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Jul
13
comment a Ramsey-type question
Oops, I meant category. The rough idea I had in mind was that if you cover the $n$-sphere with $m$ open sets, then it would be nice to show that one of them was non-trivial in some topological sense. The trouble is, they're all contractible, but perhaps one could hope that one of them intersected with minus itself is not contractible when considered as a subset of projective $n$-space. That's not quite the same as Lusternik-Schnirelman category but it seems to be in a similar ball park and I don't rule out some way of getting from one to the other.
Jul
12
comment a Ramsey-type question
What's known about the continuous version? That is, if you cover the sphere $S_n$ with m closed sets, what can be said about the structures that must be contained in at least one of the sets? If m=n+1 then we get two antipodal points, but what if m=1000? It seems that one ought to get much more. At the moment I don't even see a counterexample to the assertion that if m=n then you get a path from a point to the antipodal point, though that seems a bit optimistic. Am I asking about the Lusternik-Schnirelmann capacity of projective space or something like that?
Jul
11
comment What is the complexity of this problem?
Thank you for that interesting answer. In my comment on Dick Lipton's blog I stressed more the importance of the fact that I was looking at a fixed Cayley graph because without that the problem felt much more likely to be NP-complete. But it's nice to have pointers to actual results of that kind.
Jul
5
comment Uniformly Convex spaces
Oh, and it also won't change whether it's reflexive. So the answer to your question as asked is trivially no. Do you mean "isomorphic to a space that is uniformly convex"?
Jul
5
comment Uniformly Convex spaces
The question needs reformulating slightly, since you can just add a 2D space that's not even strictly convex and it won't change whether $\ell_1$ is finitely represented.
Jul
3
comment Examples of common false beliefs in mathematics
I've just added the missing html to fix the link.
Jul
1
comment A limiting product formula for the exponential function
I think the two suggestions above don't quite meet the demands of the question (that you should just expand and look at coefficients of separate powers).
Jul
1
comment A limiting product formula for the exponential function
I've been away from my computer all day, but the above is essentially the argument I had in mind ...
Jul
1
comment Generalizing a problem to make it easier
I have assuredly found an admirable and wholly elementary proof, but this comment box is too narrow to contain it ...
Jul
1
comment How to prove a known inequality from a book
My guess would be that to minimize the inner product, you make both vectors as "unflat" as possible (that is, you make sure that each one takes just two values and that the ratio of those two values is maximized) and as different as possible (that is, you make the big values of one vector appear where the small values of the other vector appear). Then all that's left to determine is the size of one set (where the big values of one vector occur), which ought to be doable. To prove all this, I would try a variational argument.
Jun
30
comment Name of a metric space concept
Oh yes -- it's like the uniqueness part of strict convexity but not the existence part. It's probably too much to hope that a metric space with Neil's property can be isometrically embedded into a strictly convex metric space in Bula's sense.
Jun
30
comment Name of a metric space concept
No time to check, but it looks related to Definition 2.6 in this paper: home.lu.lv/~ibula/lv/petnieciba/raksti/moravica.pdf
Jun
30
comment Zeros of functions involving polynomials and square roots
I can't think of any examples to show that the multivariable version of the question is more interesting than the single-variable version. If you fix all the variables but one, then the complex plane will, I think, split up into domains on which the resulting function is holomorphic, and therefore either constant or having isolated zeros. As Jacques Carette observes, both can occur. If you somehow add a condition to rule out constant parts, then you'll get measure zero. Or am I missing something?
Jun
29
comment Shortest closed curve to inspect a sphere
I presume you mean "closed curve in $\mathbb{R}^3$".
Jun
29
comment Generalizing a problem to make it easier
a multiple of U(x). But then I would have gained nothing by the generalization.
Jun
29
comment Generalizing a problem to make it easier
As it happens, the first example already appears on a subpage of the page linked to in the question: tricki.org/article/… (not that it would be reasonable to have expected you to check all subpages before giving your answer). As for your second example, it's not completely obvious what method you intend for evaluating S(x). Is it antidifferentiation? I myself would use the fact that S(x)(1-x) is of the form aT(x)+bU(x) where $T(x)=\sum kx^k$ and $U(x)=\sum x^k$ and that (1-x)T(x) is ...
Jun
24
comment Asymptotics for Ramsey Theory
I'm afraid I don't understand the approach you are suggesting, but the general class of difficult problem this one falls into is where any small example can be turned into a sequence of big examples, but the best small examples known are rather complicated, so one has no confidence in their being best possible or any hint of what the best possible would look like (and it's conceivable that the examples get more and more complicated as they get better and better).
Jun
23
comment Asymptotics for Ramsey Theory
Still very much open is what the minimum density actually is ...
Jun
23
comment Asymptotics for Ramsey Theory
One remark that I ought to have made is that the minimum number of monochromatic cliques is not what you might expect. You might think that the best example (when n is large) was a random colouring, but a famous result of Andrew Thomason is that if you 2-colour a graph and want to minimize the proportion of cliques of size 4 that are monochromatic, then there is a sequence of colourings that give a lower limiting density than 1/32 (which is what you get for a random colouring). This disproved a conjecture of Erdős.
Jun
23
comment Asymptotics for Ramsey Theory
Oops, when I said "original group" above I meant the set (not group) {1,2,...,n}.