Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Relative entropy rate based multiple hidden Markov Model Approximation

This paper proposes a novel relative entropy rate (RER) based approach for multiple HMM (MHMM) approximation of a class of discrete-time uncertain processes. Under different uncertainty assumptions, the model design problem is posed either as a min-max optimisation problem or stochastic minimisation problem on the RER between joint laws describing the state and output processes (rather than the more usual RER between output processes). A suitable filter is proposed for which performance results are established which bound conditional mean estimation performance and show that estimation performance improves as the RER is reduced. These filter consistency and convergence bounds are the first results characterising multiple HMM approximation performance and suggest that joint RER concepts provide a useful model selection criteria. The proposed model design process and MHMM filter are demonstrated on an important image processing dim-target detection problem.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Towards an intelligent learning system for the natural born cyborg

We propose to design a Custom Learning System that responds to the unique needs and potentials of individual students, regardless of their location, abilities, attitudes, and circumstances. This project is intentionally provocative and future-looking but it is not unrealistic or unfeasible. We propose that by combining complex learning databases with a learner’s personal data, we could provide all students with a personal, customizable, and flexible education. This paper presents the initial research undertaken for this project of which the main challenges were to broadly map the complex web of data available, to identify what logic models are required to make the data meaningful for learning, and to translate this knowledge into simple and easy-to-use interfaces. The ultimate outcome of this research will be a series of candidate user interfaces and a broad system logic model for a new smart system for personalized learning. This project is student-centered, not techno-centric, aiming to deliver innovative solutions for learners and schools. It is deliberately future-looking, allowing us to ask questions that take us beyond the limitations of today to motivate new demands on technology.

Making the most of MiLK : harnessing the contemporary culture and interests of natural born cyborg

MiLK is a mobile learning kit that allows students and teachers to author their own place-based learning events using simple web and mobile technologies. We will demonstrate how MiLK has been used by a number of teachers in various contexts to connect students, curriculum and everyday environments. This workshop will introduce participants to the various MiLK tools and processes; including mapping, designing, playing and reviewing events, group journals, discussion forums, student profiles, and class profiles. We will focus on the role of place as a potential resource for curriculum design and delivery. The MiLK Team are looking for enthusiastic mobile technology champions to join us. No previous experience or training in this area is needed. This workshop is designed to be relevant to all KLAs. During this session teachers will have an opportunity to experiment with simple tools to create dynamic resources for their own classrooms.