[math-fun] Kontsevich periodicity conjecture
The conjecture was recently proved: http://golem.ph.utexas.edu/category/2013/10/who_ordered_that.html Start with a 3×3 matrix. Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix. Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse. Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse. Theorem: Up to a bit of messing about, you’re back where you started. Does anyone know what motivated taking the element-wise inverse? I ask because it's the only nonlinear operation in AES. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
You don't; the post I linked to clarifies exactly what the conditions are on the matrix. On Wed, Nov 13, 2013 at 10:52 AM, Mike Stay <metaweta@gmail.com> wrote:
The conjecture was recently proved: http://golem.ph.utexas.edu/category/2013/10/who_ordered_that.html
Start with a 3×3 matrix. Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix. Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse. Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse. Theorem: Up to a bit of messing about, you’re back where you started.
Does anyone know what motivated taking the element-wise inverse? I ask because it's the only nonlinear operation in AES. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I can't explain why Kontsevoch's result is important, but I can say that I've seen periodicity theorems and conjectures like this before (in ordinary as opposed to noncommuting variables) involving alternation between the usual matrix inverse and the goofy element-wise matrix inverse. The subject that serves as a haven for such results is called the theory of discrete integrable systems, and if any of you figure out what the heck it's all about, please explain it to me! It seems to be germane to my interests in things like domino tilings and Somos sequences, but I've never been able to make much sense of it. (The classical theory of integrable systems is about spinning tops and solitons and other systems with lots of conserved quantities; but then some mysterious characters called "Lax pairs" enter the story, which is where I usually get lost, even before everything gets discretized.) Jim Propp On Wednesday, November 13, 2013, Mike Stay <metaweta@gmail.com> wrote:
You don't; the post I linked to clarifies exactly what the conditions are on the matrix.
On Wed, Nov 13, 2013 at 10:52 AM, Mike Stay <metaweta@gmail.com> wrote:
The conjecture was recently proved: http://golem.ph.utexas.edu/category/2013/10/who_ordered_that.html
Start with a 3×3 matrix. Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix. Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse. Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse. Theorem: Up to a bit of messing about, you’re back where you started.
Does anyone know what motivated taking the element-wise inverse? I ask because it's the only nonlinear operation in AES. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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