Chalkogenid-Disilicate der Lanthanoide vom Typ M4X3[Si2O7] (M  Ce?Er; X  S,Se)

M₄X₃[Si₂O₇]-Type Lanthanide Chalcogenide Disilicates (M ? Ce? Er; X ? S, Se) Attempts to produce single crystals of MSe₂ (or MSe^2?X) by vapour phase transport with iodine or the oxidation of MCl₂ (or MClH) with sulfur in the presence of NaCl in sealed evacuated quartz containers often yielded well-grown single crystals with the composition M₄X₃[Si₂O₇] (M ? pr, Sm, Gd, X ? Se, and M ? Nd, Er, X ? S) as by-products. The crystal structures (tetragonal, 14₁/amd (no. 141)), Z = 8, contain two crystallographically independent M₃₊ Cations that are interconnected by chalcogenide (X_2?) and disilicate anions ([Si₂O₇]^6?). (M1)³⁺ is surrounded by eight (five X^2? and three terminal O_2? of the disilicate group), (M2)³⁺ by nine (three X^2? and six terminal O^2? of the [Si₂O₇]_6? anion) chalcogenide anions. The disilicate anion itself exhibits the eclipsed conformation with non-linear Si? O? Si bridges (angles: 128 – 133°).     

-bases for cohomology of local systems on hyperplane complements

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes . The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement . The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of . The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

     

Metric stability of trees and tight spans

We prove optimal extension results for roughly isometric relations between metric ( ${\mathbb{R}}$ R -)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov–Hausdorff (GH) distance, for certain metric spanning constructions: the GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.     

PotLLL: a polynomial time version of LLL with deep insertions

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where the insertions are restricted to blocks, behave nicely in practice concerning the running time. However no proof of polynomial running time is known. In this paper PotLLL, a new variant of DeepLLL with provably polynomial running time, is presented. We compare the practical behavior of the new algorithm to classical LLL, BKZ as well as blockwise variants of DeepLLL regarding both the output quality and running time.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Glass physics: from fundamentals to applications

The European Physical Journal E -     

John Napiers Trigonometrie – ein Blick zurück

Zusammenfassung  Dieser Artikel gibt einen überblick über Napiers Beitr?ge zur sph?rischen Trigonometrie, wobei die dafür notwendigen sachlichen und historischen Grundlagen weitgehend bereitgestellt werden. Da diese Beitr?ge in der Literatur trotz ihrer erheblichen Bedeutung ganz im Schatten von Napiers Logarithmen stehen, erscheint eine solche Zusammenschau gerechtfertigt. Auch kommen einige neue oder kaum bekannte Einzelheiten zur Sprache. Mathematics subject classification (2000)  01A45 · 51-03     

Hydrogen bonding in brucinium dihydrogen citrate trihydrate at 130 K

The structure of brucinium dihydrogen citrate trihydrate (systematic name: 2,3‐dimeth­oxy‐10‐oxostrychnidinium dihydrogen citrate trihydrate), C₂₃H₂₇N₂O₄⁺·C₆H₇O₇⁻·3H₂O, has been determined at 130 K. The crystallographic asymmetric unit comprises two brucinium cations, two dihydrogen citrate anions and six water mol­ecules of solvation. The two citrate anions, which are conformationally dissimilar, associate through extensive hydrogen‐bonding inter­actions with the common undulating brucinium cation layer substructures and the water mol­ecules, forming a three‐dimensional framework polymer.