitivity of the procedure to detect TTR variants in patients with TTR amyloidosis and their relatives and to identify some common characteristics that could explain the amyloidogenicity of these variants. We found that all tested amyloidogenic mutations could be detected by our method with the exception of those for which the corresponding variant was absent in plasma samples. Most of the electrically neutral amyloidogenic TTR variants had in common a reduced conformational stability of monomers by the activity of protons and urea. For three variants, e.g. TTR-F64L, TTR-I107V and TTR-V122I, the monomers had a conformational stability close to that of normal monomers but we found experimental and structural arguments for a weakening of the monomer-monomer contact. All types of amyloidogenic mutations affected the stability of TTR tetramers.     

Integration of electron density and molecular orbital techniques to reveal questionable bonds: the test case of the direct Fe-Fe bond in Fe2(CO)9

The article illustrates the advantages of partitioning the total electron density rho(rb), its Laplacian (inverted Delta)2 rho(rb), and the energy density H(rb) in terms of orbital components. By calculating the contributions of the mathematically constructed molecular orbitals to the measurable electron density, it is possible to quantify the bonding or antibonding character of each MO. This strategy is exploited to review the controversial existence of direct Fe-Fe bonding in the triply bridged Fe2(CO)9 system. Although the bond is predicted by electron counting rules, the interaction between the two pseudo-octahedral metal centers can be repulsive because of their fully occupied t(2g) sets. Moreover, previous atoms in molecules (AIM) studies failed to show a Fe-Fe bond critical point (bcp). The present electron density orbital partitioning (EDOP) analysis shows that one sigma bonding combination of the t(2g) levels is not totally overcome by the corresponding sigma* MO, which is partially delocalized over the bridging carbonyls. This suggests the existence of some, albeit weak, direct Fe-Fe bonding.     

Neutral hexa- and pentacoordinate silicon(IV) complexes with SiO(6) and SiO(4)N skeletons

The neutral heteroleptic hexacoordinate silicon(IV) complexes 4 and 5 (SiO(6) skeletons) and the neutral pentacoordinate silicon(IV) complexes 7-9 (SiO(4)N skeletons) were synthesized, starting from the hexacoordinate precursor 2 and the pentacoordinate precursor 6, respectively. In these reactions, two monoanionic cyanato-N ligands are replaced by one dianionic bidentate O,O-chelate ligand. Compounds 4, 5, and 7-9 were characterized by single-crystal X-ray diffraction and solid-state and solution NMR spectroscopy. The chiral silicon(IV) complexes 4, 5, 7, and 8 were obtained as racemic mixtures, whereas 9 was isolated as a cocrystallizate consisting of the two diastereomers, (C,S)-9 and (A,S)-9 (ratio 1:1). The stereodynamics of 5 and 8 were studied by variable-temperature (1)H NMR experiments.