Well, I can list some topics, but other people will have more to say (the following is in roughly order of importance with the most important at the top; but lots of people will probably disagree and I might agree with their disagreements once I see them):
complex analysis
more analysis an...
Not knowing anything about you personally, I would assume this would be very difficult. I would imagine most grad complex analysis classed already assume you know a lot of stuff (like what is an analytic function, Cauchy's theorems, maximum modulus stuff, etc) and these topics are quickly...
We all know the the epsilon delta definition of a limit of a sequence. But, if I give you a sequence, to use this definition to determine if the sequence converges, you have to know (or have a good guess) about what the limit is converging to. If this is a sequene of numbers, it might not be so...
Are you sure the matrix is real? Usually if the matrix is real, people say "Orthogonal" instead of "Unitary" and instead of using a star, they use either a T or a dagger (T is to denote transpose.) Does the paper explicitly say that the matrix is real?
I don't understand why you would want to publish your problem so that other people can solve it? You came up with the problem; don't let someone else solve your problem for you. Publishing a conjecture in some kind of journal might be nice, but working hard to solve it and then publishing the...
Well, you could define a group that way, but at some point, you'll have to show that this definition of group and the "normal" definition are the same. And then there's all the stuff that Office_Shredder said.
I'm not very experienced with category theory, but it seems that one of the best...
Well, I was just pointing out that uniform convergence is still important when it comes to lebesgue integral, even though you are certainly correct that uniform convergence doesn't play nearly the same role in lebesgue integration as in riemann.
I don't think this is quite true. For example, Egorov's Theorem is a theorem about uniform convergence. Most of the time, though, this is used on a compact set, and as long as the functions are also riemann integrable, we can just use riemann integrals, but this isn't always the case and can be...
I'm not sure this is correct. For example, functions can converge in L2-norm with out converging uniformly. Indeed, functions can converge in L2-norm with out even converging ae.
True, but so what? This is not relevant in this case.
This is true, but you don't need hypotheses this strong...
As a side question, what the heck is a "frontier" of a set? This looks equivalent to the definition of boundary. Is this just another word for boundary? If so, why?
That is, why have a new word?
I am assuming p is prime. Do you agree that p divides p! and that p does not divide k! and that p does not divide (p-k)! ?
If so, then p must divide p choose k since there are no factors of p in the denominator that would "cancel" the p in the numerator.
As a side note, I would suggest not...
Yes, m and n must both be integers. Otherwise, as you pointed out, the two functions are not orthogonal. So, not only must m and n but integer differences of each other, they must both be integers.
If I understand correctly, the pumping lemma gives you a number, p, such that a bunch of stuff happens. Can you reference the exact pumping lemma you are talking about?
It is well known that \sin x / x is not Lebesgue integrable on [0, +\infty) though it is (improper) Riemann Integrable. It is also fairly easily shown (integrating by parts) that
\Bigg\lvert \int\limits_{a}^{b} \frac{\sin x}{x} dx\Bigg\rvert \leq 4
Since [a,b] is compact, the Riemann and...
You seem to be confusing "trivial" with "the solution is short to write down."
I don't think that this problem is all that trivial, and your solution is certainly not. I haven't really given it much thought, but I think most people would use the intermediate value theorem or something...
If P is the variable that you are integrating. For example, \int \sin(x^2) d(x^2) can be computed the way it seems you want to compute it. That is, \int \sin(x^2) d(x^2) = - \cos(x^2) . However, \int \sin(x^2) dx \neq - \cos(x^2). The question you are asking has a lot to do with the integration...
If you are using the \epsilon - \delta definition of continuity, then the idea is that eventually x_n will be within to x and so f(x_n) will be withi in \epsilon of f(x). But this is exactly what it means for f(x_n) to converge to f(x).
If you are trying to get estimates on something (like upper bounds) you can use Cauchy Schawrtz:
\sum_{i=0}^{n} x^{i}*y^{i} \leq (\sum_{i=0}^{n} x_i^2)^{1/2}(\sum_{i=0}^{n} y_i^2)^{1/2}
I don't understand how he can take Diff Eq without linear algebra. Has he already taken some "lesser" linear algebra? Perhaps stuck into the calc sequence?
Well, your proofs probably suck, but don't be worried about that at all. You are thinking about proving stuff, which puts you heads and shoulders above nearly everyone else. Learning to write proofs is hard, and you will not be a great proof writer when you first begin writing them. Good...
Do they offer a topology class? Or perhaps another analysis class (or do you already have to take these?) If it is between these two, I would pick the Geometry class.
Why in the world would you think this??
As a matter of fact, the exam you describe seems a lot like ordinary exams I took. Perhaps the one you describe is slightly more involved than an ordinary midterm exam, but it seems about right for a final exam.
I'm currently at GaTech and your application is much stronger than mine was. Though, I think I was helped by the fact that I did undergrad at GaTech.
I know, I know, people say it is bad to do grad school where you do undergrad, but I am married, my wife is in school, and it is just too...
That link looks more like an Advanced Calculus class and it is probably more along the lines of what is in Spivak rather than what is in a "true" analysis course. If the Table of Contents is a good indication, the link has a few topic not covered in Spivak (the multivariable stuff.)
I would...