Benford’s law is an observation about the leading digits in sets of numbers that span a wide enough range and which arise “naturally”. Its most well-known claim is that the first digit tends to be 1 about 30% of the time instead of the 11.1% (1 out of 9) that one would expect. Benford’s law … Read More “What is Benford’s Law?” »
I recently upgraded from an old phone with a 3.5mm headphone jack to one that’s too new and fancy for such antiquated features. This created a nuisance because I had only wired 3.5mm headphones available, so instead of using my phone as an MP3 player I resorted to a dedicated MP3 player from the 2010s … Read More “Randomness on Repeat” »
Given a real number $a$, the exponential $\exp(ta)$ is the solution to the initial value problem: $$\displaystyle \begin{cases} \frac{dy}{dt} = ay & \\ y(0) = 1& \end{cases} $$ This definition does not depend very much on the fact that $a$ is a real number. $a$ could perfectly well be a complex number and nothing in … Read More “Left and Right Exponentials” »
In a previous post I showed that for any closed region $\Omega$, the integral of the outward-pointing surface-normal unit vector $\hat{n}$ over its boundary is zero: $$\displaystyle \int_{\partial\Omega} \hat{n}\,dS = \int_\Omega \nabla 1 \,dx = 0 $$ This post will explore a counterintuitive consequence of this fact. Suppose instead that we want the same integral, … Read More “Integral of the Unit Normal Over a Surface” »
The Sherman-Morrison-Woodbury formula expresses the inverse of an “update” to a matrix $A$ in terms of $A^{-1}$ and the factors of the update: $$\displaystyle (A + UCV)^{-1} = A^{-1} – A^{-1} U\left( C^{-1} + VA^{-1}U \right)^{-1}VA^{-1} $$ where $A$ and $C$ are invertible square matrices and $U$ and $V$ have dimensions such that $UCV$ has … Read More “Sherman-Morrison-Woodbury for Determinants” »
Suppose we have a region $\Omega \subset \mathbb{R}^n$ that is a polytope, i.e. a non-empty intersection of half-spaces. Visually this would be a polygon in 2D or a polyhedron in 3D – “polytope” is a single term to describe such objects regardless of the number of dimensions. The important feature is that the boundary of … Read More “Polytope Identities from Vector Calculus” »
Any motion of a rigid object is equivalent to a rotation around some axis followed by a translation along that axis. We can therefore represent any motion mathematically using some object that includes information for a single rotation and a single translation. An extremely convenient object for this purpose is the dual quaternion. Part of … Read More “The Exponential and Logarithm for Dual Quaternions” »
Introduction In dynamics, one frequently needs a derivative of a vector in an inertial frame of reference in order to apply Newton’s laws when it is much more convenient to express that vector in a different frame. The Basic Kinematic Equation (also known by many other, less-informative names) relates the derivatives in two frames to … Read More “What is the Basic Kinematic Equation?” »
Every whole number (except 1) can be factorized in one and only one way into a product of prime numbers. For example: $$\displaystyle \begin{aligned} 21 &= 3\times 7 \\ 34 &= 2\times 17 \\ 35 &= 5\times 7 \end{aligned} $$ By looking at the factorizations one can easily tell when two numbers share a common … Read More “Numbers With Nothing In Common” »
You know about electricity, and you know about magnetism, and you’ve probably heard the word “electromagnetism” somewhere before to refer to electricity and magnetism combined in some sense. But why do these two seemingly different phenomena get lumped together into one word, whereas there’s no such thing (yet?) as “electrogravity”? If you know some more … Read More “Magnetism from Relativity” »