Does P≠NP over ℝ imply P≠NP ?
where ℝ is for Real number algorithms as described by Smale with a suitable formulation of P≠NP over ℝ.
Complexity Theory and Numerical Analysis, Steve Smale, 2000
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.4678&rep=rep1&type=pdf
The local-global principle you are citing comes from the fact that for any open subgroup $Hleq G$, $H^n(G,A)stackrel{text{Res}}{longrightarrow}H^n(H,A)stackrel{text{Cor}}{longrightarrow}H^n(G,A)$ is multiplication by $[G:H]$. So from that you can derive lots of local-global principles. E.g. as a generalisation of the one you cite, you can deduce that if $H_1$ and $H_2$ are two open subgroups of co-prime index such that $H^n(H_i,A)=0$ for $i=1,2$, then $H^n(G,A)=0$.
In the simply connected case, essentially everything is in principle computable, by some very early work of E.H. Brown:
bib{MR0083733}{article}{
author={Brown, Edgar H., Jr.},
title={Finite computability of Postnikov complexes},
journal={Ann. of Math. (2)},
volume={65},
date={1957},
pages={1--20},
issn={0003-486X},
review={MR{0083733 (18,753a)}},
}
In particular, if $X$ and $Y$ are finite simplicial complexes then $[Sigma^n X,Sigma^n Y]$ is computable for $ngeq 2$, and for large $n$ this gives the group of stable homotopy classes of maps. However, I do not think that there are practical algorithms for many such questions, although I am not up to date on this. Probably the simplest case that I do not know is as follows: is there a practical algorithm to compute the complex $K$-theory $K^0(X)$ for a finite simplicial complex $X$?
To add to what Charles wrote, another reference is Mac Lane and Moerdijk's Sheaves in Geometry and Logic. They prove something a bit more general, involving Lawvere-Tierney topologies on a topos. For the purposes of understanding what I'm about to write, it's not necessary to know what a Lawvere-Tierney topology is.
Mac Lane and Moerdijk's book contains the following two results:
Let $mathcal{E}$ be a topos. Then the subtoposes of $mathcal{E}$ (i.e. the reflective full subcategories with left exact reflectors) correspond canonically to the Lawvere-Tierney topologies on $mathcal{E}$.
Let $mathbf{C}$ be a small category. Then the Lawvere-Tierney topologies on $mathbf{Set}^{mathbf{C}^{mathrm{op}}}$ correspond canonically to the Grothendieck topologies on $mathbf{C}$.
Result 1 is almost part of Corollary VII.4.7. The "almost" is because they don't go the whole way in proving the one-to-one correspondence, but I guess it's not too hard to finish it off. (Edit: it also appears as Theorem A.4.4.8 of Johnstone's Sketches of an Elephant, where Lawvere-Tierney topologies are called local operators.) Result 2 is Theorem V.4.1.
I agree with the point of view that Charles advocates. When I started learning topos theory I got bogged down in detailed stuff about Grothendieck topologies, and it all seemed pretty technical and unappealing. It wasn't until years later that I learned the wonderful fact that Charles mentions: an elementary topos is Grothendieck iff it's a subtopos of some presheaf topos. I wish someone had told me that in the first place!
When Harry needed help escaping Malfoy Manor, Aberforth Dumbledore sent Dobby to apparate them out of there:
... the mirror fragment fell sparkling to the floor, and he saw a gleam of brightest blue -
Dumbledore’s eye was gazing at him out of the mirror.
“Help us!” he yelled at it in mad desperation. “We’re in the cellar of Malfoy Manor, help us!”
The eye blinked and was gone.
(Deathly Hallows, Chapter 23, "Malfoy Manor")
Dobby would never be able to tell them who had sent him to the cellar, but Harry knew what he had seen. A piercing blue eye had looked out of the mirror fragment, and then help had come.
Help will always be given at Hogwarts to those who ask for it.
(Deathly Hallows, Chapter 24, "The Wandmaker")
He wore spectacles. Behind the dirty lenses, the eyes were a piercing,
brilliant blue.
“It’s your eye I’ve been seeing in the mirror.”
There was a silence in the room. Harry and the barman looked
at each other.
“You sent Dobby.”
The barman nodded and looked around for the elf.
(Deathly Hallows, Chapter 28, "The Missing Mirror")
So, when Harry begged for help, how did he know to send a House-Elf?
How did he know that Malfoy Manor had anti-apparation set up?
OK, being an old-timey wizard he might have guessed that it'd be the case, but I'd like canon support for why he thought so.
How did Aberforth know that a House-Elf would be a correct solution to that problem?
The fact that house-elf magic is different and especially that they can apparate in anti-apparation jinx conditions doesn't seem to be a widely held knowledge in the series; and Aberforth isn't "learned" like Albus.
It is an example of chiasmus, which is
- Repetition of ideas in inverted order
- Repetition of grammatical structures in inverted order (not to be mistaken with antimetabole, in which identical words are repeated and inverted).
See here or any other reputable source. (One of my favorites, from which the above is drawn, is Silva Rhetoricae.
My interpretation is that the syntax is correct, and "termination" is necessary. The more common idiom "to come up" (to emerge, happen, arise, be noticed) is being combined with the prepositional phrase "for termination". The law regarding freedom from land obligations "came up" (in that it came to people's attention) in 1863, and specifically for the purpose of being terminated.
Semantically, it does seem rather confusing, I will admit. I think the author's intent was that the obligations themselves came up for termination, not the freedom from the obligations, but I don't know enough about Russian history to say which actually happened.