A proof of the four-colour theorem

The four-colour problem remained unsolved for more than a hundred years has played a role of the utmost importance in the development of graph theory. The four-colour theorem was confirmed in 1976, which is not completely satisfied due to: i) part of the proof using computers cannot be verified by hand; ii) even the part, supposedly hand-checkable, is extraordinarily complicated and tedious, and as far as we know, no one has entirely verified it. Seeking a hand-checkable proof of the four-colour theorem is one of world-interested problems, which is addressed in this paper. A necessary and sufficient condition for n-colour theorem in a space is: there exists a largest n-complete graph base in the same space. Examples are given to illustrate applications

Adaptive online parameter estimation algorithm of PEM fuel cells

Since most of fuel cell models are generally nonlinearly parameterized functions, existing modeling techniques rely on the optimization approaches and impose heavy computational costs. In this paper, an adaptive online parameter estimation approach for PEM fuel cells is developed in order to directly estimate unknown parameters. The general framework of this approach is that the electrochemical model is first reformulated using Taylor series expansion. Then, one recently proposed adaptive parameter estimation method is further tailored to estimate the unknown parameters. In this method, the adaptive law is directly driven by the parameter estimation errors without using any predictors or observers. Moreover, parameter estimation errors can be guaranteed to achieve exponential convergence. Besides, the online validation of regressor matrix invertibility are avoided such that computation costs can be effectively reduced. Finally, comparative simulation results demonstrate that the proposed approach can achieve better performance than least square algorithm for estimating unknown parameters of fuel cells.Postprint (published version