真正的3x+1问题
The Real 3x+1 Problem
摘要：
在这项工作中，我们引入了另一个扩展的3 n +1功能的实直线。我们提出了一个关于U-轨线的猜想，它推广了著名的3 n +1（或Collatz）猜想。然后我们证明了我们关于U的迭代的主要结果（这与这两个代数直接相关）。我们还介绍了“翻转3x+1”的功能\widetilde U，并证明了类似的结果，其轨迹。在最后一节中，我们提出了一些关于U（和\widetilde U）的迭代的有趣的问题，证明了关于U和\widetilde U的迭代的几个简单的结果，介绍了其他相关的函数，并提出了关于它们的迭代的更多的结构和问题。我们希望本文的结果、图解和问题不仅与3 n +1猜想本身有关，而且本身也有意义。
英文摘要：
In this work, we introduce another extension U of the 3n+1 function to the real line. We propose a conjecture about the U-trajectories that generalizes the famous 3n+1 (or Collatz) conjecture. We then prove our main result about the iterates of U (which is directly related to both of these conjectures). We also introduce the "flipped 3x+1" function \widetilde U and prove an analogous result for its trajectories. In the final section, we pose some interesting questions about the iterates of U (and \widetilde U), prove a couple of simple results about the iterates of U and \widetilde U, introduce other related functions and propose yet more conjectures and questions about their iterates. It's our hope that the results, conjectures and questions presented here will be not only relevant to the 3n+1 conjecture itself, but also of interest in their own right.
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