Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations

In is known that the equations [u_t ? g(u)u_x]x = ±g′(u) describe pseudo-spherical surfaces, i.e. that these equations are the integrability conditions for the structural equations of such surfaces, provided g satisfies g′ + µg = θ. In this paper we obtain self-Bäcklund transformations for these equations by a geometric method, and show how the inverse scattering method generates global solutions.     

Half-range completeness for the Fokker-Planck equation

Half-range completeness theorems are proved for eigenfunctions associated to the one-dimensional Fokker-Planck equation in a semi-infinite medium. Existence and uniqueness results for perfectly absorbing, partially absorbing, and purely specularly reflecting boundary conditions are deduced for the stationary and time-dependent problems. Similar results are obtained for a slab geometry.     

Multi-point radiation monitor

A simple, inexpensive instrument for measuring radiation fields in otherwise inaccessible locations has been developed. The RadRope consists of a series of radiation detectors inside a flexible sheath and connected to a data readout device to alert the operator to unexpected radiation fields at a remote location. The instrument is designed for use in a maritime environment and will assist inspection personnel in detecting potential illicit radioactive materials or radiological/nuclear weapons of mass destruction.     

Solid phase microextraction sampling of high explosive residues in the presence of radionuclides and radionuclide surrogate metals

The Federal Bureau of Investigation (FBI) Laboratory currently does not have on site facilities for handling radioactive evidentiary materials and there are no established FBI methods or procedures for decontaminating high explosive (HE) evidence while maintaining evidentiary value. One experimental method for the isolation of HE residue involves using solid phase microextraction (SPME) fibers to remove residue of interest. Due to their high affinity for organics, SPME fibers should have little affinity for most metals. However, no studies have measured the affinity of radionuclides for SPME fibers. The focus of this research was to examine the affinity of dissolved radionuclide (^239/240Pu, ²³⁸U, ²³⁷Np, ⁸⁵Sr, ¹³³Ba, ¹³⁷Cs, ⁶⁰Co and ²²⁶Ra) and stable radionuclide surrogate metals (Sr, Co, Ir, Re, Ni, Ba, Cs, Nb, Ru, and Nd) for SPME fibers at the exposure conditions that favor the uptake of HE residues. Our results from radiochemical and mass spectrometric analyses indicate these metals have little measurable affinity for these SPME fibers during conditions that are conducive to HE residue uptake with subsequent analysis by liquid or gas phase chromatography with mass spectrometric detection.     

Exact solutions and branching of singularities for some hyperbolic equations in two variables

Exact global propagators are constructed for the singular hyperbolic operators in two variables , λ a real parameter, and for the degenerate hyperbolic operators . Qualitative phenomena such as uniqueness in the Cauchy problem and branching of singularities vary with λ, as shown earlier by Treves and by Taniguchi and Tozaki.     

A 2-component Camassa-Holm Equation,Euler-Bernoulli Beam Problem,and Noncommutative Continued Fractions

A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed:

1. motivated by an analogy with the Camassa-Holm equation a class of isospectral deformations of the beam problem is formulated;
2. a reformulation of the matrix string problem in terms of a certain compact operator is used to obtain basic spectral properties of the inhomogeneous matrix string problem with Dirichlet boundary conditions;
3. the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a noncommutative generalization of Stieltjes’ continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel-like determinants.

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