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Elegant six-page proof reveals the emergence of random structure

quantamagazine.org

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Re: Elegant six-page proof reveals the emergence of random structure

#131

Post by photochemsyn »

G3rn0ti wrote: post #96
galaxyLogic wrote: post #47

I 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 c…

Current origin-of-life theories are pretty diverse, but a very promising approach postulates the ribosome, the protein synthesis machine composed of proteins and RNA, as the original self-replicating entity. This model supposes the abiotic synthesis of both simple amino acids and nucleosides, the building blocks of proteins and nucleic acids, and some kind of random polymerization process.

If such random polymers were able to assemble into something like the ribosome, that structure could plausibly start replicating itself using the abiotic pool of free amino acids and nucleosides. This is somewhat different from older 'RNA world' theories in that it involves both amino acids and nucleosides. Abiotic synthesis of these precursors is fairly plausible.

Re: Elegant six-page proof reveals the emergence of random structure

#132

Post by [deleted] »

rocqua wrote: post #126
scoot wrote: post #112

Earlier quoted context omitted.

That seems more like a truism: If on average one case leads to more than one other (R > 1), the number of cases will grow. The R number therefore appears to be nothing more than a measure of growth or decline. What am I missing?

This is essentially what is happening in the percolation case as well.

[deleted]

Re: Elegant six-page proof reveals the emergence of random structure

#133

Post by mjreacher »

Ar-Curunir wrote: post #117
ThinkBeat wrote: post #105

I notice that most links on physics, astronomy and math cover pre-prints. Would it not be prudent to wait for the final version. Is all the peer reviews and "fact checking" done prior to the preprint?

For a big result like this, enough experts read the paper to catch glaring mistakes. Sort of a de facto peer review

Exactly, Quanta would have the experts they interview go over the paper first if they didn't already go over it before and evaluate it before making any comments.

Plus arXiv papers are freely accessible ;)

Re: Elegant six-page proof reveals the emergence of random structure

#134

Post by fallingknife »

Ar-Curunir wrote: post #116
urthor wrote: post #71

Earlier quoted context omitted.

From the link: "Jinyoung is a first-generation college graduate, and after seven years as a secondary school teacher in South Korea, she went on to earn a mathematics Ph.D. from Rutgers University. Jinyoung’s story demonstrates the importance of role models at all levels of one’s education and the fact that it is never too late to begin anew."

And yet people will shit on DEI efforts. Incorporating more folks into the scientific endeavor helps combat things like stagnation in science precisely because it reduces academic incest.

DEI efforts discriminate against Asians like Jinyoung, so yes, I will shit on them.

Re: Elegant six-page proof reveals the emergence of random structure

#135

Post by mjreacher »

gardnr wrote: post #42

> 2 (of a scientific theory or solution to a problem) pleasingly ingenious and simple: the grand unified theory is compact and elegant in mathematical terms. A 6 page proof described as "elegant" must be incredibly dense.

As the other poster noted, for a big math result this is comparatively rather short. For a concrete example[0], a recent important result related to the Arnold conjecture is 268 pages long, with another 100 pages in appendices.

[0]: https://arxiv.org/abs/2103.01507

Re: Elegant six-page proof reveals the emergence of random structure

#136

Post by jonnycomputer »

nsane wrote: post #130
rurban wrote: post #102

As I already commented in the first submission of this article: Now that's something I can really use in one of my current problems. Calculating a minimal perfect hash by creating acyclic random graphs. This conjecture gives now tresholds when to stop trying creating random graphs and start afresh. This eg is needed for large perfect hashes in gperf or integer sets in compilers, such as eg. for C switch statements wi…

Isn't the proof for Hamiltonian cycles? From what I could understand from your link to CHM92, it only requires a cyclic graph to stop, which isn't necessarily supposed to touch all vertices. Is that right?

The Hamiltonian cycle is only a motivating example. The conjecture which was proven is much more general.

https://gilkalai.wordpress.com/2022/04/02/amazing-jinyoung-p...

Re: Elegant six-page proof reveals the emergence of random structure

#137

Post by omnicognate »

leephillips wrote: post #125
omnicognate wrote: post #120

Earlier quoted context omitted.

The chain of edges forming the cycle doesn't have to include all edges in the graph. There just has to exist a set of edges that form a Hamiltonian cycle. Adding further edges doesn't change that (but adding further vertices would). Edit: Put another way, it's not the Hamiltonian cycle itself that's increasing, it's the property of there existing a Hamiltonian cycle in the graph.

So the definition in the article, “a chain of edges that passes through every vertex exactly once”, is incorrect?

No, that's the correct definition of a Hamiltonian cycle. However, the article isn't as clear as it could be about what the "increasingness" aspect applies to.

> It’s possible to think about any property, so long as it is “increasing” — that is, if adding more edges to a graph that already contains the property will not destroy the property.

The "property" here isn't the Hamiltonian cycle itself, it's the property "a Hamiltonian cycle exists within this graph". This property, rather than the cycle itself, is increasing: if you add edges to a graph that contains a Hamiltonian cycle it will still contain a Hamiltonian cycle.

Re: Elegant six-page proof reveals the emergence of random structure

#138

Post by scoot »

joshocar wrote: post #121
scoot wrote: post #112

Earlier quoted context omitted.

That seems more like a truism: If on average one case leads to more than one other (R > 1), the number of cases will grow. The R number therefore appears to be nothing more than a measure of growth or decline. What am I missing?

As far as I understand it R is not a rate because it does not include time. Diseases that take on average a few days for you to pass on verses a few months will have different growth rates. For example, an STI might have an R factor of 3, but the rate is completely dependent on how often people have unprotected sex.

Good point, R is often referred to as a rate both in media and academic publications, but "number" or "value" appears to be more correct. Fixed.

Re: Elegant six-page proof reveals the emergence of random structure

#139

Post by dandare »

zeroonetwothree wrote: post #124
dandare wrote: post #123

It always fascinated me how basically two quarks, one electron and some bosons can create so much chemical complexity. It feels like trowing just 6 types of lego bricks into a huge bag and after a lot of random mixing obtaining whole universe of diversity.

All the complexity of computing comes from just two bits. (Some even argue that’s true of the universe itself)

Not true, the bits live in a super complex hardware, the don't create any complexity on their own.

Re: Elegant six-page proof reveals the emergence of random structure

#140

Post by [deleted] »

omnicognate wrote: post #137
leephillips wrote: post #125

Earlier quoted context omitted.

So the definition in the article, “a chain of edges that passes through every vertex exactly once”, is incorrect?

No, that's the correct definition of a Hamiltonian cycle. However, the article isn't as clear as it could be about what the "increasingness" aspect applies to. > It’s possible to think about any property, so long as it is “increasing” — that is, if adding more edges to a graph that already contains the property will not destroy the property. The "property" here isn't the Hamiltonian cycle itself, it's the property "a…

[deleted]
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