Least quadratic non-residues in algebraic number fields

Let νp denote a totally positive integer of an algebraic number field K such that νp is a least quadratic non-residue modulo a prime ideal p of K, least in the sense that N(νp) is minimal. Then the following result is shown: For x ≥ 2 and ε > 0,

$\text{|p;Np?x}\text{and}\text{N(vp)>Np}{}^{\mathrm{\epsilon }}\text{|=O}{}_{\mathrm{\epsilon }}\text{(}\text{log log}\text{3x).}$

     

Temperaturbestimmung

Ohne Zusammenfassung     

Coherence in microchip traps

We report the coherent manipulation of internal states of neutral atoms in a magnetic microchip trap. Coherence lifetimes exceeding 1 s are observed with atoms at distances of 5-130 microm from the microchip surface. The coherence lifetime in the chip trap is independent of atom-surface distance within our measurement accuracy and agrees well with the results of similar measurements in macroscopic magnetic traps. Because of the absence of surface-induced decoherence, a miniaturized atomic clock with a relative stability in the 10(-13) range can be realized. For applications in quantum information processing, we propose to use microwave near fields in the proximity of chip wires to create potentials that depend on the internal state of the atoms.     

Über das Anwachsen der Lösungen homogener algebraischer Differentialgleichungen zweiter Ordnung

It is shown that the general solution of a homogeneous second-order algebraic differential equation has the form w(z)=(g₁(z)/g₂(z)) exp (g₃(z)), where g₁, g₂ and g₃ are entire functions of finite order.     

Fusion reaction rate for a plasma boiler moving in an ambient plasma

The enhancement of the fusion reaction rate for a thermal plasma moving as a boiler through an ambient plasma is calculated. The anisotropy of the neutron emission is discussed and the calculations are applied to measurements on a small Mather type focus device.     

The conformational analysis of 1-haloalkanes by low resolution microwave spectroscopy

The low resolution microwave spectra of a series of 1-haloalkanes were studied. With the exception of 1-bromobutane only the all s-trans conformer and those gauche conformers in which a single dihedral angle is gauche were foud. By analyzing data from a homologous series, a gauche dihedral angle of 286 ± 4° was determined for the gauche conformers of the bromo and iodo species. An analysis of relative intensities of the 1-bromobutane and 1-iodobutane bands yielded standard Gibbs free energy differences.