Hedonic Calculus

 I'm of the age where my first phone housed only one game, snake. Snake is played in a square world: if you try to escape out of one edge you are teleported to the opposite edge. It seems that modern iterations of the game have hardened the boundaries of the square, subverting my entire post but cast your mind back to when snake looked like this:  In some sense we can think about one edge being the same edge  as the opposite edge. In 2 dimensions the only way to visualise this is with the snake crossing one edge to come back out the other side but thankfully we have access to 3 dimensions. I can glue one pair of edges together to form a cylinder and then glue the circular ends of the cylinder together to form a doughnut shape, called a torus . And so  this game which allows you to play snake on a torus and claims to be a fun spin off is in fact identical to the original game. Importantly, the square was 2 dimensional and so is the torus. When I talk about the torus ...

 This post is for Mum who asked for an r/explainlikeim5 style post on what I study. My PhD is very closely related to my Masters project (which will get its own post soon) but I want to talk about the broader field of which this is a part. There are quantities which depend on one thing: the easiest example is a population depending on time. A differential equation  is a relationship between something's growth rate and how its size. In a population that has some reproduction rate, the sentence 'the larger the population, the faster it grows' encodes a differential equation.  Another example is radioactive decay: if I have a block of uranium then after some amount of time, half of it will have decayed. Then after another half-life , another half will decay. This is the same as the population example, but now 'the larger the size, the faster it decays'. These are the two simplest differential equations we can describe, both written as y' = ky. There are myriad othe...

If I let a ball roll down an inclined plane then it'll reach the bottom with some velocity. If I kick it back up the plane with the same velocity it'll reach its original position. But what if it reaches the bottom of the plane and rolls along a flat surface subject to friction. Then eventually it'll come to rest but there's no way to tell where the ball started. In some sense we've lost information. [[If we knew the coefficient of friction and the relevant angles and distances we could recover the initial position but that's not the point.]] This is a matter of time reversibility . In an ideal system (no friction, energy conserved) I can 'run time backwards' and get back to my starting point. What this means in practice is that my governing equations are invariant under mapping t to -t. Looking at Newton's 2nd law, if our force is time independent (like gravity) then acceleration picks up 2 minus signs from the chain rule when I swap t to -t. On the...

Bit more of a technical one today, assuming some background in multivariable calculus. Our starting point is going to be Maxwell's equations for a magnetic field B and an electric field E in the absence of any charges: We've got these constants mu_0 and epsilon_0 baked into the equations and these are known as vacuum permeability  and vacuum permittivity  respectively. I'm not entirely sure what that means but the important point is that these are features of the universe and everyone can agree on their value. (Oh look, a gun hanging on the wall).

Picking up where the last one left off, still dedicated to BK

 This post is dedicated to BK. I've procrastinated this one because it's a huge topic and one quite important to me, learned from good teachers.

This post is dedicated to RA and this meme he sent me. This post is a little longer than usual, but hopefully it seems small in proportion to its topic.   In The Fault In Our Stars, Hazel-Grace boldy proclaims that "some infinities are bigger than others" citing the fact that there are more numbers between 0 and 10 than between 0 and 1. If I were marking her work then she would get a drawing of a donut because, although her conclusion is correct, she used woefully misinformed reasoning to reach it. Let's dig in.

 In the  Ant on a Rubber Rope post  I mentioned a discrete solution that I found very pleasing and I plan to present it here. I would normally recoil from anything that smells of 'discrete maths' but I think turns out to be far more enlightening as the gut feeling that the ant shouldn't reach the end of the rope is the same feeling that the harmonic series should converge.

I actually wrote this over a year ago but this seems like the right platform so please enjoy an alternative motivation of the extension from the real numbers to the complex numbers.