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Show your mathematics to the world!

According to Timothy Gowers:
“After a very good start, the rate of signatures to the open access petition at the White House has slowed down to the point where, unless it speeds up again, the target of 25,000 will not obviously be reached, and ceratinly not surpassed by miles — which was the hope. There have been just over 100 signatures since the beginning of today (which, by my calculations, means eight hours ago in Washington DC). If you haven’t signed, then, unless you disagree with the petition, please take the short time needed to do so. And if you have, how about spreading the word?”

According to Terence Tao:
“More signatories already than the Elsevier boycott, but still only about 75% of the minimum target (with 20 days remaining).”

Sign the petition here.

Have you signed the open-access petition?
Please read this post.

Gowers's Weblog

Update 4th June 2012. The petition has now passed 25,000 signatures. It would still be great to push on and reach a significantly higher number by the June 19th deadline.

As you may know, there is a system in the US for setting up online petitions. Any petition that reaches 25,000 signatures in 30 days will be considered by White House staff. Recently, a petition was set up asking the Obama administration to require publications resulting from research paid for by the US taxpayer to be freely available. If such a requirement were to be put in place, it would be a huge boost to the campaign to make all academic research easily accessible.

It became possible to sign the petition last Monday, since when there have been (as I write) 14,303 signatures, well over half the number required. Even if the rate of signing goes down, the target of…

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Deriving double angle trigonometric formulas using transformations. A really nice elementary proof.

Double Angle Trigonometric Formulas.

In the television show Futurama, episode “The Prisoner of Benda,” Dr. Farnsworth created a machine that swaps the minds of two individuals. Unfortunately, they discover that the machine cannot be used on the same pair of bodies again.

The question is, given a collection of jumbled up minds and bodies, how can you get them all back to normal? Well, they have to come up with some equation to prove that, with enough people switching, eventually everyone will end up in their rightful form.

It was first mentioned in an interview that head writer and executive producer David X. Cohen gave to the American Physical Society:

“In an APS News exclusive, Cohen reveals for the first time that in the 10th episode of the upcoming season, tentatively entitled “The Prisoner of Benda,” a theorem based on group theory was specifically written (and proven!) by staffer/PhD mathematician Ken Keeler to explain a plot twist. Cohen can’t help but chuckle at the irony: his television-writing rule is that entertainment trumps science, but in this special case, a mathematical theorem was penned for the sake of entertainment.”

This theorem is now known as Keeler’s Theorem, or the “Futurama Theorem”.

The Futurama Theorem proves that no matter the size of the collection or how many mind swaps have been made, all minds can be restored to their original bodies using only two extra “clean” bodies (individuals who haven’t swapped minds with anyone) are needed, since a collection of just two people with swapped minds could never get back to normal.

If you want to know more about the details on group theory (group, the symmetric group, permutation diagrams, transpositions, cycle notation, products of disjoint cycles, products of distinct transpositions, etc.) read the talk given by Dana C. Ernst about the Futurama Theorem. He has blogged twice on this talk:
1. Talk about the Futurama Theorem
2. Another talk about the Futurama Theorem

and he has given the talk three times:
1. November 3, 2011 Gordon Talk
2. December 7, 2011 PSU Talk
3. February 28, 2012 UNO Talk

I personally like the first talk. He mentions:

“The slides for the second talk are very similar to the first set, but there are a few differences:

  • The second talk is shorter. I trimmed a few things from the first talk that were not completely necessary.
  • I’ve improved the wording in a few spots.
  • In the second talk, multiplication of permutations is right to left.”

Talks 2 and 3 are almost identical.

If you read the talk, you will understand the proof that appeared in the show,

Prisoner_of_Benda_Theorem_on_Chalkboard.png

which uses nothing other than basic facts about group theory and permutations. Furthermore, you will understand applying the algorithm to the arrangement of minds in Futurama requires 13 moves. However, since Fry and Zoiberg only swapped with each other and since there is only one other cycle, we could use Fry and Zoiberg as x and y, respectively in the algorithm and thus it only requires 9 moves.

Below are the implementations of the 13 move algorithm in Sage and WolframAlpha. (Note, Sage and WolframAlpha will multiply cycles left to right):

S_11 = SymmetricGroup(11)
c7 = S_11("(1,2,3,4,5,6,7)")
c1 = S_11("(8,9)")
t1 = S_11("(8,10)")
t2 = S_11("(9,11)")
t3 = S_11("(9,10)")
t4 = S_11("(8,11)")
t5 = S_11("(10,1)")
t6 = S_11("(11,7)")
t7 = S_11("(11,6)")
t8 = S_11("(11,5)")
t9 = S_11("(11,4)")
t10 = S_11("(11,3)")
t11 = S_11("(11,2)")
t12 = S_11("(11,1)")
t13 = S_11("(10,7)")

c1*t1*t2*t3*t4*c7*t5*t6*t7*t8*t9*t10*t11*t12*t13

Click here to view the Futurama permutation in WolframAlpha

Click here to view the 13 move algorithm for the Futurama permutation solution in WolframAlpha

Here are some questions posed by Dana C. Ernst:
1. Can we do better (i.e., fewer moves to fix the cycles)?
2. Under what circumstances do we not need to introduce two “clean” people?
3. Would only using one “clean” person ever be useful?
4. Could you use fewer moves if you allowed more than two “clean” people?

We now know the answer to some these questions. Students from the University of California, San Diego have improved the result, giving a better algorithm for finding the minimum number of switches to put everyone’s head back in the right places, give optimal solutions for two particular situations, and give necessary and sufficient conditions for it being possible to represent the identity permutation as m distinct transpositions in S_n.

Here is the paper submitted on April 26, 2012.

Harvard University and MIT today announced edX, a transformational new partnership in online education. Through edX, the two institutions will collaborate to enhance campus-based teaching and learning and build a global community of online learners.

EdX will build on both universities’ experience in offering online instructional content. The technological platform recently established by MITx, which will serve as the foundation for the new learning system, was designed to offer online versions of MIT courses featuring video lesson segments, embedded quizzes, immediate feedback, student-ranked questions and answers, online laboratories and student-paced learning. Certificates of mastery will be available for those who are motivated and able to demonstrate their knowledge of the course material.

Read more about it here:
MIT and Harvard announce edX – MIT News Office.

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