Math and stuff

Proposition A multiplicatively closed subset $S$ of a ring $A$ is said to be saturated if \[\begin{align*} xy \in S \iff x \in S \text{ and } y \in S. \end{align*}\] Prove that $S$ is saturated $\iff A \setminus S$ is a union of prime ideals. If $S$ is any...

Proposition Find the number of zeros of $3\exp(z) - z$ in $\overline{D}[0, 1]$. $\frac{1}{3}\exp(z) - z$ in $\overline{D}[0, 1]$. $z^4 - 5z + 1$ in $\{ z \in \mathbb{C} : 1 \leq \abs{z} \leq 2 \}$. Solution 1 For any $\abs{z} = 1$, $\abs{3\exp(z)} \geq 3/e > 1 = \abs{-z}$....

Proposition Suppose $f$ is entire and there exists $M > 0$ such that $\abs{f(z)} > M$ for all $z \in \mathbb{C}$. Prove that $f$ is constant. Solution Since $f(z) \ne 0$ for any $z \in \mathbb{C}$, $1/f(z)$ is an entire function. Since $\abs{1/f(z)} < 1/M$, $1/f(z)$ is a bounded entire...

Proposition Suppose $f$ is entire with bounded real part. Prove that $f$ is constant. Solution Since $f$ is entire, $\exp(f)$ is entire. $\abs{\exp(f)} = \exp(\Re(f))$ is bounded. Thus by Liouville’s theorem, $f$ is constant.

Proposition Let $E_i, i = 1, \cdots, k$ be Euclidean spaces of various dimensions. If $f$ is multilinear and $i \ne j$, show that for $h = (h_1, \cdots, h_k)$, with $h_l \in E_l$, we have \(\begin{align*} \lim_{h \rightarrow 0} \frac{\abs{f(a_1, \cdots, h_i, \cdots, h_j, \cdots, a_k)}}{\abs{h}} = 0. \end{align*}\)...