Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

Development of a Java-based Unified and Flexible Natural Language Discourse System

This paper outlines the design and development of a Java-based, unified and flexible natural language dialogue system that enables users to interact using natural language, e.g. speech. A number of software development issues are considered with the aim of designing an architecture that enables different discourse components to be readily and flexibly combined in a manner that permits information to be easily shared. Use of XML schemas assists this component interaction. The paper describes how a range of Java language features were employed to support the development of the architecture, providing an illustration of how a modern programming language makes tractable the development of a complex dialogue system.