bio | website | |
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location | Berkeley | |
age | ||
visits | member for | 3 years, 10 months |
seen | yesterday | |
stats | profile views | 752 |
Feb 20 |
comment |
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 |
comment |
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 |
comment |
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 |
comment |
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 |
comment |
Algebraic characterization of real differentiation
With respect to what topology should the rational functions be dense and the operator continuous? |
Feb 17 |
comment |
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. |
Feb 15 |
comment |
Error of midpoint method for functions that are not twice-differentiable
Hi James. I improved the notation and added some details. (The function $f$ does not depend on $n$.) Let me know if that helps. |
Feb 15 |
revised |
Error of midpoint method for functions that are not twice-differentiable
added details and clearer notation |
Feb 15 |
asked | Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$? |
Feb 15 |
answered | Error of midpoint method for functions that are not twice-differentiable |
Feb 11 |
awarded | Revival |
Feb 11 |
answered | Recursive Non-Well-Orders that are Sneaky, but not THAT Sneaky. |
Feb 6 |
revised |
Sneaky Recursive Non-Well-Orders
fixed ordinal notation |
Feb 6 |
accepted | Sneaky Recursive Non-Well-Orders |
Jan 10 |
comment |
What are the most attractive Turing undecidable problems in mathematics?
Peter -- It's true that this is not an example of an intermediate Turing degree, but it is also not a reduction to the halting problem, because the problem of verifying differentiability is much more difficult than the halting problem. Because differentiability is naively a $\Pi^1_1$ property, the problem is reducible to Kleene's O. This is optimal by a 1936 paper of Mazurkiewicz. (His argument is reproduced in Kechris and Woodin's 1986 paper "Ranks of Differentiable Functions" as the original is hard to find. The original argument was set-theoretic but it effectivizes.) |
Dec 10 |
comment |
Representability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)
@Stefan in that case it would seem that if we write $K_n(\xi_n)$ in your desired form, then $K_n(\xi_n) = w\cdot X_i$ where $X_i$ must be a set containing exactly one element which has least period $n$. Does that answer your question? |
Dec 9 |
comment |
Representability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)
@Stefan Ah whoops, I meant for $\xi_n$ to have period $n$ but then went and wrote the wrong thing. What is $K_n(\xi_n)$ if $\xi_n = \underbrace{0\dots 0}_{n-1} 1 \underbrace{0\dots 0}_{n-1} 1 0\dots$? I ask because under my current reading of your question, I think $K_n(\xi_n) = \{ \xi_n \}$ (i.e. $K_n(\xi_n)$ has just one element), but I wanted to clarify that because I am still doubting whether I correctly understand your notation. |