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You've stated an axiom.
Elegant six-page proof reveals the emergence of random structure
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Re: Elegant six-page proof reveals the emergence of random structure
#92Earlier quoted context omitted.
If you think something is absurdly obvious you usually need to re-read what you've read.
No. It's not that rare for absurdly obvious things to get published to great fanfare. I'm still bitter about "De Morgan's Laws". There are two of them: 1. If two things are not both true, then one or more of them is false. 2. If neither of two things is true, then both of them are false. Of course this is obvious to everyone. Writing it down did not merit having it named after yourself. I guarantee many other people…
Interestingly, both rules are rejected in so-called "intuitionist" math. They introduce a third state any proposition can be in, as long as it hasn't been proven true or false (using other axioms). The math that derives from that is pretty nutty, like most things spawned after Godel numbering was discovered.
Re: Elegant six-page proof reveals the emergence of random structure
#93I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.
Why not simply use a tablet?
Re: Elegant six-page proof reveals the emergence of random structure
#94Earlier quoted context omitted.
If you think something is absurdly obvious you usually need to re-read what you've read.
No. It's not that rare for absurdly obvious things to get published to great fanfare. I'm still bitter about "De Morgan's Laws". There are two of them: 1. If two things are not both true, then one or more of them is false. 2. If neither of two things is true, then both of them are false. Of course this is obvious to everyone. Writing it down did not merit having it named after yourself. I guarantee many other people…
Don't forget the celebrated Bayes theorem that falls directly out of the definition of conditional probability P(A|B) = P(A & B) / P(B).
If you had to squint at it and turn that into P(B|A) = P(A & B) / P(A), you'd realize that you can simply multiply the top and bottom by P(A), then pull out the remaining P(A)/P(B).
P(A & B) / P(B)
= P(A & B) * P(A) / (P(A) * P(B))
= P(A & B) / P(A) * P(A) / P(B)
= P(B | A) * P(A) / P(B).Re: Elegant six-page proof reveals the emergence of random structure
#95So it seems this allows us to understand when phase transitions occur, I wonder if this has applications in improving high temperature superconductors.
Re: Elegant six-page proof reveals the emergence of random structure
#96I wonder if this has implications for evolution of life. Life is about DNA which is about a graph of nucleoids connecting to each other in specific structures. But DNA must have evolved out of random structures and random mutations. So if there's a threshold at which structures become inevitable, then DNA has a chance to be born?
> But DNA must have evolved out of random structures and random mutations
Before DNA there was only RNA. DNA came later but established itself because it is much better at conserving information. There are still retro vira carrying their genetic information in the form of RNA reminiscent of that RNA era.
So the course of evolution might have been like this: 1) random RNA coils slowly gained structures being able to catalyze reactions e.g. like replicating itself. 2) Replicators become catalysts for other reactions. 3) Specialized RNA molecules start entangling themselves like e.g. one of them replicating the others while another provides access to chemistry providing energy (aka "food"). 4) RNAs start encoding proteins and enzymes adding to the entanglements. 5) Membranes appear isolating the entanglements from the environment inventing cells. 6) DNAs are created from RNAs forming the modern biochemical tri-unity of DNA, RNA and proteins.
Re: Elegant six-page proof reveals the emergence of random structure
#97Entry at Gil Kalai's blog on this result:
https://gilkalai.wordpress.com/2022/04/02/amazing-jinyoung-p...
Re: Elegant six-page proof reveals the emergence of random structure
#98Wasn't there also a connection between solving constraint satisfaction (k-SAT Boolean CSP examples) where the clause length to the number of clauses makes it super easy or super hard for solvers to solve for a solution?
I vaguely remember this from reading literature when writing heuristic solvers for my CS grad course in AI search.
Re: Elegant six-page proof reveals the emergence of random structure
#99Wasn't there also a connection between solving constraint satisfaction (k-SAT Boolean CSP examples) where the clause length to the number of clauses makes it super easy or super hard for solvers to solve for a solution? I vaguely remember this from reading literature when writing heuristic solvers for my CS grad course in AI search.
To answer my own question here are some references to the problem at hand https://dl.acm.org/doi/10.1145/3491210
Re: Elegant six-page proof reveals the emergence of random structure
#100I'm trying to visualize this. I go to Wolfram Alpha and type "chance of getting 504 heads in 1000 coin flips" and see the answer is about 1/40, and when I change 504 to 505 I see the odds are about 1/41 - only slightly worse. Then I check the differences between 524 and 525 and I see that the odds are decreasing much more sharply (1/400, 1/459). The little graph they helpfully provided shows what's happening: I've mo…
I don't think it's Bernoulli, binomial rather?