Complex powers of classical SG-pseudodifferential operators

Abstract Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components. Keywords: Complex power, Weighted symbols, Noncompact manifolds     

Synthesis and Crystal Structure of the Hydrogen Polyphosphide Bis(Tetraphenylphosphonium) hydrogenheptaphosphide‐ammonia(1/3) (PPh4)2HP7 · 3NH3

(PPh₄)₂HP₇ · 3NH₃ was prepared by the reaction of K₃P₇ with a proton‐charged ion exchange resin in the presence of (PPh₄)Br in liquid ammonia, and characterized by low temperature X‐ray structure analysis. The thermally very unstable compound contains the hydrogenheptaphosphide anion HPequation/tex2gif-stack-1.gif, the nortricyclane‐like cage of which is slightly distorted by the attachment of the hydrogen atom.     

Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

Bounded imaginary powers of differential operators on manifolds with conical singularities

 We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L _p -spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers A ^z , zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Received: 12 June 2001; in final form: 3 June 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 35J70, 47A10, 35K57     

Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Fröhlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.     

Towards the Synthesis of Azoacetylenes

The synthesis of azoacetylenes (=dialkynyldiazenes) 1 and 2 has been investigated. They represent a still elusive class of chromophores with potentially very interesting applications as novel bistable photochemical molecular switches or as antitumor agents (Fig. 1). Our synthetic efforts have led us alongside three different approaches (Scheme 1). In a first route, it was envisioned to generate the azo (=diazene) bond by photolysis of N,N′‐dialkynylated 1,3,4‐thiadiazolidine‐2,5‐diones that are themselves challenging targets (Scheme 2). Attempts are described to obtain the latter by alkynylation of the parent heterocycle with substituted alkynyliodonium salts. In a conceptually similar approach, the no‐less‐challenging dialkynylated 9,10‐dihydro‐9,10‐diazanoanthracene ( 29 ) was to be generated by alkynylation of the unsubstituted hydrazine 28 (Scheme 6). In a second route, the generation of the N?N bond from Br‐substituted divinylidenehydrazines (ketene‐azines) 35 was attempted in a synthetic scheme involving an aza‐Wittig reaction between azinobis(phosphorane) 36 and (triisopropylsilyl)ketene 37 (Scheme 7). Finally, a third approach, based on the formation of the central azo bond as the key step, was explored. This route involved the extrapolation of a newly discovered condensation reaction of N,N‐disilylated anilines with nitroso compounds (Scheme 11, and Tables 1 and 2) to the transformation of N,N‐disilylated ynamine 55 and nitroso‐alkyne 54 (Scheme 13).