Christoffel-Darboux-type formulae and a recurrence for biorthogonal polynomials

It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with different degrees. This is used to produce a mixed recurrence relation, which is valid for all biorthogonal polynomials. This recurrence relation establishes several results on interlacing property of zeros of successive biorthogonal polynomials and leads to a new result on the interlace of zeros of orthogonal polynomials (of equal degrees) with respect to two distributionsdψ(x) andx ^p dψ(x), 0<p≤1, with support in either [0, 1] or [1, ∞).     

Angular dependent rayleigh scattering of Mössbauer radiation on proteins

RSMR experiments with⁵⁷Fe radiation were performed on myoglobin. An areasensitive detector was employed for simultaneous angular dependent collection of the scattered quanta up to a maximum angle 2θ of 17‡. Experimental data of polycrystalline and lyophilized myoglobin are compared with computer calculations of the scattering which are based on the atomic coordinates determined by X-ray structure analysis. Special attention has been paid to the influence of coherence effects from collectively moving parts of the protein. A simple model is introduced in order to take into account these segmental motions. Our first results indicate that the sizes of collectively moving segments are comparable with spheres of about 6 å in diameter in dry myoglobin. In myoglobin crystals, where the molecules are surrounded by large hydration shells, the movements appear to be correlated in segments with sizes comparable to helices.

     

A search technique for global optimization in a chaotic environment

We describe a new algorithm which uses the trajectories of a discrete dynamical system to sample the domain of an unconstrained objective function in search of global minima. The algorithm is unusually adept at avoiding nonoptimal local minima and successfully converging to a global minimum. Trajectories generated by the algorithm for objective functions with many local minima exhibit chaotic behavior, in the sense that they are extremely sensitive to changes in initial conditions and system parameters. In this context, chaos seems to have a beneficial effect: failure to converge to a global minimum from a given initial point can often be rectified by making arbitrarily small changes in the system parameters.     

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.