Timeline for Can (how) one distinguish germs of continuous functions by a countable set of params?
Current License: CC BY-SA 3.0
14 events
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| Oct 27, 2014 at 19:11 | comment | added | Alexander Chervov | Function on germs: IF germ has derivative, then F(germ)=derivative, ELSE zero. This function seems to me quite "nice" and "constructive". Seems we do not need AC to define it. In contrast to your another example - "reals/rational translation" - it seems to me for that example there is NO "nice" function at all, is it correct ? So may be these two examples are different ? Pay attention that germs correspond to tail of sequences which have limit - may be property of having limit change something... | |
| Oct 27, 2014 at 18:26 | comment | added | Bjørn Kjos-Hanssen | @AlexanderChervov So maybe what you want to ask is, if we consider all possible properties $P$ such as fractal dimension, differentiability, etc., then do these determine the germ? Perhaps if the collection of such properties $\mathcal P$ is complicated then there is no way to given $\mathcal P$ construct a germ satisfying all members of $\mathcal P$, without choice. In which case, maybe you could find a description of $\mathcal P$ without choice. Interesting... | |
| Oct 27, 2014 at 18:14 | comment | added | Bjørn Kjos-Hanssen | @AlexanderChervov Re: "choose representative by definition": this works as long as you're only considering one germ, but if considering all germs it requires AC. | |
| Oct 27, 2014 at 17:46 | comment | added | Alexander Chervov | Remark. It seems to me fractal dimension is "nice" function on germs. But it seems you say that there are NO "nice" functions on germs (AC is necessary).... Some contradiction in my head | |
| Oct 27, 2014 at 17:25 | comment | added | Alexander Chervov | Germs = function/equivalence relation. It seems to me we can choose representative by definition - choose an a element in subset of equivalent functions. What is wrong ? And yes, it gives answer to my question, of course, it is not the anwer I hoped to, but formally it is an answer. I'll think how to modify my question to avoid this answer, but let me do it later... | |
| Oct 25, 2014 at 19:58 | comment | added | Bjørn Kjos-Hanssen | @AlexanderChervov Well otherwise you might as well take any continuous function as parameter for its germ.. or weave its values together into a single real number | |
| Oct 25, 2014 at 19:49 | comment | added | Alexander Chervov | Why do we need to choose representative ? | |
| Oct 25, 2014 at 18:53 | comment | added | Bjørn Kjos-Hanssen | @AlexanderChervov we can look at the values $f(x)$ for $x\in\mathbb Q$, $x$ close to $0$, to determine the germ of $f$ at 0. But to pick out one and only one representative of each germ requires choice | |
| Oct 25, 2014 at 18:17 | comment | added | Alexander Chervov | It seems you are saying that germs are isomorphic as a ring to tails of sequences with additional requirement that sequences should have a limit. Indeed, take x_n->0 and map a function to its values f(x_n), passing to germs corresponds to tails. Correct ? What is the role of AC ? | |
| Oct 23, 2014 at 2:59 | comment | added | François G. Dorais | Yes, and because $E_0$ is a minimal degree above smooth, it suffices to show that this equivalence is not smooth. | |
| Oct 23, 2014 at 0:03 | comment | added | François G. Dorais | Thanks for adding the relevant tags Bjørn! Right now, I only have time for my daily sweep of MO but the relevant keywords are $E_0$ and Borel equivalence relations. | |
| Oct 22, 2014 at 19:22 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |
added 2 characters in body
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| Oct 22, 2014 at 19:19 | comment | added | Alexander Chervov | Thank you ! First sentence : analytic - you mean continuous? | |
| Oct 22, 2014 at 16:59 | history | answered | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |