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## A interesting proof on there exists irrational numbers a and b such that a^b is rational

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.

## Proof without words

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

## Stein’s Functional Analysis photocopy released

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.

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