Recent content by phreak

Homework Statement Maximize the functional \int_{-1}^1 x^3 g(x), where g is subject to the following conditions: \int^1_{-1} g(x)dx = \int^1_{-1} x g(x)dx = \int^1_{-1} x^2 g(x)dx = 0 and \int^1_{-1} |g(x)|^2 dx = 1. Homework Equations In the previous part of the problem, I computed...

Ok, I proved it. Thanks a lot for the help. (For the curious: You prove it for \alpha k rational. Then you pick r = p/q \in \Q s.t. |\alpha k - p/q | < \epsilon. You do a lot of clever approximation and you get that 1/N * the sum \le (2\epsilon+q)/N, which goes to 0 as N goes to infinity.)

Homework Statement Given k\in \mathbb{Z} \setminus \{ 0 \}, prove that \lim_{n\to \infty} \frac{1}{N} \sum_{n=1}^N e^{2\pi i k n \alpha}=0, for all \alpha \in \mathbb{R} \setminus \mathbb{Q}. Homework Equations The Attempt at a Solution Well, I had an idea, but I'm not sure how well it...

Most PhD students are funded, and at most top privates, all are funded. The hard part is getting in.

Thank you for your comments. Josyulasharma: The book this is from is 'Partial Differential Equations' by Walter Strauss.

I saw a rather easy proof of the Heisenberg Uncertainty Principle in a PDE textbook the other day, but I'm not sure if it's correct. The proof goes as following: Note that \left| \int xf(x) f'(x) \right| \le \left[ \int |xf(x)|^2 dx \right]^{1/2} \left[ \int |f'(x)|^2 dx \right]^{1/2} , by the...

If I recall correctly, it used to be better. I'm not sure when or why the change occurred.

The standard basis for R^4 is (1,0,0,0), (0,1,0,0), (0,0,1,0), and (0,0,0,1). The idea of this problem is that once you know where the basis elements go, you can uniquely determine what the matrix is (using methods described in your textbook!) So you need to find T(1,0,0,0), which you already...

It's simply not true. In physics, linear algebra and calculus work together quite often to create some rather powerful results. They're both extremely important and without either of them, you could do very little. If one wanted to be particular, though, you can get through first year physics...

The integral is finite for any interval (k,1), where 0 < k < 1, because continuous, bounded functions are always integrable. You can't compare 1/ln(x) to 1/x or 1/(x+1) for that matter because the technique you're working with only works for positive functions. We're trying to check that the...

You're working with vectors. So you need to find the dot product. W = F dot D.

To my understanding, relativity was not yet firmly accepted to be true until Einstein's later years. Hence, the committee was afraid to hand out a medal for a theory whose validity was uncertain.

Go to Michigan. MIT's not good enough for $20k/year. I made a similar decision 2 years ago, except I picked the better and more expensive school. When you take into account all the facts, and all the benefits, it's just not worth it when you can go to the library and catch up on the material...

Because in general, they're not equal. For instance, let x_n = 0 if n even, 1 if n odd. Then lim sup (-x_n) = 0, and lim sup(x_n) = 1. Now let y_n = 0 for all n. Then we have 1 = lim sup (x_n+y_n) + lim sup(-x_n) > lim sup y_n = 0.

The CEO of FedEx, hit hard like every company by the recession, could have forced a whole lot of people to be layed off. Instead, he took a 20% pay cut, gave his senior management a 10-15% pay cut, and cut the wages of everyone else by 5%. It's responsible and admirable thinking. Sometimes...