## Borel Borel Borel Distribution

Posted on 29 Sep 2017

### Acknowledgements

This blog could not have been written if it weren’t for jolly Francken members, beer and Wikipedia. Bedrankt!

## Introduction

Many a mathematician, physicist or even astronomer may have heard the name: Borel (be sure to get the emphasis right). On a Monday afternoon, some Franckenmembers decided they wanted to find out who this Borel was, mostly because his name sounded so appealing to us. He was known for the Heine-Borel theorem in real analysis (reëel) and the Borel-Weil- Bott theorem in group theory. We were surprised to find that we were not dealing with a single Borel! The Heine-Borel theorem and the Borel-Weil- Bott theorem do not descend from the same mathematician! The Borels who did this are Émile Borel and Armand Borel, respectively. In further research, we found that both of them were very ingenious and intelligent men, who contributed a lot to various aspects of mathematics and physics. I was bored enough and thought this subject has just the right level of relevance to write a blog about. This blog is dedicated to these Borels and all the surprisingly appropriate things we can do with their work and their surname. Hoogh!

## Émile Borel

### Life and achievements

If we have to celebrate Borels in some order, we might as well stick to chronological order. Émile Borel (1871-1956) was born in Saint-Affrique, Aveyron in France. At age 18, Borel won the concours générale, a prestigious French academic competition. He then went on to study mathematics, resulting in the thesis “On some points in the theory of functions”, a very specific title. Borel subsequently managed to publish 22 papers in his four years as a lecturer at the university of Lille. Borel was an extraordinarily clever man. When he was thirty, he married a 17 year old girl. Furthermore, you may know him from a number of results. He most notably proved the Heine-Borel theorem, together with Eduard Heine, which is elementary in real analysis. They proved (although we are convinced that Borel did the real work) that a subset of some Cartesian power of the set of real numbers $(\mathbb{R}^n)$ is compact if and only if it is closed and bounded. The RUG docent Arthemy Kiselev had a quite ingenious twist on this theorem. Roughly in his words: “You might say to a girl: ‘you are very compact’, and she would reply: ‘Is it because I am petite?’ Then you could say: ‘No. It is because you are closed in character and bounded in intellect.’” Gerjan Wielink turned this analogy around: “You might say to a girl: ‘you are definitely not compact’, and she would reply: ‘Is it because I am open in character or unbounded in intellect?’ Then you could say: ‘No. You’re fat.’” And then some say mathematicians don’t have a sense of humour! In case physicists stopped reading: Borel was a physicist! He originally laid the connection between hyperbolic geometry and special relativity. On top of everything, Borel had a political career. Not only was he responsible for PIS (Paris Institute of Statistics), he has also been the French minister of marine. He was a proud member of the French resistance during the Second World War. Sadly, Borel passed away, just more than 61 years ago, at age 85, in Paris. May he rest in fertile ground, and may we ever generate inspiration from his talent, wisdom and surname.

##### The Infinite Freshman Theorem

Borel was also a prominent contributor to measure and integration theory, which relates strongly to stadtistics. He devised the Borel algebra (the collection of all open sets contained in some interval), an essential concept in measure theory, that induces the Borel measure. In the probability direction, the Infinite Monkey Theorem is also a Borel work. This theorem says that if some freshman would randomly color pixels in Microsoft Paint for an infinite amount of time, it will almost surely produce the entire Francken Almanak an arbitrary number of times. ‘Almost surely’ sounds rather vague; it means that the probability of it happening is $1$. However, it is not impossible that the Francken Almanak is not produced. Statistics can behave counterintuitively when infinity is involved. Somewhat similarly, it is not entirely safe to say that your phone number will occur in the decimal expansion of $\pi$, but it almost surely will (my phone number, for instance, does not appear in the first 200 million decimal digits of $\pi$).

### The Bor(r)el distribution

Borel did even more than the results described above: he was the first to publish on game theory, and (co-)produced Borel’s lemma, Borel’s law of large numbers (Francken members are familiar with an analogous borrel law), the Borel-Kolmogorov paradox, the Borel-Cantelli lemma, the Borel- Cadthéodory theorem, the Borel summation and, last but not least, the Borel distribution. The Borel distribution, given by $$P_\mu(n)=\frac{e^{-\mu n}(\mu n)^{n-1}}{n!},$$ is shown below, for $\mu$ varying from $0$ to $1$. I made it in MATLAB.

