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 post I want to provide examples of some simple uses of Monte Carlo simulations. We'll start with the classical example of calculating the value of \pi by throwing darts. Estimating pi Suppose we take a square board and inscribe a quarter of a circle into it. We then proceed to throw darts at the…
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-time signals, we use …
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 …
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 Suppose we …
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 …
Before I read The Man from the Future by Ananyo Bhattacharya, I only knew about John von Neumann in two contexts: that computers use the von Neumann architecture, and that he appeared in a story about a mathematical problem I … Continue reading →
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