Liliana Rybarska-Rusinek

Stochastic stability analysis of competitive neural networks with different time-scales

Most computational models for competitive neural networks describe activity-connectivity interactions at different time-scales. We extend these existing models by considering stochastic processes and establish stability results based on the theory of singularly perturbed stochastic systems. Based on a reduced-order model we determine conditions that ensure the existence of the exponentially mean-square stability equilibria of the stochastic nonlinear system. It is assumed that the system is described by Ito-type equations. We derive a Lyapunov function for the coupled system and an upper bound for the parameters of the independent stochastic process.

On solving continuum-mechanics problems by fast multipole methods

It is shown analytically and numerically that the kernel-independent fast multipole method provides the accuracy and computational complexity of the analytical method if circular (spherical in 3D) equivalent surfaces are used.

A Note on Evaluation of Temporal Derivative of Hypersingular Integrals over Open Surface with Propagating Contour

The short note concerns with elasticity problems involving singular and hypersingular integrals over open surfaces, specifically cracks, with the contour propagating in time. Noting that near a smooth part of a propagating contour the state is asymptotically plane, we focus on 1D hypersingular integrals and employ complex variables. By using the theory of complex variable singular and hypersingular integrals, we show that the rule for evaluation of the temporal derivative is the same as that for proper integrals. Being applied to crack problems the rule implies that the temporal derivative may be evaluated by differentiation under the integral sign.