The Initial Value Problem for Weakly Nonlinear PDE

We will discuss an extension of the pseudospectral method developed by Wineberg, McGrath, Gabl, and Scott for the numerical integration of the KdV initial value problem. Our generalization of their algorithm can be used to solve initial value problems for a wide class of evolution equations that are "weakly nonlinear" in a sense that we will make precise. This class includes in particular the other classical soliton equations (SGE and NLS). As well as being very simple to implement, this method exhibits remarkable speed and stability, making it ideal for use with visualization tools where it makes it possible to experiment in real-time with soliton interactions and to see how a general solution decomposes into solitons. We will analyze the structure of the algorithm, discuss some of the reasons behind its robust numerical behavior, and finally describe a fixed point theorem we have found that proves that the pseudospectral stepping algorithm converges

The long-term effects of the micrometeoroid and orbital debris environments on materials used in space

The long-term effects of the orbital debris and micrometeoroid environments on materials that are current candidates for use on space vehicles are discussed. In addition, the limits of laboratory testing to determine these effects are defined and the need for space-based data is delineated. The impact effects discussed are divided into primary and secondary surfaces. Primary surfaces are those that are subject to erosion, pitting, the degradation and delamination of optical coatings, perforation of atomic oxygen erosion barriers, vapor coating of optics and the production of secondary ejecta particles. Secondary surfaces are those that are affected by the result of the perforation of primary surfaces, for example, vapor deposition on electronic components and other sensitive equipment, and the production of fragments with damage potential to internal pressurized elements. The material properties and applications that are required to prevent or lessen the effects described, are defined

Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system

The Mishchenko-Fomenko theorem on superintegrable Hamiltonian systems is generalized to superintegrable Hamiltonian systems with noncompact invariant submanifolds. It is formulated in the case of globally superintegrable Hamiltonian systems which admit global generalized action-angle coordinates. The well known Kepler system falls into two different globally superintegrable systems with compact and noncompact invariant submanifolds.Comment: 23 page

The Picard group of the loop space of the Riemann sphere

The loop space of the Riemann sphere consisting of all C^k or Sobolev W^{k,p} maps from the circle S^1 to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbeault group. The group of Mobius transformations G and its loop group LG act on this loop space. We prove that an element of the Picard group is LG-fixed if it is G-fixed; thus completely answer the question by Millson and Zombro about G-equivariant projective embedding of the loop space of the Riemann sphere.Comment: International Journal of Mathematic

Hypervelocity impact shield

A hypervelocity impact shield and method for protecting a wall structure, such as a spacecraft wall, from impact with particles of debris having densities of about 2.7 g/cu cm and impact velocities up to 16 km/s are disclosed. The shield comprises a stack of ultra thin sheets of impactor disrupting material supported and arranged by support means in spaced relationship to one another and mounted to cover the wall in a position for intercepting the particles. The sheets are of a number and spacing such that the impacting particle and the resulting particulates of the impacting particle and sheet material are successively impact-shocked to a thermal state of total melt and/or vaporization to a degree as precludes perforation of the wall. The ratio of individual sheet thickness to the theoretical diameter of particles of debris which may be of spherical form is in the range of 0.03 to 0.05. The spacing between adjacent sheets is such that the debris cloud plume of liquid and vapor resulting from an impacting particle penetrating a sheet does not puncture the next adjacent sheet prior to the arrival thereat of fragment particulates of sheet material and the debris particle produced by a previous impact

On the self-adjointness of certain reduced Laplace-Beltrami operators

The self-adjointness of the reduced Hamiltonian operators arising from the Laplace-Beltrami operator of a complete Riemannian manifold through quantum Hamiltonian reduction based on a compact isometry group is studied. A simple sufficient condition is provided that guarantees the inheritance of essential self-adjointness onto a certain class of restricted operators and allows us to conclude the self-adjointness of the reduced Laplace-Beltrami operators in a concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland type reductions of `free' Hamiltonians under polar actions of compact Lie groups follows immediately.Comment: 9 pages, minor changes, updated references in v