论文标题
哥伦布广告的非库赛环圈中的最高限制和超级限制
Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers
论文作者
论文摘要
众所周知，哥伦布序列序列的限制概念$ \ widetilde {\ mathbb {r}} $不会概括经典结果。例如〜序列$ \ frac {1} {n} \ not \ to0 $和一个序列$（x_ {n}）_ {n \ in \ Mathbb {n}} $收敛\ emph {if}，并且仅当$ x__ {n+1} -x_-x_ {n+emph {if}。在整合一般函数的整合过程中，这会产生几个深刻的后果，例如〜在序列，分析性广义功能或Sigma-radivitivity和经典限制定理中。缺乏这些结果也与以下事实有关：$ \ wideTilde {\ mathbb {r}} $一定不是完整的有序集，例如〜所有无限次数的集合都没有上符号和imblemimum。我们提出了这些问题的解决方案，引入了过度自然数量，超平等，近距离和幼稚的概念。通过这种方式，我们可以将所有经典定理推广到超平等的超级限制中。该论文探讨了可以应用于其他非架构设置的想法。
It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers $\widetilde{\mathbb{R}}$ does not generalize classical results. E.g.~the sequence $\frac{1}{n}\not\to0$ and a sequence $(x_{n})_{n\in\mathbb{N}}$ converges \emph{if} and only if $x_{n+1}-x_{n}\to0$. This has several deep consequences, e.g.~in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that $\widetilde{\mathbb{R}}$ is necessarily not a complete ordered set, e.g.~the set of all the infinitesimals does not have neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.
