090917qa

=2009 September 17th - questions/answers=

1 Mike D
Let K be a field with a non-discrete valuation, and (R,m) its valuation ring. Let R_m be the completion of R at m. Is the map R -> R_m a ring isomorphism? Is it a homeomorphism? (R_m has the m-adic topology.)

2 Mike V
Let A be an NxN "Scottish flag matrix":
 * the jth diagonal entry is 2sin(2pi.j/N)
 * 1's on the super-diagonal, -1's on the subdiagonal,
 * 1 in the bottom left corner, and -1 in the top left corner
 * all other entries 0.

What are its eigenvalues? Do they lie inside a square in the complex plan with corners $$\pm 2 \pm 2i$$?

3 Andrew D
Say K is a field of characteristic 2, algebraically closed. Geometric version: is the zero set of z(xy-z^2) in A^3 normal at (0,0,0)? Algebraic version: if f,g in A[x,y] have no common factors, must $$xf^2+yg^2$$ be squarefree?

FIRST ANSWER by Peter M.

Yes. Write $$xf^2+yg^2=p^2h$$, and differentiate both sides wrt x and y (in characteristic 2):
 * $$f^2=h_xp^2$$
 * $$g^2=h_yp^2$$

Now f and g have a common factor, p.

4 Pablo S
If N is an integer such that N^2 has only 0's and 1's in is base 10 expansion, must N be a power of 10?

5 Mike H
The Mandelbrot set is usually defined over C. It can also be defined in the quaternions or hypercomplexes. What does it look like? If we visualize this 4D set as a time-varying 3D set, will it change smoothly transitions, or will it just flicker in and out of existence?

6 Critch
Here is an attempt to create a finite topological space with nontrivial fundamental group:

Let S be the unit circle in the complex plane. Identify the open top half of the circle, $$\{z\in S | im(z)>0\}$$, to a single point T, and the open bottom half $$\{z\in S | im(z)<0}$$ to a single point B. Let X={+1,T,-1-B} be the resulting 4-point space (the open sets are {}, X, {B}, {T}, X\{+1}, X\{-1}).

I believe the indentification map S -> X is not nulhomotopic. Can someone give an elementary proof of this?

7 Dan H-L
If V and W are vector fields on a manifold M, each of which defines an "eternal flow" (a flow that is defined for all times t), does V+W define an eternal flow?