642 reputation
411
bio website
location Berkeley
age
visits member for 3 years, 10 months
seen yesterday

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.