## On the Stirling numbers of the first kind

I am teaching MAT 412/512 Introduction to Combinatorics this semester. Earlier in the semester, I introduced the Stirling numbers of the second kind, and I did not find a good opportunity to introduce the Stirling numbers of the first kind until I started introducing permutation groups and group actions to build up to the Pólya… Continue reading On the Stirling numbers of the first kind

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## On Gromov’s theorem on groups of polynomial growth

This article documents my presentation of Gromov’s theorem on groups of polynomial growth at the MIT combinatorics reading group. The presentation is based on Gromov’s 1981 paper, Groups of polynomial growth and expanding maps, Kleiner’s 2007 paper, A new proof of Gromov’s theorem on groups of polynomial growth, and Tao’s 2009 blog post, A finitary… Continue reading On Gromov’s theorem on groups of polynomial growth

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## On the basis exchange property

One of the students in my class, Undergraduate Seminar on Discrete Mathematics, asks if the exchange property of a matroid can be strengthened to the following, of which I was completely unaware. Strong basis exchange property. For every pair of bases and , there exists such that both and are still bases. As it turns… Continue reading On the basis exchange property

## Minimal Distance to Pi

Here is a problem from Week of Code 29 hosted by Hackerrank. Problem. Given two integers and (), find and print a common fraction such that and is minimal. If there are several fractions having minimal distance to , choose the one with the smallest denominator. Note that checking all possible denominators does not work… Continue reading Minimal Distance to Pi

## Shalom to Ning

I had never expected that Feb 19, 2017 would be the last day we say farewell to each. We used to talk about math puzzles, from blue-eyed islander puzzle to the hardest logic puzzle ever, every time on the bus from or to Shuk. We joked about the possibility that the apple cores we throw… Continue reading Shalom to Ning

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## First week in Israel

Here is a list of things I have experienced during the first week in Israel and some tips for those of you who plan to visit me in Haifa! Hainan airline has a direct flight from Beijing to Tel Aviv, and they have a resting area for transiting passengers in Beijing. If this is your… Continue reading First week in Israel

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## A Short Proof of the Nash-Williams' Partition Theorem

Notations. – the set of natural numbers; – the family of all subsets of of size ; – the family of all finite subsets of ; – the family of all infinite subsets of ; The infinite Ramsey theorem, in its simplest form, states that for every partition , there exists an infinite set such… Continue reading A Short Proof of the Nash-Williams' Partition Theorem

## Alternative to Beamer for Math Presentation

Although using blackboard and chalk is the best option for a math talk for various reasons, sometimes due to limit on the time, one has to make slides to save time on writing. The most common tools to create slides nowadays are LaTeX and Beamer. When I was preparing for my talk at Vancouver for… Continue reading Alternative to Beamer for Math Presentation

## 十一年

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## A Short Proof for Hausdorff Moment Problem

Hausdorff moment problem asks for necessary and sufficient conditions that a given sequence with be the sequence of moments of a random variable supported on , i.e., for all . In 1921, Hausdorff showed that is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the… Continue reading A Short Proof for Hausdorff Moment Problem