Analysis of trigonometric implicit Runge-Kutta methods

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge-Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge-Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods. (c) 2006 Elsevier B.V. All rights reserved.

Implicit stochastic Runge-Kutta methods for stochastic differential equations

In this paper we construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

Stability for parametric implicit vector equilibrium problems

In this paper, we consider a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces where a mapping f and a set K are perturbed by parameters is an element of and lambda respectively. We establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution set mapping S : Lambda(1) x A(2) -> 2(X) for such parametric implicit vector equilibrium problems. (c) 2005 Elsevier Ltd. All rights reserved.

Implicit learning is not subject to blocking

Explicit (aware) learning has been shown to evidence certain characteristics, such as extinction, blocking, occasion setting, and reliance on context. These characteristics have not been assessed in implicit (unaware) learning. The current study investigated whether implicit learning is subject to blocking. Participants completed a cued reaction time task, where they watched rapid presentations of a random sequence of 8 pairs of shapes, and responded to two target shapes. One target was always preceded by a cue. The experimental group completed a pretraining phase where half the cue, one shape, was followed by the target. Both experimental and control groups completed a training phase where both elements of the cue, two shapes, were followed by the target. Both aware and unaware participants evidenced learning, whereby responding was faster for cued than uncued targets. Aware participants in the experimental group responded faster to targets preceded by the pretrained element than by the other element of the cue. Control and unaware experimental participants were faster to respond to targets preceded by either element of the cue. As blocking was only evident in aware participants, but implicit learning was observed in all participants, it is concluded that implicit learning is not subject to blocking.