Posted on 01/20/2011 7:35:04 AM PST by decimon
Hear! Hear!
:’D
Cheers!
I hope you realize there are actually no fractals in nature. Fractals depend on having an infinitely divisible space, and actual physical space (as opposed to the abstractions we mathematicians deal with) is not infinitely divisible: no measurement finer than the Planck scale can be made.
There are a whole lot of natural recursive patterns that look like approximations to fractals (because if one works out how the recursion would go for infinitely many steps — assuming an infinitely divisible space — one gets a fractal), but there aren’t actually any fractals in the physical universe.
Off-hand, I don't know any. On the other hand, that's not really a relevant question. The mathematics underpinning RSA encryption was about 300 years old when it was finally put to practical use. The mathematics underlying general relativity had been done without any application in sight over the 30 years or so before Einstein used it (and w/o understanding general relativity, neither atomic energy nor the GPS system would be possible). The Radon transform uses in medical imaging was discovered in the 1930's w/o any practical application in mind.
Mathematical results (which oddly, as some fundamental level are all tautologies -- the weird thing is that there are non-obvious tautologies that have to be discovered) have an infinite shelf-life, and often turn out to be surprisingly useful years later.
Actually, we've been able to determine the number of partitions of a large number for years, the curious thing Ono claims to have done is give an ordinary formula for the number. (The term of art is a "closed formula", which basically is a formula made out of the operations on a calculator, with nothing like a sum whose number of terms depends on the value n, or a place where one uses one formula of something is true, and a different formula if it's not.)
It turns out p(n) is the coefficient of x^n when one computes the product of (1 - x^k)^(-1) for k=1,. . . n by writing each factor as a geometric series (1 + x^k + x^(2k) + . . .) then multiplying them and collecting terms. (If one is willing to allow infinitely many factors with k ranging over all the counting numbers one gets a product whose value is the series (1 + p(1)x + p(2)x^2 + p(3)x^3 +. . . .) .
For practical purposes, one is left with the question of whether evaluating Ono's formula has a lower computational complexity than the procedure I just described. Now that we have computers, a closed formula may not even be useful if it's slower to compute than some other algorithm that computes the same value.
At times FR really needs a “like” button for comments. This is one of them.
How do you know? Have you experienced that? How do you know no one else knows it? If the others knowing don't publicize it how do you know about their knowledge? Sort of a tree falling in the forest thing.
I prefer Churchill's comment: "There is nothing quite as exhilarating as being shot at and missed!" That, I have experienced and I agree.
It does get your blood racing, doesn’t it? The first time, I remember thinking, “What made him choose me as his target?” Just chance was my determination.
Yes, in a way ... there's something intoxicating about finding a relatively simple answer to a big question. Laughter is a common response.
Yes. Some of them were later patented. By the results of their efforts, if you know a simple technique that prevents certain defects in the final product, and your competitors' parts have those defects you can be fairly certain they don't know the cure.
Good for you. I think "new" ideas are simply new associations of old ideas. Was that your experience?
Sort of. Mostly pulling stuff from widely disparate and unrelated fields and combining and adapting them, some inspiration from The Great Inventor (it’s pretty hard to beat a couple billion years of evolution) and such.
Occasionally something really new happens, though.
You are answering the question I was asking and reinforcing my own thoughts. Associating ideas from disparate fields is what I had in mind. Intuition is also key.
I read a book called Connections, by John Burke, I think, which traced history just that way. He showed how history was not linear but that things in one area would lead to something else in another and then off society would go in that new area.
You should watch the Connections series.
Thank you. I didn't know there was one. I tracked it down and I will watch it. Thanks again.
I think you will love it. Enjoy!
Agree with HD. Thanks for taking the time ...
bttt
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