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

Poisson Summation Formula and Basel Problem

This note is based on Professor Noam Elkies’ talk at Carnegie Mellon University on December 2, 2014. A classical mathematical analysis problem, also known as the Basel problem, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735, asks the precise sum in closed form of the infinite series Here we… Continue reading Poisson Summation Formula and Basel Problem

A Note on Thue's Method

This note is based on my talk Introduction to Diophantine Approximation at Carnegie Mellon University on November 4, 2014. Due to limit of time, details of Thue’s method were not fully presented in the talk. The note is supposed to serve as a complement to that talk. Diophantine Approximation Diophantine approximation deals with the approximation… Continue reading A Note on Thue's Method

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美国数学奥林匹克观察（下）

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美国数学奥林匹克观察（上）

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Categorized as 芳华

How I wrote my first independent paper

This is what happened behind the scene. In the first draft, I was very pedantic. The resulted draft is notation-heavy and has no introductory section. It was then totally discarded. However, the structure of the proof was reused in the later drafts. My advisor suggested a useful dictionary by Trzeciak for mathematicians like me who… Continue reading How I wrote my first independent paper

Fun with Hex (Cont.)

In the previous post Fun with Hex, I asserted without proof that the Hex Game will never result a draw. In Jiri Matousek’s An Invitation to Discrete Mathematics, I found an elegant proof for the assertion, which requires only a little bit of elementary graph theory. The idea of the proof can also prove Sperner’s… Continue reading Fun with Hex (Cont.)