The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary soluti...

The use of stationary-action formulations for dynamical systems allows one to generate fundamental solutions for classes of two-point boundary-value problems (TPBVPs). One solves for stationary points of the payoff as a function of inputs, a task which is significantly different from that in optimal control problems. Both a dynamic programming principle (DPP) and a Hamilton-Jacobi partial diffe...

We investigate the error surface of the XOR problem for a 2-2-1 network with sigmoid transfer functions. It is proved that all stationary points with finite weights are saddle points with positive error or absolute minima with error zero. So, for finite weights no local minima occur. The proof results from a careful analysis of the Taylor series expansion around the stationary points. For some ...