1
Making any integer with four 2s
There's a cute math puzzle that can be interesting to folks on very different levels:
Eli Bendersky
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.
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2
Bloom Filters
The original motivation for the creation of Bloom filters is efficient set membership, using a probabilistic approach to significantly reduce the time and space required to reject items that are not members in a certain set.
3
My favorite prime number generator
Many years ago I've re-posted a Stack Overflow answer with Python code for a terse prime sieve function that generates a potentially infinite sequence of prime numbers ("potentially" because it will run out of memory eventually). Since then, I've used this code many times - mostly because it's short and clear. In this…
4
Monte-Carlo Simulations
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] .
5
Hilbert space: Treating functions as vectors
The tools of linear algebra are extremely useful when working in Euclidean space (e.g. \mathbb{R}^3 ). Wouldn’t it be great if we could apply these tools to additional mathematical constructs, such as functions and sequences? Hilbert space allows us to do exactly this - apply linear algebra to functions.
6
Understanding gradient descent
Gradient descent is a standard tool for optimizing complex functions iteratively within a computer program. Its goal is: given some arbitrary function, find a minumum. For some small subset of functions - those that are convex - there's just a single minumum which also happens to be global. For most realistic…
7
Demystifying Tupper's Formula
A book I was recently reading mentioned a mathematical curiosity I haven't seen before - Tupper's self-referential formula . There are some resources about it online, but this post is my attempt to explain how it works - along with an interactive implementation you can try in the browser.
8
RSA: Theory and Implementation
RSA has been a staple of public key cryptography for over 40 years, and is still being used today for some tasks in the newest TLS 1.3 standard. This post describes the theory behind RSA - the math that makes it work, as well as some practical considerations; it also presents a complete implementation of RSA key…
9
Convolutions, Polynomials and Flipped Kernels
This is a post about multiplying polynomials, convolution sums and the connection between them.