4 problems

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A point $x$ is a fixed point if $f(x)=x$. The two-fixed-point conjecture. If $f$ has…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A matrix is unipotent when all its eigenvalues are $1$. The $C^1$ unipotent Jacobian…

Let $n\in\mathbb{Z}_{\geq2}$ be fixed. For each $m\in\{2,3,\ldots,n\}$ and each $\theta\in\left(\mathcal{G}_{m}^{\left(\operatorname{U}\right)}\cup\{\gamma\}\right)$, let…

Let $h:\mathbb{R}^n\setminus\{\mathbf{0}\}\to\mathbb{R}^n$ be defined componentwise by … Let $\mathbf{J}$ be the Jacobian of $h$, and let … For $\boldsymbol{\beta}\in\mathcal{B}$,…