Preparation and properties of dinitrogen complexes of molybdenum and tungsten with trimethylphosphine as coligand : III. Synthesis and properties of cis-[W(N2)2(PMe3)4], trans-[W(C2H4)2(PMe3)4] and [M(N2)(PMe3)5] (M = Mo,W). The crystal and molecular structure of [Mo(N2)(PMe3)5]

The reduction of [WCl₄(PMe₃)₃] with dispersed sodium, under dinitrogen, gives cis-[W(N₂)₂(PMe₃)₄], while under ethylene trans-[W(C₂H₄)₂(PMe₃)₄] is obtained. The ethylene complex can also be prepared by displacement of the dinitrogen molecules in cis-[W(N₂)₂(PMe₃)₄] by ethylene at room temperature and pressure. Interaction of cis-[M(N₂)₂(PMe₃)₄] complexes (M = Mo, W), with PMe₃, under helium or argon, yields [M(N₂)(PMe₃)₅]. The molybdenum complex crystallizes in the orthorhombic space group Pnma, with a 22.063(6), b 12.106(4), c 9.745(4) Å. The Mo—P distance trans to the dinitrogen ligand (2.483(7) Å) is slightly longer than the average of the other four Mo—P bonds (2.460(5) Å).     

The formation and molecular structure of acetylcyclopentadienylsodium-tetrahydrofuranate

Acetylcyclopentadienylsodium has been isolated in crystalline form as a THF adduct from a reaction between cyclopentadienylsodium and methyl acetate in THF solution. The product has been characterized by means of a single-crystal X-ray diffraction study. {[C₅H₄CMeO]Na·THF}_n crystallizes in the monoclinic space group P2₁/c with unit cell parameters a 6.698(3), b 16.095(4), c 10.661(3) Å, β 92.93(3)° and D_c 1.17 g cm^?3 for Z = 4. Least-squares refinement led to a final R value of 0.080 based on 661 independent observed reflections. The coordination sphere around each sodium atom consists of the oxygen atoms from two C₅H₄CMeO ligands, the oxygen atom of the THF molecule, and an ion contact pair between the sodium and the five ring carbon atoms of the C₅H₄CMeO ligand.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Dispersive Blow-Up Ⅱ.Schr(o)dinger-Type Equations,Optical and Oceanic Rogue Waves

Addressed here is the occurrence of point singularities which owe to the fo-cusing of short or long waves,a phenomenon labeled dispersive blow-up.The context of this investigation is linear and nonlinear,strongly dispersive equations or systems of equa-tions.The present essay deals with linear and nonlinear Schr(o)dinger equations,a class of fractional order SchrSdinger equations and the linearized water wave equations,with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.     

Convex Komlós sets in Banach function spaces

In 1967 Komlós proved that for any sequence n{fn} in L₁(μ), with ‖fn‖?M<∞ (where μ is a probability measure), there exists a subsequence n{gn} of n{fn} and a function g∈L₁(μ) such that for any further subsequence n{hn} of n{gn},

     

Some observations on the substructure lattice of a Δ1 ultrapower

Given a (nontrivial) Δ₁ ultrapower ?/??, let ??_U denote the set of all Π₂‐correct substructures of ?/??; i.e., ??_U is the collection of all those subsets of |?/??| that are closed under computable (in the sense of ?/??) functions. Defining in the obvious way the lattice ??(?/??)) with domain ??_U, we obtain some preliminary results about lattice embeddings into – or realization as – an ??(?/??). The basis for these results, as far as we take the matter, consists of (1) the well‐known class of (non‐trivial) minimal ?/??'s, which function as atoms, and (2) the class of minimalfree ?/??'s, to whose nonemptiness a substantial section of the paper is devoted. It is shown that an infinite, convergent monotone sequence together with its limit point is embeddable in an ??(?/??), and that the initial segment lattices {0, 1,…, n } are not just embeddable in (as is trivial), but in fact realizable as, lattices ??(?/??). Finally, the diamond is (easily) embeddable; and if it is not realizable, then either the 1 ‐ 3 ‐ 1 lattice or the pentagon is at least embeddable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)