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1. Computing arc length in Poincare disk model of hyperbolic space

I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows: ... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2: ds^2 = \frac{4}{(1-r^2)^2} dx^2...
2. Norm induced by inner product?

On a finite-dimensional vector space over R or C, is every norm induced by an inner product? I know that this can fail for infinite-dimensional vector spaces. It just struck me that we never made a distinction between normed vector spaces and inner product spaces in my linear algebra course...
3. Deg 1 map S^3 -> RP(3) ?

It's easy to construct maps of even degree from the three-sphere to real projective three-space. Do there exist maps of odd degree?
4. Bessel functions

Homework Statement Find the expansion of 1 - x^2 on the interval 0 < x < 1 in terms of the Eigenfunctions J_0 ( \sqrt{ \lambda_k ^{(0)}} x) (where \lambda_k ^{(0)} denotes the kth root > 0 of J_0) of (x u')' + \lambda x u = 0 u(1) = 0 u and u' bounded. Homework Equations Hint from...
5. Circle x sphere = ?

circle x sphere = ??? Is the product space S^1 \times S^2 related (e.g. homeomorphic or homotopy equivalent) to a more familiar topological space? I am currently looking at maps from S^1 \times S^2 into other spaces, and I am having a really hard time visualizing what I am doing. Any thoughts...

Realistically speaking, what league of schools does a hardworking, smart but not brilliant math major have a shot at for a PhD in pure math? Let's say I might apply with 8 graduate courses at a "lower Ivy", coauthored 1-2 papers, solid recommendations. I am not getting my hopes up for Stanford...