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Diamonds and Triamonds | Azimuth
The structure of a diamond crystal is fascinating. But there's an equally fascinating form of carbon, called the triamond, that's theoretically possible but never yet seen in nature. Here it is: In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds…
baez  done 
july 2017 by karlicoss
Correlated Equilibria in Game Theory | Azimuth
Given this, it’s not surprising that Nash equilibria can be hard to find. Last September a paper came out making this precise, in a strong way:

• Yakov Babichenko and Aviad Rubinstein, Communication complexity of approximate Nash equilibria.

The authors show there’s no guaranteed method for players to find even an approximate Nash equilibrium unless they tell each other almost everything about their preferences. This makes finding the Nash equilibrium prohibitively difficult to find when there are lots of players… in general. There are particular games where it’s not difficult, and that makes these games important: for example, if you’re trying to run a government well. (A laughable notion these days, but still one can hope.)

Klarreich’s article in Quanta gives a nice readable account of this work and also a more practical alternative to the concept of Nash equilibrium. It’s called a ‘correlated equilibrium’, and it was invented by the mathematician Robert Aumann in 1974. You can see an attempt to define it here:
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july 2017 by nhaliday
Computing the Uncomputable | Azimuth
I love the more mind-blowing results of mathematical logic: • Surprises in logic. Here's a new one: • Joel David Hamkins, Any function can be computable. Let me try to explain it without assuming you're an expert on mathematical logic. That may be hard, but I'll give it a try. We need to start with…
baez  done 
june 2017 by karlicoss
Shock Breakout | Azimuth
Here you can see the brilliant flash of a supernova as its core blasts through its surface. This is an animated cartoon made by NASA based on observations of a red supergiant star that exploded in 2011. It has been sped up by a factor of 240. You can see a graph of brightness showing…
baez  done 
june 2017 by karlicoss
Information Geometry (Part 16) | Azimuth
While preparing this talk, I discovered a cool fact. I doubt it’s new, but I haven’t exactly seen it elsewhere. I came up with it while trying to give a precise and general statement of ‘Fisher’s fundamental theorem of natural selection’. I won’t start by explaining that theorem, since my version looks rather different than Fisher’s, and I came up with mine precisely because I had trouble understanding his. I’ll say a bit more about this at the end.

Here’s my version:
The square of the rate at which a population learns information is the variance of its fitness.
baez  mathtariat  evolution  bio  genetics  population-genetics  bits  interdisciplinary  models  exposition  math.DS  giants  information-theory  entropy-like  org:bleg  nibble  fisher  EGT  dynamical 
february 2017 by nhaliday

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