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I found a question on mathoverflow, it asks “is there  exists irrational numbers a and b such that a^b is rational”.

Obviously, if you let a=e, b=\log 3, then a^b=e^{\log 3}=3 is rational.

But someone gave a very interesting proof.


You don’t have to know whether \sqrt{2}^{\sqrt{2}} is rational or not, you can just get the answer. That is a clever trick.



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I found a quite interesting site http://memtropy.com/proofs-without-words/, if you can understand what it says, click it and see more.


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After a long time waiting, Elias Stein’s Functional Analysis released the photocopy version in China this week. Luckily I registered the pre-selling notification at China-pub,  so I paid for it as soon as I received the SMS.

Here are more infos of this book:

Functional Analysis:Introduction to Further Topics in Analysis (Princeton Lectures In Analysis) by Elias M.Stein  and Rami Shakarchi.

I paste the intro at china-pub here. http://product.china-pub.com/3020533

《泛函分析》是Stein的“Princeton Lectures In Analysis”四卷集中的最后一卷,这一个系列的教科书旨在全面剖析分析的核心,从泛函分析的基础开始,讲述巴纳赫空间、lp空间和分布理论,强调了 它们在调和分析中的核心地位。接着应用Baire范畴定理详解了一些重点,包括Besicovitch集合的存在性;本书的第二部分引导读者进入概率论和 布朗运动等分析的其他核心话题,以Dirichlet问题作为结束;最后几章讲述了多复变量和傅里叶分析中的振荡积分,并简述了在非线性色散方程中的计数 网格点问题中的应用。作者通篇紧紧围绕这个理论诸领域的核心思想,使得本课题的各个有机部分更加紧凑,层次分明,清晰易懂。

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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao