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Mathematics professor at Cambridge
Feb 1 |
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Why does the Riemann zeta function have non-trivial zeros?
That is a very nice argument, but it also has a magic flavour to it, since you somehow manage to bootstrap a very small error (arising from the fact that $\psi_0(x)$ is discontinuous) into a much bigger one (that the error term in PNT must be more like a square root). But perhaps the bootstrapping is done by the functional equation rather than your argument. |
Feb 1 |
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Why does the Riemann zeta function have non-trivial zeros?
It's precisely this issue -- why the error term in PNT isn't absolutely tiny -- that I want to understand. E.g. to prove that π(x) does not approximate $Li(x)$ to within $latex x^{1/3}$, the obvious method is to point to the zeros on the critical line. So I'm going round in circles. With the help of the functional equation one can say that if there are no zeros on or to the right of the critical line then there are none at all, but I don't count that as an intuitive argument. |
Jan 22 |
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Proving “almost all matrices over C are diagonalizable”.
Or you could simply upper-triangularize your matrix and do the same. |
Jan 15 |
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Improving a sequence of 1s and -1s
Quick question: does the set of 1s in a quasiperiodic sequence have to have a well-defined density? I keep trying to find a counterexample and I keep failing. |
Jan 15 |
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Has mathoverflow yet led to mathematical breakthroughs?
Apologies if I have asked the question in the wrong place -- meta isn't where it should be in my consciousness but I'll put it there now. |
Jan 15 |
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Has mathoverflow yet led to mathematical breakthroughs?
My own feeling about use of the internet is that one can more or less prove that it isn't "crucial" in the fully anal sense, because there are other means of communication. But it can be crucial in the weaker sense of being so much more efficient that it makes the difference between progress not happening and happening. Your example isn't quite like that, since you'd presumably have completed your research in reasonable time without mathoverflow, but it's quite far along a spectrum towards that. |
Jan 9 |
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Undecidable graph problems?
Yes, I was using it as a synonym for "independent", as the sometimes is used that way. |
Jan 7 |
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Partial sums of multiplicative functions
I was fairly sure that partial sums of mu were not better than the square root of n, but I didn't in fact know this argument, so thanks for giving it. I'll think about whether it can be adapted to work for the Liouville function. |
Dec 6 |
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Why is it useful to study vector bundles?
OK, in that case I think one has to turn to more sophisticated answers such as that you can use them to form K groups. If you'll excuse the indirect self-promotion, I'd recommend Burt Totaro's article on algebraic topology in the Princeton Companion to Mathematics, where he has quite a lot to say about bundles and why they are important. |
Dec 5 |
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What are the most misleading alternate definitions in taught mathematics?
I would almost prefer not even to say what a function is at all. I'd just say that if f is a function from A to B and x is an element of A then f(x) is an element of B. And that's all you need to know. Of course, I'm exaggerating a bit, and this point of view is not sufficient after a while (e.g. how would you decide whether the set of functions from A to B is countable, how would you define function spaces, etc.?) but in some situations this is the most important fact that you need from the basic definition of functions. Of course, one would also give examples, including artificial ones. |
Dec 5 |
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What are the most misleading alternate definitions in taught mathematics?
I totally agree with this and always tell students to think of "kernel of some homomorphism" as the definition and "closed under conjugation by any element of G" as a fact that can be shown to be equivalent to it. |
Dec 1 |
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k-pseudorandom measures
The problem there is that the sets we are looking at are so sparse that the U^k norm of any function is dominated by the degenerate cubes and so tells you nothing. Also, the control of one norm by another is often OK in the dense case but breaks down if you are sitting inside a sparse random set (because the constant depends on the sparsity of the random set). |
Nov 29 |
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k-pseudorandom measures
It's not true that we removed the correlation condition -- that question is still open. What we did was look at functions bounded by random as opposed to pseudorandom measures, and we obtained best possible results by considering a specially constructued norm rather than the $U^k$ norm. The paper will be posted on the arXiv soon. |
Nov 21 |
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Can infinity shorten proofs a lot?
Not the latter as I go by my middle name. Happy to make the change but have not managed to find where I can do it. (Please excuse my utter incompetence.) |
Nov 20 |
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Continued fractions using all natural integers
I think you're right -- and indeed I was worried that something like that would happen. So it seems that the quotients have to grow rather fast to make the number transcendental. |
Nov 18 |
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Can infinity shorten proofs a lot?
I'm not sure I count induction either. It seems to me that one is not relying on infinity in a serious way to sum the first million cubes: one is saying, "If it's valid for 1 then it's valid for 2, and if it's valid for 2 then it's valid for 3, and ... and if it's valid for 999999 then it's valid for 1000000." That is a ridiculously long proof, but induction allows one to shorten it by giving a rule for when it's OK to put in the "..." (A more formal way of making the point is that the axiom of infinity isn't part of first-order Peano arithmetic.) |
Nov 17 |
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Can infinity shorten proofs a lot?
You can get the inequality quite easily I think. The log of n! is the sum of log m from 1 to n, which is at least the integral from 1 to n of log x, which is nlogn-n. Done. |
Nov 16 |
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Can infinity shorten proofs a lot?
That's a good example, but I think it may already form part of the presentation (when they talk about infinite sets and the work of Cantor). I'll check though. |
Nov 16 |
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Can infinity shorten proofs a lot?
Ultimately what's wanted is a very nice demonstration for the non-mathematician of why infinity is useful even if all you care about is finitary results. |
Nov 2 |
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Are there any interesting examples of random NP-complete problems?
Hmm, I'm wondering now if there's a boring answer, which is to take a problem that is hard on average and randomly restrict it in some natural way to a small class of instances. If that works, then maybe the question is interesting only in particular cases and not as a general question. |