Stable extendibility of vector bundles over lens spaces mod 3 and the stable splitting problem

Let Lⁿ(3) denote the (2n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over Lⁿ(3) to be stably extendible to L^m(3) for every mn, and establish the formula on the power ζ^k=ζζ (k-fold) of a real vector bundle ζ over Lⁿ(3). Moreover, we answer the stable splitting problem for real vector bundles over Lⁿ(3) by means of arithmetic conditions.     

Chloride‐Bridged Dinuclear Rhodium(III) Complexes Bearing Chiral Diphosphine Ligands: Catalyst Precursors for Asymmetric Hydrogenation of Simple Olefins

Efficient rhodium(III) catalysts were developed for asymmetric hydrogenation of simple olefins. A new series of chloride‐bridged dinuclear rhodium(III) complexes 1 were synthesized from the rhodium(I) precursor [RhCl(cod)]₂, chiral diphosphine ligands, and hydrochloric acid. Complexes from the series acted as efficient catalysts for asymmetric hydrogenation of (E)‐prop‐1‐ene‐1,2‐diyldibenzene and its derivatives without any directing groups, in sharp contrast to widely used rhodium(I) catalytic systems that require a directing group for high enantioselectivity. The catalytic system was applied to asymmetric hydrogenation of allylic alcohols, alkenylboranes, and unsaturated cyclic sulfones. Control experiments support the superiority of dinuclear rhodium(III) complexes 1 over typical rhodium(I) catalytic systems.     

Olefin Metathesis Catalysts Generated In Situ from Molybdenum(VI)-Oxo Complexes by Tuning Pendant Ligands

Tailored molybdenum(VI)-oxo complexes of the form MoOCl₂(OR)₂(OEt₂) catalyse olefin metathesis upon reaction with an organosilicon reducing agent at 70 °C, in the presence of olefins. While this reactivity parallels what has recently been observed for the corresponding classical heterogeneous catalysts based on supported metal oxide under similar conditions, the well-defined nature of our starting molecular systems allows us to understand the influence of structural, spectroscopic and electronic characteristics of the catalytic precursor on the initiation and catalytic proficiency of the final species. The catalytic performances of the pre-catalysts are determined by the highly electron withdrawing (σ-donation) character of alkoxide ligands, O^tBu_F9 being the best. This activity correlates with both the ⁹⁵Mo chemical shift and the reduction potential that follows the same trend: O^tBu_F9>O^tBu_F6>O^tBu_F3.