On weighted Hilbert spaces and integration of functions of infinitely many variables

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables _x1,_x2,…∈D$x_1,x_2,\dots \in D$. The setting is based on a reproducing kernel k$k$ for functions on D$D$, a family of non-negative weights _γu${\gamma }_u$, where u$u$ varies over all finite subsets of N$\mathbb{N}$, and a probability measure ρ$\rho$ on D$D$. We consider the weighted superposition K=_∑uγuku$K={\sum }_u{\gamma }_u{k}_u$ of finite tensor products _ku$k_u$ of k$k$. Under mild assumptions we show that K$K$ is a reproducing kernel on a properly chosen domain in the sequence space D^N$D^{\mathbb{N}}$, and that the reproducing kernel Hilbert space H(K)$H\left(K\right)$ is the orthogonal sum of the spaces H(_γuku)$H\left({\gamma }_u{k}_u\right)$. Integration on H(K)$H\left(K\right)$ can be defined in two ways, via a canonical representer or with respect to the product measure ρ^N${\rho }^{\mathbb{N}}$ on D^N$D^{\mathbb{N}}$. We relate both approaches and provide sufficient conditions for the two approaches to coincide.     

Detecting ligand binding to a small RNA target via saturation transfer difference NMR experiments in D(2)O and H(2)O

A small RNA motif is used as a target for ligand-based NMR-screening by saturation transfer difference (STD) NMR experiments. The prerequisites for using a small RNA target in STD experiments, such as saturation time, frequency, and pulses, are discussed. We also show that it is of advantage to use D2O as solvent instead of H2O due to the reduced R1 relaxation rate in D2O. The 27-nucleotide model of the ribosomal A-site was known to bind the aminoglycoside paromomycin with high affinity. This binding interaction could be detected easily, proving the effectiveness of STD NMR experiments as a screening tool for RNA-ligand interactions.     

Incorporation of an Amino Function in a (1S,2S,3R)-3-Hydroxy-2-methoxy-1-cyclohexane Carboxylic Acid

(1S,2S,3R)-3-Hydroxy-2-methoxy-1-cyclohexanecarboxylic acid was synthesized for the construction of an amino acid to be used in a constrained ring didemnin B analog ( 2 ). The amine functionality was introduced into the cyclohexane ring by reductive amination using sodium triacetoxyborohydride or by oxime formation followed by reduction.     

New cleavable isocyanides for the combinatorial synthesis of α-amino acid analogue tetrazoles

3-Substituted 3-isocyano propionates as new cleavable isocyanides in combinatorial tetrazole synthesis via Ugi-reaction are introduced. The obtained 5-substituted tetrazoles are transformed into carboxylic acid isosteric 5-substituted 1H-tetrazoles under basic cleavage conditions.     

TEM analysis of the influence of dose on damage behavior in MeV Au2+-implanted silicon

Extended lattice damage created by implantation of 3.6 MeV Au²⁺ ions has been investigated using transmission electron microscopy (TEM) and Rutherford backscattering spectrometry (RBS). Systematic observations of damage for Au²⁺ ions implanted with varying doses into silicon are explained in terms of a model. The origin of two distinct bands of extended defects is explained in terms of annealing of the central region of implant-damage, during the course of the implantation. Two distinct bands of Au precipitates are observed in high-dose implanted samples. This observation is explained as being the result, in part, of segregation of gold in front of a recrystallizing front, and in part, of gettering of dopant-atoms to nodes in a dislocation network. The network arises as a result of dynamic annealing of damaged crystalline silicon.