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Mar
5
comment Linear Algebra Proofs in Combinatorics?
Actually yes ... It's possible to see who Tricki articles are by if you look at their revision history.
Mar
4
comment Counting distinct undirected, partially labelled graphs
Sorry, I meant triples closed under the given rotation (some, but not all, of which give you triangles). If the vertices are 0,1,2,3,4,5, then some examples of such triples are 01,23,45 and 03,14,25, and also the equilateral triangles 02,24,40 and 13,35,51. In fact, there's one more triple, namely 14,25,30, and any graph with rotational symmetry of order 3 is a union of these triples, so there are 32 such graphs.
Feb
27
comment Can a connected planar compactum minus a point be totally disconnected?
Is there an example of a compact connected set such that no two points can be joined by a path?
Feb
26
comment Heuristic argument for the prime number theorem?
That may indeed be exactly what I am looking for. I'll have to digest it carefully to see.
Feb
24
comment Value of “of course” in the mathematical literature
Was it by any chance me? Probably not, but it is something I have often said. But I didn't think it up for myself -- I got it from David Preiss. It's useful not just for evaluating the work of others, but also one's own work. That is, if you've just written a proof of something but don't feel quite secure about it, look for the bits where you didn't give full detail.
Feb
7
comment alternative construction of the quotient group
I don't claim it's a good way of doing things, but one could (in desperation) argue that here one is defining quotients in just one situation (what you need to define a group in terms of generators and relations) and getting all quotients out of it. But I wouldn't want to go to the wall on this one ...
Feb
3
comment Why does the Riemann zeta function have non-trivial zeros?
That is a very useful comment -- thanks!
Feb
2
comment Why does the Riemann zeta function have non-trivial zeros?
That was another of the thoughts that lay behind my question. Somehow the fact that the distribution of primes can't be "better than random" feels like a fact that ought to have an elementary proof using some Parseval-like identity. I suspect the zeta function is sort of doing that (with a Mellin transform rather than a Fourier transform), but it doesn't appear to be saying something simple like, "That function has the same L_2 norm and trivially has L_2 norm at least the square root of n."
Feb
1
comment 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
comment 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
comment Proving “almost all matrices over C are diagonalizable”.
Or you could simply upper-triangularize your matrix and do the same.
Jan
15
comment 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
comment 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
comment 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
comment Undecidable graph problems?
Yes, I was using it as a synonym for "independent", as the sometimes is used that way.
Jan
7
comment 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
comment 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
comment 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
comment 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
comment 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).