The technique of inverse Mellin transform for processes occurring in a background magnetic field

We develop the technique of inverse Mellin transform for processes occurring in a background magnetic field. We show by analyticity that the energy (momentum) derivatives of a field theory amplitude at the zero energy (momentum) is equal to the Mellin transform of the absorptive part of the amplitude. By inverting the transform, the absorptive part of the amplitude can be easily calculated. We apply this technique to calculate the photon polarization function in a background magnetic field.Comment: 3 pages, LATEX; talk presented at ICHEP02, Amsterdam, The Netherlands, 24-31 July 200

Extremal Spectral Gaps for Periodic Schr\"odinger Operators

The spectrum of a Schr\"odinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.Comment: 34 pages, 14 figure

An analytic study of nonsteady two-phase laminar boundary layer around an airfoil

Recently, NASA, FAA, and other organizations have focused their attention upon the possible effects of rain on airfoil performance. Rhode carried out early experiments and concluded that the rain impacting the aircraft increased the drag. Bergrum made numerical calculation for the rain effects on airfoils. Luers and Haines did an analytic investigation and found that heavy rain induces severe aerodynamic penalties including both a momentum penalty due to the impact of the rain and a drag and lift penalty due to rain roughening of the airfoil and fuselage. More recently, Hansman and Barsotti performed experiments and declared that performance degradation of an airfoil in heavy rain is due to the effective roughening of the surface by the water layer. Hansman and Craig did further experimental research at low Reynolds number. E. Dunham made a critical review for the potential influence of rain on airfoil performance. Dunham et al. carried out experiments for the transport type airfoil and concluded that there is a reduction of maximum lift capability with increase in drag. There is a scarcity of published literature in analytic research of two-phase boundary layer around an airfoil. Analytic research is being improved. The following assumptions are made: the fluid flow is non-steady, viscous, and incompressible; the airfoil is represented by a two-dimensional flat plate; and there is only a laminar boundary layer throughout the flow region. The boundary layer approximation is solved and discussed