I agree especially with points made by Choppy and CrysPhys: first of all, the purpose of writing the thesis, for most of us, is to strengthen the author's research chops to the point of being ready to do, and write up, competitive research - i.e. it is a learning experience for the author of...
When I was a young researcher (in pure mathematics), incompletely trained (by my own fault), I learned to use information I did not understand. I was happy to find out that was possible, as it helped me make research progress in less time. There was some loss of mental security since I had to...
In my opinion,
GL(n) acts, not on V, but on k^n, so it acts on the n dimensional space V only after choosing an ordered basis for V, (and thus also an ordered dual basis for V*).
Then if a is a covector in V*, represented by a row vector, and v is a vector in V, represented by a column vector...
some more detail:
let F = Q(t) where t^2 =2. then elements of F have form z = a+bt, for rational a and b. define z’ = a-bt to be the conjugate of z.
then claim if z = a +bt and w = c+dt, then z’w’ = (zw)’, i.e. the conjugate of a product of elements of F equals the product of their...
If it makes you feel any better, I am a retired professional mathematician, and although I can follow that proof, I think it is one of the most complicated proofs I have seen in a supposedly elementary book. With apologies to Ethan, I myself would possibly toss this book on the strength of that...
note in pasmith's nice discussion that -phi also fixes, not only 0, but all multiples of delta, i.e. all elements of K[delta] with x=0, but this fixed set does not form a subfield, i.e. is not closed under addition and multiplication. A nice property of phi, that causes its fixed set to be a...
Apologies. Since I cannot understand why you are asking, and you have not clarified it, I just did some research and suggested a paper that may be relevant.
@Mondayman. You have very good taste. I owned them all for many years. My favorite is probably Courant, for its charm, insight, and my personal history with it. In one sense I quite agree with you. I would say Apostol is the most scholarly, but also the most dry, and Spivak is the most fun...
the OP is apparently talking about the solid torus, S^1 x D, but as a subspace of the 3-sphere S^3. He is taking advantage of a "Heegaard" decomposition of S^3 into a union of two solid toruses, glued along their boundaries as manifolds, namely a common copy of S^1 x S^1. see this paper, esp...
pardon me I am not too familiar with foliations. what is a torus K1 knot? what is a leaf? for that matter what is the definition of a foliation? (presumably an equivalence relation whose classes are manifolds and with local product structure.)
what does the 3-torus have to do with the hopf...
This was the recommended text at Harvard, in 1960 for math 11, the elite honors course that today, at schools where it is still offered, like UGa or Chicago, would often use a book like Spivak's Calculus. Having read both, and knowing Spivak's background as a Harvard undergrad, it seemed likely...
I have my own issues with Wikipedia. I often find useful information there about things I am ignorant of, but do not post on things I feel knowledgable about, after one experience, when I tried to post an answer to a question about the Riemann-Roch theorem.
My elementary and concrete version...
Am I the only one to check the expansions in Bill Tre's pi posts? I was struck by the apparent error in the berry pie version, which looked off to me, although as it turns out merely (correctly) rounded up. The pearls before swine version is more in line with my approach to such things, as...