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.     

On the zeros of polynomials over arbitrary finite dimensional algebras

In [3] it was shown that a polynomial of degree n with coefficients in an associative division algebra, which is d-dimensional over its center, has either infinitely many or at most n^d zeros. In this paper we raise the same question for arbitrary m-ary F-algebras A which are d-dimensional over the algebraically closed field F. Our main result states that in the affine space of m-ary algebras of dimension d there is a non-empty Zariski-open set whose elements A have the following property: in the space of polynomial of precise degree n with coefficients in A there is a non-empty Zariski-open set whose elements have precisely n^d zeros. It is shown that all simple algebras, all semi-simple associative algebras, all semisimple Jordan algebras (char F2), all semi-simple Lie algebras (char F=0), and the generic algebra possess this property.     

Equivariant framability of involutions on homotopy spheres

Let G be the group Z₂. Denote byR ^n,k theR ^n+k with non trivial G-action on the first n coordinates. Let ^n,k be the trivial bundle with fibreR ^n,k. We say that a G-manifold M is (n,k)-framable if t(M)= =^n,k in KO_G(M) with t(M) the tangent bundle of M. We show that if G acts on a homotopy sphere ^n+k such that the fixed point set is a k-dimensional homotopy sphere then is (n,k)-framable.     

Structure and annealing of (n, γ)-Induced point defects in CaF2

Neutron activated²⁰F nuclei (T _1/2=11 s) in CaF₂ serve for two purposes: first they produce well defined point defects by their (n, )-production mechanism, second, they act as probes to detect the properties of these defects in a- and-ray detected nuclear magnetic resonance experiment. Below about 70 K an interstitial defect is observed due to a quadrupolar splitting of the²⁰F-NMR line. The defect induced electric field gradient is oriented along the 111-axis of the crystal and causes a quadrupole coupling constante ² qQ/h=–2.0(1) MHz at the²⁰F site. The defect anneals above 70 K with an Arrhenius like temperature dependence corresponding to an activation energyE _M =0.17(4) eV.     

The dispersion relation of the charge and current compensated relativistic electron beam-plasma system

Analysing the general dispersion relation numerically, the unstable regions of the relativistic electron-beam plasma system were determined. The external parameters were varied to ensure more effective instability excitations. The full charge and current compensation presumptions lead to new synchronism predictions. The return current's slow space charge wave and return current's slow cyclotron wave are synchronous with the plasma ion wave.     

Application of the JKR Method to the Measurement of Adhesion to Langmuir-Blodgett Cellulose Surfaces

The JKR method has been applied for studying adhesion between poly(dimethylsiloxane) (PDMS) caps and Langmuir–Blodgett cellulose surfaces including the substrate, hydrophobized mica, and two flat mineral surfaces, bare mica and glass. The self-adhesion of PDMS caps and oxidized PDMS caps are included as a reference to compare with literature data. The results of the measurements have been compared with previous studies using the surface force apparatus and similar systems. A satisfactory agreement is obtained for simple systems showing no, or very limited, hysteresis between loading and unloading curves. In several cases, however, a large hysteresis is found between loading and unloading curves, with a larger adhesion measured from the pull-off force than from the JKR-curve determined on loading. This is, for instance, the case for PDMS against cellulose. The situation is analogous to that found in wetting studies showing a large hysteresis between advancing and receding contact angles.     

Performance of an in-plane detection cell with integrated waveguides for UV/Vis absorbance measurements on microfluidic separation devices

A microfluidic device with integrated waveguides and a long path length detection cell for UV/Vis absorbance detection is presented. The 750 microm U-cell detection geometry was evaluated in terms of its optical performance as well as its influence on efficiency for electrophoretic separations in the microdevice. Stray light was found to have a strong effect on both, the sensitivity of the detection and the available linear range. The long path length U-cell showed a 9 times higher sensitivity when compared to a conventional capillary electrophoresis (CE) system with a 75 microm inner diameter (ID) capillary, and a 22 times higher sensitivity than with a 50 microm ID capillary. The linear range was comparable to that achieved in a 75 microm ID capillary and more than twice as large as in a 50 microm ID capillary. The use of the 750 microm U-cell did not contribute significantly to band broadening; however, a clear quantification was made difficult by the convolution of several other band broadening sources.     

Beiträge zur Chemie der Pyrrolpigmente, 78. Mitt.: Die Eigenschaften desb-Nor-bilatrien-abc-b-Nor-biladien-ac-Systems

The redox coupleb-nor-biladiene-ac (2)/b-nor-bilatriene-abc (3) exhibits an oxidative half wave potential of 800 mV in acetonitrile solution. It is demonstrated that it can be used as an extremely efficient electron carrier through bulk membranes. Whereas2 is not suitable as a ligand in carrier mediated cation transport,3 is a medium suited carrier for Cu(II) or Zn(II) ions, however, it is the most efficient one found so far for transport of Hg(II) ions. A 1:1 zinc chelate of3 is isolated and characterized. it is deduced to represent a polymer consisting of about 73 monomer units.

     

Design of organic semiconductors: tuning the electronic properties of pi-conjugated oligothiophenes with the 3,4-ethylenedioxythiophene (EDOT) building block

Hybrid oligothiophenes based on a various combinations of thiophene and 3,4-ethylenedioxythiophene (EDOT) groups have been synthesized. UV/Vis absorption spectra show that the number and relative positions of the EDOT groups considerably affect the width of the HOMO-LUMO gap and the rigidity of the conjugated system. Analysis of the crystallographic structure of two hybrid quaterthiophenes confirms that insertion of two adjacent EDOT units in the middle of the molecule leads to a self-rigidification of the conjugated systems by intramolecular SO interactions. Cyclic voltammetry data shows that the first oxidation potential of the oligomers decreases with increasing chain length and increasing number of EDOT groups for a given chain length. Electrochemical studies and theoretical calculations show that the positions of the EDOT units in the conjugated chain control the potential difference (DeltaE(p)) between the first and second oxidation steps. Moving the EDOT groups from the outer to the inner positions of the conjugated system increases DeltaE(p). Theoretical calculations confirm that this phenomenon reflects an increase of the intramolecular coulombic repulsion between positive charges in the dication. A thin-film field-effect transistor was fabricated by vacuum sublimation of a pentamer with alternating thiophene-EDOT structure, and the hole mobility was determined.