Timeline for Why is differentiating mechanics and integration art?
Current License: CC BY-SA 3.0
17 events
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| Jun 9, 2011 at 6:53 | comment | added | vonjd | @Terry: I would very much appreciate if you could give me a hint if I am on the right track with my last comment. The problem still bothers me a lot... Thank you! | |
| May 31, 2011 at 7:58 | comment | added | vonjd | @Terry: Just for the sake of intuition an oversimplification: Having a picture, basically you’re saying that it is easier to find a closed form for a sharpened version of it which is tantamount to finding a linearization of the function in every single point (but only those points respectively) - than finding a closed form for a blurred version of it because you have to include the information of the whole picture in every single point (which is a de-linearization in its nature, i.e. finding a non-linear function from all the surrounding linear approximations). Is that basically correct? | |
| May 31, 2011 at 1:13 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 31, 2011 at 1:05 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 31, 2011 at 0:53 | comment | added | Terry Tao | Unfortunately, rigorously proving lower bounds on circuit complexity is a notoriously difficult task. But at a heuristic level, once an operation takes an unbounded number of operations to compute by a "naive" method, and the amount of algebraic structure present does not seem strong enough to suggest a significantly "smarter" method than the naive method, then the default assumption would be (barring sporadic or otherwise rare coincidences) that the operation is truly complicated. This is not a proof of anything, but it does provide guidance as to what to expect with such inverse problems... | |
| May 30, 2011 at 21:00 | comment | added | Ryan Reich | I'm un-downvoting but I'm still not sold on the local/global business. However, I'm curious if you could expand on your idea of elementary functions being those of "bounded complexity". Is there some way you can look at a function (like $e^{-x^2}$) and say that this O(N) complexity of doing Riemann sums will somehow push it over the line? | |
| May 30, 2011 at 20:33 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 30, 2011 at 20:25 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 30, 2011 at 20:22 | comment | added | Terry Tao | I've expanded upon my original answer in response to the above comments. | |
| May 30, 2011 at 20:19 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 30, 2011 at 17:01 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| May 30, 2011 at 16:44 | comment | added | Ryan Reich | Yes, that's it. | |
| May 30, 2011 at 16:37 | comment | added | Mariano Suárez-Álvarez | My guess is, Ryan finds this unsatisfactory because maybe the question is referring of the algorithmic nature of the problems of integration and differentiation (one being essentially trivial when starting from a formula, the other not so much) and it is not clear how Terry's answer relates to that. | |
| May 30, 2011 at 16:24 | comment | added | Ryan Reich | @Peter: In complex analysis, it is easy to prove that a holomorphic function is infinitely differentiable using Cauchy's integral formula, but very difficult to do so just from the definition of a derivative. The integral formula uses a path that is necessarily bounded away from the point at which you want to differentiate. So is differentiation still easier than integration just because it's local? | |
| May 30, 2011 at 16:03 | comment | added | Peter Luthy | Ryan --- he's saying that computing the derivative is simpler because it only depends on local information at a point. Integration depends on information about every point on a fixed interval, simultaneously. Thus differentiation is fundamentally simpler than integration, simply from the definitions. Doesn't it seem a lot easier to organize the cutting down of a forest one tree at a time than having to cut them all down simultaneously? | |
| May 30, 2011 at 15:05 | comment | added | Ryan Reich | I'm afraid I have to vote this down. How does the local-ness of differentiation make it more mechanical than integration? It seems to me (especially after the latest edit) that the question is about symbolic calculus, but your answer is about theoretical analysis. Though it would be interesting, if you had this in mind, to know how the global nature of integration influences, say, its effect on elementary functions. | |
| May 30, 2011 at 14:48 | history | answered | Terry Tao | CC BY-SA 3.0 |