090917q

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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