@tb: No, but only if you think of a torus that way!
Of course, maybe our intuition for curvature is just not very good. After all, the cylinder S^1 x (0, 1) is flat even as an embedded submanifold of Euclidean space with the induced Riemannian metric.
@ZhenLin Oh, but all bar-resolutions you know arise from monads (that's why monads were called "standard constructions" before some people insisted on that terrible "triple" thing). For instance, the standard resolution in sheaf theory.
@ZhenLin I don't know -- I found it nice, but I already knew about spectral sequences before. I learned about them in Bott-Tu and Hilton-Stammbach. But many people are put off by exact couples.
But Chow's article is definitely worth having a look at.
Where's the hole in my argument: Let R be an integral domain (k-algebra of finite type, k algebraically closed), m a maximal ideal of R, dim R_m \ge dim R; but dim R_m = ht m \le dim R, so dim R = dim R_m.
Today I learned that cover spaces of pointed spaces correspond to subgroups of the fundamental group of the space. I went to ask if there's any related to Galois theory, and not surprising there is a deep relation which connected most of the advanced non-set theoretic courses together.
@ZhenLin Yeah, this is what was explained to me. However the words "alg. geo." were not said. I just know what is behind everything and it was apparent now how all that is connected.
@Asaf: The most amusing thing about it, really, is that classical Galois theory, from the geometric point of view, appears to be the study of covering spaces of a one-point space!
I was always good with set theory, and mediocre at best with the rest of the math... it was obvious for me. I was always enjoying large cardinalities. Infinite better than finite, etc etc.
@QED: Why don't you look at stuff you liked before and see if you can find some leads for further study? Good books almost always contain some pointers for exploring a subject further and deeper.
And similarly, does f_n converges weakly to f really mean \lim_{n \rightarrow \infty} \sup _{\| f_n - f\| \leq 1} |\phi(f_n) - \phi(f)| for all \phi \in (l^1)^\ast?
@tb: Confusing. The norm in the dual is sup \{|\phi(f)| \mid \| f \| \leq 1 \} but convergence in the dual is just convergence like in R instead of being sup \{|\phi(f) - \phi(f_n)| \mid \| f - f_n\| \leq 1 \}.
@Matt But we're not talking about convergence with respect to any norm, we're talking about convergence of real (or complex) numbers. If you evaluate a linear functional at a point in the Banach space, you get such a number. Now weak convergence f_n -> f means that for all continuous linear functionals \phi we have |\phi(f_n) - \phi(f)| -> 0
@JonasTeuwen I guess you missed the experiment the OP conducted on this site... He posted three or four questions in rapid succession essentially containing: what do you see: *** **
*le what? : D "bridging west and east through the medium of mathematics, the two great traditions of western science (cf boundaries of science) and eastern meditation (cf buddhist prescription)."
Well, I'm pretty sure hyperref uses some LaTeX specific things. I'm not sure about a substitute but I would think not as it would work with LaTeX as well. I'll check it out. One minute :-).
@tb hyperref does not work with plain TeX but you could write your own macros using tug.org/eplain.
One "type theory" is Whitehead & Russell's proposed foundation of mathematics in Principia Mathematica -- a higher-order set theory with particular syntactic constraints.
Another "type theory" is part of the abstract theory of programming language design in computer science.
There are genealogical connections between the two.
The "intuitionistic type theory" seems to touch upon both viewpoints.
So I linked to the second one you talk about, here's the page of the first one. Anyway, that question would have been better off with a just a little bit of information and background.
en.wikipedia.org/wiki/Type_theory covers both of the directions I spoke about, though it does not go very much in depth about computer-science development.
@Srivatsan: But I don't believe in age. Every time I have to drive somewhere by car I get this "Oh. I'm actually all grown up."-feeling. That's the only time when I realise that I'm a grown up. : )
@Srivatsan I just happened to notice them next to each other, and I couldn't tell whose was who's. Perhaps I just had nothing better to do at the moment. I am not sure exactly why.
Okay, now I'm happy. But this does require an argument, and after all I don't think it's easier than what I suggested in the comments there. Thanks guys!
The integral along the x+it/2 path and the integral along the x path are the same since there are no singularities (thus no residue) between them and the pieces at infinity drop out.
@robjohn I don't think measure theory would lose a single application when you add the requirement that all measures are semi-finite (every set of infinite measure contains a set of finite measure).
@Srivatsan right, exactly.
A counterexample like this just shows that the definitions include silly stuff. A good counterexample should involve things you could actually encounter.
@Srivatsan That may well be the case (and I suspect he didn't understand the answer I gave to the "duplicate" thread either). Anyway, I think the totally atomic case is the only one where you have uniqueness. And no, I wasn't planning on expanding on it, plug in the definitions and basta.
(remember that it's the same guy with the a[A] + b[B] question.)
Well, the way I see it is: his first question was good but turned out to be a duplicate. In an effort to non-duplicate it, he modified it to a "counter-* example" question. (I feel it's not that natural and all that, but leave that aside.) So I wouldn't blame that on him.
About the a[A]+b[B] one, that one suffered from a trivial counter-example, but if we cut that out by restricting 0<a<b, the resulting question is not half-bad.
I can see why you might think the op is yet to learn to walk. But I don't see where he is trying to run. =)
