Recent content by AxiomOfChoice

Here is what I have so far. I have decided to confine my attention to ##[0,\infty)##. Suppose ##f_n \to f## uniformly on a set ##E##, where each ##f_n## is continuous. (The latter is a hypothesis I inadvertently omitted from my previous posts.) Let ##\{ x_n \}## be a sequence of points in ##E##...

Ok, I think I see the strategy you are suggesting. Since \{ x_n \} is bounded (since it converges), we have -M \leq x_n \leq M for all n and for some M. I can restrict my attention to [-M,M]. On that interval, \left( 1 + \frac xn \right)^n converges uniformly to e^x, and I can apply the result...

So is it always true that if f_n \to f pointwise and x_n \to x, then f_n(x_n) \to f(x)?

What are the rules if you have a sequence f_n of real-valued functions on \mathbb R and consider the sequence f_n(x_n), where x_n is some sequence of real numbers that converges: x_n \to x. All I have found is an exercise in Baby Rudin that says that if f_n \to f uniformly on E, then f_n(x_n)...

What does it mean to be exactly numerically solvable? Forgive me for being so pedantic, but it seems that people are saying a lot of conflicting things in this thread, and I'm wondering if it isn't just an issue of defining terms. Would you be kind enough to provide a source? If you can link...

This was a very helpful response. Thanks! Although I do have one clarifying question: what sorts of operations fall under the heading of "elementary operations"? Also, in what sorts of places -- other than, say, introductions to differential Galois theory -- do we encounter the equation x\ f'(x)...

Ok. So this brings up a few questions: (1) What's the exact solution for H2+, a system with at most 2 electrons? (2) How is the exact solution different from the analytic solution? Or are you using those two words interchangeably?

I tend to agree that statistics is more employable than applied mathematics. I have a PhD in pure mathematics with an MS in physics, and my experience in hunting for industry jobs is that a lot of employers (though certainly not all) think "math" guys are smart but have a hard time believing...

Maybe, maybe not. Maybe...maybe not. I really dislike this question. It seems to presume that a person is employable simply because they managed to get a diploma from a good school. This is neither sufficient nor necessary. See above. It is as simple as this: If you want to be highly...

That's very interesting. Can you share some of them?

Yes, this is kind of what I'm getting at. If a textbook says "there is no analytic solution to this PDE," I take that to mean that someone has proved there is no analytic solution. If it is only a matter of no one knowing how to arrive at the solution at present - regardless of how long...

Yes, and I am quite familiar with the Born-Oppenheimer-to-elliptic-coordinates approach to solving the H_2^+ molecule. But this occurs only after the B-O approximation is invoked, and the approximation is (in my experience) always motivated by the observation that the full stationary Schrödinger...

Ok. But this doesn't mean such techniques will never be found or are somehow incapable of being found, right?

This is an interesting "way out"! But do I take it to mean that, if I have some partial differential equation in the variables x_1,x_2,\ldots, x_n, then I can say, "Let the solution to this PDE be f(x_1,x_2,\ldots, x_n)," and I can therefore claim to have solved the PDE analytically? That...

This seems to imply that the only PDEs that can be solved analytically are separable PDEs. Is that really the case? Again, I think the real point at issue here is what, exactly, it means to say that a PDE can be solved analytically. Does it just mean: "There exists a standard set of...