Wave scattering functions and their application to multiple scattering problems

Scattering functions arise naturally in standard treatments of the effects of a material object or surface embedded in a uniform field. The most commonly used scattering function describes the far-field modulation imparted at large distances to a spherical wavefront eminating from the scatterer. The purpose of this is to develop the properties of the spectrum of scattered plane waves as an exact generalized scattering function. The linearity of the wave equations guarantees that such a representation exists; moreover, it is possible to derive the generalized scattering function from the far-field scattering function by analytic continuation. Although these properties are known, recent theoretical developments have motivated us to reexplore the interrelations among the far-field scattering function, the Green's function and various forms of the generalized scattering function as well as the symmetry properties of the generalized scattering function imposed by reciprocity. For multiple-scattering objects that can be separated by parallel planes, a system of difference equations is developed that fully accommodates the mutual interaction among the scatterers. The mutual interaction equations were developed earlier, but we show here that they can be transformed into the form that would be obtained by using the Foldy-Lax-Twersky formalism. This reinforces the equivalence between wave-space and configuration space formulations of the scattering problems.     

Room-temperature multiwavelength erbium-doped fiber ring laser employing sinusoidal phase-modulation feedback

An effective method for achieving a room-temperature multiwavelength erbium-doped fiber ring laser is presented. Simultaneous multiwavelength lasing with 0.5-nm intervals is achieved both experimentally and theoretically by addition of sinusoidal phase modulation in the ring cavity to prevent single-wavelength oscillation.     

Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways.     

On convection in a fluid filling the cavity of a moving solid body

The problem of simultaneous motion about a fixed point O of a solid body and unevenly heated viscous incompressible fluid completely filling a finite cavity of the body is considered in linear formulation. The center of mass of the system body plus fluid in the state of mechanical equilibrium is assumed to coincide with point O. The theorem on solvability of the Cauchy problem for small unsteady equilibrium perturbations is proved, and normal perturbations and the spectrum of the problem arising in the analysis of such perturbations are investigated. It is shown that the whole spectrum consists of normal eigenvalues and lies in some half-band containing the real axis. It is shown that the respective system of root vectors is complete. Properties of the spectrum and the dependence on Rayleigh numbers are investigated. Rayleigh numbers for which the real parts of eigenvalues are positive,i.e. when the generated oscillating normal perturbations are damped in time, are evaluated in the case of the fluid being heated from below and above.