Travelling nearly lightspeed

I'm about to finish reading Why does $E=mc^2$? by Brian Cox and Jeff Forshaw (You may buy it here ). I must say it is not a very good book if you're not already a good mathematician that knows nothing about physics, otherwise I think they just tried too hard in explaining the concepts in a very abstract way without using equations that feels utterly complicated and vague language.

Having said that, I find it a very interesting book. One of the most interesting fact that I got from it is that the $c$ in Einstein's most famous formula is not the speed of light from first principle, that's just a conclusion achieved from experimentation. The $c$ is actually the speed limit at which mass-less particles must travel under the special relativity assumptions, and it must be the same speed regardless of how or who is measuring it. Since apparently photons are mass-less we have to conclude that $c$, the universal speed limit, is the speed of light.

This brings me to my point, which is just a rant about how it feels weird when they break laws of physics in action films. I'm not complaining about they don't respect them in the films, I love action films and I am expecting to see some unreal stuff in there, that's basically what I am paying for. My point is that sometimes it feels weird. I am going to put a couple of examples before getting to my point.



Consider when Steve Rogers in training some boxing in Marvel's Avengers Assemble, while he's punching the bag he recalls his war stories with his team and losing his mind he ends throwing that one punch that sends the bag flying away. The scene is obviously not real, but it still feels ok. I like to thinks that because we know Captain Rogers is capable of doing inhuman feats he has the strength to send the bag flying away and the bag follows the expected intuitive parabolic trajectory. If the trajectory is not parabolic the feel of the scene would not be the same, just as happens with the Crouching Tiger Hidden Dragon film.



Again, this doesn't mean I'm complaining, or that the film is bad, I'm just saying that the changing the laws of physics creates in the audience a weird sensation. But what happens when we're not used to some things happening and still the film decides to fuck physics just because. This happened to me when I saw Star Trek: Into Darkness.



I believe the the Star Trek films are just awesome. However the doubt still haunts me, when the Vengeance fires on the Enterprise it makes a hole on the ship. The next thing that happens is that a lot of stuff escapes from that hole, that's understandable because of the change in pressure but that doesn't mean that jumping from a ship travelling nearly at the speed of light is going to stop you. If one does jump from a ship in space, since friction is basically non existent, then one should have the same speed with respect to the ship. Especially if one's jumping nearly lightspeed because one can't jump faster.

The point is that I felt weird when people flew out and back of the ship, I think they should have flown just out perpendicular to the tangential plane of the ship at the hole. Jumping from a spaceship is essentially different than jumping off a train. At least that's what I feel, however I have never flown at lightspeed or close to that, anyways... I will continue enjoying action films with or without physics misconceptions, and the next one comes this week!

It's alive!

Today I finally finish proving the result we wanted for my PhD thesis. It has been an epic journey. I wanted to find a rate function for a large deviation principle of the density in the first finite number of sites of a semi infinite totally asymmetric simple exclusion process.

I think that unless one is familiarised with the subject my topic is full of technical vocabulary. So I'll be brief on the explanation of what these all mean.

On one hand there's the large deviation principle. Say you have a sequence of random objects, say tossing coins $X_n$ where they have the value 1 if it's heads and 0 if it's tails, and consider the average of the first $n$ tosses. \[S_n=\frac{1}{n}\sum_{k=1}^nX_k\]

By the law of large numbers we know that this average converges to the actual probability of the toss showing heads, say $p$. Mathematically, for all $\varepsilon>0$ we have that
\[\lim_{n\rightarrow\infty}\mathbb P[|S_n-p|\leq\varepsilon]=1\]

Say we don't want to find the probability of $S_n$ being close to $p$ but any other number $x$, then that limit would be zero:
\[\lim_{n\rightarrow\infty}\mathbb P[|S_n-x|\leq\varepsilon]=0\]

What if we don't take the limit? What if we just take a big natural number $n$? The probability would be close to 0, but positive. Maybe, we can write it like this:
\[\mathbb P[|S_n-x|\leq\varepsilon]\approx\exp\{-nI(x)\}\]

Theory has been developed to explicitly find the rate function. The coin tossing example is the simplest one and can actually be done by hand. However, in general this is not an easy task. The rate function allows you to now the exponential decay rate of the probabilities of rare events, so it's relatively easy to think of applications where this might come in handy.

I have some slides on this here. I think I'll explain what the TASEP is in another post, but the point here is that I proved an LDP for that and I just need to write it down to have a thesis... now I need to finish and get a new job.

Applied maths

Last week I was in Chicago for the annual meeting of the Society for Industrial and Applied Mathematics (or simply SIAM for short). I found very interesting the fact that there are actually mathematicians using real data to work on their projects and not all of them have to do with statistics. In fact, it seems that applied mathematicians have a generalised phobia for stochastic processes and statistics... which is precisely what brings me to write this.

What is applied maths? I have come up with the idea that it depends who you ask. Regardless of the speed at which one travels one always measure the speed of light at the same value... well, applied maths is not a conserved constant in the universe, it pretty much depends on where one studied and what one does for a living.

I have previously said that I studied my undergrad at ITAM and although the website portrays a very broad sense of the subject, in practise it focuses on finance and economics. However the degree is quite flexible and one can certainly focus in computer science or statistics as I did.

I can't say anything bad against ITAM, it is truly a very good university for applied maths. Having said that, I was never prepared for my masters degree at the University of Bath. Contrary to what happens in Mexico where studying maths is a weird thing, studying it in the UK is a more common thing. There aren't many universities in Mexico where one can study applied Maths while in the UK there's a massive market for that. They all agree in one thing: applied maths is solving PDEs, analytically and numerically. In Mexico, the PDE course was only an optional module. In the UK it is the expected thing to know when you're an applied mathematician.

Of course when I mean "knowing to solve PDEs analytically" I mean the easy ones: the heat and wave equations for example. 

The problem arises when somebody outside the maths department asks what applied maths is. Because even if one researches on optimising numerical methods for solving some specific PDEs, to the eyes of the laymen people the solving of the equation is pointless if it doesn't have a higher purpose. What sort of higher purpose? As I like to put it... how does one make money out of that? If one can't answer this question quickly and clearly, whatever one is doing, it is not applied enough.

I like thinking I am an applied mathematician, but being a probabilist doing analysis definitely makes me a weird object within the applied mathematicians at the SIAM meeting. Nevertheless it was quite fun and the research is fascinating... by the way, I had the chance to listen and meet Jorge Nocedal (I didn't see that one coming).