Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_[].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.     

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.     

Ammonolysis of mono(pentamethylcyclopentadienyl) titanium(IV) derivatives

Ammonolyses of mono(pentamethylcyclopentadienyl) titanium(IV) derivatives [Ti(eta5-C5Me5)X3] (X = NMe2, Me, Cl) have been carried out in solution to give polynuclear nitrido complexes. Reaction of the tris(dimethylamido) derivative [Ti(eta5-C5Me5)(NMe2)3] with excess of ammonia at 80-100 degrees C gives the cubane complex [[Ti(eta5-C5Me5)]4(mu3-N)4] (1). Treatment of the trimethyl derivative [Ti(eta5-C5Me5)Me3] with NH3 at room temperature leads to the trinuclear imido-nitrido complex [[Ti(eta/5-CsMes)(mu-NH)]3(mu3-N)] (2) via the intermediate [[Ti(eta5-C5Me5)Me]2(mu-NH)2] (3). The analogous reaction of [Ti(eta5-C5Me5)Me3] with 2,4,6-trimethylaniline (ArNH2) gives the dinuclear imido complex [[Ti(eta5-C5Me5)Me])2(mu-NAr)2] (4) which reacts with ammonia to afford [[Ti(eta5-C5Me5)(NH2)]2(mu-NAr)2] (5). Complex 2 has been used, by treatments with the tris(dimethylamido) derivatives [Ti(eta5-C5H5-nRn)(NMe2)3], as precursor of the cubane nitrido systems [[Ti4(eta5-C5Me5)3(eta5-C5H5-nRn)](mu3-N)4] [R = Me n = 5 (1), R = H n = 0 (6), R = SiMe3 n = 1 (7), R = Me n = 1 (8)] via dimethylamine elimination. Reaction of [Ti(eta5-C5Me5)Cl3] or [Ti(eta5-C5Me5)(NMe2)Cl2] with excess of ammonia at room temperature gives the dinuclear complex [[Ti2(eta5-C5Me5)2Cl3(NH3)](mu-N)] (9) where an intramolecular hydrogen bonding and a nonlineal nitrido ligand bridge the "Ti(eta5-C5Me5)Cl(NH3)" and "Ti(eta5-C5Me5)Cl2" moieties. The molecular structures of [[Ti(eta5-C5Me5)Me]2 (mu-NAr)2] (4) and [[Ti2(eta5-C5Me5)2Cl3(NH3)](mu-N)] (9) have been determined by X-ray crystallographic studies. Density functional theory calculations also have been conducted on complex 9 to confirm the existence of an intramolecular N-H...Cl hydrogen bond and to evaluate different aspects of its molecular disposition.     

Transfer hydrogenation processes to mu(3)-alkylidyne groups on the organotitanium oxide [Ti(3)Cp*O(3)

The photochemical treatment of mu(3)-alkylidyne complexes [[TiCp*(mu-O)](3)(mu(3)-CR)] (R=H (1), Me (2), Cp*=eta(5)-C(5)Me(5)) with the amines (2,6-Me(2)C(6)H(3))NH(2), Et(2)NH, and Ph(2)NH and the imine Ph(2)C=NH leads to the partial hydrogenation of the alkylidyne moiety that is supported on the organometallic oxide, [Ti(3)Cp*O(3)], and the formation of new oxoderivatives [[TiCp*(3)(mu-CHR)(R'NR")] (R"=2,6-Me(2)C(6)H(3), R'=H, R=H (3), Me (4); R'=R"=Et, R=H (5), Me (6); R'=R"=Ph, R=H (7), Me (8)) and [[TiCp*(mu-O)](3)(mu-CHR)(N=CPh(2))] (R=H (9), R=Me (10)), respectively. A sequential transfer hydrogenation process occurs when complex 1 is treated with tBuNH(2), which initially gives the mu-methylene [[TiCp*(mu-O)](3)(mu-CH(2))(HNtBu)] (11) complex and finally, the alkyl derivative [[TiCp*(mu-O)](3)(mu-NtBu)Me] (12). Furthermore, irradiation of solutions of the mu(3)-alkylidyne complexes 1 or 2 in the presence of diamines o-C(6)H(4)(NH(2))(2) and H(2)NCH(2)CH(2)NH(2) (en) affords [[TiCp*(mu-O)](3)(mu(3)-eta(2)-NC(6)H(4)NH)] (13) and [[TiCp*(mu-O)](3)(mu(3)-eta(2)-NC(2)H(4)NH)] (14) by either methane or ethane elimination, respectively. In the reaction of 1 with en, an intermediate complex [[TiCp*(mu-O)](3)(mu-CH(2))(NHCH(2)CH(2)NH(2))] (15) is detected by (1)H NMR spectroscopy. Thermal treatment of the complexes 4-10 quantitatively regenerates the starting mu(3)-alkylidyne compounds and the amine R'(2)NH or the imine Ph(2)C=NH; however, heating of solutions of 3 or 4 in [D(6)]benzene or a equimolecular mixture of both at 170 degrees C produces methane, ethane, or both, and the complex [[TiCp*(mu-O)](3)[mu(3)-eta(2)-NC(6)H(3)(Me)CH(2)]] (16). The molecular structure of 8 has been established by single-crystal X-ray analysis.     

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.