Stability inequalities and universal Schubert calculus of rank 2

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each ??sufficiently rich?? spherical building Y of type W we associate a certain cohomology theory $ H_{BK}^*(Y) $ and verify that, first, it depends only on W (i.e., all such buildings are ??homotopy equivalent??), and second, $ H_{BK}^*(Y) $ is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology ??pre-ring?? on Y. The convex ??stability?? cones in $ {\left( {{\mathbb{R}^2}} \right)^m} $ defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of [KLM1]; equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. The independence of the (co)homology theory of Y refines the result of [KLM2], which asserted that the Stability Cone depends on W rather than on Y. Quite remarkably, the cohomology ring $ H_{BK}^*(Y) $ is obtained from a certain universal algebra A _t by a kind of ??crystal limit?? that has been previously introduced by Belkale?CKumar for the cohomology of ag varieties and Grassmannians. Another degeneration of A _t leads to the homology theory H _*(Y).     

An octanuclear iron(III) isobutyrate wheel

The reaction of the μ₃‐oxido‐centred trinuclear isobutyrate cluster [Fe₃O(O₂CCHMe₂)₆(H₂O)₃]⁺ with an excess of phenol (PhOH) in chloroform produces a novel octanuclear Fe^III cluster, cyclo‐tetra‐μ₂‐hydroxido‐dodeca‐μ₂‐isobutyrato‐κ²⁴O:O′‐octa‐μ₂‐phenolato‐κ¹⁶O:O′‐octairon(III) phenol hexasolvate monohydrate, [Fe₈(C₄H₇O₂)₁₂(C₆H₅O)₈(OH)₄]·6C₆H₅OH·H₂O. The neutral cluster is located about a centre of inversion and consists of a planar ring of eight Fe^III centres with two types of bridges between adjacent Fe atoms: each Fe atom is bridged to one of its neighbours by a μ‐hydroxide and two 1,3‐bridging carboxylates, or by two phenolate and one 1,3‐bridging isobutyrate ligand. The cavity within the {Fe₈} wheel is occupied by a disordered water molecule. Intermolecular O—H...O hydrogen bonds and C—H...π interactions connect the clusters and the phenol solvent molecules to form a three‐dimensional network.     

Tuning of Photocatalytic Hydrogen Production and Photoinduced Intramolecular Electron Transfer Rates by Regioselective Bridging Ligand Substitution

Artificial photosynthesis based on supramolecular photocatalysts offers the unique possibility to study the molecular processes underlying catalytic conversion of photons into chemical fuels in great detail and to tune the properties of the photocatalyst by alterations of the molecular framework. Herein we focus on both possibilities in studying the photocatalytic reduction of protons by derivatives of the well‐known photocatalyst [(tbbpy)₂Ru(tpphz)PdCl₂](PF₆)₂ [4,4′‐di‐tert‐butyl‐2,2′‐bipyridine (tbbpy), tetrapyrido[3,2‐a:2′,3′‐c:3′′,2′′‐h:2′′′,3′′′‐j]phenazine (tpphz)]. We report on a modified photocatalyst where the crucial bridging ligand tpphz is substituted by bromine and investigate the effect of the structural variation on the catalytic properties of the complex and its ultrafast intramolecular charge transfer behavior. It is found that structural modification stabilizes the phenanthroline‐centered metal‐to‐ligand charge‐transfer state on the tpphz moiety, thereby reducing the electron transfer gradient across the entire electron‐relaying bridging ligand and at the same time accelerating nanosecond ground‐state recovery. The same structural modifications cause an overall reduction of the catalytic activity of the complex. Thus, the results highlight the potential of small structural variations in the molecular framework of supramolecular catalysts in understanding the photoinduced charge‐transfer processes and optimizing their catalytic performance.     

Tris(oxazolinyl)boratomagnesium-catalyzed cross-dehydrocoupling of organosilanes with amines, hydrazine, and ammonia

We report magnesium-catalyzed cross-dehydrocoupling of Si-H and N-H bonds to give Si-N bonds and H(2). A number of silazanes are accessible using this method, as well as silylamines from NH(3) and silylhydrazines from N(2)H(4). Kinetic studies of the overall catalytic cycle and a stoichiometric Si-N bond-forming reaction suggest nucleophilic attack by a magnesium amide as the turnover-limiting step.     

Electrostatic rigidity of polyelectrolytes from reparametrization invariance

The persistence length l_p of a polyelectrolyte chain can be represented as l_p = l_O + l_e where l_O is the bare persistence and I_e is the electrostatic contribution coming from the effects of electrostatic chain self-interactions. Using a reparametrization-invariant path integral model of semiflexible polymers we find that I_e depends on the ionic strength I as I_e ∼ I^−1/2. This result accords with experimental observations and recent Monte Carlo simulations. Reparametrization-invariance is apparently an essential constrainet in selecting acceptable models of semiflexible polymers.     

Interplay of chemical and physical factors in reacting polymer blends. Theoretical considerations

Various effects produced by copolymers in polymer blends are discussed, with an emphasis on the role of interchain interactions. Simple theoretical models are considered to study the following problems: the interplay of diffusion and macromolecular reaction in compatible and incompatible blends, the stabilizing effect of premade diblock copolymer on the system of minor phase particles in incompatible blends, the kinetics of transesterification in a homogeneous blend. The effect of diblock copolymer on the Ostwald ripening in a polymer blend is stated in more details; the possibility of narrowing the size distribution of minor phase particles is predicted.