One of the most wonderful aspects of math is how deep we can get by studying seemingly trivial questions. Consider the roots of unity, where "unity" simply means the number 1. Basics: roots of unity for real numbers Studying the roots of unity usually begins in middle school: what is the square root...

Monte Carlo simulations (or methods) is the technique of applying randomness and the Law of large numbers to the solution of various scientific and engineering problems. One of its first documented uses was by Stanislaw Ulam and John von Neumann for nuclear weapon simulations after WWII [1]. In this...

"Wuthering Heights" by Emily Brontë - good writing, but the protagonists are quite something. There's barely a likable character in the whole story - they are all either crazy, evil, stupid or a combination of these. It's also remarkable to consider how small the world of some people in those times ...

Like many programmers, I find Alexis King's Parse, don't validate article fascinating, because it gives a name to an idiom that seems familiar and important - one I've observed and used in the past without naming it explicitly. This post is a review of the "Parse, don't validate" pattern applied to ...

The following are my notes on discrete-time Fourier series (DTFS), as well as the discrete-time Fourier transform (DTFT). These topics serve as an important theoretical underpinning to the digital processing of signals by computers using the DFT (which will be covered in a future post). For discrete...

The other day, I found myself wondering how big 52! (52 factorial) is, and that led me to ponder how these could be estimated without a calculator or a computer. It turns out there’s some fairly interesting math behind being able to estimate the size (number of digits) of a factorial reasonably accu...

This is part 8 in a series of posts on writing concurrent network servers. In this part, we'll switch to Go and see how it tackles the challenges described earlier in the series. All posts in the series: Part 1 - Introduction Part 2 - Threads Part 3 - Event-driven Part 4 - libuv Part 5 - Redis case ...

This is part 7 in a series of posts on writing concurrent network servers. In this part, we discuss how the challenges described in earlier parts are tackled in the Rust programming language. All posts in the series: Part 1 - Introduction Part 2 - Threads Part 3 - Event-driven Part 4 - libuv Part 5 ...

In Physics simulations or game engines it’s sometimes useful to determine the speed with which two objects are approaching each other. This post will discuss the concept of closing speed, which is the normal component of the relative velocity of two objects. Relative velocity and its components Supp...

The Fourier series is a great tool for analyzing periodic functions. But what about functions that don’t repeat? We’ve seen that we can compute Fourier series for a non-periodic function defined on a finite interval, as long as we don’t care about its behavior beyond that interval. Let’s extend this...

The goal of this post is to answer a simple question: why are the following two definitions of the vector dot product in Euclidean space [1] equivalent for vectors \vec{a} and \vec{b}: Component definition: \vec{a}\cdot\vec{b}=\sum_{i=1}^{n}a_i b_i Geometric definition: \vec{a}\cdot\vec{b}=|\vec{a}|...

"The Nuremberg Trial" by John Tusa and Ann Tusa - a detailed, meticulously researched account of the Nuremberg Trials. There's not a whole lot of side questing in this book - it's all focused on the trials themselves. Interesting read overall, though somewhat dry and academic. "Things Become Other T...

Recently I came upon Pluggy, a Python library for developing plugin systems. It was originally developed as part of the pytest project - known for its rich plugin ecosystem - and later extracted into a standalone library. You're supposed to reach out for Pluggy if you want to add a plugin system to ...

A few months ago I wrote about using LLM agents to help restructuring one of my Python projects. It's worth beginning by saying that the rewrite has been successful by all reasonable measures; I've been able to continue maintaining that project since then without an issue. In this post, I want to di...

The trigonometric Fourier series is a beautiful mathematical theory that shows how to decompose a periodic function into an infinite sum of sinusoids. These are my notes on the subject, with some examples and the connection to linear algebra in Hilbert space. Coefficients of Fourier series Let’s ass...

This is a brief and simple [1] explanation of how to adjust the standard sinusoid sin(x) to change its amplitude, frequency and phase shift. More precisely, given the general function: \[s(x)=A\cdot sin(w\cdot x+\theta)\] We’ll see how adjusting the parameters , and affect the shape of s(x). Each se...

This week the article Wasm is not quite a stack machine has been making the rounds and has caught my eye. The post claims that WASM is not a pure stack machine because it has locals and is missing some stack manipulation operations like dup and swap. While I don't necessarily disagree, IMHO it's a b...

When I was working on the WASM backend for my Scheme compiler, I ran into several tricky situations with debugging generated WASM code. It turned out that Chrome has a very capable WASM debugger in its DevTools, so in this brief post I want to share how it can be used. The setup and harness I'll be ...

I'm happy to announce the general availability of watgo - the WebAssembly Toolkit for Go. This project is similar to wabt (C++) or wasm-tools (Rust), but in pure, zero-dependency Go. watgo comes with a CLI and a Go API to parse WAT (WebAssembly Text), validate it, and encode it into WASM binaries; i...

"Intellectuals and Society" by Thomas Sowell - a collection of essays in which Sowell criticizes "intellectuals", by which he mostly means left-leaning thinkers and opinions. Interesting, though certainly very biased. This book is from 2009 and focuses mostly on early and mid 20th century; yes, hist...

Polynomial interpolation is a method of finding a polynomial function that fits a given set of data perfectly. More concretely, suppose we have a set of n+1 distinct points [1]: \[(x_0,y_0), (x_1, y_1), (x_2, y_2)\cdots(x_n, y_n)\] And we want to find the polynomial coefficients {a_0\cdots a_n} such...

We’ll be working with the set P_n(\mathbb{R}), real polynomials of degree \leq n. Such polynomials can be expressed using n+1 scalar coefficients a_i as follows: \[p(x)=a_0+a_1 x + a_2 x^2 + \cdots + a_n x^n\] Vector space The set P_n(\mathbb{R}), along with addition of polynomials and scalar multip...

pycparser is my most widely used open source project (with ~20M daily downloads from PyPI [1]). It's a pure-Python parser for the C programming language, producing ASTs inspired by Python's own. Until very recently, it's been using PLY: Python Lex-Yacc for the core parsing. In this post, I'll descri...

One of my oldest open-source projects - Bob - has celebrated 15 a couple of months ago. Bob is a suite of implementations of the Scheme programming language in Python, including an interpreter, a compiler and a VM. Back then I was doing some hacking on CPython internals and was very curious about ho...

"The Origins of Political Order: From Prehuman Times to the French Revolution" by Francis Fukuyama - while reading this book it occurred to me that domains of study like political sciense must be incredibly difficult and frustrating. Imagine trying to match a model onto a set of data; the model has ...