“杜西结论”的推广

On the extension of the Duthe's conclusion

研究和推广"杜西结论",设α∈φn={(a1,a2,…,an)|ai∈N,i=1,2,…,n},定义杜西变换:D(α)=(|a1-a2|,|a2-a3|,…,|an-a1|),利用离散动力系统的分析方法,研究更一般的问题,得出结论:n(n=2k,(k=1,2,…))个自然数形成一个环形,再进行相邻两数大数减小数,则在有限步内n个数必都变为零,即对任意α∈φn,当n=2k,(k=1,2,…)时,存在m∈N,有Dm(α)=θ,并得出几个相关的结论.

Supposed α∈φn={（a1,a2,…,an）｜ai∈N,i=1,2,…,n},in the φn,definited an operator D,called Duthe operator,with the following property：D（α）=（｜a1-a2｜,｜a2-a3｜,…,｜an-a1｜）.Thereby using discrete dynamic system,we got a general conclusion,which extended the ＂Duthe＇s Conclusion＂to the 2k-space,as following：for any α∈φn,where n=2k,（k=1,2,…）,then existed m∈N,we had Dm（α）=θ.We also got some relative conclusions in the paper.