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Apr
9 |
accepted | Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code |
Apr
6 |
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Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
I'm glad to know of this nice paper of Hinman. As for my question, I'll accept your answer of "folklore" in a few days if no references come up before then. |
Apr
5 |
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Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
(I have a pdf now, thank you MO!) |
Apr
5 |
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Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Thank you for the reference. I couldn't find an electronic copy in the usual places, so my response will take a little longer. (If anyone has a pdf handy I would be grateful for the share.) |
Apr
5 |
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Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Liang Yu, thank you for the nice example. It is short and pleasing, but different from what I am looking for in two ways. First, the fact I wish to cite would give the exact degree of unsolvability for each $\alpha$, while this example leaves a few jumps' gap. (The gap can be closed by coding $0^{(\alpha)}$ into the $U$.) Second, I'm looking for a reference, not a proof sketch. When I need to cite folklore, I will generally write up a proof for the sake of the literature's completeness. However, I prefer to hope that someone wrote it already! |
Apr
4 |
asked | Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code |
Jul
2 |
awarded | Curious |
May
2 |
awarded | Nice Question |
May
1 |
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Is this property equivalent to Lusin's property (N) for continuous functions?
And what a marvelous book it is, both efficient and clear. I think I'll be buying a Dover copy myself. Thank you Christian! |
May
1 |
accepted | Is this property equivalent to Lusin's property (N) for continuous functions? |
Apr
29 |
asked | Is this property equivalent to Lusin's property (N) for continuous functions? |
Feb
20 |
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Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Saks does seem to have this, though it is not set aside as a theorem, at a first look-through I think all the pieces are there. It has a very pleasing parallel analysis of the narrow and wide cases in Chapter VIII, sections 1,4 and 5. Thank you! |
Feb
20 |
accepted | Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$? |
Feb
20 |
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Error of midpoint method for functions that are not twice-differentiable
I meant with the checkbox, but in any case I'm glad it helped. |
Feb
19 |
awarded | Yearling |
Feb
18 |
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Error of midpoint method for functions that are not twice-differentiable
There are certainly some hypotheses one could add to get rid of the kind of bad behavior above. For a very weak example, I don't know how to make an $O(1/n)$ convergence with an absolutely continuity restriction. If you're satisfied with my answer, though, I hope you will accept it :-) |
Feb
18 |
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Algebraic characterization of real differentiation
For example, the polynomials are dense in $C^1[a,b]$ for the topology of uniform convergence (Stone-Weierstrass). But the derivative operator is not continuous with respect to this topology. This example does not contradict your answer, nor does it directly address the OP. But the thing I am confused about is under what circumstances would one be able to use your answer to identify the derivative. |
Feb
17 |
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Algebraic characterization of real differentiation
With respect to what topology should the rational functions be dense and the operator continuous? |
Feb
17 |
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Error of midpoint method for functions that are not twice-differentiable
Now I think I understand your previous comments -- you are just as interested in the "$o$" as the "$1/n$". I added a nasty function for which the error fails to be $o(1/n)$. |
Feb
17 |
revised |
Error of midpoint method for functions that are not twice-differentiable
Added an example which illustrates that $O(1/n)$ is an exact bound. |