Approximation numbers of Sobolev imbeddings over unbounded domains

Let Ω ? $\text{R}$^N be an open set with $\text{dist}\text{(x, ?Ω) = O(¦ x ¦}{}^{\mathrm{?l}}\text{)}\text{for}\text{x ? Ω}$ and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers α_n of Sobolev imbeddings

over these quasibounded domains Ω. Here

denotes the Sobolev space obtained by completing $\text{C}{}_0{}^{\mathrm{staggered\infty }}\text{(Ω)}$ under the usual Sobolev norm. We prove $\text{α}{}_{\mathrm{n}}\text{(I}{}_{\mathrm{p,q}}{}^{\mathrm{m}}\text{)}\text{\$}\text{?}\text{n}{}^{\mathrm{?\gamma }}$, where

. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of I_p,q^m. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded domain of the above type.     

Ground-state properties and static dipole polarizabilities of the alkali dimers from K2 n to Fr2 n(n=0,+1) from scalar relativistic pseudopotential coupled cluster and density functional studies

The newly adjusted energy-consistent nine-valence-electron pseudopotentials for K to Fr are used to calculate spectroscopic properties for the neutral and positively charged alkali dimers using coupled cluster and density functional theory. For the neutral dimers the static dipole polarizability was calculated. The coupled cluster results are all in excellent agreement with experimental values. The density functionals used can give quite different spectroscopic properties especially for the dipole polarizability, with the Perdew-Wang PW91 functional performing best.     

The immunosensors for measurement of 2,4-dichlorophenoxyacetic acid based on electrochemical impedance spectroscopy

Electrochemical impedance spectroscopy (EIS) was evaluated for the direct determination of herbicide 2,4-dichlorophenoxyacetic acid (2,4-D). Specific antibody against 2,4-D was immobilised onto different gold electrodes. Several methods of antibody immobilisation by covalent linkage to modified surface were studied. Self-assembled monolayers formed using thiocompounds as cystamine, 4-aminothiophenol (ATPh), 3,3'-dithiopropionic acid di-(N-succinimidyl ester) (DTSP) and 11-mercaptoundecanoic acid (MUA) were chosen for the sensing surface activation. Three different sensor types were tested: screen-printed disc and finger-like structures and interdigitated array (IDA) electrodes produced by lithography. The measurements were carried out in a stationary arrangement, and the reaction between hapten and the immobilised antibody was observed online. Changes of impedance parameters were evaluated, and the best immobilisation technique (using 4-aminothiophenol) was chosen for further measurements. Impedance changes due to immunocomplex formation were evaluated, and the possibility of direct monitoring of 2,4-D binding to the antibody was demonstrated at a fixed frequency. For the strip sensor, the calibration curves were constructed in concentration range from 45 nmol l(-1) to 0.45 mmol l(-1) of 2,4-D.     

A top-down approach to protein structure studies using chemical cross-linking and Fourier transform mass spectrometry

In a preliminary communication we described a top-down approach to the determination of chemical cross-link location in proteins using Fourier transform mass spectrometry (FT-MS). We have since extended the approach to use a series of homobifunctional cross-linkers with the same reactive functional groups, but different cross-linker arm lengths. Correlating cross-linking data across a series of related linkers allows the distance constraint derived from a cross-link between two reactive side chains to be determined more accurately and increases the confidence in the assignment of the cross-links. In ubiquitin, there are seven lysines with primary amino groups and the amino terminus. Disuccinimidyl suberate (DSS, cross-linker arm length = 11.4 A), disuccinimidyl glutarate (DSG, cross-linker arm length = 7.5 A) and disuccinimidyl tartrate (DST, cross- linker arm length = 5.8 A) are homobifunctional cross-linking reagents that react specifically with primary amines. Using tandem mass spectrometry (MS/MS) on the singly, internally cross-linked precursor ion of ubiquitin, we found cross-links with DSS and DSG between the amino terminus and Lys 6, between Lys 6 and Lys 11, and between Lys 63 and Lys 48. Using disuccinimidyl tartrate (DST), the shortest cross-linker in the series, only the cross-links between the amino terminus and Lys 6, and between Lys 6 and Lys 11 were observed. The observed cross-links are consistent with the crystal structure of ubiquitin, if the lysine side chains and the amino terminus are assumed to have considerable flexibility. In a separate study, we probed the reactivity of the primary amino groups in ubiquitin using the amino acetylating reagent, N-hydroxy succinimidyl acetate (NHSAc), and a top-down approach to localize the acetylated lysine residues. The reactivity order obtained in that study (M1 approximate, equals K6 approximate, equals K48 approximate, equals K63) > K33 > K11 > (K27, K29), shows that the cross-link first formed in ubiquitin by reaction with DSS and DSG occurs between the most reactive residues.     

