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.     

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 .     

Stability of the gold(i)-phosphine bond. A comparison with other group 11 elements

The stability of gold phosphine complexes of the form [Au(PH(3))(n)()](+) (n = 1-4) and [AuCl(PH(3))(n)()] (n = 1-3) is analyzed in detail by applying quantum theoretical methods and compared to the coordination behavior of the lighter group 11 elements copper and silver. It is shown that, once [M(PH(3))(2)](+) or [MClPH(3)] (M = Cu, Ag, and Au) is formed, further coordination by PH(3) ligands is relatively weak; i.e., the energy gain to form [M(PH(3))(3)](+) from [M(PH(3))(2)](+) is less than 60 kJ mol(-)(1), and less than 100 kJ mol(-)(1) to form [MCl(PH(3))(2)] from [MClPH(3)]. Relativistic effects in gold significantly influence these factors and reduce the tendency for phosphine coordination beyond two-coordination. This implies that the most favored coordination number for gold is two with either a linear P-Au-P or P-Au-X arrangement (X = a strongly coordinating ligand like Cl(-)). Instead, X-Au-PH(3) units prefer to interact via close Au-Au contacts (aurophilic interactions) keeping the linear structure approximately intact, while the corresponding copper and silver compounds prefer PH(3) coordination to strongly bound M(2)Cl(2) units (M = Cu or Ag) where two chlorine atoms bridge the two metal atoms thus having the formal coordination number of three for copper or silver.     

Protein adsorption in fused-silica and polyacrylamide-coated capillaries

The model proteins cytochrome c, myoglobin, ovalbumin, and beta-lactoglobulin were investigated with regard to their adsorption properties on capillaries for electrophoresis. The model compounds were selected to cover a wide range of properties. Cytochrome c is a basic protein (isoelectric point (pI): 9.6; M(r): 11.7 kDa), beta-lactoglobulin is rather acidic (pI: 5.4, M(r): 18.4 kDa), myoglobin was chosen as a neutral reference protein (pI: 6.8-7.4, M(r): 17.8 kDa), and ovalbumin (pI: 5.1, M(r): 45.0 kDa) was selected as a relatively larger analyte. First, the pH dependence of adsorption was investigated for the bare fused silica. A clear correlation to the respective pIs was noted. For myoglobin and ovalbumin, none or negligible adsorption was found above the pI, whereas strong adsorption was noted just below this parameter. Cytochrome c and beta-lactoglobulin already showed distinct adsorption above their pIs. However, none of the proteins showed any significant adsorption more than one pH unit above the pIs. For linear polyacrylamide-coated capillaries, a decreased but not a complete lack of adsorption was observed. Here, pH-dependent adsorption was noted as well. Regeneration of the capillaries by rinsing with buffers containing 200 mM SDS was also investigated. This method was completely successful for myoglobin, but that too for only freshly-adsorbed protein. After a storage time of 24 h and due to the aging of the adsorbate, a sufficient regeneration was no longer possible.     

On the use of local basis sets for localized molecular orbitals

Two procedures are discussed for the direct variational optimization of localized molecular orbitals which are expanded in local subsets of the molecular basis set. It is shown that a Newton-Raphson approach is more efficient than an iterative diagonalization scheme. The effect of the basis-set truncation on the quality ofab-initio SCF results is investigated for Be, Li₂, HF, H₂O, NH₃, CH₄ and C₂H₆.     

Spektralphotometrische Bestimmung der Gleichgewichte aromatischer Amine mit Chlorwasserstoffsäure in Methanol

Zusammenfassung Die Ionisationskonstanten des Aniliniumions, der drei Chlor- und der drei Nitroaniliniumionen in Methanol bei 2°C, 15°C und 25°C werden mit Hilfe eines spektralphotometrischen Verfahrens bestimmt. Da die Amine im ultravioletten oder im sichtbaren Bereich starke Absorptionsbanden aufweisen, während die Ionenformen fast nicht absorbieren, ist es möglich, durch Messung der Extinktion von Lösungen, in, denen Anilin (substituiertes Anilin) mit Aniliniumion (substituiertem Aniliniumion) im Gleichgewicht steht, die Ionensationskonstante zu ermitteln.Es zeigt sich, daß die Basizität der untersuchten Amine in Methanol größer als in Wasser ist und daß die Einführung eines Chloratoms oder einer Nitrogruppe in den aromatischen Ring eine Abnahme der Basizität verursacht. Dieser Effekt ist bei den Nitroanilinen stärker als bei den Chloranilinen, und bei den o-Verbindungen stärker als bei den m- und p-Verbindungen ausgeprägt.Die Temperaturabhängigkeit der Ionisationskonstanten und damit auch die Ionisationsenthalpien der Aniliniumionen sind in Methanol ebenfalls größer als in Wasser.Mit 2 AbbildungenAus der Diplomarbeit vonW. Melhardt, Technische Hochschule Wien, 1962.