Author Archives: leonardwong

Complex analysis – Problem solving strategies.

This post is written in spirit of Terrance Tao’s post on problem solving strategies in real analysis. Our emphasis will be on concrete techniques with plenty of examples (many are taken from past prelim questions in UW).

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Principle of subordination

First we recall the statement of Schwarz’s lemma, which is a basic result in complex analysis. Let be the unit disk in . Schwarz’s lemma. Let . If for all and , then , and for all . If or … Continue reading

Posted in Complex analysis | 1 Comment

Brownian motion and the classical Dirichlet problem 0: Introduction

In this sequence of posts I will present a probabilistic solution of the classical Dirichlet problem using Brownian motion. I will follow the structure of the book Green, Brown and Probability and Kai-Lai Chung with some little changes and somewhat … Continue reading

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Exercises in Real Analysis II

This post is a sequel to Exercises in Real Analysis. 4. Let be the triangle , and be the restriction of the planar Lebesgue measure on . Suppose that . Prove that . Solution. Assume on the contrary that . … Continue reading

Posted in Analysis, Functional analysis | 2 Comments

Exercises in real analysis

In this post I type up solutions of some interesting questions in the real analysis qualifying examinations. Most questions will be from UW. This post will be updated continuously. Of course, you are encouraged to think about the question before … Continue reading

Posted in Analysis, Functional analysis | 8 Comments

Martingale Theory III: Optional stopping theorem

This is a sequel to Martingale Theory II: Conditional Expectation. Our aim here is to prove the main theorems about discrete time martingales. More advanced probability texts (e.g. those on stochastic calculus) assume that these theorems are well known to … Continue reading

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Water puzzles

Here is a typical water puzzle: You have two cups. Their capacities are 5 units and 3 units respectively. You may get water, pour water way or to another cup, but there are no marks on the cups. If you … Continue reading

Posted in Discrete Mathematics | 2 Comments