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 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
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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). Advertisements
Posted in Complex analysis
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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
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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 KaiLai Chung with some little changes and somewhat … Continue reading
Posted in Uncategorized
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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
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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
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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
Posted in Probability
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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
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