Continuous Utility Functions Through Scales

We present here a direct elementary construction of continuous utility functions on perfectly separable totally preordered sets that does not make use of the well-known Debreu’s open gap lemma. This new construction leans on the concept of a separating countable decreasing scale. Starting from a perfectly separable totally ordered structure, we give an explicit construction of a separating countable decreasing scale, from which we show how to get a continuous utility map.      

The project of a new of neural networks products line: An application

The aim of this paper is to propose a new approach to the issue ofpositioning and pricing a firm's new products line; it is shown here how to conceive an artificial neural system able to find the optimal solution. The formulation of the problem follows the model proposed by Dobson and Kalish, while the implementation is based on the Hopfield Neural Network, thus used here for a new class of problems. The model shows a capacity to converge to the optimal solution; besides its effectiveness, this technique appears to also be efficient if applied to such a production problem.