Recent content by tornado28

Thanks. That gets me the first term, and at least gives me an educated guess for the second term. I'm guessing that \frac{d}{dr} \left( d(\exp_p)_{rv} \right) = ric_p(v). When you integrate this over the whole ball you end up getting the scalar curvature which is what was supposed to happen.

I'm trying to work out the following problem: Find the first two terms of the power series expansion for the volume of a ball of radius r centered at p in a Riemannian Manifold, M with dimension n. We are given that Vol(B_r(p)) = \int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t...

You are being tripped up by the difference between a subset and a proper subset. If S is a set then U is a subset of S if each element of U is also in S. Thus every set is a subset of itself, and the empty set is a subset of every set. A proper subset is a subset that is not equal to the...

Thanks. I found another thread with information about when matrices commute. Apparently two matrices commute iff they're simultaneously diagonalizable.

Thanks morphism! (A-I)(B-I) = AB-A-B+I = AB-AB+I=I. Therefore B-I is the inverse of A-I so we have that I=(B-I)(A-I) = BA-A-B+I = BA-AB+I. Thus BA-AB = 0 as needed. How did you know to write it that way? Also, do you know any good general conditions related to matrices which commute...

I'm preparing for a qualifying exam and this problem came up on a previous qual: Let A and B be nxn matrices. Show that if A + B = AB then AB=BA.

For a school project, I would like to make a website that allows the user to draw http://en.wikipedia.org/wiki/Knot_(mathematics)" . In other words, I want to allow the user to draw many sided polygons (enough sides to approximate a curve) that are allowed to self intersect. There are two...

Ok, I understand why \|f_n-f\|_p \rightarrow 0 \Rightarrow \|f_n\|_p \rightarrow \|f\|_p , but I still don't have a solution for the other direction. In your solution, Vig, it is not necessarily true that 2^p( |f_n|^p+|f|^p-|f_n - f|^p) \geq 0 since we could, for instance, have f_n=1, f = -1...

Ha, I get it now. Thats why it works. As I pointed out earlier we can assume that f is positive on (a, b). Since f(a) = f(b) = 0 and ln(0) = -\infty we have that ln(f(x)) \rightarrow -\infty as x \rightarrow a (or b). Now apply the mean value theorem twice, once on the interval (a+ \epsilon...

Citan, that's a clever observation but how do you use it to solve the problem?

Ok, made some progress. We will assume that f(x)>0 for x\in (a,b). (We could prove using a bit of topology that provided f is not identically 0 then there is a subinterval where this is true for either f or -f.) We want to use Darboux's theorem which requires that f'_+(a) exists, so first...

Thanks so much for your help! Here's another practice qual problem if you're interested. https://www.physicsforums.com/showthread.php?p=3436359#post3436359

Let f:[a,b] \rightarrow R be a continuous function such that f(a)=f(b)=0 and f' exists on (a,b). Prove that for every real \lambda there is a c \in (a,b) such that f'(c) = \lambda f(c).

This is a functional analysis qualifying exam problem that I can't figure out. Any assistance would be appreciated since I have to take a similar qual soon. I was able to make some limited progress in the p=2 case using Holders inequality. Suppose f_n, f\in L^p where 1\le p <\infty and that...