Paul Michael Kielstra

Any function \(f:\mathbb{R}\mapsto\mathbb{R}\) can be written as the sum of an even and an odd function: \(f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}.\) Similarly, any square matrix can be written as the sum of a symmetric and an antisymmetric matrix: \(A = \frac{A + A^t}{2} + \frac{A - A^t}{2}.\) A student of mine recently asked me about this similarity, and, while I described it in lecture as a “standard mathematical trick,” I thought I should sit down and work out the pattern.

The QR algorithm for finding eigenvalues is a deceptively simple piece of applied mathematics. Start with a square matrix \(\require{mathtools} A \coloneqq A_0\). Then, for the matrix \(A_k\), carry out a QR factorization \(A_k = Q_k R_k\). Set \(A_{k+1} = R_k Q_k\) and iterate to convergence. Magically, the result will be an upper-triangular matrix with the same eigenvalues as \(A\), and you can read the eigenvalues off the diagonal.

In Luke 4:1-13, we read:

Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a periodic function with period \(2\). Then, for parameters \(a\) and \(b\), we define the Weierstrass sum to be the function \[W_{a,b}[f](x)=\sum_{n=0}^\infty a^n f(b^n x).\] In order for the sum to converge, we assume that \(0 < a < 1\); in order for the resulting function to have a fractal shape, we take \(b\) to be a positive odd integer.

Continuing what I suppose is by now a series analyzing Disney musicals, we turn to the question of Gaston. He eats, we are told, five dozen eggs every morning, which he claims makes him roughly the size of a barge. Now, it’s pretty well-established that we shouldn’t trust everything Gaston says. But is he right about this?

This is not legal advice. It consists largely of my notes from a conversation with the Berkeley Attorney for Students, who also wasn’t offering legal advice.

My body is a machine that converts oxygen into carbon dioxide. My lifestyle is a machine that converts oil and natural gas into even more of it. This is not good. While I try to reduce my carbon footprint as much as possible, there’s always something in me that reminds me that I am, at my baseline, a net loss to the planet. I could live in a cave, eating nuts and berries, and even then I would still be breathing – and I’m not going to live in a cave. I’m scared it would be cold. More importantly, I’m a pragmatist, and ideologies that tell people to live in caves tend to do poorly at the ballot box. I need another option.

When I start a new research project, I usually hardcode a random seed. This enables repeatable experiments. In order to demonstrate that I didn’t somehow “hack” this part of the project, choosing a seed that gives abnormally good results, I normally pick a word and spell it out in hexadecimal, thus:

QR codes do not expire. QR codes do not have a limit on total scans. QR codes do not allow their creators to track you when you use them, or monetize that process.

When I have computer problems, as I often do, my usual response is somewhere on the spectrum from frustration to anger. This usually works pretty well as a motivator. A few months ago, I called a friend and told him that I was chasing down a subtle bug and it was difficult to keep up my motivation. “Michael,” he immediately told me, “You have to name it.” So I did. I picked a name more or less at random, and immediately I was emotionally ready to go again. Fred was not going to keep me down. I was going to get him if it were the last thing I did. I had a correct code by the end of the next day.