Timeline for Does any research mathematics involve solving functional equations?
Current License: CC BY-SA 2.5
33 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2011 at 20:08 | history | edited | Martin Rubey |
edited tags
|
|
| Jan 28, 2011 at 19:44 | comment | added | Timothy Chow | The introduction to Small's book on functional equations gives a partial answer to your question. amazon.com/Functional-Equations-Solve-Problem-Mathematics/dp/… Small is concerned primarily with IMO-style problems but he gives a historical overview of the subject and mentions several connections to research mathematics. | |
| Jan 28, 2011 at 11:09 | comment | added | Anixx | By the way, I would not say the equation in your example is that contrived. It can be reduced to a linear equation: $xf^{[-1]}(x)+c_1=f^{[-1]}(c_2+x^2)$, where $f^{[-1]}(x)$ is inverse functiom of $f(x)$, with a condition $c_1=f(c_2)$ | |
| Jan 27, 2011 at 23:59 | vote | accept | Qiaochu Yuan | ||
| Jan 27, 2011 at 23:53 | answer | added | Terry Tao | timeline score: 46 | |
| Jan 27, 2011 at 21:21 | answer | added | Anixx | timeline score: 2 | |
| Jan 27, 2011 at 19:26 | answer | added | Mark Lewko | timeline score: 3 | |
| Jan 27, 2011 at 15:24 | comment | added | ndkrempel | @darij: Great idea - when will the newspapers start including functional equations on their puzzle pages? | |
| Jan 27, 2011 at 14:56 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 559 characters in body
|
| Jan 27, 2011 at 14:29 | answer | added | Primoz | timeline score: 5 | |
| Jan 27, 2011 at 14:23 | answer | added | ndkrempel | timeline score: 10 | |
| Jan 27, 2011 at 13:27 | answer | added | Joe Silverman | timeline score: 14 | |
| Jan 27, 2011 at 11:24 | comment | added | Qiaochu Yuan | @darij: I agree. The question I intended to ask is the second (I am not surprised that you were able to pick up on this) and I hope that addresses Terry Tao's concern. | |
| Jan 27, 2011 at 10:53 | comment | added | darij grinberg | ... is useful, admittedly. Other than that, solving func. eqn.s have always reminded me more of solving crossword puzzles or hacking servers than of real mathematics. | |
| Jan 27, 2011 at 10:52 | comment | added | darij grinberg | ... hidden group action, exploiting a symmetry), but most of the tricks (substituting 0, 1, -1, x=y, etc.; exploiting monotony proving that functions are continuous; etc.) I have never seen used anywhere outside of func. eqns., except in two places: (a) proofs that some axioms of algebraic structures can be omitted (e. g., if I remember correctly, one can omit the commutativity of + in the definition of a field) and (b) the proof that any homomorphism between fields must be injective. While (a) can be research mathematics, it seems more like an art-pour-l'art to me, without any uses. (b) ... | |
| Jan 27, 2011 at 10:48 | comment | added | darij grinberg | I think two questions should be distinguished here: (1) Does any research mathematics involve solving (nontrivial) functional equations (by any method whatsoever)? (2) Does any research mathematics involve the methods usually employed for olympiad-style elementary functional equations? While the answer to (1) is clearly a "yes" (differential equations, difference equations etc.), I believe the answer to (2) is a No in the sense that most methods I have learnt in my olympiad time are utterly useless to me now. Of course, some func. eqn. problems HAVE conceptual solutions (such as finding a ... | |
| Jan 27, 2011 at 10:35 | comment | added | Terry Tao | It's somewhat strange to exclude any example that involves a group action. If we've learned anything in the last century, it is that a majority of interesting mathematical structure that are worth considering are tied to the actions of groups (or group-like objects)... | |
| Jan 27, 2011 at 10:03 | answer | added | Qfwfq | timeline score: 4 | |
| Jan 27, 2011 at 8:55 | comment | added | gowers | I have seen differential equations described as functional equations, though it's clear that you are using a definition that excludes them ... | |
| Jan 27, 2011 at 8:48 | answer | added | JBorger | timeline score: 3 | |
| Jan 27, 2011 at 8:20 | answer | added | Martin Brandenburg | timeline score: 18 | |
| Jan 27, 2011 at 7:23 | answer | added | Denis Serre | timeline score: 17 | |
| Jan 27, 2011 at 3:36 | answer | added | Gerhard Paseman | timeline score: 2 | |
| Jan 27, 2011 at 2:23 | comment | added | Yemon Choi | @Qiaochu: thanks. I think "research mathematics" is a lot less ambiguous or subjective. | |
| Jan 27, 2011 at 2:23 | answer | added | Will Jagy | timeline score: 4 | |
| Jan 27, 2011 at 2:12 | comment | added | Theo Johnson-Freyd | At the trivial end, people care about idempotents --- functions $f$ satisfying $f(f(x)) = f(x)$. But this certainly does not get to the heart of the matter. I like the question quite a lot --- +1, by the way --- although I expect the only satisfying answer will be of the form "Yes, let me (an expert) describe for you in rich detail my work, which does deal with solving particular functional equations." (I could also imagine an answer "No, here are some no-go theorems that there cannot be an interesting theory", but I doubt such theorems exist, and would be dubious of any such claim.) | |
| Jan 27, 2011 at 1:58 | comment | added | Qiaochu Yuan | @Yemon: changed to "research." Does that sound better? | |
| Jan 27, 2011 at 1:58 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
edited title
|
| Jan 27, 2011 at 1:45 | answer | added | NebulousReveal | timeline score: 17 | |
| Jan 27, 2011 at 1:33 | comment | added | Willie Wong | maybe try searching MathSciNet for all recent papers with primary MSC 39BXX? | |
| Jan 27, 2011 at 1:23 | answer | added | Pietro Majer | timeline score: 12 | |
| Jan 27, 2011 at 1:19 | comment | added | Yemon Choi | I'm ambivalent about your actual question, but I would strongly prefer it if you changed your title. Surely you can find an adjective other than "serious" to describe what you mean? | |
| Jan 27, 2011 at 1:02 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |