Barcode restoration and division modulo a prime

Yesterday, I came across a puzzle in which we had to decode a barcode (Code 128) of which a piece had been obfuscated. The math involved in this is very elegant, so I decided to write it down, hoping that one day someone will benefit from it.

Repeated substitution and Chebyshev polynomials

Recently, I was experimenting with a process in which we start with the polynomial f0(x)=xf_0(x) = x and obtain fn(x)=fn1(1sx)f_n(x) = f_{n-1} \left(\frac{1}{s-x}\right) by substituting x1sxx \mapsto \frac{1}{s-x}, for some number sCs \in \mathbb C. By accident, I saw that both for s=1s=1 and s=1s=-1, f3=f0f_3 = f_0. Of course, for s=0s=0, f2=f0f_2 = f_0 as well. This made me wonder: for which values of ss do we have fn=f0f_n = f_0 for some n>0n>0?

Group theory 101.0

This is an introduction to group theory, covering the concepts of groups, subgroups and (homo)morphisms.

Hello, World!

I am setting up this blog to share interesting stuff I find on the internet, in my books or in my lectures.