Phase structure function of random wave fields

The phase properties of a complex Gaussian field, which can arise following scattering from random phase screens and through extended random media, are investigated. It is shown that the unwrapped phase of a complex Gaussian field constitutes a non-stationary process, such that the phase autocorrelation function does not exist. However, the phase structure function remains finite, allowing analytical results to be obtained for various field correlations. Methods for numerical simulation are discussed and their results found to be in excellent agreement with analytical predictions. More general considerations reveal that the phase structure function of a complex Gaussian field increases linearly with large separation distance. The results are relevant to phase-sensitive detection in fields undergoing strong intensity fluctuations.     

On closed boundary value problems for equations of mixed elliptic‐hyperbolic type

For partial differential equations of mixed elliptic‐hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet‐conormal types. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a ? b ? c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. © 2006 Wiley Periodicals, Inc.     

A generalization of Ryser's theorem on term rank

In his work on classes of (0, 1)-matrices with given row and column sum vectors, Herbert Ryser proved that the maximum term rank possible in a normalized class, ρ, can be realized by a matrix having ρ (independent) 1's in positions (1, ρ), (2, ρ − 1), … , (ρ, 1). We study the positions occupied by sets of t ρ independent 1's.     

Longevity risk and the Grim Reaper’s toxic tail: The survivor fan charts

This paper uses survivor fan charts to illustrate the prospective density functions of future male survival rates. The fan charts are based on a version of the Cairns–Blake–Dowd model of male mortality that provides a good fit to recent mortality data for England and Wales. They indicate that although none of us can escape the Grim Reaper, survivorship uncertainty is greatest for males aged a little over 90, confirming that there exists a ‘toxic tail’ for those institutions, such as annuity and pension providers, which are obliged to make payments to them for as long as they live. We also find that taking account of uncertainty in the parameters of the underlying mortality model leads to major increases in estimates of the widths of the fan charts.     

An efficient shooting algorithm for Evans function calculations in large systems

In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products [J.C. Alexander, R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World 2 (4) (1995) 471–507; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Ph.D. Thesis, Indiana University, Bloomington, 1998; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (235) (2001) 1071–1088; L.Q. Brin, K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, in: Seventh Workshop on Partial Differential Equations, Part I, 2001, Rio de Janeiro, Mat. Contemp. 22 (2002) 19–32; T.J. Bridges, G. Derks, G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Physica D 172 (1–4) (2002) 190–216]. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as , where n is the dimension of the system written as a first-order ODE and k (typically n/2) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing “pure” (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury–Davey [A. Davey, An automatic orthonormalization method for solving stiff boundary value problems, J. Comput. Phys. 51 (2) (1983) 343–356; L.O. Drury, Numerical solution of Orr-Sommerfeld-type equations, J. Comput. Phys. 37 (1) (1980) 133–139] and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.     

Force Deployment in a Conventional Theatre-level Military Engagement

This paper is concerned with the study of force deployment in a conventional theatre-level military engagement. First, it discusses some of the governing parameters that affect the modelling. A non-linear programming model is then developed to describe the combat dynamics. Sensitivity analysis on the optimal solution is also discussed. The merit of the model is demonstrated by applying the formulation to study the effect of various sustainment policies on the outcome of a battle.     

Restarted GMRES preconditioned by deflation

This paper presents a new preconditioning technique for the restarted GMRES algorithm. It is based on an invariant subspace approximation which is updated at each cycle. Numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm, at a low cost of operations and memory.