Convergence of path measures arising from a mean field or polaron type interaction

Summary We discuss the limiting path measures of Markov processes with either a mean field or a polaron type interaction of the paths. In the polaron type situation the strength is decaying at large distances on the time axis, and so the interaction is of short range in time. In contrast, in the mean field model, the interaction is weak, but of long range in time. Donsker and Varadhan proved that for the partition functions, there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range. The discussion of the path measures is more subtle. We treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples.Research supported by the Swiss National Foundation (21-29833.90)This article was processed by the authors using the Springer-Verlag T_EX ProbTh macro package 1991.     

(PPh4)2[WO2Cl3]2 · 2 CH2Cl2: Synthese,Schwingungsspektrum und Kristallstruktur

(PPh₄)₂[WO₂Cl₃]₂ · 2 CH₂Cl₂. Synthesis, Vibrational Spectrum, and Crystal Structure Depending on the stoichiometry and the solvent, dichloromethane or 1.2-dichloroethane, WO₂Cl₂ reacts with tetraphenylphosphonium chloride affording (PPh₄)₂[WO₂Cl₄] or (PPh₄)₂[WO₂Cl₃]₂, respectively. Both compounds are easily soluble in dichloromethane, from which they can be crystallized under incorporation of two molecules CH₂Cl₂ per formula unit. The crystalline compounds have been characterized by their IR and Raman spectra. According to the X-ray crystal structure analysis, (PPh₄)₂[WO₂Cl₃]₂ · 2 CH₂Cl₂ crystallizes in the triclinic space group P1 with one formula unit per unit cell (986 independent observed reflexions, R = 0.061). Lattice constants: a = 1100.2, b = 1116.9, c = 1238.4 pm, = 69.40, = 80.46 and = 85.62°. The crystals consist of PPh₄^⊕ ions, centrosymmetric [WO₂Cl₃]₂^2? anions and CH₂Cl₂ molecules. In the anions, the tungsten atoms are linked via two oxo bridges with WO distances of 184 and 252 pm. The distorted octahedral coordination around each tungsten atom is completed by three terminal chloro and one terminal oxo ligand (WO bond length 166 pm), the latter being in trans position to the longer WO bridging bond. (PPh₄)₂[WO₂Cl₄] · 2 CH₂Cl₂ also forms triclinic crystals that are isotypic with (PPh₄)₂[WOCl₅] · 2 CH₂Cl₂ and in which the anions must have orientational disorder.     

Rank-dependent moderate deviations of U-empirical measures in strong topologies

 We prove a rank-dependent moderate deviation principle for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S,𝒮). The result can be formulated on a suitable subset of all signed measures on (S ^m ,𝒮^{⊗ m} ). We endow this space with a topology, which is stronger than the usual τ-topology. A moderate deviation principle for Banach-space valued U-statistics is obtained as a particular application. The advantage of our result is that we obtain in the degenerate case moderate deviations in non-Gaussian situations with non-convex rate functions. Received: 22 February 2000 / Revised version: 15 November 2002 / Published online: 28 March 2003 Research partially supported by the Swiss National Foundation, Contract No. 21-298333.90. Mathematics Subject Classification (2000): Primary 60F10; Secondary 62G20, 28A35 Key words or phrases: Rank-dependent moderate deviations – Empirical measures – Strong topology – U-statistics     

Hydrogendiazid Synthese und Kristallstruktur von PPh4[N3HN3]

Hydrogen Diazide – Synthesis and Crystal Structure of PPh₄[N₃HN₃] PPh₄[N₃HN₃] has been prepared from PPh₄N₃ and Me₃SiN₃ by the reaction with water or ethanol forming colourless nonexplosive crystal needles, which were characterized by IR spectroscopy and by a crystal structure determination. Space group C2/c, Z = 12, lattice dimensions at –70 °C: a = 3782.4(3), b = 727.8(2), c = 2512.4(2) pm, β = 110.13(1)°, R = 0.0841. The [N₃HN₃]^– ion is characterized by an asymmetric N–H…N hydrogen bridge with a NN distance of 272(1) pm.     

Die Hydroaluminierung von 1,1,4,4‐Tetramethyl‐2,3‐diazabutadien mit Dialkylaluminiumhydriden – Synthese von Dialkylaluminiumhydrazoniden

The Hydroalumination of 1,1,4,4‐Tetramethyl‐2,3‐diazabutadiene by Dialkylaluminium Hydrides – Synthesis of Dialkylaluminium Hydrazonides 1,1,4,4‐Tetramethyl‐2,3‐diazabutadiene reacted with dimethylaluminium hydride by hydroalumination of only one C=N double bond. The hydrazone derivative [Me₂Al–N(CHMe₂)–N=CMe₂]₂ ( 1 ) was formed which gave a dimer possessing a six‐membered Al₂N₄ heterocycle. The hydroalumination of both C=N double bonds was not observed. Also an excess of di(tert‐butyl)‐ or bis(trimethylsilylmethyl)aluminium hydride afforded only the product of a single hydroalumination step, a second dialkylaluminium hydride molecule was attached via a coordinative interaction between its central aluminium atom and the nitrogen atom of the C=N double bond and in addition via a 3 c‐2 e Al–H–Al bond. Compounds [(Me₃C)₂Al][(Me₃C)₂AlH]N(CHMe₂)NCMe₂ ( 2 ) and [(Me₃SiCH₂)₂Al][(Me₃SiCH₂)₂AlH]N(CHMe₂)NCMe₂ ( 3 ) were formed which have five‐membered Al₂N₂H heterocycles. Thermolysis of 2 gave by C–H activation compound [(Me₃C)₂Al]₂[CH₂C(Me)=N–]₂ ( 4 ) in trace amounts which possesses two anellated AlN₂C₂ rings with a common N–N bond. In contrast, the thermal decomposition of 3 yielded by the cleavage of the N–N bond a dimeric dialkylaluminium methylideneamide ( 5 ) which has two intact C=N double bonds. Up to now our attempts to insert a C=N double bond into an Al–C bond remained unsuccessful, and only the formation of an adduct [(Me₃C)₃Al(–N=CMe₂)₂] ( 6 ) was observed upon treatment of tri(tert‐butyl)aluminium with the diazabutadiene derivative.