Recent questions tagged cmi2025-math

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statemen...

Consider the $\operatorname{ring} \mathcal{C}(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$, with pointwise addition and multiplication. For $A \subset...

Let $f, g, h$ be functions from $\mathbb{R}$ to $\mathbb{R}$ such that\[h(f(x)+g(y))=x y\]for all $x, y \in \mathbb{R}$. Show the following:$h$ is surjective.If $f$ is co...

Let $f:[0,1] \longrightarrow \mathbb{R}$ and $g: \mathbb{R} \longrightarrow \mathbb{R}$ be continuous functions. Assume that $g$ is periodic with period $1.$ Show that\[\...

Prove or disprove each of the statements below.Let $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ be a continuous function that takes both positive and negative values. T...

Prove or disprove the following statements:Suppose that $f(z)$ is a complex analytic function in the punctured unit disk $0<|z|<1$ such that $\lim _{n \rightarrow \infty}...

Let $X \subseteq \mathbb{R}^{n}$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma^{\prime}(0)$, where $\gamma:(-\epsilon, \epsilon) \longrightarrow X$ is...

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $P \in \mathbb{F}_{q}[x]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when ...

Show that the power series $\sum_{n=1}^{\infty} z^{n!}$ represents an analytic function $f(z)$ in the open unit disk $\Delta$ centred at $0.$ Show that $f(z)$ cannot be e...

It is known that there exist surjective continuous maps $I \longrightarrow I^{2}$ where $I=[0,1]$ is the unit interval.Using the above result or otherwise, show that ther...