101–110 of 112 posts
https://www.imo-official.org/participant_r.aspx?id=19624
Also, only 1 person got 3 perfect scores in the history of the event: Ciprian Manolescu (3 for 3 perfect) https://www.imo-official.org/participant_r.aspx?id=3789
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I gasped when I saw my acquaintance's name on HN. By coincidence, I'm wearing an expii t-shirt right now. He's really nice, and excited about improving math and science education. Expii could be called an "explorable explanations" Wikipedia[0]--one can write code (I think a subset of JS), embed videos etc in service of a better explanation. I'm also fond of him because his office has an X server (or would that be X c…
Maybe I'm just dense. Do you mean X11 server ( https://en.wikipedia.org/wiki/X_Window_System ) or something else?
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The standard example of unexpected esoteric physic application is the Special and General Relativity corrections of the GPS. 100 years ago nobody thought that we would ever use that small correction or that everybody would have a special-and-general-aware device in the pocket. Another cool application is the superconductors in the NMR devices. They can see what is inside your head and even do something equivalent to…
Don't get me wrong, QM for example was ridiculously useful. But, pointing to past and saying these tiny particle accelerators where useful let's make a multi billion dollar one feels like cargo cult science for the lack of a better phrase. We just keep piling higher and deeper without a clear reason to do so other than we can afford to do so. String theory is another example where lot's of effort from seemingly smart…
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IMO, high level Math and Science is generally a waste of a lot of really smart peoples time. The LHC for example is studying particle energy's so far outside of 'reasonable' that we are not going to get useful technology from there. The same is true of a lot of esoteric Math which mostly ends up divorced from anything actually useful. That's not to say funding such things is necessarily a waste, just the focus on STE…
The standard example of unexpected esoteric physic application is the Special and General Relativity corrections of the GPS. 100 years ago nobody thought that we would ever use that small correction or that everybody would have a special-and-general-aware device in the pocket. Another cool application is the superconductors in the NMR devices. They can see what is inside your head and even do something equivalent to…
Earlier quoted context omitted.
Don't get me wrong, QM for example was ridiculously useful. But, pointing to past and saying these tiny particle accelerators where useful let's make a multi billion dollar one feels like cargo cult science for the lack of a better phrase. We just keep piling higher and deeper without a clear reason to do so other than we can afford to do so. String theory is another example where lot's of effort from seemingly smart…
GH Hardy famously predicted that elementary number theory would have no practical applications. Lo and behold, today it is an indispensable part of web cryptography used by hundreds of millions every day. Waiting on the order of decades for a ROI on fundamental research is simply part of the game. I suspect the point where we experience diminishing returns from this is much to far in the future to even consider the q…
Consider, we know the first five digits of the gravitational constant. So, while it might seem like the diminishing returns are a long way off. Yet, each extra digit becomes exponentially more expensive and less useful. So, actually learning g out just 9 digits is probably a huge waste of resources.
Or in the words of a physicist, in 1920 second rate physicists where doing first rate research. Now, first rate physicistare doing second rate research.
Notation: We borrow subscript and superscript notation from D. Knuth's TeX. So, a with a subscript j is written a_j.
Consider the simple graph, based just on typing, below. The graph shows the first four values of a specific sequence a_j. This sequence is a contradiction to claim of Problem 6.
In this graph, the values of j = 1, 2, ... are plotted on the horizontal axis with 1, 2, ..., and the values of the a_j are plotted on the vertical axis from 1 to 2015. We plot a value of a_j as just 'a'. Due to constraint (ii), we indicate by 'x' points that, due to prior values of a_j, cannot have an 'a'. So, the graph is:
2015
.
.
.
a_j .
5
4
3 a a
2 x x ...
