Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

Zur Aktivierung partiell silylierter Kohlenhydrate mittels Triphenylphosphan/Azodicarbonsäureester

On the Activation of Partially Silylated Carbohydrates Using Triphenylphosphane/Diethylazodicarboxylate Reaction of methyl α-D-glucopyranoside ( 1 ) with two equivalents of t-butyldimethylchlorosilane yields methyl 2,6-bis[O-(t-butyldimethylsilyl)]-α-D-glucopyranoside ( 1a ) and methyl 3,6-bis[O(t-butyldimethylsilyl)]-α-D-glucopyranoside ( 1b ) in a ratio of 4:1. The anomeric β-pyranoside 2 affords methyl 2,6-bis[O(t-butyldimethylsilyl)]-β-D-glucopyranoside ( 2a ) and methyl 3,6-bis[O(t-butyldimethylsilyl)]-β-D-glucopyranoside ( 2b ) in nearly equal amounts. 2b is isomerized to methyl 4,6-bis[O(t-butyldimethylsilyl)]-β;-D-glucopyranoside ( 2c ) (83%) and 2a (10%) with triphenylphosphane/diethylazodicarboxylate. Structures were assigned by NMR.-analysis and CD.-analysis of the corresponding benzoates 1c , 1d and 2d and of the acetates 2e and 2f . 1a is transformed into methyl 4-azido-2, 6-bis[O(t-butyldimethylsilyl)]-4-deoxy-α-D-galactopyranoside ( 3 ) with triphenylphosphane/diethylazodicarboxylate/HN₃. 2a and 2c yield the 3-azido-allosides 5 and 7 respectively under similar conditions. The activation by triphenylphosphane/diethylazodicarboxylate is high enough to introduce also p-nitrobenzoate groups with inversion of configuration at the reaction center. By this way 1a and 2a give methyl 2, 6-bis[O(t-butyldimethylsilyl)]-4-O-p-nitrobenzoyl-α-D-galactopyranoside ( 4 ) and methyl 2, 6-bis[O-(t-butyldimethylsilyl)]-3-O?ptrobenzoyl-β-D-allopyranoside ( 6 ) respectively. For elucidation of structures the acetate derivatives 3a-7a were prepared.     

Strukturelle Abwandlungen anN-Acetylneuraminsäure, 3

From methyl-5-acetylamino-7,8-anhydro-4,9-O-bis-(t-butyldimethylsilyl)-3,5-dideoxy--D-glycero-D-galacto-2-nonulopyranosidonic acid methylester (1) the derivatives1 a and1 b were obtained by removing the 9-O-(t-butyldimethylsilyl)group withBu ₄NF, followed by acetylation. Treatment of1 b with 80% acetic acid and acetanhydride/pyridine yields the 8-epi-N-acetylneuraminic acid derivative2 a and the 7-epi-N-acetylneuraminic acid derivative3 a in a ratio of 3:1 (Scheme 1). The structure elucidation of2 b was achieved by converting2 b via the 4,9-bis-O-(tBDMSi)-8-O-tosyl-derivative2 d into the epoxide1 (Scheme 2). Using the same sequence the epoxides4 and5 were transformed into theN-acetylneuraminic acid derivative6 a and the 7,8-bis-epi-N-acetylneuraminic acid derivative7 a (Scheme 3). After treatment with sodium hydroxide and 0.025m HCl and Dowex 50 H⁺ the 8-epi-, 7-epi- and 7,8-bis-epi-N-acetylneuraminic acids2,3, and7 were obtained. These three compounds were tested withCMP-N-acetylneuraminic acid synthetase.

Herrn KollegenK. Schlögl mit den besten Wünschen zum 60. Geburtstag.     

Supramolecular Assemblies for Electronic Materials

This work presents a synergy between organic electronics and supramolecular chemistry, in which a host–guest complex is designed to function as an efficacious electronic material. Specifically, the noncovalent recognition of a fullerene, phenyl-C₆₁-butyric acid methyl ester ( PC₆₁BM ), by an alternating perylene diimide ( P )-bithiophene ( B ) conjugated macrocycle ( PBPB ) results in a greater than five-fold enhancement in electron mobility, relative to the macrocycle alone. Characterization and quantification of the binding of fullerenes by host PBPB is provided alongside evidence for intermolecular electronic communication within the host–guest complexes.     

Nucleophile‐Dependent Regio‐ and Stereoselective Ring Opening of 1‐Azoniabicyclo[3.1.0]hexane Tosylate

1‐[(1R)‐(1‐Phenylethyl)]‐1‐azoniabicyclo[3.1.0]hexane tosylate was generated as a stable bicyclic aziridinium salt from the corresponding 2‐(3‐hydroxypropyl)aziridine upon reaction with p‐toluenesulfonyl anhydride. This bicyclic aziridinium ion was then treated with various nucleophiles including halides, azide, acetate, and cyanide in CH₃CN to afford either piperidines or pyrrolidines through regio‐ and stereoselective ring opening, mediated by the characteristics of the applied nucleophile. On the basis of DFT calculations, ring‐opening reactions under thermodynamic control yield piperidines, whereas reactions under kinetic control can yield both piperidines and pyrrolidines depending on the activation energies for both pathways.     

Variational Solution of Stochastic Schrödinger Equations With Power-Type Nonlinearity

We investigate a Schrödinger problem with multiplicative Gaussian noise term and power-type nonlinearity on a bounded one-dimensional domain. In order to prove the existence and uniqueness of the variational solution, a further process will be introduced which allows to transform the stochastic nonlinear Schrödinger problem into a pathwise one. Galerkin approximations and compact embedding results are used.