In this post I go over the basics of data compression with arithmetic coding and describe the Context Tree Weighting algorithm. At the end I discuss implementation and some experimental results. Information Theory Review First a quick review of information theory. Suppose that we receive some data \(x\) drawn from a discrete random variable \(X\), whose probability distribution is \(P\). Then the Shannon information content of \(x\) is defined as

In this post I show how writing primes as the sum of two squares is related to factoring Gaussian integers. I then describe an algorithm to compute the sum of two squares representation. Gaussian Integers I have a special penchant for Gaussian integers because they are the first ring I learned about (besides the regular integers of course). Mathematically, they are the set of complex numbers \(a+bi\) with \(a,b\) integers, and they are denoted as \(\mathbb{Z}[i]\).

In this post I summarize the content and proof of Liouville’s Theorem on Conformal Rigidity, which I learned in 2018 from Professor Alex Austin (now at RIT) in his class at UCLA. Conformal Maps A conformal transformation is one that preserves angles. In two dimensions, this is equivalent to being holomorphic and having a non-vanishing derivative. There are tons of these transformations (my personal favorites are Mobius transformations). In fact, the famous Riemann mapping Theorem asserts that any simply connected domain \( U \subset \mathbb{C} \) admits a bijective conformal map \(f: U \to \mathbb{D}\) to the unit disc.