If they had the public keys to all Mt Gox's wallets whey would be able to check them for funds, but there would still be questions about who controlled the wallets and if they had found them all so it would still not be proof of fraud (but the more evidence collected the more likely there will be something confirm-able discovered)
No. 16, 'Chaos Theory' is actually called the 'Logistic Equation' and it is used in biology to model populations. It does exhibit transition to a chaotic regime for certain values of k, but it's improper to refer to it as 'Chaos Theory equation'.
Yep, if anything the Restricted 3-Body Problem (R3BP) should be considered the "chaos theory equation", since it was Poincaré study of chaotic solutions in the R3BP that led to the discovery of sensitivity to initial conditions. Smale's horseshoe map would be another apt equation, since it succinctly describes the dynamics of chaotic systems, through the twisting and warping of the map. The Lorenz attractor is another important example that highlights the concept of deterministic chaos.
I don't quite get how chaos theory changed the world. All of the other equations I get (although whether Shannon's equation changed the world or merely described the change is probably arguable.)
Understanding deterministic chaos has been the cornerstone of a wide range of fields, including physics, medicine, economics, and business. There are a wealth of everyday applications. Control systems in the real world have to deal with the underlying structure of the dynamics that govern how such systems naturally behave in the feedback loop.
There are a few different resources you can find that list practical applications of chaos theory. Here are a few that I found that I think you will find interesting:
It has very important epistemological implications: it means a phenomenon may follow a deterministic law which, due to its mathematical properties, would nevertheless make long-term predictions impossible. So even though you would have a mathematical model for the phenomenon, you wouldn't be able to make real-world long-term predictions. The problem lies in the sensitive dependence on initial conditions.
Many ideas do "float in the air", so you should take you gut feelings of "what's the next app that should be out there" seriously.
It has happened to me personally that an idea which I thought was "crazy, but perhaps crazy enough to be good" – yet never started working on it – was the same idea a start up had not long before. I could but congratulate them on their genius :)
"The legend about a low-level employee who pipes up with a shockingly brilliant yet simple suggestion that either makes or saves his employer untold millions of dollars has long been part of the canon of business legends."
No doubt. But these sorts of things also do happen.
I used a 20 dollar pen cam off eBay for diagnostics once, avoiding tens of thousands in diagnostic downtime needed otherwise and found a root cause of an otherwise undetected massive failure mode. All because the blasted thing fit. (Full disclosure: the idea started with a plant tech who wondered if there was an application for the pens - research and testing later showed it safe under our operating conditions (a small explosion was possible).)
It's a rare opportunity, but magical when it happens (and modern, cheap tech is magical for heavy industry).
