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Mathematics professor at Cambridge

Aug 3 |
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Lebesgue non measurability in the plane
See Pietro Majer's comment below. I think it deals with your objection. |

Jul 20 |
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Method for variable substitution in multiple summation
I think you need to be clearer about what the substitution should achieve. Otherwise, the question is a bit too vague. In your examples, I don't see any great simplification resulting from the substitutions, which is what confuses me. |

Jul 18 |
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Examples of non-abelian groups arising in nature without any natural action
That's the group of affine transformations of the rationals: (i,p) sends x to px+i, so (i,p)*(j,q) sends x to p(qx+j)+i=pqx + i+pj. |

Jul 17 |
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Analysis pathology
To be clear, the derivative I'm talking about is the derivative of the function that's $x^2\sin(1/x)$ when $x\ne 0$ and 0 at 0. |

Jul 17 |
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Analysis pathology
Doesn't the very standard a=2, b=1 case of Gerald's suggestion do the job? It's THE example of a derivative that isn't continuous, but since it's bounded and the discontinuity is at just one point, it's integrable. I'm not sure this counts as a research-level question ... |

Jul 15 |
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Lower bounding the maximum size of sets in a set family with union promise
So s depends on $\epsilon$ and the sets $C_1,\dots,C_k$? |

Jul 14 |
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Current status of Waring-Goldbach problem
Sorry, I take that back -- I was thinking of the result with zeros. But I think the zeros are needed for congruence reasons and it's not clear that one couldn't use some trick. E.g. if k=6 then each kth power is 1 mod 7, except that we can throw in a few $p_i=7$s to deal with that. |

Jul 14 |
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Current status of Waring-Goldbach problem
I'm pretty sure this is known for all k but I'm afraid I don't know a reference. |

Jul 14 |
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Lower bounding the maximum size of sets in a set family with union promise
With your use of the words "given" and "let" I find it hard to work out exactly how the quantification works in your question. Would it be possible to ask it in a more formal way? |

Jul 13 |
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a Ramsey-type question
Yes it was then. |

Jul 13 |
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a Ramsey-type question
By the way, I find this cluster of related questions you are asking very appealing. (My attention was first drawn to them by Noga Alon.) |

Jul 13 |
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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 |
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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 |
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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 |
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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 |
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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 |
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Examples of common false beliefs in mathematics.
I've just added the missing html to fix the link. |

Jul 1 |
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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 |
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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 |
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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 ... |