Set-valued Markov chains and negative semitrajectories of discretized dynamical systems

Computer simulation of dynamical systems involves a phase space which is the finite set of machine arithmetic. Rounding state values of the continuous system to this grid yields a spatially discrete dynamical system, often with different dynamical behaviour. Discretization of an invertible smooth system gives a system with set-valued negative semitrajectories. As the grid is refined, asymptotic behaviour of the semitrajectories follows probabilistic laws which correspond to a set-valued Markov chain, whose transition probabilities can be explicitly calculated. The results are illustrated for two-dimensional dynamical systems obtained by discretization of fractional linear transformations of the unit disc in the complex plane.

Outcome variables for osteoarthritis clinical trials: the OMERACT-OARSI set of responder criteria

Improvement in analysis and reporting results of osteoarthritis (OA) clinical trials has been recently obtained because of harmonization and standardization of the selection of outcome variables (OMERACT 3 and OARSI). Moreover, OARSI has recently proposed the OARSI responder criteria. This composite index permits presentation of results of symptom modifying clinical trials in OA based on individual patient responses (responder yes/no). The 2 organizations (OMERACT and OARSI) established. a task force aimed at evaluating: (1) the variability of observed placebo and active treatment effects using the OARSI responder criteria; and (2) the possibility of proposing a simplified set of criteria. The conclusions of the task force were presented and discussed during the OMERACT 6 conference, where a simplified set of responder criteria (OMERACT-OARSI set of criteria) was proposed.