flaubere
Hungary has a disproportionate number of the world class mathematicians of the 20th century. The numbers are even more extreme when you consider that those mathematicians are all from Budapest, all from Eötvös, and the vast majority of them Fejér's supervisees. It's totally plausible that Fejér created an environment where people who might otherwise have been middling mathematicians, decent engineers, or never have found their niche, were transformed, by collaboration and inspiration, into the likes of von Neumann, Erdős, Pólya.
Even today the Eötvös school of analysis and combinatorics is a significant powerhouse, with most members direct mathematical descendants from Fejér. And there are other branches such as Cambridge combinatorialists where the influence of Fejér's students and grand-students has been key in the formation of world class mathematicians.
However, I also did manual labour in the summer when I was ~20. There was nothing wrong with it. I would certainly recommend a brief stint of it to a young person, especially if you are training towards a sitting down/talking type job.
You are outdoors, you end the day physically tired but with mental and emotional energy for other things. You gain skills and improve your health and fitness. There's usually a good atmosphere among the workers. And you get to point at something concrete and say 'I helped put that up/knock that down/repair that'.
The vast majority I would say that he is trying to retcon his huge success and the success of some businesses he has been associated with into a coherent worldview. I believe that in 'Hackers and Painters' he actually goes through some back-of-envelope calculations that show that the money he made when Yahoo bought Viaweb corresponded closely to the real value he had created, in some sense.
It is baffling to me why he isn't able to say "I got lucky - I worked hard and created something very valuable, but I was also in the right place at the right time." Clearly there were special factors at play selling an e-commerce platform to Yahoo in 1998. He's also done intelligent and pro-social things with both his money and his time since then it appears. I don't know what the shame is in saying "I won a lottery - but I have tried to do the right thing with my good fortune."
I think if you asked Jamie Zawinski, who I think was no less technically skilled, nor less purposeful about working on interesting and important things (nor, tbh, any worse at writing thoughtful essays), he would readily admit to having been extremely lucky. I don't know what the difference is between these two personalities. I think I'd rather be jwz in similar circumstances.
Then again some people still enjoy watching professional cycling so maybe it will be fine.
However in the US the word 'state' is overloaded, so 'nation state' is used. It's confusing, because 'nation' is sometimes used to mean 'country' (which is how it's intended in 'nation state') and more traditionally used to mean a people with a shared language and history.
Is there a way of checking these automatically? Eg if you can tile a certain amount of space without gaps then it must be able to continue forever? Or can you write down a vector expression for the location of each shape and show finitely that you have exactly covered all lattice points?
Cut the 3-cube with a plane which is diagonal to all axes, eg x + y + z = c. Start at a corner and take sequential sections. First you get a small equilateral triangle, then a bigger and bigger one, until the cut goes between 3 vertices of the cube. Next you get truncated equilateral triangles, with bigger and bigger truncations. In the center of the cube the size of the truncations matches the remaining edges and you get an regular hexagon. Then the whole thing in reverse as half the sides get smaller and smaller, until you're back to triangles.
If you're not sure what it looks like at any point, you can easily solve the intersection of x + y + z = c and the equation of one face of the cube (x or y or z = 0 or 1).
Now do it for the 4-cube. Important observations: 1. again you can solve algebraically, either for the 3-cubes which bound the 4-cube or the 2-squares which bound them 2. you can also just try and imagine the intersection with the 3-cubes, since it will be one of the shapes you thought about in the previous exercise (x + y + z + w = c && w = 1 => x + y + z = c - 1) 3. c goes between 0 and 4, with [0, 2] symmetrical to [2, 4]. There are two 'regions' of behavior, c in [0, 1] and c in [1, 2], with the type of shape only changing when the plane intersects with vertices.
I would put this at better than 1/10^22. If you claimed that your religion was real and showed me an incontrovertible 'miracle' which had a chance of less than 1/10^20 of occurring by chance, I think I would accept your claim.
"What's going on, officer?"
"Supreme court just ruled on the Minecraft speedrun case."
"What's that?"
"This Youtube asshole got sued for playing a video game or something. I don't know. Hughes, what was it?"
"He mined a diamond on his second try."
"This asshole mined a diamond on his second try. Now all youse are getting released."
"So what's going on, we're gonna get a retrial or something?"
"Nah. Everyone here is going home."
The release of all incarcerated felons in the United States passes off almost without incident, although police leave in Chicago is cancelled after several homicides in the downtown area on a single night. These are believed to involve released prisoners, but this was never confirmed. Pawnshops and liquor stores in several areas of the country hire extra security for the following week after viral WhatsApp messages warn of a crime wave, but the expected spike in criminal behavior never takes place.
On the afternoon of the second day of releases two Marshalls and a adjunct professor of mathematics get out of a rented car outside FCI Marion, having flown to Chicago from the nation's capital. They are too late to stop the release of the man convicted of ripping off the 'McDonald's Monopoly' lottery game. A numerate intern at the DoJ noticed late on the first 'release day' that the chance of every single large award in the fast food prize draw between 1995 and 2000 being won by one person's associates or family members, technically exceeded the new probabilistic standard for reasonable doubt. They are ten minutes too late, the fraudster having departed by Uber. For the first time in its history the FBI Most Wanted list consists of a single name, for nine days until his recapture at Canadian border. The email alerting prison staff to his upheld conviction had been sent to the email address of a recently retired prison officer, a fact cited in his retrial as evidence that "he's just a lucky guy, you know?"