This Borel distribution is strikingly similar to another distribution: the well-known borrel distribution corresponding to the Franckenroom, which can be found on page 146 of our latest lustrumbook . Note that $P(X=n)$ means you drink at least $n$ : if you drink $2$ beers, this also counts as drinking $1$ beer. That explains the strict decreasing of the curve. Sjoerd Meesters found an explanation for the shape of this curve, but it also follows straightforwardly from the Borel distribution. Concidence? Not a real option. We may conclude that our drinking problem is well- conditioned: it behaves sufficiently in accordance with the Borel distribution. How very netses of us. A question one might ask is what the parameter $\mu$ means. The answer to this question is that $\mu$ is a measure for how uniform our drinking is. Suppose we have two members who drink the same number of beers per year (The Borel distribution has been normalized, such that the subject drinks one beer per day on average), and suppose one of them has a smaller $\mu$-value. Then that person usually drinks a few beers, and does so on many distinct days. An extreme case is $\mu = 0$, implying that the member drinks precisely one beer every day. The other member does not drink on many days, but when he does, escalation is guaranteed. Looking at the distribution in the lustrumbook, we conclude that Francken as a total has a fairly large $\mu$-value. I would estimate $\mu \approx 0.8$. This is plausible: on many days, for instance weekend days, few or no beers are drunk, and a greater number of beers is consumed on some scarce occasions, such as Borrels. Mark Redeman has been ingenious enough to derive a Maximum Likelihood Estimator (MLE) from anyone’s given streepstatistics. Very cool! This MLE is surprisingly simple: $$\hat{\mu}_{MLE} = 1 - \frac{1}{\bar{x}},$$ where $\bar{x}$ is the average number of beers drunk on a day at which the person drinks at least one beer. Using this formula, he managed to calculate the $\mu$-value of some individuals as well as that of the entire association. Which as of this writing is equal to $0.977$.

I suggest that we introduce this $\mu$-value as a prominent property of the drinking behavior of any (group of) people. In my opinion, the $\mu$-value of the entire association should always be presented at a general assembly. Furthermore, the idea that borrelcie.vodka/jij or professorfrancken.nl/profile would show not only a member’s raw streepstatistics, but his or her $\mu$-value, makes me very excited. Imagine a Borelaar shouting, at the start of some borrel: “I’m going to raise my $\mu$-value by 0.02 tonight!” upon starting a game of Chwazi. The future is here, people.

## Armand Borel

Another kneitse Borel was Armand Borel. Armand Borel (1923-2003) was a Swiss mathematician. His work finds its place more in pure mathematics than in applied mathematics and physics. However, the puns related to his surname are incredible! Borel began his mathematical career under the supervision of Heinz Hopf, a German mathematician. Hopf may be seen as the very beginning, origin and catalyst of Borel. His first significant contribution was in collaboration with Jacques Tits, a Belgian-born French mathematician. Many of you may have seen Tits: he is well-known for introducing Tits buildings, the Tits alternative and the Tits group. In general, mathematicians have the greatest respect for Tits. Another important reason for this is Tits’ hard work on the Coxeter number, Coxeter group and Coxeter graph. Borel did important work in proving the Borel-Weil- Bott-theorem and then co-devised the Borel- Moore homology theory. Although the latter may sound more like a real commandment, it is actually a homology theory for locally compact spaces. Appearances can be deceptive. Borel was extraordinarily influential. He was even considered praiseworthy enough to be awarded a real Brouwer medal. This medal was named after the famous Dutch mathematician Luitzen Egbertus Jan Brouwer, who is known for proving the Hairy Ball Theorem and for founding Intuitionism in the philosophy of mathematics. Borel seemed to have lived his whole life in order for us to make funny jokes about his surname. The people he made part of his biography constitute precisely what any given Borel needs. However, his adtchievements in mathematics prove that he was more than a surname: he was reëel.

## Are Borel and Borel related?

Émile Borel was particularly quiet about the question whether he was related to Armand Borel. A plausible explanation for that is that Borel never knew of the existence of Armand Borel. The other way around was entirely different, in that sense: Borel knew that Émile Borel was also a famous scientist. It was hence consistently a question to Borel whether Borel was related to him. Borel’s replies to these questions were not very consistent. Borel sometimes stated that Borel and Borel are not related, and at other occasions he claimed that he was Borel’s nephew.

## Conclusion

This blog undoubtedly leads us to conclude that scientific progress and Borels are inseparable! By that I do not mean that a countable basis exists for neither, but that they are time invariantly and irreversibly intertwined. If Borels never existed, that would have at the very least caused a severe delay in the progress of mathematics and physics. We should be proud of how Francken contributes to the science-catalyzing phenomenon of Borels. Borel on, twenty diamonds!