Other

Miscellaneous

This is a collection of links to things that I like or that I find worthy of being here (which is basically a tautology...).

1. Sheldon primes. My favourite TV series, The Big Bang Theory, has inspired a theorem! It is about the so-called Sheldon prime numbers: want to discover more?
2. Asteroids. Asteroids are minor bodies orbiting in our Solar System. Some of them can come close to the Earth when travelling along their orbit, thus having a non-negligible chance of impacting our planet. The problem of monitoring the NEA population is solved by using some mathematical tools developed and pioneered by Andrea Milani, in several papers about the year 2000. Want to discover more?
3. Classical music. Music has always been present in my life, currently through dance and in the past thanks to piano. I thus really enjoy listening to classical music. This is a very well-done web site about Beethoven and his works, which is definitely worth a look:

As a mathematician, it shouldn't be surprising that I put maths stuff also here even if this should be an "Other" section.

1. : the community of italian dynamicists
2. : a Q&A community of people studying maths at any level
3. : a platform whereI like playing chess when I have time
4. : a miscellany of maths articles and puzzles
5. : database of papers
Sheldon primes

The Big Bang Theory is a CBS sitcom created in 2007. It follows the escapades of five friends: Sheldon Cooper, physicist; Leonard Hofstadter, physicist; Howard Wolowitz, aerospace engineer; Raj Koothrappali, astrophysicist; and Penny, a waitress and aspiring actress who lives across the hall. If you are reading these lines, be sure to have seen at least one episode of The Big Bang Theory. If not, stop reading and open this link!

In the 73rd episode we find the following conversation:

Sheldon: "What is the best number? By the way, there's only one correct answer. [...] 73 is the 21st prime number. Its mirror, 37, is the 12th, and its mirror, 21, is the product of multiplying - hang on to your hats - seven and three. Heh? Heh? Did I lie?"
Leonard: "We get it. 73 is the Chuck Norris of numbers."
Sheldon: "Chuck Norris wishes. In binary, 73 is a palindrome: 1001001, which backwards is 1001001. Exactly the same. All Chuck Norris backwards gets you is Sirron Kcuhc."

It can be easily verified that Sheldon is correct about each property of the number 73. A number that satisfies these properties (which we define explicitly below has been called a Sheldon prime in [1]. Sheldon's assertion that 73 is the best number can be interpreted by saying that 73 is the only Sheldon prime. This is the Sheldon conjecture.

Let's be more formal. For a positive integer $n$ let $p_n$ denote the $n$-th prime number. We say that $p_n$ has the product property if the product of its base-10 digits is exactly $n$. For any positive integer $x$, we define $rev\left(x\right)$ to be the integer whose sequence of base-10 digits is the reverse of the digits of $x$. We say that $p_n$ satisfies the mirror property if $rev\left(p_n\right) = p_{rev\left(n\right)}$.

Definition. The prime $p_n$ is a Sheldon prime if it satisfies both the product property and the mirror property.

Well, 73 is a Sheldon prime: this is the content of the above conversation. The Sheldon Conjecture can be then formulated as follows: 73 is the unique Sheldon prime. In [2], the mathematicians Pomerance and Spicer proved the conjecture, which has now become a theorem!

Theorem. The number 73 is the unique Sheldon prime.

It's worth noting that the proof of the theorem uses, among others, the Prime Number Theorem, which states that

$\displaystyle{\lim_{x\rightarrow +\infty} \frac{\pi\left(x\right)}{x/\log x} = 1}$,

where $\pi\left(x\right)$ counts the number of primes that do not exceed $x$. We actually know that $\pi\left(x\right)$ is slightly larger than $\frac{x}{\log x}$ for "large" values of $x$. A result of Rosser and Schoenfeld makes the previous sentence precise: we have that $\pi\left(x\right)<\frac{x}{\log x}$ for all $x\geq 17$. This inequality immediately allows the authors to prove that no Sheldon prime exceeds $10^{45}$, and in fact, they only need the product property to show this. This is just the beginning of the story: if you want to know more, the references contain the link to get the paper.

Last thing you may want to know. Jim Parsons, the actor playing the role of Sheldon Cooper in The Big Bang Theory and which you can see in the picture on the right of this text, was born on 1973!

References

[1] J. Byrnes, C. Spicer, A. Turnquist, The Sheldon Conjecture, Math Horizons, 23(2), 2015
[2] C. Pomerance, C. Spicer, Proof of the Sheldon Conjecture, The American Mathematical Monthly, 126(8), 2019

Asteroids

Since when I was a child, I have been attracted by the stars shining on top of my head: the pictures taken by telescopes showing galaxies, nebulae, constellations was so intriguing! On the other hand, I have always loved maths. But... what does maths have to do with astronomy? Actually the two are related and there are active branches of mathematics studying this kind of problems. Thus, part of the research I did for my PhD was exactly in applied celestial mechanics, in particular about asteroid dynamics and probabilistic models for asteroid impacts.

Let's start from the beginning.

Definition. A Near-Earth Asteroid (NEA) is an asteroid orbiting around the Sun along an orbit with perihelion distance $q\leq$1.3 au.

By definition a NEA can come very close to the Earth when travelling along its orbit and the problem of its hazard assessment arises. The impact of an asteroid with the Earth is commonly considered to be very unlikely, but this is not true. We currently know about 600,000 asteroids (see this page for the updated number): more than 20,000 of them are NEA and for more than 800 NEAs there is a non-zero probability of impacting our planet in the next century (as of June 2019). NEODyS provides the impact probability for all of these potentially dangerous asteroids in the Risk List.

Of course not all the asteroids are equally threatening: indeed their size can vary from a few meters to tens of kilometers, with extremely different consequences in case of impact. For instance, the impact of an asteroid of about ten kilometers in diameter is capable of erasing the life on the Earth, as happened in case of Chicxulub, the asteroid that caused the extinction of the dinosaurs (see the picture above). The good news is that time-scale of this extreme events is a few tens of millions of years, wider not only than the duration of a human life, but even than the rise and fall of entire civilizations. Nevertheless, even the impact of a small asteroid of a few meters in diameter is a potentially dangerous event and must be properly monitored.

Some details more! When an asteroid has just been discovered, its orbit is weakly constrained by the available astrometric observations and it might be the case that an impact with the Earth in the near future, e.g. within the next 100 years, cannot be excluded. If additional observations are obtained, the uncertainty of the orbit shrinks and the impact may become incompatible with the available information. Thus a crucial issue is to be able to identify the cases that could have a threatening Earth close encounter within a century, as soon as new asteroids are discovered or as new observations are added to prior discoveries. The main goal of impact monitoring is to solicit astrometric follow up to either confirm or dismiss the announced risk cases. This is achieved by communicating the impact date, the impact probability and the estimated impact energy. This activity required an automated system that continually monitors the NEA catalogue. CLOMON-2 and Sentry are two independent impact monitoring systems that have been operational at the University of Pisa since 1999 and at JPL since 2002, providing the list of asteroids with a non-zero probability to impact the Earth within a century.

A cautionary note: beware of conspiracy and anti-scientific web pages!