Synthesis of Highly Fluorinated Arene Complexes of [Rh(Chelating Phosphine)]+ Cations,and their use in Synthesis and Catalysis

The synthesis of rhodium complexes with weakly binding highly fluorinated benzene ligands is described: 1,2,3-F₃C₆H₃, 1,2,3,4-F₄C₆H₂ and 1,2,3,4,5-F₅C₆H are shown to bind with cationic [Rh(Cy₂P(CH₂)_xPCy₂)]⁺ fragments (x=1, 2). Their structures and reactivity with alkenes, and use in catalysis for promoting the Tishchenko reaction of a simple aldehyde, are demonstrated. Key to the synthesis of these complexes is the highly concentrated reaction conditions and use of the [Al{OC(CF₃)₃}₄]⁻ anion.     

Gravity flow of coarse cohesionless granular materials in conical hoppers

This paper is concerned with the flow of cohesionless granular materials in conical hoppers. The granules are considered to be coarse such that the effects of interstitial fluid can be neglected. Two analytical approaches based upon the method of integral relations are presented. In the first the detailed variations of shear and normal stresses over the cross-section of the hopper are considered. The second is a very simple analysis which makes use of a mean normal stress averaged over the cross-section. The two analyses yield predictions for the flow rate which are almost identical. After the inclusion of an empirical correction factor to account for non-uniform exit velocity profiles and bulk density reductions near the outlet, the flow rate predictions and their variations with internal and wall friction angles and with hopper wall slope are found to agree well with experimental measurements.

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Résumé Cet article traite de l'écoulement de matières granuleuses sans cohésion dans des trémies coniques. On admet que les granules sont assez gros pour que les effets dûs au fluide intersistiel restent négligeables. 2 méthodes d'approche analytiques basées sur la méthode des relations intégrales sont présentées. La première considère en détail la distribution des contraintes normales et de cisaillement sur la section de la trémie. La seconde méthode est une simple analyse basée sur la moyenne des contraintes normales sur cette section. Les 2 analyses fournissent des prédictions presque identiques sur le taux d'écoulement. En introduisant un facteur de correction empirique pour tenir compte de la non-uniformité de profils de vitesses d'écoulement ainsi que de la réduction de densité à la sortie, l'analyse fournit des taux d'écoulement et leurs variations avec la friction interne, les angles de friction aux parois et l'inclinaison des parois de la trémie. Les résultants obtenus sont en accord avec des mesures expérimentales.

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On leave at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge.     

Single crystal growth and the elastic behaviour of In2Bi

Ultrasound velocities have been measured in In₂Bi crystals grown by zone-levelling. The elastic constants are reported. The elastic behaviour is remarkably isotropic: a pure shear and two almost pure waves propagate in all directions other than pure mode directions.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.