1 axax
123456789
j
a_1 = 3
a_2 = 1
/
| 3 if j is odd
|
a_j = (a_j - b) + (a_{j + 1} - b) >= 2
sum_{j = m + 1}^{m + 2k} (a_j - b) >= 2 k
| sum_{j=m+1}^n (a_j - b) |
= sum_{j=m+1}^n (a_j - b)
= 2 (1007^2) > 1007^2
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GH Hardy famously predicted that elementary number theory would have no practical applications. Lo and behold, today it is an indispensable part of web cryptography used by hundreds of millions every day. Waiting on the order of decades for a ROI on fundamental research is simply part of the game. I suspect the point where we experience diminishing returns from this is much to far in the future to even consider the q…
Elementary number theory is the opposite of what I am talking about. RSA is from the kiddie pool of that field. Consider, we know the first five digits of the gravitational constant. So, while it might seem like the diminishing returns are a long way off. Yet, each extra digit becomes exponentially more expensive and less useful. So, actually learning g out just 9 digits is probably a huge waste of resources. Or in t…
In number theory, what's considered 'elementary' now was cutting edge in the times of Diophantus, all the way to Fermat, to Euler, to Gauss (etc). The fact that children are now routinely conversant in it, I think, is another point in favor of the importance of making such discoveries in the first place.
My point is that applications that were never envisioned for these (at the time) centuries-old-facts, are now commonplace and indispensable.
I think that there is a bit of survivorship bias that warps our understanding of old science. We remember only the great discoveries because those are the most likely to be republished and read.
Also, in the case of math, it is my impression that an amazing amount of very significant progress is being made in the present era.
For Problem 6 given in the OP, the claim is false: Notation: We borrow subscript and superscript notation from D. Knuth's TeX. So, a with a subscript j is written a_j. Consider the simple graph, based just on typing, below. The graph shows the first four values of a specific sequence a_j. This sequence is a contradiction to claim of Problem 6. In this graph, the values of j = 1, 2, ... are plotted on the horizontal a…
The theorem seems entirely correct to me. You can prove it with these sub-steps:
(1) the set of all j + a_j is the set of nonnegative integers minus a finite number of gaps
(2) thus for large enough n you can express \sum_{j=1}^n (j + a_j) as a quadratic function of n, plus a residual term e(n) of magnitude at most 1007^2/2
(3) more precisely, \sum_{j=m+1}^n a_j = g (n-m) + e(n) - e(m) where g is the number of gaps in (1)
Then choosing b = g solves the problem.
Hope that helps.
Earlier quoted context omitted.
Elementary number theory is the opposite of what I am talking about. RSA is from the kiddie pool of that field. Consider, we know the first five digits of the gravitational constant. So, while it might seem like the diminishing returns are a long way off. Yet, each extra digit becomes exponentially more expensive and less useful. So, actually learning g out just 9 digits is probably a huge waste of resources. Or in t…
>Elementary number theory is the opposite of what I am talking about. In number theory, what's considered 'elementary' now was cutting edge in the times of Diophantus, all the way to Fermat, to Euler, to Gauss (etc). The fact that children are now routinely conversant in it, I think, is another point in favor of the importance of making such discoveries in the first place. My point is that applications that were neve…
As to survivorship bias, that's huge but it's not just based on good ideas. Copernicus was ~4,500 years late to proposing the sun was the center of the solar system. But, the pop story looks better when Darwin is breaking new ground instead of simply collecting more evidence in support of an old theory.
As to Amazing progress, I would hope the ~1,000,000 active mathematicians are not all wasting their time. But again, the point is we don't need to maximizing the number of Mathematicians, we are well into diminishing returns.
it is a good news. because American become more confident in educating their kids in maths and abandon the idea that you can be lawyer if you are not good at math. on the other hand, Chinese can stop believing that math is highest form of intelligence and pursue career as lawyer etc.
??? Wouldn't it be better for mankind ... if as many people like these kids as possible pursued math and science more vigorously ??? Especially kids like these, who have what I would term "creative" intelligence. Why would we want those people in law ??? That gives us far less than having them in math and science. And we will get the MOST by having them in art AND math and science. Then we'd really get their creativi…