We consider the one-dimensional motion of a particle randomly accelerated by Gaussian white noise on the line segment 0ϽxϽ1. The reflections of the particle from the boundaries at xϭ0,1 are inelastic. The velocities just before and after reflection are related by v f ϭϪrv i , where r is the coefficient of restitution. Cornell, Swift, and Bray ͓Phys. Rev. Lett. 81, 1142 ͑1998͔͒ have argued that there is an inelastic collapse transition in this system. For rϾr c ϭe Ϫ/ͱ3 the particle moves throughout the interval 0ϽxϽ1, while for rϽr c the particle is localized at xϭ0 or xϭ1. In this paper the equilibrium distribution function P(x,v) is analyzed for rϾr c by solving the steady-state Fokker-Planck equation, and the results are compared with numerical simulations.
The use of neuron-like networks (NN) for pattern recognition has a well-established history and numerous current applications. Most such applications are to static patterns while the theory developed for temporally changing visual patterns usually assumes rigid objects with well-defined boundaries. In applications such as analysis of cardiac movement, however, the object is flexible and the images are often imperfect. The authors current model for NN activity captures the dynamic nature of the signal processing of the neural dendritic tree, allowing both faster learning of dynamic patterns and a very reduced number of receptors required for distinguishing diverse types of motion or changes. The design of the NN model is presented and a training algorithm which exhibits in practice extremely fast convergence (as few as I5 iterations) to near optimal recognition behavior is introduced.
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