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Mar
28 |
comment |
Recursively defined numbers vs geometric series
@JasonStarr. I do mean that $h$ is a rational function of $d_1$ and $d_2$. It can be assumed to be symmetric. In the context of Kontsevich's formula $h(d_1,d_2)\!=\!f(d_1\!+\!d_2,d_1d_2)/g(d_1\!+\!d_2)$ for some polynomials $f$ and $g$. Since the number of $(d_1,d_2)$ with $d_1\!+\!d_2\!<\!2d(\epsilon)$ is finite, there is a bound on the corresponding $h(d_1,d_2)$, but it depends on $\epsilon$. |
Mar
28 |
comment |
Recursively defined numbers vs geometric series
@JasonStarr. . With a specific and somewhat ugly $h$, this recursion is equivalent to Kontsevich's formula for $2^dn_d/(3dâˆ’1)!$; there is more in my post from 2 days ago. While only this specific $h$ is relevant in this context, I find the general question interesting and more natural. I realize that $n_1$ can equivalently be taken to be any positive real. |
Mar
28 |
comment |
Recursively defined numbers vs geometric series
@JasonStarr. Since this sequence is bounded above and below, I do not find it too surprising that it has many increasing subsequences. Of course, it could be just decreasing. This would be fine for my purposes too, since I actually want to know that $\sqrt[d]{n_d}$ has a limit (in relation to my question from 2 days ago). |
Mar
28 |
asked | Recursively defined numbers vs geometric series |
Mar
28 |
comment |
Asymptotic analysis of generating functions
@JayPantone. The paper is hep-th/9412175. My $F$ is (2.54) after normalizing by $a^d$, with $a$ as in (2.59). The expansion is (2.56), along with (2.58). The limsup statement is a weaker version of (2.59). The ODE is (2.55). I'll post a note on arXiv on this with related observations and speculations from 3 years ago soon, now that I know that even the limsup conclusion is not a triviality. One of the speculations is my next MO posting. |
Mar
28 |
comment |
Asymptotic analysis of generating functions
@GeraldEdgar. Thank you for the suggestion; it looks like a nice book. Unfortunately, none of the three cases of Chapter 5 covers polylogarithmic singularities. Thm 5.3.1 is about algebraic singularities. On the other hand, looking at this book suggested to me how to think about the first part of my question and to realize that the answer to it is no. There could be a subsequence of $a_d$ with values of say $d^{-3}$. This would mean the limsup is infinite, but the expansion could be valid (if these value appear rarely enough). So, even the weaker conclusion requires some clever use of the ODE. |
Mar
25 |
asked | Asymptotic analysis of generating functions |
Nov
19 |
awarded | Commentator |
Nov
19 |
comment |
Reference request for cohomology of coverings
@BenWieland. It sounds like there is no reasonable statement along the lines of the original question then. This is very enlightening. Many thanks, Ben, as well as Matthias and Dan. |
Nov
18 |
comment |
Reference request for cohomology of coverings
This answer was meant to give an alternative description of Matthias's answer for cyclic coverings. As Ben points out, this does not suffice for the inductive step and there is no hope of extending this construction even to ${\mathbb{Z}}^2$-coverings without additional assumptions. |
Nov
18 |
comment |
Reference request for cohomology of coverings
Is there a statement in the spirit of the original one which is true? It is true if the rank of $G$ is at most one. Is it true if the abelianization of the commutator subgroup of $\pi_1(B)$ is finite? or perhaps even the commutator subgroup itself is finite? |
Nov
18 |
comment |
Reference request for cohomology of coverings
Thank you, Ben. I think I see where my thinking with the induction went wrong. $H^*(\hat{B})^{\mathbb Z}$ comes from $H^*(\hat{B}/{\mathbb Z})$, but only $H^*(\hat{B}/{\mathbb Z})^{\mathbb Z}$ comes from $H^*(\hat{B}/{\mathbb Z}^2)$. More relevantly, one needs to show that $H^*(\hat{B})^{\mathbb{Z}^2}$ comes from $H^*(\hat{B}/{\mathbb{Z}})^{\mathbb{Z}}$, and there was nothing in the inductive step about some auxiliary group. |
Nov
16 |
answered | Reference request for cohomology of coverings |
May
31 |
awarded | Critic |
Apr
14 |
comment |
On the proof of Robert Lipshitz's formula on Maslov index.
Why not just e-mail, Robert? |
Mar
31 |
answered | Difference between parallel transport and derivative of the exponential map |
Feb
19 |
awarded | Enthusiast |
Jan
29 |
answered | Analytic curve on Riemann surface |
Jan
28 |
revised |
Tubular neighborhoods of chains
added 36 characters in body |
Jan
28 |
revised |
Tubular neighborhoods of chains
added 156 characters in body; deleted 103 characters in body |