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May
25
comment Major mathematical advances past age fifty
As I understand it, Apery, who has already been mentioned, remains a good example even if you restrict the question in this way.
May
24
comment Making N (set of all positive integers) a group
Ah -- I had a look at the Wikipedia article on those but should have looked more carefully.
May
23
comment Some models for random graphs that I am curious about
Gil, this feels like things that Bela Bollobas, Svante Janson and Oliver Riordan have thought about. If you don't mind using old-fashioned methods, then contacting one of them by email might be a good starting point.
May
22
comment Is there an intuitive explanation for an extremal property of Chebyshev polynomials?
I feel as though none of the answers so far quite answers the question. There are many beautiful characterizations of the Chebyshev polynomials, but what does any of them have to do with the minimization problem in the question? But perhaps the answer to the OP is that the Chebyshev polynomials were thought of for a different reason, and it then became clear what else they could do. (I don't know whether that is the actual story -- does anyone know their history here?)
May
20
comment Fundamental theorems
I don't know how standard they are, but a Google search reveals that at least some people refer to fundamental theorems of Galois theory, space curves, projective geometry and Riemannian geometry.
May
19
comment Another mixed mean inequality
I like the question and am tempted to have a go -- but no time right now.
May
18
comment Order types of positive reals
A small remark: I once gave a graduate-level course in which I wanted to do transfinite induction over the countable ordinals but didn't want to spend time developing the theory of ordinals. So I defined the countable ordinals as equivalence classes of well-ordered subsets of the reals, which is the kind of thing one would like to do for the ordinals themselves but cannot because of set-theoretic paradoxes. It worked nicely and was completely rigorous.
May
15
comment Is this strange problem NP-complete?
Or perhaps that should be nlogn.
May
15
comment Is this strange problem NP-complete?
That is indeed the point of the question -- how easy is it to determine whether it can be simplified in this particular way? Obviously it can be simplified in the normal way in linear time.
May
15
comment Is this strange problem NP-complete?
I actually checked that statement, but what you write demonstrates that my check came to the wrong conclusion. I'm fairly sure, but not certain, that irrelevant common factors can exist.
May
15
comment Is this strange problem NP-complete?
What is the obvious algorithm? The only obvious algorithm I can see appears to take exponential time.
May
12
comment What does it mean for a mathematical statement to be true?
If someone asks you why it is true, then you will deduce it from something more basic. And if they ask you why that is true, you will deduce it from something more basic still. What you won't say (unless you want to leave your questioner very dissatisfied) is that it's just a Platonic fact about the mathematical universe. Obviously these "why" questions have to come to an end eventually, and ... bingo ... there are your postulates.
May
8
comment Measuring how “heavily linked” a node is in a graph
Alternatively, take a lead from Google and work out the eigenvector with largest eigenvalue. That gives you a pretty good measure of which vertices are most linked, taking into account the linking of the neighbours.
May
7
comment Does this approach for the Poincaré conjecture work?
I should add that it is clear from a glance at the paper that he is not a crackpot: it looks like real mathematics that's wrong, rather than weird stuff that doesn't even count as mathematics.
May
7
comment Does this approach for the Poincaré conjecture work?
I agree that the God sentence is not evidence enough on its own. But there is another important piece of evidence: the paper is short and has been in the public domain since January. If it were correct, it would surely have been hailed as a spectacular twist in the story of the Poincaré conjecture. Given that it hasn't, I am predisposed not to believe the paper, and against that background the God sentence increases my scepticism.
May
6
comment Does this approach for the Poincaré conjecture work?
I'm not hugely encouraged by the final sentence of the introduction to the paper: "This essay is devoted to the meditation of God’s Word among the inhabitants of the earth."
May
6
comment Examples of common false beliefs in mathematics
I'd think of that not as a belief that people are likely to have so much as a statement that looks moderately plausible and doesn't have an obvious counterexample. In other words, it's not something that people unthinkingly assume, because they don't tend to think about it at all. But perhaps I'm wrong about that and it is routinely used as a lemma by inexperienced algebraists.
May
5
comment What practical applications does set theory have?
I think of this proof as "morally" a pure existence proof, but it happens that it can be made constructive. (I'd say the same about any proof that begins, "Let q_1,q_2,... be an enumeration of the rationals," when it doesn't matter in the slightest what the enumeration is.) I don't have a formalization of this view though.
May
5
comment Examples of common false beliefs in mathematics
By an amusing coincidence, I came across this for the first time a couple of days ago. There was a Cambridge exam question in 2008 where you had to show that products and subspaces of separable metric spaces were separable, and then you were given a topological space and asked to show that its square was separable, and that a certain subspace of it was not separable. I had to stare at it for about a minute before I understood why I had not just proved a contradiction.
May
4
comment Help with a double sum, please
I enthusiastically second that.