B. M. Herbst

On the extension of the Painlevé property to difference equations

It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z ), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

Offline signature verification using the discrete radon transform and a hidden Markov model

We developed a system that automatically authenticates offline handwritten signatures using the discrete Radon transform (DRT) and a hidden Markov model (HMM). Given the robustness of our algorithm and the fact that only global features are considered, satisfactory results are obtained. Using a database of 924 signatures from 22 writers, our system achieves an equal error rate (EER) of 18% when only high-quality forgeries (skilled forgeries) are considered and an EER of 4.5% in the case of only casual forgeries. These signatures were originally captured offline. Using another database of 4800 signatures from 51 writers, our system achieves an EER of 12.2% when only skilled forgeries are considered. These signatures were originally captured online and then digitally converted into static signature images. These results compare well with the results of other algorithms that consider only global features.

On homoclinic structure and numerically induced chaos for the nonlinear Schro¨dinger equation

It has recently been demonstrated that standard discretizations of the cubic nonlinear Schrodinger (NLS) equation may lead to spurious numerical behavior. In particular, the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the NLS equation. In this paper, an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated. Differences between an integrable discretization and standard discretizations are highlighted.

Singular value decomposition, eigenfaces, and 3D reconstructions

Singular value decomposition (SVD) is one of the most important and useful factoriza- tions in linear algebra. We describe how SVD is applied to problems involving image processing—in particular, how SVD aids the calculation of so-called eigenfaces, which pro- vide an efficient representation of facial images in face recognition. Although the eigenface technique was developed for ordinary grayscale images, the technique is not limited to these images. Imagine an image where the different shades of gray convey the physical three- dimensional structure of a face. Although the eigenface technique can again be applied, the problem is finding the three-dimensional image in the first place. We therefore also show how SVD can be used to reconstruct three-dimensional objects from a two-dimensional video stream.

Numerical experience with the nonlinear Schrödinger equation

Abstract Increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear Schrodinger equation without changing the initial condition u ( x , 0)=sech x , leads to bound states of an increasing number of solitons. This results in very steep gradients in space and time and so provides a more severe test of numerical methods than before. In particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice. Various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples.

Estimating the pen trajectories of static signatures using hidden Markov models

Static signatures originate as handwritten images on documents and by definition do not contain any dynamic information. This lack of information makes static signature verification systems significantly less reliable than their dynamic counterparts. This study involves extracting dynamic information from static images, specifically the pen trajectory while the signature was created. We assume that a dynamic version of the static image is available (typically obtained during an earlier registration process). We then derive a hidden Markov model from the static image and match it to the dynamic version of the image. This match results in the estimated pen trajectory of the static image.

Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings

A Hamiltonian difference scheme associated with the integrable nonlinear Schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution. The chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the solutions are not subject to even constraints, the traditional mechanism of crossings of the unperturbed homoclinic orbits/manifolds is not observed.

Dubuc-Deslauriers Subdivision for Finite Sequences and Interpolation Wavelets on an Interval

In this paper we consider a method of adapting Dubuc--Deslauriers subdivision, which is defined for bi-infinite sequences, to accommodate sequences of finite length. After deriving certain useful properties of the Dubuc--Deslauriers refinable function on

Equidistributing principles in moving finite element methods

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Numerical simulation of quasi-periodic solutions of the sine-Gordon equation

, we define a multiscale finite sequence of functions on a bounded interval, which are then proved to be refinable. Using this fact, the resulting adapted interpolatory subdivision scheme for finite sequences is then shown to be convergent. Corresponding interpolation wavelets on an interval are defined, and explicit formulations of the resulting decomposition and reconstruction algorithms are calculated. Finally, we give two numerical examples on signature smoothing and two-dimensional feature extraction of the subdivision and wavelet algorithms.