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

# Category: Mathematics

## 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

## 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.)

## Communicating Information through Randomness

This week Staren sent me a puzzle on WeChat. After we discovered a solution for the puzzle, I tried to backtrack the source of it. I found it appeared in the blog post Yet another prisoner puzzle by Oliver Nash. The author seemed to work at a quantitative trading company SIG at that time. Given… Continue reading Communicating Information through Randomness

## Crazy Telescoping

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation. I learnt this technique when I was doing maths olympiad. However until last year I learnt this buzz word ‘telescoping’ since I received my education in China and we call it ‘the method of… Continue reading Crazy Telescoping

## On Hardy–Littlewood maximal function of singular measure

In this exposition, we explore the behavior of the Hardy–Littlewood maximal function of measures that are singular with respect to Lebesgue measure. We are going to prove that for every positive Borel measure that is singular with respect to Lebesgue measure , for all , where is a torus and is the Hardy–Littilewood maximal function… Continue reading On Hardy–Littlewood maximal function of singular measure

## Alternating Fourier Coefficients

Suppose is a periodic function from to with period . Let be its Fourier coefficients, namely for all . Prove for all it is almost surely that function is in where is an infinite sequence of independent and identical random variables indexed by with equals either or with probability . I heard this problem from… Continue reading Alternating Fourier Coefficients

## Number of non-isomorphic graphs

This expository essay is to test my understanding of the techniques used in More Bricks – More Walls?, Thirty-three Miniatures by Jiří Matoušek’s. We shall prove the sequence is unimodal, i.e., it is first nondecreasing and then, from some point on, non-increasing, where is the number of non-isomorphic graphs with vertices and edges. In particular,… Continue reading Number of non-isomorphic graphs