[math-fun] the 4th rep-4 dragon
I'm sure it's in Jörg's menagerie, but Julian had to lead me by the nose. http://gosper.org/4thdragon.png What a strange, camouflaged creature. --rwg
Hey! They're all rep-2s! gosper.org/1stdragon.png, gosper.org/2nddragon.png,gosper.org/4flopfour.png --rwg On Thu, Dec 15, 2016 at 6:47 PM, Bill Gosper <billgosper@gmail.com> wrote:
I'm sure it's in Jörg's menagerie, but Julian had to lead me by the nose. http://gosper.org/4thdragon.png What a strange, camouflaged creature. --rwg
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate? If this question can be made rigorous, how might that be done? (And if so, what is the rigorous answer, or at least a method of approaching it?) —Dan
Relatedly: Why is it so easy to obtain a formula for f(n)-f(n-1) given a formula for f(n), but so hard to obtain a formula for f(1)+f(2)+...+f(n) given a formula for f(n)? :-) Jim Propp On Friday, December 16, 2016, Dan Asimov <asimov@msri.org> wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Two comments: 1. The Risch algorithm is "complete"; i.e., it is an *algorithm*: https://en.wikipedia.org/wiki/Risch_algorithm (I'm assuming that you aren't asking about the computational complexity of the Risch anti-derivative algorithm.) 2. Some closed form integrals may require polynomial root-finding (or root expressing), so you may not like the look of the resulting expression. E.g., you might find floating point approximations to certain numbers instead of a "perfectly precise" answer. At 01:54 PM 12/16/2016, Dan Asimov wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
ÂDan
Numerically speaking, integration is in contrast easier than differentiation ... WFL On 12/16/16, Henry Baker <hbaker1@pipeline.com> wrote:
Two comments:
1. The Risch algorithm is "complete"; i.e., it is an *algorithm*:
https://en.wikipedia.org/wiki/Risch_algorithm
(I'm assuming that you aren't asking about the computational complexity of the Risch anti-derivative algorithm.)
2. Some closed form integrals may require polynomial root-finding (or root expressing), so you may not like the look of the resulting expression. E.g., you might find floating point approximations to certain numbers instead of a "perfectly precise" answer.
At 01:54 PM 12/16/2016, Dan Asimov wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I might approach the question by looking at it algebraically. 1. Differentiation has a no-nonsense compositional rule (i.e., the chain rule), and integration has no equivalent. 2. If you look at what you're integrating as formal field extensions to, say, the complexes, then the integral itself often cannot be represented in that field. (Example: integrals of functions on R(x), like 1/x, require you to extend your field transcendentally with a logarithm.) So not only do you have the problem of computing the integral, you need to compute where the integral even lives. Differentiation has no such surprises. Robert P.S. Analytically, we might say that integrals are easier to reason about. They're much more well-behaved than derivatives. On Fri, Dec 16, 2016 at 1:54 PM, Dan Asimov <asimov@msri.org> wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
* Bill Gosper <billgosper@gmail.com> [Dec 17. 2016 09:27]:
I'm sure it's in Jörg's menagerie,
Since yesterday 2 pm it is. See the last page of http://jjj.de/tmp-xmas/all-r16-q-lr-decomp.pdf The file contains all curves of this sort (Dekking's "folding morphisms") of order 16. Curves whose order are divisors of 16 appear as well. The curves with 2-old rotational symmetry are omitted as there are always curves with "simple" L-systems giving the same shapes, covered in my prior search. The search for all curves of small orders is still running (since yesterday noon), but I already found about 2 million such curves with more than 80,000 different shapes. There might be still more curves if turns are allowed at the ends of the rules for R and L, or the maps for R and L are swapped. I am looking at this right now. Best regards, jj
but Julian had to lead me by the nose. http://gosper.org/4thdragon.png What a strange, camouflaged creature. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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