Three Electrons in a Two-Dimensional Parabolic Trap: The Relative Motion Solution

In agreement with the Kohn theorem the relative motion (rel) of three electrons in a two-dimensional parabolic trap separates from the centre-of-mass (CM) motion. By introducing new coordinates the Hamiltonian for relative motion in the approximation of non-interacting electrons can be taken to the normal form. The eigenstates of the normalized Hamiltonian are products of the Fock-Darwin states for normal modes. The energy levels for relative motion are obtained by diagonalizing the exact Hamiltonian in the eigenbasis for the non-interacting case. In this basis the interaction matrix elements can be obtained in the analytical form. Since the rank of the Hamiltonian matrix is significantly reduced, the calculations are faster and more accurate than those for the full (CM + rel) motion. This advantage is especially important for the calculations of excited states and the analysis of energy spectra.     

On Quantum Team Games

Recently Liu and Simaan (2004) convex static multi-team classical games have been introduced. Here they are generalized to both nonconvex, dynamic and quantum games. Puu's incomplete information dynamical systems are modified and applied to Cournot team game. The replicator dynamics of the quantum prisoner's dilemma game is also studied.     

Determination of trace levels of diosmin in a pharmaceutical preparation by adsorptive stripping voltammetry at a glassy carbon electrode.

A systematic study on the electrochemical behavior of diosmin in Britton-Robinson buffer (pH 2.0-10.0) at a glassy carbon electrode (GCE) was made. The oxidation process of the drug was found to be quasi-reversible with an adsorption-controlled step. The adsorption stripping response was evaluated with respect to various experimental conditions, such as the pH of the supporting electrolyte, the accumulation potential and the accumulation time. The observed anodic peak current at +0.73 V vs. Ag/AgCl reference electrode increased linearly over two orders of magnitude from 5.0x10(-8) M to 9.0x10(-6) M. A limit of detection down to 3.5x10(-8) M of diosmin at the GCE was achieved with a mean recovery of 97+/-2.1%. Based on the electrochemical data, an open-circuit accumulation step in a stirred sample solution of BR at pH 3.0 was developed. The proposed method was successfully applied to the determination of the drug in pharmaceutical formulations. The results compared favorably with the data obtained via spectrophotometric and HPLC methods.     

High pressure X-ray diffraction study of CdAl2Se4 and Raman study of AAl2Se4 (A=Hg, Zn) and CdAl2X4 (X=Se, S)

We report the results of an X-ray diffraction study of CdAl₂Se₄ and of Raman studies of HgAl₂Se₄ and ZnAl₂Se₄ at room temperature, and of CdAl₂S₄ and CdAl₂Se₄ at 80 K at high pressure. The ambient pressure phase of CdAl₂Se₄ is stable up to a pressure of 9.1 GPa above which a phase transition to a disordered rock salt phase is observed. A fit of the volume pressure data to a Birch-Murnaghan type equation of state yields a bulk modulus of 52.1 GPa. The relative volume change at the phase transition at ∼9 GPa is about 10%. The analysis of the Raman data of HgAl₂Se₄ and ZnAl₂Se₄ reveals a general trend observed for different defect chalcopyrite materials. The line widths of the Raman peaks change at intermediate pressures between 4 and 6 GPa as an indication of the pressure induced two stage order-disorder transition observed in these materials. In addition, we include results of a low temperature Raman study of CdAl₂S₄ and CdAl₂Se₄, which shows a very weak temperature dependence of the Raman-active phonon modes.     

Harmonic mappings of the Sierpinski gasket to the circle

Harmonic mappings from the Sierpinski gasket to the circle are described explicitly in terms of boundary values and topological data. In particular, all such mappings minimize energy within a given homotopy class. Explicit formulas are also given for the energy of the mapping and its normal derivatives at boundary points.

     

Stability of Mappings with Bounded Distortion on the Heisenberg Group

A stability theorem is proved for mappings with bounded distortion over John domains in the Heisenberg group furnished with the Carnot–Caratheodory metric.     

Barotropic Instability of the Bickley Jet at High Reynolds Numbers

We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case     , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because     and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber  α²≤ 9/2  ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition.     

Arithmetic on Groups of Positive Rationals

In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory of genera in algebraic number fields. We establish generalizations of both Dirichlet's theorem on primes in arithmetic progressions and the theory of genera for Abelian extensions of the rationals.