Recent content by mathwonk

This has several excellent answers and I cannot improve on them. I just have this persistent nagging voice wanting to express this abstractly, if perhaps not actually usefully for a beginner. Namely, all answers point out, one way or another, that exponentiation is a function that changes...

@Surf_and_Sky: Have you talked to anyone at the U (in Salt Lake)? That is a very dynamic place, with lots of very bright people. That's where I think you are most likely to get relevant advice, and from people especially aware of the local options. If you cannot get help there, then maybe you...

Re: post #5. Here is the usual proof that every set has more subsets than elements. If S is any set (e.g. the set of rational numbers) and 2^S is the set of its subsets, and if S had as many elements as subsets, then there would be a surjective map f:S-->>2^S. We claim this is impossible...

Thank you for drawing attention to this. Many years ago, one of my friends told me as a joke on himself, that he had been referee for one of Faltings early 1980's papers, and had scolded the author for being bold enough to suggest that he hoped his work might contribute to a solution of...

by the way, if you decide to try this class, the fact that you know his source for the lecture material and for the exams, should make preparation for them rather straightforward. I.e. together with like minded fellow students, make sure you learn to do every single question on those MIT exams...

The way you describe it, the prof sounds very unreasonable indeed, but we have not heard his version. And at some point you should decide, whether your goal is to maximize your learning, or maximize your grade. It also matters of course whether you are in line for a B- in this class or an F...

To illustrate what a la-la land the current grading practice at Harvard inhabits, the Harvard college student handbook already defines an A or A- as work of "excellent quality, demonstrating full mastery of the subject", with an A reserved for "extraordinary distinction". And speaking of grade...

That article was interesting. Some statements should be taken with salt however, e.g. the suggestion that grade inflation is related to the higher caliber of recent students at schools like Florida and Georgia. At UGA, they created a "Hope" scholarship for students entering with a B average...

I think this is a difficult problem. As a student at Harvard in the early 1960’s, when maybe 15% of grades were A’s, I got a lot of well deserved D’s, until I eventually learned from them to improve my study skills and concentration. I feel now that receiving low grades for mediocre work was...

It seems indeed this is the usual meaning of trivial, i.e. equivalent to a product, but without a choice of trivialization. I was assuming otherwise, but it was just what seemed plausible to me, not citing any source. I should have remembered my own frequent advice not to disagree about the...

well yes. I had assumed you meant this to be an F-bundle over A, in which case those two projections give, in my opinion, one trivialization of it as such. but yes, they also allow it to be considered as a trivial A- bundle over F, which can be thought of as a second trivialization. You just...

I essentially agree with you @cianfa72. i.e. technically a trivial bundle is a trivializable bundle plus a trivialization. the difference however does not matter for many purposes. more generally, a complex line bundle is a locally trivializable C- bundle. most of the time the properties we...

yes thank you. I probably should have said k = real numbers in my discussion above.

Given any (finite dimensional) vector space V, it has a dual space V* consisting of linear functions f:V-->k, where k is the field of scalars. (Take k = real numbers, so we can take square roots to define length.) The most natural feature of V associated to such an f, is the "kernel" of f, i.e...

Yes. If the question referred to the center of mass of a hemispherical shell, (embedded in 3 -space), then since as I believe Archimedes knew, the surface area of a hemisphere equals that of the circumscribing cylinder, and the same holds for horizontal bands cut from the two figures by pairs...