## 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

## 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

## 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

## An Upper Bound on Stirling Number of the Second Kind

We shall show an upper bound on the Stirling number of the second kind, a byproduct of a homework exercise of Probabilistic Combinatorics offered by Prof. Tom Bohman. Definition. A Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of objects into non-empty subsets and… Continue reading An Upper Bound on Stirling Number of the Second Kind

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## A Probabilistic Proof of Isoperimetric Inequality

This note is based on Nicolas Garcia Trillos’ talk, Some Problems and Techniques in Geometric Probability, at Carnegie Mellon University on January 29, 2015. In particular, we demonstrate a probabilistic proof of the isoperimetric inequality. The proof can also be found in Integral Geometry and Geometric Probability by Luis A. Santaló. Theorem. For a convex set… Continue reading A Probabilistic Proof of Isoperimetric Inequality

## On Extension of Rationals

This note is based on my talk An Expedition to the World of -adic Numbers at Carnegie Mellon University on January 15, 2014. Construction from Cauchy Sequences A standard approach to construct the real numbers from the rational numbers is a procedure called completion, which forces all Cauchy sequences in a metric space by adding… Continue reading On Extension of Rationals