Abstract.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 Painlevé 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 Painlevé 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.
This paper is dedicated to Edward L. Reiss on the occasion of his 60th birthday.Abstract. It has recently been demonstrated that standard discretizations of the cubic nonlinear Schr/bdinger (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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.