Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Electron-electron interaction in strong electromagnetic fields: the two-electron contribution to the ground-state energy in He-like uranium

Radiative recombination transitions into the ground state of cooled bare and hydrogenlike uranium ions were measured at the storage ring ESR. By comparing the corresponding x-ray centroid energies, this technique allows for a direct measurement of the electron-electron contribution to the ionization potential in the heaviest He-like ions. For the two-electron contribution to the ionization potential of He-like uranium we obtain a value of 2248+/-9 eV. This represents the most accurate determination of two-electron effects in the domain of high-Z He-like ions, and the accuracy reaches already the size of the specific two-electron radiative QED corrections.     

Synthesis of ring-oxidized retinoids as substrates of mouse class I alcohol dehydrogenase (ADH1)

Ring-oxidized retinoids have been synthesized stereoselectively using the Stille cross-coupling reaction. Kinetic constants of mouse class I alcohol dehydrogenase (ADH1) with these retinoids were determined.     

Alkaline side-coordination strategy for the design of nickel(II) and nickel(III) bis(1,2-diselenolene) complex based materials

The deprotonated form of the pyrazine-2,3-diselenol (pds) ligand, pds(2-), reacts with Ni(II) inorganic salts to form the nickel compounds [Ni(II)(pds)(2)](nBu(4)N)(2) (1), [Ni(II)(pds)(2)]Na(2).2H(2)O (2), and [Ni(III)(pds)(2)](2)Na(2).4H(2)O (3), depending on the reaction conditions. They are characterized by NMR, EPR, UV-vis, and IR spectroscopies, elemental analysis, cyclic voltammetry, and X-ray crystallography. The crystal structure of compound 3 shows the formation of segregated stacks of Ni(pds)(2-) units, with a strong dimerization along the stacks. The stacked fashion of the crystal packing was expected since the supramolecular forces of the alkaline side coordination to the pyrazine moieties dominate, as happens in the recently reported analogous copper system [Cu(III)(pds)(2)]Na.2H(2)O. The structure of 2 further emphasizes the alkaline coordination as the dominating supramolecular event, and an orthogonal array of 2D layers is observed. The absence of alkaline cations in complex 1 is reflected in a crystal packing with isolated complex Ni(pds)(2)(2-) units. The dimerization found in the paramagnetic Ni(III) complex 3 promotes a very strong antiferromagnetic interaction, leading to a singlet ground state.     

Highly stereoselective aldol reactions of titanium enolates from lactate-derived chiral ketones

[reaction: see text] Highly stereoselective titanium-mediated aldol reactions based on lactate-derived ketones are reported. The stereochemical outcome of the process depends on the protecting group (PMB or Bn) and the Lewis acid (i-PrOTiCl(3) or TiCl(4)) used in the enolization step, the corresponding anti-syn or syn-syn aldols being prepared in high yields and with diastereomeric ratios up to 99:1.     

Geometrical-numerical approach to diffraction phenomena

The calculation of diffracted fields is considered by means of a geometrical analysis of the incoming wave into semiperiodic zones in the aperture plane, followed by a numerical process for addition of the contributions corresponding to the semiperiodic zones. This general approach constitutes a novel interpretation of diffraction phenomena that permits exact evaluation of the mathematical expressions of diffraction theory and overcomes the limitations of any approximation. The method is illustrated by analysis of two important configuration in optics: the pinhole camera, for which we deduce the optimum radius for imaging, and the diffraction of a spherical converging wave through a circular aperture, from which we determine the limit of the validity of the Fraunhofer approximation (i.e., of the Airy pattern) and the influence of the obliquity factor.     

Fourier Transform Spectra of the E(2)Pi(u)-X(2)Pi(g)((3/2)) System of CuCl(2)

High-resolution Fourier transform spectra of the laser-induced fluorescence of (63)Cu(37)Cl(2) produced in a cell have been recorded following excitation of a single vibronic level of the E(2)Pi(u) electronic state. Fluorescence occurs in combination bands to a broad spread of levels in the ground electronic state. A global vibronic model is proposed for the ground state based on an effective Hamiltonian, which fits the experimental data (2782 fluorescence lines, lower state quantum numbers: v(1) = 0-6, v(2) = 0-2, v(3) = 0-6, and J = 4(1/2)-80(1/2)) to 0.019 cm(-1) rms error. Vibrational, rotational and Renner-Teller parameters are obtained (e.g., omega(2) = 95.195(36) cm(-1), B(e) = 0.055106(3) cm(-1), epsilon = -0.1893(28)). A revised value for the equilibrium internuclear distance Cu-Cl is deduced: r(e)(Cu-Cl) = 0.20341(3) nm. The energy diagram of vibronic levels in the ground state is plotted up to 4000 cm(-1). Copyright 2000 Academic Press.     

1s lamb shift in hydrogenlike uranium measured on cooled, decelerated ion beams

The Lyman- alpha transitions of hydrogenlike uranium associated with electron capture were measured in collisions of stored bare U (92+) ions with gaseous targets at the storage ring ESR. By applying the deceleration technique, the experiment was performed at slow collision energies in order to reduce the uncertainties associated with Doppler corrections. From the measured centroid energies, a ground state Lamb shift of 468 eV+/-13 eV is deduced which gives the most precise test of quantum electrodynamics for a single electron system in the strong field regime. In particular, the technique applied paves the way towards the 1 eV precision regime.