Characterisations of function spaces of generalised smoothness

We investigate function spaces of generalised smoothness of Besov and Triebel–Lizorkin type. Equivalent quasi-norms in terms of maximal functions and local means are given. An atomic decomposition theorem for this type of spaces is proved. Mathematics Subject Classification (2000) 46E35     

6-Thia-1-aza-5λ5-phosphabicyclo[3.2.0]hept-3-ene und Isothiocyanatophosphin-sulfide aus (Methylidenamino)phosphinen,Schwefelkohlenstoff und Acetylen- oder Ethylen-Derivaten

6-Thia-l-aza-5λ⁵-phosphabicyclo[3.2.0]hept-3-enes and Isothiocyanatophosphine-sulfides from (Methylidenamino)phosphines, Carbon Disulfide, and Acetylene- or Ethylene Derivatives (Methylidenamino)phosphines react with carbon disulfide to 1,3,2-thiazaphosphete derivatives 2 . Acetylenedicarboxylates give with 2 6-thia-l-aza-5λ⁵-phosphabicyclo[3.2.0]hept-3-enes 4a , b , and acrylates yield isothio-cyanatophosphin-sulfides 5a – c . The structures of 4a and 5a are proven by an X-ray diffraction analysis.     

Out of Equilibrium Correlations in the XY Chain

We study the transversal spin–spin correlations in the non-equilibrium steady state of the XY chain constructed by coupling a finite cutout of the chain to the two infinite parts to its left and right acting as thermal reservoirs at different temperatures. We prove that the spatial decay of these correlations is at least exponentially fast.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.