Rational Cherednik algebras and Hilbert schemes

Let H_c be the rational Cherednik algebra of type A_n-1 with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators, with associated graded ring where W is the nth symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod;• the associated graded Z-algebra has grB-lqgr?coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.This can be regarded as saying that U_c simultaneously gives a non-commutative deformation of h⊕h^*/W and of its resolution of singularities Hilb(n)→h⊕h^*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of H_c and its relationship to Hilb(n).     

Syntheses and structures of bowl-shaped triarylphosphines and their palladium(II) complexes

Novel bowl-shaped triarylphosphines, tris(2,2″,6,6″-tetraalkyl[1,1′:3′,1″-terphenyl]-5′-yl)phosphines (TRMP: alkyl = methyl; TRIP: alkyl = isopropyl), were prepared by lithiation of the corresponding m-terphenyl bromides followed by reaction with PCl₃. X-ray crystallographic analysis revealed that the phosphorus center of TRMP is embedded in the shallow bowl-shaped cavity of 16 Å diameter and 2.1 Å depth formed by three radially extended m-terphenyl units. Its cone angle was estimated to be as large as 174°. In the crystal structure of TRIP, the depth of the cavity and cone angle increased to 3.3 Å and 206°, respectively, because of the different arrangement of the m-terphenyl units. In contrast with TRMP, which can form the mononuclear complex, PdCl₂(TRMP)₂ (6), in the reaction with PdCl₂, treatment of TRIP (1 or 3 eq.) with PdCl₂ produced the trinuclear palladium(II) chloride complex, [(PdCl₂)₃(TRIP)₂] (8), as a single product. X-ray crystallography established the structure of 8, where the trimer of PdCl₂ is terminated by two TRIP ligands. The formation of the different types of PdCl₂ complexes was explained in terms of the difference in the cavity shape of TRMP and TRIP.     

New states in heavy Cd isotopes and evidence for weakening of the N = 82 shell structure

A chemically selective laser ion source has been used in a β-decay study of heavy Ag isotopes into even-even Cd nuclides. Gamma-spectroscopic techniques in time-resolving event-by-event and multiscaling modes have permitted the identification of the first 2⁺ and 4⁺ levels in ¹²⁶Cd₇₈, ¹²⁸Cd₈₀, and tentatively the 2⁺ state in ¹³⁰Cd₈₂. From a comparison of these new states in ₄₈Cd with the E(2⁺) and E(4⁺)/E(2⁺) level systematics of ₄₆Pd and ₅₂Te isotopes and several recent model predictions, possible evidence for a weakening of the spherical N = 82 neutron-shell below double-magic ¹³²Sn is obtained. Received: 13 July 2000 / Accepted: 12 October 2000     

Existence of Hopf subalgebras of GK-dimension two

Let H be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If H is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then H contains a Hopf subalgebra of Gelfand-Kirillov dimension two.     

Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations

The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem.     

A note on the split common fixed-point problem for quasi-nonexpansive operators

Based on the very recent work by Censor and Segal (2009) [1], and inspired by Xu (2006) [9], Zhao and Yang (2005) [10], and Bauschke and Combettes (2001) [2], we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.     

Three-component entanglements consisting of three crescent-shaped bidentate ligands coordinated to an octahedral metal centre

3,3'-biisoquinoline ligands (biiq) L, bearing aromatic substituents on their 8 and 8' positions, have been used to generate interwoven systems consisting of three crescent-shaped ligands disposed around an octahedral metal centre. Mono-ligand complexes of the type [ReL(CO)3py]+ (py: pyridine) have also been prepared, leading to sterically non-hindering complexes in spite of the endotopic nature of the chelate used. The three-component entanglements have been prepared by using either FeII or RuII as gathering metal centre. The synthetic procedure is simple and efficient, affording fully characterised complexes as their PF6 or SbCl6 salts. X-ray crystallography clearly shows that the crescent-shaped ligands do not repel each other in the tris-chelate complexes. In an analogous way, the ReI complexes show open structures with no steric repulsion between the L ligand and the ancillary CO or py groups. The FeL3 or RuL3 compounds are very unusual in the sense that, contrary to all the other tris-bidentate chelate complexes made till now, the three organic components are tangled up, in a situation which will be very favourable to the formation of new non trivial topologies of the catenane type.     

Synthesis and characterisation of nonclassical ruthenium hydride complexes containing chelating bidentate and tridentate phosphine ligands

The synthesis and characterisation of nonclassical ruthenium hydride complexes containing bidentate PP and tridentate PCP and PNP pincer-type ligands are described. The mononuclear and dinuclear ruthenium complexes presented have been synthesised in moderate to high yields by the direct hydrogenation route (one-pot synthesis) or in a two-step procedure. In both cases [Ru(cod)(metallyl)(2)] served as a readily available precursor. The influences of the coordination geometry and the ligand framework on the structure, binding, and chemical properties of the M--H(2) fragments were studied by X-ray crystal structure analysis, spectroscopic methods, and reactivity towards N(2), D(2), and deuterated solvents.