Bounded rank-one perturbations in sampling theory

Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from a family of bounded rank-one perturbations of a self-adjoint operator that has only discrete spectrum of multiplicity one.     

Hyperasymptotics and hyperterminants: Exceptional cases

A new method is introduced for the computation of hyperterminants. It is based on recurrence relations, and can also be used to compute the parameter derivatives of the hyperterminants. These parameter derivatives are needed in hyperasymptotic expansions in exceptional cases. Numerical illustrations and an application are included.     

Total curvatures of geodesic spheres associated to quadratic curvature invariants

Total scalar curvatures of geodesic spheres obtained by integrating the second-order scalar invariants of the curvature tensor are investigated. The first terms in their power-series expansions are derived and these results are used to characterize the two-point homogeneous spaces among Riemannian manifolds with adapted holonomy. Dedicated to Professor L. VanheckeMathematics Subject Classification (2000) 53C25, 53C30     

Resonance expansions in quantum mechanics

The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

New methods for calculating the dew/bubble curves of classical model fluids

We describe two new methods for locating the dew/bubble curves of fluids. One is a numerical method and the other an analytical method based on the use of series expansions. The utility of these two methods is illustrated by application to a simple one-component fluid model and to several model polydisperse fluids. The numerical method is based on a new geometric representation of the equilibrium conditions-similar in spirit to the geometric representations often used for solving the equilibrium conditions of pure fluids. Our calculations show that the series-expansion technique can be quite effective at producing accurate representations of the phase boundaries.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

The Theorem of Mather on Generic Projections for Singular Varieties

The theorem of Mather on generic projections of smooth algebraic varieties is also proved for the singular ones.     

A conditionally linearized Monte Carlo filter in non-linear structural dynamics

State and parameter estimations of non-linear dynamical systems, based on incomplete and noisy measurements, are considered using Monte Carlo simulations. Given the measurements, the proposed method obtains the marginalized posterior distribution of an appropriately chosen (ideally small) subset of the state vector using a particle filter. Samples (particles) of the marginalized states are then used to construct a family of conditionally linearized system of equations and thus obtain the posterior distribution of the states using a bank of Kalman filters. Discrete process equations for the marginalized states are derived through truncated Ito-Taylor expansions. Increased analyticity and reduced dispersion of weights computed over a smaller sample space of marginalized states are the key features of the filter that help achieve smaller sample variance of the estimates. Numerical illustrations are provided for state/parameter estimations of a Duffing oscillator and a 3-DOF non-linear oscillator. Performance of the filter in parameter estimation is also assessed using measurements obtained through experiments on simple models in the laboratory. Despite an added computational cost, the results verify that the proposed filter generally produces estimates with lower sample variance over the standard sequential importance sampling (SIS) filter.