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Yann Ponty - CNRS@LIX
Musings
What Feynman saw in a flower...

Because understanding the process does not necessarily make the result any less appealing! (otherwise, would there be any hope left for lasting companionship? ;) )
Beautiful work by Fraser Davidson.

If you haven't seen one of these...


... then you obviously never TA'ed Bioinformatics ;) (Lucky bastard!). Full credits/kudos to XKCD.

On another note (and yes, this will be my last xkcd strip, otherwise my page would simply end up being a mirror):

Full credits again to XKCD.

The best (i.e. most selective) Journal ever!!!

Ever complained about the academic daily routine having become a quest for the best journal that would nevertheless accept one's half-cooked manuscript? Ever felt that citations and impact factors were way too overrated, and that selectivity should prevail? Look no further, as it is my pleasure to share my recent discovery of (arguably) the best journal ever!

With an acceptance rate way lower than 1%, the journal of universal rejection may rightfully pride itself of enforcing the strictest of academic standards (Nature beware!). Created in 2009, the journal solicits submissions in poetry, prose, visual art, and research. Despite such a wide scope, its widely competent editorial board can be trusted to lay an expert eye on your submission, and helpfully motivate its final decision.

Submit Now!

P vs NP: The irrefutable proof ;)

Today is the day I almost died laughing while reading Scott Aaronson's Shtetl-Optimized blog dedicated to Complexity Theory. Ok, I know, this sort of humor alone may be regarded as a conclusive diagnostic by future generations of psychopathologists, but I'd still like to share his beautiful, human-centric, argument for why P != NP. Basically, he is asked the question:

Can you summarize to a (curious/smart) dummy the P vs NP question and its many implications?

He starts by stating the question in four informal ways, one of which is:

Is it harder to solve a math problem yourself than to check a solution by someone else?

and adds

This is where you insert a comment about the delicious irony, that P vs. NP itself is a perfect example of a monstrously-hard problem for which we could nevertheless recognize a solution if we saw one---and hence, part of the explanation for why it's so hard to prove P!=NP is that P!=NP...

I woke up the following morning at the hospital, and must be clinicaly monitored during any future access to Scott's blog, but I encourage you to check it out for a daily dose of CS wittiness.

Amicable numbers

Amicable numbers are members of the number theory zoology (See their Wikipedia page), which, like trousers and all sorts of useful stuff, come handy in pairs. Formally, one starts by defining the restricted divisor function s(n) to be the sum of all divisors of n, itself excepted.
For instance for n = 220, one finds

s(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

Then, amicable numbers are pairs of natural numbers (p,q) such that s(p) = q and s(q) = p.
To make this notion explicit, we get back to the example above and find

s(284) = 1 + 2 + 4 + 71 + 142 = 220

and therefore (220,284) are amicable numbers.

Now there is something about this definition that may sound arbitrarily limited to a computer scientist. Indeed, consider the (infinite) directed graph whose vertices are natural numbers, and whose edges are pairs (n,s(n)). Here is a picture, rendered using the allmighty GraphViz (neato mode), showing the resulting network/graph for numbers below 50 (blue/red arcs indicate decreasing/increasing values of s, number 1 omitted for readability, golden nodes are perfect numbers):

And here is the larger graph of numbers below 300 (Full-screen PDF version):

One then easily sees that amicable numbers are in bijection with cycles of length 2 in the graph. Indeed, zooming in on the above graph, one may easily spot the first amicable pair:

This raises the question

Why not consider more general iterations of the s(.) function?

Indeed, one can alternatively, yet equivalently, define amicable numbers as pairs

(n,s(n)) such that s(s(n)) = n and s(n) ≠ n

which naturally generalizes into what is called sociable numbers, i.e. k-tuples of numbers

(n, s(n), s(s(n)), … , sk-1(n)) such that sk(n) = n and sj(n) ≠ n, ∀j∈[1,k-1].

These sequences are also called periodic aliquot sequences. Here is one of the biggest periodic sequence known to date (k=28):

These numbers are currently of some interest in number theory:
  For k=1, sociable numbers are perfect numbers.
  For k=2, sociable numbers are amicable numbers.
  In general, what are aliquot sequence of cycle length k? Do they even exist for large values of k?
Whether there exists a cycle-free infinite aliquot sequence is currently open!

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