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Jun
23
comment Asymptotics for Ramsey Theory
Oops, when I said "original group" above I meant the set (not group) {1,2,...,n}.
Jun
23
comment Asymptotics for Ramsey Theory
PS Would it be useful if I expanded a bit on the regularity-lemma argument?
Jun
23
comment Asymptotics for Ramsey Theory
The trick with progressions is to work in the cyclic group of order n instead of the interval {1,2,...,n}. Then, at least if n is prime, the number of progressions of a given step size does not depend on the step size. Of course, not all progressions in the cyclic group correspond to progressions in the original group, but there are various tricks for dealing with this. (One, for instance, is to prove a lemma that every progression of length k in the cyclic group contains a "genuine" subprogression of length about $\sqrt{k}$, which comes from a pigeonhole argument.)
Jun
22
comment Digital Pen for Math: Your Experiences?
+1 for use of the word "quid" even if the rest of the answer had been uninteresting (which it wasn't).
Jun
14
comment How many definitions are there of the Jones polynomial?
I don't know enough to know whether you would count this as a genuinely different definition, but anyway I think Tutte deserves more publicity than he tends to get for his polynomial, which predates the Jones polynomial by many years. Given a knot, one can convert it into a graph and the Tutte polynomial of that graph restricts to the Jones polynomial: en.wikipedia.org/wiki/Tutte_polynomial, omup.jp/modules/papers/knot/chap01.pdf
Jun
13
comment Arithmetic closed subsets
Essentially the same question would be to describe sets of finitely supported non-negative-integer-valued sequences that are closed under the operation of adding disjointly supported sequences together (pointwise). Such sets don't seem to have a particularly interesting description.
Jun
11
comment Is this sequence of polynomials well-known?
Actually, I've just looked it up (though your second link takes me to the same page as the first) and I see that the answer is that even "unnatural" sequences such as those obtained by reading a triangle along its rows are included in OEIS, and that subsequences are recognised too. That makes it a more useful resource than I had realized.
Jun
11
comment Is this sequence of polynomials well-known?
Out of interest, what made you think that that sequence would yield something? And why not e.g. 1,1,1,1,3,2,1,7,12,6,1,15?
Jun
2
comment Examples of theorems misapplied to non-mathematical contexts
Wow, do you think he's making a bid for the Templeton Prize?
Jun
1
comment Why is differentiating mechanics and integration art?
The reason, I think, is that the analogy between the two situations isn't all that good. When you solve numerically you are trying to find a number. That coincides with what you are looking for when you solve a quadratic equation, whereas when you are antidifferentiating you are looking for a function, and numerical methods just give you function values.
May
31
comment Why is differentiating mechanics and integration art?
I'm not sure I buy your argument that the linearization "is somehow a simplification". True, linear functions are simple, but the linearization we perform when differentiating varies from point to point in a nonlinear way, and it's the entire derivative we're interested in here rather than just the derivative at a point. (This is particularly clear if we look at functions of more than one variable, when the derivative is a decidedly more complicated object than the original function.)
May
25
comment nontrivial theorems with trivial proofs
Some might call that a trivial theorem with a non-trivial proof ...
May
25
comment n-partite n-clique
I haven't checked, but it seems likely to me that a random graph will be a counterexample: you need a very high edge probability to get an n-clique and I think probably a lot lower to satisfy your conditions with high probability.
May
25
comment n-partite n-clique
FWIW here is a reformulation (I think). Let G be a graph with vertex set the edges of the complete bipartite graph K(n,n). Suppose that each vertex of G is contained in a perfect matching in K(n,n) and is joined to all the other edges in that perfect matching. Prove that G contains an n-clique.
May
25
comment Is anything known about this braid group quotient?
In a very elementary way I observed that when thinking about it. Using the third relation (the one that takes a string and passes it round all the other strings) and applying it to each string in turn, you get not a single twist but a double twist.
May
23
comment Is anything known about this braid group quotient?
Looking again at the paper by Gillette and van Buskirk, I see that (if I understand correctly) my group is the mapping class group but not the spherical braid group: the mapping class group adds an extra relation that allows you to twist the entire bottom of the braid through a full turn. So it's really the mapping class group that I'm interested in, though obviously the two are closely related.
May
23
comment Is anything known about this braid group quotient?
Ian Agol says in a comment below that Mosher showed that mapping class groups are automatic and hence that there is a polynomial-time algorithm for the word problem. Am I missing something here, or does that mean that the problem you mention was solved in 1995 (the date of Mosher's paper)?
May
18
comment Efficient computation of the least fraction with square denominator greater than the square root of 2.
Is it really the very best approximation you want to find, or would you be content with a non-trivially good one? (I'm not saying I have an answer even to that, but it feels quite a lot more feasible.)
May
14
comment Almost orthogonal vectors
Almost a year later I've just seen your comment. I had in mind that each point you put in rules out a spherical cap of spherical radius $\pi/2-\epsilon$. Since this has exponentially small volume, you can just greedily add exponentially many points. I think we must be talking at cross purposes though ...
May
14
comment A Bijection Between the Reals and Infinite Binary Strings
I should add that you need to encode whether the digit is before or after the decimal point.