Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than

Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .

     

Geometric Dynamics

A kinematic differential system on a Riemann (or semi-Riemann) manifold induces a Lorentz-Udrite world-force law, i.e., any local group with one parameter (any local flow) on a Riemann (or semi-Riemann) manifold induces the dynamics of the given vector field or of an associated particle, which will be called geometric dynamics.The cases of Riemann-Jacobi or Riemann-Jacobi-Lagrange structures are imposed by the behavior of an external tensor field of type (1,1). The case of the Finsler-Jacobi structure appears if the initial metric is chosen such that the energy of the given vector field is constant (Sec. 1). At the end of Sec. 1 are formulated open problems regarding some extensions of geometric dynamics.Adequate structures on the tangent bundle describe the geometric dynamics in the Hamilton language (Sec. 2).Section 3 proves the existence of a Finsler-Jacobi structure induced by an almost contact metric structure.The theory is applied to electromagnetic dynamical systems (the starting point of our theory), offering new principles of unification of the gravitation and the electromagnetism. Also, here, one enounces open problems regarding the geometric dynamics induced by the electric intensity and magnetizing force (Sec. 4).From the geometrical point of view, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds ensuring that all trajectories of a given vector field are geodesics. Having T₁M²ⁿ⁺¹ in mind, the problem of creating a wider class of Riemannian manifolds, in which there exists a vector field such that (1) all trajectories of the vector field are geodesics; (2) the flow defined by is incompressible; (3) the condition which corresponds to the property that is the associate vector field of the contact structure is satisfied;was studied intensively by S. Sasaki. The results were not satisfactory, but Sasaki discovered (, , )-structures [10].AMS Subject Classification (1991): 70H35, 53C22, 58F25, 83C22     

Spectroscopy of XY(5)Z (C(4v)) Molecules: A Tensorial Formalism Adapted to the O(3) superset O(h) superset C(4v) Chain

A tensorial formalism adapted to the case of XY(5)Z symmetric tops has been developed as an extension of the usual one for the octahedral molecules. We use the O(3) superset O(h) superset C(4&v) group chain. All the coupling coefficients and formulas for the computation of matrix elements are given for this chain. Such relations are also deduced in the C(4v) group itself. Copyright 2000 Academic Press.     

Zur Bestimmung der Thonerde im Wein

Ohne Zusammenfassung     

Coulomb effects in an optically excited electron-hole plasma of a disordered semiconductor

In this paper we extend the work of Altshuler and Aronov [1] on the effect of Coulomb interaction in disordered metals to the case of an optically generated quasi-equilibrium electron-hole plasma in a disordered semiconductor in view of its possible implications for nonlinear optics. The nonlinearity considered here arises through the optically excited plasma density. The plasma is bipolar and not necessarily degenerate as in the metallic case. The density of states and the optical spectra due to freecarrier interband transitions are numerically computed in the presence and in the absence of the Altshuler-Aronov Coulomb-disorder singularity.     

Zur Herstellung w?ssriger schwefliger S?ure

Ohne Zusammenfassung     

An experimental and kinetic modeling study on the oxidation of 1,3-dioxolane

The modern catalytic or enzymatic advances allow the production of novel biofuel. Among them, 1,3-dioxolane can be produced from formaldehyde and ethylene glycol, both can be obtained from biomass. In this study, the oxidation of 1,3-dioxolane is studied at stoichiometric conditions. The ignition delay times of 1,3-dioxolane/O₂/inert mixtures were measured in a shock tube and in a rapid compression machine at pressures of 20 to 40 bar and temperatures ranging from 630 to 1300 K. The pressure profiles recorded in the rapid compression machine show a first stage of ignition enlightening the influence of the low temperature chemistry of combustion. Furthermore, mole fraction profiles of the stable intermediates produced during the oxidation of 1,3-dioxolane were measured in a jet-stirred reactor at 10 bar. Following these observations, a detailed kinetic model was developed with reaction rate coefficients and thermochemical data calculated by theoretical calculations or estimated by analogies to suitable molecules. In order to get an insight into the most important reaction pathways brute force sensitivity analysis and reaction pathway analysis were performed with the proposed model and discussed. It became clear that in the fuel-in-air case for the alkylhydroperoxide of 1,3-dioxolane the ring opening beta-scission pathway is favored against the further alkane-like second addition to molecular oxygen, which leads to a limited negative temperature coefficient.     

Study of the heating effect contribution to the nonlinear dielectric response of a supercooled liquid

We present a detailed study of the heating effects in dielectric measurements carried out on a liquid. Such effects come from the dissipation of the electric power in the liquid and give contribution to the nonlinear third harmonics susceptibility χ(3), which depends on the frequency and temperature. This study is used to evaluate a possible "spurious" contribution to the recently measured nonlinear susceptibility of an archetypical glassforming liquid (glycerol). Those measurements have been shown to give a direct evaluation of the number of dynamically correlated molecules temperature dependence close to the glass transition temperature T(g) ≈ 190 K [Crauste-Thibierge et al., Phys. Rev. Lett. 104, 165703 (2010)]. We show that the heating contribution is totally negligible (i) below 204 K at any frequency; (ii) for any temperature at the frequency where the third harmonics response χ(3) is maximum. Besides, this heating contribution does not scale as a function of f/f(α), with f(α)(T) the relaxation frequency of the liquid. In the high frequency range, when f/f(α) ≥ 1, we find that the heating contribution is damped because the dipoles cannot follow instantaneously the temperature modulation due to the heating phenomenon. An estimate of the magnitude of this damping is given.