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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.)

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2 Mike V

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

  • the $ j $th 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 $?

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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?

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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...

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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?

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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?

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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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