Applications of geometric means on symmetric cones

Let V be a Euclidean Jordan algebra, and let be the corresponding symmetric cone. The geometric mean of two elements a and b in is defined as a unique solution, which belongs to of the quadratic equation where P is the quadratic representation of V. In this paper, we show that for any a in the sequence of iterate of the function defined by converges to a. As applications, we obtain that the geometric mean of can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the L?wner-Heinz inequality on the symmetric cone Furthermore, we obtain a formula which leads a Golden-Thompson type inequality for the spectral norm on V. Received October 5, 1999 / Revised March 6, 2000 / Published online October 30, 2000     

Europium-doped nanocrystalline Y2O3−La2O3 solid solutions with bixbyite structure

Nanocrystals of Y₂O₃ and La₂O₃ solid solutions were synthesized with a cubic bixbyite structure containing La₂O₃ content of up to 50 mol%. This is comparatively higher than that in bulk materials of the same structure, where La₂O₃ content of only 20 mol% can be obtained. A set of europium-doped (Y_1−xLa_x)₂O₃ (x=0.1, 0.2, 0.3, 0.4) cubic bixbyite solid solutions with crystallites of approximately 10 nm in size was prepared using the polymer complex solution method. Structural analysis was performed using X-ray diffraction measurements, Rietveld full profile refinement, and from Eu³⁺ luminescence emission. The energy levels of the Eu³⁺ ion, second order crystal field parameters, and crystal field strength were obtained for all compositions of solid solutions. We show that the crystal field parameters linearly depend on unit cell parameter and that these dependencies may be considered as part of an overall dependence for the entire sesquioxide family.     

On the quantum Fermi accelerator and its relevance to ‘quantum chaos’

We discuss a quantum version of the Fermi acceleration model, which consists of a particle bouncing between a fixed and oscillating wall. The actual movement of the particle crucially depends on the boundary conditions of the Schrödinger equation. Under Dirichlet boundary conditions, the quantum system displays a regular behaviour, but its classical limit exhibits some unphysical attributes. Only for certain initial conditions does it correspond to the stable motion of a ball bouncing once for an integer number of wall oscillations. In the classical model that situation gives rise to regular islands imbedded in the chaotic sea.     

Increasing Self-Described Sequences

We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation In [1] we exhibited the simple solution f (t)=Ct, for some (0,1), of the associated functional-differential equation and we proved that provided <2/(2+d()), where we have the asymtotic equivalence F(m)~ Cm.In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by |F^–1(m)|=F(m)², that is

On the Singular Values of Generalized Toeplitz Matrices

For a bounded function defined on , let be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries are equal to

Early stages of laser graphitization of diamond

A mechanism for photographitization of a free diamond surface is proposed. The quantum-kinetic rate of this process is determined. The graphitization rate is close to zero if the activation energy of the graphitization process is taken as being equal to the binding energy of a carbon atom with the surface (i.e. equal to the sublimation energy of a carbon atom). On the contrary, if the activation energy is close to the energy of C–C bonds, the graphitization process may occur at a noticeable rate and be observed under ‘relatively smooth’ experimental conditions. The temperature rise leads to a considerable increase in the graphitization rates. Preliminary experimental data on the low-rate laser ablation of diamond are presented to support the proposed model of photographitization. An early stage of laser-induced graphitization in the bulk of diamond is also considered. It is found that the nucleation of a ‘tiny graphite drop’ is possible in the bulk of the diamond inside the focal area of a laser beam; the ‘graphite drop’ growth causing the appearance of mechanical stresses in the surrounding regions. The maximum size of the graphite drop is determined, which, when exceeded, leads to mechanical damage of the sample and to a change in the mechanism of laser graphitization. An evident mechanical criterion for laser-induced damage of diamond is proposed. Received: 2 October 2002 / Accepted: 5 October 2002 / Published online: 29 January 2003 RID="*" ID="*"Corresponding author. E-mail: stvn@stankin.ru     

Apollonian isometries of planar domains are Möbius mappings

The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the Möbius mappings. In this article we show that this is the case in all open subsets of the plane with at least three boundary points.     

Lattice dynamical study of body-centred tetragonal indium

The phonon dispersion curves, phonon frequency distribution function as well as the lattice specific heat of body-centred tetragonal indium have been deduced using a lattice dynamical model which includes central, angular and volume forces. Six elastic constants, four zone boundary frequencies and an equilibrium condition were used in the evaluation of the force constants. It is shown that this model is elastically consistent and satisfies the symmetry requirements of the lattice, the phonon frequencies of indium deduced from it are in very good agreement with the experimental values of Reichardt and Smith and the theoretical values of Garrett and Swihart, and theθ _D values compare well with the experimental values over a wide temperature range. The apparent discrepancies in the phonon dispersion curves and theθ _D-T curves obtained from deficient models, importance of umklapp processes and the significance of angular forces in the lattice dynamical models are discussed.     

Numerical modeling of a He-Zn laser with helical hollow cathode

A kinetic model is proposed for a He-Zn laser with a helical hollow cathode oscillating on the Zn II 758.8 nm line: The dependence of the laser output power and gain on the active medium parameters, excitation pulse shape and cavity is investigated theoretically. The results from the numerical calculations are in a good agreement with the experimental data.