Modellrechnungen an einigen Siliziumwasserstoffverbindungen vom Typ SiHn,SiH n + und SiH n − nach der Einzentrenmethode

Zusammenfassung Es werden molekulare Struktur, Energie des Grundzustandes, Bindungsabstände, Bindungsenergie, Ionisierungsenergie und Protonenaffinität der Siliziumwasserstoffverbindungen SiH _n , SiH _n ⁺ und SiH_n ^– (n=3, 4 oder 5) nach der Einzentrenmethode berechnet.

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OCE-calculations on some silicon hydrides of the type SiH _n , SiH _n ⁺ and SiH _n ^–

OCE-Calculations are reported for molecular structures, ground state energies, bond distances, binding energies, ionization potentials and proton affinities of the silicon hydrides SiH _n , SiH _n ⁺ and SiH _n ^– (n=3, 4 or 5).

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Résumé Calcul par la méthode monocentrique de la structure moléculaire, de l'énergie de l'état fondamental, des longueurs de liaison, des énergies de liaison, des potentiels d'ionisation et des affinités protoniques pour les hydrures de silicium SiH _n , SiH _n ⁺ et SiH _n ^– (n=3, 4 ou 5).

     

Die Lösung der Prae-Maxwellschen Gleichungen mit Hilfe vollständiger Lösungssysteme

This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R ³. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .     

Rhodium(I) complexes with 1′-(diphenylphosphino)ferrocenecarboxylic acid as active and recyclable catalysts for 1-hexene hydroformylation

The rhodium complex trans-[Rh(CO)(Hdpf-κP)(dpf-κ²O,P)] (1), (Hdpf = 1′-(diphenylphosphino)ferrocenecarboxylic acid) was used as an efficient and recyclable catalyst for 1-hexene hydroformylation producing ca. 80% of aldehydes at 10 atm CO/H₂ and 80 °C. After the reaction, unchanged complex 1 was separated from the reaction mixture and used again three times with the same catalytic activity. The effect of modifying ligands, phosphines and phosphites, on the reactivity of 1 was investigated. The active catalytic systems containing 1 or trans-[Rh(CO)(L)(dpf-κ²O,P)] (2) were formed in situ from acetylacetonato rhodium(I) precursors [Rh(CO)₂(acac)] (3) or [RhL(CO)(acac)] (4) and Hdpf or Medpf (L = phosphine, Medpf = methyl ester of Hdpf).     

Surface patterns of block copolymers in thin layers after vapor treatment

A systematic study of formation of surface patterns in block copolymer thin layers after their exposure to solvent vapors was performed. The studied effect involves layers of thickness approximately equal to the ordering size of polymers - about 45 nm. Experiments were performed on three styrene - methacrylate derivative block copolymers, synthesized by living anionic polymerization: poly(4-octylstyrene)-block-poly(butyl methacrylate), poly(4-fluorostyrene)-block-poly(butyl methacrylate) and poly(p-octylstyrene)-block-poly(methyl methacrylate). The polymers were exposed to vapors of chloroform, 1,4-dioxane, hexane, acetone and tetrahydrofuran.     

Stabilization energies of the hydrogen-bonded and stacked structures of nucleic acid base pairs in the crystal geometries of CG, AT, and AC DNA steps and in the NMR geometry of the 5'-d(GCGAAGC)-3' hairpin: Complete basis set calculations at the MP2 and CCSD(T) levels

Stabilization energies of the H-bonded and stacked structures of a DNA base pair were studied in the crystal structures of adenine-thymine, cytosine-guanine, and adenine-cytosine steps as well as in the 5'-d(GCGAAGC)-3' hairpin (utilizing the NMR geometry). Stabilization energies were determined as the sum of the complete basis set (CBS) limit of MP2 stabilization energies and the Delta E(CCSD(T)) - Delta E(MP2) correction term evaluated with the 6-31G*(0.25) basis set. The CBS limit was determined by a two-point extrapolation using the aug-cc-pVXZ basis sets for X = D and T. While the H-bonding energies are comparable to those of base pairs in a crystal and a vacuum, the stacking energies are considerably smaller in a crystal. Despite this, the stacking is still important and accounts for a significant part of the overall stabilization. It contributes equally to the stability of DNA as does H-bonding for AT-rich DNAs, while in the case of GC-rich DNAs it forms about one-third of the total stabilization. Interstrand stacking reaches surprisingly large values, well comparable to the intrastrand ones, and thus contributes significantly to the overall stabilization. The hairpin structure is characterized by significant stacking, and both guanine...cytosine pairs possess stacking energies larger than 11.5 kcal/mol. A high portion of stabilization in the studied hairpin comes from stacking (similar to that found for AT-rich DNAs) despite the fact that it contains two GC Watson-Crick pairs having very large H-bonding stabilization. The DFT/B3LYP/6-31G** method yields satisfactory values of interaction energies for H-bonded structures, while it fails completely for stacking.