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