~~Update~~
So I met with the assistant dean today since I had to turn in a full written report by tomorrow morning. She was pretty upset and clearly doesn't like me and all in all it was a pretty awful experience. She says there's a possibility of a lawsuit and that I might be fired and she...
@Office_Shredder, Yah I certainly wouldn't lie, if I have to make a full report then I'll have to incriminate the student and that sucks but I wouldn't know what else to do. No one knows about my PF account and I doubt this is serious enough for people to start going online searching for info...
Hopefully this is the correct stack exchange for this, if not feel free to tell me where to move it where it would be more appropriate.
So I am a teaching assistant and masters student in the math dept. at a large state university. I was holding a discussion section this morning for the...
I imagine myself flipping a coin repeatedly and recording the outcomes. With only the WLLN being true, I expect to periodically encounter long strings of mostly heads or mostly tails, causing the running average to fall outside some epsilon's distance from the mean. These strings would occur...
French universities generally require that you pass a language exam at the B2 level (intermediate-advanced), you can find examples of these tests online along with information on the scores required to pass. L'Ecole Polytechnique is I believe a rather prestigious university.
With regard to...
You could check out the Times Higher Ed World University Rankings for Europe. I don't think that European universities are quite as stratified as American ones are, but I get the impression that many of the best are in German speaking cities, and as usual in the major cities: Berlin, Munich...
With a bachelors in computer engineering and a masters in computer science, you would be a good candidate for a phd in quantum computing. There is currently interesting work to be done in designing the hardware/firmware required to make quantum computers a reality. In fact I think Booz Allen...
This is something which has been on my mind for a while now after considerable exploration (online) of the job market.
I see a lot of postings for various math jobs in industry requiring a masters in applied math. So let's say the focus of my degree was on discrete math and I did my thesis...
Homework Statement
Show that there is no non-abelian group G such that Z(G)=\mathbb{Z}_2, which satisfies the short exact \mathbb{Z}_2\rightarrow G\rightarrow\mathbb{Z}_2^3.
The Attempt at a Solution
I have knowledge of group theory up through proofs of the Sylow theorems. I know the center...
Number theory is an extremely fascinating and beautiful subject, but only at the graduate level, introductory number theory classes suck and should be abolished for fear of turning people off to the subject, take optics.
He says by "naive" he means elementary. Incidentally, I took an elementary number theory class and didn't particularly like it. Which is only interesting because now I intend to become a number theorist.
So going from there we use the fact that the order topology is Hausdorff and find two open disjoint sets one containing f(x) and the other containing g(x), from here I would like to map back under f^-1 and g^-1 to two open sets in X and from here take the intersection of them which would contain...
Proving a Set is Closed (Topology)
Homework Statement
Let Y be an ordered set in the order topology with f,g:X\rightarrow Y continuous. Show that the set A = \{x:f(x)\leq g(x)\} is closed in X.
Homework Equations
The Attempt at a Solution
I cannot for the life of me figure...
Comparing Partions of a Natural Number
Homework Statement
Let r(n) denote the number of ordered triples of natural numbers (a,b,c) such that a + 2b + 3c = n, for n\geq 0. Prove that this is equal to the number of ways of writing n = x + y + z with 0\leq x \leq y \leq z for x,y,z natural...
It certainly doesn't hurt to be a prodigy, but remember you can make yourself smarter be working hard, the brain has a high degree of neuroplasticity, especially when you're young. By working hard you can give yourself the brain a prodigy might have been born with.
Given the domain ℂ\[-1,1] and the function, f(z)=\frac{z}{(z-1)(z+1)}, defined on this domain, the Residue Theorem shows that for \alpha a positive parametrization of the circle of radius two centered at the origin, that:
\int_{\alpha}f(z)=\int_{\alpha}\frac{z}{(z-1)(z+1)} = 2\pi i
Can I...
I am also intending to become a mathematician and there is almost always some mathematical problem or concept which is lingering in the back of my mind. So 'all the time' is actually a more serious answer than at first it might appear. I wouldn't be able to count the number of times that I've...
I'm trying to prove that log|z| is not the real part of an analytic function defined on an annulus centered at zero. Due to the Cauchy-Riemann Equations, I've been under the impression that given a harmonic function, such as log|z|, its role as the real part of an analytic function is unique...
Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a...
Homework Statement
Evaluate the integral
\int_0^{2\pi}log|e^{i\theta}-1|d\theta
Homework Equations
The Attempt at a Solution
So I'm essentially integrating log|z| around a circle of radius 1 centered at -1. Evaluating at the endpoints gives a singularity, but I feel like that...
Ah yes, thank you, and while I have your attention, what does it mean to say that a function is defined in a neighborhood of the unit disk? I know what it means for it to be defined in a neighborhood of a point, but I can't figure out what this means. I can't tell if it means some neighborhood...
I vaguely recollect that the following statement is true:
Let f be analytic on a connected set D, then if f is constant on some nonempty open subset of D then it is constant on all of D.
Can anyone confirm that this is true and is it a specific theorem? Thanks.
Homework Statement
Show that:
\frac{1}{2\pi}\int_0^{2\pi}log|re^{i\theta} - z_0|d\theta = \left\{\begin{matrix}
log|z_0| & if & |z_0| < r \\
log|r| & if & |z_0| > r
\end{matrix}\right.
Homework Equations
The function log|z| is harmonic in the slit plane since it is the real part of...
Ah yes, I apologize. This problem is tough. You won't be able to prove it's uniformly continuous, thus recall that proving point-wise continuity allows delta to depend not only on epsilon but also on the point at which you are trying to prove continuity.
Thus choose a point x \in (-1,1) and...
You know it's funny, on wikipedia both versions of the Intermediate Value Theorem require f to be continuous as a hypothesis, are you sure your question isn't asking you to prove f^{-1} is continuous?