Euler–Mahonian statistics via polyhedral geometry

A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler–Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler–Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.     

Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ^∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.     

Faddeev random phase approximation applied to molecules

The Faddeev Random Phase Approximation (FRPA) is a Green’s function method which couples collective degrees of freedom to the single particle motion by resumming an infinite number of Feynman diagrams. The Faddeev technique is applied to describe the two-particle-one-hole (2p1h) and two-hole-one-particle (2h1p) Green’s function in terms of non-interacting propagators and kernels for the particle-particle (pp) and particle-hole (ph) interactions. This results in an equal treatment of the intermediary pp and ph channels. In FRPA both the pp and ph phonons are calculated on the random phase approximation (RPA) level. In this work the equations that lead to the FRPA eigenvalue problem are derived. The method is then applied to atoms, small molecules and the Hubbard model, for which the ground state energy and the ionization energies are calculated. Special attention is directed to the RPA instability in the dissociation limit of diatomic molecules and in the Hubbard model. Several solutions are proposed to overcome this problem.     

Delayed-Choice Experiments and the Metaphysics of Entanglement

Delayed-choice experiments in quantum mechanics are often taken to undermine a realistic interpretation of the quantum state. More specifically, Healey has recently argued that the phenomenon of delayed-choice entanglement swapping is incompatible with the view that entanglement is a physical relation between quantum systems. This paper argues against these claims. It first reviews two paradigmatic delayed-choice experiments and analyzes their metaphysical implications. It then applies the results of this analysis to the case of entanglement swapping, showing that such experiments pose no threat to realism about entanglement.     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.     

Chalcogen–chalcogen-bond activation by an ambiphilic,doubly reduced organoborane

The 9,10-dimethyl-9,10-dihydro-9,10-diboraanthracene dianion salt Li₂[1] instantaneously reacts with the dichalcogens O₂(tBu)₂, S₂Me₂, S₂(pTol)₂, Se₂Ph₂, and Te₂Ph₂ under reductive cleavage of the chalcogen–chalcogen bonds to give lithium chalcogenides and neutral 1. In solution, the products are involved in rapid association/dissociation equilibria. In the crystalline state, 1 and Li[O(tBu)] form a diadduct, whereas Li[SMe] gives a B?B-bridged monoadduct. In the three remaining cases, the crystal lattices contain monoadducts of 1 with one tri- and one tetracoordinate boron atom.     

Ring-expansion approaches for the total synthesis of salimabromide

We describe the evolution of a synthetic strategy for the construction of the marine polyketide salimabromide. Combining a bicyclo[3.1.0]hexan-2-one ring-expansion to build up a functionalized naphthalene and an unprecedented rearrangement/cyclization cascade, enabled synthesis of a dearomatized tricyclic subunit of the target compound. Alternatively, an intramolecular keteniminium [2 + 2]-cycloaddition and subsequent Baeyer–Villiger ring-expansion gave access to the sterically encumbered architecture of salimabromide. Sequential oxidation of the carbon framework finally enabled the total synthesis of this unusual natural product.     

Phase Formation Studies of Lead(II) Copper(II) Oxotellurates: The Crystal Structures of Dimorphic PbCuTeO5, PbCuTe2O6, and [Pb2Cu2(Te4O11)](NO3)2

Two modifications of the oxotellurate(VI) PbCuTeO₅ were isolated as single crystals from product mixtures obtained from solid state reactions, whereas single crystals of the oxotellurates(IV) PbCuTe₂O₆ and [Pb₂Cu₂(Te₄O₁₁)](NO₃)₂ were grown under hydrothermal conditions. The crystal structures of all compounds comprise of characteristic coordination polyhedra, viz. nearly square [CuO₄] plaquettes for divalent copper, octahedral [TeO₆] units for hexavalent tellurium, trigonal‐pyramidal [TeO₃] and bisphenoidal [TeO₄] groups for tetravalent tellurium, and distorted [PbO_x] polyhedra for divalent lead. PbCuTeO₅ is dimorphic and crystallizes in a monoclinic and a triclinic modification, related by a translationengleiche group‐subgroup relation of index 2. PbCuTe₂O₆ represents the ideal composition of the rare mineral choloalite. The characteristic feature of the crystal structure of [Pb₂Cu₂(Te₄O₁₁)](NO₃)₂ is its layered set‐up, comprised of cationic [Pb₂Cu₂(Te₄O₁₁)]²⁺ ribbons (width approximately 6.7 Å) sandwiched between nitrate anions that are only weakly bound to the cationic layers.     

Reduction vs. Addition: The Reaction of an Aluminyl Anion with 1,3,5,7‐Cyclooctatetraene

The potassium aluminyl complex K[Al(NON^Ar)] (NON=NON^Ar=[O(SiMe₂NAr)₂]^2?, Ar=2,6‐iPr₂C₆H₃) reacts with 1,3,5,7‐cyclooctatetraene (COT) to give K[Al(NON^Ar)(COT)]. The COT‐ligand is present in the asymmetric unit as a planar μ₂‐η²:η⁸‐bridge between Al and K, with additional K???π‐aryl interactions to neighboring molecules that generate a helical chain. DFT calculations indicate significant aromatic character, consistent with reduction to [COT]^2?. Addition of 18‐crown‐6 causes a rearrangement of the C₈‐carbocycle to form the isomeric 9‐aluminabicyclo[4.2.1]nona‐2,4,7‐triene anion.