Hedonic Calculus

 When I meet other PhD students and the unavoidable question (analogous to Durham's 'What college are you in?'), "What area of research are you in?" is asked, my answer varies between 'Analysis', 'Partial differential equations', 'Laplacian eigenvalues' and 'Spectral theory'. For now I'm sticking with the latter. But to someone who has never studied maths (and to many who have), it's not at all clear from the name what spectral theory entails. 

 This post has absolutely nothing to do with maths but I wasn't really sure where else to put this. I want to share my experience of flying home with a bike after bikepacking as there doesn't seem to be a single comprehensive source of information on this and I relied on anecdotes and Reddit threads for my information. This isn't meant to be a complete guide, just my experience. I booked the flight 2 days in advance once I'd arrived in Prague which alone cost £70. I paid for the largest luggage allowance which was 32kg and cost a further £80. The day before the flight I visited a bike shop to ask if they had a cardboard box for bikes. They didn't but directed me to another bike shop which did. I asked them to hold the box until the next day.  I disassembled the bike outside the bike shop: removed the pedals, seat post, wheels, pannier rack and handlebars. The bike shop lent me a wrench for the pedals. I packed the empty pannier bags into the box and put the rest of ...

I plan to write a few posts demonstrating the ways in which probability is misunderstood, the first of which highlights the consequences of probabilities being misrepresented.

 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.