Mostly a mathematician.
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.
NSF ACCESS is a fantastic high-performance computing program. If you’re affiliated with a university or other institution, you can upload a CV, write a paragraph describing your project, and get approved for several thousand CPU-hours within just a few days. If you don’t have existing grants it’s totally free. It even works if you’re a grad student, although then you’ll need your advisor to provide a form letter saying that you’re legit. But they even have suggested language for that.
When working at scale, it is often slow or impossible to form full-sized matrices. Instead, we make use of black-box functions to calculate matrix-vector products \(Ab\). There is thus a need for solvers which accept these functions instead of requiring matrix representations of our linear maps – and, in particular, for conjugate gradient solvers for the symmetric positive definite case.
Define \[\langle p, q \rangle = \int_{-1}^1 p(x)q(x)\,dx\] to be an inner product on \( \mathscr{P}_n(\mathbb{R}) \), the space of polynomials of degree at most \(n\) with real coefficients. Note that \(\varphi(p)=p(0)\) is a linear functional on \( \mathscr{P}_n(\mathbb{R}) \). Therefore, by the Riesz representation theorem, there must exist a unique polynomial \(q_n(x)\) such that \[\int_{-1}^1 p(x)q(x)\,dx = p(0)\] for all \(p\) with degree at most \(n\).