51–60 of 178 posts
Jinyoung Park's Path to Math video on IAS is a really wonderful 3 minute story of how she came to study threshold behavior in random discrete structures, just like this work. Highly recommended. https://www.ias.edu/ideas/paths-math-jinyoung-park
> 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.
The proof section is 3.5 pages. That's really pretty short, as far as modern math results go. I get your point, though.
Jinyoung Park's Path to Math video on IAS is a really wonderful 3 minute story of how she came to study threshold behavior in random discrete structures, just like this work. Highly recommended. https://www.ias.edu/ideas/paths-math-jinyoung-park
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The proof section is 3.5 pages. That's really pretty short, as far as modern math results go. I get your point, though.
The abstract is a one liner and the intro has in the first line a formula or two. No idea if this is the tone of math papers but I call this dense :)
Comparatively one of the papers I looked at the other day was something like 100 pages long with a ~30 page proof section with very little free whitespace packed full of complex mathematics.
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I feel like there’s a market for a publishing service that could partner with popular blogs and websites. They could either allow articles to be selected by the user or the partner site, which could then be auto-paginated and printed being being posted to the subscriber. Hell, sell a service where you will curate articles based on used interests. Add some relevant Twitter threads for letters to the editor. Partner si…
Such a service exists. Here are two examples: https://www.myscreenbreak.com/ https://waldenpond.press/
Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
The interesting thing a out graph properties (like the emergence of connectivity, giant components, Hamiltonian paths etc.) is that they happen as "phase transitions". A classic example is cuckoo hashing: You want to know how many edges the random graph of hashes can have before it contains a cycle, since that's when you need to rehash into a larger table. You might expect that this number is "pretty random" in that…
I'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'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…
The emergence of random structure in graphs, however, is different. The chance of a specific structure, such as a cycle of some length or a spanning tree of certain dimension, can go from not particularly likely (<20%) to significantly likely (95%+) in just a single additional node. Those transition thresholds at which the percentage changes in an intuitively surprising way are the subject matter of interest here.
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[Edited for content a second time: I quoted an article I was just reading by chance, presenting data that are exemplary of another point, about the relation between generic statements and actual differences: I now remove it realizing I missed a detail in that quote that excluded the professional context.]