5 problems

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ and $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}\in(\mathbb{R}^k_{>0})^\downarrow$. Define … For $\mathcal G\in\mathbb T_d(\mathbf a)$, write…

Let $F=\{f_i\}_{i\in I}$ and $G=\{g_i\}_{i\in I}$ be unit-norm frames in $\mathbb{R}^n$. Suppose that $F$ and $G$ have the same scalability poset, and that $G$ is a tight frame. St…

Let $F=\{f_i\}_{i\in I}\subseteq\mathbb{R}^2$ be a frame with factor poset $\mathbb{F}_F$, and let $EC(\mathbb{F}_F)$ denote its empty cover. Prime-subframe bound conjecture. … Her…

Let $H_n$ be an $n$-dimensional Hilbert space, let $I$ be an index set with $|I|=k=mn$ for some $m\in\mathbb{N}$, and let $F=\{f_i\}_{i\in I}\subseteq H_n\setminus\{0\}$ be a tight…

Let $P_1$ and $P_2$ be factor posets, and let $calD(P)$ denote the smallest dimension in which a frame with factor poset $P$ has an inverse. Lower-bound conjecture. … The preceding…