Math and stuff

Proposition Let $A, B$ be rings, let $M$ be an $A$-module, $P$ a $B$-module, and $N$ an $(A, B)$-bimodule. Then $M \otimes_A N$ is naturally a $B$-module, $N \otimes_B P$ an $A$-module, and we have \[\begin{align*} (M \otimes_A N) \otimes_B P \cong M \otimes_A (N \otimes_B P). \end{align*}\] Solution We...

Proposition Show that the Mobius transformation $f(z) = \frac{1 + z}{1 - z}$ maps the unit circle (minus the point $z = 1$) onto the imaginary axis. Solution Let $z$ be a point on the unit circle minus the point $z = 1$. Then $\abs{z} = 1$. Let $w =...

Proposition Prove that any Mobius transformation different from the identity map can have at most two fixed points. Solution Let $f(z) = (az + b) / (cz + d)$ with $ad - bc \ne 0$. If $c = 0$, it is easy to see that there exists exactly one $z...

Proposition Suppose that $f$ is holomorphic in the region $G$ and $f(G)$ is a subset of the unit circle. Show that $f$ is constant. Solution If $f(G) = \{ 1 \}$, we are done. Suppose otherwise. Let $g \in G$ such that $f(g) \ne 1$. Let $U$ be a neighborhood...

Proposition If $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$, define a vector field $\mathbf{f}$ by $\mathbf{f}(p) = f(p)_p \in \mathbb{R}^n_p$. Show that every vector field $F$ on $\mathbb{R}^n$ is of the form $\mathbf{f}$ for some $f$. Show that $\div \mathbf{f} = \trace f’$. Solution 1 Define $f_i:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f_i(p) = F^i(p)$...