Julian's musings

Strong induction and ordinary induction


One of my UKMT Mentoring scheme mentees was asking me about induction, and we were discussing how strong induction and ordinary induction are related to each other. In the end, I ended up writing this piece, which I’m sharing here for general interest.

Implicit differentiation I


I’ve been thinking about implicit differentiation with my colleagues recently. How do we teach it (at high school level), and what subtleties are involved? It started by trying to understand what we mean by the equation

\begin{equation} \frac{dy}{dx}=1\biggm/\frac{dx}{dy}. \label{eq:recip} \end{equation}

Some questions raised by this include:

(a) What does this equation mean?

(b) How can we explain this to students and also why it is true?

(c) Where would this result be useful to them (besides in artificial exam questions)?

In this post, I will offer some thoughts on (a) and (b), but I’m still fairly stuck on (c).

Comments on Ellenberg and Gijswijt's capset paper


I recently had the fun of reading Ellenberg and Gijswijt’s paper on the capset problem, where they bound the size of a subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression by $c^n$ with $c<q$.

The paper is beautifully written, and amazingly needs only relatively elementary undergraduate algebra. (It is generalised to the Galois field $\mathbb{F}_q$, but if we take $q$ to be prime, then even that is unnecessary to understand the argument.)

I was somewhat stuck on two small points at the start of the proof of Proposition 4, and thought I would share my realisation of the argument here for others’ benefit.