In this paper, evolution and visualization of the existence of saddle points of nonlinear functionals or multi-variable functions in finite dimensional spaces are presented. New algorithms are developed based on the mountain pass lemma and link thery in nonlinear analysis. Further more, a simple comparison of the steepest descent algorithm and the genetic algorithm is given. The process of the saddle point finding is visualised in an inteactive graphical interface.

This paper presents a new design optimization technique by artificial neural networks. Based on the theory of partial ordering and cones in Banach spaces, a new kind of neural networks with functional links is proposed. Learning algorithm lies heavily on the ordering structure of the sample space with an alternative feed-forward and backpropagation technique. Experiments are described with comparison with traditional functional link networks and wavelet networks.

In this paper, we give some new results on the existence and properties of solutions of singular elliptic boundary value problems of second order. We discuss the difference of weak solutions and classical solutions.