Bis[N,N′‐diisopropylbenzamidinato(−)]silicon(II): Lewis Acid/Base Reactions with Triorganylboranes

Reaction of the donor‐stabilized silylene 1 (which is three‐coordinate in the solid state and four‐coordinate in solution) with BEt₃ and BPh₃ leads to the formation of the Lewis acid/base complexes 2 and 3 , respectively, which are the first five‐coordinate silicon compounds with an Si?B bond. These compounds were structurally characterized by crystal structure analyses and by multinuclear NMR spectroscopic studies in the solid state and in solution. Additionally, the bonding situation in 2 and 3 was analyzed by quantum chemical studies.     

Modulable and Highly Diastereoselective Carbometalations of Cyclopropenes

The copper‐catalyzed carbomagnesiation reaction of cyclopropenyl esters 1 leads to various substituted cyclopropanes species 3 in good yields with very high diastereoselectivities. The reaction proceeds through a syn‐chelated carbomagnesiation reaction and could be extended to various cyclopropenylmethyl ester derivatives 5 . The potential of this approach was illustrated by the preparation of two consecutive all‐carbon quaternary stereocenters. However, the carbometalation reaction needs to be performed at temperature ranging from ?35 to ?20 °C to avoid subsequent fragmentation reaction into stereodefined β,γ‐nonconjugated unsaturated esters 4 . Alternatively, the carbocupration reaction with organocopper species could also be performed to leads to configurationally stable cyclopropyl copper species 2[Cu] . Additionally, when the Lewis acid character of the copper center is decreased (i.e., RCuCNLi), the reaction proceed with an anti‐selectivity. The diastereodivergent behavior of these organometallic species is of synthetic interest, since both diastereomers syn‐ 3 and anti‐ 3 can be obtained, at will, from the same precursor cyclopropenyl esters 1 .     

Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.     

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.     

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.