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2009 September 17th - questions

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1 Mike D

Suppose that K is a field that is complete with respect to a non-discrete valuation v, and (R,m) its valuation ring. Complete means that v-cauchy sequences convergerge. A v-cauchy sequence is a sequence f_n such that for every large g in the value group, there exists N such that for i,j > N, v(f_i-f_j) > g.

Let R_m be the completion of R at m. Is the map R \to R_m a ring isomorphism? Is it a homeomorphism? (R_m has the m-adic topology.)


2 Mike V

Let A be an N\times N "Scottish flag matrix":

  • the jth diagonal entry is 2\sin(2\pi 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?


4 Pablo S

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

Scott: Just checked it's true for N up to 100 billion...


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

A finite topological space which should have 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 | 0 > im(z)\} 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\setminus \{+1\}, X\setminus\{-1\}).

Can anyone see directly why the map S \to X is not contractible, without fancy theorems?


7 Dan H-L

If V,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?


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