2009 September 17th - questions
1 Mike D
Suppose that is a field that is complete with respect to a non-discrete valuation , and its valuation ring. Complete means that -cauchy sequences convergerge. A -cauchy sequence is a sequence such that for every large in the value group, there exists such that for , .
Let be the completion of at . Is the map a ring isomorphism? Is it a homeomorphism? ( has the -adic topology.)
2 Mike V
Let be an "Scottish flag matrix":
- the th diagonal entry is
- 's on the super-diagonal, 's on the subdiagonal,
- in the bottom left corner, and in the top left corner
- all other entries .
What are its eigenvalues? Do they lie inside a square in the complex plan with corners ?
3 Andrew D
Say is a field of characteristic 2, algebraically closed.
Geometric version: is the zero set of in normal at ?
Algebraic version: if have no common factors, must be squarefree?
4 Pablo S
If is an integer such that has only 's and 's in its base expansion, must be a power of ?
Scott: Just checked it's true for 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?
A finite topological space which should have nontrivial fundamental group:
Let be the unit circle in the complex plane. Identify the open top half of the circle, , to a single point , and the open bottom half to a single point . Let be the resulting 4-point space (the open sets are ).
Can anyone see directly why the map is not contractible, without fancy theorems?
7 Dan H-L
If are vector fields on a manifold , each of which defines an "eternal flow" (a flow that is defined for all times ), does define an eternal flow?