On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.     

Closed manifolds coming from Artinian complete intersections

We reformulate the integrality property of the Poincaré inner product in the middle dimension, for an arbitrary Poincaré -algebra, in classical terms (discriminant and local invariants). When the algebra is -connected, we show that this property is the only obstruction to realizing it by a smooth closed manifold, in dimension . We analyse the homogeneous artinian complete intersections over realized by smooth closed manifolds of dimension , and their signatures.

     

Computing the flow around a moving profile by numerically integrating the Navier–Stokes equations

A vorticity velocity formulation is proposed for the solution of the equations for viscous flow around a moving profile. A non-inertial reference frame is used and the velocities are computed from a Poincaré integral formula. The studies are directed towards the need to understand helicopter blade aerodynamics. Worked examples are given which validate the method and programme for laminar flows, at least for low Reynolds numbers. © 1998 John Wiley & Sons, Ltd.     

Spatial properties of Wohlhart symmetric mechanism

The aim of this paper is to present a new analytical resolution to find closure equations of Wohlhart symmetric mechanism and to give some interesting spatial properties of this mechanism. These properties are used to put in evidence possible industrial applications for this type of mechanism. First, it concerns the analytic resolution of the twelve equations system obtained from the closure equation, system particularized for the case of Wohlhart symmetric mechanism. The matrix form of closure equations is written for a single loop overconstrained mechanism composed by six revolute joints, mechanisms called 6R. A kinematical analysis is made, putting in evidence some interesting kinematical and geometrical properties. These properties allow us to consider a new spatial disposition for this mechanism, by imposing three non-successive joints to remain in a determined plan. Thus, we obtain a translator, a possible future industrial application for this type of overconstrained mechanism.     

Hirzebruch–Milnor classes of complete intersections

We prove a new formula for the Hirzebruch–Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern–Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusiński and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern–Milnor classes of complete intersections with isolated singularities.     

Springs, rings, and spirals of rutile-structured tin oxide nanobelts

Single-crystalline springs, rings, and spirals have been discovered for the first time in rutile-structured SnO2. The formation process is proposed for minimizing the electrostatic energy due to the polar charges on the (011) surfaces. These structures are not only ideal systems for fundamental understanding of the polarization effect on the morphology at the nanoscale level but also have potential applications as nanoscale sensors, resonators, and transducers.     

Capillary condensation in porous alumina observed by positronium lifetime spectroscopy

The PALS method based on time distribution measurements has been used to study capillary condensation of different gases adsorbed in microporous alumina powder. The isotherms exhibit features which are associated with a shifted gas-liquid transition. The sorption and desorption processes are irreversible presenting a hysteresis effect.Suggestions on some new aspects of the capillary condensation dynamics are made.     

Compact eye-safe laser sources based on OPOs with KTP or PPKTP crystals

We report on compact eye-safe nanosecond laser sources emitting in the 1.5 μm wavelength range based on non-critically phase-matched parametric interaction in optical parametric oscillators (OPOs) with KTP and periodically poled KTP (PPKTP) crystals, pumped by the fundamental frequency of Nd:YAG lasers. As much as 250 μJ signal pulse energy at 1.5 μm wavelength, 6.5 ns FWHM pulse-width, has been obtained in a PPKTP-OPO, extracavity pumped by a Nd:YAG microlaser oscillator–amplifier at 650 μJ pump pulse energy, 8 ns pulse-width. A single signal pulse of 2.7-mJ output energy at 1.57 μm wavelength, less than 5 ns pulse-width, was generated in a KTP-OPO, intracavity pumped by a passively Q-switched Nd:YAG laser.