## Category: Dynamical systems

### “Guided” and “quantised” dynamical systems

Evegenios Kakariadis and I have recently posted our paper “On operator algebras associated with monomial ideals in noncommuting variables” on the arxiv. The subject of the paper is several operator algebras (at the outset, there are seven algebras, but later we prove that some are isomorphic to others) that one can associate with each monomial ideal, in such a way that these algebras encode various aspects of the relations defining the ideal.

I refer you to the abstract and intro of that paper for more information about we do there. In this post I would like to discuss at some length an issue that came up writing the paper, and the paper itself was not an appropriate place to have this discussion.

### Where have all the functional equations gone (part II)

I’ll start off exactly where I stopped in the previous post: I will tell you my solution to the problem my PDEs lecturer (and later master’s thesis advisor) Paneah gave us:

Problem: Find all continuously differentiable solutions to the following functional equation:

(FE) $f(t) = f\left(\frac{t+1}{2} \right) + f \left(\frac{t-1}{2} \right) \,\, , \,\, t \in [-1,1] .$

Before writing a solution, let me say that I think it is a fun exercise for undergraduate students, and only calculus is required for solving it, so if you want to try it now is your chance.