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, ∞).     

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.     

The diffraction of plane gravitational waves by the edge of an ice cover

The diffraction of plane surface gravitational waves by the edge of an ice cover lying on the surface of an incompressible fluid of infinitely great depth is considered. The ice cover is simulated by a thin elastic plate. The wave reflection and transmission coefficients are determined when it interacts with the ice cover. A wave field is constructed in the fluid under the conditions that a periodic lumped force and a lumped moment act on the edge of the ice cover. It is shown that as the incident wavelength increases the reflection coefficient tends to zero and the transmission coefficient tends to unity.     

Sound field variations caused by intense internal waves in a shallow sea with a weak thermocline

Results of an experiment on investigating the fluctuations of a sound signal with a frequency of 295 Hz on a stationary 32-km-long acoustic path in a shallow sea are presented. Hydrological conditions on the path were characterized by the presence of a weak thermocline and a frequent occurrence of intense internal wave trains. The space-time characteristics of these internal waves were measured in detail. Experimental evidence of the repetition of the forms of internal solitons in the variations of the amplitude and phase of an acoustic signal was obtained.     

Coherent transients at dressed levels

Possibilities for the control of the parameters of free-polarization decay (FPD), optical nutation, and photon echo (PE) using the dressing field are studied. Coherent transients are generated with the Stark switching technique and are detected in the radiation of the probe field polarized orthogonally to the dressing field. The evolution-operator technique is employed in the calculations. The experiments are performed at the R(4, 3) transition of the 0 ? 1 v₃ ¹³CH₃F vibrational band with the radiation of a cw CO₂ laser. It is theoretically and experimentally demonstrated that FPD and PE are suppressed upon an increase in the intensity of the dressing field. The observed shapes of the transient FPD and PE signals and their variations with the dressing field intensity are in qualitative agreement with the results of the calculations.     

On a class of operators related to second-order differential equations

In the present paper, we find a class of linear homogeneous differential equations of order n + 1 (n > 1) whose fundamental system of solutions is constructed from the fundamental system of solutions of a second-order differential equation. The spectral properties of differential operators generated by these differential expressions are investigated. In particular, sufficient conditions are obtained for the coefficients of a second-order differential equation under which the case of maximal deficiency indices is realized. Dedicated to the memory of B. M. Levitan