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 Dec 15 awarded Enthusiast Dec 8 answered Cardinality of Equivalence Classes of Cauchy Sequences 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 answered What are the most misleading alternate definitions in taught mathematics? 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 5 answered Why is it useful to study vector bundles? 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). Nov 30 awarded Nice Answer Nov 29 comment 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 29 awarded Nice Answer Nov 29 revised Typical value of totient function added 468 characters in body Nov 28 awarded Popular Question Nov 28 awarded Self-Learner Nov 28 answered Typical value of totient function Nov 28 awarded Editor Nov 28 answered What is the first interesting theorem in (insert subject here)? Nov 28 asked Typical value of totient function Nov 28 revised Intuitive explanation to Probability question added 42 characters in body Nov 28 answered Intuitive explanation to Probability question